Mercurial > hg > mpdl-group
changeset 23:e845310098ba
diverse Korrekturen
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/.classpath Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,20 @@ +<?xml version="1.0" encoding="UTF-8"?> +<classpath> + <classpathentry kind="src" path="src"/> + <classpathentry kind="con" path="org.eclipse.jdt.launching.JRE_CONTAINER/org.eclipse.jdt.internal.launching.macosx.MacOSXType/1.6.0 (MacOS X Default)"> + <attributes> + <attribute name="owner.project.facets" value="java"/> + </attributes> + </classpathentry> + <classpathentry kind="con" path="org.eclipse.jst.server.core.container/org.eclipse.jst.server.tomcat.runtimeTarget/Apache Tomcat v6.0"> + <attributes> + <attribute name="owner.project.facets" value="jst.web"/> + </attributes> + </classpathentry> + <classpathentry kind="con" path="org.eclipse.jst.j2ee.internal.web.container"/> + <classpathentry kind="con" path="org.eclipse.jst.j2ee.internal.module.container"/> + <classpathentry combineaccessrules="false" kind="src" path="/mpiwg-mpdl-xml"/> + <classpathentry combineaccessrules="false" kind="src" path="/mpiwg-mpdl-lt"/> + <classpathentry combineaccessrules="false" kind="src" path="/mpiwg-mpdl-cms"/> + <classpathentry kind="output" path="build/classes"/> +</classpath>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/.externalToolBuilders/mpiwg-mpdl-cms-web-build.launch Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,13 @@ +<?xml version="1.0" encoding="UTF-8" standalone="no"?> +<launchConfiguration type="org.eclipse.ant.AntBuilderLaunchConfigurationType"> +<booleanAttribute key="org.eclipse.ant.ui.ATTR_TARGETS_UPDATED" value="true"/> +<booleanAttribute key="org.eclipse.ant.ui.DEFAULT_VM_INSTALL" value="false"/> +<booleanAttribute key="org.eclipse.debug.ui.ATTR_LAUNCH_IN_BACKGROUND" value="false"/> +<stringAttribute key="org.eclipse.jdt.launching.CLASSPATH_PROVIDER" value="org.eclipse.ant.ui.AntClasspathProvider"/> +<booleanAttribute key="org.eclipse.jdt.launching.DEFAULT_CLASSPATH" value="true"/> +<stringAttribute key="org.eclipse.jdt.launching.PROJECT_ATTR" value="mpiwg-mpdl-cms-web"/> +<booleanAttribute key="org.eclipse.ui.externaltools.ATTR_BUILDER_ENABLED" value="false"/> +<stringAttribute key="org.eclipse.ui.externaltools.ATTR_LOCATION" value="${workspace_loc:/mpiwg-mpdl-cms-web/build/build.xml}"/> +<stringAttribute key="org.eclipse.ui.externaltools.ATTR_RUN_BUILD_KINDS" value=""/> +<booleanAttribute key="org.eclipse.ui.externaltools.ATTR_TRIGGERS_CONFIGURED" value="true"/> +</launchConfiguration>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/.project Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,46 @@ +<?xml version="1.0" encoding="UTF-8"?> +<projectDescription> + <name>mpiwg-mpdl-cms-web</name> + <comment></comment> + <projects> + </projects> + <buildSpec> + <buildCommand> + <name>org.eclipse.wst.jsdt.core.javascriptValidator</name> + <arguments> + </arguments> + </buildCommand> + <buildCommand> + <name>org.eclipse.jdt.core.javabuilder</name> + <arguments> + </arguments> + </buildCommand> + <buildCommand> + <name>org.eclipse.wst.common.project.facet.core.builder</name> + <arguments> + </arguments> + </buildCommand> + <buildCommand> + <name>org.eclipse.wst.validation.validationbuilder</name> + <arguments> + </arguments> + </buildCommand> + <buildCommand> + <name>org.eclipse.ui.externaltools.ExternalToolBuilder</name> + <triggers>full,incremental,</triggers> + <arguments> + <dictionary> + <key>LaunchConfigHandle</key> + <value><project>/.externalToolBuilders/mpiwg-mpdl-cms-web-build.launch</value> + </dictionary> + </arguments> + </buildCommand> + </buildSpec> + <natures> + <nature>org.eclipse.jem.workbench.JavaEMFNature</nature> + <nature>org.eclipse.wst.common.modulecore.ModuleCoreNature</nature> + <nature>org.eclipse.wst.common.project.facet.core.nature</nature> + <nature>org.eclipse.jdt.core.javanature</nature> + <nature>org.eclipse.wst.jsdt.core.jsNature</nature> + </natures> +</projectDescription>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/.settings/.jsdtscope Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,12 @@ +<?xml version="1.0" encoding="UTF-8"?> +<classpath> + <classpathentry kind="src" path="WebContent"/> + <classpathentry kind="con" path="org.eclipse.wst.jsdt.launching.JRE_CONTAINER"/> + <classpathentry kind="con" path="org.eclipse.wst.jsdt.launching.WebProject"> + <attributes> + <attribute name="hide" value="true"/> + </attributes> + </classpathentry> + <classpathentry kind="con" path="org.eclipse.wst.jsdt.launching.baseBrowserLibrary"/> + <classpathentry kind="output" path=""/> +</classpath>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/.settings/org.eclipse.core.resources.prefs Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,5 @@ +#Mon Sep 24 15:59:11 CEST 2012 +eclipse.preferences.version=1 +encoding//src/de/mpg/mpiwg/berlin/mpdl/servlets/cms/DocumentOperation.java=UTF-8 +encoding//src/de/mpg/mpiwg/berlin/mpdl/servlets/cms/GetDocInfo.java=UTF-8 +encoding//src/de/mpg/mpiwg/berlin/mpdl/servlets/cms/GetDocument.java=UTF-8
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/.settings/org.eclipse.jdt.core.prefs Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,12 @@ +#Mon Sep 12 15:41:45 CEST 2011 +eclipse.preferences.version=1 +org.eclipse.jdt.core.compiler.codegen.inlineJsrBytecode=enabled +org.eclipse.jdt.core.compiler.codegen.targetPlatform=1.6 +org.eclipse.jdt.core.compiler.codegen.unusedLocal=preserve +org.eclipse.jdt.core.compiler.compliance=1.6 +org.eclipse.jdt.core.compiler.debug.lineNumber=generate +org.eclipse.jdt.core.compiler.debug.localVariable=generate +org.eclipse.jdt.core.compiler.debug.sourceFile=generate +org.eclipse.jdt.core.compiler.problem.assertIdentifier=error +org.eclipse.jdt.core.compiler.problem.enumIdentifier=error +org.eclipse.jdt.core.compiler.source=1.6
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/.settings/org.eclipse.wst.common.component Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,10 @@ +<?xml version="1.0" encoding="UTF-8"?> +<project-modules id="moduleCoreId" project-version="1.5.0"> + <wb-module deploy-name="mpiwg-mpdl-xml-web"> + <wb-resource deploy-path="/" source-path="/WebContent"/> + <wb-resource deploy-path="/WEB-INF/classes" source-path="/src"/> + <wb-resource deploy-path="/WEB-INF/lib" source-path="/WebContent/WEB-INF/lib"/> + <property name="java-output-path" value="/mpiwg-mpdl-xml-web/build/classes"/> + <property name="context-root" value="mpiwg-mpdl-xml-web"/> + </wb-module> +</project-modules>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/.settings/org.eclipse.wst.common.project.facet.core.xml Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,10 @@ +<?xml version="1.0" encoding="UTF-8"?> +<faceted-project> + <runtime name="Apache Tomcat v6.0"/> + <fixed facet="jst.web"/> + <fixed facet="wst.jsdt.web"/> + <fixed facet="java"/> + <installed facet="java" version="1.6"/> + <installed facet="jst.web" version="2.5"/> + <installed facet="wst.jsdt.web" version="1.0"/> +</faceted-project>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/.settings/org.eclipse.wst.jsdt.ui.superType.container Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,1 @@ +org.eclipse.wst.jsdt.launching.baseBrowserLibrary \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/.settings/org.eclipse.wst.jsdt.ui.superType.name Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,1 @@ +Window \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/.settings/org.eclipse.wst.ws.service.policy.prefs Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,3 @@ +#Mon Sep 12 15:41:45 CEST 2011 +eclipse.preferences.version=1 +org.eclipse.wst.ws.service.policy.projectEnabled=false
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/META-INF/MANIFEST.MF Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,3 @@ +Manifest-Version: 1.0 +Class-Path: +
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/WEB-INF/classes/constants.properties Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,5 @@ +documentsDir=/Users/jwillenborg/mpdl/data/xml/documents +luceneDocumentsDir=/Users/jwillenborg/mpdl/data/xml/lucene/documents +luceneNodesDir=/Users/jwillenborg/mpdl/data/xml/lucene/nodes +dataDir=/Users/jwillenborg/mpdl/data/lt +confDir=/Users/jwillenborg/mpdl/data/collectionConfs
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/WEB-INF/lib/commons-collections-LICENSE.txt Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,202 @@ + + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. Definitions. + + "License" shall mean the terms and conditions for use, reproduction, + and distribution as defined by Sections 1 through 9 of this document. + + "Licensor" shall mean the copyright owner or entity authorized by + the copyright owner that is granting the License. + + "Legal Entity" shall mean the union of the acting entity and all + other entities that control, are controlled by, or are under common + control with that entity. For the purposes of this definition, + "control" means (i) the power, direct or indirect, to cause the + direction or management of such entity, whether by contract or + otherwise, or (ii) ownership of fifty percent (50%) or more of the + outstanding shares, or (iii) beneficial ownership of such entity. + + "You" (or "Your") shall mean an individual or Legal Entity + exercising permissions granted by this License. + + "Source" form shall mean the preferred form for making modifications, + including but not limited to software source code, documentation + source, and configuration files. + + "Object" form shall mean any form resulting from mechanical + transformation or translation of a Source form, including but + not limited to compiled object code, generated documentation, + and conversions to other media types. + + "Work" shall mean the work of authorship, whether in Source or + Object form, made available under the License, as indicated by a + copyright notice that is included in or attached to the work + (an example is provided in the Appendix below). + + "Derivative Works" shall mean any work, whether in Source or Object + form, that is based on (or derived from) the Work and for which the + editorial revisions, annotations, elaborations, or other modifications + represent, as a whole, an original work of authorship. For the purposes + of this License, Derivative Works shall not include works that remain + separable from, or merely link (or bind by name) to the interfaces of, + the Work and Derivative Works thereof. + + "Contribution" shall mean any work of authorship, including + the original version of the Work and any modifications or additions + to that Work or Derivative Works thereof, that is intentionally + submitted to Licensor for inclusion in the Work by the copyright owner + or by an individual or Legal Entity authorized to submit on behalf of + the copyright owner. For the purposes of this definition, "submitted" + means any form of electronic, verbal, or written communication sent + to the Licensor or its representatives, including but not limited to + communication on electronic mailing lists, source code control systems, + and issue tracking systems that are managed by, or on behalf of, the + Licensor for the purpose of discussing and improving the Work, but + excluding communication that is conspicuously marked or otherwise + designated in writing by the copyright owner as "Not a Contribution." + + "Contributor" shall mean Licensor and any individual or Legal Entity + on behalf of whom a Contribution has been received by Licensor and + subsequently incorporated within the Work. + + 2. Grant of Copyright License. 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If You + institute patent litigation against any entity (including a + cross-claim or counterclaim in a lawsuit) alleging that the Work + or a Contribution incorporated within the Work constitutes direct + or contributory patent infringement, then any patent licenses + granted to You under this License for that Work shall terminate + as of the date such litigation is filed. + + 4. Redistribution. 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The contents + of the NOTICE file are for informational purposes only and + do not modify the License. You may add Your own attribution + notices within Derivative Works that You distribute, alongside + or as an addendum to the NOTICE text from the Work, provided + that such additional attribution notices cannot be construed + as modifying the License. + + You may add Your own copyright statement to Your modifications and + may provide additional or different license terms and conditions + for use, reproduction, or distribution of Your modifications, or + for any such Derivative Works as a whole, provided Your use, + reproduction, and distribution of the Work otherwise complies with + the conditions stated in this License. + + 5. Submission of Contributions. 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However, in accepting such obligations, You may act only + on Your own behalf and on Your sole responsibility, not on behalf + of any other Contributor, and only if You agree to indemnify, + defend, and hold each Contributor harmless for any liability + incurred by, or claims asserted against, such Contributor by reason + of your accepting any such warranty or additional liability. + + END OF TERMS AND CONDITIONS + + APPENDIX: How to apply the Apache License to your work. + + To apply the Apache License to your work, attach the following + boilerplate notice, with the fields enclosed by brackets "[]" + replaced with your own identifying information. (Don't include + the brackets!) The text should be enclosed in the appropriate + comment syntax for the file format. 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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/WEB-INF/lib/commons-collections-NOTICE.txt Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,5 @@ +Apache Commons Collections +Copyright 2001-2008 The Apache Software Foundation + +This product includes software developed by +The Apache Software Foundation (http://www.apache.org/).
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/WEB-INF/lib/commons-httpclient-LICENSE.txt Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,176 @@ + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. Definitions. + + "License" shall mean the terms and conditions for use, reproduction, + and distribution as defined by Sections 1 through 9 of this document. + + "Licensor" shall mean the copyright owner or entity authorized by + the copyright owner that is granting the License. + + "Legal Entity" shall mean the union of the acting entity and all + other entities that control, are controlled by, or are under common + control with that entity. For the purposes of this definition, + "control" means (i) the power, direct or indirect, to cause the + direction or management of such entity, whether by contract or + otherwise, or (ii) ownership of fifty percent (50%) or more of the + outstanding shares, or (iii) beneficial ownership of such entity. + + "You" (or "Your") shall mean an individual or Legal Entity + exercising permissions granted by this License. + + "Source" form shall mean the preferred form for making modifications, + including but not limited to software source code, documentation + source, and configuration files. + + "Object" form shall mean any form resulting from mechanical + transformation or translation of a Source form, including but + not limited to compiled object code, generated documentation, + and conversions to other media types. + + "Work" shall mean the work of authorship, whether in Source or + Object form, made available under the License, as indicated by a + copyright notice that is included in or attached to the work + (an example is provided in the Appendix below). + + "Derivative Works" shall mean any work, whether in Source or Object + form, that is based on (or derived from) the Work and for which the + editorial revisions, annotations, elaborations, or other modifications + represent, as a whole, an original work of authorship. For the purposes + of this License, Derivative Works shall not include works that remain + separable from, or merely link (or bind by name) to the interfaces of, + the Work and Derivative Works thereof. + + "Contribution" shall mean any work of authorship, including + the original version of the Work and any modifications or additions + to that Work or Derivative Works thereof, that is intentionally + submitted to Licensor for inclusion in the Work by the copyright owner + or by an individual or Legal Entity authorized to submit on behalf of + the copyright owner. For the purposes of this definition, "submitted" + means any form of electronic, verbal, or written communication sent + to the Licensor or its representatives, including but not limited to + communication on electronic mailing lists, source code control systems, + and issue tracking systems that are managed by, or on behalf of, the + Licensor for the purpose of discussing and improving the Work, but + excluding communication that is conspicuously marked or otherwise + designated in writing by the copyright owner as "Not a Contribution." + + "Contributor" shall mean Licensor and any individual or Legal Entity + on behalf of whom a Contribution has been received by Licensor and + subsequently incorporated within the Work. + + 2. Grant of Copyright License. Subject to the terms and conditions of + this License, each Contributor hereby grants to You a perpetual, + worldwide, non-exclusive, no-charge, royalty-free, irrevocable + copyright license to reproduce, prepare Derivative Works of, + publicly display, publicly perform, sublicense, and distribute the + Work and such Derivative Works in Source or Object form. + + 3. Grant of Patent License. Subject to the terms and conditions of + this License, each Contributor hereby grants to You a perpetual, + worldwide, non-exclusive, no-charge, royalty-free, irrevocable + (except as stated in this section) patent license to make, have made, + use, offer to sell, sell, import, and otherwise transfer the Work, + where such license applies only to those patent claims licensable + by such Contributor that are necessarily infringed by their + Contribution(s) alone or by combination of their Contribution(s) + with the Work to which such Contribution(s) was submitted. If You + institute patent litigation against any entity (including a + cross-claim or counterclaim in a lawsuit) alleging that the Work + or a Contribution incorporated within the Work constitutes direct + or contributory patent infringement, then any patent licenses + granted to You under this License for that Work shall terminate + as of the date such litigation is filed. + + 4. Redistribution. You may reproduce and distribute copies of the + Work or Derivative Works thereof in any medium, with or without + modifications, and in Source or Object form, provided that You + meet the following conditions: + + (a) You must give any other recipients of the Work or + Derivative Works a copy of this License; and + + (b) You must cause any modified files to carry prominent notices + stating that You changed the files; and + + (c) You must retain, in the Source form of any Derivative Works + that You distribute, all copyright, patent, trademark, and + attribution notices from the Source form of the Work, + excluding those notices that do not pertain to any part of + the Derivative Works; and + + (d) If the Work includes a "NOTICE" text file as part of its + distribution, then any Derivative Works that You distribute must + include a readable copy of the attribution notices contained + within such NOTICE file, excluding those notices that do not + pertain to any part of the Derivative Works, in at least one + of the following places: within a NOTICE text file distributed + as part of the Derivative Works; within the Source form or + documentation, if provided along with the Derivative Works; or, + within a display generated by the Derivative Works, if and + wherever such third-party notices normally appear. The contents + of the NOTICE file are for informational purposes only and + do not modify the License. You may add Your own attribution + notices within Derivative Works that You distribute, alongside + or as an addendum to the NOTICE text from the Work, provided + that such additional attribution notices cannot be construed + as modifying the License. + + You may add Your own copyright statement to Your modifications and + may provide additional or different license terms and conditions + for use, reproduction, or distribution of Your modifications, or + for any such Derivative Works as a whole, provided Your use, + reproduction, and distribution of the Work otherwise complies with + the conditions stated in this License. + + 5. Submission of Contributions. Unless You explicitly state otherwise, + any Contribution intentionally submitted for inclusion in the Work + by You to the Licensor shall be under the terms and conditions of + this License, without any additional terms or conditions. + Notwithstanding the above, nothing herein shall supersede or modify + the terms of any separate license agreement you may have executed + with Licensor regarding such Contributions. + + 6. Trademarks. This License does not grant permission to use the trade + names, trademarks, service marks, or product names of the Licensor, + except as required for reasonable and customary use in describing the + origin of the Work and reproducing the content of the NOTICE file. + + 7. Disclaimer of Warranty. Unless required by applicable law or + agreed to in writing, Licensor provides the Work (and each + Contributor provides its Contributions) on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or + implied, including, without limitation, any warranties or conditions + of TITLE, NON-INFRINGEMENT, MERCHANTABILITY, or FITNESS FOR A + PARTICULAR PURPOSE. You are solely responsible for determining the + appropriateness of using or redistributing the Work and assume any + risks associated with Your exercise of permissions under this License. + + 8. Limitation of Liability. In no event and under no legal theory, + whether in tort (including negligence), contract, or otherwise, + unless required by applicable law (such as deliberate and grossly + negligent acts) or agreed to in writing, shall any Contributor be + liable to You for damages, including any direct, indirect, special, + incidental, or consequential damages of any character arising as a + result of this License or out of the use or inability to use the + Work (including but not limited to damages for loss of goodwill, + work stoppage, computer failure or malfunction, or any and all + other commercial damages or losses), even if such Contributor + has been advised of the possibility of such damages. + + 9. Accepting Warranty or Additional Liability. While redistributing + the Work or Derivative Works thereof, You may choose to offer, + and charge a fee for, acceptance of support, warranty, indemnity, + or other liability obligations and/or rights consistent with this + License. However, in accepting such obligations, You may act only + on Your own behalf and on Your sole responsibility, not on behalf + of any other Contributor, and only if You agree to indemnify, + defend, and hold each Contributor harmless for any liability + incurred by, or claims asserted against, such Contributor by reason + of your accepting any such warranty or additional liability. + + END OF TERMS AND CONDITIONS
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/WEB-INF/lib/commons-httpclient-NOTICE.txt Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,5 @@ +Apache Jakarta HttpClient +Copyright 1999-2007 The Apache Software Foundation + +This product includes software developed by +The Apache Software Foundation (http://www.apache.org/).
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/WEB-INF/lib/commons-logging-LICENSE.txt Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,202 @@ + + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. Definitions. + + "License" shall mean the terms and conditions for use, reproduction, + and distribution as defined by Sections 1 through 9 of this document. + + "Licensor" shall mean the copyright owner or entity authorized by + the copyright owner that is granting the License. + + "Legal Entity" shall mean the union of the acting entity and all + other entities that control, are controlled by, or are under common + control with that entity. For the purposes of this definition, + "control" means (i) the power, direct or indirect, to cause the + direction or management of such entity, whether by contract or + otherwise, or (ii) ownership of fifty percent (50%) or more of the + outstanding shares, or (iii) beneficial ownership of such entity. + + "You" (or "Your") shall mean an individual or Legal Entity + exercising permissions granted by this License. + + "Source" form shall mean the preferred form for making modifications, + including but not limited to software source code, documentation + source, and configuration files. + + "Object" form shall mean any form resulting from mechanical + transformation or translation of a Source form, including but + not limited to compiled object code, generated documentation, + and conversions to other media types. + + "Work" shall mean the work of authorship, whether in Source or + Object form, made available under the License, as indicated by a + copyright notice that is included in or attached to the work + (an example is provided in the Appendix below). + + "Derivative Works" shall mean any work, whether in Source or Object + form, that is based on (or derived from) the Work and for which the + editorial revisions, annotations, elaborations, or other modifications + represent, as a whole, an original work of authorship. For the purposes + of this License, Derivative Works shall not include works that remain + separable from, or merely link (or bind by name) to the interfaces of, + the Work and Derivative Works thereof. + + "Contribution" shall mean any work of authorship, including + the original version of the Work and any modifications or additions + to that Work or Derivative Works thereof, that is intentionally + submitted to Licensor for inclusion in the Work by the copyright owner + or by an individual or Legal Entity authorized to submit on behalf of + the copyright owner. For the purposes of this definition, "submitted" + means any form of electronic, verbal, or written communication sent + to the Licensor or its representatives, including but not limited to + communication on electronic mailing lists, source code control systems, + and issue tracking systems that are managed by, or on behalf of, the + Licensor for the purpose of discussing and improving the Work, but + excluding communication that is conspicuously marked or otherwise + designated in writing by the copyright owner as "Not a Contribution." + + "Contributor" shall mean Licensor and any individual or Legal Entity + on behalf of whom a Contribution has been received by Licensor and + subsequently incorporated within the Work. + + 2. Grant of Copyright License. Subject to the terms and conditions of + this License, each Contributor hereby grants to You a perpetual, + worldwide, non-exclusive, no-charge, royalty-free, irrevocable + copyright license to reproduce, prepare Derivative Works of, + publicly display, publicly perform, sublicense, and distribute the + Work and such Derivative Works in Source or Object form. + + 3. Grant of Patent License. Subject to the terms and conditions of + this License, each Contributor hereby grants to You a perpetual, + worldwide, non-exclusive, no-charge, royalty-free, irrevocable + (except as stated in this section) patent license to make, have made, + use, offer to sell, sell, import, and otherwise transfer the Work, + where such license applies only to those patent claims licensable + by such Contributor that are necessarily infringed by their + Contribution(s) alone or by combination of their Contribution(s) + with the Work to which such Contribution(s) was submitted. If You + institute patent litigation against any entity (including a + cross-claim or counterclaim in a lawsuit) alleging that the Work + or a Contribution incorporated within the Work constitutes direct + or contributory patent infringement, then any patent licenses + granted to You under this License for that Work shall terminate + as of the date such litigation is filed. + + 4. Redistribution. You may reproduce and distribute copies of the + Work or Derivative Works thereof in any medium, with or without + modifications, and in Source or Object form, provided that You + meet the following conditions: + + (a) You must give any other recipients of the Work or + Derivative Works a copy of this License; and + + (b) You must cause any modified files to carry prominent notices + stating that You changed the files; and + + (c) You must retain, in the Source form of any Derivative Works + that You distribute, all copyright, patent, trademark, and + attribution notices from the Source form of the Work, + excluding those notices that do not pertain to any part of + the Derivative Works; and + + (d) If the Work includes a "NOTICE" text file as part of its + distribution, then any Derivative Works that You distribute must + include a readable copy of the attribution notices contained + within such NOTICE file, excluding those notices that do not + pertain to any part of the Derivative Works, in at least one + of the following places: within a NOTICE text file distributed + as part of the Derivative Works; within the Source form or + documentation, if provided along with the Derivative Works; or, + within a display generated by the Derivative Works, if and + wherever such third-party notices normally appear. The contents + of the NOTICE file are for informational purposes only and + do not modify the License. You may add Your own attribution + notices within Derivative Works that You distribute, alongside + or as an addendum to the NOTICE text from the Work, provided + that such additional attribution notices cannot be construed + as modifying the License. + + You may add Your own copyright statement to Your modifications and + may provide additional or different license terms and conditions + for use, reproduction, or distribution of Your modifications, or + for any such Derivative Works as a whole, provided Your use, + reproduction, and distribution of the Work otherwise complies with + the conditions stated in this License. + + 5. Submission of Contributions. Unless You explicitly state otherwise, + any Contribution intentionally submitted for inclusion in the Work + by You to the Licensor shall be under the terms and conditions of + this License, without any additional terms or conditions. + Notwithstanding the above, nothing herein shall supersede or modify + the terms of any separate license agreement you may have executed + with Licensor regarding such Contributions. + + 6. Trademarks. This License does not grant permission to use the trade + names, trademarks, service marks, or product names of the Licensor, + except as required for reasonable and customary use in describing the + origin of the Work and reproducing the content of the NOTICE file. + + 7. Disclaimer of Warranty. Unless required by applicable law or + agreed to in writing, Licensor provides the Work (and each + Contributor provides its Contributions) on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or + implied, including, without limitation, any warranties or conditions + of TITLE, NON-INFRINGEMENT, MERCHANTABILITY, or FITNESS FOR A + PARTICULAR PURPOSE. You are solely responsible for determining the + appropriateness of using or redistributing the Work and assume any + risks associated with Your exercise of permissions under this License. + + 8. Limitation of Liability. In no event and under no legal theory, + whether in tort (including negligence), contract, or otherwise, + unless required by applicable law (such as deliberate and grossly + negligent acts) or agreed to in writing, shall any Contributor be + liable to You for damages, including any direct, indirect, special, + incidental, or consequential damages of any character arising as a + result of this License or out of the use or inability to use the + Work (including but not limited to damages for loss of goodwill, + work stoppage, computer failure or malfunction, or any and all + other commercial damages or losses), even if such Contributor + has been advised of the possibility of such damages. + + 9. Accepting Warranty or Additional Liability. While redistributing + the Work or Derivative Works thereof, You may choose to offer, + and charge a fee for, acceptance of support, warranty, indemnity, + or other liability obligations and/or rights consistent with this + License. However, in accepting such obligations, You may act only + on Your own behalf and on Your sole responsibility, not on behalf + of any other Contributor, and only if You agree to indemnify, + defend, and hold each Contributor harmless for any liability + incurred by, or claims asserted against, such Contributor by reason + of your accepting any such warranty or additional liability. + + END OF TERMS AND CONDITIONS + + APPENDIX: How to apply the Apache License to your work. + + To apply the Apache License to your work, attach the following + boilerplate notice, with the fields enclosed by brackets "[]" + replaced with your own identifying information. (Don't include + the brackets!) The text should be enclosed in the appropriate + comment syntax for the file format. We also recommend that a + file or class name and description of purpose be included on the + same "printed page" as the copyright notice for easier + identification within third-party archives. + + Copyright [yyyy] [name of copyright owner] + + Licensed under the Apache License, Version 2.0 (the "License"); + you may not use this file except in compliance with the License. + You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + + Unless required by applicable law or agreed to in writing, software + distributed under the License is distributed on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + See the License for the specific language governing permissions and + limitations under the License.
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/WEB-INF/lib/commons-logging-NOTICE.txt Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,6 @@ +Apache Commons Logging +Copyright 2003-2007 The Apache Software Foundation + +This product includes software developed by +The Apache Software Foundation (http://www.apache.org/). +
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/WEB-INF/lib/quartz-LICENSE.txt Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,202 @@ + + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. Definitions. + + "License" shall mean the terms and conditions for use, reproduction, + and distribution as defined by Sections 1 through 9 of this document. + + "Licensor" shall mean the copyright owner or entity authorized by + the copyright owner that is granting the License. + + "Legal Entity" shall mean the union of the acting entity and all + other entities that control, are controlled by, or are under common + control with that entity. For the purposes of this definition, + "control" means (i) the power, direct or indirect, to cause the + direction or management of such entity, whether by contract or + otherwise, or (ii) ownership of fifty percent (50%) or more of the + outstanding shares, or (iii) beneficial ownership of such entity. + + "You" (or "Your") shall mean an individual or Legal Entity + exercising permissions granted by this License. + + "Source" form shall mean the preferred form for making modifications, + including but not limited to software source code, documentation + source, and configuration files. + + "Object" form shall mean any form resulting from mechanical + transformation or translation of a Source form, including but + not limited to compiled object code, generated documentation, + and conversions to other media types. + + "Work" shall mean the work of authorship, whether in Source or + Object form, made available under the License, as indicated by a + copyright notice that is included in or attached to the work + (an example is provided in the Appendix below). + + "Derivative Works" shall mean any work, whether in Source or Object + form, that is based on (or derived from) the Work and for which the + editorial revisions, annotations, elaborations, or other modifications + represent, as a whole, an original work of authorship. For the purposes + of this License, Derivative Works shall not include works that remain + separable from, or merely link (or bind by name) to the interfaces of, + the Work and Derivative Works thereof. + + "Contribution" shall mean any work of authorship, including + the original version of the Work and any modifications or additions + to that Work or Derivative Works thereof, that is intentionally + submitted to Licensor for inclusion in the Work by the copyright owner + or by an individual or Legal Entity authorized to submit on behalf of + the copyright owner. For the purposes of this definition, "submitted" + means any form of electronic, verbal, or written communication sent + to the Licensor or its representatives, including but not limited to + communication on electronic mailing lists, source code control systems, + and issue tracking systems that are managed by, or on behalf of, the + Licensor for the purpose of discussing and improving the Work, but + excluding communication that is conspicuously marked or otherwise + designated in writing by the copyright owner as "Not a Contribution." + + "Contributor" shall mean Licensor and any individual or Legal Entity + on behalf of whom a Contribution has been received by Licensor and + subsequently incorporated within the Work. + + 2. Grant of Copyright License. Subject to the terms and conditions of + this License, each Contributor hereby grants to You a perpetual, + worldwide, non-exclusive, no-charge, royalty-free, irrevocable + copyright license to reproduce, prepare Derivative Works of, + publicly display, publicly perform, sublicense, and distribute the + Work and such Derivative Works in Source or Object form. + + 3. Grant of Patent License. Subject to the terms and conditions of + this License, each Contributor hereby grants to You a perpetual, + worldwide, non-exclusive, no-charge, royalty-free, irrevocable + (except as stated in this section) patent license to make, have made, + use, offer to sell, sell, import, and otherwise transfer the Work, + where such license applies only to those patent claims licensable + by such Contributor that are necessarily infringed by their + Contribution(s) alone or by combination of their Contribution(s) + with the Work to which such Contribution(s) was submitted. If You + institute patent litigation against any entity (including a + cross-claim or counterclaim in a lawsuit) alleging that the Work + or a Contribution incorporated within the Work constitutes direct + or contributory patent infringement, then any patent licenses + granted to You under this License for that Work shall terminate + as of the date such litigation is filed. + + 4. Redistribution. You may reproduce and distribute copies of the + Work or Derivative Works thereof in any medium, with or without + modifications, and in Source or Object form, provided that You + meet the following conditions: + + (a) You must give any other recipients of the Work or + Derivative Works a copy of this License; and + + (b) You must cause any modified files to carry prominent notices + stating that You changed the files; and + + (c) You must retain, in the Source form of any Derivative Works + that You distribute, all copyright, patent, trademark, and + attribution notices from the Source form of the Work, + excluding those notices that do not pertain to any part of + the Derivative Works; and + + (d) If the Work includes a "NOTICE" text file as part of its + distribution, then any Derivative Works that You distribute must + include a readable copy of the attribution notices contained + within such NOTICE file, excluding those notices that do not + pertain to any part of the Derivative Works, in at least one + of the following places: within a NOTICE text file distributed + as part of the Derivative Works; within the Source form or + documentation, if provided along with the Derivative Works; or, + within a display generated by the Derivative Works, if and + wherever such third-party notices normally appear. The contents + of the NOTICE file are for informational purposes only and + do not modify the License. You may add Your own attribution + notices within Derivative Works that You distribute, alongside + or as an addendum to the NOTICE text from the Work, provided + that such additional attribution notices cannot be construed + as modifying the License. + + You may add Your own copyright statement to Your modifications and + may provide additional or different license terms and conditions + for use, reproduction, or distribution of Your modifications, or + for any such Derivative Works as a whole, provided Your use, + reproduction, and distribution of the Work otherwise complies with + the conditions stated in this License. + + 5. Submission of Contributions. Unless You explicitly state otherwise, + any Contribution intentionally submitted for inclusion in the Work + by You to the Licensor shall be under the terms and conditions of + this License, without any additional terms or conditions. + Notwithstanding the above, nothing herein shall supersede or modify + the terms of any separate license agreement you may have executed + with Licensor regarding such Contributions. + + 6. Trademarks. This License does not grant permission to use the trade + names, trademarks, service marks, or product names of the Licensor, + except as required for reasonable and customary use in describing the + origin of the Work and reproducing the content of the NOTICE file. + + 7. Disclaimer of Warranty. Unless required by applicable law or + agreed to in writing, Licensor provides the Work (and each + Contributor provides its Contributions) on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or + implied, including, without limitation, any warranties or conditions + of TITLE, NON-INFRINGEMENT, MERCHANTABILITY, or FITNESS FOR A + PARTICULAR PURPOSE. You are solely responsible for determining the + appropriateness of using or redistributing the Work and assume any + risks associated with Your exercise of permissions under this License. + + 8. Limitation of Liability. In no event and under no legal theory, + whether in tort (including negligence), contract, or otherwise, + unless required by applicable law (such as deliberate and grossly + negligent acts) or agreed to in writing, shall any Contributor be + liable to You for damages, including any direct, indirect, special, + incidental, or consequential damages of any character arising as a + result of this License or out of the use or inability to use the + Work (including but not limited to damages for loss of goodwill, + work stoppage, computer failure or malfunction, or any and all + other commercial damages or losses), even if such Contributor + has been advised of the possibility of such damages. + + 9. Accepting Warranty or Additional Liability. While redistributing + the Work or Derivative Works thereof, You may choose to offer, + and charge a fee for, acceptance of support, warranty, indemnity, + or other liability obligations and/or rights consistent with this + License. However, in accepting such obligations, You may act only + on Your own behalf and on Your sole responsibility, not on behalf + of any other Contributor, and only if You agree to indemnify, + defend, and hold each Contributor harmless for any liability + incurred by, or claims asserted against, such Contributor by reason + of your accepting any such warranty or additional liability. + + END OF TERMS AND CONDITIONS + + APPENDIX: How to apply the Apache License to your work. + + To apply the Apache License to your work, attach the following + boilerplate notice, with the fields enclosed by brackets "[]" + replaced with your own identifying information. (Don't include + the brackets!) The text should be enclosed in the appropriate + comment syntax for the file format. We also recommend that a + file or class name and description of purpose be included on the + same "printed page" as the copyright notice for easier + identification within third-party archives. + + Copyright [yyyy] [name of copyright owner] + + Licensed under the Apache License, Version 2.0 (the "License"); + you may not use this file except in compliance with the License. + You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + + Unless required by applicable law or agreed to in writing, software + distributed under the License is distributed on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + See the License for the specific language governing permissions and + limitations under the License.
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/WEB-INF/lib/saxon.txt Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,3 @@ +Saxon: + +Release 9.1.0.5 (free version): releases < 9.1.0.7 support saxon extension functions
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/WEB-INF/web.xml Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,173 @@ +<?xml version="1.0" encoding="UTF-8"?> +<web-app xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://java.sun.com/xml/ns/javaee" xmlns:web="http://java.sun.com/xml/ns/javaee/web-app_2_5.xsd" xsi:schemaLocation="http://java.sun.com/xml/ns/javaee http://java.sun.com/xml/ns/javaee/web-app_2_5.xsd" id="WebApp_ID" version="2.5"> + <display-name>mpiwg-mpdl-cms-web</display-name> + <welcome-file-list> + <welcome-file>index.html</welcome-file> + </welcome-file-list> + <servlet> + <description>GetDocument</description> + <display-name>GetDocument</display-name> + <servlet-name>GetDocument</servlet-name> + <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.cms.GetDocument</servlet-class> + </servlet> + <servlet-mapping> + <servlet-name>GetDocument</servlet-name> + <url-pattern>/doc/GetDocument</url-pattern> + </servlet-mapping> + <servlet> + <description>DocumentOperation</description> + <display-name>DocumentOperation</display-name> + <servlet-name>DocumentOperation</servlet-name> + <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.cms.DocumentOperation</servlet-class> + </servlet> + <servlet-mapping> + <servlet-name>DocumentOperation</servlet-name> + <url-pattern>/doc/DocumentOperation</url-pattern> + </servlet-mapping> + <servlet> + <description>GetPage</description> + <display-name>GetPage</display-name> + <servlet-name>GetPage</servlet-name> + <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.cms.GetPage</servlet-class> + </servlet> + <servlet-mapping> + <servlet-name>GetPage</servlet-name> + <url-pattern>/query/GetPage</url-pattern> + </servlet-mapping> + <servlet> + <description>QueryDocuments</description> + <display-name>QueryDocuments</display-name> + <servlet-name>QueryDocuments</servlet-name> + <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.cms.QueryDocuments</servlet-class> + </servlet> + <servlet-mapping> + <servlet-name>QueryDocuments</servlet-name> + <url-pattern>/query/QueryDocuments</url-pattern> + </servlet-mapping> + <servlet> + <description>QueryDocument</description> + <display-name>QueryDocument</display-name> + <servlet-name>QueryDocument</servlet-name> + <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.cms.QueryDocument</servlet-class> + </servlet> + <servlet-mapping> + <servlet-name>QueryDocument</servlet-name> + <url-pattern>/query/QueryDocument</url-pattern> + </servlet-mapping> + <servlet> + <description>XQueryDocument</description> + <display-name>XQueryDocument</display-name> + <servlet-name>XQueryDocument</servlet-name> + <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.cms.XQueryDocument</servlet-class> + </servlet> + <servlet-mapping> + <servlet-name>XQueryDocument</servlet-name> + <url-pattern>/query/XQueryDocument</url-pattern> + </servlet-mapping> + <servlet> + <description>MoreLikeThis</description> + <display-name>MoreLikeThis</display-name> + <servlet-name>MoreLikeThis</servlet-name> + <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.cms.MoreLikeThis</servlet-class> + </servlet> + <servlet-mapping> + <servlet-name>MoreLikeThis</servlet-name> + <url-pattern>/query/MoreLikeThis</url-pattern> + </servlet-mapping> + <servlet> + <description>GetToken</description> + <display-name>GetToken</display-name> + <servlet-name>GetToken</servlet-name> + <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.cms.GetToken</servlet-class> + </servlet> + <servlet-mapping> + <servlet-name>GetToken</servlet-name> + <url-pattern>/query/GetToken</url-pattern> + </servlet-mapping> + <servlet> + <description>GetDocInfo</description> + <display-name>GetDocInfo</display-name> + <servlet-name>GetDocInfo</servlet-name> + <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.cms.GetDocInfo</servlet-class> + </servlet> + <servlet-mapping> + <servlet-name>GetDocInfo</servlet-name> + <url-pattern>/query/GetDocInfo</url-pattern> + </servlet-mapping> + <servlet> + <description>GetDocumentJobs</description> + <display-name>GetDocumentJobs</display-name> + <servlet-name>GetDocumentJobs</servlet-name> + <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.cms.GetDocumentJobs</servlet-class> + </servlet> + <servlet-mapping> + <servlet-name>GetDocumentJobs</servlet-name> + <url-pattern>/doc/GetDocumentJobs</url-pattern> + </servlet-mapping> + + <servlet> + <description>GetDictionaryEntries</description> + <display-name>GetDictionaryEntries</display-name> + <servlet-name>GetDictionaryEntries</servlet-name> + <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.lt.GetDictionaryEntries</servlet-class> + </servlet> + <servlet-mapping> + <servlet-name>GetDictionaryEntries</servlet-name> + <url-pattern>/lt/GetDictionaryEntries</url-pattern> + </servlet-mapping> + <servlet> + <description>GetLemmas</description> + <display-name>GetLemmas</display-name> + <servlet-name>GetLemmas</servlet-name> + <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.lt.GetLemmas</servlet-class> + </servlet> + <servlet-mapping> + <servlet-name>GetLemmas</servlet-name> + <url-pattern>/lt/GetLemmas</url-pattern> + </servlet-mapping> + <servlet> + <description>GetForms</description> + <display-name>GetForms</display-name> + <servlet-name>GetForms</servlet-name> + <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.lt.GetForms</servlet-class> + </servlet> + <servlet-mapping> + <servlet-name>GetForms</servlet-name> + <url-pattern>/lt/GetForms</url-pattern> + </servlet-mapping> + <servlet> + <description>Tokenize</description> + <display-name>Tokenize</display-name> + <servlet-name>Tokenize</servlet-name> + <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.lt.Tokenize</servlet-class> + </servlet> + <servlet-mapping> + <servlet-name>Tokenize</servlet-name> + <url-pattern>/text/Tokenize</url-pattern> + </servlet-mapping> + <servlet> + <description>Normalize</description> + <display-name>Normalize</display-name> + <servlet-name>Normalize</servlet-name> + <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.lt.Normalize</servlet-class> + </servlet> + <servlet-mapping> + <servlet-name>Normalize</servlet-name> + <url-pattern>/text/Normalize</url-pattern> + </servlet-mapping> + <servlet> + <description>Transcode</description> + <display-name>Transcode</display-name> + <servlet-name>Transcode</servlet-name> + <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.lt.Transcode</servlet-class> + </servlet> + <servlet-mapping> + <servlet-name>Transcode</servlet-name> + <url-pattern>/text/Transcode</url-pattern> + </servlet-mapping> + + <listener> + <listener-class>de.mpg.mpiwg.berlin.mpdl.servlets.cms.MpiwgMpdlCmsWebServletContextListener</listener-class> + </listener> + +</web-app> \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/cms-services.html Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,563 @@ +<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0//EN"> +<html> + <head> + <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> + <title>Max Planck Institute for the History of Science - CMS Services</title> + </head> +<body> +<h2>Max Planck Institute for the History of Science - CMS Services</h2> + +<ul> + <li><b>Url: /mpiwg-mpdl-cms-web/doc/GetDocument</b> + <ul> + <li>Request parameters + <ul> + <li>id (required) + <ul> + <li>identifier of the document (e.g. "/tei/de/dt-ptolemaeus-tei-merge2.xml")</li> + </ul> + </li> + </ul> + </li> + <li>Response output + <ul> + <li>Link to document job</li> + <li>Example: <a href="doc/GetDocument?id=/tei/de/dt-ptolemaeus-tei-merge2.xml">Get test document</a></li> + </ul> + </li> + </ul> + </li> + + <li><b>Url: /mpiwg-mpdl-cms-web/doc/DocumentOperation</b> + <ul> + <li>Request parameters + <ul> + <li>srcUrl (optional) + <ul> + <li>url of the source document (if operation is create)</li> + </ul> + </li> + <li>docId (required) + <ul> + <li>document identifier of the document (e.g. "/tei/de/dt-ptolemaeus-tei-merge2.xml")</li> + </ul> + </li> + <li>elementNames (optional) + <ul> + <li>list of xml element names which should be indexed</li> + <li>default: "s head caption variables description"</li> + </ul> + </li> + <li>operation (required) + <ul> + <li>create (if docId already exists then it is updated)</li> + <li>delete</li> + </ul> + </li> + <li>outputFormat (optional) + <ul> + <li>xml</li> + <li>html</li> + <li>string</li> + <li>default: xml</li> + </ul> + </li> + </ul> + </li> + <li>Response output + <ul> + <li>Link to document job</li> + <li>Example: <a href="doc/DocumentOperation?srcUrl=http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/getDoc?doc=/tei/de/dt-ptolemaeus-tei-merge2.xml&docId=/tei/de/dt-ptolemaeus-tei-merge2.xml&operation=create">Create test document</a></li> + </ul> + </li> + </ul> + </li> + + <li><b>Url: /mpiwg-mpdl-cms-web/query/GetPage</b> + <ul> + <li>Request parameters + <ul> + <li>docId (required) + <ul> + <li>document identifier of the document (e.g. "/tei/de/dt-ptolemaeus-tei-merge2.xml")</li> + </ul> + </li> + <li>page (optional) + <ul> + <li>page number in document</li> + <li>default: 1</li> + </ul> + </li> + <li>normalization (optional) + <ul> + <li>orig (original)</li> + <li>reg (regularized)</li> + <li>norm (regularized and normalized)</li> + <li>default: norm</li> + </ul> + </li> + <li>highlightQuery (optional) + <ul> + <li>Lucene query string for which words should be highlighted</li> + <li>default: empty</li> + </ul> + </li> + <li>highlightQueryType (optional) + <ul> + <li>form (form)</li> + <li>morph (morphological)</li> + <li>default: form</li> + </ul> + </li> + <li>highlightElem (optional) + <ul> + <li>element name which should be highlighted</li> + <li>default: empty</li> + </ul> + </li> + <li>highlightElemPos (optional) + <ul> + <li>element position which should be highlighted</li> + <li>default: 1</li> + </ul> + </li> + <li>mode (optional) + <ul> + <li>untokenized</li> + <li>tokenized (page result is enriched with word elements: e.g. <w lang="deu" form="satz" formRegularized="satz" formNormalized="satz" forms="satz, sätze, satzes" lemmas="satz">Satz</w>)</li> + <li>default: untokenized</li> + </ul> + </li> + <li>outputFormat (optional) + <ul> + <li>html</li> + <li>xml (pure xml)</li> + <li>xmlDisplay (xml like display through html)</li> + <li>default: html</li> + </ul> + </li> + <li>cssUrl (optional) + <ul> + <li>url of css file</li> + <li>default: <a href="css/page.css">/mpiwg-mpdl-cms-web/css/page.css</a></li> + </ul> + </li> + </ul> + </li> + <li>Response output + <ul> + <li>Document page</li> + <li>Example: <a href="query/GetPage?docId=/echo/la/Benedetti_1585_163127KK.xml&page=13">Page 13 in Benedetti document</a></li> + <li>Example: <a href="query/GetPage?docId=/echo/la/Benedetti_1585_163127KK.xml&page=13&mode=tokenized">Page 13 in Benedetti document, tokenized</a></li> + <li>Example: <a href="query/GetPage?docId=/echo/la/Benedetti_1585_163127KK.xml&page=13&highlightElem=s&highlightElemPos=1">Page 13 in Benedetti document, first sentence highlighted</a></li> + <li>Example: <a href="query/GetPage?docId=/echo/la/Benedetti_1585_163127KK.xml&page=13&outputFormat=xmlDisplay&highlightElem=s&highlightElemPos=1">Page 13 in Benedetti document, first sentence highlighted (XML output)</a></li> + <li>Example: <a href="query/GetPage?docId=/echo/la/Benedetti_1585_163127KK.xml&page=13&normalization=reg&highlightQuery=tokenReg:relatiuum">Page 13 in Benedetti document, regularized form "relatiuum" is highlighted</a></li> + <li>Example: <a href="query/GetPage?docId=/echo/la/Benedetti_1585_163127KK.xml&page=13&highlightQueryType=norm&highlightQuery=tokenMorph:praeclarus&language=lat">Page 13 in Benedetti document, morphological forms of "praeclarus" are highlighted</a></li> + </ul> + </li> + </ul> + </li> + + <li><b>Url: /mpiwg-mpdl-cms-web/query/QueryDocuments</b> + <ul><li><a href="query/index.html">Web query interface</a></li></ul> + <ul> + <li>Request parameters + <ul> + <li>query (required) + <ul> + <li><a href="http://lucene.apache.org/core/3_6_0/queryparsersyntax.html">Lucene query</a> (e.g. +author:ben* +tokenMorph:quantitas +tokenMorph:sum) + <ul> + <li>supported fields + <ul> + <li>docId</li> + <li>identifier</li> + <li>author</li> + <li>title</li> + <li>language</li> + <li>publisher (place)</li> + <li>date (year)</li> + <li>subject</li> + <li>rights</li> + <li>license</li> + <li>accessRights</li> + <li>tokenOrig</li> + <li>tokenReg</li> + <li>tokenNorm</li> + <li>tokenMorph</li> + <li>default: tokenOrig</li> + </ul> + </li> + </ul> + </li> + </ul> + </li> + <li>sortBy (optional) + <ul> + <li>fieldNames separated by blank + <ul> + <li>supported fields + <ul> + <li>author</li> + <li>title</li> + <li>language</li> + <li>publisher (place)</li> + <li>docId</li> + <li>date (year)</li> + <li>lastModified</li> + <li>schemaName</li> + </ul> + </li> + <li>default: result is sorted by score (the most relevant result first)</li> + </ul> + </li> + </ul> + </li> + <li>language (optional) + <ul> + <li>ISO 639-3 specifier</li> + <li>default: automatically derived from query</li> + </ul> + </li> + <li>translate (optional) + <ul> + <li>expansion of the query: translation of the query terms to other languages</li> + <li>true</li> + <li>false</li> + <li>default: false</li> + </ul> + </li> + <li>page (optional) + <ul> + <li>result page number</li> + <li>default: 1</li> + </ul> + </li> + <li>pageSize (optional) + <ul> + <li>result page size</li> + <li>default: 10</li> + </ul> + </li> + <li>outputFormat (optional) + <ul> + <li>xml</li> + <li>html</li> + <li>default: html</li> + </ul> + </li> + </ul> + </li> + <li>Response output + <ul> + <li>Documents matching the query</li> + <li>Example: <a href="query/QueryDocuments?query=tokenOrig:tempore">all documents which contains "tempore" in original content</a></li> + <li>Example: <a href="query/QueryDocuments?query=tokenOrig:relatiuũ">all documents which contains "relatiuũ" in original content</a></li> + <li>Example: <a href="query/QueryDocuments?query=tokenReg:relatiuum">all documents which contains "relatiuum" in regularized content</a></li> + <li>Example: <a href="query/QueryDocuments?query=tokenNorm:relativum">all documents which contains "relativum" in normalized content</a></li> + <li>Example: <a href="query/QueryDocuments?query=tokenOrig:Sabaudiæ&language=lat">all latin documents which contains "sabaudiæ" in content</a></li> + <li>Example: <a href="query/QueryDocuments?query=tokenMorph:sum&language=lat&page=2">all latin documents which morphological contains "sum" in content (result page 2)</a></li> + <li>Example: <a href="query/QueryDocuments?query=tokenOrig:buch&translate=true">all documents which contains translated "buch" in content</a></li> + </ul> + </li> + </ul> + </li> + + <li><b>Url: /mpiwg-mpdl-cms-web/query/QueryDocument</b> + <ul> + <li>Request parameters + <ul> + <li>docId (required) + <ul> + <li>document identifier (e.g. "/echo/la/Benedetti_1585.xml"</li> + </ul> + </li> + <li>query (required) + <ul> + <li><a href="http://lucene.apache.org/core/3_6_0/queryparsersyntax.html">Lucene query</a> (e.g. +tokenMorph:quantitas +tokenMorph:sum) + <ul> + <li>supported fields + <ul> + <li>language</li> + <li>pageNumber</li> + <li>lineNumber</li> + <li>elementName</li> + <li>elementDocPosition (position in document, e.g. the 7th element in document)</li> + <li>elementAbsolutePosition (position in all elements with this name, e.g. the 5th sentence in document)</li> + <li>elementPagePosition (position of this element within a page, e.g. the 8th sentence on page 5)</li> + <li>elementPosition (position of this element in parent element, e.g. the 3rd sentence element in a paragraph)</li> + <li>xmlId</li> + <li>xpath</li> + <li>tokenOrig</li> + <li>tokenReg</li> + <li>tokenNorm</li> + <li>tokenMorph</li> + <li>default: tokenOrig</li> + </ul> + </li> + </ul> + </li> + </ul> + </li> + <li>page (optional) + <ul> + <li>result page number</li> + <li>default: 1</li> + </ul> + </li> + <li>pageSize (optional) + <ul> + <li>result page size</li> + <li>default: 10</li> + </ul> + </li> + <li>outputFormat (optional) + <ul> + <li>xml</li> + <li>html</li> + <li>default: xml</li> + </ul> + </li> + </ul> + </li> + <li>Response output + <ul> + <li>nodes/elements in document which matches the query, sorted by position in document, word tokens are extracted and hits are highlighted</li> + <li>Example: <a href="query/QueryDocument?docId=/echo/la/Benedetti_1585_163127KK.xml&query=tokenOrig:tempore">in /echo/la/Benedetti_1585_163127KK.xml: all nodes which contains "tempore" in original content (XML output)</a></li> + <li>Example: <a href="query/QueryDocument?docId=/echo/la/Benedetti_1585_163127KK.xml&query=tokenOrig:tempore&outputFormat=html">in /echo/la/Benedetti_1585_163127KK.xml: all nodes which contains "tempore" in original content (HTML output)</a></li> + <li>Example: <a href="query/QueryDocument?docId=/echo/la/Benedetti_1585_163127KK.xml&query=tokenReg:relatiuum">in /echo/la/Benedetti_1585.xml: all nodes which contains "relatiuum" in regularized content (XML output)</a></li> + <li>Example: <a href="query/QueryDocument?docId=/echo/la/Benedetti_1585_163127KK.xml&query=tokenMorph:praeclarus&outputFormat=html">in /echo/la/Benedetti_1585_163127KK.xml: all nodes which contains morphological "praeclarus" in content (HTML output)</a></li> + <li>Example: <a href="query/QueryDocument?docId=/tei/en/Test_1789.xml&query=tokenOrig:sentence&outputFormat=html">in /tei/en/Test_1789.xml: all nodes which contains "sentence" in original content (HTML output)</a></li> + <li>Example: <a href="query/QueryDocument?docId=/echo/zh/SongYingxing_1637.xml&query=xmlId:N400022&outputFormat=xml">in /echo/zh/SongYingxing_1637.xml: node with xmlId "N400022"</a></li> + </ul> + </li> + </ul> + </li> + + <li><b>Url: /mpiwg-mpdl-cms-web/query/XQueryDocument</b> + <ul> + <li>Request parameters + <ul> + <li>docId (required) + <ul> + <li>document identifier (e.g. "/echo/la/Benedetti_1585.xml"</li> + </ul> + </li> + <li>query (required) + <ul> + <li><a href="http://www.saxonica.com/documentation/javadoc/net/sf/saxon/s9api/package-summary.html">Saxon's</a> <a href="http://www.w3.org/TR/xquery">XQuery</a>/<a href="http://www.w3.org/TR/xpath20">XPath</a></li> + </ul> + </li> + <li>page (optional) + <ul> + <li>result page number</li> + <li>default: 1</li> + </ul> + </li> + <li>pageSize (optional) + <ul> + <li>result page size</li> + <li>default: 10</li> + </ul> + </li> + <li>outputFormat (optional) + <ul> + <li>xml</li> + <li>html</li> + <li>default: xml</li> + </ul> + </li> + </ul> + </li> + <li>Response output + <ul> + <li>Example: <a href="query/XQueryDocument?docId=/echo/la/Benedetti_1585_163127KK.xml&query=count(//*:s)">in /echo/la/Benedetti_1585_163127KK.xml: count all sentences (XML output)</a></li> + <li>Example: <a href="query/XQueryDocument?docId=/echo/la/Benedetti_1585_163127KK.xml&query=//*:s">in /echo/la/Benedetti_1585_163127KK.xml: all sentences (XML output)</a></li> + <li>Example: <a href="query/XQueryDocument?docId=/tei/en/Test_1789.xml&query=//*:note[@n = 1 or @n=2]&outputFormat=html">in /tei/en/Test_1789.xml: note with number 1 or 2 (HTML output)</a></li> + <li>Example: <a href="query/XQueryDocument?docId=/tei/en/Test_1789.xml&query=//*:note/@n">in /tei/en/Test_1789.xml: "n" attributes of all notes (XML output)</a></li> + </ul> + </li> + </ul> + </li> + + <li><b>Url: /wspCmsWebApp/query/MoreLikeThis</b> + <ul> + <li>Request parameters + <ul> + <li>docId (required) + <ul> + <li>document identifier (e.g. "/echo/la/Benedetti_1585_163127KK.xml"</li> + </ul> + </li> + <li>page (optional) + <ul> + <li>result page number</li> + <li>default: 1</li> + </ul> + </li> + <li>pageSize (optional) + <ul> + <li>result page size</li> + <li>default: 10</li> + </ul> + </li> + <li>outputFormat (optional) + <ul> + <li>xml</li> + <li>html</li> + <li>json</li> + <li>default: html</li> + </ul> + </li> + <li>addInf (optional) + <ul> + <li>result with additional information such as person names etc.</li> + <li>true</li> + <li>false</li> + <li>default: false</li> + </ul> + </li> + </ul> + </li> + <li>Response output + <ul> + <li>Similar documents</li> + <li>Example: <a href="query/MoreLikeThis?docId=/echo/la/Benedetti_1585_163127KK.xml">similar documents of "/echo/la/Benedetti_1585_163127KK.xml"</a></li> + </ul> + </li> + </ul> + </li> + + <li><b>Url: /mpiwg-mpdl-cms-web/query/GetToken</b> + <ul> + <li>Request parameters + <ul> + <li>docId (optional) + <ul> + <li>document identifier (e.g. "/echo/la/Benedetti_1585_163127KK.xml"</li> + </ul> + </li> + <li>attribute (optional) + <ul> + <li>identifier</li> + <li>author</li> + <li>title</li> + <li>language</li> + <li>date</li> + <li>rights</li> + <li>license</li> + <li>accessRights</li> + <li>tokenOrig</li> + <li>tokenReg</li> + <li>tokenNorm</li> + <li>tokenMorph</li> + <li>default: tokenOrig</li> + </ul> + </li> + <li>query (optional) + <ul> + <li>beginning characters of token (e.g. "a")</li> + <li>default: empty (beginning alphabetically at first position in all tokens)</li> + </ul> + </li> + <li>count (optional) + <ul> + <li>count of tokens</li> + <li>default: 100</li> + </ul> + </li> + <li>outputFormat (optional) + <ul> + <li>xml</li> + <li>default: xml</li> + </ul> + </li> + </ul> + </li> + <li>Response output + <ul> + <li>Tokens of the attribute</li> + <li>Example: <a href="query/GetToken?attribute=docId">all token in docId field (starting from beginning)</a></li> + <li>Example: <a href="query/GetToken?docId=/echo/la/Benedetti_1585_163127KK.xml&attribute=tokenMorph&query=a">in /echo/la/Benedetti_1585_163127KK.xml: all token beginning with "a" in morphological content (starting from "a")</a></li> + <li>Example: <a href="query/GetToken?attribute=tokenMorph&query=a">in all documents: all token beginning with "a" in morphological content (starting from "a")</a></li> + </ul> + </li> + </ul> + </li> + + <li><b>Url: /mpiwg-mpdl-cms-web/query/GetDocInfo</b> + <ul> + <li>Request parameters + <ul> + <li>docId (required) + <ul> + <li>document identifier (e.g. "/echo/la/Benedetti_1585_163127KK.xml"</li> + </ul> + </li> + <li>field (optional) + <ul> + <li>author</li> + <li>title</li> + <li>language</li> + <li>date</li> + <li>rights</li> + <li>license</li> + <li>accessRights</li> + <li>countPages</li> + <li>lastModified</li> + <li>schema</li> + <li>echoId</li> + <li>toc (table of contents)</li> + <li>figures</li> + <li>notes</li> + <li>notesHandwritten</li> + <li>pages</li> + <li>places</li> + <li>default: null (all fields except automatic generated lists such as toc, figures, etc.)</li> + </ul> + </li> + <li>outputFormat (optional) + <ul> + <li>xml</li> + <li>default: xml</li> + </ul> + </li> + </ul> + </li> + <li>Response output + <ul> + <li>Metadata information of the document (document fields)</li> + <li>Example: <a href="query/GetDocInfo?docId=/echo/la/Benedetti_1585_163127KK.xml">Meta info of document /echo/la/Benedetti_1585_163127KK.xml</a></li> + </ul> + </li> + </ul> + </li> + + <li><b>Url: /mpiwg-mpdl-cms-web/doc/GetDocumentJobs</b> + <ul> + <li>Request parameters + <ul> + <li>id (optional) + <ul> + <li>id of the document job</li> + </ul> + </li> + <li>outputFormat (optional) + <ul> + <li>xml</li> + <li>default: xml</li> + </ul> + </li> + </ul> + </li> + <li>Response output + <ul> + <li>all document jobs (when parameter id is not set) or document job with the specified id</li> + <li>Example: <a href="doc/GetDocumentJobs">get all document jobs</a></li> + </ul> + </li> + </ul> + </li> + +</ul> + +</body></html> \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/css/getDictionaryEntries.css Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,95 @@ +.body { + margin:3px; + color: #000000; + background-color: #ebebeb; + min-width:1100px; +} +div.title { + font-size: 30px; + font-weight: bold; + margin-bottom: 10px; +} +input.query { + border-radius: 5px; +} +span.inputType { + font-size: 22px; + font-weight:bold; + display:block; + margin-top: 10px; +} + +.tree ul, +.tree li { + padding: 0; + margin: 0; + margin-left: 10px; + list-style: none; +} +.tree input { + position: absolute; + opacity: 0; +} +.tree input + label + ul { + margin: 0 0 0 22px; +} +.tree input + label + ul { + display: none; +} +.tree label, +.tree label::before { + cursor: pointer; +} +.tree input:disabled + label { + cursor: default; + opacity: .6; +} +.tree input:checked:not(:disabled) + label + ul { + display: block; +} +.tree label, +.tree label::before { + background: url("/mpiwg-mpdl-lt-web/images/treecheckbox.png") no-repeat; +} +.tree label, +.tree a, +.tree label::before { + display: inline-block; + height: 16px; + line-height: 16px;, + vertical-align: middle; +} +.tree label { + background-position: 18px 0; +} +.tree label::before { + content: ""; + width: 16px; + margin: 0 22px 0 0; + vertical-align: middle; + background-position: 0 -32px; +} +.tree input:checked + label::before { + background-position: 0 -16px; +} + +/* webkit adjacent element selector bugfix */ +@media screen and (-webkit-min-device-pixel-ratio:0) +{ + .tree + { + -webkit-animation: webkit-adjacent-element-selector-bugfix infinite 1s; + } + + @-webkit-keyframes webkit-adjacent-element-selector-bugfix + { + from + { + padding: 0; + } + to + { + padding: 0; + } + } +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/css/page.css Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,269 @@ +/* general */ +.it { font-style:italic } /* DESpecs: _ _ and attribute "it" */ +.bf { font-weight:bold } /* DESpecs: <bf> */ +.sc { font-variant:small-caps } /* DESpecs: <sc> */ +.sub { vertical-align:sub; font-size:.8em } /* DESpecs: <_> */ /* line-height:1em; ? */ +.super { vertical-align:super; font-size:.8em } /* DESpecs: <^> */ +.sm { font-size:smaller } /* DESpecs: <sm>, if within <p> */ +.ul { text-decoration:underline } /* DESpecs: <ul> */ +.ol { text-decoration:overline } /* DESpecs: <ol> */ +.st { text-decoration:line-through } /* DESpecs: <st> */ +.red { color:red } /* DESpecs: <red> */ +.sp { letter-spacing:0.3em } /* DESpecs: <sp> */ /* "font-stretch:wider;" is deprecated */ +.center { text-align:center } +.fr { font-family:'Lucida blackletter'} /* preliminary representation of Fraktur */ +.rom { font-family:'default'} + +/* unused */ +.setoff { margin-left:2cm; margin-right:2cm } + +/* highlighting */ +.highlight { background-color:#D3D3D3; } /* LightGrey */ +div.highlight.elem { background-color:#D3D3D3; } /* LightGrey */ +span.highlight.elem { background-color:#D3D3D3; } /* LightGrey */ +span.highlight.hit { background-color:#77DD77; } /* pastell green */ +span.hit.highlight { background-color:#77DD77; } /* pastell green */ + +/* about */ +span.about {float:right;} + +/* query results */ +span.query { font-weight:bold; font-size:150%; } +span.result { display:none } +tr.hit { vertical-align: top; } +td.hitNum { display:inline-block; } +td.hitLink { margin-left:1%; display:inline-block; width:30%; } +td.hitContent { display:inline-block; width:90%; } + +/* xml presentation */ +span.xml.elementName { font-weight:bold;color:purple; } +span.xml.attributeName { font-weight:bold; } +span.xml.attributeValue { color:blue; } +span.xml.comment { color:green; font-style:italic; } +ul.xml.element { margin-left:0px;padding-left:8px } +ul.xml.element.highlight { background-color:#D3D3D3; } /* LightGrey */ + +/* divs */ +div.float.none { + margin-left:10%; +} +div.float.right { + float:right; + clear:both; + margin-right:10% +} + +/* page breaks */ +span.pb { + display:none; +} + +/* Figures */ +span.figure { + display: block; + width: 200px; + margin-top: 0.5em; + margin-bottom: 0.5em; + padding: 5px; + border: 1px dashed silver; +} +span.figureNumber { +} +span.figureNum { + display:none; +} +span.figureNumText { + display: block; +} +span.caption { +} +span.description { + display: block; +} +span.variables { + display: block; +} + +span.handwritten { +} + +/* toc elements */ +div.tocItem { + float:left; + width:90%; +} +span.tocPageRef { + float:right; + vertical-align:top; +} + +/* paragraphs */ +div.p { + margin: 1.12em 0px; +} + +/* initials */ +span.initial { + float: left; + font-family: Georgia; + font-size: 250%; + line-height: 0.8em; + margin-right: 10px; + padding-top: 1px; +} + +/* small caps */ +span.sc { font-weight:bold; } +span.sc.it {font-weight:bold; font-style:italic; } + +/* variables and numbers */ +span.var { font-style:italic; } +span.var.segment { font-style:italic; text-decoration:overline; } +span.var.line { font-style:italic; text-decoration:underline; } +span.var.gnomon { font-style:italic; text-decoration:line-through; } +span.num { color:maroon } + +/* foreign languages */ +span.foreign { color:#006400; } +span.foreign a:link { color:#006400; } +span.foreign a:visited { color:#1B370D; } +span.foreign.el span.foreign.grc span.foreign.greek { color:#006400; } +span.foreign.en span.foreign.english { color:#006400; } +span.foreign.fr span.foreign.french { color:#006400; } +span.foreign.la span.foreign.latin { color:#006400; } + +/* quotes */ +span.q { font-style:italic; } +div.quote { margin-left:10px; font-style:italic; } +div.blockquote { margin-left:10px; font-style:normal; } +div.set-off { margin-left:10px; font-style:italic; } + +/* dictionary / terminology */ +span.term { color:purple; } /* Blue */ +span.entry { display:block; margin-bottom:20px;} +span.entryDiv { margin-left:2em; display:block; margin-bottom:0.5em; margin-top:0.5em;} +span.orth { font-weight:bold; } +span.mentioned { font-style:italic; } + +/* GIS elements */ +span.place { color:#0000FF; } /* Blue */ +span.person { color:#0000FF; } /* Blue */ + +/* sentence */ +span.s.highlight { background-color:#D3D3D3; } /* LightGrey */ +span.s:hover { background-color:#D3D3D3; } /* LightGrey */ +span.s.it { font-style:italic; } + +/* ref */ +span.ref a:link {text-decoration: underline; color: blue;} +span.ref a:visited {text-decoration: none; color: #800080;} +span.ref a:hover {text-decoration: underline; color: blue;} + +/* Links */ +a.dictionary:link {text-decoration: none; color: #892B06;} +a.dictionary:visited {text-decoration: none; color: #892B06;} +a.dictionary:hover {text-decoration: underline; color: #892B06;} + +/* Notes */ +span.note { font-style:italic; } +span.note span.noteBody span.noteSign {display:none;} +span.note.handwritten span.noteSign {display:none;} +span.note.handwritten span.noteBody span.noteSign {display:none;} +span.note span.left { position:absolute; left:2em; width: 5em; font-style:normal; } +span.note span.right { position:absolute; left:30em; width: 5em; font-style:normal; } +hr.notesBottom { + border:none; + border-top:2px dashed black; + height:2px; + color:#000000; + background:transparent; +} +a.note { + color: blue; + margin-right: 0.2em; +} +a.note:before { + content: url('../images/linkback.png'); + margin-left: 0.1em; + margin-right: 0.2em; +} +a.note:link { text-decoration: none; } +a.note:visited { text-decoration: none; } +a.note:hover { text-decoration: underline; } +a.noteRef { + color: blue; + margin-right: 0.2em; +} +a.noteRef:before { + content: url('../images/linkto.png'); + margin-left: 0.1em; + margin-right: 0.2em; +} +a.noteRef:link { text-decoration: none; } +a.noteRef:visited { text-decoration: none; } +a.noteRef:hover { text-decoration: underline; } +/* +a:link {text-decoration: none;} +a:visited {text-decoration: none} +a:active {text-decoration: none} +a:hover {text-decoration: underline; color: red;} + */ + +/* table */ +table { margin-left:30px; } + +table.toc { width:100%; margin-left:1px; } + +/* page styling for generating PDF documents with Flying Saucer */ +div.pageHeaderTitle { + font-weight:bold; + text-align:center; +} +body { + counter-reset: pn; +} +div.page:before { + float:right; + font:11pt sans-serif; + font-weight:bold; + content:"[Page " counter(pn) "]"; + counter-increment:pn; + page:pdfPage; + clear:both; +} +div.page { + page-break-after:always; + page:pdfPage; + clear:both; +} +@page pdfPage { + size:A4; + margin-top:0.7cm; + margin-bottom:0.7cm; + margin-left:0.7cm; + margin-right:0.7cm; + padding:0.2cm; + border:thin solid #808080; + @top-left { font: 11pt sans-serif; padding-left: 0.2cm; padding-right: 1cm; font-weight:bold;}; + @top-right { font: 11pt sans-serif; white-space: nowrap; font-weight:bold;}; + @bottom-left { font: 11pt sans-serif; white-space: nowrap; font-weight:bold;}; + @bottom-right { font: 11pt sans-serif; white-space: nowrap; content: counter(page);}; +} + +div.tocPage { + page-break-after:always; + page:tocPage; +} +@page tocPage { + size:A4; + margin-top:0.7cm; + margin-bottom:0.7cm; + margin-left:0.7cm; + margin-right:0.7cm; + padding:0.2cm; + border:thin solid #808080; + @top-left { font: 11pt sans-serif; padding-left: 0.2cm; padding-right: 1cm; font-weight:bold;}; + @top-right { font: 11pt sans-serif; white-space: nowrap; font-weight:bold;}; + @bottom-left { font: 11pt sans-serif; white-space: nowrap; font-weight:bold;}; + @bottom-right { font: 11pt sans-serif; white-space: nowrap; content: counter(page);}; +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/css/pageNorm.css Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,12 @@ +span.w a.dictionary { + display:none; +} +span.w span.dictionary { + display:none; +} +span.w span.orig { + display:none; +} +span.w span.reg { + display:none; +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/css/pageNormDict.css Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,15 @@ +span.w a.dictionary span.orig { + display:none; +} +span.w a.dictionary span.reg { + display:none; +} +span.w span.dictionary span.orig { + display:none; +} +span.w span.dictionary span.reg { + display:none; +} +span.w span.nodictionary { + display:none; +} \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/css/pageOrig.css Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,12 @@ +span.w a.dictionary { + display:none; +} +span.w span.dictionary { + display:none; +} +span.w span.reg { + display:none; +} +span.w span.norm { + display:none; +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/css/pageOrigDict.css Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,15 @@ +span.w a.dictionary span.reg { + display:none; +} +span.w a.dictionary span.norm { + display:none; +} +span.w span.dictionary span.reg { + display:none; +} +span.w span.dictionary span.norm { + display:none; +} +span.w span.nodictionary { + display:none; +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/css/pageReg.css Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,12 @@ +span.w a.dictionary { + display:none; +} +span.w span.dictionary { + display:none; +} +span.w span.orig { + display:none; +} +span.w span.norm { + display:none; +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/css/pageRegDict.css Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,15 @@ +span.w a.dictionary span.orig { + display:none; +} +span.w a.dictionary span.norm { + display:none; +} +span.w span.dictionary span.orig { + display:none; +} +span.w span.dictionary span.norm { + display:none; +} +span.w span.nodictionary { + display:none; +}
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/index.html Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,15 @@ +<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0//EN"> +<html> + <head> + <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> + <title>Max Planck Institute for the History of Science - Language technology & CMS services</title> + </head> +<body> +<h2>Max Planck Institute for the History of Science - Language technology & CMS services</h2> + +<ul> + <li><a href="lt-services.html">Language technology services</a></li> + <li><a href="cms-services.html">CMS technology services</a></li> +</ul> + +</body></html> \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/lt-services.html Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,408 @@ +<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0//EN"> +<html> + <head> + <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> + <title>Max Planck Institute for the History of Science - Language technology services</title> + </head> +<body> +<table align="right"> +<tr> +<td> + [<i>This software is dedicated to <a href="http://mpdl-proto.mpiwg-berlin.mpg.de/mpdl/info.xql?info=malcolm">Dr. Malcolm Hyman</a></i>] + <img src="/mpiwg-mpdl-cms-web/images/info.png" width="15" height="15" border="0" alt="Info"/><br/> + [<i>It is based on <a href="http://archimedes.fas.harvard.edu/">Donatus and Pollux</a></i>] + <img src="/mpiwg-mpdl-cms-web/images/info.png" width="15" height="15" border="0" alt="Info"/> +</td> +</tr> +</table> +<h2>Max Planck Institute for the History of Science - Language technology services</h2> + +<ul> + <li><b>Url: /mpiwg-mpdl-cms-web/lt/GetDictionaryEntries</b> + <ul> + <li>Request parameters + <ul> + <li>query (required) + <ul> + <li>by one form or lemma (e.g. "revolution")</li> + <li>by a list of forms or lemmas (e.g. "revolution equality brotherliness")</li> + <li>by a prefix range: entries starting with a prefix (e.g. "a*")</li> + </ul> + </li> + <li>queryDisplay (optional) + <ul> + <li>display of the query</li> + <li>default: content of parameter "query"</li> + </ul> + </li> + <li>inputType (optional) + <ul> + <li>"form"</li> + <li>"lemma"</li> + <li>default: "form"</li> + </ul> + </li> + <li>language (optional) + <ul> + <li>ISO 639-3 specifier</li> + <li>default: "eng"</li> + </ul> + </li> + <li>dictionary (optional) + <ul> + <li>dictionary name, e.g. "webster"</li> + <li>default: "all" (all dictionaries for the specified language)</li> + </ul> + </li> + <li>outputType (optional) + <ul> + <li>this parameter can occur many times (e.g. "outputType=morphCompact&outputType=dictCompact") + <ul> + <li>"morphCompact"</li> + <li>"dictCompact"</li> + <li>"wikiCompact"</li> + <li>"allCompact" (all output types compact)</li> + <li>"morphFull"</li> + <li>"dictFull"</li> + <li>"wikiFull"</li> + <li>"allFull" (all output types full)</li> + </ul> + </li> + <li>default: "allCompact"</li> + </ul> + </li> + <li>outputFormat (optional) + <ul> + <li>"html"</li> + <li>"xml"</li> + <li>default: "xml"</li> + </ul> + </li> + <li>normalization (optional) + <ul> + <li>"none"</li> + <li>"norm"</li> + <li>default: "norm"</li> + </ul> + </li> + <li>resultPageNumber (optional) + <ul> + <li>works only for range queries</li> + <li>page number of the result (e.g. "2": result entries from position 51 to 100)</li> + <li>default: "1"</li> + </ul> + </li> + <li>resultPageSize (optional) + <ul> + <li>works only for range queries</li> + <li>page size of the result (e.g. "100": each result page has a size of 100)</li> + <li>default: "50"</li> + </ul> + </li> + </ul> + </li> + <li>Response output + <ul> + <li>dependent of outputFormat, outputType and resultPage: morphology, dictionary and Wikipedia entries in Xml or Html format</li> + <li>Example: <a href="lt/GetDictionaryEntries?query=a*&language=lat&outputFormat=html">query=a*&language=lat&outputFormat=html</a></li> + <li>Example: <a href="lt/GetDictionaryEntries?query=a*&dictionary=ls">query=a*&dictionary=ls</a></li> + <li>Example: <a href="lt/GetDictionaryEntries?query=revolution&language=eng">query=revolution&language=lat</a></li> + <li>Example: <a href="lt/GetDictionaryEntries?query=multa&language=lat&outputFormat=html&outputType=allCompact">query=multa&language=lat&outputFormat=html&outputType=allCompact</a></li> + </ul> + </li> + </ul> + </li> + + <li><b>Url: /mpiwg-mpdl-cms-web/lt/GetLemmas</b> + <ul> + <li>Request parameters + <ul> + <li>query (required) + <ul> + <li>one form or lemma (e.g. "revolution") or</li> + <li>blank separated list of forms or lemmas (e.g. "revolution equality brotherliness")</li> + </ul> + </li> + <li>inputType (optional) + <ul> + <li>"form"</li> + <li>"lemma"</li> + <li>default: "form"</li> + </ul> + </li> + <li>language (optional) + <ul> + <li>ISO 639-3 specifier</li> + <li>default: "eng"</li> + </ul> + </li> + <li>outputType (optional) + <ul> + <li>"compact"</li> + <li>"full"</li> + <li>default: "compact"</li> + </ul> + </li> + <li>outputFormat (optional) + <ul> + <li>"html"</li> + <li>"xml"</li> + <li>"string" (lemma names separated by a blank)</li> + <li>default: "xml"</li> + </ul> + </li> + <li>normalization (optional) + <ul> + <li>"none"</li> + <li>"norm"</li> + <li>default: "norm"</li> + </ul> + </li> + </ul> + </li> + <li>Response output + <ul> + <li>dependent of outputFormat and outputType: lemma entries in Xml or Html or string format</li> + <li>Example: <a href="lt/GetLemmas?query=multa&language=lat&outputFormat=html">query=multa&language=lat&outputFormat=html</a></li> + </ul> + </li> + </ul> + </li> + + <li><b>Url: /mpiwg-mpdl-cms-web/lt/GetForms</b> + <ul> + <li>Request parameters + <ul> + <li>query (required) + <ul> + <li>one lemma (e.g. "revolution") or</li> + <li>blank separated list of forms (e.g. "revolution equality brotherliness")</li> + </ul> + </li> + <li>language (optional) + <ul> + <li>ISO 639-3 specifier</li> + <li>default: "eng"</li> + </ul> + </li> + <li>outputType (optional) + <ul> + <li>"compact"</li> + <li>"full"</li> + <li>default: "compact"</li> + </ul> + </li> + <li>outputFormat (optional) + <ul> + <li>"html"</li> + <li>"xml"</li> + <li>"string" (lemma names separated by a blank)</li> + <li>default: "xml"</li> + </ul> + </li> + <li>normalization (optional) + <ul> + <li>"none"</li> + <li>"norm"</li> + <li>default: "norm"</li> + </ul> + </li> + </ul> + </li> + <li>Response output + <ul> + <li>dependent of outputFormat and outputType: form entries in Xml or Html or string format</li> + <li>Example: <a href="lt/GetForms?query=edo sum&language=lat&outputFormat=string">query=edo sum&language=lat&outputFormat=string</a></li> + </ul> + </li> + </ul> + </li> + + <li><b>Url: /mpiwg-mpdl-cms-web/text/Tokenize</b> + <ul> + <li>Request parameters + <ul> + <li>inputString or srcUrl (required) + <ul> + <li>inputString + <ul> + <li>string which should be tokenized + <ul> + <li>unstructured text</li> + <li>XML fragment/document</li> + </ul> + </li> + </ul> + </li> + <li>srcUrl + <ul> + <li>source URL + <ul> + <li>unstructured text</li> + <li>XML fragment/document</li> + </ul> + </li> + </ul> + </li> + </ul> + </li> + <li>language (optional) + <ul> + <li>ISO 639-3 specifier</li> + <li>if input is XML and an element contains the attribute "xml:lang" this value is used for this element</li> + <li>default: "eng"</li> + </ul> + </li> + <li>normalization (optional) + <ul> + <li>"none" (no normalization)</li> + <li>"reg" (regularized)</li> + <li>"norm" (regularized + normalized)</li> + <li>default: "norm"</li> + </ul> + </li> + <li>normalizationType (optional) + <ul> + <li>"dictionary"</li> + <li>"display"</li> + <li>default: "dictionary"</li> + </ul> + </li> + <li>elements (optional) + <ul> + <li>list of xml element names which should be tokenized (e.g. "s head")</li> + <li>default: empty list (which means: all elements are tokenized)</li> + </ul> + </li> + <li>stopElements (optional) + <ul> + <li>list of xml element names which are stop elements(e.g. "var emph"): stop elements: its tokens should not get word tags (when output format is "xml") or its tokens should be removed (if output format is "string")</li> + <li>default: empty list</li> + </ul> + </li> + <li>highlightTerms (optional) + <ul> + <li>list of word forms which should be highlighted. Each matched word form is surrounded by <hi></hi>. The matching function is dependent of the normalization. E.g. if normalization = "norm" then the normalized word form is fetched and highlighted.</li> + <li>default: empty list</li> + </ul> + </li> + <li>outputFormat (optional) + <ul> + <li>"xml"</li> + <li>"string"</li> + <li>default: "xml"</li> + </ul> + </li> + <li>outputOptions (optional) + <ul> + <li>output options separated by blanks (e.g. "withForms withLemmas") + <ul> + <li>"withForms"</li> + <li>"withLemmas"</li> + <li>default: empty list</li> + </ul> + </li> + </ul> + </li> + </ul> + </li> + <li>Response output + <ul> + <li>outputFormat=xml + <ul> + <li>tokenized inputString or document (enriched by element <w>) + <ul> + <li>Example: <s><w lang="deu" form="dies" formRegularized="dies" formNormalized="dies" forms="dies, dieser, dieses, diesen" lemmas="dieser">Dies</w> <w + lang="deu" form="ist" formRegularized="ist" formNormalized="ist" forms="bin, bist, ist, seid, sind, sein, war, warst, wart" lemmas="sein">ist</w> <w + lang="deu" form="ein" formRegularized="ein" formNormalized="ein" forms="ein, eines, einer" lemmas="ein">ein</w> <w lang="deu" form="satz" + formRegularized="satz" formNormalized="satz" forms="satz, sätze, satzes" lemmas="satz">Satz</w></s> + </li> + </ul> + </li> + </ul> + <li>outputFormat=string + <ul> + <li>word tokens of inputString or document (separated by Blank)</li> + </ul> + <li>Example: <a href="text/Tokenize?inputString=edo sum philoſophi&language=lat&outputFormat=xml">inputString=edo sum philoſophi&language=lat&outputFormat=xml</a></li> + <li>Example: <a href="text/Tokenize?language=lat&srcUrl=http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/page-query-result.xql?document=/echo/la/Benedetti_1585.xml%26mode=pureXml%26pn=13">language=lat&srcUrl=http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/page-query-result.xql?document=/echo/la/Benedetti_1585.xml%26mode=pureXml%26pn=13</a></li> + <li>Example: <a href="text/Tokenize?language=lat&highlightTerms=eorumque&srcUrl=http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/page-query-result.xql?document=/echo/la/Benedetti_1585.xml%26mode=pureXml%26pn=13">language=lat&highlightTerms=eorumque&srcUrl=http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/page-query-result.xql?document=/echo/la/Benedetti_1585.xml%26mode=pureXml%26pn=13</a></li> + <li>Example: <a href="text/Tokenize?language=lat&outputOptions=withForms withLemmas&srcUrl=http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/page-query-result.xql?document=/echo/la/Benedetti_1585.xml%26mode=pureXml%26pn=13">language=lat&outputOptions=withForms withLemmas&srcUrl=http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/page-query-result.xql?document=/echo/la/Benedetti_1585.xml%26mode=pureXml%26pn=13</a></li> + <li>Example: <a href="text/Tokenize?language=lat&outputFormat=string&normalization=orig&srcUrl=http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/page-query-result.xql?document=/echo/la/Benedetti_1585.xml%26mode=pureXml%26pn=13">language=lat&outputFormat=string&normalization=orig&srcUrl=http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/page-query-result.xql?document=/echo/la/Benedetti_1585.xml%26mode=pureXml%26pn=13</a></li> + <li>Example: <a href="text/Tokenize?language=lat&outputFormat=string&outputOptions=withLemmas&srcUrl=http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/page-query-result.xql?document=/echo/la/Benedetti_1585.xml%26mode=pureXml%26pn=13">language=lat&outputFormat=string&outputOptions=withLemmas&srcUrl=http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/page-query-result.xql?document=/echo/la/Benedetti_1585.xml%26mode=pureXml%26pn=13</a></li> + </ul> + </li> + </ul> + </li> + + <li><b>Url: /mpiwg-mpdl-cms-web/text/Normalize</b> + <ul> + <li>Request parameters + <ul> + <li>inputString (required) + <ul> + <li>string which should be normalized</li> + </ul> + </li> + <li>language (optional) + <ul> + <li>ISO 639-3 specifier</li> + <li>default: "eng"</li> + </ul> + </li> + <li>type (optional) + <ul> + <li>"dictionary"</li> + <li>"display"</li> + <li>default: "display"</li> + </ul> + </li> + </ul> + </li> + <li>Response output + <ul> + <li>normalized string</li> + <li>Example: <a href="text/Normalize?inputString=philoſophi&language=lat">inputString=philoſophi&language=lat</a></li> + </ul> + </li> + </ul> + </li> + + <li><b>Url: /mpiwg-mpdl-cms-web/text/Transcode</b> + <ul> + <li>Request parameters + <ul> + <li>inputString (required) + <ul> + <li>string which should be transcoded</li> + </ul> + </li> + <li>srcEncoding (required) + <ul> + <li>"betacode"</li> + <li>"buckwalter"</li> + <li>"unicode"</li> + </ul> + </li> + <li>destEncoding (optional) + <ul> + <li>"betacode"</li> + <li>"buckwalter"</li> + <li>"unicode"</li> + <li>default: "unicode"</li> + </ul> + </li> + </ul> + </li> + <li>Response output + <ul> + <li>transcoded string</li> + <li>Example: <a href="text/Transcode?inputString=kai/&srcEncoding=betacode&destEncoding=unicode">inputString=kai/&srcEncoding=betacode&destEncoding=unicode</a></li> + </ul> + </li> + </ul> + </li> +</ul> + + +</body></html> \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/query/index.html Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,313 @@ +<html> +<head> +<title>MPIWG Query project</title> +<script type="text/javascript"> +<!-- +function Browse() { + var luceneQuery = document.getElementById("luceneQueryId1"); + var schema1checked = document.getElementById("schema1Id").checked; + var schema2checked = document.getElementById("schema2Id").checked; + var schema3checked = document.getElementById("schema3Id").checked; + var schema4checked = document.getElementById("schema4Id").checked; + var luceneSchemaQueryStr = GetLuceneSchemaQuery(); + if (schema1checked && schema2checked && schema3checked && schema4checked) + luceneSchemaQueryStr = "*"; + luceneQuery.value = luceneSchemaQueryStr; +} +function GetLuceneSchemaQuery() { + var schemaFieldValue = "schemaName"; + var schema1Val = document.getElementById("schema1Id").value; + var schema2Val = document.getElementById("schema2Id").value; + var schema3Val = document.getElementById("schema3Id").value; + var schema4Val = document.getElementById("schema4Id").value; + var schema1checked = document.getElementById("schema1Id").checked; + var schema2checked = document.getElementById("schema2Id").checked; + var schema3checked = document.getElementById("schema3Id").checked; + var schema4checked = document.getElementById("schema4Id").checked; + var luceneQueryStr = "+" + schemaFieldValue + ":" + "("; + if (schema1checked) + luceneQueryStr = luceneQueryStr + schema1Val; + if (schema2checked) + luceneQueryStr = luceneQueryStr + " " + schema2Val; + if (schema3checked) + luceneQueryStr = luceneQueryStr + " " + schema3Val; + if (schema4checked) + luceneQueryStr = luceneQueryStr + " " + schema4Val; + luceneQueryStr = luceneQueryStr + ")"; + return luceneQueryStr; +} +function GetLuceneBoolOp(relOp) { + if (relOp == "mustContain") + return "+"; + else if (relOp == "mustNotContain") + return "-"; + else + return ""; +} +function AttributeQuery() { + var luceneQuery = document.getElementById("luceneQueryId2"); + var sortBy = document.getElementById("sortById2"); + var field1Val = document.getElementById("field1Id").value; + var field1QueryVal = document.getElementById("field1QueryId").value; + var relOp1Value = document.getElementById("relOp1Id").value; + var relOp1ValueLucene = GetLuceneBoolOp(relOp1Value); + var field2Val = document.getElementById("field2Id").value; + var field2QueryVal = document.getElementById("field2QueryId").value; + var relOp2Value = document.getElementById("relOp2Id").value; + var relOp2ValueLucene = GetLuceneBoolOp(relOp2Value); + var luceneQueryStr = "*:* " + relOp1ValueLucene + field1Val + ":"; // *:* so that the minus-Operator works correct when it is alone + luceneQueryStr = luceneQueryStr + field1QueryVal; + if (field2QueryVal != "") + luceneQueryStr = luceneQueryStr + " " + relOp2ValueLucene + field2Val + ":" + field2QueryVal; + var schemaQueryStr = GetLuceneSchemaQuery(); + var schema1checked = document.getElementById("schema1Id").checked; + var schema2checked = document.getElementById("schema2Id").checked; + var schema3checked = document.getElementById("schema3Id").checked; + var schema4checked = document.getElementById("schema4Id").checked; + if (schema1checked && schema2checked && schema3checked && schema4checked) + schemaQueryStr = ""; + luceneQueryStr = luceneQueryStr + " " + schemaQueryStr; + luceneQuery.value = luceneQueryStr; + sortBy.value = "author"; +} +function FulltextQuery() { + var luceneQuery = document.getElementById("luceneQueryId3"); + var translateElem = document.getElementById("translateId3"); + translateElem.value = "false"; + var isTranslateQuery = document.getElementById("isTranslateQueryId").checked; + if (isTranslateQuery) + translateElem.value = "true"; + var fulltextQueryFieldValue = "tokenOrig"; + var isMorphQuery = document.getElementById("isMorphQueryId").checked; + if (isMorphQuery) + fulltextQueryFieldValue = "tokenMorph"; + var fulltextQueryVal = document.getElementById("fulltextQueryId").value; + var luceneQueryStr = "+" + fulltextQueryFieldValue + ":(" + fulltextQueryVal + ")"; + changeIsMorph(); + var schemaQueryStr = GetLuceneSchemaQuery(); + var schema1checked = document.getElementById("schema1Id").checked; + var schema2checked = document.getElementById("schema2Id").checked; + var schema3checked = document.getElementById("schema3Id").checked; + var schema4checked = document.getElementById("schema4Id").checked; + if (schema1checked && schema2checked && schema3checked && schema4checked) + schemaQueryStr = ""; + luceneQueryStr = luceneQueryStr + " " + schemaQueryStr; + luceneQuery.value = luceneQueryStr; +} +function changeIsMorph() { + var isMorphQuery = document.getElementById("isMorphQueryId").checked; + var language = document.getElementById("languageId3"); + if (isMorphQuery) { + var languageValue = document.getElementById("languageId").value; + if (languageValue == "none") { + if (language != null) + language.parentNode.removeChild(language); + } else if (language == null) { + var queryDocumentsForm = document.getElementById("queryDocumentsId"); + var languageElement = document.createElement("input"); + languageElement.setAttribute("type", "hidden"); + languageElement.setAttribute("name", "language"); + languageElement.setAttribute("id", "languageId3"); + languageElement.value = languageValue; + queryDocumentsForm.appendChild(languageElement); + } else if (language != null) { + language.value = languageValue; + } + } else { + if (language != null) + language.parentNode.removeChild(language); + } +} +function checkCR(event) { + var keyCode = event.keyCode + if (keyCode == 13) + return false; +} + +--> +</script> +</head> +<body> + <table height="60px"> + <colgroup> + <col width="90%"/> + <col width="10%"/> + </colgroup> + <tr> + <td align="left" valign="top"> + <text style="font-weight:bold;font-size:30px">Query documents <a href="info.html"><img src="../images/info.png" valign="bottom" width="18" height="18" border="0" alt="Info Query Documents"/></a></text> + </td> + <td align="left" valign="top"> + Release 0.5, July, 2012 + </td> + </tr> + </table> + <hr/> + <table> + <tr> + <td valign="top" height="60px"> + <table> + <tr> + <td valign="top"><b>Documents:</b></td> + <td> + <text style="margin-left:15px;">Archimedes DTD</text><br/> + <text style="margin-left:15px;">(until 2008)</text> + </td> + <td valign="top"> + <text style="margin-left:1px;"></text><input type="checkbox" name="schema1" id="schema1Id" value="archimedes" checked="checked"/> + </td> + <td> + <text style="margin-left:20px;">Echo Schema</text><br/> + <text style="margin-left:20px;">(since 2009)</text> + </td> + <td valign="top"> + <text style="margin-left:1px;"></text><input type="checkbox" name="schema2" id="schema2Id" value="echo" checked="checked"/> + </td> + <td> + <text style="margin-left:20px;">TEI Schema</text><br/> + <text style="margin-left:20px;">(since 2011)</text> + </td> + <td valign="top"> + <text style="margin-left:1px;"></text><input type="checkbox" name="schema3" id="schema3Id" value="TEI" checked="checked"/> + </td> + <td> + <text style="margin-left:20px;">Diverse</text><br/> + <text style="margin-left:20px;">(since 2011)</text> + </td> + <td valign="top"> + <text style="margin-left:1px;"></text><input type="checkbox" name="schema4" id="schema4Id" value="html" checked="checked"/> + </td> + <td valign="top"> + <form name="queryDocuments" action="/mpiwg-mpdl-cms-web/query/QueryDocuments" method="get"> + <input type="hidden" name="query" id="luceneQueryId1"/> + <input type="hidden" name="sortBy" id="sortById1" value="author"/> + <text style="margin-left:40px;"></text><button type="submit" onclick="Browse()">Browse</button> + </form> + </td> + </tr> + <tr> + </tr> + </table> + </td> + </tr> + + <tr> + <td valign="top" height="70px"> + <table> + <tr> + <td> + <select name="field1" id="field1Id"> + <option value ="author" selected="true">Author</option> + <option value ="title">Title</option> + <option value ="publisher">Place</option> + <option value ="date">Year</option> + <option value ="docId">Id</option> + <option value ="language">Language</option> + </select> + </td> + <td> + <select name="relOp1" id="relOp1Id"> + <option value ="mustContain" selected="true">must contain</option> + <option value ="mayContain">may contain</option> + <option value ="mustNotContain">does not contain</option> + </select> + </td> + <td> + <input type="text" size="40" name="field1Query" id="field1QueryId" onkeypress="return checkCR(event)"/> + </td> + <td valign="middle"> + <form name="queryDocuments" action="/mpiwg-mpdl-cms-web/query/QueryDocuments" method="get"> + <input type="hidden" name="query" id="luceneQueryId2"/> + <input type="hidden" name="sortBy" id="sortById2"/> + <button type="submit" onclick="AttributeQuery()">Query</button> + </form> + </td> + </tr> + + <tr> + <td> + <select name="field2" id="field2Id"> + <option value ="author">Author</option> + <option value ="title" selected="true">Title</option> + <option value ="publisher">Place</option> + <option value ="date">Year</option> + <option value ="docId">Id</option> + <option value ="language">Language</option> + </select> + </td> + <td> + <select name="relOp2" id="relOp2Id"> + <option value ="mustContain" selected="true">must contain</option> + <option value ="mayContain">may contain</option> + <option value ="mustNotContain">does not contain</option> + </select> + </td> + <td> + <input type="text" size="40" name="field2Query" id="field2QueryId" value="" onkeypress="return checkCR(event)"/> + </td> + </tr> + </table> + </td> + </tr> + + <tr> + <td valign="top" height="60px"> + <table> + <tr> + <td> + Document contains <input type="text" size="40" name="fulltextQuery" id="fulltextQueryId" onkeypress="return checkCR(event)"/> + </td> + <td valign="middle"> + <form id="queryDocumentsId" name="queryDocuments" action="/mpiwg-mpdl-cms-web/query/QueryDocuments" method="get"> + <input type="hidden" name="query" id="luceneQueryId3"/> + <input type="hidden" name="language" id="languageId3"/> + <input type="hidden" name="translate" id="translateId3"/> + <button type="submit" onclick="FulltextQuery()">Query</button> + </form> + </td> + </tr> + </table> + <table> + <tr> + <td valign="top"> + <text style="margin-left:30px;">morphological</text> + </td> + <td valign="top"> + <text style="margin-left:1px;"></text><input type="checkbox" name="isMorphQuery" id="isMorphQueryId" value="isMorphQuery" onchange="ChangeIsMorph()"/> + </td> + <td valign="top"> + <text style="margin-left:3px;">language: </text> + </td> + <td valign="top" style="margin-left:3px;"> + <text style="margin-left:1px;"></text> + <select name="language" id="languageId"> + <option value ="none" selected="true">None</option> + <option value ="ara">Arabic</option> + <option value ="zho">Chinese</option> + <option value ="nld">Dutch</option> + <option value ="eng">English</option> + <option value ="fra">French</option> + <option value ="deu">German</option> + <option value ="grc">Greek</option> + <option value ="ita">Italian</option> + <option value ="lat">Latin</option> + </select> + </td> + <td valign="top"> + <text style="margin-left:10px;">translate</text><br/> + </td> + <td valign="top"> + <text style="margin-left:1px;"></text><input type="checkbox" name="isTranslateQuery" id="isTranslateQueryId" value="translate"/> + </td> + </tr> + </table> + </td> + </tr> + + </table> + <hr/> + <p/> + See the <a href="../cms-services.html">CMS Services Page</a>, if you find a bug <a href="https://it-dev.mpiwg-berlin.mpg.de/tracs/mpdl-project-software/newticket">let us know</a> + <br/>Last software update: July, 2012 +</body> +</html> \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/query/info.html Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,21 @@ +<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0//EN"> +<html> + <head> + <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> + <title>Max Planck Institute for the History of Science - CMS services - Documentation</title> + </head> +<body> +<h2>Max Planck Institute for the History of Science - CMS services- Documentation</h2> + +<h4>Document bases</h4> +<ul> + <li><b>Archimedes DTD (until 2008): </b>more than 120 documents (4KB - 18MB), coming from Echo collections "Archimedes" and "Historical Travel Guides", languages: latin, italian, english, german, french, dutch, greek, arabic, chinese<br/></li> + <li><b>Echo Schema (since 2009): </b>planned more than 100 documents in different languages<br/></li> + <li><b>TEI Schema (since 2011): </b>different languages, mainly by external document providers<br/></li> + <li><b>Diverse (since 2011): </b>different languages, no schema prerequesites, mainly HTML documents</li> +</ul> + +<h4>Metadata and fulltext search</h4> +Query power of Lucene, see: <a href="http://lucene.apache.org/core/3_6_0/queryparsersyntax.html">Lucene query syntax</a> + +</body></html> \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/WebContent/xsl/generateId.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,34 @@ +<?xml version="1.0"?> + +<xsl:stylesheet version="2.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform"> + +<xsl:output method="xml" encoding="utf-8"/> + +<xsl:template match="@*|node()"> + <xsl:copy> + <xsl:apply-templates select="@*|node()"/> + </xsl:copy> +</xsl:template> + +<xsl:template match="*:s|*:head"> + <xsl:variable name="elemName" select="name()"/> + <xsl:variable name="docPos"> + <xsl:choose> + <xsl:when test="$elemName = 's'"> + <xsl:value-of select="count(./preceding::*:s) + 1"/> + </xsl:when> + <xsl:when test="$elemName = 'head'"> + <xsl:value-of select="count(./preceding::*:head) + 1"/> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:copy> + <xsl:attribute name="xml:id"> + <xsl:value-of select="concat($elemName, $docPos)"/> + </xsl:attribute> + <xsl:apply-templates select="@*|node()"/> + </xsl:copy> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/build/build.xml Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,57 @@ +<!DOCTYPE project> +<project name="mpiwg-mpdl-cms-web" default="dist" basedir="../"> + <description>mpiwg-mpdl-cms-web</description> + <!-- global properties --> + <property name="baseLibFile" location="../mpiwg-mpdl-cms/dist/mpiwg-mpdl-cms.jar"/> + <property name="src" location="src"/> + <property name="lib" location="WebContent/WEB-INF/lib"/> + <property name="libTomcat" location="/Applications/java/apache-tomcat-7.0.26/lib"/> + <property name="webappTomcat" location="/Applications/java/apache-tomcat-7.0.26/webapps"/> + <property name="build" location="build/classes"/> + <property name="dist" location="dist"/> + + <path id="classpath"> + <fileset dir="${lib}" includes="**/*.jar"/> + <fileset dir="${libTomcat}" includes="**/*.jar"/> + </path> + + <target name="init"> + <!-- Create time stamp --> + <tstamp/> + <mkdir dir="${build}"/> + <mkdir dir="${dist}"/> + <copy file="${baseLibFile}" todir="${lib}"/> + </target> + + <target name="compile" depends="init" description="compile"> + <javac srcdir="${src}" destdir="${build}" classpathref="classpath" includeantruntime="false"/> + </target> + + <target name="dist" depends="compile" description="generate the distribution"> + <delete file="WebContent/WEB-INF/classes/constants.properties"/> + <copy file="conf/constants.properties" tofile="WebContent/WEB-INF/classes/constants.properties"/> + <jar jarfile="${dist}/mpiwg-mpdl-cms-web.jar" basedir="${build}"/> + <copy file="dist/mpiwg-mpdl-cms-web.jar" todir="${lib}"/> + <war destfile="dist/mpiwg-mpdl-cms-web.war" webxml="WebContent/WEB-INF/web.xml"> + <fileset dir="WebContent"/> + <lib dir="WebContent/WEB-INF/lib"/> + </war> + <copy file="dist/mpiwg-mpdl-cms-web.war" todir="${webappTomcat}"/> + </target> + + <target name="dist-remote-thrax" depends="compile" description="generate the distribution"> + <delete file="WebContent/WEB-INF/classes/constants.properties"/> + <copy file="conf/constants-thrax.properties" tofile="WebContent/WEB-INF/classes/constants.properties"/> + <jar jarfile="dist-remote/mpiwg-mpdl-cms-web.jar" basedir="${build}"/> + <copy file="dist-remote/mpiwg-mpdl-cms-web.jar" todir="${lib}"/> + <war destfile="dist-remote/mpiwg-mpdl-cms-web.war" webxml="WebContent/WEB-INF/web.xml"> + <fileset dir="WebContent"/> + <lib dir="WebContent/WEB-INF/lib"/> + </war> + </target> + + <target name="clean" description="clean" > + <delete dir="${build}"/> + <delete file="${dist}/mpiwg-mpdl-cms-web.jar"/> + </target> +</project> \ No newline at end of file
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/conf/constants-thrax.properties Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,5 @@ +documentsDir=/usr/local/java/mpdl-data/xml/documents +luceneDocumentsDir=/usr/local/java/mpdl-data/xml/lucene/documents +luceneNodesDir=/usr/local/java/mpdl-data/xml/lucene/nodes +dataDir=/usr/local/java/mpdl-data/lt +confDir=/usr/local/java/mpdl-data/collectionConfs
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/conf/constants.properties Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,5 @@ +documentsDir=/Users/jwillenborg/mpdl/data/xml/documents +luceneDocumentsDir=/Users/jwillenborg/mpdl/data/xml/lucene/documents +luceneNodesDir=/Users/jwillenborg/mpdl/data/xml/lucene/nodes +dataDir=/Users/jwillenborg/mpdl/data/lt +confDir=/Users/jwillenborg/mpdl/data/collectionConfs
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Binary file software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/.DS_Store has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/exception/ApplicationException.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,15 @@ +package de.mpg.mpiwg.berlin.mpdl.exception; + +public class ApplicationException extends Exception { + private static final long serialVersionUID = 1L; + + public ApplicationException(Exception e) { + super(e); + } + + public ApplicationException(String str) { + super(str); + } + +} +
Binary file software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/.DS_Store has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/cms/DocumentOperation.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,102 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.cms; + +import java.io.IOException; +import java.io.PrintWriter; + +import javax.servlet.ServletConfig; +import javax.servlet.ServletException; +import javax.servlet.http.HttpServlet; +import javax.servlet.http.HttpServletRequest; +import javax.servlet.http.HttpServletResponse; + +import de.mpg.mpiwg.berlin.mpdl.cms.scheduler.CmsChainScheduler; +import de.mpg.mpiwg.berlin.mpdl.cms.scheduler.CmsDocOperation; +import de.mpg.mpiwg.berlin.mpdl.servlets.util.ServletUtil; + +public class DocumentOperation extends HttpServlet { + private static final long serialVersionUID = 1L; + + public DocumentOperation() { + super(); + } + + public void init(ServletConfig config) throws ServletException { + super.init(config); + } + + protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + request.setCharacterEncoding("utf-8"); + response.setCharacterEncoding("utf-8"); + String srcUrlStr = request.getParameter("srcUrl"); + String docId = request.getParameter("docId"); // id in file system: e.g. /tei/en/Test_1789.xml + String mainLanguage = request.getParameter("mainLanguage"); // main language of that document + String elementNames = request.getParameter("elementNames"); + String operation = request.getParameter("operation"); + String outputFormat = request.getParameter("outputFormat"); + if (outputFormat == null) + outputFormat = "xml"; + if (outputFormat.equals("xml")) { + response.setContentType("text/xml"); + } else { + response.setContentType("text/html"); + } + CmsDocOperation docOperation = new CmsDocOperation(operation, srcUrlStr, null, docId); + if (mainLanguage != null) + docOperation.setMainLanguage(mainLanguage); + String[] elementNamesArray = null; + if (elementNames != null) + elementNamesArray = elementNames.split(" "); + docOperation.setElementNames(elementNamesArray); + try { + if (docId == null || docId.isEmpty()) { + write(response, "Parameter: \"docId\" is not set. Please set parameter \"docId\"."); + return; + } + if (operation.equals("create") || operation.equals("delete")) { + CmsChainScheduler scheduler = CmsChainScheduler.getInstance(); + docOperation = scheduler.doOperation(docOperation); + String jobId = "" + docOperation.getOrderId(); + String baseUrl = ServletUtil.getInstance().getBaseUrl(request); + String docJobUrlStr = baseUrl + "/doc/GetDocumentJobs?id=" + jobId; + if (outputFormat.equals("xml")) { + write(response, "<result>"); + write(response, "<docJob>"); + write(response, "<id>" + jobId + "</id>"); + write(response, "<url>" + docJobUrlStr + "</url>"); + write(response, "</docJob>"); + write(response, "</result>"); + } else if (outputFormat.equals("html")) { + write(response, "<html>"); + write(response, "<h2>" + "Document operation result" + "</h2>"); + write(response, "See your document operation " + jobId + " <a href=\"" + docJobUrlStr + "\">" + "here" + "</a>"); + write(response, "</html>"); + } else if (outputFormat.equals("string")) { + write(response, docJobUrlStr); + } + } else { + String errorStr = "Error: Operation: " + operation + " is not supported"; + if (outputFormat.equals("xml")) { + write(response, "<error>" + errorStr + "</error>"); + } else { + write(response, "<html>"); + write(response, "<h2>" + "Error" + "</h2>"); + write(response, errorStr); + } + } + PrintWriter out = response.getWriter(); + out.close(); + } catch (Exception e) { + throw new ServletException(e); + } + } + + protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + // TODO + } + + private void write(HttpServletResponse response, String str) throws IOException { + PrintWriter out = response.getWriter(); + out.write(str); + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/cms/GetDocInfo.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,172 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.cms; + +import java.io.File; +import java.io.IOException; +import java.io.PrintWriter; +import java.util.Date; + +import javax.servlet.ServletConfig; +import javax.servlet.ServletContext; +import javax.servlet.ServletException; +import javax.servlet.http.HttpServlet; +import javax.servlet.http.HttpServletRequest; +import javax.servlet.http.HttpServletResponse; + +import org.apache.commons.io.FileUtils; + +import de.mpg.mpiwg.berlin.mpdl.cms.document.DocumentHandler; +import de.mpg.mpiwg.berlin.mpdl.cms.document.MetadataRecord; +import de.mpg.mpiwg.berlin.mpdl.cms.lucene.IndexHandler; +import de.mpg.mpiwg.berlin.mpdl.cms.transform.TocTransformer; +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.util.Util; + +public class GetDocInfo extends HttpServlet { + private static final long serialVersionUID = 1L; + private TocTransformer tocTransformer; + + public GetDocInfo() { + super(); + } + + public void init(ServletConfig config) throws ServletException { + super.init(config); + ServletContext context = getServletContext(); + tocTransformer = (TocTransformer) context.getAttribute("tocTransformer"); + } + + protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + request.setCharacterEncoding("utf-8"); + response.setCharacterEncoding("utf-8"); + String docId = request.getParameter("docId"); + String field = request.getParameter("field"); + String outputFormat = request.getParameter("outputFormat"); + if (outputFormat == null) + outputFormat = "xml"; + String cssUrl = request.getParameter("cssUrl"); + if (cssUrl == null) { + String baseUrl = getBaseUrl(request); + cssUrl = baseUrl + "/css/page.css"; + } + try { + IndexHandler indexHandler = IndexHandler.getInstance(); + MetadataRecord mdRecord = indexHandler.getDocMetadata(docId); + if (outputFormat.equals("xml")) + response.setContentType("text/xml"); + else if (outputFormat.equals("html")) + response.setContentType("text/html"); + PrintWriter out = response.getWriter(); + if (mdRecord != null && outputFormat.equals("xml")) { + out.print("<doc>"); + out.print("<id>" + docId + "</id>"); + String identifier = mdRecord.getIdentifier(); + if ((field == null || (field != null && field.equals("identifier"))) && identifier != null) + out.print("<identifier>" + identifier + "</identifier>"); + String uri = mdRecord.getUri(); + if ((field == null || (field != null && field.equals("uri"))) && uri != null) + out.print("<uri>" + uri + "</uri>"); + String collectionNames = mdRecord.getCollectionNames(); + if ((field == null || (field != null && field.equals("collectionNames"))) && collectionNames != null) + out.print("<collectionNames>" + collectionNames + "</collectionNames>"); + String author = mdRecord.getCreator(); + if ((field == null || (field != null && field.equals("author"))) && author != null) + out.print("<author>" + author + "</author>"); + String title = mdRecord.getTitle(); + if ((field == null || (field != null && field.equals("title"))) && title != null) + out.print("<title>" + title + "</title>"); + String language = mdRecord.getLanguage(); + if ((field == null || (field != null && field.equals("language"))) && language != null) + out.print("<language>" + language + "</language>"); + String publisher = mdRecord.getPublisher(); + if ((field == null || (field != null && field.equals("publisher"))) && publisher != null) + out.print("<publisher>" + publisher + "</publisher>"); + String date = mdRecord.getYear(); + if ((field == null || (field != null && field.equals("date"))) && date != null) + out.print("<date>" + date + "</date>"); + String subject = mdRecord.getSubject(); + if ((field == null || (field != null && field.equals("subject"))) && subject != null) + out.print("<subject>" + subject + "</subject>"); + String rights = mdRecord.getRights(); + if ((field == null || (field != null && field.equals("rights"))) && rights != null) + out.print("<rights>" + rights + "</rights>"); + String license = mdRecord.getLicense(); + if ((field == null || (field != null && field.equals("license"))) && license != null) + out.print("<license>" + license + "</license>"); + String accessRights = mdRecord.getAccessRights(); + if ((field == null || (field != null && field.equals("accessRights"))) && accessRights != null) + out.print("<accessRights>" + accessRights + "</accessRights>"); + String echoId = mdRecord.getEchoId(); + if ((field == null || (field != null && field.equals("echoId"))) && echoId != null) + out.print("<echoId>" + echoId + "</echoId>"); + String echoPageImageDir = mdRecord.getEchoPageImageDir(); + if ((field == null || (field != null && field.equals("echoPageImageDir"))) && echoPageImageDir != null) + out.print("<echoPageImageDir>" + echoPageImageDir + "</echoPageImageDir>"); + String echoFiguresDir = mdRecord.getEchoFiguresDir(); + if ((field == null || (field != null && field.equals("echoFiguresDir"))) && echoFiguresDir != null) + out.print("<echoFiguresDir>" + echoFiguresDir + "</echoFiguresDir>"); + String mpiwgDocId = mdRecord.getMpiwgDocId(); + if ((field == null || (field != null && field.equals("mpiwgDocId"))) && mpiwgDocId != null) + out.print("<mpiwgDocId>" + mpiwgDocId + "</mpiwgDocId>"); + if (field == null || (field != null && ! field.equals("toc") && ! field.equals("figures") && ! field.equals("notes") && ! field.equals("notesHandwritten") && ! field.equals("pages") && ! field.equals("places"))) + out.print("<system>"); + int pageCount = mdRecord.getPageCount(); + if (field == null || (field != null && field.equals("countPages"))) + out.print("<countPages>" + pageCount + "</countPages>"); + Date lastModified = mdRecord.getLastModified(); + if ((field == null || (field != null && field.equals("lastModified"))) && lastModified != null) { + String lastModifiedStr = new Util().toXsDate(lastModified); + out.print("<lastModified>" + lastModifiedStr + "</lastModified>"); + } + String schemaName = mdRecord.getSchemaName(); + if ((field == null || (field != null && field.equals("schema"))) && schemaName != null) + out.print("<schema>" + schemaName + "</schema>"); + if (field == null || (field != null && ! field.equals("toc") && ! field.equals("figures") && ! field.equals("notes") && ! field.equals("notesHandwritten") && ! field.equals("pages") && ! field.equals("places"))) + out.print("</system>"); + if (field != null && (field.equals("toc") || field.equals("figures") || field.equals("notes") || field.equals("notesHandwritten") || field.equals("pages") || field.equals("places"))) { + DocumentHandler docHandler = new DocumentHandler(); + String docDir = docHandler.getDocDir(docId); + String tocFileName = docDir + "/toc.xml"; + File tocFile = new File(tocFileName); + String tocStr = FileUtils.readFileToString(tocFile, "utf-8"); + String tocXmlStr = tocTransformer.transform(tocStr, field, outputFormat); + out.print(tocXmlStr); + } + out.print("</doc>"); + } else if (mdRecord != null && outputFormat.equals("html")) { + if (field != null && (field.equals("toc") || field.equals("figures") || field.equals("notes") || field.equals("notesHandwritten") || field.equals("pages") || field.equals("places"))) { + DocumentHandler docHandler = new DocumentHandler(); + String docDir = docHandler.getDocDir(docId); + String tocFileName = docDir + "/toc.xml"; + File tocFile = new File(tocFileName); + String tocStr = FileUtils.readFileToString(tocFile, "utf-8"); + String tocHtmlStr = tocTransformer.transform(tocStr, field, outputFormat); + String title = docId + ", " + field; + String head = "<head><title>" + title + "</title><link rel=\"stylesheet\" type=\"text/css\" href=\"" + cssUrl + "\"/></head>"; + String result = "<html>" + head + "<body>" + tocHtmlStr + "</body>" + "</html>"; + out.print(result); + } + } else { + out.print("<result>" + "no document found with id: " + docId + "</result>"); + } + out.close(); + } catch (ApplicationException e) { + throw new ServletException(e); + } + } + + protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + doGet(request, response); + } + + private String getBaseUrl(HttpServletRequest request) { + return getServerUrl(request) + request.getContextPath(); + } + + private String getServerUrl(HttpServletRequest request) { + if ( ( request.getServerPort() == 80 ) || ( request.getServerPort() == 443 ) ) + return request.getScheme() + "://" + request.getServerName(); + else + return request.getScheme() + "://" + request.getServerName() + ":" + request.getServerPort(); + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/cms/GetDocument.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,71 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.cms; + +import java.io.BufferedInputStream; +import java.io.File; +import java.io.FileInputStream; +import java.io.IOException; +import java.io.OutputStream; +import java.net.URLConnection; + +import javax.servlet.ServletConfig; +import javax.servlet.ServletException; +import javax.servlet.http.HttpServlet; +import javax.servlet.http.HttpServletRequest; +import javax.servlet.http.HttpServletResponse; + +import de.mpg.mpiwg.berlin.mpdl.cms.document.DocumentHandler; + +public class GetDocument extends HttpServlet { + private static final long serialVersionUID = 1L; + private OutputStream out = null; + + public GetDocument() { + super(); + } + + public void init(ServletConfig config) throws ServletException { + super.init(config); + } + + protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + request.setCharacterEncoding("utf-8"); + response.setCharacterEncoding("utf-8"); + String id = request.getParameter("id"); + DocumentHandler docHandler = new DocumentHandler(); + String fullFileName = docHandler.getDocFullFileName(id); + File file = new File(fullFileName); + if (file.exists()) { + write(response, file); + } else { + write(response, "Document: " + id + " does not exist"); + } + } + + protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + // TODO + } + + private void write(HttpServletResponse response, File file) throws IOException { + String fileName = file.getName(); + BufferedInputStream is = new BufferedInputStream(new FileInputStream(file)); + String contentType = URLConnection.guessContentTypeFromName(fileName); // other methods: URLConnection.guessContentTypeFromStream(is); or MIMEUtils.getMIMEType(file); + if (contentType != null) + response.setContentType(contentType); + response.setHeader("Content-Disposition", "filename=" + fileName); + out = response.getOutputStream(); + byte[] buf = new byte[20000*1024]; // 20MB buffer + int bytesRead; + while ((bytesRead = is.read(buf)) != -1) { + out.write(buf, 0, bytesRead); + } + is.close(); + out.flush(); + } + + private void write(HttpServletResponse response, String str) throws IOException { + out = response.getOutputStream(); + byte[] strBytes = str.getBytes("utf-8"); + out.write(strBytes, 0, strBytes.length); + out.flush(); + } +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/cms/GetDocumentJobs.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,244 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.cms; + +import java.io.IOException; +import java.io.PrintWriter; +import java.util.ArrayList; +import java.util.Date; + +import javax.servlet.ServletConfig; +import javax.servlet.ServletException; +import javax.servlet.http.HttpServlet; +import javax.servlet.http.HttpServletRequest; +import javax.servlet.http.HttpServletResponse; + +import de.mpg.mpiwg.berlin.mpdl.cms.scheduler.CmsChainScheduler; +import de.mpg.mpiwg.berlin.mpdl.cms.scheduler.CmsDocOperation; + +public class GetDocumentJobs extends HttpServlet { + private static final long serialVersionUID = 1L; + public GetDocumentJobs() { + super(); + } + + public void init(ServletConfig config) throws ServletException { + super.init(config); + } + + protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + request.setCharacterEncoding("utf-8"); + response.setCharacterEncoding("utf-8"); + response.setContentType("text/xml"); + String jobIdStr = request.getParameter("id"); + String outputFormat = request.getParameter("outputFormat"); + if (outputFormat == null) + outputFormat = "xml"; + try { + boolean getAllJobs = false; + if (jobIdStr == null) + getAllJobs = true; + CmsChainScheduler scheduler = CmsChainScheduler.getInstance(); + ArrayList<CmsDocOperation> docOperations = new ArrayList<CmsDocOperation>(); + if (getAllJobs) { + docOperations = scheduler.getDocOperations(); + } else { + int jobId = Integer.parseInt(jobIdStr); + CmsDocOperation docOperation = scheduler.getDocOperation(jobId); + if (docOperation != null) + docOperations.add(docOperation); + } + PrintWriter out = response.getWriter(); + String resultStr = ""; + if (outputFormat.equals("xml")) { + response.setContentType("text/xml"); + resultStr = createXmlString(docOperations); + } else if (outputFormat.equals("html")) { + response.setContentType("text/html"); + resultStr = createHtmlString(docOperations); + } else if (outputFormat.equals("error")) { + response.setContentType("text/html"); + resultStr = createErrorString(docOperations); + } + out.print(resultStr); + out.close(); + } catch (Exception e) { + throw new ServletException(e); + } + } + + String createXmlString(ArrayList<CmsDocOperation> docOperations) { + StringBuilder result = new StringBuilder(); + if (docOperations != null && ! docOperations.isEmpty()) { + result.append("<docJobs>"); + for (int i=0; i<docOperations.size(); i++) { + CmsDocOperation docOperation = docOperations.get(i); + result.append("<job>"); + int jobId = docOperation.getOrderId(); + result.append("<id>" + jobId + "</id>"); + result.append("<name>" + docOperation.getName() + "</name>"); + result.append("<status>"); + Date start = docOperation.getStart(); + String startStr = "No start time available because job is scheduled into server queue where other jobs have been started earlier"; + if (start != null) + startStr = start.toString(); + result.append("<started>" + startStr + "</started>"); + Date end = docOperation.getEnd(); + String endStr = "No end time available because job is not finished yet"; + if (end != null) + endStr = end.toString(); + result.append("<finished>" + endStr + "</finished>"); + String status = docOperation.getStatus(); + result.append("<description>" + status + "</description>"); + String errorMessage = docOperation.getErrorMessage(); + if (errorMessage == null) + errorMessage = "no error"; + result.append("<error>" + errorMessage + "</error>"); + result.append("</status>"); + if (docOperation.getName().equals("delete")) { + result.append("<destination>"); + result.append("<docId>" + docOperation.getDocIdentifier() + "</docId>"); + result.append("</destination>"); + } else if (docOperation.getName().equals("create")) { + result.append("<source>"); + result.append("<url>" + docOperation.getSrcUrl() + "</url>"); + result.append("<uploadFileName>" + docOperation.getUploadFileName() + "</uploadFileName>"); + result.append("</source>"); + result.append("<destination>"); + result.append("<docId>" + docOperation.getDocIdentifier() + "</docId>"); + result.append("</destination>"); + } + String desc = "Document operations are maintained on server asychronously. Each operation is scheduled into a server job queue " + + "and is executed when all previous started jobs in the queue are worked off. Each operation needs some execution time dependent " + + "on the size and the number of pages of the document, the speed of the network connection and the performance of the " + + "server."; + result.append("<description>" + desc + "</description>"); + result.append("</job>"); + } + result.append("</docJobs>"); + } else { + String message = "there are no scheduled jobs (neither finished, queued or executed)"; + result.append("<message>" + message + "</message>"); + } + return result.toString(); + } + + String createHtmlString(ArrayList<CmsDocOperation> docOperations) { + // TODO all + StringBuilder result = new StringBuilder(); + result.append("<html>"); + result.append("<head>"); + result.append("<title>" + "Document operation status" + "</title>"); + result.append("</head>"); + result.append("<body>"); + result.append("<table>"); + result.append("<h1>" + "Document operation status" + "</h1>"); + if (docOperations != null && ! docOperations.isEmpty()) { + for (int i=0; i<docOperations.size(); i++) { + result.append("<tr>"); + result.append("<td>"); + CmsDocOperation docOperation = docOperations.get(i); + result.append("<b>Operation: </b>" + docOperation.getName()); + int jobId = docOperation.getOrderId(); + result.append("<p/>"); + result.append("<b>Job id: </b>" + jobId); + result.append("<p/>"); + result.append("<b>Job status: </b>"); + result.append("<ul>"); + Date start = docOperation.getStart(); + String startStr = "No start time available because job is scheduled into server queue where other jobs have been started earlier"; + if (start != null) + startStr = start.toString(); + result.append("<li>"); + result.append("<b>Started: </b>"); + result.append("<started>" + startStr + "</started>"); + result.append("</li>"); + Date end = docOperation.getEnd(); + String endStr = "No end time available because job is not finished yet"; + if (end != null) + endStr = end.toString(); + result.append("<li>"); + result.append("<b>Finished: </b>"); + result.append("<finished>" + endStr + "</finished>"); + result.append("</li>"); + String status = docOperation.getStatus(); + result.append("<li>"); + result.append("<b>Description: </b>"); + result.append("<description>" + status + "</description>"); + result.append("</li>"); + String errorMessage = docOperation.getErrorMessage(); + result.append("<li>"); + if (errorMessage != null) { + result.append("<font color=\"#FF0000\"><b>Error: </b></font>"); + result.append("<error>" + errorMessage + "</error>"); + } else { + result.append("<b>Error: </b>"); + result.append("<error>" + "no error" + "</error>"); + } + result.append("</li>"); + result.append("</ul>"); + if (docOperation.getName().equals("delete")) { + result.append("<b>Destination</b>"); + result.append("<ul>"); + result.append("<li>"); + result.append("<b>Document identifier: </b>"); + result.append("<docId>" + docOperation.getDocIdentifier() + "</docId>"); + result.append("</li>"); + result.append("</ul>"); + } else if (docOperation.getName().equals("create")) { + result.append("<b>Source</b>"); + result.append("<ul>"); + result.append("<li>"); + result.append("<url>" + docOperation.getSrcUrl() + "</url>"); + String uploadFileName = docOperation.getUploadFileName(); + if (uploadFileName != null) { + result.append("<li>"); + result.append("<uploadFileName>" + uploadFileName + "</uploadFileName>"); + result.append("</li>"); + } + result.append("</li>"); + result.append("</ul>"); + result.append("<b>Destination</b>"); + result.append("<ul>"); + result.append("<li>"); + result.append("<b>Document identifier: </b>"); + result.append("<docId>" + docOperation.getDocIdentifier() + "</docId>"); + result.append("</li>"); + result.append("</ul>"); + } + String desc = "Document operations are maintained on server asychronously. Each operation is scheduled into a server job queue " + + "and is executed when all previous started jobs in the queue are worked off. Each operation needs some execution time dependent " + + "on the size and the number of pages of the document, the speed of the network connection and the performance of the " + + "server."; + result.append("<b>Description: </b>" + desc); + result.append("<p/>"); + result.append("<hr/>"); + } + result.append("</td>"); + result.append("</tr>"); + } else { + String message = "there are no scheduled jobs (neither finished, queued or executed)"; + result.append("<message>" + message + "</message>"); + } + result.append("</table>"); + result.append("</body>"); + result.append("</html>"); + return result.toString(); + } + + String createErrorString(ArrayList<CmsDocOperation> docOperations) { + String error = ""; + if (docOperations != null && ! docOperations.isEmpty()) { + for (int i=0; i<docOperations.size(); i++) { + CmsDocOperation docOperation = docOperations.get(i); + String errorMessage = docOperation.getErrorMessage(); + if (errorMessage != null) + error = errorMessage; + } + } + return error; + } + + protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + doGet(request, response); + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/cms/GetPage.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,215 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.cms; + +import java.io.File; +import java.io.IOException; +import java.io.PrintWriter; +import java.io.StringReader; + +import javax.servlet.ServletConfig; +import javax.servlet.ServletContext; +import javax.servlet.ServletException; +import javax.servlet.http.HttpServlet; +import javax.servlet.http.HttpServletRequest; +import javax.servlet.http.HttpServletResponse; + +import org.apache.commons.io.FileUtils; +import org.xml.sax.InputSource; +import org.xml.sax.SAXException; +import org.xml.sax.XMLReader; + +import com.sun.org.apache.xerces.internal.parsers.SAXParser; + +import de.mpg.mpiwg.berlin.mpdl.cms.document.DocumentHandler; +import de.mpg.mpiwg.berlin.mpdl.cms.document.MetadataRecord; +import de.mpg.mpiwg.berlin.mpdl.cms.lucene.IndexHandler; +import de.mpg.mpiwg.berlin.mpdl.cms.transform.HighlightContentHandler; +import de.mpg.mpiwg.berlin.mpdl.cms.transform.PageTransformer; +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.WordContentHandler; + +public class GetPage extends HttpServlet { + private static final long serialVersionUID = 1L; + private PageTransformer pageTransformer; + + public GetPage() { + super(); + } + + public void init(ServletConfig config) throws ServletException { + super.init(config); + ServletContext context = getServletContext(); + pageTransformer = (PageTransformer) context.getAttribute("pageTransformer"); + } + + protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + String result = ""; + request.setCharacterEncoding("utf-8"); + response.setCharacterEncoding("utf-8"); + String docId = request.getParameter("docId"); + String pageStr = request.getParameter("page"); + String normalization = request.getParameter("normalization"); + String highlightQuery = request.getParameter("highlightQuery"); + String highlightQueryType = request.getParameter("highlightQueryType"); + if (highlightQueryType == null) + highlightQueryType = "form"; + String highlightElem = request.getParameter("highlightElem"); + String highlightElemPosStr = request.getParameter("highlightElemPos"); + int highlightElemPos = -1; + if (highlightElemPosStr != null) + highlightElemPos = Integer.parseInt(highlightElemPosStr); + String mode = request.getParameter("mode"); + if (mode == null) + mode = "untokenized"; + String outputFormat = request.getParameter("outputFormat"); + if (outputFormat == null) + outputFormat = "html"; + String cssUrl = request.getParameter("cssUrl"); + String baseUrl = getBaseUrl(request); + if (cssUrl == null) { + cssUrl = baseUrl + "/css/page.css"; + } + int page = 1; + if (pageStr != null) + page = Integer.parseInt(pageStr); + if (outputFormat.equals("xml")) + response.setContentType("text/xml"); + else if (outputFormat.equals("html") || outputFormat.equals("xmlDisplay")) + response.setContentType("text/html"); + // normalization + if (normalization == null || ! (normalization.equals("orig") || normalization.equals("reg") || normalization.equals("norm"))) + normalization = "norm"; + if (outputFormat.equals("xmlDisplay")) + normalization = "orig"; + PrintWriter out = response.getWriter(); + try { + IndexHandler indexHandler = IndexHandler.getInstance(); + MetadataRecord mdRecord = indexHandler.getDocMetadata(docId); + DocumentHandler docHandler = new DocumentHandler(); + String docDir = docHandler.getDocDir(docId); + String docPageDir = docDir + "/" + "pages"; + String pageFileName = docPageDir + "/page-" + page + "-morph.xml"; + File pageFile = new File(pageFileName); + if (page == 1 && ! (new File(docPageDir)).exists()) { + String docFileName = docHandler.getDocFullFileName(docId); + pageFile = new File(docFileName); // when no page breaks are in the document then the whole document is the first page + } + if (! pageFile.exists()) { + out.print("There is no page: " + page + " in document"); + out.close(); + return; + } + String pageHtmlFileName = docPageDir + "/page-" + page + ".html"; + File pageHtmlFile = new File(pageHtmlFileName); + String fragmentMorphStr = FileUtils.readFileToString(pageFile, "utf-8"); + if (! pageHtmlFile.exists()) // TODO rausnehmen sobald alle Dokumente neu indexiert wurden + fragmentMorphStr = enrichWordsOrigRegNorm(fragmentMorphStr); + if (outputFormat.equals("html") || outputFormat.equals("xmlDisplay")) { + String schemaName = mdRecord.getSchemaName(); + String title = docId + ", Page: " + page; + String xmlHeader = "<?xml version=\"1.0\" encoding=\"utf-8\"?>"; + String cssShowWordFileName = "pageNormDict.css"; + if (outputFormat.equals("xmlDisplay")) + cssShowWordFileName = "pageOrig.css"; // xml display shows always the original text + else if (normalization.equals("orig") && mode.equals("untokenized")) + cssShowWordFileName = "pageOrig.css"; + else if (normalization.equals("orig") && mode.equals("tokenized")) + cssShowWordFileName = "pageOrigDict.css"; + else if (normalization.equals("reg") && mode.equals("untokenized")) + cssShowWordFileName = "pageReg.css"; + else if (normalization.equals("reg") && mode.equals("tokenized")) + cssShowWordFileName = "pageRegDict.css"; + else if (normalization.equals("norm") && mode.equals("untokenized")) + cssShowWordFileName = "pageNorm.css"; + String showWordCssUrl = baseUrl + "/css/" + cssShowWordFileName; + String mainCssLink = "<link rel=\"stylesheet\" type=\"text/css\" href=\"" + cssUrl + "\"/>"; + String showWordCssLink = "<link rel=\"stylesheet\" type=\"text/css\" href=\"" + showWordCssUrl + "\"/>"; + String head = "<head>" + "<title>" + title + "</title>" + showWordCssLink + mainCssLink + "</head>"; + String namespace = ""; + String pageHtmlStr = null; + if (pageHtmlFile.exists() && outputFormat.equals("html") && (highlightElem == null && highlightQuery == null)) { + pageHtmlStr = FileUtils.readFileToString(pageHtmlFile, "utf-8"); + } else { + if (highlightElem != null || highlightQuery != null) { + String hiQueryType = "orig"; + if (highlightQueryType.equals("morph")) + hiQueryType = "morph"; + else + hiQueryType = normalization; + String language = mdRecord.getLanguage(); + fragmentMorphStr = highlight(fragmentMorphStr, highlightElem, highlightElemPos, hiQueryType, highlightQuery, language); + } + pageHtmlStr = pageTransformer.transform(fragmentMorphStr, mdRecord, page, outputFormat); + } + if (schemaName != null && schemaName.equals("echo")) { + namespace = "xmlns:echo=\"http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/\" xmlns:de=\"http://www.mpiwg-berlin.mpg.de/ns/de/1.0/\" " + + "xmlns:dcterms=\"http://purl.org/dc/terms\" " + "xmlns:xhtml=\"http://www.w3.org/1999/xhtml\" xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" " + + "xmlns:xlink=\"http://www.w3.org/1999/xlink\""; + } + result = xmlHeader + "<html " + namespace + ">" + head + "<body>" + pageHtmlStr + "</body>" + "</html>"; + } else { + String pageFileNameOrig = docPageDir + "/page-" + page + ".xml"; + File pageFileOrig = new File(pageFileNameOrig); + if (pageFileOrig.exists()) + result = FileUtils.readFileToString(pageFileOrig, "utf-8"); + else + result = ""; + } + out.print(result); + out.close(); + } catch (ApplicationException e) { + throw new ServletException(e); + } + } + + protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + doGet(request, response); + } + + private String getBaseUrl(HttpServletRequest request) { + return getServerUrl(request) + request.getContextPath(); + } + + private String getServerUrl(HttpServletRequest request) { + if ( ( request.getServerPort() == 80 ) || ( request.getServerPort() == 443 ) ) + return request.getScheme() + "://" + request.getServerName(); + else + return request.getScheme() + "://" + request.getServerName() + ":" + request.getServerPort(); + } + + private String enrichWordsOrigRegNorm(String xmlStr) throws ApplicationException { + try { + WordContentHandler wordContentHandler = new WordContentHandler(); + XMLReader xmlParser = new SAXParser(); + xmlParser.setContentHandler(wordContentHandler); + StringReader strReader = new StringReader(xmlStr); + InputSource inputSource = new InputSource(strReader); + xmlParser.parse(inputSource); + String result = wordContentHandler.getResult(); + return result; + } catch (SAXException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + private String highlight(String xmlStr, String highlightElem, int highlightElemPos, String highlightQueryType, String highlightQuery, String language) throws ApplicationException { + String result = null; + try { + HighlightContentHandler highlightContentHandler = new HighlightContentHandler(highlightElem, highlightElemPos, highlightQueryType, highlightQuery, language); + highlightContentHandler.setFirstPageBreakReachedMode(true); + XMLReader xmlParser = new SAXParser(); + xmlParser.setContentHandler(highlightContentHandler); + StringReader stringReader = new StringReader(xmlStr); + InputSource inputSource = new InputSource(stringReader); + xmlParser.parse(inputSource); + result = highlightContentHandler.getResult().toString(); + } catch (SAXException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + return result; + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/cms/GetToken.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,87 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.cms; + +import java.io.IOException; +import java.io.PrintWriter; +import java.util.ArrayList; + +import javax.servlet.ServletConfig; +import javax.servlet.ServletException; +import javax.servlet.http.HttpServlet; +import javax.servlet.http.HttpServletRequest; +import javax.servlet.http.HttpServletResponse; + +import org.apache.lucene.index.Term; + +import de.mpg.mpiwg.berlin.mpdl.cms.document.Token; +import de.mpg.mpiwg.berlin.mpdl.cms.lucene.IndexHandler; +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; + +public class GetToken extends HttpServlet { + private static final long serialVersionUID = 1L; + public GetToken() { + super(); + } + + public void init(ServletConfig config) throws ServletException { + super.init(config); + } + + protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + request.setCharacterEncoding("utf-8"); + response.setCharacterEncoding("utf-8"); + String docId = request.getParameter("docId"); + String attribute = request.getParameter("attribute"); + if (attribute == null) + attribute = "tokenOrig"; + String query = request.getParameter("query"); + String countStr = request.getParameter("count"); + if (countStr == null) + countStr = "100"; + int count = Integer.parseInt(countStr); + String outputFormat = request.getParameter("outputFormat"); + if (outputFormat == null) + outputFormat = "xml"; + try { + IndexHandler indexHandler = IndexHandler.getInstance(); + ArrayList<Token> token = null; + if (docId == null) + token = indexHandler.getToken(attribute, query, count); + else + token = indexHandler.getToken(docId, attribute, query, count); + if (outputFormat.equals("xml")) + response.setContentType("text/xml"); + else if (outputFormat.equals("html")) + response.setContentType("text/html"); + else + response.setContentType("text/xml"); + PrintWriter out = response.getWriter(); + out.print("<result>"); + out.print("<attribute>" + attribute + "</attribute>"); + out.print("<query>" + query + "</query>"); + out.print("<count>" + count + "</count>"); + out.print("<result>"); + if (token != null) { + for (int i=0; i<token.size(); i++) { + Token t = token.get(i); + Term term = t.getTerm(); + int freq = t.getFreq(); + out.print("<token>"); + out.print("<text>" + term.text() + "</text>"); + if (freq != -1) + out.print("<freq>" + freq + "</freq>"); + out.print("</token>"); + } + } + out.print("</result>"); + out.print("</result>"); + out.close(); + } catch (ApplicationException e) { + throw new ServletException(e); + } + } + + protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + doGet(request, response); + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/cms/MoreLikeThis.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,140 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.cms; + +import java.io.IOException; +import java.io.PrintWriter; +import java.util.ArrayList; +import java.util.Date; + +import javax.servlet.ServletException; +import javax.servlet.http.HttpServlet; +import javax.servlet.http.HttpServletRequest; +import javax.servlet.http.HttpServletResponse; + +import org.apache.lucene.document.Fieldable; + +import de.mpg.mpiwg.berlin.mpdl.cms.document.Document; +import de.mpg.mpiwg.berlin.mpdl.cms.document.Hits; +import de.mpg.mpiwg.berlin.mpdl.cms.lucene.IndexHandler; + +public class MoreLikeThis extends HttpServlet { + private static final long serialVersionUID = 1L; + + public MoreLikeThis() { + super(); + } + + protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + request.setCharacterEncoding("utf-8"); + response.setCharacterEncoding("utf-8"); + String outputFormat = request.getParameter("outputFormat"); + if (outputFormat == null) + outputFormat = "html"; + String docId = request.getParameter("docId"); + String pageStr = request.getParameter("page"); + if (pageStr == null) + pageStr = "1"; + int page = Integer.parseInt(pageStr); + String pageSizeStr = request.getParameter("pageSize"); + if (pageSizeStr == null) + pageSizeStr = "10"; + int pageSize = Integer.parseInt(pageSizeStr); + int from = (page * pageSize) - pageSize; // e.g. 0 + int to = page * pageSize - 1; // e.g. 9 + PrintWriter out = response.getWriter(); + IndexHandler indexHandler; + try { + Date begin = new Date(); + indexHandler = IndexHandler.getInstance(); + Hits hits = indexHandler.moreLikeThis(docId, from, to); + ArrayList<Document> docs = null; + if (hits != null) + docs = hits.getHits(); + int hitsSize = -1; + int docsSize = -1; + if (hits != null) + hitsSize = hits.getSize(); + if (docs != null) + docsSize = docs.size(); + Date end = new Date(); + long elapsedTime = end.getTime() - begin.getTime(); + if (outputFormat.equals("xml")) + response.setContentType("text/xml"); + else if (outputFormat.equals("html")) + response.setContentType("text/html"); + else + response.setContentType("text/xml"); + if (outputFormat.equals("xml")) { + out.print("<result>"); + out.print("<query>"); + out.print("<docId>" + docId + "</docId>"); + out.print("<resultPage>" + page + "</resultPage>"); + out.print("<resultPageSize>" + pageSize + "</resultPageSize>"); + out.print("</query>"); + out.print("<hitsSize>" + hitsSize + "</hitsSize>"); + out.print("<hits>"); + for (int i=0; i<docsSize; i++) { + Document doc = docs.get(i); + out.print("<doc>"); + String similarDocId = doc.getFieldable("docId").stringValue(); + out.print("<docId>" + similarDocId + "</docId>"); + Fieldable docCollectionNamesField = doc.getFieldable("collectionNames"); + if (docCollectionNamesField != null) { + String docCollectionNames = docCollectionNamesField.stringValue(); + out.print("<collectionName>" + docCollectionNames + "</collectionName>"); + } + ArrayList<String> hitFragments = doc.getHitFragments(); + if (hitFragments != null) { + out.print("<hitFragments>"); + for (int j=0; j<hitFragments.size(); j++) { + String hitFragment = hitFragments.get(j); + out.print("<hitFragment>" + hitFragment + "</hitFragment>"); + } + out.print("</hitFragments>"); + } + out.print("</doc>"); + } + out.print("</hits>"); + out.print("<executionTime>" + elapsedTime + "</executionTime>"); + out.print("</result>"); + } else if (outputFormat.equals("html")) { + StringBuilder htmlStrBuilder = new StringBuilder(); + htmlStrBuilder.append("<?xml version=\"1.0\" encoding=\"UTF-8\"?><!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\"\"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">"); + htmlStrBuilder.append("<html>"); + htmlStrBuilder.append("<head>"); + htmlStrBuilder.append("<title>Similar documents of: " + docId + "</title>"); + htmlStrBuilder.append("</head>"); + htmlStrBuilder.append("<body>"); + htmlStrBuilder.append("<h4>Similar documents of: " + docId + "</h4>"); + int fromDisplay = from + 1; + int toDisplay = to + 1; + if (hitsSize < to) + toDisplay = hitsSize; + htmlStrBuilder.append("Result: " + fromDisplay + " - " + toDisplay + " of " + hitsSize + " documents" + "</td>"); + htmlStrBuilder.append("<ul>"); + for (int i=0; i<docsSize; i++) { + Document doc = docs.get(i); + String similarDocId = doc.getFieldable("docId").stringValue(); + int num = (page - 1) * pageSize + i + 1; + htmlStrBuilder.append("<li>" + num + ". " + similarDocId + "</li>"); + } + htmlStrBuilder.append("</ul>"); + htmlStrBuilder.append("<p/>"); + htmlStrBuilder.append("Elapsed time: " + elapsedTime + " ms"); + htmlStrBuilder.append("</body>"); + htmlStrBuilder.append("</html>"); + out.print(htmlStrBuilder.toString()); + } + out.close(); + } catch (Exception e) { + throw new ServletException(e); + } + } + + /** + * @see HttpServlet#doPost(HttpServletRequest request, HttpServletResponse response) + */ + protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + // TODO Auto-generated method stub + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/cms/MpiwgMpdlCmsWebServletContextListener.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,67 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.cms; + +import javax.servlet.ServletContext; +import javax.servlet.ServletContextEvent; +import javax.servlet.ServletContextListener; + +import de.mpg.mpiwg.berlin.mpdl.lt.dict.db.LexHandler; +import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.MorphologyCache; +import de.mpg.mpiwg.berlin.mpdl.servlets.cms.MpiwgMpdlCmsWebServletContextListener; +import de.mpg.mpiwg.berlin.mpdl.xml.transform.FragmentTransformer; +import de.mpg.mpiwg.berlin.mpdl.xml.xquery.XQueryEvaluator; +import de.mpg.mpiwg.berlin.mpdl.cms.general.Constants; +import de.mpg.mpiwg.berlin.mpdl.cms.lucene.IndexHandler; +import de.mpg.mpiwg.berlin.mpdl.cms.scheduler.CmsChainScheduler; +import de.mpg.mpiwg.berlin.mpdl.cms.transform.PageTransformer; +import de.mpg.mpiwg.berlin.mpdl.cms.transform.TocTransformer; + +public class MpiwgMpdlCmsWebServletContextListener implements ServletContextListener { + private ServletContext context = null; + private FragmentTransformer fragmentTransformer = null; + private PageTransformer pageTransformer = null; + private TocTransformer tocTransformer = null; + private XQueryEvaluator xQueryEvaluator = null; + public void contextInitialized(ServletContextEvent event) { + try { + this.context = event.getServletContext(); + String documentsDirectory = Constants.getInstance().getDocumentsDir(); + String luceneDocumentsDirectory = Constants.getInstance().getLuceneDocumentsDir(); + String luceneNodesDirectory = Constants.getInstance().getLuceneNodesDir(); + context.setAttribute("documentDirectory", documentsDirectory); + context.setAttribute("luceneDocumentsDirectory", luceneDocumentsDirectory); + context.setAttribute("luceneNodesDirectory", luceneNodesDirectory); + System.out.println(MpiwgMpdlCmsWebServletContextListener.class.getName() + ": contextInitialized (documentsDirectory= \"" + documentsDirectory + "\", set in constants.properties)"); + System.out.println(MpiwgMpdlCmsWebServletContextListener.class.getName() + ": contextInitialized (luceneDocumentsDirectory= \"" + luceneDocumentsDirectory + "\", set in constants.properties)"); + System.out.println(MpiwgMpdlCmsWebServletContextListener.class.getName() + ": contextInitialized (luceneNodesDirectory= \"" + luceneNodesDirectory + "\", set in constants.properties)"); + fragmentTransformer = new FragmentTransformer(); + context.setAttribute("fragmentTransformer", fragmentTransformer); + System.out.println(MpiwgMpdlCmsWebServletContextListener.class.getName() + ": contextInitialized (fragmentTransformer)"); + pageTransformer = new PageTransformer(); + context.setAttribute("pageTransformer", pageTransformer); + System.out.println(MpiwgMpdlCmsWebServletContextListener.class.getName() + ": contextInitialized (pageTransformer)"); + tocTransformer = new TocTransformer(); + context.setAttribute("tocTransformer", tocTransformer); + System.out.println(MpiwgMpdlCmsWebServletContextListener.class.getName() + ": contextInitialized (tocTransformer)"); + xQueryEvaluator = new XQueryEvaluator(); + context.setAttribute("xQueryEvaluator", xQueryEvaluator); + System.out.println(MpiwgMpdlCmsWebServletContextListener.class.getName() + ": contextInitialized (xQueryEvaluator)"); + } catch (Exception e) { + e.printStackTrace(); + } + } + + public void contextDestroyed(ServletContextEvent event) { + try { + this.context = null; + LexHandler.getInstance().end(); + MorphologyCache.getInstance().end(); + IndexHandler.getInstance().end(); + CmsChainScheduler scheduler = CmsChainScheduler.getInstance(); + scheduler.end(); + Thread.sleep(1000); // with this, also the scheduler worker threads could be closed + System.out.println(MpiwgMpdlCmsWebServletContextListener.class.getName() + ": contextDestroyed"); + } catch (Exception e) { + e.printStackTrace(); + } + } +} \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/cms/QueryDocument.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,350 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.cms; + +import java.io.IOException; +import java.io.PrintWriter; +import java.io.StringReader; +import java.util.ArrayList; + +import javax.servlet.ServletConfig; +import javax.servlet.ServletContext; +import javax.servlet.ServletException; +import javax.servlet.http.HttpServlet; +import javax.servlet.http.HttpServletRequest; +import javax.servlet.http.HttpServletResponse; + +import org.apache.lucene.document.Fieldable; +import org.xml.sax.InputSource; +import org.xml.sax.SAXException; +import org.xml.sax.XMLReader; + +import com.sun.org.apache.xerces.internal.parsers.SAXParser; + +import de.mpg.mpiwg.berlin.mpdl.cms.document.Document; +import de.mpg.mpiwg.berlin.mpdl.cms.document.Hits; +import de.mpg.mpiwg.berlin.mpdl.cms.document.MetadataRecord; +import de.mpg.mpiwg.berlin.mpdl.cms.lucene.IndexHandler; +import de.mpg.mpiwg.berlin.mpdl.cms.transform.HighlightContentHandler; +import de.mpg.mpiwg.berlin.mpdl.cms.transform.PageTransformer; +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; + +public class QueryDocument extends HttpServlet { + private static final long serialVersionUID = 1L; + private PageTransformer pageTransformer = null; + + public QueryDocument() { + super(); + } + + public void init(ServletConfig config) throws ServletException { + super.init(config); + ServletContext context = getServletContext(); + pageTransformer = (PageTransformer) context.getAttribute("pageTransformer"); + } + + protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + doGet(request, response); + } + + protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + request.setCharacterEncoding("utf-8"); + response.setCharacterEncoding("utf-8"); + String docId = request.getParameter("docId"); + String query = request.getParameter("query"); + String[] normFunctions = {"none"}; + if (query.contains("tokenReg")) // TODO ordentlich behandeln + normFunctions[0] = "reg"; + else if (query.contains("tokenNorm")) // TODO ordentlich behandeln + normFunctions[0] = "norm"; + String[] outputOptions = {}; + if (query.contains("tokenMorph")) { // TODO ordentlich behandeln + outputOptions = new String[1]; + outputOptions[0] = "withLemmas"; + } + String pageStr = request.getParameter("page"); + if (pageStr == null) + pageStr = "1"; + int page = Integer.parseInt(pageStr); + String pageSizeStr = request.getParameter("pageSize"); + if (pageSizeStr == null) + pageSizeStr = "10"; + int pageSize = Integer.parseInt(pageSizeStr); + int from = (page * pageSize) - pageSize; // e.g. 0 + int to = page * pageSize - 1; // e.g. 9 + String outputFormat = request.getParameter("outputFormat"); + if (outputFormat == null) + outputFormat = "xml"; + try { + IndexHandler indexHandler = IndexHandler.getInstance(); + Hits hits = indexHandler.queryDocument(docId, query, from, to); + MetadataRecord docMetadataRecord = indexHandler.getDocMetadata(docId); + if (outputFormat.equals("xml")) + response.setContentType("text/xml"); + else if (outputFormat.equals("html")) + response.setContentType("text/html"); + else + response.setContentType("text/xml"); + PrintWriter out = response.getWriter(); + String resultStr = ""; + if (outputFormat.equals("xml")) + resultStr = createXmlString(docMetadataRecord, query, page, pageSize, normFunctions, outputOptions, hits); + else if (outputFormat.equals("html")) + resultStr = createHtmlString(docMetadataRecord, query, page, pageSize, normFunctions, outputOptions, hits, request); + out.print(resultStr); + out.close(); + } catch (ApplicationException e) { + throw new ServletException(e); + } + } + + private String createXmlString(MetadataRecord docMetadataRecord, String query, int page, int pageSize, String[] normFunctions, String[] outputOptions, Hits hits) throws ApplicationException { + String docId = docMetadataRecord.getDocId(); + ArrayList<Document> docs = null; + if (hits != null) + docs = hits.getHits(); + int hitsSize = -1; + int docsSize = -1; + if (hits != null) + hitsSize = hits.getSize(); + if (docs != null) + docsSize = docs.size(); + StringBuilder xmlStrBuilder = new StringBuilder(); + xmlStrBuilder.append("<document>"); + xmlStrBuilder.append("<id>" + docId + "</id>"); + xmlStrBuilder.append("<query>"); + xmlStrBuilder.append("<queryText>" + query + "</queryText>"); + xmlStrBuilder.append("<resultPage>" + page + "</resultPage>"); + xmlStrBuilder.append("<resultPageSize>" + pageSize + "</resultPageSize>"); + xmlStrBuilder.append("</query>"); + xmlStrBuilder.append("<hitsSize>" + hitsSize + "</hitsSize>"); + xmlStrBuilder.append("<hits>"); + for (int i=0; i<docsSize; i++) { + Document doc = docs.get(i); + int num = (page - 1) * pageSize + i + 1; + xmlStrBuilder.append("<hit>"); + xmlStrBuilder.append("<num>" + num + "</num>"); + String pageNumber = null; + Fieldable fPageNumber = doc.getFieldable("pageNumber"); + if (fPageNumber != null) { + pageNumber = fPageNumber.stringValue(); + xmlStrBuilder.append("<pageNumber>" + pageNumber + "</pageNumber>"); + } + String elementPagePosition = null; + Fieldable fElementPagePosition = doc.getFieldable("elementPagePosition"); + if (fElementPagePosition != null) { + elementPagePosition = fElementPagePosition.stringValue(); + xmlStrBuilder.append("<pagePosition>" + elementPagePosition + "</pagePosition>"); + } + String lineNumber = null; + Fieldable fLineNumber = doc.getFieldable("lineNumber"); + if (fLineNumber != null) { + lineNumber = fLineNumber.stringValue(); + xmlStrBuilder.append("<lineNumber>" + lineNumber + "</lineNumber>"); + } + String elementPosition = null; + Fieldable fElementPosition = doc.getFieldable("elementAbsolutePosition"); + if (fElementPosition != null) { + elementPosition = fElementPosition.stringValue(); + xmlStrBuilder.append("<absolutePosition>" + elementPosition + "</absolutePosition>"); + } + String xpath = null; + Fieldable fXPath = doc.getFieldable("xpath"); + if (fXPath != null) { + xpath = fXPath.stringValue(); + xmlStrBuilder.append("<xpath>" + xpath + "</xpath>"); + } + String xmlId = null; + Fieldable fXmlId = doc.getFieldable("xmlId"); + if (fXmlId != null) { + xmlId = fXmlId.stringValue(); + xmlStrBuilder.append("<xmlId>" + xmlId + "</xmlId>"); + } + String language = null; + Fieldable fLanguage = doc.getFieldable("language"); + if (fLanguage != null) { + language = fLanguage.stringValue(); + xmlStrBuilder.append("<language>" + language + "</language>"); + } + String xmlContentTokenized = null; + Fieldable fXmlContentTokenized = doc.getFieldable("xmlContentTokenized"); + if (fXmlContentTokenized != null) { + String highlightQueryType = "orig"; + if (withLemmas(outputOptions)) { + highlightQueryType = "morph"; + } else if (normFunctions != null) { + String normFunction = normFunctions[0]; + highlightQueryType = normFunction; + if (normFunction.equals("none")) { + highlightQueryType = "orig"; + } + } + xmlContentTokenized = fXmlContentTokenized.stringValue(); + String xmlPre = "<content xmlns:xhtml=\"http://www.w3.org/1999/xhtml\" xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">"; + String xmlPost = "</content>"; + String xmlInputStr = xmlPre + xmlContentTokenized + xmlPost; + String docLanguage = docMetadataRecord.getLanguage(); + String highlightedXmlStr = highlight(xmlInputStr, highlightQueryType, query, docLanguage); + if (highlightedXmlStr == null) + highlightedXmlStr = "<content>" + xmlContentTokenized + "</content>"; + xmlStrBuilder.append(highlightedXmlStr); + } + xmlStrBuilder.append("</hit>"); + } + xmlStrBuilder.append("</hits>"); + xmlStrBuilder.append("</document>"); + return xmlStrBuilder.toString(); + } + + private String createHtmlString(MetadataRecord docMetadataRecord, String query, int page, int pageSize, String[] normFunctions, String[] outputOptions, Hits hits, HttpServletRequest request) throws ApplicationException { + String docId = docMetadataRecord.getDocId(); + ArrayList<Document> docs = null; + if (hits != null) + docs = hits.getHits(); + int hitsSize = -1; + int docsSize = -1; + if (hits != null) + hitsSize = hits.getSize(); + if (docs != null) + docsSize = docs.size(); + String highlightQueryType = "orig"; + String normalizationStr = ""; + String highlightQueryTypeStr = ""; + if (withLemmas(outputOptions)) { + highlightQueryTypeStr = "&highlightQueryType=norm"; + highlightQueryType = "norm"; + } else if (normFunctions != null) { + String normFunction = normFunctions[0]; + normalizationStr = "&normalization=" + normFunction; + highlightQueryType = normFunction; + if (normFunction.equals("none")) { + normalizationStr = "&normalization=" + "orig"; + highlightQueryType = "orig"; + } + } + StringBuilder xmlStrBuilder = new StringBuilder(); + xmlStrBuilder.append("<html>"); + xmlStrBuilder.append("<head>"); + xmlStrBuilder.append("<title>Document: \"" + docId + " " + query + "\"</title>"); + String baseUrl = getBaseUrl(request); + String cssUrl = baseUrl + "/css/page.css"; + String cssShowWordFileName = "pageOrig.css"; + if (highlightQueryType.equals("reg")) + cssShowWordFileName = "pageReg.css"; + else if (highlightQueryType.equals("norm")) + cssShowWordFileName = "pageNorm.css"; + String showWordCssUrl = baseUrl + "/css/" + cssShowWordFileName; + xmlStrBuilder.append("<link rel=\"stylesheet\" type=\"text/css\" href=\"" + showWordCssUrl + "\"/>"); + xmlStrBuilder.append("<link rel=\"stylesheet\" type=\"text/css\" href=\"" + cssUrl + "\"/>"); + xmlStrBuilder.append("</head>"); + xmlStrBuilder.append("<body>"); + xmlStrBuilder.append("<span class=\"about\">[<span class=\"it\">This is a MPIWG CMS technology service</span>] <a href=\"/mpiwg-mpdl-cms-web/index.html\"><img src=\"/mpiwg-mpdl-cms-web/images/info.png\" valign=\"bottom\" width=\"15\" height=\"15\" border=\"0\" alt=\"MPIWG CMS service\"/></a></span>"); + xmlStrBuilder.append("<span class=\"query\">Query: " + query + "</span>"); + xmlStrBuilder.append("<span class=\"result\">"); + xmlStrBuilder.append("<span class=\"resultPage\">" + page + "</span>"); + xmlStrBuilder.append("<span class=\"resultPageSize\">" + pageSize + "</span>"); + xmlStrBuilder.append("<span class=\"hitsSize\">" + hitsSize + "</span>"); + xmlStrBuilder.append("</span>"); + xmlStrBuilder.append("<table>"); + for (int i=0; i<docsSize; i++) { + xmlStrBuilder.append("<tr class=\"hit\">"); + Document doc = docs.get(i); + int num = (page - 1) * pageSize + i + 1; + xmlStrBuilder.append("<td class=\"hitNum\">" + num + ". " + "</td>"); + xmlStrBuilder.append("<td class=\"hitLink\">"); + String posStr = ""; + String pageNumber = ""; + Fieldable fPageNumber = doc.getFieldable("pageNumber"); + if (fPageNumber != null) { + pageNumber = fPageNumber.stringValue(); + posStr = posStr + "Page " + pageNumber + ", "; + } + String elementName = null; + String presElementName = ""; + Fieldable fElementName = doc.getFieldable("elementName"); + if (fElementName != null) { + elementName = fElementName.stringValue(); + presElementName = getPresentationName(elementName); + } + String elementPagePosition = ""; + Fieldable fElementPagePosition = doc.getFieldable("elementPagePosition"); + if (fElementPagePosition != null) { + elementPagePosition = fElementPagePosition.stringValue(); + posStr = posStr + presElementName + " " + elementPagePosition + ":"; + } + String language = docMetadataRecord.getLanguage(); + String getPageLink = baseUrl + "/query/GetPage?docId=" + docId + "&page=" + pageNumber + normalizationStr + "&highlightElem=" + elementName + "&highlightElemPos=" + elementPagePosition + highlightQueryTypeStr + "&highlightQuery=" + query + "&language=" + language; + xmlStrBuilder.append("<a href=\"" + getPageLink + "\">" + posStr + "</a>"); + xmlStrBuilder.append("</td>"); + String xmlContentTokenized = null; + Fieldable fXmlContentTokenized = doc.getFieldable("xmlContentTokenized"); + if (fXmlContentTokenized != null) { + xmlContentTokenized = fXmlContentTokenized.stringValue(); + String highlightedXmlStr = highlight(xmlContentTokenized, highlightQueryType, query, language); + String highlightHtmlStr = pageTransformer.transform(highlightedXmlStr, docMetadataRecord, -1, "html"); // TODO performance: do not highlight each single node but highlight them all in one step + xmlStrBuilder.append("<td class=\"hitContent\">"); + xmlStrBuilder.append(highlightHtmlStr); + xmlStrBuilder.append("</td>"); + } + xmlStrBuilder.append("</tr>"); + } + xmlStrBuilder.append("</table>"); + xmlStrBuilder.append("</body>"); + xmlStrBuilder.append("</html>"); + return xmlStrBuilder.toString(); + } + + private String highlight(String xmlStr, String highlightQueryType, String highlightQuery, String language) throws ApplicationException { + String result = null; + try { + HighlightContentHandler highlightContentHandler = new HighlightContentHandler(null, -1, highlightQueryType, highlightQuery, language); + highlightContentHandler.setFirstPageBreakReachedMode(true); + XMLReader xmlParser = new SAXParser(); + xmlParser.setContentHandler(highlightContentHandler); + StringReader stringReader = new StringReader(xmlStr); + InputSource inputSource = new InputSource(stringReader); + xmlParser.parse(inputSource); + result = highlightContentHandler.getResult().toString(); + } catch (SAXException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + return result; + } + + private String getPresentationName(String elemName) { + String retStr = null; + if (elemName != null) { + if (elemName.equals("s")) { + retStr = "Sentence"; + } else { + // first char to uppercase + char[] stringArray = elemName.toCharArray(); + stringArray[0] = Character.toUpperCase(stringArray[0]); + retStr = new String(stringArray); + } + } + return retStr; + } + + private String getBaseUrl(HttpServletRequest request) { + return getServerUrl(request) + request.getContextPath(); + } + + private String getServerUrl(HttpServletRequest request) { + if ( ( request.getServerPort() == 80 ) || ( request.getServerPort() == 443 ) ) + return request.getScheme() + "://" + request.getServerName(); + else + return request.getScheme() + "://" + request.getServerName() + ":" + request.getServerPort(); + } + + private boolean withLemmas(String[] outputOptions) { + boolean result = false; + for (int i=0; i< outputOptions.length; i++) { + String function = outputOptions[i]; + if (function.equals("withLemmas")) + return true; + } + return result; + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/cms/QueryDocuments.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,333 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.cms; + +import java.io.File; +import java.io.IOException; +import java.io.PrintWriter; +import java.util.ArrayList; +import java.util.Date; + +import javax.servlet.ServletConfig; +import javax.servlet.ServletException; +import javax.servlet.http.HttpServlet; +import javax.servlet.http.HttpServletRequest; +import javax.servlet.http.HttpServletResponse; + +import org.apache.lucene.document.Fieldable; +import org.apache.lucene.search.Query; + +import de.mpg.mpiwg.berlin.mpdl.cms.document.Document; +import de.mpg.mpiwg.berlin.mpdl.cms.document.DocumentHandler; +import de.mpg.mpiwg.berlin.mpdl.cms.document.Hits; +import de.mpg.mpiwg.berlin.mpdl.cms.lucene.IndexHandler; + +public class QueryDocuments extends HttpServlet { + private static final long serialVersionUID = 1L; + public QueryDocuments() { + super(); + } + + public void init(ServletConfig config) throws ServletException { + super.init(config); + } + + protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + request.setCharacterEncoding("utf-8"); + response.setCharacterEncoding("utf-8"); + String query = request.getParameter("query"); + String sortBy = request.getParameter("sortBy"); + String[] sortFields = null; + if (sortBy != null && ! sortBy.trim().isEmpty()) + sortFields = sortBy.split(" "); + String language = request.getParameter("language"); // optional: is also detected by translator service + if (language != null && language.equals("none")) + language = null; + String translate = request.getParameter("translate"); + String pageStr = request.getParameter("page"); + if (pageStr == null) + pageStr = "1"; + int page = Integer.parseInt(pageStr); + String pageSizeStr = request.getParameter("pageSize"); + if (pageSizeStr == null) + pageSizeStr = "10"; + int pageSize = Integer.parseInt(pageSizeStr); + int from = (page * pageSize) - pageSize; // e.g. 0 + int to = page * pageSize - 1; // e.g. 9 + String outputFormat = request.getParameter("outputFormat"); + if (outputFormat == null) + outputFormat = "html"; + if (outputFormat.equals("xml")) + response.setContentType("text/xml"); + else if (outputFormat.equals("html")) + response.setContentType("text/html"); + else + response.setContentType("text/xml"); + PrintWriter out = response.getWriter(); + if (query == null) { + out.print("no query specified: please set parameter \"query\""); + out.close(); + return; + } + try { + Date begin = new Date(); + IndexHandler indexHandler = IndexHandler.getInstance(); + boolean translateBool = false; + if (translate != null && translate.equals("true")) + translateBool = true; + boolean withHitHighlights = false; + if (query.contains("tokenOrig:") || query.contains("tokenMorph:") || query.contains("tokenReg:") || query.contains("tokenNorm:")) + withHitHighlights = true; + Hits hits = indexHandler.queryDocuments(query, sortFields, language, from, to, withHitHighlights, translateBool); + ArrayList<Document> docs = null; + if (hits != null) + docs = hits.getHits(); + int hitsSize = -1; + int docsSize = -1; + if (hits != null) + hitsSize = hits.getSize(); + if (docs != null) + docsSize = docs.size(); + Date end = new Date(); + long elapsedTime = end.getTime() - begin.getTime(); + if (outputFormat.equals("xml")) { + out.print("<result>"); + out.print("<query>"); + out.print("<queryText>" + query + "</queryText>"); + out.print("<resultPage>" + page + "</resultPage>"); + out.print("<resultPageSize>" + pageSize + "</resultPageSize>"); + out.print("</query>"); + out.print("<hitsSize>" + hitsSize + "</hitsSize>"); + out.print("<hits>"); + for (int i=0; i<docsSize; i++) { + Document doc = docs.get(i); + out.print("<doc>"); + String docId = doc.getFieldable("docId").stringValue(); + out.print("<docId>" + docId + "</docId>"); + Fieldable docCollectionNamesField = doc.getFieldable("collectionNames"); + if (docCollectionNamesField != null) { + String docCollectionNames = docCollectionNamesField.stringValue(); + out.print("<collectionName>" + docCollectionNames + "</collectionName>"); + } + ArrayList<String> hitFragments = doc.getHitFragments(); + if (hitFragments != null) { + out.print("<hitFragments>"); + for (int j=0; j<hitFragments.size(); j++) { + String hitFragment = hitFragments.get(j); + out.print("<hitFragment>" + hitFragment + "</hitFragment>"); + } + out.print("</hitFragments>"); + } + out.print("</doc>"); + } + out.print("</hits>"); + out.print("<executionTime>" + elapsedTime + "</executionTime>"); + out.print("</result>"); + } else if (outputFormat.equals("html")) { + StringBuilder htmlStrBuilder = new StringBuilder(); + String baseUrl = getBaseUrl(request); + String cssUrl = baseUrl + "/css/page.css"; + htmlStrBuilder.append("<?xml version=\"1.0\" encoding=\"UTF-8\"?><!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\"\"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">"); + htmlStrBuilder.append("<html>"); + htmlStrBuilder.append("<head>"); + htmlStrBuilder.append("<title>Query: " + query + "</title>"); + htmlStrBuilder.append("<link rel=\"stylesheet\" type=\"text/css\" href=\"" + cssUrl + "\"/>"); + htmlStrBuilder.append("</head>"); + htmlStrBuilder.append("<body>"); + htmlStrBuilder.append("<table align=\"right\" valign=\"top\">"); + htmlStrBuilder.append("<td>[<i>This is a MPIWG CMS technology service</i>] <a href=\"/mpiwg-mpdl-cms-web/index.html\"><img src=\"/mpiwg-mpdl-cms-web/images/info.png\" valign=\"bottom\" width=\"15\" height=\"15\" border=\"0\" alt=\"MPIWG CMS service\"/></a></td>"); + htmlStrBuilder.append("</table>"); + htmlStrBuilder.append("<p/>"); + String luceneQueryStr = query; + Query luceneQuery = hits.getQuery(); + if (query != null) + luceneQueryStr = luceneQuery.toString(); + String sortByStr = sortBy; + if (sortBy == null) + sortByStr = ""; + htmlStrBuilder.append("<h4>Query: " + luceneQueryStr + ", sorted by: " + sortByStr + "</h4>"); + htmlStrBuilder.append("<form action=\"QueryDocuments\" method=\"get\">"); + htmlStrBuilder.append("<input type=\"hidden\" name=\"query\" value=\"" + query + "\"/>"); + if (translate != null) + htmlStrBuilder.append("<input type=\"hidden\" name=\"translate\" value=\"" + translate + "\"/>"); + if (language != null) + htmlStrBuilder.append("<input type=\"hidden\" name=\"language\" value=\"" + language + "\"/>"); + htmlStrBuilder.append("<input type=\"hidden\" name=\"page\" id=\"pageId\" value=\"" + page + "\"/>"); + htmlStrBuilder.append("<input type=\"hidden\" name=\"sortBy\" id=\"sortById\" value=\"" + sortByStr + "\"/>"); + htmlStrBuilder.append("<input type=\"submit\" id=\"submitId\" style=\"position: absolute; left: -9999px\"/>"); + htmlStrBuilder.append("<table>"); + htmlStrBuilder.append("<colgroup>"); + htmlStrBuilder.append("<col width=\"3%\"/>"); + htmlStrBuilder.append("<col width=\"7%\"/>"); + htmlStrBuilder.append("<col width=\"3%\"/>"); + htmlStrBuilder.append("<col width=\"12%\"/>"); + htmlStrBuilder.append("<col width=\"70%\"/>"); + htmlStrBuilder.append("</colgroup>"); + htmlStrBuilder.append("<tr>"); + int countPages = hitsSize / 10 + 1; + if (hitsSize % 10 == 0) // modulo operator: e.g. 280 % 10 is 0 + countPages = hitsSize / 10; + int pageLeft = page - 1; + if (page == 1) + pageLeft = 1; + int pageRight = page + 1; + if (page == countPages) + pageRight = countPages; + htmlStrBuilder.append("<td align=\"left\" valign=\"top\"><button onclick=\"document.getElementById('pageId').value=" + pageLeft + "\" style=\"background:none;border:none;\"><img src=\"../images/left.gif\"/></button></td>"); + htmlStrBuilder.append("<td align=\"middle\" valign=\"top\" nowrap=\"true\">" + page + " / " + countPages + "</td>"); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\"><button onclick=\"document.getElementById('pageId').value=" + pageRight + "\" style=\"background:none;border:none;\"><img src=\"../images/right.gif\"/></button></td>"); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\" nowrap=\"true\">Page: <input type=\"text\" size=\"3\" value=\"" + page + "\" id=\"pageTextId\" onkeydown=\"if (event.keyCode == 13) {document.getElementById('pageId').value=document.getElementById('pageTextId').value; document.getElementById('submitId').click();}\"/></td>"); + int fromDisplay = from + 1; + int toDisplay = to + 1; + if (hitsSize < to) + toDisplay = hitsSize; + htmlStrBuilder.append("<td align=\"right\" valign=\"top\">" + fromDisplay + " - " + toDisplay + " of " + hitsSize + " documents" + "</td>"); + htmlStrBuilder.append("</tr>"); + htmlStrBuilder.append("</table>"); + htmlStrBuilder.append("<p/>"); + htmlStrBuilder.append("<table width=\"100%\" align=\"right\" border=\"2\" rules=\"groups\">"); + htmlStrBuilder.append("<colgroup>"); + htmlStrBuilder.append("<col width=\"3%\"/>"); + htmlStrBuilder.append("<col width=\"15%\"/>"); + htmlStrBuilder.append("<col width=\"45%\"/>"); + htmlStrBuilder.append("<col width=\"10%\"/>"); + htmlStrBuilder.append("<col width=\"5%\"/>"); + htmlStrBuilder.append("<col width=\"10%\"/>"); + htmlStrBuilder.append("<col width=\"6%\"/>"); + htmlStrBuilder.append("<col width=\"5%\"/>"); + htmlStrBuilder.append("<col width=\"4%\"/>"); + htmlStrBuilder.append("</colgroup>"); + htmlStrBuilder.append("<thead>"); + htmlStrBuilder.append("<tr>"); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\" style=\"font-weight:bold;\">" + "No" + "</td>"); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\">" + "<button onclick=\"document.getElementById('sortById').value='author'\" style=\"padding:0px;font-weight:bold;font-size:14px;background:none;border:none;\">" + "Author" + "</button></td>"); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\">" + "<button onclick=\"document.getElementById('sortById').value='title'\" style=\"padding:0px;font-weight:bold;font-size:14px;background:none;border:none;\">" + "Title" + "</button></td>"); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\">" + "<button onclick=\"document.getElementById('sortById').value='publisher'\" style=\"padding:0px;font-weight:bold;font-size:14px;background:none;border:none;\">" + "Place" + "</button></td>"); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\">" + "<button onclick=\"document.getElementById('sortById').value='date'\" style=\"padding:0px;font-weight:bold;font-size:14px;background:none;border:none;\">" + "Year" + "</button></td>"); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\">" + "<button onclick=\"document.getElementById('sortById').value='docId'\" style=\"padding:0px;font-weight:bold;font-size:14px;background:none;border:none;\">" + "Id" + "</button></td>"); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\">" + "<button onclick=\"document.getElementById('sortById').value='lastModified'\" style=\"padding:0px;font-weight:bold;font-size:14px;background:none;border:none;\">" + "Last modified" + "</button></td>"); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\">" + "<button onclick=\"document.getElementById('sortById').value='language'\" style=\"padding:0px;font-weight:bold;font-size:14px;background:none;border:none;\">" + "Language" + "</button></td>"); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\">" + "<button onclick=\"document.getElementById('sortById').value='schemaName'\" style=\"padding:0px;font-weight:bold;font-size:14px;background:none;border:none;\">" + "Schema" + "</button></td>"); + htmlStrBuilder.append("</tr>"); + htmlStrBuilder.append("</thead>"); + htmlStrBuilder.append("<tbody>"); + for (int i=0; i<docsSize; i++) { + Document doc = docs.get(i); + htmlStrBuilder.append("<tr valign=\"top\">"); + int num = (page - 1) * pageSize + i + 1; + htmlStrBuilder.append("<td align=\"left\" valign=\"top\">" + num + ". " + "</td>"); + Fieldable authorField = doc.getFieldable("author"); + String author = ""; + if (authorField != null) + author = authorField.stringValue(); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\">" + author + "</td>"); + Fieldable titleField = doc.getFieldable("title"); + String title = ""; + if (titleField != null) + title = titleField.stringValue(); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\">" + title + "</td>"); + Fieldable publisherField = doc.getFieldable("publisher"); + String publisher = ""; + if (publisherField != null) + publisher = publisherField.stringValue(); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\">" + publisher + "</td>"); + Fieldable yearField = doc.getFieldable("date"); + String year = ""; + if (yearField != null) + year = yearField.stringValue(); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\">" + year + "</td>"); + String docId = doc.getFieldable("docId").stringValue(); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\">" + docId + "</td>"); + Fieldable lastModifiedField = doc.getFieldable("lastModified"); + String lastModified = ""; + if (lastModifiedField != null) + lastModified = lastModifiedField.stringValue(); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\">" + lastModified + "</td>"); + Fieldable languageField = doc.getFieldable("language"); + String lang = ""; + if (languageField != null) + lang = languageField.stringValue(); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\" style=\"padding-left:5px\">" + lang + "</td>"); + Fieldable schemaNameField = doc.getFieldable("schemaName"); + String schemaName = ""; + if (schemaNameField != null) + schemaName = schemaNameField.stringValue(); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\">" + schemaName + "</td>"); + htmlStrBuilder.append("</tr>"); + // Link row + htmlStrBuilder.append("<tr valign=\"top\">"); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\"></td>"); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\" colspan=\"8\">"); + Fieldable echoIdField = doc.getFieldable("echoId"); + String echoId = null; + if (echoIdField != null) + echoId = echoIdField.stringValue(); + String urlDocuView = "http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=" + echoId; + if (echoId == null) + urlDocuView = "http://echo.mpiwg-berlin.mpg.de/"; + htmlStrBuilder.append("<img src=\"/mpiwg-mpdl-cms-web/images/book.png\" width=\"15\" height=\"15\" border=\"0\"/>" + " <a href=\"" + urlDocuView + "\">Full view</a>"); + htmlStrBuilder.append(", " + "<a href=\"/mpiwg-mpdl-cms-web/query/GetPage?docId=" + docId + "\">Lite view</a>"); + htmlStrBuilder.append("</td>"); + htmlStrBuilder.append("</tr>"); + htmlStrBuilder.append("<tr valign=\"top\">"); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\"></td>"); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\" colspan=\"8\">"); + htmlStrBuilder.append("<img src=\"/mpiwg-mpdl-cms-web/images/download.png\" width=\"15\" height=\"15\" border=\"0\"/>" + " Download: <a href=\"/mpiwg-mpdl-cms-web/doc/GetDocument?id=" + docId + "\">XML</a>"); + DocumentHandler docHandler = new DocumentHandler(); + String destFileNamePdf = docHandler.getFullFileName(docId, "pdf"); + String destFileNameHtml = docHandler.getFullFileName(docId, "html"); + int lastDot = docId.lastIndexOf("."); + String docIdWithoutExtension = docId.substring(0, lastDot); + File destFilePdf = new File(destFileNamePdf); + File destFileHtml = new File(destFileNameHtml); + if (destFilePdf.exists()) + htmlStrBuilder.append(", " + "<a href=\"/mpiwg-mpdl-cms-web/doc/GetDocument?id=" + docIdWithoutExtension + ".pdf" + "\">PDF</a>"); + if (destFileHtml.exists()) + htmlStrBuilder.append(", " + "<a href=\"/mpiwg-mpdl-cms-web/doc/GetDocument?id=" + docIdWithoutExtension + ".html" + "\">HTML</a>"); + htmlStrBuilder.append("</td>"); + htmlStrBuilder.append("</tr>"); + // hit fragments row + ArrayList<String> hitFragments = doc.getHitFragments(); + if (hitFragments != null) { + StringBuilder hitFragmentsStrBuilder = new StringBuilder(); + hitFragmentsStrBuilder.append("<b>Hit summary: </b>"); + hitFragmentsStrBuilder.append("(...) "); + for (int j=0; j<hitFragments.size(); j++) { + String hitFragment = hitFragments.get(j); + hitFragmentsStrBuilder.append(hitFragment + " (...) "); + } + htmlStrBuilder.append("<tr valign=\"top\">"); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\"></td>"); + htmlStrBuilder.append("<td align=\"left\" valign=\"top\" colspan=\"8\">" + hitFragmentsStrBuilder.toString() + "</td>"); + htmlStrBuilder.append("</tr>"); + } + } + htmlStrBuilder.append("</tbody>"); + htmlStrBuilder.append("</table>"); + htmlStrBuilder.append("</form>"); + htmlStrBuilder.append("<p/>"); + htmlStrBuilder.append("Elapsed time: " + elapsedTime + " ms"); + htmlStrBuilder.append("</body>"); + htmlStrBuilder.append("</html>"); + out.print(htmlStrBuilder.toString()); + } + out.close(); + } catch (Exception e) { + throw new ServletException(e); + } + } + + protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + doGet(request, response); + } + + private String getBaseUrl(HttpServletRequest request) { + return getServerUrl(request) + request.getContextPath(); + } + + private String getServerUrl(HttpServletRequest request) { + if ( ( request.getServerPort() == 80 ) || ( request.getServerPort() == 443 ) ) + return request.getScheme() + "://" + request.getServerName(); + else + return request.getScheme() + "://" + request.getServerName() + ":" + request.getServerPort(); + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/cms/XQueryDocument.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,228 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.cms; + +import java.io.IOException; +import java.io.PrintWriter; +import java.net.URL; +import java.util.ArrayList; + +import javax.servlet.ServletConfig; +import javax.servlet.ServletContext; +import javax.servlet.ServletException; +import javax.servlet.http.HttpServlet; +import javax.servlet.http.HttpServletRequest; +import javax.servlet.http.HttpServletResponse; + +import de.mpg.mpiwg.berlin.mpdl.cms.document.DocumentHandler; +import de.mpg.mpiwg.berlin.mpdl.util.StringUtils; +import de.mpg.mpiwg.berlin.mpdl.xml.xquery.Hit; +import de.mpg.mpiwg.berlin.mpdl.xml.xquery.Hits; +import de.mpg.mpiwg.berlin.mpdl.cms.document.MetadataRecord; +import de.mpg.mpiwg.berlin.mpdl.cms.lucene.IndexHandler; +import de.mpg.mpiwg.berlin.mpdl.cms.transform.PageTransformer; +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.xml.xquery.XQueryEvaluator; + +public class XQueryDocument extends HttpServlet { + private static final long serialVersionUID = 1L; + private XQueryEvaluator xQueryEvaluator = null; + private PageTransformer pageTransformer = null; + + public XQueryDocument() { + super(); + } + + public void init(ServletConfig config) throws ServletException { + super.init(config); + ServletContext context = getServletContext(); + xQueryEvaluator = (XQueryEvaluator) context.getAttribute("xQueryEvaluator"); + pageTransformer = (PageTransformer) context.getAttribute("pageTransformer"); + } + + protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + doGet(request, response); + } + + protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + request.setCharacterEncoding("utf-8"); + response.setCharacterEncoding("utf-8"); + String docId = request.getParameter("docId"); + String query = request.getParameter("query"); + String pageStr = request.getParameter("page"); + if (pageStr == null) + pageStr = "1"; + int page = Integer.parseInt(pageStr); + String pageSizeStr = request.getParameter("pageSize"); + if (pageSizeStr == null) + pageSizeStr = "10"; + int pageSize = Integer.parseInt(pageSizeStr); + int from = (page * pageSize) - pageSize; // e.g. 0 + int to = page * pageSize - 1; // e.g. 9 + String outputFormat = request.getParameter("outputFormat"); + if (outputFormat == null) + outputFormat = "xml"; + try { + DocumentHandler docHandler = new DocumentHandler(); + String docFileName = docHandler.getDocFullFileName(docId); + URL docFileUrl = new URL("file:" + docFileName); + Hits hits = null; + String errorStr = null; + try { + hits = xQueryEvaluator.evaluate(docFileUrl, query, from, to); + } catch (ApplicationException e) { + errorStr = e.getLocalizedMessage(); + } + IndexHandler indexHandler = IndexHandler.getInstance(); + MetadataRecord docMetadataRecord = indexHandler.getDocMetadata(docId); + if (outputFormat.equals("xml")) + response.setContentType("text/xml"); + else if (outputFormat.equals("html")) + response.setContentType("text/html"); + else + response.setContentType("text/xml"); + PrintWriter out = response.getWriter(); + String resultStr = ""; + if (errorStr == null) { + if (outputFormat.equals("xml")) + resultStr = createXmlString(docMetadataRecord, query, page, pageSize, hits); + else if (outputFormat.equals("html")) + resultStr = createHtmlString(docMetadataRecord, query, page, pageSize, hits, request); + } else { + resultStr = "Saxon XQuery error: " + errorStr; + } + out.print(resultStr); + out.close(); + } catch (ApplicationException e) { + throw new ServletException(e); + } + } + + private String createXmlString(MetadataRecord docMetadataRecord, String query, int page, int pageSize, Hits hits) throws ApplicationException { + String docId = docMetadataRecord.getDocId(); + ArrayList<Hit> hitsArray = null; + if (hits != null) + hitsArray = hits.getHits(); + int hitsSize = -1; + int hitsArraySize = -1; + if (hits != null) + hitsSize = hits.getSize(); + if (hitsArray != null) + hitsArraySize = hitsArray.size(); + StringBuilder xmlStrBuilder = new StringBuilder(); + xmlStrBuilder.append("<document>"); + xmlStrBuilder.append("<id>" + docId + "</id>"); + xmlStrBuilder.append("<query>"); + String queryXml = StringUtils.deresolveXmlEntities(query); + xmlStrBuilder.append("<queryText>" + queryXml + "</queryText>"); + xmlStrBuilder.append("<resultPage>" + page + "</resultPage>"); + xmlStrBuilder.append("<resultPageSize>" + pageSize + "</resultPageSize>"); + xmlStrBuilder.append("</query>"); + xmlStrBuilder.append("<hitsSize>" + hitsSize + "</hitsSize>"); + xmlStrBuilder.append("<hits>"); + for (int i=0; i<hitsArraySize; i++) { + int num = (page - 1) * pageSize + i + 1; + xmlStrBuilder.append("<hit n=\"" + num + "\">"); + Hit hit = hitsArray.get(i); + String name = hit.getName(); + String typeStr = "ELEMENT"; + int type = hit.getType(); + if (type == Hit.TYPE_ATTRIBUTE) + typeStr = "ATTRIBUTE"; + else if (type == Hit.TYPE_ATOMIC_VALUE) + typeStr = "ATOMIV_VALUE"; + int docPage = hit.getPage(); + int hitPagePosition = hit.getHitPagePosition(); + String xmlContent = hit.getContent(); + if (name != null) + xmlStrBuilder.append("<name>" + name + "</name>"); + xmlStrBuilder.append("<type>" + typeStr + "</type>"); + if (docPage != -1) { + xmlStrBuilder.append("<page>" + docPage + "</page>"); + xmlStrBuilder.append("<posInPage>" + hitPagePosition + "</posInPage>"); + } + xmlStrBuilder.append("<content>" + xmlContent + "</content>"); + xmlStrBuilder.append("</hit>"); + } + xmlStrBuilder.append("</hits>"); + xmlStrBuilder.append("</document>"); + return xmlStrBuilder.toString(); + } + + private String createHtmlString(MetadataRecord docMetadataRecord, String query, int page, int pageSize, Hits hits, HttpServletRequest request) throws ApplicationException { + String docId = docMetadataRecord.getDocId(); + ArrayList<Hit> hitsArray = null; + if (hits != null) + hitsArray = hits.getHits(); + int hitsArraySize = -1; + if (hitsArray != null) + hitsArraySize = hitsArray.size(); + int hitsSize = hits.getSize(); + StringBuilder xmlStrBuilder = new StringBuilder(); + xmlStrBuilder.append("<html>"); + xmlStrBuilder.append("<head>"); + xmlStrBuilder.append("<title>Document: \"" + query + "\"</title>"); + String baseUrl = getBaseUrl(request); + String cssUrl = baseUrl + "/css/page.css"; + xmlStrBuilder.append("<link rel=\"stylesheet\" type=\"text/css\" href=\"" + cssUrl + "\"/>"); + xmlStrBuilder.append("</head>"); + xmlStrBuilder.append("<body>"); + xmlStrBuilder.append("<span class=\"about\">[<span class=\"it\">This is a MPIWG CMS technology service</span>] <a href=\"/mpiwg-mpdl-cms-web/index.html\"><img src=\"/mpiwg-mpdl-cms-web/images/info.png\" valign=\"bottom\" width=\"15\" height=\"15\" border=\"0\" alt=\"MPIWG CMS service\"/></a></span>"); + xmlStrBuilder.append("<span class=\"query\">XQuery: " + query + "</span>"); + xmlStrBuilder.append("<span class=\"result\">"); + xmlStrBuilder.append("<span class=\"resultPage\">" + page + "</span>"); + xmlStrBuilder.append("<span class=\"resultPageSize\">" + pageSize + "</span>"); + xmlStrBuilder.append("<span class=\"hitsSize\">" + hitsSize + "</span>"); + xmlStrBuilder.append("</span>"); + xmlStrBuilder.append("<table>"); + if (hitsSize == 1 && hitsArray.get(0).getType() == Hit.TYPE_ATOMIC_VALUE) { + Hit hit = hitsArray.get(0); + String xmlContent = hit.getContent(); + xmlContent = StringUtils.deresolveXmlEntities(xmlContent); + xmlStrBuilder.append(xmlContent); + } else { + for (int i=0; i<hitsArraySize; i++) { + xmlStrBuilder.append("<tr class=\"hit\">"); + Hit hit = hitsArray.get(i); + int docPage = hit.getPage(); + String hitName = hit.getName(); + int hitType = hit.getType(); + int hitPagePosition = hit.getHitPagePosition(); + String getPageLink = baseUrl + "/query/GetPage?docId=" + docId + "&page=" + docPage + "&outputFormat=" + "xmlDisplay" + "&highlightElem=" + hitName + "&highlightElemPos=" + hitPagePosition; + String hitPres = hitName + "[" + hitPagePosition + "]"; + if (hitType == Hit.TYPE_ATTRIBUTE) { + hitPres = "@" + hitName; + getPageLink = baseUrl + "/query/GetPage?docId=" + docId + "&page=" + docPage; + } + String posStr = "Page " + docPage + ", " + hitPres + ":"; + int num = (page - 1) * pageSize + i + 1; + xmlStrBuilder.append("<td class=\"hitNum\">" + num + ". " + "</td>"); + xmlStrBuilder.append("<td class=\"hitLink\">"); + if (docPage != -1) { + xmlStrBuilder.append("<a href=\"" + getPageLink + "\">" + posStr + "</a>"); + } + xmlStrBuilder.append("</td>"); + String xmlContent = hit.getContent(); + String htmlContent = pageTransformer.transform(xmlContent, docMetadataRecord, -1, "xmlDisplay"); // TODO performance: do not transform each single hit but transform then all in one step + xmlStrBuilder.append("<td class=\"hitContent\">"); + xmlStrBuilder.append(htmlContent); + xmlStrBuilder.append("</td>"); + xmlStrBuilder.append("</tr>"); + } + } + xmlStrBuilder.append("</table>"); + xmlStrBuilder.append("</body>"); + xmlStrBuilder.append("</html>"); + return xmlStrBuilder.toString(); + } + + private String getBaseUrl(HttpServletRequest request) { + return getServerUrl(request) + request.getContextPath(); + } + + private String getServerUrl(HttpServletRequest request) { + if ( ( request.getServerPort() == 80 ) || ( request.getServerPort() == 443 ) ) + return request.getScheme() + "://" + request.getServerName(); + else + return request.getScheme() + "://" + request.getServerName() + ":" + request.getServerPort(); + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/lt/GetDictionaryEntries.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,485 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.lt; + +import java.io.IOException; +import java.io.PrintWriter; +import java.util.ArrayList; +import java.util.Arrays; +import java.util.Collections; +import java.util.Date; + +import javax.servlet.ServletConfig; +import javax.servlet.ServletException; +import javax.servlet.http.HttpServlet; +import javax.servlet.http.HttpServletRequest; +import javax.servlet.http.HttpServletResponse; + +import org.apache.commons.lang3.StringEscapeUtils; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.lt.dict.app.Lexica; +import de.mpg.mpiwg.berlin.mpdl.lt.dict.app.Lexicon; +import de.mpg.mpiwg.berlin.mpdl.lt.dict.app.LexiconEntry; +import de.mpg.mpiwg.berlin.mpdl.lt.dict.db.LexHandler; +import de.mpg.mpiwg.berlin.mpdl.lt.general.Language; +import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.Form; +import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.Lemma; +import de.mpg.mpiwg.berlin.mpdl.lt.text.norm.Normalizer; +import de.mpg.mpiwg.berlin.mpdl.servlets.util.ServletUtil; + +public class GetDictionaryEntries extends HttpServlet { + private static final long serialVersionUID = 1L; + private LexHandler lexHandler; + + public GetDictionaryEntries() { + super(); + } + + public void init(ServletConfig config) throws ServletException { + super.init(config); + try { + lexHandler = LexHandler.getInstance(); + } catch (ApplicationException e) { + throw new ServletException(e); + } + } + + protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + Date begin = new Date(); + request.setCharacterEncoding("utf-8"); + response.setCharacterEncoding("utf-8"); + String query = request.getParameter("query"); + String queryDisplay = request.getParameter("queryDisplay"); + String language = request.getParameter("language"); + String inputType = request.getParameter("inputType"); + String outputFormat = request.getParameter("outputFormat"); + String[] outputTypesArray = request.getParameterValues("outputType"); + String dictionary = request.getParameter("dictionary"); + String normalization = request.getParameter("normalization"); + String normalizationType = request.getParameter("normalizationType"); + String resultPageNumber = request.getParameter("resultPageNumber"); + String resultPageSize = request.getParameter("resultPageSize"); + if (query == null) + query = "a*"; + boolean isRangeQuery = false; + if (query.endsWith("*")) + isRangeQuery = true; + if (queryDisplay == null) + queryDisplay = query; + if (language == null) + language = "eng"; + if (inputType == null || ! (inputType.equals("form") || inputType.equals("lemma"))) + inputType = "form"; + if (outputFormat == null || ! (outputFormat.equals("xml") || outputFormat.equals("html") || outputFormat.equals("htmlFragment"))) + outputFormat = "xml"; + ArrayList<String> outputTypes = new ArrayList<String>(); + if (outputTypesArray == null) { + outputTypes.add("allCompact"); + } else if (isRangeQuery) { + outputTypes.add("dictCompact"); + } else { + outputTypes = new ArrayList<String>(Arrays.asList(outputTypesArray)); + } + if (normalization == null || ! (normalization.equals("none") || normalization.equals("reg") || normalization.equals("reg norm"))) + normalization = "norm"; + if (normalizationType == null || ! (normalizationType.equals("display") || normalizationType.equals("dictionary"))) + normalizationType = "dictionary"; + String xmlDict = "all"; + if (dictionary != null) + xmlDict = dictionary; + int pageNumber = 1; + int pageSize = 50; + if (resultPageNumber != null) + pageNumber = new Integer(resultPageNumber); + if (resultPageSize != null) + pageSize = new Integer(resultPageSize); + String xmlQueryString = "<query>"; + xmlQueryString = xmlQueryString + "<name>" + query + "</name>"; + xmlQueryString = xmlQueryString + "<display>" + queryDisplay + "</display>"; + xmlQueryString = xmlQueryString + "<language>" + language + "</language>"; + xmlQueryString = xmlQueryString + "<inputType>" + inputType + "</inputType>"; + xmlQueryString = xmlQueryString + "<outputFormat>" + outputFormat + "</outputFormat>"; + xmlQueryString = xmlQueryString + "<outputTypes>" + outputTypes + "</outputTypes>"; + xmlQueryString = xmlQueryString + "<dictionary>" + xmlDict + "</dictionary>"; + xmlQueryString = xmlQueryString + "<normalization>" + normalization + "</normalization>"; + xmlQueryString = xmlQueryString + "<normalizationType>" + normalizationType + "</normalizationType>"; + if (isRangeQuery) { + xmlQueryString = xmlQueryString + "<resultPageNumber>" + pageNumber + "</resultPageNumber>"; + xmlQueryString = xmlQueryString + "<resultPageSize>" + pageSize + "</resultPageSize>"; + } + xmlQueryString = xmlQueryString + "</query>"; + try { + if (outputFormat.equals("xml")) + response.setContentType("text/xml"); + else if (outputFormat.equals("html") || outputFormat.equals("htmlFragment")) + response.setContentType("text/html"); + else + response.setContentType("text/xml"); + PrintWriter out = response.getWriter(); + if (query == null || query.isEmpty()) { + out.print("request parameter \"query\" is empty. Please specify a query."); + out.close(); + return; + } + ArrayList<Lemma> lemmas = null; + ArrayList<Lexicon> dictionaries = null; + if (isRangeQuery) { + String queryTmp = query.substring(0, query.length() - 1); // without last star + if (dictionary != null) + dictionaries = lexHandler.getLexEntriesByLexiconBeginningWith(dictionary, queryTmp, pageNumber, pageSize); + else + dictionaries = lexHandler.getLexEntriesBeginningWith(language, queryTmp, pageNumber, pageSize); + } else { + String lang = language; + if (dictionary != null) { + Lexicon lexicon = Lexica.getInstance().getLexicon(dictionary); + if (lexicon != null) + lang = lexicon.getSourceLanguage(); + } + int normMode = Normalizer.DICTIONARY; + if (normalization.equals("none")) + normMode = Normalizer.NONE; + else if (normalizationType.equals("display")) + normMode = Normalizer.DISPLAY; + lemmas = lexHandler.getLemmas(query, inputType, lang, normMode, true); + dictionaries = lexHandler.getLexEntries(lemmas, lang, dictionary, query); + } + String baseUrl = ServletUtil.getInstance().getBaseUrl(request); + Date end = new Date(); + String elapsedTime = String.valueOf(end.getTime() - begin.getTime()); + String result = ""; + if (outputFormat == null || outputFormat.equals("xml")) + result = createXmlOutputString(query, lemmas, dictionaries, outputTypes, baseUrl, xmlQueryString, elapsedTime); + else if (outputFormat.equals("html") || outputFormat.equals("htmlFragment")) + result = createHtmlOutputString(query, queryDisplay, language, lemmas, dictionaries, pageNumber, pageSize, isRangeQuery, outputFormat, outputTypes, elapsedTime); + else + result = createXmlOutputString(query, lemmas, dictionaries, outputTypes, baseUrl, xmlQueryString, elapsedTime); + out.print(result); + out.close(); + } catch (ApplicationException e) { + throw new ServletException(e); + } + } + + protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + doGet(request, response); + } + + private String createXmlOutputString(String query, ArrayList<Lemma> lemmas, ArrayList<Lexicon> lexicons, ArrayList<String> outputTypes, String baseUrl, String xmlQueryString, String elapsedTime) { + boolean outputTypeMorphCompact = false; + if (outputTypes.contains("allCompact") || outputTypes.contains("morphCompact")) + outputTypeMorphCompact = true; + boolean outputTypeMorphFull = false; + if (outputTypes.contains("allFull") || outputTypes.contains("morphFull")) + outputTypeMorphFull = true; + boolean outputTypeDictCompact = false; + if (outputTypes.contains("allCompact") || outputTypes.contains("dictCompact")) + outputTypeDictCompact = true; + boolean outputTypeDictFull = false; + if (outputTypes.contains("allFull") || outputTypes.contains("dictFull")) + outputTypeDictFull = true; + boolean outputTypeWikiCompact = false; + if (outputTypes.contains("allCompact") || outputTypes.contains("wikiCompact")) + outputTypeWikiCompact = true; + boolean outputTypeWikiFull = false; + if (outputTypes.contains("allFull") || outputTypes.contains("wikiFull")) + outputTypeWikiFull = true; + String result = "<result>"; + result = result + "<provider>" + "MPIWG MPDL language technology service (see: " + "" + baseUrl + "), Max Planck Institute for the History of Science, Berlin." + "</provider>"; + result = result + xmlQueryString; + result = result + "<elapsed-time-ms>" + elapsedTime + "</elapsed-time-ms>"; + if ((outputTypeMorphCompact || outputTypeMorphFull) && lemmas != null && ! lemmas.isEmpty()) { + result = result + "<morphology>"; + for (int i=0; i<lemmas.size(); i++) { + Lemma lemma = lemmas.get(i); + String lemmaName = lemma.getLemmaName(); + String language = lemma.getLanguage(); + result = result + "<lemma>"; + result = result + "<name>" + lemmaName + "</name>"; + if (outputTypeMorphFull) { + String lemmaProvider = lemma.getProvider(); + result = result + "<provider>" + lemmaProvider + "</provider>"; + result = result + "<language>" + language + "</language>"; + } + if (Language.getInstance().isArabic(language) || Language.getInstance().isLatin(language)) { + String remoteUrl = "http://www.perseus.tufts.edu/hopper/morph?l=" + lemmaName + "&la=" + language; + result = result + "<remoteUrl>" + remoteUrl + "</remoteUrl>"; + } else if (Language.getInstance().isGreek(language)) { + String remoteUrl = "http://www.perseus.tufts.edu/hopper/morph?l=" + lemmaName + "&la=" + "greek"; + result = result + "<remoteUrl>" + remoteUrl + "</remoteUrl>"; + } + if (outputTypeMorphFull) { + ArrayList<Form> forms = lemma.getFormsList(); + Collections.sort(forms); + if (forms != null && ! forms.isEmpty()) { + result = result + "<forms>"; + for (int j=0; j<forms.size(); j++) { + result = result + "<form>"; + Form f = forms.get(j); + String formName = f.getFormName(); + String formProvider = f.getProvider(); + result = result + "<provider>" + formProvider + "</provider>"; + result = result + "<language>" + language + "</language>"; + result = result + "<name>" + formName + "</name>"; + result = result + "</form>"; + } + result = result + "</forms>"; + } + } + result = result + "</lemma>"; + } + result = result + "</morphology>"; + } + if ((outputTypeDictCompact || outputTypeDictFull) && lexicons != null) { + result = result + "<dictionaries>"; + for (int i=0; i<lexicons.size(); i++) { + Lexicon lexicon = lexicons.get(i); + if (outputTypeDictFull) + result = result + lexicon.toXmlString(); + else if (outputTypeDictCompact) + result = result + lexicon.toXmlStringCompact(); + } + result = result + "</dictionaries>"; + } + if ((outputTypeWikiCompact || outputTypeWikiFull) && lemmas != null && ! lemmas.isEmpty()) { + result = result + "<wikipedia>"; + String language = null; + for (int i=0; i<lemmas.size(); i++) { + Lemma lemma = lemmas.get(i); + String lemmaName = lemma.getLemmaName(); + language = Language.getInstance().getLanguageId(lemma.getLanguage()); + result = result + "<article>"; + result = result + "<name>" + lemmaName + "</name>"; + String wikiHrefExact = "http://" + language + ".wikipedia.org/wiki/" + lemmaName; + String wikiHrefSearch = "http://" + language + ".wikipedia.org/wiki/index.php?search=" + lemmaName; + result = result + "<remoteUrl>" + wikiHrefExact + "</remoteUrl>"; + result = result + "<remoteUrlSearch>" + wikiHrefSearch + "</remoteUrlSearch>"; + result = result + "</article>"; + } + if (language != null && Language.getInstance().isGerman(language) && query != null) { + String[] queryFormNames = query.split(" "); + for (int j=0; j<queryFormNames.length; j++) { + String queryFormName = queryFormNames[j]; + result = result + "<article>"; + result = result + "<name>" + queryFormName + "</name>"; + String wikiHrefExact = "http://" + language + ".wikipedia.org/wiki/" + queryFormName; + String wikiHrefSearch = "http://" + language + ".wikipedia.org/wiki/index.php?search=" + queryFormName; + result = result + "<remoteUrl>" + wikiHrefExact + "</remoteUrl>"; + result = result + "<remoteUrlSearch>" + wikiHrefSearch + "</remoteUrlSearch>"; + result = result + "</article>"; + } + } + result = result + "</wikipedia>"; + } + result = result + "</result>"; + return result; + } + + private String createHtmlOutputString(String query, String queryDisplay, String language, ArrayList<Lemma> lemmas, ArrayList<Lexicon> lexicons, int pageNumber, int pageSize, boolean isRangeQuery, String outputFormat, ArrayList<String> outputTypes, String elapsedTime) { + boolean outputTypeMorphCompact = false; + if (outputTypes.contains("allCompact") || outputTypes.contains("morphCompact")) + outputTypeMorphCompact = true; + boolean outputTypeMorphFull = false; + if (outputTypes.contains("allFull") || outputTypes.contains("morphFull")) + outputTypeMorphFull = true; + boolean outputTypeDictCompact = false; + if (outputTypes.contains("allCompact") || outputTypes.contains("dictCompact")) + outputTypeDictCompact = true; + boolean outputTypeDictFull = false; + if (outputTypes.contains("allFull") || outputTypes.contains("dictFull")) + outputTypeDictFull = true; + boolean outputTypeWikiCompact = false; + if (outputTypes.contains("allCompact") || outputTypes.contains("wikiCompact")) + outputTypeWikiCompact = true; + boolean outputTypeWikiFull = false; + if (outputTypes.contains("allFull") || outputTypes.contains("wikiFull")) + outputTypeWikiFull = true; + String result = ""; + result = result + "<html>"; + result = result + "<head>"; + result = result + "<title>WordInfo for: \"" + queryDisplay + "\"</title>"; + result = result + "<link rel=\"stylesheet\" type=\"text/css\" href=\"/mpiwg-mpdl-cms-web/css/getDictionaryEntries.css\"/>"; + result = result + getJavascriptFunctions(); + result = result + "</head>"; + result = result + "<body class=\"body\">"; + result = result + "<table align=\"right\" valign=\"top\">"; + result = result + "<td>[<i>This is a MPIWG MPDL language technology service</i>] <a href=\"/mpiwg-mpdl-cms-web/index.html\"><img src=\"/mpiwg-mpdl-cms-web/images/info.png\" valign=\"bottom\" width=\"15\" height=\"15\" border=\"0\" alt=\"MPIWG MPDL language technology service\"/></a></td>"; + result = result + "</table>"; + result = result + "<p/>"; + result = result + "<div class=\"title\">WordInfo</div>"; + + result = result + "<form name=\"getDictionaryEntries\" action=\"GetDictionaryEntries\" method=\"get\">"; + result = result + "Query: <input name=\"query\" class=\"query\" type=\"text\" size=\"20\" value=\"" + query + "\"/>"; + String htmlSelectBox = Language.getInstance().getHtmlSelectBox(language); + result = result + " Language: " + htmlSelectBox; + result = result + "<input type=\"hidden\" name=\"outputFormat\" value=\"" + outputFormat + "\"/>"; + for (int i=0; i<outputTypes.size(); i++) { + String type = outputTypes.get(i); + result = result + "<input type=\"hidden\" name=\"outputType\" value=\"" + type + "\"/>"; + } + result = result + " <button type=\"submit\" onclick=\"var query = document.getDictionaryEntries.query.value; if(query.substring(query.length-1, query.length) == '*') {document.getDictionaryEntries.outputType.value = 'dictCompact';}\">Query</button>"; + result = result + "</form>"; + result = result + "<hr/>"; + + if ((outputTypeMorphCompact || outputTypeMorphFull) && lemmas != null && ! lemmas.isEmpty()) { + result = result + "<span class=\"inputType\">Morphology</span>"; + result = result + "<span class=\"tree\">"; + result = result + "<ul>"; + result = result + "<li>" + "<b>" + "Lemmata" + "</b>"; + result = result + "<ul>"; + for (int i=0; i<lemmas.size(); i++) { + Lemma lemma = lemmas.get(i); + String lemmaName = lemma.getLemmaName(); + String providerText = ""; + String externalLinkText = ""; + String formsText = ""; + if (outputTypeMorphFull) { + String lemmaProvider = lemma.getProvider(); + providerText = " (data provider: " + lemmaProvider + ")"; + } + if (Language.getInstance().isArabic(language) || Language.getInstance().isLatin(language)) + externalLinkText = " (external link: <a href=\"http://www.perseus.tufts.edu/hopper/morph?l=" + lemmaName + "&la=" + language + "\">" + lemmaName + "</a>)"; + else if (Language.getInstance().isGreek(language)) + externalLinkText = " (external link: <a href=\"http://www.perseus.tufts.edu/hopper/morph?l=" + lemmaName + "&la=" + "greek" + "\">" + lemmaName + "</a>)"; + ArrayList<Form> forms = lemma.getFormsList(); + Collections.sort(forms); + if (forms != null && ! forms.isEmpty()) { + formsText = formsText + "<ul>"; + formsText = formsText + "<b>" + "Forms: " + "</b>"; + for (int j=0; j<forms.size(); j++) { + Form f = forms.get(j); + String formName = f.getFormName(); + String formProvider = f.getProvider(); + formsText = formsText + formName + " (data provider: " + formProvider + "), "; + } + formsText = formsText.substring(0, formsText.length() - 2); // without last comma and blank + formsText = formsText + "</ul>"; + } + if (outputTypeMorphFull) { + result = result + "<li><input type=\"checkbox\" checked=\"checked\"/><label>" + lemmaName + providerText + externalLinkText + "</label>" + formsText + "</li>"; + } else if (outputTypeMorphCompact) { + result = result + "<li><input type=\"checkbox\"/><label>" + "<a href=\"GetDictionaryEntries?query=" + lemmaName + "&language=" + language + "&inputType=lemma" + "&outputFormat=html" + "&outputType=morphFull" + "\">" + lemmaName + "</a>" + providerText + externalLinkText + "</label>" + formsText + "</li>"; + } + } + result = result + "</ul>"; + result = result + "</li>"; + result = result + "</ul>"; + result = result + "</span>"; + } + if ((outputTypeDictCompact || outputTypeDictFull) && lexicons != null && ! lexicons.isEmpty()) { + if (isRangeQuery) { + int pageNumberUp = 1; + if (pageNumber > 1) + pageNumberUp = pageNumber - 1; + int pageNumberDown = pageNumber + 1; + int from = (pageNumber * pageSize) - pageSize + 1; + int to = pageNumber * pageSize; + result = result + "<table align=\"right\">"; + result = result + "<tr>"; + result = result + "<td valign=\"middle\">Page: </td>"; + result = result + "<td valign=\"middle\"><a href=\"/mpiwg-mpdl-cms-web/lt/GetDictionaryEntries?query=" + query + "&language=" + language + "&resultPageNumber=" + pageNumberUp + "&outputFormat=html\"><img alt=\"page up\" src=\"/mpiwg-mpdl-cms-web/images/left.gif\" height=\"20\" width=\"20\"></a></td>"; + result = result + "<td valign=\"middle\">" + pageNumber + "</td>"; + result = result + "<td valign=\"middle\"><a href=\"/mpiwg-mpdl-cms-web/lt/GetDictionaryEntries?query=" + query + "&language=" + language + "&resultPageNumber=" + pageNumberDown + "&outputFormat=html\"><img alt=\"page down\" src=\"/mpiwg-mpdl-cms-web/images/right.gif\" height=\"20\" width=\"20\"></a></td>"; + result = result + "<td valign=\"middle\">(" + from + " - " + to + ")</td>"; + result = result + "</tr>"; + result = result + "</table>"; + } + result = result + "<span class=\"inputType\">Dictionary</span>"; + result = result + "<span class=\"tree\">"; + result = result + "<ul>"; + for (int i=0; i<lexicons.size(); i++) { + Lexicon lexicon = lexicons.get(i); + result = result + "<li>"; + result = result + "<b>" + lexicon.getDescription() + "</b>"; + result = result + "<ul>"; + ArrayList<LexiconEntry> entries = lexicon.getEntries(); + for (int j=0; j<entries.size(); j++) { + String entryContent = ""; + LexiconEntry entry = entries.get(j); + if (lexicon.isLocalLexicon()) { + if (entry.isXmlValid()) { + String repairedEntry = entry.getRepairedEntry(); + repairedEntry = repairedEntry.replaceAll("<repaired-entry>", ""); + repairedEntry = repairedEntry.replaceAll("</repaired-entry>", ""); + entryContent = entryContent + repairedEntry; // valid unicode content of the original entry + } else { + entryContent = entryContent + "[Remark: <i> this dictionary entry has no valid XML/HTML content in database so a text version of this entry is shown.</i>]: <br/>"; + String originalEntry = entry.getOriginalEntry(); + originalEntry = originalEntry.replaceAll("<original-entry>", ""); + originalEntry = originalEntry.replaceAll("</original-entry>", ""); + originalEntry = StringEscapeUtils.escapeXml(originalEntry); // create text version of the invalid xml content + entryContent = entryContent + originalEntry; + } + if (entry.getRemoteUrl() != null) { + entryContent = entryContent + "<div>(external link: <a href=\"" + entry.getRemoteUrl() + "\">" + entry.getFormName() + "</a>)</div>"; + } + } else { + if (entry.getRemoteUrl() != null) { + entryContent = entryContent + "external link: <a href=\"" + entry.getRemoteUrl() + "\">" + entry.getFormName() + "</a>"; + } + } + String formName = entry.getFormName(); + String dictName = lexicon.getName(); + if (outputTypeDictFull) { + result = result + "<li><input type=\"checkbox\" checked=\"checked\"/><label><b>" + formName + "</b></label><ul><li>" + entryContent + "</li></ul></li>"; + } else if (outputTypeDictCompact) { + result = result + "<li><input type=\"checkbox\"/><label>" + "<a href=\"GetDictionaryEntries?query=" + formName + "&language=" + language + "&dictionary=" + dictName + "&inputType=lemma" + "&outputFormat=html" + "&outputType=dictFull" + "\">" + formName + "</a>" + "</label><ul><li>" + entryContent + "</li></ul></li>"; + } + } + result = result + "</ul>"; + result = result + "</li>"; // lexicon entry + } + result = result + "</ul>"; + result = result + "</span>"; + } + if ((outputTypeWikiCompact || outputTypeWikiFull) && lemmas != null && ! lemmas.isEmpty()) { + String langId = Language.getInstance().getLanguageId(language); + String wikiUrl = langId + ".wikipedia.org"; + result = result + "<span class=\"inputType\">Wikipedia</span>"; + result = result + "<span class=\"tree\">"; + result = result + "<ul>"; + result = result + "<li>" + "<b>" + wikiUrl + "</b>"; + result = result + "<ul>"; + for (int i=0; i<lemmas.size(); i++) { + Lemma lemma = lemmas.get(i); + String lemmaName = lemma.getLemmaName(); + String wikiHrefExact = "http://" + wikiUrl + "/wiki/" + lemmaName; + String wikiHrefSearch = "http://" + wikiUrl + "/wiki/index.php?search=" + lemmaName; + result = result + "<li><input type=\"checkbox\" checked=\"checked\"/><label>" + "External link: <a href=\"" + wikiHrefExact + "\">" + lemmaName + "</a> (or search for <a href=\"" + wikiHrefSearch + "\">" + lemmaName + "</a>)" + "</label>" + "</li>"; + } + if (language != null && Language.getInstance().isGerman(language) && query != null) { + String[] queryFormNames = query.split(" "); + for (int j=0; j<queryFormNames.length; j++) { + String queryFormName = queryFormNames[j]; + result = result + "<li>"; + String wikiHrefExact = "http://" + wikiUrl + "/wiki/" + queryFormName; + String wikiHrefSearch = "http://" + wikiUrl + "/wiki/index.php?search=" + queryFormName; + result = result + "<li><input type=\"checkbox\" checked=\"checked\"/><label>" + "External link: <a href=\"" + wikiHrefExact + "\">" + queryFormName + "</a> (or search for <a href=\"" + wikiHrefSearch + "\">" + queryFormName + "</a>)" + "</label>" + "</li>"; + result = result + "</li>"; + } + } + result = result + "</ul>"; + result = result + "</li>"; + result = result + "</ul>"; + result = result + "</span>"; + } + result = result + "<p/>"; + result = result + "[* external links may not function]"; + result = result + "<hr/>"; + result = result + "<p/>"; + result = result + "Elapsed time: " + elapsedTime + " ms, see the <a href=\"/mpiwg-mpdl-cms-web/index.html\">service description</a> of this page, if you find a bug <a href=\"https://it-dev.mpiwg-berlin.mpg.de/tracs/mpdl-project-software/newticket\">let us know</a>"; + result = result + "</body>"; + result = result + "</html>"; + return result; + } + + private String getJavascriptFunctions() { + String result = "<script type=\"text/javascript\">"; + result = result + "<!-- "; + result = result + "function strEndsWith(str, suffix) {"; + result = result + " return str.match(suffix + \"$\") == suffix;"; + result = result + "}"; + result = result + " -->"; + result = result + "</script>"; + return result; + } +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/lt/GetForms.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,214 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.lt; + +import java.io.IOException; +import java.io.PrintWriter; +import java.util.ArrayList; +import java.util.Collections; +import java.util.Date; +import java.util.Hashtable; + +import javax.servlet.ServletConfig; +import javax.servlet.ServletException; +import javax.servlet.http.HttpServlet; +import javax.servlet.http.HttpServletRequest; +import javax.servlet.http.HttpServletResponse; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.lt.dict.db.LexHandler; +import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.Form; +import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.Lemma; +import de.mpg.mpiwg.berlin.mpdl.lt.text.norm.Normalizer; + +public class GetForms extends HttpServlet { + private static final long serialVersionUID = 1L; + private LexHandler lexHandler; + + public GetForms() { + super(); + } + + public void init(ServletConfig config) throws ServletException { + super.init(config); + try { + lexHandler = LexHandler.getInstance(); + } catch (ApplicationException e) { + throw new ServletException(e); + } + } + + protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + Date begin = new Date(); + request.setCharacterEncoding("utf-8"); + response.setCharacterEncoding("utf-8"); + String query = request.getParameter("query"); + String language = request.getParameter("language"); + String outputFormat = request.getParameter("outputFormat"); + String outputType = request.getParameter("outputType"); + String normalization = request.getParameter("normalization"); + if (language == null) + language = "eng"; + if (outputFormat == null || ! (outputFormat.equals("xml") || outputFormat.equals("html") || outputFormat.equals("string"))) + outputFormat = "xml"; + if (outputType == null || ! (outputType.equals("compact") || outputType.equals("full"))) + outputType = "compact"; + if (normalization == null || ! (normalization.equals("none") || normalization.equals("reg") || normalization.equals("reg norm"))) + normalization = "norm"; + int normalizationType = Normalizer.DICTIONARY; + if (normalization != null && normalization.equals("none")) + normalizationType = Normalizer.NONE; + + String xmlQueryString = "<query><name>" + query + "</name>" + "<language>" + language + "</language>" + + "<outputFormat>" + outputFormat + "</outputFormat>" + "<outputType>" + outputType + "</outputType>" + "<normalization>" + normalization + "</normalization>" + "</query>"; + try { + if (outputFormat.equals("xml")) + response.setContentType("text/xml"); + else if (outputFormat.equals("html") || outputFormat.equals("string")) + response.setContentType("text/html"); + else + response.setContentType("text/xml"); + PrintWriter out = response.getWriter(); + if (query == null || query.isEmpty()) { + out.print("request parameter query is empty. Please specify a query."); + out.close(); + return; + } + ArrayList<Lemma> lemmas = lexHandler.getLemmas(query, "lemma", language, normalizationType, true); + Hashtable<String, Form> formsHashtable = new Hashtable<String, Form>(); + ArrayList<Form> forms = new ArrayList<Form>(); + if (lemmas != null && ! lemmas.isEmpty()) { + for (int i=0; i<lemmas.size(); i++) { + Lemma lemma = lemmas.get(i); + ArrayList<Form> lemmaForms = lemma.getFormsList(); + for (int j=0; j < lemmaForms.size(); j++) { + Form form = lemmaForms.get(j); + formsHashtable.put(form.getFormName(), form); + } + } + } + forms.addAll(formsHashtable.values()); + Collections.sort(forms); + String baseUrl = getBaseUrl(request); + Date end = new Date(); + String elapsedTime = String.valueOf(end.getTime() - begin.getTime()); + String result = ""; + if (outputFormat == null || outputFormat.equals("xml")) + result = createXmlOutputString(query, forms, outputType, baseUrl, xmlQueryString, elapsedTime); + else if (outputFormat.equals("html")) + result = createHtmlOutputString(query, forms, outputType, elapsedTime); + else if (outputFormat.equals("string")) + result = createStringOutputString(forms); + else + result = createXmlOutputString(query, forms, outputType, baseUrl, xmlQueryString, elapsedTime); + out.print(result); + out.close(); + } catch (ApplicationException e) { + throw new ServletException(e); + } + } + + protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + doGet(request, response); + } + + private String getBaseUrl( HttpServletRequest request ) { + if (request.getServerPort() == 80 || request.getServerPort() == 443) + return request.getScheme() + "://" + request.getServerName() + request.getContextPath(); + else + return request.getScheme() + "://" + request.getServerName() + ":" + request.getServerPort() + request.getContextPath(); + } + + + private String createXmlOutputString(String query, ArrayList<Form> forms, String outputType, String baseUrl, String xmlQueryString, String elapsedTime) { + String result = "<result>"; + result = result + "<provider>" + "MPIWG MPDL language technology service (see: " + "" + baseUrl + "), Max Planck Institute for the History of Science, Berlin." + "</provider>"; + result = result + xmlQueryString; + result = result + "<elapsed-time-ms>" + elapsedTime + "</elapsed-time-ms>"; + if (forms != null && ! forms.isEmpty()) { + result = result + "<morphology>"; + result = result + "<forms>"; + for (int i=0; i<forms.size(); i++) { + result = result + "<form>"; + Form f = forms.get(i); + String formName = f.getFormName(); + String language = f.getLanguage(); + String formProvider = f.getProvider(); + String lemmaName = f.getLemmaName(); + result = result + "<provider>" + formProvider + "</provider>"; + result = result + "<language>" + language + "</language>"; + result = result + "<lemmaName>" + lemmaName + "</lemmaName>"; + result = result + "<formName>" + formName + "</formName>"; + result = result + "</form>"; + } + result = result + "</forms>"; + result = result + "</morphology>"; + } + result = result + "</result>"; + return result; + } + + private String createHtmlOutputString(String query, ArrayList<Form> forms, String outputType, String elapsedTime) { + String result = ""; + result = result + "<html>"; + result = result + "<head>"; + result = result + "<title>Lemmas for: \"" + query + "\"</title>"; + result = result + "</head>"; + result = result + "<body>"; + result = result + "<table align=\"right\" valign=\"top\">"; + result = result + "<td>[<i>This is a MPIWG MPDL language technology service</i>] <a href=\"/mpiwg-mpdl-lt-web/index.html\"><img src=\"/mpiwg-mpdl-lt-web/images/info.png\" valign=\"bottom\" width=\"15\" height=\"15\" border=\"0\" alt=\"MPIWG MPDL language technology service\"/></a></td>"; + result = result + "</table>"; + result = result + "<p/>"; + result = result + "<h1>Forms for: \"" + query + "\"</h1>"; + if (forms != null && ! forms.isEmpty()) { + result = result + "<h3>Morphology</h3>"; + result = result + "<ul>"; + result = result + "<p/>"; + if (outputType != null && outputType.equals("full")) { + for (int i=0; i<forms.size(); i++) { + result = result + "<li>"; + Form f = forms.get(i); + String formName = f.getFormName(); + String formProvider = f.getProvider(); + String language = f.getLanguage(); + String lemmaName = f.getLemmaName(); + result = result + formName + " (data provider: " + formProvider + ", language: " + language + ", lemmaName: " + lemmaName + ")"; + result = result + "</li>"; + } + } else if (outputType == null || outputType.equals("compact")) { + result = result + "<li>"; + for (int i=0; i<forms.size(); i++) { + Form f = forms.get(i); + String formName = f.getFormName(); + result = result + formName + ", "; + } + result = result.substring(0, result.length() - 2); // without last comma and blank + result = result + "</li>"; + } else if (outputType.equals("string")) { + for (int i=0; i<forms.size(); i++) { + Form f = forms.get(i); + String formName = f.getFormName(); + result = result + formName + " "; + } + result = result.substring(0, result.length() - 1); // without last blank + } + } + result = result + "<hr/>"; + result = result + "<p/>"; + result = result + "Elapsed time: " + elapsedTime + " ms, see the <a href=\"/mpiwg-mpdl-lt-web/index.html\">service description</a> of this page, if you find a bug <a href=\"https://it-dev.mpiwg-berlin.mpg.de/tracs/mpdl-project-software/newticket\">let us know</a>"; + result = result + "</body>"; + result = result + "</html>"; + return result; + } + + private String createStringOutputString(ArrayList<Form> forms) { + String result = ""; + for (int i=0; i<forms.size(); i++) { + Form f = forms.get(i); + String formName = f.getFormName(); + result = result + formName + " "; + } + result = result.substring(0, result.length() - 1); // without last blank + return result; + } + + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/lt/GetLemmas.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,231 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.lt; + +import java.io.IOException; +import java.io.PrintWriter; +import java.util.ArrayList; +import java.util.Collections; +import java.util.Date; + +import javax.servlet.ServletConfig; +import javax.servlet.ServletException; +import javax.servlet.http.HttpServlet; +import javax.servlet.http.HttpServletRequest; +import javax.servlet.http.HttpServletResponse; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.lt.dict.db.LexHandler; +import de.mpg.mpiwg.berlin.mpdl.lt.general.Language; +import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.Form; +import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.Lemma; +import de.mpg.mpiwg.berlin.mpdl.lt.text.norm.Normalizer; + +public class GetLemmas extends HttpServlet { + private static final long serialVersionUID = 1L; + private LexHandler lexHandler; + + public GetLemmas() { + super(); + } + + public void init(ServletConfig config) throws ServletException { + super.init(config); + try { + lexHandler = LexHandler.getInstance(); + } catch (ApplicationException e) { + throw new ServletException(e); + } + } + + protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + Date begin = new Date(); + request.setCharacterEncoding("utf-8"); + response.setCharacterEncoding("utf-8"); + String query = request.getParameter("query"); + String language = request.getParameter("language"); + String inputType = request.getParameter("inputType"); + String outputFormat = request.getParameter("outputFormat"); + String outputType = request.getParameter("outputType"); + String normalization = request.getParameter("normalization"); + if (language == null) + language = "eng"; + if (inputType == null || ! (inputType.equals("form") || inputType.equals("lemma"))) + inputType = "form"; + if (outputFormat == null || ! (outputFormat.equals("xml") || outputFormat.equals("html") || outputFormat.equals("string"))) + outputFormat = "xml"; + if (outputType == null || ! (outputType.equals("compact") || outputType.equals("full"))) + outputType = "compact"; + if (normalization == null || ! (normalization.equals("none") || normalization.equals("reg") || normalization.equals("reg norm"))) + normalization = "norm"; + int normalizationType = Normalizer.DICTIONARY; + if (normalization != null && normalization.equals("none")) + normalizationType = Normalizer.NONE; + + String xmlQueryString = "<query><name>" + query + "</name>" + "<language>" + language + "</language>" + "<inputType>" + inputType + "</inputType>" + + "<outputFormat>" + outputFormat + "</outputFormat>" + "<outputType>" + outputType + "</outputType>" + "<normalization>" + normalization + "</normalization>" + "</query>"; + try { + if (outputFormat.equals("xml")) + response.setContentType("text/xml"); + else if (outputFormat.equals("html") || outputFormat.equals("string")) + response.setContentType("text/html"); + else + response.setContentType("text/xml"); + PrintWriter out = response.getWriter(); + if (query == null || query.isEmpty()) { + out.print("request parameter query is empty. Please specify a query."); + out.close(); + return; + } + ArrayList<Lemma> lemmas = lexHandler.getLemmas(query, inputType, language, normalizationType, true); + String baseUrl = getBaseUrl(request); + Date end = new Date(); + String elapsedTime = String.valueOf(end.getTime() - begin.getTime()); + String result = ""; + if (outputFormat == null || outputFormat.equals("xml")) + result = createXmlOutputString(query, lemmas, outputType, baseUrl, xmlQueryString, elapsedTime); + else if (outputFormat.equals("html")) + result = createHtmlOutputString(query, lemmas, outputType, elapsedTime); + else if (outputFormat.equals("string")) + result = createStringOutputString(lemmas); + else + result = createXmlOutputString(query, lemmas, outputType, baseUrl, xmlQueryString, elapsedTime); + out.print(result); + out.close(); + } catch (ApplicationException e) { + throw new ServletException(e); + } + } + + protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + doGet(request, response); + } + + private String getBaseUrl( HttpServletRequest request ) { + if (request.getServerPort() == 80 || request.getServerPort() == 443) + return request.getScheme() + "://" + request.getServerName() + request.getContextPath(); + else + return request.getScheme() + "://" + request.getServerName() + ":" + request.getServerPort() + request.getContextPath(); + } + + + private String createXmlOutputString(String query, ArrayList<Lemma> lemmas, String outputType, String baseUrl, String xmlQueryString, String elapsedTime) { + String result = "<result>"; + result = result + "<provider>" + "MPIWG MPDL language technology service (see: " + "" + baseUrl + "), Max Planck Institute for the History of Science, Berlin." + "</provider>"; + result = result + xmlQueryString; + result = result + "<elapsed-time-ms>" + elapsedTime + "</elapsed-time-ms>"; + if (lemmas != null && ! lemmas.isEmpty()) { + result = result + "<morphology>"; + for (int i=0; i<lemmas.size(); i++) { + Lemma lemma = lemmas.get(i); + String lemmaName = lemma.getLemmaName(); + String language = lemma.getLanguage(); + result = result + "<lemma>"; + result = result + "<name>" + lemmaName + "</name>"; + if (outputType != null && outputType.equals("full")) { + String lemmaProvider = lemma.getProvider(); + result = result + "<provider>" + lemmaProvider + "</provider>"; + result = result + "<language>" + language + "</language>"; + } + if (Language.getInstance().isArabic(language) || Language.getInstance().isLatin(language)) { + String remoteUrl = "http://www.perseus.tufts.edu/hopper/morph?l=" + lemmaName + "&la=" + language; + result = result + "<remoteUrl>" + remoteUrl + "</remoteUrl>"; + } else if (Language.getInstance().isGreek(language)) { + String remoteUrl = "http://www.perseus.tufts.edu/hopper/morph?l=" + lemmaName + "&la=" + "greek"; + result = result + "<remoteUrl>" + remoteUrl + "</remoteUrl>"; + } + if (outputType != null && outputType.equals("full")) { + ArrayList<Form> forms = lemma.getFormsList(); + Collections.sort(forms); + if (forms != null && ! forms.isEmpty()) { + result = result + "<forms>"; + for (int j=0; j<forms.size(); j++) { + result = result + "<form>"; + Form f = forms.get(j); + String formName = f.getFormName(); + String formProvider = f.getProvider(); + result = result + "<provider>" + formProvider + "</provider>"; + result = result + "<language>" + language + "</language>"; + result = result + "<name>" + formName + "</name>"; + result = result + "</form>"; + } + result = result + "</forms>"; + } + } + result = result + "</lemma>"; + } + result = result + "</morphology>"; + } + result = result + "</result>"; + return result; + } + + private String createHtmlOutputString(String query, ArrayList<Lemma> lemmas, String outputType, String elapsedTime) { + String result = ""; + result = result + "<html>"; + result = result + "<head>"; + result = result + "<title>Lemmas for: \"" + query + "\"</title>"; + result = result + "</head>"; + result = result + "<body>"; + result = result + "<table align=\"right\" valign=\"top\">"; + result = result + "<td>[<i>This is a MPIWG language technology service</i>] <a href=\"/mpiwg-mpdl-cms-web/lt-services.html\"><img src=\"/mpiwg-mpdl-cms-web/images/info.png\" valign=\"bottom\" width=\"15\" height=\"15\" border=\"0\" alt=\"MPIWG language technology service\"/></a></td>"; + result = result + "</table>"; + result = result + "<p/>"; + result = result + "<h1>Lemmas for: \"" + query + "\"</h1>"; + if (lemmas != null && ! lemmas.isEmpty()) { + result = result + "<h3>Morphology</h3>"; + result = result + "<ul>"; + result = result + "<p/>"; + for (int i=0; i<lemmas.size(); i++) { + Lemma lemma = lemmas.get(i); + String lemmaName = lemma.getLemmaName(); + String language = lemma.getLanguage(); + result = result + "<li>"; + result = result + lemmaName; + if (outputType != null && outputType.equals("full")) { + String lemmaProvider = lemma.getProvider(); + result = result + " (data provider: " + lemmaProvider + ")"; + } + if (Language.getInstance().isArabic(language) || Language.getInstance().isLatin(language)) + result = result + " (external link: <a href=\"http://www.perseus.tufts.edu/hopper/morph?l=" + lemmaName + "&la=" + language + "\">" + lemmaName + "</a>)"; + else if (Language.getInstance().isGreek(language)) + result = result + " (external link: <a href=\"http://www.perseus.tufts.edu/hopper/morph?l=" + lemmaName + "&la=" + "greek" + "\">" + lemmaName + "</a>)"; + if (outputType != null && outputType.equals("full")) { + ArrayList<Form> forms = lemma.getFormsList(); + Collections.sort(forms); + if (forms != null && ! forms.isEmpty()) { + result = result + "<ul>"; + for (int j=0; j<forms.size(); j++) { + Form f = forms.get(j); + String formName = f.getFormName(); + String formProvider = f.getProvider(); + result = result + formName + " (data provider: " + formProvider + "), "; + } + result = result.substring(0, result.length() - 2); // without last comma and blank + result = result + "</ul>"; + } + } + result = result + "</li>"; + } + result = result + "</ul>"; + } + result = result + "[* external links may not function]"; + result = result + "<hr/>"; + result = result + "<p/>"; + result = result + "Elapsed time: " + elapsedTime + " ms, see the <a href=\"/mpiwg-mpdl-lt-web/index.html\">service description</a> of this page, if you find a bug <a href=\"https://it-dev.mpiwg-berlin.mpg.de/tracs/mpdl-project-software/newticket\">let us know</a>"; + result = result + "</body>"; + result = result + "</html>"; + return result; + } + + private String createStringOutputString(ArrayList<Lemma> lemmas) { + String result = ""; + for (int i=0; i<lemmas.size(); i++) { + Lemma l = lemmas.get(i); + String lemmaName = l.getLemmaName(); + result = result + lemmaName + " "; + } + result = result.substring(0, result.length() - 1); // without last blank + return result; + } + + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/lt/MpiwgMpdlLtWebServletContextListener.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,43 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.lt; + +import javax.servlet.ServletContext; +import javax.servlet.ServletContextEvent; +import javax.servlet.ServletContextListener; + +import de.mpg.mpiwg.berlin.mpdl.lt.dict.db.LexHandler; +import de.mpg.mpiwg.berlin.mpdl.lt.general.Constants; +import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.MorphologyCache; + +public class MpiwgMpdlLtWebServletContextListener implements ServletContextListener { + private ServletContext context = null; + + public void contextInitialized(ServletContextEvent event) { + try { + /* + this.context = event.getServletContext(); + // String dataDirectory = System.getProperty("catalina.base") + "/webapps/mpiwg-mpdl-lt-web/WEB-INF/data"; + String dataDirectory = Constants.getInstance().getDataDir(); + context.setAttribute("dataDirectory", dataDirectory); + System.out.println(MpiwgMpdlLtWebServletContextListener.class.getName() + ": contextInitialized (data directory= \"" + dataDirectory + "\", set in constants.properties)"); + MorphologyCache.getInstance(); + LexHandler.getInstance(); + */ + } catch (Exception e) { + e.printStackTrace(); + } + } + + public void contextDestroyed(ServletContextEvent event) { + try { + /* + this.context = null; + LexHandler.getInstance().end(); + MorphologyCache.getInstance().end(); + String dataDirectory = Constants.getInstance().getDataDir(); + System.out.println(MpiwgMpdlLtWebServletContextListener.class.getName() + ": contextDestroyed (databases in directory: \"" + dataDirectory + "\" are closed)"); + */ + } catch (Exception e) { + e.printStackTrace(); + } + } +} \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/lt/Normalize.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,68 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.lt; + +import java.io.IOException; +import java.io.PrintWriter; + +import javax.servlet.ServletConfig; +import javax.servlet.ServletException; +import javax.servlet.http.HttpServlet; +import javax.servlet.http.HttpServletRequest; +import javax.servlet.http.HttpServletResponse; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.lt.text.norm.Normalizer; + +public class Normalize extends HttpServlet { + private static final long serialVersionUID = 1L; + + public Normalize() { + super(); + } + + public void init(ServletConfig config) throws ServletException { + super.init(config); + } + + protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + request.setCharacterEncoding("utf-8"); + response.setCharacterEncoding("utf-8"); + String inputString = request.getParameter("inputString"); + String language = request.getParameter("language"); + String method = request.getParameter("method"); + String typeStr = request.getParameter("type"); + if (language == null) + language = "eng"; + if (method == null) + method = "norm"; + String[] methods = method.split(" "); + if (typeStr == null) + typeStr = "display"; + int type = Normalizer.DISPLAY; + if (typeStr.equals("dictionary")) + type = Normalizer.DICTIONARY; + else if (typeStr.equals("search")) + type = Normalizer.SEARCH; + String result = null; + try { + response.setContentType("text/html"); + PrintWriter out = response.getWriter(); + if (inputString == null || inputString.isEmpty()) { + out.print("request parameter \"inputString\" is empty. Please specify \"inputString\""); + out.close(); + return; + } + Normalizer normalizer = new Normalizer(methods, language); + normalizer.setNormMode(type); + result = normalizer.normalize(inputString); + if (result != null) + out.print(result); + out.close(); + } catch (ApplicationException e) { + throw new ServletException(e); + } + } + + protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + doGet(request, response); + } +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/lt/Tokenize.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,212 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.lt; + +import java.io.BufferedInputStream; +import java.io.IOException; +import java.io.InputStream; +import java.io.PrintWriter; +import java.io.StringReader; +import java.net.URL; +import java.util.ArrayList; +import java.util.Hashtable; + +import javax.servlet.ServletConfig; +import javax.servlet.ServletException; +import javax.servlet.http.HttpServlet; +import javax.servlet.http.HttpServletRequest; +import javax.servlet.http.HttpServletResponse; + +import org.apache.commons.io.IOUtils; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.lt.dict.app.Lexicon; +import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.Token; +import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.Tokenizer; +import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.XmlTokenizer; + +public class Tokenize extends HttpServlet { + private static final long serialVersionUID = 1L; + + public Tokenize() { + super(); + } + + public void init(ServletConfig config) throws ServletException { + super.init(config); + } + + protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + doGet(request, response); + } + + protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + request.setCharacterEncoding("utf-8"); + response.setCharacterEncoding("utf-8"); + String inputString = request.getParameter("inputString"); + String srcUrlStr = request.getParameter("srcUrl"); + String language = request.getParameter("language"); + String normalization = request.getParameter("normalization"); + String stopElements = request.getParameter("stopElements"); + String elements = request.getParameter("elements"); + String highlightTerms = request.getParameter("highlightTerms"); + String outputFormat = request.getParameter("outputFormat"); + String outputOptionsStr = request.getParameter("outputOptions"); + if (language == null) + language = "eng"; + if (normalization == null) + normalization = "norm"; + String[] normFunctions = normalization.split(" "); + String[] stopElementsArray = null; + if (stopElements != null && ! stopElements.equals("")) + stopElementsArray = stopElements.split(" "); + String[] elementsArray = null; + if (elements != null && ! elements.equals("")) + elementsArray = elements.split(" "); + if (highlightTerms == null) + highlightTerms = ""; + String[] highlightTermsArray = highlightTerms.split(" "); + if (outputFormat == null) + outputFormat = "xml"; + if (outputOptionsStr == null) + outputOptionsStr = ""; + String[] outputOptions = outputOptionsStr.split(" "); + String result = null; + try { + if (outputFormat.equals("xml")) { + response.setContentType("text/xml"); + } else if (outputFormat.equals("string")) { + response.setContentType("text/html"); + } else { + response.setContentType("text/xml"); + } + response.setCharacterEncoding("utf-8"); + PrintWriter out = response.getWriter(); + String inputText = null; // contains string or xml text + if ((inputString == null || inputString.isEmpty()) && (srcUrlStr == null || srcUrlStr.isEmpty())) { + out.print("request parameter \"inputString\" or \"srcUrl\" is empty. Please specify \"inputString\""); + out.close(); + return; + } else { + if (srcUrlStr != null && ! srcUrlStr.isEmpty()) { + URL srcUrl = new URL(srcUrlStr); + InputStream inputStream = srcUrl.openStream(); + BufferedInputStream in = new BufferedInputStream(inputStream); + inputText = IOUtils.toString(in, "utf-8"); + in.close(); + } else if (inputString != null && ! inputString.isEmpty()) { + inputText = inputString; + } + } + inputText = inputText.trim(); + // Tokenize + boolean inputTextIsXml = false; + if (inputText != null && inputText.startsWith("<") && inputText.endsWith(">")) // TODO check properly for xml type of the inputText + inputTextIsXml = true; + if (! inputTextIsXml) { + inputText = "<result>" + inputText + "</result>"; + } + StringReader xmlInputStringReader = new StringReader(inputText); + XmlTokenizer xmlTokenizer = new XmlTokenizer(xmlInputStringReader); + xmlTokenizer.setDocIdentifier(srcUrlStr); // TODO + xmlTokenizer.setLanguage(language); + xmlTokenizer.setNormFunctions(normFunctions); + xmlTokenizer.setOutputFormat(outputFormat); + xmlTokenizer.setOutputOptions(outputOptions); + if (stopElementsArray != null && stopElementsArray.length > 0) + xmlTokenizer.setStopElements(stopElementsArray); + if (elementsArray != null && elementsArray.length > 0) + xmlTokenizer.setElements(elementsArray); + if (highlightTermsArray != null && highlightTermsArray.length > 0) + xmlTokenizer.setHighlightTerms(highlightTermsArray); + xmlTokenizer.tokenize(); + if (outputFormat != null && outputFormat.equals("xml")) { + result = xmlTokenizer.getXmlResult(); + } else { // outputFormat == string + result = xmlTokenizer.getStringResult(); + } + if (result != null) + out.print(result); + out.close(); + } catch (ApplicationException e) { + throw new ServletException(e); + } + } + + private ArrayList<String> getToken(String inputString, String language, String[] normFunctions) throws ApplicationException { + ArrayList<String> retTokens = null; + try { + StringReader reader = new StringReader(inputString); + Tokenizer tokenizer = new Tokenizer(reader); + tokenizer.setLanguage(language); + tokenizer.setNormFunctions(normFunctions); + ArrayList<Token> tokens = tokenizer.getTokens(); + if (tokens != null) { + retTokens = new ArrayList<String>(); + for (int i=0; i<tokens.size(); i++) { + Token t = tokens.get(i); + String tokenStr = t.getContentOrig(); + if (useNormFunction(normFunctions)) + tokenStr = t.getContentNorm(); + retTokens.add(tokenStr); + } + } + tokenizer.end(); + tokenizer.close(); + } catch (IOException e) { + throw new ApplicationException(e); + } + return retTokens; + } + + private String createXmlOutputString(ArrayList<String> tokens, Hashtable<String, ArrayList<Lexicon>> tokensDictionaries, String baseUrl, String elapsedTime) { + StringBuilder result = new StringBuilder(); + result.append("<result>"); + result.append("<provider>" + "MPIWG MPDL language technology service (see: " + "" + baseUrl + "), Max Planck Institute for the History of Science, Berlin." + "</provider>"); + result.append("<elapsed-time-ms>" + elapsedTime + "</elapsed-time-ms>"); + if (tokens != null && ! tokens.isEmpty()) { + result.append("<tokens>"); + for (int i=0; i<tokens.size(); i++) { + String token = tokens.get(i); + result.append("<token>"); + result.append("<name>" + token + "</name>"); + if (tokensDictionaries != null && ! tokensDictionaries.isEmpty()) { + ArrayList<Lexicon> tokenDictionaries = tokensDictionaries.get(token); + if (tokenDictionaries != null) { + result.append("<dictionaries>"); + for (int j=0; j<tokenDictionaries.size(); j++) { + Lexicon lexicon = tokenDictionaries.get(j); + result.append(lexicon.toXmlString()); + } + result.append("</dictionaries>"); + } + } + result.append("</token>"); + } + result.append("</tokens>"); + } + result.append("</result>"); + return result.toString(); + } + + private String createStringOutputString(ArrayList<String> tokens) { + StringBuilder result = new StringBuilder(); + if (tokens != null && ! tokens.isEmpty()) { + for (int i=0; i<tokens.size(); i++) { + String token = tokens.get(i); + result.append(token + " "); + } + result.setLength(result.length() - 1); // without last blank + } + return result.toString(); + } + + private boolean useNormFunction(String[] normFunctions) { + boolean useNorm = false; + for (int i=0; i< normFunctions.length; i++) { + String function = normFunctions[i]; + if (function.equals("norm")) + return true; + } + return useNorm; + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/lt/Transcode.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,77 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.lt; + +import java.io.IOException; +import java.io.PrintWriter; + +import javax.servlet.ServletConfig; +import javax.servlet.ServletException; +import javax.servlet.http.HttpServlet; +import javax.servlet.http.HttpServletRequest; +import javax.servlet.http.HttpServletResponse; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.lt.text.transcode.Transcoder; + +public class Transcode extends HttpServlet { + private static final long serialVersionUID = 1L; + private Transcoder transcoder; + + public Transcode() { + super(); + } + + public void init(ServletConfig config) throws ServletException { + super.init(config); + transcoder = Transcoder.getInstance(); + } + + protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + request.setCharacterEncoding("utf-8"); + response.setCharacterEncoding("utf-8"); + String inputString = request.getParameter("inputString"); + String srcEncoding = request.getParameter("srcEncoding"); + String destEncoding = request.getParameter("destEncoding"); + if (destEncoding == null) + destEncoding = "unicode"; + String result = null; + try { + response.setContentType("text/html"); + PrintWriter out = response.getWriter(); + if (inputString == null || inputString.isEmpty()) { + out.print("request parameter \"inputString\" is empty. Please specify \"inputString\""); + out.close(); + return; + } + if (srcEncoding == null || srcEncoding.isEmpty()) { + out.print("request parameter \"srcEncoding\" is empty. Please specify \"srcEncoding\""); + out.close(); + return; + } + if (destEncoding == null || destEncoding.isEmpty()) { + out.print("request parameter \"destEncoding\" is empty. Please specify \"destEncoding\""); + out.close(); + return; + } + if (srcEncoding.equals("buckwalter") && destEncoding.equals("unicode")) { + result = transcoder.transcodeFromBuckwalter2Unicode(inputString); + } else if (srcEncoding.equals("betacode") && destEncoding.equals("unicode")) { + result = transcoder.transcodeFromBetaCode2Unicode(inputString); + } else if (srcEncoding.equals("unicode") && destEncoding.equals("betacode")) { + result = transcoder.transcodeFromUnicode2BetaCode(inputString); + } else if (srcEncoding.equals("unicode") && destEncoding.equals("buckwalter")) { + result = transcoder.transcodeFromUnicode2Buckwalter(inputString); + } else { + result = "your combination of \"srcEncoding\" and \"destEncoding\" is not yet supported"; + } + if (result != null) + out.print(result); + out.close(); + } catch (ApplicationException e) { + throw new ServletException(e); + } + } + + protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { + doGet(request, response); + } +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/util/ServletUtil.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,27 @@ +package de.mpg.mpiwg.berlin.mpdl.servlets.util; + +import javax.servlet.http.HttpServletRequest; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; + +public class ServletUtil { + private static ServletUtil instance; + + public static ServletUtil getInstance() throws ApplicationException { + if (instance == null) { + instance = new ServletUtil(); + } + return instance; + } + + public ServletUtil() { + } + + public String getBaseUrl( HttpServletRequest request ) { + if (request.getServerPort() == 80 || request.getServerPort() == 443) + return request.getScheme() + "://" + request.getServerName() + request.getContextPath(); + else + return request.getScheme() + "://" + request.getServerName() + ":" + request.getServerPort() + request.getContextPath(); + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/.classpath Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,32 @@ +<?xml version="1.0" encoding="UTF-8"?> +<classpath> + <classpathentry kind="src" path="src"/> + <classpathentry kind="con" path="org.eclipse.jdt.launching.JRE_CONTAINER"/> + <classpathentry combineaccessrules="false" kind="src" path="/mpiwg-mpdl-xml"/> + <classpathentry combineaccessrules="false" kind="src" path="/mpiwg-mpdl-lt"/> + <classpathentry kind="lib" path="lib/lucene-core-3.5.0.jar" sourcepath="lib/lucene-core-3.5.0-sources.jar"> + <attributes> + <attribute name="javadoc_location" value="jar:platform:/resource/mpiwg-mpdl-cms/lib/lucene-core-3.5.0-javadoc.jar!/"/> + </attributes> + </classpathentry> + <classpathentry kind="lib" path="lib/quartz-1.6.5.jar"/> + <classpathentry kind="lib" path="lib/commons-codec-1.3.jar"/> + <classpathentry kind="lib" path="lib/commons-io-2.0.1.jar"/> + <classpathentry kind="lib" path="lib/commons-lang3-3.0.1.jar"/> + <classpathentry kind="lib" path="lib/commons-collections-3.2.1.jar"/> + <classpathentry kind="lib" path="lib/commons-logging-1.1.1.jar"/> + <classpathentry kind="lib" path="lib/saxon9-s9api.jar"/> + <classpathentry kind="lib" path="lib/saxon9.jar"/> + <classpathentry kind="lib" path="lib/commons-httpclient-3.1.jar"/> + <classpathentry kind="lib" path="lib/lucene-queries-3.5.0.jar"/> + <classpathentry kind="lib" path="lib/httpclient-4.1.2.jar"/> + <classpathentry kind="lib" path="lib/httpcore-4.1.2.jar"/> + <classpathentry kind="lib" path="lib/json_simple-1.1.jar"/> + <classpathentry kind="lib" path="lib/microsoft-translator-java-api-0.4.jar"/> + <classpathentry kind="lib" path="lib/lucene-highlighter-3.5.0.jar" sourcepath="lib/lucene-highlighter-3.5.0-javadoc.jar"/> + <classpathentry kind="lib" path="lib/lucene-memory-3.5.0.jar"/> + <classpathentry kind="lib" path="lib/iText-2.0.8.jar"/> + <classpathentry kind="lib" path="lib/core-renderer.jar"/> + <classpathentry kind="lib" path="lib/saxon9-dom.jar"/> + <classpathentry kind="output" path="bin"/> +</classpath>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/.externalToolBuilders/mpiwg-mpdl-cms-build.launch Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,13 @@ +<?xml version="1.0" encoding="UTF-8" standalone="no"?> +<launchConfiguration type="org.eclipse.ant.AntBuilderLaunchConfigurationType"> +<booleanAttribute key="org.eclipse.ant.ui.ATTR_TARGETS_UPDATED" value="true"/> +<booleanAttribute key="org.eclipse.ant.ui.DEFAULT_VM_INSTALL" value="false"/> +<booleanAttribute key="org.eclipse.debug.ui.ATTR_LAUNCH_IN_BACKGROUND" value="false"/> +<stringAttribute key="org.eclipse.jdt.launching.CLASSPATH_PROVIDER" value="org.eclipse.ant.ui.AntClasspathProvider"/> +<booleanAttribute key="org.eclipse.jdt.launching.DEFAULT_CLASSPATH" value="true"/> +<stringAttribute key="org.eclipse.jdt.launching.PROJECT_ATTR" value="mpiwg-mpdl-cms"/> +<booleanAttribute key="org.eclipse.ui.externaltools.ATTR_BUILDER_ENABLED" value="false"/> +<stringAttribute key="org.eclipse.ui.externaltools.ATTR_LOCATION" value="${workspace_loc:/mpiwg-mpdl-cms/build/build.xml}"/> +<stringAttribute key="org.eclipse.ui.externaltools.ATTR_RUN_BUILD_KINDS" value="full,incremental,"/> +<booleanAttribute key="org.eclipse.ui.externaltools.ATTR_TRIGGERS_CONFIGURED" value="true"/> +</launchConfiguration>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/.project Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,27 @@ +<?xml version="1.0" encoding="UTF-8"?> +<projectDescription> + <name>mpiwg-mpdl-cms</name> + <comment></comment> + <projects> + </projects> + <buildSpec> + <buildCommand> + <name>org.eclipse.jdt.core.javabuilder</name> + <arguments> + </arguments> + </buildCommand> + <buildCommand> + <name>org.eclipse.ui.externaltools.ExternalToolBuilder</name> + <triggers>full,incremental,</triggers> + <arguments> + <dictionary> + <key>LaunchConfigHandle</key> + <value><project>/.externalToolBuilders/mpiwg-mpdl-cms-build.launch</value> + </dictionary> + </arguments> + </buildCommand> + </buildSpec> + <natures> + <nature>org.eclipse.jdt.core.javanature</nature> + </natures> +</projectDescription>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/.settings/org.eclipse.core.resources.prefs Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,6 @@ +#Thu May 17 12:31:24 CEST 2012 +eclipse.preferences.version=1 +encoding//src/de/mpg/mpiwg/berlin/mpdl/cms/lucene/IndexHandler.java=UTF-8 +encoding//src/de/mpg/mpiwg/berlin/mpdl/cms/test/TestLocal.java=UTF-8 +encoding//src/de/mpg/mpiwg/berlin/mpdl/cms/transform/GetFragmentsContentHandler.java=UTF-8 +encoding//src/de/mpg/mpiwg/berlin/mpdl/cms/transform/HighlightContentHandler.java=UTF-8
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/bin/constants.properties Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,5 @@ +dataDir=/Users/jwillenborg/mpdl/data/lt +documentsDir=/Users/jwillenborg/mpdl/data/xml/documents +luceneDocumentsDir=/Users/jwillenborg/mpdl/data/xml/lucene/documents +luceneNodesDir=/Users/jwillenborg/mpdl/data/xml/lucene/nodes +confDir=/Users/jwillenborg/mpdl/data/collectionConfs \ No newline at end of file
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/bin/de/mpg/mpiwg/berlin/mpdl/cms/scheduler/quartz.properties Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,9 @@ +org.quartz.scheduler.instanceName = MpdlScheduler +org.quartz.scheduler.instanceId = auto +org.quartz.scheduler.rmi.export = false +org.quartz.scheduler.rmi.proxy = false + +org.quartz.threadPool.class = org.quartz.simpl.SimpleThreadPool +org.quartz.threadPool.threadCount = 3 + +org.quartz.jobStore.class = org.quartz.simpl.RAMJobStore
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/bin/de/mpg/mpiwg/berlin/mpdl/cms/transform/chars.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,10 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform"> + +<xsl:output method="xml" encoding="utf-8" omit-xml-declaration="yes"/> + +<xsl:template match="text()"> + <xsl:sequence select="normalize-space(.)"/> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/bin/de/mpg/mpiwg/berlin/mpdl/cms/transform/pageArchimedes.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,324 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xs="http://www.w3.org/2001/XMLSchema" + xmlns:functx="http://www.functx.com" + xmlns:saxon="http://saxon.sf.net/" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xs functx saxon dc dcterms math svg xhtml" + > + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="echoFiguresDir"></xsl:param> +<xsl:param name="echoPageImgDir"></xsl:param> + +<xsl:variable name="dictionaryServiceName" select="'http://mpdl-service.mpiwg-berlin.mpg.de/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries'"/> + +<xsl:template match="*:archimedes"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'archimedes'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:expan|*:emph|*:q|*:quote|*:blockquote|*:set-off|*:reg|*:var|*:num"> + <span> + <xsl:attribute name="class"><xsl:value-of select="name()"/></xsl:attribute> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="text"> + <xsl:variable name="contentStr" select="normalize-space(string(.))"/> + <xsl:variable name="figures" select=".//*:figure|.//*:handwritten"/> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'text'"/></xsl:attribute> + <xsl:apply-templates/> + <xsl:if test="$contentStr = '' and empty($figures)"> <!-- test if div only contains empty nodes but no figures --> + <div> + <xsl:attribute name="class"><xsl:value-of select="'emptyPage'"/></xsl:attribute> + <xsl:value-of select="'[Empty page]'"/> + </div> + </xsl:if> + </div> + <!-- Notes --> + <xsl:variable name="bottomNotes" select="//*:note[contains(@position, 'foot') or empty(string(@position))]"/> + <xsl:variable name="countBottomNotes" select="count($bottomNotes)"/> + <xsl:if test="$countBottomNotes > 0"> + <span class="notes"> + <xsl:for-each select="$bottomNotes"> + <xsl:variable name="noteSign"><xsl:value-of select="count(./preceding::*:note) + 1"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'noteBody foot'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:apply-templates/> + </span> + </span> + </xsl:for-each> + </span> + </xsl:if> +</xsl:template> + +<xsl:template match="*:div"> + <xsl:if test="node() or normalize-space()"> <!-- test: contains nodes or is not empty --> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'div'"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:if> +</xsl:template> + +<xsl:template match="*:p"> + <xsl:variable name="type" select="string(@type)"/> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:choose> + <xsl:when test="$type = 'head'"> + <xsl:attribute name="class"><xsl:value-of select="'head bf'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:otherwise> + <xsl:attribute name="class"><xsl:value-of select="'p'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </div> +</xsl:template> + +<xsl:template match="*:pb"> + <xsl:variable name="number"> + <xsl:choose> + <xsl:when test="not(empty(@n))"><xsl:value-of select="@n"/></xsl:when> + <xsl:otherwise><xsl:value-of select="count(./preceding::*:pb) + 1"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="file"> + <xsl:choose> + <xsl:when test="not(empty(@xlink:href))"><xsl:value-of select="replace(@xlink:href, '/', '.')"/></xsl:when> + <xsl:otherwise><xsl:value-of select="$number"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'pb'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'src'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="concat('pageimg/', string($file))"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'n'"/></xsl:attribute> + <xsl:value-of select="$number"/> + </span> + </span> + </span> +</xsl:template> + +<xsl:template match="*:head"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'head bf'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<!-- MathML --> +<xsl:template match="math:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<!-- SVG --> +<xsl:template match="svg:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:figure"> + <span class="figure"> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:variable name="fileHref" select="replace(@xlink:href, '/', '.')"/> + <xsl:variable name="fileName" select="concat($echoFiguresDir, '/', $fileHref)"/> + <xsl:variable name="digilibUrl" select="concat('http://digilib.mpiwg-berlin.mpg.de/digitallibrary/jquery/digilib.html?fn=', $fileName)"/> + <xsl:variable name="scalerUrl" select="concat('http://digilib.mpiwg-berlin.mpg.de/digitallibrary/servlet/Scaler?fn=', $fileName, '&dh=200&dw=200')"/> + <xsl:variable name="ancestorWords" select="./ancestor::*:w"/> <!-- if figure is in a word then no link within a link should be produced --> + <xsl:choose> + <xsl:when test="empty($ancestorWords)"> + <a href="{$digilibUrl}"> + <img> + <xsl:attribute name="src"><xsl:value-of select="$scalerUrl"/></xsl:attribute> + </img> + </a> + </xsl:when> + <xsl:otherwise> + <img> + <xsl:attribute name="src"><xsl:value-of select="$scalerUrl"/></xsl:attribute> + </img> + </xsl:otherwise> + </xsl:choose> + <xsl:if test="not(empty(@number))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNumber'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNum'"/></xsl:attribute> + <xsl:value-of select="string(@number)"/> + </span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNumText'"/></xsl:attribute> + <xsl:value-of select="concat('[', 'Figure', ' ', string(@number), ']')"/> + </span> + </span> + </xsl:if> + </span> +</xsl:template> + +<xsl:template match="*:lb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'lb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="*:cb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'cb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="note"> + <xsl:variable name="place"><xsl:value-of select="string(@position)"/></xsl:variable> + <xsl:variable name="noteSign"><xsl:value-of select="count(./preceding::*:note) + 1"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:if test="$place != '' and $place != 'foot'"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('noteBody ', $place)"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:apply-templates/> + </span> + </xsl:if> + </span> +</xsl:template> + +<xsl:template match="foreign"> + <xsl:variable name="lang" select="@lang"/> + <xsl:variable name="xmllang" select="@xml:lang"/> + <xsl:variable name="language"> + <xsl:choose> + <xsl:when test="not(empty($xmllang))"><xsl:value-of select="$xmllang"/></xsl:when> + <xsl:when test="not(empty($lang))"><xsl:value-of select="$lang"/></xsl:when> + <xsl:otherwise><xsl:value-of select="''"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat('foreign ', $language)"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="gap"> + <span class="gap"> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'gap'"/></xsl:attribute> + <xsl:if test="not(empty(@extent))"><span class="extent"><xsl:value-of select="string(@extent)"/></span></xsl:if> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="s"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'s'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- words --> +<xsl:template match="*:w"> + <xsl:variable name="wordLanguage" select="@lang"/> + <xsl:variable name="form" select="encode-for-uri(string(@form))"/> + <xsl:variable name="dictionary" select="string(@dictionary)"/> + <xsl:variable name="displayWordOrig"> + <xsl:choose> + <xsl:when test="not(empty(*:orig))"><xsl:apply-templates select="*:orig/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordReg"> + <xsl:choose> + <xsl:when test="not(empty(*:reg))"><xsl:apply-templates select="*:reg/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordNorm"> + <xsl:choose> + <xsl:when test="not(empty(*:norm))"><xsl:apply-templates select="*:norm/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordUrlEncoded" select="encode-for-uri($displayWordOrig)"/> + <xsl:variable name="dictionaryPart"> + <xsl:choose> + <xsl:when test="$dictionary = 'true'"> + <a class="dictionary"> + <xsl:attribute name="href"><xsl:value-of select="concat($dictionaryServiceName, '?query=', $form, '&queryDisplay=', $displayWordUrlEncoded, '&language=', $wordLanguage, '&outputFormat=html', '&outputType=morphCompact&outputType=dictFull')"/></xsl:attribute> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </a> + </xsl:when> + <xsl:otherwise> + <span class="dictionary"> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span class="w"> + <xsl:sequence select="$dictionaryPart"/> + <span class="nodictionary orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="nodictionary reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="nodictionary norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> +</xsl:template> + +<xsl:template match="*:hi"> + <xsl:choose> + <xsl:when test="@type = 'elem'"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/bin/de/mpg/mpiwg/berlin/mpdl/cms/transform/pageEcho.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,536 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xs="http://www.w3.org/2001/XMLSchema" + xmlns:functx="http://www.functx.com" + xmlns:saxon="http://saxon.sf.net/" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" + xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:mml="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xs functx saxon dc dcterms echo de math mml svg xhtml" + > + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="echoFiguresDir"></xsl:param> +<xsl:param name="echoPageImgDir"></xsl:param> + +<xsl:variable name="dictionaryServiceName" select="'http://mpdl-service.mpiwg-berlin.mpg.de/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries'"/> + +<xsl:template match="*:echo"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'echo'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:text"> + <xsl:variable name="contentStr" select="normalize-space(string(.))"/> + <xsl:variable name="figures" select=".//*:figure|.//*:handwritten"/> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'text'"/></xsl:attribute> + <xsl:apply-templates/> + <xsl:if test="$contentStr = '' and empty($figures)"> <!-- test if div only contains empty nodes but no figures --> + <div> + <xsl:attribute name="class"><xsl:value-of select="'emptyPage'"/></xsl:attribute> + <xsl:value-of select="'[Empty page]'"/> + </div> + </xsl:if> + </div> + <!-- Notes --> + <xsl:variable name="bottomNotes" select="//*:note[contains(@position, 'foot') or empty(string(@position))]"/> + <xsl:variable name="countBottomNotes" select="count($bottomNotes)"/> + <xsl:if test="$countBottomNotes > 0"> + <span class="notes"> + <xsl:for-each select="$bottomNotes"> + <xsl:variable name="noteSign"><xsl:value-of select="count(./preceding::*:note) + 1"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'noteBody foot'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:apply-templates/> + </span> + </span> + </xsl:for-each> + </span> + </xsl:if> +</xsl:template> + +<xsl:template match="*:head"> + <xsl:variable name="style" select="@style"/> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'head bf'"/></xsl:attribute> + <xsl:choose> + <xsl:when test="not(empty($style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="$style"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:when> + <xsl:otherwise> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </div> +</xsl:template> + +<xsl:template match="*:figure"> + <xsl:variable name="class"> + <xsl:choose> + <xsl:when test="empty(@position)"><xsl:value-of select="name()"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat(name(), ' ', string(@position))"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="image" select="*:image"/> + <xsl:variable name="caption" select="*:caption"/> + <xsl:variable name="description" select="*:description"/> + <xsl:variable name="variables" select="*:variables"/> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:if test="not(empty($image))"> + <xsl:variable name="file" select="$image/@file"/> + <xsl:variable name="pn" select="$image/@pn"/> + <xsl:variable name="wx" select="$image/@wx"/> + <xsl:variable name="wy" select="$image/@wy"/> + <xsl:variable name="ww" select="$image/@ww"/> + <xsl:variable name="wh" select="$image/@wh"/> + <xsl:variable name="digilibServiceName" select="'http://digilib.mpiwg-berlin.mpg.de/digitallibrary/jquery/digilib.html'"/> + <xsl:variable name="scalerServiceName" select="'http://digilib.mpiwg-berlin.mpg.de/digitallibrary/servlet/Scaler'"/> + <xsl:variable name="fileName"> + <xsl:choose> + <xsl:when test="empty($wx) and empty($file)"><xsl:value-of select="$echoFiguresDir"/></xsl:when> + <xsl:when test="empty($wx) and not(empty($file))"><xsl:value-of select="concat($echoFiguresDir, '/', $file)"/></xsl:when> + <xsl:when test="not(empty($wx)) and empty($file)"><xsl:value-of select="$echoPageImgDir"/></xsl:when> + <xsl:when test="not(empty($wx)) and not(empty($file))"><xsl:value-of select="concat($echoPageImgDir, '/', $file)"/></xsl:when> + <xsl:otherwise><xsl:value-of select="$echoFiguresDir"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="digilibUrl"> + <xsl:choose> + <xsl:when test="empty($pn) and empty($wx)"><xsl:value-of select="concat($digilibServiceName, '?fn=', $fileName)"/></xsl:when> + <xsl:when test="not(empty($pn)) and empty($wx)"><xsl:value-of select="concat($digilibServiceName, '?fn=', $fileName, '&pn=', $pn)"/></xsl:when> + <xsl:when test="empty($pn) and not(empty($wx))"><xsl:value-of select="concat($digilibServiceName, '?fn=', $fileName, '&wx=', $wx, '&wy=', $wy, '&ww=', $ww, '&wh=', $wh)"/></xsl:when> + <xsl:when test="not(empty($pn)) and not(empty($wx))"><xsl:value-of select="concat($digilibServiceName, '?fn=', $fileName, '&pn=', $pn, '&wx=', $wx, '&wy=', $wy, '&ww=', $ww, '&wh=', $wh)"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat($digilibServiceName, '?fn=', $fileName)"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="scalerUrl"> + <xsl:choose> + <xsl:when test="empty($pn) and empty($wx)"><xsl:value-of select="concat($scalerServiceName, '?fn=', $fileName, '&dh=200')"/></xsl:when> + <xsl:when test="not(empty($pn)) and empty($wx)"><xsl:value-of select="concat($scalerServiceName, '?fn=', $fileName, '&pn=', $pn, '&dh=200')"/></xsl:when> + <xsl:when test="empty($pn) and not(empty($wx))"><xsl:value-of select="concat($scalerServiceName, '?fn=', $fileName, '&wx=', $wx, '&wy=', $wy, '&ww=', $ww, '&wh=', $wh, '&dh=200')"/></xsl:when> + <xsl:when test="not(empty($pn)) and not(empty($wx))"><xsl:value-of select="concat($scalerServiceName, '?fn=', $fileName, '&pn=', $pn, '&wx=', $wx, '&wy=', $wy, '&ww=', $ww, '&wh=', $wh, '&dh=200')"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat($digilibServiceName, '?fn=', $fileName, '&dh=200')"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <a href="{$digilibUrl}"> + <img> + <xsl:attribute name="src"><xsl:value-of select="$scalerUrl"/></xsl:attribute> + </img> + </a> + </xsl:if> + <xsl:if test="not(empty(@number))"> + <xsl:variable name="type"><xsl:value-of select="'Figure'"/></xsl:variable> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNumber'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNum'"/></xsl:attribute> + <xsl:value-of select="string(@number)"/> + </span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNumText'"/></xsl:attribute> + <xsl:value-of select="concat('[', $type, ' ', string(@number), ']')"/> + </span> + </span> + </xsl:if> + <xsl:if test="not(empty($caption))"> + <xsl:for-each select="$caption"> + <span class="caption"><xsl:apply-templates/></span> + </xsl:for-each> + </xsl:if> + <xsl:if test="not(empty($description))"> + <xsl:for-each select="$description"> + <span class="description"><xsl:apply-templates/></span> + </xsl:for-each> + </xsl:if> + <xsl:if test="not(empty($variables))"> + <xsl:for-each select="$variables"> + <span class="variables"><xsl:apply-templates/></span> + </xsl:for-each> + </xsl:if> + </span> +</xsl:template> + +<!-- MathML --> +<xsl:template match="math:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="mml:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<!-- XHTML: remove the xhtml namespace --> +<xsl:template match="xhtml:*"> + <xsl:variable name="hasLabel" select="not(empty(@xhtml:label))"/> + <xsl:variable name="isTable" select="local-name() = 'table'"/> + <xsl:choose> + <xsl:when test="(not($hasLabel)) or ($isTable and $hasLabel)"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- SVG --> +<xsl:template match="svg:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:place|*:person"> + <xsl:element name="span"> + <xsl:attribute name="class"><xsl:value-of select="name()"/></xsl:attribute> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:div"> + <xsl:if test="node() or normalize-space()"> <!-- test: contains nodes or is not empty --> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'div'"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:if> +</xsl:template> + +<xsl:template match="*:p"> + <xsl:variable name="style" select="@style"/> + <xsl:variable name="class" select="'p'"/> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:choose> + <xsl:when test="not(empty($style))"> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:when> + <xsl:otherwise> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </div> +</xsl:template> + +<xsl:template match="*:pb"> + <xsl:variable name="number"> + <xsl:choose> + <xsl:when test="not(empty(@n))"><xsl:value-of select="@n"/></xsl:when> + <xsl:otherwise><xsl:value-of select="count(./preceding::*:pb) + 1"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="file"> + <xsl:choose> + <xsl:when test="not(empty(@file))"><xsl:value-of select="@file"/></xsl:when> + <xsl:otherwise><xsl:value-of select="$number"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'pb'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'src'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="concat('pageimg/', string($file))"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'n'"/></xsl:attribute> + <xsl:value-of select="$number"/> + </span> + <xsl:if test="not(empty(@o))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="'o'"/></xsl:attribute> + <xsl:value-of select="@o"/> + </span> + </xsl:if> + <xsl:if test="not(empty(@rhead))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="'rhead'"/></xsl:attribute> + <xsl:value-of select="@rhead"/> + </span> + </xsl:if> + </span> + </span> +</xsl:template> + +<xsl:template match="*:lb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'lb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="*:cb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'cb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="*:expan"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'expan'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:note"> + <xsl:variable name="place"><xsl:value-of select="string(@position)"/></xsl:variable> + <xsl:variable name="noteSign"><xsl:value-of select="count(./preceding::*:note) + 1"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:if test="$place != '' and $place != 'foot'"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('noteBody ', $place)"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:apply-templates/> + </span> + </xsl:if> + </span> +</xsl:template> + +<xsl:template match="*:handwritten"> + <xsl:variable name="noteSign"><xsl:value-of select="string(@number)"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note handwritten'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'noteBody'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'noteNumText'"/></xsl:attribute> + <xsl:value-of select="concat('[Handwritten note ', string(@number), ']')"/> + </span> + </span> + </span> +</xsl:template> + +<xsl:template match="*:emph"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'emph'"/></xsl:attribute> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="string(@style)"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:when> + <xsl:otherwise> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </span> +</xsl:template> + +<xsl:template match="*:foreign"> + <xsl:variable name="lang" select="@lang"/> + <xsl:variable name="xmllang" select="@xml:lang"/> + <xsl:variable name="language"> + <xsl:choose> + <xsl:when test="not(empty($xmllang))"><xsl:value-of select="$xmllang"/></xsl:when> + <xsl:when test="not(empty($lang))"><xsl:value-of select="$lang"/></xsl:when> + <xsl:otherwise><xsl:value-of select="''"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="class"> + <xsl:choose> + <xsl:when test="$language = ''"><xsl:value-of select="'foreign'"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat('foreign ', $language)"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:q"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'q'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:quote"> + <span class="quote"> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:blockquote"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'blockquote'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:set-off"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'set-off'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:reg"> + <span class="reg"> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'reg'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:var"> + <xsl:variable name="class"> + <xsl:choose> + <xsl:when test="empty(@type)"><xsl:value-of select="'var'"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat('var ', string(@type))"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:num"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'num'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:gap"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'gap'"/></xsl:attribute> + <xsl:if test="not(empty(@extent))"><span class="extent"><xsl:value-of select="string(@extent)"/></span></xsl:if> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:s"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'s'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- words --> +<xsl:template match="*:w"> + <xsl:variable name="wordLanguage" select="@lang"/> + <xsl:variable name="form" select="encode-for-uri(string(@form))"/> + <xsl:variable name="dictionary" select="string(@dictionary)"/> + <xsl:variable name="displayWordOrig"> + <xsl:choose> + <xsl:when test="not(empty(*:orig))"><xsl:apply-templates select="*:orig/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordReg"> + <xsl:choose> + <xsl:when test="not(empty(*:reg))"><xsl:apply-templates select="*:reg/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordNorm"> + <xsl:choose> + <xsl:when test="not(empty(*:norm))"><xsl:apply-templates select="*:norm/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordUrlEncoded" select="encode-for-uri($displayWordOrig)"/> + <xsl:variable name="dictionaryPart"> + <xsl:choose> + <xsl:when test="$dictionary = 'true'"> + <a class="dictionary"> + <xsl:attribute name="href"><xsl:value-of select="concat($dictionaryServiceName, '?query=', $form, '&queryDisplay=', $displayWordUrlEncoded, '&language=', $wordLanguage, '&outputFormat=html', '&outputType=morphCompact&outputType=dictFull')"/></xsl:attribute> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </a> + </xsl:when> + <xsl:otherwise> + <span class="dictionary"> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span class="w"> + <xsl:sequence select="$dictionaryPart"/> + <span class="nodictionary orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="nodictionary reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="nodictionary norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> +</xsl:template> + +<xsl:template match="*:hi"> + <xsl:choose> + <xsl:when test="@type = 'elem'"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/bin/de/mpg/mpiwg/berlin/mpdl/cms/transform/pageTei.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,650 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xs="http://www.w3.org/2001/XMLSchema" + xmlns:functx="http://www.functx.com" + xmlns:saxon="http://saxon.sf.net/" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xs functx saxon dc dcterms math svg xhtml" + > + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="echoFiguresDir"></xsl:param> +<xsl:param name="echoPageImgDir"></xsl:param> + +<xsl:variable name="dictionaryServiceName" select="'http://mpdl-service.mpiwg-berlin.mpg.de/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries'"/> + +<xsl:template match="*:TEI"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'TEI'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:expan|*:emph|*:q|*:quote|*:reg|*:num"> + <xsl:element name="span"> + <xsl:attribute name="class"><xsl:value-of select="name()"/></xsl:attribute> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:teiHeader"> + <!-- do nothing --> +</xsl:template> + +<xsl:template match="*:text"> + <xsl:variable name="contentStr" select="normalize-space(string(.))"/> + <xsl:variable name="figures" select=".//*:figure|.//*:handwritten"/> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'text'"/></xsl:attribute> + <xsl:apply-templates/> + <xsl:if test="$contentStr = '' and empty($figures)"> <!-- test if div only contains empty nodes but no figures --> + <div> + <xsl:attribute name="class"><xsl:value-of select="'emptyPage'"/></xsl:attribute> + <xsl:value-of select="'[Empty page]'"/> + </div> + </xsl:if> + </div> + <!-- Dictionary --> + <xsl:if test="not(empty(//*:entry))"> + <xsl:for-each select="//*:entry"> + <xsl:sort select="*:form/*:orth"/> + <xsl:variable name="position" select="position()"/> + <span class="entry"> + <xsl:apply-templates select="*:form"/> + <xsl:apply-templates select="*:sense"/> + <xsl:if test="not(empty(*:figure))"> + <span class="entryDiv"> + <span class="bf"><xsl:value-of select="'Figures: '"/></span> + <span class="entryDiv"> + <xsl:for-each select="*:figure"> + <xsl:variable name="href" select="*:graphic/@url"/> + <xsl:variable name="figDesc" select="string(*:figDesc)"/> + <xsl:if test="$href != ''"> + <div class="figure" style="margin-left:10px;"> + <a href="{$href}"><img alt="Figure: {$figDesc}" src="{$href}" width="200" height="200"/></a> + <br/> + <xsl:value-of select="'[Figure]: '"/><xsl:apply-templates select="*:head"/> + </div> + </xsl:if> + </xsl:for-each> + </span> + </span> + </xsl:if> + <xsl:if test="not(empty(*:xr))"> + <span class="entryDiv"> + <span class="bf"><xsl:value-of select="'References: '"/></span> + <span class="entryDiv"> + <xsl:for-each select="*:xr/*:ref"> + <li><xsl:apply-templates select="."/></li> + </xsl:for-each> + </span> + </span> + </xsl:if> + </span> + </xsl:for-each> + </xsl:if> + <!-- Notes --> + <xsl:variable name="bottomNotes" select="//*:note[contains(@place, 'foot') or empty(string(@place))]"/> + <xsl:variable name="countBottomNotes" select="count($bottomNotes)"/> + <xsl:if test="$countBottomNotes > 0"> + <span class="notes"> + <xsl:for-each select="$bottomNotes"> + <xsl:variable name="noteSign"><xsl:value-of select="count(./preceding::*:note) + 1"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'noteBody foot'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:apply-templates/> + </span> + </span> + </xsl:for-each> + </span> + </xsl:if> +</xsl:template> + +<xsl:template match="*:div"> + <xsl:if test="node() or normalize-space()"> <!-- test: contains nodes or is not empty --> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'div'"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:if> +</xsl:template> + +<xsl:template match="*:p"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'p'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:pb"> + <xsl:variable name="number"> + <xsl:choose> + <xsl:when test="not(empty(@n))"><xsl:value-of select="@n"/></xsl:when> + <xsl:otherwise><xsl:value-of select="count(./preceding::*:pb) + 1"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="file"> + <xsl:choose> + <xsl:when test="not(empty(@facs))"><xsl:value-of select="@facs"/></xsl:when> + <xsl:otherwise><xsl:value-of select="$number"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'pb'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'src'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="concat('pageimg/', string($file))"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'n'"/></xsl:attribute> + <xsl:value-of select="$number"/> + </span> + </span> + </span> +</xsl:template> + +<xsl:template match="*:head"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'head bf'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:figure"> + <span class="figure"> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:variable name="url"> + <xsl:choose> + <xsl:when test="not(empty(@facs))"><xsl:value-of select="string(@facs)"/></xsl:when> + <xsl:otherwise><xsl:value-of select="string(graphic/@url)"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="fileName" select="concat($echoFiguresDir, '/', $url)"/> + <xsl:variable name="digilibUrl"> + <xsl:choose> + <xsl:when test="starts-with($url, 'http')"><xsl:value-of select="$url"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat('http://digilib.mpiwg-berlin.mpg.de/digitallibrary/jquery/digilib.html?fn=', $fileName)"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="scalerUrl"> + <xsl:choose> + <xsl:when test="starts-with($url, 'http')"><xsl:value-of select="$url"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat('http://digilib.mpiwg-berlin.mpg.de/digitallibrary/servlet/Scaler?fn=', $fileName, '&dh=200&dw=200')"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <a href="{$digilibUrl}"> + <img> + <xsl:attribute name="src"><xsl:value-of select="$scalerUrl"/></xsl:attribute> + </img> + </a> + <xsl:if test="not(empty(@number))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNumber'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNum'"/></xsl:attribute> + <xsl:value-of select="string(@number)"/> + </span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNumText'"/></xsl:attribute> + <xsl:value-of select="concat('[', 'Figure', ' ', string(@number), ']')"/> + </span> + </span> + </xsl:if> + <xsl:if test="not(empty(*:head))"> + <span class="caption"> + <xsl:for-each select="*:head"> + <xsl:apply-templates/> + </xsl:for-each> + </span> + </xsl:if> + </span> +</xsl:template> + +<!-- segmentation --> +<xsl:template match="*:seg"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:choose> + <xsl:when test="@rend = 'highlight'"> + <xsl:attribute name="class"><xsl:value-of select="'seg highlight'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="@rend = 'highlightPoint'"> + <xsl:attribute name="class"><xsl:value-of select="'seg highlightPoint'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:otherwise> + <xsl:attribute name="class"><xsl:value-of select="'seg'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </span> +</xsl:template> + +<!-- choice --> +<xsl:template match="*:choice"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'choice'"/></xsl:attribute> + <xsl:if test="not(empty(*:orig))"> + <xsl:apply-templates select="*:orig"/> + </xsl:if> + <xsl:if test="not(empty(*:abbr))"> + <xsl:apply-templates select="*:abbr"/> + </xsl:if> + <xsl:if test="not(empty(*:am))"> + <xsl:apply-templates select="*:am"/> + </xsl:if> + <xsl:if test="not(empty(*:sic))"> + <xsl:apply-templates select="*:sic"/> + </xsl:if> + </span> +</xsl:template> + +<!-- hi (highlighted) --> +<xsl:template match="*:hi"> + <xsl:choose> + <xsl:when test="not(empty(@type)) and @type = 'elem'"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:when> + <xsl:when test="not(empty(@type)) and @type != 'elem'"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:when> + <xsl:when test="@rend = 'initial'"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'initial'"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:when> + <xsl:when test="@rend = 'bold'"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'bf'"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="@rend"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- name (of type: place, person, ...) --> +<xsl:template match="*:name"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:choose> + <xsl:when test="@type = 'place'"> + <xsl:attribute name="class"><xsl:value-of select="'place'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="'Place'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="@type = 'person'"> + <xsl:attribute name="class"><xsl:value-of select="'person'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="'Person'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="@type = 'org'"> + <xsl:attribute name="class"><xsl:value-of select="'organization'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="'Organization'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:otherwise> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="@type"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </span> +</xsl:template> + +<!-- place, person, ...) --> +<xsl:template match="*:placeName"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'place'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="concat('Place: ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> +<xsl:template match="*:persName"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'person'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="'Person'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- term --> +<xsl:template match="*:term"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'term'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="'Terminology entry'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- line --> +<xsl:template match="*:lg"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'lg'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:l"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'l'"/></xsl:attribute> + <xsl:apply-templates/> + </span> + <br/> +</xsl:template> + +<!-- reference --> +<xsl:template match="*:ref"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'ref'"/></xsl:attribute> + <xsl:if test="not(empty(@target))"> + <a class="ref"> + <xsl:attribute name="href"><xsl:value-of select="@target"/></xsl:attribute> + <xsl:apply-templates/> + </a> + </xsl:if> + <xsl:if test="empty(@target)"> + <xsl:apply-templates/> + </xsl:if> + </span> +</xsl:template> + +<!-- table --> +<xsl:template match="*:table"> + <table> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty(head))"> + <caption align="top"><xsl:apply-templates select="*:head"/></caption> + </xsl:if> + <xsl:apply-templates select="*:row"/> + </table> +</xsl:template> + +<xsl:template match="*:row"> + <tr> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:choose> + <xsl:when test="@role = 'label'"> + <xsl:attribute name="style"><xsl:value-of select="'font-weight:bold;'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="@role = 'data' or empty(@role)"> + <xsl:apply-templates/> + </xsl:when> + <xsl:otherwise> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </tr> +</xsl:template> + +<xsl:template match="*:cell"> + <td> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:choose> + <xsl:when test="@role = 'label' and empty(@cols)"> + <xsl:attribute name="style"><xsl:value-of select="'font-weight:bold;'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="@role = 'label' and not(empty(@cols))"> + <xsl:attribute name="colspan"><xsl:value-of select="@cols"/></xsl:attribute> + <xsl:attribute name="style"><xsl:value-of select="'font-weight:bold;'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="(@role = 'data' or empty(@role)) and empty(@cols)"> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="(@role = 'data' or empty(@role)) and not(empty(@cols))"> + <xsl:attribute name="colspan"><xsl:value-of select="@cols"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:otherwise> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </td> +</xsl:template> + +<!-- dictionary --> +<xsl:template match="*:entry"> + <!-- empty: handled in text tag --> +</xsl:template> + +<xsl:template match="*:form"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'form'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:orth"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'orth'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:sense"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'sense'"/></xsl:attribute> + <span class="entryDiv"><xsl:apply-templates select="*:def"/></span> + <span class="entryDiv"><xsl:apply-templates select="*:etym"/></span> + </span> +</xsl:template> + +<xsl:template match="*:def"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'def'"/></xsl:attribute> + <span class="bf"><xsl:value-of select="'Definition: '"/></span> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:mentioned"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'mentioned'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:etym"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'etym'"/></xsl:attribute> + <span class="bf"><xsl:value-of select="'Etymology: '"/></span> + <span class="entryDiv"> + <xsl:for-each select="*:cit"> + <li><xsl:apply-templates select="*:quote"/><xsl:value-of select="' ('"/><xsl:apply-templates select="*:def"/><xsl:value-of select="')'"/></li> + </xsl:for-each> + </span> + </span> +</xsl:template> + +<xsl:template match="*:cit"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'cit'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- MathML --> +<xsl:template match="math:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<!-- SVG --> +<xsl:template match="svg:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:lb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'lb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="*:cb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'cb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="*:note"> + <xsl:variable name="place"><xsl:value-of select="string(@place)"/></xsl:variable> + <xsl:variable name="noteSign"><xsl:value-of select="count(./preceding::*:note) + 1"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:if test="$place != '' and $place != 'foot'"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('noteBody ', $place)"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:apply-templates/> + </span> + </xsl:if> + </span> +</xsl:template> + +<xsl:template match="*:foreign"> + <xsl:variable name="xmllang" select="@xml:lang"/> + <xsl:variable name="language"> + <xsl:choose> + <xsl:when test="not(empty($xmllang))"><xsl:value-of select="$xmllang"/></xsl:when> + <xsl:otherwise><xsl:value-of select="''"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat('foreign ', $language)"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:gap"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'gap'"/></xsl:attribute> + <xsl:if test="not(empty(@extent))"><span class="extent"><xsl:value-of select="string(@extent)"/></span></xsl:if> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:s"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'s'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- words --> +<xsl:template match="*:w"> + <xsl:variable name="wordLanguage" select="@lang"/> + <xsl:variable name="form" select="encode-for-uri(string(@form))"/> + <xsl:variable name="dictionary" select="string(@dictionary)"/> + <xsl:variable name="displayWordOrig"> + <xsl:choose> + <xsl:when test="not(empty(*:orig))"><xsl:apply-templates select="*:orig/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordReg"> + <xsl:choose> + <xsl:when test="not(empty(*:reg))"><xsl:apply-templates select="*:reg/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordNorm"> + <xsl:choose> + <xsl:when test="not(empty(*:norm))"><xsl:apply-templates select="*:norm/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordUrlEncoded" select="encode-for-uri($displayWordOrig)"/> + <xsl:variable name="dictionaryPart"> + <xsl:choose> + <xsl:when test="$dictionary = 'true'"> + <a class="dictionary"> + <xsl:attribute name="href"><xsl:value-of select="concat($dictionaryServiceName, '?query=', $form, '&queryDisplay=', $displayWordUrlEncoded, '&language=', $wordLanguage, '&outputFormat=html', '&outputType=morphCompact&outputType=dictFull')"/></xsl:attribute> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </a> + </xsl:when> + <xsl:otherwise> + <span class="dictionary"> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span class="w"> + <xsl:sequence select="$dictionaryPart"/> + <span class="nodictionary orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="nodictionary reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="nodictionary norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/bin/de/mpg/mpiwg/berlin/mpdl/cms/transform/pageXhtml.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,260 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xs="http://www.w3.org/2001/XMLSchema" + xmlns:functx="http://www.functx.com" + xmlns:saxon="http://saxon.sf.net/" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xs functx saxon dc dcterms math svg xhtml" + > + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="echoFiguresDir"></xsl:param> +<xsl:param name="echoPageImgDir"></xsl:param> + +<xsl:variable name="dictionaryServiceName" select="'http://mpdl-service.mpiwg-berlin.mpg.de/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries'"/> + +<xsl:template match="*:html"> + <xsl:apply-templates/> +</xsl:template> + +<xsl:template match="body"> + <xsl:apply-templates mode="html"/> +</xsl:template> + +<!-- html mode: transform html texts --> +<xsl:template match="@*|node()" mode="html"> + <xsl:copy> + <xsl:apply-templates select="@*|node()" mode="html"/> + </xsl:copy> +</xsl:template> + +<xsl:template match="s" mode="html"> + <xsl:apply-templates select="."/> +</xsl:template> + +<xsl:template match="w" mode="html"> + <xsl:apply-templates select="."/> +</xsl:template> + + +<xsl:template match="*:expan|*:emph|*:q|*:quote|*:blockquote|*:set-off|*:reg|*:var|*:num|*:figure"> + <span> + <xsl:attribute name="class"><xsl:value-of select="name()"/></xsl:attribute> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:div"> + <xsl:if test="node() or normalize-space()"> <!-- test: contains nodes or is not empty --> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'div'"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:if> +</xsl:template> + +<xsl:template match="*:p"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'p'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="pb"> + <xsl:variable name="number" select="count(./preceding::*:pb) + 1"/> + <xsl:variable name="file" select="$number"/> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'pb'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'src'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="concat('pageimg/', string($file))"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'n'"/></xsl:attribute> + <xsl:value-of select="$number"/> + </span> + </span> + </span> +</xsl:template> + +<xsl:template match="*:head"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'head bf'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<!-- MathML --> +<xsl:template match="math:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<!-- SVG --> +<xsl:template match="svg:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:lb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'lb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="*:cb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'cb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="br"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'br'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="note"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="foreign"> + <xsl:variable name="lang" select="@lang"/> + <xsl:variable name="xmllang" select="@xml:lang"/> + <xsl:variable name="language"> + <xsl:choose> + <xsl:when test="not(empty($xmllang))"><xsl:value-of select="$xmllang"/></xsl:when> + <xsl:when test="not(empty($lang))"><xsl:value-of select="$lang"/></xsl:when> + <xsl:otherwise><xsl:value-of select="''"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat('foreign ', $language)"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="gap"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'gap'"/></xsl:attribute> + <xsl:if test="not(empty(@extent))"><span class="extent"><xsl:value-of select="string(@extent)"/></span></xsl:if> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- XHTML: remove the xhtml namespace --> +<xsl:template match="xhtml:*"> + <xsl:variable name="hasLabel" select="string(@xlink:label) != ''"/> + <xsl:variable name="isTable" select="name() = 'table'"/> + <xsl:choose> + <xsl:when test="(not($hasLabel)) or ($isTable and $hasLabel)"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="s"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'s'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- words --> +<xsl:template match="*:w"> + <xsl:variable name="wordLanguage" select="@lang"/> + <xsl:variable name="form" select="encode-for-uri(string(@form))"/> + <xsl:variable name="dictionary" select="string(@dictionary)"/> + <xsl:variable name="displayWordOrig"> + <xsl:choose> + <xsl:when test="not(empty(*:orig))"><xsl:apply-templates select="*:orig/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordReg"> + <xsl:choose> + <xsl:when test="not(empty(*:reg))"><xsl:apply-templates select="*:reg/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordNorm"> + <xsl:choose> + <xsl:when test="not(empty(*:norm))"><xsl:apply-templates select="*:norm/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordUrlEncoded" select="encode-for-uri($displayWordOrig)"/> + <xsl:variable name="dictionaryPart"> + <xsl:choose> + <xsl:when test="$dictionary = 'true'"> + <a class="dictionary"> + <xsl:attribute name="href"><xsl:value-of select="concat($dictionaryServiceName, '?query=', $form, '&queryDisplay=', $displayWordUrlEncoded, '&language=', $wordLanguage, '&outputFormat=html', '&outputType=morphCompact&outputType=dictFull')"/></xsl:attribute> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </a> + </xsl:when> + <xsl:otherwise> + <span class="dictionary"> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span class="w"> + <xsl:sequence select="$dictionaryPart"/> + <span class="nodictionary orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="nodictionary reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="nodictionary norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> +</xsl:template> + +<xsl:template match="*:hi"> + <xsl:choose> + <xsl:when test="@type = 'elem'"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/bin/de/mpg/mpiwg/berlin/mpdl/cms/transform/pageXml.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,86 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink dcterms math svg xhtml" + > + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="echoFiguresDir"></xsl:param> +<xsl:param name="echoPageImgDir"></xsl:param> + +<!-- transform to browser like xml display --> +<xsl:template match="element()"> + <xsl:variable name="elementName" select="name()"/> + <xsl:variable name="elementPresentation"> + <xsl:choose> + <xsl:when test="element() = node() or text() != ''"> + <xsl:value-of select="'<'"/> + <span class="xml elementName"><xsl:value-of select="$elementName"/></span> + <xsl:apply-templates select="attribute()"/> + <xsl:value-of select="'>'"/> + <xsl:apply-templates select="element()|text()|comment()|processing-instruction()"/> + <xsl:value-of select="'</'"/> + <span class="xml elementName"><xsl:value-of select="$elementName"/></span> + <xsl:value-of select="'>'"/> + </xsl:when> + <xsl:otherwise> + <xsl:value-of select="'<'"/> + <span class="xml elementName"><xsl:value-of select="$elementName"/></span> + <xsl:apply-templates select="attribute()"/> + <xsl:value-of select="'/>'"/> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <ul class="xml element"> + <xsl:sequence select="$elementPresentation"/> + </ul> +</xsl:template> + +<xsl:template match="attribute()"> + <xsl:variable name="attributeName" select="name()"/> + <span class="xml attributeName"> + <xsl:value-of select="' '"/> + <xsl:value-of select="$attributeName"/> + </span> + <xsl:value-of select="'="'"/> + <span class="xml attributeValue"><xsl:value-of select="."/></span><xsl:value-of select="'"'"/> +</xsl:template> + +<xsl:template match="comment()"> + <span class="xml comment"> + <xsl:value-of select="'<!-- '"/><xsl:value-of select="."/><xsl:value-of select="' -->'"/> + </span> +</xsl:template> + +<xsl:template match="processing-instruction()"> +</xsl:template> + +<xsl:template match="*:w"> + <xsl:apply-templates select="*:orig/node()"/> +</xsl:template> + +<xsl:template match="*:hi"> + <xsl:choose> + <xsl:when test="@type = 'elem'"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/bin/de/mpg/mpiwg/berlin/mpdl/cms/transform/queryDocument.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,93 @@ +<?xml version="1.0"?> + +<xsl:stylesheet version="2.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform"> + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="query"></xsl:param> +<xsl:param name="flags"></xsl:param> +<xsl:param name="outputFormat"></xsl:param> + +<xsl:variable name="apos">'</xsl:variable> +<xsl:variable name="xpathQuery"> + <xsl:choose> + <xsl:when test="$flags = ''"><xsl:value-of select="concat('//*:s[matches(.,', $apos, $query, $apos, ')]', '|', '//*:head[matches(.,', $apos, $query, $apos, ')]')"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat('//*:s[matches(.,', $apos, $query, $apos, ',', $apos, $flags, $apos, ')]', '|', '//*:head[matches(.,', $apos, $query, $apos, ',', $apos, $flags, $apos, ')]')"/></xsl:otherwise> + </xsl:choose> +</xsl:variable> +<xsl:variable name="queryResult" select="saxon:evaluate($xpathQuery)" xmlns:saxon="http://saxon.sf.net/"/> +<xsl:variable name="queryResultSize"><xsl:value-of select="count($queryResult)"/></xsl:variable> +<xsl:variable name="queryResultPageSize" select="20"/> +<xsl:variable name="queryResultPages"> + <xsl:choose> + <xsl:when test="$queryResultSize = 0"><xsl:value-of select="count($queryResult)"/></xsl:when> + <xsl:otherwise><xsl:value-of select="$queryResultSize idiv $queryResultPageSize + 1"/></xsl:otherwise> + </xsl:choose> +</xsl:variable> +<xsl:variable name="pageBreaks" select="saxon:evaluate('//*:pb')" xmlns:saxon="http://saxon.sf.net/"/> +<xsl:variable name="queryResultHtml"> +<xsl:choose> + <xsl:when test="$outputFormat = 'html'"> + <table> + <thead> + <tr> + <th align="left" valign="top"> + <button name="order-by" value="author" style="padding:0px;font-weight:bold;font-size:14px;background:none;border:none;">No.</button> + </th> + <th align="left" valign="top"> + <button name="order-by" value="author" style="padding:0px;font-weight:bold;font-size:14px;background:none;border:none;">Path</button> + </th> + <th align="left" valign="top"> + <button name="order-by" value="author" style="padding:0px;font-weight:bold;font-size:14px;background:none;border:none;">Hit</button> + </th> + <th align="left" valign="top"> + <button name="order-by" value="author" style="padding:0px;font-weight:bold;font-size:14px;background:none;border:none;">Page</button> + </th> + </tr> + </thead> + <tbody> + <xsl:for-each select="$queryResult"> + <xsl:variable name="hit" select="."/> + <tr> + <td align="left" valign="top"><xsl:value-of select="position()"/></td> + <td align="left" valign="top"><xsl:value-of select="saxon:path(.)" xmlns:saxon="http://saxon.sf.net/"/></td> + <td align="left" valign="top"><xsl:value-of select="."/></td> + <td align="left" valign="top"><xsl:value-of select="count(./preceding::*:pb)"/></td> <!-- better performance: count($pageBreaks[. << $hit]) --> + </tr> + </xsl:for-each> + </tbody> + </table> + </xsl:when> + <xsl:otherwise> + <result> + <query><xsl:value-of select="$query"/></query> + <flags><xsl:value-of select="$flags"/></flags> + <size><xsl:value-of select="$queryResultSize"/></size> + <page-size><xsl:value-of select="$queryResultPageSize"/></page-size> + <pages><xsl:value-of select="$queryResultPages"/></pages> + <pn>1</pn> + <hits> + <xsl:for-each select="$queryResult"> + <xsl:variable name="hit" select="."/> + <hit> + <hitType><xsl:value-of select="'s'"/></hitType> + <pos><xsl:value-of select="position()"/></pos> + <pn><xsl:value-of select="count(./preceding::*:pb)"/></pn> + <hitId>xmlId</hitId> + <hitPos>4711</hitPos> + <hitString><xsl:value-of select="."/></hitString> + <hitSurroundsPB>false</hitSurroundsPB> + </hit> + </xsl:for-each> + </hits> + <query-forms></query-forms> + <query-regularizations></query-regularizations> + </result> + </xsl:otherwise> +</xsl:choose> +</xsl:variable> +<xsl:template match="/"> + <xsl:sequence select="$queryResultHtml"/> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/bin/de/mpg/mpiwg/berlin/mpdl/cms/transform/replaceAnchor.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,82 @@ +<?xml version="1.0"?> + +<xsl:stylesheet + version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xhtml" +> + +<xsl:output method="xml" encoding="utf-8"/> + +<xsl:template match="@*|node()"> + <xsl:copy> + <xsl:apply-templates select="@*|node()"/> + </xsl:copy> +</xsl:template> + +<!-- insert figure number --> +<xsl:template match="*:figure"> + <xsl:variable name="number" select="count(./preceding::*:figure) + 1"/> + <xsl:element name="{name()}" namespace="{namespace-uri()}"><!-- remove namespace in element and its subnodes --> + <xsl:apply-templates select="@*"/> + <xsl:attribute name="number"><xsl:value-of select="$number"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<!-- insert figure number --> +<xsl:template match="*:handwritten"> + <xsl:variable name="number" select="count(./preceding::*:handwritten) + 1"/> + <xsl:element name="{name()}" namespace="{namespace-uri()}"><!-- remove namespace in element and its subnodes --> + <xsl:apply-templates select="@*"/> + <xsl:attribute name="number"><xsl:value-of select="$number"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:anchor"> + <xsl:variable name="type" select="@type"/> + <xsl:variable name="href" select="@xlink:href"/> + <xsl:choose> + <xsl:when test="$type = 'figure'"> + <xsl:variable name="figure" select="//*:figure[@xlink:label = $href]"/> + <xsl:apply-templates select="$figure"/> + </xsl:when> + <xsl:when test="$type = 'handwritten'"> + <xsl:variable name="handwritten" select="//*:handwritten[@xlink:label = $href]"/> + <xsl:apply-templates select="$handwritten"/> + </xsl:when> + <xsl:when test="$type = 'note'"> + <xsl:variable name="note" select="//*:note[@xlink:label = $href]"/> + <xsl:sequence select="$note"/> + </xsl:when> + <xsl:when test="$type = 'table'"> + <xsl:variable name="table" select="//xhtml:table[@xlink:label = $href]"/> + <xsl:sequence select="$table"/> + </xsl:when> + <xsl:otherwise> + <xsl:copy> + <xsl:apply-templates select="@*|node()"/> + </xsl:copy> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- remove the anchored divs: div which contain a labeled figure, note, handwritten or table --> +<xsl:template match="*:div"> + <xsl:variable name="containingElems" select="*:figure|*:note|*:handwritten|xhtml:table"/> + <xsl:variable name="hasLabel" select="not(empty($containingElems[1]/@xlink:label))"/> + <xsl:choose> + <xsl:when test="$hasLabel"></xsl:when> + <xsl:otherwise> + <xsl:copy> + <xsl:apply-templates select="@*|node()"/> + </xsl:copy> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/bin/de/mpg/mpiwg/berlin/mpdl/cms/transform/toc.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,153 @@ +<?xml version="1.0"?> + +<xsl:stylesheet + version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xhtml" +> + +<xsl:output method="xml" encoding="utf-8" indent="yes"/> + +<xsl:template match="/"> + <xsl:variable name="toc" select="//*:div[@type = 'section' or @type = 'chapter' or @type = 'part']"/> + <xsl:variable name="figures" select="//*:figure"/> + <xsl:variable name="notes" select="//*:note"/> + <xsl:variable name="notesHandwritten" select="//*:handwritten"/> + <xsl:variable name="pages" select="//*:pb"/> + <xsl:variable name="places" select="//*:place"/> + <list> + <xsl:if test="not(empty($toc))"> + <list type ="toc"> + <xsl:attribute name="count"><xsl:value-of select="count($toc)"/></xsl:attribute> + <head>Table of contents</head> + <xsl:apply-templates select="$toc"/> + </list> + </xsl:if> + <xsl:if test="not(empty($figures))"> + <list type="figures"> + <xsl:attribute name="count"><xsl:value-of select="count($figures)"/></xsl:attribute> + <head>Figures</head> + <xsl:apply-templates select="$figures"/> + </list> + </xsl:if> + <xsl:if test="not(empty($notes))"> + <list type="notes"> + <xsl:attribute name="count"><xsl:value-of select="count($notes)"/></xsl:attribute> + <head>Notes</head> + <xsl:apply-templates select="$notes"/> + </list> + </xsl:if> + <xsl:if test="not(empty($notesHandwritten))"> + <list type="notesHandwritten"> + <xsl:attribute name="count"><xsl:value-of select="count($notesHandwritten)"/></xsl:attribute> + <head>Handwritten notes</head> + <xsl:apply-templates select="$notesHandwritten"/> + </list> + </xsl:if> + <xsl:if test="not(empty($pages))"> + <list type="pages"> + <xsl:attribute name="count"><xsl:value-of select="count($pages)"/></xsl:attribute> + <head>Pages</head> + <xsl:apply-templates select="$pages"/> + </list> + </xsl:if> + <xsl:if test="not(empty($places))"> + <list type="places"> + <xsl:attribute name="count"><xsl:value-of select="count($places)"/></xsl:attribute> + <head>Places</head> + <xsl:apply-templates select="$places"/> + </list> + </xsl:if> + </list> +</xsl:template> + +<xsl:template match="*:figure"> + <xsl:variable name="page" select="count(./preceding::*:pb)"/> + <xsl:variable name="number" select="@number"/> + <item> + <xsl:if test="not(empty($number))"><xsl:attribute name="n"><xsl:value-of select="$number"/></xsl:attribute></xsl:if> + <xsl:apply-templates select="*:caption"/> + <xsl:apply-templates select="*:description"/> + <xsl:apply-templates select="*:variables"/> + <xsl:if test="not(empty($page))"><ref><xsl:value-of select="$page"/></ref></xsl:if> + </item> +</xsl:template> + +<xsl:template match="*:caption"><xsl:value-of select="' '"/> + <xsl:apply-templates/> +</xsl:template> + +<xsl:template match="*:description"> + <xsl:apply-templates/><xsl:value-of select="' '"/> +</xsl:template> + +<xsl:template match="*:variables"> + <xsl:apply-templates/><xsl:value-of select="' '"/> +</xsl:template> + +<xsl:template match="*:note"> + <xsl:variable name="page" select="count(./preceding::*:pb)"/> + <xsl:variable name="number" select="@number"/> + <item> + <xsl:if test="not(empty($number))"><xsl:attribute name="n"><xsl:value-of select="$number"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty($page))"><ref><xsl:value-of select="$page"/></ref></xsl:if> + </item> +</xsl:template> + +<xsl:template match="*:handwritten"> + <xsl:variable name="page" select="count(./preceding::*:pb)"/> + <xsl:variable name="number" select="@number"/> + <item> + <xsl:if test="not(empty($number))"><xsl:attribute name="n"><xsl:value-of select="$number"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty($page))"><ref><xsl:value-of select="$page"/></ref></xsl:if> + </item> +</xsl:template> + +<xsl:template match="*:place"> + <xsl:variable name="page" select="count(./preceding::*:pb)"/> + <xsl:variable name="id" select="@id"/> + <item> + <xsl:if test="not(empty($id))"><xsl:attribute name="id"><xsl:value-of select="$id"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty($page))"><ref><xsl:value-of select="$page"/></ref></xsl:if> + </item> +</xsl:template> + +<xsl:template match="*:div"> + <xsl:variable name="level"><xsl:number level="multiple" count="*:div[@type = 'section' or @type = 'chapter' or @type = 'part']" format="1."/></xsl:variable> + <xsl:variable name="depth" select="string-length(replace($level, '[^\\.]', ''))"/> + <xsl:variable name="pb" select="./preceding::*:pb[1]"/> + <xsl:variable name="o" select="$pb/@o"/> + <xsl:variable name="oNorm" select="$pb/@o-norm"/> + <xsl:variable name="page" select="count(./preceding::*:pb)"/> + <item> + <xsl:if test="not(empty($level))"><xsl:attribute name="n"><xsl:value-of select="$level"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty($level))"><xsl:attribute name="lv"><xsl:value-of select="$depth"/></xsl:attribute></xsl:if> + <xsl:apply-templates select="*:head"/> + <xsl:if test="not(empty($page))"> + <ref> + <xsl:if test="not(empty($o))"><xsl:attribute name="o"><xsl:value-of select="$o"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty($oNorm))"><xsl:attribute name="o-norm"><xsl:value-of select="$oNorm"/></xsl:attribute></xsl:if> + <xsl:value-of select="$page"/> + </ref> + </xsl:if> + </item> +</xsl:template> + +<xsl:template match="*:head"> + <xsl:apply-templates/><xsl:value-of select="' '"/> +</xsl:template> + +<xsl:template match="*:pb"> + <xsl:variable name="page" select="count(./preceding::*:pb) + 1"/> + <item> + <xsl:attribute name="n"><xsl:value-of select="$page"/></xsl:attribute> + <xsl:if test="not(empty(@o))"><xsl:attribute name="o"><xsl:value-of select="@o"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty(@o-norm))"><xsl:attribute name="o-norm"><xsl:value-of select="@o-norm"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty(@file))"><xsl:attribute name="file"><xsl:value-of select="@file"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty(@facs))"><xsl:attribute name="file"><xsl:value-of select="@facs"/></xsl:attribute></xsl:if> + </item> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/bin/de/mpg/mpiwg/berlin/mpdl/cms/transform/tocOut.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,45 @@ +<?xml version="1.0"?> + +<xsl:stylesheet + version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" +> + +<xsl:output method="xhtml"/> + +<xsl:param name="type"></xsl:param> +<xsl:param name="outputFormat"></xsl:param> + +<xsl:template match="/"> + <xsl:choose> + <xsl:when test="$outputFormat = 'html'"> + <xsl:apply-templates select="//*:list[@type = $type]/item" mode="html"/> + </xsl:when> + <xsl:otherwise> + <xsl:sequence select="//*:list[@type = $type]"/> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="*:item" mode="html"> + <xsl:variable name="n" select="@n"/> + <xsl:variable name="level" select="@lv"/> + <div> + <xsl:attribute name="class"><xsl:value-of select="'tocItem'"/></xsl:attribute> + <xsl:value-of select="concat($n, ' ')"/> + <xsl:apply-templates mode="html"/> + </div> +</xsl:template> + +<xsl:template match="*:ref" mode="html"> + <xsl:variable name="pageOrig" select="@o"/> + <xsl:variable name="page" select="text()"/> + <xsl:variable name="pageHref" select="concat('#page', $page)"/> + <span> + <xsl:attribute name="class"><xsl:value-of select="'tocPageRef'"/></xsl:attribute> + <xsl:value-of select="'Page: '"/> + <a href="{$pageHref}"><xsl:apply-templates mode="html"/></a> + </span> +</xsl:template> + +</xsl:stylesheet>
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/build/build.xml Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,39 @@ +<!DOCTYPE project> +<project name="mpiwg-mpdl-cms" default="compile" basedir="../"> + <description>mpiwg-mpdl-cms</description> + <!-- global properties --> + <property name="src" location="src"/> + <property name="lib" location="lib"/> + <property name="build" location="build/classes"/> + <property name="dist" location="dist"/> + + <path id="classpath"> + <fileset dir="${lib}" includes="**/*.jar"/> + </path> + + <target name="init"> + <!-- Create time stamp --> + <tstamp/> + <mkdir dir="${build}"/> + <mkdir dir="${dist}"/> + </target> + + <target name="compile" depends="init" description="compile"> + <copy file="/Users/jwillenborg/java/mpiwg-mpdl-xml/dist/mpiwg-mpdl-xml.jar" todir="${lib}"/> + <copy file="/Users/jwillenborg/java/mpiwg-mpdl-lt/dist/mpiwg-mpdl-lt.jar" todir="${lib}"/> + <javac srcdir="${src}" destdir="${build}" classpathref="classpath" includeantruntime="false"/> + <copy todir="${build}"> + <fileset dir="${src}"> + <include name="**/*.xsl"/> + <include name="**/*.properties"/> + </fileset> + </copy> + <jar jarfile="${dist}/mpiwg-mpdl-cms.jar" basedir="${build}"/> + <copy file="${dist}/mpiwg-mpdl-cms.jar" todir="/Users/jwillenborg/java/mpiwg-mpdl-cms-web/WebContent/WEB-INF/lib"/> + </target> + + <target name="clean" description="clean" > + <delete dir="${build}"/> + <delete file="${dist}/mpiwg-mpdl-cms.jar"/> + </target> +</project> \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/build/classes/constants.properties Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,5 @@ +dataDir=/Users/jwillenborg/mpdl/data/lt +documentsDir=/Users/jwillenborg/mpdl/data/xml/documents +luceneDocumentsDir=/Users/jwillenborg/mpdl/data/xml/lucene/documents +luceneNodesDir=/Users/jwillenborg/mpdl/data/xml/lucene/nodes +confDir=/Users/jwillenborg/mpdl/data/collectionConfs \ No newline at end of file
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Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/document/PdfHandlerUserAgent.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/document/Token.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/general/Constants.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/harvester/PathExtractor.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/lucene/IndexHandler.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/scheduler/CmsChainScheduler.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/scheduler/CmsChainSchedulerListener.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/scheduler/CmsDocJob.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/scheduler/CmsDocOperation.class has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/scheduler/quartz.properties Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,9 @@ +org.quartz.scheduler.instanceName = MpdlScheduler +org.quartz.scheduler.instanceId = auto +org.quartz.scheduler.rmi.export = false +org.quartz.scheduler.rmi.proxy = false + +org.quartz.threadPool.class = org.quartz.simpl.SimpleThreadPool +org.quartz.threadPool.threadCount = 3 + +org.quartz.jobStore.class = org.quartz.simpl.RAMJobStore
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/test/TestLocal.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/GetFragmentsContentHandler$1.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/GetFragmentsContentHandler$Element.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/GetFragmentsContentHandler.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/HighlightContentHandler.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/HighlightTransformer.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/PageTransformer.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/QueryTransformer.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/ReplaceAnchorTransformer.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/TocTransformer.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/XslResourceTransformer.class has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/chars.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,10 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform"> + +<xsl:output method="xml" encoding="utf-8" omit-xml-declaration="yes"/> + +<xsl:template match="text()"> + <xsl:sequence select="normalize-space(.)"/> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/highlight.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,29 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform"> + +<xsl:output method="html" encoding="utf-8"/> + +<xsl:template match="*:hi"> + <xsl:variable name="type" select="string(@type)"/> + <xsl:choose> + <xsl:when test="$type = 'elem'"> + <div style="background-color:#D3D3D3"><xsl:apply-templates/></div> + </xsl:when> + <xsl:when test="$type = 'hit'"> + <span style="background-color:#77DD77"><xsl:apply-templates/></span> + </xsl:when> + <xsl:otherwise> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="*[. except self::*:hi]"> + <xsl:element name="span"> + <xsl:attribute name="class"><xsl:value-of select="name()"/></xsl:attribute> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/pageArchimedes.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,324 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xs="http://www.w3.org/2001/XMLSchema" + xmlns:functx="http://www.functx.com" + xmlns:saxon="http://saxon.sf.net/" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xs functx saxon dc dcterms math svg xhtml" + > + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="echoFiguresDir"></xsl:param> +<xsl:param name="echoPageImgDir"></xsl:param> + +<xsl:variable name="dictionaryServiceName" select="'http://mpdl-service.mpiwg-berlin.mpg.de/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries'"/> + +<xsl:template match="*:archimedes"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'archimedes'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:expan|*:emph|*:q|*:quote|*:blockquote|*:set-off|*:reg|*:var|*:num"> + <span> + <xsl:attribute name="class"><xsl:value-of select="name()"/></xsl:attribute> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="text"> + <xsl:variable name="contentStr" select="normalize-space(string(.))"/> + <xsl:variable name="figures" select=".//*:figure|.//*:handwritten"/> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'text'"/></xsl:attribute> + <xsl:apply-templates/> + <xsl:if test="$contentStr = '' and empty($figures)"> <!-- test if div only contains empty nodes but no figures --> + <div> + <xsl:attribute name="class"><xsl:value-of select="'emptyPage'"/></xsl:attribute> + <xsl:value-of select="'[Empty page]'"/> + </div> + </xsl:if> + </div> + <!-- Notes --> + <xsl:variable name="bottomNotes" select="//*:note[contains(@position, 'foot') or empty(string(@position))]"/> + <xsl:variable name="countBottomNotes" select="count($bottomNotes)"/> + <xsl:if test="$countBottomNotes > 0"> + <span class="notes"> + <xsl:for-each select="$bottomNotes"> + <xsl:variable name="noteSign"><xsl:value-of select="count(./preceding::*:note) + 1"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'noteBody foot'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:apply-templates/> + </span> + </span> + </xsl:for-each> + </span> + </xsl:if> +</xsl:template> + +<xsl:template match="*:div"> + <xsl:if test="node() or normalize-space()"> <!-- test: contains nodes or is not empty --> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'div'"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:if> +</xsl:template> + +<xsl:template match="*:p"> + <xsl:variable name="type" select="string(@type)"/> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:choose> + <xsl:when test="$type = 'head'"> + <xsl:attribute name="class"><xsl:value-of select="'head bf'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:otherwise> + <xsl:attribute name="class"><xsl:value-of select="'p'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </div> +</xsl:template> + +<xsl:template match="*:pb"> + <xsl:variable name="number"> + <xsl:choose> + <xsl:when test="not(empty(@n))"><xsl:value-of select="@n"/></xsl:when> + <xsl:otherwise><xsl:value-of select="count(./preceding::*:pb) + 1"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="file"> + <xsl:choose> + <xsl:when test="not(empty(@xlink:href))"><xsl:value-of select="replace(@xlink:href, '/', '.')"/></xsl:when> + <xsl:otherwise><xsl:value-of select="$number"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'pb'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'src'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="concat('pageimg/', string($file))"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'n'"/></xsl:attribute> + <xsl:value-of select="$number"/> + </span> + </span> + </span> +</xsl:template> + +<xsl:template match="*:head"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'head bf'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<!-- MathML --> +<xsl:template match="math:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<!-- SVG --> +<xsl:template match="svg:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:figure"> + <span class="figure"> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:variable name="fileHref" select="replace(@xlink:href, '/', '.')"/> + <xsl:variable name="fileName" select="concat($echoFiguresDir, '/', $fileHref)"/> + <xsl:variable name="digilibUrl" select="concat('http://digilib.mpiwg-berlin.mpg.de/digitallibrary/jquery/digilib.html?fn=', $fileName)"/> + <xsl:variable name="scalerUrl" select="concat('http://digilib.mpiwg-berlin.mpg.de/digitallibrary/servlet/Scaler?fn=', $fileName, '&dh=200&dw=200')"/> + <xsl:variable name="ancestorWords" select="./ancestor::*:w"/> <!-- if figure is in a word then no link within a link should be produced --> + <xsl:choose> + <xsl:when test="empty($ancestorWords)"> + <a href="{$digilibUrl}"> + <img> + <xsl:attribute name="src"><xsl:value-of select="$scalerUrl"/></xsl:attribute> + </img> + </a> + </xsl:when> + <xsl:otherwise> + <img> + <xsl:attribute name="src"><xsl:value-of select="$scalerUrl"/></xsl:attribute> + </img> + </xsl:otherwise> + </xsl:choose> + <xsl:if test="not(empty(@number))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNumber'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNum'"/></xsl:attribute> + <xsl:value-of select="string(@number)"/> + </span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNumText'"/></xsl:attribute> + <xsl:value-of select="concat('[', 'Figure', ' ', string(@number), ']')"/> + </span> + </span> + </xsl:if> + </span> +</xsl:template> + +<xsl:template match="*:lb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'lb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="*:cb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'cb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="note"> + <xsl:variable name="place"><xsl:value-of select="string(@position)"/></xsl:variable> + <xsl:variable name="noteSign"><xsl:value-of select="count(./preceding::*:note) + 1"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:if test="$place != '' and $place != 'foot'"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('noteBody ', $place)"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:apply-templates/> + </span> + </xsl:if> + </span> +</xsl:template> + +<xsl:template match="foreign"> + <xsl:variable name="lang" select="@lang"/> + <xsl:variable name="xmllang" select="@xml:lang"/> + <xsl:variable name="language"> + <xsl:choose> + <xsl:when test="not(empty($xmllang))"><xsl:value-of select="$xmllang"/></xsl:when> + <xsl:when test="not(empty($lang))"><xsl:value-of select="$lang"/></xsl:when> + <xsl:otherwise><xsl:value-of select="''"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat('foreign ', $language)"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="gap"> + <span class="gap"> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'gap'"/></xsl:attribute> + <xsl:if test="not(empty(@extent))"><span class="extent"><xsl:value-of select="string(@extent)"/></span></xsl:if> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="s"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'s'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- words --> +<xsl:template match="*:w"> + <xsl:variable name="wordLanguage" select="@lang"/> + <xsl:variable name="form" select="encode-for-uri(string(@form))"/> + <xsl:variable name="dictionary" select="string(@dictionary)"/> + <xsl:variable name="displayWordOrig"> + <xsl:choose> + <xsl:when test="not(empty(*:orig))"><xsl:apply-templates select="*:orig/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordReg"> + <xsl:choose> + <xsl:when test="not(empty(*:reg))"><xsl:apply-templates select="*:reg/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordNorm"> + <xsl:choose> + <xsl:when test="not(empty(*:norm))"><xsl:apply-templates select="*:norm/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordUrlEncoded" select="encode-for-uri($displayWordOrig)"/> + <xsl:variable name="dictionaryPart"> + <xsl:choose> + <xsl:when test="$dictionary = 'true'"> + <a class="dictionary"> + <xsl:attribute name="href"><xsl:value-of select="concat($dictionaryServiceName, '?query=', $form, '&queryDisplay=', $displayWordUrlEncoded, '&language=', $wordLanguage, '&outputFormat=html', '&outputType=morphCompact&outputType=dictFull')"/></xsl:attribute> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </a> + </xsl:when> + <xsl:otherwise> + <span class="dictionary"> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span class="w"> + <xsl:sequence select="$dictionaryPart"/> + <span class="nodictionary orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="nodictionary reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="nodictionary norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> +</xsl:template> + +<xsl:template match="*:hi"> + <xsl:choose> + <xsl:when test="@type = 'elem'"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/pageEcho.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,536 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xs="http://www.w3.org/2001/XMLSchema" + xmlns:functx="http://www.functx.com" + xmlns:saxon="http://saxon.sf.net/" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" + xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:mml="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xs functx saxon dc dcterms echo de math mml svg xhtml" + > + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="echoFiguresDir"></xsl:param> +<xsl:param name="echoPageImgDir"></xsl:param> + +<xsl:variable name="dictionaryServiceName" select="'http://mpdl-service.mpiwg-berlin.mpg.de/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries'"/> + +<xsl:template match="*:echo"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'echo'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:text"> + <xsl:variable name="contentStr" select="normalize-space(string(.))"/> + <xsl:variable name="figures" select=".//*:figure|.//*:handwritten"/> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'text'"/></xsl:attribute> + <xsl:apply-templates/> + <xsl:if test="$contentStr = '' and empty($figures)"> <!-- test if div only contains empty nodes but no figures --> + <div> + <xsl:attribute name="class"><xsl:value-of select="'emptyPage'"/></xsl:attribute> + <xsl:value-of select="'[Empty page]'"/> + </div> + </xsl:if> + </div> + <!-- Notes --> + <xsl:variable name="bottomNotes" select="//*:note[contains(@position, 'foot') or empty(string(@position))]"/> + <xsl:variable name="countBottomNotes" select="count($bottomNotes)"/> + <xsl:if test="$countBottomNotes > 0"> + <span class="notes"> + <xsl:for-each select="$bottomNotes"> + <xsl:variable name="noteSign"><xsl:value-of select="count(./preceding::*:note) + 1"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'noteBody foot'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:apply-templates/> + </span> + </span> + </xsl:for-each> + </span> + </xsl:if> +</xsl:template> + +<xsl:template match="*:head"> + <xsl:variable name="style" select="@style"/> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'head bf'"/></xsl:attribute> + <xsl:choose> + <xsl:when test="not(empty($style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="$style"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:when> + <xsl:otherwise> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </div> +</xsl:template> + +<xsl:template match="*:figure"> + <xsl:variable name="class"> + <xsl:choose> + <xsl:when test="empty(@position)"><xsl:value-of select="name()"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat(name(), ' ', string(@position))"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="image" select="*:image"/> + <xsl:variable name="caption" select="*:caption"/> + <xsl:variable name="description" select="*:description"/> + <xsl:variable name="variables" select="*:variables"/> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:if test="not(empty($image))"> + <xsl:variable name="file" select="$image/@file"/> + <xsl:variable name="pn" select="$image/@pn"/> + <xsl:variable name="wx" select="$image/@wx"/> + <xsl:variable name="wy" select="$image/@wy"/> + <xsl:variable name="ww" select="$image/@ww"/> + <xsl:variable name="wh" select="$image/@wh"/> + <xsl:variable name="digilibServiceName" select="'http://digilib.mpiwg-berlin.mpg.de/digitallibrary/jquery/digilib.html'"/> + <xsl:variable name="scalerServiceName" select="'http://digilib.mpiwg-berlin.mpg.de/digitallibrary/servlet/Scaler'"/> + <xsl:variable name="fileName"> + <xsl:choose> + <xsl:when test="empty($wx) and empty($file)"><xsl:value-of select="$echoFiguresDir"/></xsl:when> + <xsl:when test="empty($wx) and not(empty($file))"><xsl:value-of select="concat($echoFiguresDir, '/', $file)"/></xsl:when> + <xsl:when test="not(empty($wx)) and empty($file)"><xsl:value-of select="$echoPageImgDir"/></xsl:when> + <xsl:when test="not(empty($wx)) and not(empty($file))"><xsl:value-of select="concat($echoPageImgDir, '/', $file)"/></xsl:when> + <xsl:otherwise><xsl:value-of select="$echoFiguresDir"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="digilibUrl"> + <xsl:choose> + <xsl:when test="empty($pn) and empty($wx)"><xsl:value-of select="concat($digilibServiceName, '?fn=', $fileName)"/></xsl:when> + <xsl:when test="not(empty($pn)) and empty($wx)"><xsl:value-of select="concat($digilibServiceName, '?fn=', $fileName, '&pn=', $pn)"/></xsl:when> + <xsl:when test="empty($pn) and not(empty($wx))"><xsl:value-of select="concat($digilibServiceName, '?fn=', $fileName, '&wx=', $wx, '&wy=', $wy, '&ww=', $ww, '&wh=', $wh)"/></xsl:when> + <xsl:when test="not(empty($pn)) and not(empty($wx))"><xsl:value-of select="concat($digilibServiceName, '?fn=', $fileName, '&pn=', $pn, '&wx=', $wx, '&wy=', $wy, '&ww=', $ww, '&wh=', $wh)"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat($digilibServiceName, '?fn=', $fileName)"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="scalerUrl"> + <xsl:choose> + <xsl:when test="empty($pn) and empty($wx)"><xsl:value-of select="concat($scalerServiceName, '?fn=', $fileName, '&dh=200')"/></xsl:when> + <xsl:when test="not(empty($pn)) and empty($wx)"><xsl:value-of select="concat($scalerServiceName, '?fn=', $fileName, '&pn=', $pn, '&dh=200')"/></xsl:when> + <xsl:when test="empty($pn) and not(empty($wx))"><xsl:value-of select="concat($scalerServiceName, '?fn=', $fileName, '&wx=', $wx, '&wy=', $wy, '&ww=', $ww, '&wh=', $wh, '&dh=200')"/></xsl:when> + <xsl:when test="not(empty($pn)) and not(empty($wx))"><xsl:value-of select="concat($scalerServiceName, '?fn=', $fileName, '&pn=', $pn, '&wx=', $wx, '&wy=', $wy, '&ww=', $ww, '&wh=', $wh, '&dh=200')"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat($digilibServiceName, '?fn=', $fileName, '&dh=200')"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <a href="{$digilibUrl}"> + <img> + <xsl:attribute name="src"><xsl:value-of select="$scalerUrl"/></xsl:attribute> + </img> + </a> + </xsl:if> + <xsl:if test="not(empty(@number))"> + <xsl:variable name="type"><xsl:value-of select="'Figure'"/></xsl:variable> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNumber'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNum'"/></xsl:attribute> + <xsl:value-of select="string(@number)"/> + </span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNumText'"/></xsl:attribute> + <xsl:value-of select="concat('[', $type, ' ', string(@number), ']')"/> + </span> + </span> + </xsl:if> + <xsl:if test="not(empty($caption))"> + <xsl:for-each select="$caption"> + <span class="caption"><xsl:apply-templates/></span> + </xsl:for-each> + </xsl:if> + <xsl:if test="not(empty($description))"> + <xsl:for-each select="$description"> + <span class="description"><xsl:apply-templates/></span> + </xsl:for-each> + </xsl:if> + <xsl:if test="not(empty($variables))"> + <xsl:for-each select="$variables"> + <span class="variables"><xsl:apply-templates/></span> + </xsl:for-each> + </xsl:if> + </span> +</xsl:template> + +<!-- MathML --> +<xsl:template match="math:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="mml:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<!-- XHTML: remove the xhtml namespace --> +<xsl:template match="xhtml:*"> + <xsl:variable name="hasLabel" select="not(empty(@xhtml:label))"/> + <xsl:variable name="isTable" select="local-name() = 'table'"/> + <xsl:choose> + <xsl:when test="(not($hasLabel)) or ($isTable and $hasLabel)"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- SVG --> +<xsl:template match="svg:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:place|*:person"> + <xsl:element name="span"> + <xsl:attribute name="class"><xsl:value-of select="name()"/></xsl:attribute> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:div"> + <xsl:if test="node() or normalize-space()"> <!-- test: contains nodes or is not empty --> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'div'"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:if> +</xsl:template> + +<xsl:template match="*:p"> + <xsl:variable name="style" select="@style"/> + <xsl:variable name="class" select="'p'"/> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:choose> + <xsl:when test="not(empty($style))"> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:when> + <xsl:otherwise> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </div> +</xsl:template> + +<xsl:template match="*:pb"> + <xsl:variable name="number"> + <xsl:choose> + <xsl:when test="not(empty(@n))"><xsl:value-of select="@n"/></xsl:when> + <xsl:otherwise><xsl:value-of select="count(./preceding::*:pb) + 1"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="file"> + <xsl:choose> + <xsl:when test="not(empty(@file))"><xsl:value-of select="@file"/></xsl:when> + <xsl:otherwise><xsl:value-of select="$number"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'pb'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'src'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="concat('pageimg/', string($file))"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'n'"/></xsl:attribute> + <xsl:value-of select="$number"/> + </span> + <xsl:if test="not(empty(@o))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="'o'"/></xsl:attribute> + <xsl:value-of select="@o"/> + </span> + </xsl:if> + <xsl:if test="not(empty(@rhead))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="'rhead'"/></xsl:attribute> + <xsl:value-of select="@rhead"/> + </span> + </xsl:if> + </span> + </span> +</xsl:template> + +<xsl:template match="*:lb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'lb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="*:cb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'cb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="*:expan"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'expan'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:note"> + <xsl:variable name="place"><xsl:value-of select="string(@position)"/></xsl:variable> + <xsl:variable name="noteSign"><xsl:value-of select="count(./preceding::*:note) + 1"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:if test="$place != '' and $place != 'foot'"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('noteBody ', $place)"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:apply-templates/> + </span> + </xsl:if> + </span> +</xsl:template> + +<xsl:template match="*:handwritten"> + <xsl:variable name="noteSign"><xsl:value-of select="string(@number)"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note handwritten'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'noteBody'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'noteNumText'"/></xsl:attribute> + <xsl:value-of select="concat('[Handwritten note ', string(@number), ']')"/> + </span> + </span> + </span> +</xsl:template> + +<xsl:template match="*:emph"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'emph'"/></xsl:attribute> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="string(@style)"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:when> + <xsl:otherwise> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </span> +</xsl:template> + +<xsl:template match="*:foreign"> + <xsl:variable name="lang" select="@lang"/> + <xsl:variable name="xmllang" select="@xml:lang"/> + <xsl:variable name="language"> + <xsl:choose> + <xsl:when test="not(empty($xmllang))"><xsl:value-of select="$xmllang"/></xsl:when> + <xsl:when test="not(empty($lang))"><xsl:value-of select="$lang"/></xsl:when> + <xsl:otherwise><xsl:value-of select="''"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="class"> + <xsl:choose> + <xsl:when test="$language = ''"><xsl:value-of select="'foreign'"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat('foreign ', $language)"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:q"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'q'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:quote"> + <span class="quote"> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:blockquote"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'blockquote'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:set-off"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'set-off'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:reg"> + <span class="reg"> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'reg'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:var"> + <xsl:variable name="class"> + <xsl:choose> + <xsl:when test="empty(@type)"><xsl:value-of select="'var'"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat('var ', string(@type))"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:num"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'num'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:gap"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'gap'"/></xsl:attribute> + <xsl:if test="not(empty(@extent))"><span class="extent"><xsl:value-of select="string(@extent)"/></span></xsl:if> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:s"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'s'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- words --> +<xsl:template match="*:w"> + <xsl:variable name="wordLanguage" select="@lang"/> + <xsl:variable name="form" select="encode-for-uri(string(@form))"/> + <xsl:variable name="dictionary" select="string(@dictionary)"/> + <xsl:variable name="displayWordOrig"> + <xsl:choose> + <xsl:when test="not(empty(*:orig))"><xsl:apply-templates select="*:orig/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordReg"> + <xsl:choose> + <xsl:when test="not(empty(*:reg))"><xsl:apply-templates select="*:reg/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordNorm"> + <xsl:choose> + <xsl:when test="not(empty(*:norm))"><xsl:apply-templates select="*:norm/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordUrlEncoded" select="encode-for-uri($displayWordOrig)"/> + <xsl:variable name="dictionaryPart"> + <xsl:choose> + <xsl:when test="$dictionary = 'true'"> + <a class="dictionary"> + <xsl:attribute name="href"><xsl:value-of select="concat($dictionaryServiceName, '?query=', $form, '&queryDisplay=', $displayWordUrlEncoded, '&language=', $wordLanguage, '&outputFormat=html', '&outputType=morphCompact&outputType=dictFull')"/></xsl:attribute> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </a> + </xsl:when> + <xsl:otherwise> + <span class="dictionary"> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span class="w"> + <xsl:sequence select="$dictionaryPart"/> + <span class="nodictionary orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="nodictionary reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="nodictionary norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> +</xsl:template> + +<xsl:template match="*:hi"> + <xsl:choose> + <xsl:when test="@type = 'elem'"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/pageEchoSpan.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,112 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xs="http://www.w3.org/2001/XMLSchema" + xmlns:functx="http://www.functx.com" + xmlns:saxon="http://saxon.sf.net/" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:mml="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xs functx saxon dc dcterms echo math mml svg xhtml" + > + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="mode"></xsl:param> +<xsl:param name="normalization"></xsl:param> + +<xsl:variable name="dictionaryServiceName" select="'http://mpdl-service.mpiwg-berlin.mpg.de/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries'"/> + +<xsl:template match="*:echo"> + <span class="page"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="*:head|*:div|*:p|*:s|*:pb|*:cb|*:expan|*:emph|*:quote|*:q|*:blockquote|*:set-off|*:reg|*:var|*:num|*:gap|*:anchor|*:note|*:figure|*:caption|*:description|*:handwritten|*:place|*:person" mode="text"> + <xsl:element name="span"> + <xsl:attribute name="class"><xsl:value-of select="name()"/></xsl:attribute> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> +</xsl:template> + +<!-- MathML --> +<xsl:template match="math:*" mode="text"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> +</xsl:template> + +<!-- SVG --> +<xsl:template match="svg:*" mode="text"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:lb" mode="text"> + <br/><xsl:apply-templates mode="text"/> +</xsl:template> + +<!-- XHTML: remove the xhtml namespace --> +<xsl:template match="xhtml:*" mode="text"> + <xsl:variable name="hasLabel" select="string(@xlink:label) != ''"/> + <xsl:variable name="isTable" select="name() = 'table'"/> + <xsl:choose> + <xsl:when test="(not($hasLabel)) or ($isTable and $hasLabel)"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> +</xsl:template> + + +<!-- words --> +<xsl:template match="*:w" mode="text"> + <xsl:variable name="wordLanguage" select="@lang"/> + <xsl:variable name="form" select="encode-for-uri(string(@form))"/> + <xsl:variable name="formNotUrlEncoded" select="string(@form)"/> + <xsl:variable name="formRegularized" select="string(@formRegularized)"/> + <xsl:variable name="formNormalized" select="string(@formNormalized)"/> + <xsl:variable name="lemmas" select="string(@lemmas)"/> + <xsl:variable name="dictionary" select="string(@dictionary)"/> + <xsl:variable name="displayWord"> + <xsl:choose> + <xsl:when test="$normalization = 'orig'"><xsl:apply-templates mode="text"/></xsl:when> + <xsl:when test="$normalization = 'reg' and $formRegularized = ''"><xsl:apply-templates mode="text"/></xsl:when> + <xsl:when test="$normalization = 'reg' and $formRegularized != ''"><xsl:sequence select="$formRegularized"/></xsl:when> + <xsl:when test="$normalization = 'norm'"><xsl:apply-templates mode="text"/></xsl:when> + <xsl:otherwise><xsl:apply-templates mode="text"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordUrlEncoded" select="encode-for-uri($displayWord)"/> + <xsl:choose> + <xsl:when test="$dictionary = 'true' and $mode = 'tokenized'"> + <span class="w"> + <xsl:copy-of select="@*"/> + <xsl:attribute name="href"><xsl:value-of select="concat($dictionaryServiceName, '?query=', $form, '&queryDisplay=', $displayWordUrlEncoded, '&language=', $wordLanguage, '&outputFormat=html', '&outputType=morphCompact&outputType=dictFull')"/></xsl:attribute> + <xsl:sequence select="$displayWord"/> + </span> + </xsl:when> + <xsl:otherwise> + <xsl:sequence select="$displayWord"/> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="text()" mode="text"> + <xsl:value-of select="."/> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/pageTei.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,650 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xs="http://www.w3.org/2001/XMLSchema" + xmlns:functx="http://www.functx.com" + xmlns:saxon="http://saxon.sf.net/" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xs functx saxon dc dcterms math svg xhtml" + > + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="echoFiguresDir"></xsl:param> +<xsl:param name="echoPageImgDir"></xsl:param> + +<xsl:variable name="dictionaryServiceName" select="'http://mpdl-service.mpiwg-berlin.mpg.de/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries'"/> + +<xsl:template match="*:TEI"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'TEI'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:expan|*:emph|*:q|*:quote|*:reg|*:num"> + <xsl:element name="span"> + <xsl:attribute name="class"><xsl:value-of select="name()"/></xsl:attribute> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:teiHeader"> + <!-- do nothing --> +</xsl:template> + +<xsl:template match="*:text"> + <xsl:variable name="contentStr" select="normalize-space(string(.))"/> + <xsl:variable name="figures" select=".//*:figure|.//*:handwritten"/> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'text'"/></xsl:attribute> + <xsl:apply-templates/> + <xsl:if test="$contentStr = '' and empty($figures)"> <!-- test if div only contains empty nodes but no figures --> + <div> + <xsl:attribute name="class"><xsl:value-of select="'emptyPage'"/></xsl:attribute> + <xsl:value-of select="'[Empty page]'"/> + </div> + </xsl:if> + </div> + <!-- Dictionary --> + <xsl:if test="not(empty(//*:entry))"> + <xsl:for-each select="//*:entry"> + <xsl:sort select="*:form/*:orth"/> + <xsl:variable name="position" select="position()"/> + <span class="entry"> + <xsl:apply-templates select="*:form"/> + <xsl:apply-templates select="*:sense"/> + <xsl:if test="not(empty(*:figure))"> + <span class="entryDiv"> + <span class="bf"><xsl:value-of select="'Figures: '"/></span> + <span class="entryDiv"> + <xsl:for-each select="*:figure"> + <xsl:variable name="href" select="*:graphic/@url"/> + <xsl:variable name="figDesc" select="string(*:figDesc)"/> + <xsl:if test="$href != ''"> + <div class="figure" style="margin-left:10px;"> + <a href="{$href}"><img alt="Figure: {$figDesc}" src="{$href}" width="200" height="200"/></a> + <br/> + <xsl:value-of select="'[Figure]: '"/><xsl:apply-templates select="*:head"/> + </div> + </xsl:if> + </xsl:for-each> + </span> + </span> + </xsl:if> + <xsl:if test="not(empty(*:xr))"> + <span class="entryDiv"> + <span class="bf"><xsl:value-of select="'References: '"/></span> + <span class="entryDiv"> + <xsl:for-each select="*:xr/*:ref"> + <li><xsl:apply-templates select="."/></li> + </xsl:for-each> + </span> + </span> + </xsl:if> + </span> + </xsl:for-each> + </xsl:if> + <!-- Notes --> + <xsl:variable name="bottomNotes" select="//*:note[contains(@place, 'foot') or empty(string(@place))]"/> + <xsl:variable name="countBottomNotes" select="count($bottomNotes)"/> + <xsl:if test="$countBottomNotes > 0"> + <span class="notes"> + <xsl:for-each select="$bottomNotes"> + <xsl:variable name="noteSign"><xsl:value-of select="count(./preceding::*:note) + 1"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'noteBody foot'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:apply-templates/> + </span> + </span> + </xsl:for-each> + </span> + </xsl:if> +</xsl:template> + +<xsl:template match="*:div"> + <xsl:if test="node() or normalize-space()"> <!-- test: contains nodes or is not empty --> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'div'"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:if> +</xsl:template> + +<xsl:template match="*:p"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'p'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:pb"> + <xsl:variable name="number"> + <xsl:choose> + <xsl:when test="not(empty(@n))"><xsl:value-of select="@n"/></xsl:when> + <xsl:otherwise><xsl:value-of select="count(./preceding::*:pb) + 1"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="file"> + <xsl:choose> + <xsl:when test="not(empty(@facs))"><xsl:value-of select="@facs"/></xsl:when> + <xsl:otherwise><xsl:value-of select="$number"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'pb'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'src'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="concat('pageimg/', string($file))"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'n'"/></xsl:attribute> + <xsl:value-of select="$number"/> + </span> + </span> + </span> +</xsl:template> + +<xsl:template match="*:head"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'head bf'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:figure"> + <span class="figure"> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:variable name="url"> + <xsl:choose> + <xsl:when test="not(empty(@facs))"><xsl:value-of select="string(@facs)"/></xsl:when> + <xsl:otherwise><xsl:value-of select="string(graphic/@url)"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="fileName" select="concat($echoFiguresDir, '/', $url)"/> + <xsl:variable name="digilibUrl"> + <xsl:choose> + <xsl:when test="starts-with($url, 'http')"><xsl:value-of select="$url"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat('http://digilib.mpiwg-berlin.mpg.de/digitallibrary/jquery/digilib.html?fn=', $fileName)"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="scalerUrl"> + <xsl:choose> + <xsl:when test="starts-with($url, 'http')"><xsl:value-of select="$url"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat('http://digilib.mpiwg-berlin.mpg.de/digitallibrary/servlet/Scaler?fn=', $fileName, '&dh=200&dw=200')"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <a href="{$digilibUrl}"> + <img> + <xsl:attribute name="src"><xsl:value-of select="$scalerUrl"/></xsl:attribute> + </img> + </a> + <xsl:if test="not(empty(@number))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNumber'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNum'"/></xsl:attribute> + <xsl:value-of select="string(@number)"/> + </span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNumText'"/></xsl:attribute> + <xsl:value-of select="concat('[', 'Figure', ' ', string(@number), ']')"/> + </span> + </span> + </xsl:if> + <xsl:if test="not(empty(*:head))"> + <span class="caption"> + <xsl:for-each select="*:head"> + <xsl:apply-templates/> + </xsl:for-each> + </span> + </xsl:if> + </span> +</xsl:template> + +<!-- segmentation --> +<xsl:template match="*:seg"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:choose> + <xsl:when test="@rend = 'highlight'"> + <xsl:attribute name="class"><xsl:value-of select="'seg highlight'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="@rend = 'highlightPoint'"> + <xsl:attribute name="class"><xsl:value-of select="'seg highlightPoint'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:otherwise> + <xsl:attribute name="class"><xsl:value-of select="'seg'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </span> +</xsl:template> + +<!-- choice --> +<xsl:template match="*:choice"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'choice'"/></xsl:attribute> + <xsl:if test="not(empty(*:orig))"> + <xsl:apply-templates select="*:orig"/> + </xsl:if> + <xsl:if test="not(empty(*:abbr))"> + <xsl:apply-templates select="*:abbr"/> + </xsl:if> + <xsl:if test="not(empty(*:am))"> + <xsl:apply-templates select="*:am"/> + </xsl:if> + <xsl:if test="not(empty(*:sic))"> + <xsl:apply-templates select="*:sic"/> + </xsl:if> + </span> +</xsl:template> + +<!-- hi (highlighted) --> +<xsl:template match="*:hi"> + <xsl:choose> + <xsl:when test="not(empty(@type)) and @type = 'elem'"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:when> + <xsl:when test="not(empty(@type)) and @type != 'elem'"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:when> + <xsl:when test="@rend = 'initial'"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'initial'"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:when> + <xsl:when test="@rend = 'bold'"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'bf'"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="@rend"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- name (of type: place, person, ...) --> +<xsl:template match="*:name"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:choose> + <xsl:when test="@type = 'place'"> + <xsl:attribute name="class"><xsl:value-of select="'place'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="'Place'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="@type = 'person'"> + <xsl:attribute name="class"><xsl:value-of select="'person'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="'Person'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="@type = 'org'"> + <xsl:attribute name="class"><xsl:value-of select="'organization'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="'Organization'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:otherwise> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="@type"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </span> +</xsl:template> + +<!-- place, person, ...) --> +<xsl:template match="*:placeName"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'place'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="concat('Place: ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> +<xsl:template match="*:persName"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'person'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="'Person'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- term --> +<xsl:template match="*:term"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'term'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="'Terminology entry'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- line --> +<xsl:template match="*:lg"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'lg'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:l"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'l'"/></xsl:attribute> + <xsl:apply-templates/> + </span> + <br/> +</xsl:template> + +<!-- reference --> +<xsl:template match="*:ref"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'ref'"/></xsl:attribute> + <xsl:if test="not(empty(@target))"> + <a class="ref"> + <xsl:attribute name="href"><xsl:value-of select="@target"/></xsl:attribute> + <xsl:apply-templates/> + </a> + </xsl:if> + <xsl:if test="empty(@target)"> + <xsl:apply-templates/> + </xsl:if> + </span> +</xsl:template> + +<!-- table --> +<xsl:template match="*:table"> + <table> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty(head))"> + <caption align="top"><xsl:apply-templates select="*:head"/></caption> + </xsl:if> + <xsl:apply-templates select="*:row"/> + </table> +</xsl:template> + +<xsl:template match="*:row"> + <tr> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:choose> + <xsl:when test="@role = 'label'"> + <xsl:attribute name="style"><xsl:value-of select="'font-weight:bold;'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="@role = 'data' or empty(@role)"> + <xsl:apply-templates/> + </xsl:when> + <xsl:otherwise> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </tr> +</xsl:template> + +<xsl:template match="*:cell"> + <td> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:choose> + <xsl:when test="@role = 'label' and empty(@cols)"> + <xsl:attribute name="style"><xsl:value-of select="'font-weight:bold;'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="@role = 'label' and not(empty(@cols))"> + <xsl:attribute name="colspan"><xsl:value-of select="@cols"/></xsl:attribute> + <xsl:attribute name="style"><xsl:value-of select="'font-weight:bold;'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="(@role = 'data' or empty(@role)) and empty(@cols)"> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="(@role = 'data' or empty(@role)) and not(empty(@cols))"> + <xsl:attribute name="colspan"><xsl:value-of select="@cols"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:otherwise> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </td> +</xsl:template> + +<!-- dictionary --> +<xsl:template match="*:entry"> + <!-- empty: handled in text tag --> +</xsl:template> + +<xsl:template match="*:form"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'form'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:orth"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'orth'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:sense"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'sense'"/></xsl:attribute> + <span class="entryDiv"><xsl:apply-templates select="*:def"/></span> + <span class="entryDiv"><xsl:apply-templates select="*:etym"/></span> + </span> +</xsl:template> + +<xsl:template match="*:def"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'def'"/></xsl:attribute> + <span class="bf"><xsl:value-of select="'Definition: '"/></span> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:mentioned"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'mentioned'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:etym"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'etym'"/></xsl:attribute> + <span class="bf"><xsl:value-of select="'Etymology: '"/></span> + <span class="entryDiv"> + <xsl:for-each select="*:cit"> + <li><xsl:apply-templates select="*:quote"/><xsl:value-of select="' ('"/><xsl:apply-templates select="*:def"/><xsl:value-of select="')'"/></li> + </xsl:for-each> + </span> + </span> +</xsl:template> + +<xsl:template match="*:cit"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'cit'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- MathML --> +<xsl:template match="math:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<!-- SVG --> +<xsl:template match="svg:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:lb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'lb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="*:cb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'cb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="*:note"> + <xsl:variable name="place"><xsl:value-of select="string(@place)"/></xsl:variable> + <xsl:variable name="noteSign"><xsl:value-of select="count(./preceding::*:note) + 1"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:if test="$place != '' and $place != 'foot'"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('noteBody ', $place)"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:apply-templates/> + </span> + </xsl:if> + </span> +</xsl:template> + +<xsl:template match="*:foreign"> + <xsl:variable name="xmllang" select="@xml:lang"/> + <xsl:variable name="language"> + <xsl:choose> + <xsl:when test="not(empty($xmllang))"><xsl:value-of select="$xmllang"/></xsl:when> + <xsl:otherwise><xsl:value-of select="''"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat('foreign ', $language)"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:gap"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'gap'"/></xsl:attribute> + <xsl:if test="not(empty(@extent))"><span class="extent"><xsl:value-of select="string(@extent)"/></span></xsl:if> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:s"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'s'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- words --> +<xsl:template match="*:w"> + <xsl:variable name="wordLanguage" select="@lang"/> + <xsl:variable name="form" select="encode-for-uri(string(@form))"/> + <xsl:variable name="dictionary" select="string(@dictionary)"/> + <xsl:variable name="displayWordOrig"> + <xsl:choose> + <xsl:when test="not(empty(*:orig))"><xsl:apply-templates select="*:orig/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordReg"> + <xsl:choose> + <xsl:when test="not(empty(*:reg))"><xsl:apply-templates select="*:reg/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordNorm"> + <xsl:choose> + <xsl:when test="not(empty(*:norm))"><xsl:apply-templates select="*:norm/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordUrlEncoded" select="encode-for-uri($displayWordOrig)"/> + <xsl:variable name="dictionaryPart"> + <xsl:choose> + <xsl:when test="$dictionary = 'true'"> + <a class="dictionary"> + <xsl:attribute name="href"><xsl:value-of select="concat($dictionaryServiceName, '?query=', $form, '&queryDisplay=', $displayWordUrlEncoded, '&language=', $wordLanguage, '&outputFormat=html', '&outputType=morphCompact&outputType=dictFull')"/></xsl:attribute> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </a> + </xsl:when> + <xsl:otherwise> + <span class="dictionary"> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span class="w"> + <xsl:sequence select="$dictionaryPart"/> + <span class="nodictionary orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="nodictionary reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="nodictionary norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/pageXhtml.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,260 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xs="http://www.w3.org/2001/XMLSchema" + xmlns:functx="http://www.functx.com" + xmlns:saxon="http://saxon.sf.net/" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xs functx saxon dc dcterms math svg xhtml" + > + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="echoFiguresDir"></xsl:param> +<xsl:param name="echoPageImgDir"></xsl:param> + +<xsl:variable name="dictionaryServiceName" select="'http://mpdl-service.mpiwg-berlin.mpg.de/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries'"/> + +<xsl:template match="*:html"> + <xsl:apply-templates/> +</xsl:template> + +<xsl:template match="body"> + <xsl:apply-templates mode="html"/> +</xsl:template> + +<!-- html mode: transform html texts --> +<xsl:template match="@*|node()" mode="html"> + <xsl:copy> + <xsl:apply-templates select="@*|node()" mode="html"/> + </xsl:copy> +</xsl:template> + +<xsl:template match="s" mode="html"> + <xsl:apply-templates select="."/> +</xsl:template> + +<xsl:template match="w" mode="html"> + <xsl:apply-templates select="."/> +</xsl:template> + + +<xsl:template match="*:expan|*:emph|*:q|*:quote|*:blockquote|*:set-off|*:reg|*:var|*:num|*:figure"> + <span> + <xsl:attribute name="class"><xsl:value-of select="name()"/></xsl:attribute> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:div"> + <xsl:if test="node() or normalize-space()"> <!-- test: contains nodes or is not empty --> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'div'"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:if> +</xsl:template> + +<xsl:template match="*:p"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'p'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="pb"> + <xsl:variable name="number" select="count(./preceding::*:pb) + 1"/> + <xsl:variable name="file" select="$number"/> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'pb'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'src'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="concat('pageimg/', string($file))"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'n'"/></xsl:attribute> + <xsl:value-of select="$number"/> + </span> + </span> + </span> +</xsl:template> + +<xsl:template match="*:head"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'head bf'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<!-- MathML --> +<xsl:template match="math:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<!-- SVG --> +<xsl:template match="svg:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:lb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'lb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="*:cb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'cb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="br"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'br'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="note"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="foreign"> + <xsl:variable name="lang" select="@lang"/> + <xsl:variable name="xmllang" select="@xml:lang"/> + <xsl:variable name="language"> + <xsl:choose> + <xsl:when test="not(empty($xmllang))"><xsl:value-of select="$xmllang"/></xsl:when> + <xsl:when test="not(empty($lang))"><xsl:value-of select="$lang"/></xsl:when> + <xsl:otherwise><xsl:value-of select="''"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat('foreign ', $language)"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="gap"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'gap'"/></xsl:attribute> + <xsl:if test="not(empty(@extent))"><span class="extent"><xsl:value-of select="string(@extent)"/></span></xsl:if> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- XHTML: remove the xhtml namespace --> +<xsl:template match="xhtml:*"> + <xsl:variable name="hasLabel" select="string(@xlink:label) != ''"/> + <xsl:variable name="isTable" select="name() = 'table'"/> + <xsl:choose> + <xsl:when test="(not($hasLabel)) or ($isTable and $hasLabel)"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="s"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'s'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- words --> +<xsl:template match="*:w"> + <xsl:variable name="wordLanguage" select="@lang"/> + <xsl:variable name="form" select="encode-for-uri(string(@form))"/> + <xsl:variable name="dictionary" select="string(@dictionary)"/> + <xsl:variable name="displayWordOrig"> + <xsl:choose> + <xsl:when test="not(empty(*:orig))"><xsl:apply-templates select="*:orig/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordReg"> + <xsl:choose> + <xsl:when test="not(empty(*:reg))"><xsl:apply-templates select="*:reg/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordNorm"> + <xsl:choose> + <xsl:when test="not(empty(*:norm))"><xsl:apply-templates select="*:norm/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordUrlEncoded" select="encode-for-uri($displayWordOrig)"/> + <xsl:variable name="dictionaryPart"> + <xsl:choose> + <xsl:when test="$dictionary = 'true'"> + <a class="dictionary"> + <xsl:attribute name="href"><xsl:value-of select="concat($dictionaryServiceName, '?query=', $form, '&queryDisplay=', $displayWordUrlEncoded, '&language=', $wordLanguage, '&outputFormat=html', '&outputType=morphCompact&outputType=dictFull')"/></xsl:attribute> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </a> + </xsl:when> + <xsl:otherwise> + <span class="dictionary"> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span class="w"> + <xsl:sequence select="$dictionaryPart"/> + <span class="nodictionary orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="nodictionary reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="nodictionary norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> +</xsl:template> + +<xsl:template match="*:hi"> + <xsl:choose> + <xsl:when test="@type = 'elem'"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/pageXml.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,86 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink dcterms math svg xhtml" + > + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="echoFiguresDir"></xsl:param> +<xsl:param name="echoPageImgDir"></xsl:param> + +<!-- transform to browser like xml display --> +<xsl:template match="element()"> + <xsl:variable name="elementName" select="name()"/> + <xsl:variable name="elementPresentation"> + <xsl:choose> + <xsl:when test="element() = node() or text() != ''"> + <xsl:value-of select="'<'"/> + <span class="xml elementName"><xsl:value-of select="$elementName"/></span> + <xsl:apply-templates select="attribute()"/> + <xsl:value-of select="'>'"/> + <xsl:apply-templates select="element()|text()|comment()|processing-instruction()"/> + <xsl:value-of select="'</'"/> + <span class="xml elementName"><xsl:value-of select="$elementName"/></span> + <xsl:value-of select="'>'"/> + </xsl:when> + <xsl:otherwise> + <xsl:value-of select="'<'"/> + <span class="xml elementName"><xsl:value-of select="$elementName"/></span> + <xsl:apply-templates select="attribute()"/> + <xsl:value-of select="'/>'"/> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <ul class="xml element"> + <xsl:sequence select="$elementPresentation"/> + </ul> +</xsl:template> + +<xsl:template match="attribute()"> + <xsl:variable name="attributeName" select="name()"/> + <span class="xml attributeName"> + <xsl:value-of select="' '"/> + <xsl:value-of select="$attributeName"/> + </span> + <xsl:value-of select="'="'"/> + <span class="xml attributeValue"><xsl:value-of select="."/></span><xsl:value-of select="'"'"/> +</xsl:template> + +<xsl:template match="comment()"> + <span class="xml comment"> + <xsl:value-of select="'<!-- '"/><xsl:value-of select="."/><xsl:value-of select="' -->'"/> + </span> +</xsl:template> + +<xsl:template match="processing-instruction()"> +</xsl:template> + +<xsl:template match="*:w"> + <xsl:apply-templates select="*:orig/node()"/> +</xsl:template> + +<xsl:template match="*:hi"> + <xsl:choose> + <xsl:when test="@type = 'elem'"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/queryDocument.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,93 @@ +<?xml version="1.0"?> + +<xsl:stylesheet version="2.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform"> + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="query"></xsl:param> +<xsl:param name="flags"></xsl:param> +<xsl:param name="outputFormat"></xsl:param> + +<xsl:variable name="apos">'</xsl:variable> +<xsl:variable name="xpathQuery"> + <xsl:choose> + <xsl:when test="$flags = ''"><xsl:value-of select="concat('//*:s[matches(.,', $apos, $query, $apos, ')]', '|', '//*:head[matches(.,', $apos, $query, $apos, ')]')"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat('//*:s[matches(.,', $apos, $query, $apos, ',', $apos, $flags, $apos, ')]', '|', '//*:head[matches(.,', $apos, $query, $apos, ',', $apos, $flags, $apos, ')]')"/></xsl:otherwise> + </xsl:choose> +</xsl:variable> +<xsl:variable name="queryResult" select="saxon:evaluate($xpathQuery)" xmlns:saxon="http://saxon.sf.net/"/> +<xsl:variable name="queryResultSize"><xsl:value-of select="count($queryResult)"/></xsl:variable> +<xsl:variable name="queryResultPageSize" select="20"/> +<xsl:variable name="queryResultPages"> + <xsl:choose> + <xsl:when test="$queryResultSize = 0"><xsl:value-of select="count($queryResult)"/></xsl:when> + <xsl:otherwise><xsl:value-of select="$queryResultSize idiv $queryResultPageSize + 1"/></xsl:otherwise> + </xsl:choose> +</xsl:variable> +<xsl:variable name="pageBreaks" select="saxon:evaluate('//*:pb')" xmlns:saxon="http://saxon.sf.net/"/> +<xsl:variable name="queryResultHtml"> +<xsl:choose> + <xsl:when test="$outputFormat = 'html'"> + <table> + <thead> + <tr> + <th align="left" valign="top"> + <button name="order-by" value="author" style="padding:0px;font-weight:bold;font-size:14px;background:none;border:none;">No.</button> + </th> + <th align="left" valign="top"> + <button name="order-by" value="author" style="padding:0px;font-weight:bold;font-size:14px;background:none;border:none;">Path</button> + </th> + <th align="left" valign="top"> + <button name="order-by" value="author" style="padding:0px;font-weight:bold;font-size:14px;background:none;border:none;">Hit</button> + </th> + <th align="left" valign="top"> + <button name="order-by" value="author" style="padding:0px;font-weight:bold;font-size:14px;background:none;border:none;">Page</button> + </th> + </tr> + </thead> + <tbody> + <xsl:for-each select="$queryResult"> + <xsl:variable name="hit" select="."/> + <tr> + <td align="left" valign="top"><xsl:value-of select="position()"/></td> + <td align="left" valign="top"><xsl:value-of select="saxon:path(.)" xmlns:saxon="http://saxon.sf.net/"/></td> + <td align="left" valign="top"><xsl:value-of select="."/></td> + <td align="left" valign="top"><xsl:value-of select="count(./preceding::*:pb)"/></td> <!-- better performance: count($pageBreaks[. << $hit]) --> + </tr> + </xsl:for-each> + </tbody> + </table> + </xsl:when> + <xsl:otherwise> + <result> + <query><xsl:value-of select="$query"/></query> + <flags><xsl:value-of select="$flags"/></flags> + <size><xsl:value-of select="$queryResultSize"/></size> + <page-size><xsl:value-of select="$queryResultPageSize"/></page-size> + <pages><xsl:value-of select="$queryResultPages"/></pages> + <pn>1</pn> + <hits> + <xsl:for-each select="$queryResult"> + <xsl:variable name="hit" select="."/> + <hit> + <hitType><xsl:value-of select="'s'"/></hitType> + <pos><xsl:value-of select="position()"/></pos> + <pn><xsl:value-of select="count(./preceding::*:pb)"/></pn> + <hitId>xmlId</hitId> + <hitPos>4711</hitPos> + <hitString><xsl:value-of select="."/></hitString> + <hitSurroundsPB>false</hitSurroundsPB> + </hit> + </xsl:for-each> + </hits> + <query-forms></query-forms> + <query-regularizations></query-regularizations> + </result> + </xsl:otherwise> +</xsl:choose> +</xsl:variable> +<xsl:template match="/"> + <xsl:sequence select="$queryResultHtml"/> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/replaceAnchor.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,82 @@ +<?xml version="1.0"?> + +<xsl:stylesheet + version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xhtml" +> + +<xsl:output method="xml" encoding="utf-8"/> + +<xsl:template match="@*|node()"> + <xsl:copy> + <xsl:apply-templates select="@*|node()"/> + </xsl:copy> +</xsl:template> + +<!-- insert figure number --> +<xsl:template match="*:figure"> + <xsl:variable name="number" select="count(./preceding::*:figure) + 1"/> + <xsl:element name="{name()}" namespace="{namespace-uri()}"><!-- remove namespace in element and its subnodes --> + <xsl:apply-templates select="@*"/> + <xsl:attribute name="number"><xsl:value-of select="$number"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<!-- insert figure number --> +<xsl:template match="*:handwritten"> + <xsl:variable name="number" select="count(./preceding::*:handwritten) + 1"/> + <xsl:element name="{name()}" namespace="{namespace-uri()}"><!-- remove namespace in element and its subnodes --> + <xsl:apply-templates select="@*"/> + <xsl:attribute name="number"><xsl:value-of select="$number"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:anchor"> + <xsl:variable name="type" select="@type"/> + <xsl:variable name="href" select="@xlink:href"/> + <xsl:choose> + <xsl:when test="$type = 'figure'"> + <xsl:variable name="figure" select="//*:figure[@xlink:label = $href]"/> + <xsl:apply-templates select="$figure"/> + </xsl:when> + <xsl:when test="$type = 'handwritten'"> + <xsl:variable name="handwritten" select="//*:handwritten[@xlink:label = $href]"/> + <xsl:apply-templates select="$handwritten"/> + </xsl:when> + <xsl:when test="$type = 'note'"> + <xsl:variable name="note" select="//*:note[@xlink:label = $href]"/> + <xsl:sequence select="$note"/> + </xsl:when> + <xsl:when test="$type = 'table'"> + <xsl:variable name="table" select="//xhtml:table[@xlink:label = $href]"/> + <xsl:sequence select="$table"/> + </xsl:when> + <xsl:otherwise> + <xsl:copy> + <xsl:apply-templates select="@*|node()"/> + </xsl:copy> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- remove the anchored divs: div which contain a labeled figure, note, handwritten or table --> +<xsl:template match="*:div"> + <xsl:variable name="containingElems" select="*:figure|*:note|*:handwritten|xhtml:table"/> + <xsl:variable name="hasLabel" select="not(empty($containingElems[1]/@xlink:label))"/> + <xsl:choose> + <xsl:when test="$hasLabel"></xsl:when> + <xsl:otherwise> + <xsl:copy> + <xsl:apply-templates select="@*|node()"/> + </xsl:copy> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/toc.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,152 @@ +<?xml version="1.0"?> + +<xsl:stylesheet + version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xhtml" +> + +<xsl:output method="xml" encoding="utf-8" indent="yes"/> + +<xsl:template match="/"> + <xsl:variable name="toc" select="//*:div[@type = 'section' or @type = 'chapter' or @type = 'part']"/> + <xsl:variable name="figures" select="//*:figure"/> + <xsl:variable name="notes" select="//*:note"/> + <xsl:variable name="notesHandwritten" select="//*:handwritten"/> + <xsl:variable name="pages" select="//*:pb"/> + <xsl:variable name="places" select="//*:place"/> + <list> + <xsl:if test="not(empty($toc))"> + <list type ="toc"> + <xsl:attribute name="count"><xsl:value-of select="count($toc)"/></xsl:attribute> + <head>Table of contents</head> + <xsl:apply-templates select="$toc"/> + </list> + </xsl:if> + <xsl:if test="not(empty($figures))"> + <list type="figures"> + <xsl:attribute name="count"><xsl:value-of select="count($figures)"/></xsl:attribute> + <head>Figures</head> + <xsl:apply-templates select="$figures"/> + </list> + <xsl:if test="not(empty($notes))"> + <list type="notes"> + <xsl:attribute name="count"><xsl:value-of select="count($notes)"/></xsl:attribute> + <head>Notes</head> + <xsl:apply-templates select="$notes"/> + </list> + </xsl:if> + <xsl:if test="not(empty($notesHandwritten))"> + <list type="notesHandwritten"> + <xsl:attribute name="count"><xsl:value-of select="count($notesHandwritten)"/></xsl:attribute> + <head>Handwritten notes</head> + <xsl:apply-templates select="$notesHandwritten"/> + </list> + </xsl:if> + <xsl:if test="not(empty($pages))"> + <list type="pages"> + <xsl:attribute name="count"><xsl:value-of select="count($pages)"/></xsl:attribute> + <head>Pages</head> + <xsl:apply-templates select="$pages"/> + </list> + </xsl:if> + <xsl:if test="not(empty($places))"> + <list type="places"> + <xsl:attribute name="count"><xsl:value-of select="count($places)"/></xsl:attribute> + <head>Places</head> + <xsl:apply-templates select="$places"/> + </list> + </xsl:if> + </list> +</xsl:template> + +<xsl:template match="*:figure"> + <xsl:variable name="page" select="count(./preceding::*:pb)"/> + <xsl:variable name="number" select="@number"/> + <item> + <xsl:if test="not(empty($number))"><xsl:attribute name="n"><xsl:value-of select="$number"/></xsl:attribute></xsl:if> + <xsl:apply-templates select="*:caption"/> + <xsl:apply-templates select="*:description"/> + <xsl:apply-templates select="*:variables"/> + <xsl:if test="not(empty($page))"><ref><xsl:value-of select="$page"/></ref></xsl:if> + </item> +</xsl:template> + +<xsl:template match="*:caption"><xsl:value-of select="' '"/> + <xsl:apply-templates/> +</xsl:template> + +<xsl:template match="*:description"> + <xsl:apply-templates/><xsl:value-of select="' '"/> +</xsl:template> + +<xsl:template match="*:variables"> + <xsl:apply-templates/><xsl:value-of select="' '"/> +</xsl:template> + +<xsl:template match="*:note"> + <xsl:variable name="page" select="count(./preceding::*:pb)"/> + <xsl:variable name="number" select="@number"/> + <item> + <xsl:if test="not(empty($number))"><xsl:attribute name="n"><xsl:value-of select="$number"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty($page))"><ref><xsl:value-of select="$page"/></ref></xsl:if> + </item> +</xsl:template> + +<xsl:template match="*:handwritten"> + <xsl:variable name="page" select="count(./preceding::*:pb)"/> + <xsl:variable name="number" select="@number"/> + <item> + <xsl:if test="not(empty($number))"><xsl:attribute name="n"><xsl:value-of select="$number"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty($page))"><ref><xsl:value-of select="$page"/></ref></xsl:if> + </item> +</xsl:template> + +<xsl:template match="*:place"> + <xsl:variable name="page" select="count(./preceding::*:pb)"/> + <xsl:variable name="id" select="@id"/> + <item> + <xsl:if test="not(empty($id))"><xsl:attribute name="id"><xsl:value-of select="$id"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty($page))"><ref><xsl:value-of select="$page"/></ref></xsl:if> + </item> +</xsl:template> + +<xsl:template match="*:div"> + <xsl:variable name="level"><xsl:number level="multiple" count="*:div[@type = 'section' or @type = 'chapter' or @type = 'part']" format="1."/></xsl:variable> + <xsl:variable name="depth" select="string-length(replace($level, '[^\\.]', ''))"/> + <xsl:variable name="pb" select="./preceding::*:pb[1]"/> + <xsl:variable name="o" select="$pb/@o"/> + <xsl:variable name="oNorm" select="$pb/@o-norm"/> + <xsl:variable name="page" select="count(./preceding::*:pb)"/> + <item> + <xsl:if test="not(empty($level))"><xsl:attribute name="n"><xsl:value-of select="$level"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty($level))"><xsl:attribute name="lv"><xsl:value-of select="$depth"/></xsl:attribute></xsl:if> + <xsl:apply-templates select="*:head"/> + <xsl:if test="not(empty($page))"> + <ref> + <xsl:if test="not(empty($o))"><xsl:attribute name="o"><xsl:value-of select="$o"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty($oNorm))"><xsl:attribute name="o-norm"><xsl:value-of select="$oNorm"/></xsl:attribute></xsl:if> + <xsl:value-of select="$page"/> + </ref> + </xsl:if> + </item> +</xsl:template> + +<xsl:template match="*:head"> + <xsl:apply-templates/><xsl:value-of select="' '"/> +</xsl:template> + +<xsl:template match="*:pb"> + <xsl:variable name="page" select="count(./preceding::*:pb) + 1"/> + <item> + <xsl:attribute name="n"><xsl:value-of select="$page"/></xsl:attribute> + <xsl:if test="not(empty(@o))"><xsl:attribute name="o"><xsl:value-of select="@o"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty(@o-norm))"><xsl:attribute name="o-norm"><xsl:value-of select="@o-norm"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty(@file))"><xsl:attribute name="file"><xsl:value-of select="@file"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty(@facs))"><xsl:attribute name="file"><xsl:value-of select="@facs"/></xsl:attribute></xsl:if> + </item> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/tocHtml.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,36 @@ +<?xml version="1.0"?> + +<xsl:stylesheet + version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" +> + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="type"></xsl:param> + +<xsl:template match="/"> + <xsl:apply-templates select="//*:list[@type = $type]/head"/> + <xsl:apply-templates select="//*:list[@type = $type]/item"/> +</xsl:template> + +<xsl:template match="*:item"> + <xsl:variable name="n" select="@n"/> + <xsl:variable name="level" select="@lv"/> + <div> + <xsl:attribute name="class"><xsl:value-of select="'item'"/></xsl:attribute> + <xsl:value-of select="concat($n, ' ')"/> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:ref"> + <xsl:variable name="o" select="@o"/> + <span> + <xsl:attribute name="class"><xsl:value-of select="'ref'"/></xsl:attribute> + <xsl:value-of select="'Page: '"/> + <xsl:apply-templates/> + </span> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/build/classes/de/mpg/mpiwg/berlin/mpdl/cms/transform/tocOut.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,45 @@ +<?xml version="1.0"?> + +<xsl:stylesheet + version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" +> + +<xsl:output method="xhtml"/> + +<xsl:param name="type"></xsl:param> +<xsl:param name="outputFormat"></xsl:param> + +<xsl:template match="/"> + <xsl:choose> + <xsl:when test="$outputFormat = 'html'"> + <xsl:apply-templates select="//*:list[@type = $type]/item" mode="html"/> + </xsl:when> + <xsl:otherwise> + <xsl:sequence select="//*:list[@type = $type]"/> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="*:item" mode="html"> + <xsl:variable name="n" select="@n"/> + <xsl:variable name="level" select="@lv"/> + <div> + <xsl:attribute name="class"><xsl:value-of select="'tocItem'"/></xsl:attribute> + <xsl:value-of select="concat($n, ' ')"/> + <xsl:apply-templates mode="html"/> + </div> +</xsl:template> + +<xsl:template match="*:ref" mode="html"> + <xsl:variable name="pageOrig" select="@o"/> + <xsl:variable name="page" select="text()"/> + <xsl:variable name="pageHref" select="concat('#page', $page)"/> + <span> + <xsl:attribute name="class"><xsl:value-of select="'tocPageRef'"/></xsl:attribute> + <xsl:value-of select="'Page: '"/> + <a href="{$pageHref}"><xsl:apply-templates mode="html"/></a> + </span> +</xsl:template> + +</xsl:stylesheet>
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/lib/commons-collections-LICENSE.txt Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,202 @@ + + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. Definitions. + + "License" shall mean the terms and conditions for use, reproduction, + and distribution as defined by Sections 1 through 9 of this document. + + "Licensor" shall mean the copyright owner or entity authorized by + the copyright owner that is granting the License. + + "Legal Entity" shall mean the union of the acting entity and all + other entities that control, are controlled by, or are under common + control with that entity. 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If You + institute patent litigation against any entity (including a + cross-claim or counterclaim in a lawsuit) alleging that the Work + or a Contribution incorporated within the Work constitutes direct + or contributory patent infringement, then any patent licenses + granted to You under this License for that Work shall terminate + as of the date such litigation is filed. + + 4. Redistribution. You may reproduce and distribute copies of the + Work or Derivative Works thereof in any medium, with or without + modifications, and in Source or Object form, provided that You + meet the following conditions: + + (a) You must give any other recipients of the Work or + Derivative Works a copy of this License; and + + (b) You must cause any modified files to carry prominent notices + stating that You changed the files; and + + (c) You must retain, in the Source form of any Derivative Works + that You distribute, all copyright, patent, trademark, and + attribution notices from the Source form of the Work, + excluding those notices that do not pertain to any part of + the Derivative Works; and + + (d) If the Work includes a "NOTICE" text file as part of its + distribution, then any Derivative Works that You distribute must + include a readable copy of the attribution notices contained + within such NOTICE file, excluding those notices that do not + pertain to any part of the Derivative Works, in at least one + of the following places: within a NOTICE text file distributed + as part of the Derivative Works; within the Source form or + documentation, if provided along with the Derivative Works; or, + within a display generated by the Derivative Works, if and + wherever such third-party notices normally appear. The contents + of the NOTICE file are for informational purposes only and + do not modify the License. You may add Your own attribution + notices within Derivative Works that You distribute, alongside + or as an addendum to the NOTICE text from the Work, provided + that such additional attribution notices cannot be construed + as modifying the License. + + You may add Your own copyright statement to Your modifications and + may provide additional or different license terms and conditions + for use, reproduction, or distribution of Your modifications, or + for any such Derivative Works as a whole, provided Your use, + reproduction, and distribution of the Work otherwise complies with + the conditions stated in this License. + + 5. Submission of Contributions. Unless You explicitly state otherwise, + any Contribution intentionally submitted for inclusion in the Work + by You to the Licensor shall be under the terms and conditions of + this License, without any additional terms or conditions. + Notwithstanding the above, nothing herein shall supersede or modify + the terms of any separate license agreement you may have executed + with Licensor regarding such Contributions. + + 6. Trademarks. This License does not grant permission to use the trade + names, trademarks, service marks, or product names of the Licensor, + except as required for reasonable and customary use in describing the + origin of the Work and reproducing the content of the NOTICE file. + + 7. Disclaimer of Warranty. Unless required by applicable law or + agreed to in writing, Licensor provides the Work (and each + Contributor provides its Contributions) on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or + implied, including, without limitation, any warranties or conditions + of TITLE, NON-INFRINGEMENT, MERCHANTABILITY, or FITNESS FOR A + PARTICULAR PURPOSE. You are solely responsible for determining the + appropriateness of using or redistributing the Work and assume any + risks associated with Your exercise of permissions under this License. + + 8. Limitation of Liability. In no event and under no legal theory, + whether in tort (including negligence), contract, or otherwise, + unless required by applicable law (such as deliberate and grossly + negligent acts) or agreed to in writing, shall any Contributor be + liable to You for damages, including any direct, indirect, special, + incidental, or consequential damages of any character arising as a + result of this License or out of the use or inability to use the + Work (including but not limited to damages for loss of goodwill, + work stoppage, computer failure or malfunction, or any and all + other commercial damages or losses), even if such Contributor + has been advised of the possibility of such damages. + + 9. Accepting Warranty or Additional Liability. While redistributing + the Work or Derivative Works thereof, You may choose to offer, + and charge a fee for, acceptance of support, warranty, indemnity, + or other liability obligations and/or rights consistent with this + License. However, in accepting such obligations, You may act only + on Your own behalf and on Your sole responsibility, not on behalf + of any other Contributor, and only if You agree to indemnify, + defend, and hold each Contributor harmless for any liability + incurred by, or claims asserted against, such Contributor by reason + of your accepting any such warranty or additional liability. + + END OF TERMS AND CONDITIONS + + APPENDIX: How to apply the Apache License to your work. + + To apply the Apache License to your work, attach the following + boilerplate notice, with the fields enclosed by brackets "[]" + replaced with your own identifying information. (Don't include + the brackets!) The text should be enclosed in the appropriate + comment syntax for the file format. We also recommend that a + file or class name and description of purpose be included on the + same "printed page" as the copyright notice for easier + identification within third-party archives. + + Copyright [yyyy] [name of copyright owner] + + Licensed under the Apache License, Version 2.0 (the "License"); + you may not use this file except in compliance with the License. + You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + + Unless required by applicable law or agreed to in writing, software + distributed under the License is distributed on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + See the License for the specific language governing permissions and + limitations under the License.
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/lib/commons-collections-NOTICE.txt Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,5 @@ +Apache Commons Collections +Copyright 2001-2008 The Apache Software Foundation + +This product includes software developed by +The Apache Software Foundation (http://www.apache.org/).
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/lib/commons-httpclient-LICENSE.txt Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,176 @@ + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. Definitions. + + "License" shall mean the terms and conditions for use, reproduction, + and distribution as defined by Sections 1 through 9 of this document. + + "Licensor" shall mean the copyright owner or entity authorized by + the copyright owner that is granting the License. + + "Legal Entity" shall mean the union of the acting entity and all + other entities that control, are controlled by, or are under common + control with that entity. For the purposes of this definition, + "control" means (i) the power, direct or indirect, to cause the + direction or management of such entity, whether by contract or + otherwise, or (ii) ownership of fifty percent (50%) or more of the + outstanding shares, or (iii) beneficial ownership of such entity. + + "You" (or "Your") shall mean an individual or Legal Entity + exercising permissions granted by this License. + + "Source" form shall mean the preferred form for making modifications, + including but not limited to software source code, documentation + source, and configuration files. + + "Object" form shall mean any form resulting from mechanical + transformation or translation of a Source form, including but + not limited to compiled object code, generated documentation, + and conversions to other media types. + + "Work" shall mean the work of authorship, whether in Source or + Object form, made available under the License, as indicated by a + copyright notice that is included in or attached to the work + (an example is provided in the Appendix below). + + "Derivative Works" shall mean any work, whether in Source or Object + form, that is based on (or derived from) the Work and for which the + editorial revisions, annotations, elaborations, or other modifications + represent, as a whole, an original work of authorship. For the purposes + of this License, Derivative Works shall not include works that remain + separable from, or merely link (or bind by name) to the interfaces of, + the Work and Derivative Works thereof. + + "Contribution" shall mean any work of authorship, including + the original version of the Work and any modifications or additions + to that Work or Derivative Works thereof, that is intentionally + submitted to Licensor for inclusion in the Work by the copyright owner + or by an individual or Legal Entity authorized to submit on behalf of + the copyright owner. For the purposes of this definition, "submitted" + means any form of electronic, verbal, or written communication sent + to the Licensor or its representatives, including but not limited to + communication on electronic mailing lists, source code control systems, + and issue tracking systems that are managed by, or on behalf of, the + Licensor for the purpose of discussing and improving the Work, but + excluding communication that is conspicuously marked or otherwise + designated in writing by the copyright owner as "Not a Contribution." + + "Contributor" shall mean Licensor and any individual or Legal Entity + on behalf of whom a Contribution has been received by Licensor and + subsequently incorporated within the Work. + + 2. Grant of Copyright License. Subject to the terms and conditions of + this License, each Contributor hereby grants to You a perpetual, + worldwide, non-exclusive, no-charge, royalty-free, irrevocable + copyright license to reproduce, prepare Derivative Works of, + publicly display, publicly perform, sublicense, and distribute the + Work and such Derivative Works in Source or Object form. + + 3. Grant of Patent License. Subject to the terms and conditions of + this License, each Contributor hereby grants to You a perpetual, + worldwide, non-exclusive, no-charge, royalty-free, irrevocable + (except as stated in this section) patent license to make, have made, + use, offer to sell, sell, import, and otherwise transfer the Work, + where such license applies only to those patent claims licensable + by such Contributor that are necessarily infringed by their + Contribution(s) alone or by combination of their Contribution(s) + with the Work to which such Contribution(s) was submitted. If You + institute patent litigation against any entity (including a + cross-claim or counterclaim in a lawsuit) alleging that the Work + or a Contribution incorporated within the Work constitutes direct + or contributory patent infringement, then any patent licenses + granted to You under this License for that Work shall terminate + as of the date such litigation is filed. + + 4. Redistribution. You may reproduce and distribute copies of the + Work or Derivative Works thereof in any medium, with or without + modifications, and in Source or Object form, provided that You + meet the following conditions: + + (a) You must give any other recipients of the Work or + Derivative Works a copy of this License; and + + (b) You must cause any modified files to carry prominent notices + stating that You changed the files; and + + (c) You must retain, in the Source form of any Derivative Works + that You distribute, all copyright, patent, trademark, and + attribution notices from the Source form of the Work, + excluding those notices that do not pertain to any part of + the Derivative Works; and + + (d) If the Work includes a "NOTICE" text file as part of its + distribution, then any Derivative Works that You distribute must + include a readable copy of the attribution notices contained + within such NOTICE file, excluding those notices that do not + pertain to any part of the Derivative Works, in at least one + of the following places: within a NOTICE text file distributed + as part of the Derivative Works; within the Source form or + documentation, if provided along with the Derivative Works; or, + within a display generated by the Derivative Works, if and + wherever such third-party notices normally appear. The contents + of the NOTICE file are for informational purposes only and + do not modify the License. You may add Your own attribution + notices within Derivative Works that You distribute, alongside + or as an addendum to the NOTICE text from the Work, provided + that such additional attribution notices cannot be construed + as modifying the License. + + You may add Your own copyright statement to Your modifications and + may provide additional or different license terms and conditions + for use, reproduction, or distribution of Your modifications, or + for any such Derivative Works as a whole, provided Your use, + reproduction, and distribution of the Work otherwise complies with + the conditions stated in this License. + + 5. Submission of Contributions. Unless You explicitly state otherwise, + any Contribution intentionally submitted for inclusion in the Work + by You to the Licensor shall be under the terms and conditions of + this License, without any additional terms or conditions. + Notwithstanding the above, nothing herein shall supersede or modify + the terms of any separate license agreement you may have executed + with Licensor regarding such Contributions. + + 6. Trademarks. This License does not grant permission to use the trade + names, trademarks, service marks, or product names of the Licensor, + except as required for reasonable and customary use in describing the + origin of the Work and reproducing the content of the NOTICE file. + + 7. Disclaimer of Warranty. Unless required by applicable law or + agreed to in writing, Licensor provides the Work (and each + Contributor provides its Contributions) on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or + implied, including, without limitation, any warranties or conditions + of TITLE, NON-INFRINGEMENT, MERCHANTABILITY, or FITNESS FOR A + PARTICULAR PURPOSE. You are solely responsible for determining the + appropriateness of using or redistributing the Work and assume any + risks associated with Your exercise of permissions under this License. + + 8. Limitation of Liability. In no event and under no legal theory, + whether in tort (including negligence), contract, or otherwise, + unless required by applicable law (such as deliberate and grossly + negligent acts) or agreed to in writing, shall any Contributor be + liable to You for damages, including any direct, indirect, special, + incidental, or consequential damages of any character arising as a + result of this License or out of the use or inability to use the + Work (including but not limited to damages for loss of goodwill, + work stoppage, computer failure or malfunction, or any and all + other commercial damages or losses), even if such Contributor + has been advised of the possibility of such damages. + + 9. Accepting Warranty or Additional Liability. While redistributing + the Work or Derivative Works thereof, You may choose to offer, + and charge a fee for, acceptance of support, warranty, indemnity, + or other liability obligations and/or rights consistent with this + License. However, in accepting such obligations, You may act only + on Your own behalf and on Your sole responsibility, not on behalf + of any other Contributor, and only if You agree to indemnify, + defend, and hold each Contributor harmless for any liability + incurred by, or claims asserted against, such Contributor by reason + of your accepting any such warranty or additional liability. + + END OF TERMS AND CONDITIONS
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/lib/commons-httpclient-NOTICE.txt Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,5 @@ +Apache Jakarta HttpClient +Copyright 1999-2007 The Apache Software Foundation + +This product includes software developed by +The Apache Software Foundation (http://www.apache.org/).
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/lib/commons-logging-LICENSE.txt Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,202 @@ + + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. Definitions. + + "License" shall mean the terms and conditions for use, reproduction, + and distribution as defined by Sections 1 through 9 of this document. + + "Licensor" shall mean the copyright owner or entity authorized by + the copyright owner that is granting the License. + + "Legal Entity" shall mean the union of the acting entity and all + other entities that control, are controlled by, or are under common + control with that entity. For the purposes of this definition, + "control" means (i) the power, direct or indirect, to cause the + direction or management of such entity, whether by contract or + otherwise, or (ii) ownership of fifty percent (50%) or more of the + outstanding shares, or (iii) beneficial ownership of such entity. + + "You" (or "Your") shall mean an individual or Legal Entity + exercising permissions granted by this License. + + "Source" form shall mean the preferred form for making modifications, + including but not limited to software source code, documentation + source, and configuration files. + + "Object" form shall mean any form resulting from mechanical + transformation or translation of a Source form, including but + not limited to compiled object code, generated documentation, + and conversions to other media types. + + "Work" shall mean the work of authorship, whether in Source or + Object form, made available under the License, as indicated by a + copyright notice that is included in or attached to the work + (an example is provided in the Appendix below). + + "Derivative Works" shall mean any work, whether in Source or Object + form, that is based on (or derived from) the Work and for which the + editorial revisions, annotations, elaborations, or other modifications + represent, as a whole, an original work of authorship. For the purposes + of this License, Derivative Works shall not include works that remain + separable from, or merely link (or bind by name) to the interfaces of, + the Work and Derivative Works thereof. + + "Contribution" shall mean any work of authorship, including + the original version of the Work and any modifications or additions + to that Work or Derivative Works thereof, that is intentionally + submitted to Licensor for inclusion in the Work by the copyright owner + or by an individual or Legal Entity authorized to submit on behalf of + the copyright owner. For the purposes of this definition, "submitted" + means any form of electronic, verbal, or written communication sent + to the Licensor or its representatives, including but not limited to + communication on electronic mailing lists, source code control systems, + and issue tracking systems that are managed by, or on behalf of, the + Licensor for the purpose of discussing and improving the Work, but + excluding communication that is conspicuously marked or otherwise + designated in writing by the copyright owner as "Not a Contribution." + + "Contributor" shall mean Licensor and any individual or Legal Entity + on behalf of whom a Contribution has been received by Licensor and + subsequently incorporated within the Work. + + 2. Grant of Copyright License. Subject to the terms and conditions of + this License, each Contributor hereby grants to You a perpetual, + worldwide, non-exclusive, no-charge, royalty-free, irrevocable + copyright license to reproduce, prepare Derivative Works of, + publicly display, publicly perform, sublicense, and distribute the + Work and such Derivative Works in Source or Object form. + + 3. Grant of Patent License. Subject to the terms and conditions of + this License, each Contributor hereby grants to You a perpetual, + worldwide, non-exclusive, no-charge, royalty-free, irrevocable + (except as stated in this section) patent license to make, have made, + use, offer to sell, sell, import, and otherwise transfer the Work, + where such license applies only to those patent claims licensable + by such Contributor that are necessarily infringed by their + Contribution(s) alone or by combination of their Contribution(s) + with the Work to which such Contribution(s) was submitted. If You + institute patent litigation against any entity (including a + cross-claim or counterclaim in a lawsuit) alleging that the Work + or a Contribution incorporated within the Work constitutes direct + or contributory patent infringement, then any patent licenses + granted to You under this License for that Work shall terminate + as of the date such litigation is filed. + + 4. Redistribution. You may reproduce and distribute copies of the + Work or Derivative Works thereof in any medium, with or without + modifications, and in Source or Object form, provided that You + meet the following conditions: + + (a) You must give any other recipients of the Work or + Derivative Works a copy of this License; and + + (b) You must cause any modified files to carry prominent notices + stating that You changed the files; and + + (c) You must retain, in the Source form of any Derivative Works + that You distribute, all copyright, patent, trademark, and + attribution notices from the Source form of the Work, + excluding those notices that do not pertain to any part of + the Derivative Works; and + + (d) If the Work includes a "NOTICE" text file as part of its + distribution, then any Derivative Works that You distribute must + include a readable copy of the attribution notices contained + within such NOTICE file, excluding those notices that do not + pertain to any part of the Derivative Works, in at least one + of the following places: within a NOTICE text file distributed + as part of the Derivative Works; within the Source form or + documentation, if provided along with the Derivative Works; or, + within a display generated by the Derivative Works, if and + wherever such third-party notices normally appear. The contents + of the NOTICE file are for informational purposes only and + do not modify the License. You may add Your own attribution + notices within Derivative Works that You distribute, alongside + or as an addendum to the NOTICE text from the Work, provided + that such additional attribution notices cannot be construed + as modifying the License. + + You may add Your own copyright statement to Your modifications and + may provide additional or different license terms and conditions + for use, reproduction, or distribution of Your modifications, or + for any such Derivative Works as a whole, provided Your use, + reproduction, and distribution of the Work otherwise complies with + the conditions stated in this License. + + 5. Submission of Contributions. Unless You explicitly state otherwise, + any Contribution intentionally submitted for inclusion in the Work + by You to the Licensor shall be under the terms and conditions of + this License, without any additional terms or conditions. + Notwithstanding the above, nothing herein shall supersede or modify + the terms of any separate license agreement you may have executed + with Licensor regarding such Contributions. + + 6. Trademarks. This License does not grant permission to use the trade + names, trademarks, service marks, or product names of the Licensor, + except as required for reasonable and customary use in describing the + origin of the Work and reproducing the content of the NOTICE file. + + 7. Disclaimer of Warranty. Unless required by applicable law or + agreed to in writing, Licensor provides the Work (and each + Contributor provides its Contributions) on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or + implied, including, without limitation, any warranties or conditions + of TITLE, NON-INFRINGEMENT, MERCHANTABILITY, or FITNESS FOR A + PARTICULAR PURPOSE. You are solely responsible for determining the + appropriateness of using or redistributing the Work and assume any + risks associated with Your exercise of permissions under this License. + + 8. Limitation of Liability. In no event and under no legal theory, + whether in tort (including negligence), contract, or otherwise, + unless required by applicable law (such as deliberate and grossly + negligent acts) or agreed to in writing, shall any Contributor be + liable to You for damages, including any direct, indirect, special, + incidental, or consequential damages of any character arising as a + result of this License or out of the use or inability to use the + Work (including but not limited to damages for loss of goodwill, + work stoppage, computer failure or malfunction, or any and all + other commercial damages or losses), even if such Contributor + has been advised of the possibility of such damages. + + 9. Accepting Warranty or Additional Liability. While redistributing + the Work or Derivative Works thereof, You may choose to offer, + and charge a fee for, acceptance of support, warranty, indemnity, + or other liability obligations and/or rights consistent with this + License. However, in accepting such obligations, You may act only + on Your own behalf and on Your sole responsibility, not on behalf + of any other Contributor, and only if You agree to indemnify, + defend, and hold each Contributor harmless for any liability + incurred by, or claims asserted against, such Contributor by reason + of your accepting any such warranty or additional liability. + + END OF TERMS AND CONDITIONS + + APPENDIX: How to apply the Apache License to your work. + + To apply the Apache License to your work, attach the following + boilerplate notice, with the fields enclosed by brackets "[]" + replaced with your own identifying information. (Don't include + the brackets!) The text should be enclosed in the appropriate + comment syntax for the file format. We also recommend that a + file or class name and description of purpose be included on the + same "printed page" as the copyright notice for easier + identification within third-party archives. + + Copyright [yyyy] [name of copyright owner] + + Licensed under the Apache License, Version 2.0 (the "License"); + you may not use this file except in compliance with the License. + You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + + Unless required by applicable law or agreed to in writing, software + distributed under the License is distributed on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + See the License for the specific language governing permissions and + limitations under the License.
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/lib/commons-logging-NOTICE.txt Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,6 @@ +Apache Commons Logging +Copyright 2003-2007 The Apache Software Foundation + +This product includes software developed by +The Apache Software Foundation (http://www.apache.org/). +
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/lib/quartz-LICENSE.txt Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,202 @@ + + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. Definitions. + + "License" shall mean the terms and conditions for use, reproduction, + and distribution as defined by Sections 1 through 9 of this document. + + "Licensor" shall mean the copyright owner or entity authorized by + the copyright owner that is granting the License. + + "Legal Entity" shall mean the union of the acting entity and all + other entities that control, are controlled by, or are under common + control with that entity. For the purposes of this definition, + "control" means (i) the power, direct or indirect, to cause the + direction or management of such entity, whether by contract or + otherwise, or (ii) ownership of fifty percent (50%) or more of the + outstanding shares, or (iii) beneficial ownership of such entity. + + "You" (or "Your") shall mean an individual or Legal Entity + exercising permissions granted by this License. + + "Source" form shall mean the preferred form for making modifications, + including but not limited to software source code, documentation + source, and configuration files. + + "Object" form shall mean any form resulting from mechanical + transformation or translation of a Source form, including but + not limited to compiled object code, generated documentation, + and conversions to other media types. + + "Work" shall mean the work of authorship, whether in Source or + Object form, made available under the License, as indicated by a + copyright notice that is included in or attached to the work + (an example is provided in the Appendix below). + + "Derivative Works" shall mean any work, whether in Source or Object + form, that is based on (or derived from) the Work and for which the + editorial revisions, annotations, elaborations, or other modifications + represent, as a whole, an original work of authorship. For the purposes + of this License, Derivative Works shall not include works that remain + separable from, or merely link (or bind by name) to the interfaces of, + the Work and Derivative Works thereof. + + "Contribution" shall mean any work of authorship, including + the original version of the Work and any modifications or additions + to that Work or Derivative Works thereof, that is intentionally + submitted to Licensor for inclusion in the Work by the copyright owner + or by an individual or Legal Entity authorized to submit on behalf of + the copyright owner. For the purposes of this definition, "submitted" + means any form of electronic, verbal, or written communication sent + to the Licensor or its representatives, including but not limited to + communication on electronic mailing lists, source code control systems, + and issue tracking systems that are managed by, or on behalf of, the + Licensor for the purpose of discussing and improving the Work, but + excluding communication that is conspicuously marked or otherwise + designated in writing by the copyright owner as "Not a Contribution." + + "Contributor" shall mean Licensor and any individual or Legal Entity + on behalf of whom a Contribution has been received by Licensor and + subsequently incorporated within the Work. + + 2. Grant of Copyright License. Subject to the terms and conditions of + this License, each Contributor hereby grants to You a perpetual, + worldwide, non-exclusive, no-charge, royalty-free, irrevocable + copyright license to reproduce, prepare Derivative Works of, + publicly display, publicly perform, sublicense, and distribute the + Work and such Derivative Works in Source or Object form. + + 3. Grant of Patent License. Subject to the terms and conditions of + this License, each Contributor hereby grants to You a perpetual, + worldwide, non-exclusive, no-charge, royalty-free, irrevocable + (except as stated in this section) patent license to make, have made, + use, offer to sell, sell, import, and otherwise transfer the Work, + where such license applies only to those patent claims licensable + by such Contributor that are necessarily infringed by their + Contribution(s) alone or by combination of their Contribution(s) + with the Work to which such Contribution(s) was submitted. If You + institute patent litigation against any entity (including a + cross-claim or counterclaim in a lawsuit) alleging that the Work + or a Contribution incorporated within the Work constitutes direct + or contributory patent infringement, then any patent licenses + granted to You under this License for that Work shall terminate + as of the date such litigation is filed. + + 4. Redistribution. You may reproduce and distribute copies of the + Work or Derivative Works thereof in any medium, with or without + modifications, and in Source or Object form, provided that You + meet the following conditions: + + (a) You must give any other recipients of the Work or + Derivative Works a copy of this License; and + + (b) You must cause any modified files to carry prominent notices + stating that You changed the files; and + + (c) You must retain, in the Source form of any Derivative Works + that You distribute, all copyright, patent, trademark, and + attribution notices from the Source form of the Work, + excluding those notices that do not pertain to any part of + the Derivative Works; and + + (d) If the Work includes a "NOTICE" text file as part of its + distribution, then any Derivative Works that You distribute must + include a readable copy of the attribution notices contained + within such NOTICE file, excluding those notices that do not + pertain to any part of the Derivative Works, in at least one + of the following places: within a NOTICE text file distributed + as part of the Derivative Works; within the Source form or + documentation, if provided along with the Derivative Works; or, + within a display generated by the Derivative Works, if and + wherever such third-party notices normally appear. The contents + of the NOTICE file are for informational purposes only and + do not modify the License. You may add Your own attribution + notices within Derivative Works that You distribute, alongside + or as an addendum to the NOTICE text from the Work, provided + that such additional attribution notices cannot be construed + as modifying the License. + + You may add Your own copyright statement to Your modifications and + may provide additional or different license terms and conditions + for use, reproduction, or distribution of Your modifications, or + for any such Derivative Works as a whole, provided Your use, + reproduction, and distribution of the Work otherwise complies with + the conditions stated in this License. + + 5. Submission of Contributions. Unless You explicitly state otherwise, + any Contribution intentionally submitted for inclusion in the Work + by You to the Licensor shall be under the terms and conditions of + this License, without any additional terms or conditions. + Notwithstanding the above, nothing herein shall supersede or modify + the terms of any separate license agreement you may have executed + with Licensor regarding such Contributions. + + 6. Trademarks. This License does not grant permission to use the trade + names, trademarks, service marks, or product names of the Licensor, + except as required for reasonable and customary use in describing the + origin of the Work and reproducing the content of the NOTICE file. + + 7. Disclaimer of Warranty. Unless required by applicable law or + agreed to in writing, Licensor provides the Work (and each + Contributor provides its Contributions) on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or + implied, including, without limitation, any warranties or conditions + of TITLE, NON-INFRINGEMENT, MERCHANTABILITY, or FITNESS FOR A + PARTICULAR PURPOSE. You are solely responsible for determining the + appropriateness of using or redistributing the Work and assume any + risks associated with Your exercise of permissions under this License. + + 8. Limitation of Liability. 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We also recommend that a + file or class name and description of purpose be included on the + same "printed page" as the copyright notice for easier + identification within third-party archives. + + Copyright [yyyy] [name of copyright owner] + + Licensed under the Apache License, Version 2.0 (the "License"); + you may not use this file except in compliance with the License. + You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + + Unless required by applicable law or agreed to in writing, software + distributed under the License is distributed on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + See the License for the specific language governing permissions and + limitations under the License.
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/lib/saxon.txt Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,3 @@ +Saxon: + +Release 9.1.0.5 (free version): releases < 9.1.0.7 support saxon extension functions
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/constants.properties Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,5 @@ +dataDir=/Users/jwillenborg/mpdl/data/lt +documentsDir=/Users/jwillenborg/mpdl/data/xml/documents +luceneDocumentsDir=/Users/jwillenborg/mpdl/data/xml/lucene/documents +luceneNodesDir=/Users/jwillenborg/mpdl/data/xml/lucene/nodes +confDir=/Users/jwillenborg/mpdl/data/collectionConfs \ No newline at end of file
Binary file software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/.DS_Store has changed
Binary file software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/.DS_Store has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/confmanager/CollectionReader.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,122 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.confmanager; + +import java.io.File; +import java.io.IOException; +import java.util.ArrayList; +import java.util.HashMap; +import java.util.List; + +import javax.xml.parsers.DocumentBuilder; +import javax.xml.parsers.DocumentBuilderFactory; +import javax.xml.parsers.ParserConfigurationException; + +import de.mpg.mpiwg.berlin.mpdl.cms.general.Constants; +import de.mpg.mpiwg.berlin.mpdl.cms.harvester.PathExtractor; +import org.w3c.dom.Document; +import org.w3c.dom.Node; +import org.w3c.dom.NodeList; +import org.xml.sax.SAXException; + +public class CollectionReader { + + ConfManagerResultWrapper cmrw; + private HashMap<String, ConfManagerResultWrapper> wrapperContainer; + private static CollectionReader collectionReader; + + private CollectionReader() { + wrapperContainer = new HashMap<String, ConfManagerResultWrapper>(); + readConfFiles(); + } + + public static CollectionReader getInstance() { + if (collectionReader == null) + collectionReader = new CollectionReader(); + return collectionReader; + } + + private void readConfFiles(){ + System.out.println("---------------"); + System.out.println("reading configuration files..."); + + // holt alle konfiguratiuonsdateien aus dem konf-Ordner + PathExtractor ext = new PathExtractor(); + List<String> configsList = ext.extractPathLocally(Constants.getInstance().getConfDir()); + System.out.println("Anzahl der konfugirationsdateien : " + configsList.size()); + DocumentBuilderFactory docFactory = DocumentBuilderFactory.newInstance(); + // docFactory.setNamespaceAware(true); + DocumentBuilder builder = null; + + File configFile = null; + for (String configXml : configsList) { + System.out.println("reading : " + configXml); + try { + builder = docFactory.newDocumentBuilder(); + } catch (ParserConfigurationException e) { + e.printStackTrace(); + } + configFile = new File(configXml); + Document document = null; + try { + document = builder.parse(configFile); + } catch (SAXException e) { + e.printStackTrace(); + } catch (IOException e) { + e.printStackTrace(); + } + cmrw = new ConfManagerResultWrapper(); + + NodeList idlist = document.getElementsByTagName("collectionId"); + // darf jeweils nur ein node enthalten sein + Node idNode = idlist.item(0); + if(idNode != null){ + if (!idNode.getTextContent().equals("")) { + cmrw.setCollectionId(idNode.getTextContent()); + } + } + NodeList nodeliste = document.getElementsByTagName("mainLanguage"); + // darf jeweils nur ein node enthalten sein + Node langNode = nodeliste.item(0); + if(langNode != null){ + if (!langNode.getTextContent().equals("")) { + cmrw.setMainLanguage(langNode.getTextContent()); + } + } + NodeList collNamelist = document.getElementsByTagName("name"); + // darf jeweils nur ein node enthalten sein + Node nameNode = collNamelist.item(0); + if(nameNode != null){ + if (!nameNode.getTextContent().equals("")) { + cmrw.setCollectionName(nameNode.getTextContent()); + } + } + + NodeList fieldNodes = document.getElementsByTagName("field"); + ArrayList<String> fields = new ArrayList<String>(); + fields = new ArrayList<String>(); + if(fieldNodes != null){ + for (int i = 0; i < fieldNodes.getLength(); i++) { + if (!fieldNodes.item(i).getTextContent().equals("")) { + fields.add((fieldNodes.item(i).getTextContent().trim())); + } + } + } + cmrw.setFields(fields); + + NodeList nodeli = document.getElementsByTagName("collectionDataUrl"); + Node dataNode = nodeli.item(0); + if(dataNode != null){ + if (!dataNode.getTextContent().trim().equals("")) { + cmrw.setCollectionDataUrl(dataNode.getTextContent()); + } + } + + wrapperContainer.put(cmrw.getCollectionId(), cmrw); + } + } + + public ConfManagerResultWrapper getResultWrapper(String collectionId) { + ConfManagerResultWrapper cmrw = wrapperContainer.get(collectionId); + return cmrw; + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/confmanager/ConfManager.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,139 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.confmanager; + +import java.io.File; +import java.io.FileOutputStream; +import java.net.URL; +import java.util.ArrayList; +import java.util.HashMap; +import java.util.List; + +import javax.xml.parsers.DocumentBuilder; +import javax.xml.parsers.DocumentBuilderFactory; +import javax.xml.xpath.XPathExpressionException; + +import de.mpg.mpiwg.berlin.mpdl.cms.harvester.PathExtractor; + +import de.mpg.mpiwg.berlin.mpdl.xml.xquery.XQueryEvaluator; + +import de.mpg.mpiwg.berlin.mpdl.cms.general.Constants; +import org.w3c.dom.Document; +import org.w3c.dom.Node; +import org.w3c.dom.NodeList; + +import com.sun.org.apache.xml.internal.serialize.XMLSerializer; + +public class ConfManager { + private ConfManagerResultWrapper cmrw; + private HashMap<String, ConfManagerResultWrapper> wrapperContainer; + private static ConfManager confManager; + + private ConfManager() { + wrapperContainer = new HashMap<String, ConfManagerResultWrapper>(); + try { + checkCollectionConfFiles(); + } catch (XPathExpressionException e) { + e.printStackTrace(); + } + } + + public static ConfManager getInstance(){ + if(confManager == null) + confManager = new ConfManager(); + return confManager; + } + + /** + * checks if an update of a project is necessary by checking configuration + * file + * + * @throws XPathExpressionException + */ + private void checkCollectionConfFiles() throws XPathExpressionException { + System.out.println("---------------"); + System.out.println("checking configuration files..."); + // holt alle Konfiguratiuonsdateien aus dem konf-Ordner + PathExtractor ext = new PathExtractor(); + List<String> configsList = ext.extractPathLocally(Constants.getInstance().getConfDir()); + System.out.println("Anzahl der Konfigurationsdateien : " + configsList.size()); + try { + File configFile = null; + // Ueberprueft alle Konf-dateien auf update und fuehrt es bei Bedarf aus + for (String configXml : configsList) { + System.out.println("checking : " + configXml); + XQueryEvaluator xQueryEvaluator = new XQueryEvaluator(); + configFile = new File(configXml); + URL srcUrl = configFile.toURI().toURL(); + String update = xQueryEvaluator.evaluateAsString(srcUrl, "//collection/update/text()"); + if (update != null && update.equals("true")) { + System.out.println("update tag is set on : " + update); + cmrw = new ConfManagerResultWrapper(); + String collectionId = xQueryEvaluator.evaluateAsString(srcUrl, "//collectionId/text()"); + if (collectionId != null) { + cmrw.setCollectionId(collectionId); + } + String mainLanguage = xQueryEvaluator.evaluateAsString(srcUrl, "//mainLanguage/text()"); + if (mainLanguage != null) { + cmrw.setMainLanguage(mainLanguage); + } + String name = xQueryEvaluator.evaluateAsString(srcUrl, "//name/text()"); + if (name != null) { + cmrw.setCollectionName(name); + } + String fieldsStr = xQueryEvaluator.evaluateAsStringValueJoined(srcUrl, "//field"); + ArrayList<String> fields = new ArrayList<String>(); + if (fields != null) { + fieldsStr = fieldsStr.trim(); + String[] fieldsArray = fieldsStr.split(" "); + for (int i=0; i<fieldsArray.length; i++) { + String field = fieldsArray[i]; + fields.add(field); + } + cmrw.setFields(fields); + } + String collectionDataUrl = xQueryEvaluator.evaluateAsString(srcUrl, "//specifyUrl/collectionDataUrl/text()"); + if (collectionDataUrl != null) { + cmrw.setCollectionDataUrl(collectionDataUrl); + String excludesStr = xQueryEvaluator.evaluateAsStringValueJoined(srcUrl, "//specifyUrl/exclude"); + if(collectionDataUrl.endsWith("/")) + extractUrlsFromCollections(collectionDataUrl, cmrw, excludesStr); + else{ + List<String> collectionUrls = new ArrayList<String>(); + collectionUrls.add(collectionDataUrl); + cmrw.setCollectionUrls(collectionUrls); + } + // flag im Konfigurations-File auf false setzen durch serialisierung in das File + DocumentBuilderFactory docFactory = DocumentBuilderFactory.newInstance(); + DocumentBuilder builder = docFactory.newDocumentBuilder(); + Document configFileDocument = builder.parse(configFile); + NodeList updateNodeList = configFileDocument.getElementsByTagName("update"); + Node n = updateNodeList.item(0); + n.setTextContent("false"); + FileOutputStream os = new FileOutputStream(configFile); + XMLSerializer ser = new XMLSerializer(os, null); + ser.serialize(configFileDocument); // Vorsicht: wenn es auf true ist: es wird alles neu indexiert + } + wrapperContainer.put(collectionId, cmrw); + } + } + } catch(Exception e) { + e.printStackTrace(); + } + } + + /** + * der Extractor holt alle Projekt zugehoerigen Urls + */ + private void extractUrlsFromCollections(String collectionDataUrl, ConfManagerResultWrapper cmrw, String excludesStr) { + System.out.println("collecting urls of resources that need update..."); + if(!collectionDataUrl.equals("")){ + PathExtractor extractor = new PathExtractor(); + List<String> collectionUrls = extractor.initExtractor(collectionDataUrl, excludesStr); + cmrw.setCollectionUrls(collectionUrls); + } + } + + public ConfManagerResultWrapper getResultWrapper(String collectionId) { + ConfManagerResultWrapper cmrw = wrapperContainer.get(collectionId); + return cmrw; + } +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/confmanager/ConfManagerResultWrapper.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,93 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.confmanager; + +import java.util.ArrayList; +import java.util.List; + +public class ConfManagerResultWrapper { + + private List<String> collectionUrls; + private String collectionId; + private String collectionName; + private String collectionDataUrl; + private String mainLanguage; + private ArrayList<String> fields; + private List<String> formats; + private List<String> excludeField; + + + public ConfManagerResultWrapper(){ + collectionUrls = new ArrayList<String>(); + collectionId = ""; + collectionName = ""; + collectionDataUrl = ""; + mainLanguage = ""; + fields = new ArrayList<String>(); + formats = new ArrayList<String>(); + excludeField = new ArrayList<String>(); + } + + public List<String> getCollectionUrls() { + return collectionUrls; + } + + public String getCollectionId() { + return collectionId; + } + + public String getCollectionName() { + return collectionName; + } + + public void setCollectionId(String collectionId){ + this.collectionId = collectionId; + } + + public void setCollectionUrls(List<String> collectionUrls){ + this.collectionUrls = collectionUrls; + } + + public void setCollectionName(String collectionName) { + this.collectionName = collectionName; + } + + public String getCollectiondataUrl() { + return collectionDataUrl; + } + + public void setCollectionDataUrl(String collectiondataUrl) { + this.collectionDataUrl = collectiondataUrl; + } + + public List<String> getFormats() { + return formats; + } + + public void setFormats(List<String> formats) { + this.formats = formats; + } + + public List<String> getExcludeField() { + return excludeField; + } + + public void setExcludeField(List<String> excludeField) { + this.excludeField = excludeField; + } + + public String getMainLanguage() { + return mainLanguage; + } + + public void setMainLanguage(String mainLanguage) { + this.mainLanguage = mainLanguage; + } + + public ArrayList<String> getFields() { + return fields; + } + + public void setFields(ArrayList<String> fields) { + this.fields = fields; + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/document/Document.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,38 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.document; + +import java.util.ArrayList; + +import org.apache.lucene.document.Fieldable; + +public class Document { + private org.apache.lucene.document.Document document; + private ArrayList<String> hitFragments; + + public Document(org.apache.lucene.document.Document luceneDocument) { + this.document = luceneDocument; + } + + public org.apache.lucene.document.Document getDocument() { + return document; + } + + public Fieldable getFieldable(String field) { + if (document != null) + return document.getFieldable(field); + else + return null; + } + + public void setDocument(org.apache.lucene.document.Document document) { + this.document = document; + } + + public ArrayList<String> getHitFragments() { + return hitFragments; + } + + public void setHitFragments(ArrayList<String> hitFragments) { + this.hitFragments = hitFragments; + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/document/DocumentHandler.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,863 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.document; + +import java.io.BufferedOutputStream; +import java.io.File; +import java.io.FileNotFoundException; +import java.io.FileOutputStream; +import java.io.IOException; +import java.io.OutputStreamWriter; +import java.io.StringReader; +import java.net.MalformedURLException; +import java.net.URI; +import java.net.URISyntaxException; +import java.net.URL; +import java.util.Arrays; +import java.util.Date; +import java.util.Hashtable; +import java.util.Iterator; +import java.util.List; +import java.util.logging.Logger; + +import net.sf.saxon.s9api.Axis; +import net.sf.saxon.s9api.QName; +import net.sf.saxon.s9api.XdmNode; +import net.sf.saxon.s9api.XdmNodeKind; +import net.sf.saxon.s9api.XdmSequenceIterator; + +import org.apache.commons.httpclient.HttpClient; +import org.apache.commons.httpclient.HttpException; +import org.apache.commons.httpclient.methods.GetMethod; +import org.apache.commons.io.FileUtils; +import org.apache.http.HttpResponse; +import org.apache.http.client.methods.HttpGet; +import org.apache.http.impl.client.DefaultHttpClient; +import org.apache.http.params.BasicHttpParams; +import org.apache.http.params.HttpConnectionParams; +import org.apache.http.params.HttpParams; +import org.xml.sax.InputSource; +import org.xml.sax.SAXException; +import org.xml.sax.XMLReader; + +import com.sun.org.apache.xerces.internal.parsers.SAXParser; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.WordContentHandler; +import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.XmlTokenizer; +import de.mpg.mpiwg.berlin.mpdl.util.StringUtils; +import de.mpg.mpiwg.berlin.mpdl.util.Util; +import de.mpg.mpiwg.berlin.mpdl.xml.xquery.XQueryEvaluator; +import de.mpg.mpiwg.berlin.mpdl.cms.general.Constants; +import de.mpg.mpiwg.berlin.mpdl.cms.lucene.IndexHandler; +import de.mpg.mpiwg.berlin.mpdl.cms.scheduler.CmsDocOperation; +import de.mpg.mpiwg.berlin.mpdl.cms.transform.GetFragmentsContentHandler; +import de.mpg.mpiwg.berlin.mpdl.cms.transform.PageTransformer; +import de.mpg.mpiwg.berlin.mpdl.cms.transform.XslResourceTransformer; + +/** + * Handler for documents (singleton). + */ +public class DocumentHandler { + private static Logger LOGGER = Logger.getLogger(DocumentHandler.class.getName()); + private static List<String> EXCLUDED_PROJECT_DOCS = + Arrays.asList("/echo/zh/Min_chan_luyi_1_7MCGW0WG.xml", // the Saxon transfomer has heavy problems with some characters in CJK Unified Ideographs Extension B, e.g.: line 309 (second reg on page 16) + "/echo/zh/Min_chan_luyi_2_U7Y9NQ9V.xml", + "/echo/zh/Min_chan_luyi_3_2FP9M172.xml", + "/echo/zh/Min_chan_luyi_4_FXA6FSFH.xml", + "/echo/zh/Min_chan_luyi_5_VG6NY5XD.xml", + "/echo/zh/Xifa_shenji.xml", + "/echo/zh/Yulei_tushuo_1_HXX4MGZW.xml", + "/echo/zh/Yulei_tushuo_2_FN1CTY5C.xml"); + private long beginOfOperation; + private long endOfOperation; + + public void doOperation(CmsDocOperation docOperation) throws ApplicationException { + String operationName = docOperation.getName(); + if (operationName.equals("create")) { + create(docOperation); + } else if (operationName.equals("delete")) { + delete(docOperation); + } else if (operationName.equals("importDirectory")) { + importDirectory(docOperation); + } else if (operationName.equals("createPdf")) { + createPdf(docOperation); + } else if (operationName.equals("createAllPdfInDirectory")) { + createAllPdfInDirectory(docOperation); + } + } + + private void importDirectory(CmsDocOperation docOperation) throws ApplicationException { + try { + LOGGER.info("Start of DocumentHandler. This operation could be time consuming because documents are indexed (normal indexing times are 1-10 minutes for a document)"); + beginOperation(); + String localDocumentsUrlStr = docOperation.getSrcUrl(); // start directory: file:/a/local/directory + String collectionNames = docOperation.getCollectionNames(); // e.g. "echo" + File localDocumentsDir = new File(new URI(localDocumentsUrlStr)); + boolean docDirExists = localDocumentsDir.exists(); + if (! docDirExists) + throw new ApplicationException("Document directory:" + localDocumentsUrlStr + " does not exists. Please use a directory that exists and perform the operation again."); + String[] fileExtensions = {"xml"}; + Iterator<File> iterFiles = FileUtils.iterateFiles(localDocumentsDir, fileExtensions, true); + int i = 0; + while(iterFiles.hasNext()) { + i++; + File xmlFile = iterFiles.next(); + String xmlFileStr = xmlFile.getPath(); + int relativePos = (int) localDocumentsDir.getPath().length(); + String docId = xmlFileStr.substring(relativePos); // relative path name starting from localDocumentsDir, e.g. /tei/de/Test_1789.xml + String xmlFileUrlStr = xmlFile.toURI().toURL().toString(); + CmsDocOperation createDocOperation = new CmsDocOperation("create", xmlFileUrlStr, null, docId); + createDocOperation.setCollectionNames(collectionNames); + try { + doOperation(createDocOperation); + Date now = new Date(); + LOGGER.info("Document " + i + ": " + docId + " successfully imported (" + now.toString() + ")"); + } catch (Exception e) { + LOGGER.info("Document " + i + ": " + docId + " has problems:"); + e.printStackTrace(); + } + } + endOperation(); + LOGGER.info("The DocumentHandler needed: " + (endOfOperation - beginOfOperation) + " ms" ); + } catch (Exception e) { + throw new ApplicationException(e); + } + } + + private void createAllPdfInDirectory(CmsDocOperation docOperation) throws ApplicationException { + try { + LOGGER.info("Start of generating Pdf-Documents. This operation could be time consuming because Pdf generation needs some time."); + beginOperation(); + String localDocumentsUrlStr = docOperation.getSrcUrl(); // start directory: file:/a/local/directory + String collectionNames = docOperation.getCollectionNames(); // e.g. "echo" + File localDocumentsDir = new File(new URI(localDocumentsUrlStr)); + boolean docDirExists = localDocumentsDir.exists(); + if (! docDirExists) + throw new ApplicationException("Document directory:" + localDocumentsUrlStr + " does not exists. Please use a directory that exists and perform the operation again."); + String[] fileExtensions = {"xml"}; + Iterator<File> iterFiles = FileUtils.iterateFiles(localDocumentsDir, fileExtensions, true); + int i = 0; + while(iterFiles.hasNext()) { + i++; + File xmlFile = iterFiles.next(); + String xmlFileStr = xmlFile.getPath(); + int relativePos = (int) localDocumentsDir.getPath().length(); + String docId = xmlFileStr.substring(relativePos); // relative path name starting from localDocumentsDir, e.g. /tei/de/Test_1789.xml + CmsDocOperation createPdfOperation = new CmsDocOperation("createPdf", null, null, docId); + createPdfOperation.setCollectionNames(collectionNames); + try { + doOperation(createPdfOperation); + Date now = new Date(); + LOGGER.info("Pdf document " + i + ": " + docId + " successfully created (" + now.toString() + ")"); + } catch (Exception e) { + LOGGER.info("Pdf document " + i + ": " + docId + " has problems:"); + e.printStackTrace(); + } + } + endOperation(); + LOGGER.info("The Pdf generation needed: " + (endOfOperation - beginOfOperation) + " ms" ); + } catch (Exception e) { + throw new ApplicationException(e); + } + } + + private boolean isProjectDoc(String docId) { + boolean isProjectDoc = true; + if (EXCLUDED_PROJECT_DOCS.contains(docId)) + return false; + return isProjectDoc; + } + + private void create(CmsDocOperation docOperation) throws ApplicationException { + try { + String operationName = docOperation.getName(); + String srcUrlStr = docOperation.getSrcUrl(); + String docId = docOperation.getDocIdentifier(); + if (! isProjectDoc(docId)) { + LOGGER.info("Operation: " + operationName + " not performed on: " + docId + ". Cause: document is excluded as project doc"); + return; + } + String mainLanguage = docOperation.getMainLanguage(); + if (mainLanguage == null) { + mainLanguage = getMainLanguage(docId); + } + String[] elementNames = docOperation.getElementNames(); + if (elementNames == null) { + String[] defaultElementNames = {"s", "head", "caption", "variables", "description"}; + docOperation.setElementNames(defaultElementNames); // default + } + String docDirName = getDocDir(docId); + String docDestFileName = getDocFullFileName(docId); + URL srcUrl = null; + String protocol = null; + if (srcUrlStr != null && ! srcUrlStr.equals("empty")) { + srcUrl = new URL(srcUrlStr); + protocol = srcUrl.getProtocol(); + } + File docDestFile = new File(docDestFileName); + // parse validation on file + XQueryEvaluator xQueryEvaluator = new XQueryEvaluator(); + XdmNode docNode = xQueryEvaluator.parse(srcUrl); // if it is not parseable an exception with a detail message is thrown + String docType = getNodeType(docNode); // archimedes, echo, TEI, html ... + docType = docType.trim(); + if (docType == null) { + docOperation.setErrorMessage("file type of: " + srcUrlStr + "is not supported"); + return; + } + // perform operation on file system + if (protocol.equals("file")) { + docOperation.setStatus("upload file: " + srcUrlStr + " to CMS"); + } else { + docOperation.setStatus("download file from: " + srcUrlStr + " to CMS"); + } + FileUtils.copyURLToFile(srcUrl, docDestFile, 100000, 100000); + + // replace anchor in echo documents and also add the number attribute to figures + String docDestFileNameUpgrade = docDestFileName + ".upgrade"; + File docDestFileUpgrade = new File(docDestFileNameUpgrade); + XslResourceTransformer replaceAnchorTransformer = new XslResourceTransformer("replaceAnchor.xsl"); + String docDestFileUrlStr = docDestFile.getPath(); + String result = replaceAnchorTransformer.transform(docDestFileUrlStr); + FileUtils.writeStringToFile(docDestFileUpgrade, result, "utf-8"); + + MetadataRecord mdRecord = new MetadataRecord(); + mdRecord.setDocId(docId); + mdRecord.setCollectionNames(docOperation.getCollectionNames()); + mdRecord.setType("text/xml"); + + // generate toc file (toc, figure, handwritten) + XslResourceTransformer tocTransformer = new XslResourceTransformer("toc.xsl"); + File tocFile = new File(docDirName + "/toc.xml"); + String tocResult = tocTransformer.transform(docDestFileNameUpgrade); + FileUtils.writeStringToFile(tocFile, tocResult, "utf-8"); + + // Get metadata info of the xml document + docOperation.setStatus("extract metadata of: " + srcUrlStr + " to CMS"); + XQueryEvaluator xQueryEvaluator2 = new XQueryEvaluator(); + mdRecord = getMetadataRecord(docDestFileUpgrade, docType, mdRecord, xQueryEvaluator2); + String mdRecordLanguage = mdRecord.getLanguage(); + if (mdRecordLanguage == null && mainLanguage != null) + mdRecord.setLanguage(mainLanguage); + + // save all pages as single xml files (untokenized and tokenized) + docOperation.setStatus("extract page fragments of: " + srcUrlStr + " to CMS"); + File docDir = new File(docDirName + "/pages"); + FileUtils.deleteQuietly(docDir); // first delete pages directory + Hashtable<Integer, StringBuilder> pageFragments = getFragments(docDestFileNameUpgrade, "pb"); + int pageCount = pageFragments.size(); + if (pageCount == 0) { + // no pb element is found: then the whole document is the first page + String docXmlStr = FileUtils.readFileToString(docDestFileUpgrade, "utf-8"); + docXmlStr = docXmlStr.replaceAll("<\\?xml.*?\\?>", ""); // remove the xml declaration if it exists + pageFragments = new Hashtable<Integer, StringBuilder>(); + pageFragments.put(new Integer(1), new StringBuilder(docXmlStr)); + pageCount = 1; + } + PageTransformer pageTransformer = new PageTransformer(); + for (int page=1; page<=pageCount; page++) { + String fragment = pageFragments.get(new Integer(page)).toString(); + fragment = "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n" + fragment; + String docPageFileName = docDirName + "/pages/page-" + page + ".xml"; + File docPageFile = new File(docPageFileName); + FileUtils.writeStringToFile(docPageFile, fragment, "utf-8"); + String language = mdRecord.getLanguage(); + String tokenizedXmlStr = tokenizeWithLemmas(fragment, language); // xml fragment enriched with <w> elements + tokenizedXmlStr = "<?xml version=\"1.0\" encoding=\"utf-8\"?>" + tokenizedXmlStr; + tokenizedXmlStr = enrichWordsOrigRegNorm(tokenizedXmlStr); // xml string: enrich <w> elements with normalization info (orig, reg, norm) + String docPageTokenizedFileName = docDirName + "/pages/page-" + page + "-morph.xml"; + File docPageTokenizedFile = new File(docPageTokenizedFileName); + FileUtils.writeStringToFile(docPageTokenizedFile, tokenizedXmlStr, "utf-8"); + String docPageHtmlFileName = docDirName + "/pages/page-" + page + ".html"; + File docPageHtmlFile = new File(docPageHtmlFileName); + String htmlStr = pageTransformer.transform(tokenizedXmlStr, mdRecord, page, "html"); + FileUtils.writeStringToFile(docPageHtmlFile, htmlStr, "utf-8"); + } + + // perform operation on Lucene + docOperation.setStatus(operationName + " document: " + docId + " in CMS"); + docOperation.setMdRecord(mdRecord); + IndexHandler indexHandler = IndexHandler.getInstance(); + indexHandler.indexDocument(docOperation); + + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + private void delete(CmsDocOperation docOperation) throws ApplicationException { + String operationName = docOperation.getName(); + String docIdentifier = docOperation.getDocIdentifier(); + if (docIdentifier == null || docIdentifier.trim().equals("")) + throw new ApplicationException("Your document identifier is empty. Please specify a document identifier for your document."); + String docDirStr = getDocDir(docIdentifier); + File docDir = new File(docDirStr); + boolean docExists = docDir.exists(); + if (! docExists) { + throw new ApplicationException("Document:" + docIdentifier + " does not exists. Please use a name that exists and perform the operation \"Delete\" again."); + } + // perform operation on file system + docOperation.setStatus(operationName + " document: " + docIdentifier + " in CMS"); + FileUtils.deleteQuietly(docDir); + + // perform operation on Lucene + IndexHandler indexHandler = IndexHandler.getInstance(); + indexHandler.deleteDocument(docOperation); + + } + + private void createPdf(CmsDocOperation docOperation) throws ApplicationException { + String docId = docOperation.getDocIdentifier(); + String operationName = docOperation.getName(); + if (docId == null || docId.trim().equals("")) + throw new ApplicationException("Your document identifier is empty. Please specify a document identifier for your document."); + if (! isProjectDoc(docId)) { + LOGGER.info("Operation: " + operationName + " not performed on: " + docId + ". Cause: document is excluded as project doc"); + return; + } + IndexHandler indexHandler = IndexHandler.getInstance(); + MetadataRecord mdRecord = indexHandler.getDocMetadata(docId); + docOperation.setStatus("create PDF and HTML versions of the document: " + docId); + PdfHandler pdfHandler = PdfHandler.getInstance(); + pdfHandler.createFile(true, true, mdRecord); // generate Pdf + Html document + } + + private MetadataRecord getMetadataRecord(File xmlFile, String schemaName, MetadataRecord mdRecord, XQueryEvaluator xQueryEvaluator) throws ApplicationException { + if (schemaName == null) + return mdRecord; + try { + URL srcUrl = xmlFile.toURI().toURL(); + if (schemaName.equals("archimedes")) + mdRecord = getMetadataRecordArch(xQueryEvaluator, srcUrl, mdRecord); + else if (schemaName.equals("echo")) + mdRecord = getMetadataRecordEcho(xQueryEvaluator, srcUrl, mdRecord); + else if (schemaName.equals("TEI")) + mdRecord = getMetadataRecordTei(xQueryEvaluator, srcUrl, mdRecord); + else if (schemaName.equals("html")) + mdRecord = getMetadataRecordHtml(xQueryEvaluator, srcUrl, mdRecord); + else + mdRecord.setSchemaName("diverse"); // all other cases: set docType to schemaName + } catch (MalformedURLException e) { + throw new ApplicationException(e); + } + mdRecord.setLastModified(new Date()); + return mdRecord; + } + + private MetadataRecord getMetadataRecordArch(XQueryEvaluator xQueryEvaluator, URL srcUrl, MetadataRecord mdRecord) throws ApplicationException { + String metadataXmlStr = xQueryEvaluator.evaluateAsString(srcUrl, "/archimedes//info"); + if (metadataXmlStr != null) { + String identifier = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/info/locator"); + if (identifier != null) + identifier = StringUtils.deresolveXmlEntities(identifier); + String creator = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/info/author"); + if (creator != null) + creator = StringUtils.deresolveXmlEntities(creator); + String title = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/info/title"); + if (title != null) + title = StringUtils.deresolveXmlEntities(title); + String language = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/info/lang[1]"); + if (language != null) + language = StringUtils.deresolveXmlEntities(language); + String place = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/info/place"); + if (place != null) + place = StringUtils.deresolveXmlEntities(place); + String yearStr = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/info/date"); + Date date = null; + if (yearStr != null && ! yearStr.equals("")) { + yearStr = StringUtils.deresolveXmlEntities(yearStr); + yearStr = new Util().toYearStr(yearStr); // test if possible etc + if (yearStr != null) { + try { + date = new Util().toDate(yearStr + "-01-01T00:00:00.000Z"); + } catch (Exception e) { + // nothing + } + } + } + String rights = "open access"; + String license = "http://echo.mpiwg-berlin.mpg.de/policy/oa_basics/declaration"; + String accessRights = "free"; + + mdRecord.setIdentifier(identifier); + mdRecord.setLanguage(language); + mdRecord.setCreator(creator); + mdRecord.setTitle(title); + mdRecord.setPublisher(place); + mdRecord.setRights(rights); + mdRecord.setDate(date); + mdRecord.setLicense(license); + mdRecord.setAccessRights(accessRights); + + // get echo metadata + String echoDir = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/info/echodir"); + String docId = mdRecord.getDocId(); + String echoIdTmp = docId; + if (docId != null && ! docId.isEmpty()) { + int start = docId.lastIndexOf("/"); + if (start != -1) + start = start + 1; + else + start = 0; + int end = docId.lastIndexOf("."); + if (end == -1) + end = docId.length(); + echoIdTmp = docId.substring(start, end); + } + String echoId = "/permanent/archimedes/" + echoIdTmp; + if (echoDir != null && ! echoDir.isEmpty()) { + echoId = echoDir; + } + mdRecord = getEchoMetadata(xQueryEvaluator, echoId, mdRecord); + } + String pageCountStr = xQueryEvaluator.evaluateAsString(srcUrl, "count(//pb)"); + int pageCount = Integer.valueOf(pageCountStr); + mdRecord.setPageCount(pageCount); + mdRecord.setSchemaName("archimedes"); + return mdRecord; + } + + private MetadataRecord getMetadataRecordEcho(XQueryEvaluator xQueryEvaluator, URL srcUrl, MetadataRecord mdRecord) throws ApplicationException { + String metadataXmlStr = xQueryEvaluator.evaluateAsString(srcUrl, "/*:echo/*:metadata"); + if (metadataXmlStr != null) { + String identifier = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/*:metadata/*:identifier"); + if (identifier != null) { + identifier = StringUtils.deresolveXmlEntities(identifier); + } + String creator = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/*:metadata/*:creator"); + if (creator != null) + creator = StringUtils.deresolveXmlEntities(creator); + String title = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/*:metadata/*:title"); + if (title != null) + title = StringUtils.deresolveXmlEntities(title); + String language = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/*:metadata/*:language[1]"); + if (language != null) + language = StringUtils.deresolveXmlEntities(language); + String yearStr = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/*:metadata/*:date"); + Date date = null; + if (yearStr != null && ! yearStr.equals("")) { + yearStr = StringUtils.deresolveXmlEntities(yearStr); + yearStr = new Util().toYearStr(yearStr); // test if possible etc + if (yearStr != null) { + try { + date = new Util().toDate(yearStr + "-01-01T00:00:00.000Z"); + } catch (Exception e) { + // nothing + } + } + } + String rights = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/*:metadata/*:rights"); + if (rights != null) + rights = StringUtils.deresolveXmlEntities(rights); + String license = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/*:metadata/*:license"); + if (license != null) + license = StringUtils.deresolveXmlEntities(license); + String accessRights = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/*:metadata/*:accessRights"); + if (accessRights != null) + accessRights = StringUtils.deresolveXmlEntities(accessRights); + + mdRecord.setIdentifier(identifier); + mdRecord.setLanguage(language); + mdRecord.setCreator(creator); + mdRecord.setTitle(title); + mdRecord.setRights(rights); + mdRecord.setDate(date); + mdRecord.setLicense(license); + mdRecord.setAccessRights(accessRights); + + // get echo metadata + String echoDir = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/*:metadata/*:echodir"); + String echoIdTmp = identifier; + if (identifier != null && ! identifier.isEmpty()) { + int start = identifier.indexOf("ECHO:"); + if (start != -1) + start = start + 5; + else + start = 0; + int end = identifier.lastIndexOf("."); + if (end == -1) + end = identifier.length(); + echoIdTmp = identifier.substring(start, end); + } + String echoId = "/permanent/library/" + echoIdTmp; + if (echoDir != null && ! echoDir.isEmpty()) { + echoId = echoDir; + } + mdRecord = getEchoMetadata(xQueryEvaluator, echoId, mdRecord); + } + String pageCountStr = xQueryEvaluator.evaluateAsString(srcUrl, "count(//*:pb)"); + int pageCount = Integer.valueOf(pageCountStr); + mdRecord.setPageCount(pageCount); + mdRecord.setSchemaName("echo"); + return mdRecord; + } + + private MetadataRecord getMetadataRecordTei(XQueryEvaluator xQueryEvaluator, URL srcUrl, MetadataRecord mdRecord) throws ApplicationException { + String metadataXmlStr = xQueryEvaluator.evaluateAsString(srcUrl, "/*:TEI/*:teiHeader"); + if (metadataXmlStr != null) { + String identifier = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/*:teiHeader/*:fileDesc/*:publicationStmt/*:idno"); + if (identifier != null) + identifier = StringUtils.deresolveXmlEntities(identifier); + String creator = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/*:teiHeader/*:fileDesc/*:titleStmt/*:author"); + if (creator != null) + creator = StringUtils.deresolveXmlEntities(creator); + String title = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/*:teiHeader/*:fileDesc/*:titleStmt/*:title"); + if (title != null) + title = StringUtils.deresolveXmlEntities(title); + String language = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "string(/*:teiHeader/*:profileDesc/*:langUsage/*:language[1]/@ident)"); + if (language != null && language.isEmpty()) + language = null; + if (language != null) + language = StringUtils.deresolveXmlEntities(language); + String place = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/*:teiHeader/*:fileDesc/*:publicationStmt/*:pubPlace"); + if (place != null) + place = StringUtils.deresolveXmlEntities(place); + String yearStr = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/*:teiHeader/*:fileDesc/*:publicationStmt/*:date"); + Date date = null; + if (yearStr != null && ! yearStr.equals("")) { + yearStr = StringUtils.deresolveXmlEntities(yearStr); + yearStr = new Util().toYearStr(yearStr); // test if possible etc + if (yearStr != null) { + try { + date = new Util().toDate(yearStr + "-01-01T00:00:00.000Z"); + } catch (Exception e) { + // nothing + } + } + } + String subject = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "string(/*:teiHeader/*:profileDesc/*:textClass/*:keywords/*:term)"); + if (subject != null) + subject = StringUtils.deresolveXmlEntities(subject); + String rights = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "/*:teiHeader/*:fileDesc/*:publicationStmt/*:availability"); + if (rights == null) + rights = "open access"; + rights = StringUtils.deresolveXmlEntities(rights); + String license = "http://echo.mpiwg-berlin.mpg.de/policy/oa_basics/declaration"; + String accessRights = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "string(/*:teiHeader/*:fileDesc/*:publicationStmt/*:availability/@status)"); + if (accessRights == null) + accessRights = "free"; + accessRights = StringUtils.deresolveXmlEntities(accessRights); + + mdRecord.setIdentifier(identifier); + mdRecord.setLanguage(language); + mdRecord.setCreator(creator); + mdRecord.setTitle(title); + mdRecord.setPublisher(place); + mdRecord.setRights(rights); + mdRecord.setDate(date); + mdRecord.setSubject(subject); + mdRecord.setLicense(license); + mdRecord.setAccessRights(accessRights); + + // get echo metadata + mdRecord = getEchoMetadata(xQueryEvaluator, identifier, mdRecord); // identifier is echoDir + } + String pageCountStr = xQueryEvaluator.evaluateAsString(srcUrl, "count(//*:pb)"); + int pageCount = Integer.valueOf(pageCountStr); + mdRecord.setPageCount(pageCount); + mdRecord.setSchemaName("TEI"); + return mdRecord; + } + + private MetadataRecord getMetadataRecordHtml(XQueryEvaluator xQueryEvaluator, URL srcUrl, MetadataRecord mdRecord) throws ApplicationException { + String metadataXmlStr = xQueryEvaluator.evaluateAsString(srcUrl, "/html/head"); + if (metadataXmlStr != null) { + String identifier = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "string(/meta[@name = 'DC.identifier']/@content)"); + if (identifier != null && ! identifier.isEmpty()) + identifier = StringUtils.deresolveXmlEntities(identifier); + String creator = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "string(/meta[@name = 'DC.creator']/@content)"); + if (creator != null && ! creator.isEmpty()) + creator = StringUtils.deresolveXmlEntities(creator); + String title = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "string(/meta[@name = 'DC.title']/@content)"); + if (title != null && ! title.isEmpty()) + title = StringUtils.deresolveXmlEntities(title); + String language = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "string(/meta[@name = 'DC.language']/@content)"); + if (language != null && language.isEmpty()) + language = null; + if (language != null && ! language.isEmpty()) + language = StringUtils.deresolveXmlEntities(language); + String publisher = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "string(/meta[@name = 'DC.publisher']/@content)"); + if (publisher != null) + publisher = StringUtils.deresolveXmlEntities(publisher); + String yearStr = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "string(/meta[@name = 'DC.date']/@content)"); + Date date = null; + if (yearStr != null && ! yearStr.equals("")) { + yearStr = StringUtils.deresolveXmlEntities(yearStr); + yearStr = new Util().toYearStr(yearStr); // test if possible etc + if (yearStr != null) { + try { + date = new Util().toDate(yearStr + "-01-01T00:00:00.000Z"); + } catch (Exception e) { + // nothing + } + } + } + String subject = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "string(/meta[@name = 'DC.subject']/@content)"); + if (subject != null) + subject = StringUtils.deresolveXmlEntities(subject); + String rights = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "string(/meta[@name = 'DC.rights']/@content)"); + if (rights != null && ! rights.isEmpty()) + rights = StringUtils.deresolveXmlEntities(rights); + String license = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "string(/meta[@name = 'DC.license']/@content)"); + if (license != null && ! license.isEmpty()) + license = StringUtils.deresolveXmlEntities(license); + String accessRights = xQueryEvaluator.evaluateAsStringValueJoined(metadataXmlStr, "string(/meta[@name = 'DC.accessRights']/@content)"); + if (accessRights != null && ! accessRights.isEmpty()) + accessRights = StringUtils.deresolveXmlEntities(accessRights); + + mdRecord.setIdentifier(identifier); + mdRecord.setLanguage(language); + mdRecord.setCreator(creator); + mdRecord.setTitle(title); + mdRecord.setPublisher(publisher); + mdRecord.setRights(rights); + mdRecord.setDate(date); + mdRecord.setSubject(subject); + mdRecord.setLicense(license); + mdRecord.setAccessRights(accessRights); + + // get echo metadata + mdRecord = getEchoMetadata(xQueryEvaluator, identifier, mdRecord); // identifier is echoDir + } + String pageCountStr = xQueryEvaluator.evaluateAsString(srcUrl, "count(//pb)"); + int pageCount = Integer.valueOf(pageCountStr); + mdRecord.setPageCount(pageCount); + mdRecord.setSchemaName("html"); + return mdRecord; + } + + private MetadataRecord getEchoMetadata(XQueryEvaluator xQueryEvaluator, String echoDir, MetadataRecord mdRecord) throws ApplicationException { + if (echoDir == null) + return mdRecord; + String urLTexter = "http://digilib.mpiwg-berlin.mpg.de/digitallibrary/servlet/Texter?fn=" + echoDir + "/index.meta"; + String echoIndexMetaStr = performGetRequest(urLTexter); + String echoPageImageDir = null; + String echoFiguresDir = null; + String mpiwgDocId = null; + if (echoIndexMetaStr != null) { + if (echoIndexMetaStr.equals("XXXXTimeoutXXXX")) + return null; + echoPageImageDir = xQueryEvaluator.evaluateAsStringValueJoined(echoIndexMetaStr, "/resource/meta/texttool/image"); + if (echoPageImageDir != null) + echoPageImageDir = echoDir + "/" + echoPageImageDir; + else + echoPageImageDir = echoDir + "/" + "pageimg"; // default + echoFiguresDir = xQueryEvaluator.evaluateAsStringValueJoined(echoIndexMetaStr, "/resource/meta/texttool/figures"); + if (echoFiguresDir != null) + echoFiguresDir = echoDir + "/" + echoFiguresDir; + else + echoFiguresDir = echoDir + "/" + "figures"; // default + mpiwgDocId = xQueryEvaluator.evaluateAsStringValueJoined(echoIndexMetaStr, "/resource/meta/dri[@type = 'mpiwg']"); + } + mdRecord.setEchoId(echoDir); + mdRecord.setEchoPageImageDir(echoPageImageDir); + mdRecord.setEchoFiguresDir(echoFiguresDir); + mdRecord.setMpiwgDocId(mpiwgDocId); + return mdRecord; + } + + private String getNodeType(XdmNode node) { + String nodeType = null; + XdmSequenceIterator iter = node.axisIterator(Axis.CHILD); + if (iter != null) { + while (iter.hasNext()) { + XdmNode firstChild = (XdmNode) iter.next(); + if (firstChild != null) { + XdmNodeKind nodeKind = firstChild.getNodeKind(); + if (nodeKind.ordinal() == XdmNodeKind.ELEMENT.ordinal()) { + QName nodeQName = firstChild.getNodeName(); + nodeType = nodeQName.getLocalName(); + } + } + } + } + return nodeType; + } + + public String getDocFullFileName(String docId) { + String docDir = getDocDir(docId); + String docFileName = getDocFileName(docId); + String docFullFileName = docDir + "/" + docFileName; + return docFullFileName; + } + + public String getFullFileName(String docId, String type) { + String docDir = getDocDir(docId); + String docFileName = getDocFileName(docId); + int lastDot = docFileName.lastIndexOf("."); + String docFileNameWithoutExtension = docFileName.substring(0, lastDot); + String fullFileName = docDir + "/" + docFileNameWithoutExtension + ".xml"; + if (type != null && ! type.equals("toc")) { + fullFileName = docDir + "/" + docFileNameWithoutExtension + "." + type; + } else if (type != null && type.equals("toc")) { + fullFileName = docDir + "/toc.xml"; + } + return fullFileName; + } + + public String getDocDir(String docId) { + String documentsDirectory = Constants.getInstance().getDocumentsDir(); + String subDir = docId; + if (docId.contains(".")) { + int index = docId.lastIndexOf("."); + subDir = docId.substring(0, index); + } + if (! subDir.startsWith("/")) + subDir = "/" + subDir; + String docDir = documentsDirectory + subDir; + return docDir; + } + + public String getDocFileName(String docId) { + String docFileName = docId; + int index = docId.lastIndexOf("/"); + if (index != -1) { + docFileName = docId.substring(index + 1); + } + return docFileName; + } + + private String getMainLanguage(String docId) { + String mainLang = null; + int to = docId.lastIndexOf("/"); + if (to != -1) { + String preStr = docId.substring(0, to); + int from = preStr.lastIndexOf("/"); + if (from != -1) + mainLang = preStr.substring(from + 1, to); + } + return mainLang; + } + + private Hashtable<Integer, StringBuilder> getFragments(String fileName, String milestoneElementName) throws ApplicationException { + try { + GetFragmentsContentHandler getFragmentsContentHandler = new GetFragmentsContentHandler(milestoneElementName); + XMLReader xmlParser = new SAXParser(); + xmlParser.setContentHandler(getFragmentsContentHandler); + StringReader bla = new StringReader(FileUtils.readFileToString(new File(fileName), "utf-8")); + InputSource inputSource = new InputSource(bla); + xmlParser.parse(inputSource); + Hashtable<Integer, StringBuilder> resultFragments = getFragmentsContentHandler.getResultPages(); + return resultFragments; + } catch (SAXException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + private String tokenizeWithLemmas(String xmlStr, String language) throws ApplicationException { + StringReader strReader = new StringReader(xmlStr); + XmlTokenizer xmlTokenizer = new XmlTokenizer(strReader); + xmlTokenizer.setLanguage(language); + String[] outputOptionsWithLemmas = {"withLemmas"}; // so all tokens are fetched with lemmas (costs performance) + String[] nwbElements = {"lb", "br", "cb", "hi"}; // non word breaking elements; TODO examine bugs with emph, figure e.g. page 30 in /echo/la/Cataneo_1600.xml + xmlTokenizer.setNWBElements(nwbElements); + xmlTokenizer.setOutputOptions(outputOptionsWithLemmas); + xmlTokenizer.tokenize(); + String retStr = xmlTokenizer.getXmlResult(); + return retStr; + } + + private String enrichWordsOrigRegNorm(String xmlStr) throws ApplicationException { + try { + WordContentHandler wordContentHandler = new WordContentHandler(); + XMLReader xmlParser = new SAXParser(); + xmlParser.setContentHandler(wordContentHandler); + StringReader strReader = new StringReader(xmlStr); + InputSource inputSource = new InputSource(strReader); + xmlParser.parse(inputSource); + String result = wordContentHandler.getResult(); + return result; + } catch (SAXException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + private String performGetRequest(String url) throws ApplicationException { + String resultStr = null; + try { + boolean urlIsOk = checkUri(url, 2000); // if url doesn't answer after 2 seconds + if (! urlIsOk) + return "XXXXTimeoutXXXX"; + HttpClient httpClient = new HttpClient(); + GetMethod method = new GetMethod(url); + httpClient.executeMethod(method); + byte[] resultBytes = method.getResponseBody(); + resultStr = new String(resultBytes, "utf-8"); + method.releaseConnection(); + } catch (HttpException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + return resultStr; + } + + private boolean checkUri(String uriStr, int timeoutMilliseconds) throws ApplicationException { + boolean isOk = true; + try { + URI uri = new URI(uriStr); + HttpGet httpGet = new HttpGet(uri); + HttpParams httpParameters = new BasicHttpParams(); + // Set the timeout in milliseconds until a connection is established. + // The default value is zero, that means the timeout is not used. + int timeoutConnection = 2000; + HttpConnectionParams.setConnectionTimeout(httpParameters, timeoutConnection); + // Set the default socket timeout (SO_TIMEOUT) + // in milliseconds which is the timeout for waiting for data. + int timeoutSocket = 2000; + HttpConnectionParams.setSoTimeout(httpParameters, timeoutSocket); + DefaultHttpClient httpClient = new DefaultHttpClient(httpParameters); + HttpResponse response = httpClient.execute(httpGet); + } catch (IOException e) { + isOk = false; // if timeout exception is thrown + } catch (URISyntaxException e) { + throw new ApplicationException(e); + } + return isOk; + } + + /** + * Write string into destFile. If directory for that destFile does not exist + * it creates this directory including parent directories. + * @param str string to write + * @param destFileName destination file name + * @throws ApplicationException + */ + private void saveFile(String str, String destFileName) throws ApplicationException { + OutputStreamWriter out = null; + try { + if (str == null) + return; // do nothing + File destFile = new File(destFileName); + File destDir = new File(destFile.getParent()); + if (! destDir.exists()) { + destDir.mkdirs(); // create the directory including parent directories which do not exist + } + out = new OutputStreamWriter(new BufferedOutputStream(new FileOutputStream(destFile)), "utf-8"); + out.write(str); + out.flush(); + } catch (FileNotFoundException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } finally { + try { + if (out != null) + out.close(); + } catch (Exception e) { + // nothing: always close the stream at the end of the method + } + } + } + + private void beginOperation() { + beginOfOperation = new Date().getTime(); + } + + private void endOperation() { + endOfOperation = new Date().getTime(); + } + +} \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/document/Hits.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,60 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.document; + +import java.util.ArrayList; + +import org.apache.lucene.search.Query; + +public class Hits { + private ArrayList<Document> hits; + private Query query; + private int from; + private int to; + private int size = 0; + + public Hits(ArrayList<Document> hits, int from, int to) { + this.hits = hits; + this.from = from; + this.to = to; + } + + public int getSize() { + return size; + } + + public void setSize(int size) { + this.size = size; + } + + public Query getQuery() { + return query; + } + + public void setQuery(Query query) { + this.query = query; + } + + public ArrayList<Document> getHits() { + return hits; + } + + public void setHits(ArrayList<Document> hits) { + this.hits = hits; + } + + public int getFrom() { + return from; + } + + public void setFrom(int from) { + this.from = from; + } + + public int getTo() { + return to; + } + + public void setTo(int to) { + this.to = to; + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/document/MetadataRecord.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,217 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.document; + +import java.util.Calendar; +import java.util.Date; + +public class MetadataRecord { + private String docId; // local id: document identifier in index system, e.g. /echo/la/Benedetti_1585.xml + private String identifier; // local id: identifier field in documents metadata: e.g. /echo:echo/echo:metadata/dcterms:identifier + private String uri; // global id: document URI (uniform resource identifier), e.g. http://de.wikipedia.org/wiki/Ramones + private String language; + private String creator; // author + private String title; + private String description; // abstract etc. + private String subject; // subject keywords from the title or description or content or subject lists (thesaurus etc.) + private String publisher; // publisher with place: e.g. Springer, New York + private String type; // mime type: e.g. text/xml // TODO ist eigentlich das Feld "format" --> zus. instance variable "format" definieren + private String rights; // e.g. open access + private Date date; // creation date, modification date, etc. + private String license; // e.g. http://echo.mpiwg-berlin.mpg.de/policy/oa_basics/declaration + private String accessRights; // e.g. free + private String collectionNames; // e.g. "collection1 collection7" + private String schemaName; // e.g. TEI, echo, html, or archimedes + private Date lastModified; + private int pageCount; + private String echoId; // document identifier in echo system: directory name, e.g. /permanent/library/163127KK + private String echoPageImageDir; // document page image directory in echo system: directory name, e.g. /permanent/library/163127KK/pageimg + private String echoFiguresDir; // document figures directory in echo system: directory name, e.g. /permanent/library/163127KK/figures + private String mpiwgDocId; // mpiwg docId in echo system: e.g. MPIWG:U7FWNX06 + + public String getDocId() { + return docId; + } + + public void setDocId(String docId) { + this.docId = docId; + } + + public String getUri() { + return uri; + } + + public void setUri(String uri) { + this.uri = uri; + } + + public String getRights() { + return rights; + } + + public void setRights(String rights) { + this.rights = rights; + } + + public int getPageCount() { + return pageCount; + } + + public void setPageCount(int pageCount) { + this.pageCount = pageCount; + } + + public String getLicense() { + return license; + } + + public void setLicense(String license) { + this.license = license; + } + + public String getAccessRights() { + return accessRights; + } + + public void setAccessRights(String accessRights) { + this.accessRights = accessRights; + } + + public String getCreator() { + return creator; + } + + public void setCreator(String creator) { + this.creator = creator; + } + + public String getTitle() { + return title; + } + + public void setTitle(String title) { + this.title = title; + } + + public Date getDate() { + return date; + } + + public void setDate(Date date) { + this.date = date; + } + + public String getYear() { + String year = null; + if (date != null) { + Calendar cal = Calendar.getInstance(); + cal.setTime(date); + int iYear = cal.get(Calendar.YEAR); + year = "" + iYear; + } + return year; + } + + public String getDescription() { + return description; + } + + public void setDescription(String description) { + this.description = description; + } + + public String getSubject() { + return subject; + } + + public void setSubject(String subject) { + this.subject = subject; + } + + public String getCollectionNames() { + return collectionNames; + } + + public void setCollectionNames(String collectionNames) { + this.collectionNames = collectionNames; + } + + public String getIdentifier() { + return identifier; + } + + public void setIdentifier(String identifier) { + this.identifier = identifier; + } + + public String getLanguage() { + return language; + } + + public void setLanguage(String language) { + this.language = language; + } + + public String getPublisher() { + return publisher; + } + + public void setPublisher(String publisher) { + this.publisher = publisher; + } + + public String getType() { + return type; + } + + public void setType(String type) { + this.type = type; + } + + public String getSchemaName() { + return schemaName; + } + + public void setSchemaName(String schemaName) { + this.schemaName = schemaName; + } + + public Date getLastModified() { + return lastModified; + } + + public void setLastModified(Date lastModified) { + this.lastModified = lastModified; + } + + public String getEchoId() { + return echoId; + } + + public void setEchoId(String echoId) { + this.echoId = echoId; + } + + public String getEchoPageImageDir() { + return echoPageImageDir; + } + + public void setEchoPageImageDir(String echoPageImageDir) { + this.echoPageImageDir = echoPageImageDir; + } + + public String getEchoFiguresDir() { + return echoFiguresDir; + } + + public void setEchoFiguresDir(String echoFiguresDir) { + this.echoFiguresDir = echoFiguresDir; + } + + public String getMpiwgDocId() { + return mpiwgDocId; + } + + public void setMpiwgDocId(String mpiwgDocId) { + this.mpiwgDocId = mpiwgDocId; + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/document/PdfHandler.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,387 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.document; + +import java.io.File; +import java.io.FileNotFoundException; +import java.io.FileOutputStream; +import java.io.IOException; +import java.io.OutputStream; +import java.io.UnsupportedEncodingException; +import java.util.Hashtable; + +import org.apache.commons.httpclient.HttpClient; +import org.apache.commons.httpclient.HttpException; +import org.apache.commons.httpclient.methods.GetMethod; +import org.apache.commons.io.FileUtils; +import org.xhtmlrenderer.layout.SharedContext; +import org.xhtmlrenderer.pdf.ITextFontResolver; +import org.xhtmlrenderer.pdf.ITextRenderer; +import org.xhtmlrenderer.util.XRRuntimeException; + +import com.lowagie.text.DocumentException; +import com.lowagie.text.pdf.BaseFont; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.cms.general.Constants; +import de.mpg.mpiwg.berlin.mpdl.cms.transform.TocTransformer; + +public class PdfHandler { + private static PdfHandler instance; + private static String CSS_SHOW_WORD_URL = "http://thrax-dev.rz-berlin.mpg.de/mpiwg-mpdl-cms-web/css/pageOrig.css"; + private static String CSS_URL = "http://thrax-dev.rz-berlin.mpg.de/mpiwg-mpdl-cms-web/css/page.css"; + // private static String CSS_DOCUVIEWER_URL = "http://echo.mpiwg-berlin.mpg.de/ECHOdocuView/template/docuviewer_css"; + private ITextRenderer renderer = new ITextRenderer(); + private Hashtable<String, String> fontFileNames; + private DocumentHandler docHandler; + private TocTransformer tocTransformer; + + public static PdfHandler getInstance() throws ApplicationException { + if (instance == null) { + instance = new PdfHandler(); + instance.init(); + } + return instance; + } + + public void init() throws ApplicationException { + renderer = new ITextRenderer(); + SharedContext rendererSharedContext = renderer.getSharedContext(); + PdfHandlerUserAgent mpdlUserAgent = new PdfHandlerUserAgent(); // user agent to get a callback handle to the web access of images (getImageResource(url)) + mpdlUserAgent.setSharedContext(rendererSharedContext); + rendererSharedContext.setUserAgentCallback(mpdlUserAgent); + fontFileNames = new Hashtable<String, String>(); + String fontJunicodeFileName = Constants.getInstance().getDocumentsDir() + "/../fonts/Junicode-Regular.ttf"; + String fontJunicodeBoldFileName = Constants.getInstance().getDocumentsDir() + "/../fonts/Junicode-Bold.ttf"; + String fontJunicodeItalicFileName = Constants.getInstance().getDocumentsDir() + "/../fonts/Junicode-Italic.ttf"; + String fontJunicodeBoldItalicFileName = Constants.getInstance().getDocumentsDir() + "/../fonts/Junicode-BoldItalic.ttf"; + String fontSunExtAFileName = Constants.getInstance().getDocumentsDir() + "/../fonts/Sun-ExtA.ttf"; // chinese symbols + String fontSunExtBFileName = Constants.getInstance().getDocumentsDir() + "/../fonts/Sun-ExtB.ttf"; // chinese symbols + String fontDejaVuFileName = Constants.getInstance().getDocumentsDir() + "/../fonts/DejaVuSans.ttf"; // arabic symbols + setFont(fontJunicodeFileName); + setFont(fontJunicodeBoldFileName); + setFont(fontJunicodeItalicFileName); + setFont(fontJunicodeBoldItalicFileName); // if set then some not bold italic characters are shown bold (e.g. in Benedetti_1585.xml) + setFont(fontSunExtAFileName); + setFont(fontSunExtBFileName); + setFont(fontDejaVuFileName); + docHandler = new DocumentHandler(); + tocTransformer = new TocTransformer(); + } + + public void createFile(boolean pdf, boolean html, MetadataRecord mdRecord) throws ApplicationException { + OutputStream osPdf = null; + OutputStream osHtml = null; + OutputStream osHtmlPdf = null; + String docId = mdRecord.getDocId(); + String language = mdRecord.getLanguage(); + if (docId == null) + throw new ApplicationException("Pdf/Html-Generation failed: no docId given in mdRecord"); + String docDir = docHandler.getDocDir(docId); + String docFileName = docHandler.getDocFileName(docId); + int lastDot = docFileName.lastIndexOf("."); + String docFileNameWithoutExtension = docFileName.substring(0, lastDot); + String docIdExtension = docFileName.substring(lastDot + 1); + String destFileNamePdf = docDir + "/" + docFileNameWithoutExtension + ".pdf"; + String destFileNameHtml = docDir + "/" + docFileNameWithoutExtension + ".html"; + if (docIdExtension != null && docIdExtension.equals("html")) { + destFileNameHtml = docDir + "/" + docFileNameWithoutExtension + "-gen.html"; + } + String destFileNameHtmlPdfTmp = docDir + "/" + docFileNameWithoutExtension + "-4Pdf.html"; + try { + // start document + if (pdf) { + osPdf = new FileOutputStream(new File(destFileNamePdf)); + osHtmlPdf = new FileOutputStream(new File(destFileNameHtmlPdfTmp)); + } + if (html) + osHtml = new FileOutputStream(new File(destFileNameHtml)); + int countPages = mdRecord.getPageCount(); + // style page + String pageStyleHtml = "float:left; clear:both; border: thin solid #808080; width: 21.0cm; margin-top: 0.2cm; margin-bottom: 1cm; margin-left: 0.7cm; margin-right: 0.7cm; padding: 0.2cm;"; + // firstPage + String firstPageHtml = getFirstPageHtml(mdRecord); + String mdRecordStr = getMdRecordString(mdRecord); + String htmlHeadStr = getHtmlHead(null, mdRecordStr); + String fontStyle = getFontStyle(language); + if(pdf) { + write("<html>" + htmlHeadStr + "<body style=\"" + fontStyle + "\">", osHtmlPdf); + // first page + write(firstPageHtml, osHtmlPdf); + } + if (html) { + write("<html>" + htmlHeadStr + "<body style=\"" + fontStyle + "\">", osHtml); + // first page + write("<div style=\"" + pageStyleHtml + "\">", osHtml); + write(firstPageHtml, osHtml); + write("</div>", osHtml); + } + // table of content of document + String htmlToc = getTocHtml(mdRecord); + if (html && htmlToc != null) { + write("<div style=\"" + pageStyleHtml + "\">", osHtml); + write(htmlToc, osHtml); + write("</div>", osHtml); + } + if(pdf && htmlToc != null) { + write(htmlToc, osHtmlPdf); + } + // all pages of the document + for(int i=1; i<=countPages; i++) { + String htmlPageFragment = getPageFragmentHtml(mdRecord, i); + htmlPageFragment = "<div id=\"page" + i + "\" class=\"page\">" + htmlPageFragment + "</div>"; + if (html) { + write("<div style=\"" + "clear:both; text-align:right; width:21.0cm; font-weight:bold;" + "\">", osHtml); + write("</div>", osHtml); + write("<div style=\"" + pageStyleHtml + "\">", osHtml); + // String htmlPageFragmentWithImageUrl = htmlPageFragment.replaceAll("<img src=\".+?\"/>", "<img height=\"24\" width=\"24\" src=\"http://" + CMS_WEB_SERVER_PDF + "/" + CMS_WEB_APP_PDF + "/images/camera.png\"/>"); + write(htmlPageFragment, osHtml); + write("</div>", osHtml); + } + if(pdf) { + // String htmlPageFragmentWithImageUrl = htmlPageFragment.replaceAll("<img src=\".+?\"/>", "<img height=\"24\" width=\"24\" src=\"http://" + CMS_WEB_SERVER_PDF + "/" + CMS_WEB_APP_PDF + "/images/camera.png\"/>"); + write(htmlPageFragment, osHtmlPdf); + } + } + if (html) { + write("</body></html>", osHtml); + } + // create PDF document + if(pdf) { + write("</body></html>", osHtmlPdf); + osHtmlPdf.close(); + renderer.setDocument(new File(destFileNameHtmlPdfTmp)); + renderer.layout(); // takes the most time + renderer.createPDF(osPdf); + } + } catch (Exception e) { + init(); + String message = e.getMessage(); + if (message != null && message.indexOf("digilib") > 0 && message.indexOf("500") > 0) { + throw new ApplicationException("fetch image is not possible: " + message); + } + throw new ApplicationException(e); + } finally { + try { + osHtmlPdf.close(); + osPdf.close(); + osHtml.close(); + FileUtils.deleteQuietly(new File(destFileNameHtmlPdfTmp)); + } catch (IOException e) { + // nothing + } + } + } + + private String getFirstPageHtml(MetadataRecord mdRecord) { + String firstPageHtml = null; + try { + // Url to Echo viewer + String echoId = mdRecord.getEchoId(); + String urlDocuView = "http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=" + echoId + "&viewMode=indexonly"; + String wholeFirstPageHtml = performGetRequest(urlDocuView); + int from = wholeFirstPageHtml.indexOf("<body>"); + int to = wholeFirstPageHtml.indexOf("</body>"); + if (from != -1 && to != -1) { + firstPageHtml = "<div>" + wholeFirstPageHtml.substring(from + 6, to) + "</div>"; + } + } catch (ApplicationException e) { + // nothing + } + return firstPageHtml; + } + + private String getFirstPageHtml(MetadataRecord mdRecord, boolean shortPage) { + String author = mdRecord.getCreator(); + String title = mdRecord.getTitle(); + String year = mdRecord.getYear(); + String firstPageHtml = "<div class=\"firstPage\">"; + firstPageHtml = firstPageHtml + "<h2 style=\"text-align:center\">" + "Max Planck Institute for the History of Science" + "</h2>"; + firstPageHtml = firstPageHtml + "<p style=\"text-align:center\">" + "Max-Planck-Institut fŸr Wissenschaftsgeschichte" + "</p>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + if (! shortPage) { + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + } + if (author != null) { + firstPageHtml = firstPageHtml + "<h2 style=\"text-align:center\">" + author + "</h2>"; + } + if (title != null) { + firstPageHtml = firstPageHtml + "<h2 style=\"text-align:center\">" + title + "</h2>"; + } + if (year != null) { + firstPageHtml = firstPageHtml + "<h2 style=\"text-align:center\">" + year + "</h2>"; + } + if (! shortPage) { + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + } + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + firstPageHtml = firstPageHtml + "<br></br>"; + // Url to Echo viewer + String echoId = mdRecord.getEchoId(); + String urlDocuView = "http://echo.mpiwg-berlin.mpg.de/ECHOdocuView"; + String document = "?url=" + echoId; + String urlDoc = urlDocuView + document; + String echoLink = "<a href=\"" + urlDoc + "\">" + urlDocuView + " <br></br>" + document + "</a>"; + if (echoId == null) + echoLink = "<a href=\"http://echo.mpiwg-berlin.mpg.de\">" + "http://echo.mpiwg-berlin.mpg.de" + "</a>"; + firstPageHtml = firstPageHtml + "<p style=\"font:11pt sans-serif;\">Document link: <br></br>" + echoLink + "</p>"; + firstPageHtml = firstPageHtml + "</div>"; + return firstPageHtml; + } + + private String getTocHtml(MetadataRecord mdRecord) throws ApplicationException { + String htmlStr = null; + try { + String docId = mdRecord.getDocId(); + String tocFileName = docHandler.getFullFileName(docId, "toc"); + File tocFile = new File(tocFileName); + String tocStr = FileUtils.readFileToString(tocFile, "utf-8"); + String htmlToc = tocTransformer.transform(tocStr, "toc", "html"); + if (htmlToc != null) { + htmlStr = "<div class=\"tocPage\">" + "<text style=\"font-weight:bold; font-size:20pt; margin-left:2%; \">Table of contents</text>" + htmlToc + "</div>"; + } + } catch (IOException e) { + throw new ApplicationException(e); + } + return htmlStr; + } + + private String getPageFragmentHtml(MetadataRecord mdRecord, int pageNumber) throws ApplicationException { + String pageHtmlStrFragment = null; + try { + String docId = mdRecord.getDocId(); + String docDir = docHandler.getDocDir(docId); + String docPageDir = docDir + "/" + "pages"; + String pageFileName = docPageDir + "/page-" + pageNumber + ".html"; + File pageFile = new File(pageFileName); + pageHtmlStrFragment = FileUtils.readFileToString(pageFile, "utf-8"); + } catch (IOException e) { + throw new ApplicationException(e); + } + return pageHtmlStrFragment; + } + + private String getMdRecordString(MetadataRecord mdRecord) { + String author = mdRecord.getCreator(); + String title = mdRecord.getTitle(); + String year = mdRecord.getYear(); + String mdRecordStr = ""; + if (mdRecord != null) { + if (author != null && ! author.equals("")) + mdRecordStr = mdRecordStr + author; + if (title != null && ! title.equals("")) + mdRecordStr = mdRecordStr + ". " + title; + if (year != null && ! year.equals("")) + mdRecordStr = mdRecordStr + ". " + year + "."; + if (mdRecordStr.isEmpty()) { + String docId = mdRecord.getDocId(); + mdRecordStr = mdRecordStr + docId; + } + } + return mdRecordStr; + } + + private String getHtmlHead(String stylePageStr, String titleStr) { + String htmlStr = "<head>"; + if (stylePageStr != null) + htmlStr = htmlStr + "<style type=\"text/css\">" + stylePageStr + "</style>"; + htmlStr = htmlStr + "<title>" + titleStr + "</title>"; + htmlStr = htmlStr + "<link rel=\"stylesheet\" type=\"text/css\" href=\"" + CSS_SHOW_WORD_URL + "\"/>"; + htmlStr = htmlStr + "<link rel=\"stylesheet\" type=\"text/css\" href=\"" + CSS_URL + "\"/>"; + htmlStr = htmlStr + "</head>"; + return htmlStr; + } + + private String getFontStyle(String language) { + String fontFamily = "Junicode"; + if (language.equals("ar")) + fontFamily = "DejaVu Sans"; + else if (language.equals("zh") || language.equals("zho-Hant")) + fontFamily = "Sun-ExtA, Sun-ExtB"; + return "font-size:11pt; font-family:" + fontFamily + ";"; + } + + private void write(String str, OutputStream out) throws ApplicationException { + try { + byte[] bytes = str.getBytes("utf-8"); + out.write(bytes, 0, bytes.length); + out.flush(); + } catch (UnsupportedEncodingException e) { + throw new ApplicationException(e); + } catch (FileNotFoundException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + private void setFont(String fontFileName) throws ApplicationException { + try { + String existingFontFileName = fontFileNames.get(fontFileName); + if (existingFontFileName == null) { + fontFileNames.put(fontFileName, fontFileName); + ITextFontResolver fontResolver = renderer.getFontResolver(); + fontResolver.addFont(fontFileName, BaseFont.IDENTITY_H, BaseFont.NOT_EMBEDDED); // Identy_H is Unicode Horizontal; not_embedded means not embedded in the PDF doc + } + } catch (XRRuntimeException e) { + init(); + String message = e.getMessage(); + if (message.indexOf("digilib") > 0 && message.indexOf("500") > 0) { + throw new ApplicationException("fetch image is not possible: please try again later"); + } + throw new ApplicationException(e); + } catch (IOException e) { + init(); + String message = e.getMessage(); + if (message.indexOf("digilib") > 0 && message.indexOf("500") > 0) { + throw new ApplicationException("fetch image is not possible: " + message); + } + throw new ApplicationException(e); + } catch (DocumentException e) { + init(); + String message = e.getMessage(); + if (message.indexOf("digilib") > 0 && message.indexOf("500") > 0) { + throw new ApplicationException("fetch image is not possible: " + message); + } + throw new ApplicationException(e); + } + } + + private String performGetRequest(String url) throws ApplicationException { + String resultStr = null; + try { + HttpClient httpClient = new HttpClient(); + GetMethod method = new GetMethod(url); + httpClient.executeMethod(method); + byte[] resultBytes = method.getResponseBody(); + resultStr = new String(resultBytes, "utf-8"); + method.releaseConnection(); + } catch (HttpException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + return resultStr; + } +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/document/PdfHandlerUserAgent.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,149 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.document; + +import java.io.IOException; +import java.io.InputStream; +import java.net.URL; +import java.util.logging.Logger; + +import org.xhtmlrenderer.layout.SharedContext; +import org.xhtmlrenderer.pdf.ITextFSImage; +import org.xhtmlrenderer.pdf.ITextOutputDevice; +import org.xhtmlrenderer.pdf.PDFAsImage; +import org.xhtmlrenderer.resource.ImageResource; +import org.xhtmlrenderer.swing.NaiveUserAgent; + +import com.lowagie.text.Image; +import com.lowagie.text.Rectangle; +import com.lowagie.text.pdf.PdfReader; + +public class PdfHandlerUserAgent extends NaiveUserAgent { + private static final int IMAGE_CACHE_CAPACITY = 32; + private static final float DEFAULT_DOTS_PER_POINT = 20f * 4f / 3f; + private static Logger LOGGER = Logger.getLogger(PdfHandlerUserAgent.class.getName()); + private SharedContext sharedContext; + private ITextOutputDevice outputDevice; + + public PdfHandlerUserAgent() { + super(IMAGE_CACHE_CAPACITY); + outputDevice = new ITextOutputDevice(DEFAULT_DOTS_PER_POINT); + } + + @SuppressWarnings("unchecked") + public ImageResource getImageResource(String inputUri) { + ImageResource resource = null; + String uri = resolveURI(inputUri); + resource = (ImageResource) _imageCache.get(uri); + if (resource == null) { + InputStream is = resolveAndOpenStream(uri); + if (is != null) { + try { + URL url = new URL(uri); + if (url.getPath() != null && url.getPath().toLowerCase().endsWith(".pdf")) { + PdfReader reader = outputDevice.getReader(url); + PDFAsImage image = new PDFAsImage(url); + Rectangle rect = reader.getPageSizeWithRotation(1); + image.setInitialWidth(rect.getWidth()*outputDevice.getDotsPerPoint()); + image.setInitialHeight(rect.getHeight()*outputDevice.getDotsPerPoint()); + resource = new ImageResource(image); + } else { + Image image = getImage(url); + if (image == null) + return null; + scaleToOutputResolution(image); + resource = new ImageResource(new ITextFSImage(image)); + } + _imageCache.put(uri, resource); + } catch (IOException e) { + LOGGER.severe("Can't get image file: unexpected problem for URI: '" + uri + "': " + e.getMessage()); + } finally { + try { + if (is != null) + is.close(); + } catch (IOException e) { + // ignore + } + } + } + } + if (resource == null) { + resource = new ImageResource(null); + } + return resource; + } + + private void scaleToOutputResolution(Image image) { + float factor = sharedContext.getDotsPerPixel(); + image.scaleAbsolute(image.getPlainWidth() * factor, image.getPlainHeight() * factor); + } + + public SharedContext getSharedContext() { + return sharedContext; + } + + public void setSharedContext(SharedContext sharedContext) { + this.sharedContext = sharedContext; + } + + private Image getImage(URL url) { + Image image = null; + try { + image = Image.getInstance(url); + } catch (Exception e) { + try { + Thread.sleep(1000); + } catch (InterruptedException ee) { + // nothing + } + LOGGER.severe("first retry to get image for URL '" + url.toString() + "': " + e.getMessage()); + try { + image = Image.getInstance(url); + } catch (Exception e2) { + try { + Thread.sleep(1000); + } catch (InterruptedException ee) { + // nothing + } + LOGGER.severe("second retry to get image for URL '" + url.toString() + "': " + e.getMessage()); + try { + image = Image.getInstance(url); + } catch (Exception e3) { + LOGGER.severe("third retry to get image for URL '" + url.toString() + "': " + e.getMessage()); + return null; + } + } + } + return image; + } + + protected InputStream resolveAndOpenStream(String inputUri) { + InputStream is = null; + String uri = resolveURI(inputUri); + try { + is = new URL(uri).openStream(); + } catch (Exception e) { + try { + Thread.sleep(1000); + } catch (InterruptedException ee) { + // nothing + } + LOGGER.severe("first retry to open stream for URL '" + uri + "': " + e.getMessage()); + try { + is = new URL(uri).openStream(); + } catch (Exception e2) { + try { + Thread.sleep(1000); + } catch (InterruptedException ee) { + // nothing + } + LOGGER.severe("second retry to open stream for URL '" + uri + "': " + e.getMessage()); + try { + is = new URL(uri).openStream(); + } catch (Exception e3) { + LOGGER.severe("third retry to open stream for URL '" + uri + "': " + e.getMessage()); + return null; + } + } + } + return is; + } +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/document/Token.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,29 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.document; + +import org.apache.lucene.index.Term; + +public class Token { + private Term term; + private int freq = -1; + + public Token(Term term) { + this.term = term; + } + + public Term getTerm() { + return term; + } + + public void setTerm(Term term) { + this.term = term; + } + + public int getFreq() { + return freq; + } + + public void setFreq(int freq) { + this.freq = freq; + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/general/Constants.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,55 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.general; + +import java.net.URL; +import java.util.Properties; + +import de.mpg.mpiwg.berlin.mpdl.util.Util; + +public class Constants { + private static Constants instance; + private Properties properties; + + public static Constants getInstance() { + if (instance == null) { + instance = new Constants(); + instance.init(); + } + return instance; + } + + private void init() { + URL url = Constants.class.getClassLoader().getResource("constants.properties"); + if (url != null) { + String propertiesFileName = url.toString().substring(5); + properties = (new Util()).getProperties(propertiesFileName); + } + } + + public String getDocumentsDir() { + if (properties != null) + return properties.getProperty("documentsDir"); + else + return "no properties file"; + } + + public String getLuceneDocumentsDir() { + if (properties != null) + return properties.getProperty("luceneDocumentsDir"); + else + return "no properties file"; + } + + public String getLuceneNodesDir() { + if (properties != null) + return properties.getProperty("luceneNodesDir"); + else + return "no properties file"; + } + + public String getConfDir() { + if (properties != null) + return properties.getProperty("confDir"); + else + return "no properties file"; + } +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/harvester/PathExtractor.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,137 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.harvester; + +import java.io.File; +import java.io.IOException; +import java.util.ArrayList; +import java.util.List; + +import javax.xml.stream.XMLInputFactory; +import javax.xml.stream.XMLStreamConstants; +import javax.xml.stream.XMLStreamException; +import javax.xml.stream.XMLStreamReader; + +import org.apache.http.HttpEntity; +import org.apache.http.HttpResponse; +import org.apache.http.client.HttpClient; +import org.apache.http.client.methods.HttpGet; +import org.apache.http.impl.client.DefaultHttpClient; + +public class PathExtractor { + private List<String> ressourceLoc; + String excludes; + + public PathExtractor() { + + } + + public List<String> initExtractor(String startingUri, String excludes) { + this.excludes = excludes; + ressourceLoc = new ArrayList<String>(); + // parameter necessery, because it's recursive, thus changing the uri + extractDocLocations(startingUri); + System.out.println("extracing resource locations done."); + return this.ressourceLoc; + } + + /** + * recursive Method to extract the path of the resources + * + * @param startUrl + */ + private void extractDocLocations(String startUrl) { + HttpClient client = new DefaultHttpClient(); + HttpGet httpget = new HttpGet(startUrl); + HttpResponse resp = null; + try { + resp = client.execute(httpget); + } catch (IOException e) { + e.printStackTrace(); + } + HttpEntity entity = resp.getEntity(); + if (entity != null) { + XMLInputFactory iFactory = XMLInputFactory.newInstance(); + XMLStreamReader reader = null; + try { + reader = iFactory.createXMLStreamReader(entity.getContent()); + } catch (IllegalStateException e1) { + e1.printStackTrace(); + } catch (XMLStreamException e1) { + e1.printStackTrace(); + } catch (IOException e1) { + e1.printStackTrace(); + } + + try { + while (true) { + int event = reader.next(); + if (event == XMLStreamConstants.END_DOCUMENT) { + reader.close(); + break; + } + if (event == XMLStreamConstants.START_ELEMENT) { + if ((reader.getAttributeValue(null, "name")) != null) { + if (reader.getLocalName().equals("collection") && !(startUrl.endsWith(reader.getAttributeValue(null, "name")))) { + if(!(this.excludes.contains(reader.getAttributeValue(null, "name").toLowerCase()))){ + if (reader.getAttributeValue(null, "name").startsWith("/")) { + client.getConnectionManager().closeExpiredConnections(); + extractDocLocations(startUrl + reader.getAttributeValue(null, "name")); + } else { + client.getConnectionManager().closeExpiredConnections(); + if (!startUrl.endsWith("/")) { + extractDocLocations(startUrl + "/" + reader.getAttributeValue(null, "name")); + } else { + extractDocLocations(startUrl + reader.getAttributeValue(null, "name")); + } + } + } + } + if (reader.getLocalName().equals("resource")) { + if (!startUrl.endsWith("/")) { + ressourceLoc.add(startUrl + "/" + reader.getAttributeValue(null, "name")); + } else { + ressourceLoc.add(startUrl + reader.getAttributeValue(null, "name")); + } + } + } + } + if (event == XMLStreamConstants.ATTRIBUTE) { + // System.out.println("localName : "+reader.getLocalName()); + } + } + } catch (XMLStreamException e) { + e.printStackTrace(); + } + } + } + + /** + * extrahiert ebenso wie extractDocLocations(String startUri) Pfade, tut dies + * aber local und nicht über HTTP + * + * @return + */ + public List<String> extractPathLocally(String startUrl) { + List<String> pathList = new ArrayList<String>(); + + // home verzeichnis pfad über system variable + // String loc = System.getenv("HOME")+"/wsp/configs"; + // out.println("hom variable + conf datei : "+loc); + File f = new File(startUrl); + // out.println("readable : "+Boolean.toString(f.canRead())); + // out.println("readable : "+f.isDirectory()); + if (f.isDirectory()) { + File[] filelist = f.listFiles(); + for (File file : filelist) { + if (file.getName().toLowerCase().contains("config")) { + if (!startUrl.endsWith("/")) { + pathList.add(startUrl + "/" + file.getName()); + } else { + pathList.add(startUrl + file.getName()); + } + } + } + } + return pathList; + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/lucene/IndexHandler.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,1463 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.lucene; + +import java.io.File; +import java.io.FileInputStream; +import java.io.FileReader; +import java.io.IOException; +import java.io.InputStreamReader; +import java.io.StringReader; +import java.util.ArrayList; +import java.util.Date; +import java.util.HashMap; +import java.util.HashSet; +import java.util.Map; + +import org.apache.commons.io.FileUtils; +import org.apache.lucene.analysis.Analyzer; +import org.apache.lucene.analysis.KeywordAnalyzer; +import org.apache.lucene.analysis.PerFieldAnalyzerWrapper; +import org.apache.lucene.analysis.TokenStream; +import org.apache.lucene.analysis.standard.StandardAnalyzer; +import org.apache.lucene.document.Document; +import org.apache.lucene.document.Field; +import org.apache.lucene.document.FieldSelector; +import org.apache.lucene.document.Fieldable; +import org.apache.lucene.document.SetBasedFieldSelector; +import org.apache.lucene.index.IndexReader; +import org.apache.lucene.index.IndexWriter; +import org.apache.lucene.index.IndexWriterConfig; +import org.apache.lucene.index.IndexWriterConfig.OpenMode; +import org.apache.lucene.index.Term; +import org.apache.lucene.index.TermEnum; +import org.apache.lucene.index.TermFreqVector; +import org.apache.lucene.queryParser.QueryParser; +import org.apache.lucene.search.BooleanClause; +import org.apache.lucene.search.BooleanQuery; +import org.apache.lucene.search.FuzzyQuery; +import org.apache.lucene.search.IndexSearcher; +import org.apache.lucene.search.MatchAllDocsQuery; +import org.apache.lucene.search.PhraseQuery; +import org.apache.lucene.search.PrefixQuery; +import org.apache.lucene.search.Query; +import org.apache.lucene.search.SearcherManager; +import org.apache.lucene.search.Sort; +import org.apache.lucene.search.SortField; +import org.apache.lucene.search.TermQuery; +import org.apache.lucene.search.TermRangeQuery; +import org.apache.lucene.search.TopDocs; +import org.apache.lucene.search.highlight.Highlighter; +import org.apache.lucene.search.highlight.QueryScorer; +import org.apache.lucene.search.highlight.SimpleHTMLFormatter; +import org.apache.lucene.search.highlight.TextFragment; +import org.apache.lucene.search.highlight.TokenSources; +import org.apache.lucene.search.similar.MoreLikeThis; +import org.apache.lucene.store.FSDirectory; +import org.apache.lucene.util.Version; +import org.xml.sax.InputSource; +import org.xml.sax.SAXException; +import org.xml.sax.XMLReader; + +import com.sun.org.apache.xerces.internal.parsers.SAXParser; + +import de.mpg.mpiwg.berlin.mpdl.cms.confmanager.CollectionReader; +import de.mpg.mpiwg.berlin.mpdl.cms.confmanager.ConfManagerResultWrapper; +import de.mpg.mpiwg.berlin.mpdl.cms.document.DocumentHandler; +import de.mpg.mpiwg.berlin.mpdl.cms.document.Hits; +import de.mpg.mpiwg.berlin.mpdl.cms.document.MetadataRecord; +import de.mpg.mpiwg.berlin.mpdl.cms.document.Token; +import de.mpg.mpiwg.berlin.mpdl.cms.general.Constants; +import de.mpg.mpiwg.berlin.mpdl.cms.scheduler.CmsDocOperation; +import de.mpg.mpiwg.berlin.mpdl.cms.transform.XslResourceTransformer; +import de.mpg.mpiwg.berlin.mpdl.cms.translator.MicrosoftTranslator; +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.lt.dict.db.LexHandler; +import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.Form; +import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.Lemma; +import de.mpg.mpiwg.berlin.mpdl.lt.text.norm.Normalizer; +import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.WordContentHandler; +import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.XmlTokenizer; +import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.XmlTokenizerContentHandler; +import de.mpg.mpiwg.berlin.mpdl.util.StringUtils; +import de.mpg.mpiwg.berlin.mpdl.util.Util; + +public class IndexHandler { + private static IndexHandler instance; + private IndexWriter documentsIndexWriter; + private IndexWriter nodesIndexWriter; + private SearcherManager documentsSearcherManager; + private SearcherManager nodesSearcherManager; + private IndexReader documentsIndexReader; + private PerFieldAnalyzerWrapper documentsPerFieldAnalyzer; + private PerFieldAnalyzerWrapper nodesPerFieldAnalyzer; + + + public static IndexHandler getInstance() throws ApplicationException { + if (instance == null) { + instance = new IndexHandler(); + instance.init(); + } + return instance; + } + + private void init() throws ApplicationException { + documentsIndexWriter = getDocumentsWriter(); + documentsIndexWriter.setMaxFieldLength(1000000); + nodesIndexWriter = getNodesWriter(); + nodesIndexWriter.setMaxFieldLength(1000000); + documentsSearcherManager = getNewSearcherManager(documentsIndexWriter); + nodesSearcherManager = getNewSearcherManager(nodesIndexWriter); + documentsIndexReader = getDocumentsReader(); + } + + public void indexDocument(CmsDocOperation docOperation) throws ApplicationException { + try { + // first delete document in documentsIndex and nodesIndex + deleteDocumentLocal(docOperation); + indexDocumentLocal(docOperation); + documentsIndexWriter.commit(); + nodesIndexWriter.commit(); + } catch (Exception e) { + try { + documentsIndexWriter.rollback(); + nodesIndexWriter.rollback(); + } catch (Exception ex) { + // nothing + } + throw new ApplicationException(e); + } + } + + private void indexDocumentLocal(CmsDocOperation docOperation) throws ApplicationException { + FileReader fr = null; + try { + MetadataRecord mdRecord = docOperation.getMdRecord(); + String docId = mdRecord.getDocId(); + DocumentHandler docHandler = new DocumentHandler(); + String docFileName = docHandler.getDocFullFileName(docId) + ".upgrade"; + // add document to documentsIndex + Document doc = new Document(); + Field docIdField = new Field("docId", docId, Field.Store.YES, Field.Index.ANALYZED); + doc.add(docIdField); + String docIdSortedStr = docId.toLowerCase(); // so that sorting is lower case + Field docIdFieldSorted = new Field("docIdSorted", docIdSortedStr, Field.Store.YES, Field.Index.NOT_ANALYZED); + doc.add(docIdFieldSorted); + String identifier = mdRecord.getIdentifier(); + if (identifier != null) { + Field identifierField = new Field("identifier", identifier, Field.Store.YES, Field.Index.ANALYZED); + doc.add(identifierField); + } + String uri = docOperation.getSrcUrl(); + if (uri != null) { + Field uriField = new Field("uri", uri, Field.Store.YES, Field.Index.ANALYZED); + doc.add(uriField); + } + String collectionNames = docOperation.getCollectionNames(); + if (collectionNames != null) { + Field collectionNamesField = new Field("collectionNames", collectionNames, Field.Store.YES, Field.Index.ANALYZED); + doc.add(collectionNamesField); + } + if (mdRecord.getCreator() != null) { + String authorStr = mdRecord.getCreator(); + Field authorField = new Field("author", authorStr, Field.Store.YES, Field.Index.ANALYZED, Field.TermVector.WITH_POSITIONS_OFFSETS); + doc.add(authorField); + if (authorStr != null) + authorStr = authorStr.toLowerCase(); // so that sorting is lower case + Field authorFieldSorted = new Field("authorSorted", authorStr, Field.Store.YES, Field.Index.NOT_ANALYZED); + doc.add(authorFieldSorted); + } + if (mdRecord.getTitle() != null) { + String titleStr = mdRecord.getTitle(); + Field titleField = new Field("title", titleStr, Field.Store.YES, Field.Index.ANALYZED, Field.TermVector.WITH_POSITIONS_OFFSETS); + doc.add(titleField); + if (titleStr != null) + titleStr = titleStr.toLowerCase(); // so that sorting is lower case + Field titleFieldSorted = new Field("titleSorted", titleStr, Field.Store.YES, Field.Index.NOT_ANALYZED); + doc.add(titleFieldSorted); + } + if (mdRecord.getLanguage() != null) { + String langStr = mdRecord.getLanguage(); + if (langStr != null) + langStr = langStr.toLowerCase(); // all language codes are lower case + Field languageField = new Field("language",langStr, Field.Store.YES, Field.Index.ANALYZED); + doc.add(languageField); + Field languageFieldSorted = new Field("languageSorted", langStr, Field.Store.YES, Field.Index.NOT_ANALYZED); + doc.add(languageFieldSorted); + } + if (mdRecord.getPublisher() != null) { + String publisherStr = mdRecord.getPublisher(); + Field publisherField = new Field("publisher", publisherStr, Field.Store.YES, Field.Index.ANALYZED); + doc.add(publisherField); + if (publisherStr != null) + publisherStr = publisherStr.toLowerCase(); // so that sorting is lower case + Field publisherFieldSorted = new Field("publisherSorted", publisherStr, Field.Store.YES, Field.Index.NOT_ANALYZED); + doc.add(publisherFieldSorted); + } + if (mdRecord.getYear() != null) { + Field dateField = new Field("date", mdRecord.getYear(), Field.Store.YES, Field.Index.ANALYZED); + doc.add(dateField); + Field dateFieldSorted = new Field("dateSorted", mdRecord.getYear(), Field.Store.YES, Field.Index.NOT_ANALYZED); + doc.add(dateFieldSorted); + } + if (mdRecord.getSubject() != null) { + Field subjectField = new Field("subject", mdRecord.getSubject(), Field.Store.YES, Field.Index.ANALYZED); + doc.add(subjectField); + } + if (mdRecord.getRights() != null) { + Field rightsField = new Field("rights", mdRecord.getRights(), Field.Store.YES, Field.Index.ANALYZED); + doc.add(rightsField); + } + if (mdRecord.getLicense() != null) { + Field licenseField = new Field("license", mdRecord.getLicense(), Field.Store.YES, Field.Index.ANALYZED); + doc.add(licenseField); + } + if (mdRecord.getAccessRights() != null) { + Field accessRightsField = new Field("accessRights", mdRecord.getAccessRights(), Field.Store.YES, Field.Index.ANALYZED); + doc.add(accessRightsField); + } + String echoId = mdRecord.getEchoId(); + if (echoId != null) { + Field echoIdField = new Field("echoId", echoId, Field.Store.YES, Field.Index.ANALYZED); + doc.add(echoIdField); + } + String echoPageImageDir = mdRecord.getEchoPageImageDir(); + if (echoPageImageDir != null) { + Field echoPageImageDirField = new Field("echoPageImageDir", echoPageImageDir, Field.Store.YES, Field.Index.ANALYZED); + doc.add(echoPageImageDirField); + } + String echoFiguresDir = mdRecord.getEchoFiguresDir(); + if (echoFiguresDir != null) { + Field echoFiguresDirField = new Field("echoFiguresDir", echoFiguresDir, Field.Store.YES, Field.Index.ANALYZED); + doc.add(echoFiguresDirField); + } + String mpiwgDocId = mdRecord.getMpiwgDocId(); + if (mpiwgDocId != null) { + Field mpiwgDocIdField = new Field("mpiwgDocId", mpiwgDocId, Field.Store.YES, Field.Index.ANALYZED); + doc.add(mpiwgDocIdField); + } + if (mdRecord.getLastModified() != null) { + Date lastModified = mdRecord.getLastModified(); + String xsDateStr = new Util().toXsDate(lastModified); + Field lastModifiedField = new Field("lastModified", xsDateStr, Field.Store.YES, Field.Index.ANALYZED); + doc.add(lastModifiedField); + long time = lastModified.getTime(); + String timeStr = String.valueOf(time); + Field lastModifiedFieldSorted = new Field("lastModifiedSorted", timeStr, Field.Store.YES, Field.Index.NOT_ANALYZED); + doc.add(lastModifiedFieldSorted); + } + if (mdRecord.getSchemaName() != null) { + String schemNameStr = mdRecord.getSchemaName(); + Field schemaField = new Field("schemaName", schemNameStr, Field.Store.YES, Field.Index.ANALYZED); + doc.add(schemaField); + if (schemNameStr != null) + schemNameStr = schemNameStr.toLowerCase(); // so that sorting is lower case + Field schemaFieldSorted = new Field("schemaNameSorted", schemNameStr, Field.Store.YES, Field.Index.NOT_ANALYZED); + doc.add(schemaFieldSorted); + } + + String language = mdRecord.getLanguage(); + InputStreamReader docFileReader = new InputStreamReader(new FileInputStream(docFileName), "utf-8"); + // to guarantee that utf-8 is used (if not done, it does not work on Tomcat which has another default charset) + XmlTokenizer docXmlTokenizer = new XmlTokenizer(docFileReader); + docXmlTokenizer.setDocIdentifier(docId); + docXmlTokenizer.setLanguage(language); + docXmlTokenizer.setOutputFormat("string"); + String[] outputOptionsWithLemmas = { "withLemmas" }; // so all tokens are + // fetched with lemmas (costs performance) + docXmlTokenizer.setOutputOptions(outputOptionsWithLemmas); + String[] normFunctionNone = { "none" }; + docXmlTokenizer.setNormFunctions(normFunctionNone); + docXmlTokenizer.tokenize(); + + int pageCount = docXmlTokenizer.getPageCount(); + if (pageCount == 0) + pageCount = 1; // each document at least has one page + String pageCountStr = String.valueOf(pageCount); + Field pageCountField = new Field("pageCount", pageCountStr, Field.Store.YES, Field.Index.ANALYZED); + doc.add(pageCountField); + + String[] outputOptionsEmpty = {}; + docXmlTokenizer.setOutputOptions(outputOptionsEmpty); + // must be set to null so that the normalization function works + String docTokensOrig = docXmlTokenizer.getStringResult(); + String[] normFunctionReg = { "reg" }; + docXmlTokenizer.setNormFunctions(normFunctionReg); + String docTokensReg = docXmlTokenizer.getStringResult(); + String[] normFunctionNorm = { "norm" }; + docXmlTokenizer.setNormFunctions(normFunctionNorm); + String docTokensNorm = docXmlTokenizer.getStringResult(); + docXmlTokenizer.setOutputOptions(outputOptionsWithLemmas); + String docTokensMorph = docXmlTokenizer.getStringResult(); + + Field tokenOrigField = new Field("tokenOrig", docTokensOrig, Field.Store.YES, Field.Index.ANALYZED, Field.TermVector.WITH_POSITIONS_OFFSETS); + Field tokenRegField = new Field("tokenReg", docTokensReg, Field.Store.YES, Field.Index.ANALYZED, Field.TermVector.WITH_POSITIONS_OFFSETS); + Field tokenNormField = new Field("tokenNorm", docTokensNorm, Field.Store.YES, Field.Index.ANALYZED, Field.TermVector.WITH_POSITIONS_OFFSETS); + Field tokenMorphField = new Field("tokenMorph", docTokensMorph, Field.Store.YES, Field.Index.ANALYZED, Field.TermVector.WITH_POSITIONS_OFFSETS); + doc.add(tokenOrigField); + doc.add(tokenRegField); + doc.add(tokenNormField); + doc.add(tokenMorphField); + + // save original content of the doc file + File docFile = new File(docFileName); + String contentXml = FileUtils.readFileToString(docFile, "utf-8"); + Field contentXmlField = new Field("xmlContent", contentXml, Field.Store.YES, Field.Index.ANALYZED, Field.TermVector.WITH_POSITIONS_OFFSETS); + doc.add(contentXmlField); + + // generate original chars content + XslResourceTransformer charsTransformer = new XslResourceTransformer("chars.xsl"); + String content = charsTransformer.transform(docFileName); + Field contentField = new Field("content", content, Field.Store.YES, Field.Index.ANALYZED, Field.TermVector.WITH_POSITIONS_OFFSETS); + doc.add(contentField); + + documentsIndexWriter.addDocument(doc); + + // add all elements with the specified names of the document to nodesIndex + String[] elementNamesArray = docOperation.getElementNames(); + String elementNames = ""; + for (int i = 0; i < elementNamesArray.length; i++) { + String elemName = elementNamesArray[i]; + elementNames = elementNames + elemName + " "; + } + elementNames = elementNames.substring(0, elementNames.length() - 1); + ArrayList<XmlTokenizerContentHandler.Element> elements = docXmlTokenizer.getElements(elementNames); + for (int i = 0; i < elements.size(); i++) { + XmlTokenizerContentHandler.Element element = elements.get(i); + Document nodeDoc = new Document(); + nodeDoc.add(docIdField); + String nodeLanguage = element.lang; + if (nodeLanguage == null) + nodeLanguage = language; + String nodePageNumber = String.valueOf(element.pageNumber); + String nodeLineNumber = String.valueOf(element.lineNumber); + String nodeElementName = String.valueOf(element.name); + String nodeElementDocPosition = String.valueOf(element.docPosition); + String nodeElementAbsolutePosition = String.valueOf(element.position); + String nodeElementPagePosition = String.valueOf(element.pagePosition); + String nodeElementPosition = String.valueOf(element.elemPosition); + String nodeXmlId = element.xmlId; + String nodeXpath = element.xpath; + String nodeXmlContent = element.toXmlString(); + String nodeTokensOrig = element.getTokensStr("orig"); + String nodeTokensReg = element.getTokensStr("reg"); + String nodeTokensNorm = element.getTokensStr("norm"); + String nodeTokensMorph = element.getTokensStr("morph"); + if (nodeLanguage != null) { + Field nodeLanguageField = new Field("language", nodeLanguage, Field.Store.YES, Field.Index.ANALYZED); + nodeDoc.add(nodeLanguageField); + } + Field nodePageNumberField = new Field("pageNumber", nodePageNumber, Field.Store.YES, Field.Index.ANALYZED); + nodeDoc.add(nodePageNumberField); + Field nodeLineNumberField = new Field("lineNumber", nodeLineNumber, Field.Store.YES, Field.Index.ANALYZED); + nodeDoc.add(nodeLineNumberField); + Field nodeElementNameField = new Field("elementName", nodeElementName, Field.Store.YES, Field.Index.ANALYZED); + nodeDoc.add(nodeElementNameField); + Field nodeElementDocPositionField = new Field("elementDocPosition", nodeElementDocPosition, Field.Store.YES, Field.Index.ANALYZED); + nodeDoc.add(nodeElementDocPositionField); + Field nodeElementDocPositionFieldSorted = new Field("elementDocPositionSorted", nodeElementDocPosition, Field.Store.YES, Field.Index.NOT_ANALYZED); + nodeDoc.add(nodeElementDocPositionFieldSorted); + Field nodeElementAbsolutePositionField = new Field("elementAbsolutePosition", nodeElementAbsolutePosition, Field.Store.YES, Field.Index.ANALYZED); + nodeDoc.add(nodeElementAbsolutePositionField); + Field nodeElementPagePositionField = new Field("elementPagePosition", nodeElementPagePosition, Field.Store.YES, Field.Index.ANALYZED); + nodeDoc.add(nodeElementPagePositionField); + Field nodeElementPositionField = new Field("elementPosition", nodeElementPosition, Field.Store.YES, Field.Index.ANALYZED); + nodeDoc.add(nodeElementPositionField); + if (nodeXmlId != null) { + Field nodeXmlIdField = new Field("xmlId", nodeXmlId, Field.Store.YES, Field.Index.ANALYZED); + nodeDoc.add(nodeXmlIdField); + } + if (nodeXpath != null) { + Field nodeXpathField = new Field("xpath", nodeXpath, Field.Store.YES, Field.Index.ANALYZED); + nodeDoc.add(nodeXpathField); + } + if (nodeXmlContent != null) { + Field nodeXmlContentField = new Field("xmlContent", nodeXmlContent, Field.Store.YES, Field.Index.ANALYZED, Field.TermVector.WITH_POSITIONS_OFFSETS); + nodeDoc.add(nodeXmlContentField); + } + if (nodeXmlContent != null) { + String nodeXmlContentTokenized = toTokenizedXmlString(nodeXmlContent, nodeLanguage); + byte[] blabla = nodeXmlContentTokenized.getBytes("utf-8"); // TODO why is tokenizedXmlStr not already utf-8 on page 444 Benedetti ? + nodeXmlContentTokenized = new String(blabla, "utf-8"); + nodeXmlContentTokenized = enrichWordsOrigRegNorm(nodeXmlContentTokenized); + Field nodeXmlContentTokenizedField = new Field("xmlContentTokenized", nodeXmlContentTokenized, Field.Store.YES, Field.Index.ANALYZED, Field.TermVector.WITH_POSITIONS_OFFSETS); + nodeDoc.add(nodeXmlContentTokenizedField); + } + if (nodeTokensOrig != null) { + Field nodeTokenOrigField = new Field("tokenOrig", nodeTokensOrig, Field.Store.YES, Field.Index.ANALYZED, Field.TermVector.WITH_POSITIONS_OFFSETS); + nodeDoc.add(nodeTokenOrigField); + } + if (nodeTokensReg != null) { + Field nodeTokenRegField = new Field("tokenReg", nodeTokensReg, Field.Store.YES, Field.Index.ANALYZED, Field.TermVector.WITH_POSITIONS_OFFSETS); + nodeDoc.add(nodeTokenRegField); + } + if (nodeTokensNorm != null) { + Field nodeTokenNormField = new Field("tokenNorm", nodeTokensNorm, Field.Store.YES, Field.Index.ANALYZED, Field.TermVector.WITH_POSITIONS_OFFSETS); + nodeDoc.add(nodeTokenNormField); + } + if (nodeTokensMorph != null) { + Field nodeTokenMorphField = new Field("tokenMorph", nodeTokensMorph, Field.Store.YES, Field.Index.ANALYZED, Field.TermVector.WITH_POSITIONS_OFFSETS); + nodeDoc.add(nodeTokenMorphField); + } + + nodesIndexWriter.addDocument(nodeDoc); + } + } catch (Exception e) { + throw new ApplicationException(e); + } finally { + try { + if (fr != null) + fr.close(); + } catch (Exception e) { + // nothing + } + } + } + + public void deleteDocument(CmsDocOperation docOperation) throws ApplicationException { + try { + deleteDocumentLocal(docOperation); + documentsIndexWriter.commit(); + nodesIndexWriter.commit(); + } catch (Exception e) { + try { + documentsIndexWriter.rollback(); + nodesIndexWriter.rollback(); + } catch (Exception ex) { + // nothing + } + throw new ApplicationException(e); + } + } + + private void deleteDocumentLocal(CmsDocOperation docOperation) throws ApplicationException { + String docId = docOperation.getDocIdentifier(); + try { + Term termIdentifier = new Term("docId", docId); + documentsIndexWriter.deleteDocuments(termIdentifier); + nodesIndexWriter.deleteDocuments(termIdentifier); + } catch (Exception e) { + throw new ApplicationException(e); + } + } + + public Hits queryDocuments(String queryStr, String[] sortFieldNames, String language, int from, int to, boolean withHitFragments, boolean translate) throws ApplicationException { + Hits hits = null; + IndexSearcher searcher = null; + try { + makeDocumentsSearcherManagerUpToDate(); + searcher = documentsSearcherManager.acquire(); + String defaultQueryFieldName = "tokenOrig"; + QueryParser queryParser = new QueryParser(Version.LUCENE_35, defaultQueryFieldName, documentsPerFieldAnalyzer); + Query query = null; + if (queryStr.equals("*")) { + query = new MatchAllDocsQuery(); + } else { + query = queryParser.parse(queryStr); + } + Query morphQuery = buildMorphQuery(query, language, false, translate); + Query highlighterQuery = buildMorphQuery(query, language, true, translate); + if (query instanceof PhraseQuery || query instanceof PrefixQuery || query instanceof FuzzyQuery || query instanceof TermRangeQuery) { + highlighterQuery = query; // TODO wenn sie rekursiv enthalten sind + } + String beginHitMark = "!!!BEGIN_HIT!!!"; + String endHitMark = "!!!END_HIT!!!"; + SimpleHTMLFormatter htmlFormatter = new SimpleHTMLFormatter(beginHitMark, endHitMark); // marks should not contain xml special chars + QueryScorer queryScorer = new QueryScorer(highlighterQuery); + Highlighter highlighter = new Highlighter(htmlFormatter, queryScorer); + TopDocs resultDocs = null; + if (sortFieldNames != null) { + Sort sort = buildSort(sortFieldNames, "doc"); // build sort criteria + resultDocs = searcher.search(morphQuery, 10000, sort); + } else { + resultDocs = searcher.search(morphQuery, 10000); + } + resultDocs.setMaxScore(1); + int toTmp = to; + if (resultDocs.scoreDocs.length <= to) + toTmp = resultDocs.scoreDocs.length - 1; + if (resultDocs != null) { + ArrayList<de.mpg.mpiwg.berlin.mpdl.cms.document.Document> docs = new ArrayList<de.mpg.mpiwg.berlin.mpdl.cms.document.Document>(); + for (int i=from; i<=toTmp; i++) { + int docID = resultDocs.scoreDocs[i].doc; + FieldSelector docFieldSelector = getDocFieldSelector(); + Document luceneDoc = searcher.doc(docID, docFieldSelector); + de.mpg.mpiwg.berlin.mpdl.cms.document.Document doc = new de.mpg.mpiwg.berlin.mpdl.cms.document.Document(luceneDoc); + if (withHitFragments) { + ArrayList<String> hitFragments = new ArrayList<String>(); + Fieldable docContentField = luceneDoc.getFieldable("content"); + if (docContentField != null) { + String docContent = docContentField.stringValue(); + TokenStream tokenStream = TokenSources.getAnyTokenStream(this.documentsIndexReader, docID, docContentField.name(), luceneDoc, documentsPerFieldAnalyzer); + // highlighter.setMaxDocCharsToAnalyze(100000); // the first 100000 chars are fetched maximal, but performance is not really better + TextFragment[] textfragments = highlighter.getBestTextFragments(tokenStream, docContent, false, 5); + if (textfragments.length > 0) { + for (int j=0; j<textfragments.length; j++) { + String textFragment = textfragments[j].toString().trim(); + textFragment = StringUtils.deresolveXmlEntities(textFragment); + textFragment = textFragment.replaceAll(beginHitMark, "<span class=\"highlight hit\">"); + textFragment = textFragment.replaceAll(endHitMark, "</span>"); + hitFragments.add(checkHitFragment(textFragment)); + } + } + } + if (! hitFragments.isEmpty()) + doc.setHitFragments(hitFragments); + } + docs.add(doc); + } + if (docs != null) { + hits = new Hits(docs, from, to); + hits.setSize(resultDocs.scoreDocs.length); + hits.setQuery(morphQuery); + } + } + } catch (Exception e) { + throw new ApplicationException(e); + } finally { + try { + if (searcher != null) + documentsSearcherManager.release(searcher); + } catch (IOException e) { + // nothing + } + } + // Do not use searcher after this! + searcher = null; + return hits; + } + + public Hits queryDocument(String docId, String queryStr, int from, int to) throws ApplicationException { + Hits hits = null; + IndexSearcher searcher = null; + MetadataRecord docMetadataRecord = getDocMetadata(docId); + if (docMetadataRecord == null) + return null; // no document with that docId is in index + try { + makeNodesSearcherManagerUpToDate(); + searcher = nodesSearcherManager.acquire(); + String fieldNameDocId = "docId"; + Query queryDocId = new QueryParser(Version.LUCENE_35, fieldNameDocId, nodesPerFieldAnalyzer).parse(docId); + String defaultQueryFieldName = "tokenOrig"; + Query query = new QueryParser(Version.LUCENE_35, defaultQueryFieldName, nodesPerFieldAnalyzer).parse(queryStr); + String language = docMetadataRecord.getLanguage(); + if (language == null || language.equals("")) { + String collectionNames = docMetadataRecord.getCollectionNames(); + ConfManagerResultWrapper collectionInfo = CollectionReader.getInstance().getResultWrapper(collectionNames); + if (collectionInfo != null) { + String mainLang = collectionInfo.getMainLanguage(); + if (mainLang != null) + language = mainLang; + } + } + Query morphQuery = buildMorphQuery(query, language); + BooleanQuery queryDoc = new BooleanQuery(); + queryDoc.add(queryDocId, BooleanClause.Occur.MUST); + queryDoc.add(morphQuery, BooleanClause.Occur.MUST); + String[] sortFieldNames = {"elementDocPosition"}; + Sort sortByPosition = buildSort(sortFieldNames, "node"); + TopDocs topDocs = searcher.search(queryDoc, 100000, sortByPosition); + topDocs.setMaxScore(1); + int toTmp = to; + if (topDocs.scoreDocs.length <= to) + toTmp = topDocs.scoreDocs.length - 1; + if (topDocs != null) { + ArrayList<de.mpg.mpiwg.berlin.mpdl.cms.document.Document> docs = new ArrayList<de.mpg.mpiwg.berlin.mpdl.cms.document.Document>(); + for (int i=from; i<=toTmp; i++) { + int docID = topDocs.scoreDocs[i].doc; + FieldSelector nodeFieldSelector = getNodeFieldSelector(); + Document luceneDoc = searcher.doc(docID, nodeFieldSelector); + de.mpg.mpiwg.berlin.mpdl.cms.document.Document doc = new de.mpg.mpiwg.berlin.mpdl.cms.document.Document(luceneDoc); + docs.add(doc); + } + if (docs != null) { + hits = new Hits(docs, from, to); + hits.setSize(topDocs.scoreDocs.length); + } + } + searcher.close(); + } catch (Exception e) { + throw new ApplicationException(e); + } finally { + try { + if (searcher != null) + documentsSearcherManager.release(searcher); + } catch (IOException e) { + // nothing + } + } + // Do not use searcher after this! + searcher = null; + return hits; + } + + public Hits moreLikeThis(String docId, int from, int to) throws ApplicationException { + Hits hits = null; + ArrayList<de.mpg.mpiwg.berlin.mpdl.cms.document.Document> wspDocs = null; + IndexSearcher searcher1 = null; + IndexSearcher searcher2 = null; + try { + makeDocumentsSearcherManagerUpToDate(); + searcher1 = documentsSearcherManager.acquire(); + String fieldNameDocId = "docId"; + Query queryDocId = new QueryParser(Version.LUCENE_35, fieldNameDocId, documentsPerFieldAnalyzer).parse(docId); + TopDocs topDocs = searcher1.search(queryDocId, 100000); + topDocs.setMaxScore(1); + int docID = -1; + if (topDocs != null && topDocs.scoreDocs != null && topDocs.scoreDocs.length > 0) { + docID = topDocs.scoreDocs[0].doc; + } + makeDocumentsSearcherManagerUpToDate(); + searcher2 = documentsSearcherManager.acquire(); + MoreLikeThis mlt = new MoreLikeThis(documentsIndexReader); // TODO documentsIndexReader is ok ? + mlt.setFieldNames(new String[]{"content"}); // similarity function works against these fields + mlt.setMinWordLen(2); + mlt.setBoost(true); + Query queryMoreLikeThis = mlt.like(docID); + TopDocs moreLikeThisDocs = searcher2.search(queryMoreLikeThis, 10); + moreLikeThisDocs.setMaxScore(10); + if (moreLikeThisDocs != null) { + if (wspDocs == null) + wspDocs = new ArrayList<de.mpg.mpiwg.berlin.mpdl.cms.document.Document>(); + for (int i=0; i<moreLikeThisDocs.scoreDocs.length; i++) { + int docIdent = moreLikeThisDocs.scoreDocs[i].doc; + Document luceneDoc = searcher2.doc(docIdent); + de.mpg.mpiwg.berlin.mpdl.cms.document.Document wspDoc = new de.mpg.mpiwg.berlin.mpdl.cms.document.Document(luceneDoc); + wspDocs.add(wspDoc); + } + } + if (wspDocs != null) { + hits = new Hits(wspDocs, from, to); + hits.setSize(moreLikeThisDocs.scoreDocs.length); + } + } catch (Exception e) { + throw new ApplicationException(e); + } finally { + try { + if (searcher1 != null) + documentsSearcherManager.release(searcher1); + if (searcher2 != null) + documentsSearcherManager.release(searcher2); + } catch (IOException e) { + // nothing + } + } + // Do not use searcher after this! + searcher1 = null; + searcher2 = null; + return hits; + } + + public MetadataRecord getDocMetadata(String docId) throws ApplicationException { + MetadataRecord mdRecord = null; + Document doc = getDocument(docId); + if (doc != null) { + String identifier = null; + Fieldable identifierField = doc.getFieldable("identifier"); + if (identifierField != null) + identifier = identifierField.stringValue(); + String uri = null; + Fieldable uriField = doc.getFieldable("uri"); + if (uriField != null) + uri = uriField.stringValue(); + String collectionNames = null; + Fieldable collectionNamesField = doc.getFieldable("collectionNames"); + if (collectionNamesField != null) + collectionNames = collectionNamesField.stringValue(); + String author = null; + Fieldable authorField = doc.getFieldable("author"); + if (authorField != null) + author = authorField.stringValue(); + String title = null; + Fieldable titleField = doc.getFieldable("title"); + if (titleField != null) + title = titleField.stringValue(); + String language = null; + Fieldable languageField = doc.getFieldable("language"); + if (languageField != null) + language = languageField.stringValue(); + else { + ConfManagerResultWrapper collectionInfo = CollectionReader.getInstance().getResultWrapper(collectionNames); + if (collectionInfo != null) { + String mainLang = collectionInfo.getMainLanguage(); + if (mainLang != null) + language = mainLang; + } + } + String publisher = null; + Fieldable publisherField = doc.getFieldable("publisher"); + if (publisherField != null) + publisher = publisherField.stringValue(); + Date yearDate = null; + Fieldable dateField = doc.getFieldable("date"); + if (dateField != null) { + String dateStr = dateField.stringValue(); + if (dateStr != null && !dateStr.equals("")) { + dateStr = StringUtils.deresolveXmlEntities(dateStr); + String yearStr = new Util().toYearStr(dateStr); // test if possible + // etc + if (yearStr != null) { + yearDate = new Util().toDate(yearStr + "-01-01T00:00:00.000Z"); + } + } + } + String rights = null; + Fieldable rightsField = doc.getFieldable("rights"); + if (rightsField != null) + rights = rightsField.stringValue(); + String license = null; + Fieldable licenseField = doc.getFieldable("license"); + if (licenseField != null) + license = licenseField.stringValue(); + String accessRights = null; + Fieldable accessRightsField = doc.getFieldable("accessRights"); + if (accessRightsField != null) + accessRights = accessRightsField.stringValue(); + String echoId = null; + Fieldable echoIdField = doc.getFieldable("echoId"); + if (echoIdField != null) + echoId = echoIdField.stringValue(); + String echoPageImageDir = null; + Fieldable echoPageImageDirField = doc.getFieldable("echoPageImageDir"); + if (echoPageImageDirField != null) + echoPageImageDir = echoPageImageDirField.stringValue(); + String echoFiguresDir = null; + Fieldable echoFiguresDirField = doc.getFieldable("echoFiguresDir"); + if (echoFiguresDirField != null) + echoFiguresDir = echoFiguresDirField.stringValue(); + String mpiwgDocId = null; + Fieldable mpiwgDocIdField = doc.getFieldable("mpiwgDocId"); + if (mpiwgDocIdField != null) + mpiwgDocId = mpiwgDocIdField.stringValue(); + int pageCount = -1; + Fieldable pageCountField = doc.getFieldable("pageCount"); + if (pageCountField != null) { + String pageCountStr = pageCountField.stringValue(); + pageCount = Integer.valueOf(pageCountStr); + } + String schemaName = null; + Fieldable schemaNameField = doc.getFieldable("schemaName"); + if (schemaNameField != null) + schemaName = schemaNameField.stringValue(); + Date lastModified = null; + Fieldable lastModifiedField = doc.getFieldable("lastModified"); + if (lastModifiedField != null) { + String lastModifiedXSDateStr = lastModifiedField.stringValue(); + lastModified = new Util().toDate(lastModifiedXSDateStr); + } + mdRecord = new MetadataRecord(); + mdRecord.setDocId(docId); + mdRecord.setUri(uri); + mdRecord.setIdentifier(identifier); + mdRecord.setCollectionNames(collectionNames); + mdRecord.setCreator(author); + mdRecord.setTitle(title); + mdRecord.setDate(yearDate); + mdRecord.setLanguage(language); + mdRecord.setPublisher(publisher); + mdRecord.setLicense(license); + mdRecord.setRights(rights); + mdRecord.setAccessRights(accessRights); + mdRecord.setEchoId(echoId); + mdRecord.setEchoPageImageDir(echoPageImageDir); + mdRecord.setEchoFiguresDir(echoFiguresDir); + mdRecord.setMpiwgDocId(mpiwgDocId); + mdRecord.setPageCount(pageCount); + mdRecord.setSchemaName(schemaName); + mdRecord.setLastModified(lastModified); + } + return mdRecord; + } + + public ArrayList<Token> getToken(String fieldName, String value, int count) throws ApplicationException { + ArrayList<Token> retToken = null; + int counter = 0; + TermEnum terms = null; + try { + if (value == null) + value = ""; + Term term = new Term(fieldName, value); + makeIndexReaderUpToDate(); + terms = documentsIndexReader.terms(term); + while (terms != null && fieldName != null && fieldName.equals(terms.term().field()) && counter < count) { + if (retToken == null) + retToken = new ArrayList<Token>(); + Term termContent = terms.term(); + Token token = new Token(termContent); + retToken.add(token); + counter++; + if (!terms.next()) + break; + } + } catch (Exception e) { + throw new ApplicationException(e); + } finally { + if (terms != null) { + try { + terms.close(); + } catch (IOException e) { + // nothing + } + } + } + return retToken; + } + + public ArrayList<Token> getToken(String docId, String fieldName, String value, int count) throws ApplicationException { + ArrayList<Token> retToken = null; + if (value == null) + value = ""; + int counter = 0; + IndexSearcher searcher = null; + try { + makeDocumentsSearcherManagerUpToDate(); + makeIndexReaderUpToDate(); + searcher = documentsSearcherManager.acquire(); + Query queryDocId = new TermQuery(new Term("docId", docId)); + TopDocs topDocs = searcher.search(queryDocId, 1); + if (topDocs != null) { + int docIdInt = topDocs.scoreDocs[0].doc; + TermFreqVector termFreqVector = documentsIndexReader.getTermFreqVector(docIdInt, fieldName); + if (termFreqVector != null) { + String[] terms = termFreqVector.getTerms(); + int[] freqs = termFreqVector.getTermFrequencies(); + boolean success = false; + if (terms != null) { + retToken = new ArrayList<Token>(); + for (int i = 0; i < terms.length; i++) { + String termStr = terms[i]; + if (termStr.startsWith(value)) + success = true; + if (success) { + counter++; + int freq = freqs[i]; + Term t = new Term(fieldName, termStr); + Token tok = new Token(t); + tok.setFreq(freq); + retToken.add(tok); + } + if (counter >= count) + break; + } + } + } + } + } catch (Exception e) { + throw new ApplicationException(e); + } finally { + try { + if (searcher != null) + documentsSearcherManager.release(searcher); + } catch (IOException e) { + // nothing + } + } + // Do not use searcher after this! + searcher = null; + return retToken; + } + + public void end() throws ApplicationException { + try { + if (documentsIndexWriter != null) + documentsIndexWriter.close(); + if (nodesIndexWriter != null) + nodesIndexWriter.close(); + if (documentsSearcherManager != null) + documentsSearcherManager.close(); + if (nodesSearcherManager != null) + nodesSearcherManager.close(); + if (documentsIndexReader != null) + documentsIndexReader.close(); + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + private Query buildMorphQuery(Query query, String language) throws ApplicationException { + return buildMorphQuery(query, language, false, false); + } + + private Query buildMorphQuery(Query query, String language, boolean withAllForms, boolean translate) throws ApplicationException { + Query morphQuery = null; + if (query instanceof TermQuery) { + TermQuery termQuery = (TermQuery) query; + morphQuery = buildMorphQuery(termQuery, language, withAllForms, translate); + } else if (query instanceof BooleanQuery) { + BooleanQuery booleanQuery = (BooleanQuery) query; + morphQuery = buildMorphQuery(booleanQuery, language, withAllForms, translate); + } else { + morphQuery = query; // all other cases: PrefixQuery, PhraseQuery, FuzzyQuery, TermRangeQuery, ... + } + return morphQuery; + } + + private Query buildMorphQuery(TermQuery inputTermQuery, String fromLang, boolean withAllForms, boolean translate) throws ApplicationException { + String[] toLanguages = {"deu", "eng", "fra"}; // TODO + String fromLanguage = null; + String inputTerm = inputTermQuery.getTerm().text(); + if (fromLang == null) { + String detectedLang = MicrosoftTranslator.detectLanguageCode(inputTerm); + if (detectedLang != null) + fromLanguage = detectedLang; + } else { + fromLanguage = fromLang; + } + LexHandler lexHandler = LexHandler.getInstance(); + String fieldName = inputTermQuery.getTerm().field(); + ArrayList<TermQuery> queryTerms = new ArrayList<TermQuery>(); + if (fieldName != null && fieldName.equals("tokenMorph")) { + ArrayList<Lemma> lemmas = lexHandler.getLemmas(inputTerm, "form", fromLanguage, Normalizer.DICTIONARY, true); + if (lemmas == null) { // if no lemmas are found then do a query in tokenOrig TODO should this really be done ? + if (translate) { + String[] terms = {inputTerm}; + ArrayList<String> translatedTerms = MicrosoftTranslator.translate(terms, fromLanguage, toLanguages); + for (int i=0; i<translatedTerms.size(); i++) { + String translatedTerm = translatedTerms.get(i); + Term translatedTermTokenOrig = new Term("tokenOrig", translatedTerm); + TermQuery translatedTermQueryInTokenOrig = new TermQuery(translatedTermTokenOrig); + queryTerms.add(translatedTermQueryInTokenOrig); + } + } else { + Term termTokenOrig = new Term("tokenOrig", inputTerm); + TermQuery termQueryInTokenOrig = new TermQuery(termTokenOrig); + queryTerms.add(termQueryInTokenOrig); + } + } else { + if (translate) { + ArrayList<String> morphTerms = new ArrayList<String>(); + for (int i=0; i<lemmas.size(); i++) { + Lemma lemma = lemmas.get(i); + if (withAllForms) { // all word forms are put into the query as boolean or clauses: needed in fragments search when all forms should be highlighted + ArrayList<Form> forms = lemma.getFormsList(); + for (int j=0; j<forms.size(); j++) { + Form form = forms.get(j); + String formName = form.getFormName(); + morphTerms.add(formName); + } + } else { + String lemmaName = lemma.getLemmaName(); + morphTerms.add(lemmaName); + } + } + String[] morphTermsArray = morphTerms.toArray(new String[morphTerms.size()]); + ArrayList<String> translatedMorphTerms = MicrosoftTranslator.translate(morphTermsArray, fromLanguage, toLanguages); + for (int i=0; i<translatedMorphTerms.size(); i++) { + String translatedMorphTermStr = translatedMorphTerms.get(i); + Term translatedMorphTerm = new Term(fieldName, translatedMorphTermStr); + TermQuery translatedMorphTermQuery = new TermQuery(translatedMorphTerm); + queryTerms.add(translatedMorphTermQuery); + } + } else { + for (int i = 0; i < lemmas.size(); i++) { + Lemma lemma = lemmas.get(i); + if (withAllForms) { // all word forms are put into the query as boolean or clauses: needed in fragments search when all forms should be highlighted + ArrayList<Form> forms = lemma.getFormsList(); + for (int j=0; j<forms.size(); j++) { + Form form = forms.get(j); + Term formTerm = new Term(fieldName, form.getFormName()); + TermQuery morphTermQuery = new TermQuery(formTerm); + queryTerms.add(morphTermQuery); + } + } else { + Term lemmaTerm = new Term(fieldName, lemma.getLemmaName()); + TermQuery morphTermQuery = new TermQuery(lemmaTerm); + queryTerms.add(morphTermQuery); + } + } + } + } + } else { + // if it is not the morph field then do a normal query + if (translate) { + String inputTermQueryField = inputTermQuery.getTerm().field(); + String inputTermQueryStr = inputTermQuery.getTerm().text(); + String[] terms = {inputTermQueryStr}; + ArrayList<String> translatedTerms = MicrosoftTranslator.translate(terms, fromLanguage, toLanguages); + for (int i=0; i<translatedTerms.size(); i++) { + String translatedTerm = translatedTerms.get(i); + Term translatedTermTokenOrig = new Term(inputTermQueryField, translatedTerm); + TermQuery translatedTermQueryInTokenOrig = new TermQuery(translatedTermTokenOrig); + queryTerms.add(translatedTermQueryInTokenOrig); + } + } else { + queryTerms.add(inputTermQuery); + } + //TODO ?? perhaps other fields should also be queried morphological e.g. title etc. + } + Query retQuery = buildBooleanShouldQuery(queryTerms); + return retQuery; + } + + private Query buildBooleanShouldQuery(ArrayList<TermQuery> queryTerms) throws ApplicationException { + BooleanQuery retBooleanQuery = new BooleanQuery(); + for (int i = 0; i < queryTerms.size(); i++) { + TermQuery termQuery = queryTerms.get(i); + retBooleanQuery.add(termQuery, BooleanClause.Occur.SHOULD); + } + return retBooleanQuery; + } + + private Query buildMorphQuery(BooleanQuery query, String language, boolean withAllForms, boolean translate) throws ApplicationException { + BooleanQuery morphBooleanQuery = new BooleanQuery(); + BooleanClause[] booleanClauses = query.getClauses(); + for (int i = 0; i < booleanClauses.length; i++) { + BooleanClause boolClause = booleanClauses[i]; + Query q = boolClause.getQuery(); + Query morphQuery = buildMorphQuery(q, language, withAllForms, translate); + BooleanClause.Occur occur = boolClause.getOccur(); + morphBooleanQuery.add(morphQuery, occur); + } + return morphBooleanQuery; + } + + public ArrayList<String> fetchTerms(String queryStr) throws ApplicationException { + ArrayList<String> terms = null; + String defaultQueryFieldName = "tokenOrig"; + try { + Query query = new QueryParser(Version.LUCENE_35, defaultQueryFieldName, nodesPerFieldAnalyzer).parse(queryStr); + terms = fetchTerms(query); + } catch (Exception e) { + throw new ApplicationException(e); + } + return terms; + } + + /** + * recursively fetch all terms of the query + * + * @param query + * @return + */ + private ArrayList<String> fetchTerms(Query query) throws ApplicationException { + ArrayList<String> terms = new ArrayList<String>(); + if (query instanceof TermQuery) { + TermQuery termQuery = (TermQuery) query; + String termQueryStr = termQuery.getTerm().text(); + terms.add(termQueryStr); + } else if (query instanceof BooleanQuery) { + BooleanQuery booleanQuery = (BooleanQuery) query; + terms = fetchTerms(booleanQuery); + } else { + String queryStr = query.toString(); + terms.add(queryStr); // all other cases: PrefixQuery, PhraseQuery, + // FuzzyQuery, TermRangeQuery, ... + } + return terms; + } + + private ArrayList<String> fetchTerms(BooleanQuery query) throws ApplicationException { + ArrayList<String> terms = new ArrayList<String>(); + BooleanClause[] booleanClauses = query.getClauses(); + for (int i = 0; i < booleanClauses.length; i++) { + BooleanClause boolClause = booleanClauses[i]; + Query q = boolClause.getQuery(); + ArrayList<String> qTerms = fetchTerms(q); + BooleanClause.Occur occur = boolClause.getOccur(); + if (occur == BooleanClause.Occur.SHOULD || occur == BooleanClause.Occur.MUST) + terms.addAll(qTerms); + } + return terms; + } + + public ArrayList<String> fetchTerms(String queryStr, String language) throws ApplicationException { + ArrayList<String> terms = null; + String defaultQueryFieldName = "tokenOrig"; + try { + Query query = new QueryParser(Version.LUCENE_35, defaultQueryFieldName, nodesPerFieldAnalyzer).parse(queryStr); + terms = fetchTerms(query, language); + } catch (Exception e) { + throw new ApplicationException(e); + } + return terms; + } + + /** + * recursively fetch all terms of the query + * + * @param query + * @return + */ + private ArrayList<String> fetchTerms(Query query, String language) throws ApplicationException { + ArrayList<String> terms = new ArrayList<String>(); + if (query instanceof TermQuery) { + TermQuery termQuery = (TermQuery) query; + terms = fetchTerms(termQuery, language); + } else if (query instanceof BooleanQuery) { + BooleanQuery booleanQuery = (BooleanQuery) query; + terms = fetchTerms(booleanQuery, language); + } else { + String queryStr = query.toString(); + terms.add(queryStr); + // all other cases: PrefixQuery, PhraseQuery, FuzzyQuery, TermRangeQuery, ... + } + return terms; + } + + private ArrayList<String> fetchTerms(TermQuery termQuery, String language) throws ApplicationException { + if (language == null) + language = "eng"; + ArrayList<String> terms = new ArrayList<String>(); + Term termQueryTerm = termQuery.getTerm(); + String term = termQuery.getTerm().text(); + String fieldName = termQueryTerm.field(); + if (fieldName != null && fieldName.equals("tokenMorph")) { + LexHandler lexHandler = LexHandler.getInstance(); + ArrayList<Lemma> lemmas = lexHandler.getLemmas(term, "form", language, Normalizer.DICTIONARY, true); + // TODO : language über den translator service holen + if (lemmas == null) { + terms.add(term); + } else { + for (int i = 0; i < lemmas.size(); i++) { + Lemma lemma = lemmas.get(i); + ArrayList<Form> forms = lemma.getFormsList(); + for (int j = 0; j < forms.size(); j++) { + Form form = forms.get(j); + String formName = form.getFormName(); + terms.add(formName); + } + } + } + } else { + terms.add(term); + } + return terms; + } + + private ArrayList<String> fetchTerms(BooleanQuery query, String language) throws ApplicationException { + ArrayList<String> terms = new ArrayList<String>(); + BooleanClause[] booleanClauses = query.getClauses(); + for (int i = 0; i < booleanClauses.length; i++) { + BooleanClause boolClause = booleanClauses[i]; + Query q = boolClause.getQuery(); + ArrayList<String> qTerms = fetchTerms(q, language); + BooleanClause.Occur occur = boolClause.getOccur(); + if (occur == BooleanClause.Occur.SHOULD || occur == BooleanClause.Occur.MUST) + terms.addAll(qTerms); + } + return terms; + } + + private Document getDocument(String docId) throws ApplicationException { + Document doc = null; + IndexSearcher searcher = null; + try { + makeDocumentsSearcherManagerUpToDate(); + searcher = documentsSearcherManager.acquire(); + String fieldNameDocId = "docId"; + Query queryDocId = new QueryParser(Version.LUCENE_35, fieldNameDocId, documentsPerFieldAnalyzer).parse(docId); + TopDocs topDocs = searcher.search(queryDocId, 100000); + topDocs.setMaxScore(1); + if (topDocs != null && topDocs.scoreDocs != null && topDocs.scoreDocs.length > 0) { + int docID = topDocs.scoreDocs[0].doc; + FieldSelector docFieldSelector = getDocFieldSelector(); + doc = searcher.doc(docID, docFieldSelector); + } + searcher.close(); + } catch (Exception e) { + throw new ApplicationException(e); + } finally { + try { + if (searcher != null) + documentsSearcherManager.release(searcher); + } catch (IOException e) { + // nothing + } + } + // Do not use searcher after this! + searcher = null; + return doc; + } + + private IndexWriter getDocumentsWriter() throws ApplicationException { + IndexWriter writer = null; + String luceneDocsDirectoryStr = Constants.getInstance().getLuceneDocumentsDir(); + File luceneDocsDirectory = new File(luceneDocsDirectoryStr); + try { + Map<String, Analyzer> documentsFieldAnalyzers = new HashMap<String, Analyzer>(); + documentsFieldAnalyzers.put("docId", new KeywordAnalyzer()); + documentsFieldAnalyzers.put("identifier", new KeywordAnalyzer()); + documentsFieldAnalyzers.put("uri", new KeywordAnalyzer()); + documentsFieldAnalyzers.put("collectionNames", new StandardAnalyzer(Version.LUCENE_35)); + documentsFieldAnalyzers.put("author", new StandardAnalyzer(Version.LUCENE_35)); + documentsFieldAnalyzers.put("title", new StandardAnalyzer(Version.LUCENE_35)); + documentsFieldAnalyzers.put("language", new StandardAnalyzer(Version.LUCENE_35)); + documentsFieldAnalyzers.put("publisher", new StandardAnalyzer(Version.LUCENE_35)); + documentsFieldAnalyzers.put("date", new StandardAnalyzer(Version.LUCENE_35)); + documentsFieldAnalyzers.put("subject", new StandardAnalyzer(Version.LUCENE_35)); + documentsFieldAnalyzers.put("rights", new StandardAnalyzer(Version.LUCENE_35)); + documentsFieldAnalyzers.put("license", new StandardAnalyzer(Version.LUCENE_35)); + documentsFieldAnalyzers.put("accessRights", new StandardAnalyzer(Version.LUCENE_35)); + documentsFieldAnalyzers.put("echoId", new KeywordAnalyzer()); + documentsFieldAnalyzers.put("echoPageImageDir", new KeywordAnalyzer()); + documentsFieldAnalyzers.put("echoFiguresDir", new KeywordAnalyzer()); + documentsFieldAnalyzers.put("mpiwgDocId", new KeywordAnalyzer()); + documentsFieldAnalyzers.put("type", new KeywordAnalyzer()); // e.g. mime type "text/xml" + documentsFieldAnalyzers.put("pageCount", new KeywordAnalyzer()); + documentsFieldAnalyzers.put("schemaName", new StandardAnalyzer(Version.LUCENE_35)); + documentsFieldAnalyzers.put("lastModified", new KeywordAnalyzer()); + documentsFieldAnalyzers.put("tokenOrig", new StandardAnalyzer(Version.LUCENE_35)); + documentsFieldAnalyzers.put("tokenReg", new StandardAnalyzer(Version.LUCENE_35)); + documentsFieldAnalyzers.put("tokenNorm", new StandardAnalyzer(Version.LUCENE_35)); + documentsFieldAnalyzers.put("tokenMorph", new StandardAnalyzer(Version.LUCENE_35)); + documentsFieldAnalyzers.put("xmlContent", new StandardAnalyzer(Version.LUCENE_35)); + documentsFieldAnalyzers.put("content", new StandardAnalyzer(Version.LUCENE_35)); + documentsPerFieldAnalyzer = new PerFieldAnalyzerWrapper(new StandardAnalyzer(Version.LUCENE_35), documentsFieldAnalyzers); + IndexWriterConfig conf = new IndexWriterConfig(Version.LUCENE_35, documentsPerFieldAnalyzer); + conf.setOpenMode(OpenMode.CREATE_OR_APPEND); + conf.setRAMBufferSizeMB(300); // 300 MB because some documents are big; 16 MB is default + FSDirectory fsDirectory = FSDirectory.open(luceneDocsDirectory); + writer = new IndexWriter(fsDirectory, conf); + writer.commit(); // when directory is empty this creates init files + } catch (IOException e) { + throw new ApplicationException(e); + } + return writer; + } + + private IndexWriter getNodesWriter() throws ApplicationException { + IndexWriter writer = null; + String luceneNodesDirectoryStr = Constants.getInstance().getLuceneNodesDir(); + File luceneNodesDirectory = new File(luceneNodesDirectoryStr); + try { + Map<String, Analyzer> nodesFieldAnalyzers = new HashMap<String, Analyzer>(); + nodesFieldAnalyzers.put("docId", new KeywordAnalyzer()); + nodesFieldAnalyzers.put("language", new StandardAnalyzer(Version.LUCENE_35)); // language (through xml:id): e.g. "lat" + nodesFieldAnalyzers.put("pageNumber", new KeywordAnalyzer()); // page number (through element pb): e.g. "13" + nodesFieldAnalyzers.put("lineNumber", new KeywordAnalyzer()); // line number on the page (through element lb): e.g. "17" + nodesFieldAnalyzers.put("elementName", new KeywordAnalyzer()); // element name: e.g. "tei:s" + nodesFieldAnalyzers.put("elementDocPosition", new KeywordAnalyzer()); // absolute position in document: e.g. "4711" + nodesFieldAnalyzers.put("elementPosition", new KeywordAnalyzer()); // position in parent node (in relation to other nodes of the same name): e.g. "5" + nodesFieldAnalyzers.put("elementAbsolutePosition", new KeywordAnalyzer()); // absolute position in document (in relation to other nodes of the same name): e.g. "213" + nodesFieldAnalyzers.put("elementPagePosition", new KeywordAnalyzer()); // position in relation to other nodes of the same name: e.g. "213" + nodesFieldAnalyzers.put("xmlId", new KeywordAnalyzer()); // xml id: e.g. "4711bla" + nodesFieldAnalyzers.put("xpath", new KeywordAnalyzer()); // xpath: e.g. "/echo[1]/text[1]/p[1]/s[5]" + nodesFieldAnalyzers.put("tokenOrig", new StandardAnalyzer(Version.LUCENE_35)); + nodesFieldAnalyzers.put("tokenReg", new StandardAnalyzer(Version.LUCENE_35)); + nodesFieldAnalyzers.put("tokenNorm", new StandardAnalyzer(Version.LUCENE_35)); + nodesFieldAnalyzers.put("tokenMorph", new StandardAnalyzer(Version.LUCENE_35)); + nodesFieldAnalyzers.put("xmlContent", new StandardAnalyzer(Version.LUCENE_35)); + nodesPerFieldAnalyzer = new PerFieldAnalyzerWrapper(new StandardAnalyzer(Version.LUCENE_35), nodesFieldAnalyzers); + IndexWriterConfig conf = new IndexWriterConfig(Version.LUCENE_35, nodesPerFieldAnalyzer); + conf.setOpenMode(OpenMode.CREATE_OR_APPEND); + conf.setRAMBufferSizeMB(300); // 300 MB because some documents are big; 16 MB is default + FSDirectory fsDirectory = FSDirectory.open(luceneNodesDirectory); + writer = new IndexWriter(fsDirectory, conf); + writer.commit(); + } catch (IOException e) { + throw new ApplicationException(e); + } + return writer; + } + + private Sort buildSort(String[] sortFieldNames, String type) { + Sort sort = new Sort(); + ArrayList<SortField> sortFields = new ArrayList<SortField>(); + for (int i=0; i<sortFieldNames.length; i++) { + String sortFieldName = sortFieldNames[i]; + int sortFieldType = getDocSortFieldType(sortFieldName); + if (type.equals("node")) + sortFieldType = getNodeSortFieldType(sortFieldName); + String realSortFieldName = getDocSortFieldName(sortFieldName); + SortField sortField = new SortField(realSortFieldName, sortFieldType); + sortFields.add(sortField); + } + if (sortFieldNames.length == 1) { + SortField sortField1 = sortFields.get(0); + sort.setSort(sortField1); + } else if (sortFieldNames.length == 2) { + SortField sortField1 = sortFields.get(0); + SortField sortField2 = sortFields.get(1); + sort.setSort(sortField1, sortField2); + } else if (sortFieldNames.length == 2) { + SortField sortField1 = sortFields.get(0); + SortField sortField2 = sortFields.get(1); + SortField sortField3 = sortFields.get(2); + sort.setSort(sortField1, sortField2, sortField3); + } + return sort; + } + + private String getDocSortFieldName(String fieldName) { + String sortFieldName = fieldName + "Sorted"; + return sortFieldName; + } + + private int getDocSortFieldType(String fieldName) { + int type = SortField.STRING; + if (fieldName.equals("lastModified")) + type = SortField.LONG; + return type; + } + + private int getNodeSortFieldType(String fieldName) { + int type = SortField.STRING; + if (fieldName.equals("pageNumber") || fieldName.equals("lineNumber") || fieldName.equals("elementDocPosition")) + type = SortField.INT; + return type; + } + + private FieldSelector getDocFieldSelector() { + HashSet<String> fields = new HashSet<String>(); + fields.add("docId"); + fields.add("identifier"); + fields.add("uri"); + fields.add("collectionNames"); + fields.add("author"); + fields.add("title"); + fields.add("language"); + fields.add("publisher"); + fields.add("date"); + fields.add("subject"); + fields.add("rights"); + fields.add("license"); + fields.add("echoId"); + fields.add("echoPageImageDir"); + fields.add("echoFiguresDir"); + fields.add("mpiwgDocId"); + fields.add("type"); + fields.add("pageCount"); + fields.add("schemaName"); + fields.add("lastModified"); + fields.add("content"); + FieldSelector fieldSelector = new SetBasedFieldSelector(fields, fields); + return fieldSelector; + } + + private FieldSelector getNodeFieldSelector() { + HashSet<String> fields = new HashSet<String>(); + fields.add("docId"); + fields.add("language"); + fields.add("pageNumber"); + fields.add("lineNumber"); + fields.add("elementName"); + fields.add("elementDocPosition"); + fields.add("elementPosition"); + fields.add("elementAbsolutePosition"); + fields.add("elementPagePosition"); + fields.add("xmlId"); + fields.add("xpath"); + fields.add("xmlContent"); + fields.add("xmlContentTokenized"); + FieldSelector fieldSelector = new SetBasedFieldSelector(fields, fields); + return fieldSelector; + } + + private SearcherManager getNewSearcherManager(IndexWriter indexWriter) throws ApplicationException { + SearcherManager searcherManager = null; + try { + searcherManager = new SearcherManager(indexWriter, true, null, null); + } catch (IOException e) { + throw new ApplicationException(e); + } + return searcherManager; + } + + private IndexReader getDocumentsReader() throws ApplicationException { + IndexReader reader = null; + String luceneDocsDirectoryStr = Constants.getInstance().getLuceneDocumentsDir(); + File luceneDocsDirectory = new File(luceneDocsDirectoryStr); + try { + FSDirectory fsDirectory = FSDirectory.open(luceneDocsDirectory); + reader = IndexReader.open(fsDirectory, true); + } catch (IOException e) { + throw new ApplicationException(e); + } + return reader; + } + + private void makeIndexReaderUpToDate() throws ApplicationException { + try { + boolean isCurrent = documentsIndexReader.isCurrent(); + if (!isCurrent) { + documentsIndexReader = IndexReader.openIfChanged(documentsIndexReader); + } + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + private void makeDocumentsSearcherManagerUpToDate() throws ApplicationException { + try { + boolean isCurrent = documentsSearcherManager.isSearcherCurrent(); + if (!isCurrent) { + documentsSearcherManager.maybeReopen(); + } + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + private void makeNodesSearcherManagerUpToDate() throws ApplicationException { + try { + boolean isCurrent = nodesSearcherManager.isSearcherCurrent(); + if (!isCurrent) { + nodesSearcherManager.maybeReopen(); + } + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + private String toTokenizedXmlString(String xmlStr, String language) throws ApplicationException { + String xmlPre = "<tokenized xmlns:xhtml=\"http://www.w3.org/1999/xhtml\" xmlns:de=\"http://www.mpiwg-berlin.mpg.de/ns/de/1.0/\" xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">"; + String xmlPost = "</tokenized>"; + String xmlStrTmp = xmlPre + xmlStr + xmlPost; + StringReader xmlInputStringReader = new StringReader(xmlStrTmp); + XmlTokenizer xmlTokenizer = new XmlTokenizer(xmlInputStringReader); + xmlTokenizer.setLanguage(language); + String[] outputOptions = { "withLemmas" }; + xmlTokenizer.setOutputOptions(outputOptions); + xmlTokenizer.tokenize(); + String result = xmlTokenizer.getXmlResult(); + return result; + } + + private String enrichWordsOrigRegNorm(String xmlStr) throws ApplicationException { + try { + WordContentHandler wordContentHandler = new WordContentHandler(); + XMLReader xmlParser = new SAXParser(); + xmlParser.setContentHandler(wordContentHandler); + StringReader strReader = new StringReader(xmlStr); + InputSource inputSource = new InputSource(strReader); + xmlParser.parse(inputSource); + String result = wordContentHandler.getResult(); + return result; + } catch (SAXException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + private String escapeLuceneChars(String inputStr) { + String luceneCharsStr = "+-&|!(){}[]^~*?:\\"; // Lucene escape symbols + StringBuilder retStrBuilder = new StringBuilder(); + for (int i = 0; i < inputStr.length(); i++) { + char c = inputStr.charAt(i); + if (luceneCharsStr.contains(String.valueOf(c))) + retStrBuilder.append("\\"); + retStrBuilder.append(c); + } + return retStrBuilder.toString(); + } + + /** + * sorgt für sinnvolle satzanfänge + * + * @param fragment + */ + private String checkHitFragment(String fragment) { + if (fragment.startsWith(".") + || fragment.startsWith(":") + || fragment.startsWith(",") + || fragment.startsWith("-") + || fragment.startsWith(";") + || fragment.startsWith("?") + || fragment.startsWith(")") + || fragment.startsWith("!")) { + fragment = fragment.substring(1, fragment.length()); + // finds first occurence of a given string out.println("first index of point : "+StringUtils.indexOfAny(fragment, ".")); + } + return fragment; + } + +} \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/scheduler/CmsChainScheduler.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,243 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.scheduler; + +import java.net.URL; +import java.util.ArrayList; +import java.util.Collection; +import java.util.Date; +import java.util.HashMap; +import java.util.Iterator; +import java.util.List; +import java.util.PriorityQueue; +import java.util.Queue; + +import java.util.logging.Logger; +import org.quartz.JobDataMap; +import org.quartz.JobDetail; +import org.quartz.JobExecutionContext; +import org.quartz.JobListener; +import org.quartz.SchedulerException; +import org.quartz.SimpleTrigger; +import org.quartz.Trigger; +import org.quartz.impl.StdSchedulerFactory; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; + +public class CmsChainScheduler { + private static CmsChainScheduler instance; + private static String CRUD_JOB = "MPDL_CRUD_JOB"; + private static String CRUD_TRIGGER = "MPDL_CRUD_TRIGGER"; + private static String CRUD_GROUP = "MPDL_CRUD_GROUP"; + private static Logger LOGGER = Logger.getLogger(CmsDocJob.class.getName()); + private org.quartz.Scheduler scheduler; + private JobListener jobListener; + private Queue<CmsDocOperation> docOperationQueue = new PriorityQueue<CmsDocOperation>(); + private HashMap<Integer, CmsDocOperation> finishedDocOperations = new HashMap<Integer, CmsDocOperation>(); + private boolean operationInProgress = false; + private int jobOrderId = 0; + + public static CmsChainScheduler getInstance() throws ApplicationException { + if (instance == null) { + instance = new CmsChainScheduler(); + instance.init(); + } + return instance; + } + + public CmsDocOperation doOperation(CmsDocOperation docOperation) throws ApplicationException { + jobOrderId++; + docOperation.setOrderId(jobOrderId); + queueOperation(docOperation); + scheduleNextOperation(); + return docOperation; + } + + public void finishOperation(CmsDocOperation docOperation) throws ApplicationException { + operationInProgress = false; + Date now = new Date(); + docOperation.setEnd(now); + docOperation.setStatus("finished"); + int jobId = new Integer(docOperation.getOrderId()); + finishedDocOperations.put(jobId, docOperation); + log(docOperation); + // schedule next job if there is one + scheduleNextOperation(); + } + + private void log(CmsDocOperation docOperation) { + Date startTime = docOperation.getStart(); + Date endTime = docOperation.getEnd(); + long executionTime = -1; + if (startTime != null && endTime != null) + executionTime = (endTime.getTime() - startTime.getTime()); + String jobInfo = "Document operation " + docOperation.toString() + ": started at: " + startTime + + " and ended at: " + endTime + " (needed time: " + executionTime + " ms)"; + LOGGER.info(jobInfo); + } + + public synchronized void scheduleNextOperation() throws ApplicationException { + if (isOperationInProgress()) { + // nothing, operation has to wait + } else { + CmsDocOperation docOperation = docOperationQueue.poll(); + if (docOperation == null) { + // if queue is empty then do nothing (there are no more operations to execute) + } else { + Date now = new Date(); + operationInProgress = true; + docOperation.setStart(now); + scheduleJob(docOperation, now); + } + } + } + + public ArrayList<CmsDocOperation> getDocOperations() throws ApplicationException { + ArrayList<CmsDocOperation> docOperations = new ArrayList<CmsDocOperation>(); + try { + // first: all finished jobs + Collection<CmsDocOperation> finiDocOperations = finishedDocOperations.values(); + docOperations.addAll(finiDocOperations); + // second: all currently executed jobs + if (operationInProgress) { + List<JobExecutionContext> currentJobs = (List<JobExecutionContext>) scheduler.getCurrentlyExecutingJobs(); + Iterator<JobExecutionContext> iter = currentJobs.iterator(); + while (iter.hasNext()) { + JobExecutionContext jobExecutionContext = iter.next(); + CmsDocOperation docOperation = getDocOperation(jobExecutionContext); + if (docOperation != null) { + docOperations.add(docOperation); + } + } + } + // third: all queued jobs + Iterator<CmsDocOperation> iter = docOperationQueue.iterator(); + while (iter.hasNext()) { + CmsDocOperation docOperation = iter.next(); + docOperations.add(docOperation); + } + } catch (SchedulerException e) { + LOGGER.severe(e.getMessage()); + throw new ApplicationException(e); + } + return docOperations; + } + + public CmsDocOperation getDocOperation(int jobId) throws ApplicationException { + CmsDocOperation docOperation = null; + try { + // first try: looks into currently executing jobs + if (operationInProgress) { + List<JobExecutionContext> currentJobs = (List<JobExecutionContext>) scheduler.getCurrentlyExecutingJobs(); + Iterator<JobExecutionContext> iter = currentJobs.iterator(); + while (iter.hasNext()) { + JobExecutionContext jobExecutionContext = iter.next(); + docOperation = getDocOperation(jobExecutionContext); + if (docOperation != null) { + int dopOpJobId = docOperation.getOrderId(); + if (jobId == dopOpJobId) + return docOperation; + } + } + } + // second try: look into finished jobs + docOperation = finishedDocOperations.get(new Integer(jobId)); + if (docOperation != null) { + return docOperation; + } + // third try: look into queued jobs + Iterator<CmsDocOperation> iter = docOperationQueue.iterator(); + while (iter.hasNext()) { + docOperation = iter.next(); + if (docOperation.getOrderId() == jobId) + return docOperation; + } + } catch (SchedulerException e) { + LOGGER.severe(e.getMessage()); + throw new ApplicationException(e); + } + // if not found return null + return null; + } + + public CmsDocOperation getDocOperation(JobExecutionContext jobExecutionContext) { + CmsDocOperation docOperation = null; + if (jobExecutionContext != null) { + JobDetail job = jobExecutionContext.getJobDetail(); + JobDataMap parameters = job.getJobDataMap(); + docOperation = (CmsDocOperation) parameters.get("operation"); + } + return docOperation; + } + + private void queueOperation(CmsDocOperation docOperation) { + int operationsBefore = docOperationQueue.size(); + if (operationsBefore == 0) + docOperation.setStatus("waiting in operation queue"); + else + docOperation.setStatus("waiting in operation queue: " + operationsBefore + " operations heve to be executed before this operation"); + docOperationQueue.offer(docOperation); + } + + private synchronized boolean isOperationInProgress() { + return operationInProgress; + } + + private void scheduleJob(CmsDocOperation docOperation, Date fireTime) throws ApplicationException { + try { + int jobId = docOperation.getOrderId(); + String jobName = CRUD_JOB + "-id-" + jobId + "-timeId-" + fireTime; + JobDetail job = new JobDetail(jobName, CRUD_GROUP, CmsDocJob.class); + JobDataMap parameters = new JobDataMap(); + parameters.put("operation", docOperation); + job.setJobDataMap(parameters); + job.addJobListener(jobListener.getName()); + String triggerName = CRUD_TRIGGER + "-id-" + jobId + "-timeId-" + fireTime; + Trigger trigger = new SimpleTrigger(triggerName, CRUD_GROUP, fireTime); + scheduler.scheduleJob(job, trigger); + String jobInfo = "Schedule document operation: " + docOperation.toString() + ": done at: " + fireTime.toString(); + LOGGER.info(jobInfo); + } catch (SchedulerException e) { + LOGGER.severe(e.getMessage()); + throw new ApplicationException(e); + } + } + + private void init() throws ApplicationException { + try { + if (scheduler == null) { + String quartzPath = getQuartzPath(); + StdSchedulerFactory schedulerFactory = new StdSchedulerFactory(quartzPath); + scheduler = schedulerFactory.getScheduler(); + jobListener = new CmsChainSchedulerListener(); + scheduler.addJobListener(jobListener); + scheduler.start(); + LOGGER.info("Started Quartz scheduler factory: " + quartzPath); + } + } catch (SchedulerException e) { + LOGGER.severe(e.getMessage()); + throw new ApplicationException(e); + } + } + + public void end() throws ApplicationException { + try { + if (scheduler != null) { + scheduler.shutdown(); + } + String quartzPath = getQuartzPath(); + LOGGER.info("Ended Quartz scheduler factory: " + quartzPath); + } catch (SchedulerException e) { + LOGGER.severe(e.getMessage()); + throw new ApplicationException(e); + } + } + + private String getQuartzPath() { + URL quartzUrl = CmsChainScheduler.class.getResource("quartz.properties"); + String quartzPath = quartzUrl.getPath(); + if (quartzPath.indexOf(".jar!") != -1) { + int beginIndex = quartzPath.indexOf(".jar!") + 6; + quartzPath = quartzPath.substring(beginIndex); + } + return quartzPath; + } +} \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/scheduler/CmsChainSchedulerListener.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,53 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.scheduler; + +import java.util.logging.Logger; +import org.quartz.JobDataMap; +import org.quartz.JobDetail; +import org.quartz.JobExecutionContext; +import org.quartz.JobExecutionException; +import org.quartz.JobListener; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; + +public class CmsChainSchedulerListener implements JobListener { + private static Logger LOGGER = Logger.getLogger(CmsDocJob.class.getName()); + + public String getName() { + return "MpdlJobChainingListener"; + } + + public void jobToBeExecuted(JobExecutionContext inContext) { + } + + public void jobExecutionVetoed(JobExecutionContext inContext) { + String message = "Quartz: JobChainingListener: Job execution was vetoed."; + LOGGER.fine(message); + } + + public void jobWasExecuted(JobExecutionContext inContext, JobExecutionException inException) { + // after finishing his job it tries to schedule the next operation (if there is one in the queue) + CmsDocOperation docOperation = null; + try { + CmsChainScheduler mpdlChainScheduler = CmsChainScheduler.getInstance(); + docOperation = getDocOperation(inContext); + mpdlChainScheduler.finishOperation(docOperation); + } catch (ApplicationException e) { + if (docOperation != null) { + docOperation.setErrorMessage(e.getMessage()); + } + LOGGER.severe(e.getMessage()); + } + } + + private CmsDocOperation getDocOperation(JobExecutionContext context) { + CmsDocOperation docOperation = null; + if (context != null) { + JobDetail job = context.getJobDetail(); + JobDataMap parameters = job.getJobDataMap(); + docOperation = (CmsDocOperation) parameters.get("operation"); + } + return docOperation; + } + + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/scheduler/CmsDocJob.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,75 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.scheduler; + +import java.util.Date; + +import java.util.logging.Logger; +import org.quartz.Job; +import org.quartz.JobDataMap; +import org.quartz.JobDetail; +import org.quartz.JobExecutionContext; +import org.quartz.JobExecutionException; + +import de.mpg.mpiwg.berlin.mpdl.cms.document.DocumentHandler; +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; + +public class CmsDocJob implements Job { + public static String STATUS_BEGIN = "started"; + private static Logger LOGGER = Logger.getLogger(CmsDocJob.class.getName()); + private JobExecutionContext currentExecutedContext; + + public void execute(JobExecutionContext context) throws JobExecutionException { + this.currentExecutedContext = context; + CmsDocOperation docOperation = getDocOperation(); + try { + docOperation.setStatus(STATUS_BEGIN); + String operationName = docOperation.getName(); + if (operationName.equals("create")) { + DocumentHandler docHandler = new DocumentHandler(); + docHandler.doOperation(docOperation); + } else if (operationName.equals("delete")) { + DocumentHandler docHandler = new DocumentHandler(); + docHandler.doOperation(docOperation); + } else if (operationName.equals("importDirectory")) { + DocumentHandler docHandler = new DocumentHandler(); + docHandler.doOperation(docOperation); + } + Date startingTime = docOperation.getStart(); + String jobInfo = "Document operation " + docOperation.toString() + ": started at: " + startingTime; + LOGGER.info(jobInfo); + this.currentExecutedContext = null; + } catch (Exception e) { + try { + // Quartz will automatically unschedule all triggers associated with this job so that it does not run again + CmsChainScheduler mpdlChainScheduler = CmsChainScheduler.getInstance(); + mpdlChainScheduler.finishOperation(docOperation); + String errorMessage = e.getMessage(); + if (errorMessage == null) { + Throwable t = e.getCause(); + if (t == null) { + errorMessage = e.toString(); + } else { + errorMessage = t.getMessage(); + } + } + docOperation.setErrorMessage(errorMessage); + LOGGER.severe(errorMessage); + JobExecutionException jobExecutionException = new JobExecutionException(e); + jobExecutionException.setUnscheduleAllTriggers(true); + throw jobExecutionException; + } catch (ApplicationException ex) { + // nothing + } + } + } + + private CmsDocOperation getDocOperation() { + CmsDocOperation docOperation = null; + if (currentExecutedContext != null) { + JobDetail job = currentExecutedContext.getJobDetail(); + JobDataMap parameters = job.getJobDataMap(); + docOperation = (CmsDocOperation) parameters.get("operation"); + } + return docOperation; + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/scheduler/CmsDocOperation.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,160 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.scheduler; + +import java.util.Date; + +import de.mpg.mpiwg.berlin.mpdl.cms.document.MetadataRecord; + +public class CmsDocOperation implements Comparable<CmsDocOperation> { + private int id; + private Date start; + private Date end; + private String name; + private String status; + private String errorMessage; + private String uploadFileName; + private String srcUrl; + private String docIdentifier; + private String mainLanguage; + private String[] elementNames; // element names which should be indexed (e.g. "s head, caption") + private String collectionNames; // collections to which this document belongs separated by blanks (e.g. "collection1 collection7") + private MetadataRecord mdRecord; + + public CmsDocOperation(String name, String srcUrl, String uploadFileName, String docIdentifier) { + this.name = name; + this.srcUrl = srcUrl; + this.uploadFileName = uploadFileName; + this.docIdentifier = docIdentifier; + } + + public int compareTo(CmsDocOperation op) { + Integer opOrderId = new Integer(op.id); + Integer thisOrderId = new Integer(id); + return thisOrderId.compareTo(opOrderId); + } + + public boolean isFinished() { + if (status != null && status.equals("finished")) + return true; + else + return false; + } + + public boolean isError() { + if (errorMessage != null && errorMessage.length() > 0) + return true; + else + return false; + } + + public int getOrderId() { + return id; + } + + public void setOrderId(int orderId) { + this.id = orderId; + } + + public String getStatus() { + return status; + } + + public void setStatus(String status) { + this.status = status; + } + + public Date getStart() { + return start; + } + + public void setStart(Date start) { + this.start = start; + } + + public Date getEnd() { + return end; + } + + public void setEnd(Date end) { + this.end = end; + } + + public String getName() { + return name; + } + + public void setName(String name) { + this.name = name; + } + + public String getErrorMessage() { + return errorMessage; + } + + public void setErrorMessage(String errorMessage) { + this.errorMessage = errorMessage; + } + + public String getDocIdentifier() { + return docIdentifier; + } + + public void setDocIdentifier(String docIdentifier) { + this.docIdentifier = docIdentifier; + } + + public String getMainLanguage() { + return mainLanguage; + } + + public void setMainLanguage(String mainLanguage) { + this.mainLanguage = mainLanguage; + } + + public String[] getElementNames() { + return elementNames; + } + + public void setElementNames(String[] elementNames) { + this.elementNames = elementNames; + } + + public String getCollectionNames() { + return collectionNames; + } + + public void setCollectionNames(String collectionNames) { + this.collectionNames = collectionNames; + } + + public String getSrcUrl() { + return srcUrl; + } + + public void setSrcUrl(String srcUrl) { + this.srcUrl = srcUrl; + } + + public String getUploadFileName() { + return uploadFileName; + } + + public void setUploadFileName(String uploadFileName) { + this.uploadFileName = uploadFileName; + } + + public MetadataRecord getMdRecord() { + return mdRecord; + } + + public void setMdRecord(MetadataRecord mdRecord) { + this.mdRecord = mdRecord; + } + + public String toString() { + if (name.equals("delete")) + return name + "(" + id + ", " + docIdentifier + ")"; + else + return name + "(" + id + ", " + uploadFileName + ", " + docIdentifier + ")"; + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/scheduler/quartz.properties Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,9 @@ +org.quartz.scheduler.instanceName = MpdlScheduler +org.quartz.scheduler.instanceId = auto +org.quartz.scheduler.rmi.export = false +org.quartz.scheduler.rmi.proxy = false + +org.quartz.threadPool.class = org.quartz.simpl.SimpleThreadPool +org.quartz.threadPool.threadCount = 3 + +org.quartz.jobStore.class = org.quartz.simpl.RAMJobStore
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/test/TestLocal.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,412 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.test; + +import java.io.BufferedInputStream; +import java.io.File; +import java.io.FileReader; +import java.io.IOException; +import java.io.InputStream; +import java.io.PrintWriter; +import java.io.StringReader; +import java.io.StringWriter; +import java.net.URL; +import java.util.ArrayList; +import java.util.Date; +import java.util.Hashtable; + +import javax.xml.transform.stream.StreamSource; + +import net.sf.saxon.s9api.Processor; +import net.sf.saxon.s9api.QName; +import net.sf.saxon.s9api.Serializer; +import net.sf.saxon.s9api.XdmNode; +import net.sf.saxon.s9api.XsltCompiler; +import net.sf.saxon.s9api.XsltExecutable; + +import org.apache.commons.io.FileUtils; +import org.apache.commons.io.IOUtils; +import org.apache.lucene.document.Document; +import org.apache.lucene.document.Fieldable; +import org.apache.lucene.index.Term; +import org.xml.sax.InputSource; +import org.xml.sax.SAXException; +import org.xml.sax.XMLReader; + +import com.sun.org.apache.xerces.internal.parsers.SAXParser; + +import de.mpg.mpiwg.berlin.mpdl.cms.lucene.IndexHandler; +import de.mpg.mpiwg.berlin.mpdl.cms.scheduler.CmsChainScheduler; +import de.mpg.mpiwg.berlin.mpdl.cms.scheduler.CmsDocOperation; +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.lt.dict.db.LexHandler; +import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.MorphologyCache; +import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.WordContentHandler; +import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.XmlTokenizer; +import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.XmlTokenizerContentHandler; +import de.mpg.mpiwg.berlin.mpdl.util.Util; +import de.mpg.mpiwg.berlin.mpdl.xml.xquery.XQueryEvaluator; +import de.mpg.mpiwg.berlin.mpdl.cms.transform.GetFragmentsContentHandler; +import de.mpg.mpiwg.berlin.mpdl.cms.transform.HighlightContentHandler; +import de.mpg.mpiwg.berlin.mpdl.cms.transform.PageTransformer; +import de.mpg.mpiwg.berlin.mpdl.cms.transform.XslResourceTransformer; +import de.mpg.mpiwg.berlin.mpdl.cms.document.DocumentHandler; +import de.mpg.mpiwg.berlin.mpdl.cms.document.Hits; +import de.mpg.mpiwg.berlin.mpdl.cms.document.MetadataRecord; +import de.mpg.mpiwg.berlin.mpdl.cms.document.Token; + +public class TestLocal { + private IndexHandler indexer; + + public static void main(String[] args) throws ApplicationException { + try { + TestLocal test = new TestLocal(); + test.init(); + // test.importAllDocuments(); + // test.testTransform(); + // test.testXml(); + // test.generateToc(); + test.testCalls(); + // test.generatePdf(); + // test.xquery(); + // test.createToc(); + // test.testScheduler(); + // test.getDocInfo(); + // test.testChars(); + test.end(); + } catch (Exception e) { + e.printStackTrace(); + } + } + + private void init() throws ApplicationException { + indexer = IndexHandler.getInstance(); + } + + private void end() throws ApplicationException { + indexer.end(); + } + + private void testXml() throws ApplicationException { + try { + DocumentHandler docHandler = new DocumentHandler(); + String docDirName = docHandler.getDocDir("/echo/la/Benedetti_1585_163127KK.xml"); + String pageXmlFileName = docDirName + "/pages" + "/page-" + "444" + ".xml"; + File pageXmlFile = new File(pageXmlFileName); + String pageXmlStr = null; + if (pageXmlFile.exists()) + pageXmlStr = FileUtils.readFileToString(pageXmlFile, "utf-8"); + System.out.println(pageXmlStr); + String tokStr = tokenizeWithLemmas(pageXmlStr, "lat"); + System.out.println(tokStr); + tokStr = "<?xml version=\"1.0\" encoding=\"utf-8\"?>" + tokStr; + byte[] blablabla = tokStr.getBytes("utf-8"); + String blablu = new String(blablabla, "utf-8"); + String bla = enrichWordsOrigRegNorm(blablu); + System.out.println(bla); + + XQueryEvaluator xQueryEvaluator = new XQueryEvaluator(); + URL url = new URL("file:/var/yp/Test_1789.xml"); + XdmNode docNode = xQueryEvaluator.parse(url); // if it is not parseable an exception with a detail message is thrown + + File srcFile = new File("/Users/jwillenborg/mpdl/data/xml/documents/tei/de/dt-ptolemaeus-tei-merge2.xml"); + FileReader docFileReader = new FileReader(srcFile); + XmlTokenizer docXmlTokenizer = new XmlTokenizer(docFileReader); + docXmlTokenizer.setDocIdentifier("/tei/de/dt-ptolemaeus-tei-merge2.xml"); + docXmlTokenizer.tokenize(); + ArrayList<XmlTokenizerContentHandler.Element> elements = docXmlTokenizer.getElements("s"); + String blabla = ""; + } catch (Exception e) { + e.printStackTrace(); + } + } + + private void importAllDocuments() throws ApplicationException { + DocumentHandler docHandler = new DocumentHandler(); + CmsDocOperation docOperation = new CmsDocOperation("importDirectory", "file:/Users/jwillenborg/test/documents", null, null); + docOperation.setCollectionNames("echo"); + docHandler.doOperation(docOperation); + } + + private void generatePdf() throws ApplicationException { + long begin = new Date().getTime(); + DocumentHandler docHandler = new DocumentHandler(); + CmsDocOperation docOperation = new CmsDocOperation("createPdf", null, null, "/echo/la/Benedetti_1585_163127KK.xml"); + docHandler.doOperation(docOperation); + docOperation = new CmsDocOperation("createPdf", null, null, "/tei/de/dt-ptolemaeus-tei-merge2.xml"); + // docHandler.doOperation(docOperation); + long end = new Date().getTime(); + System.out.println("Needed time: " + (end - begin)); + } + + private void testChars() throws ApplicationException { + String docId = "/test/benedetti/page-444.xml"; + String docSrcUrlStr = "file:/Users/jwillenborg/texts/mpdl/documents" + docId; + DocumentHandler docHandler = new DocumentHandler(); + CmsDocOperation docOperation = new CmsDocOperation("create", docSrcUrlStr, null, docId); + docHandler.doOperation(docOperation); + } + + private void testCalls() throws ApplicationException { + Date before = new Date(); + System.out.println("Indexing start: " + before.getTime()); + String docIdGoerz = "/tei/de/dt-ptolemaeus-tei-merge2.xml"; + String docSrcUrlStr = "file:/Users/jwillenborg/texts/mpdl/documents" + docIdGoerz; + DocumentHandler docHandler = new DocumentHandler(); + CmsDocOperation docOperation = new CmsDocOperation("create", docSrcUrlStr, null, docIdGoerz); + // docHandler.doOperation(docOperation); + String docIdSchulz = "/tei/de/Schulz_2009.xml"; + docSrcUrlStr = "file:/Users/jwillenborg/texts/mpdl/documents" + docIdSchulz; + docHandler = new DocumentHandler(); + docOperation = new CmsDocOperation("create", docSrcUrlStr, null, docIdSchulz); + // docHandler.doOperation(docOperation); + String docIdBenedetti = "/echo/la/Benedetti_1585_163127KK.xml"; + docSrcUrlStr = "file:/Users/jwillenborg/texts/mpdl/documents" + docIdBenedetti; + docHandler = new DocumentHandler(); + docOperation = new CmsDocOperation("create", docSrcUrlStr, null, docIdBenedetti); + // docHandler.doOperation(docOperation); + String docIdSongYingxing = "/echo/zh/SongYingxing_1637.xml"; + docSrcUrlStr = "file:/Users/jwillenborg/texts/mpdl/documents" + docIdSongYingxing; + docHandler = new DocumentHandler(); + docOperation = new CmsDocOperation("create", docSrcUrlStr, null, docIdSongYingxing); + // docHandler.doOperation(docOperation); + String docIdMonte = "/archimedes/la/monte_mecha_036_la_1577.xml"; + docSrcUrlStr = "file:/Users/jwillenborg/texts/mpdl/documents" + docIdMonte; + docHandler = new DocumentHandler(); + docOperation = new CmsDocOperation("create", docSrcUrlStr, null, docIdMonte); + // docHandler.doOperation(docOperation); + String docIdEinstein = "/diverse/de/Einst_Antwo_de_1912.xml"; + docSrcUrlStr = "file:/Users/jwillenborg/texts/mpdl/documents" + docIdEinstein; + docHandler = new DocumentHandler(); + docOperation = new CmsDocOperation("create", docSrcUrlStr, null, docIdEinstein); + // docHandler.doOperation(docOperation); + String docIdEinsteinUeber = "/diverse/de/Einst_Ueber_de_1907_02.xml"; + docSrcUrlStr = "file:/Users/jwillenborg/texts/mpdl/documents" + docIdEinsteinUeber; + docHandler = new DocumentHandler(); + docOperation = new CmsDocOperation("create", docSrcUrlStr, null, docIdEinsteinUeber); + // docHandler.doOperation(docOperation); + String docIdTest = "/echo/zh/Yulei_tushuo_2_FN1CTY5C.xml"; + docSrcUrlStr = "file:/Users/jwillenborg/texts/mpdl/documents" + docIdTest; + docHandler = new DocumentHandler(); + docOperation = new CmsDocOperation("create", docSrcUrlStr, null, docIdTest); + // docHandler.doOperation(docOperation); + // indexer.deleteDocument(docIdGoerz); + // indexer.deleteDocument(docIdBenedetti); + MorphologyCache.getInstance().end(); + LexHandler.getInstance().end(); + } + + private Hashtable<Integer, StringBuilder> getFragments(String fileName) throws ApplicationException { + try { + GetFragmentsContentHandler getFragmentsContentHandler = new GetFragmentsContentHandler(); + XMLReader xmlParser = new SAXParser(); + xmlParser.setContentHandler(getFragmentsContentHandler); + InputSource inputSource = new InputSource(fileName); + xmlParser.parse(inputSource); + Hashtable<Integer, StringBuilder> resultFragments = getFragmentsContentHandler.getResultPages(); + return resultFragments; + } catch (SAXException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + private void testTransform() throws ApplicationException { + Date begin = new Date(); + XslResourceTransformer xslResourceTransformer = new XslResourceTransformer("pageXml.xsl"); + xslResourceTransformer = new XslResourceTransformer("pageTei.xsl"); + xslResourceTransformer = new XslResourceTransformer("pageArchimedes.xsl"); + xslResourceTransformer = new XslResourceTransformer("pageXhtml.xsl"); + xslResourceTransformer = new XslResourceTransformer("pageEcho.xsl"); + Date end = new Date(); + System.out.println("Needed time: " + (end.getTime() - begin.getTime()) + " ms"); + begin = new Date(); + String docFilePath = "/Users/jwillenborg/mpdl/data/xml/documents/echo/la/Benedetti_1585/pages/page-13-morph.xml"; + xslResourceTransformer = new XslResourceTransformer("pageEcho.xsl"); + String result = xslResourceTransformer.transform(docFilePath); + xslResourceTransformer = new XslResourceTransformer("pageEcho.xsl"); + result = xslResourceTransformer.transform(docFilePath); + xslResourceTransformer = new XslResourceTransformer("pageEcho.xsl"); + result = xslResourceTransformer.transform(docFilePath); + xslResourceTransformer = new XslResourceTransformer("pageEcho.xsl"); + result = xslResourceTransformer.transform(docFilePath); + xslResourceTransformer = new XslResourceTransformer("pageEcho.xsl"); + result = xslResourceTransformer.transform(docFilePath); + end = new Date(); + System.out.println("Needed time: " + (end.getTime() - begin.getTime()) + " ms"); + xslResourceTransformer = new XslResourceTransformer("pageEcho.xsl"); + begin = new Date(); + result = xslResourceTransformer.transform(docFilePath); + result = xslResourceTransformer.transform(docFilePath); + result = xslResourceTransformer.transform(docFilePath); + result = xslResourceTransformer.transform(docFilePath); + result = xslResourceTransformer.transform(docFilePath); + end = new Date(); + System.out.println("Needed time: " + (end.getTime() - begin.getTime()) + " ms"); + try { + Processor processor = new Processor(false); + XsltCompiler xsltCompiler = processor.newXsltCompiler(); + URL xslUrl = XslResourceTransformer.class.getResource("pageEcho.xsl"); + StreamSource xslStreamSource = new StreamSource(xslUrl.openStream()); + XsltExecutable xsltExecutable = xsltCompiler.compile(xslStreamSource); + net.sf.saxon.s9api.XsltTransformer xsltTransformer = xsltExecutable.load(); + Serializer serializer = new Serializer(); + serializer.setOutputWriter(new StringWriter()); + begin = new Date(); + for (int i=0; i<=5; i++) { + StreamSource xmlDoc = new StreamSource(docFilePath); + xsltTransformer.setSource(xmlDoc); // needs some time for bigger documents + xsltTransformer.setDestination(serializer); + xsltTransformer.transform(); // needs some time for bigger documents + result = serializer.getOutputDestination().toString(); + } + end = new Date(); + System.out.println("Needed time: " + (end.getTime() - begin.getTime()) + " ms"); + } catch (Exception e) { + + } + } + + private String tokenizeXmlFragment() throws ApplicationException { + String result = null; + try { + String xmlFragment = new String(FileUtils.readFileToByteArray(new File("/Users/jwillenborg/tmp/testFragment2.xml")), "utf-8"); + String srcUrlStr = "http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/page-query-result.xql?document=/echo/la/Benedetti_1585.xml&mode=pureXml&pn=13"; + URL srcUrl = new URL(srcUrlStr); + InputStream inputStream = srcUrl.openStream(); + BufferedInputStream in = new BufferedInputStream(inputStream); + xmlFragment = IOUtils.toString(in, "utf-8"); + in.close(); + + XmlTokenizer xmlTokenizer = new XmlTokenizer(new StringReader(xmlFragment)); + xmlTokenizer.setLanguage("lat"); + String[] stopElements = {"var"}; + // xmlTokenizer.setOutputFormat("string"); + String[] outputOptions = {"withLemmas"}; + xmlTokenizer.setOutputOptions(outputOptions); + xmlTokenizer.setStopElements(stopElements); + xmlTokenizer.tokenize(); + result = xmlTokenizer.getXmlResult(); + System.out.println(result); + } catch (Exception e) { + throw new ApplicationException(e); + } + return result; + } + + private String normalizeWords(String xmlStr) throws ApplicationException { + try { + WordContentHandler wordContentHandler = new WordContentHandler(); + XMLReader xmlParser = new SAXParser(); + xmlParser.setContentHandler(wordContentHandler); + StringReader strReader = new StringReader(xmlStr); + InputSource inputSource = new InputSource(strReader); + xmlParser.parse(inputSource); + String result = wordContentHandler.getResult(); + return result; + } catch (SAXException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + private String tokenizeWithLemmas(String xmlStr, String language) throws ApplicationException { + StringReader strReader = new StringReader(xmlStr); + XmlTokenizer xmlTokenizer = new XmlTokenizer(strReader); + xmlTokenizer.setLanguage(language); + String[] outputOptionsWithLemmas = {"withLemmas"}; // so all tokens are fetched with lemmas (costs performance) + xmlTokenizer.setOutputOptions(outputOptionsWithLemmas); + xmlTokenizer.tokenize(); + String retStr = xmlTokenizer.getXmlResult(); + return retStr; + } + + private String enrichWordsOrigRegNorm(String xmlStr) throws ApplicationException { + try { + WordContentHandler wordContentHandler = new WordContentHandler(); + XMLReader xmlParser = new SAXParser(); + xmlParser.setContentHandler(wordContentHandler); + StringReader strReader = new StringReader(xmlStr); + InputSource inputSource = new InputSource(strReader); + xmlParser.parse(inputSource); + String result = wordContentHandler.getResult(); + return result; + } catch (SAXException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + private String highlight(String xmlStr, String highlightElem, int highlightElemPos, String highlightQueryType, String highlightQuery, String language) throws ApplicationException { + String result = null; + try { + xmlStr = normalizeWords(xmlStr); + HighlightContentHandler highlightContentHandler = new HighlightContentHandler(highlightElem, highlightElemPos, highlightQueryType, highlightQuery, language); + highlightContentHandler.setFirstPageBreakReachedMode(true); + XMLReader xmlParser = new SAXParser(); + xmlParser.setContentHandler(highlightContentHandler); + StringReader stringReader = new StringReader(xmlStr); + InputSource inputSource = new InputSource(stringReader); + xmlParser.parse(inputSource); + result = highlightContentHandler.getResult().toString(); + } catch (SAXException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + return result; + } + + private void testScheduler() throws ApplicationException { + CmsDocOperation docOperation = new CmsDocOperation("update", "http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/getDoc?doc=/echo/zh/SongYingxing_1637.xml", null, "/echo/zh/SongYingxing_1637.xml"); + String[] elemNames = {"s", "head"}; + docOperation.setElementNames(elemNames); + CmsChainScheduler scheduler = CmsChainScheduler.getInstance(); + docOperation = scheduler.doOperation(docOperation); + String bla = ""; + } + + private void xquery() throws ApplicationException { + try { + XQueryEvaluator xQueryEvaluator = new XQueryEvaluator(); + URL srcUrl = new URL("file:/Users/jwillenborg/tmp/blablabla/Benedetti_1585.xml"); + String getTocEntries = "let $tocEntries := //echo:div[@type = 'section' or @type = 'chapter']"; + String getFigures = + "let $allFigures := //*:figure " + + "let $figures := " + + " for $figure at $pos in $allFigures "+ + " let $caption := string-join($figure/*:caption/text(), ' ') " + + " let $description := string-join($figure/*:description/text(), ' ') " + + " let $variables := string-join($figure/*:variables/text(), ' ') " + + " let $retFigure := " + + " element {'figure'}" + + " { attribute {'number'} {$pos}, " + + " element {'caption'} {$caption}, " + + " element {'description'} {$description}, " + + " element {'variables'} {$variables} }" + + " return " + + " $retFigure " + + "return $figures"; + String result = xQueryEvaluator.evaluateAsString(srcUrl, getFigures); + String bla = result; + } catch (Exception e) { + throw new ApplicationException(e); + } + } + + private void createToc() throws ApplicationException { + String docDirName = "/Users/jwillenborg/mpdl/data/xml/documents/echo/zh/SongYingxing_1637"; + XslResourceTransformer tocTransformer = new XslResourceTransformer("toc.xsl"); + File tocFile = new File(docDirName + "/toc.xml"); + String docDestFileName = docDirName + "/SongYingxing_1637.xml"; + String tocResult = tocTransformer.transform(docDestFileName); + String bla = ""; + } + + private void getDocInfo() throws ApplicationException { + IndexHandler indexHandler = IndexHandler.getInstance(); + MetadataRecord mdRecord = indexHandler.getDocMetadata("/echo/la/Benedetti_1585_163127KK.xml"); + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/transform/GetFragmentsContentHandler.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,203 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.transform; + +import java.util.ArrayList; +import java.util.Hashtable; + +import org.xml.sax.*; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.util.StringUtils; + +public class GetFragmentsContentHandler implements ContentHandler { + private String xmlnsString = ""; + private int currentMilestonePosition = 0; + private Element currentElement; + private Element currentMilestoneElement; + private ArrayList<Element> elementQueue; // queue of opened elements (before they were closed): to connect them to a parent hierarchy + private Hashtable<Integer, StringBuilder> resultFragments = new Hashtable<Integer, StringBuilder>(); + private String milestoneElemenName = "pb"; // default is pb + + public GetFragmentsContentHandler() throws ApplicationException { + } + + public GetFragmentsContentHandler(String milestoneElemenName) throws ApplicationException { + this.milestoneElemenName = milestoneElemenName; + } + + public Hashtable<Integer, StringBuilder> getResultPages() { + return resultFragments; + } + + public int getMilestoneCount() { + return currentMilestonePosition; + } + + public void startDocument() throws SAXException { + } + + public void endDocument() throws SAXException { + // write the closePath after the last minus 1 milestone element (the closing path after the last milestone element is automatically written by the normal closing tags) + if (currentMilestoneElement != null) { + String msClosePath = currentMilestoneElement.getClosePath(); + write(msClosePath, currentMilestoneElement.milestonePosition - 1); + } + resultFragments.remove(new Integer(0)); // this fragment is filled but should not + } + + public void characters(char[] c, int start, int length) throws SAXException { + char[] cCopy = new char[length]; + System.arraycopy(c, start, cCopy, 0, length); + String charactersStr = String.valueOf(cCopy); + if (charactersStr != null && ! charactersStr.equals("")) { + if (currentMilestonePosition > 0) { + charactersStr = StringUtils.deresolveXmlEntities(charactersStr); + write(charactersStr); + } + } + } + + public void ignorableWhitespace(char[] c, int start, int length) throws SAXException { + } + + public void processingInstruction(String target, String data) throws SAXException { + } + + public void setDocumentLocator(Locator locator) { + } + + public void startPrefixMapping(String prefix, String uri) throws SAXException { + if (prefix != null && prefix.equals("")) + xmlnsString += "xmlns" + "=\"" + uri + "\" "; + else + xmlnsString += "xmlns:" + prefix + "=\"" + uri + "\" "; + } + + public void endPrefixMapping(String prefix) throws SAXException { + } + + public void skippedEntity(String name) throws SAXException { + } + + public void startElement(String uri, String localName, String name, Attributes attrs) throws SAXException { + if (elementQueue == null) + elementQueue = new ArrayList<Element>(); + Element newElement = new Element(name); + if (currentElement != null) { + newElement.parent = currentElement; + } + currentElement = newElement; + if (localName != null && localName.equals(milestoneElemenName)) { + currentMilestonePosition++; + if (currentMilestoneElement != null) { + String msClosePath = currentMilestoneElement.getClosePath(); + write(msClosePath, currentMilestoneElement.milestonePosition - 1); + } + currentMilestoneElement = currentElement; + currentMilestoneElement.milestonePosition = currentMilestonePosition; + String msOpenPath = currentMilestoneElement.getOpenPath(); + write(msOpenPath); + } + int attrSize = attrs.getLength(); + String attrString = ""; + for (int i=0; i<attrSize; i++) { + String attrQName = attrs.getQName(i); + String attrValue = attrs.getValue(i); + attrValue = StringUtils.forXML(attrValue); + attrString = attrString + " " + attrQName + "=\"" + attrValue + "\""; + } + if (attrString != null && ! attrString.isEmpty()) { + attrString = attrString.trim(); + currentElement.attrString = attrString; + } + if (xmlnsString != null && ! xmlnsString.isEmpty()) { + xmlnsString = xmlnsString.trim(); + currentElement.xmlnsString = xmlnsString; + } + if (currentMilestonePosition > 0) { + write("<" + name); + if (xmlnsString != null && ! xmlnsString.isEmpty()) + write(" " + xmlnsString); + if (attrString != null && ! attrString.isEmpty()) + write(" " + attrString); + write(">"); + } + xmlnsString = ""; + elementQueue.add(currentElement); + } + + public void endElement(String uri, String localName, String name) throws SAXException { + if (currentMilestonePosition > 0) { + write("</" + name + ">"); + } + if (elementQueue != null && elementQueue.size() > 0) { + int lastIndex = elementQueue.size() - 1; + elementQueue.remove(lastIndex); + } + if (elementQueue != null && elementQueue.size() > 0) { + int lastIndex = elementQueue.size() - 1; + currentElement = elementQueue.get(lastIndex); + } else { + currentElement = null; + } + } + + private void write(String outStr) throws SAXException { + StringBuilder resultFragment = resultFragments.get(new Integer(currentMilestonePosition)); + if (resultFragment == null) { + resultFragment = new StringBuilder(); + resultFragments.put(new Integer(currentMilestonePosition), resultFragment); + } + resultFragment.append(outStr); + } + + private void write(String outStr, int milestoneNumber) throws SAXException { + StringBuilder resultFragment = resultFragments.get(new Integer(milestoneNumber)); + if (resultFragment == null) { + resultFragment = new StringBuilder(); + resultFragments.put(new Integer(milestoneNumber), resultFragment); + } + resultFragment.append(outStr); + } + + public class Element { + public String name; + private String xmlnsString; + private String attrString; + private int milestonePosition; + private Element parent; + + private Element(String name) { + this.name = name; + } + + private String getOpenPath() { + StringBuilder ancestorsStrBuilder = new StringBuilder(); + if (parent != null) { + ancestorsStrBuilder.append(parent.getOpenPath()); + ancestorsStrBuilder.append("<"); + ancestorsStrBuilder.append(parent.name); + if (parent.xmlnsString != null && ! parent.xmlnsString.isEmpty()) { + ancestorsStrBuilder.append(" "); + ancestorsStrBuilder.append(parent.xmlnsString); + } + if (parent.attrString != null && ! parent.attrString.isEmpty()) { + ancestorsStrBuilder.append(" " + parent.attrString); + } + ancestorsStrBuilder.append(">"); + } + return ancestorsStrBuilder.toString(); + } + + private String getClosePath() { + StringBuilder ancestorsStrBuilder = new StringBuilder(); + if (parent != null) { + ancestorsStrBuilder.append("</"); + ancestorsStrBuilder.append(parent.name); + ancestorsStrBuilder.append(">"); + ancestorsStrBuilder.append(parent.getClosePath()); + } + return ancestorsStrBuilder.toString(); + } + + } +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/transform/HighlightContentHandler.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,215 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.transform; + +import java.util.ArrayList; + +import org.xml.sax.*; + +import de.mpg.mpiwg.berlin.mpdl.cms.lucene.IndexHandler; +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.util.StringUtils; + +public class HighlightContentHandler implements ContentHandler { + private String xmlnsString = ""; + private String highlightElemName; + private int highlightElemPos = 1; + private int currentHighlightElemPos = 0; + private boolean highlightElemMode = false; + private int highlightElemModeOpenTags = 0; + private String highlightQueryType = "orig"; // orig, reg, norm or morph + private String highlightQuery; // complex Lucene query + private String highlightQueryForms; // highlight terms separated by a blank + private boolean highlightHitMode = false; + private int highlightHitModeOpenTags = 0; + private boolean firstPageBreakReachedMode = false; // in a page fragment: if a page break element is surrounded by an element (e.g. "s") then this element should not increment the currentHighlightElemPos + private boolean firstPageBreakReached = true; + private StringBuilder result = new StringBuilder(); + + public HighlightContentHandler() throws ApplicationException { + } + + public HighlightContentHandler(String highlightElemName, int highlightElemPos) throws ApplicationException { + this.highlightElemName = highlightElemName; + this.highlightElemPos = highlightElemPos; + } + + public HighlightContentHandler(String highlightElemName, int highlightElemPos, String highlightQueryType, String highlightQuery, String language) throws ApplicationException { + this.highlightElemName = highlightElemName; + this.highlightElemPos = highlightElemPos; + this.highlightQueryType = highlightQueryType; + this.highlightQuery = highlightQuery; + if (highlightQuery != null) { + IndexHandler indexHandler = IndexHandler.getInstance(); + ArrayList<String> queryTerms = indexHandler.fetchTerms(highlightQuery, language); // all query terms in query (also morphological terms) + highlightQueryForms = toString(queryTerms); + } + } + + public void setFirstPageBreakReachedMode(boolean firstPageBreakReachedMode) { + this.firstPageBreakReachedMode = firstPageBreakReachedMode; + if (firstPageBreakReachedMode) + this.firstPageBreakReached = false; // is first set to false and later if a page break is found (by startElement) it is set to true + } + + public StringBuilder getResult() { + return result; + } + + public void startDocument() throws SAXException { + } + + public void endDocument() throws SAXException { + } + + public void characters(char[] c, int start, int length) throws SAXException { + char[] cCopy = new char[length]; + System.arraycopy(c, start, cCopy, 0, length); + String charactersStr = String.valueOf(cCopy); + if (charactersStr != null && ! charactersStr.equals("")) { + charactersStr = StringUtils.deresolveXmlEntities(charactersStr); + write(charactersStr); + } + } + + public void ignorableWhitespace(char[] c, int start, int length) throws SAXException { + } + + public void processingInstruction(String target, String data) throws SAXException { + } + + public void setDocumentLocator(Locator locator) { + } + + public void startPrefixMapping(String prefix, String uri) throws SAXException { + xmlnsString += "xmlns:" + prefix + "=\"" + uri + "\" "; + if (prefix != null && prefix.equals("")) + xmlnsString = "xmlns" + "=\"" + uri + "\" "; + } + + public void endPrefixMapping(String prefix) throws SAXException { + } + + public void skippedEntity(String name) throws SAXException { + } + + public void startElement(String uri, String localName, String name, Attributes attrs) throws SAXException { + int attrSize = attrs.getLength(); + String attrString = ""; + for (int i=0; i<attrSize; i++) { + String attrQName = attrs.getQName(i); + String attrValue = attrs.getValue(i); + attrValue = StringUtils.forXML(attrValue); + attrString = attrString + " " + attrQName + "=\"" + attrValue + "\""; + } + if (attrString != null && ! attrString.isEmpty()) { + attrString = attrString.trim(); + } + if (xmlnsString != null && ! xmlnsString.isEmpty()) { + xmlnsString = xmlnsString.trim(); + } + if (localName.equals("pb")) + firstPageBreakReached = true; + // start highlight element at position + if (highlightElemName != null && highlightElemName.equals(localName) && firstPageBreakReached) { + currentHighlightElemPos++; + if (currentHighlightElemPos == highlightElemPos && highlightElemModeOpenTags == 0) { + highlightElemMode = true; + write("<hi type=\"elem\">"); + } + } + if (highlightElemMode) { + highlightElemModeOpenTags++; + } + // start highlight query + if (highlightQuery != null && localName.equals("w")) { + boolean matched = false; + String attrQName = "form"; + if (highlightQueryType.equals("orig")) + attrQName = "form"; + else if (highlightQueryType.equals("reg")) + attrQName = "formRegularized"; + else if (highlightQueryType.equals("norm")) + attrQName = "formNormalized"; + else if (highlightQueryType.equals("morph")) + attrQName = "lemmas"; + String attrValue = getAttrValue(attrs, attrQName); + if (highlightQueryType.equals("reg") && attrValue == null) + attrValue = getAttrValue(attrs, "form"); // if no regularized form exist it takes the form + if (attrValue != null) { + String[] forms = highlightQueryForms.split(" "); + for (int i=0; i<forms.length; i++) { + if (! matched) { + String form = forms[i]; + if (form.endsWith("*")) { // TODO support middle wildcard queries: bla*bla bla?bla + form = form.replace("*", ""); + matched = attrValue.startsWith(form); + } else { + matched = attrValue.equals(form); + } + } + } + } + if ((highlightElemName == null && matched && highlightHitModeOpenTags == 0) || (highlightElemName != null && highlightElemMode && matched && highlightHitModeOpenTags == 0)) { + highlightHitMode = true; + write("<hi type=\"hit\">"); + } + } + if (highlightHitMode) { + highlightHitModeOpenTags++; + } + write("<" + name); + if (xmlnsString != null && ! xmlnsString.isEmpty()) + write(" " + xmlnsString); + if (attrString != null && ! attrString.isEmpty()) + write(" " + attrString); + write(">"); + xmlnsString = ""; + } + + public void endElement(String uri, String localName, String name) throws SAXException { + write("</" + name + ">"); + // end highlight element at position + if (highlightElemMode) { + if (highlightElemModeOpenTags == 1) { + highlightElemMode = false; + write("</hi>"); + } + highlightElemModeOpenTags--; + } + // end highlight query + if (highlightHitMode) { + if (highlightHitModeOpenTags == 1) { + highlightHitMode = false; + write("</hi>"); + } + highlightHitModeOpenTags--; + } + } + + private String toString(ArrayList<String> queryForms) { + String queryFormsStr = ""; + for (int i=0; i<queryForms.size(); i++) { + String form = queryForms.get(i); + queryFormsStr = queryFormsStr + form + " "; + } + if (queryForms == null || queryForms.size() == 0) + return null; + else + return queryFormsStr.substring(0, queryFormsStr.length() -1); + } + + private void write(String outStr) throws SAXException { + result.append(outStr); + } + + private String getAttrValue(Attributes attrs, String attrQName) { + String retValue = null; + int attrSize = attrs.getLength(); + for (int i=0; i<attrSize; i++) { + String attrQNameTmp = attrs.getQName(i); + String attrValue = attrs.getValue(i); + if (attrQNameTmp.equals(attrQName)) + return attrValue; + } + return retValue; + } +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/transform/PageTransformer.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,118 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.transform; + +import java.io.IOException; +import java.io.StringReader; +import java.io.StringWriter; +import java.net.URL; + +import javax.xml.transform.stream.StreamSource; + +import de.mpg.mpiwg.berlin.mpdl.cms.document.MetadataRecord; +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; + +import net.sf.saxon.s9api.Processor; +import net.sf.saxon.s9api.QName; +import net.sf.saxon.s9api.SaxonApiException; +import net.sf.saxon.s9api.Serializer; +import net.sf.saxon.s9api.XdmAtomicValue; +import net.sf.saxon.s9api.XdmValue; +import net.sf.saxon.s9api.XsltCompiler; +import net.sf.saxon.s9api.XsltExecutable; +import net.sf.saxon.s9api.XsltTransformer; + +public class PageTransformer { + private int counter = 0; + private Processor processor; + private XsltCompiler xsltCompiler; + private XsltTransformer pageTeiTransformer; + private XsltTransformer pageEchoTransformer; + private XsltTransformer pageArchimedesTransformer; + private XsltTransformer pageXhtmlTransformer; + private XsltTransformer pageXmlTransformer; + + public PageTransformer() throws ApplicationException { + init(); + } + + private void init() throws ApplicationException { + try { + processor = new Processor(false); + xsltCompiler = processor.newXsltCompiler(); + URL getFragmentXslUrl = PageTransformer.class.getResource("pageTei.xsl"); + StreamSource xslStreamSource = new StreamSource(getFragmentXslUrl.openStream()); + XsltExecutable xsltExecutable = xsltCompiler.compile(xslStreamSource); + pageTeiTransformer = xsltExecutable.load(); + getFragmentXslUrl = PageTransformer.class.getResource("pageEcho.xsl"); + xslStreamSource = new StreamSource(getFragmentXslUrl.openStream()); + xsltExecutable = xsltCompiler.compile(xslStreamSource); + pageEchoTransformer = xsltExecutable.load(); + getFragmentXslUrl = PageTransformer.class.getResource("pageArchimedes.xsl"); + xslStreamSource = new StreamSource(getFragmentXslUrl.openStream()); + xsltExecutable = xsltCompiler.compile(xslStreamSource); + pageArchimedesTransformer = xsltExecutable.load(); + getFragmentXslUrl = PageTransformer.class.getResource("pageXhtml.xsl"); + xslStreamSource = new StreamSource(getFragmentXslUrl.openStream()); + xsltExecutable = xsltCompiler.compile(xslStreamSource); + pageXhtmlTransformer = xsltExecutable.load(); + getFragmentXslUrl = PageTransformer.class.getResource("pageXml.xsl"); + xslStreamSource = new StreamSource(getFragmentXslUrl.openStream()); + xsltExecutable = xsltCompiler.compile(xslStreamSource); + pageXmlTransformer = xsltExecutable.load(); + } catch (SaxonApiException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + public String transform(String inputStr, MetadataRecord mdRecord, int page, String outputFormat) throws ApplicationException { + String pageFragment = null; + String schemaName = mdRecord.getSchemaName(); + if (schemaName != null) + schemaName = schemaName.toLowerCase(); + try { + XsltTransformer transformer = null; + if (schemaName != null && schemaName.equals("tei") && outputFormat.equals("html")) + transformer = pageTeiTransformer; + else if (schemaName != null && schemaName.equals("echo") && outputFormat.equals("html")) + transformer = pageEchoTransformer; + else if (schemaName != null && schemaName.equals("archimedes") && outputFormat.equals("html")) + transformer = pageArchimedesTransformer; + else if (schemaName != null && (schemaName.equals("html") || schemaName.equals("xhtml")) && outputFormat.equals("html")) + transformer = pageXhtmlTransformer; + else if (outputFormat.equals("xmlDisplay")) + transformer = pageXmlTransformer; + else + transformer = pageTeiTransformer; + StringReader inputStrReader = new StringReader(inputStr); + StreamSource xmlDoc = new StreamSource(inputStrReader); + Serializer serializer = new Serializer(); + serializer.setOutputWriter(new StringWriter()); + serializer.setOutputProperty(Serializer.Property.SAXON_STYLESHEET_VERSION, "2.0"); + serializer.setOutputProperty(Serializer.Property.METHOD, "xml"); + serializer.setOutputProperty(Serializer.Property.MEDIA_TYPE, "text/html"); + serializer.setOutputProperty(Serializer.Property.INDENT, "no"); + serializer.setOutputProperty(Serializer.Property.OMIT_XML_DECLARATION, "yes"); + serializer.setOutputProperty(Serializer.Property.ENCODING, "utf-8"); + transformer.setSource(xmlDoc); + transformer.setDestination(serializer); + String echoFiguresDir = mdRecord.getEchoFiguresDir(); + QName echoFiguresDirQName = new QName("echoFiguresDir"); + XdmValue echoFiguresDirXdmValue = new XdmAtomicValue(echoFiguresDir); + transformer.setParameter(echoFiguresDirQName, echoFiguresDirXdmValue); + String echoPageImgDir = mdRecord.getEchoPageImageDir(); + QName echoPageImgDirQName = new QName("echoPageImgDir"); + XdmValue echoPageImgDirXdmValue = new XdmAtomicValue(echoPageImgDir); + transformer.setParameter(echoPageImgDirQName, echoPageImgDirXdmValue); + transformer.transform(); + pageFragment = serializer.getOutputDestination().toString(); + } catch (Exception e) { + throw new ApplicationException(e); + } + counter++; + if (counter == 100) + init(); + return pageFragment; + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/transform/QueryTransformer.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,70 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.transform; + +import java.io.IOException; +import java.io.StringWriter; +import java.net.URL; + +import javax.xml.transform.stream.StreamSource; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; + +import net.sf.saxon.s9api.Processor; +import net.sf.saxon.s9api.QName; +import net.sf.saxon.s9api.SaxonApiException; +import net.sf.saxon.s9api.Serializer; +import net.sf.saxon.s9api.XdmAtomicValue; +import net.sf.saxon.s9api.XdmValue; +import net.sf.saxon.s9api.XsltCompiler; +import net.sf.saxon.s9api.XsltExecutable; +import net.sf.saxon.s9api.XsltTransformer; + +public class QueryTransformer { + private Processor processor; + private XsltCompiler xsltCompiler; + private XsltTransformer xsltQueryDocument; + + public QueryTransformer() throws ApplicationException { + init(); + } + + private void init() throws ApplicationException { + try { + processor = new Processor(false); + xsltCompiler = processor.newXsltCompiler(); + URL queryDocumentXslUrl = QueryTransformer.class.getResource("queryDocument.xsl"); + StreamSource xslStreamSource = new StreamSource(queryDocumentXslUrl.openStream()); + XsltExecutable xsltExecutable = xsltCompiler.compile(xslStreamSource); + xsltQueryDocument = xsltExecutable.load(); + } catch (SaxonApiException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + public String queryDocument(String xmlFileName, String query, String flags, String outputFormat) throws ApplicationException { + String pageFragment = null; + try { + StreamSource xmlDoc = new StreamSource(xmlFileName); + Serializer serializer = new Serializer(); + serializer.setOutputWriter(new StringWriter()); + serializer.setOutputProperty(Serializer.Property.INDENT, "yes"); + xsltQueryDocument.setSource(xmlDoc); // needs some time for bigger documents + xsltQueryDocument.setDestination(serializer); + QName queryQName = new QName("query"); + XdmValue queryXdmValue = new XdmAtomicValue(query); + QName flagsQName = new QName("flags"); + XdmValue flagsXdmValue = new XdmAtomicValue(flags); + QName outputFormatQName = new QName("outputFormat"); + XdmValue outputFormatXdmValue = new XdmAtomicValue(outputFormat); + xsltQueryDocument.setParameter(queryQName, queryXdmValue); + xsltQueryDocument.setParameter(flagsQName, flagsXdmValue); + xsltQueryDocument.setParameter(outputFormatQName, outputFormatXdmValue); + xsltQueryDocument.transform(); // needs some time for bigger documents + pageFragment = serializer.getOutputDestination().toString(); + } catch (Exception e) { + throw new ApplicationException(e); + } + return pageFragment; + } +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/transform/TocTransformer.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,74 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.transform; + +import java.io.IOException; +import java.io.StringReader; +import java.io.StringWriter; +import java.net.URL; + +import javax.xml.transform.stream.StreamSource; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; + +import net.sf.saxon.s9api.Processor; +import net.sf.saxon.s9api.QName; +import net.sf.saxon.s9api.SaxonApiException; +import net.sf.saxon.s9api.Serializer; +import net.sf.saxon.s9api.XdmAtomicValue; +import net.sf.saxon.s9api.XdmValue; +import net.sf.saxon.s9api.XsltCompiler; +import net.sf.saxon.s9api.XsltExecutable; +import net.sf.saxon.s9api.XsltTransformer; + +public class TocTransformer { + private Processor processor; + private XsltCompiler xsltCompiler; + private XsltTransformer tocTransformer; + + public TocTransformer() throws ApplicationException { + init(); + } + + private void init() throws ApplicationException { + try { + processor = new Processor(false); + xsltCompiler = processor.newXsltCompiler(); + URL tocXslUrl = TocTransformer.class.getResource("tocOut.xsl"); + StreamSource xslStreamSource = new StreamSource(tocXslUrl.openStream()); + XsltExecutable xsltExecutable = xsltCompiler.compile(xslStreamSource); + tocTransformer = xsltExecutable.load(); + } catch (SaxonApiException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + public String transform(String inputStr, String type, String outputFormat) throws ApplicationException { + String result = null; + try { + StringReader inputStrReader = new StringReader(inputStr); + StreamSource xmlDoc = new StreamSource(inputStrReader); + Serializer serializer = new Serializer(); + serializer.setOutputWriter(new StringWriter()); + serializer.setOutputProperty(Serializer.Property.SAXON_STYLESHEET_VERSION, "2.0"); + serializer.setOutputProperty(Serializer.Property.METHOD, "xhtml"); + serializer.setOutputProperty(Serializer.Property.INDENT, "no"); + serializer.setOutputProperty(Serializer.Property.OMIT_XML_DECLARATION, "yes"); + serializer.setOutputProperty(Serializer.Property.ENCODING, "utf-8"); + tocTransformer.setSource(xmlDoc); + tocTransformer.setDestination(serializer); + QName typeQName = new QName("type"); + XdmValue typeXdmValue = new XdmAtomicValue(type); + QName outputFormatQName = new QName("outputFormat"); + XdmValue outputFormatXdmValue = new XdmAtomicValue(outputFormat); + tocTransformer.setParameter(typeQName, typeXdmValue); + tocTransformer.setParameter(outputFormatQName, outputFormatXdmValue); + tocTransformer.transform(); + result = serializer.getOutputDestination().toString(); + } catch (Exception e) { + throw new ApplicationException(e); + } + return result; + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/transform/XslResourceTransformer.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,94 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.transform; + +import java.io.IOException; +import java.io.StringReader; +import java.io.StringWriter; +import java.net.URL; + +import javax.xml.transform.stream.StreamSource; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; + +import net.sf.saxon.s9api.Processor; +import net.sf.saxon.s9api.QName; +import net.sf.saxon.s9api.SaxonApiException; +import net.sf.saxon.s9api.Serializer; +import net.sf.saxon.s9api.XdmValue; +import net.sf.saxon.s9api.XsltCompiler; +import net.sf.saxon.s9api.XsltExecutable; +import net.sf.saxon.s9api.XsltTransformer; + +public class XslResourceTransformer { + private Processor processor; + private XsltCompiler xsltCompiler; + private Serializer serializer; + private XsltTransformer xsltTransformer; + + public XslResourceTransformer(String resourceName) throws ApplicationException { + init(resourceName); + } + + private void init(String resourceName) throws ApplicationException { + try { + processor = new Processor(false); + xsltCompiler = processor.newXsltCompiler(); + URL xslUrl = XslResourceTransformer.class.getResource(resourceName); + StreamSource xslStreamSource = new StreamSource(xslUrl.openStream()); + XsltExecutable xsltExecutable = xsltCompiler.compile(xslStreamSource); + xsltTransformer = xsltExecutable.load(); + } catch (SaxonApiException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + public String transform(String xmlFileName) throws ApplicationException { + String result = null; + try { + StreamSource xmlDoc = new StreamSource(xmlFileName); + serializer = new Serializer(); + serializer.setOutputWriter(new StringWriter()); + xsltTransformer.setSource(xmlDoc); // needs some time for bigger documents + xsltTransformer.setDestination(serializer); + xsltTransformer.transform(); // needs some time for bigger documents + result = serializer.getOutputDestination().toString(); + } catch (Exception e) { + throw new ApplicationException(e); + } + return result; + } + + public String transformStr(String xmlStr) throws ApplicationException { + String retStr = null; + try { + StringReader inputStrReader = new StringReader(xmlStr); + StreamSource xmlDoc = new StreamSource(inputStrReader); + serializer = new Serializer(); + serializer.setOutputWriter(new StringWriter()); + xsltTransformer.setSource(xmlDoc); // needs some time for bigger documents + xsltTransformer.setDestination(serializer); + xsltTransformer.transform(); // needs some time for bigger documents + retStr = serializer.getOutputDestination().toString(); + } catch (Exception e) { + throw new ApplicationException(e); + } + return retStr; + } + + public void setParameter(QName name, XdmValue value) throws ApplicationException { + try { + xsltTransformer.setParameter(name, value); + } catch (Exception e) { + throw new ApplicationException(e); + } + } + + public void setOutputProperty(Serializer.Property property, String value) throws ApplicationException { + try { + serializer.setOutputProperty(property, value); + } catch (Exception e) { + throw new ApplicationException(e); + } + } +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/transform/chars.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,10 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform"> + +<xsl:output method="xml" encoding="utf-8" omit-xml-declaration="yes"/> + +<xsl:template match="text()"> + <xsl:sequence select="normalize-space(.)"/> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/transform/pageArchimedes.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,324 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xs="http://www.w3.org/2001/XMLSchema" + xmlns:functx="http://www.functx.com" + xmlns:saxon="http://saxon.sf.net/" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xs functx saxon dc dcterms math svg xhtml" + > + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="echoFiguresDir"></xsl:param> +<xsl:param name="echoPageImgDir"></xsl:param> + +<xsl:variable name="dictionaryServiceName" select="'http://mpdl-service.mpiwg-berlin.mpg.de/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries'"/> + +<xsl:template match="*:archimedes"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'archimedes'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:expan|*:emph|*:q|*:quote|*:blockquote|*:set-off|*:reg|*:var|*:num"> + <span> + <xsl:attribute name="class"><xsl:value-of select="name()"/></xsl:attribute> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="text"> + <xsl:variable name="contentStr" select="normalize-space(string(.))"/> + <xsl:variable name="figures" select=".//*:figure|.//*:handwritten"/> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'text'"/></xsl:attribute> + <xsl:apply-templates/> + <xsl:if test="$contentStr = '' and empty($figures)"> <!-- test if div only contains empty nodes but no figures --> + <div> + <xsl:attribute name="class"><xsl:value-of select="'emptyPage'"/></xsl:attribute> + <xsl:value-of select="'[Empty page]'"/> + </div> + </xsl:if> + </div> + <!-- Notes --> + <xsl:variable name="bottomNotes" select="//*:note[contains(@position, 'foot') or empty(string(@position))]"/> + <xsl:variable name="countBottomNotes" select="count($bottomNotes)"/> + <xsl:if test="$countBottomNotes > 0"> + <span class="notes"> + <xsl:for-each select="$bottomNotes"> + <xsl:variable name="noteSign"><xsl:value-of select="count(./preceding::*:note) + 1"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'noteBody foot'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:apply-templates/> + </span> + </span> + </xsl:for-each> + </span> + </xsl:if> +</xsl:template> + +<xsl:template match="*:div"> + <xsl:if test="node() or normalize-space()"> <!-- test: contains nodes or is not empty --> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'div'"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:if> +</xsl:template> + +<xsl:template match="*:p"> + <xsl:variable name="type" select="string(@type)"/> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:choose> + <xsl:when test="$type = 'head'"> + <xsl:attribute name="class"><xsl:value-of select="'head bf'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:otherwise> + <xsl:attribute name="class"><xsl:value-of select="'p'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </div> +</xsl:template> + +<xsl:template match="*:pb"> + <xsl:variable name="number"> + <xsl:choose> + <xsl:when test="not(empty(@n))"><xsl:value-of select="@n"/></xsl:when> + <xsl:otherwise><xsl:value-of select="count(./preceding::*:pb) + 1"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="file"> + <xsl:choose> + <xsl:when test="not(empty(@xlink:href))"><xsl:value-of select="replace(@xlink:href, '/', '.')"/></xsl:when> + <xsl:otherwise><xsl:value-of select="$number"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'pb'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'src'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="concat('pageimg/', string($file))"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'n'"/></xsl:attribute> + <xsl:value-of select="$number"/> + </span> + </span> + </span> +</xsl:template> + +<xsl:template match="*:head"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'head bf'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<!-- MathML --> +<xsl:template match="math:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<!-- SVG --> +<xsl:template match="svg:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:figure"> + <span class="figure"> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:variable name="fileHref" select="replace(@xlink:href, '/', '.')"/> + <xsl:variable name="fileName" select="concat($echoFiguresDir, '/', $fileHref)"/> + <xsl:variable name="digilibUrl" select="concat('http://digilib.mpiwg-berlin.mpg.de/digitallibrary/jquery/digilib.html?fn=', $fileName)"/> + <xsl:variable name="scalerUrl" select="concat('http://digilib.mpiwg-berlin.mpg.de/digitallibrary/servlet/Scaler?fn=', $fileName, '&dh=200&dw=200')"/> + <xsl:variable name="ancestorWords" select="./ancestor::*:w"/> <!-- if figure is in a word then no link within a link should be produced --> + <xsl:choose> + <xsl:when test="empty($ancestorWords)"> + <a href="{$digilibUrl}"> + <img> + <xsl:attribute name="src"><xsl:value-of select="$scalerUrl"/></xsl:attribute> + </img> + </a> + </xsl:when> + <xsl:otherwise> + <img> + <xsl:attribute name="src"><xsl:value-of select="$scalerUrl"/></xsl:attribute> + </img> + </xsl:otherwise> + </xsl:choose> + <xsl:if test="not(empty(@number))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNumber'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNum'"/></xsl:attribute> + <xsl:value-of select="string(@number)"/> + </span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNumText'"/></xsl:attribute> + <xsl:value-of select="concat('[', 'Figure', ' ', string(@number), ']')"/> + </span> + </span> + </xsl:if> + </span> +</xsl:template> + +<xsl:template match="*:lb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'lb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="*:cb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'cb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="note"> + <xsl:variable name="place"><xsl:value-of select="string(@position)"/></xsl:variable> + <xsl:variable name="noteSign"><xsl:value-of select="count(./preceding::*:note) + 1"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:if test="$place != '' and $place != 'foot'"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('noteBody ', $place)"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:apply-templates/> + </span> + </xsl:if> + </span> +</xsl:template> + +<xsl:template match="foreign"> + <xsl:variable name="lang" select="@lang"/> + <xsl:variable name="xmllang" select="@xml:lang"/> + <xsl:variable name="language"> + <xsl:choose> + <xsl:when test="not(empty($xmllang))"><xsl:value-of select="$xmllang"/></xsl:when> + <xsl:when test="not(empty($lang))"><xsl:value-of select="$lang"/></xsl:when> + <xsl:otherwise><xsl:value-of select="''"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat('foreign ', $language)"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="gap"> + <span class="gap"> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'gap'"/></xsl:attribute> + <xsl:if test="not(empty(@extent))"><span class="extent"><xsl:value-of select="string(@extent)"/></span></xsl:if> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="s"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'s'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- words --> +<xsl:template match="*:w"> + <xsl:variable name="wordLanguage" select="@lang"/> + <xsl:variable name="form" select="encode-for-uri(string(@form))"/> + <xsl:variable name="dictionary" select="string(@dictionary)"/> + <xsl:variable name="displayWordOrig"> + <xsl:choose> + <xsl:when test="not(empty(*:orig))"><xsl:apply-templates select="*:orig/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordReg"> + <xsl:choose> + <xsl:when test="not(empty(*:reg))"><xsl:apply-templates select="*:reg/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordNorm"> + <xsl:choose> + <xsl:when test="not(empty(*:norm))"><xsl:apply-templates select="*:norm/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordUrlEncoded" select="encode-for-uri($displayWordOrig)"/> + <xsl:variable name="dictionaryPart"> + <xsl:choose> + <xsl:when test="$dictionary = 'true'"> + <a class="dictionary"> + <xsl:attribute name="href"><xsl:value-of select="concat($dictionaryServiceName, '?query=', $form, '&queryDisplay=', $displayWordUrlEncoded, '&language=', $wordLanguage, '&outputFormat=html', '&outputType=morphCompact&outputType=dictFull')"/></xsl:attribute> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </a> + </xsl:when> + <xsl:otherwise> + <span class="dictionary"> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span class="w"> + <xsl:sequence select="$dictionaryPart"/> + <span class="nodictionary orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="nodictionary reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="nodictionary norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> +</xsl:template> + +<xsl:template match="*:hi"> + <xsl:choose> + <xsl:when test="@type = 'elem'"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/transform/pageEcho.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,536 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xs="http://www.w3.org/2001/XMLSchema" + xmlns:functx="http://www.functx.com" + xmlns:saxon="http://saxon.sf.net/" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" + xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:mml="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xs functx saxon dc dcterms echo de math mml svg xhtml" + > + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="echoFiguresDir"></xsl:param> +<xsl:param name="echoPageImgDir"></xsl:param> + +<xsl:variable name="dictionaryServiceName" select="'http://mpdl-service.mpiwg-berlin.mpg.de/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries'"/> + +<xsl:template match="*:echo"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'echo'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:text"> + <xsl:variable name="contentStr" select="normalize-space(string(.))"/> + <xsl:variable name="figures" select=".//*:figure|.//*:handwritten"/> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'text'"/></xsl:attribute> + <xsl:apply-templates/> + <xsl:if test="$contentStr = '' and empty($figures)"> <!-- test if div only contains empty nodes but no figures --> + <div> + <xsl:attribute name="class"><xsl:value-of select="'emptyPage'"/></xsl:attribute> + <xsl:value-of select="'[Empty page]'"/> + </div> + </xsl:if> + </div> + <!-- Notes --> + <xsl:variable name="bottomNotes" select="//*:note[contains(@position, 'foot') or empty(string(@position))]"/> + <xsl:variable name="countBottomNotes" select="count($bottomNotes)"/> + <xsl:if test="$countBottomNotes > 0"> + <span class="notes"> + <xsl:for-each select="$bottomNotes"> + <xsl:variable name="noteSign"><xsl:value-of select="count(./preceding::*:note) + 1"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'noteBody foot'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:apply-templates/> + </span> + </span> + </xsl:for-each> + </span> + </xsl:if> +</xsl:template> + +<xsl:template match="*:head"> + <xsl:variable name="style" select="@style"/> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'head bf'"/></xsl:attribute> + <xsl:choose> + <xsl:when test="not(empty($style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="$style"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:when> + <xsl:otherwise> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </div> +</xsl:template> + +<xsl:template match="*:figure"> + <xsl:variable name="class"> + <xsl:choose> + <xsl:when test="empty(@position)"><xsl:value-of select="name()"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat(name(), ' ', string(@position))"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="image" select="*:image"/> + <xsl:variable name="caption" select="*:caption"/> + <xsl:variable name="description" select="*:description"/> + <xsl:variable name="variables" select="*:variables"/> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:if test="not(empty($image))"> + <xsl:variable name="file" select="$image/@file"/> + <xsl:variable name="pn" select="$image/@pn"/> + <xsl:variable name="wx" select="$image/@wx"/> + <xsl:variable name="wy" select="$image/@wy"/> + <xsl:variable name="ww" select="$image/@ww"/> + <xsl:variable name="wh" select="$image/@wh"/> + <xsl:variable name="digilibServiceName" select="'http://digilib.mpiwg-berlin.mpg.de/digitallibrary/jquery/digilib.html'"/> + <xsl:variable name="scalerServiceName" select="'http://digilib.mpiwg-berlin.mpg.de/digitallibrary/servlet/Scaler'"/> + <xsl:variable name="fileName"> + <xsl:choose> + <xsl:when test="empty($wx) and empty($file)"><xsl:value-of select="$echoFiguresDir"/></xsl:when> + <xsl:when test="empty($wx) and not(empty($file))"><xsl:value-of select="concat($echoFiguresDir, '/', $file)"/></xsl:when> + <xsl:when test="not(empty($wx)) and empty($file)"><xsl:value-of select="$echoPageImgDir"/></xsl:when> + <xsl:when test="not(empty($wx)) and not(empty($file))"><xsl:value-of select="concat($echoPageImgDir, '/', $file)"/></xsl:when> + <xsl:otherwise><xsl:value-of select="$echoFiguresDir"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="digilibUrl"> + <xsl:choose> + <xsl:when test="empty($pn) and empty($wx)"><xsl:value-of select="concat($digilibServiceName, '?fn=', $fileName)"/></xsl:when> + <xsl:when test="not(empty($pn)) and empty($wx)"><xsl:value-of select="concat($digilibServiceName, '?fn=', $fileName, '&pn=', $pn)"/></xsl:when> + <xsl:when test="empty($pn) and not(empty($wx))"><xsl:value-of select="concat($digilibServiceName, '?fn=', $fileName, '&wx=', $wx, '&wy=', $wy, '&ww=', $ww, '&wh=', $wh)"/></xsl:when> + <xsl:when test="not(empty($pn)) and not(empty($wx))"><xsl:value-of select="concat($digilibServiceName, '?fn=', $fileName, '&pn=', $pn, '&wx=', $wx, '&wy=', $wy, '&ww=', $ww, '&wh=', $wh)"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat($digilibServiceName, '?fn=', $fileName)"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="scalerUrl"> + <xsl:choose> + <xsl:when test="empty($pn) and empty($wx)"><xsl:value-of select="concat($scalerServiceName, '?fn=', $fileName, '&dh=200')"/></xsl:when> + <xsl:when test="not(empty($pn)) and empty($wx)"><xsl:value-of select="concat($scalerServiceName, '?fn=', $fileName, '&pn=', $pn, '&dh=200')"/></xsl:when> + <xsl:when test="empty($pn) and not(empty($wx))"><xsl:value-of select="concat($scalerServiceName, '?fn=', $fileName, '&wx=', $wx, '&wy=', $wy, '&ww=', $ww, '&wh=', $wh, '&dh=200')"/></xsl:when> + <xsl:when test="not(empty($pn)) and not(empty($wx))"><xsl:value-of select="concat($scalerServiceName, '?fn=', $fileName, '&pn=', $pn, '&wx=', $wx, '&wy=', $wy, '&ww=', $ww, '&wh=', $wh, '&dh=200')"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat($digilibServiceName, '?fn=', $fileName, '&dh=200')"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <a href="{$digilibUrl}"> + <img> + <xsl:attribute name="src"><xsl:value-of select="$scalerUrl"/></xsl:attribute> + </img> + </a> + </xsl:if> + <xsl:if test="not(empty(@number))"> + <xsl:variable name="type"><xsl:value-of select="'Figure'"/></xsl:variable> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNumber'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNum'"/></xsl:attribute> + <xsl:value-of select="string(@number)"/> + </span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNumText'"/></xsl:attribute> + <xsl:value-of select="concat('[', $type, ' ', string(@number), ']')"/> + </span> + </span> + </xsl:if> + <xsl:if test="not(empty($caption))"> + <xsl:for-each select="$caption"> + <span class="caption"><xsl:apply-templates/></span> + </xsl:for-each> + </xsl:if> + <xsl:if test="not(empty($description))"> + <xsl:for-each select="$description"> + <span class="description"><xsl:apply-templates/></span> + </xsl:for-each> + </xsl:if> + <xsl:if test="not(empty($variables))"> + <xsl:for-each select="$variables"> + <span class="variables"><xsl:apply-templates/></span> + </xsl:for-each> + </xsl:if> + </span> +</xsl:template> + +<!-- MathML --> +<xsl:template match="math:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="mml:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<!-- XHTML: remove the xhtml namespace --> +<xsl:template match="xhtml:*"> + <xsl:variable name="hasLabel" select="not(empty(@xhtml:label))"/> + <xsl:variable name="isTable" select="local-name() = 'table'"/> + <xsl:choose> + <xsl:when test="(not($hasLabel)) or ($isTable and $hasLabel)"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- SVG --> +<xsl:template match="svg:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:place|*:person"> + <xsl:element name="span"> + <xsl:attribute name="class"><xsl:value-of select="name()"/></xsl:attribute> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:div"> + <xsl:if test="node() or normalize-space()"> <!-- test: contains nodes or is not empty --> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'div'"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:if> +</xsl:template> + +<xsl:template match="*:p"> + <xsl:variable name="style" select="@style"/> + <xsl:variable name="class" select="'p'"/> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:choose> + <xsl:when test="not(empty($style))"> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:when> + <xsl:otherwise> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </div> +</xsl:template> + +<xsl:template match="*:pb"> + <xsl:variable name="number"> + <xsl:choose> + <xsl:when test="not(empty(@n))"><xsl:value-of select="@n"/></xsl:when> + <xsl:otherwise><xsl:value-of select="count(./preceding::*:pb) + 1"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="file"> + <xsl:choose> + <xsl:when test="not(empty(@file))"><xsl:value-of select="@file"/></xsl:when> + <xsl:otherwise><xsl:value-of select="$number"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'pb'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'src'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="concat('pageimg/', string($file))"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'n'"/></xsl:attribute> + <xsl:value-of select="$number"/> + </span> + <xsl:if test="not(empty(@o))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="'o'"/></xsl:attribute> + <xsl:value-of select="@o"/> + </span> + </xsl:if> + <xsl:if test="not(empty(@rhead))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="'rhead'"/></xsl:attribute> + <xsl:value-of select="@rhead"/> + </span> + </xsl:if> + </span> + </span> +</xsl:template> + +<xsl:template match="*:lb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'lb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="*:cb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'cb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="*:expan"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'expan'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:note"> + <xsl:variable name="place"><xsl:value-of select="string(@position)"/></xsl:variable> + <xsl:variable name="noteSign"><xsl:value-of select="count(./preceding::*:note) + 1"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:if test="$place != '' and $place != 'foot'"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('noteBody ', $place)"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:apply-templates/> + </span> + </xsl:if> + </span> +</xsl:template> + +<xsl:template match="*:handwritten"> + <xsl:variable name="noteSign"><xsl:value-of select="string(@number)"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note handwritten'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'noteBody'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'noteNumText'"/></xsl:attribute> + <xsl:value-of select="concat('[Handwritten note ', string(@number), ']')"/> + </span> + </span> + </span> +</xsl:template> + +<xsl:template match="*:emph"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'emph'"/></xsl:attribute> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="string(@style)"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:when> + <xsl:otherwise> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </span> +</xsl:template> + +<xsl:template match="*:foreign"> + <xsl:variable name="lang" select="@lang"/> + <xsl:variable name="xmllang" select="@xml:lang"/> + <xsl:variable name="language"> + <xsl:choose> + <xsl:when test="not(empty($xmllang))"><xsl:value-of select="$xmllang"/></xsl:when> + <xsl:when test="not(empty($lang))"><xsl:value-of select="$lang"/></xsl:when> + <xsl:otherwise><xsl:value-of select="''"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="class"> + <xsl:choose> + <xsl:when test="$language = ''"><xsl:value-of select="'foreign'"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat('foreign ', $language)"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:q"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'q'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:quote"> + <span class="quote"> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:blockquote"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'blockquote'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:set-off"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'set-off'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:reg"> + <span class="reg"> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'reg'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:var"> + <xsl:variable name="class"> + <xsl:choose> + <xsl:when test="empty(@type)"><xsl:value-of select="'var'"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat('var ', string(@type))"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:num"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'num'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:gap"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'gap'"/></xsl:attribute> + <xsl:if test="not(empty(@extent))"><span class="extent"><xsl:value-of select="string(@extent)"/></span></xsl:if> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:s"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'s'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- words --> +<xsl:template match="*:w"> + <xsl:variable name="wordLanguage" select="@lang"/> + <xsl:variable name="form" select="encode-for-uri(string(@form))"/> + <xsl:variable name="dictionary" select="string(@dictionary)"/> + <xsl:variable name="displayWordOrig"> + <xsl:choose> + <xsl:when test="not(empty(*:orig))"><xsl:apply-templates select="*:orig/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordReg"> + <xsl:choose> + <xsl:when test="not(empty(*:reg))"><xsl:apply-templates select="*:reg/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordNorm"> + <xsl:choose> + <xsl:when test="not(empty(*:norm))"><xsl:apply-templates select="*:norm/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordUrlEncoded" select="encode-for-uri($displayWordOrig)"/> + <xsl:variable name="dictionaryPart"> + <xsl:choose> + <xsl:when test="$dictionary = 'true'"> + <a class="dictionary"> + <xsl:attribute name="href"><xsl:value-of select="concat($dictionaryServiceName, '?query=', $form, '&queryDisplay=', $displayWordUrlEncoded, '&language=', $wordLanguage, '&outputFormat=html', '&outputType=morphCompact&outputType=dictFull')"/></xsl:attribute> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </a> + </xsl:when> + <xsl:otherwise> + <span class="dictionary"> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span class="w"> + <xsl:sequence select="$dictionaryPart"/> + <span class="nodictionary orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="nodictionary reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="nodictionary norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> +</xsl:template> + +<xsl:template match="*:hi"> + <xsl:choose> + <xsl:when test="@type = 'elem'"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/transform/pageTei.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,650 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xs="http://www.w3.org/2001/XMLSchema" + xmlns:functx="http://www.functx.com" + xmlns:saxon="http://saxon.sf.net/" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xs functx saxon dc dcterms math svg xhtml" + > + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="echoFiguresDir"></xsl:param> +<xsl:param name="echoPageImgDir"></xsl:param> + +<xsl:variable name="dictionaryServiceName" select="'http://mpdl-service.mpiwg-berlin.mpg.de/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries'"/> + +<xsl:template match="*:TEI"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'TEI'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:expan|*:emph|*:q|*:quote|*:reg|*:num"> + <xsl:element name="span"> + <xsl:attribute name="class"><xsl:value-of select="name()"/></xsl:attribute> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:teiHeader"> + <!-- do nothing --> +</xsl:template> + +<xsl:template match="*:text"> + <xsl:variable name="contentStr" select="normalize-space(string(.))"/> + <xsl:variable name="figures" select=".//*:figure|.//*:handwritten"/> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'text'"/></xsl:attribute> + <xsl:apply-templates/> + <xsl:if test="$contentStr = '' and empty($figures)"> <!-- test if div only contains empty nodes but no figures --> + <div> + <xsl:attribute name="class"><xsl:value-of select="'emptyPage'"/></xsl:attribute> + <xsl:value-of select="'[Empty page]'"/> + </div> + </xsl:if> + </div> + <!-- Dictionary --> + <xsl:if test="not(empty(//*:entry))"> + <xsl:for-each select="//*:entry"> + <xsl:sort select="*:form/*:orth"/> + <xsl:variable name="position" select="position()"/> + <span class="entry"> + <xsl:apply-templates select="*:form"/> + <xsl:apply-templates select="*:sense"/> + <xsl:if test="not(empty(*:figure))"> + <span class="entryDiv"> + <span class="bf"><xsl:value-of select="'Figures: '"/></span> + <span class="entryDiv"> + <xsl:for-each select="*:figure"> + <xsl:variable name="href" select="*:graphic/@url"/> + <xsl:variable name="figDesc" select="string(*:figDesc)"/> + <xsl:if test="$href != ''"> + <div class="figure" style="margin-left:10px;"> + <a href="{$href}"><img alt="Figure: {$figDesc}" src="{$href}" width="200" height="200"/></a> + <br/> + <xsl:value-of select="'[Figure]: '"/><xsl:apply-templates select="*:head"/> + </div> + </xsl:if> + </xsl:for-each> + </span> + </span> + </xsl:if> + <xsl:if test="not(empty(*:xr))"> + <span class="entryDiv"> + <span class="bf"><xsl:value-of select="'References: '"/></span> + <span class="entryDiv"> + <xsl:for-each select="*:xr/*:ref"> + <li><xsl:apply-templates select="."/></li> + </xsl:for-each> + </span> + </span> + </xsl:if> + </span> + </xsl:for-each> + </xsl:if> + <!-- Notes --> + <xsl:variable name="bottomNotes" select="//*:note[contains(@place, 'foot') or empty(string(@place))]"/> + <xsl:variable name="countBottomNotes" select="count($bottomNotes)"/> + <xsl:if test="$countBottomNotes > 0"> + <span class="notes"> + <xsl:for-each select="$bottomNotes"> + <xsl:variable name="noteSign"><xsl:value-of select="count(./preceding::*:note) + 1"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'noteBody foot'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:apply-templates/> + </span> + </span> + </xsl:for-each> + </span> + </xsl:if> +</xsl:template> + +<xsl:template match="*:div"> + <xsl:if test="node() or normalize-space()"> <!-- test: contains nodes or is not empty --> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'div'"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:if> +</xsl:template> + +<xsl:template match="*:p"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'p'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:pb"> + <xsl:variable name="number"> + <xsl:choose> + <xsl:when test="not(empty(@n))"><xsl:value-of select="@n"/></xsl:when> + <xsl:otherwise><xsl:value-of select="count(./preceding::*:pb) + 1"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="file"> + <xsl:choose> + <xsl:when test="not(empty(@facs))"><xsl:value-of select="@facs"/></xsl:when> + <xsl:otherwise><xsl:value-of select="$number"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'pb'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'src'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="concat('pageimg/', string($file))"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'n'"/></xsl:attribute> + <xsl:value-of select="$number"/> + </span> + </span> + </span> +</xsl:template> + +<xsl:template match="*:head"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'head bf'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:figure"> + <span class="figure"> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:variable name="url"> + <xsl:choose> + <xsl:when test="not(empty(@facs))"><xsl:value-of select="string(@facs)"/></xsl:when> + <xsl:otherwise><xsl:value-of select="string(graphic/@url)"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="fileName" select="concat($echoFiguresDir, '/', $url)"/> + <xsl:variable name="digilibUrl"> + <xsl:choose> + <xsl:when test="starts-with($url, 'http')"><xsl:value-of select="$url"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat('http://digilib.mpiwg-berlin.mpg.de/digitallibrary/jquery/digilib.html?fn=', $fileName)"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="scalerUrl"> + <xsl:choose> + <xsl:when test="starts-with($url, 'http')"><xsl:value-of select="$url"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat('http://digilib.mpiwg-berlin.mpg.de/digitallibrary/servlet/Scaler?fn=', $fileName, '&dh=200&dw=200')"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <a href="{$digilibUrl}"> + <img> + <xsl:attribute name="src"><xsl:value-of select="$scalerUrl"/></xsl:attribute> + </img> + </a> + <xsl:if test="not(empty(@number))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNumber'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNum'"/></xsl:attribute> + <xsl:value-of select="string(@number)"/> + </span> + <span> + <xsl:attribute name="class"><xsl:value-of select="'figureNumText'"/></xsl:attribute> + <xsl:value-of select="concat('[', 'Figure', ' ', string(@number), ']')"/> + </span> + </span> + </xsl:if> + <xsl:if test="not(empty(*:head))"> + <span class="caption"> + <xsl:for-each select="*:head"> + <xsl:apply-templates/> + </xsl:for-each> + </span> + </xsl:if> + </span> +</xsl:template> + +<!-- segmentation --> +<xsl:template match="*:seg"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:choose> + <xsl:when test="@rend = 'highlight'"> + <xsl:attribute name="class"><xsl:value-of select="'seg highlight'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="@rend = 'highlightPoint'"> + <xsl:attribute name="class"><xsl:value-of select="'seg highlightPoint'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:otherwise> + <xsl:attribute name="class"><xsl:value-of select="'seg'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </span> +</xsl:template> + +<!-- choice --> +<xsl:template match="*:choice"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'choice'"/></xsl:attribute> + <xsl:if test="not(empty(*:orig))"> + <xsl:apply-templates select="*:orig"/> + </xsl:if> + <xsl:if test="not(empty(*:abbr))"> + <xsl:apply-templates select="*:abbr"/> + </xsl:if> + <xsl:if test="not(empty(*:am))"> + <xsl:apply-templates select="*:am"/> + </xsl:if> + <xsl:if test="not(empty(*:sic))"> + <xsl:apply-templates select="*:sic"/> + </xsl:if> + </span> +</xsl:template> + +<!-- hi (highlighted) --> +<xsl:template match="*:hi"> + <xsl:choose> + <xsl:when test="not(empty(@type)) and @type = 'elem'"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:when> + <xsl:when test="not(empty(@type)) and @type != 'elem'"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:when> + <xsl:when test="@rend = 'initial'"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'initial'"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:when> + <xsl:when test="@rend = 'bold'"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'bf'"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="@rend"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- name (of type: place, person, ...) --> +<xsl:template match="*:name"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:choose> + <xsl:when test="@type = 'place'"> + <xsl:attribute name="class"><xsl:value-of select="'place'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="'Place'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="@type = 'person'"> + <xsl:attribute name="class"><xsl:value-of select="'person'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="'Person'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="@type = 'org'"> + <xsl:attribute name="class"><xsl:value-of select="'organization'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="'Organization'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:otherwise> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="@type"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </span> +</xsl:template> + +<!-- place, person, ...) --> +<xsl:template match="*:placeName"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'place'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="concat('Place: ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> +<xsl:template match="*:persName"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'person'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="'Person'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- term --> +<xsl:template match="*:term"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'term'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="'Terminology entry'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- line --> +<xsl:template match="*:lg"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'lg'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="*:l"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'l'"/></xsl:attribute> + <xsl:apply-templates/> + </span> + <br/> +</xsl:template> + +<!-- reference --> +<xsl:template match="*:ref"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'ref'"/></xsl:attribute> + <xsl:if test="not(empty(@target))"> + <a class="ref"> + <xsl:attribute name="href"><xsl:value-of select="@target"/></xsl:attribute> + <xsl:apply-templates/> + </a> + </xsl:if> + <xsl:if test="empty(@target)"> + <xsl:apply-templates/> + </xsl:if> + </span> +</xsl:template> + +<!-- table --> +<xsl:template match="*:table"> + <table> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty(head))"> + <caption align="top"><xsl:apply-templates select="*:head"/></caption> + </xsl:if> + <xsl:apply-templates select="*:row"/> + </table> +</xsl:template> + +<xsl:template match="*:row"> + <tr> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:choose> + <xsl:when test="@role = 'label'"> + <xsl:attribute name="style"><xsl:value-of select="'font-weight:bold;'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="@role = 'data' or empty(@role)"> + <xsl:apply-templates/> + </xsl:when> + <xsl:otherwise> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </tr> +</xsl:template> + +<xsl:template match="*:cell"> + <td> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:choose> + <xsl:when test="@role = 'label' and empty(@cols)"> + <xsl:attribute name="style"><xsl:value-of select="'font-weight:bold;'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="@role = 'label' and not(empty(@cols))"> + <xsl:attribute name="colspan"><xsl:value-of select="@cols"/></xsl:attribute> + <xsl:attribute name="style"><xsl:value-of select="'font-weight:bold;'"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="(@role = 'data' or empty(@role)) and empty(@cols)"> + <xsl:apply-templates/> + </xsl:when> + <xsl:when test="(@role = 'data' or empty(@role)) and not(empty(@cols))"> + <xsl:attribute name="colspan"><xsl:value-of select="@cols"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:when> + <xsl:otherwise> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:apply-templates/> + </xsl:otherwise> + </xsl:choose> + </td> +</xsl:template> + +<!-- dictionary --> +<xsl:template match="*:entry"> + <!-- empty: handled in text tag --> +</xsl:template> + +<xsl:template match="*:form"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'form'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:orth"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'orth'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:sense"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'sense'"/></xsl:attribute> + <span class="entryDiv"><xsl:apply-templates select="*:def"/></span> + <span class="entryDiv"><xsl:apply-templates select="*:etym"/></span> + </span> +</xsl:template> + +<xsl:template match="*:def"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'def'"/></xsl:attribute> + <span class="bf"><xsl:value-of select="'Definition: '"/></span> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:mentioned"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'mentioned'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:etym"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'etym'"/></xsl:attribute> + <span class="bf"><xsl:value-of select="'Etymology: '"/></span> + <span class="entryDiv"> + <xsl:for-each select="*:cit"> + <li><xsl:apply-templates select="*:quote"/><xsl:value-of select="' ('"/><xsl:apply-templates select="*:def"/><xsl:value-of select="')'"/></li> + </xsl:for-each> + </span> + </span> +</xsl:template> + +<xsl:template match="*:cit"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'cit'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- MathML --> +<xsl:template match="math:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<!-- SVG --> +<xsl:template match="svg:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:lb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'lb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="*:cb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'cb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="*:note"> + <xsl:variable name="place"><xsl:value-of select="string(@place)"/></xsl:variable> + <xsl:variable name="noteSign"><xsl:value-of select="count(./preceding::*:note) + 1"/></xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:if test="$place != '' and $place != 'foot'"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('noteBody ', $place)"/></xsl:attribute> + <span class="noteSign"><xsl:value-of select="$noteSign"/></span> + <xsl:apply-templates/> + </span> + </xsl:if> + </span> +</xsl:template> + +<xsl:template match="*:foreign"> + <xsl:variable name="xmllang" select="@xml:lang"/> + <xsl:variable name="language"> + <xsl:choose> + <xsl:when test="not(empty($xmllang))"><xsl:value-of select="$xmllang"/></xsl:when> + <xsl:otherwise><xsl:value-of select="''"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat('foreign ', $language)"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:gap"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'gap'"/></xsl:attribute> + <xsl:if test="not(empty(@extent))"><span class="extent"><xsl:value-of select="string(@extent)"/></span></xsl:if> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:s"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'s'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- words --> +<xsl:template match="*:w"> + <xsl:variable name="wordLanguage" select="@lang"/> + <xsl:variable name="form" select="encode-for-uri(string(@form))"/> + <xsl:variable name="dictionary" select="string(@dictionary)"/> + <xsl:variable name="displayWordOrig"> + <xsl:choose> + <xsl:when test="not(empty(*:orig))"><xsl:apply-templates select="*:orig/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordReg"> + <xsl:choose> + <xsl:when test="not(empty(*:reg))"><xsl:apply-templates select="*:reg/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordNorm"> + <xsl:choose> + <xsl:when test="not(empty(*:norm))"><xsl:apply-templates select="*:norm/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordUrlEncoded" select="encode-for-uri($displayWordOrig)"/> + <xsl:variable name="dictionaryPart"> + <xsl:choose> + <xsl:when test="$dictionary = 'true'"> + <a class="dictionary"> + <xsl:attribute name="href"><xsl:value-of select="concat($dictionaryServiceName, '?query=', $form, '&queryDisplay=', $displayWordUrlEncoded, '&language=', $wordLanguage, '&outputFormat=html', '&outputType=morphCompact&outputType=dictFull')"/></xsl:attribute> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </a> + </xsl:when> + <xsl:otherwise> + <span class="dictionary"> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span class="w"> + <xsl:sequence select="$dictionaryPart"/> + <span class="nodictionary orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="nodictionary reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="nodictionary norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/transform/pageXhtml.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,260 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xs="http://www.w3.org/2001/XMLSchema" + xmlns:functx="http://www.functx.com" + xmlns:saxon="http://saxon.sf.net/" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xs functx saxon dc dcterms math svg xhtml" + > + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="echoFiguresDir"></xsl:param> +<xsl:param name="echoPageImgDir"></xsl:param> + +<xsl:variable name="dictionaryServiceName" select="'http://mpdl-service.mpiwg-berlin.mpg.de/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries'"/> + +<xsl:template match="*:html"> + <xsl:apply-templates/> +</xsl:template> + +<xsl:template match="body"> + <xsl:apply-templates mode="html"/> +</xsl:template> + +<!-- html mode: transform html texts --> +<xsl:template match="@*|node()" mode="html"> + <xsl:copy> + <xsl:apply-templates select="@*|node()" mode="html"/> + </xsl:copy> +</xsl:template> + +<xsl:template match="s" mode="html"> + <xsl:apply-templates select="."/> +</xsl:template> + +<xsl:template match="w" mode="html"> + <xsl:apply-templates select="."/> +</xsl:template> + + +<xsl:template match="*:expan|*:emph|*:q|*:quote|*:blockquote|*:set-off|*:reg|*:var|*:num|*:figure"> + <span> + <xsl:attribute name="class"><xsl:value-of select="name()"/></xsl:attribute> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="*:div"> + <xsl:if test="node() or normalize-space()"> <!-- test: contains nodes or is not empty --> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'div'"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:if> +</xsl:template> + +<xsl:template match="*:p"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'p'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<xsl:template match="pb"> + <xsl:variable name="number" select="count(./preceding::*:pb) + 1"/> + <xsl:variable name="file" select="$number"/> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'pb'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'src'"/></xsl:attribute> + <xsl:attribute name="title"><xsl:value-of select="concat('pageimg/', string($file))"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="'n'"/></xsl:attribute> + <xsl:value-of select="$number"/> + </span> + </span> + </span> +</xsl:template> + +<xsl:template match="*:head"> + <div> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'head bf'"/></xsl:attribute> + <xsl:apply-templates/> + </div> +</xsl:template> + +<!-- MathML --> +<xsl:template match="math:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<!-- SVG --> +<xsl:template match="svg:*"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:lb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'lb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="*:cb"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'cb'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="br"> + <br> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'br'"/></xsl:attribute> + </br> +</xsl:template> + +<xsl:template match="note"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="foreign"> + <xsl:variable name="lang" select="@lang"/> + <xsl:variable name="xmllang" select="@xml:lang"/> + <xsl:variable name="language"> + <xsl:choose> + <xsl:when test="not(empty($xmllang))"><xsl:value-of select="$xmllang"/></xsl:when> + <xsl:when test="not(empty($lang))"><xsl:value-of select="$lang"/></xsl:when> + <xsl:otherwise><xsl:value-of select="''"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat('foreign ', $language)"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<xsl:template match="gap"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'gap'"/></xsl:attribute> + <xsl:if test="not(empty(@extent))"><span class="extent"><xsl:value-of select="string(@extent)"/></span></xsl:if> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- XHTML: remove the xhtml namespace --> +<xsl:template match="xhtml:*"> + <xsl:variable name="hasLabel" select="string(@xlink:label) != ''"/> + <xsl:variable name="isTable" select="name() = 'table'"/> + <xsl:choose> + <xsl:when test="(not($hasLabel)) or ($isTable and $hasLabel)"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates/> + </xsl:element> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="s"> + <span> + <xsl:if test="not(empty(@xml:id))"><xsl:attribute name="id"><xsl:value-of select="@xml:id"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="'s'"/></xsl:attribute> + <xsl:apply-templates/> + </span> +</xsl:template> + +<!-- words --> +<xsl:template match="*:w"> + <xsl:variable name="wordLanguage" select="@lang"/> + <xsl:variable name="form" select="encode-for-uri(string(@form))"/> + <xsl:variable name="dictionary" select="string(@dictionary)"/> + <xsl:variable name="displayWordOrig"> + <xsl:choose> + <xsl:when test="not(empty(*:orig))"><xsl:apply-templates select="*:orig/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordReg"> + <xsl:choose> + <xsl:when test="not(empty(*:reg))"><xsl:apply-templates select="*:reg/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordNorm"> + <xsl:choose> + <xsl:when test="not(empty(*:norm))"><xsl:apply-templates select="*:norm/node()"/></xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordUrlEncoded" select="encode-for-uri($displayWordOrig)"/> + <xsl:variable name="dictionaryPart"> + <xsl:choose> + <xsl:when test="$dictionary = 'true'"> + <a class="dictionary"> + <xsl:attribute name="href"><xsl:value-of select="concat($dictionaryServiceName, '?query=', $form, '&queryDisplay=', $displayWordUrlEncoded, '&language=', $wordLanguage, '&outputFormat=html', '&outputType=morphCompact&outputType=dictFull')"/></xsl:attribute> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </a> + </xsl:when> + <xsl:otherwise> + <span class="dictionary"> + <span class="orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span class="w"> + <xsl:sequence select="$dictionaryPart"/> + <span class="nodictionary orig"><xsl:sequence select="$displayWordOrig"/></span> + <span class="nodictionary reg"><xsl:sequence select="$displayWordReg"/></span> + <span class="nodictionary norm"><xsl:sequence select="$displayWordNorm"/></span> + </span> +</xsl:template> + +<xsl:template match="*:hi"> + <xsl:choose> + <xsl:when test="@type = 'elem'"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/transform/pageXml.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,86 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink dcterms math svg xhtml" + > + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="echoFiguresDir"></xsl:param> +<xsl:param name="echoPageImgDir"></xsl:param> + +<!-- transform to browser like xml display --> +<xsl:template match="element()"> + <xsl:variable name="elementName" select="name()"/> + <xsl:variable name="elementPresentation"> + <xsl:choose> + <xsl:when test="element() = node() or text() != ''"> + <xsl:value-of select="'<'"/> + <span class="xml elementName"><xsl:value-of select="$elementName"/></span> + <xsl:apply-templates select="attribute()"/> + <xsl:value-of select="'>'"/> + <xsl:apply-templates select="element()|text()|comment()|processing-instruction()"/> + <xsl:value-of select="'</'"/> + <span class="xml elementName"><xsl:value-of select="$elementName"/></span> + <xsl:value-of select="'>'"/> + </xsl:when> + <xsl:otherwise> + <xsl:value-of select="'<'"/> + <span class="xml elementName"><xsl:value-of select="$elementName"/></span> + <xsl:apply-templates select="attribute()"/> + <xsl:value-of select="'/>'"/> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <ul class="xml element"> + <xsl:sequence select="$elementPresentation"/> + </ul> +</xsl:template> + +<xsl:template match="attribute()"> + <xsl:variable name="attributeName" select="name()"/> + <span class="xml attributeName"> + <xsl:value-of select="' '"/> + <xsl:value-of select="$attributeName"/> + </span> + <xsl:value-of select="'="'"/> + <span class="xml attributeValue"><xsl:value-of select="."/></span><xsl:value-of select="'"'"/> +</xsl:template> + +<xsl:template match="comment()"> + <span class="xml comment"> + <xsl:value-of select="'<!-- '"/><xsl:value-of select="."/><xsl:value-of select="' -->'"/> + </span> +</xsl:template> + +<xsl:template match="processing-instruction()"> +</xsl:template> + +<xsl:template match="*:w"> + <xsl:apply-templates select="*:orig/node()"/> +</xsl:template> + +<xsl:template match="*:hi"> + <xsl:choose> + <xsl:when test="@type = 'elem'"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </div> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('highlight ', @type)"/></xsl:attribute> + <xsl:apply-templates/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/transform/queryDocument.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,93 @@ +<?xml version="1.0"?> + +<xsl:stylesheet version="2.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform"> + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="query"></xsl:param> +<xsl:param name="flags"></xsl:param> +<xsl:param name="outputFormat"></xsl:param> + +<xsl:variable name="apos">'</xsl:variable> +<xsl:variable name="xpathQuery"> + <xsl:choose> + <xsl:when test="$flags = ''"><xsl:value-of select="concat('//*:s[matches(.,', $apos, $query, $apos, ')]', '|', '//*:head[matches(.,', $apos, $query, $apos, ')]')"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat('//*:s[matches(.,', $apos, $query, $apos, ',', $apos, $flags, $apos, ')]', '|', '//*:head[matches(.,', $apos, $query, $apos, ',', $apos, $flags, $apos, ')]')"/></xsl:otherwise> + </xsl:choose> +</xsl:variable> +<xsl:variable name="queryResult" select="saxon:evaluate($xpathQuery)" xmlns:saxon="http://saxon.sf.net/"/> +<xsl:variable name="queryResultSize"><xsl:value-of select="count($queryResult)"/></xsl:variable> +<xsl:variable name="queryResultPageSize" select="20"/> +<xsl:variable name="queryResultPages"> + <xsl:choose> + <xsl:when test="$queryResultSize = 0"><xsl:value-of select="count($queryResult)"/></xsl:when> + <xsl:otherwise><xsl:value-of select="$queryResultSize idiv $queryResultPageSize + 1"/></xsl:otherwise> + </xsl:choose> +</xsl:variable> +<xsl:variable name="pageBreaks" select="saxon:evaluate('//*:pb')" xmlns:saxon="http://saxon.sf.net/"/> +<xsl:variable name="queryResultHtml"> +<xsl:choose> + <xsl:when test="$outputFormat = 'html'"> + <table> + <thead> + <tr> + <th align="left" valign="top"> + <button name="order-by" value="author" style="padding:0px;font-weight:bold;font-size:14px;background:none;border:none;">No.</button> + </th> + <th align="left" valign="top"> + <button name="order-by" value="author" style="padding:0px;font-weight:bold;font-size:14px;background:none;border:none;">Path</button> + </th> + <th align="left" valign="top"> + <button name="order-by" value="author" style="padding:0px;font-weight:bold;font-size:14px;background:none;border:none;">Hit</button> + </th> + <th align="left" valign="top"> + <button name="order-by" value="author" style="padding:0px;font-weight:bold;font-size:14px;background:none;border:none;">Page</button> + </th> + </tr> + </thead> + <tbody> + <xsl:for-each select="$queryResult"> + <xsl:variable name="hit" select="."/> + <tr> + <td align="left" valign="top"><xsl:value-of select="position()"/></td> + <td align="left" valign="top"><xsl:value-of select="saxon:path(.)" xmlns:saxon="http://saxon.sf.net/"/></td> + <td align="left" valign="top"><xsl:value-of select="."/></td> + <td align="left" valign="top"><xsl:value-of select="count(./preceding::*:pb)"/></td> <!-- better performance: count($pageBreaks[. << $hit]) --> + </tr> + </xsl:for-each> + </tbody> + </table> + </xsl:when> + <xsl:otherwise> + <result> + <query><xsl:value-of select="$query"/></query> + <flags><xsl:value-of select="$flags"/></flags> + <size><xsl:value-of select="$queryResultSize"/></size> + <page-size><xsl:value-of select="$queryResultPageSize"/></page-size> + <pages><xsl:value-of select="$queryResultPages"/></pages> + <pn>1</pn> + <hits> + <xsl:for-each select="$queryResult"> + <xsl:variable name="hit" select="."/> + <hit> + <hitType><xsl:value-of select="'s'"/></hitType> + <pos><xsl:value-of select="position()"/></pos> + <pn><xsl:value-of select="count(./preceding::*:pb)"/></pn> + <hitId>xmlId</hitId> + <hitPos>4711</hitPos> + <hitString><xsl:value-of select="."/></hitString> + <hitSurroundsPB>false</hitSurroundsPB> + </hit> + </xsl:for-each> + </hits> + <query-forms></query-forms> + <query-regularizations></query-regularizations> + </result> + </xsl:otherwise> +</xsl:choose> +</xsl:variable> +<xsl:template match="/"> + <xsl:sequence select="$queryResultHtml"/> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/transform/replaceAnchor.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,82 @@ +<?xml version="1.0"?> + +<xsl:stylesheet + version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xhtml" +> + +<xsl:output method="xml" encoding="utf-8"/> + +<xsl:template match="@*|node()"> + <xsl:copy> + <xsl:apply-templates select="@*|node()"/> + </xsl:copy> +</xsl:template> + +<!-- insert figure number --> +<xsl:template match="*:figure"> + <xsl:variable name="number" select="count(./preceding::*:figure) + 1"/> + <xsl:element name="{name()}" namespace="{namespace-uri()}"><!-- remove namespace in element and its subnodes --> + <xsl:apply-templates select="@*"/> + <xsl:attribute name="number"><xsl:value-of select="$number"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<!-- insert figure number --> +<xsl:template match="*:handwritten"> + <xsl:variable name="number" select="count(./preceding::*:handwritten) + 1"/> + <xsl:element name="{name()}" namespace="{namespace-uri()}"><!-- remove namespace in element and its subnodes --> + <xsl:apply-templates select="@*"/> + <xsl:attribute name="number"><xsl:value-of select="$number"/></xsl:attribute> + <xsl:apply-templates/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:anchor"> + <xsl:variable name="type" select="@type"/> + <xsl:variable name="href" select="@xlink:href"/> + <xsl:choose> + <xsl:when test="$type = 'figure'"> + <xsl:variable name="figure" select="//*:figure[@xlink:label = $href]"/> + <xsl:apply-templates select="$figure"/> + </xsl:when> + <xsl:when test="$type = 'handwritten'"> + <xsl:variable name="handwritten" select="//*:handwritten[@xlink:label = $href]"/> + <xsl:apply-templates select="$handwritten"/> + </xsl:when> + <xsl:when test="$type = 'note'"> + <xsl:variable name="note" select="//*:note[@xlink:label = $href]"/> + <xsl:sequence select="$note"/> + </xsl:when> + <xsl:when test="$type = 'table'"> + <xsl:variable name="table" select="//xhtml:table[@xlink:label = $href]"/> + <xsl:sequence select="$table"/> + </xsl:when> + <xsl:otherwise> + <xsl:copy> + <xsl:apply-templates select="@*|node()"/> + </xsl:copy> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- remove the anchored divs: div which contain a labeled figure, note, handwritten or table --> +<xsl:template match="*:div"> + <xsl:variable name="containingElems" select="*:figure|*:note|*:handwritten|xhtml:table"/> + <xsl:variable name="hasLabel" select="not(empty($containingElems[1]/@xlink:label))"/> + <xsl:choose> + <xsl:when test="$hasLabel"></xsl:when> + <xsl:otherwise> + <xsl:copy> + <xsl:apply-templates select="@*|node()"/> + </xsl:copy> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/transform/toc.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,153 @@ +<?xml version="1.0"?> + +<xsl:stylesheet + version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xhtml" +> + +<xsl:output method="xml" encoding="utf-8" indent="yes"/> + +<xsl:template match="/"> + <xsl:variable name="toc" select="//*:div[@type = 'section' or @type = 'chapter' or @type = 'part']"/> + <xsl:variable name="figures" select="//*:figure"/> + <xsl:variable name="notes" select="//*:note"/> + <xsl:variable name="notesHandwritten" select="//*:handwritten"/> + <xsl:variable name="pages" select="//*:pb"/> + <xsl:variable name="places" select="//*:place"/> + <list> + <xsl:if test="not(empty($toc))"> + <list type ="toc"> + <xsl:attribute name="count"><xsl:value-of select="count($toc)"/></xsl:attribute> + <head>Table of contents</head> + <xsl:apply-templates select="$toc"/> + </list> + </xsl:if> + <xsl:if test="not(empty($figures))"> + <list type="figures"> + <xsl:attribute name="count"><xsl:value-of select="count($figures)"/></xsl:attribute> + <head>Figures</head> + <xsl:apply-templates select="$figures"/> + </list> + </xsl:if> + <xsl:if test="not(empty($notes))"> + <list type="notes"> + <xsl:attribute name="count"><xsl:value-of select="count($notes)"/></xsl:attribute> + <head>Notes</head> + <xsl:apply-templates select="$notes"/> + </list> + </xsl:if> + <xsl:if test="not(empty($notesHandwritten))"> + <list type="notesHandwritten"> + <xsl:attribute name="count"><xsl:value-of select="count($notesHandwritten)"/></xsl:attribute> + <head>Handwritten notes</head> + <xsl:apply-templates select="$notesHandwritten"/> + </list> + </xsl:if> + <xsl:if test="not(empty($pages))"> + <list type="pages"> + <xsl:attribute name="count"><xsl:value-of select="count($pages)"/></xsl:attribute> + <head>Pages</head> + <xsl:apply-templates select="$pages"/> + </list> + </xsl:if> + <xsl:if test="not(empty($places))"> + <list type="places"> + <xsl:attribute name="count"><xsl:value-of select="count($places)"/></xsl:attribute> + <head>Places</head> + <xsl:apply-templates select="$places"/> + </list> + </xsl:if> + </list> +</xsl:template> + +<xsl:template match="*:figure"> + <xsl:variable name="page" select="count(./preceding::*:pb)"/> + <xsl:variable name="number" select="@number"/> + <item> + <xsl:if test="not(empty($number))"><xsl:attribute name="n"><xsl:value-of select="$number"/></xsl:attribute></xsl:if> + <xsl:apply-templates select="*:caption"/> + <xsl:apply-templates select="*:description"/> + <xsl:apply-templates select="*:variables"/> + <xsl:if test="not(empty($page))"><ref><xsl:value-of select="$page"/></ref></xsl:if> + </item> +</xsl:template> + +<xsl:template match="*:caption"><xsl:value-of select="' '"/> + <xsl:apply-templates/> +</xsl:template> + +<xsl:template match="*:description"> + <xsl:apply-templates/><xsl:value-of select="' '"/> +</xsl:template> + +<xsl:template match="*:variables"> + <xsl:apply-templates/><xsl:value-of select="' '"/> +</xsl:template> + +<xsl:template match="*:note"> + <xsl:variable name="page" select="count(./preceding::*:pb)"/> + <xsl:variable name="number" select="@number"/> + <item> + <xsl:if test="not(empty($number))"><xsl:attribute name="n"><xsl:value-of select="$number"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty($page))"><ref><xsl:value-of select="$page"/></ref></xsl:if> + </item> +</xsl:template> + +<xsl:template match="*:handwritten"> + <xsl:variable name="page" select="count(./preceding::*:pb)"/> + <xsl:variable name="number" select="@number"/> + <item> + <xsl:if test="not(empty($number))"><xsl:attribute name="n"><xsl:value-of select="$number"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty($page))"><ref><xsl:value-of select="$page"/></ref></xsl:if> + </item> +</xsl:template> + +<xsl:template match="*:place"> + <xsl:variable name="page" select="count(./preceding::*:pb)"/> + <xsl:variable name="id" select="@id"/> + <item> + <xsl:if test="not(empty($id))"><xsl:attribute name="id"><xsl:value-of select="$id"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty($page))"><ref><xsl:value-of select="$page"/></ref></xsl:if> + </item> +</xsl:template> + +<xsl:template match="*:div"> + <xsl:variable name="level"><xsl:number level="multiple" count="*:div[@type = 'section' or @type = 'chapter' or @type = 'part']" format="1."/></xsl:variable> + <xsl:variable name="depth" select="string-length(replace($level, '[^\\.]', ''))"/> + <xsl:variable name="pb" select="./preceding::*:pb[1]"/> + <xsl:variable name="o" select="$pb/@o"/> + <xsl:variable name="oNorm" select="$pb/@o-norm"/> + <xsl:variable name="page" select="count(./preceding::*:pb)"/> + <item> + <xsl:if test="not(empty($level))"><xsl:attribute name="n"><xsl:value-of select="$level"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty($level))"><xsl:attribute name="lv"><xsl:value-of select="$depth"/></xsl:attribute></xsl:if> + <xsl:apply-templates select="*:head"/> + <xsl:if test="not(empty($page))"> + <ref> + <xsl:if test="not(empty($o))"><xsl:attribute name="o"><xsl:value-of select="$o"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty($oNorm))"><xsl:attribute name="o-norm"><xsl:value-of select="$oNorm"/></xsl:attribute></xsl:if> + <xsl:value-of select="$page"/> + </ref> + </xsl:if> + </item> +</xsl:template> + +<xsl:template match="*:head"> + <xsl:apply-templates/><xsl:value-of select="' '"/> +</xsl:template> + +<xsl:template match="*:pb"> + <xsl:variable name="page" select="count(./preceding::*:pb) + 1"/> + <item> + <xsl:attribute name="n"><xsl:value-of select="$page"/></xsl:attribute> + <xsl:if test="not(empty(@o))"><xsl:attribute name="o"><xsl:value-of select="@o"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty(@o-norm))"><xsl:attribute name="o-norm"><xsl:value-of select="@o-norm"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty(@file))"><xsl:attribute name="file"><xsl:value-of select="@file"/></xsl:attribute></xsl:if> + <xsl:if test="not(empty(@facs))"><xsl:attribute name="file"><xsl:value-of select="@facs"/></xsl:attribute></xsl:if> + </item> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/transform/tocOut.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,45 @@ +<?xml version="1.0"?> + +<xsl:stylesheet + version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" +> + +<xsl:output method="xhtml"/> + +<xsl:param name="type"></xsl:param> +<xsl:param name="outputFormat"></xsl:param> + +<xsl:template match="/"> + <xsl:choose> + <xsl:when test="$outputFormat = 'html'"> + <xsl:apply-templates select="//*:list[@type = $type]/item" mode="html"/> + </xsl:when> + <xsl:otherwise> + <xsl:sequence select="//*:list[@type = $type]"/> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="*:item" mode="html"> + <xsl:variable name="n" select="@n"/> + <xsl:variable name="level" select="@lv"/> + <div> + <xsl:attribute name="class"><xsl:value-of select="'tocItem'"/></xsl:attribute> + <xsl:value-of select="concat($n, ' ')"/> + <xsl:apply-templates mode="html"/> + </div> +</xsl:template> + +<xsl:template match="*:ref" mode="html"> + <xsl:variable name="pageOrig" select="@o"/> + <xsl:variable name="page" select="text()"/> + <xsl:variable name="pageHref" select="concat('#page', $page)"/> + <span> + <xsl:attribute name="class"><xsl:value-of select="'tocPageRef'"/></xsl:attribute> + <xsl:value-of select="'Page: '"/> + <a href="{$pageHref}"><xsl:apply-templates mode="html"/></a> + </span> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/cms/translator/MicrosoftTranslator.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,120 @@ +package de.mpg.mpiwg.berlin.mpdl.cms.translator; + +import java.util.ArrayList; + +import com.memetix.mst.language.Language; +import com.memetix.mst.translate.Translate; +import com.memetix.mst.detect.Detect; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; + +public class MicrosoftTranslator { + private static final String KEY = "474A4E72DB217E37031EC190ACB4159378A6917C"; // free key to use with Microsoft server + private static String[] DEST_LANGUAGES = {"eng", "ger", "fra"}; + + private MicrosoftTranslator() { + // nothing: prevent instantiation + } + + public static String[] getDestLanguages() { + return DEST_LANGUAGES; + } + + public static String translate(String query, String toLanguageStr) throws ApplicationException { + if (toLanguageStr == null) + throw new ApplicationException("MicrosoftTranslator: toLanguageStr is null"); + String translation = null; + try { + Translate.setKey(KEY); // Set the API key once per JVM. It is set statically and applies to all services + String langId = de.mpg.mpiwg.berlin.mpdl.lt.general.Language.getInstance().getLanguageId(toLanguageStr); // e.g. "de" is delivered from "deu" + Language toLanguage = Language.fromString(langId); + translation = Translate.execute(query, toLanguage); + translation = translation.toLowerCase(); + } catch (Exception e) { + throw new ApplicationException(e); + } + return translation; + } + + public static String translate(String query, String fromLanguageStr, String toLanguageStr) throws ApplicationException { + if (fromLanguageStr == null) + throw new ApplicationException("MicrosoftTranslator: toLanguageStr is null"); + if (toLanguageStr == null) + throw new ApplicationException("MicrosoftTranslator: fromLanguageStr is null"); + String translation = null; + try { + Translate.setKey(KEY); + String fromLangId = de.mpg.mpiwg.berlin.mpdl.lt.general.Language.getInstance().getLanguageId(fromLanguageStr); // e.g. "de" is delivered from "deu" + String toLangId = de.mpg.mpiwg.berlin.mpdl.lt.general.Language.getInstance().getLanguageId(toLanguageStr); // e.g. "de" is delivered from "deu" + Language fromLanguage = Language.fromString(fromLangId); + Language toLanguage = Language.fromString(toLangId); + translation = Translate.execute(query, fromLanguage, toLanguage); + translation = translation.toLowerCase(); + } catch (Exception e) { + throw new ApplicationException(e); + } + return translation; + } + + public static ArrayList<String> translate(String[] query, String fromLanguageStr, String[] toLanguagesStr) throws ApplicationException { + ArrayList<String> translations = new ArrayList<String>(); + for (int i=0; i<toLanguagesStr.length; i++) { + String toLanguageStr = toLanguagesStr[i]; + String[] translationsArray = translate(query, fromLanguageStr, toLanguageStr); + for (int j=0; j<translationsArray.length; j++) { + String translation = translationsArray[j]; + translations.add(translation); + } + } + return translations; + } + + public static String[] translate(String[] query, String fromLanguageStr, String toLanguageStr) throws ApplicationException { + if (fromLanguageStr == null) + throw new ApplicationException("MicrosoftTranslator: toLanguageStr is null"); + if (toLanguageStr == null) + throw new ApplicationException("MicrosoftTranslator: fromLanguageStr is null"); + String[] translation = null; + try { + Translate.setKey(KEY); + String fromLangId = de.mpg.mpiwg.berlin.mpdl.lt.general.Language.getInstance().getLanguageId(fromLanguageStr); // e.g. "de" is delivered from "deu" + String toLangId = de.mpg.mpiwg.berlin.mpdl.lt.general.Language.getInstance().getLanguageId(toLanguageStr); // e.g. "de" is delivered from "deu" + Language fromLanguage = Language.fromString(fromLangId); + Language toLanguage = Language.fromString(toLangId); + translation = Translate.execute(query, fromLanguage, toLanguage); + for (int i=0; i<translation.length; i++) { + translation[i] = translation[i].toLowerCase(); + } + } catch (Exception e) { + throw new ApplicationException(e); + } + return translation; + } + + public static String detectLanguageCode(String query) throws ApplicationException { + String langCode = null; + try { + Detect.setKey(KEY); + Language detectedLanguage = Detect.execute(query); + if (detectedLanguage != null) + langCode = detectedLanguage.toString(); + } catch (Exception e) { + throw new ApplicationException(e); + } + return langCode; + } + + public static String detectLanguageName(String query) throws ApplicationException { + String langName = null; + try { + Detect.setKey(KEY); + Language detectedLanguage = Detect.execute(query); + if (detectedLanguage != null) + langName = detectedLanguage.getName(Language.GERMAN); + } catch (Exception e) { + throw new ApplicationException(e); + } + return langName; + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-cms/src/de/mpg/mpiwg/berlin/mpdl/exception/ApplicationException.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,14 @@ +package de.mpg.mpiwg.berlin.mpdl.exception; + +public class ApplicationException extends Exception { + + public ApplicationException(Exception e) { + super(e); + } + + public ApplicationException(String str) { + super(str); + } + +} +
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/.classpath Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,18 +0,0 @@ -<?xml version="1.0" encoding="UTF-8"?> -<classpath> - <classpathentry kind="src" path="src"/> - <classpathentry kind="con" path="org.eclipse.jdt.launching.JRE_CONTAINER/org.eclipse.jdt.internal.launching.macosx.MacOSXType/1.6.0 (MacOS X Default)"> - <attributes> - <attribute name="owner.project.facets" value="java"/> - </attributes> - </classpathentry> - <classpathentry kind="con" path="org.eclipse.jst.server.core.container/org.eclipse.jst.server.tomcat.runtimeTarget/Apache Tomcat v6.0"> - <attributes> - <attribute name="owner.project.facets" value="jst.web"/> - </attributes> - </classpathentry> - <classpathentry kind="con" path="org.eclipse.jst.j2ee.internal.web.container"/> - <classpathentry kind="con" path="org.eclipse.jst.j2ee.internal.module.container"/> - <classpathentry combineaccessrules="false" kind="src" path="/mpiwg-mpdl-lt"/> - <classpathentry kind="output" path="build/classes"/> -</classpath>
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/.externalToolBuilders/Ant-Build.launch Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,12 +0,0 @@ -<?xml version="1.0" encoding="UTF-8" standalone="no"?> -<launchConfiguration type="org.eclipse.ant.AntBuilderLaunchConfigurationType"> -<booleanAttribute key="org.eclipse.ant.ui.ATTR_TARGETS_UPDATED" value="true"/> -<booleanAttribute key="org.eclipse.ant.ui.DEFAULT_VM_INSTALL" value="false"/> -<booleanAttribute key="org.eclipse.debug.ui.ATTR_LAUNCH_IN_BACKGROUND" value="false"/> -<stringAttribute key="org.eclipse.jdt.launching.CLASSPATH_PROVIDER" value="org.eclipse.ant.ui.AntClasspathProvider"/> -<booleanAttribute key="org.eclipse.jdt.launching.DEFAULT_CLASSPATH" value="true"/> -<stringAttribute key="org.eclipse.jdt.launching.PROJECT_ATTR" value="mpiwg-mpdl-xml-web"/> -<stringAttribute key="org.eclipse.ui.externaltools.ATTR_LOCATION" value="${workspace_loc:/mpiwg-mpdl-xml-web/build/build.xml}"/> -<stringAttribute key="org.eclipse.ui.externaltools.ATTR_RUN_BUILD_KINDS" value="full,incremental,"/> -<booleanAttribute key="org.eclipse.ui.externaltools.ATTR_TRIGGERS_CONFIGURED" value="true"/> -</launchConfiguration>
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/.project Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,46 +0,0 @@ -<?xml version="1.0" encoding="UTF-8"?> -<projectDescription> - <name>mpiwg-mpdl-lt-web</name> - <comment></comment> - <projects> - </projects> - <buildSpec> - <buildCommand> - <name>org.eclipse.wst.jsdt.core.javascriptValidator</name> - <arguments> - </arguments> - </buildCommand> - <buildCommand> - <name>org.eclipse.jdt.core.javabuilder</name> - <arguments> - </arguments> - </buildCommand> - <buildCommand> - <name>org.eclipse.wst.common.project.facet.core.builder</name> - <arguments> - </arguments> - </buildCommand> - <buildCommand> - <name>org.eclipse.wst.validation.validationbuilder</name> - <arguments> - </arguments> - </buildCommand> - <buildCommand> - <name>org.eclipse.ui.externaltools.ExternalToolBuilder</name> - <triggers>full,incremental,</triggers> - <arguments> - <dictionary> - <key>LaunchConfigHandle</key> - <value><project>/.externalToolBuilders/Ant-Build.launch</value> - </dictionary> - </arguments> - </buildCommand> - </buildSpec> - <natures> - <nature>org.eclipse.jem.workbench.JavaEMFNature</nature> - <nature>org.eclipse.wst.common.modulecore.ModuleCoreNature</nature> - <nature>org.eclipse.wst.common.project.facet.core.nature</nature> - <nature>org.eclipse.jdt.core.javanature</nature> - <nature>org.eclipse.wst.jsdt.core.jsNature</nature> - </natures> -</projectDescription>
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/.settings/.jsdtscope Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,12 +0,0 @@ -<?xml version="1.0" encoding="UTF-8"?> -<classpath> - <classpathentry kind="src" path="WebContent"/> - <classpathentry kind="con" path="org.eclipse.wst.jsdt.launching.JRE_CONTAINER"/> - <classpathentry kind="con" path="org.eclipse.wst.jsdt.launching.WebProject"> - <attributes> - <attribute name="hide" value="true"/> - </attributes> - </classpathentry> - <classpathentry kind="con" path="org.eclipse.wst.jsdt.launching.baseBrowserLibrary"/> - <classpathentry kind="output" path=""/> -</classpath>
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/.settings/org.eclipse.jdt.core.prefs Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,12 +0,0 @@ -#Mon Sep 12 15:41:45 CEST 2011 -eclipse.preferences.version=1 -org.eclipse.jdt.core.compiler.codegen.inlineJsrBytecode=enabled -org.eclipse.jdt.core.compiler.codegen.targetPlatform=1.6 -org.eclipse.jdt.core.compiler.codegen.unusedLocal=preserve -org.eclipse.jdt.core.compiler.compliance=1.6 -org.eclipse.jdt.core.compiler.debug.lineNumber=generate -org.eclipse.jdt.core.compiler.debug.localVariable=generate -org.eclipse.jdt.core.compiler.debug.sourceFile=generate -org.eclipse.jdt.core.compiler.problem.assertIdentifier=error -org.eclipse.jdt.core.compiler.problem.enumIdentifier=error -org.eclipse.jdt.core.compiler.source=1.6
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/.settings/org.eclipse.wst.common.component Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,10 +0,0 @@ -<?xml version="1.0" encoding="UTF-8"?> -<project-modules id="moduleCoreId" project-version="1.5.0"> - <wb-module deploy-name="mpiwg-mpdl-xml-web"> - <wb-resource deploy-path="/" source-path="/WebContent"/> - <wb-resource deploy-path="/WEB-INF/classes" source-path="/src"/> - <wb-resource deploy-path="/WEB-INF/lib" source-path="/WebContent/WEB-INF/lib"/> - <property name="java-output-path" value="/mpiwg-mpdl-xml-web/build/classes"/> - <property name="context-root" value="mpiwg-mpdl-xml-web"/> - </wb-module> -</project-modules>
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/.settings/org.eclipse.wst.common.project.facet.core.xml Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,10 +0,0 @@ -<?xml version="1.0" encoding="UTF-8"?> -<faceted-project> - <runtime name="Apache Tomcat v6.0"/> - <fixed facet="jst.web"/> - <fixed facet="wst.jsdt.web"/> - <fixed facet="java"/> - <installed facet="java" version="1.6"/> - <installed facet="jst.web" version="2.5"/> - <installed facet="wst.jsdt.web" version="1.0"/> -</faceted-project>
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/.settings/org.eclipse.wst.jsdt.ui.superType.container Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1 +0,0 @@ -org.eclipse.wst.jsdt.launching.baseBrowserLibrary \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/.settings/org.eclipse.wst.jsdt.ui.superType.name Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1 +0,0 @@ -Window \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/.settings/org.eclipse.wst.ws.service.policy.prefs Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,3 +0,0 @@ -#Mon Sep 12 15:41:45 CEST 2011 -eclipse.preferences.version=1 -org.eclipse.wst.ws.service.policy.projectEnabled=false
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/WebContent/META-INF/MANIFEST.MF Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,3 +0,0 @@ -Manifest-Version: 1.0 -Class-Path: -
Binary file software/mpdl-services/mpiwg-mpdl-lt-web/WebContent/WEB-INF/classes/.DS_Store has changed
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/WebContent/WEB-INF/classes/constants.properties Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1 +0,0 @@ -dataDir=/Users/jwillenborg/mpdl/data/lt \ No newline at end of file
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--- a/software/mpdl-services/mpiwg-mpdl-lt-web/WebContent/WEB-INF/web.xml Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,70 +0,0 @@ -<?xml version="1.0" encoding="UTF-8"?> -<web-app xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://java.sun.com/xml/ns/javaee" xmlns:web="http://java.sun.com/xml/ns/javaee/web-app_2_5.xsd" xsi:schemaLocation="http://java.sun.com/xml/ns/javaee http://java.sun.com/xml/ns/javaee/web-app_2_5.xsd" id="WebApp_ID" version="2.5"> - <display-name>mpiwg-mpdl-xml-web</display-name> - <welcome-file-list> - <welcome-file>index.html</welcome-file> - </welcome-file-list> - <servlet> - <description>GetDictionaryEntries</description> - <display-name>GetDictionaryEntries</display-name> - <servlet-name>GetDictionaryEntries</servlet-name> - <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.lt.GetDictionaryEntries</servlet-class> - </servlet> - <servlet-mapping> - <servlet-name>GetDictionaryEntries</servlet-name> - <url-pattern>/lt/GetDictionaryEntries</url-pattern> - </servlet-mapping> - <servlet> - <description>GetLemmas</description> - <display-name>GetLemmas</display-name> - <servlet-name>GetLemmas</servlet-name> - <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.lt.GetLemmas</servlet-class> - </servlet> - <servlet-mapping> - <servlet-name>GetLemmas</servlet-name> - <url-pattern>/lt/GetLemmas</url-pattern> - </servlet-mapping> - <servlet> - <description>GetForms</description> - <display-name>GetForms</display-name> - <servlet-name>GetForms</servlet-name> - <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.lt.GetForms</servlet-class> - </servlet> - <servlet-mapping> - <servlet-name>GetForms</servlet-name> - <url-pattern>/lt/GetForms</url-pattern> - </servlet-mapping> - <servlet> - <description>Tokenize</description> - <display-name>Tokenize</display-name> - <servlet-name>Tokenize</servlet-name> - <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.lt.Tokenize</servlet-class> - </servlet> - <servlet-mapping> - <servlet-name>Tokenize</servlet-name> - <url-pattern>/text/Tokenize</url-pattern> - </servlet-mapping> - <servlet> - <description>Normalize</description> - <display-name>Normalize</display-name> - <servlet-name>Normalize</servlet-name> - <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.lt.Normalize</servlet-class> - </servlet> - <servlet-mapping> - <servlet-name>Normalize</servlet-name> - <url-pattern>/text/Normalize</url-pattern> - </servlet-mapping> - <servlet> - <description>Transcode</description> - <display-name>Transcode</display-name> - <servlet-name>Transcode</servlet-name> - <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.lt.Transcode</servlet-class> - </servlet> - <servlet-mapping> - <servlet-name>Transcode</servlet-name> - <url-pattern>/text/Transcode</url-pattern> - </servlet-mapping> - <listener> - <listener-class>de.mpg.mpiwg.berlin.mpdl.servlets.lt.MpiwgMpdlLtWebServletContextListener</listener-class> - </listener> -</web-app> \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/WebContent/css/getDictionaryEntries.css Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,95 +0,0 @@ -.body { - margin:3px; - color: #000000; - background-color: #ebebeb; - min-width:1100px; -} -div.title { - font-size: 30px; - font-weight: bold; - margin-bottom: 10px; -} -input.query { - border-radius: 5px; -} -span.inputType { - font-size: 22px; - font-weight:bold; - display:block; - margin-top: 10px; -} - -.tree ul, -.tree li { - padding: 0; - margin: 0; - margin-left: 10px; - list-style: none; -} -.tree input { - position: absolute; - opacity: 0; -} -.tree input + label + ul { - margin: 0 0 0 22px; -} -.tree input + label + ul { - display: none; -} -.tree label, -.tree label::before { - cursor: pointer; -} -.tree input:disabled + label { - cursor: default; - opacity: .6; -} -.tree input:checked:not(:disabled) + label + ul { - display: block; -} -.tree label, -.tree label::before { - background: url("/mpiwg-mpdl-lt-web/images/treecheckbox.png") no-repeat; -} -.tree label, -.tree a, -.tree label::before { - display: inline-block; - height: 16px; - line-height: 16px;, - vertical-align: middle; -} -.tree label { - background-position: 18px 0; -} -.tree label::before { - content: ""; - width: 16px; - margin: 0 22px 0 0; - vertical-align: middle; - background-position: 0 -32px; -} -.tree input:checked + label::before { - background-position: 0 -16px; -} - -/* webkit adjacent element selector bugfix */ -@media screen and (-webkit-min-device-pixel-ratio:0) -{ - .tree - { - -webkit-animation: webkit-adjacent-element-selector-bugfix infinite 1s; - } - - @-webkit-keyframes webkit-adjacent-element-selector-bugfix - { - from - { - padding: 0; - } - to - { - padding: 0; - } - } -}
Binary file software/mpdl-services/mpiwg-mpdl-lt-web/WebContent/images/dictionaryMorph.gif has changed
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--- a/software/mpdl-services/mpiwg-mpdl-lt-web/WebContent/index.html Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,400 +0,0 @@ -<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0//EN"> -<html> -<head> -<meta http-equiv="Content-Type" content="text/html; charset=utf-8"> -<title>Max Planck Institute for the History of Science - Mpdl: Language technology services</title> -</head> -<body> -<table align="right"> -<tr> -<td> - [<i>This software is dedicated to <a href="http://mpdl-proto.mpiwg-berlin.mpg.de/mpdl/info.xql?info=malcolm">Dr. Malcolm Hyman</a></i>] - <img src="/mpiwg-mpdl-lt-web/images/info.png" width="15" height="15" border="0" alt="Info"/><br/> - [<i>It is based on <a href="http://archimedes.fas.harvard.edu/">Donatus and Pollux</a></i>] - <img src="/mpiwg-mpdl-lt-web/images/info.png" width="15" height="15" border="0" alt="Info"/> -</td> -</tr> -</table> -<h2>Max Planck Institute for the History of Science - Mpdl: Language technology services</h2> -<ul> - <li><b>Url: /mpiwg-mpdl-lt-web/lt/GetDictionaryEntries</b> - <ul> - <li>Request parameters - <ul> - <li>query (required) - <ul> - <li>by one form or lemma (e.g. "revolution")</li> - <li>by a list of forms or lemmas (e.g. "revolution equality brotherliness")</li> - <li>by a prefix range: entries starting with a prefix (e.g. "a*")</li> - </ul> - </li> - <li>queryDisplay (optional) - <ul> - <li>display of the query</li> - <li>default: content of parameter "query"</li> - </ul> - </li> - <li>inputType (optional) - <ul> - <li>"form"</li> - <li>"lemma"</li> - <li>default: "form"</li> - </ul> - </li> - <li>language (optional) - <ul> - <li>ISO 639-3 specifier</li> - <li>default: "eng"</li> - </ul> - </li> - <li>dictionary (optional) - <ul> - <li>dictionary name, e.g. "webster"</li> - <li>default: "all" (all dictionaries for the specified language)</li> - </ul> - </li> - <li>outputType (optional) - <ul> - <li>this parameter can occur many times (e.g. "outputType=morphCompact&outputType=dictCompact") - <ul> - <li>"morphCompact"</li> - <li>"dictCompact"</li> - <li>"wikiCompact"</li> - <li>"allCompact" (all output types compact)</li> - <li>"morphFull"</li> - <li>"dictFull"</li> - <li>"wikiFull"</li> - <li>"allFull" (all output types full)</li> - </ul> - </li> - <li>default: "allCompact"</li> - </ul> - </li> - <li>outputFormat (optional) - <ul> - <li>"html"</li> - <li>"xml"</li> - <li>default: "xml"</li> - </ul> - </li> - <li>normalization (optional) - <ul> - <li>"none"</li> - <li>"norm"</li> - <li>default: "norm"</li> - </ul> - </li> - <li>resultPageNumber (optional) - <ul> - <li>works only for range queries</li> - <li>page number of the result (e.g. "2": result entries from position 51 to 100)</li> - <li>default: "1"</li> - </ul> - </li> - <li>resultPageSize (optional) - <ul> - <li>works only for range queries</li> - <li>page size of the result (e.g. "100": each result page has a size of 100)</li> - <li>default: "50"</li> - </ul> - </li> - </ul> - </li> - <li>Response output - <ul> - <li>dependent of outputFormat, outputType and resultPage: morphology, dictionary and Wikipedia entries in Xml or Html format</li> - <li>Example: <a href="lt/GetDictionaryEntries?query=a*&language=lat&outputFormat=html">query=a*&language=lat&outputFormat=html</a></li> - <li>Example: <a href="lt/GetDictionaryEntries?query=a*&dictionary=ls">query=a*&dictionary=ls</a></li> - <li>Example: <a href="lt/GetDictionaryEntries?query=revolution&language=eng">query=revolution&language=lat</a></li> - <li>Example: <a href="lt/GetDictionaryEntries?query=multa&language=lat&outputFormat=html&outputType=allCompact">query=multa&language=lat&outputFormat=html&outputType=allCompact</a></li> - </ul> - </li> - </ul> - </li> - - <li><b>Url: /mpiwg-mpdl-lt-web/lt/GetLemmas</b> - <ul> - <li>Request parameters - <ul> - <li>query (required) - <ul> - <li>one form or lemma (e.g. "revolution") or</li> - <li>blank separated list of forms or lemmas (e.g. "revolution equality brotherliness")</li> - </ul> - </li> - <li>inputType (optional) - <ul> - <li>"form"</li> - <li>"lemma"</li> - <li>default: "form"</li> - </ul> - </li> - <li>language (optional) - <ul> - <li>ISO 639-3 specifier</li> - <li>default: "eng"</li> - </ul> - </li> - <li>outputType (optional) - <ul> - <li>"compact"</li> - <li>"full"</li> - <li>default: "compact"</li> - </ul> - </li> - <li>outputFormat (optional) - <ul> - <li>"html"</li> - <li>"xml"</li> - <li>"string" (lemma names separated by a blank)</li> - <li>default: "xml"</li> - </ul> - </li> - <li>normalization (optional) - <ul> - <li>"none"</li> - <li>"norm"</li> - <li>default: "norm"</li> - </ul> - </li> - </ul> - </li> - <li>Response output - <ul> - <li>dependent of outputFormat and outputType: lemma entries in Xml or Html or string format</li> - <li>Example: <a href="lt/GetLemmas?query=multa&language=lat&outputFormat=html">query=multa&language=lat&outputFormat=html</a></li> - </ul> - </li> - </ul> - </li> - - <li><b>Url: /mpiwg-mpdl-lt-web/lt/GetForms</b> - <ul> - <li>Request parameters - <ul> - <li>query (required) - <ul> - <li>one lemma (e.g. "revolution") or</li> - <li>blank separated list of forms (e.g. "revolution equality brotherliness")</li> - </ul> - </li> - <li>language (optional) - <ul> - <li>ISO 639-3 specifier</li> - <li>default: "eng"</li> - </ul> - </li> - <li>outputType (optional) - <ul> - <li>"compact"</li> - <li>"full"</li> - <li>default: "compact"</li> - </ul> - </li> - <li>outputFormat (optional) - <ul> - <li>"html"</li> - <li>"xml"</li> - <li>"string" (lemma names separated by a blank)</li> - <li>default: "xml"</li> - </ul> - </li> - <li>normalization (optional) - <ul> - <li>"none"</li> - <li>"norm"</li> - <li>default: "norm"</li> - </ul> - </li> - </ul> - </li> - <li>Response output - <ul> - <li>dependent of outputFormat and outputType: form entries in Xml or Html or string format</li> - <li>Example: <a href="lt/GetForms?query=edo sum&language=lat&outputFormat=string">query=edo sum&language=lat&outputFormat=string</a></li> - </ul> - </li> - </ul> - </li> - - <li><b>Url: /mpiwg-mpdl-lt-web/text/Tokenize</b> - <ul> - <li>Request parameters - <ul> - <li>inputString or srcUrl (required) - <ul> - <li>inputString - <ul> - <li>string which should be tokenized - <ul> - <li>unstructured text</li> - <li>XML fragment/document</li> - </ul> - </li> - </ul> - </li> - <li>srcUrl - <ul> - <li>source URL - <ul> - <li>unstructured text</li> - <li>XML fragment/document</li> - </ul> - </li> - </ul> - </li> - </ul> - </li> - <li>language (optional) - <ul> - <li>ISO 639-3 specifier</li> - <li>if input is XML and an element contains the attribute "xml:lang" this value is used for this element</li> - <li>default: "eng"</li> - </ul> - </li> - <li>normalization (optional) - <ul> - <li>"none"</li> - <li>"norm"</li> - <li>default: "norm"</li> - </ul> - </li> - <li>normalizationType (optional) - <ul> - <li>"dictionary"</li> - <li>"display"</li> - <li>default: "dictionary"</li> - </ul> - </li> - <li>dictionary (optional) - <ul> - <li>"yes"</li> - <li>"no"</li> - <li>default: "yes"</li> - </ul> - </li> - <li>stopElements (optional) - <ul> - <li>list of xml element names which should not be tokenized (e.g. "var")</li> - <li>default: empty list</li> - </ul> - </li> - <li>outputFormat (optional) - <ul> - <li>"xml"</li> - <li>"string"</li> - <li>default: "xml"</li> - </ul> - </li> - <li>outputOptions (optional) - <ul> - <li>output options separated with blanks (e.g. "withForms withLemmas") - <ul> - <li>"withForms"</li> - <li>"withLemmas"</li> - <li>default: empty list</li> - </ul> - </li> - </ul> - </li> - </ul> - </li> - <li>Response output - <ul> - <li>outputFormat=xml - <ul> - <li>tokenized inputString or document (enriched by element <w>) - <ul> - <li>Example: <s><w lang="deu" form="dies" forms="dies, dieser, dieses, diesen" lemmas="dieser">Dies</w> <w - lang="deu" form="ist" forms="bin, bist, ist, seid, sind, sein, war, warst, wart" lemmas="sein">ist</w> <w - lang="deu" form="ein" forms="ein, eines, einer" lemmas="ein">ein</w> <w lang="deu" form="satz" - forms="satz, sätze, satzes" lemmas="satz">Satz</w></s> - </li> - </ul> - </li> - </ul> - <li>outputFormat=string - <ul> - <li>word tokens of inputString or document (separated by Blank)</li> - </ul> - <li>Example: <a href="text/Tokenize?inputString=edo sum philoſophi&language=lat&outputFormat=xml">inputString=edo sum philoſophi&language=lat&outputFormat=xml</a></li> - <li>Example: <a href="text/Tokenize?srcUrl=http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/page-query-result.xql?document=/echo/la/Benedetti_1585.xml%26mode=pureXml%26pn=13&language=lat">srcUrl=http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/page-query-result.xql?document=/echo/la/Benedetti_1585.xml%26mode=pureXml%26pn=13&language=lat</a></li> - <li>Example: <a href="text/Tokenize?srcUrl=http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/page-query-result.xql?document=/echo/la/Benedetti_1585.xml%26mode=pureXml%26pn=13&language=lat&outputOptions=withForms withLemmas">srcUrl=http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/page-query-result.xql?document=/echo/la/Benedetti_1585.xml%26mode=pureXml%26pn=13&language=lat&outputOptions=withForms withLemmas</a></li> - </ul> - </li> - </ul> - </li> - - <li><b>Url: /mpiwg-mpdl-lt-web/text/Normalize</b> - <ul> - <li>Request parameters - <ul> - <li>inputString (required) - <ul> - <li>string which should be normalized</li> - </ul> - </li> - <li>language (optional) - <ul> - <li>ISO 639-3 specifier</li> - <li>default: "eng"</li> - </ul> - </li> - <li>type (optional) - <ul> - <li>"dictionary"</li> - <li>"display"</li> - <li>default: "display"</li> - </ul> - </li> - </ul> - </li> - <li>Response output - <ul> - <li>normalized string</li> - <li>Example: <a href="text/Normalize?inputString=philoſophi&language=lat">inputString=philoſophi&language=lat</a></li> - </ul> - </li> - </ul> - </li> - - <li><b>Url: /mpiwg-mpdl-lt-web/text/Transcode</b> - <ul> - <li>Request parameters - <ul> - <li>inputString (required) - <ul> - <li>string which should be transcoded</li> - </ul> - </li> - <li>srcEncoding (required) - <ul> - <li>"betacode"</li> - <li>"buckwalter"</li> - <li>"unicode"</li> - </ul> - </li> - <li>destEncoding (optional) - <ul> - <li>"betacode"</li> - <li>"buckwalter"</li> - <li>"unicode"</li> - <li>default: "unicode"</li> - </ul> - </li> - </ul> - </li> - <li>Response output - <ul> - <li>transcoded string</li> - <li>Example: <a href="text/Transcode?inputString=kai/&srcEncoding=betacode&destEncoding=unicode">inputString=kai/&srcEncoding=betacode&destEncoding=unicode</a></li> - </ul> - </li> - </ul> - </li> -</ul> - -</body> -</html> - -
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/build/build.xml Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,57 +0,0 @@ -<!DOCTYPE project> -<project name="mpiwg-mpdl-lt-web" default="dist" basedir="../"> - <description>mpiwg-mpdl-lt-web</description> - <!-- global properties --> - <property name="baseLibFile" location="../mpiwg-mpdl-lt/dist/mpiwg-mpdl-lt.jar"/> - <property name="src" location="src"/> - <property name="lib" location="WebContent/WEB-INF/lib"/> - <property name="libTomcat" location="/Applications/java/apache-tomcat-6.0.18/lib"/> - <property name="webappTomcat" location="/Applications/java/apache-tomcat-6.0.18/webapps"/> - <property name="build" location="build/classes"/> - <property name="dist" location="dist"/> - - <path id="classpath"> - <fileset dir="${lib}" includes="**/*.jar"/> - <fileset dir="${libTomcat}" includes="**/*.jar"/> - </path> - - <target name="init"> - <!-- Create time stamp --> - <tstamp/> - <mkdir dir="${build}"/> - <mkdir dir="${dist}"/> - <copy file="${baseLibFile}" todir="${lib}"/> - </target> - - <target name="compile" depends="init" description="compile"> - <javac srcdir="${src}" destdir="${build}" classpathref="classpath" includeantruntime="false"/> - </target> - - <target name="dist" depends="compile" description="generate the distribution"> - <delete file="WebContent/WEB-INF/classes/constants.properties"/> - <copy file="conf/constants.properties" tofile="WebContent/WEB-INF/classes/constants.properties"/> - <jar jarfile="${dist}/mpiwg-mpdl-lt-web.jar" basedir="${build}"/> - <copy file="dist/mpiwg-mpdl-lt-web.jar" todir="${lib}"/> - <war destfile="dist/mpiwg-mpdl-lt-web.war" webxml="WebContent/WEB-INF/web.xml"> - <fileset dir="WebContent"/> - <lib dir="WebContent/WEB-INF/lib"/> - </war> - <copy file="dist/mpiwg-mpdl-lt-web.war" todir="${webappTomcat}"/> - </target> - - <target name="dist-remote-mpdl-system" depends="compile" description="generate the distribution"> - <delete file="WebContent/WEB-INF/classes/constants.properties"/> - <copy file="conf/constants-mpdl-system.properties" tofile="WebContent/WEB-INF/classes/constants.properties"/> - <jar jarfile="dist-remote/mpiwg-mpdl-lt-web.jar" basedir="${build}"/> - <copy file="dist-remote/mpiwg-mpdl-lt-web.jar" todir="${lib}"/> - <war destfile="dist-remote/mpiwg-mpdl-lt-web.war" webxml="WebContent/WEB-INF/web.xml"> - <fileset dir="WebContent"/> - <lib dir="WebContent/WEB-INF/lib"/> - </war> - </target> - - <target name="clean" description="clean" > - <delete dir="${build}"/> - <delete file="${dist}/mpiwg-mpdl-lt-web.jar"/> - </target> -</project> \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/conf/constants-mpdl-system.properties Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1 +0,0 @@ -dataDir=/usr/local/tomcat-mpdl/mpdl-data/lt \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/conf/constants.properties Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1 +0,0 @@ -dataDir=/Users/jwillenborg/mpdl/data/lt \ No newline at end of file
Binary file software/mpdl-services/mpiwg-mpdl-lt-web/src/de/mpg/mpiwg/berlin/mpdl/.DS_Store has changed
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/src/de/mpg/mpiwg/berlin/mpdl/exception/ApplicationException.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,14 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.exception; - -public class ApplicationException extends Exception { - - public ApplicationException(Exception e) { - super(e); - } - - public ApplicationException(String str) { - super(str); - } - -} -
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/lt/GetDictionaryEntries.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,485 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.servlets.lt; - -import java.io.IOException; -import java.io.PrintWriter; -import java.util.ArrayList; -import java.util.Arrays; -import java.util.Collections; -import java.util.Date; - -import javax.servlet.ServletConfig; -import javax.servlet.ServletException; -import javax.servlet.http.HttpServlet; -import javax.servlet.http.HttpServletRequest; -import javax.servlet.http.HttpServletResponse; - -import org.apache.commons.lang3.StringEscapeUtils; - -import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; -import de.mpg.mpiwg.berlin.mpdl.lt.dict.app.Lexica; -import de.mpg.mpiwg.berlin.mpdl.lt.dict.app.Lexicon; -import de.mpg.mpiwg.berlin.mpdl.lt.dict.app.LexiconEntry; -import de.mpg.mpiwg.berlin.mpdl.lt.dict.db.LexHandler; -import de.mpg.mpiwg.berlin.mpdl.lt.general.Language; -import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.Form; -import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.Lemma; -import de.mpg.mpiwg.berlin.mpdl.lt.text.norm.Normalizer; -import de.mpg.mpiwg.berlin.mpdl.servlets.util.ServletUtil; - -public class GetDictionaryEntries extends HttpServlet { - private static final long serialVersionUID = 1L; - private LexHandler lexHandler; - - public GetDictionaryEntries() { - super(); - } - - public void init(ServletConfig config) throws ServletException { - super.init(config); - try { - lexHandler = LexHandler.getInstance(); - } catch (ApplicationException e) { - throw new ServletException(e); - } - } - - protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { - Date begin = new Date(); - request.setCharacterEncoding("utf-8"); - response.setCharacterEncoding("utf-8"); - String query = request.getParameter("query"); - String queryDisplay = request.getParameter("queryDisplay"); - String language = request.getParameter("language"); - String inputType = request.getParameter("inputType"); - String outputFormat = request.getParameter("outputFormat"); - String[] outputTypesArray = request.getParameterValues("outputType"); - String dictionary = request.getParameter("dictionary"); - String normalization = request.getParameter("normalization"); - String normalizationType = request.getParameter("normalizationType"); - String resultPageNumber = request.getParameter("resultPageNumber"); - String resultPageSize = request.getParameter("resultPageSize"); - if (query == null) - query = "a*"; - boolean isRangeQuery = false; - if (query.endsWith("*")) - isRangeQuery = true; - if (queryDisplay == null) - queryDisplay = query; - if (language == null) - language = "eng"; - if (inputType == null || ! (inputType.equals("form") || inputType.equals("lemma"))) - inputType = "form"; - if (outputFormat == null || ! (outputFormat.equals("xml") || outputFormat.equals("html") || outputFormat.equals("htmlFragment"))) - outputFormat = "xml"; - ArrayList<String> outputTypes = new ArrayList<String>(); - if (outputTypesArray == null) { - outputTypes.add("allCompact"); - } else if (isRangeQuery) { - outputTypes.add("dictCompact"); - } else { - outputTypes = new ArrayList<String>(Arrays.asList(outputTypesArray)); - } - if (normalization == null || ! (normalization.equals("none") || normalization.equals("reg") || normalization.equals("reg norm"))) - normalization = "norm"; - if (normalizationType == null || ! (normalizationType.equals("display") || normalizationType.equals("dictionary"))) - normalizationType = "dictionary"; - String xmlDict = "all"; - if (dictionary != null) - xmlDict = dictionary; - int pageNumber = 1; - int pageSize = 50; - if (resultPageNumber != null) - pageNumber = new Integer(resultPageNumber); - if (resultPageSize != null) - pageSize = new Integer(resultPageSize); - String xmlQueryString = "<query>"; - xmlQueryString = xmlQueryString + "<name>" + query + "</name>"; - xmlQueryString = xmlQueryString + "<display>" + queryDisplay + "</display>"; - xmlQueryString = xmlQueryString + "<language>" + language + "</language>"; - xmlQueryString = xmlQueryString + "<inputType>" + inputType + "</inputType>"; - xmlQueryString = xmlQueryString + "<outputFormat>" + outputFormat + "</outputFormat>"; - xmlQueryString = xmlQueryString + "<outputTypes>" + outputTypes + "</outputTypes>"; - xmlQueryString = xmlQueryString + "<dictionary>" + xmlDict + "</dictionary>"; - xmlQueryString = xmlQueryString + "<normalization>" + normalization + "</normalization>"; - xmlQueryString = xmlQueryString + "<normalizationType>" + normalizationType + "</normalizationType>"; - if (isRangeQuery) { - xmlQueryString = xmlQueryString + "<resultPageNumber>" + pageNumber + "</resultPageNumber>"; - xmlQueryString = xmlQueryString + "<resultPageSize>" + pageSize + "</resultPageSize>"; - } - xmlQueryString = xmlQueryString + "</query>"; - try { - if (outputFormat.equals("xml")) - response.setContentType("text/xml"); - else if (outputFormat.equals("html") || outputFormat.equals("htmlFragment")) - response.setContentType("text/html"); - else - response.setContentType("text/xml"); - PrintWriter out = response.getWriter(); - if (query == null || query.isEmpty()) { - out.print("request parameter \"query\" is empty. Please specify a query."); - out.close(); - return; - } - ArrayList<Lemma> lemmas = null; - ArrayList<Lexicon> dictionaries = null; - if (isRangeQuery) { - String queryTmp = query.substring(0, query.length() - 1); // without last star - if (dictionary != null) - dictionaries = lexHandler.getLexEntriesByLexiconBeginningWith(dictionary, queryTmp, pageNumber, pageSize); - else - dictionaries = lexHandler.getLexEntriesBeginningWith(language, queryTmp, pageNumber, pageSize); - } else { - String lang = language; - if (dictionary != null) { - Lexicon lexicon = Lexica.getInstance().getLexicon(dictionary); - if (lexicon != null) - lang = lexicon.getSourceLanguage(); - } - int normMode = Normalizer.DICTIONARY; - if (normalization.equals("none")) - normMode = Normalizer.NONE; - else if (normalizationType.equals("display")) - normMode = Normalizer.DISPLAY; - lemmas = lexHandler.getLemmas(query, inputType, lang, normMode); - dictionaries = lexHandler.getLexEntries(lemmas, lang, dictionary, query); - } - String baseUrl = ServletUtil.getInstance().getBaseUrl(request); - Date end = new Date(); - String elapsedTime = String.valueOf(end.getTime() - begin.getTime()); - String result = ""; - if (outputFormat == null || outputFormat.equals("xml")) - result = createXmlOutputString(query, lemmas, dictionaries, outputTypes, baseUrl, xmlQueryString, elapsedTime); - else if (outputFormat.equals("html") || outputFormat.equals("htmlFragment")) - result = createHtmlOutputString(query, queryDisplay, language, lemmas, dictionaries, pageNumber, pageSize, isRangeQuery, outputFormat, outputTypes, elapsedTime); - else - result = createXmlOutputString(query, lemmas, dictionaries, outputTypes, baseUrl, xmlQueryString, elapsedTime); - out.print(result); - out.close(); - } catch (ApplicationException e) { - throw new ServletException(e); - } - } - - protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { - - } - - private String createXmlOutputString(String query, ArrayList<Lemma> lemmas, ArrayList<Lexicon> lexicons, ArrayList<String> outputTypes, String baseUrl, String xmlQueryString, String elapsedTime) { - boolean outputTypeMorphCompact = false; - if (outputTypes.contains("allCompact") || outputTypes.contains("morphCompact")) - outputTypeMorphCompact = true; - boolean outputTypeMorphFull = false; - if (outputTypes.contains("allFull") || outputTypes.contains("morphFull")) - outputTypeMorphFull = true; - boolean outputTypeDictCompact = false; - if (outputTypes.contains("allCompact") || outputTypes.contains("dictCompact")) - outputTypeDictCompact = true; - boolean outputTypeDictFull = false; - if (outputTypes.contains("allFull") || outputTypes.contains("dictFull")) - outputTypeDictFull = true; - boolean outputTypeWikiCompact = false; - if (outputTypes.contains("allCompact") || outputTypes.contains("wikiCompact")) - outputTypeWikiCompact = true; - boolean outputTypeWikiFull = false; - if (outputTypes.contains("allFull") || outputTypes.contains("wikiFull")) - outputTypeWikiFull = true; - String result = "<result>"; - result = result + "<provider>" + "MPIWG MPDL language technology service (see: " + "" + baseUrl + "), Max Planck Institute for the History of Science, Berlin." + "</provider>"; - result = result + xmlQueryString; - result = result + "<elapsed-time-ms>" + elapsedTime + "</elapsed-time-ms>"; - if ((outputTypeMorphCompact || outputTypeMorphFull) && lemmas != null && ! lemmas.isEmpty()) { - result = result + "<morphology>"; - for (int i=0; i<lemmas.size(); i++) { - Lemma lemma = lemmas.get(i); - String lemmaName = lemma.getLemmaName(); - String language = lemma.getLanguage(); - result = result + "<lemma>"; - result = result + "<name>" + lemmaName + "</name>"; - if (outputTypeMorphFull) { - String lemmaProvider = lemma.getProvider(); - result = result + "<provider>" + lemmaProvider + "</provider>"; - result = result + "<language>" + language + "</language>"; - } - if (Language.getInstance().isArabic(language) || Language.getInstance().isLatin(language)) { - String remoteUrl = "http://www.perseus.tufts.edu/hopper/morph?l=" + lemmaName + "&la=" + language; - result = result + "<remoteUrl>" + remoteUrl + "</remoteUrl>"; - } else if (Language.getInstance().isGreek(language)) { - String remoteUrl = "http://www.perseus.tufts.edu/hopper/morph?l=" + lemmaName + "&la=" + "greek"; - result = result + "<remoteUrl>" + remoteUrl + "</remoteUrl>"; - } - if (outputTypeMorphFull) { - ArrayList<Form> forms = lemma.getFormsList(); - Collections.sort(forms); - if (forms != null && ! forms.isEmpty()) { - result = result + "<forms>"; - for (int j=0; j<forms.size(); j++) { - result = result + "<form>"; - Form f = forms.get(j); - String formName = f.getFormName(); - String formProvider = f.getProvider(); - result = result + "<provider>" + formProvider + "</provider>"; - result = result + "<language>" + language + "</language>"; - result = result + "<name>" + formName + "</name>"; - result = result + "</form>"; - } - result = result + "</forms>"; - } - } - result = result + "</lemma>"; - } - result = result + "</morphology>"; - } - if ((outputTypeDictCompact || outputTypeDictFull) && lexicons != null) { - result = result + "<dictionaries>"; - for (int i=0; i<lexicons.size(); i++) { - Lexicon lexicon = lexicons.get(i); - if (outputTypeDictFull) - result = result + lexicon.toXmlString(); - else if (outputTypeDictCompact) - result = result + lexicon.toXmlStringCompact(); - } - result = result + "</dictionaries>"; - } - if ((outputTypeWikiCompact || outputTypeWikiFull) && lemmas != null && ! lemmas.isEmpty()) { - result = result + "<wikipedia>"; - String language = null; - for (int i=0; i<lemmas.size(); i++) { - Lemma lemma = lemmas.get(i); - String lemmaName = lemma.getLemmaName(); - language = Language.getInstance().getLanguageId(lemma.getLanguage()); - result = result + "<article>"; - result = result + "<name>" + lemmaName + "</name>"; - String wikiHrefExact = "http://" + language + ".wikipedia.org/wiki/" + lemmaName; - String wikiHrefSearch = "http://" + language + ".wikipedia.org/wiki/index.php?search=" + lemmaName; - result = result + "<remoteUrl>" + wikiHrefExact + "</remoteUrl>"; - result = result + "<remoteUrlSearch>" + wikiHrefSearch + "</remoteUrlSearch>"; - result = result + "</article>"; - } - if (language != null && Language.getInstance().isGerman(language) && query != null) { - String[] queryFormNames = query.split(" "); - for (int j=0; j<queryFormNames.length; j++) { - String queryFormName = queryFormNames[j]; - result = result + "<article>"; - result = result + "<name>" + queryFormName + "</name>"; - String wikiHrefExact = "http://" + language + ".wikipedia.org/wiki/" + queryFormName; - String wikiHrefSearch = "http://" + language + ".wikipedia.org/wiki/index.php?search=" + queryFormName; - result = result + "<remoteUrl>" + wikiHrefExact + "</remoteUrl>"; - result = result + "<remoteUrlSearch>" + wikiHrefSearch + "</remoteUrlSearch>"; - result = result + "</article>"; - } - } - result = result + "</wikipedia>"; - } - result = result + "</result>"; - return result; - } - - private String createHtmlOutputString(String query, String queryDisplay, String language, ArrayList<Lemma> lemmas, ArrayList<Lexicon> lexicons, int pageNumber, int pageSize, boolean isRangeQuery, String outputFormat, ArrayList<String> outputTypes, String elapsedTime) { - boolean outputTypeMorphCompact = false; - if (outputTypes.contains("allCompact") || outputTypes.contains("morphCompact")) - outputTypeMorphCompact = true; - boolean outputTypeMorphFull = false; - if (outputTypes.contains("allFull") || outputTypes.contains("morphFull")) - outputTypeMorphFull = true; - boolean outputTypeDictCompact = false; - if (outputTypes.contains("allCompact") || outputTypes.contains("dictCompact")) - outputTypeDictCompact = true; - boolean outputTypeDictFull = false; - if (outputTypes.contains("allFull") || outputTypes.contains("dictFull")) - outputTypeDictFull = true; - boolean outputTypeWikiCompact = false; - if (outputTypes.contains("allCompact") || outputTypes.contains("wikiCompact")) - outputTypeWikiCompact = true; - boolean outputTypeWikiFull = false; - if (outputTypes.contains("allFull") || outputTypes.contains("wikiFull")) - outputTypeWikiFull = true; - String result = ""; - result = result + "<html>"; - result = result + "<head>"; - result = result + "<title>WordInfo for: \"" + queryDisplay + "\"</title>"; - result = result + "<link rel=\"stylesheet\" type=\"text/css\" href=\"/mpiwg-mpdl-lt-web/css/getDictionaryEntries.css\"/>"; - result = result + getJavascriptFunctions(); - result = result + "</head>"; - result = result + "<body class=\"body\">"; - result = result + "<table align=\"right\" valign=\"top\">"; - result = result + "<td>[<i>This is a MPIWG MPDL language technology service</i>] <a href=\"/mpiwg-mpdl-lt-web/index.html\"><img src=\"/mpiwg-mpdl-lt-web/images/info.png\" valign=\"bottom\" width=\"15\" height=\"15\" border=\"0\" alt=\"MPIWG MPDL language technology service\"/></a></td>"; - result = result + "</table>"; - result = result + "<p/>"; - result = result + "<div class=\"title\">WordInfo</div>"; - - result = result + "<form name=\"getDictionaryEntries\" action=\"GetDictionaryEntries\" method=\"get\">"; - result = result + "Query: <input name=\"query\" class=\"query\" type=\"text\" size=\"20\" value=\"" + query + "\"/>"; - String htmlSelectBox = Language.getInstance().getHtmlSelectBox(language); - result = result + " Language: " + htmlSelectBox; - result = result + "<input type=\"hidden\" name=\"outputFormat\" value=\"" + outputFormat + "\"/>"; - for (int i=0; i<outputTypes.size(); i++) { - String type = outputTypes.get(i); - result = result + "<input type=\"hidden\" name=\"outputType\" value=\"" + type + "\"/>"; - } - result = result + " <button type=\"submit\" onclick=\"var query = document.getDictionaryEntries.query.value; if(query.substring(query.length-1, query.length) == '*') {document.getDictionaryEntries.outputType.value = 'dictCompact';}\">Query</button>"; - result = result + "</form>"; - result = result + "<hr/>"; - - if ((outputTypeMorphCompact || outputTypeMorphFull) && lemmas != null && ! lemmas.isEmpty()) { - result = result + "<span class=\"inputType\">Morphology</span>"; - result = result + "<span class=\"tree\">"; - result = result + "<ul>"; - result = result + "<li>" + "<b>" + "Lemmata" + "</b>"; - result = result + "<ul>"; - for (int i=0; i<lemmas.size(); i++) { - Lemma lemma = lemmas.get(i); - String lemmaName = lemma.getLemmaName(); - String providerText = ""; - String externalLinkText = ""; - String formsText = ""; - if (outputTypeMorphFull) { - String lemmaProvider = lemma.getProvider(); - providerText = " (data provider: " + lemmaProvider + ")"; - } - if (Language.getInstance().isArabic(language) || Language.getInstance().isLatin(language)) - externalLinkText = " (external link: <a href=\"http://www.perseus.tufts.edu/hopper/morph?l=" + lemmaName + "&la=" + language + "\">" + lemmaName + "</a>)"; - else if (Language.getInstance().isGreek(language)) - externalLinkText = " (external link: <a href=\"http://www.perseus.tufts.edu/hopper/morph?l=" + lemmaName + "&la=" + "greek" + "\">" + lemmaName + "</a>)"; - ArrayList<Form> forms = lemma.getFormsList(); - Collections.sort(forms); - if (forms != null && ! forms.isEmpty()) { - formsText = formsText + "<ul>"; - formsText = formsText + "<b>" + "Forms: " + "</b>"; - for (int j=0; j<forms.size(); j++) { - Form f = forms.get(j); - String formName = f.getFormName(); - String formProvider = f.getProvider(); - formsText = formsText + formName + " (data provider: " + formProvider + "), "; - } - formsText = formsText.substring(0, formsText.length() - 2); // without last comma and blank - formsText = formsText + "</ul>"; - } - if (outputTypeMorphFull) { - result = result + "<li><input type=\"checkbox\" checked=\"checked\"/><label>" + lemmaName + providerText + externalLinkText + "</label>" + formsText + "</li>"; - } else if (outputTypeMorphCompact) { - result = result + "<li><input type=\"checkbox\"/><label>" + "<a href=\"GetDictionaryEntries?query=" + lemmaName + "&language=" + language + "&inputType=lemma" + "&outputFormat=html" + "&outputType=morphFull" + "\">" + lemmaName + "</a>" + providerText + externalLinkText + "</label>" + formsText + "</li>"; - } - } - result = result + "</ul>"; - result = result + "</li>"; - result = result + "</ul>"; - result = result + "</span>"; - } - if ((outputTypeDictCompact || outputTypeDictFull) && lexicons != null && ! lexicons.isEmpty()) { - if (isRangeQuery) { - int pageNumberUp = 1; - if (pageNumber > 1) - pageNumberUp = pageNumber - 1; - int pageNumberDown = pageNumber + 1; - int from = (pageNumber * pageSize) - pageSize + 1; - int to = pageNumber * pageSize; - result = result + "<table align=\"right\">"; - result = result + "<tr>"; - result = result + "<td valign=\"middle\">Page: </td>"; - result = result + "<td valign=\"middle\"><a href=\"/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries?query=" + query + "&language=" + language + "&resultPageNumber=" + pageNumberUp + "&outputFormat=html\"><img alt=\"page up\" src=\"/mpiwg-mpdl-lt-web/images/left.gif\" height=\"20\" width=\"20\"></a></td>"; - result = result + "<td valign=\"middle\">" + pageNumber + "</td>"; - result = result + "<td valign=\"middle\"><a href=\"/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries?query=" + query + "&language=" + language + "&resultPageNumber=" + pageNumberDown + "&outputFormat=html\"><img alt=\"page down\" src=\"/mpiwg-mpdl-lt-web/images/right.gif\" height=\"20\" width=\"20\"></a></td>"; - result = result + "<td valign=\"middle\">(" + from + " - " + to + ")</td>"; - result = result + "</tr>"; - result = result + "</table>"; - } - result = result + "<span class=\"inputType\">Dictionary</span>"; - result = result + "<span class=\"tree\">"; - result = result + "<ul>"; - for (int i=0; i<lexicons.size(); i++) { - Lexicon lexicon = lexicons.get(i); - result = result + "<li>"; - result = result + "<b>" + lexicon.getDescription() + "</b>"; - result = result + "<ul>"; - ArrayList<LexiconEntry> entries = lexicon.getEntries(); - for (int j=0; j<entries.size(); j++) { - String entryContent = ""; - LexiconEntry entry = entries.get(j); - if (lexicon.isLocalLexicon()) { - if (entry.isXmlValid()) { - String repairedEntry = entry.getRepairedEntry(); - repairedEntry = repairedEntry.replaceAll("<repaired-entry>", ""); - repairedEntry = repairedEntry.replaceAll("</repaired-entry>", ""); - entryContent = entryContent + repairedEntry; // valid unicode content of the original entry - } else { - entryContent = entryContent + "[Remark: <i> this dictionary entry has no valid XML/HTML content in database so a text version of this entry is shown.</i>]: <br/>"; - String originalEntry = entry.getOriginalEntry(); - originalEntry = originalEntry.replaceAll("<original-entry>", ""); - originalEntry = originalEntry.replaceAll("</original-entry>", ""); - originalEntry = StringEscapeUtils.escapeXml(originalEntry); // create text version of the invalid xml content - entryContent = entryContent + originalEntry; - } - if (entry.getRemoteUrl() != null) { - entryContent = entryContent + "<div>(external link: <a href=\"" + entry.getRemoteUrl() + "\">" + entry.getFormName() + "</a>)</div>"; - } - } else { - if (entry.getRemoteUrl() != null) { - entryContent = entryContent + "external link: <a href=\"" + entry.getRemoteUrl() + "\">" + entry.getFormName() + "</a>"; - } - } - String formName = entry.getFormName(); - String dictName = lexicon.getName(); - if (outputTypeDictFull) { - result = result + "<li><input type=\"checkbox\" checked=\"checked\"/><label><b>" + formName + "</b></label><ul><li>" + entryContent + "</li></ul></li>"; - } else if (outputTypeDictCompact) { - result = result + "<li><input type=\"checkbox\"/><label>" + "<a href=\"GetDictionaryEntries?query=" + formName + "&language=" + language + "&dictionary=" + dictName + "&inputType=lemma" + "&outputFormat=html" + "&outputType=dictFull" + "\">" + formName + "</a>" + "</label><ul><li>" + entryContent + "</li></ul></li>"; - } - } - result = result + "</ul>"; - result = result + "</li>"; // lexicon entry - } - result = result + "</ul>"; - result = result + "</span>"; - } - if ((outputTypeWikiCompact || outputTypeWikiFull) && lemmas != null && ! lemmas.isEmpty()) { - String langId = Language.getInstance().getLanguageId(language); - String wikiUrl = langId + ".wikipedia.org"; - result = result + "<span class=\"inputType\">Wikipedia</span>"; - result = result + "<span class=\"tree\">"; - result = result + "<ul>"; - result = result + "<li>" + "<b>" + wikiUrl + "</b>"; - result = result + "<ul>"; - for (int i=0; i<lemmas.size(); i++) { - Lemma lemma = lemmas.get(i); - String lemmaName = lemma.getLemmaName(); - String wikiHrefExact = "http://" + wikiUrl + "/wiki/" + lemmaName; - String wikiHrefSearch = "http://" + wikiUrl + "/wiki/index.php?search=" + lemmaName; - result = result + "<li><input type=\"checkbox\" checked=\"checked\"/><label>" + "External link: <a href=\"" + wikiHrefExact + "\">" + lemmaName + "</a> (or search for <a href=\"" + wikiHrefSearch + "\">" + lemmaName + "</a>)" + "</label>" + "</li>"; - } - if (language != null && Language.getInstance().isGerman(language) && query != null) { - String[] queryFormNames = query.split(" "); - for (int j=0; j<queryFormNames.length; j++) { - String queryFormName = queryFormNames[j]; - result = result + "<li>"; - String wikiHrefExact = "http://" + wikiUrl + "/wiki/" + queryFormName; - String wikiHrefSearch = "http://" + wikiUrl + "/wiki/index.php?search=" + queryFormName; - result = result + "<li><input type=\"checkbox\" checked=\"checked\"/><label>" + "External link: <a href=\"" + wikiHrefExact + "\">" + queryFormName + "</a> (or search for <a href=\"" + wikiHrefSearch + "\">" + queryFormName + "</a>)" + "</label>" + "</li>"; - result = result + "</li>"; - } - } - result = result + "</ul>"; - result = result + "</li>"; - result = result + "</ul>"; - result = result + "</span>"; - } - result = result + "<p/>"; - result = result + "[* external links may not function]"; - result = result + "<hr/>"; - result = result + "<p/>"; - result = result + "Elapsed time: " + elapsedTime + " ms, see the <a href=\"/mpiwg-mpdl-lt-web/index.html\">service description</a> of this page, if you find a bug <a href=\"https://it-dev.mpiwg-berlin.mpg.de/tracs/mpdl-project-software/newticket\">let us know</a>"; - result = result + "</body>"; - result = result + "</html>"; - return result; - } - - private String getJavascriptFunctions() { - String result = "<script type=\"text/javascript\">"; - result = result + "<!-- "; - result = result + "function strEndsWith(str, suffix) {"; - result = result + " return str.match(suffix + \"$\") == suffix;"; - result = result + "}"; - result = result + " -->"; - result = result + "</script>"; - return result; - } -}
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/lt/GetForms.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,214 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.servlets.lt; - -import java.io.IOException; -import java.io.PrintWriter; -import java.util.ArrayList; -import java.util.Collections; -import java.util.Date; -import java.util.Hashtable; - -import javax.servlet.ServletConfig; -import javax.servlet.ServletException; -import javax.servlet.http.HttpServlet; -import javax.servlet.http.HttpServletRequest; -import javax.servlet.http.HttpServletResponse; - -import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; -import de.mpg.mpiwg.berlin.mpdl.lt.dict.db.LexHandler; -import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.Form; -import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.Lemma; -import de.mpg.mpiwg.berlin.mpdl.lt.text.norm.Normalizer; - -public class GetForms extends HttpServlet { - private static final long serialVersionUID = 1L; - private LexHandler lexHandler; - - public GetForms() { - super(); - } - - public void init(ServletConfig config) throws ServletException { - super.init(config); - try { - lexHandler = LexHandler.getInstance(); - } catch (ApplicationException e) { - throw new ServletException(e); - } - } - - protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { - Date begin = new Date(); - request.setCharacterEncoding("utf-8"); - response.setCharacterEncoding("utf-8"); - String query = request.getParameter("query"); - String language = request.getParameter("language"); - String outputFormat = request.getParameter("outputFormat"); - String outputType = request.getParameter("outputType"); - String normalization = request.getParameter("normalization"); - if (language == null) - language = "eng"; - if (outputFormat == null || ! (outputFormat.equals("xml") || outputFormat.equals("html") || outputFormat.equals("string"))) - outputFormat = "xml"; - if (outputType == null || ! (outputType.equals("compact") || outputType.equals("full"))) - outputType = "compact"; - if (normalization == null || ! (normalization.equals("none") || normalization.equals("reg") || normalization.equals("reg norm"))) - normalization = "norm"; - int normalizationType = Normalizer.DICTIONARY; - if (normalization != null && normalization.equals("none")) - normalizationType = Normalizer.NONE; - - String xmlQueryString = "<query><name>" + query + "</name>" + "<language>" + language + "</language>" + - "<outputFormat>" + outputFormat + "</outputFormat>" + "<outputType>" + outputType + "</outputType>" + "<normalization>" + normalization + "</normalization>" + "</query>"; - try { - if (outputFormat.equals("xml")) - response.setContentType("text/xml"); - else if (outputFormat.equals("html") || outputFormat.equals("string")) - response.setContentType("text/html"); - else - response.setContentType("text/xml"); - PrintWriter out = response.getWriter(); - if (query == null || query.isEmpty()) { - out.print("request parameter query is empty. Please specify a query."); - out.close(); - return; - } - ArrayList<Lemma> lemmas = lexHandler.getLemmas(query, "lemma", language, normalizationType); - Hashtable<String, Form> formsHashtable = new Hashtable<String, Form>(); - ArrayList<Form> forms = new ArrayList<Form>(); - if (lemmas != null && ! lemmas.isEmpty()) { - for (int i=0; i<lemmas.size(); i++) { - Lemma lemma = lemmas.get(i); - ArrayList<Form> lemmaForms = lemma.getFormsList(); - for (int j=0; j < lemmaForms.size(); j++) { - Form form = lemmaForms.get(j); - formsHashtable.put(form.getFormName(), form); - } - } - } - forms.addAll(formsHashtable.values()); - Collections.sort(forms); - String baseUrl = getBaseUrl(request); - Date end = new Date(); - String elapsedTime = String.valueOf(end.getTime() - begin.getTime()); - String result = ""; - if (outputFormat == null || outputFormat.equals("xml")) - result = createXmlOutputString(query, forms, outputType, baseUrl, xmlQueryString, elapsedTime); - else if (outputFormat.equals("html")) - result = createHtmlOutputString(query, forms, outputType, elapsedTime); - else if (outputFormat.equals("string")) - result = createStringOutputString(forms); - else - result = createXmlOutputString(query, forms, outputType, baseUrl, xmlQueryString, elapsedTime); - out.print(result); - out.close(); - } catch (ApplicationException e) { - throw new ServletException(e); - } - } - - protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { - - } - - private String getBaseUrl( HttpServletRequest request ) { - if (request.getServerPort() == 80 || request.getServerPort() == 443) - return request.getScheme() + "://" + request.getServerName() + request.getContextPath(); - else - return request.getScheme() + "://" + request.getServerName() + ":" + request.getServerPort() + request.getContextPath(); - } - - - private String createXmlOutputString(String query, ArrayList<Form> forms, String outputType, String baseUrl, String xmlQueryString, String elapsedTime) { - String result = "<result>"; - result = result + "<provider>" + "MPIWG MPDL language technology service (see: " + "" + baseUrl + "), Max Planck Institute for the History of Science, Berlin." + "</provider>"; - result = result + xmlQueryString; - result = result + "<elapsed-time-ms>" + elapsedTime + "</elapsed-time-ms>"; - if (forms != null && ! forms.isEmpty()) { - result = result + "<morphology>"; - result = result + "<forms>"; - for (int i=0; i<forms.size(); i++) { - result = result + "<form>"; - Form f = forms.get(i); - String formName = f.getFormName(); - String language = f.getLanguage(); - String formProvider = f.getProvider(); - String lemmaName = f.getLemmaName(); - result = result + "<provider>" + formProvider + "</provider>"; - result = result + "<language>" + language + "</language>"; - result = result + "<lemmaName>" + lemmaName + "</lemmaName>"; - result = result + "<formName>" + formName + "</formName>"; - result = result + "</form>"; - } - result = result + "</forms>"; - result = result + "</morphology>"; - } - result = result + "</result>"; - return result; - } - - private String createHtmlOutputString(String query, ArrayList<Form> forms, String outputType, String elapsedTime) { - String result = ""; - result = result + "<html>"; - result = result + "<head>"; - result = result + "<title>Lemmas for: \"" + query + "\"</title>"; - result = result + "</head>"; - result = result + "<body>"; - result = result + "<table align=\"right\" valign=\"top\">"; - result = result + "<td>[<i>This is a MPIWG MPDL language technology service</i>] <a href=\"/mpiwg-mpdl-lt-web/index.html\"><img src=\"/mpiwg-mpdl-lt-web/images/info.png\" valign=\"bottom\" width=\"15\" height=\"15\" border=\"0\" alt=\"MPIWG MPDL language technology service\"/></a></td>"; - result = result + "</table>"; - result = result + "<p/>"; - result = result + "<h1>Forms for: \"" + query + "\"</h1>"; - if (forms != null && ! forms.isEmpty()) { - result = result + "<h3>Morphology</h3>"; - result = result + "<ul>"; - result = result + "<p/>"; - if (outputType != null && outputType.equals("full")) { - for (int i=0; i<forms.size(); i++) { - result = result + "<li>"; - Form f = forms.get(i); - String formName = f.getFormName(); - String formProvider = f.getProvider(); - String language = f.getLanguage(); - String lemmaName = f.getLemmaName(); - result = result + formName + " (data provider: " + formProvider + ", language: " + language + ", lemmaName: " + lemmaName + ")"; - result = result + "</li>"; - } - } else if (outputType == null || outputType.equals("compact")) { - result = result + "<li>"; - for (int i=0; i<forms.size(); i++) { - Form f = forms.get(i); - String formName = f.getFormName(); - result = result + formName + ", "; - } - result = result.substring(0, result.length() - 2); // without last comma and blank - result = result + "</li>"; - } else if (outputType.equals("string")) { - for (int i=0; i<forms.size(); i++) { - Form f = forms.get(i); - String formName = f.getFormName(); - result = result + formName + " "; - } - result = result.substring(0, result.length() - 1); // without last blank - } - } - result = result + "<hr/>"; - result = result + "<p/>"; - result = result + "Elapsed time: " + elapsedTime + " ms, see the <a href=\"/mpiwg-mpdl-lt-web/index.html\">service description</a> of this page, if you find a bug <a href=\"https://it-dev.mpiwg-berlin.mpg.de/tracs/mpdl-project-software/newticket\">let us know</a>"; - result = result + "</body>"; - result = result + "</html>"; - return result; - } - - private String createStringOutputString(ArrayList<Form> forms) { - String result = ""; - for (int i=0; i<forms.size(); i++) { - Form f = forms.get(i); - String formName = f.getFormName(); - result = result + formName + " "; - } - result = result.substring(0, result.length() - 1); // without last blank - return result; - } - - -}
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/lt/GetLemmas.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,231 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.servlets.lt; - -import java.io.IOException; -import java.io.PrintWriter; -import java.util.ArrayList; -import java.util.Collections; -import java.util.Date; - -import javax.servlet.ServletConfig; -import javax.servlet.ServletException; -import javax.servlet.http.HttpServlet; -import javax.servlet.http.HttpServletRequest; -import javax.servlet.http.HttpServletResponse; - -import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; -import de.mpg.mpiwg.berlin.mpdl.lt.dict.db.LexHandler; -import de.mpg.mpiwg.berlin.mpdl.lt.general.Language; -import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.Form; -import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.Lemma; -import de.mpg.mpiwg.berlin.mpdl.lt.text.norm.Normalizer; - -public class GetLemmas extends HttpServlet { - private static final long serialVersionUID = 1L; - private LexHandler lexHandler; - - public GetLemmas() { - super(); - } - - public void init(ServletConfig config) throws ServletException { - super.init(config); - try { - lexHandler = LexHandler.getInstance(); - } catch (ApplicationException e) { - throw new ServletException(e); - } - } - - protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { - Date begin = new Date(); - request.setCharacterEncoding("utf-8"); - response.setCharacterEncoding("utf-8"); - String query = request.getParameter("query"); - String language = request.getParameter("language"); - String inputType = request.getParameter("inputType"); - String outputFormat = request.getParameter("outputFormat"); - String outputType = request.getParameter("outputType"); - String normalization = request.getParameter("normalization"); - if (language == null) - language = "eng"; - if (inputType == null || ! (inputType.equals("form") || inputType.equals("lemma"))) - inputType = "form"; - if (outputFormat == null || ! (outputFormat.equals("xml") || outputFormat.equals("html") || outputFormat.equals("string"))) - outputFormat = "xml"; - if (outputType == null || ! (outputType.equals("compact") || outputType.equals("full"))) - outputType = "compact"; - if (normalization == null || ! (normalization.equals("none") || normalization.equals("reg") || normalization.equals("reg norm"))) - normalization = "norm"; - int normalizationType = Normalizer.DICTIONARY; - if (normalization != null && normalization.equals("none")) - normalizationType = Normalizer.NONE; - - String xmlQueryString = "<query><name>" + query + "</name>" + "<language>" + language + "</language>" + "<inputType>" + inputType + "</inputType>" + - "<outputFormat>" + outputFormat + "</outputFormat>" + "<outputType>" + outputType + "</outputType>" + "<normalization>" + normalization + "</normalization>" + "</query>"; - try { - if (outputFormat.equals("xml")) - response.setContentType("text/xml"); - else if (outputFormat.equals("html") || outputFormat.equals("string")) - response.setContentType("text/html"); - else - response.setContentType("text/xml"); - PrintWriter out = response.getWriter(); - if (query == null || query.isEmpty()) { - out.print("request parameter query is empty. Please specify a query."); - out.close(); - return; - } - ArrayList<Lemma> lemmas = lexHandler.getLemmas(query, inputType, language, normalizationType); - String baseUrl = getBaseUrl(request); - Date end = new Date(); - String elapsedTime = String.valueOf(end.getTime() - begin.getTime()); - String result = ""; - if (outputFormat == null || outputFormat.equals("xml")) - result = createXmlOutputString(query, lemmas, outputType, baseUrl, xmlQueryString, elapsedTime); - else if (outputFormat.equals("html")) - result = createHtmlOutputString(query, lemmas, outputType, elapsedTime); - else if (outputFormat.equals("string")) - result = createStringOutputString(lemmas); - else - result = createXmlOutputString(query, lemmas, outputType, baseUrl, xmlQueryString, elapsedTime); - out.print(result); - out.close(); - } catch (ApplicationException e) { - throw new ServletException(e); - } - } - - protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { - - } - - private String getBaseUrl( HttpServletRequest request ) { - if (request.getServerPort() == 80 || request.getServerPort() == 443) - return request.getScheme() + "://" + request.getServerName() + request.getContextPath(); - else - return request.getScheme() + "://" + request.getServerName() + ":" + request.getServerPort() + request.getContextPath(); - } - - - private String createXmlOutputString(String query, ArrayList<Lemma> lemmas, String outputType, String baseUrl, String xmlQueryString, String elapsedTime) { - String result = "<result>"; - result = result + "<provider>" + "MPIWG MPDL language technology service (see: " + "" + baseUrl + "), Max Planck Institute for the History of Science, Berlin." + "</provider>"; - result = result + xmlQueryString; - result = result + "<elapsed-time-ms>" + elapsedTime + "</elapsed-time-ms>"; - if (lemmas != null && ! lemmas.isEmpty()) { - result = result + "<morphology>"; - for (int i=0; i<lemmas.size(); i++) { - Lemma lemma = lemmas.get(i); - String lemmaName = lemma.getLemmaName(); - String language = lemma.getLanguage(); - result = result + "<lemma>"; - result = result + "<name>" + lemmaName + "</name>"; - if (outputType != null && outputType.equals("full")) { - String lemmaProvider = lemma.getProvider(); - result = result + "<provider>" + lemmaProvider + "</provider>"; - result = result + "<language>" + language + "</language>"; - } - if (Language.getInstance().isArabic(language) || Language.getInstance().isLatin(language)) { - String remoteUrl = "http://www.perseus.tufts.edu/hopper/morph?l=" + lemmaName + "&la=" + language; - result = result + "<remoteUrl>" + remoteUrl + "</remoteUrl>"; - } else if (Language.getInstance().isGreek(language)) { - String remoteUrl = "http://www.perseus.tufts.edu/hopper/morph?l=" + lemmaName + "&la=" + "greek"; - result = result + "<remoteUrl>" + remoteUrl + "</remoteUrl>"; - } - if (outputType != null && outputType.equals("full")) { - ArrayList<Form> forms = lemma.getFormsList(); - Collections.sort(forms); - if (forms != null && ! forms.isEmpty()) { - result = result + "<forms>"; - for (int j=0; j<forms.size(); j++) { - result = result + "<form>"; - Form f = forms.get(j); - String formName = f.getFormName(); - String formProvider = f.getProvider(); - result = result + "<provider>" + formProvider + "</provider>"; - result = result + "<language>" + language + "</language>"; - result = result + "<name>" + formName + "</name>"; - result = result + "</form>"; - } - result = result + "</forms>"; - } - } - result = result + "</lemma>"; - } - result = result + "</morphology>"; - } - result = result + "</result>"; - return result; - } - - private String createHtmlOutputString(String query, ArrayList<Lemma> lemmas, String outputType, String elapsedTime) { - String result = ""; - result = result + "<html>"; - result = result + "<head>"; - result = result + "<title>Lemmas for: \"" + query + "\"</title>"; - result = result + "</head>"; - result = result + "<body>"; - result = result + "<table align=\"right\" valign=\"top\">"; - result = result + "<td>[<i>This is a MPIWG MPDL language technology service</i>] <a href=\"/mpiwg-mpdl-lt-web/index.html\"><img src=\"/mpiwg-mpdl-lt-web/images/info.png\" valign=\"bottom\" width=\"15\" height=\"15\" border=\"0\" alt=\"MPIWG MPDL language technology service\"/></a></td>"; - result = result + "</table>"; - result = result + "<p/>"; - result = result + "<h1>Lemmas for: \"" + query + "\"</h1>"; - if (lemmas != null && ! lemmas.isEmpty()) { - result = result + "<h3>Morphology</h3>"; - result = result + "<ul>"; - result = result + "<p/>"; - for (int i=0; i<lemmas.size(); i++) { - Lemma lemma = lemmas.get(i); - String lemmaName = lemma.getLemmaName(); - String language = lemma.getLanguage(); - result = result + "<li>"; - result = result + lemmaName; - if (outputType != null && outputType.equals("full")) { - String lemmaProvider = lemma.getProvider(); - result = result + " (data provider: " + lemmaProvider + ")"; - } - if (Language.getInstance().isArabic(language) || Language.getInstance().isLatin(language)) - result = result + " (external link: <a href=\"http://www.perseus.tufts.edu/hopper/morph?l=" + lemmaName + "&la=" + language + "\">" + lemmaName + "</a>)"; - else if (Language.getInstance().isGreek(language)) - result = result + " (external link: <a href=\"http://www.perseus.tufts.edu/hopper/morph?l=" + lemmaName + "&la=" + "greek" + "\">" + lemmaName + "</a>)"; - if (outputType != null && outputType.equals("full")) { - ArrayList<Form> forms = lemma.getFormsList(); - Collections.sort(forms); - if (forms != null && ! forms.isEmpty()) { - result = result + "<ul>"; - for (int j=0; j<forms.size(); j++) { - Form f = forms.get(j); - String formName = f.getFormName(); - String formProvider = f.getProvider(); - result = result + formName + " (data provider: " + formProvider + "), "; - } - result = result.substring(0, result.length() - 2); // without last comma and blank - result = result + "</ul>"; - } - } - result = result + "</li>"; - } - result = result + "</ul>"; - } - result = result + "[* external links may not function]"; - result = result + "<hr/>"; - result = result + "<p/>"; - result = result + "Elapsed time: " + elapsedTime + " ms, see the <a href=\"/mpiwg-mpdl-lt-web/index.html\">service description</a> of this page, if you find a bug <a href=\"https://it-dev.mpiwg-berlin.mpg.de/tracs/mpdl-project-software/newticket\">let us know</a>"; - result = result + "</body>"; - result = result + "</html>"; - return result; - } - - private String createStringOutputString(ArrayList<Lemma> lemmas) { - String result = ""; - for (int i=0; i<lemmas.size(); i++) { - Lemma l = lemmas.get(i); - String lemmaName = l.getLemmaName(); - result = result + lemmaName + " "; - } - result = result.substring(0, result.length() - 1); // without last blank - return result; - } - - -}
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/lt/MpiwgMpdlLtWebServletContextListener.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,39 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.servlets.lt; - -import javax.servlet.ServletContext; -import javax.servlet.ServletContextEvent; -import javax.servlet.ServletContextListener; - -import de.mpg.mpiwg.berlin.mpdl.lt.dict.db.LexHandler; -import de.mpg.mpiwg.berlin.mpdl.lt.general.Constants; -import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.MorphologyCache; -import de.mpg.mpiwg.berlin.mpdl.lt.text.reg.RegularizationManager; - -public class MpiwgMpdlLtWebServletContextListener implements ServletContextListener { - private ServletContext context = null; - - public void contextInitialized(ServletContextEvent event) { - try { - this.context = event.getServletContext(); - // String dataDirectory = System.getProperty("catalina.base") + "/webapps/mpiwg-mpdl-lt-web/WEB-INF/data"; - String dataDirectory = Constants.getInstance().getDataDir(); - context.setAttribute("dataDirectory", dataDirectory); - System.out.println(MpiwgMpdlLtWebServletContextListener.class.getName() + ": contextInitialized (data directory= \"" + dataDirectory + "\", set in constants.properties)"); - } catch (Exception e) { - e.printStackTrace(); - } - } - - public void contextDestroyed(ServletContextEvent event) { - try { - this.context = null; - LexHandler.getInstance().end(); - MorphologyCache.getInstance().end(); - RegularizationManager.getInstance().end(); - String dataDirectory = Constants.getInstance().getDataDir(); - System.out.println(MpiwgMpdlLtWebServletContextListener.class.getName() + ": contextDestroyed (databases in directory: \"" + dataDirectory + "\" are closed)"); - } catch (Exception e) { - e.printStackTrace(); - } - } -} \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/lt/Normalize.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,68 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.servlets.lt; - -import java.io.IOException; -import java.io.PrintWriter; - -import javax.servlet.ServletConfig; -import javax.servlet.ServletException; -import javax.servlet.http.HttpServlet; -import javax.servlet.http.HttpServletRequest; -import javax.servlet.http.HttpServletResponse; - -import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; -import de.mpg.mpiwg.berlin.mpdl.lt.text.norm.Normalizer; - -public class Normalize extends HttpServlet { - private static final long serialVersionUID = 1L; - - public Normalize() { - super(); - } - - public void init(ServletConfig config) throws ServletException { - super.init(config); - } - - protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { - request.setCharacterEncoding("utf-8"); - response.setCharacterEncoding("utf-8"); - String inputString = request.getParameter("inputString"); - String language = request.getParameter("language"); - String method = request.getParameter("method"); - String typeStr = request.getParameter("type"); - if (language == null) - language = "eng"; - if (method == null) - method = "norm"; - String[] methods = method.split(" "); - if (typeStr == null) - typeStr = "display"; - int type = Normalizer.DISPLAY; - if (typeStr.equals("dictionary")) - type = Normalizer.DICTIONARY; - else if (typeStr.equals("search")) - type = Normalizer.SEARCH; - String result = null; - try { - response.setContentType("text/html"); - PrintWriter out = response.getWriter(); - if (inputString == null || inputString.isEmpty()) { - out.print("request parameter \"inputString\" is empty. Please specify \"inputString\""); - out.close(); - return; - } - Normalizer normalizer = new Normalizer(methods, language); - normalizer.setNormMode(type); - result = normalizer.normalize(inputString); - if (result != null) - out.print(result); - out.close(); - } catch (ApplicationException e) { - throw new ServletException(e); - } - } - - protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { - - } -}
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/lt/Tokenize.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,234 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.servlets.lt; - -import java.io.BufferedInputStream; -import java.io.IOException; -import java.io.InputStream; -import java.io.PrintWriter; -import java.io.StringReader; -import java.net.URL; -import java.util.ArrayList; -import java.util.Date; -import java.util.Hashtable; - -import javax.servlet.ServletConfig; -import javax.servlet.ServletException; -import javax.servlet.http.HttpServlet; -import javax.servlet.http.HttpServletRequest; -import javax.servlet.http.HttpServletResponse; - -import org.apache.commons.io.IOUtils; -import org.apache.lucene.analysis.tokenattributes.CharTermAttribute; - -import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; -import de.mpg.mpiwg.berlin.mpdl.lt.dict.app.Lexicon; -import de.mpg.mpiwg.berlin.mpdl.lt.dict.db.LexHandler; -import de.mpg.mpiwg.berlin.mpdl.lt.morph.app.Lemma; -import de.mpg.mpiwg.berlin.mpdl.lt.text.norm.Normalizer; -import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.Token; -import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.Tokenizer; -import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.XmlTokenizer; -import de.mpg.mpiwg.berlin.mpdl.servlets.util.ServletUtil; - -public class Tokenize extends HttpServlet { - private static final long serialVersionUID = 1L; - - public Tokenize() { - super(); - } - - public void init(ServletConfig config) throws ServletException { - super.init(config); - } - - protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { - Date begin = new Date(); - request.setCharacterEncoding("utf-8"); - response.setCharacterEncoding("utf-8"); - String inputString = request.getParameter("inputString"); - String srcUrlStr = request.getParameter("srcUrl"); - String language = request.getParameter("language"); - String normalization = request.getParameter("normalization"); - String dictionary = request.getParameter("dictionary"); - String stopElements = request.getParameter("stopElements"); - String outputFormat = request.getParameter("outputFormat"); - String outputOptionsStr = request.getParameter("outputOptions"); - if (language == null) - language = "eng"; - if (normalization == null) - normalization = "norm"; - String[] normFunctions = normalization.split(" "); - if (dictionary == null) - dictionary = "yes"; - if (stopElements == null) - stopElements = ""; - String[] stopElementsArray = stopElements.split(" "); - if (outputFormat == null) - outputFormat = "xml"; - if (outputOptionsStr == null) - outputOptionsStr = ""; - String[] outputOptions = outputOptionsStr.split(" "); - String result = null; - try { - if (outputFormat.equals("xml")) { - response.setContentType("text/xml"); - } else if (outputFormat.equals("string")) { - response.setContentType("text/html"); - } else { - response.setContentType("text/xml"); - } - response.setCharacterEncoding("utf-8"); - PrintWriter out = response.getWriter(); - String inputText = null; // contains string or xml text - if ((inputString == null || inputString.isEmpty()) && (srcUrlStr == null || srcUrlStr.isEmpty())) { - out.print("request parameter \"inputString\" or \"srcUrl\" is empty. Please specify \"inputString\""); - out.close(); - return; - } else { - if (srcUrlStr != null && ! srcUrlStr.isEmpty()) { - URL srcUrl = new URL(srcUrlStr); - InputStream inputStream = srcUrl.openStream(); - BufferedInputStream in = new BufferedInputStream(inputStream); - inputText = IOUtils.toString(in, "utf-8"); - in.close(); - } else if (inputString != null && ! inputString.isEmpty()) { - inputText = inputString; - } - } - inputText = inputText.trim(); - // Tokenize - boolean inputTextIsXml = false; - if (inputText != null && inputText.startsWith("<") && inputText.endsWith(">")) // TODO check properly for xml type of the inputText - inputTextIsXml = true; - if (! inputTextIsXml) { - ArrayList<String> tokens = getToken(inputText, language, normFunctions); - Hashtable<String, ArrayList<Lexicon>> tokensDictionaries = null; - if (dictionary.equals("yes")) { - tokensDictionaries = new Hashtable<String, ArrayList<Lexicon>>(); - LexHandler lexHandler = LexHandler.getInstance(); - for (int i = 0; i < tokens.size(); i++) { - String token = tokens.get(i); - ArrayList<Lemma> lemmas = lexHandler.getLemmas(token, "form", language, Normalizer.NONE); - ArrayList<Lexicon> dictionaries = lexHandler.getLexEntries(lemmas, language, null, null); - tokensDictionaries.put(token, dictionaries); - } - } - String baseUrl = ServletUtil.getInstance().getBaseUrl(request); - Date end = new Date(); - String elapsedTime = String.valueOf(end.getTime() - begin.getTime()); - if (outputFormat.equals("xml")) - result = createXmlOutputString(tokens, tokensDictionaries, baseUrl, elapsedTime); - else if (outputFormat.equals("string")) - result = createStringOutputString(tokens); - else - result = "<result><error>outputFormat: \"" + outputFormat + "\" is not supported</error></result>"; - } else { - StringReader xmlInputStringReader = new StringReader(inputText); - XmlTokenizer xmlTokenizer = new XmlTokenizer(xmlInputStringReader); - xmlTokenizer.setLanguage(language); - xmlTokenizer.setNormFunctions(normFunctions); - xmlTokenizer.setOutputOptions(outputOptions); - if (stopElementsArray != null) - xmlTokenizer.setStopElements(stopElementsArray); - result = xmlTokenizer.tokenize(); - } - if (result != null) - out.print(result); - out.close(); - } catch (ApplicationException e) { - throw new ServletException(e); - } - } - - private ArrayList<String> getToken(String inputString, String language, String[] normFunctions) throws ApplicationException { - ArrayList<String> retTokens = null; - try { - StringReader reader = new StringReader(inputString); - Tokenizer tokenizer = new Tokenizer(reader); - tokenizer.setLanguage(language); - tokenizer.setNormFunctions(normFunctions); - ArrayList<Token> tokens = tokenizer.getTokens(); - if (tokens != null) { - retTokens = new ArrayList<String>(); - for (int i=0; i<tokens.size(); i++) { - Token t = tokens.get(i); - retTokens.add(t.getContent()); - } - } - tokenizer.end(); - tokenizer.close(); - } catch (IOException e) { - throw new ApplicationException(e); - } - return retTokens; - } - - private String createXmlOutputString(ArrayList<String> tokens, Hashtable<String, ArrayList<Lexicon>> tokensDictionaries, String baseUrl, String elapsedTime) { - String result = "<result>"; - result = result + "<provider>" + "MPIWG MPDL language technology service (see: " + "" + baseUrl + "), Max Planck Institute for the History of Science, Berlin." + "</provider>"; - result = result + "<elapsed-time-ms>" + elapsedTime + "</elapsed-time-ms>"; - if (tokens != null && ! tokens.isEmpty()) { - result = result + "<tokens>"; - for (int i=0; i<tokens.size(); i++) { - String token = tokens.get(i); - result = result + "<token>"; - result = result + "<name>" + token + "</name>"; - if (tokensDictionaries != null && ! tokensDictionaries.isEmpty()) { - ArrayList<Lexicon> tokenDictionaries = tokensDictionaries.get(token); - if (tokenDictionaries != null) { - result = result + "<dictionaries>"; - for (int j=0; j<tokenDictionaries.size(); j++) { - Lexicon lexicon = tokenDictionaries.get(j); - result = result + lexicon.toXmlString(); - } - result = result + "</dictionaries>"; - } - } - result = result + "</token>"; - } - result = result + "</tokens>"; - } - result = result + "</result>"; - return result; - } - - private String createStringOutputString(ArrayList<String> tokens) { - String result = ""; - if (tokens != null && ! tokens.isEmpty()) { - for (int i=0; i<tokens.size(); i++) { - String token = tokens.get(i); - result = result + token + " "; - } - result = result.substring(0, result.length() - 1); // without last blank - } - return result; - } - - private ArrayList<String> getTokenOld(String inputString, String language, String[] normFunctions) throws ApplicationException { - ArrayList<String> tokens = new ArrayList<String>(); - try { - StringReader reader = new StringReader(inputString); - Tokenizer tokenizer = new Tokenizer(reader); - tokenizer.setLanguage(language); - tokenizer.setNormFunctions(normFunctions); - // tokenizer.reset(); - /* - result = new MpdlFilter(result); // filter to remove the hyphen in a token etc. - result = new LowerCaseFilter(result); - result = new StopFilter(result, stopSet); - */ - CharTermAttribute charTermAttribute = tokenizer.getAttribute(CharTermAttribute.class); - // Token token = tokenizer.getAttribute(Token.class); - while (tokenizer.incrementToken()) { - // String tokenStr = token.toString(); - String term = charTermAttribute.toString(); - tokens.add(term); - } - tokenizer.end(); - tokenizer.close(); - } catch (IOException e) { - throw new ApplicationException(e); - } - return tokens; - } - -}
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/lt/Transcode.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,77 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.servlets.lt; - -import java.io.IOException; -import java.io.PrintWriter; - -import javax.servlet.ServletConfig; -import javax.servlet.ServletException; -import javax.servlet.http.HttpServlet; -import javax.servlet.http.HttpServletRequest; -import javax.servlet.http.HttpServletResponse; - -import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; -import de.mpg.mpiwg.berlin.mpdl.lt.text.transcode.Transcoder; - -public class Transcode extends HttpServlet { - private static final long serialVersionUID = 1L; - private Transcoder transcoder; - - public Transcode() { - super(); - } - - public void init(ServletConfig config) throws ServletException { - super.init(config); - transcoder = Transcoder.getInstance(); - } - - protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { - request.setCharacterEncoding("utf-8"); - response.setCharacterEncoding("utf-8"); - String inputString = request.getParameter("inputString"); - String srcEncoding = request.getParameter("srcEncoding"); - String destEncoding = request.getParameter("destEncoding"); - if (destEncoding == null) - destEncoding = "unicode"; - String result = null; - try { - response.setContentType("text/html"); - PrintWriter out = response.getWriter(); - if (inputString == null || inputString.isEmpty()) { - out.print("request parameter \"inputString\" is empty. Please specify \"inputString\""); - out.close(); - return; - } - if (srcEncoding == null || srcEncoding.isEmpty()) { - out.print("request parameter \"srcEncoding\" is empty. Please specify \"srcEncoding\""); - out.close(); - return; - } - if (destEncoding == null || destEncoding.isEmpty()) { - out.print("request parameter \"destEncoding\" is empty. Please specify \"destEncoding\""); - out.close(); - return; - } - if (srcEncoding.equals("buckwalter") && destEncoding.equals("unicode")) { - result = transcoder.transcodeFromBuckwalter2Unicode(inputString); - } else if (srcEncoding.equals("betacode") && destEncoding.equals("unicode")) { - result = transcoder.transcodeFromBetaCode2Unicode(inputString); - } else if (srcEncoding.equals("unicode") && destEncoding.equals("betacode")) { - result = transcoder.transcodeFromUnicode2BetaCode(inputString); - } else if (srcEncoding.equals("unicode") && destEncoding.equals("buckwalter")) { - result = transcoder.transcodeFromUnicode2Buckwalter(inputString); - } else { - result = "your combination of \"srcEncoding\" and \"destEncoding\" is not yet supported"; - } - if (result != null) - out.print(result); - out.close(); - } catch (ApplicationException e) { - throw new ServletException(e); - } - } - - protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { - - } -}
--- a/software/mpdl-services/mpiwg-mpdl-lt-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/util/ServletUtil.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,27 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.servlets.util; - -import javax.servlet.http.HttpServletRequest; - -import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; - -public class ServletUtil { - private static ServletUtil instance; - - public static ServletUtil getInstance() throws ApplicationException { - if (instance == null) { - instance = new ServletUtil(); - } - return instance; - } - - public ServletUtil() { - } - - public String getBaseUrl( HttpServletRequest request ) { - if (request.getServerPort() == 80 || request.getServerPort() == 443) - return request.getScheme() + "://" + request.getServerName() + request.getContextPath(); - else - return request.getScheme() + "://" + request.getServerName() + ":" + request.getServerPort() + request.getContextPath(); - } - -}
--- a/software/mpdl-services/mpiwg-mpdl-lt/.classpath Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/.classpath Tue Nov 27 12:35:19 2012 +0100 @@ -7,8 +7,13 @@ <classpathentry kind="lib" path="lib/saxon9-s9api.jar"/> <classpathentry kind="lib" path="lib/saxon9.jar"/> <classpathentry kind="lib" path="lib/commons-lang3-3.0.1.jar"/> - <classpathentry kind="lib" path="lib/lucene-core-3.4.0.jar"/> <classpathentry kind="lib" path="lib/commons-io-2.0.1.jar"/> <classpathentry kind="lib" path="lib/mpiwg-mpdl-xml.jar"/> + <classpathentry kind="lib" path="lib/lucene-core-3.5.0.jar"> + <attributes> + <attribute name="javadoc_location" value="jar:platform:/resource/mpiwg-mpdl-lt/lib/lucene-core-3.5.0-javadoc.jar!/"/> + </attributes> + </classpathentry> + <classpathentry combineaccessrules="false" kind="src" path="/mpiwg-mpdl-xml"/> <classpathentry kind="output" path="bin"/> </classpath>
--- a/software/mpdl-services/mpiwg-mpdl-lt/.externalToolBuilders/Ant_Build.launch Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,18 +0,0 @@ -<?xml version="1.0" encoding="UTF-8" standalone="no"?> -<launchConfiguration type="org.eclipse.ant.AntBuilderLaunchConfigurationType"> -<booleanAttribute key="org.eclipse.ant.ui.ATTR_TARGETS_UPDATED" value="true"/> -<booleanAttribute key="org.eclipse.ant.ui.DEFAULT_VM_INSTALL" value="false"/> -<listAttribute key="org.eclipse.debug.core.MAPPED_RESOURCE_PATHS"> -<listEntry value="/mpiwg-mpdl-lt"/> -</listAttribute> -<listAttribute key="org.eclipse.debug.core.MAPPED_RESOURCE_TYPES"> -<listEntry value="4"/> -</listAttribute> -<booleanAttribute key="org.eclipse.debug.ui.ATTR_LAUNCH_IN_BACKGROUND" value="false"/> -<stringAttribute key="org.eclipse.jdt.launching.CLASSPATH_PROVIDER" value="org.eclipse.ant.ui.AntClasspathProvider"/> -<booleanAttribute key="org.eclipse.jdt.launching.DEFAULT_CLASSPATH" value="true"/> -<stringAttribute key="org.eclipse.jdt.launching.PROJECT_ATTR" value="mpiwg-mpdl-lt"/> -<stringAttribute key="org.eclipse.ui.externaltools.ATTR_LOCATION" value="${workspace_loc:/mpiwg-mpdl-lt/build/build.xml}"/> -<stringAttribute key="org.eclipse.ui.externaltools.ATTR_RUN_BUILD_KINDS" value="full,incremental,"/> -<booleanAttribute key="org.eclipse.ui.externaltools.ATTR_TRIGGERS_CONFIGURED" value="true"/> -</launchConfiguration>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/.externalToolBuilders/mpiwg-mpdl-lt-build.launch Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,18 @@ +<?xml version="1.0" encoding="UTF-8" standalone="no"?> +<launchConfiguration type="org.eclipse.ant.AntBuilderLaunchConfigurationType"> +<booleanAttribute key="org.eclipse.ant.ui.ATTR_TARGETS_UPDATED" value="true"/> +<booleanAttribute key="org.eclipse.ant.ui.DEFAULT_VM_INSTALL" value="false"/> +<listAttribute key="org.eclipse.debug.core.MAPPED_RESOURCE_PATHS"> +<listEntry value="/mpiwg-mpdl-lt"/> +</listAttribute> +<listAttribute key="org.eclipse.debug.core.MAPPED_RESOURCE_TYPES"> +<listEntry value="4"/> +</listAttribute> +<booleanAttribute key="org.eclipse.debug.ui.ATTR_LAUNCH_IN_BACKGROUND" value="false"/> +<stringAttribute key="org.eclipse.jdt.launching.CLASSPATH_PROVIDER" value="org.eclipse.ant.ui.AntClasspathProvider"/> +<booleanAttribute key="org.eclipse.jdt.launching.DEFAULT_CLASSPATH" value="true"/> +<stringAttribute key="org.eclipse.jdt.launching.PROJECT_ATTR" value="mpiwg-mpdl-lt"/> +<stringAttribute key="org.eclipse.ui.externaltools.ATTR_LOCATION" value="${workspace_loc:/mpiwg-mpdl-lt/build/build.xml}"/> +<stringAttribute key="org.eclipse.ui.externaltools.ATTR_RUN_BUILD_KINDS" value="full,incremental,"/> +<booleanAttribute key="org.eclipse.ui.externaltools.ATTR_TRIGGERS_CONFIGURED" value="true"/> +</launchConfiguration>
--- a/software/mpdl-services/mpiwg-mpdl-lt/.project Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/.project Tue Nov 27 12:35:19 2012 +0100 @@ -16,7 +16,7 @@ <arguments> <dictionary> <key>LaunchConfigHandle</key> - <value><project>/.externalToolBuilders/Ant_Build.launch</value> + <value><project>/.externalToolBuilders/mpiwg-mpdl-lt-build.launch</value> </dictionary> </arguments> </buildCommand>
--- a/software/mpdl-services/mpiwg-mpdl-lt/.settings/org.eclipse.core.resources.prefs Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/.settings/org.eclipse.core.resources.prefs Tue Nov 27 12:35:19 2012 +0100 @@ -1,3 +1,8 @@ -#Fri Oct 07 18:44:17 CEST 2011 +#Wed May 16 11:47:32 CEST 2012 eclipse.preferences.version=1 +encoding//src/de/mpg/mpiwg/berlin/mpdl/lt/dict/db/LexHandler.java=UTF-8 +encoding//src/de/mpg/mpiwg/berlin/mpdl/lt/text/tokenize/Tokenizer.java=UTF-8 +encoding//src/de/mpg/mpiwg/berlin/mpdl/lt/text/tokenize/TokenizerNew.java=UTF-8 +encoding//src/de/mpg/mpiwg/berlin/mpdl/lt/text/tokenize/WordContentHandler.java=UTF-8 +encoding//src/de/mpg/mpiwg/berlin/mpdl/lt/text/tokenize/XmlTokenizerContentHandler.java=UTF-8 encoding//src/de/mpg/mpiwg/berlin/mpdl/test/TestLocal.java=UTF-8
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/bin/constants.properties Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,1 @@ +dataDir=/Users/jwillenborg/mpdl/data/lt \ No newline at end of file
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Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/.DS_Store has changed
Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/.DS_Store has changed
Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/Normalizer.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexAR.class has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexAR.lex Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,96 @@ +/* + * Normalization rules for Arabic text + * [this is a JFlex specification] + * + * Wolfgang Schmidle + * version 2011-02-28 + * + */ + +package de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang; + +%% + +%public +%class MpdlNormalizerLexAR +%type java.lang.String +%unicode + +// Arabic: ar + +%states DISP, DICT, SEARCH + +%{ + private String original = ""; + private String normalized = ""; + private int problem = 0; + + private void add (String norm) { + original += yytext(); + normalized += norm; + } + + private static final String LB = "[\u002d\u00ad] "; +%} + +hyphen = [-\u{00ad}] // hyphen and soft hyphen +LB = {hyphen} \u0020 +// lb = ({hyphen} \u0020)? + +END = \n + +%% + +// jump over empty xml elements +"<"[^><]+"/>" { add(yytext()); } +"-<"[^><]+"/>" { add(yytext()); } +"<"[^><]+"></"[^><]+">" { add(yytext()); } +"-<"[^><]+"></"[^><]+">" { add(yytext()); } + +@ { problem = 1; add(yytext()); } +{LB} { add(yytext()); } +. { add(yytext()); } + + +<DISP> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized; + } + } +} + +<DICT> { + +{END} { + switch (problem) { + case 1: return ""; + default: return normalized.replaceAll(LB, ""); + } + } +} + +<SEARCH> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized.replaceAll(LB, ""); + } + } +} + + +/* + +Annahmen: +- die Routine wird wortweise aufgerufen, mit einem \n am Ende des Strings +- Zeilenumbrüche innerhalb des Wortes sind als hyphen/soft hyphen plus space markiert + +TO DO: + +AR: fehlt noch + +*/
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexAll.lex Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,143 @@ +/* + * Normalization rules for all languages + * [this is a JFlex specification] + * + * Wolfgang Schmidle + * 2011-01-25 + * + */ + +package de.mpg.mpiwg.berlin.mpdl.lt.analyzer.lang; + +%% + +%public +%class MpdlNormalizerLexAll +%type java.lang.String +%unicode +// %debug + +%states LA, ZH + +%{ + int cv = 0; // consonant = 1, vowel = 2, everything else = 0 +%} + +VOWEL=[AEIOUaeiouÆæęà èòùœ] +CONS=[BCDFGHKLMNPQRSTVWXZbcdfghklmnpqrstvwxzſß] +LR=[lLrR] +QUE=(que)? +END=\n + +%% + +<LA> { + +// 1. simple replacements + +// 1.1 single characters +Å¿ { cv = 1; return "s"; } +ß { cv = 1; return "ss"; } +[æę] { cv = 2; return "ae"; } +Æ { cv = 2; return "AE"; } +Å“ { cv = 2; return "oe"; } +// 1.2 character combinations +ij { cv = 2; return "ii"; } + +// 2. diacritics + +// 2.1 superfluous diacritics in single words +^ hîc {END} { return "hic"; } + +// 2.2 superfluous diacritics at the end of a word +// 2.2.1 common cases +à / {QUE} {END} { return "a"; } +à m / {QUE} {END} { return "am"; } +à s / {QUE} {END} { return "as"; } // (-à sque will likely never occur) +// à / [ms]? {QUE} {END} { return "a"; } +è / {QUE} {END} { return "e"; } +ò / {QUE} {END} { return "o"; } +òd / {QUE} {END} { return "od"; } +ùm / {QUE} {END} { return "um"; } +ùs / {QUE} {END} { return "us"; } + +// 2.3 superfluous diacritics within a word +// 2.3.1 common cases +aë { cv = 2; return "ae"; } +oë { cv = 2; return "oe"; } +// 2.3.2 rare cases +oï { cv = 2; return "oi"; } +uï { cv = 2; return "ui"; } +// 2.3.3 extremely rare cases +uü { cv = 2; return "uu"; } + +// 3. rules for u and v + +// 3.1 rules for u + +u/{VOWEL} { + switch(cv) { + case 2: return "v"; + default: cv = 2; return "u"; + } + } +U/{VOWEL} { + switch(cv) { + case 2: return "V"; + default: cv = 2; return "U"; + } + } + +// 3.2 rules for v + +qv { cv = 1; return "qu"; } // the replaced v still counts as consonant +Qv { cv = 1; return "Qu"; } +QV { cv = 1; return "QU"; } + +{LR}v { + switch(cv) { + case 1: return yytext().replace("v", "u"); + default: cv = 1; return yytext(); + } + } +{LR}V { + switch(cv) { + case 1: return yytext().replace("V", "U"); + default: cv = 1; return yytext(); + } + } + +v/{CONS} { cv = 1; return "u"; } +V/{CONS} { cv = 1; return "U"; } + + +// default + +{VOWEL} { cv = 2; return yytext(); } +{CONS} { cv = 1; return yytext(); } +\n { cv = 0; return ""; } +. { cv = 0; return yytext(); } + +} + +<ZH> { + +// Codepoint < FFFF + +ç«’ { return "奇"; } // 7AD2 --> 5947 +æ—¹ { return "時"; } // 65F9 --> 6642 +æ´ { return "æ·"; } // 6B74 --> 6B77 +ï¨ { return "ç²¾"; } // FA1D --> 7CBE (FA1D is a compatibility ideograph) + +// Codepoint > FFFF + +悁{ return "庶"; } // 2F88D --> 5EB6 (2F88D is a compatibility ideograph) + + +} + + +// default (can be overridden by individual languages) + +\n { return ""; } +. { return yytext(); }
Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexDE.class has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexDE.lex Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,140 @@ +/* + * Normalization rules for German text + * [this is a JFlex specification] + * + * Wolfgang Schmidle + * version 2011-07-12 + * + */ + +package de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang; + +%% + +%public +%class MpdlNormalizerLexDE +%type java.lang.String +%unicode + +// German: de, deu, ger + +%states DISP, DICT, SEARCH +%state CELEX, GRIMM + +%{ + private String original = ""; + private String normalized = ""; + private int problem = 0; + + private void add (String norm) { + original += yytext(); + normalized += norm; + } + + private static final String LB = "[\u002d\u00ad] "; +%} + +hyphen = [-\u{00ad}] // hyphen and soft hyphen +LB = {hyphen} \u0020 +// lb = ({hyphen} \u0020)? + +END = \n + +Alphabet = [abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ] + +%% + +// jump over empty xml elements +"<"[^><]+"/>" { add(yytext()); } +"-<"[^><]+"/>" { add(yytext()); } +"<"[^><]+"></"[^><]+">" { add(yytext()); } +"-<"[^><]+"></"[^><]+">" { add(yytext()); } + +ſ { add("s"); } + +// Fraktur + +<DISP, DICT, SEARCH, +GRIMM> { + +uͦ {add("u"); } +aͤ {add("ä"); } +oͤ {add("ö"); } +uͤ {add("ü"); } + +} + +<CELEX> { + +// normalize ä ö ü ß only for Celex! + +ä | Ä | aͤ { add("ae"); } +ö | Ö | oͤ { add("oe"); } +ü | Ü | uͤ { add("ue"); } +uͦ {add("u"); } +ß { add("ss"); } + +{Alphabet} { add(yytext()); } + +. { problem = 1; add(yytext()); } + +} + +<GRIMM> { + +ß { add("sz"); } + +} + + +// default + +@ { problem = 1; add(yytext()); } +{LB} { add(yytext()); } +. { add(yytext()); } + + +<DISP> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized; + } + } +} + +<DICT, CELEX, GRIMM> { + +{END} { + switch (problem) { + case 1: return ""; + default: return normalized.replaceAll(LB, ""); + } + } +} + +<SEARCH> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized.replaceAll(LB, "").toLowerCase(); + } + } +} + + + +/* + +Annahmen: +- die Routine wird wortweise aufgerufen, mit einem \n am Ende des Strings +- Zeilenumbrüche innerhalb des Wortes sind als hyphen/soft hyphen plus space markiert + +TO DO: + +DE: Trennung von Deutsch und Fraktur? +DE: Celex: hyphens weg? + +*/
Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexEL.class has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexEL.lex Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,145 @@ +/* + * Normalization rules for Greek text + * [this is a JFlex specification] + * + * Wolfgang Schmidle + * version 2011-08-03 + * + */ + +package de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang; + +%% + +%public +%class MpdlNormalizerLexEL +%type java.lang.String +%unicode + +// Greek: el, grc + +%states DISP, DICT, SEARCH +%state SIGMA + +%{ + private String original = ""; + private String normalized = ""; + private int problem = 0; + + private void add (String norm) { + original += yytext(); + normalized += norm; + } + + private static final String LB = "[\u002d\u00ad] "; +%} + +hyphen = [-\u{00ad}] // hyphen and soft hyphen +LB = {hyphen} \u0020 +// lb = ({hyphen} \u0020)? + +END = \n + +wordend = [νÏÏ‚]? {END} + +Latin = [abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ] + + +%% + +// jump over empty xml elements +"<"[^><]+"/>" { add(yytext()); } +"-<"[^><]+"/>" { add(yytext()); } +"<"[^><]+"></"[^><]+">" { add(yytext()); } +"-<"[^><]+"></"[^><]+">" { add(yytext()); } + + +// always replace tonos by oxia +// (although this should really be corrected in the text rather than normalized) +ά { add("ά"); } +Î { add("Î"); } +ή { add("ή"); } +ί { add("ί"); } +ÏŒ { add("ÏŒ"); } +Ï { add("Ï"); } +ÏŽ { add("ÏŽ"); } + + +<DICT, SEARCH, SIGMA> { + +á½° / {wordend} { add("á½±"); } +á¾² / {wordend} { add("á¾´"); } +á½² / {wordend} { add("á½³"); } +á½´ / {wordend} { add("á½µ"); } +á¿‚ / {wordend} { add("á¿„"); } +á½¶ / {wordend} { add("á½·"); } +ὸ / {wordend} { add("á½¹"); } +ὺ / {wordend} { add("á½»"); } +á½¼ / {wordend} { add("á½½"); } +ῲ / {wordend} { add("á¿´"); } + +// other candidates: Ὰ Ὲ Ὴ Ὶ Ὺ Ὸ Ὼ + +} + +<SIGMA> { + +Ï‚ { add("σ"); } + +} + +// default + +@ { problem = 1; add(yytext()); } +{Latin} { problem = 1; add(yytext()); } + +{LB} { add(yytext()); } +. { add(yytext()); } + + +<DISP> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized; + } + } +} + +<DICT, SIGMA> { + +{END} { + switch (problem) { + case 1: return ""; + default: return normalized.replaceAll(LB, ""); + } + } +} + +<SEARCH> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized.replaceAll(LB, "").toLowerCase(); + } + } +} + + +/* + +Annahmen: +- die Routine wird wortweise aufgerufen, mit einem \n am Ende des Strings +- Zeilenumbrüche innerhalb des Wortes sind als hyphen/soft hyphen plus space markiert + +TO DO: + +EL: tonos --> oxia wieder rausnehmen, weil es im Text geändert werden muss? +EL: gibt es noch weitere Fälle, wo legitimerweise ein Gravis vorkommen kann? +EL: kommen Großbuchstaben mit Gravis bei uns jemals vor, und sollen sie normalisiert werden? +EL: neuer State BETACODE ? +EL: nicht falsche Zeichen definieren, sondern erlaubte Zeichen + +*/
Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexEN.class has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexEN.lex Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,105 @@ +/* + * Normalization rules for English text + * [this is a JFlex specification] + * + * Wolfgang Schmidle + * version 2011-07-12 + * + */ + +package de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang; + +%% + +%public +%class MpdlNormalizerLexEN +%type java.lang.String +%unicode + +// 1.5 English: en + +%states DISP, DICT, SEARCH + +%{ + private String original = ""; + private String normalized = ""; + private int problem = 0; + + private void add (String norm) { + original += yytext(); + normalized += norm; + } + + private static final String LB = "[\u002d\u00ad] "; +%} + +hyphen = [-\u{00ad}] // hyphen and soft hyphen +LB = {hyphen} \u0020 +// lb = ({hyphen} \u0020)? + +END = \n + +%% + +// jump over empty xml elements +"<"[^><]+"/>" { add(yytext()); } +"-<"[^><]+"/>" { add(yytext()); } +"<"[^><]+"></"[^><]+">" { add(yytext()); } +"-<"[^><]+"></"[^><]+">" { add(yytext()); } + +<DISP, DICT, SEARCH> { + +ſ { add("s"); } + +} + + +// default + +@ { problem = 1; add(yytext()); } +{LB} { add(yytext()); } +. { add(yytext()); } + + +<DISP> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized; + } + } +} + +<DICT> { + +{END} { + switch (problem) { + case 1: return ""; + default: return normalized.replaceAll(LB, ""); + } + } +} + +<SEARCH> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized.replaceAll(LB, "").toLowerCase(); + } + } +} + + +/* + +Annahmen: +- die Routine wird wortweise aufgerufen, mit einem \n am Ende des Strings +- Zeilenumbrüche innerhalb des Wortes sind als hyphen/soft hyphen plus space markiert + +TO DO: + +EN: vollständig? + +*/
Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexFR.class has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexFR.lex Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,125 @@ +/* + * Normalization rules for French text + * [this is a JFlex specification] + * + * Wolfgang Schmidle + * version 2011-07-12 + * + */ + +package de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang; + +%% + +%public +%class MpdlNormalizerLexFR +%type java.lang.String +%unicode + +// French: fr + +%states DISP, DICT, SEARCH +%state CELEX + +%{ + private String original = ""; + private String normalized = ""; + private int problem = 0; + + private void add (String norm) { + original += yytext(); + normalized += norm; + } + + private static final String LB = "[\u002d\u00ad] "; +%} + +hyphen = [-\u{00ad}] // hyphen and soft hyphen +LB = {hyphen} \u0020 +// lb = ({hyphen} \u0020)? + +END = \n + +Alphabet = [abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ] + +%% + +// jump over empty xml elements +"<"[^><]+"/>" { add(yytext()); } +"-<"[^><]+"/>" { add(yytext()); } +"<"[^><]+"></"[^><]+">" { add(yytext()); } +"-<"[^><]+"></"[^><]+">" { add(yytext()); } + +<DISP, DICT, SEARCH, CELEX> { + +Å¿ { add("s"); } +ß { add("ss"); } +æ { add("ae"); } + +} + +<CELEX> { + +[œŒ] { add("oe"); } +[áà â] { add("a"); } +[éèê] { add("e"); } +[Ãìî] { add("i"); } +[óòô] { add("o"); } +[úùû] { add("u"); } +’ { add(""); } + +{Alphabet} { add(yytext()); } + +. { problem = 1; add(yytext()); } // in particular "@" + +} + +// default + +@ { problem = 1; add(yytext()); } +{LB} { add(yytext()); } +. { add(yytext()); } + + +<DISP> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized; + } + } +} + +<DICT, CELEX> { + +{END} { + switch (problem) { + case 1: return ""; + default: return normalized.replaceAll(LB, ""); + } + } +} + +<SEARCH> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized.replaceAll(LB, "").toLowerCase(); + } + } +} + + +/* + +Annahmen: +- die Routine wird wortweise aufgerufen, mit einem \n am Ende des Strings +- Zeilenumbrüche innerhalb des Wortes sind als hyphen/soft hyphen plus space markiert + +TO DO: + +FR: richtig? vollständig? + +*/
Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexIT.class has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexIT.lex Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,189 @@ +/* + * Normalization rules for Italian text + * [this is a JFlex specification] + * + * Wolfgang Schmidle + * version 2011-07-12 + * + */ + +package de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang; + +%% + +%public +%class MpdlNormalizerLexIT +%type java.lang.String +%unicode + +// Italian: it, ita + +%states DISP, DICT, SEARCH + +%{ + private static final int CONS = 1; + private static final int VOWEL = 2; + private int cv = 0; // consonant = 1, vowel = 2, everything else = 0 + + private String original = ""; + private String normalized = ""; + private int problem = 0; + + private void add (String norm) { + original += yytext(); + normalized += norm; + } + + private static final String LB = "[\u002d\u00ad] "; +%} + +Vowel = [AEIOUaeiouÆæęà èòùœ] +Cons = [BCDFGHKLMNPQRSTVWXZbcdfghklmnpqrstvwxzſß] +LR = [lLrR] + + +hyphen = [\u002d\u00ad] // hyphen and soft hyphen +LB = {hyphen} \u0020 +lb = ({hyphen} \u0020)? + +END = \n + +prefixCons = (in{lb}ter | per | Å¿u{lb}per | Å¿er) + +%% + +// jump over empty xml elements +"<"[^><]+"/>" { add(yytext()); } +"-<"[^><]+"/>" { add(yytext()); } +"<"[^><]+"></"[^><]+">" { add(yytext()); } +"-<"[^><]+"></"[^><]+">" { add(yytext()); } + +<DICT, SEARCH> { + +À { add("Ã"); } +È { add("É"); } +ÃŒ { add("Ã"); } +Ã’ { add("Ó"); } +Ù { add("Ú"); } +à { add("á"); } +è { add("é"); } +ì { add("Ã"); } +ò { add("ó"); } +ù { add("ú"); } + +} + +<DISP, DICT, SEARCH> { + +Å¿ { cv = CONS; add("s"); } +ß { cv = CONS; add("ss"); } +æ { cv = VOWEL; add("ae"); } +Æ { cv = VOWEL; add("AE"); } +Å“ { cv = VOWEL; add("oe"); } +Å’ { cv = VOWEL; add("OE"); } + +ij { cv = VOWEL; add("ii"); } + +tio { cv = VOWEL; add("zio"); } +TIO { cv = VOWEL; add("ZIO"); } + +// h-Regeln aus Arboreal: +^ ha / {END} { add(yytext()); } +^ hai / {END} { add(yytext()); } +^ han{lb}no / {END} { add(yytext()); } +^ ho / {END} { add(yytext()); } +^ h { add(""); } + + +// u/v rules are taken from MpdlNormalizerLexLA.lex + +// 1. rules for u --> v + +^ {prefixCons} / {lb} { cv = VOWEL; add(yytext().replace("Å¿", "s")); } + +^ [uU] / {Vowel} { cv = VOWEL; add(yytext().replaceAll("u", "v").replaceAll("U", "V")); } + + +[uU] / {Vowel} { + switch(cv) { + case VOWEL: add(yytext().replace("u", "v").replace("U", "V")); break; + default: cv = VOWEL; add(yytext()); break; + } + } + +// 2. rules for v --> u + +qv { cv = CONS; add("qu"); } // the replaced v still counts as consonant +Qv { cv = CONS; add("Qu"); } +QV { cv = CONS; add("QU"); } + +{LR} [vV] { + switch(cv) { + case CONS: add(yytext().replace("v", "u").replace("V", "U")); break; + default: cv = CONS; add(yytext()); break; + } + } + +v / {lb} {Cons} { cv = CONS; add("u"); } +V / {lb} {Cons} { cv = CONS; add("U"); } + +// 3. override default rule for . + +{Vowel} { cv = VOWEL; add(yytext()); } +{Cons} { cv = CONS; add(yytext()); } +@ { problem = 1; cv = 0; add(yytext()); } +{LB} { add(yytext()); } +. { cv = 0; add(yytext()); } + +} + + +<DISP> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized; + } + } +} + +<DICT> { + +{END} { + switch (problem) { + case 1: return ""; + default: return normalized.replaceAll(LB, ""); + } + } +} + +<SEARCH> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized.replaceAll(LB, "").toLowerCase(); + } + } +} + + +/* + +Annahmen: +- die Routine wird wortweise aufgerufen, mit einem \n am Ende des Strings +- Zeilenumbrüche innerhalb des Wortes sind als hyphen/soft hyphen plus space markiert + +TO DO: + +IT: all these rules are taken from Arboreal; do we need them all? +IT: richtig? vollständig? +IT: Sind die u/v-Regeln wirklich genau wie in LA ? insbesondere: gleiche Vokal-Klasse? +IT: Änderungen in den lateinischen u/v-Regeln übernehmen? +IT: italienische Beispielwörter für die u/v-Regeln angeben +IT: Brauchen wir die Gravis-Regeln aus Arboreal in DICT wirklich? +IT: wenn ja: gehört À --> à etc. in die Wörterbuch-Schicht? Und einschränken auf letzte Silbe? +IT: ist prefixCons = (inter | per | Å¿uper | Å¿er) auch für Italienisch gültig? + +*/
Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexLA.class has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexLA.lex Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,234 @@ +/* + * Normalization rules for Latin text + * [this is a JFlex specification] + * + * Wolfgang Schmidle + * version 2011-07-12 + * + */ + +package de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang; + +%% + +%public +%class MpdlNormalizerLexLA +%type java.lang.String +%unicode + +// Latin: la, lat + +%states DISP, DICT, SEARCH +%states RENAISSANCE_DISP, RENAISSANCE_DICT, RENAISSANCE_SEARCH + +%{ + private static final int CONS = 1; + private static final int VOWEL = 2; + private int cv = 0; // consonant = 1, vowel = 2, everything else = 0 + + private String original = ""; + private String normalized = ""; + private int problem = 0; + + private void add (String norm) { + original += yytext(); + normalized += norm; + } + + private static final String LB = "[\u002d\u00ad] "; +%} + +Vowel = [AEIOUaeiouÆæęœ] // without à èòù etc. +Cons = [BCDFGHKLMNPQRSTVWXZbcdfghklmnpqrstvwxzſß] +// y counts neither as Vowel nor as Cons, see the default rule below: [yY] { cv = 0; add(yytext()); } + +LR = [lLrR] + +hyphen = [\u002d\u00ad] // hyphen and soft hyphen +LB = {hyphen} \u0020 +lb = ({hyphen} \u0020)? + +END = \n + +que = (que)? // optional -que +enclitic = (que | ve | ne) +prefixCons = (in{lb}ter | per | Å¿u{lb}per | Å¿er) // "Å¿er" for forms of Å¿ervare + +%% + + +// jump over empty xml elements +"<"[^><]+"/>" { cv = 0; add(yytext()); } +"-<"[^><]+"/>" { cv = 0; add(yytext()); } +"<"[^><]+"></"[^><]+">" { cv = 0; add(yytext()); } +"-<"[^><]+"></"[^><]+">" { cv = 0; add(yytext()); } + +// TEST, siehe Benedetti Seite 444 +ð†‘ { add("X"); } // (U+10191; D800+DD91) + + +<DISP, DICT, SEARCH, +RENAISSANCE_DISP, RENAISSANCE_DICT, RENAISSANCE_SEARCH> { + +// 1. simple replacements + +// 1.1 single characters +Å¿ { cv = CONS; add("s"); } +ß { cv = CONS; add("ss"); } +[æę] { cv = VOWEL; add("ae"); } +Æ { cv = VOWEL; add("AE"); } +Å“ { cv = VOWEL; add("oe"); } + +// 1.2 character combinations +ij { cv = VOWEL; add("ii"); } + +// 2. superfluous diacritics + +// 2.1 acute accent +qÌue / {END} { add("que"); } // G +á / [mrst]? {enclitic} {END} { add("a"); } // G +é / [mrst]? {enclitic} {END} { add("e"); } // G +à / [mrst]? {enclitic} {END} { add("i"); } // G +ó / [mrst]? {enclitic} {END} { add("o"); } // G +ú / [mrst]? {enclitic} {END} { add("u"); } // G + +úe / {END} { add("ve"); } // W ?? + +// 2.2 grave accent +à / {que} {END} { add("a"); } // W G +à m / {que} {END} { add("am"); } // W (G) +à s / {que} {END} { add("as"); } // W (G) (-à sque will likely never occur) +è / {que} {END} { add("e"); } // W G +ò / {que} {END} { add("o"); } // W G +òd / {que} {END} { add("od"); } // W (G) +ùm / {que} {END} { add("um"); } // W (G) +ùs / {que} {END} { add("us"); } // W G + +ès / {que} {END} { add("es"); } // (G) +^ quì / {END} { add("qui"); } // W ?? +^ Quì / {END} { add("Qui"); } // W ?? +à c / {END} { add("ac"); } // W ?? +èr / {END} { add("er"); } // W ?? +èt / {END} { add("et"); } // W ?? +ù / {END} { add("u"); } // W ?? +ùl / {END} { add("ul"); } // W ?? + +// 2.3 circumflex accent +^ hîc / {END} { add("hic"); } // W G +^ Hîc / {END} { add("Hic"); } // W G +^ ô / {END} { add("o"); } // G +â / {que} {END} { add("a"); } // W G +ûs / {END} { add("us"); } // W G +âr { add("ar"); } // W (G) --> this is only a rough approximation! + +// 2.4 trema +// 2.4.1 common cases +aë { cv = VOWEL; add("ae"); } +oë { cv = VOWEL; add("oe"); } +// 2.4.2 rare cases +oï { cv = VOWEL; add("oi"); } +uï { cv = VOWEL; add("ui"); } +// 2.4.3 extremely rare cases +uü { cv = VOWEL; add("uu"); } + + +// 3. rules for u and v + +// 3.1 rules for u --> v + +// peruenias --> pervenias, interuallum --> intervallum +^ {prefixCons} / {lb} { cv = VOWEL; add(yytext().replace("Å¿", "s")); } // not cv = CONS ! + +// uellet --> vellet +^ [uU] / {Vowel} { cv = VOWEL; add(yytext().replaceAll("u", "v").replaceAll("U", "V")); } + +// diuidatur --> dividatur +// ut, volui: unchanged +// no rule for veruina because we cannot distinguish it from volui +[uU] / {Vowel} { + switch(cv) { + case VOWEL: add(yytext().replace("u", "v").replace("U", "V")); break; + default: cv = VOWEL; add(yytext()); break; + } + } + +// 3.2 rules for v --> u + +// qvam --> quam +qv { cv = CONS; add("qu"); } // the replaced v still counts as consonant +Qv { cv = CONS; add("Qu"); } +QV { cv = CONS; add("QU"); } + +// febrvarius --> februarius +// curva: unchanged +{LR} [vV] { + switch(cv) { + case CONS: add(yytext().replace("v", "u").replace("V", "U")); break; + default: cv = CONS; add(yytext()); break; + } + } + +// februarivs --> februarius +v / {lb} {Cons} { cv = CONS; add("u"); } +V / {lb} {Cons} { cv = CONS; add("U"); } + +// 3.3 override default rule for . + +{Vowel} { cv = VOWEL; add(yytext()); } +{Cons} { cv = CONS; add(yytext()); } +[yY] { cv = 0; add(yytext()); } + +@ { problem = 1; cv = 0; add(yytext()); } +{LB} { add(yytext()); } +. { problem = 1; cv = 0; add(yytext()); } // in particular from Arboreal: "〈" (2329), "〉" (232A), Ç, ç + +} + + +<DISP, RENAISSANCE_DISP> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized; + } + } +} + +<DICT, RENAISSANCE_DICT> { + +{END} { + switch (problem) { + case 1: return ""; + default: return normalized.replaceAll(LB, ""); + } + } +} + +<SEARCH, RENAISSANCE_SEARCH> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized.replaceAll(LB, "").toLowerCase(); + } + } +} + + +/* + +Annahmen: +- die Routine wird wortweise aufgerufen, mit einem \n am Ende des Strings +- Zeilenumbrüche innerhalb des Wortes sind als hyphen/soft hyphen plus space markiert + + +TO DO: + +LA: Nochmal überlegen, ob man Ææęà èòùœ in der Vokal-Klasse weglassen kann. Sie schaden aber auch nicht. (Oder doch !?) Unterscheide Vokal-Klassen vor und nach dem u ? +LA: Diakritika nochmal mit Paul durchgehen +LA: Die Disambiguierungen durch die Diakritika fehlen noch. +LA: ist J wirklich ein Problemfall? +LA: gibt es Wörter wie super-rv... oder super-lv... in Klein- oder Großbuchstaben? + +*/
Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexNL.class has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexNL.lex Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,105 @@ +/* + * Normalization rules for Dutch text + * [this is a JFlex specification] + * + * Wolfgang Schmidle + * version 2011-07-12 + * + */ + +package de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang; + +%% + +%public +%class MpdlNormalizerLexNL +%type java.lang.String +%unicode + +// Dutch: nl + +%states DISP, DICT, SEARCH + +%{ + private String original = ""; + private String normalized = ""; + private int problem = 0; + + private void add (String norm) { + original += yytext(); + normalized += norm; + } + + private static final String LB = "[\u002d\u00ad] "; +%} + +hyphen = [-\u{00ad}] // hyphen and soft hyphen +LB = {hyphen} \u0020 +// lb = ({hyphen} \u0020)? + +END = \n + +%% + +// jump over empty xml elements +"<"[^><]+"/>" { add(yytext()); } +"-<"[^><]+"/>" { add(yytext()); } +"<"[^><]+"></"[^><]+">" { add(yytext()); } +"-<"[^><]+"></"[^><]+">" { add(yytext()); } + +<DISP, DICT, SEARCH> { + +ſ { add("s"); } + +} + + +// default + +@ { problem = 1; add(yytext()); } +{LB} { add(yytext()); } +. { add(yytext()); } + + +<DISP> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized; + } + } +} + +<DICT> { + +{END} { + switch (problem) { + case 1: return ""; + default: return normalized.replaceAll(LB, ""); + } + } +} + +<SEARCH> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized.replaceAll(LB, "").toLowerCase(); + } + } +} + + +/* + +Annahmen: +- die Routine wird wortweise aufgerufen, mit einem \n am Ende des Strings +- Zeilenumbrüche innerhalb des Wortes sind als hyphen/soft hyphen plus space markiert + +TO DO: + +NL: vollständig? + +*/
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexTemplate.lex Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,89 @@ +/* + * Template for normalization rules + * [this is a JFlex specification] + * + * Wolfgang Schmidle + * version 2011-07-12 + * + */ + +package de.mpg.mpiwg.berlin.mpdl.lt.analyzer.lang; + +%% + +%public +%class MpdlNormalizerLexTemplate +%type java.lang.String +%unicode + +// Language: list of ISO codes + +%states DISP, DICT, SEARCH + +%{ + private String original = ""; + private String normalized = ""; + private int problem = 0; + + private void add (String norm) { + original += yytext(); + normalized += norm; + } + + private static final String LB = "[\u002d\u00ad] "; +%} + +hyphen = [-\u{00ad}] // hyphen and soft hyphen +LB = {hyphen} \u0020 +// lb = ({hyphen} \u0020)? + +END = \n + +%% + +<DISP, DICT, SEARCH> { + +Å¿ { add("s"); } // sample rule + +} + + +// default rules + +@ { problem = 1; add(yytext()); } +{LB} { add(yytext()); } +. { add(yytext()); } + + +// at the end, determine which string to return + +<DISP> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized; + } + } +} + +<DICT> { + +{END} { + switch (problem) { + case 1: return ""; + default: return normalized.replaceAll(LB, ""); + } + } +} + +<SEARCH> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized.replaceAll(LB, "").toLowerCase(); + } + } +} +
Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexZH.class has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexZH.lex Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,120 @@ +/* + * Normalization rules for Chinese text + * [this is a JFlex specification] + * + * Wolfgang Schmidle + * version 2011-02-28 + * + */ + +package de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang; + +%% + +%public +%class MpdlNormalizerLexZH +%type java.lang.String +%unicode + +// classical Chinese: zh, zho, zho-Hant + +%states DISP, DICT, SEARCH + +%{ + private String original = ""; + private String normalized = ""; + private int problem = 0; + + private void add (String norm) { + original += yytext(); + normalized += norm; + } +%} + +ZWS = [\u{200b}] + +END = \n + +%% + +// Normalization in Chinese means that character variants will be replaced by their standard characters +// if there is no doubt about what the standard character is. + +// The input is supposed to be a single Chinese character, but strings of characters are also handled correctly. + +<DISP, DICT, SEARCH> { + +// Codepoint < FFFF + +倂 { add("ä½µ"); } // 5002 --> 4F75 +å‚ | åœ { add("åŸ"); } // 5081, 53DC --> 53DF +ç«’ { add("奇"); } // 7AD2 --> 5947 +å¹· { add("å¹¶"); } // 5E77 --> 5E76 +牀 { add("床"); } // 7240 --> 5E8A +æ—¹ { add("時"); } // 65F9 --> 6642 +æ´ { add("æ·"); } // 6B74 --> 6B77 +爲 { add("為"); } // 7232 --> 70BA +éš‚ { add("é™°"); } // 9682 --> 9670 +é£ { add("é¢"); } // 9763 --> 9762 +ï¨ { add("ç²¾"); } // FA1D --> 7CBE (FA1D is a compatibility ideograph) + +// Codepoint > FFFF + +// note that [ABC] is not equivalent to A | B | C for codepoints above FFFF due to their internal encoding: +// for example, 悁(U+2F88D) is represented as a sequence of two codepoints: D87E DC8D +// i.e. never use [ABC] but A | B | C + +悁{ add("庶"); } // 2F88D --> 5EB6 (2F88D is a compatibility ideograph) + +} + +<DICT, SEARCH> { + +// remove Zero Width Space (if there is any in the the input string) + +{ZWS} { add(""); } + +} + +// default + +@ { problem = 1; add(yytext()); } +. { add(yytext()); } + + +<DISP, SEARCH> { + +{END} { + switch (problem) { + case 1: return original; + default: return normalized; + } + } +} + +<DICT> { + +{END} { + switch (problem) { + case 1: return ""; + default: return normalized; + } + } +} + + +/* + +Annahmen: +- die Routine wird zeichenweise (oder mit mehr als einem Zeichen) aufgerufen, mit einem \n am Ende des Strings +- es gibt keine Zeilenumbrüche + +TO DO: + +ZH: Liste ergänzen +ZH: was ist, wenn man wirklich die Variante, die im Text steht, nachschlagen will? Dann muss man das Zeichen wohl selbst rauskopieren. +ZH: sollen lateinische Buchstaben bewirken, dass problem = 1 ist? +ZH: sollen Zeilenumbrüche rausgenommen werden, auch wenn sie in korrekt markiertem Text nicht vorkommen? +ZH: was ist, wenn beijing übergeben wird und einen Zeilenumbruch enthält? Verlässt sich der Wrapper darauf, dass die Zeichenzahl gleich bleibt, oder macht er ein hyphen rein? was macht <place> oder <reg>? + +*/
Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/tokenize/Token.class has changed
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Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/tokenize/XmlTokenizerContentHandler$Element.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/tokenize/XmlTokenizerContentHandler.class has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/transcode/Betacode2Unicode.lex Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,332 @@ +package de.mpg.mpiwg.berlin.mpdl.lt.general; + +%% +%{ + /* + * Betacode to Unicode conversion + */ + + private String toUnicode(int in) { + char c = (char) in; + Character ch = new Character(c); + String retString = ch.toString(); + return retString; + } + +%} + +%class Betacode2UnicodeLex +%public +%type java.lang.String +%unicode +%% + + +"<"[^>]+">" { return yytext(); } + +"*j" { return "H"; } +"j" { return "h"; } +"*v" { return "F"; } +"v" { return "f"; } +"*s" { return toUnicode(0x03a3); } + +"!" { return "."; } +":" { return toUnicode(0x00B7); } /* MPDL update */ + +"a)" { return toUnicode(0x1F00); } +"a(" { return toUnicode(0x1F01); } +"a)\\" { return toUnicode(0x1F02); } +"a(\\" { return toUnicode(0x1F03); } +"a)/" { return toUnicode(0x1F04); } +"a(/" { return toUnicode(0x1F05); } +"a)=" { return toUnicode(0x1F06); } +"a(=" { return toUnicode(0x1F07); } +"*)a" { return toUnicode(0x1F08); } +"*(a" { return toUnicode(0x1F09); } +"*)\\a" { return toUnicode(0x1F0A); } +"*(\\a" { return toUnicode(0x1F0B); } +"*)/a" { return toUnicode(0x1F0C); } +"*(/a" { return toUnicode(0x1F0D); } +"*)=a" { return toUnicode(0x1F0E); } +"*(=a" { return toUnicode(0x1F0F); } +"e)" { return toUnicode(0x1F10); } +"e(" { return toUnicode(0x1F11); } +"e)\\" { return toUnicode(0x1F12); } +"e(\\" { return toUnicode(0x1F13); } +"e)/" { return toUnicode(0x1F14); } +"e(/" { return toUnicode(0x1F15); } +"*)e" { return toUnicode(0x1F18); } +"*(e" { return toUnicode(0x1F19); } +"*)\\e" { return toUnicode(0x1F1A); } +"*(\\e" { return toUnicode(0x1F1B); } +"*)/e" { return toUnicode(0x1F1C); } +"*(/e" { return toUnicode(0x1F1D); } +"h)" { return toUnicode(0x1F20); } +"h(" { return toUnicode(0x1F21); } +"h)\\" { return toUnicode(0x1F22); } +"h(\\" { return toUnicode(0x1F23); } +"h)/" { return toUnicode(0x1F24); } +"h(/" { return toUnicode(0x1F25); } +"h)=" { return toUnicode(0x1F26); } +"h(=" { return toUnicode(0x1F27); } +"*)h" { return toUnicode(0x1F28); } +"*(h" { return toUnicode(0x1F29); } +"*)\\h" { return toUnicode(0x1F2A); } +"*(\\h" { return toUnicode(0x1F2B); } +"*)/h" { return toUnicode(0x1F2C); } +"*(/h" { return toUnicode(0x1F2D); } +"*)=h" { return toUnicode(0x1F2E); } +"*(=h" { return toUnicode(0x1F2F); } +"i)" { return toUnicode(0x1F30); } +"i(" { return toUnicode(0x1F31); } +"i)\\" { return toUnicode(0x1F32); } +"i(\\" { return toUnicode(0x1F33); } +"i)/" { return toUnicode(0x1F34); } +"i(/" { return toUnicode(0x1F35); } +"i)=" { return toUnicode(0x1F36); } +"i(=" { return toUnicode(0x1F37); } +"*)i" { return toUnicode(0x1F38); } +"*(i" { return toUnicode(0x1F39); } +"*)\\i" { return toUnicode(0x1F3A); } +"*(\\i" { return toUnicode(0x1F3B); } +"*)/i" { return toUnicode(0x1F3C); } +"*(/i" { return toUnicode(0x1F3D); } +"*)=i" { return toUnicode(0x1F3E); } +"*(=i" { return toUnicode(0x1F3F); } +"o)" { return toUnicode(0x1F40); } +"o(" { return toUnicode(0x1F41); } +"o)\\" { return toUnicode(0x1F42); } +"o(\\" { return toUnicode(0x1F43); } +"o)/" { return toUnicode(0x1F44); } +"o(/" { return toUnicode(0x1F45); } +"*)o" { return toUnicode(0x1F48); } +"*(o" { return toUnicode(0x1F49); } +"*)\\o" { return toUnicode(0x1F4A); } +"*(\\o" { return toUnicode(0x1F4B); } +"*)/o" { return toUnicode(0x1F4C); } +"*(/o" { return toUnicode(0x1F4D); } +"u)" { return toUnicode(0x1F50); } +"u(" { return toUnicode(0x1F51); } +"u)\\" { return toUnicode(0x1F52); } +"u(\\" { return toUnicode(0x1F53); } +"u)/" { return toUnicode(0x1F54); } +"u(/" { return toUnicode(0x1F55); } +"u)=" { return toUnicode(0x1F56); } +"u(=" { return toUnicode(0x1F57); } +"*(u" { return toUnicode(0x1F59); } +"*(\\u" { return toUnicode(0x1F5B); } +"*(/u" { return toUnicode(0x1F5D); } +"*(=u" { return toUnicode(0x1F5F); } +"w)" { return toUnicode(0x1F60); } +"w(" { return toUnicode(0x1F61); } +"w)\\" { return toUnicode(0x1F62); } +"w(\\" { return toUnicode(0x1F63); } +"w)/" { return toUnicode(0x1F64); } +"w(/" { return toUnicode(0x1F65); } +"w)=" { return toUnicode(0x1F66); } +"w(=" { return toUnicode(0x1F67); } +"*)w" { return toUnicode(0x1F68); } +"*(w" { return toUnicode(0x1F69); } +"*)\\w" { return toUnicode(0x1F6A); } +"*(\\w" { return toUnicode(0x1F6B); } +"*)/w" { return toUnicode(0x1F6C); } +"*(/w" { return toUnicode(0x1F6D); } +"*)=w" { return toUnicode(0x1F6E); } +"*(=w" { return toUnicode(0x1F6F); } +"a\\" { return toUnicode(0x1F70); } +"a/" { return toUnicode(0x1F71); } +"e\\" { return toUnicode(0x1F72); } +"e/" { return toUnicode(0x1F73); } +"h\\" { return toUnicode(0x1F74); } +"h/" { return toUnicode(0x1F75); } +"i\\" { return toUnicode(0x1F76); } +"i/" { return toUnicode(0x1F77); } +"o\\" { return toUnicode(0x1F78); } +"o/" { return toUnicode(0x1F79); } +"u\\" { return toUnicode(0x1F7A); } +"u/" { return toUnicode(0x1F7B); } +"w\\" { return toUnicode(0x1F7C); } +"w/" { return toUnicode(0x1F7D); } +"a)|" { return toUnicode(0x1F80); } +"a(|" { return toUnicode(0x1F81); } +"a)\\|" { return toUnicode(0x1F82); } +"a(\\|" { return toUnicode(0x1F83); } +"a)/|" { return toUnicode(0x1F84); } +"a(/|" { return toUnicode(0x1F85); } +"a)=|" { return toUnicode(0x1F86); } +"a(=|" { return toUnicode(0x1F87); } +"*)|a" { return toUnicode(0x1F88); } +"*(|a" { return toUnicode(0x1F89); } +"*)\\|a" { return toUnicode(0x1F8A); } +"*(\\|a" { return toUnicode(0x1F8B); } +"*)/|a" { return toUnicode(0x1F8C); } +"*(/|a" { return toUnicode(0x1F8D); } +"*)=|a" { return toUnicode(0x1F8E); } +"*(=|a" { return toUnicode(0x1F8F); } +"h)|" { return toUnicode(0x1F90); } +"h(|" { return toUnicode(0x1F91); } +"h)\\|" { return toUnicode(0x1F92); } +"h(\\|" { return toUnicode(0x1F93); } +"h)/|" { return toUnicode(0x1F94); } +"h(/|" { return toUnicode(0x1F95); } +"h)=|" { return toUnicode(0x1F96); } +"h(=|" { return toUnicode(0x1F97); } +"*)|h" { return toUnicode(0x1F98); } +"*(|h" { return toUnicode(0x1F99); } +"*)\\|h" { return toUnicode(0x1F9A); } +"*(\\|h" { return toUnicode(0x1F9B); } +"*)/|h" { return toUnicode(0x1F9C); } +"*(/|h" { return toUnicode(0x1F9D); } +"*)=|h" { return toUnicode(0x1F9E); } +"*(=|h" { return toUnicode(0x1F9F); } +"w)|" { return toUnicode(0x1FA0); } +"w(|" { return toUnicode(0x1FA1); } +"w)\\|" { return toUnicode(0x1FA2); } +"w(\\|" { return toUnicode(0x1FA3); } +"w)/|" { return toUnicode(0x1FA4); } +"w(/|" { return toUnicode(0x1FA5); } +"w)=|" { return toUnicode(0x1FA6); } +"w(=|" { return toUnicode(0x1FA7); } +"*)|w" { return toUnicode(0x1FA8); } +"*(|w" { return toUnicode(0x1FA9); } +"*)\\|w" { return toUnicode(0x1FAA); } +"*(\\|w" { return toUnicode(0x1FAB); } +"*)/|w" { return toUnicode(0x1FAC); } +"*(/|w" { return toUnicode(0x1FAD); } +"*)=|w" { return toUnicode(0x1FAE); } +"*(=|w" { return toUnicode(0x1FAF); } +"a^" { return toUnicode(0x1FB0); } +"a_" { return toUnicode(0x1FB1); } +"a\\|" { return toUnicode(0x1FB2); } +"a|" { return toUnicode(0x1FB3); } +"a/|" { return toUnicode(0x1FB4); } +"a=" { return toUnicode(0x1FB6); } +"a=|" { return toUnicode(0x1FB7); } +"*a^" { return toUnicode(0x1FB8); } +"*a_" { return toUnicode(0x1FB9); } +"*a\\" { return toUnicode(0x1FBA); } +"*a/" { return toUnicode(0x1FBB); } +"*a|" { return toUnicode(0x1FBC); } +"h\\|" { return toUnicode(0x1FC2); } +"h|" { return toUnicode(0x1FC3); } +"h/|" { return toUnicode(0x1FC4); } +"h=" { return toUnicode(0x1FC6); } +"h=|" { return toUnicode(0x1FC7); } +"*e\\" { return toUnicode(0x1FC8); } +"*e/" { return toUnicode(0x1FC9); } +"*h\\" { return toUnicode(0x1FCA); } +"*h/" { return toUnicode(0x1FCB); } +"*h|" { return toUnicode(0x1FCC); } +"i^" { return toUnicode(0x1FD0); } +"i_" { return toUnicode(0x1FD1); } +"i+\\" { return toUnicode(0x1FD2); } +"i+/" { return toUnicode(0x1FD3); } +"i=" { return toUnicode(0x1FD6); } +"i+=" { return toUnicode(0x1FD7); } +"*i^" { return toUnicode(0x1FD8); } +"*i_" { return toUnicode(0x1FD9); } +"*i\\" { return toUnicode(0x1FDA); } +"*i/" { return toUnicode(0x1FDB); } +"u^" { return toUnicode(0x1FE0); } +"u_" { return toUnicode(0x1FE1); } +"u+\\" { return toUnicode(0x1FE2); } +"u+/" { return toUnicode(0x1FE3); } +"r)" { return toUnicode(0x1FE4); } +"r(" { return toUnicode(0x1FE5); } +"u=" { return toUnicode(0x1FE6); } +"u+=" { return toUnicode(0x1FE7); } +"*u^" { return toUnicode(0x1FE8); } +"*u_" { return toUnicode(0x1FE9); } +"*u\\" { return toUnicode(0x1FEA); } +"*u/" { return toUnicode(0x1FEB); } +"*(r" { return toUnicode(0x1FEC); } +"w\\|" { return toUnicode(0x1FF2); } +"w|" { return toUnicode(0x1FF3); } +"w/|" { return toUnicode(0x1FF4); } +"*w\\" { return toUnicode(0x1FFA); } +"*w/" { return toUnicode(0x1FFB); } +"*w|" { return toUnicode(0x1FFC); } +"w=" { return toUnicode(0x1FF6); } +"w=|" { return toUnicode(0x1FF7); } +"*o\\" { return toUnicode(0x1FF8); } +"*o/" { return toUnicode(0x1FF9); } + +"\\" { return toUnicode(0x0300); } +"/" { return toUnicode(0x0301); } +"_" { return toUnicode(0x0304); } +"^" { return toUnicode(0x0306); } +"+" { return toUnicode(0x0308); } +"=" { return toUnicode(0x0302); } +")" { return toUnicode(0x0313); } +"(" { return toUnicode(0x0314); } +"?" { return toUnicode(0x0323); } +"|" { return toUnicode(0x0345); } + +"a" { return toUnicode(0x03b1); } /* MPDL update */ +"*a" { return toUnicode(0x0391); } /* MPDL update */ +"b" { return toUnicode(0x03b2); } /* MPDL update */ +"*b" { return toUnicode(0x0392); } /* MPDL update */ +"g" { return toUnicode(0x03b3); } /* MPDL update */ +"*g" { return toUnicode(0x0393); } /* MPDL update */ +"d" { return toUnicode(0x03b4); } /* MPDL update */ +"*d" { return toUnicode(0x0394); } /* MPDL update */ +"e" { return toUnicode(0x03b5); } /* MPDL update */ +"*e" { return toUnicode(0x0395); } /* MPDL update */ +"z" { return toUnicode(0x03b6); } /* MPDL update */ +"*z" { return toUnicode(0x0396); } /* MPDL update */ +"h" { return toUnicode(0x03b7); } /* MPDL update */ +"*h" { return toUnicode(0x0397); } /* MPDL update */ +"q" { return toUnicode(0x03b8); } /* MPDL update */ +"*q" { return toUnicode(0x0398); } /* MPDL update */ +"i" { return toUnicode(0x03b9); } /* MPDL update */ +"*i" { return toUnicode(0x0399); } /* MPDL update */ +"k" { return toUnicode(0x03ba); } /* MPDL update */ +"*k" { return toUnicode(0x039a); } /* MPDL update */ +"l" { return toUnicode(0x03bb); } /* MPDL update */ +"*l" { return toUnicode(0x039b); } /* MPDL update */ +"m" { return toUnicode(0x03bc); } /* MPDL update */ +"*m" { return toUnicode(0x039c); } /* MPDL update */ +"n" { return toUnicode(0x03bd); } /* MPDL update */ +"*n" { return toUnicode(0x039d); } /* MPDL update */ +"c" { return toUnicode(0x03be); } /* MPDL update */ +"*c" { return toUnicode(0x039e); } /* MPDL update */ +"o" { return toUnicode(0x03bf); } /* MPDL update */ +"*o" { return toUnicode(0x039f); } /* MPDL update */ +"p" { return toUnicode(0x03c0); } /* MPDL update */ +"*p" { return toUnicode(0x03a0); } /* MPDL update */ +"r" { return toUnicode(0x03c1); } /* MPDL update */ +"*r" { return toUnicode(0x03a1); } /* MPDL update */ + +"*s" { return toUnicode(0x03a3); } /* MPDL update */ +"s1" { return toUnicode(0x03c3); } /* mdh 2002-01-07 */ +"s"/\-\- { return toUnicode(0x03c2); } +"s"/\> }[a-z\?\!0-9*=\/()\'\-] { return toUnicode(0x03c3); } /* MPDL update */ +"s"/\< { return toUnicode(0x03c2); } /* MPDL update */ +"s"/[\[\]][a-z\?\!0-9*=\/()\'\-] { return toUnicode(0x03c3); } /* MPDL update */ +"s"/\??[^a-z0-9*=\/()\'\-\[\?] { return toUnicode(0x03c2); } +"s" { return toUnicode(0x03c3); } /* MPDL update */ + +"t" { return toUnicode(0x03c4); } /* MPDL update */ +"*t" { return toUnicode(0x03a4); } /* MPDL update */ +"u" { return toUnicode(0x03c5); } /* MPDL update */ +"*u" { return toUnicode(0x03a5); } /* MPDL update */ +"f" { return toUnicode(0x03c6); } /* MPDL update */ +"*f" { return toUnicode(0x03a6); } /* MPDL update */ +"x" { return toUnicode(0x03c7); } /* MPDL update */ +"*x" { return toUnicode(0x03a7); } /* MPDL update */ +"y" { return toUnicode(0x03c8); } /* MPDL update */ +"*y" { return toUnicode(0x03a8); } /* MPDL update */ +"w" { return toUnicode(0x03c9); } /* MPDL update */ +"*w" { return toUnicode(0x03a9); } /* MPDL update */ + +[\&_]"vert;" { return "|"; } +[\&_]"lpar;" { return "("; } +[\&_]"rpar;" { return ")"; } +[\_\&]"lt;" { return "<"; } +[\_\&]"gt;" { return ">"; } +"'" { return "'"; } /* MPDL update */ + +"&"[a-zA-Z]+";" { return yytext(); } + +. { return yytext(); } +\n { return yytext(); } \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/transcode/Betacode2Unicode.lex.old Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,318 @@ +package de.mpg.mpiwg.berlin.mpdl.lt.general; + +%% +%{ + /* + * Betacode to Unicode conversion + */ + + private int isUpper = 0; + + private String toUnicodeGreek(int in) { + String retStr = toUnicode(in - (isUpper * 0x0020)); + isUpper = 0; + return retStr; + } + + private String toUnicode(int in) { + char c = (char) in; + Character ch = new Character(c); + String retString = ch.toString(); + return retString; + } + +%} + +%class Betacode2UnicodeLex +%public +%type java.lang.String +%unicode +%% + + +"<"[^>]+">" { return yytext(); } + +"*j" { return "H"; } +"j" { return "h"; } +"*v" { return "F"; } +"v" { return "f"; } +"*s" { return toUnicode(0x03a3); } + +"!" { return "."; } +":" { return toUnicode(0x00B7); } /* MPDL update */ + +"a)" { return toUnicode(0x1F00); } +"a(" { return toUnicode(0x1F01); } +"a)\\" { return toUnicode(0x1F02); } +"a(\\" { return toUnicode(0x1F03); } +"a)/" { return toUnicode(0x1F04); } +"a(/" { return toUnicode(0x1F05); } +"a)=" { return toUnicode(0x1F06); } +"a(=" { return toUnicode(0x1F07); } +"*)a" { return toUnicode(0x1F08); } +"*(a" { return toUnicode(0x1F09); } +"*)\\a" { return toUnicode(0x1F0A); } +"*(\\a" { return toUnicode(0x1F0B); } +"*)/a" { return toUnicode(0x1F0C); } +"*(/a" { return toUnicode(0x1F0D); } +"*)=a" { return toUnicode(0x1F0E); } +"*(=a" { return toUnicode(0x1F0F); } +"e)" { return toUnicode(0x1F10); } +"e(" { return toUnicode(0x1F11); } +"e)\\" { return toUnicode(0x1F12); } +"e(\\" { return toUnicode(0x1F13); } +"e)/" { return toUnicode(0x1F14); } +"e(/" { return toUnicode(0x1F15); } +"*)e" { return toUnicode(0x1F18); } +"*(e" { return toUnicode(0x1F19); } +"*)\\e" { return toUnicode(0x1F1A); } +"*(\\e" { return toUnicode(0x1F1B); } +"*)/e" { return toUnicode(0x1F1C); } +"*(/e" { return toUnicode(0x1F1D); } +"h)" { return toUnicode(0x1F20); } +"h(" { return toUnicode(0x1F21); } +"h)\\" { return toUnicode(0x1F22); } +"h(\\" { return toUnicode(0x1F23); } +"h)/" { return toUnicode(0x1F24); } +"h(/" { return toUnicode(0x1F25); } +"h)=" { return toUnicode(0x1F26); } +"h(=" { return toUnicode(0x1F27); } +"*)h" { return toUnicode(0x1F28); } +"*(h" { return toUnicode(0x1F29); } +"*)\\h" { return toUnicode(0x1F2A); } +"*(\\h" { return toUnicode(0x1F2B); } +"*)/h" { return toUnicode(0x1F2C); } +"*(/h" { return toUnicode(0x1F2D); } +"*)=h" { return toUnicode(0x1F2E); } +"*(=h" { return toUnicode(0x1F2F); } +"i)" { return toUnicode(0x1F30); } +"i(" { return toUnicode(0x1F31); } +"i)\\" { return toUnicode(0x1F32); } +"i(\\" { return toUnicode(0x1F33); } +"i)/" { return toUnicode(0x1F34); } +"i(/" { return toUnicode(0x1F35); } +"i)=" { return toUnicode(0x1F36); } +"i(=" { return toUnicode(0x1F37); } +"*)i" { return toUnicode(0x1F38); } +"*(i" { return toUnicode(0x1F39); } +"*)\\i" { return toUnicode(0x1F3A); } +"*(\\i" { return toUnicode(0x1F3B); } +"*)/i" { return toUnicode(0x1F3C); } +"*(/i" { return toUnicode(0x1F3D); } +"*)=i" { return toUnicode(0x1F3E); } +"*(=i" { return toUnicode(0x1F3F); } +"o)" { return toUnicode(0x1F40); } +"o(" { return toUnicode(0x1F41); } +"o)\\" { return toUnicode(0x1F42); } +"o(\\" { return toUnicode(0x1F43); } +"o)/" { return toUnicode(0x1F44); } +"o(/" { return toUnicode(0x1F45); } +"*)o" { return toUnicode(0x1F48); } +"*(o" { return toUnicode(0x1F49); } +"*)\\o" { return toUnicode(0x1F4A); } +"*(\\o" { return toUnicode(0x1F4B); } +"*)/o" { return toUnicode(0x1F4C); } +"*(/o" { return toUnicode(0x1F4D); } +"u)" { return toUnicode(0x1F50); } +"u(" { return toUnicode(0x1F51); } +"u)\\" { return toUnicode(0x1F52); } +"u(\\" { return toUnicode(0x1F53); } +"u)/" { return toUnicode(0x1F54); } +"u(/" { return toUnicode(0x1F55); } +"u)=" { return toUnicode(0x1F56); } +"u(=" { return toUnicode(0x1F57); } +"*(u" { return toUnicode(0x1F59); } +"*(\\u" { return toUnicode(0x1F5B); } +"*(/u" { return toUnicode(0x1F5D); } +"*(=u" { return toUnicode(0x1F5F); } +"w)" { return toUnicode(0x1F60); } +"w(" { return toUnicode(0x1F61); } +"w)\\" { return toUnicode(0x1F62); } +"w(\\" { return toUnicode(0x1F63); } +"w)/" { return toUnicode(0x1F64); } +"w(/" { return toUnicode(0x1F65); } +"w)=" { return toUnicode(0x1F66); } +"w(=" { return toUnicode(0x1F67); } +"*)w" { return toUnicode(0x1F68); } +"*(w" { return toUnicode(0x1F69); } +"*)\\w" { return toUnicode(0x1F6A); } +"*(\\w" { return toUnicode(0x1F6B); } +"*)/w" { return toUnicode(0x1F6C); } +"*(/w" { return toUnicode(0x1F6D); } +"*)=w" { return toUnicode(0x1F6E); } +"*(=w" { return toUnicode(0x1F6F); } +"a\\" { return toUnicode(0x1F70); } +"a/" { return toUnicode(0x1F71); } +"e\\" { return toUnicode(0x1F72); } +"e/" { return toUnicode(0x1F73); } +"h\\" { return toUnicode(0x1F74); } +"h/" { return toUnicode(0x1F75); } +"i\\" { return toUnicode(0x1F76); } +"i/" { return toUnicode(0x1F77); } +"o\\" { return toUnicode(0x1F78); } +"o/" { return toUnicode(0x1F79); } +"u\\" { return toUnicode(0x1F7A); } +"u/" { return toUnicode(0x1F7B); } +"w\\" { return toUnicode(0x1F7C); } +"w/" { return toUnicode(0x1F7D); } +"a)|" { return toUnicode(0x1F80); } +"a(|" { return toUnicode(0x1F81); } +"a)\\|" { return toUnicode(0x1F82); } +"a(\\|" { return toUnicode(0x1F83); } +"a)/|" { return toUnicode(0x1F84); } +"a(/|" { return toUnicode(0x1F85); } +"a)=|" { return toUnicode(0x1F86); } +"a(=|" { return toUnicode(0x1F87); } +"*)|a" { return toUnicode(0x1F88); } +"*(|a" { return toUnicode(0x1F89); } +"*)\\|a" { return toUnicode(0x1F8A); } +"*(\\|a" { return toUnicode(0x1F8B); } +"*)/|a" { return toUnicode(0x1F8C); } +"*(/|a" { return toUnicode(0x1F8D); } +"*)=|a" { return toUnicode(0x1F8E); } +"*(=|a" { return toUnicode(0x1F8F); } +"h)|" { return toUnicode(0x1F90); } +"h(|" { return toUnicode(0x1F91); } +"h)\\|" { return toUnicode(0x1F92); } +"h(\\|" { return toUnicode(0x1F93); } +"h)/|" { return toUnicode(0x1F94); } +"h(/|" { return toUnicode(0x1F95); } +"h)=|" { return toUnicode(0x1F96); } +"h(=|" { return toUnicode(0x1F97); } +"*)|h" { return toUnicode(0x1F98); } +"*(|h" { return toUnicode(0x1F99); } +"*)\\|h" { return toUnicode(0x1F9A); } +"*(\\|h" { return toUnicode(0x1F9B); } +"*)/|h" { return toUnicode(0x1F9C); } +"*(/|h" { return toUnicode(0x1F9D); } +"*)=|h" { return toUnicode(0x1F9E); } +"*(=|h" { return toUnicode(0x1F9F); } +"w)|" { return toUnicode(0x1FA0); } +"w(|" { return toUnicode(0x1FA1); } +"w)\\|" { return toUnicode(0x1FA2); } +"w(\\|" { return toUnicode(0x1FA3); } +"w)/|" { return toUnicode(0x1FA4); } +"w(/|" { return toUnicode(0x1FA5); } +"w)=|" { return toUnicode(0x1FA6); } +"w(=|" { return toUnicode(0x1FA7); } +"*)|w" { return toUnicode(0x1FA8); } +"*(|w" { return toUnicode(0x1FA9); } +"*)\\|w" { return toUnicode(0x1FAA); } +"*(\\|w" { return toUnicode(0x1FAB); } +"*)/|w" { return toUnicode(0x1FAC); } +"*(/|w" { return toUnicode(0x1FAD); } +"*)=|w" { return toUnicode(0x1FAE); } +"*(=|w" { return toUnicode(0x1FAF); } +"a^" { return toUnicode(0x1FB0); } +"a_" { return toUnicode(0x1FB1); } +"a\\|" { return toUnicode(0x1FB2); } +"a|" { return toUnicode(0x1FB3); } +"a/|" { return toUnicode(0x1FB4); } +"a=" { return toUnicode(0x1FB6); } +"a=|" { return toUnicode(0x1FB7); } +"*a^" { return toUnicode(0x1FB8); } +"*a_" { return toUnicode(0x1FB9); } +"*a\\" { return toUnicode(0x1FBA); } +"*a/" { return toUnicode(0x1FBB); } +"*a|" { return toUnicode(0x1FBC); } +"h\\|" { return toUnicode(0x1FC2); } +"h|" { return toUnicode(0x1FC3); } +"h/|" { return toUnicode(0x1FC4); } +"h=" { return toUnicode(0x1FC6); } +"h=|" { return toUnicode(0x1FC7); } +"*e\\" { return toUnicode(0x1FC8); } +"*e/" { return toUnicode(0x1FC9); } +"*h\\" { return toUnicode(0x1FCA); } +"*h/" { return toUnicode(0x1FCB); } +"*h|" { return toUnicode(0x1FCC); } +"i^" { return toUnicode(0x1FD0); } +"i_" { return toUnicode(0x1FD1); } +"i+\\" { return toUnicode(0x1FD2); } +"i+/" { return toUnicode(0x1FD3); } +"i=" { return toUnicode(0x1FD6); } +"i+=" { return toUnicode(0x1FD7); } +"*i^" { return toUnicode(0x1FD8); } +"*i_" { return toUnicode(0x1FD9); } +"*i\\" { return toUnicode(0x1FDA); } +"*i/" { return toUnicode(0x1FDB); } +"u^" { return toUnicode(0x1FE0); } +"u_" { return toUnicode(0x1FE1); } +"u+\\" { return toUnicode(0x1FE2); } +"u+/" { return toUnicode(0x1FE3); } +"r)" { return toUnicode(0x1FE4); } +"r(" { return toUnicode(0x1FE5); } +"u=" { return toUnicode(0x1FE6); } +"u+=" { return toUnicode(0x1FE7); } +"*u^" { return toUnicode(0x1FE8); } +"*u_" { return toUnicode(0x1FE9); } +"*u\\" { return toUnicode(0x1FEA); } +"*u/" { return toUnicode(0x1FEB); } +"*(r" { return toUnicode(0x1FEC); } +"w\\|" { return toUnicode(0x1FF2); } +"w|" { return toUnicode(0x1FF3); } +"w/|" { return toUnicode(0x1FF4); } +"*w\\" { return toUnicode(0x1FFA); } +"*w/" { return toUnicode(0x1FFB); } +"*w|" { return toUnicode(0x1FFC); } +"w=" { return toUnicode(0x1FF6); } +"w=|" { return toUnicode(0x1FF7); } +"*o\\" { return toUnicode(0x1FF8); } +"*o/" { return toUnicode(0x1FF9); } + +"*" isUpper = 1; + +"\\" { return toUnicode(0x0300); } +"/" { return toUnicode(0x0301); } +"_" { return toUnicode(0x0304); } +"^" { return toUnicode(0x0306); } +"+" { return toUnicode(0x0308); } +"=" { return toUnicode(0x0302); } +")" { return toUnicode(0x0313); } +"(" { return toUnicode(0x0314); } +"?" { return toUnicode(0x0323); } +"|" { return toUnicode(0x0345); } + +"a" { return toUnicodeGreek(0x03b1); } +"b" { return toUnicodeGreek(0x03b2); } +"g" { return toUnicodeGreek(0x03b3); } +"d" { return toUnicodeGreek(0x03b4); } +"e" { return toUnicodeGreek(0x03b5); } +"z" { return toUnicodeGreek(0x03b6); } +"h" { return toUnicodeGreek(0x03b7); } +"q" { return toUnicodeGreek(0x03b8); } +"i" { return toUnicodeGreek(0x03b9); } +"k" { return toUnicodeGreek(0x03ba); } +"l" { return toUnicodeGreek(0x03bb); } +"m" { return toUnicodeGreek(0x03bc); } +"n" { return toUnicodeGreek(0x03bd); } +"c" { return toUnicodeGreek(0x03be); } +"o" { return toUnicodeGreek(0x03bf); } +"p" { return toUnicodeGreek(0x03c0); } +"r" { return toUnicodeGreek(0x03c1); } + +"s1" { return toUnicode(0x03c3); } /* mdh 2002-01-07 */ +"s"/\-\- { return toUnicode(0x03c2); } +"s"/\> }[a-z\?\!0-9*=\/()\'\-] { return toUnicodeGreek(0x03c3); } +"s"/\< { return toUnicodeGreek(0x03c2); } /* MPDL update */ +"s"/[\[\]][a-z\?\!0-9*=\/()\'\-] { return toUnicodeGreek(0x03c3); } +"s"/\??[^a-z0-9*=\/()\'\-\[\?] { return toUnicode(0x03c2); } +"s" { return toUnicodeGreek(0x03c3); } + +"t" { return toUnicodeGreek(0x03c4); } +"u" { return toUnicodeGreek(0x03c5); } +"f" { return toUnicodeGreek(0x03c6); } +"x" { return toUnicodeGreek(0x03c7); } +"y" { return toUnicodeGreek(0x03c8); } +"w" { return toUnicodeGreek(0x03c9); } + +[\&_]"vert;" { return "|"; } +[\&_]"lpar;" { return "("; } +[\&_]"rpar;" { return ")"; } +[\_\&]"lt;" { return "<"; } +[\_\&]"gt;" { return ">"; } +"'" { return "'"; } /* MPDL update */ + +"&"[a-zA-Z]+";" { return yytext(); } + +. { return yytext(); } +\n { return yytext(); } \ No newline at end of file
Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/transcode/Betacode2UnicodeLex.class has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/transcode/Buckwalter2Unicode.lex Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,121 @@ +package de.mpg.mpiwg.berlin.mpdl.lt.general; + +%% +%{ + /* + * Betacode to Unicode conversion + */ + +%} + +%class Buckwalter2UnicodeLex +%public +%type java.lang.String +%unicode +%% + + +"<"[^>]+">" { return yytext(); } + +"'" { return "\u0621"; } /* Hamza */ +"|" { return "\u0622"; } /* ALEF WITH MADDA ABOVE from AraMorph */ +">" { return "\u0623"; } /* Hamza */ +"&" { return "\u0624"; } /* Hamza */ +"<" { return "\u0625"; } /* Alif + HamzaBelow */ +"}" { return "\u0626"; } /* Ya + HamzaAbove */ +"A" { return "\u0627"; } /* Alif */ +"b" { return "\u0628"; } /* Ba */ +"p" { return "\u0629"; } /* TaMarbuta */ +"t" { return "\u062A"; } /* Ta */ +"v" { return "\u062B"; } /* Tha */ +"j" { return "\u062C"; } /* Jeem */ +"H" { return "\u062D"; } /* HHa */ +"x" { return "\u062E"; } /* Kha */ +"d" { return "\u062F"; } /* Dal */ +"*" { return "\u0630"; } /* Thal */ +"r" { return "\u0631"; } /* Ra */ +"z" { return "\u0632"; } /* Zain */ +"s" { return "\u0633"; } /* Seen */ +"$" { return "\u0634"; } /* Sheen */ +"S" { return "\u0635"; } /* Sad */ +"D" { return "\u0636"; } /* DDad */ +"T" { return "\u0637"; } /* TTa */ +"Z" { return "\u0638"; } /* DTha */ +"E" { return "\u0639"; } /* Ain */ +"g" { return "\u063A"; } /* Ghain */ + +"_" { return "\u0640"; } /* Tatweel */ +"f" { return "\u0641"; } /* Fa */ +"q" { return "\u0642"; } /* Qaf */ +"k" { return "\u0643"; } /* Kaf */ +"l" { return "\u0644"; } /* Lam */ +"m" { return "\u0645"; } /* Meem */ +"n" { return "\u0646"; } /* Noon */ +"h" { return "\u0647"; } /* Ha */ +"w" { return "\u0648"; } /* Waw */ +"Y" { return "\u0649"; } /* AlifMaksura */ +"y" { return "\u064A"; } /* Ya */ +"F" { return "\u064B"; } /* Fathatan */ +"N" { return "\u064C"; } /* Dammatan */ +"K" { return "\u064D"; } /* Kasratan */ +"a" { return "\u064E"; } /* Fatha */ +"u" { return "\u064F"; } /* Damma */ +"i" { return "\u0650"; } /* Kasra */ +"~" { return "\u0651"; } /* Shadda */ +"o" { return "\u0652"; } /* Sukun */ +"^" { return "\u0653"; } /* Maddah */ +"#" { return "\u0654"; } /* HamzaAbove */ + +"`" { return "\u0670"; } /* AlifKhanjareeya */ +"{" { return "\u0671"; } /* Alif + HamzatWasl */ + +"P" { return "\u067E"; } /* PEH from AraMorph */ +"J" { return "\u0686"; } /* TCHEH from AraMorph */ +"V" { return "\u06A4"; } /* VEH from AraMorph */ +"G" { return "\u06AF"; } /* GAF from AraMorph */ +"R" { return "\u0698"; } /* JEH from AraMorph */ +"?" { return "\u061F"; } /* QUESTION MARK from AraMorph */ + +":" { return "\u06DC"; } /* SmallHighSeen */ +"@" { return "\u06DF"; } /* SmallHighRoundedZero */ + +"[" { return "\u06E2"; } /* SmallHighMeemIsolatedForm */ +";" { return "\u06E3"; } /* SmallLowSeen */ +"," { return "\u06E5"; } /* SmallWaw */ +"." { return "\u06E6"; } /* SmallYa */ +"!" { return "\u06E8"; } /* SmallHighNoon */ +"-" { return "\u06EA"; } /* EmptyCentreLowStop */ +"+" { return "\u06EB"; } /* EmptyCentreHighStop */ +"%" { return "\u06EC"; } /* RoundedHighStopWithFilledCentre */ +"]" { return "\u06ED"; } /* SmallLowMeem */ + +[\&_]"vert;" { return "|"; } +[\&_]"lpar;" { return "("; } +[\&_]"rpar;" { return ")"; } +[\_\&]"lt;" { return "<"; } +[\_\&]"gt;" { return ">"; } +"'" { return "'"; } + +"&"[a-zA-Z]+";" { return yytext(); } + +. { return yytext(); } +\n { return yytext(); } + +/* make problemes */ +/* "\\"" { return "\u06E0"; } SmallHighUprightRectangularZero */ + + +/* double entries */ +/* "," { return "\u060C"; } COMMA from AraMorph */ +/* ";" { return "\u061B"; } SEMICOLON from AraMorph */ + +/* not in buckwalter contained */ +/* \u0679 : ARABIC LETTER TTEH */ +/* \u0688 : ARABIC LETTER DDAL */ +/* \u06A9 : ARABIC LETTER KEHEH */ +/* \u0691 : ARABIC LETTER RREH */ +/* \u06BA : ARABIC LETTER NOON GHUNNA */ +/* \u06BE : ARABIC LETTER HEH DOACHASHMEE */ +/* \u06C1 : ARABIC LETTER HEH GOAL */ +/* \u06D2 : ARABIC LETTER YEH BARREE */ +
Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/transcode/Buckwalter2UnicodeLex.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/transcode/Transcoder.class has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/transcode/Unicode2Betacode.lex Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,319 @@ +package de.mpg.mpiwg.berlin.mpdl.lt.general; + +%% + +%class Unicode2BetacodeLex +%public +%type java.lang.String +%unicode +%% + + +"<"[^>]\u+">" { return yytext(); } + +"H" { return "*j"; } +"h" { return "j"; } +"F" { return "*v"; } +"f" { return "v"; } +"\u03a3" { return "*s"; } + +"." { return "!"; } +"\u00B7" { return ":"; } /* MPDL update */ + +"\u1F00" { return "a)"; } +"\u1F01" { return "a("; } +"\u1F02" { return "a)\\"; } +"\u1F03" { return "a(\\"; } +"\u1F04" { return "a)/"; } +"\u1F05" { return "a(/"; } +"\u1F06" { return "a)="; } +"\u1F07" { return "a(="; } +"\u1F08" { return "*)a"; } +"\u1F09" { return "*(a"; } +"\u1F0A" { return "*)\\a"; } +"\u1F0B" { return "*(\\a"; } +"\u1F0C" { return "*)/a"; } +"\u1F0D" { return "*(/a"; } +"\u1F0E" { return "*)=a"; } +"\u1F0F" { return "*(=a"; } +"\u1F10" { return "e)"; } +"\u1F11" { return "e("; } +"\u1F12" { return "e)\\"; } +"\u1F13" { return "e(\\"; } +"\u1F14" { return "e)/"; } +"\u1F15" { return "e(/"; } +"\u1F18" { return "*)e"; } +"\u1F19" { return "*(e"; } +"\u1F1A" { return "*)\\e"; } +"\u1F1B" { return "*(\\e"; } +"\u1F1C" { return "*)/e"; } +"\u1F1D" { return "*(/e"; } +"\u1F20" { return "h)"; } +"\u1F21" { return "h("; } +"\u1F22" { return "h)\\"; } +"\u1F23" { return "h(\\"; } +"\u1F24" { return "h)/"; } +"\u1F25" { return "h(/"; } +"\u1F26" { return "h)="; } +"\u1F27" { return "h(="; } +"\u1F28" { return "*)h"; } +"\u1F29" { return "*(h"; } +"\u1F2A" { return "*)\\h"; } +"\u1F2B" { return "*(\\h"; } +"\u1F2C" { return "*)/h"; } +"\u1F2D" { return "*(/h"; } +"\u1F2E" { return "*)=h"; } +"\u1F2F" { return "*(=h"; } +"\u1F30" { return "i)"; } +"\u1F31" { return "i("; } +"\u1F32" { return "i)\\"; } +"\u1F33" { return "i(\\"; } +"\u1F34" { return "i)/"; } +"\u1F35" { return "i(/"; } +"\u1F36" { return "i)="; } +"\u1F37" { return "i(="; } +"\u1F38" { return "*)i"; } +"\u1F39" { return "*(i"; } +"\u1F3A" { return "*)\\i"; } +"\u1F3B" { return "*(\\i"; } +"\u1F3C" { return "*)/i"; } +"\u1F3D" { return "*(/i"; } +"\u1F3E" { return "*)=i"; } +"\u1F3F" { return "*(=i"; } +"\u1F40" { return "o)"; } +"\u1F41" { return "o("; } +"\u1F42" { return "o)\\"; } +"\u1F43" { return "o(\\"; } +"\u1F44" { return "o)/"; } +"\u1F45" { return "o(/"; } +"\u1F48" { return "*)o"; } +"\u1F49" { return "*(o"; } +"\u1F4A" { return "*)\\o"; } +"\u1F4B" { return "*(\\o"; } +"\u1F4C" { return "*)/o"; } +"\u1F4D" { return "*(/o"; } +"\u1F50" { return "u)"; } +"\u1F51" { return "u("; } +"\u1F52" { return "u)\\"; } +"\u1F53" { return "u(\\"; } +"\u1F54" { return "u)/"; } +"\u1F55" { return "u(/"; } +"\u1F56" { return "u)="; } +"\u1F57" { return "u(="; } +"\u1F59" { return "*(u"; } +"\u1F5B" { return "*(\\u"; } +"\u1F5D" { return "*(/u"; } +"\u1F5F" { return "*(=u"; } +"\u1F60" { return "w)"; } +"\u1F61" { return "w("; } +"\u1F62" { return "w)\\"; } +"\u1F63" { return "w(\\"; } +"\u1F64" { return "w)/"; } +"\u1F65" { return "w(/"; } +"\u1F66" { return "w)="; } +"\u1F67" { return "w(="; } +"\u1F68" { return "*)w"; } +"\u1F69" { return "*(w"; } +"\u1F6A" { return "*)\\w"; } +"\u1F6B" { return "*(\\w"; } +"\u1F6C" { return "*)/w"; } +"\u1F6D" { return "*(/w"; } +"\u1F6E" { return "*)=w"; } +"\u1F6F" { return "*(=w"; } +"\u1F70" { return "a\\"; } +"\u1F71" { return "a/"; } +"\u1F72" { return "e\\"; } +"\u1F73" { return "e/"; } +"\u1F74" { return "h\\"; } +"\u1F75" { return "h/"; } +"\u1F76" { return "i\\"; } +"\u1F77" { return "i/"; } +"\u1F78" { return "o\\"; } +"\u1F79" { return "o/"; } +"\u1F7A" { return "u\\"; } +"\u1F7B" { return "u/"; } +"\u1F7C" { return "w\\"; } +"\u1F7D" { return "w/"; } +"\u1F80" { return "a)|"; } +"\u1F81" { return "a(|"; } +"\u1F82" { return "a)\\|"; } +"\u1F83" { return "a(\\|"; } +"\u1F84" { return "a)/|"; } +"\u1F85" { return "a(/|"; } +"\u1F86" { return "a)=|"; } +"\u1F87" { return "a(=|"; } +"\u1F88" { return "*)|a"; } +"\u1F89" { return "*(|a"; } +"\u1F8A" { return "*)\\|a"; } +"\u1F8B" { return "*(\\|a"; } +"\u1F8C" { return "*)/|a"; } +"\u1F8D" { return "*(/|a"; } +"\u1F8E" { return "*)=|a"; } +"\u1F8F" { return "*(=|a"; } +"\u1F90" { return "h)|"; } +"\u1F91" { return "h(|"; } +"\u1F92" { return "h)\\|"; } +"\u1F93" { return "h(\\|"; } +"\u1F94" { return "h)/|"; } +"\u1F95" { return "h(/|"; } +"\u1F96" { return "h)=|"; } +"\u1F97" { return "h(=|"; } +"\u1F98" { return "*)|h"; } +"\u1F99" { return "*(|h"; } +"\u1F9A" { return "*)\\|h"; } +"\u1F9B" { return "*(\\|h"; } +"\u1F9C" { return "*)/|h"; } +"\u1F9D" { return "*(/|h"; } +"\u1F9E" { return "*)=|h"; } +"\u1F9F" { return "*(=|h"; } +"\u1FA0" { return "w)|"; } +"\u1FA1" { return "w(|"; } +"\u1FA2" { return "w)\\|"; } +"\u1FA3" { return "w(\\|"; } +"\u1FA4" { return "w)/|"; } +"\u1FA5" { return "w(/|"; } +"\u1FA6" { return "w)=|"; } +"\u1FA7" { return "w(=|"; } +"\u1FA8" { return "*)|w"; } +"\u1FA9" { return "*(|w"; } +"\u1FAA" { return "*)\\|w"; } +"\u1FAB" { return "*(\\|w"; } +"\u1FAC" { return "*)/|w"; } +"\u1FAD" { return "*(/|w"; } +"\u1FAE" { return "*)=|w"; } +"\u1FAF" { return "*(=|w"; } +"\u1FB0" { return "a^"; } +"\u1FB1" { return "a_"; } +"\u1FB2" { return "a\\|"; } +"\u1FB3" { return "a|"; } +"\u1FB4" { return "a/|"; } +"\u1FB6" { return "a="; } +"\u1FB7" { return "a=|"; } +"\u1FB8" { return "*a^"; } +"\u1FB9" { return "*a_"; } +"\u1FBA" { return "*a\\"; } +"\u1FBB" { return "*a/"; } +"\u1FBC" { return "*a|"; } +"\u1FC2" { return "h\\|"; } +"\u1FC3" { return "h|"; } +"\u1FC4" { return "h/|"; } +"\u1FC6" { return "h="; } +"\u1FC7" { return "h=|"; } +"\u1FC8" { return "*e\\"; } +"\u1FC9" { return "*e/"; } +"\u1FCA" { return "*h\\"; } +"\u1FCB" { return "*h/"; } +"\u1FCC" { return "*h|"; } +"\u1FD0" { return "i^"; } +"\u1FD1" { return "i_"; } +"\u1FD2" { return "i+\\"; } +"\u1FD3" { return "i+/"; } +"\u1FD6" { return "i="; } +"\u1FD7" { return "i+="; } +"\u1FD8" { return "*i^"; } +"\u1FD9" { return "*i_"; } +"\u1FDA" { return "*i\\"; } +"\u1FDB" { return "*i/"; } +"\u1FE0" { return "u^"; } +"\u1FE1" { return "u_"; } +"\u1FE2" { return "u+\\"; } +"\u1FE3" { return "u+/"; } +"\u1FE4" { return "r)"; } +"\u1FE5" { return "r("; } +"\u1FE6" { return "u="; } +"\u1FE7" { return "u+="; } +"\u1FE8" { return "*u^"; } +"\u1FE9" { return "*u_"; } +"\u1FEA" { return "*u\\"; } +"\u1FEB" { return "*u/"; } +"\u1FEC" { return "*(r"; } +"\u1FF2" { return "w\\|"; } +"\u1FF3" { return "w|"; } +"\u1FF4" { return "w/|"; } +"\u1FFA" { return "*w\\"; } +"\u1FFB" { return "*w/"; } +"\u1FFC" { return "*w|"; } +"\u1FF6" { return "w="; } +"\u1FF7" { return "w=|"; } +"\u1FF8" { return "*o\\"; } +"\u1FF9" { return "*o/"; } + +"\u0300" { return "\\"; } +"\u0301" { return "/"; } +"\u0304" { return "_"; } +"\u0306" { return "^"; } +"\u0308" { return "+"; } +"\u0302" { return "="; } +"\u0313" { return ")"; } +"\u0314" { return "("; } +"\u0323" { return "?"; } +"\u0345" { return "|"; } + +"\u03b1" { return "a"; } /* MPDL update */ +"\u0391" { return "*a"; } /* MPDL update */ +"\u03b2" { return "b"; } /* MPDL update */ +"\u0392" { return "*b"; } /* MPDL update */ +"\u03b3" { return "g"; } /* MPDL update */ +"\u0393" { return "*g"; } /* MPDL update */ +"\u03b4" { return "d"; } /* MPDL update */ +"\u0394" { return "*d"; } /* MPDL update */ +"\u03b5" { return "e"; } /* MPDL update */ +"\u0395" { return "*e"; } /* MPDL update */ +"\u03b6" { return "z"; } /* MPDL update */ +"\u0396" { return "*z"; } /* MPDL update */ +"\u03b7" { return "h"; } /* MPDL update */ +"\u0397" { return "*h"; } /* MPDL update */ +"\u03b8" { return "q"; } /* MPDL update */ +"\u0398" { return "*q"; } /* MPDL update */ +"\u03b9" { return "i"; } /* MPDL update */ +"\u0399" { return "*i"; } /* MPDL update */ +"\u03ba" { return "k"; } /* MPDL update */ +"\u039a" { return "*k"; } /* MPDL update */ +"\u03bb" { return "l"; } /* MPDL update */ +"\u039b" { return "*l"; } /* MPDL update */ +"\u03bc" { return "m"; } /* MPDL update */ +"\u039c" { return "*m"; } /* MPDL update */ +"\u03bd" { return "n"; } /* MPDL update */ +"\u039d" { return "*n"; } /* MPDL update */ +"\u03be" { return "c"; } /* MPDL update */ +"\u039e" { return "*c"; } /* MPDL update */ +"\u03bf" { return "o"; } /* MPDL update */ +"\u039f" { return "*o"; } /* MPDL update */ +"\u03c0" { return "p"; } /* MPDL update */ +"\u03a0" { return "*p"; } /* MPDL update */ +"\u03c1" { return "r"; } /* MPDL update */ +"\u03a1" { return "*r"; } /* MPDL update */ + +"\u03a3" { return "*s"; } /* MPDL update */ +"\u03c3" { return "s1"; } /* mdh 2002-01-07 */ +"\u03c2"/\-\- { return "s"; } +"\u03c3"/\> }[a-z\?\!0-9*=\/()\'\-] { return "s"; } /* MPDL update */ +"\u03c2"/\< { return "s"; } /* MPDL update */ +"\u03c3"/[\[\]][a-z\?\!0-9*=\/()\'\-] { return "s"; } /* MPDL update */ +"\u03c2"/\??[^a-z0-9*=\/()\'\-\[\?] { return "s"; } +"\u03c3" { return "s"; } /* MPDL update */ + +"\u03c4" { return "t"; } /* MPDL update */ +"\u03a4" { return "*t"; } /* MPDL update */ +"\u03c5" { return "u"; } /* MPDL update */ +"\u03a5" { return "*u"; } /* MPDL update */ +"\u03c6" { return "f"; } /* MPDL update */ +"\u03a6" { return "*f"; } /* MPDL update */ +"\u03c7" { return "x"; } /* MPDL update */ +"\u03a7" { return "*x"; } /* MPDL update */ +"\u03c8" { return "y"; } /* MPDL update */ +"\u03a8" { return "*y"; } /* MPDL update */ +"\u03c9" { return "w"; } /* MPDL update */ +"\u03a9" { return "*w"; } /* MPDL update */ + +[\&_]"vert;" { return "|"; } +[\&_]"lpar;" { return "("; } +[\&_]"rpar;" { return ")"; } +[\_\&]"lt;" { return "<"; } +[\_\&]"gt;" { return ">"; } +"'" { return "'"; } /* MPDL update */ + +"&"[a-zA-Z]+";" { return yytext(); } + +. { return yytext(); } +\n { return yytext(); } \ No newline at end of file
Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/transcode/Unicode2BetacodeLex.class has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/transcode/Unicode2Buckwalter.lex Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,121 @@ +package de.mpg.mpiwg.berlin.mpdl.lt.general; + +%% +%{ + /* + * Betacode to Unicode conversion + */ + +%} + +%class Unicode2BuckwalterLex +%public +%type java.lang.String +%unicode +%% + + +"<"[^>]+">" { return yytext(); } + +"\u0621" { return "'"; } /* Hamza */ +"\u0622" { return "|"; } /* ALEF WITH MADDA ABOVE from AraMorph */ +"\u0623" { return ">"; } /* Hamza */ +"\u0624" { return "&"; } /* Hamza */ +"\u0625" { return "<"; } /* Alif + HamzaBelow */ +"\u0626" { return "}"; } /* Ya + HamzaAbove */ +"\u0627" { return "A"; } /* Alif */ +"\u0628" { return "b"; } /* Ba */ +"\u0629" { return "p"; } /* TaMarbuta */ +"\u062A" { return "t"; } /* Ta */ +"\u062B" { return "v"; } /* Tha */ +"\u062C" { return "j"; } /* Jeem */ +"\u062D" { return "H"; } /* HHa */ +"\u062E" { return "x"; } /* Kha */ +"\u062F" { return "d"; } /* Dal */ +"\u0630" { return "*"; } /* Thal */ +"\u0631" { return "r"; } /* Ra */ +"\u0632" { return "z"; } /* Zain */ +"\u0633" { return "s"; } /* Seen */ +"\u0634" { return "$"; } /* Sheen */ +"\u0635" { return "S"; } /* Sad */ +"\u0636" { return "D"; } /* DDad */ +"\u0637" { return "T"; } /* TTa */ +"\u0638" { return "Z"; } /* DTha */ +"\u0639" { return "E"; } /* Ain */ +"\u063A" { return "g"; } /* Ghain */ + +"\u0640" { return "_"; } /* Tatweel */ +"\u0641" { return "f"; } /* Fa */ +"\u0642" { return "q"; } /* Qaf */ +"\u0643" { return "k"; } /* Kaf */ +"\u0644" { return "l"; } /* Lam */ +"\u0645" { return "m"; } /* Meem */ +"\u0646" { return "n"; } /* Noon */ +"\u0647" { return "h"; } /* Ha */ +"\u0648" { return "w"; } /* Waw */ +"\u0649" { return "Y"; } /* AlifMaksura */ +"\u064A" { return "y"; } /* Ya */ +"\u064B" { return "F"; } /* Fathatan */ +"\u064C" { return "N"; } /* Dammatan */ +"\u064D" { return "K"; } /* Kasratan */ +"\u064E" { return "a"; } /* Fatha */ +"\u064F" { return "u"; } /* Damma */ +"\u0650" { return "i"; } /* Kasra */ +"\u0651" { return "~"; } /* Shadda */ +"\u0652" { return "o"; } /* Sukun */ +"\u0653" { return "^"; } /* Maddah */ +"\u0654" { return "#"; } /* HamzaAbove */ + +"\u0670" { return "`"; } /* AlifKhanjareeya */ +"\u0671" { return "{"; } /* Alif + HamzatWasl */ + +"\u067E" { return "P"; } /* PEH from AraMorph */ +"\u0686" { return "J"; } /* TCHEH from AraMorph */ +"\u06A4" { return "V"; } /* VEH from AraMorph */ +"\u06AF" { return "G"; } /* GAF from AraMorph */ +"\u0698" { return "R"; } /* JEH from AraMorph */ +"\u061F" { return "?"; } /* QUESTION MARK from AraMorph */ + +"\u06DC" { return ":"; } /* SmallHighSeen */ +"\u06DF" { return "@"; } /* SmallHighRoundedZero */ + +"\u06E2" { return "["; } /* SmallHighMeemIsolatedForm */ +"\u06E3" { return ";"; } /* SmallLowSeen */ +"\u06E5" { return ","; } /* SmallWaw */ +"\u06E6" { return "."; } /* SmallYa */ +"\u06E8" { return "!"; } /* SmallHighNoon */ +"\u06EA" { return "-"; } /* EmptyCentreLowStop */ +"\u06EB" { return "+"; } /* EmptyCentreHighStop */ +"\u06EC" { return "%"; } /* RoundedHighStopWithFilledCentre */ +"\u06ED" { return "]"; } /* SmallLowMeem */ + +[\&_]"vert;" { return "|"; } +[\&_]"lpar;" { return "("; } +[\&_]"rpar;" { return ")"; } +[\_\&]"lt;" { return "<"; } +[\_\&]"gt;" { return ">"; } +"'" { return "'"; } + +"&"[a-zA-Z]+";" { return yytext(); } + +. { return yytext(); } +\n { return yytext(); } + +/* make problemes */ +/* "\u06E0" { return "\\""; } SmallHighUprightRectangularZero */ + + +/* double entries */ +/* "\u060C" { return ","; } COMMA from AraMorph */ +/* "\u061B" { return ";"; } SEMICOLON from AraMorph */ + +/* not in buckwalter contained */ +/* \u0679 : ARABIC LETTER TTEH */ +/* \u0688 : ARABIC LETTER DDAL */ +/* \u06A9 : ARABIC LETTER KEHEH */ +/* \u0691 : ARABIC LETTER RREH */ +/* \u06BA : ARABIC LETTER NOON GHUNNA */ +/* \u06BE : ARABIC LETTER HEH DOACHASHMEE */ +/* \u06C1 : ARABIC LETTER HEH GOAL */ +/* \u06D2 : ARABIC LETTER YEH BARREE */ +
Binary file software/mpdl-services/mpiwg-mpdl-lt/bin/de/mpg/mpiwg/berlin/mpdl/lt/text/transcode/Unicode2BuckwalterLex.class has changed
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--- a/software/mpdl-services/mpiwg-mpdl-lt/build/build.xml Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/build/build.xml Tue Nov 27 12:35:19 2012 +0100 @@ -1,5 +1,5 @@ <!DOCTYPE project> -<project name="mpiwg-mpdl-lt" default="dist" basedir="../"> +<project name="mpiwg-mpdl-lt" default="compile" basedir="../"> <description>mpiwg-mpdl-lt</description> <!-- global properties --> <property name="src" location="src"/> @@ -25,9 +25,6 @@ <include name="**/*.properties"/> </fileset> </copy> - </target> - - <target name="dist" depends="compile" description="generate the distribution"> <jar jarfile="${dist}/mpiwg-mpdl-lt.jar" basedir="${build}"/> <copy todir="/Users/jwillenborg/java/mpiwg-mpdl-lt-web/WebContent/WEB-INF/lib"> <fileset dir="${lib}"> @@ -35,6 +32,7 @@ </fileset> </copy> <copy file="${dist}/mpiwg-mpdl-lt.jar" todir="/Users/jwillenborg/java/mpiwg-mpdl-lt-web/WebContent/WEB-INF/lib"/> + <copy file="${dist}/mpiwg-mpdl-lt.jar" todir="/Users/jwillenborg/java/mpiwg-mpdl-cms-web/WebContent/WEB-INF/lib"/> </target> <target name="clean" description="clean" >
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/build/classes/constants.properties Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,1 @@ +dataDir=/Users/jwillenborg/mpdl/data/lt \ No newline at end of file
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--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/dict/app/Lexica.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/dict/app/Lexica.java Tue Nov 27 12:35:19 2012 +0100 @@ -1,6 +1,7 @@ package de.mpg.mpiwg.berlin.mpdl.lt.dict.app; import java.util.ArrayList; +import java.util.Collections; import java.util.HashMap; import java.util.Iterator; import java.util.Set; @@ -21,6 +22,7 @@ public class Lexica { private static Lexica instance; private static HashMap<String, Lexicon> localLexica = new HashMap<String, Lexicon>(); + private static HashMap<String, String[]> localLangLexiconNames = new HashMap<String, String[]>(); private static HashMap<String, Lexicon> remoteLexica = new HashMap<String, Lexicon>(); public static Lexica getInstance() { @@ -33,27 +35,25 @@ private void init() { Lexicon autenrieth = new Lexicon("autenrieth", "el"); - autenrieth.setDescription("Autenrieth, a Homeric lexicon"); + autenrieth.setDescription("Autenrieth: A Homeric lexicon"); autenrieth.setQueryUrl("http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0073:entry="); Lexicon baretti = new Lexicon("baretti", "it"); - baretti.setDescription("Baretti, a dictionary of the English and Italian languages"); + baretti.setDescription("Baretti: A dictionary of the English and Italian languages"); Lexicon bonitz = new Lexicon("bonitz", "el"); - bonitz.setDescription("Bonitz, index Aristotelicus"); + bonitz.setDescription("Bonitz: Index Aristotelicus"); Lexicon cooper = new Lexicon("cooper", "la"); - cooper.setDescription("Cooper, Thesaurus Linguae Romanae et Brittanicae"); + cooper.setDescription("Cooper: Thesaurus Linguae Romanae et Brittanicae"); Lexicon florio = new Lexicon("florio", "it"); - florio.setDescription("Florio, a worlde of wordes, or most copious, dictionarie in Italian and English"); + florio.setDescription("Florio: a worlde of wordes, or most copious, dictionarie in Italian and English"); Lexicon ls = new Lexicon("ls", "la"); - ls.setDescription("Lewis and Short, Latin dictionary"); + ls.setDescription("Lewis and Short: Latin dictionary"); ls.setQueryUrl("http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0059:entry="); Lexicon lsj = new Lexicon("lsj", "el"); - lsj.setDescription("Liddell-Scott-Jones, a Greek-English lexicon"); + lsj.setDescription("Liddell-Scott-Jones: A Greek-English lexicon"); lsj.setQueryUrl("http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0057:entry="); Lexicon salmone = new Lexicon("salmone", "ar"); - salmone.setDescription("Salmone, an advanced learner's Arabic-English dictionary"); + salmone.setDescription("Salmone: An advanced learner's Arabic-English dictionary"); salmone.setQueryUrl("http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:2002.02.0005:entry="); - Lexicon salmoneUnicode = new Lexicon("salmoneUnicode", "ar"); - salmoneUnicode.setDescription("Salmone, an advanced learner's Arabic-English dictionary"); Lexicon webster = new Lexicon("webster", "en"); webster.setDescription("Webster's revised unabridged dictionary (1913)"); localLexica.put("autenrieth", autenrieth); @@ -65,14 +65,49 @@ localLexica.put("lsj", lsj); localLexica.put("salmone", salmone); localLexica.put("webster", webster); + localLangLexiconNames = new HashMap<String, String[]>(); + String[] arLangLexNames = {"salmone"}; + localLangLexiconNames.put("ar", arLangLexNames); + String[] elLangLexNames = {"autenrieth", "bonitz", "lsj"}; + localLangLexiconNames.put("el", elLangLexNames); + String[] enLangLexNames = {"webster"}; + localLangLexiconNames.put("en", enLangLexNames); + String[] itLangLexNames = {"baretti", "florio"}; + localLangLexiconNames.put("it", itLangLexNames); + String[] laLangLexNames = {"cooper", "ls"}; + localLangLexiconNames.put("la", laLangLexNames); Lexicon dwds = new Lexicon("dwds", "de"); - dwds.setDescription("Deutsches Wšrterbuch der deutschen Sprache"); + dwds.setDescription("DWDS: Deutsches Wšrterbuch der deutschen Sprache"); dwds.setQueryUrl("http://www.dwds.de/search/?qu="); dwds.setType("remote"); + Lexicon leoDE = new Lexicon("leoDE", "de"); + leoDE.setDescription("LEO: German - English Dictionary"); + leoDE.setQueryUrl("http://dict.leo.org/ende?lang=en&search="); + leoDE.setType("remote"); + Lexicon dictLeipzigDE = new Lexicon("dictLeipzigDE", "de"); + dictLeipzigDE.setDescription("UniversitŠt Leipzig: German - English Dictionary"); + dictLeipzigDE.setQueryUrl("http://dict.uni-leipzig.de/index.php?wort="); + dictLeipzigDE.setType("remote"); + Lexicon lingueeDE = new Lexicon("lingueeDE", "de"); + lingueeDE.setDescription("Linguee: German - English Dictionary"); + lingueeDE.setQueryUrl("http://www.linguee.de/deutsch-englisch/search?query="); + lingueeDE.setType("remote"); Lexicon slater = new Lexicon("slater", "el"); - slater.setDescription("William J. Slater, Lexicon to Pindar"); + slater.setDescription("William J. Slater: Lexicon to Pindar"); slater.setQueryUrl("http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0072:entry="); slater.setType("remote"); + Lexicon leoEN = new Lexicon("leoEN", "en"); + leoEN.setDescription("LEO: German - English Dictionary"); + leoEN.setQueryUrl("http://dict.leo.org/ende?lang=en&search="); + leoEN.setType("remote"); + Lexicon lingueeEN = new Lexicon("lingueeEN", "en"); + lingueeEN.setDescription("Linguee: German - English Dictionary"); + lingueeEN.setQueryUrl("http://www.linguee.de/deutsch-englisch/search?query="); + lingueeEN.setType("remote"); + Lexicon leoFR = new Lexicon("leoFR", "fr"); + leoFR.setDescription("LEO: German - French Dictionary"); + leoFR.setQueryUrl("http://dict.leo.org/frde?lang=en&search="); + leoFR.setType("remote"); Lexicon artflFr = new Lexicon("artfl-fr", "fr"); artflFr.setDescription("The ARTFL project: Dictionnaires d'autrefois: French dictionaries of the 17th, 18th, 19th and 20th centuries"); artflFr.setQueryUrl("http://machaut.uchicago.edu/?resource=frengdict&action=search&french="); @@ -81,37 +116,53 @@ artflFrEn.setDescription("The ARTFL project: French - English dictionary"); artflFrEn.setQueryUrl("http://artflx.uchicago.edu/cgi-bin/dicos/pubdico1look.pl?strippedhw="); artflFrEn.setType("remote"); + Lexicon leoIT = new Lexicon("leoIT", "it"); + leoIT.setDescription("LEO: German - Italian Dictionary"); + leoIT.setQueryUrl("http://dict.leo.org/itde?lang=en&search="); + leoIT.setType("remote"); Lexicon lewis = new Lexicon("lewis", "la"); - lewis.setDescription("Charlton T. Lewis, an Elementary Latin Dictionary"); + lewis.setDescription("Charlton T. Lewis: An Elementary Latin Dictionary"); lewis.setQueryUrl("http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0060:entry="); lewis.setType("remote"); Lexicon wikiwoordenboek = new Lexicon("wikiwoordenboek", "nl"); wikiwoordenboek.setDescription("Wiktionary: WikiWoordenboek"); wikiwoordenboek.setQueryUrl("http://nl.wiktionary.org/wiki/"); wikiwoordenboek.setType("remote"); + Lexicon leoCH = new Lexicon("leoCH", "ch"); + leoCH.setDescription("LEO: German - Chinese Dictionary"); + leoCH.setQueryUrl("http://dict.leo.org/chde?lang=en&search="); + leoCH.setType("remote"); Lexicon ctp = new Lexicon("ctp", "zh"); - ctp.setDescription("Chinese Text Project"); + ctp.setDescription("Chinese Text Project: Chinese - English Dictionary"); ctp.setQueryUrl("http://ctext.org/dictionary.pl?if=en&char="); ctp.setType("remote"); Lexicon linyutan = new Lexicon("linyutan", "zh"); - linyutan.setDescription("Lin Yutang"); + linyutan.setDescription("Lin Yutang: Chinese - English Dictionary"); linyutan.setQueryUrl("http://humanum.arts.cuhk.edu.hk/cgi-bin/agrep-lindict?query="); linyutan.setType("remote"); Lexicon chineseUnicode = new Lexicon("chinese-unicode", "zh"); - chineseUnicode.setDescription("Unicode"); + chineseUnicode.setDescription("Unicode.org: Chinese - English"); chineseUnicode.setQueryUrl("http://www.unicode.org/cgi-bin/GetUnihanData.pl?codepoint="); chineseUnicode.setType("remote"); Lexicon chineseWiktionary = new Lexicon("chinese-wiktionary", "zh"); - chineseWiktionary.setDescription("Wiktionary"); + chineseWiktionary.setDescription("Wiktionary: Chinese - English"); chineseWiktionary.setQueryUrl("http://en.wiktionary.org/wiki/"); chineseWiktionary.setType("remote"); remoteLexica.put("dwds", dwds); + remoteLexica.put("leoDE", leoDE); + remoteLexica.put("dictLeipzigDE", dictLeipzigDE); + remoteLexica.put("lingueeDE", lingueeDE); remoteLexica.put("slater", slater); + remoteLexica.put("leoEN", leoEN); + remoteLexica.put("lingueeEN", lingueeEN); + remoteLexica.put("leoFR", leoFR); remoteLexica.put("artfl-fr", artflFr); remoteLexica.put("artfl-fr-en", artflFrEn); + remoteLexica.put("leoIT", leoIT); remoteLexica.put("lewis", lewis); remoteLexica.put("wikiwoordenboek", wikiwoordenboek); remoteLexica.put("ctp", ctp); + remoteLexica.put("leoCH", leoCH); remoteLexica.put("linyutan", linyutan); remoteLexica.put("chinese-unicode", chineseUnicode); remoteLexica.put("chinese-wiktionary", chineseWiktionary); @@ -142,6 +193,12 @@ return retLexicons; } + public String[] getLocalLexiconNames(String lang) { + String language = Language.getInstance().getLanguageId(lang); + String[] retLexiconNames = localLangLexiconNames.get(language); + return retLexiconNames; + } + public ArrayList<Lexicon> getRemoteLexicons(String lang) { String language = Language.getInstance().getLanguageId(lang); ArrayList<Lexicon> retLexicons = null; @@ -157,6 +214,8 @@ retLexicons.add(lexicon); } } + if (retLexicons != null) + Collections.sort(retLexicons); return retLexicons; }
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/dict/app/Lexicon.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/dict/app/Lexicon.java Tue Nov 27 12:35:19 2012 +0100 @@ -10,6 +10,7 @@ import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; import de.mpg.mpiwg.berlin.mpdl.lt.general.Language; import de.mpg.mpiwg.berlin.mpdl.lt.text.transcode.Transcoder; +import de.mpg.mpiwg.berlin.mpdl.util.StringUtils; public class Lexicon implements Comparable<Lexicon> { private String name; @@ -27,7 +28,10 @@ } public int compareTo(Lexicon l) { - return name.compareTo(l.name); + if (description != null) + return description.compareTo(l.description); + else + return name.compareTo(l.name); } public String getName() { @@ -195,8 +199,11 @@ } result = result + "</content>"; } - if (entry.getRemoteUrl() != null) - result = result + "<remoteUrl>" + entry.getRemoteUrl() + "</remoteUrl>"; + if (entry.getRemoteUrl() != null) { + String remoteUrl = entry.getRemoteUrl(); + remoteUrl = StringEscapeUtils.escapeXml(remoteUrl); + result = result + "<remoteUrl>" + remoteUrl + "</remoteUrl>"; + } result = result + "</entry>"; } result = result + "</entries>";
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/dict/db/DbEnvLex.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/dict/db/DbEnvLex.java Tue Nov 27 12:35:19 2012 +0100 @@ -31,6 +31,8 @@ envConfig = new EnvironmentConfig(); dbConfig = new DatabaseConfig(); envPath = new File(dataDir); + envConfig.setReadOnly(true); + dbConfig.setReadOnly(true); env = new Environment(envPath, envConfig); } catch (DatabaseException e) { throw new ApplicationException(e);
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/dict/db/LexHandler.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/dict/db/LexHandler.java Tue Nov 27 12:35:19 2012 +0100 @@ -49,7 +49,9 @@ dbEnvLexica.closeDatabase(lexiconName); } dbEnvLexica.close(); - LOGGER.info("Lexicon db cache: closed"); + LOGGER.info("Lexicon cache: db closed"); + dbEnvLexica = null; + instance = null; } /** @@ -60,7 +62,7 @@ * @return lemmas * @throws ApplicationException */ - public ArrayList<Lemma> getLemmas(String query, String type, String language, int normMode) throws ApplicationException { + public ArrayList<Lemma> getLemmas(String query, String type, String language, int normMode, boolean atLeastOneLemmaWithWordForm) throws ApplicationException { ArrayList<Lemma> lexLemmas = new ArrayList<Lemma>(); // get lemmas of all forms in query MorphologyCache morphologyCache = MorphologyCache.getInstance(); @@ -79,8 +81,10 @@ if (lemmas != null && ! lemmas.isEmpty()) { lexLemmas.addAll(lemmas); } else { - Lemma l = new Lemma("created dynamically cause no lemma is available", language, queryForm); // at least the word form is added for finding it in the lexicon - lexLemmas.add(l); + if (atLeastOneLemmaWithWordForm) { + Lemma l = new Lemma("created dynamically cause no lemma is available", language, queryForm); // at least the word form is added for finding it in the lexicon + lexLemmas.add(l); + } } } Collections.sort(lexLemmas); @@ -170,12 +174,12 @@ boolean hasLexEntry = false; if (language.equals("zh")) // each chinese character always has a lexicon entry return true; - ArrayList<Lexicon> statLexicons = Lexica.getInstance().getLocalLexicons(language); - if (statLexicons != null) { - for (int i=0; i<statLexicons.size(); i++) { - Lexicon lexicon = statLexicons.get(i).clone(); // clone without lexicon entries - LexiconEntry lexEntry = readEntry(lexicon.getName(), formName); - if (lexEntry != null) { + String[] lexiconNames = Lexica.getInstance().getLocalLexiconNames(language); + if (lexiconNames != null) { + for (int i=0; i<lexiconNames.length; i++) { + String lexiconName = lexiconNames[i]; + hasLexEntry = hasKey(lexiconName, formName); + if (hasLexEntry) { return true; } } @@ -241,6 +245,28 @@ return lexEntry; } + private boolean hasKey(String lexiconName, String formName) throws ApplicationException { + boolean hasKey = false; + try { + String keyStr = formName; + DatabaseEntry dbEntryKey = new DatabaseEntry(keyStr.getBytes("utf-8")); + Database lexDB = dbEnvLexica.getLexiconDB(lexiconName); + Cursor cursor = lexDB.openCursor(null, null); + DatabaseEntry foundValue = new DatabaseEntry(); + foundValue.setPartial(0, 0, true); // more performance: the value is not fetched: only the key is fetched + OperationStatus operationStatus = cursor.getSearchKey(dbEntryKey, foundValue, LockMode.DEFAULT); + if (operationStatus == OperationStatus.SUCCESS) { + hasKey = true; + } + cursor.close(); + } catch (DatabaseException e) { + throw new ApplicationException(e); + } catch (UnsupportedEncodingException e) { + throw new ApplicationException(e); + } + return hasKey; + } + private LexiconEntry readEntry(String lexiconName, String formName) throws ApplicationException { LexiconEntry retLexEntry = null; try { @@ -337,7 +363,7 @@ String lexiconName = lexicon.getName(); dbEnvLexica.openDatabase(lexiconName); } - LOGGER.info("Lexicon db cache: opened"); + LOGGER.info("Lexicon cache: db opened read only"); } private void readSampleData() throws ApplicationException {
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/morph/app/MorphologyCache.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/morph/app/MorphologyCache.java Tue Nov 27 12:35:19 2012 +0100 @@ -25,11 +25,13 @@ private static String DB_DIR_DONATUS = DATA_DIR + "/dataBerkeleyDB/donatus"; public static int QUERY_MODE = 0; public static int DOCUMENT_MODE = 1; + private static long MIN_RAM = 500000000; private static int MAX_HASHTABLE_SIZE = Constants.MORPHOLOGY_CACHE_SIZE; + private Date touchTimer; protected int mode = QUERY_MODE; private Hashtable<String, Hashtable<String, Lemma>> forms = new Hashtable<String, Hashtable<String, Lemma>>(); // cache of forms: hashKey is formName private Hashtable<String, Lemma> lemmas = new Hashtable<String, Lemma>(); // cache of lemmas: hashKey is lemmaName - private DBMorphHandler dbMorphHandlerStatic; // handles static morph data (BerkeleyDB) + private DBMorphHandler dbMorphHandler; // handles morph data (BerkeleyDB) private Date beginOfOperation; private Date endOfOperation; @@ -42,13 +44,20 @@ } private void init() throws ApplicationException { + long maxMemory = Runtime.getRuntime().maxMemory(); + if (maxMemory < MIN_RAM) { + String message = "Morphology cache: at least " + MIN_RAM + " is needed as heap space: please start java with parameter -Xmx with more than this value)"; + LOGGER.severe(message); + throw new ApplicationException(message); + } + touchTimer = new Date(); instance.beginOperation(); - dbMorphHandlerStatic = new DBMorphHandler(DB_DIR_DONATUS); - dbMorphHandlerStatic.start(); - dbMorphHandlerStatic.openDatabases(); + dbMorphHandler = new DBMorphHandler(DB_DIR_DONATUS); + dbMorphHandler.startReadOnly(); + dbMorphHandler.openDatabases(); instance.endOperation(); Double elapsedTime = new Util().getSecondWithMillisecondsBetween(instance.beginOfOperation, instance.endOfOperation); - LOGGER.info("Morphology db cache: opened (needed " + elapsedTime + " seconds)"); + LOGGER.info("Morphology cache: morphology db opened read only (needed " + elapsedTime + " seconds, heap space: " + maxMemory + " bytes)"); } public int getMode() { @@ -60,16 +69,14 @@ } public void end() throws ApplicationException { - dbMorphHandlerStatic.closeDatabases(); - LOGGER.info("Morphology db cache: closed"); + dbMorphHandler.closeDatabases(); + LOGGER.info("Morphology cache: db closed"); + forms = null; + lemmas = null; + dbMorphHandler = null; + instance = null; } - /* - public ArrayList<Lemma> getLemmasByFormName(String lang, String formName, boolean normalize) throws ApplicationException { - return getLemmasByFormName(lang, formName, normalize, Normalizer.DISPLAY); - } - */ - public ArrayList<Lemma> getLemmasByFormName(String lang, String formNameArg, int normMode) throws ApplicationException { String language = Language.getInstance().getLanguageId(lang); ArrayList<Lemma> retFormLemmas = null; @@ -84,8 +91,16 @@ ArrayList<Lemma> dbFormLemmas = readLemmasByFormName(language, formName); // put lemmas into local cache int localHashTableSize = forms.size(); - if (localHashTableSize >= MAX_HASHTABLE_SIZE) { - clearCache(); + Date now = new Date(); + if (now.getTime() - touchTimer.getTime() > 900000) { // is true each 0,25 hours: then free memory is fetched (needs some time) + touchTimer = new Date(); + long freeMemory = Runtime.getRuntime().freeMemory(); + LOGGER.info(touchTimer + ": Morphology cache: free memory in heap space: " + freeMemory + " bytes"); + if (freeMemory < MIN_RAM || localHashTableSize >= MAX_HASHTABLE_SIZE) { // if freeMemory is less then MIN_RAM then clear cache to get some new memory + clearCache(); + freeMemory = Runtime.getRuntime().freeMemory(); + LOGGER.info(touchTimer + ": Morphology cache: cache cleared, free memory in heap space: " + freeMemory + " bytes"); + } } if (dbFormLemmas != null && ! dbFormLemmas.isEmpty()) { formLemmasHashtable = new Hashtable<String, Lemma>(); @@ -271,13 +286,13 @@ private ArrayList<Lemma> readLemmasByFormName(String lang, String formName) throws ApplicationException { String language = Language.getInstance().getLanguageId(lang); - ArrayList<Lemma> lemmasStatic = dbMorphHandlerStatic.readLemmas(language, formName); + ArrayList<Lemma> lemmasStatic = dbMorphHandler.readLemmas(language, formName); return lemmasStatic; } private ArrayList<Form> readFormsByLemmaName(String lang, String lemmaName) throws ApplicationException { String language = Language.getInstance().getLanguageId(lang); - ArrayList<Form> formsStatic = dbMorphHandlerStatic.readForms(language, lemmaName); + ArrayList<Form> formsStatic = dbMorphHandler.readForms(language, lemmaName); return formsStatic; }
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/morph/app/SimpleMorphContentHandler.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/morph/app/SimpleMorphContentHandler.java Tue Nov 27 12:35:19 2012 +0100 @@ -1,5 +1,7 @@ package de.mpg.mpiwg.berlin.mpdl.lt.morph.app; +import java.util.ArrayList; + import org.xml.sax.*; @@ -7,6 +9,8 @@ private Element currentElement; private Lemma lemma; private Form form; + private ArrayList<Lemma> lemmas; + private ArrayList<Form> forms; public SimpleMorphContentHandler() { } @@ -19,6 +23,14 @@ return lemma; } + public ArrayList<Form> getForms() { + return forms; + } + + public ArrayList<Lemma> getLemmas() { + return lemmas; + } + public void startDocument() throws SAXException { } @@ -95,8 +107,14 @@ currentElement = new Element(name); if (name.equals("form")) { form = new Form(); + if (forms == null) + forms = new ArrayList<Form>(); + forms.add(form); } else if (name.equals("lemma")) { lemma = new Lemma(); + if (lemmas == null) + lemmas = new ArrayList<Lemma>(); + lemmas.add(lemma); } }
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/morph/db/DBMorphHandler.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/morph/db/DBMorphHandler.java Tue Nov 27 12:35:19 2012 +0100 @@ -5,7 +5,6 @@ import java.io.StringReader; import java.io.UnsupportedEncodingException; import java.util.ArrayList; -import java.util.Hashtable; import org.xml.sax.InputSource; import org.xml.sax.SAXException; @@ -33,6 +32,12 @@ this.dbDirectory = dbDir; } + public void startReadOnly() throws ApplicationException { + morphDbEnv = new DbEnvMorph(); + morphDbEnv.setDataDir(dbDirectory); + morphDbEnv.initReadOnly(); // open databases in read only mode + } + public void start() throws ApplicationException { morphDbEnv = new DbEnvMorph(); morphDbEnv.setDataDir(dbDirectory); @@ -134,39 +139,16 @@ DatabaseEntry dbEntryKey = new DatabaseEntry(bHashKey); DatabaseEntry foundFormValue = new DatabaseEntry(); OperationStatus operationStatus = cursor.getSearchKey(dbEntryKey, foundFormValue, LockMode.DEFAULT); + StringBuilder formsXmlString = new StringBuilder(); + formsXmlString.append("<forms>"); while (operationStatus == OperationStatus.SUCCESS) { byte[] foundFormValueBytes = foundFormValue.getData(); String foundFormValueStr = new String(foundFormValueBytes, "utf-8"); - Form f = parseXmlFormString(foundFormValueStr); - retForms.add(f); + formsXmlString.append(foundFormValueStr); operationStatus = cursor.getNextDup(dbEntryKey, foundFormValue, LockMode.DEFAULT); } - cursor.close(); - } catch (DatabaseException e) { - throw new ApplicationException(e); - } catch (UnsupportedEncodingException e) { - throw new ApplicationException(e); - } - return retForms; - } - - // TODO diese Methode wird nicht verwendet bis jetzt - public Hashtable<String, Form> readForms() throws ApplicationException { - Hashtable<String, Form> retForms = new Hashtable<String, Form>(); - try { - Database lemmaDB = morphDbEnv.getLemmaDB(); - Cursor cursor = lemmaDB.openCursor(null, null); - DatabaseEntry dbEntryKey = new DatabaseEntry(); - DatabaseEntry foundFormValue = new DatabaseEntry(); - OperationStatus operationStatus = cursor.getFirst(dbEntryKey, foundFormValue, LockMode.DEFAULT); - while (operationStatus == OperationStatus.SUCCESS) { - byte[] foundFormValueBytes = foundFormValue.getData(); - String foundFormValueStr = new String(foundFormValueBytes, "utf-8"); - Form f = parseXmlFormString(foundFormValueStr); - String formHashKey = f.getLanguage() + "###" + f.getFormName(); - retForms.put(formHashKey, f); - operationStatus = cursor.getNext(dbEntryKey, foundFormValue, LockMode.DEFAULT); - } + formsXmlString.append("</forms>"); + retForms = parseXmlFormsString(formsXmlString.toString()); cursor.close(); } catch (DatabaseException e) { throw new ApplicationException(e); @@ -177,7 +159,7 @@ } public ArrayList<Lemma> readLemmas(String language, String formName) throws ApplicationException { - ArrayList<Lemma> retForms = new ArrayList<Lemma>(); + ArrayList<Lemma> retLemmas = new ArrayList<Lemma>(); String lang = Language.getInstance().getLanguageId(language); String hashKey = lang + "###" + formName; try { @@ -191,7 +173,7 @@ byte[] foundLemmaValueBytes = foundLemmaValue.getData(); String foundLemmaValueStr = new String(foundLemmaValueBytes, "utf-8"); Lemma l = parseXmlLemmaString(foundLemmaValueStr); - retForms.add(l); + retLemmas.add(l); operationStatus = cursor.getNextDup(dbEntryKey, foundLemmaValue, LockMode.DEFAULT); } cursor.close(); @@ -200,11 +182,11 @@ } catch (UnsupportedEncodingException e) { throw new ApplicationException(e); } - return retForms; + return retLemmas; } - private Form parseXmlFormString(String xmlString) throws ApplicationException { - Form form = null; + private ArrayList<Form> parseXmlFormsString(String xmlString) throws ApplicationException { + ArrayList<Form> forms = null; try { XMLReader xmlParser = new SAXParser(); SimpleMorphContentHandler morphContentHandler = new SimpleMorphContentHandler(); @@ -212,13 +194,13 @@ Reader reader = new StringReader(xmlString); InputSource input = new InputSource(reader); xmlParser.parse(input); - form = morphContentHandler.getForm(); + forms = morphContentHandler.getForms(); } catch (SAXException e) { throw new ApplicationException(e); } catch (IOException e) { throw new ApplicationException(e); } - return form; + return forms; } private Lemma parseXmlLemmaString(String xmlString) throws ApplicationException {
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/morph/db/DbEnvMorph.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/morph/db/DbEnvMorph.java Tue Nov 27 12:35:19 2012 +0100 @@ -26,6 +26,20 @@ this.dataDir = dataDir; } + public void initReadOnly() throws ApplicationException { + try { + envConfig = new EnvironmentConfig(); + dbConfig = new DatabaseConfig(); + envPath = new File(dataDir); + envConfig.setReadOnly(true); + dbConfig.setReadOnly(true); + dbConfig.setSortedDuplicates(true); + env = new Environment(envPath, envConfig); + } catch (DatabaseException e) { + throw new ApplicationException(e); + } + } + public void init() throws ApplicationException { try { envConfig = new EnvironmentConfig();
Binary file software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/.DS_Store has changed
Binary file software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/.DS_Store has changed
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/Normalizer.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/Normalizer.java Tue Nov 27 12:35:19 2012 +0100 @@ -2,7 +2,6 @@ import java.io.IOException; import java.io.StringReader; -import java.util.ArrayList; import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; import de.mpg.mpiwg.berlin.mpdl.lt.general.Language; @@ -15,8 +14,6 @@ import de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang.MpdlNormalizerLexLA; import de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang.MpdlNormalizerLexNL; import de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang.MpdlNormalizerLexZH; -import de.mpg.mpiwg.berlin.mpdl.lt.text.reg.Regularization; -import de.mpg.mpiwg.berlin.mpdl.lt.text.reg.RegularizationManager; public class Normalizer { public static int NONE = -1; // no normalization @@ -57,16 +54,6 @@ String normStr = s; if (useSpecialNormFunction()) normStr = removeSpecialNWDMarks(normStr); - if (useRegFunction()) { - // try to regularize the string to the norm form over predefined regularizations - RegularizationManager regManager = RegularizationManager.getInstance(); - ArrayList<Regularization> regs = regManager.findRegsByOrig(language, s); - if (regs != null && regs.size() > 0) { - Regularization reg = regs.get(0); // only one: the first one - String regNormStr = reg.getNorm(); - normStr = regNormStr; - } - } if (useNormFunction()) { // normalize the string by string replacements if (normMode == DICTIONARY) { @@ -82,16 +69,6 @@ return normStr; } - private boolean useRegFunction() { - boolean useReg = false; - for (int i=0; i< normFunctions.length; i++) { - String function = normFunctions[i]; - if (function.equals("reg")) - return true; - } - return useReg; - } - private boolean useNormFunction() { boolean useNorm = false; for (int i=0; i< normFunctions.length; i++) { @@ -188,7 +165,7 @@ if (mode == DISPLAY) mpdlNormalizerLex.yybegin(MpdlNormalizerLexFR.DISP); else if (mode == DICTIONARY) - mpdlNormalizerLex.yybegin(MpdlNormalizerLexFR.DICT_ASCII); + mpdlNormalizerLex.yybegin(MpdlNormalizerLexFR.DICT); else if (mode == SEARCH) mpdlNormalizerLex.yybegin(MpdlNormalizerLexFR.SEARCH); while (token != null) {
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexAR.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexAR.java Tue Nov 27 12:35:19 2012 +0100 @@ -1,4 +1,4 @@ -/* The following code was generated by JFlex 1.4.3 on 21.07.11 11:22 */ +/* The following code was generated by JFlex 1.4.3 on 28.03.12 18:57 */ /* * Normalization rules for Arabic text @@ -15,8 +15,8 @@ /** * This class is a scanner generated by * <a href="http://www.jflex.de/">JFlex</a> 1.4.3 - * on 21.07.11 11:22 from the specification file - * <tt>MpdlNormalizerLexAR.lex</tt> + * on 28.03.12 18:57 from the specification file + * <tt>/Users/jwillenborg/test/jflexNew/MpdlNormalizerLexAR.lex</tt> */ public class MpdlNormalizerLexAR { @@ -46,9 +46,9 @@ * Translates characters to character classes */ private static final String ZZ_CMAP_PACKED = - "\12\0\1\3\25\0\1\2\14\0\1\1\2\0\1\1\17\0\1\4"+ - "\40\0\1\1\2\0\1\1\20\0\1\1\5\0\1\1\1\0\1\1"+ - "\uff82\0"; + "\12\0\1\3\25\0\1\2\14\0\1\7\1\0\1\6\1\1\13\0"+ + "\1\4\1\0\1\5\1\0\1\10\40\0\1\1\2\0\1\1\20\0"+ + "\1\1\5\0\1\1\1\0\1\1\uff82\0"; /** * Translates characters to character classes @@ -61,10 +61,11 @@ private static final int [] ZZ_ACTION = zzUnpackAction(); private static final String ZZ_ACTION_PACKED_0 = - "\4\0\2\1\1\2\1\3\1\4\1\5"; + "\4\0\4\1\1\2\1\3\1\4\1\5\5\0\1\1"+ + "\2\0"; private static int [] zzUnpackAction() { - int [] result = new int[10]; + int [] result = new int[20]; int offset = 0; offset = zzUnpackAction(ZZ_ACTION_PACKED_0, offset, result); return result; @@ -89,11 +90,12 @@ private static final int [] ZZ_ROWMAP = zzUnpackRowMap(); private static final String ZZ_ROWMAP_PACKED_0 = - "\0\0\0\5\0\12\0\17\0\24\0\31\0\24\0\24"+ - "\0\24\0\24"; + "\0\0\0\11\0\22\0\33\0\44\0\55\0\66\0\77"+ + "\0\44\0\44\0\44\0\44\0\110\0\66\0\121\0\132"+ + "\0\143\0\121\0\154\0\165"; private static int [] zzUnpackRowMap() { - int [] result = new int[10]; + int [] result = new int[20]; int offset = 0; offset = zzUnpackRowMap(ZZ_ROWMAP_PACKED_0, offset, result); return result; @@ -116,12 +118,18 @@ private static final int [] ZZ_TRANS = zzUnpackTrans(); private static final String ZZ_TRANS_PACKED_0 = - "\1\5\1\6\1\5\1\0\1\7\1\5\1\6\1\5"+ - "\1\10\1\7\1\5\1\6\1\5\1\11\1\7\1\5"+ - "\1\6\1\5\1\12\1\7\7\0\1\5\2\0"; + "\1\5\1\6\1\5\1\0\1\7\2\5\1\10\1\11"+ + "\1\5\1\6\1\5\1\12\1\7\2\5\1\10\1\11"+ + "\1\5\1\6\1\5\1\13\1\7\2\5\1\10\1\11"+ + "\1\5\1\6\1\5\1\14\1\7\2\5\1\10\1\11"+ + "\13\0\1\5\6\0\4\15\2\0\3\15\2\0\1\5"+ + "\1\0\1\16\4\0\4\15\1\0\1\17\1\20\2\15"+ + "\4\0\1\21\4\0\4\15\1\0\1\22\1\20\2\15"+ + "\6\0\1\23\2\0\4\24\2\0\7\24\1\0\1\5"+ + "\3\24"; private static int [] zzUnpackTrans() { - int [] result = new int[30]; + int [] result = new int[126]; int offset = 0; offset = zzUnpackTrans(ZZ_TRANS_PACKED_0, offset, result); return result; @@ -159,10 +167,10 @@ private static final int [] ZZ_ATTRIBUTE = zzUnpackAttribute(); private static final String ZZ_ATTRIBUTE_PACKED_0 = - "\4\0\1\11\1\1\4\11"; + "\4\0\1\11\3\1\4\11\5\0\1\1\2\0"; private static int [] zzUnpackAttribute() { - int [] result = new int[10]; + int [] result = new int[20]; int offset = 0; offset = zzUnpackAttribute(ZZ_ATTRIBUTE_PACKED_0, offset, result); return result; @@ -272,7 +280,7 @@ char [] map = new char[0x10000]; int i = 0; /* index in packed string */ int j = 0; /* index in unpacked array */ - while (i < 42) { + while (i < 52) { int count = packed.charAt(i++); char value = packed.charAt(i++); do map[j++] = value; while (--count > 0);
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexAR.lex Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexAR.lex Tue Nov 27 12:35:19 2012 +0100 @@ -7,7 +7,7 @@ * */ -package de.mpg.mpiwg.berlin.mpdl.lt.analyzer.lang; +package de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang; %% @@ -41,6 +41,12 @@ %% +// jump over empty xml elements +"<"[^><]+"/>" { add(yytext()); } +"-<"[^><]+"/>" { add(yytext()); } +"<"[^><]+"></"[^><]+">" { add(yytext()); } +"-<"[^><]+"></"[^><]+">" { add(yytext()); } + @ { problem = 1; add(yytext()); } {LB} { add(yytext()); } . { add(yytext()); }
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexDE.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexDE.java Tue Nov 27 12:35:19 2012 +0100 @@ -1,11 +1,11 @@ -/* The following code was generated by JFlex 1.4.3 on 05.09.11 10:34 */ +/* The following code was generated by JFlex 1.4.3 on 28.03.12 18:57 */ /* * Normalization rules for German text * [this is a JFlex specification] * * Wolfgang Schmidle - * version 2011-08-10 + * version 2011-07-12 * */ @@ -15,8 +15,8 @@ /** * This class is a scanner generated by * <a href="http://www.jflex.de/">JFlex</a> 1.4.3 - * on 05.09.11 10:34 from the specification file - * <tt>MpdlNormalizerLexDE.lex</tt> + * on 28.03.12 18:57 from the specification file + * <tt>/Users/jwillenborg/test/jflexNew/MpdlNormalizerLexDE.lex</tt> */ public class MpdlNormalizerLexDE { @@ -27,13 +27,12 @@ private static final int ZZ_BUFFERSIZE = 16384; /** lexical states */ - public static final int SEARCH = 10; - public static final int DICT_ASCII = 6; - public static final int SEARCH_ASCII = 12; + public static final int SEARCH = 6; public static final int DICT = 4; public static final int YYINITIAL = 0; + public static final int CELEX = 8; public static final int DISP = 2; - public static final int GRIMM = 8; + public static final int GRIMM = 10; /** * ZZ_LEXSTATE[l] is the state in the DFA for the lexical state l @@ -42,18 +41,18 @@ * l is of the form l = 2*k, k a non negative integer */ private static final int ZZ_LEXSTATE[] = { - 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 + 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5 }; /** * Translates characters to character classes */ private static final String ZZ_CMAP_PACKED = - "\12\0\1\3\25\0\1\2\14\0\1\1\2\0\1\1\17\0\1\20"+ - "\32\4\6\0\1\11\2\4\1\5\12\4\1\13\5\4\1\7\5\4"+ - "\1\1\1\0\1\1\106\0\1\14\21\0\1\15\5\0\1\16\2\0"+ - "\1\17\4\0\1\14\21\0\1\15\5\0\1\16\202\0\1\6\u01e4\0"+ - "\1\12\1\0\1\10\ufc99\0"; + "\12\0\1\3\25\0\1\2\14\0\1\11\1\0\1\10\1\1\13\0"+ + "\1\6\1\0\1\7\1\0\1\24\32\4\6\0\1\15\2\4\1\5"+ + "\12\4\1\17\5\4\1\13\5\4\1\1\1\0\1\1\106\0\1\20"+ + "\21\0\1\21\5\0\1\22\2\0\1\23\4\0\1\20\21\0\1\21"+ + "\5\0\1\22\202\0\1\12\u01e4\0\1\16\1\0\1\14\ufc99\0"; /** * Translates characters to character classes @@ -66,12 +65,12 @@ private static final int [] ZZ_ACTION = zzUnpackAction(); private static final String ZZ_ACTION_PACKED_0 = - "\7\0\2\1\1\2\1\3\1\4\3\1\1\5\1\3"+ - "\3\1\1\6\1\7\1\10\1\11\1\12\1\13\1\14"+ - "\1\15\1\16\1\17"; + "\6\0\4\1\1\2\1\3\1\4\3\1\1\5\1\6"+ + "\3\3\3\1\1\7\1\10\1\11\1\12\1\13\2\0"+ + "\1\14\1\15\1\16\1\17\3\0\1\1\2\0"; private static int [] zzUnpackAction() { - int [] result = new int[30]; + int [] result = new int[41]; int offset = 0; offset = zzUnpackAction(ZZ_ACTION_PACKED_0, offset, result); return result; @@ -96,13 +95,15 @@ private static final int [] ZZ_ROWMAP = zzUnpackRowMap(); private static final String ZZ_ROWMAP_PACKED_0 = - "\0\0\0\21\0\42\0\63\0\104\0\125\0\146\0\167"+ - "\0\210\0\167\0\167\0\167\0\231\0\252\0\273\0\167"+ - "\0\210\0\314\0\335\0\356\0\167\0\167\0\167\0\167"+ - "\0\167\0\167\0\167\0\167\0\167\0\167"; + "\0\0\0\25\0\52\0\77\0\124\0\151\0\176\0\223"+ + "\0\250\0\275\0\176\0\176\0\176\0\322\0\347\0\374"+ + "\0\176\0\176\0\223\0\250\0\275\0\u0111\0\u0126\0\u013b"+ + "\0\176\0\176\0\176\0\176\0\176\0\u0150\0\250\0\176"+ + "\0\176\0\176\0\176\0\u0165\0\u017a\0\u018f\0\u0165\0\u01a4"+ + "\0\u01b9"; private static int [] zzUnpackRowMap() { - int [] result = new int[30]; + int [] result = new int[41]; int offset = 0; offset = zzUnpackRowMap(ZZ_ROWMAP_PACKED_0, offset, result); return result; @@ -125,25 +126,29 @@ private static final int [] ZZ_TRANS = zzUnpackTrans(); private static final String ZZ_TRANS_PACKED_0 = - "\1\10\1\11\1\10\1\0\1\10\1\11\1\12\1\11"+ - "\1\10\1\11\6\10\1\13\1\10\1\11\1\10\1\14"+ - "\1\10\1\11\1\12\1\15\1\10\1\16\1\10\1\17"+ - "\4\10\1\13\1\10\1\11\1\10\1\20\1\10\1\11"+ - "\1\12\1\15\1\10\1\16\1\10\1\17\4\10\2\13"+ - "\1\21\1\13\1\20\1\10\1\11\1\12\1\22\1\13"+ - "\1\23\1\13\1\24\1\25\1\26\1\27\1\30\1\13"+ - "\1\10\1\11\1\10\1\20\1\10\1\11\1\12\1\15"+ - "\1\10\1\16\1\10\1\17\3\10\1\31\1\13\1\10"+ - "\1\11\1\10\1\32\1\10\1\11\1\12\1\15\1\10"+ - "\1\16\1\10\1\17\4\10\2\13\1\21\1\13\1\32"+ - "\1\10\1\11\1\12\1\22\1\13\1\23\1\13\1\24"+ - "\1\25\1\26\1\27\1\30\1\13\23\0\1\10\20\0"+ - "\1\10\5\0\1\33\1\0\1\34\10\0\1\10\7\0"+ - "\1\35\20\0\1\36\10\0\1\10\5\0\1\33\1\0"+ - "\1\27\10\0\1\10\7\0\1\25\20\0\1\26\6\0"; + "\1\7\1\10\1\7\1\0\1\7\1\10\1\11\2\7"+ + "\1\12\1\13\1\10\1\7\1\10\6\7\1\14\1\7"+ + "\1\10\1\7\1\15\1\7\1\10\1\11\2\7\1\12"+ + "\1\13\1\16\1\7\1\17\1\7\1\20\4\7\1\14"+ + "\1\7\1\10\1\7\1\21\1\7\1\10\1\11\2\7"+ + "\1\12\1\13\1\16\1\7\1\17\1\7\1\20\4\7"+ + "\1\14\1\7\1\10\1\7\1\22\1\7\1\10\1\11"+ + "\2\7\1\12\1\13\1\16\1\7\1\17\1\7\1\20"+ + "\4\7\2\14\1\23\1\14\1\21\1\7\1\10\1\24"+ + "\2\14\1\25\1\13\1\26\1\14\1\27\1\14\1\30"+ + "\1\31\1\32\1\33\1\34\1\14\1\7\1\10\1\7"+ + "\1\21\1\7\1\10\1\11\2\7\1\12\1\13\1\16"+ + "\1\7\1\17\1\7\1\20\3\7\1\35\1\14\27\0"+ + "\1\7\22\0\6\36\2\0\15\36\2\0\1\7\3\0"+ + "\1\37\20\0\1\7\11\0\1\40\1\0\1\41\10\0"+ + "\1\7\13\0\1\42\24\0\1\43\10\0\1\7\11\0"+ + "\1\40\1\0\1\33\10\0\1\7\13\0\1\31\24\0"+ + "\1\32\6\0\6\36\1\0\1\44\1\45\14\36\6\0"+ + "\1\46\16\0\6\36\1\0\1\47\1\45\14\36\10\0"+ + "\1\50\14\0\6\51\2\0\23\51\1\0\1\7\15\51"; private static int [] zzUnpackTrans() { - int [] result = new int[255]; + int [] result = new int[462]; int offset = 0; offset = zzUnpackTrans(ZZ_TRANS_PACKED_0, offset, result); return result; @@ -181,10 +186,11 @@ private static final int [] ZZ_ATTRIBUTE = zzUnpackAttribute(); private static final String ZZ_ATTRIBUTE_PACKED_0 = - "\7\0\1\11\1\1\3\11\3\1\1\11\4\1\12\11"; + "\6\0\1\11\3\1\3\11\3\1\2\11\6\1\5\11"+ + "\2\0\4\11\3\0\1\1\2\0"; private static int [] zzUnpackAttribute() { - int [] result = new int[30]; + int [] result = new int[41]; int offset = 0; offset = zzUnpackAttribute(ZZ_ATTRIBUTE_PACKED_0, offset, result); return result; @@ -252,8 +258,6 @@ private boolean zzEOFDone; /* user code: */ - public static final int CELEX = DICT_ASCII; - private String original = ""; private String normalized = ""; private int problem = 0; @@ -296,7 +300,7 @@ char [] map = new char[0x10000]; int i = 0; /* index in packed string */ int j = 0; /* index in unpacked array */ - while (i < 88) { + while (i < 98) { int count = packed.charAt(i++); char value = packed.charAt(i++); do map[j++] = value; while (--count > 0); @@ -563,7 +567,7 @@ zzMarkedPos = zzMarkedPosL; switch (zzAction < 0 ? zzAction : ZZ_ACTION[zzAction]) { - case 10: + case 11: { add("sz"); } case 16: break; @@ -571,7 +575,7 @@ { problem = 1; add(yytext()); } case 17: break; - case 6: + case 7: { add("ae"); } case 18: break; @@ -590,11 +594,11 @@ { add("ü"); } case 21: break; - case 8: + case 9: { add("ue"); } case 22: break; - case 11: + case 6: { switch (problem) { case 1: return original; default: return normalized.replaceAll(LB, "").toLowerCase(); @@ -613,11 +617,11 @@ { add(yytext()); } case 26: break; - case 9: + case 10: { add("ss"); } case 27: break; - case 7: + case 8: { add("oe"); } case 28: break;
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexDE.lex Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexDE.lex Tue Nov 27 12:35:19 2012 +0100 @@ -7,7 +7,7 @@ * */ -package de.mpg.mpiwg.berlin.mpdl.lt.analyzer.lang; +package de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang; %% @@ -44,6 +44,12 @@ %% +// jump over empty xml elements +"<"[^><]+"/>" { add(yytext()); } +"-<"[^><]+"/>" { add(yytext()); } +"<"[^><]+"></"[^><]+">" { add(yytext()); } +"-<"[^><]+"></"[^><]+">" { add(yytext()); } + Å¿ { add("s"); } // Fraktur
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexEL.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexEL.java Tue Nov 27 12:35:19 2012 +0100 @@ -1,4 +1,4 @@ -/* The following code was generated by JFlex 1.4.3 on 05.09.11 10:35 */ +/* The following code was generated by JFlex 1.4.3 on 28.03.12 18:57 */ /* * Normalization rules for Greek text @@ -15,8 +15,8 @@ /** * This class is a scanner generated by * <a href="http://www.jflex.de/">JFlex</a> 1.4.3 - * on 05.09.11 10:35 from the specification file - * <tt>MpdlNormalizerLexEL.lex</tt> + * on 28.03.12 18:57 from the specification file + * <tt>/Users/jwillenborg/test/jflexNew/MpdlNormalizerLexEL.lex</tt> */ public class MpdlNormalizerLexEL { @@ -47,12 +47,13 @@ * Translates characters to character classes */ private static final String ZZ_CMAP_PACKED = - "\12\0\1\3\25\0\1\2\14\0\1\1\2\0\1\1\17\0\1\5"+ - "\32\5\6\0\1\6\2\5\1\6\20\5\1\6\5\5\1\1\1\0"+ - "\1\1\u032e\0\1\7\1\10\1\11\1\12\15\0\1\4\3\0\1\4"+ - "\1\30\11\0\1\13\1\14\1\15\u1ba1\0\1\16\1\0\1\20\1\0"+ - "\1\21\1\0\1\23\1\0\1\24\1\0\1\25\1\0\1\26\65\0"+ - "\1\17\17\0\1\22\57\0\1\27\ue00d\0"; + "\12\0\1\3\25\0\1\2\14\0\1\12\1\0\1\11\1\1\13\0"+ + "\1\7\1\0\1\10\1\0\1\5\32\5\6\0\1\6\2\5\1\6"+ + "\20\5\1\6\5\5\1\1\1\0\1\1\u032e\0\1\13\1\14\1\15"+ + "\1\16\15\0\1\4\3\0\1\4\1\34\11\0\1\17\1\20\1\21"+ + "\u1ba1\0\1\22\1\0\1\24\1\0\1\25\1\0\1\27\1\0\1\30"+ + "\1\0\1\31\1\0\1\32\65\0\1\23\17\0\1\26\57\0\1\33"+ + "\ue00d\0"; /** * Translates characters to character classes @@ -65,14 +66,14 @@ private static final int [] ZZ_ACTION = zzUnpackAction(); private static final String ZZ_ACTION_PACKED_0 = - "\5\0\2\1\2\2\1\3\1\4\1\5\1\6\1\7"+ - "\1\10\1\11\1\12\1\13\12\1\1\14\1\15\1\16"+ - "\1\0\1\17\1\0\1\20\1\0\1\21\1\0\1\22"+ - "\1\0\1\23\1\0\1\24\1\0\1\25\1\0\1\26"+ - "\1\0\1\27\1\0"; + "\5\0\2\1\2\2\2\1\1\3\1\4\1\5\1\6"+ + "\1\7\1\10\1\11\1\12\1\13\12\1\1\14\1\15"+ + "\2\0\1\16\1\0\1\17\1\0\1\20\1\0\1\21"+ + "\1\0\1\22\1\0\1\23\1\0\1\24\1\0\1\25"+ + "\1\0\1\26\1\0\1\27\4\0\1\1\2\0"; private static int [] zzUnpackAction() { - int [] result = new int[50]; + int [] result = new int[60]; int offset = 0; offset = zzUnpackAction(ZZ_ACTION_PACKED_0, offset, result); return result; @@ -97,16 +98,17 @@ private static final int [] ZZ_ROWMAP = zzUnpackRowMap(); private static final String ZZ_ROWMAP_PACKED_0 = - "\0\0\0\31\0\62\0\113\0\144\0\175\0\226\0\175"+ - "\0\226\0\175\0\175\0\175\0\175\0\175\0\175\0\175"+ - "\0\175\0\175\0\257\0\310\0\341\0\372\0\u0113\0\u012c"+ - "\0\u0145\0\u015e\0\u0177\0\u0190\0\175\0\175\0\175\0\u01a9"+ - "\0\175\0\u01c2\0\175\0\u01db\0\175\0\u01f4\0\175\0\u020d"+ - "\0\175\0\u0226\0\175\0\u023f\0\175\0\u0258\0\175\0\u0271"+ - "\0\175\0\u028a"; + "\0\0\0\35\0\72\0\127\0\164\0\221\0\256\0\221"+ + "\0\256\0\313\0\350\0\221\0\221\0\221\0\221\0\221"+ + "\0\221\0\221\0\221\0\221\0\u0105\0\u0122\0\u013f\0\u015c"+ + "\0\u0179\0\u0196\0\u01b3\0\u01d0\0\u01ed\0\u020a\0\221\0\221"+ + "\0\u0227\0\313\0\221\0\u0244\0\221\0\u0261\0\221\0\u027e"+ + "\0\221\0\u029b\0\221\0\u02b8\0\221\0\u02d5\0\221\0\u02f2"+ + "\0\221\0\u030f\0\221\0\u032c\0\221\0\u0349\0\u0366\0\u0383"+ + "\0\u03a0\0\u0366\0\u03bd\0\u03da"; private static int [] zzUnpackRowMap() { - int [] result = new int[50]; + int [] result = new int[60]; int offset = 0; offset = zzUnpackRowMap(ZZ_ROWMAP_PACKED_0, offset, result); return result; @@ -130,30 +132,36 @@ private static final String ZZ_TRANS_PACKED_0 = "\1\6\1\7\1\6\1\0\1\6\1\10\1\11\1\12"+ - "\1\13\1\14\1\15\1\16\1\17\1\20\14\6\1\7"+ - "\1\6\1\21\1\6\1\10\1\11\1\12\1\13\1\14"+ - "\1\15\1\16\1\17\1\20\14\6\1\7\1\6\1\22"+ - "\1\6\1\10\1\11\1\12\1\13\1\14\1\15\1\16"+ - "\1\17\1\20\1\23\1\24\1\25\1\26\1\27\1\30"+ - "\1\31\1\32\1\33\1\34\2\6\1\7\1\6\1\35"+ - "\1\6\1\10\1\11\1\12\1\13\1\14\1\15\1\16"+ - "\1\17\1\20\1\23\1\24\1\25\1\26\1\27\1\30"+ - "\1\31\1\32\1\33\1\34\2\6\1\7\1\6\1\22"+ - "\1\6\1\10\1\11\1\12\1\13\1\14\1\15\1\16"+ - "\1\17\1\20\1\23\1\24\1\25\1\26\1\27\1\30"+ - "\1\31\1\32\1\33\1\34\1\36\33\0\1\6\31\0"+ - "\1\37\1\40\23\0\1\40\3\0\1\41\1\42\23\0"+ - "\1\42\3\0\1\43\1\44\23\0\1\44\3\0\1\45"+ - "\1\46\23\0\1\46\3\0\1\47\1\50\23\0\1\50"+ - "\3\0\1\51\1\52\23\0\1\52\3\0\1\53\1\54"+ - "\23\0\1\54\3\0\1\55\1\56\23\0\1\56\3\0"+ - "\1\57\1\60\23\0\1\60\3\0\1\61\1\62\23\0"+ - "\1\62\3\0\1\37\30\0\1\41\30\0\1\43\30\0"+ - "\1\45\30\0\1\47\30\0\1\51\30\0\1\53\30\0"+ - "\1\55\30\0\1\57\30\0\1\61\25\0"; + "\2\6\1\13\1\14\1\15\1\16\1\17\1\20\1\21"+ + "\1\22\14\6\1\7\1\6\1\23\1\6\1\10\1\11"+ + "\1\12\2\6\1\13\1\14\1\15\1\16\1\17\1\20"+ + "\1\21\1\22\14\6\1\7\1\6\1\24\1\6\1\10"+ + "\1\11\1\12\2\6\1\13\1\14\1\15\1\16\1\17"+ + "\1\20\1\21\1\22\1\25\1\26\1\27\1\30\1\31"+ + "\1\32\1\33\1\34\1\35\1\36\2\6\1\7\1\6"+ + "\1\37\1\6\1\10\1\11\1\12\2\6\1\13\1\14"+ + "\1\15\1\16\1\17\1\20\1\21\1\22\1\25\1\26"+ + "\1\27\1\30\1\31\1\32\1\33\1\34\1\35\1\36"+ + "\2\6\1\7\1\6\1\24\1\6\1\10\1\11\1\12"+ + "\2\6\1\13\1\14\1\15\1\16\1\17\1\20\1\21"+ + "\1\22\1\25\1\26\1\27\1\30\1\31\1\32\1\33"+ + "\1\34\1\35\1\36\1\40\37\0\1\6\32\0\7\41"+ + "\2\0\24\41\2\0\1\6\4\0\1\42\30\0\1\43"+ + "\1\44\27\0\1\44\3\0\1\45\1\46\27\0\1\46"+ + "\3\0\1\47\1\50\27\0\1\50\3\0\1\51\1\52"+ + "\27\0\1\52\3\0\1\53\1\54\27\0\1\54\3\0"+ + "\1\55\1\56\27\0\1\56\3\0\1\57\1\60\27\0"+ + "\1\60\3\0\1\61\1\62\27\0\1\62\3\0\1\63"+ + "\1\64\27\0\1\64\3\0\1\65\1\66\27\0\1\66"+ + "\7\41\1\0\1\67\1\70\23\41\3\0\1\43\34\0"+ + "\1\45\34\0\1\47\34\0\1\51\34\0\1\53\34\0"+ + "\1\55\34\0\1\57\34\0\1\61\34\0\1\63\34\0"+ + "\1\65\40\0\1\71\25\0\7\41\1\0\1\72\1\70"+ + "\23\41\11\0\1\73\23\0\7\74\2\0\33\74\1\0"+ + "\1\6\24\74"; private static int [] zzUnpackTrans() { - int [] result = new int[675]; + int [] result = new int[1015]; int offset = 0; offset = zzUnpackTrans(ZZ_TRANS_PACKED_0, offset, result); return result; @@ -191,13 +199,13 @@ private static final int [] ZZ_ATTRIBUTE = zzUnpackAttribute(); private static final String ZZ_ATTRIBUTE_PACKED_0 = - "\5\0\1\11\1\1\1\11\1\1\11\11\12\1\3\11"+ + "\5\0\1\11\1\1\1\11\3\1\11\11\12\1\2\11"+ + "\2\0\1\11\1\0\1\11\1\0\1\11\1\0\1\11"+ "\1\0\1\11\1\0\1\11\1\0\1\11\1\0\1\11"+ - "\1\0\1\11\1\0\1\11\1\0\1\11\1\0\1\11"+ - "\1\0\1\11\1\0"; + "\1\0\1\11\1\0\1\11\4\0\1\1\2\0"; private static int [] zzUnpackAttribute() { - int [] result = new int[50]; + int [] result = new int[60]; int offset = 0; offset = zzUnpackAttribute(ZZ_ATTRIBUTE_PACKED_0, offset, result); return result; @@ -307,7 +315,7 @@ char [] map = new char[0x10000]; int i = 0; /* index in packed string */ int j = 0; /* index in unpacked array */ - while (i < 112) { + while (i < 122) { int count = packed.charAt(i++); char value = packed.charAt(i++); do map[j++] = value; while (--count > 0);
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexEL.lex Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexEL.lex Tue Nov 27 12:35:19 2012 +0100 @@ -7,7 +7,7 @@ * */ -package de.mpg.mpiwg.berlin.mpdl.lt.analyzer.lang; +package de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang; %% @@ -47,6 +47,12 @@ %% +// jump over empty xml elements +"<"[^><]+"/>" { add(yytext()); } +"-<"[^><]+"/>" { add(yytext()); } +"<"[^><]+"></"[^><]+">" { add(yytext()); } +"-<"[^><]+"></"[^><]+">" { add(yytext()); } + // always replace tonos by oxia // (although this should really be corrected in the text rather than normalized)
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexEN.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexEN.java Tue Nov 27 12:35:19 2012 +0100 @@ -1,4 +1,4 @@ -/* The following code was generated by JFlex 1.4.3 on 21.07.11 11:22 */ +/* The following code was generated by JFlex 1.4.3 on 28.03.12 18:57 */ /* * Normalization rules for English text @@ -15,8 +15,8 @@ /** * This class is a scanner generated by * <a href="http://www.jflex.de/">JFlex</a> 1.4.3 - * on 21.07.11 11:22 from the specification file - * <tt>MpdlNormalizerLexEN.lex</tt> + * on 28.03.12 18:57 from the specification file + * <tt>/Users/jwillenborg/test/jflexNew/MpdlNormalizerLexEN.lex</tt> */ public class MpdlNormalizerLexEN { @@ -46,9 +46,9 @@ * Translates characters to character classes */ private static final String ZZ_CMAP_PACKED = - "\12\0\1\3\25\0\1\2\14\0\1\1\2\0\1\1\17\0\1\5"+ - "\40\0\1\1\2\0\1\1\20\0\1\1\5\0\1\1\1\0\1\1"+ - "\u0101\0\1\4\ufe80\0"; + "\12\0\1\3\25\0\1\2\14\0\1\7\1\0\1\6\1\1\13\0"+ + "\1\4\1\0\1\5\1\0\1\11\40\0\1\1\2\0\1\1\20\0"+ + "\1\1\5\0\1\1\1\0\1\1\u0101\0\1\10\ufe80\0"; /** * Translates characters to character classes @@ -61,10 +61,11 @@ private static final int [] ZZ_ACTION = zzUnpackAction(); private static final String ZZ_ACTION_PACKED_0 = - "\4\0\2\1\1\2\1\3\1\4\1\5\1\6"; + "\4\0\4\1\1\2\1\3\1\4\1\5\1\6\5\0"+ + "\1\1\2\0"; private static int [] zzUnpackAction() { - int [] result = new int[11]; + int [] result = new int[21]; int offset = 0; offset = zzUnpackAction(ZZ_ACTION_PACKED_0, offset, result); return result; @@ -89,11 +90,12 @@ private static final int [] ZZ_ROWMAP = zzUnpackRowMap(); private static final String ZZ_ROWMAP_PACKED_0 = - "\0\0\0\6\0\14\0\22\0\30\0\36\0\30\0\30"+ - "\0\30\0\30\0\30"; + "\0\0\0\12\0\24\0\36\0\50\0\62\0\74\0\106"+ + "\0\50\0\50\0\50\0\50\0\50\0\120\0\74\0\132"+ + "\0\144\0\156\0\132\0\170\0\202"; private static int [] zzUnpackRowMap() { - int [] result = new int[11]; + int [] result = new int[21]; int offset = 0; offset = zzUnpackRowMap(ZZ_ROWMAP_PACKED_0, offset, result); return result; @@ -116,13 +118,18 @@ private static final int [] ZZ_TRANS = zzUnpackTrans(); private static final String ZZ_TRANS_PACKED_0 = - "\1\5\1\6\1\5\1\0\1\5\1\7\1\5\1\6"+ - "\1\5\1\10\1\11\1\7\1\5\1\6\1\5\1\12"+ - "\1\11\1\7\1\5\1\6\1\5\1\13\1\11\1\7"+ - "\10\0\1\5\3\0"; + "\1\5\1\6\1\5\1\0\1\7\2\5\1\10\1\5"+ + "\1\11\1\5\1\6\1\5\1\12\1\7\2\5\1\10"+ + "\1\13\1\11\1\5\1\6\1\5\1\14\1\7\2\5"+ + "\1\10\1\13\1\11\1\5\1\6\1\5\1\15\1\7"+ + "\2\5\1\10\1\13\1\11\14\0\1\5\7\0\4\16"+ + "\2\0\4\16\2\0\1\5\1\0\1\17\5\0\4\16"+ + "\1\0\1\20\1\21\3\16\4\0\1\22\5\0\4\16"+ + "\1\0\1\23\1\21\3\16\6\0\1\24\3\0\4\25"+ + "\2\0\10\25\1\0\1\5\4\25"; private static int [] zzUnpackTrans() { - int [] result = new int[36]; + int [] result = new int[140]; int offset = 0; offset = zzUnpackTrans(ZZ_TRANS_PACKED_0, offset, result); return result; @@ -160,10 +167,10 @@ private static final int [] ZZ_ATTRIBUTE = zzUnpackAttribute(); private static final String ZZ_ATTRIBUTE_PACKED_0 = - "\4\0\1\11\1\1\5\11"; + "\4\0\1\11\3\1\5\11\5\0\1\1\2\0"; private static int [] zzUnpackAttribute() { - int [] result = new int[11]; + int [] result = new int[21]; int offset = 0; offset = zzUnpackAttribute(ZZ_ATTRIBUTE_PACKED_0, offset, result); return result; @@ -273,7 +280,7 @@ char [] map = new char[0x10000]; int i = 0; /* index in packed string */ int j = 0; /* index in unpacked array */ - while (i < 46) { + while (i < 56) { int count = packed.charAt(i++); char value = packed.charAt(i++); do map[j++] = value; while (--count > 0);
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexEN.lex Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexEN.lex Tue Nov 27 12:35:19 2012 +0100 @@ -7,7 +7,7 @@ * */ -package de.mpg.mpiwg.berlin.mpdl.lt.analyzer.lang; +package de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang; %% @@ -41,6 +41,12 @@ %% +// jump over empty xml elements +"<"[^><]+"/>" { add(yytext()); } +"-<"[^><]+"/>" { add(yytext()); } +"<"[^><]+"></"[^><]+">" { add(yytext()); } +"-<"[^><]+"></"[^><]+">" { add(yytext()); } + <DISP, DICT, SEARCH> { Å¿ { add("s"); }
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexFR.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexFR.java Tue Nov 27 12:35:19 2012 +0100 @@ -1,11 +1,11 @@ -/* The following code was generated by JFlex 1.4.3 on 05.09.11 10:35 */ +/* The following code was generated by JFlex 1.4.3 on 28.03.12 18:57 */ /* * Normalization rules for French text * [this is a JFlex specification] * * Wolfgang Schmidle - * version 2011-08-10 + * version 2011-07-12 * */ @@ -15,8 +15,8 @@ /** * This class is a scanner generated by * <a href="http://www.jflex.de/">JFlex</a> 1.4.3 - * on 05.09.11 10:35 from the specification file - * <tt>MpdlNormalizerLexFR.lex</tt> + * on 28.03.12 18:57 from the specification file + * <tt>/Users/jwillenborg/test/jflexNew/MpdlNormalizerLexFR.lex</tt> */ public class MpdlNormalizerLexFR { @@ -27,10 +27,10 @@ private static final int ZZ_BUFFERSIZE = 16384; /** lexical states */ - public static final int DICT_ASCII = 8; public static final int SEARCH = 6; public static final int DICT = 4; public static final int YYINITIAL = 0; + public static final int CELEX = 8; public static final int DISP = 2; /** @@ -47,11 +47,11 @@ * Translates characters to character classes */ private static final String ZZ_CMAP_PACKED = - "\12\0\1\3\25\0\1\2\14\0\1\1\2\0\1\1\17\0\1\20"+ - "\32\4\6\0\1\5\2\4\1\5\20\4\1\5\5\4\1\1\1\0"+ - "\1\1\141\0\1\7\3\12\3\0\1\10\1\0\3\13\1\0\3\14"+ - "\3\0\3\15\4\0\3\16\126\0\2\11\53\0\1\6\u1e99\0\1\17"+ - "\udfe6\0"; + "\12\0\1\3\25\0\1\2\14\0\1\11\1\0\1\10\1\1\13\0"+ + "\1\6\1\0\1\7\1\0\1\24\32\4\6\0\1\5\2\4\1\5"+ + "\20\4\1\5\5\4\1\1\1\0\1\1\141\0\1\13\3\16\3\0"+ + "\1\14\1\0\3\17\1\0\3\20\3\0\3\21\4\0\3\22\126\0"+ + "\2\15\53\0\1\12\u1e99\0\1\23\udfe6\0"; /** * Translates characters to character classes @@ -64,12 +64,12 @@ private static final int [] ZZ_ACTION = zzUnpackAction(); private static final String ZZ_ACTION_PACKED_0 = - "\5\0\2\1\1\2\1\3\1\4\1\5\1\6\1\7"+ - "\1\10\1\2\1\11\1\12\1\13\1\14\1\15\1\16"+ - "\1\17"; + "\5\0\4\1\1\2\1\3\1\4\1\5\1\6\1\7"+ + "\1\10\3\2\1\11\1\12\1\13\1\14\1\15\1\16"+ + "\1\17\5\0\1\1\2\0"; private static int [] zzUnpackAction() { - int [] result = new int[22]; + int [] result = new int[34]; int offset = 0; offset = zzUnpackAction(ZZ_ACTION_PACKED_0, offset, result); return result; @@ -94,12 +94,14 @@ private static final int [] ZZ_ROWMAP = zzUnpackRowMap(); private static final String ZZ_ROWMAP_PACKED_0 = - "\0\0\0\21\0\42\0\63\0\104\0\125\0\146\0\125"+ - "\0\125\0\125\0\125\0\125\0\125\0\125\0\146\0\125"+ - "\0\125\0\125\0\125\0\125\0\125\0\125"; + "\0\0\0\25\0\52\0\77\0\124\0\151\0\176\0\223"+ + "\0\250\0\151\0\151\0\151\0\151\0\151\0\151\0\151"+ + "\0\176\0\223\0\250\0\151\0\151\0\151\0\151\0\151"+ + "\0\151\0\151\0\275\0\223\0\322\0\347\0\374\0\322"+ + "\0\u0111\0\u0126"; private static int [] zzUnpackRowMap() { - int [] result = new int[22]; + int [] result = new int[34]; int offset = 0; offset = zzUnpackRowMap(ZZ_ROWMAP_PACKED_0, offset, result); return result; @@ -122,17 +124,23 @@ private static final int [] ZZ_TRANS = zzUnpackTrans(); private static final String ZZ_TRANS_PACKED_0 = - "\1\6\1\7\1\6\1\0\1\6\1\7\12\6\1\10"+ - "\1\6\1\7\1\6\1\11\1\6\1\7\1\12\1\13"+ - "\1\14\7\6\1\10\1\6\1\7\1\6\1\15\1\6"+ - "\1\7\1\12\1\13\1\14\7\6\1\10\1\6\1\7"+ - "\1\6\1\16\1\6\1\7\1\12\1\13\1\14\7\6"+ - "\2\10\1\17\1\10\1\15\1\6\1\7\1\12\1\13"+ - "\1\14\1\20\1\21\1\22\1\23\1\24\1\25\1\26"+ - "\1\10\23\0\1\6\16\0"; + "\1\6\1\7\1\6\1\0\1\6\1\7\1\10\2\6"+ + "\1\11\12\6\1\12\1\6\1\7\1\6\1\13\1\6"+ + "\1\7\1\10\2\6\1\11\1\14\1\15\1\16\7\6"+ + "\1\12\1\6\1\7\1\6\1\17\1\6\1\7\1\10"+ + "\2\6\1\11\1\14\1\15\1\16\7\6\1\12\1\6"+ + "\1\7\1\6\1\20\1\6\1\7\1\10\2\6\1\11"+ + "\1\14\1\15\1\16\7\6\2\12\1\21\1\12\1\17"+ + "\1\6\1\7\1\22\2\12\1\23\1\14\1\15\1\16"+ + "\1\24\1\25\1\26\1\27\1\30\1\31\1\32\1\12"+ + "\27\0\1\6\22\0\6\33\2\0\15\33\2\0\1\6"+ + "\3\0\1\34\16\0\6\33\1\0\1\35\1\36\14\33"+ + "\6\0\1\37\16\0\6\33\1\0\1\40\1\36\14\33"+ + "\10\0\1\41\14\0\6\42\2\0\23\42\1\0\1\6"+ + "\15\42"; private static int [] zzUnpackTrans() { - int [] result = new int[119]; + int [] result = new int[315]; int offset = 0; offset = zzUnpackTrans(ZZ_TRANS_PACKED_0, offset, result); return result; @@ -170,10 +178,11 @@ private static final int [] ZZ_ATTRIBUTE = zzUnpackAttribute(); private static final String ZZ_ATTRIBUTE_PACKED_0 = - "\5\0\1\11\1\1\7\11\1\1\7\11"; + "\5\0\1\11\3\1\7\11\3\1\7\11\5\0\1\1"+ + "\2\0"; private static int [] zzUnpackAttribute() { - int [] result = new int[22]; + int [] result = new int[34]; int offset = 0; offset = zzUnpackAttribute(ZZ_ATTRIBUTE_PACKED_0, offset, result); return result; @@ -283,7 +292,7 @@ char [] map = new char[0x10000]; int i = 0; /* index in packed string */ int j = 0; /* index in unpacked array */ - while (i < 82) { + while (i < 92) { int count = packed.charAt(i++); char value = packed.charAt(i++); do map[j++] = value; while (--count > 0);
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexFR.lex Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexFR.lex Tue Nov 27 12:35:19 2012 +0100 @@ -7,7 +7,7 @@ * */ -package de.mpg.mpiwg.berlin.mpdl.lt.analyzer.lang; +package de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang; %% @@ -44,6 +44,12 @@ %% +// jump over empty xml elements +"<"[^><]+"/>" { add(yytext()); } +"-<"[^><]+"/>" { add(yytext()); } +"<"[^><]+"></"[^><]+">" { add(yytext()); } +"-<"[^><]+"></"[^><]+">" { add(yytext()); } + <DISP, DICT, SEARCH, CELEX> { Å¿ { add("s"); }
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexIT.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexIT.java Tue Nov 27 12:35:19 2012 +0100 @@ -1,4 +1,4 @@ -/* The following code was generated by JFlex 1.4.3 on 21.07.11 11:22 */ +/* The following code was generated by JFlex 1.4.3 on 28.03.12 18:57 */ /* * Normalization rules for Italian text @@ -15,8 +15,8 @@ /** * This class is a scanner generated by * <a href="http://www.jflex.de/">JFlex</a> 1.4.3 - * on 21.07.11 11:22 from the specification file - * <tt>MpdlNormalizerLexIT.lex</tt> + * on 28.03.12 18:57 from the specification file + * <tt>/Users/jwillenborg/test/jflexNew/MpdlNormalizerLexIT.lex</tt> */ public class MpdlNormalizerLexIT { @@ -46,15 +46,16 @@ * Translates characters to character classes */ private static final String ZZ_CMAP_PACKED = - "\12\0\1\6\25\0\1\5\14\0\1\4\22\0\1\52\1\1\3\2"+ - "\1\1\3\2\1\41\1\0\1\2\1\3\2\2\1\42\1\2\1\50"+ - "\1\3\1\2\1\40\1\45\1\51\2\2\1\0\1\2\6\0\1\44"+ - "\3\2\1\12\2\2\1\43\1\7\1\36\1\2\1\3\1\2\1\10"+ - "\1\37\1\14\1\46\1\13\1\2\1\11\1\16\1\47\2\2\1\0"+ - "\1\2\62\0\1\4\22\0\1\17\5\0\1\33\1\0\1\20\3\0"+ - "\1\21\5\0\1\22\6\0\1\23\5\0\1\31\1\24\5\0\1\32"+ - "\1\0\1\25\3\0\1\26\5\0\1\27\6\0\1\30\37\0\1\1"+ - "\70\0\1\35\1\34\53\0\1\15\ufe80\0"; + "\12\0\1\6\25\0\1\5\14\0\1\22\1\0\1\21\14\0\1\17"+ + "\1\0\1\20\1\0\1\56\1\1\3\2\1\1\3\2\1\45\1\0"+ + "\1\2\1\3\2\2\1\46\1\2\1\54\1\3\1\2\1\44\1\51"+ + "\1\55\2\2\1\0\1\2\6\0\1\50\3\2\1\12\2\2\1\47"+ + "\1\7\1\42\1\2\1\3\1\2\1\10\1\43\1\14\1\52\1\13"+ + "\1\2\1\11\1\16\1\53\2\2\1\0\1\2\62\0\1\4\22\0"+ + "\1\23\5\0\1\37\1\0\1\24\3\0\1\25\5\0\1\26\6\0"+ + "\1\27\5\0\1\35\1\30\5\0\1\36\1\0\1\31\3\0\1\32"+ + "\5\0\1\33\6\0\1\34\37\0\1\1\70\0\1\41\1\40\53\0"+ + "\1\15\ufe80\0"; /** * Translates characters to character classes @@ -67,17 +68,18 @@ private static final int [] ZZ_ACTION = zzUnpackAction(); private static final String ZZ_ACTION_PACKED_0 = - "\11\0\1\1\1\2\2\3\1\1\1\4\1\2\1\3"+ - "\1\5\1\2\1\6\1\7\1\10\1\11\1\12\5\3"+ - "\1\13\1\2\1\3\1\5\1\2\1\14\1\15\1\16"+ - "\1\17\1\20\1\21\1\22\1\23\1\24\1\25\1\26"+ - "\1\27\1\30\4\0\1\31\1\32\1\33\1\0\1\34"+ - "\1\0\1\35\1\36\1\0\1\37\1\40\1\41\4\0"+ - "\1\42\6\0\1\43\1\44\4\0\1\45\1\0\1\46"+ - "\10\0\1\47\4\0\1\45\2\0\1\50"; + "\13\0\1\1\1\2\2\3\1\1\1\4\1\2\1\3"+ + "\1\5\1\2\2\1\1\6\1\7\1\10\1\11\1\12"+ + "\5\3\1\13\1\2\1\3\1\5\1\2\1\14\1\15"+ + "\1\16\1\17\1\20\1\21\1\22\1\23\1\24\1\25"+ + "\1\26\1\27\1\30\5\0\1\31\1\32\1\33\1\0"+ + "\1\34\1\0\1\35\1\36\1\0\1\37\1\40\1\41"+ + "\4\0\1\42\10\0\1\43\1\44\4\0\1\45\1\0"+ + "\1\46\6\0\1\32\3\0\1\47\5\0\1\45\3\0"+ + "\1\50"; private static int [] zzUnpackAction() { - int [] result = new int[100]; + int [] result = new int[111]; int offset = 0; offset = zzUnpackAction(ZZ_ACTION_PACKED_0, offset, result); return result; @@ -102,22 +104,23 @@ private static final int [] ZZ_ROWMAP = zzUnpackRowMap(); private static final String ZZ_ROWMAP_PACKED_0 = - "\0\0\0\53\0\126\0\201\0\254\0\327\0\u0102\0\u012d"+ - "\0\u0158\0\0\0\0\0\0\0\u0183\0\u01ae\0\0\0\u01d9"+ - "\0\u0204\0\0\0\u022f\0\0\0\0\0\0\0\0\0\0"+ - "\0\u025a\0\u0285\0\u02b0\0\u02db\0\u0306\0\0\0\u0331\0\u035c"+ - "\0\u0387\0\u03b2\0\u03dd\0\0\0\0\0\0\0\0\0\0"+ - "\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\u0408"+ - "\0\u0433\0\u045e\0\u0489\0\0\0\0\0\0\0\u04b4\0\0"+ - "\0\u04df\0\0\0\0\0\u050a\0\0\0\0\0\0\0\u0535"+ - "\0\u0560\0\u058b\0\u05b6\0\0\0\u05e1\0\u060c\0\u0637\0\u0662"+ - "\0\u068d\0\0\0\0\0\0\0\u06b8\0\u06e3\0\u070e\0\u035c"+ - "\0\u0739\0\u0764\0\0\0\u078f\0\u07ba\0\u07e5\0\0\0\u0810"+ - "\0\u083b\0\u0866\0\u0891\0\0\0\u08bc\0\u08e7\0\u0912\0\u093d"+ - "\0\0\0\u0968\0\u0993\0\0"; + "\0\0\0\57\0\136\0\215\0\274\0\353\0\u011a\0\u0149"+ + "\0\u0178\0\u01a7\0\u01d6\0\u0205\0\u0205\0\u0205\0\u0234\0\u0263"+ + "\0\u0205\0\u0292\0\u02c1\0\u0205\0\u02f0\0\u01a7\0\u031f\0\u0205"+ + "\0\u0205\0\u0205\0\u0205\0\u0205\0\u034e\0\u037d\0\u03ac\0\u03db"+ + "\0\u040a\0\u0205\0\u0439\0\u0468\0\u0497\0\u04c6\0\u04f5\0\u0205"+ + "\0\u0205\0\u0205\0\u0205\0\u0205\0\u0205\0\u0205\0\u0205\0\u0205"+ + "\0\u0205\0\u0205\0\u0205\0\u0524\0\u0553\0\u0582\0\u05b1\0\u05e0"+ + "\0\u0205\0\u0205\0\u0205\0\u060f\0\u0205\0\u063e\0\u0205\0\u0205"+ + "\0\u066d\0\u0205\0\u0205\0\u0205\0\u069c\0\u06cb\0\u06fa\0\u0729"+ + "\0\u0205\0\u0758\0\u0787\0\u07b6\0\u07e5\0\u0814\0\u0205\0\u0843"+ + "\0\u0872\0\u0205\0\u0205\0\u08a1\0\u08d0\0\u08ff\0\u0468\0\u092e"+ + "\0\u095d\0\u0205\0\u098c\0\u09bb\0\u09ea\0\u0205\0\u0a19\0\u0a48"+ + "\0\u0843\0\u0a77\0\u0aa6\0\u0ad5\0\u0205\0\u0b04\0\u0b33\0\u0b62"+ + "\0\u0b91\0\u0bc0\0\u0205\0\u0bef\0\u0c1e\0\u0c4d\0\u0205"; private static int [] zzUnpackRowMap() { - int [] result = new int[100]; + int [] result = new int[111]; int offset = 0; offset = zzUnpackRowMap(ZZ_ROWMAP_PACKED_0, offset, result); return result; @@ -140,67 +143,76 @@ private static final int [] ZZ_TRANS = zzUnpackTrans(); private static final String ZZ_TRANS_PACKED_0 = - "\53\0\1\12\1\13\1\14\1\15\1\16\1\12\1\17"+ - "\1\20\1\14\1\21\1\13\1\15\1\14\1\22\1\23"+ - "\5\12\2\13\1\12\2\13\1\24\1\25\1\26\1\27"+ - "\1\30\1\12\1\13\1\31\2\13\1\14\1\13\1\23"+ - "\1\32\1\33\1\34\1\35\1\36\1\12\1\13\1\14"+ - "\1\15\1\16\1\12\1\17\1\37\1\14\1\21\1\13"+ - "\1\15\1\40\1\41\1\42\5\12\2\13\1\12\2\13"+ - "\1\24\1\25\1\26\1\27\1\30\1\12\1\13\1\31"+ - "\2\13\1\43\1\13\1\42\1\32\1\33\1\34\1\35"+ - "\1\36\1\12\1\13\1\14\1\15\1\16\1\12\1\44"+ - "\1\20\1\14\1\21\1\13\1\15\1\14\1\22\1\23"+ - "\1\45\1\46\1\47\1\50\1\51\1\52\1\53\1\54"+ - "\1\55\1\56\1\24\1\25\1\26\1\27\1\30\1\12"+ - "\1\13\1\31\2\13\1\14\1\13\1\23\1\32\1\33"+ - "\1\34\1\35\1\36\1\12\1\13\1\14\1\15\1\16"+ - "\1\12\1\44\1\37\1\14\1\21\1\13\1\15\1\40"+ - "\1\41\1\42\1\45\1\46\1\47\1\50\1\51\1\52"+ - "\1\53\1\54\1\55\1\56\1\24\1\25\1\26\1\27"+ - "\1\30\1\12\1\13\1\31\2\13\1\43\1\13\1\42"+ - "\1\32\1\33\1\34\1\35\1\36\1\12\1\13\1\14"+ - "\1\15\1\16\1\12\1\57\1\20\1\14\1\21\1\13"+ - "\1\15\1\14\1\22\1\23\1\45\1\46\1\47\1\50"+ - "\1\51\1\52\1\53\1\54\1\55\1\56\1\24\1\25"+ - "\1\26\1\27\1\30\1\12\1\13\1\31\2\13\1\14"+ - "\1\13\1\23\1\32\1\33\1\34\1\35\1\36\1\12"+ - "\1\13\1\14\1\15\1\16\1\12\1\57\1\37\1\14"+ - "\1\21\1\13\1\15\1\40\1\41\1\42\1\45\1\46"+ - "\1\47\1\50\1\51\1\52\1\53\1\54\1\55\1\56"+ - "\1\24\1\25\1\26\1\27\1\30\1\12\1\13\1\31"+ - "\2\13\1\43\1\13\1\42\1\32\1\33\1\34\1\35"+ - "\1\36\7\0\1\60\4\0\1\61\1\62\42\0\1\63"+ - "\114\0\1\64\1\0\1\64\6\0\1\65\103\0\1\66"+ - "\23\0\1\67\44\0\1\70\5\0\1\70\2\0\1\70"+ - "\3\0\1\70\5\0\2\70\1\0\2\70\1\0\3\70"+ - "\2\0\1\70\1\0\2\70\1\0\2\70\46\0\1\71"+ - "\60\0\1\72\5\0\2\73\1\74\3\0\2\73\1\0"+ - "\3\73\13\0\1\73\6\0\1\73\2\0\1\73\2\0"+ - "\4\73\50\0\1\75\1\0\1\76\3\0\2\77\1\100"+ - "\3\0\2\77\1\0\3\77\13\0\1\77\6\0\1\77"+ - "\2\0\1\77\2\0\4\77\11\0\1\101\25\0\1\66"+ - "\26\0\1\102\52\0\1\102\3\0\1\103\35\0\1\104"+ - "\5\0\1\104\2\0\1\104\3\0\1\104\5\0\2\104"+ - "\1\0\2\104\1\0\3\104\2\0\1\104\1\0\2\104"+ - "\1\0\2\104\44\0\1\105\4\0\1\106\16\0\1\107"+ - "\54\0\1\110\52\0\1\110\3\0\1\111\40\0\1\112"+ - "\105\0\1\113\55\0\1\114\15\0\1\115\52\0\1\116"+ - "\51\0\1\117\4\0\1\120\54\0\1\121\43\0\1\122"+ - "\7\0\1\120\44\0\1\123\52\0\1\123\1\124\1\125"+ - "\46\0\1\126\4\0\1\61\54\0\1\127\43\0\1\130"+ - "\7\0\1\61\40\0\2\73\4\0\2\73\1\0\3\73"+ - "\13\0\1\73\6\0\1\73\2\0\1\73\2\0\4\73"+ - "\3\0\2\77\4\0\2\77\1\0\3\77\13\0\1\77"+ - "\6\0\1\77\2\0\1\77\2\0\4\77\6\0\1\131"+ - "\51\0\1\132\53\0\1\133\53\0\1\134\50\0\1\135"+ - "\3\0\1\136\47\0\1\137\52\0\1\140\56\0\1\120"+ - "\46\0\1\141\61\0\1\120\43\0\1\142\104\0\1\143"+ - "\24\0\1\61\55\0\1\61\46\0\1\136\50\0\1\144"+ - "\44\0"; + "\17\0\1\12\2\0\1\13\34\0\1\14\1\15\1\16"+ + "\1\17\1\20\1\14\1\21\1\22\1\16\1\23\1\15"+ + "\1\17\1\16\1\24\1\25\1\26\2\14\1\27\5\14"+ + "\2\15\1\14\2\15\1\30\1\31\1\32\1\33\1\34"+ + "\1\14\1\15\1\35\2\15\1\16\1\15\1\25\1\36"+ + "\1\37\1\40\1\41\1\42\1\14\1\15\1\16\1\17"+ + "\1\20\1\14\1\21\1\43\1\16\1\23\1\15\1\17"+ + "\1\44\1\45\1\46\1\26\2\14\1\27\5\14\2\15"+ + "\1\14\2\15\1\30\1\31\1\32\1\33\1\34\1\14"+ + "\1\15\1\35\2\15\1\47\1\15\1\46\1\36\1\37"+ + "\1\40\1\41\1\42\1\14\1\15\1\16\1\17\1\20"+ + "\1\14\1\50\1\22\1\16\1\23\1\15\1\17\1\16"+ + "\1\24\1\25\1\26\2\14\1\27\1\51\1\52\1\53"+ + "\1\54\1\55\1\56\1\57\1\60\1\61\1\62\1\30"+ + "\1\31\1\32\1\33\1\34\1\14\1\15\1\35\2\15"+ + "\1\16\1\15\1\25\1\36\1\37\1\40\1\41\1\42"+ + "\1\14\1\15\1\16\1\17\1\20\1\14\1\50\1\43"+ + "\1\16\1\23\1\15\1\17\1\44\1\45\1\46\1\26"+ + "\2\14\1\27\1\51\1\52\1\53\1\54\1\55\1\56"+ + "\1\57\1\60\1\61\1\62\1\30\1\31\1\32\1\33"+ + "\1\34\1\14\1\15\1\35\2\15\1\47\1\15\1\46"+ + "\1\36\1\37\1\40\1\41\1\42\1\14\1\15\1\16"+ + "\1\17\1\20\1\14\1\63\1\22\1\16\1\23\1\15"+ + "\1\17\1\16\1\24\1\25\1\26\2\14\1\27\1\51"+ + "\1\52\1\53\1\54\1\55\1\56\1\57\1\60\1\61"+ + "\1\62\1\30\1\31\1\32\1\33\1\34\1\14\1\15"+ + "\1\35\2\15\1\16\1\15\1\25\1\36\1\37\1\40"+ + "\1\41\1\42\1\14\1\15\1\16\1\17\1\20\1\14"+ + "\1\63\1\43\1\16\1\23\1\15\1\17\1\44\1\45"+ + "\1\46\1\26\2\14\1\27\1\51\1\52\1\53\1\54"+ + "\1\55\1\56\1\57\1\60\1\61\1\62\1\30\1\31"+ + "\1\32\1\33\1\34\1\14\1\15\1\35\2\15\1\47"+ + "\1\15\1\46\1\36\1\37\1\40\1\41\1\42\7\0"+ + "\1\64\4\0\1\65\1\66\46\0\1\67\51\0\17\70"+ + "\2\0\36\70\17\0\1\12\171\0\1\71\1\0\1\71"+ + "\6\0\1\72\113\0\1\73\23\0\1\74\50\0\1\75"+ + "\5\0\1\75\2\0\1\75\3\0\1\75\11\0\2\75"+ + "\1\0\2\75\1\0\3\75\2\0\1\75\1\0\2\75"+ + "\1\0\2\75\12\0\1\72\11\0\1\12\104\0\1\76"+ + "\64\0\1\77\5\0\2\100\1\101\3\0\2\100\1\0"+ + "\3\100\4\0\1\101\12\0\1\100\6\0\1\100\2\0"+ + "\1\100\2\0\4\100\54\0\1\102\1\0\1\103\3\0"+ + "\2\104\1\105\3\0\2\104\1\0\3\104\4\0\1\105"+ + "\12\0\1\104\6\0\1\104\2\0\1\104\2\0\4\104"+ + "\11\0\1\106\31\0\1\73\26\0\1\107\56\0\1\107"+ + "\3\0\1\110\41\0\1\111\5\0\1\111\2\0\1\111"+ + "\3\0\1\111\11\0\2\111\1\0\2\111\1\0\3\111"+ + "\2\0\1\111\1\0\2\111\1\0\2\111\50\0\1\112"+ + "\4\0\1\113\16\0\1\114\60\0\1\115\56\0\1\115"+ + "\3\0\1\116\44\0\1\117\15\0\1\117\34\0\17\70"+ + "\1\0\1\120\1\121\35\70\43\0\1\122\61\0\1\123"+ + "\15\0\1\124\56\0\1\125\55\0\1\126\4\0\1\127"+ + "\10\0\1\126\47\0\1\130\47\0\1\131\7\0\1\127"+ + "\5\0\1\131\42\0\1\132\56\0\1\132\1\133\1\134"+ + "\52\0\1\135\4\0\1\65\10\0\1\135\47\0\1\136"+ + "\47\0\1\137\7\0\1\65\5\0\1\137\53\0\1\140"+ + "\37\0\17\70\1\0\1\141\1\121\35\70\2\0\2\100"+ + "\4\0\2\100\1\0\3\100\17\0\1\100\6\0\1\100"+ + "\2\0\1\100\2\0\4\100\3\0\2\104\4\0\2\104"+ + "\1\0\3\104\17\0\1\104\6\0\1\104\2\0\1\104"+ + "\2\0\4\104\6\0\1\142\55\0\1\143\15\0\1\143"+ + "\41\0\1\144\57\0\1\145\54\0\1\146\3\0\1\147"+ + "\11\0\1\146\41\0\1\150\56\0\1\151\72\0\1\152"+ + "\46\0\1\127\52\0\1\153\65\0\1\127\47\0\1\154"+ + "\114\0\1\155\24\0\1\65\61\0\1\65\42\0\17\156"+ + "\2\0\36\156\10\0\1\147\54\0\1\157\50\0\17\156"+ + "\1\0\1\72\36\156"; private static int [] zzUnpackTrans() { - int [] result = new int[2494]; + int [] result = new int[3196]; int offset = 0; offset = zzUnpackTrans(ZZ_TRANS_PACKED_0, offset, result); return result; @@ -238,14 +250,15 @@ private static final int [] ZZ_ATTRIBUTE = zzUnpackAttribute(); private static final String ZZ_ATTRIBUTE_PACKED_0 = - "\1\10\7\0\1\1\3\11\2\1\1\11\2\1\1\11"+ - "\1\1\5\11\5\1\1\11\5\1\14\11\4\0\3\11"+ + "\10\0\1\1\2\0\3\11\2\1\1\11\2\1\1\11"+ + "\3\1\5\11\5\1\1\11\5\1\14\11\5\0\3\11"+ "\1\0\1\11\1\0\2\11\1\0\3\11\4\0\1\11"+ - "\5\0\3\11\4\0\1\1\1\0\1\11\3\0\1\11"+ - "\4\0\1\11\4\0\1\11\2\0\1\11"; + "\5\0\1\11\2\0\2\11\4\0\1\1\1\0\1\11"+ + "\3\0\1\11\2\0\1\1\3\0\1\11\5\0\1\11"+ + "\3\0\1\11"; private static int [] zzUnpackAttribute() { - int [] result = new int[100]; + int [] result = new int[111]; int offset = 0; offset = zzUnpackAttribute(ZZ_ATTRIBUTE_PACKED_0, offset, result); return result; @@ -362,7 +375,7 @@ char [] map = new char[0x10000]; int i = 0; /* index in packed string */ int j = 0; /* index in unpacked array */ - while (i < 172) { + while (i < 184) { int count = packed.charAt(i++); char value = packed.charAt(i++); do map[j++] = value; while (--count > 0);
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexIT.lex Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexIT.lex Tue Nov 27 12:35:19 2012 +0100 @@ -7,7 +7,7 @@ * */ -package de.mpg.mpiwg.berlin.mpdl.lt.analyzer.lang; +package de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang; %% @@ -52,6 +52,12 @@ %% +// jump over empty xml elements +"<"[^><]+"/>" { add(yytext()); } +"-<"[^><]+"/>" { add(yytext()); } +"<"[^><]+"></"[^><]+">" { add(yytext()); } +"-<"[^><]+"></"[^><]+">" { add(yytext()); } + <DICT, SEARCH> { À { add("Ã"); }
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexLA.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexLA.java Tue Nov 27 12:35:19 2012 +0100 @@ -1,4 +1,4 @@ -/* The following code was generated by JFlex 1.4.3 on 05.09.11 10:35 */ +/* The following code was generated by JFlex 1.4.3 on 28.03.12 18:57 */ /* * Normalization rules for Latin text @@ -15,8 +15,8 @@ /** * This class is a scanner generated by * <a href="http://www.jflex.de/">JFlex</a> 1.4.3 - * on 05.09.11 10:35 from the specification file - * <tt>MpdlNormalizerLexLA.lex</tt> + * on 28.03.12 18:57 from the specification file + * <tt>/Users/jwillenborg/test/jflexNew/MpdlNormalizerLexLA.lex</tt> */ public class MpdlNormalizerLexLA { @@ -27,10 +27,10 @@ private static final int ZZ_BUFFERSIZE = 16384; /** lexical states */ - public static final int RENAISSANCE_DICT = 8; - public static final int SEARCH = 10; - public static final int RENAISSANCE_DISP = 4; - public static final int DICT = 6; + public static final int RENAISSANCE_DICT = 10; + public static final int RENAISSANCE_DISP = 8; + public static final int SEARCH = 6; + public static final int DICT = 4; public static final int YYINITIAL = 0; public static final int RENAISSANCE_SEARCH = 12; public static final int DISP = 2; @@ -42,23 +42,24 @@ * l is of the form l = 2*k, k a non negative integer */ private static final int ZZ_LEXSTATE[] = { - 0, 0, 1, 2, 1, 2, 3, 4, 3, 4, 5, 6, 5, 6 + 0, 0, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6 }; /** * Translates characters to character classes */ private static final String ZZ_CMAP_PACKED = - "\12\0\1\6\25\0\1\5\14\0\1\4\22\0\1\0\1\1\3\2"+ - "\1\1\2\2\1\53\1\1\1\0\1\2\1\3\2\2\1\1\1\2"+ - "\1\46\1\3\2\2\1\64\1\65\2\2\1\66\1\2\6\0\1\57"+ - "\1\2\1\47\1\43\1\11\2\2\1\51\1\14\1\27\1\2\1\50"+ - "\1\40\1\13\1\61\1\17\1\7\1\16\1\32\1\15\1\10\1\12"+ - "\2\2\1\66\1\2\62\0\1\4\30\0\1\25\30\0\1\23\1\37"+ - "\1\31\1\55\3\0\1\24\1\0\1\41\1\33\1\0\1\60\1\45"+ - "\1\34\1\52\1\62\2\0\1\42\1\35\1\54\4\0\1\44\1\36"+ - "\1\56\1\63\34\0\1\24\71\0\1\26\53\0\1\20\u0181\0\1\30"+ - "\ud4fe\0\1\21\u0590\0\1\22\u226e\0"; + "\12\0\1\6\25\0\1\5\14\0\1\24\1\0\1\23\14\0\1\21"+ + "\1\0\1\22\1\0\1\0\1\1\3\2\1\1\2\2\1\57\1\1"+ + "\1\0\1\2\1\3\2\2\1\1\1\2\1\52\1\3\2\2\1\70"+ + "\1\71\2\2\1\72\1\2\6\0\1\63\1\2\1\53\1\47\1\11"+ + "\2\2\1\55\1\14\1\33\1\2\1\54\1\44\1\13\1\65\1\17"+ + "\1\7\1\16\1\36\1\15\1\10\1\12\2\2\1\72\1\2\62\0"+ + "\1\4\30\0\1\31\30\0\1\27\1\43\1\35\1\61\3\0\1\30"+ + "\1\0\1\45\1\37\1\0\1\64\1\51\1\40\1\56\1\66\2\0"+ + "\1\46\1\41\1\60\4\0\1\50\1\42\1\62\1\67\34\0\1\30"+ + "\71\0\1\32\53\0\1\20\u0181\0\1\34\ud4fe\0\1\25\u0590\0\1\26"+ + "\u226e\0"; /** * Translates characters to character classes @@ -71,21 +72,21 @@ private static final int [] ZZ_ACTION = zzUnpackAction(); private static final String ZZ_ACTION_PACKED_0 = - "\12\0\1\1\1\2\2\3\1\1\1\4\1\3\1\2"+ - "\1\3\1\2\1\5\1\1\1\6\1\7\1\10\1\11"+ + "\14\0\1\1\1\2\2\3\1\1\1\4\1\3\1\2"+ + "\1\3\1\2\1\5\3\1\1\6\1\7\1\10\1\11"+ "\11\1\1\3\2\1\3\2\1\3\1\12\1\3\2\2"+ - "\1\3\1\5\3\3\1\1\1\2\1\13\1\14\4\0"+ + "\1\3\1\5\3\3\1\1\1\2\1\13\1\14\5\0"+ "\1\15\1\16\1\17\1\20\1\0\1\21\1\22\1\23"+ "\1\24\1\0\1\25\20\0\1\26\3\0\1\27\3\0"+ "\1\30\1\0\1\31\3\0\1\32\1\33\1\34\1\0"+ - "\1\35\1\36\2\0\1\37\20\0\1\40\1\0\1\41"+ + "\1\35\1\36\2\0\1\37\22\0\1\40\1\0\1\41"+ "\1\0\1\42\1\0\1\43\1\44\1\45\1\46\1\0"+ "\1\47\1\0\1\50\1\0\1\51\1\0\1\52\4\0"+ - "\1\53\10\0\1\54\6\0\1\55\3\0\1\56\1\57"+ - "\1\60\2\0\1\61\5\0\1\53"; + "\1\53\10\0\1\12\1\0\1\54\6\0\1\55\3\0"+ + "\1\56\1\57\1\60\3\0\1\61\5\0\1\53\1\0"; private static int [] zzUnpackAction() { - int [] result = new int[179]; + int [] result = new int[190]; int offset = 0; offset = zzUnpackAction(ZZ_ACTION_PACKED_0, offset, result); return result; @@ -110,32 +111,33 @@ private static final int [] ZZ_ROWMAP = zzUnpackRowMap(); private static final String ZZ_ROWMAP_PACKED_0 = - "\0\0\0\67\0\156\0\245\0\334\0\u0113\0\u014a\0\u0181"+ - "\0\u01b8\0\u01ef\0\u0226\0\u0226\0\u0226\0\u025d\0\u0294\0\u0226"+ - "\0\u02cb\0\u0302\0\u0339\0\u0370\0\u0226\0\u01ef\0\u0226\0\u0226"+ - "\0\u0226\0\u0226\0\u03a7\0\u03de\0\u0415\0\u044c\0\u0483\0\u04ba"+ - "\0\u04f1\0\u0528\0\u055f\0\u0596\0\u05cd\0\u0604\0\u063b\0\u0672"+ - "\0\u06a9\0\u06e0\0\u0226\0\u0717\0\u074e\0\u0785\0\u07bc\0\u07f3"+ - "\0\u082a\0\u0861\0\u0898\0\u08cf\0\u0906\0\u0226\0\u0226\0\u093d"+ - "\0\u0974\0\u09ab\0\u09e2\0\u0226\0\u0226\0\u0226\0\u0226\0\u0a19"+ - "\0\u0226\0\u0226\0\u0226\0\u0226\0\u0a50\0\u0226\0\u0a87\0\u0abe"+ - "\0\u0af5\0\u0b2c\0\u0b63\0\u0b9a\0\u0bd1\0\u0c08\0\u0c3f\0\u0c76"+ - "\0\u0cad\0\u0ce4\0\u0d1b\0\u0d52\0\u0d89\0\u0dc0\0\u0226\0\u0df7"+ - "\0\u0e2e\0\u0e65\0\u0226\0\u0e9c\0\u0ed3\0\u0f0a\0\u0226\0\u0f41"+ - "\0\u0226\0\u0f78\0\u0faf\0\u0fe6\0\u0226\0\u0226\0\u0226\0\u101d"+ - "\0\u0226\0\u0226\0\u1054\0\u108b\0\u0226\0\u10c2\0\u10f9\0\u1130"+ - "\0\u1167\0\u119e\0\u11d5\0\u120c\0\u1243\0\u127a\0\u0226\0\u12b1"+ - "\0\u12e8\0\u131f\0\u1356\0\u138d\0\u08cf\0\u0226\0\u13c4\0\u0226"+ - "\0\u13fb\0\u0226\0\u1432\0\u0226\0\u0226\0\u0226\0\u0226\0\u1469"+ - "\0\u0226\0\u14a0\0\u0226\0\u14d7\0\u0226\0\u150e\0\u0226\0\u1545"+ - "\0\u157c\0\u15b3\0\u07bc\0\u15ea\0\u1621\0\u1658\0\u168f\0\u16c6"+ - "\0\u16fd\0\u0226\0\u1734\0\u176b\0\u0226\0\u17a2\0\u17d9\0\u1810"+ - "\0\u1847\0\u187e\0\u18b5\0\u0226\0\u18ec\0\u1923\0\u195a\0\u0226"+ - "\0\u0226\0\u0226\0\u1991\0\u19c8\0\u0226\0\u19ff\0\u1a36\0\u1a6d"+ - "\0\u1aa4\0\u1adb\0\u0226"; + "\0\0\0\73\0\166\0\261\0\354\0\u0127\0\u0162\0\u019d"+ + "\0\u01d8\0\u0213\0\u024e\0\u0289\0\u02c4\0\u02c4\0\u02c4\0\u02ff"+ + "\0\u033a\0\u02c4\0\u0375\0\u03b0\0\u03eb\0\u0426\0\u02c4\0\u0213"+ + "\0\u0461\0\u0289\0\u02c4\0\u02c4\0\u02c4\0\u02c4\0\u049c\0\u04d7"+ + "\0\u0512\0\u054d\0\u0588\0\u05c3\0\u05fe\0\u0639\0\u0674\0\u06af"+ + "\0\u06ea\0\u0725\0\u0760\0\u079b\0\u07d6\0\u0811\0\u02c4\0\u084c"+ + "\0\u0887\0\u08c2\0\u08fd\0\u0938\0\u0973\0\u09ae\0\u09e9\0\u0a24"+ + "\0\u0a5f\0\u02c4\0\u02c4\0\u0a9a\0\u0ad5\0\u0b10\0\u0b4b\0\u0b86"+ + "\0\u02c4\0\u02c4\0\u02c4\0\u02c4\0\u0bc1\0\u02c4\0\u02c4\0\u02c4"+ + "\0\u02c4\0\u0bfc\0\u02c4\0\u0c37\0\u0c72\0\u0cad\0\u0ce8\0\u0d23"+ + "\0\u0d5e\0\u0d99\0\u0dd4\0\u0e0f\0\u0e4a\0\u0e85\0\u0ec0\0\u0efb"+ + "\0\u0f36\0\u0f71\0\u0fac\0\u02c4\0\u0fe7\0\u1022\0\u105d\0\u02c4"+ + "\0\u1098\0\u10d3\0\u110e\0\u02c4\0\u1149\0\u02c4\0\u1184\0\u11bf"+ + "\0\u11fa\0\u02c4\0\u02c4\0\u02c4\0\u1235\0\u02c4\0\u02c4\0\u1270"+ + "\0\u12ab\0\u02c4\0\u12e6\0\u1321\0\u135c\0\u1397\0\u13d2\0\u140d"+ + "\0\u1448\0\u1483\0\u14be\0\u02c4\0\u14f9\0\u1534\0\u156f\0\u15aa"+ + "\0\u15e5\0\u1620\0\u165b\0\u0a24\0\u02c4\0\u1696\0\u02c4\0\u16d1"+ + "\0\u02c4\0\u170c\0\u02c4\0\u02c4\0\u02c4\0\u02c4\0\u1747\0\u02c4"+ + "\0\u1782\0\u02c4\0\u17bd\0\u02c4\0\u17f8\0\u02c4\0\u1833\0\u186e"+ + "\0\u18a9\0\u08fd\0\u18e4\0\u191f\0\u195a\0\u1995\0\u19d0\0\u1a0b"+ + "\0\u02c4\0\u1a46\0\u1a81\0\u14f9\0\u1abc\0\u02c4\0\u1af7\0\u1b32"+ + "\0\u1b6d\0\u1ba8\0\u1be3\0\u1c1e\0\u02c4\0\u1c59\0\u1c94\0\u1ccf"+ + "\0\u02c4\0\u02c4\0\u02c4\0\u1d0a\0\u1d45\0\u1d80\0\u02c4\0\u1dbb"+ + "\0\u1df6\0\u1e31\0\u1e6c\0\u1ea7\0\u02c4\0\u1ee2"; private static int [] zzUnpackRowMap() { - int [] result = new int[179]; + int [] result = new int[190]; int offset = 0; offset = zzUnpackRowMap(ZZ_ROWMAP_PACKED_0, offset, result); return result; @@ -158,110 +160,119 @@ private static final int [] ZZ_TRANS = zzUnpackTrans(); private static final String ZZ_TRANS_PACKED_0 = - "\21\0\1\12\45\0\1\13\1\14\1\15\1\16\1\17"+ - "\1\13\1\20\1\21\1\22\1\14\1\23\1\15\1\24"+ - "\1\15\1\16\1\15\1\25\1\26\1\13\1\27\1\30"+ - "\1\31\1\32\2\13\1\33\1\15\1\34\1\35\1\36"+ - "\1\37\1\40\1\15\1\41\1\42\1\15\1\43\1\13"+ - "\1\44\1\15\1\16\1\15\1\13\1\15\1\13\1\45"+ - "\1\46\1\47\1\13\1\50\2\13\1\51\1\52\1\53"+ - 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int[7965]; int offset = 0; offset = zzUnpackTrans(ZZ_TRANS_PACKED_0, offset, result); return result; @@ -299,18 +310,18 @@ private static final int [] ZZ_ATTRIBUTE = zzUnpackAttribute(); private static final String ZZ_ATTRIBUTE_PACKED_0 = - "\10\0\1\1\1\0\3\11\2\1\1\11\4\1\1\11"+ - "\1\1\4\11\20\1\1\11\12\1\2\11\4\0\4\11"+ + "\10\0\1\1\3\0\3\11\2\1\1\11\4\1\1\11"+ + "\3\1\4\11\20\1\1\11\12\1\2\11\5\0\4\11"+ "\1\0\4\11\1\0\1\11\20\0\1\11\3\0\1\11"+ "\3\0\1\11\1\0\1\11\3\0\3\11\1\0\2\11"+ - "\2\0\1\11\11\0\1\11\6\0\1\11\1\0\1\11"+ + "\2\0\1\11\11\0\1\11\10\0\1\11\1\0\1\11"+ "\1\0\1\11\1\0\4\11\1\0\1\11\1\0\1\11"+ "\1\0\1\11\1\0\1\11\4\0\1\1\5\0\1\11"+ - "\2\0\1\11\6\0\1\11\3\0\3\11\2\0\1\11"+ - "\5\0\1\11"; + "\2\0\1\1\1\0\1\11\6\0\1\11\3\0\3\11"+ + "\3\0\1\11\5\0\1\11\1\0"; private static int [] zzUnpackAttribute() { - int [] result = new int[179]; + int [] result = new int[190]; int offset = 0; offset = zzUnpackAttribute(ZZ_ATTRIBUTE_PACKED_0, offset, result); return result; @@ -427,7 +438,7 @@ char [] map = new char[0x10000]; int i = 0; /* index in packed string */ int j = 0; /* index in unpacked array */ - while (i < 190) { + while (i < 202) { int count = packed.charAt(i++); char value = packed.charAt(i++); do map[j++] = value; while (--count > 0);
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexLA.lex Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexLA.lex Tue Nov 27 12:35:19 2012 +0100 @@ -7,7 +7,7 @@ * */ -package de.mpg.mpiwg.berlin.mpdl.lt.analyzer.lang; +package de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang; %% @@ -57,6 +57,12 @@ %% +// jump over empty xml elements +"<"[^><]+"/>" { cv = 0; add(yytext()); } +"-<"[^><]+"/>" { cv = 0; add(yytext()); } +"<"[^><]+"></"[^><]+">" { cv = 0; add(yytext()); } +"-<"[^><]+"></"[^><]+">" { cv = 0; add(yytext()); } + // TEST, siehe Benedetti Seite 444 ð†‘ { add("X"); } // (U+10191; D800+DD91)
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexNL.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexNL.java Tue Nov 27 12:35:19 2012 +0100 @@ -1,4 +1,4 @@ -/* The following code was generated by JFlex 1.4.3 on 21.07.11 11:22 */ +/* The following code was generated by JFlex 1.4.3 on 28.03.12 18:57 */ /* * Normalization rules for Dutch text @@ -15,8 +15,8 @@ /** * This class is a scanner generated by * <a href="http://www.jflex.de/">JFlex</a> 1.4.3 - * on 21.07.11 11:22 from the specification file - * <tt>MpdlNormalizerLexNL.lex</tt> + * on 28.03.12 18:57 from the specification file + * <tt>/Users/jwillenborg/test/jflexNew/MpdlNormalizerLexNL.lex</tt> */ public class MpdlNormalizerLexNL { @@ -46,9 +46,9 @@ * Translates characters to character classes */ private static final String ZZ_CMAP_PACKED = - "\12\0\1\3\25\0\1\2\14\0\1\1\2\0\1\1\17\0\1\5"+ - "\40\0\1\1\2\0\1\1\20\0\1\1\5\0\1\1\1\0\1\1"+ - "\u0101\0\1\4\ufe80\0"; + "\12\0\1\3\25\0\1\2\14\0\1\7\1\0\1\6\1\1\13\0"+ + "\1\4\1\0\1\5\1\0\1\11\40\0\1\1\2\0\1\1\20\0"+ + "\1\1\5\0\1\1\1\0\1\1\u0101\0\1\10\ufe80\0"; /** * Translates characters to character classes @@ -61,10 +61,11 @@ private static final int [] ZZ_ACTION = zzUnpackAction(); private static final String ZZ_ACTION_PACKED_0 = - "\4\0\2\1\1\2\1\3\1\4\1\5\1\6"; + "\4\0\4\1\1\2\1\3\1\4\1\5\1\6\5\0"+ + "\1\1\2\0"; private static int [] zzUnpackAction() { - int [] result = new int[11]; + int [] result = new int[21]; int offset = 0; offset = zzUnpackAction(ZZ_ACTION_PACKED_0, offset, result); return result; @@ -89,11 +90,12 @@ private static final int [] ZZ_ROWMAP = zzUnpackRowMap(); private static final String ZZ_ROWMAP_PACKED_0 = - "\0\0\0\6\0\14\0\22\0\30\0\36\0\30\0\30"+ - "\0\30\0\30\0\30"; + "\0\0\0\12\0\24\0\36\0\50\0\62\0\74\0\106"+ + "\0\50\0\50\0\50\0\50\0\50\0\120\0\74\0\132"+ + "\0\144\0\156\0\132\0\170\0\202"; private static int [] zzUnpackRowMap() { - int [] result = new int[11]; + int [] result = new int[21]; int offset = 0; offset = zzUnpackRowMap(ZZ_ROWMAP_PACKED_0, offset, result); return result; @@ -116,13 +118,18 @@ private static final int [] ZZ_TRANS = zzUnpackTrans(); private static final String ZZ_TRANS_PACKED_0 = - "\1\5\1\6\1\5\1\0\1\5\1\7\1\5\1\6"+ - "\1\5\1\10\1\11\1\7\1\5\1\6\1\5\1\12"+ - "\1\11\1\7\1\5\1\6\1\5\1\13\1\11\1\7"+ - "\10\0\1\5\3\0"; + "\1\5\1\6\1\5\1\0\1\7\2\5\1\10\1\5"+ + "\1\11\1\5\1\6\1\5\1\12\1\7\2\5\1\10"+ + "\1\13\1\11\1\5\1\6\1\5\1\14\1\7\2\5"+ + "\1\10\1\13\1\11\1\5\1\6\1\5\1\15\1\7"+ + "\2\5\1\10\1\13\1\11\14\0\1\5\7\0\4\16"+ + "\2\0\4\16\2\0\1\5\1\0\1\17\5\0\4\16"+ + "\1\0\1\20\1\21\3\16\4\0\1\22\5\0\4\16"+ + "\1\0\1\23\1\21\3\16\6\0\1\24\3\0\4\25"+ + "\2\0\10\25\1\0\1\5\4\25"; private static int [] zzUnpackTrans() { - int [] result = new int[36]; + int [] result = new int[140]; int offset = 0; offset = zzUnpackTrans(ZZ_TRANS_PACKED_0, offset, result); return result; @@ -160,10 +167,10 @@ private static final int [] ZZ_ATTRIBUTE = zzUnpackAttribute(); private static final String ZZ_ATTRIBUTE_PACKED_0 = - "\4\0\1\11\1\1\5\11"; + "\4\0\1\11\3\1\5\11\5\0\1\1\2\0"; private static int [] zzUnpackAttribute() { - int [] result = new int[11]; + int [] result = new int[21]; int offset = 0; offset = zzUnpackAttribute(ZZ_ATTRIBUTE_PACKED_0, offset, result); return result; @@ -273,7 +280,7 @@ char [] map = new char[0x10000]; int i = 0; /* index in packed string */ int j = 0; /* index in unpacked array */ - while (i < 46) { + while (i < 56) { int count = packed.charAt(i++); char value = packed.charAt(i++); do map[j++] = value; while (--count > 0);
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexNL.lex Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexNL.lex Tue Nov 27 12:35:19 2012 +0100 @@ -7,7 +7,7 @@ * */ -package de.mpg.mpiwg.berlin.mpdl.lt.analyzer.lang; +package de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang; %% @@ -41,6 +41,12 @@ %% +// jump over empty xml elements +"<"[^><]+"/>" { add(yytext()); } +"-<"[^><]+"/>" { add(yytext()); } +"<"[^><]+"></"[^><]+">" { add(yytext()); } +"-<"[^><]+"></"[^><]+">" { add(yytext()); } + <DISP, DICT, SEARCH> { Å¿ { add("s"); }
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexZH.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexZH.java Tue Nov 27 12:35:19 2012 +0100 @@ -1,4 +1,4 @@ -/* The following code was generated by JFlex 1.4.3 on 21.07.11 11:22 */ +/* The following code was generated by JFlex 1.4.3 on 28.03.12 18:57 */ /* * Normalization rules for Chinese text @@ -15,8 +15,8 @@ /** * This class is a scanner generated by * <a href="http://www.jflex.de/">JFlex</a> 1.4.3 - * on 21.07.11 11:22 from the specification file - * <tt>MpdlNormalizerLexZH.lex</tt> + * on 28.03.12 18:57 from the specification file + * <tt>/Users/jwillenborg/test/jflexNew/MpdlNormalizerLexZH.lex</tt> */ public class MpdlNormalizerLexZH {
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexZH.lex Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/norm/lang/MpdlNormalizerLexZH.lex Tue Nov 27 12:35:19 2012 +0100 @@ -7,7 +7,7 @@ * */ -package de.mpg.mpiwg.berlin.mpdl.lt.analyzer.lang; +package de.mpg.mpiwg.berlin.mpdl.lt.text.norm.lang; %%
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/reg/DBRegularizationHandler.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,146 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.lt.text.reg; - -import java.io.UnsupportedEncodingException; -import java.util.ArrayList; - -import com.sleepycat.je.Cursor; -import com.sleepycat.je.Database; -import com.sleepycat.je.DatabaseEntry; -import com.sleepycat.je.DatabaseException; -import com.sleepycat.je.LockMode; -import com.sleepycat.je.OperationStatus; - -import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; -import de.mpg.mpiwg.berlin.mpdl.lt.general.Language; - -public class DBRegularizationHandler { - private String dbDirectory; - private DbEnvRegularization regDbEnv; - - public DBRegularizationHandler(String dbDir) { - this.dbDirectory = dbDir; - } - - public void start() throws ApplicationException { - regDbEnv = new DbEnvRegularization(); - regDbEnv.setDataDir(dbDirectory); - regDbEnv.init(); // open databases in read/write mode - } - - public void openDatabases() throws ApplicationException { - regDbEnv.openDatabases(); - } - - public void closeDatabases() throws ApplicationException { - regDbEnv.close(); - } - - public void deleteData() throws ApplicationException { - regDbEnv.removeDatabases(); - } - - public void writeOrigReg(Regularization reg) throws ApplicationException { - try { - String language = Language.getInstance().getLanguageId(reg.getLanguage()); - String keyStr = language + "###" + reg.getOrig(); - String valueStr = reg.getXmlString(); - DatabaseEntry dbEntryKey = new DatabaseEntry(keyStr.getBytes("utf-8")); - DatabaseEntry dbEntryValue = new DatabaseEntry(valueStr.getBytes("utf-8")); - Database origDB = regDbEnv.getOrigDB(); - origDB.put(null, dbEntryKey, dbEntryValue); - } catch (DatabaseException e) { - throw new ApplicationException(e); - } catch (UnsupportedEncodingException e) { - throw new ApplicationException(e); - } - } - - public void writeNormReg(Regularization reg) throws ApplicationException { - try { - String language = Language.getInstance().getLanguageId(reg.getLanguage()); - String keyStr = language + "###" + reg.getNorm(); - String valueStr = reg.getXmlString(); - DatabaseEntry dbEntryKey = new DatabaseEntry(keyStr.getBytes("utf-8")); - DatabaseEntry dbEntryValue = new DatabaseEntry(valueStr.getBytes("utf-8")); - Database normDB = regDbEnv.getNormDB(); - normDB.put(null, dbEntryKey, dbEntryValue); - } catch (DatabaseException e) { - throw new ApplicationException(e); - } catch (UnsupportedEncodingException e) { - throw new ApplicationException(e); - } - } - - public void deleteReg(Regularization reg) throws ApplicationException { - try { - String language = Language.getInstance().getLanguageId(reg.getLanguage()); - String keyStrOrig = language + "###" + reg.getOrig(); - DatabaseEntry dbEntryKey = new DatabaseEntry(keyStrOrig.getBytes("utf-8")); - Database origDB = regDbEnv.getOrigDB(); - origDB.delete(null, dbEntryKey); - String keyStrNorm = reg.getLanguage() + "###" + reg.getNorm(); - dbEntryKey = new DatabaseEntry(keyStrNorm.getBytes("utf-8")); - Database normDB = regDbEnv.getNormDB(); - normDB.delete(null, dbEntryKey); - } catch (DatabaseException e) { - throw new ApplicationException(e); - } catch (UnsupportedEncodingException e) { - throw new ApplicationException(e); - } - } - - public ArrayList<Regularization> readRegsByOrig(String lang, String orig) throws ApplicationException { - String language = Language.getInstance().getLanguageId(lang); - ArrayList<Regularization> retRegs = new ArrayList<Regularization>(); - String hashKey = language + "###" + orig; - try { - Database origDB = regDbEnv.getOrigDB(); - Cursor cursor = origDB.openCursor(null, null); - byte[] bHashKey = hashKey.getBytes("utf-8"); - DatabaseEntry dbEntryKey = new DatabaseEntry(bHashKey); - DatabaseEntry foundValue = new DatabaseEntry(); - OperationStatus operationStatus = cursor.getSearchKey(dbEntryKey, foundValue, LockMode.DEFAULT); - while (operationStatus == OperationStatus.SUCCESS) { - byte[] foundValueBytes = foundValue.getData(); - String foundValueStr = new String(foundValueBytes, "utf-8"); - Regularization reg = Regularization.getInstance(foundValueStr); - retRegs.add(reg); - operationStatus = cursor.getNextDup(dbEntryKey, foundValue, LockMode.DEFAULT); - } - cursor.close(); - } catch (DatabaseException e) { - throw new ApplicationException(e); - } catch (UnsupportedEncodingException e) { - throw new ApplicationException(e); - } - return retRegs; - } - - public ArrayList<Regularization> readRegsByNorm(String lang, String norm) throws ApplicationException { - String language = Language.getInstance().getLanguageId(lang); - ArrayList<Regularization> retRegs = new ArrayList<Regularization>(); - String hashKey = language + "###" + norm; - try { - Database normDB = regDbEnv.getNormDB(); - Cursor cursor = normDB.openCursor(null, null); - byte[] bHashKey = hashKey.getBytes("utf-8"); - DatabaseEntry dbEntryKey = new DatabaseEntry(bHashKey); - DatabaseEntry foundValue = new DatabaseEntry(); - OperationStatus operationStatus = cursor.getSearchKey(dbEntryKey, foundValue, LockMode.DEFAULT); - while (operationStatus == OperationStatus.SUCCESS) { - byte[] foundValueBytes = foundValue.getData(); - String foundValueStr = new String(foundValueBytes, "utf-8"); - Regularization reg = Regularization.getInstance(foundValueStr); - retRegs.add(reg); - operationStatus = cursor.getNextDup(dbEntryKey, foundValue, LockMode.DEFAULT); - } - cursor.close(); - } catch (DatabaseException e) { - throw new ApplicationException(e); - } catch (UnsupportedEncodingException e) { - throw new ApplicationException(e); - } - return retRegs; - } - -} \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/reg/DbEnvRegularization.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,100 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.lt.text.reg; - -import java.io.File; - -import com.sleepycat.je.Database; -import com.sleepycat.je.DatabaseConfig; -import com.sleepycat.je.DatabaseException; -import com.sleepycat.je.Environment; -import com.sleepycat.je.EnvironmentConfig; - -import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; - -public class DbEnvRegularization { - private String dataDir; - private File envPath; - private Environment env; - private EnvironmentConfig envConfig; - private DatabaseConfig dbConfig; - private Database origDB; - private Database normDB; - - public DbEnvRegularization() { - } - - public void setDataDir(String dataDir) { - this.dataDir = dataDir; - } - - public void init() throws ApplicationException { - try { - envConfig = new EnvironmentConfig(); - dbConfig = new DatabaseConfig(); - envConfig.setReadOnly(false); - dbConfig.setReadOnly(false); - envConfig.setAllowCreate(true); - dbConfig.setAllowCreate(true); - envConfig.setTransactional(true); - dbConfig.setTransactional(true); - // allow duplicates for keys - dbConfig.setSortedDuplicates(true); - envPath = new File(dataDir); - env = new Environment(envPath, envConfig); - } catch (DatabaseException e) { - throw new ApplicationException(e); - } - } - - public void openDatabases() throws ApplicationException { - try { - // open databases (and create them if they do not exist) - origDB = env.openDatabase(null, "OrigDB", dbConfig); - normDB = env.openDatabase(null, "NormDB", dbConfig); - } catch (DatabaseException e) { - throw new ApplicationException(e); - } - } - - public void removeDatabases() throws ApplicationException { - try { - if (origDB != null) - origDB.close(); - if (normDB != null) - normDB.close(); - env.removeDatabase(null, "OrigDB"); - env.removeDatabase(null, "NormDB"); - origDB = null; - normDB = null; - } catch (DatabaseException e) { - throw new ApplicationException(e); - } - } - - public Environment getEnv() { - return env; - } - - public Database getNormDB() { - return normDB; - } - - public Database getOrigDB() { - return origDB; - } - - public void close() throws ApplicationException { - if (env != null) { - try { - if (origDB != null) - origDB.close(); - if (normDB != null) - normDB.close(); - if (env != null) - env.close(); - } catch (DatabaseException e) { - throw new ApplicationException(e); - } - } - } -} -
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/reg/Regularization.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,89 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.lt.text.reg; - -import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; -import de.mpg.mpiwg.berlin.mpdl.util.StringUtils; -import de.mpg.mpiwg.berlin.mpdl.xml.xquery.XQueryEvaluator; - -public class Regularization { - private String language; - private String orig; - private String norm; - private String source; - private int sourcePosition; - - public Regularization(String language, String orig, String norm, String source) { - this.language = language; - this.orig = orig; - this.norm = norm; - this.source = source; - } - - public static Regularization getInstance(String xmlStr) throws ApplicationException { - XQueryEvaluator xQueryEvaluator = new XQueryEvaluator(); - String language = xQueryEvaluator.evaluateAsStringValue(xmlStr, "//language"); - String orig = xQueryEvaluator.evaluateAsStringValue(xmlStr, "//orig"); - String norm = xQueryEvaluator.evaluateAsStringValue(xmlStr, "//norm"); - String source = xQueryEvaluator.evaluateAsStringValue(xmlStr, "//source"); - String sourcePosStr = xQueryEvaluator.evaluateAsStringValue(xmlStr, "//source/@position"); - int sourcePos = new Integer(sourcePosStr); - Regularization reg = new Regularization(language, orig, norm, source); - reg.setSourcePosition(sourcePos); - return reg; - } - - public String getLanguage() { - return language; - } - - public void setLanguage(String language) { - this.language = language; - } - - public String getOrig() { - return orig; - } - - public void setOrig(String orig) { - this.orig = orig; - } - - public String getNorm() { - return norm; - } - - public void setNorm(String norm) { - this.norm = norm; - } - - public String getSource() { - return source; - } - - public void setSource(String source) { - this.source = source; - } - - public int getSourcePosition() { - return sourcePosition; - } - - public void setSourcePosition(int sourcePosition) { - this.sourcePosition = sourcePosition; - } - - public String getXmlString() { - String xmlString = "<reg>\n"; - if (language != null) - xmlString += " <language>" + language + "</language>\n"; - if (orig != null) - xmlString += " <orig>" + StringUtils.deresolveXmlEntities(orig) + "</orig>\n"; - if (norm != null) - xmlString += " <norm>" + StringUtils.deresolveXmlEntities(norm) + "</norm>\n"; - if (source != null) - xmlString += " <source position=\"" + sourcePosition + "\">" + StringUtils.deresolveXmlEntities(source) + "</source>\n"; - xmlString += "</reg>\n"; - return xmlString; - } - - -}
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/reg/RegularizationManager.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,118 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.lt.text.reg; - -import java.util.ArrayList; -import java.util.Date; -import java.util.Hashtable; -import java.util.logging.Logger; - -import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; -import de.mpg.mpiwg.berlin.mpdl.lt.general.Constants; -import de.mpg.mpiwg.berlin.mpdl.lucene.util.LuceneUtil; -import de.mpg.mpiwg.berlin.mpdl.util.Util; - -public class RegularizationManager { - private static RegularizationManager instance; - private static String DATA_DIR = Constants.getInstance().getDataDir(); - private static String REGULARIZATION_DB_DIR = DATA_DIR + "/dataBerkeleyDB/regularization"; - private static Logger LOGGER = Logger.getLogger(RegularizationManager.class.getName()); - private DBRegularizationHandler dbRegHandler; - private Hashtable<String, ArrayList<Regularization>> regsOrig; - private Hashtable<String, ArrayList<Regularization>> regsNorm; - private Date beginOfOperation; - private Date endOfOperation; - - public static RegularizationManager getInstance() throws ApplicationException { - if (instance == null) { - instance = new RegularizationManager(); - instance.init(); - } - return instance; - } - - public static void main(String[] args) throws ApplicationException { - getInstance(); - instance.beginOperation(); - System.out.print("Start ..."); - ArrayList<Regularization> regs = instance.findRegsByNorm("la", "Illiusque"); - ArrayList<Regularization> regs2 = instance.findRegsByNorm("la", "Itaque"); - Regularization bla = regs.get(0); - Regularization bla2 = regs2.get(0); - - instance.end(); - instance.endOperation(); - Double elapsedTime = new Util().getSecondWithMillisecondsBetween(instance.beginOfOperation, instance.endOfOperation); - System.out.println("End."); - System.out.println("Needed time: " + elapsedTime + " seconds"); - } - - private void init() throws ApplicationException { - regsOrig = new Hashtable<String, ArrayList<Regularization>>(); - regsNorm = new Hashtable<String, ArrayList<Regularization>>(); - dbRegHandler = new DBRegularizationHandler(REGULARIZATION_DB_DIR); - dbRegHandler.start(); - dbRegHandler.openDatabases(); - LOGGER.info("Regularization db cache: opened"); - } - - public ArrayList<Regularization> findRegsByOrig(String language, String orig) throws ApplicationException { - orig = orig.toLowerCase(); - String hashKey = language + "###" + orig; - ArrayList<Regularization> regs = regsOrig.get(hashKey); - if (regs == null) { - regs = dbRegHandler.readRegsByOrig(language, orig); - if (regs == null || regs.isEmpty()) - regsOrig.put(hashKey, new ArrayList<Regularization>()); - else - regsOrig.put(hashKey, regs); - } - return regs; - } - - public ArrayList<Regularization> findRegsByNorm(String language, String norm) throws ApplicationException { - norm = norm.toLowerCase(); - String hashKey = language + "###" + norm; - ArrayList<Regularization> regs = regsNorm.get(hashKey); - if (regs == null || regs.isEmpty()) { - regs = dbRegHandler.readRegsByNorm(language, norm); - if (regs == null) - regsNorm.put(hashKey, new ArrayList<Regularization>()); - else - regsNorm.put(hashKey, regs); - } - return regs; - } - - public ArrayList<String> getRegOrigsByNormLuceneQueryString(String language, String luceneQueryString) throws ApplicationException { - ArrayList<String> regForms = new ArrayList<String>(); - LuceneUtil luceneUtil = LuceneUtil.getInstance(); - ArrayList<String> variants = luceneUtil.getVariantsFromLuceneQuery(luceneQueryString); - if (variants != null) { - for (int i=0; i<variants.size(); i++) { - String variant = variants.get(i); - ArrayList<Regularization> regs = findRegsByNorm(language, variant); - if (regs != null) { - for (int j=0; j<regs.size(); j++) { - Regularization reg = regs.get(j); - String orig = reg.getOrig(); - regForms.add(orig); - } - } - } - } - return regForms; - } - - public void end() throws ApplicationException { - dbRegHandler.closeDatabases(); - LOGGER.info("Regularization db cache: closed"); - } - - private void beginOperation() { - beginOfOperation = new Date(); - } - - private void endOperation() { - endOfOperation = new Date(); - } - -} \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/tokenize/Token.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/tokenize/Token.java Tue Nov 27 12:35:19 2012 +0100 @@ -1,18 +1,43 @@ package de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize; public class Token { - private String content; - private int start; - private int end; + private String docId; + private String language; + private int pageNumber; + private int lineNumber; + private String elementName; // e.g. "TEI:s" + private int elementPosition; + private int elementPagePosition; + private String xmlId; + private String xpath; + private String content; // original text content + private String contentOrig; // word form + private String contentReg; // regularized text content + private String contentNorm; // normalized word form + private String contentMorph; // lemmas separated by blank + private int start; // start position + private int end; // end position public Token(int start, int end, String content) { this.start = start; this.end = end; this.content = content; + this.contentOrig = toWordForm(); } - public String getContent() { - return content; + public Token(String docId, String language, int pageNumber, int lineNumber, int elementPosition, String elementName, String xmlId, String xpath, String contentOrig, String contentReg, String contentNorm, String contentMorph) { + this.docId = docId; + this.language = language; + this.pageNumber = pageNumber; + this.lineNumber = lineNumber; + this.elementPosition = elementPosition; + this.elementName = elementName; + this.xmlId = xmlId; + this.xpath = xpath; + this.contentOrig = contentOrig; + this.contentReg = contentReg; + this.contentNorm = contentNorm; + this.contentMorph = contentMorph; } public int getStart() { @@ -23,12 +48,157 @@ return end; } + public String toWordForm() { + if (content != null) + return content.toLowerCase(); + else + return null; + } + public String toString() { String retStr = ""; - if (content != null) - retStr += content; + if (contentOrig != null) + retStr += contentOrig; retStr = retStr + "(" + start + "," + end + ")"; return retStr; } - + + public String toXmlString() { + StringBuilder retStr = new StringBuilder(); + retStr.append("<token>"); + if (docId != null) + retStr.append("<docId>" + docId + "</docId>"); + if (language != null) + retStr.append("<language>" + language + "</language>"); + retStr.append("<pageNumber>" + pageNumber + "</pageNumber>"); + retStr.append("<elementPosition>" + elementPosition + "</elementPosition>"); + retStr.append("<elementPagePosition>" + elementPagePosition + "</elementPagePosition>"); + if (elementName != null) + retStr.append("<elementName>" + elementName + "</elementName>"); + if (contentOrig != null) + retStr.append("<contentOrig>" + contentOrig + "</contentOrig>"); + retStr.append("</token>"); + return retStr.toString(); + } + + public String getContent() { + return content; + } + + public void setContent(String content) { + this.content = content; + this.contentOrig = toWordForm(); + } + + public String getContentOrig() { + return contentOrig; + } + + public void setContentOrig(String contentOrig) { + this.contentOrig = contentOrig; + } + + public String getContentReg() { + return contentReg; + } + + public void setContentReg(String contentReg) { + this.contentReg = contentReg; + } + + public String getContentNorm() { + return contentNorm; + } + + public void setContentNorm(String contentNorm) { + this.contentNorm = contentNorm; + } + + public String getContentMorph() { + return contentMorph; + } + + public void setContentMorph(String contentMorph) { + this.contentMorph = contentMorph; + } + + public String getDocId() { + return docId; + } + + public void setDocId(String docId) { + this.docId = docId; + } + + public String getLanguage() { + return language; + } + + public void setLanguage(String language) { + this.language = language; + } + + public int getPageNumber() { + return pageNumber; + } + + public void setPageNumber(int pageNumber) { + this.pageNumber = pageNumber; + } + + public int getLineNumber() { + return lineNumber; + } + + public void setLineNumber(int lineNumber) { + this.lineNumber = lineNumber; + } + + public int getPosition() { + return elementPosition; + } + + public void setElementPosition(int elementPosition) { + this.elementPosition = elementPosition; + } + + public int getPagePosition() { + return elementPagePosition; + } + + public void setElementPagePosition(int elementPagePosition) { + this.elementPagePosition = elementPagePosition; + } + + public String getElementName() { + return elementName; + } + + public void setElementName(String elementName) { + this.elementName = elementName; + } + + public String getXmlId() { + return xmlId; + } + + public void setXmlId(String xmlId) { + this.xmlId = xmlId; + } + + public String getXpath() { + return xpath; + } + + public void setXpath(String xpath) { + this.xpath = xpath; + } + + public void setStart(int start) { + this.start = start; + } + + public void setEnd(int end) { + this.end = end; + } } \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/tokenize/Tokenizer.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/tokenize/Tokenizer.java Tue Nov 27 12:35:19 2012 +0100 @@ -17,13 +17,13 @@ import de.mpg.mpiwg.berlin.mpdl.lt.text.norm.Normalizer; public class Tokenizer extends org.apache.lucene.analysis.Tokenizer { - // variables are copied from Lucene 3.4. CharTokenizer + // variables are copied from Lucene 3.6. CharTokenizer private int offset = 0, bufferIndex = 0, dataLen = 0, finalOffset = 0; private static int MAX_WORD_LEN = 4096; // old value was 255 private static int IO_BUFFER_SIZE = 4096; private CharTermAttribute termAtt = addAttribute(CharTermAttribute.class); private OffsetAttribute offsetAtt = addAttribute(OffsetAttribute.class); - private CharacterUtils charUtils = CharacterUtils.getInstance(Version.LUCENE_34); + private CharacterUtils charUtils = CharacterUtils.getInstance(Version.LUCENE_35); private CharacterBuffer ioBuffer = CharacterUtils.newCharacterBuffer(IO_BUFFER_SIZE); // application variables private String language = "eng"; // default: english @@ -58,14 +58,17 @@ OffsetAttribute offsetAttribute = getAttribute(OffsetAttribute.class); while (incrementToken()) { String term = charTermAttribute.toString(); + term = removeElementMarksAndSpecialSymbols(term); // e.g. also "-" is deleted so that the normalizer works properly int start = offsetAttribute.startOffset(); int end = offsetAttribute.endOffset(); - String normedTerm = normalizer.normalize(term); - Token token = new Token(start, end, normedTerm); + Token token = new Token(start, end, term); + String wordForm = token.getContentOrig(); // word form (lower case) + String normedTerm = normalizer.normalize(wordForm); + token.setContentNorm(normedTerm); tokens.add(token); } - end(); // TODO needed ? - close(); // TODO needed ? + end(); + close(); // close the input reader } catch (IOException e) { throw new ApplicationException(e); } @@ -98,10 +101,10 @@ case '+': isTokenChar = false; break; case '#': isTokenChar = false; break; case '"': isTokenChar = false; break; - case 'ã': isTokenChar = false; break; - case 'Ò': isTokenChar = false; break; - case 'Ç': isTokenChar = false; break; - case 'È': isTokenChar = false; break; + case '„': isTokenChar = false; break; + case '“': isTokenChar = false; break; + case '«': isTokenChar = false; break; + case '»': isTokenChar = false; break; case '\'': isTokenChar = false; break; case '\t': isTokenChar = false; break; // do not break words which have tabs in it case '\n': isTokenChar = false; break; // do not break words which are on another line @@ -121,13 +124,14 @@ } /* - * Code is copied from Lucene 3.4. CharTokenizer.incrementToken() + * Code is copied from Lucene 3.6. CharTokenizer.incrementToken() with bug correction for supplementary chars * @see org.apache.lucene.analysis.TokenStream#incrementToken() */ public boolean incrementToken() throws IOException { clearAttributes(); int length = 0; int start = -1; // this variable is always initialized + int end = -1; char[] buffer = termAtt.buffer(); while (true) { if (bufferIndex >= dataLen) { @@ -146,13 +150,16 @@ } // use CharacterUtils here to support < 3.1 UTF-16 code unit behavior if the char based methods are gone int c = charUtils.codePointAt(ioBuffer.getBuffer(), bufferIndex); - bufferIndex += Character.charCount(c); + int charCount = Character.charCount(c); + bufferIndex += charCount; if (isTokenChar(c)) { // if it's a token char if (length == 0) { // start of token - start = offset + bufferIndex - 1; + start = offset + bufferIndex - charCount; // supplementary chars could have length 1 or 2 + end = start; } else if (length >= buffer.length-1) { // check if a supplementary could run out of bounds buffer = termAtt.resizeBuffer(2 + length); // make sure a supplementary fits in the buffer } + end += charCount; length += Character.toChars(normalize(c), buffer, length); // buffer it, normalized if (length >= MAX_WORD_LEN) // buffer overflow! make sure to check for >= surrogate pair could break == test break; @@ -160,12 +167,12 @@ break; // return 'em } termAtt.setLength(length); - offsetAtt.setOffset(correctOffset(start), finalOffset = correctOffset(start + length)); + offsetAtt.setOffset(correctOffset(start), finalOffset = correctOffset(end)); return true; } /* - * Code is copied from Lucene 3.4. CharTokenizer.end() + * Code is copied from Lucene 3.6. CharTokenizer.end() * @see org.apache.lucene.analysis.TokenStream#end() */ @Override @@ -175,7 +182,7 @@ } /* - * Code is copied from Lucene 3.4. CharTokenizer.reset() + * Code is copied from Lucene 3.6. CharTokenizer.reset() * @see org.apache.lucene.analysis.Tokenizer#reset(java.io.Reader) */ @Override @@ -190,7 +197,7 @@ } private ArrayList<Token> getTokensByChineseTokenizer(Reader input, String[] normFunctions) throws ApplicationException { - StandardTokenizer chineseTokenizer = new StandardTokenizer(Version.LUCENE_34, input); // is recommended instead of ChineseTokenizer which is deprecated + StandardTokenizer chineseTokenizer = new StandardTokenizer(Version.LUCENE_35, input); // is recommended instead of ChineseTokenizer which is deprecated ArrayList<Token> tokens = new ArrayList<Token>(); try { reset(input); @@ -199,20 +206,26 @@ OffsetAttribute offsetAttribute = chineseTokenizer.getAttribute(OffsetAttribute.class); while (chineseTokenizer.incrementToken()) { String term = charTermAttribute.toString(); - String normedTerm = normalizer.normalize(term); int start = offsetAttribute.startOffset(); int end = offsetAttribute.endOffset(); - Token token = new Token(start, end, normedTerm); + Token token = new Token(start, end, term); + String normedTerm = normalizer.normalize(term); + token.setContentNorm(normedTerm); tokens.add(token); } - chineseTokenizer.end(); // TODO needed ? - chineseTokenizer.close(); // TODO needed ? - end(); // TODO needed ? - close(); // TODO needed ? + chineseTokenizer.end(); + chineseTokenizer.close(); + end(); + close(); } catch (IOException e) { throw new ApplicationException(e); } return tokens; } + private String removeElementMarksAndSpecialSymbols(String inputStr) { + String retStr = inputStr.replaceAll("\u2424|\u2425| |\n|\t|-|\u00AD", ""); + return retStr; + } + } \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/tokenize/TokenizerNew.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,162 @@ +package de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize; + +import java.io.IOException; +import java.io.StringReader; +import java.util.ArrayList; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.lt.general.Language; +import de.mpg.mpiwg.berlin.mpdl.lt.text.norm.Normalizer; + +/** + * own simple implementation of Tokenizer + * @author jwillenborg + * + */ +public class TokenizerNew { + private StringReader input; + private String language = "eng"; // default: english + private Normalizer normalizer; + private ArrayList<Token> tokens; + + public TokenizerNew(StringReader input, String language) { + this.input = input; + String[] normFunctions = {"norm"}; + this.language = language; + this.normalizer = new Normalizer(normFunctions, language); + } + + public ArrayList<Token> tokenize() throws ApplicationException { + if (Language.getInstance().isChinese(language)) + return tokenizeChinese(); + else + return tokenizeAll(); + } + + private ArrayList<Token> tokenizeAll() throws ApplicationException { + tokens = new ArrayList<Token>(); + try { + int cInt = -1; + StringBuilder tokenStr = new StringBuilder(); + int pos = 0; + boolean isTokenChar = false; // last state + int tokenStart = -1; + int tokenEnd = -1; + while ((cInt = input.read()) != -1) { + char c = (char) cInt; + if (isTokenChar(c)) { + if (! isTokenChar) { + tokenStr = new StringBuilder(); + tokenStart = pos; + } + tokenStr.append(c); + isTokenChar = true; + } else { + if (isTokenChar) { + tokenEnd = pos - 1; + String tStr = tokenStr.toString(); + addToken(tokenStart, tokenEnd, tStr); + } + isTokenChar = false; + } + pos++; + } + // add last token if last char of input is a token char + if (isTokenChar) { + tokenEnd = pos - 1; + String tStr = tokenStr.toString(); + addToken(tokenStart, tokenEnd, tStr); + } + input.close(); + } catch (IOException e) { + throw new ApplicationException(e); + } + if (tokens.isEmpty()) + return null; + else + return tokens; + } + + /** + * each token character is a single token + * @return + * @throws ApplicationException + */ + private ArrayList<Token> tokenizeChinese() throws ApplicationException { + tokens = new ArrayList<Token>(); + try { + int cInt = -1; + int pos = 0; + while ((cInt = input.read()) != -1) { + char c = (char) cInt; + if (isTokenChar(c)) { + String tStr = String.valueOf(c); + addToken(pos, pos + 1, tStr); + } + pos++; + } + input.close(); + } catch (IOException e) { + throw new ApplicationException(e); + } + if (tokens.isEmpty()) + return null; + else + return tokens; + } + + private boolean isTokenChar(char c) { + boolean isTokenChar = true; + switch (c) { + case ' ': isTokenChar = false; break; + case '.': isTokenChar = false; break; + case ',': isTokenChar = false; break; + case '!': isTokenChar = false; break; + case '?': isTokenChar = false; break; + case ';': isTokenChar = false; break; + case ':': isTokenChar = false; break; + case '(': isTokenChar = false; break; + case ')': isTokenChar = false; break; + case '[': isTokenChar = false; break; + case ']': isTokenChar = false; break; + case '{': isTokenChar = false; break; + case '}': isTokenChar = false; break; + case '<': isTokenChar = false; break; + case '>': isTokenChar = false; break; + case '/': isTokenChar = false; break; + case '=': isTokenChar = false; break; + case '&': isTokenChar = false; break; + case '+': isTokenChar = false; break; + case '#': isTokenChar = false; break; + case '"': isTokenChar = false; break; + case '„': isTokenChar = false; break; + case '“': isTokenChar = false; break; + case '«': isTokenChar = false; break; + case '»': isTokenChar = false; break; + case '\'': isTokenChar = false; break; + case '\t': isTokenChar = false; break; // do not break words which have tabs in it + case '\n': isTokenChar = false; break; // do not break words which are on another line + case '\u2425': isTokenChar = false; break; // special char for marking xml elements + } + return isTokenChar; + } + + private void addToken(int startPos, int endPos, String tokenStr) throws ApplicationException { + tokenStr = removeElementMarks(tokenStr); + tokenStr = removeSpecialSymbols(tokenStr); // e.g. "-" is deleted so that the normalizer works properly + String tStrNormed = normalizer.normalize(tokenStr); + Token token = new Token(startPos, endPos, tokenStr); + token.setContentNorm(tStrNormed); + tokens.add(token); + } + + private String removeElementMarks(String inputStr) { + String retStr = inputStr.replaceAll("\u2424|\u2425", ""); + return retStr; + } + + private String removeSpecialSymbols(String inputStr) { + String retStr = inputStr.replaceAll(" |\n|\t|-|\u00AD", ""); // blank, newline, tab, minus, soft hyphen + return retStr; + } +} \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/tokenize/WordContentHandler.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,159 @@ +package de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize; + +import org.xml.sax.*; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.lt.text.norm.Normalizer; +import de.mpg.mpiwg.berlin.mpdl.util.StringUtils; + +public class WordContentHandler implements ContentHandler { + private static String DEFAULT_LANGUAGE = "eng"; + private String xmlnsString = ""; + private StringBuilder resultStrBuilder = new StringBuilder(); + private String language = DEFAULT_LANGUAGE; + private String formRegularized; + private int wordLevelCounter = 0; + private String wordElemContent = ""; + private String wordElementName = "w"; + + public String getResult() { + return resultStrBuilder.toString(); + } + + public void startDocument() throws SAXException { + } + + public void endDocument() throws SAXException { + } + + public void characters(char[] c, int start, int length) throws SAXException { + char[] cCopy = new char[length]; + System.arraycopy(c, start, cCopy, 0, length); + String charactersStr = String.valueOf(cCopy); + if (charactersStr != null && ! charactersStr.equals("")) { + charactersStr = StringUtils.deresolveXmlEntities(charactersStr); + write(charactersStr); + } + } + + public void ignorableWhitespace(char[] c, int start, int length) throws SAXException { + } + + public void processingInstruction(String target, String data) throws SAXException { + } + + public void setDocumentLocator(Locator locator) { + } + + public void startPrefixMapping(String prefix, String uri) throws SAXException { + if (prefix != null && prefix.equals("")) + xmlnsString += "xmlns" + "=\"" + uri + "\" "; + else + xmlnsString += "xmlns:" + prefix + "=\"" + uri + "\" "; + } + + public void endPrefixMapping(String prefix) throws SAXException { + } + + public void skippedEntity(String name) throws SAXException { + } + + public void startElement(String uri, String localName, String name, Attributes attrs) throws SAXException { + int attrSize = attrs.getLength(); + String attrString = ""; + for (int i=0; i<attrSize; i++) { + String attrQName = attrs.getQName(i); + String attrValue = attrs.getValue(i); + attrValue = StringUtils.forXML(attrValue); + if (localName != null && localName.equals(wordElementName) && (attrQName.equals("lang") || attrQName.equals("xml:lang"))) + language = attrValue; + if (localName != null && localName.equals(wordElementName) && attrQName.equals("formRegularized") && attrValue != null && ! attrValue.isEmpty()) + formRegularized = attrValue; + attrString = attrString + " " + attrQName + "=\"" + attrValue + "\""; + } + if (attrString != null && ! attrString.isEmpty()) { + attrString = attrString.trim(); + } + if (xmlnsString != null && ! xmlnsString.isEmpty()) { + xmlnsString = xmlnsString.trim(); + } + // start all elements but no word elements within word elements (level > 0) + if (localName != null && (! localName.equals(wordElementName) || (localName.equals(wordElementName) && wordLevelCounter == 0))) { + write("<" + name); + if (xmlnsString != null && ! xmlnsString.isEmpty()) + write(" " + xmlnsString); + if (attrString != null && ! attrString.isEmpty()) + write(" " + attrString); + write(">"); + } + xmlnsString = ""; + if (localName != null && localName.equals(wordElementName)) { + wordLevelCounter++; + } + } + + public void endElement(String uri, String localName, String name) throws SAXException { + try { + if (localName != null && localName.equals(wordElementName)) { + wordLevelCounter--; + } + // special handling of word elements (with level 0): insert orig, reg and norm attributes + if (localName != null && localName.equals(wordElementName) && wordLevelCounter == 0) { + // handle formRegularized + String newWordElemContentReg = ""; + if (formRegularized == null || formRegularized.isEmpty()) { + newWordElemContentReg = wordElemContent; + } else if (formRegularized.contains("- ")) { + String[] wordParts = formRegularized.split("- "); + for (int i=0; i<wordParts.length - 1; i++) { + String wp = wordParts[i]; + newWordElemContentReg = newWordElemContentReg + wp + "-<lb/>"; + } + newWordElemContentReg = newWordElemContentReg + wordParts[wordParts.length - 1]; // last one + } else if (formRegularized.contains(" ")) { + String[] wordParts = formRegularized.split(" "); + for (int i=0; i<wordParts.length - 1; i++) { + String wp = wordParts[i]; + newWordElemContentReg = newWordElemContentReg + wp + "<lb/>"; + } + newWordElemContentReg = newWordElemContentReg + wordParts[wordParts.length - 1]; // last one + } else { + newWordElemContentReg = formRegularized; + } + // handle normalized word form + String[] norm = {"norm"}; + Normalizer normNormalizer = new Normalizer(norm, language); + String newWordElemContentNorm = null; + if (formRegularized == null) + newWordElemContentNorm = normNormalizer.normalize(wordElemContent); + else + newWordElemContentNorm = normNormalizer.normalize(newWordElemContentReg); + // write full word content (including lb etc.) into elements orig, reg and norm + write("<orig>" + wordElemContent + "</orig>"); + write("<reg>" + newWordElemContentReg + "</reg>"); + write("<norm>" + newWordElemContentNorm + "</norm>"); + write("</" + name + ">"); + formRegularized = null; + wordElemContent = ""; + } else if (localName != null && localName.equals(wordElementName) && wordLevelCounter > 0) { + // nothing: remove word elements within word elements (level > 0) + } else { + write("</" + name + ">"); + } + } catch (ApplicationException e) { + throw new SAXException(e); + } + } + + private void write(String outStr) throws SAXException { + if (wordLevelCounter > 0) + writeWordElemContent(outStr); + else + resultStrBuilder.append(outStr); + } + + private void writeWordElemContent(String outStr) throws SAXException { + wordElemContent = wordElemContent + outStr; + } + +}
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/tokenize/XmlTokenizer.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/tokenize/XmlTokenizer.java Tue Nov 27 12:35:19 2012 +0100 @@ -2,6 +2,8 @@ import java.io.IOException; import java.io.Reader; +import java.util.ArrayList; +import java.util.Collections; import org.xml.sax.InputSource; import org.xml.sax.SAXException; @@ -11,19 +13,29 @@ import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; import de.mpg.mpiwg.berlin.mpdl.lt.general.Language; +import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.XmlTokenizerContentHandler.Element; public class XmlTokenizer { + private XmlTokenizerContentHandler xmlTokenizerContentHandler; private Reader input; + private String docId; private String language = "eng"; // default: english private String[] normFunctions = {"specialNorm"}; // default: use special norm function - private String[] nwbElements = {"lb", "br", "cb", "figure", "image", "handwritten", "anchor", "emph", "note"}; // non word breaking elements, default: these elements - private String[] stopElements = {}; // default: no stop elements + private String[] nwbElements = {"lb", "br", "cb", "hi"}; // non word breaking elements, default: these elements + private String[] stopElements = {}; // stop elements: its tokens should not get word tags (when output format is "xml") or its tokens should be removed (if output format is "string") + private String[] elements = {}; + private String[] highlightTerms = {}; // highlight terms, default: no highlight terms + private String outputFormat = "xml"; // default: xml private String[] outputOptions = {}; public XmlTokenizer(Reader input) { this.input = input; } + public void setDocIdentifier(String docId) { + this.docId = docId; + } + public void setLanguage(String lang) { String language = Language.getInstance().getLanguageId(lang); this.language = language; @@ -41,28 +53,149 @@ this.stopElements = stopElements; } + public void setElements(String[] elements) { + this.elements = elements; + } + + public void setOutputFormat(String outputFormat) { + this.outputFormat = outputFormat; + } + public void setOutputOptions(String[] outputOptions) { this.outputOptions = outputOptions; } - public String tokenize() throws ApplicationException { - String retString = null; + public void setHighlightTerms(String[] highlightTerms) { + this.highlightTerms = highlightTerms; + } + + public void tokenize() throws ApplicationException { try { - XmlTokenizerContentHandler dictContentHandler = new XmlTokenizerContentHandler(normFunctions, language); - dictContentHandler.setStopElements(stopElements); - dictContentHandler.setNWBElements(nwbElements); - dictContentHandler.setOutputOptions(outputOptions); + xmlTokenizerContentHandler = new XmlTokenizerContentHandler(language); + xmlTokenizerContentHandler.setDocIdentifier(docId); + xmlTokenizerContentHandler.setStopElements(stopElements); + xmlTokenizerContentHandler.setNWBElements(nwbElements); + xmlTokenizerContentHandler.setHighlightTerms(highlightTerms); + xmlTokenizerContentHandler.setNormFunctions(normFunctions); + xmlTokenizerContentHandler.setOutputOptions(outputOptions); + xmlTokenizerContentHandler.setOutputFormat(outputFormat); XMLReader xmlParser = new SAXParser(); - xmlParser.setContentHandler(dictContentHandler); + xmlParser.setContentHandler(xmlTokenizerContentHandler); InputSource inputSource = new InputSource(input); xmlParser.parse(inputSource); - retString = dictContentHandler.getXmlFragment(); } catch (SAXException e) { throw new ApplicationException(e); } catch (IOException e) { throw new ApplicationException(e); } - return retString; + } + + public String getXmlResult() throws ApplicationException { + return xmlTokenizerContentHandler.getResultString(); + } + + public ArrayList<Token> getResultTokens() { + return xmlTokenizerContentHandler.getResultTokens(); + } + + public int getPageCount() { + return xmlTokenizerContentHandler.getPageCount(); + } + + public ArrayList<Element> getElements(String elementNamesStr) { + ArrayList<Element> retElements = new ArrayList<Element>(); + String[] elementNames = elementNamesStr.split(" "); + for (int i=0; i<elementNames.length; i++) { + String elementName = elementNames[i]; + ArrayList<Element> elements = xmlTokenizerContentHandler.getElements(elementName); + if (elements != null) + retElements.addAll(elements); + Collections.sort(retElements); + } + return retElements; + } + + public String getStringResult() throws ApplicationException { + StringBuilder result = new StringBuilder(); + ArrayList<Token> resultTokens = new ArrayList<Token>(); + if (elements != null && elements.length > 0) { + for (int i=0; i<elements.length; i++) { + String elemName = elements[i]; + ArrayList<XmlTokenizerContentHandler.Element> elems = getElements(elemName); + for (int j=0; j<elems.size(); j++) { + XmlTokenizerContentHandler.Element elem = elems.get(j); + resultTokens.addAll(elem.getTokens()); + } + } + } else { + resultTokens = xmlTokenizerContentHandler.getResultTokens(); // all tokens + } + if (resultTokens != null) { + for (int i=0; i<resultTokens.size(); i++) { + Token token = resultTokens.get(i); + if (! withLemmas(outputOptions)) { + if (useNormFunction()) { + String contentNorm = token.getContentNorm(); + if (contentNorm != null) + result.append(contentNorm + " "); + } else if (useRegFunction()) { + String contentReg = token.getContentReg(); + if (contentReg != null) + result.append(contentReg + " "); + else { + String contentOrig = token.getContentOrig(); + if (contentOrig != null) + result.append(contentOrig + " "); + } + } else { + String contentOrig = token.getContentOrig(); + if (contentOrig != null) + result.append(contentOrig + " "); + } + } else { + String contentMorph = token.getContentMorph(); + if (contentMorph != null) + result.append(contentMorph + " "); + } + } + } + return result.toString(); + } + + private boolean withLemmas(String[] outputOptions) { + boolean result = false; + if (outputOptions != null) { + for (int i=0; i< outputOptions.length; i++) { + String function = outputOptions[i]; + if (function.equals("withLemmas")) + return true; + } + } + return result; + } + + private boolean useNormFunction() { + boolean useNorm = false; + if (normFunctions != null) { + for (int i=0; i< normFunctions.length; i++) { + String function = normFunctions[i]; + if (function.equals("norm")) + return true; + } + } + return useNorm; + } + + private boolean useRegFunction() { + boolean useReg = false; + if (normFunctions != null) { + for (int i=0; i< normFunctions.length; i++) { + String function = normFunctions[i]; + if (function.equals("reg")) + return true; + } + } + return useReg; } } \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/tokenize/XmlTokenizerContentHandler.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/lt/text/tokenize/XmlTokenizerContentHandler.java Tue Nov 27 12:35:19 2012 +0100 @@ -3,6 +3,7 @@ import java.io.StringReader; import java.util.ArrayList; import java.util.Collections; +import java.util.Enumeration; import java.util.Hashtable; import org.xml.sax.*; @@ -21,27 +22,39 @@ private static int COMPLEX_ELEMENT_MARK_SIZE = COMPLEX_ELEMENT_MARK.length(); private static int ELEMENT_TYPE_CHARACTERS = 1; private static int ELEMENT_TYPE_COMPLEX = 2; - private String[] normalizeFunctions = {}; // default: without normalize functions - private String[] nwbElements = {}; // non word breaking elements, default: these elements + private String docId; + private String language; + private String[] nwbElements = {}; // non word breaking elements, default: no nwb elements private String[] stopElements = {}; // default: no stop elements + private String outputFormat = "xml"; // default: xml private String[] outputOptions = {}; + private boolean withForms = false; + private boolean withLemmas = false; + private String[] highlightTerms = {}; // highlight terms, default: no highlight terms + private String[] normFunctions = {}; // default: no norm function + private boolean useNormFunction = false; + private boolean useRegFunction = false; private String xmlnsString = ""; - private String language; - private String outputXmlFragment = ""; + private StringBuilder result = new StringBuilder(); + private ArrayList<Token> resultTokens = new ArrayList<Token>(); + private Hashtable<String, ArrayList<Element>> elements = new Hashtable<String, ArrayList<Element>>(); private Element rootElement; private Element currentElement; + private int currentPosition = 0; + private int currentPageNumber = 0; + private int currentLineNumber = 0; + private Hashtable<String, Integer> currentPositions = new Hashtable<String, Integer>(); + private Hashtable<String, Integer> currentPagePositions = new Hashtable<String, Integer>(); private ArrayList<Element> elementQueue; - public XmlTokenizerContentHandler(String[] normalizeFunctions, String language) throws ApplicationException { - if (normalizeFunctions == null) { - String[] emptyFunctions = {}; - this.normalizeFunctions = emptyFunctions; - } else { - this.normalizeFunctions = normalizeFunctions; - } + public XmlTokenizerContentHandler(String language) throws ApplicationException { this.language = language; } + public void setDocIdentifier(String docId) { + this.docId = docId; + } + public void setNWBElements(String[] nwbElements) { this.nwbElements = nwbElements; } @@ -50,24 +63,66 @@ this.stopElements = stopElements; } - public void setOutputOptions(String[] outputOptions) { - this.outputOptions = outputOptions; + public void setHighlightTerms(String[] highlightTerms) { + this.highlightTerms = highlightTerms; + } + + public void setNormFunctions(String[] normFunctions) { + this.normFunctions = normFunctions; + if (this.normFunctions != null) { + for (int i=0; i< this.normFunctions.length; i++) { + String function = normFunctions[i]; + if (function.equals("norm")) + this.useNormFunction = true; + else if (function.equals("reg")) + this.useRegFunction = true; + } + } + } + + public void setOutputFormat(String outputFormat) { + this.outputFormat = outputFormat; } - public String getXmlFragment() { - return outputXmlFragment; + public void setOutputOptions(String[] outputOptions) { + this.outputOptions = outputOptions; + for (int i=0; i< this.outputOptions.length; i++) { + String function = outputOptions[i]; + if (function.equals("withForms")) + this.withForms = true; + else if (function.equals("withLemmas")) + this.withLemmas = true; + } + } + + public String getResultString() { + return result.toString(); + } + + public ArrayList<Token> getResultTokens() { + return resultTokens; } + public ArrayList<Element> getElements(String elementName) { + return elements.get(elementName); + } + + public int getPageCount() { + return currentPageNumber; + } + public void startDocument() throws SAXException { } public void endDocument() throws SAXException { try { - String rootElemToStr = rootElement.toXmlString(); + String rootElemToStr = rootElement.buildString(); write(rootElemToStr); write("\n"); } catch (NullPointerException e) { throw new SAXException(e); + } catch (ApplicationException e) { + throw new SAXException(e); } } @@ -78,6 +133,7 @@ if (charactersStr != null && ! charactersStr.equals("")) { if (currentElement != null) { Element charElement = new Element("characters", ELEMENT_TYPE_CHARACTERS); + charElement.pageNumber = currentPageNumber; charElement.value = StringUtils.deresolveXmlEntities(charactersStr); if (currentElement.composites == null) currentElement.composites = new ArrayList<Element>(); @@ -96,9 +152,10 @@ } public void startPrefixMapping(String prefix, String uri) throws SAXException { - xmlnsString += "xmlns:" + prefix + "=\"" + uri + "\" "; if (prefix != null && prefix.equals("")) - xmlnsString = "xmlns" + prefix + "=\"" + uri + "\" "; + xmlnsString += "xmlns" + "=\"" + uri + "\" "; + else + xmlnsString += "xmlns:" + prefix + "=\"" + uri + "\" "; } public void endPrefixMapping(String prefix) throws SAXException { @@ -117,8 +174,32 @@ if (currentElement.lang != null) newElement.lang = currentElement.lang; // language is inherited to childs currentElement.composites.add(newElement); + newElement.parent = currentElement; } currentElement = newElement; + if (localName != null && localName.equals("pb")) { + currentPageNumber++; + setCurrentPagePosition(localName, 0); + } + currentElement.pageNumber = currentPageNumber; + if (localName != null && localName.equals("lb")) { + currentLineNumber++; + } + currentElement.lineNumber = currentLineNumber; + currentPosition++; + currentElement.docPosition = currentPosition; + int newElemPosition = incrementCurrentPosition(localName); + currentElement.position = newElemPosition; + + currentElement.elemPosition = getElementPosition(currentElement); + Element parent = currentElement.parent; + if (parent == null) { + currentElement.xpath = "/" + currentElement.name + "[" + currentElement.elemPosition + "]"; + } else { + currentElement.xpath = parent.xpath + "/" + currentElement.name + "[" + currentElement.elemPosition + "]"; + } + int newElemPagePosition = incrementCurrentPagePosition(localName); + currentElement.pagePosition = newElemPagePosition; int attrSize = attrs.getLength(); String attrString = ""; for (int i=0; i<attrSize; i++) { @@ -126,8 +207,12 @@ String attrValue = attrs.getValue(i); attrValue = StringUtils.forXML(attrValue); attrString = attrString + " " + attrQName + "=\"" + attrValue + "\""; - if (attrQName != null && (attrQName.toLowerCase().equals("xml:lang") || attrQName.toLowerCase().equals("lang"))) + if (attrQName != null && (attrQName.toLowerCase().equals("xml:lang") || attrQName.toLowerCase().equals("lang"))) { currentElement.lang = attrValue; // if xml:lang is set, it is set to the new element and overwrites values inherited by the father + } + if (attrQName != null && (attrQName.toLowerCase().equals("xml:id") || attrQName.toLowerCase().equals("id"))) { + currentElement.xmlId = attrValue; + } } currentElement.attrString = attrString; if (! xmlnsString.equals("")) { @@ -153,112 +238,255 @@ } } - private boolean withForms() { + private int incrementCurrentPosition(String elemName) { + Integer currentElemPos = currentPositions.get(elemName); + if (currentElemPos == null) { + currentElemPos = new Integer(0); + } + currentElemPos++; + currentPositions.put(elemName, currentElemPos); + return currentElemPos.intValue(); + } + + private int getElementPosition(Element elem) { + int pos = 0; + Element parent = elem.parent; + if (parent == null) { + pos = 1; + } else { + pos = 0; + ArrayList<Element> composites = parent.composites; + if (composites != null) { + for (int i=0; i<composites.size(); i++) { + Element e = composites.get(i); + if (e.isComplex() && e.name.equals(elem.name)) { + pos++; + } + if (e == elem) + break; + } + } else { + pos = 1; + } + } + return pos; + } + + private int incrementCurrentPagePosition(String elemName) { + Integer currentElemPagePos = currentPagePositions.get(elemName); + if (currentElemPagePos == null) { + currentElemPagePos = new Integer(0); + } + currentElemPagePos++; + currentPagePositions.put(elemName, currentElemPagePos); + return currentElemPagePos.intValue(); + } + + private void setCurrentPagePosition(String elemName, int pos) { + Integer newPagePosition = new Integer(pos); + Enumeration<String> elemKeys = currentPagePositions.keys(); + while (elemKeys.hasMoreElements()) { + String elemKey = elemKeys.nextElement(); + currentPagePositions.put(elemKey, newPagePosition); + } + } + + private boolean isHighlightTerm(String term) { + if (term == null) + return false; boolean result = false; - for (int i=0; i< outputOptions.length; i++) { - String function = outputOptions[i]; - if (function.equals("withForms")) + for (int i=0; i< highlightTerms.length; i++) { + String t = highlightTerms[i].toLowerCase(); + String termLowerCase = term.toLowerCase(); + if (t.equals(termLowerCase)) return true; } return result; } - private boolean withLemmas() { + private boolean isHighlightTerm(String[] terms) { + if (terms == null) + return false; boolean result = false; - for (int i=0; i< outputOptions.length; i++) { - String function = outputOptions[i]; - if (function.equals("withLemmas")) - return true; + for (int i=0; i< highlightTerms.length; i++) { + String t = highlightTerms[i].toLowerCase(); + for (int j=0; j<terms.length; j++) { + String termLowerCase = terms[j].toLowerCase(); + if (t.equals(termLowerCase)) + return true; + } } return result; } private void write(String outStr) throws SAXException { - outputXmlFragment += outStr; + result.append(outStr); } - private class Element { + public class Element implements Comparable<Element> { private int type; - private String name; + public String name; private String xmlnsString; private String attrString; private String value; - private String lang; // normally value of attribute xml:lang or the inherited xml:lang value of the father node + public String lang; // value of attribute xml:lang or the inherited xml:lang value of the father node + public String xmlId; + public String xpath; + public int pageNumber; + public int lineNumber; + public int docPosition; // absolute position in document + public int position; // position within all elements with this name + public int elemPosition; // position in element e.g. the 6 sentence in paragraph + public int pagePosition; // position in page + private ArrayList<Token> tokens = new ArrayList<Token>(); private ArrayList<Element> composites; + private Element parent; + private boolean isStopElement = false; + private boolean isWordDelimiterElement = true; // default: is word delimiter element private Element(String name) { this.type = ELEMENT_TYPE_COMPLEX; - this.name = name; + setName(name); } private Element(String name, int type) { this.type = type; - this.name = name; + setName(name); } - private boolean isComplex() { + private void setName(String name) { + this.name = name; + for (int i=0; i<stopElements.length; i++) { + String stopElementName = stopElements[i]; + if (name.equals(stopElementName)) { + this.isStopElement = true; + break; + } + } + for (int i=0; i<nwbElements.length; i++) { + String nwbElementName = nwbElements[i]; + if (name.equals(nwbElementName)) { + this.isWordDelimiterElement = false; + break; + } + } + } + + public int compareTo(Element elem) { + return (new Integer(position)).compareTo(new Integer(elem.position)); + } + + private boolean isComplex() { boolean isComplex = false; if (type == ELEMENT_TYPE_COMPLEX) isComplex = true; return isComplex; } - private boolean isWordDelimiterElement() { - boolean isWordDelimiterElement = true; - for (int i=0; i<nwbElements.length; i++) { - String nwbElementName = nwbElements[i]; - if (name.equals(nwbElementName)) { - isWordDelimiterElement = false; - break; + public ArrayList<Token> getTokens() { + ArrayList<Token> retTokens = new ArrayList<Token>(); + if (isComplex()) { + if (composites != null) { + for (int i=0; i<composites.size(); i++) { + Element elem = composites.get(i); + if (elem.tokens != null) + retTokens.addAll(elem.tokens); + } } - } - return isWordDelimiterElement; + } + if (tokens != null) + retTokens.addAll(tokens); + return retTokens; + } + + public String getTokensStr(String type) { + ArrayList<Token> elementTokens = getTokens(); + String tokenStr = getTokensStr(type, elementTokens); + return tokenStr; } - private boolean isStopElement() { - boolean isStopElement = false; - for (int i=0; i<stopElements.length; i++) { - String stopElementName = stopElements[i]; - if (name.equals(stopElementName)) { - isStopElement = true; - break; - } + private String getTokensStr(String type, ArrayList<Token> tokens) { + StringBuilder tokenStr = new StringBuilder(); + for (int j=0; j<tokens.size(); j++) { + Token token = tokens.get(j); + String content = null; + if (type.equals("orig")) + content = token.getContentOrig(); + else if (type.equals("reg")) + content = token.getContentReg(); + else if (type.equals("norm")) + content = token.getContentNorm(); + else if (type.equals("morph")) + content = token.getContentMorph(); + if (content != null) + tokenStr.append(content + " "); } - return isStopElement; + return tokenStr.toString(); } - private String toXmlString() throws SAXException { - String retString = ""; + public String toXmlString() throws ApplicationException { + StringBuilder retStrBuilder = new StringBuilder(); + if (! isComplex()) { + retStrBuilder.append(value); + } else { + String xmlNsString = this.xmlnsString; + if (xmlNsString == null || xmlNsString.equals("")) { + retStrBuilder.append("<" + name + attrString + ">"); + } else { + retStrBuilder.append("<" + name + " " + xmlNsString + attrString + ">"); + } + if (composites != null) { + for (int i=0; i<composites.size(); i++) { + Element composite = composites.get(i); + if (! composite.isComplex()) { + if (composite.value != null && ! composite.value.equals("")) { + String compositeValueStr = StringUtils.removeNlTabBlanks(composite.value); // remove all newlines, they are no separators for words. And: if there are many Blanks/Tabs make them to one Blank + retStrBuilder.append(compositeValueStr); + } + } else { + retStrBuilder.append(composite.toXmlString()); + } + } + } + retStrBuilder.append("</" + name + ">"); + } + return retStrBuilder.toString(); + } + + private String buildString() throws ApplicationException { + StringBuilder retStrBuilder = new StringBuilder(); String elemLanguage = language; // default value for the document/page if (lang != null) elemLanguage = lang; // value of the element if available // write this element if (! isComplex()) { - retString += value; + retStrBuilder.append(value); } else { - String xmlNsString = this.xmlnsString; - if (xmlNsString == null || xmlNsString.equals("")) { - retString = retString + "<" + name + attrString + ">"; - } else { - retString = retString + "<" + name + " " + xmlNsString + attrString + ">"; + if (outputFormat != null && outputFormat.equals("xml")) { + String xmlNsString = this.xmlnsString; + if (xmlNsString == null || xmlNsString.equals("")) { + retStrBuilder.append("<" + name + attrString + ">"); + } else { + retStrBuilder.append("<" + name + " " + xmlNsString + attrString + ">"); + } + } else { // outputFormat == string + // nothing } if (composites != null) { - String compositesCharsWithMarks = ""; + StringBuilder compositesCharsWithMarks = new StringBuilder(); ArrayList<Element> complexElements = new ArrayList<Element>(); for (int i=0; i<composites.size(); i++) { Element composite = composites.get(i); if (! composite.isComplex()) { if (composite.value != null && ! composite.value.equals("")) { - String compositeValueStr = composite.value; - compositeValueStr = compositeValueStr.replaceAll("\n", ""); // remove all newlines, they are no separators for words. - compositeValueStr = compositeValueStr.replaceAll("[ \t]+", " "); // if there are many Blanks/Tabs make them to one Blank - compositesCharsWithMarks = compositesCharsWithMarks + compositeValueStr; + String compositeValueStr = StringUtils.removeNlTabBlanks(composite.value); // remove all newlines, they are no separators for words. And: if there are many Blanks/Tabs make them to one Blank + compositesCharsWithMarks.append(compositeValueStr); } } else { - if (! composite.isWordDelimiterElement()) { - compositesCharsWithMarks = compositesCharsWithMarks + COMPLEX_ELEMENT_NWD_MARK; // add a special mark symbol at the position of the "not word delimiter element" (e.g. <lb>) + if (! composite.isWordDelimiterElement) { + compositesCharsWithMarks.append(COMPLEX_ELEMENT_NWD_MARK); // add a special mark symbol at the position of the "not word delimiter element" (e.g. <lb>) } else { - compositesCharsWithMarks = compositesCharsWithMarks + COMPLEX_ELEMENT_MARK; // add a special mark symbol at the position of the "word delimiter element" (e.g. <var>) + compositesCharsWithMarks.append(COMPLEX_ELEMENT_MARK); // add a special mark symbol at the position of the "word delimiter element" (e.g. <var>) } complexElements.add(composite); } @@ -270,120 +498,235 @@ for (int i=0; i<complexElements.size(); i++) { int indexComplexElemCompositesCharsWithMarks = compositesCharsWithMarksWithWordTags.indexOf(COMPLEX_ELEMENT_MARK); Element complexElem = complexElements.get(i); - String complexElementStr = complexElem.toXmlString(); + String complexElementStr = complexElem.buildString(); String firstPiece = ""; if (indexComplexElemCompositesCharsWithMarks > 0) { firstPiece = compositesCharsWithMarksWithWordTags.substring(0, indexComplexElemCompositesCharsWithMarks); compositesCharsWithMarksWithWordTags = compositesCharsWithMarksWithWordTags.substring(indexComplexElemCompositesCharsWithMarks); } - retString = retString + firstPiece + complexElementStr; + retStrBuilder.append(firstPiece + complexElementStr); compositesCharsWithMarksWithWordTags = compositesCharsWithMarksWithWordTags.substring(COMPLEX_ELEMENT_MARK_SIZE); } - retString = retString + compositesCharsWithMarksWithWordTags; // last one must also be added + retStrBuilder.append(compositesCharsWithMarksWithWordTags); // last one must also be added } else { - retString = retString + compositesCharsWithMarksWithWordTags; // last one must also be added + retStrBuilder.append(compositesCharsWithMarksWithWordTags); // last one must also be added } } - retString = retString + "</" + name + ">"; + if (outputFormat != null && outputFormat.equals("xml")) { + retStrBuilder.append("</" + name + ">"); + } else { // outputFormat == string + // nothing + } + // put element into elements name hashtable + ArrayList<Element> elems = elements.get(name); + if (elems == null) { + elems = new ArrayList<Element>(); + elements.put(name, elems); + } + elems.add(this); } - return retString; + return retStrBuilder.toString(); } - private String insertWordTags(String charactersStrDeresolved, String language) throws SAXException { + private String insertWordTags(StringBuilder charactersStrDeresolvedBuilder, String language) throws ApplicationException { + String charactersStrDeresolved = charactersStrDeresolvedBuilder.toString(); String charactersStr = StringUtils.resolveXmlEntities(charactersStrDeresolved); - String retStr = ""; - try { - Tokenizer tokenizer = new Tokenizer(new StringReader(charactersStr)); - tokenizer.setLanguage(language); - tokenizer.setNormFunctions(normalizeFunctions); - ArrayList<Token> tokens = tokenizer.getTokens(); - int endPos = 0; - for (int i=0; i < tokens.size(); i++) { - Token token = tokens.get(i); - String wordForm = token.getContent(); - int startPos = token.getStart(); - String beforeStr = charactersStr.substring(endPos, startPos); - endPos = token.getEnd(); - String beforeStrDeresolved = StringUtils.deresolveXmlEntities(beforeStr); - String origWordForm = charactersStr.substring(startPos, endPos); - String wordTag = insertWordTags(wordForm, language, origWordForm); - retStr = retStr + beforeStrDeresolved + wordTag; + StringBuilder retStrBuilder = new StringBuilder(); + Tokenizer tokenizer = new Tokenizer(new StringReader(charactersStr)); + tokenizer.setLanguage(language); + String[] normFunction = {"norm"}; + tokenizer.setNormFunctions(normFunction); + ArrayList<Token> tokens = tokenizer.getTokens(); + int endPos = 0; + for (int i=0; i < tokens.size(); i++) { + Token token = tokens.get(i); + int startPos = token.getStart(); + String beforeStr = charactersStr.substring(endPos, startPos); + endPos = token.getEnd(); + String beforeStrDeresolved = StringUtils.deresolveXmlEntities(beforeStr); + String origWordForm = charactersStr.substring(startPos, endPos); + String wordTag = insertWordTags(token, language, origWordForm); + if (outputFormat != null && outputFormat.equals("xml")) { + retStrBuilder.append(beforeStrDeresolved + wordTag); + } else { // outputFormat == string + String beforeStrDeresolvedToBlanks = toBlanks(beforeStrDeresolved); + retStrBuilder.append(beforeStrDeresolvedToBlanks + wordTag); + } + } + String lastAfterStr = charactersStr.substring(endPos); + String lastAfterStrDeresolved = StringUtils.deresolveXmlEntities(lastAfterStr); + if (outputFormat != null && outputFormat.equals("xml")) { + retStrBuilder.append(lastAfterStrDeresolved); + } else { // outputFormat == string + String lastAfterStrDeresolvedToBlanks = toBlanks(lastAfterStrDeresolved); + retStrBuilder.append(lastAfterStrDeresolvedToBlanks); + } + return retStrBuilder.toString(); + } + + private String insertWordTags(Token token, String language, String origWordForm) throws ApplicationException { + if (origWordForm != null && origWordForm.equals(COMPLEX_ELEMENT_NWD_MARK)) { + return origWordForm; + } + String wordTag = null; + token.setDocId(docId); + token.setLanguage(lang); + token.setPageNumber(pageNumber); + token.setLineNumber(lineNumber); + token.setElementPosition(position); + token.setElementPagePosition(pagePosition); + token.setElementName(name); + token.setXmlId(xmlId); + token.setXpath("xpath"); // TODO + if (name != null && name.equals("reg")) { + if (attrString != null && attrString.contains("norm=\"")) { + int regIndexBegin = attrString.indexOf("norm=\""); + int regIndexEnd = attrString.indexOf("\"", regIndexBegin + 7); + String reg = attrString.substring(regIndexBegin + 6, regIndexEnd); + token.setContentReg(reg); + String[] normFunction = {"norm"}; + Normalizer normalizer = new Normalizer(normFunction, language); + String normStr = normalizer.normalize(reg); + token.setContentNorm(normStr); } - String lastAfterStr = charactersStr.substring(endPos); - String lastAfterStrDeresolved = StringUtils.deresolveXmlEntities(lastAfterStr); - retStr = retStr + lastAfterStrDeresolved; - } catch (ApplicationException e) { - throw new SAXException(e); + } + if (language == null) { + token.setContentOrig(origWordForm); // TODO necessary ? + tokens.add(token); + resultTokens.add(token); + return origWordForm; + } + if (isStopElement && outputFormat != null && outputFormat.equals("xml")) + return origWordForm; + if (isStopElement && outputFormat != null && outputFormat.equals("string")) + return toBlanks(origWordForm); + String wordFormNorm = token.getContentNorm(); + String origWordFormDeresolved = StringUtils.deresolveXmlEntities(origWordForm); + ArrayList<Lemma> lemmas = null; + Boolean hasDctionaryEntries = null; + String lemmasStr = ""; + if (withForms || withLemmas) { + LexHandler lexHandler = LexHandler.getInstance(); + lemmas = lexHandler.getLemmas(wordFormNorm, "form", language, Normalizer.DICTIONARY, false); // Performance: needs 15 % of the indexing time + if (lemmas != null) { + for (int i=0; i < lemmas.size(); i++) { + Lemma lemma = lemmas.get(i); + String lemmaName = lemma.getLemmaName(); + lemmasStr = lemmasStr + lemmaName + " "; + } + } + lemmasStr = lemmasStr.trim(); + token.setContentMorph(lemmasStr); + hasDctionaryEntries = false; + ArrayList<String> lexEntries = lexHandler.getLexEntryKeys(wordFormNorm, language, Normalizer.DICTIONARY); // Performance: needs 15 % of the indexing time + if (lexEntries != null) + hasDctionaryEntries = true; } + if (outputFormat != null && outputFormat.equals("xml")) { + wordTag = insertWordTags(origWordFormDeresolved, token, language, lemmas, hasDctionaryEntries); // Performance: needs 10 % of the indexing time + String tokenWordForm = token.getContentOrig(); // word form is in contentOrig + if (useRegFunction) + tokenWordForm = token.getContentReg(); + else if (useNormFunction) + tokenWordForm = token.getContentNorm(); + else if (withLemmas) + tokenWordForm = token.getContentMorph(); + boolean isHighlightTerm = false; + if (highlightTerms.length > 0 && ! withLemmas) { + isHighlightTerm = isHighlightTerm(tokenWordForm); + } else { + if (highlightTerms.length > 0 && lemmas != null) { + String[] lemmasArray = lemmasStr.split(" "); + isHighlightTerm = isHighlightTerm(lemmasArray); + } + } + if (isHighlightTerm) { + wordTag = "<hi>" + wordTag + "</hi>"; + } + } else { // outputFormat == string + String inWordFormWithoutSpecialSymbols = removeSpecialSymbols(origWordForm); // without hyphen, blanks, newline, tab + if (withLemmas) { + if (lemmas != null) { + String blanksAndNWBMarksOfOrigWord = toBlanks(origWordFormDeresolved); // to rescue the NWB marks of the origWord and put it to the beginning of the lemmasStr + wordTag = blanksAndNWBMarksOfOrigWord + lemmasStr; + token.setContentMorph(lemmasStr); + } else { + wordTag = inWordFormWithoutSpecialSymbols; + } + } else { + wordTag = inWordFormWithoutSpecialSymbols; + } + tokens.add(token); + resultTokens.add(token); + } + return wordTag; + } + + private String removeSpecialSymbols(String inputStr) { + String retStr = inputStr.replaceAll(" |\n|\t|-|\u00AD", ""); // blank, newline, tab, minus, soft hyphen return retStr; } - private String insertWordTags(String wordForm, String language, String origWordForm) throws ApplicationException { - String wordTag = null; - if (origWordForm != null && origWordForm.equals(COMPLEX_ELEMENT_NWD_MARK)) - return origWordForm; - if (isStopElement()) - return origWordForm; - wordForm = removeSpecialSymbols(wordForm); - wordForm = wordForm.toLowerCase(); - String origWordFormDeresolved = StringUtils.deresolveXmlEntities(origWordForm); - ArrayList<Lemma> lemmas = null; - if (withForms() || withLemmas()) { - LexHandler lexHandler = LexHandler.getInstance(); - lemmas = lexHandler.getLemmas(wordForm, "form", language, Normalizer.NONE); - } - wordTag = insertWordTags(origWordFormDeresolved, wordForm, language, null, lemmas); - return wordTag; - } - /** * * @param origWordToken could contain nwd marks - * @param wordForm contains no nwd marks + * @param token * @param language - * @param origWordFormNormalized * @param lemmas * @return for each substring between nwd marks create a word tag */ - private String insertWordTags(String origWordToken, String wordForm, String language, String origWordFormNormalized, ArrayList<Lemma> lemmas) { + private String insertWordTags(String origWordToken, Token token, String language, ArrayList<Lemma> lemmas, Boolean hasDictionaryEntries) { if (origWordToken.isEmpty()) return origWordToken; if (origWordToken.equals(COMPLEX_ELEMENT_NWD_MARK)) return COMPLEX_ELEMENT_NWD_MARK; String retWordTags = ""; String origWordTokenTmp = origWordToken; - while (! origWordTokenTmp.isEmpty()) { - if (origWordTokenTmp.startsWith(COMPLEX_ELEMENT_NWD_MARK)) { // single nwd mark - origWordTokenTmp = origWordTokenTmp.substring(1); - retWordTags = retWordTags + COMPLEX_ELEMENT_NWD_MARK; - } else { - int indexUpToNWD = origWordTokenTmp.indexOf(COMPLEX_ELEMENT_NWD_MARK); - if (indexUpToNWD != -1) { // not end of string reached - String origWordTokenFragment = origWordTokenTmp.substring(0, indexUpToNWD); - String origWordTokenFragmentWithTags = getWordTag(origWordTokenFragment, wordForm, language, origWordFormNormalized, lemmas); - retWordTags = retWordTags + origWordTokenFragmentWithTags + COMPLEX_ELEMENT_NWD_MARK; - origWordTokenTmp = origWordTokenTmp.substring(indexUpToNWD + 1); - } else { // end of string reached - String origWordTokenFragment = origWordTokenTmp.substring(0, origWordTokenTmp.length()); - String origWordTokenFragmentWithTags = getWordTag(origWordTokenFragment, wordForm, language, origWordFormNormalized, lemmas); - retWordTags = retWordTags + origWordTokenFragmentWithTags; - origWordTokenTmp = ""; // finente - } - } + if (outputFormat != null && outputFormat.equals("xml")) { + retWordTags = getWordTag(origWordToken, token, language, lemmas, hasDictionaryEntries); + /* + while (! origWordTokenTmp.isEmpty()) { + if (origWordTokenTmp.startsWith(COMPLEX_ELEMENT_NWD_MARK)) { // single nwd mark + origWordTokenTmp = origWordTokenTmp.substring(1); + retWordTags = retWordTags + COMPLEX_ELEMENT_NWD_MARK; + } else { + int indexUpToNWD = origWordTokenTmp.indexOf(COMPLEX_ELEMENT_NWD_MARK); + if (indexUpToNWD != -1) { // not end of string reached + String origWordTokenFragment = origWordTokenTmp.substring(0, indexUpToNWD); + String origWordTokenFragmentWithTags = getWordTag(origWordTokenFragment, token, language, lemmas, hasDictionaryEntries); + retWordTags = retWordTags + origWordTokenFragmentWithTags + COMPLEX_ELEMENT_NWD_MARK; + origWordTokenTmp = origWordTokenTmp.substring(indexUpToNWD + 1); + } else { // end of string reached + String origWordTokenFragment = origWordTokenTmp.substring(0, origWordTokenTmp.length()); + String origWordTokenFragmentWithTags = getWordTag(origWordTokenFragment, token, language, lemmas, hasDictionaryEntries); + retWordTags = retWordTags + origWordTokenFragmentWithTags; + origWordTokenTmp = ""; // finente + } + } + } + */ + } else { + // nothing } return retWordTags; } - private String getWordTag(String origWordForm, String wordForm, String language, String origWordFormNormalized, ArrayList<Lemma> lemmas) { + private String getWordTag(String origWordForm, Token token, String language, ArrayList<Lemma> lemmas, Boolean hasDictionaryEntries) { if (origWordForm == null || origWordForm.isEmpty()) return ""; + String wordForm = token.getContentOrig(); // word form is in contentOrig + String regularizedWordForm = token.getContentReg(); + String normalizedWordForm = token.getContentNorm(); String langISOCode = Language.getInstance().getISO639Code(language); - String retStr = "<w form=\"" + wordForm + "\"" + " lang=\"" + langISOCode + "\""; - if (origWordFormNormalized != null) - retStr = retStr + " formNormalized=\"" + origWordFormNormalized + "\""; + StringBuilder retStrBuilder = new StringBuilder(); + retStrBuilder.append("<w" + " lang=\"" + langISOCode + "\"" + " form=\"" + wordForm + "\""); + if (regularizedWordForm != null) + retStrBuilder.append(" formRegularized=\"" + regularizedWordForm + "\""); + if (normalizedWordForm != null) + retStrBuilder.append(" formNormalized=\"" + normalizedWordForm + "\""); if (lemmas != null) { String lemmasStr = ""; - String formsStr = ""; + StringBuilder formsStrBuilder = new StringBuilder(); Collections.sort(lemmas); Hashtable<String, Form> formsHashtable = new Hashtable<String, Form>(); for (int i=0; i < lemmas.size(); i++) { @@ -403,24 +746,40 @@ Form form = forms.get(i); String formName = form.getFormName(); formName = StringUtils.forXML(formName); - formsStr = formsStr + formName + " "; + formsStrBuilder.append(formName + " "); } + String formsStr = formsStrBuilder.toString(); if (formsStr.endsWith(" ")) formsStr = formsStr.substring(0, formsStr.length() - 1); if (lemmasStr.endsWith(" ")) lemmasStr = lemmasStr.substring(0, lemmasStr.length() - 1); - if (withForms()) - retStr = retStr + " forms=\"" + formsStr + "\""; - if (withLemmas()) - retStr = retStr + " lemmas=\"" + lemmasStr + "\""; + if (withForms) + retStrBuilder.append(" forms=\"" + formsStr + "\""); + if (withLemmas) + retStrBuilder.append(" lemmas=\"" + lemmasStr + "\""); } - retStr = retStr + ">" + origWordForm + "</w>"; - return retStr; + if (hasDictionaryEntries != null && hasDictionaryEntries) { + retStrBuilder.append(" dictionary=\"" + "true" + "\""); + } else if (hasDictionaryEntries != null && ! hasDictionaryEntries) { + retStrBuilder.append(" dictionary=\"" + "false" + "\""); + } + retStrBuilder.append(">"); + retStrBuilder.append(origWordForm); // origWordForm could contain nwd marks (these are transformed back to elements later in method buildString) + retStrBuilder.append("</w>"); + return retStrBuilder.toString(); } - private String removeSpecialSymbols(String inputStr) { - String retStr = inputStr.replaceAll(" |\n|\t|-|\u2424|\u2425", ""); - return retStr; + private String toBlanks(String inputStr) { + int size = inputStr.length(); + StringBuilder retStrBuilder = new StringBuilder(); + for (int j=0; j < size; j++) { + char c = inputStr.charAt(j); + if (c == COMPLEX_ELEMENT_NWD_MARK.charAt(0) || c == COMPLEX_ELEMENT_MARK.charAt(0)) + retStrBuilder.append(c); + else + retStrBuilder.append(" "); + } + return retStrBuilder.toString(); } }
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/test/TestLocal.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/test/TestLocal.java Tue Nov 27 12:35:19 2012 +0100 @@ -10,6 +10,11 @@ import org.apache.commons.io.FileUtils; import org.apache.commons.io.IOUtils; +import org.xml.sax.InputSource; +import org.xml.sax.SAXException; +import org.xml.sax.XMLReader; + +import com.sun.org.apache.xerces.internal.parsers.SAXParser; import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; import de.mpg.mpiwg.berlin.mpdl.lt.dict.app.Lexicon; @@ -19,6 +24,8 @@ import de.mpg.mpiwg.berlin.mpdl.lt.text.norm.Normalizer; import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.Token; import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.Tokenizer; +import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.TokenizerNew; +import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.WordContentHandler; import de.mpg.mpiwg.berlin.mpdl.lt.text.tokenize.XmlTokenizer; public class TestLocal { @@ -27,10 +34,11 @@ public static void main(String[] args) throws ApplicationException { try { TestLocal test = new TestLocal(); - test.init(); - test.testCalls(); - // test.tokenizeString(); + // test.init(); + // test.testCalls(); + test.tokenizeString(); // test.tokenizeXmlFragment(); + // test.normalizeWords(13); // test.getLexEntriesByLexiconBeginningWith("ls", "a"); // test.end(); } catch (Exception e) { @@ -47,22 +55,12 @@ } private ArrayList<Token> tokenizeString() throws ApplicationException { - ArrayList<Token> tokens = new ArrayList<Token>(); - try { - StringReader reader = new StringReader("edo philoÅ¿ophi"); - // StringReader reader = new StringReader("扞盗則æŽå…—å·ž"); - Tokenizer tokenizer = new Tokenizer(reader); - tokenizer.setLanguage("lat"); - // tokenizer.setLanguage("zho"); - String[] normFunctions = new String[1]; - normFunctions[0] = "norm"; - tokenizer.setNormFunctions(normFunctions); - tokens = tokenizer.getTokens(); - tokenizer.end(); - tokenizer.close(); - } catch (IOException e) { - throw new ApplicationException(e); - } + // StringReader reader = new StringReader("edo philoÅ¿ophi ä bla"); + // StringReader reader = new StringReader(""); + StringReader reader = new StringReader("扞盗則æŽå…—å·ž"); + TokenizerNew tokenizer = new TokenizerNew(reader, "zh"); + ArrayList<Token> tokens = tokenizer.tokenize(); + System.out.println(tokens); return tokens; } @@ -79,13 +77,13 @@ XmlTokenizer xmlTokenizer = new XmlTokenizer(new StringReader(xmlFragment)); xmlTokenizer.setLanguage("lat"); - String[] normFunctions = new String[1]; - normFunctions[0] = "norm"; - String[] stopElements = new String[1]; - stopElements[0] = "var"; - xmlTokenizer.setNormFunctions(normFunctions); + String[] stopElements = {"var"}; + // xmlTokenizer.setOutputFormat("string"); + String[] outputOptions = {"withLemmas"}; + xmlTokenizer.setOutputOptions(outputOptions); xmlTokenizer.setStopElements(stopElements); - result = xmlTokenizer.tokenize(); + xmlTokenizer.tokenize(); + result = xmlTokenizer.getXmlResult(); System.out.println(result); } catch (Exception e) { throw new ApplicationException(e); @@ -107,7 +105,7 @@ } private void getLexEntries(String query, String language, String inputType , String outputType, String outputFormat, String dictionaryName, int normMode) throws ApplicationException { - ArrayList<Lemma> lemmas = lexHandler.getLemmas(query, inputType, language, normMode); + ArrayList<Lemma> lemmas = lexHandler.getLemmas(query, inputType, language, normMode, false); ArrayList<Lexicon> dictionaries = lexHandler.getLexEntries(lemmas, language, dictionaryName, query); // String result = lexHandler.getLexEntries(query, language, inputType, outputType, outputFormat, dictionaryName, normalization); String result = ""; @@ -129,4 +127,25 @@ ArrayList<Lexicon> lexEntries = lexHandler.getLexEntriesBeginningWith(language, prefix, 1, 50); System.out.println(lexEntries); } + + private String normalizeWords(int page) throws ApplicationException { + try { + String docDir = "/Users/jwillenborg/mpdl/data/xml/documents/echo/la/Benedetti_1585"; + String pageFileName = docDir + "/" + "pages" + "/page-" + page + "-morph.xml"; + File pageFile = new File(pageFileName); + String fragment = FileUtils.readFileToString(pageFile, "utf-8"); + WordContentHandler wordContentHandler = new WordContentHandler(); + XMLReader xmlParser = new SAXParser(); + xmlParser.setContentHandler(wordContentHandler); + StringReader strReader = new StringReader(fragment); + InputSource inputSource = new InputSource(strReader); + xmlParser.parse(inputSource); + String result = wordContentHandler.getResult(); + return result; + } catch (SAXException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + } }
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/util/StringUtils.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,491 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.util; - -import java.io.UnsupportedEncodingException; -import java.net.URLEncoder; -import java.text.CharacterIterator; -import java.text.StringCharacterIterator; -import java.util.regex.Matcher; -import java.util.regex.Pattern; - -public class StringUtils { - - /** - * Puts a zwsp between two ideographic characters (e.g. in CJK Unified Ideographs) - * @param str - * @return - */ - public static String zwsp(String str) { - // based on Unicode 3.2 - String ideographic = "[\u3300-\u33ff\u3400-\u4dbf\u4e00-\u9fff\uf900-\ufaff]"; - String regex = "(" + ideographic + ")(" + ideographic + ")"; - String retStr = str.replaceAll(regex, "$1\u200b$2"); - retStr = retStr.replaceAll(regex, "$1\u200b$2"); - return retStr; - } - - - public static String deleteSpecialXmlEntities(String inputStr) { - inputStr = inputStr.replaceAll("<", ""); - inputStr = inputStr.replaceAll(">", ""); - inputStr = inputStr.replaceAll("&lt;", ""); - inputStr = inputStr.replaceAll("&gt;", ""); - return inputStr; - } - - public static String resolveXmlEntities(String inputStr) { - inputStr = inputStr.replaceAll("&", "&"); - inputStr = inputStr.replaceAll("<", "<"); - inputStr = inputStr.replaceAll(">", ">"); - inputStr = inputStr.replaceAll(""", "\""); - inputStr = inputStr.replaceAll("'", "'"); - return inputStr; - } - - public static String deresolveXmlEntities(String inputStr) { - StringBuffer buf = new StringBuffer(); - for (int i = 0; i < inputStr.length(); i++) { - char c = inputStr.charAt(i); - String replace = new String(); - switch (c) { - case '&': replace = "&"; break; - case '<': replace = "<"; break; - case '>': replace = ">"; break; - case '"': replace = """; break; - // case '\'': replace = "'"; break; // causes problems in DictionarizerContentHandler - default: replace += c; break; - } - buf.append(replace); - } - return buf.toString(); - } - - /** - * Escape characters for text appearing in HTML markup. - * - * <P>This method exists as a defence against Cross Site Scripting (XSS) hacks. - * The idea is to neutralize control characters commonly used by scripts, such that - * they will not be executed by the browser. This is done by replacing the control - * characters with their escaped equivalents. - * See {@link hirondelle.web4j.security.SafeText} as well. - * - * <P>The following characters are replaced with corresponding - * HTML character entities : - * <table border='1' cellpadding='3' cellspacing='0'> - * <tr><th> Character </th><th>Replacement</th></tr> - * <tr><td> < </td><td> < </td></tr> - * <tr><td> > </td><td> > </td></tr> - * <tr><td> & </td><td> & </td></tr> - * <tr><td> " </td><td> "</td></tr> - * <tr><td> \t </td><td> 	</td></tr> - * <tr><td> ! </td><td> !</td></tr> - * <tr><td> # </td><td> #</td></tr> - * <tr><td> $ </td><td> $</td></tr> - * <tr><td> % </td><td> %</td></tr> - * <tr><td> ' </td><td> '</td></tr> - * <tr><td> ( </td><td> (</td></tr> - * <tr><td> ) </td><td> )</td></tr> - * <tr><td> * </td><td> *</td></tr> - * <tr><td> + </td><td> + </td></tr> - * <tr><td> , </td><td> , </td></tr> - * <tr><td> - </td><td> - </td></tr> - * <tr><td> . </td><td> . </td></tr> - * <tr><td> / </td><td> / </td></tr> - * <tr><td> : </td><td> :</td></tr> - * <tr><td> ; </td><td> ;</td></tr> - * <tr><td> = </td><td> =</td></tr> - * <tr><td> ? </td><td> ?</td></tr> - * <tr><td> @ </td><td> @</td></tr> - * <tr><td> [ </td><td> [</td></tr> - * <tr><td> \ </td><td> \</td></tr> - * <tr><td> ] </td><td> ]</td></tr> - * <tr><td> ^ </td><td> ^</td></tr> - * <tr><td> _ </td><td> _</td></tr> - * <tr><td> ` </td><td> `</td></tr> - * <tr><td> { </td><td> {</td></tr> - * <tr><td> | </td><td> |</td></tr> - * <tr><td> } </td><td> }</td></tr> - * <tr><td> ~ </td><td> ~</td></tr> - * </table> - * - * <P>Note that JSTL's {@code <c:out>} escapes <em>only the first - * five</em> of the above characters. - */ - public static String forHTML(String aText){ - final StringBuilder result = new StringBuilder(); - final StringCharacterIterator iterator = new StringCharacterIterator(aText); - char character = iterator.current(); - while (character != CharacterIterator.DONE ){ - if (character == '<') { - result.append("<"); - } - else if (character == '>') { - result.append(">"); - } - else if (character == '&') { - result.append("&"); - } - else if (character == '\"') { - result.append("""); - } - else if (character == '\t') { - addCharEntity(9, result); - } - else if (character == '!') { - addCharEntity(33, result); - } - else if (character == '#') { - addCharEntity(35, result); - } - else if (character == '$') { - addCharEntity(36, result); - } - else if (character == '%') { - addCharEntity(37, result); - } - else if (character == '\'') { - addCharEntity(39, result); - } - else if (character == '(') { - addCharEntity(40, result); - } - else if (character == ')') { - addCharEntity(41, result); - } - else if (character == '*') { - addCharEntity(42, result); - } - else if (character == '+') { - addCharEntity(43, result); - } - else if (character == ',') { - addCharEntity(44, result); - } - else if (character == '-') { - addCharEntity(45, result); - } - else if (character == '.') { - addCharEntity(46, result); - } - else if (character == '/') { - addCharEntity(47, result); - } - else if (character == ':') { - addCharEntity(58, result); - } - else if (character == ';') { - addCharEntity(59, result); - } - else if (character == '=') { - addCharEntity(61, result); - } - else if (character == '?') { - addCharEntity(63, result); - } - else if (character == '@') { - addCharEntity(64, result); - } - else if (character == '[') { - addCharEntity(91, result); - } - else if (character == '\\') { - addCharEntity(92, result); - } - else if (character == ']') { - addCharEntity(93, result); - } - else if (character == '^') { - addCharEntity(94, result); - } - else if (character == '_') { - addCharEntity(95, result); - } - else if (character == '`') { - addCharEntity(96, result); - } - else if (character == '{') { - addCharEntity(123, result); - } - else if (character == '|') { - addCharEntity(124, result); - } - else if (character == '}') { - addCharEntity(125, result); - } - else if (character == '~') { - addCharEntity(126, result); - } - else { - //the char is not a special one - //add it to the result as is - result.append(character); - } - character = iterator.next(); - } - return result.toString(); - } - - - /** - * Escape all ampersand characters in a URL. - * - * <P>Replaces all <tt>'&'</tt> characters with <tt>'&'</tt>. - * - *<P>An ampersand character may appear in the query string of a URL. - * The ampersand character is indeed valid in a URL. - * <em>However, URLs usually appear as an <tt>HREF</tt> attribute, and - * such attributes have the additional constraint that ampersands - * must be escaped.</em> - * - * <P>The JSTL <c:url> tag does indeed perform proper URL encoding of - * query parameters. But it does not, in general, produce text which - * is valid as an <tt>HREF</tt> attribute, simply because it does - * not escape the ampersand character. This is a nuisance when - * multiple query parameters appear in the URL, since it requires a little - * extra work. - */ - public static String forHrefAmpersand(String aURL){ - return aURL.replace("&", "&"); - } - - /** - * Synonym for <tt>URLEncoder.encode(String, "UTF-8")</tt>. - * - * <P>Used to ensure that HTTP query strings are in proper form, by escaping - * special characters such as spaces. - * - * <P>It is important to note that if a query string appears in an <tt>HREF</tt> - * attribute, then there are two issues - ensuring the query string is valid HTTP - * (it is URL-encoded), and ensuring it is valid HTML (ensuring the - * ampersand is escaped). - */ - public static String forURL(String aURLFragment){ - String result = null; - try { - result = URLEncoder.encode(aURLFragment, "UTF-8"); - } - catch (UnsupportedEncodingException ex){ - throw new RuntimeException("UTF-8 not supported", ex); - } - return result; - } - - /** - * Escape characters for text appearing as XML data, between tags. - * - * <P>The following characters are replaced with corresponding character entities : - * <table border='1' cellpadding='3' cellspacing='0'> - * <tr><th> Character </th><th> Encoding </th></tr> - * <tr><td> < </td><td> < </td></tr> - * <tr><td> > </td><td> > </td></tr> - * <tr><td> & </td><td> & </td></tr> - * <tr><td> " </td><td> "</td></tr> - * <tr><td> ' </td><td> '</td></tr> - * </table> - * - * <P>Note that JSTL's {@code <c:out>} escapes the exact same set of - * characters as this method. <span class='highlight'>That is, {@code <c:out>} - * is good for escaping to produce valid XML, but not for producing safe - * HTML.</span> - */ - public static String forXML(String aText){ - final StringBuilder result = new StringBuilder(); - final StringCharacterIterator iterator = new StringCharacterIterator(aText); - char character = iterator.current(); - while (character != CharacterIterator.DONE ){ - if (character == '<') { - result.append("<"); - } - else if (character == '>') { - result.append(">"); - } - else if (character == '\"') { - result.append("""); - } - else if (character == '\'') { - result.append("'"); - } - else if (character == '&') { - result.append("&"); - } - else { - //the char is not a special one - //add it to the result as is - result.append(character); - } - character = iterator.next(); - } - return result.toString(); - } - - /** - * Return <tt>aText</tt> with all <tt>'<'</tt> and <tt>'>'</tt> characters - * replaced by their escaped equivalents. - */ - public static String toDisableTags(String aText){ - final StringBuilder result = new StringBuilder(); - final StringCharacterIterator iterator = new StringCharacterIterator(aText); - char character = iterator.current(); - while (character != CharacterIterator.DONE ){ - if (character == '<') { - result.append("<"); - } - else if (character == '>') { - result.append(">"); - } - else { - //the char is not a special one - //add it to the result as is - result.append(character); - } - character = iterator.next(); - } - return result.toString(); - } - - - /** - * Replace characters having special meaning in regular expressions - * with their escaped equivalents, preceded by a '\' character. - * - * <P>The escaped characters include : - *<ul> - *<li>. - *<li>\ - *<li>?, * , and + - *<li>& - *<li>: - *<li>{ and } - *<li>[ and ] - *<li>( and ) - *<li>^ and $ - *</ul> - */ - public static String forRegex(String aRegexFragment){ - final StringBuilder result = new StringBuilder(); - - final StringCharacterIterator iterator = - new StringCharacterIterator(aRegexFragment) - ; - char character = iterator.current(); - while (character != CharacterIterator.DONE ){ - /* - * All literals need to have backslashes doubled. - */ - if (character == '.') { - result.append("\\."); - } - else if (character == '\\') { - result.append("\\\\"); - } - else if (character == '?') { - result.append("\\?"); - } - else if (character == '*') { - result.append("\\*"); - } - else if (character == '+') { - result.append("\\+"); - } - else if (character == '&') { - result.append("\\&"); - } - else if (character == ':') { - result.append("\\:"); - } - else if (character == '{') { - result.append("\\{"); - } - else if (character == '}') { - result.append("\\}"); - } - else if (character == '[') { - result.append("\\["); - } - else if (character == ']') { - result.append("\\]"); - } - else if (character == '(') { - result.append("\\("); - } - else if (character == ')') { - result.append("\\)"); - } - else if (character == '^') { - result.append("\\^"); - } - else if (character == '$') { - result.append("\\$"); - } - else { - //the char is not a special one - //add it to the result as is - result.append(character); - } - character = iterator.next(); - } - return result.toString(); - } - - /** - * Escape <tt>'$'</tt> and <tt>'\'</tt> characters in replacement strings. - * - * <P>Synonym for <tt>Matcher.quoteReplacement(String)</tt>. - * - * <P>The following methods use replacement strings which treat - * <tt>'$'</tt> and <tt>'\'</tt> as special characters: - * <ul> - * <li><tt>String.replaceAll(String, String)</tt> - * <li><tt>String.replaceFirst(String, String)</tt> - * <li><tt>Matcher.appendReplacement(StringBuffer, String)</tt> - * </ul> - * - * <P>If replacement text can contain arbitrary characters, then you - * will usually need to escape that text, to ensure special characters - * are interpreted literally. - */ - public static String forReplacementString(String aInput){ - return Matcher.quoteReplacement(aInput); - } - - /** - * Disable all <tt><SCRIPT></tt> tags in <tt>aText</tt>. - * - * <P>Insensitive to case. - */ - public static String forScriptTagsOnly(String aText){ - String result = null; - Matcher matcher = SCRIPT.matcher(aText); - result = matcher.replaceAll("<SCRIPT>"); - matcher = SCRIPT_END.matcher(result); - result = matcher.replaceAll("</SCRIPT>"); - return result; - } - - // PRIVATE // - - private StringUtils(){ - //empty - prevent construction - } - - private static final Pattern SCRIPT = Pattern.compile( - "<SCRIPT>", Pattern.CASE_INSENSITIVE - ); - private static final Pattern SCRIPT_END = Pattern.compile( - "</SCRIPT>", Pattern.CASE_INSENSITIVE - ); - - private static void addCharEntity(Integer aIdx, StringBuilder aBuilder){ - String padding = ""; - if( aIdx <= 9 ){ - padding = "00"; - } - else if( aIdx <= 99 ){ - padding = "0"; - } - else { - //no prefix - } - String number = padding + aIdx.toString(); - aBuilder.append("&#" + number + ";"); - } - }
--- a/software/mpdl-services/mpiwg-mpdl-lt/src/de/mpg/mpiwg/berlin/mpdl/util/Util.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,32 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.util; - -import java.io.File; -import java.io.FileInputStream; -import java.io.IOException; -import java.util.Date; -import java.util.Properties; - -public class Util { - - public Properties getProperties(String fullFileName) { - Properties props = new Properties(); - try { - File file = new File(fullFileName); - FileInputStream in = new FileInputStream(file); - props.load(in); - } catch (IOException e) { - e.printStackTrace(); - } - return props; - } - - public Double getSecondWithMillisecondsBetween(Date begin, Date end) { - long beginMS = begin.getTime(); - long endMS = end.getTime(); - long elapsedSeconds = (endMS - beginMS) / 1000; - long elapsedMilliSecondsAfterSeconds1 = (endMS - beginMS) - (elapsedSeconds * 1000); - Double seconds = new Double(elapsedSeconds + "." + elapsedMilliSecondsAfterSeconds1); - return seconds; - } - -}
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/.classpath Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,18 +0,0 @@ -<?xml version="1.0" encoding="UTF-8"?> -<classpath> - <classpathentry kind="src" path="src"/> - <classpathentry kind="con" path="org.eclipse.jdt.launching.JRE_CONTAINER/org.eclipse.jdt.internal.launching.macosx.MacOSXType/1.6.0 (MacOS X Default)"> - <attributes> - <attribute name="owner.project.facets" value="java"/> - </attributes> - </classpathentry> - <classpathentry kind="con" path="org.eclipse.jst.server.core.container/org.eclipse.jst.server.tomcat.runtimeTarget/Apache Tomcat v6.0"> - <attributes> - <attribute name="owner.project.facets" value="jst.web"/> - </attributes> - </classpathentry> - <classpathentry kind="con" path="org.eclipse.jst.j2ee.internal.web.container"/> - <classpathentry kind="con" path="org.eclipse.jst.j2ee.internal.module.container"/> - <classpathentry combineaccessrules="false" kind="src" path="/mpiwg-mpdl-xml"/> - <classpathentry kind="output" path="build/classes"/> -</classpath>
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/.externalToolBuilders/New_Builder.launch Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,12 +0,0 @@ -<?xml version="1.0" encoding="UTF-8" standalone="no"?> -<launchConfiguration type="org.eclipse.ant.AntBuilderLaunchConfigurationType"> -<booleanAttribute key="org.eclipse.ant.ui.ATTR_TARGETS_UPDATED" value="true"/> -<booleanAttribute key="org.eclipse.ant.ui.DEFAULT_VM_INSTALL" value="false"/> -<booleanAttribute key="org.eclipse.debug.ui.ATTR_LAUNCH_IN_BACKGROUND" value="false"/> -<stringAttribute key="org.eclipse.jdt.launching.CLASSPATH_PROVIDER" value="org.eclipse.ant.ui.AntClasspathProvider"/> -<booleanAttribute key="org.eclipse.jdt.launching.DEFAULT_CLASSPATH" value="true"/> -<stringAttribute key="org.eclipse.jdt.launching.PROJECT_ATTR" value="mpiwg-mpdl-xml-web"/> -<stringAttribute key="org.eclipse.ui.externaltools.ATTR_LOCATION" value="${workspace_loc:/mpiwg-mpdl-xml-web/build/build.xml}"/> -<stringAttribute key="org.eclipse.ui.externaltools.ATTR_RUN_BUILD_KINDS" value="full,incremental,"/> -<booleanAttribute key="org.eclipse.ui.externaltools.ATTR_TRIGGERS_CONFIGURED" value="true"/> -</launchConfiguration>
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/.project Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,46 +0,0 @@ -<?xml version="1.0" encoding="UTF-8"?> -<projectDescription> - <name>mpiwg-mpdl-xml-web</name> - <comment></comment> - <projects> - </projects> - <buildSpec> - <buildCommand> - <name>org.eclipse.wst.jsdt.core.javascriptValidator</name> - <arguments> - </arguments> - </buildCommand> - <buildCommand> - <name>org.eclipse.jdt.core.javabuilder</name> - <arguments> - </arguments> - </buildCommand> - <buildCommand> - <name>org.eclipse.wst.common.project.facet.core.builder</name> - <arguments> - </arguments> - </buildCommand> - <buildCommand> - <name>org.eclipse.wst.validation.validationbuilder</name> - <arguments> - </arguments> - </buildCommand> - <buildCommand> - <name>org.eclipse.ui.externaltools.ExternalToolBuilder</name> - <triggers>full,incremental,</triggers> - <arguments> - <dictionary> - <key>LaunchConfigHandle</key> - <value><project>/.externalToolBuilders/New_Builder.launch</value> - </dictionary> - </arguments> - </buildCommand> - </buildSpec> - <natures> - <nature>org.eclipse.jem.workbench.JavaEMFNature</nature> - <nature>org.eclipse.wst.common.modulecore.ModuleCoreNature</nature> - <nature>org.eclipse.wst.common.project.facet.core.nature</nature> - <nature>org.eclipse.jdt.core.javanature</nature> - <nature>org.eclipse.wst.jsdt.core.jsNature</nature> - </natures> -</projectDescription>
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/.settings/.jsdtscope Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,12 +0,0 @@ -<?xml version="1.0" encoding="UTF-8"?> -<classpath> - <classpathentry kind="src" path="WebContent"/> - <classpathentry kind="con" path="org.eclipse.wst.jsdt.launching.JRE_CONTAINER"/> - <classpathentry kind="con" path="org.eclipse.wst.jsdt.launching.WebProject"> - <attributes> - <attribute name="hide" value="true"/> - </attributes> - </classpathentry> - <classpathentry kind="con" path="org.eclipse.wst.jsdt.launching.baseBrowserLibrary"/> - <classpathentry kind="output" path=""/> -</classpath>
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/.settings/org.eclipse.jdt.core.prefs Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,12 +0,0 @@ -#Mon Sep 12 15:41:45 CEST 2011 -eclipse.preferences.version=1 -org.eclipse.jdt.core.compiler.codegen.inlineJsrBytecode=enabled -org.eclipse.jdt.core.compiler.codegen.targetPlatform=1.6 -org.eclipse.jdt.core.compiler.codegen.unusedLocal=preserve -org.eclipse.jdt.core.compiler.compliance=1.6 -org.eclipse.jdt.core.compiler.debug.lineNumber=generate -org.eclipse.jdt.core.compiler.debug.localVariable=generate -org.eclipse.jdt.core.compiler.debug.sourceFile=generate -org.eclipse.jdt.core.compiler.problem.assertIdentifier=error -org.eclipse.jdt.core.compiler.problem.enumIdentifier=error -org.eclipse.jdt.core.compiler.source=1.6
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/.settings/org.eclipse.wst.common.component Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,10 +0,0 @@ -<?xml version="1.0" encoding="UTF-8"?> -<project-modules id="moduleCoreId" project-version="1.5.0"> - <wb-module deploy-name="mpiwg-mpdl-xml-web"> - <wb-resource deploy-path="/" source-path="/WebContent"/> - <wb-resource deploy-path="/WEB-INF/classes" source-path="/src"/> - <wb-resource deploy-path="/WEB-INF/lib" source-path="/WebContent/WEB-INF/lib"/> - <property name="java-output-path" value="/mpiwg-mpdl-xml-web/build/classes"/> - <property name="context-root" value="mpiwg-mpdl-xml-web"/> - </wb-module> -</project-modules>
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/.settings/org.eclipse.wst.common.project.facet.core.xml Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,10 +0,0 @@ -<?xml version="1.0" encoding="UTF-8"?> -<faceted-project> - <runtime name="Apache Tomcat v6.0"/> - <fixed facet="jst.web"/> - <fixed facet="wst.jsdt.web"/> - <fixed facet="java"/> - <installed facet="java" version="1.6"/> - <installed facet="jst.web" version="2.5"/> - <installed facet="wst.jsdt.web" version="1.0"/> -</faceted-project>
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/.settings/org.eclipse.wst.jsdt.ui.superType.container Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1 +0,0 @@ -org.eclipse.wst.jsdt.launching.baseBrowserLibrary \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/.settings/org.eclipse.wst.jsdt.ui.superType.name Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1 +0,0 @@ -Window \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/.settings/org.eclipse.wst.ws.service.policy.prefs Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,3 +0,0 @@ -#Mon Sep 12 15:41:45 CEST 2011 -eclipse.preferences.version=1 -org.eclipse.wst.ws.service.policy.projectEnabled=false
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/WebContent/META-INF/MANIFEST.MF Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,3 +0,0 @@ -Manifest-Version: 1.0 -Class-Path: -
Binary file software/mpdl-services/mpiwg-mpdl-xml-web/WebContent/WEB-INF/classes/.DS_Store has changed
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/WebContent/WEB-INF/classes/constants.properties Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1 +0,0 @@ -docDir=/usr/local/tomcat-mpdl/mpdl-data/xml/documents \ No newline at end of file
Binary file software/mpdl-services/mpiwg-mpdl-xml-web/WebContent/WEB-INF/lib/commons-io-2.0.1.jar has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml-web/WebContent/WEB-INF/lib/mpiwg-mpdl-xml-web.jar has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml-web/WebContent/WEB-INF/lib/mpiwg-mpdl-xml.jar has changed
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/WebContent/WEB-INF/lib/saxon.txt Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,3 +0,0 @@ -Saxon: - -Release 9.1.0.5 (free version): releases < 9.1.0.7 support saxon extension functions
Binary file software/mpdl-services/mpiwg-mpdl-xml-web/WebContent/WEB-INF/lib/saxon9-s9api.jar has changed
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/WebContent/WEB-INF/web.xml Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,40 +0,0 @@ -<?xml version="1.0" encoding="UTF-8"?> -<web-app xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://java.sun.com/xml/ns/javaee" xmlns:web="http://java.sun.com/xml/ns/javaee/web-app_2_5.xsd" xsi:schemaLocation="http://java.sun.com/xml/ns/javaee http://java.sun.com/xml/ns/javaee/web-app_2_5.xsd" id="WebApp_ID" version="2.5"> - <display-name>mpiwg-mpdl-xml-web</display-name> - <welcome-file-list> - <welcome-file>index.html</welcome-file> - </welcome-file-list> - <servlet> - <description>Transform</description> - <display-name>Transform</display-name> - <servlet-name>Transform</servlet-name> - <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.xml.Transform</servlet-class> - </servlet> - <servlet-mapping> - <servlet-name>Transform</servlet-name> - <url-pattern>/transform/Transform</url-pattern> - </servlet-mapping> - <servlet> - <description>GetFragment</description> - <display-name>GetFragment</display-name> - <servlet-name>GetFragment</servlet-name> - <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.xml.GetFragment</servlet-class> - </servlet> - <servlet-mapping> - <servlet-name>GetFragment</servlet-name> - <url-pattern>/transform/GetFragment</url-pattern> - </servlet-mapping> - <servlet> - <description>XQuery</description> - <display-name>XQuery</display-name> - <servlet-name>XQuery</servlet-name> - <servlet-class>de.mpg.mpiwg.berlin.mpdl.servlets.xml.XQuery</servlet-class> - </servlet> - <servlet-mapping> - <servlet-name>XQuery</servlet-name> - <url-pattern>/xquery/XQuery</url-pattern> - </servlet-mapping> - <listener> - <listener-class>de.mpg.mpiwg.berlin.mpdl.servlets.xml.MpiwgMpdlXmlWebServletContextListener</listener-class> - </listener> -</web-app> \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/WebContent/index.html Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,130 +0,0 @@ -<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0//EN"> -<html><head> -<meta http-equiv="Content-Type" content="text/html; charset=utf-8"> -<title>Max Planck Institute for the History of Science - Mpdl: XML Services</title> -</head> -<body> -<h2>Max Planck Institute for the History of Science - Mpdl: XML Services</h2> -<p>Available Services</p> - -<ul> - <li><b>Url: /mpiwg-mpdl-xml-web/transform/Transform</b> - <ul> - <li>Request parameters - <ul> - <li>srcUrl (required) - <ul> - <li>url of the Xml source document</li> - </ul> - </li> - <li>xslUrl (required) - <ul> - <li>url of the Xsl document which does the transformation of the Xml document</li> - </ul> - </li> - <li>parameters (optional) - <ul> - <li>parameters separated with blanks (e.g. "yourParam1=yourValue1 yourParam2=yourValue2")</li> - <li>default: no parameters</li> - </ul> - </li> - <li>outputProperties (optional) - <ul> - <li>output properties separated with blanks (e.g. "encoding=utf-8 indent=yes") - <ul> - <li>"method=xhtml"</li> - <li>"indent=yes"</li> - <li>"media-type=text/html"</li> - <li>"encoding=utf-8"</li> - <li>default: "method=xml indent=yes media-type=text/xml encoding=utf-8"</li> - </ul> - </li> - </ul> - </li> - </ul> - </li> - <li>Response output - <ul> - <li>transformed Xml document</li> - <li>Example: <a href="transform/Transform?srcUrl=http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/getDoc?doc=/tei/en/Test_1789.xml&xslUrl=http://mpdl-test.mpiwg-berlin.mpg.de:8080/mpiwg-mpdl-xml-web/xsl/generateId.xsl">Generate ids incrementally for sentences in example document</a></li> - </ul> - </li> - </ul> - </li> - - <li><b>Url: /mpiwg-mpdl-xml-web/transform/GetFragment</b> - <ul> - <li>Request parameters - <ul> - <li>docId (required) - <ul> - <li>document identifier of the Xml source document (e.g. "/tei/la/Test_1789.xml")</li> - </ul> - </li> - <li>ms1Name (required) - <ul> - <li>starting milestone element name (e.g. "pb")</li> - </ul> - </li> - <li>ms1Pos (required) - <ul> - <li>starting milestone position (e.g. "13")</li> - </ul> - </li> - <li>ms2Name (required) - <ul> - <li>ending milestone element name (e.g. "pb")</li> - </ul> - </li> - <li>ms2Pos (required) - <ul> - <li>ending milestone position (e.g. "14")</li> - </ul> - </li> - </ul> - </li> - <li>Response output - <ul> - <li>fragment between the two milestones in the Xml document</li> - <li>Example: <a href="transform/GetFragment?docId=/tei/en/Test_1789.xml&ms1Name=pb&ms1Pos=2&ms2Name=pb&ms2Pos=3">get the fragment between the second and the third page break</a></li> - </ul> - </li> - </ul> - </li> - - <li><b>Url: /mpiwg-mpdl-xml-web/xquery/XQuery</b> - <ul> - <li>Request parameters - <ul> - <li>inputString or srcUrl (required) - <ul> - <li>inputString - <ul> - <li>XML string</li> - </ul> - </li> - <li>srcUrl - <ul> - <li>source URL of XML document</li> - </ul> - </li> - </ul> - </li> - <li>xquery (required) - <ul> - <li>XQuery (or XPath) source code which should be executed</li> - </ul> - </li> - </ul> - </li> - <li>Response output - <ul> - <li>XML result of the XQuery</li> - <li>Example: <a href="xquery/XQuery?srcUrl=http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/getDoc?doc=/archimedes/la/achil_propo_087_la_1545.xml&xquery=//s">XPath: all sentences of example document</a></li> - </ul> - </li> - </ul> - </li> -</ul> - -</body></html> \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/WebContent/xsl/generateId.xsl Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,34 +0,0 @@ -<?xml version="1.0"?> - -<xsl:stylesheet version="2.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform"> - -<xsl:output method="xml" encoding="utf-8"/> - -<xsl:template match="@*|node()"> - <xsl:copy> - <xsl:apply-templates select="@*|node()"/> - </xsl:copy> -</xsl:template> - -<xsl:template match="*:s|*:head"> - <xsl:variable name="elemName" select="name()"/> - <xsl:variable name="docPos"> - <xsl:choose> - <xsl:when test="$elemName = 's'"> - <xsl:value-of select="count(./preceding::*:s) + 1"/> - </xsl:when> - <xsl:when test="$elemName = 'head'"> - <xsl:value-of select="count(./preceding::*:head) + 1"/> - </xsl:when> - <xsl:otherwise></xsl:otherwise> - </xsl:choose> - </xsl:variable> - <xsl:copy> - <xsl:attribute name="xml:id"> - <xsl:value-of select="concat($elemName, $docPos)"/> - </xsl:attribute> - <xsl:apply-templates select="@*|node()"/> - </xsl:copy> -</xsl:template> - -</xsl:stylesheet>
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/build/build.xml Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,57 +0,0 @@ -<!DOCTYPE project> -<project name="mpiwg-mpdl-xml-web" default="dist" basedir="../"> - <description>mpiwg-mpdl-xml-web</description> - <!-- global properties --> - <property name="baseLibFile" location="../mpiwg-mpdl-xml/dist/mpiwg-mpdl-xml.jar"/> - <property name="src" location="src"/> - <property name="lib" location="WebContent/WEB-INF/lib"/> - <property name="libTomcat" location="/Applications/java/apache-tomcat-6.0.18/lib"/> - <property name="webappTomcat" location="/Applications/java/apache-tomcat-6.0.18/webapps"/> - <property name="build" location="build/classes"/> - <property name="dist" location="dist"/> - - <path id="classpath"> - <fileset dir="${lib}" includes="**/*.jar"/> - <fileset dir="${libTomcat}" includes="**/*.jar"/> - </path> - - <target name="init"> - <!-- Create time stamp --> - <tstamp/> - <mkdir dir="${build}"/> - <mkdir dir="${dist}"/> - <copy file="${baseLibFile}" todir="${lib}"/> - </target> - - <target name="compile" depends="init" description="compile"> - <javac srcdir="${src}" destdir="${build}" classpathref="classpath" includeantruntime="false"/> - </target> - - <target name="dist" depends="compile" description="generate the distribution"> - <delete file="WebContent/WEB-INF/classes/constants.properties"/> - <copy file="conf/constants.properties" tofile="WebContent/WEB-INF/classes/constants.properties"/> - <jar jarfile="${dist}/mpiwg-mpdl-xml-web.jar" basedir="${build}"/> - <copy file="dist/mpiwg-mpdl-xml-web.jar" todir="${lib}"/> - <war destfile="dist/mpiwg-mpdl-xml-web.war" webxml="WebContent/WEB-INF/web.xml"> - <fileset dir="WebContent"/> - <lib dir="WebContent/WEB-INF/lib"/> - </war> - <copy file="dist/mpiwg-mpdl-xml-web.war" todir="${webappTomcat}"/> - </target> - - <target name="dist-remote-mpdl-system" depends="compile" description="generate the distribution"> - <delete file="WebContent/WEB-INF/classes/constants.properties"/> - <copy file="conf/constants-mpdl-system.properties" tofile="WebContent/WEB-INF/classes/constants.properties"/> - <jar jarfile="dist-remote/mpiwg-mpdl-xml-web.jar" basedir="${build}"/> - <copy file="dist-remote/mpiwg-mpdl-xml-web.jar" todir="${lib}"/> - <war destfile="dist-remote/mpiwg-mpdl-xml-web.war" webxml="WebContent/WEB-INF/web.xml"> - <fileset dir="WebContent"/> - <lib dir="WebContent/WEB-INF/lib"/> - </war> - </target> - - <target name="clean" description="clean" > - <delete dir="${build}"/> - <delete file="${dist}/mpiwg-mpdl-xml-web.jar"/> - </target> -</project> \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/conf/constants-mpdl-system.properties Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1 +0,0 @@ -docDir=/usr/local/tomcat-mpdl/mpdl-data/xml/documents \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/conf/constants.properties Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1 +0,0 @@ -docDir=/Users/jwillenborg/mpdl/data/xml/documents \ No newline at end of file
Binary file software/mpdl-services/mpiwg-mpdl-xml-web/dist-remote/mpiwg-mpdl-xml-web.jar has changed
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Binary file software/mpdl-services/mpiwg-mpdl-xml-web/src/de/mpg/mpiwg/berlin/mpdl/.DS_Store has changed
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/src/de/mpg/mpiwg/berlin/mpdl/exception/ApplicationException.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,15 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.exception; - -public class ApplicationException extends Exception { - private static final long serialVersionUID = 1L; - - public ApplicationException(Exception e) { - super(e); - } - - public ApplicationException(String str) { - super(str); - } - -} -
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/xml/GetFragment.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,55 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.servlets.xml; - -import java.io.IOException; -import java.io.PrintWriter; - -import javax.servlet.ServletConfig; -import javax.servlet.ServletContext; -import javax.servlet.ServletException; -import javax.servlet.http.HttpServlet; -import javax.servlet.http.HttpServletRequest; -import javax.servlet.http.HttpServletResponse; - -import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; -import de.mpg.mpiwg.berlin.mpdl.xml.transform.FragmentTransformer; - -public class GetFragment extends HttpServlet { - private static final long serialVersionUID = 1L; - private FragmentTransformer fragmentTransformer; - private String documentDirectory; - public GetFragment() { - super(); - } - - public void init(ServletConfig config) throws ServletException { - super.init(config); - ServletContext context = getServletContext(); - fragmentTransformer = (FragmentTransformer) context.getAttribute("fragmentTransformer"); - documentDirectory = (String) context.getAttribute("documentDirectory"); - } - - protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { - request.setCharacterEncoding("utf-8"); - response.setCharacterEncoding("utf-8"); - String docId = request.getParameter("docId"); - String ms1Name = request.getParameter("ms1Name"); - int ms1Pos = new Integer(request.getParameter("ms1Pos")); - String ms2Name = request.getParameter("ms2Name"); - int ms2Pos = new Integer(request.getParameter("ms2Pos")); - try { - String xmlFileName = documentDirectory + docId; - String result = fragmentTransformer.getFragment(xmlFileName, ms1Name, ms1Pos, ms2Name, ms2Pos); - response.setContentType("text/xml"); - PrintWriter out = response.getWriter(); - out.print(result); - out.close(); - } catch (ApplicationException e) { - throw new ServletException(e); - } - } - - protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { - // TODO Auto-generated method stub - } - -}
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/xml/MpiwgMpdlXmlWebServletContextListener.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,33 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.servlets.xml; - -import javax.servlet.ServletContext; -import javax.servlet.ServletContextEvent; -import javax.servlet.ServletContextListener; - -import de.mpg.mpiwg.berlin.mpdl.xml.general.Constants; -import de.mpg.mpiwg.berlin.mpdl.xml.transform.FragmentTransformer; - -public class MpiwgMpdlXmlWebServletContextListener implements ServletContextListener { - private ServletContext context = null; - private FragmentTransformer fragmentTransformer = null; - - public void contextInitialized(ServletContextEvent event) { - try { - this.context = event.getServletContext(); - fragmentTransformer = new FragmentTransformer(); - context.setAttribute("fragmentTransformer", fragmentTransformer); - String docDirectory = Constants.getInstance().getDocumentDir(); - context.setAttribute("documentDirectory", docDirectory); - System.out.println(MpiwgMpdlXmlWebServletContextListener.class.getName() + ": contextInitialized (document directory= \"" + docDirectory + "\", set in constants.properties)"); - // String documentDirectory = System.getProperty("catalina.base") + "/webapps/mpiwg-mpdl-xml-web/documents"; - } catch (Exception e) { - e.printStackTrace(); - } - } - - public void contextDestroyed(ServletContextEvent e) { - this.context = null; - this.fragmentTransformer = null; - System.out.println(MpiwgMpdlXmlWebServletContextListener.class.getName() + ": contextDestroyed"); - } -} \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/xml/Transform.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,48 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.servlets.xml; - -import java.io.IOException; -import java.io.PrintWriter; - -import javax.servlet.ServletConfig; -import javax.servlet.ServletException; -import javax.servlet.http.HttpServlet; -import javax.servlet.http.HttpServletRequest; -import javax.servlet.http.HttpServletResponse; - -import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; -import de.mpg.mpiwg.berlin.mpdl.xml.transform.BasicTransformer; - -public class Transform extends HttpServlet { - private static final long serialVersionUID = 1L; - public Transform() { - super(); - } - - public void init(ServletConfig config) throws ServletException { - super.init(config); - } - - protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { - request.setCharacterEncoding("utf-8"); - response.setCharacterEncoding("utf-8"); - String srcUrl = request.getParameter("srcUrl"); - String xslUrl = request.getParameter("xslUrl"); - String parameters = request.getParameter("parameters"); - String outputProperties = request.getParameter("outputProperties"); - try { - BasicTransformer basicTransformer = new BasicTransformer(); - String result = basicTransformer.transform(srcUrl, xslUrl, parameters, outputProperties); - response.setContentType("text/xml"); - PrintWriter out = response.getWriter(); - out.print(result); - out.close(); - } catch (ApplicationException e) { - throw new ServletException(e); - } - } - - protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { - // TODO Auto-generated method stub - } - -}
--- a/software/mpdl-services/mpiwg-mpdl-xml-web/src/de/mpg/mpiwg/berlin/mpdl/servlets/xml/XQuery.java Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,52 +0,0 @@ -package de.mpg.mpiwg.berlin.mpdl.servlets.xml; - -import java.io.IOException; -import java.io.PrintWriter; -import java.net.URL; - -import javax.servlet.ServletConfig; -import javax.servlet.ServletException; -import javax.servlet.http.HttpServlet; -import javax.servlet.http.HttpServletRequest; -import javax.servlet.http.HttpServletResponse; - -import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; -import de.mpg.mpiwg.berlin.mpdl.xml.xquery.XQueryEvaluator; - -public class XQuery extends HttpServlet { - private static final long serialVersionUID = 1L; - private XQueryEvaluator xqueryEvaluator; - - public XQuery() { - super(); - } - - public void init(ServletConfig config) throws ServletException { - super.init(config); - xqueryEvaluator = new XQueryEvaluator(); - } - - protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException { - String result = null; - request.setCharacterEncoding("utf-8"); - response.setCharacterEncoding("utf-8"); - String inputString = request.getParameter("inputString"); - String srcUrlStr = request.getParameter("srcUrl"); - String xqueryStr = request.getParameter("xquery"); - try { - if (inputString != null) { - result = xqueryEvaluator.evaluateAsString(inputString, xqueryStr); - } else if (srcUrlStr != null) { - URL srcUrl = new URL(srcUrlStr); - result = xqueryEvaluator.evaluateAsString(srcUrl, xqueryStr); - } - response.setContentType("text/xml"); - PrintWriter out = response.getWriter(); - out.print("<result>" + result + "</result>"); - out.close(); - } catch (ApplicationException e) { - throw new ServletException(e); - } - } - -}
--- a/software/mpdl-services/mpiwg-mpdl-xml/.classpath Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-xml/.classpath Tue Nov 27 12:35:19 2012 +0100 @@ -2,8 +2,8 @@ <classpath> <classpathentry kind="src" path="src"/> <classpathentry kind="con" path="org.eclipse.jdt.launching.JRE_CONTAINER"/> + <classpathentry exported="true" kind="lib" path="lib/commons-io-2.0.1.jar"/> <classpathentry kind="lib" path="lib/saxon9-s9api.jar"/> - <classpathentry kind="lib" path="lib/commons-io-2.0.1.jar"/> <classpathentry kind="lib" path="lib/saxon9.jar"/> <classpathentry kind="output" path="bin"/> </classpath>
--- a/software/mpdl-services/mpiwg-mpdl-xml/.externalToolBuilders/New_Builder (1).launch Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,12 +0,0 @@ -<?xml version="1.0" encoding="UTF-8" standalone="no"?> -<launchConfiguration type="org.eclipse.ant.AntBuilderLaunchConfigurationType"> -<booleanAttribute key="org.eclipse.ant.ui.ATTR_TARGETS_UPDATED" value="true"/> -<booleanAttribute key="org.eclipse.ant.ui.DEFAULT_VM_INSTALL" value="false"/> -<booleanAttribute key="org.eclipse.debug.ui.ATTR_LAUNCH_IN_BACKGROUND" value="false"/> -<stringAttribute key="org.eclipse.jdt.launching.CLASSPATH_PROVIDER" value="org.eclipse.ant.ui.AntClasspathProvider"/> -<booleanAttribute key="org.eclipse.jdt.launching.DEFAULT_CLASSPATH" value="true"/> -<stringAttribute key="org.eclipse.jdt.launching.PROJECT_ATTR" value="mpiwg-mpdl-xml"/> -<stringAttribute key="org.eclipse.ui.externaltools.ATTR_LOCATION" value="${workspace_loc:/mpiwg-mpdl-xml/build/build.xml}"/> -<stringAttribute key="org.eclipse.ui.externaltools.ATTR_RUN_BUILD_KINDS" value="full,incremental,"/> -<booleanAttribute key="org.eclipse.ui.externaltools.ATTR_TRIGGERS_CONFIGURED" value="true"/> -</launchConfiguration>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/.externalToolBuilders/mpiwg-mpdl-xml-build.launch Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,13 @@ +<?xml version="1.0" encoding="UTF-8" standalone="no"?> +<launchConfiguration type="org.eclipse.ant.AntBuilderLaunchConfigurationType"> +<booleanAttribute key="org.eclipse.ant.ui.ATTR_TARGETS_UPDATED" value="true"/> +<booleanAttribute key="org.eclipse.ant.ui.DEFAULT_VM_INSTALL" value="false"/> +<booleanAttribute key="org.eclipse.debug.ui.ATTR_LAUNCH_IN_BACKGROUND" value="false"/> +<stringAttribute key="org.eclipse.jdt.launching.CLASSPATH_PROVIDER" value="org.eclipse.ant.ui.AntClasspathProvider"/> +<booleanAttribute key="org.eclipse.jdt.launching.DEFAULT_CLASSPATH" value="true"/> +<stringAttribute key="org.eclipse.jdt.launching.PROJECT_ATTR" value="mpiwg-mpdl-xml"/> +<booleanAttribute key="org.eclipse.ui.externaltools.ATTR_BUILDER_ENABLED" value="false"/> +<stringAttribute key="org.eclipse.ui.externaltools.ATTR_LOCATION" value="${workspace_loc:/mpiwg-mpdl-xml/build/build.xml}"/> +<stringAttribute key="org.eclipse.ui.externaltools.ATTR_RUN_BUILD_KINDS" value="full,incremental,"/> +<booleanAttribute key="org.eclipse.ui.externaltools.ATTR_TRIGGERS_CONFIGURED" value="true"/> +</launchConfiguration>
--- a/software/mpdl-services/mpiwg-mpdl-xml/.project Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-xml/.project Tue Nov 27 12:35:19 2012 +0100 @@ -16,7 +16,7 @@ <arguments> <dictionary> <key>LaunchConfigHandle</key> - <value><project>/.externalToolBuilders/New_Builder (1).launch</value> + <value><project>/.externalToolBuilders/mpiwg-mpdl-xml-build.launch</value> </dictionary> </arguments> </buildCommand>
--- a/software/mpdl-services/mpiwg-mpdl-xml/.settings/org.eclipse.core.resources.prefs Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-xml/.settings/org.eclipse.core.resources.prefs Tue Nov 27 12:35:19 2012 +0100 @@ -1,3 +1,3 @@ -#Mon Nov 07 15:08:16 CET 2011 +#Tue Apr 03 10:39:13 CEST 2012 eclipse.preferences.version=1 encoding//src/de/mpg/mpiwg/berlin/mpdl/xml/test/TestLocal.java=UTF-8
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/bin/constants.properties Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,1 @@ +docDir=/Users/jwillenborg/mpdl/data/xml/documents \ No newline at end of file
Binary file software/mpdl-services/mpiwg-mpdl-xml/bin/de/mpg/mpiwg/berlin/mpdl/.DS_Store has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml/bin/de/mpg/mpiwg/berlin/mpdl/exception/ApplicationException.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml/bin/de/mpg/mpiwg/berlin/mpdl/util/StringUtils.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml/bin/de/mpg/mpiwg/berlin/mpdl/util/Util.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml/bin/de/mpg/mpiwg/berlin/mpdl/xml/.DS_Store has changed
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Binary file software/mpdl-services/mpiwg-mpdl-xml/bin/de/mpg/mpiwg/berlin/mpdl/xml/test/TestLocal.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml/bin/de/mpg/mpiwg/berlin/mpdl/xml/tmp/GetFragmentContentHandler.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml/bin/de/mpg/mpiwg/berlin/mpdl/xml/tmp/XmlExamples.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml/bin/de/mpg/mpiwg/berlin/mpdl/xml/transform/BasicTransformer.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml/bin/de/mpg/mpiwg/berlin/mpdl/xml/transform/FragmentTransformer.class has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/bin/de/mpg/mpiwg/berlin/mpdl/xml/transform/getFragment.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,59 @@ +<?xml version="1.0"?> +<!-- Delivers the fragment between two milestones. Takes no care about namespaces. --> + +<xsl:stylesheet version="2.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform" xmlns:fn="local-function"> + +<xsl:function name="fn:contains"> + <xsl:param name="sequence" as="node()*"/> + <xsl:param name="node" as="node()?"/> + <xsl:sequence select="some $nodeInSequence in $sequence satisfies $nodeInSequence is $node"/> +</xsl:function> + +<xsl:output method="xml" encoding="utf-8"/> + +<!-- ms1Name and ms2Name have to be given without namespace: e.g. "pb" --> +<xsl:param name="ms1Name"></xsl:param> +<xsl:param name="ms1Position"></xsl:param> +<xsl:param name="ms2Name"></xsl:param> +<xsl:param name="ms2Position"></xsl:param> + +<xsl:variable name="ms1XPath" select="concat('subsequence(//*:', $ms1Name, ', ', $ms1Position, ', 1)')"/> +<xsl:variable name="ms2XPath" select="concat('subsequence(//*:', $ms2Name, ', ', $ms2Position, ', 1)')"/> +<xsl:variable name="ms1" select="saxon:evaluate($ms1XPath)" xmlns:saxon="http://saxon.sf.net/"/> +<xsl:variable name="ms2" select="saxon:evaluate($ms2XPath)" xmlns:saxon="http://saxon.sf.net/"/> +<xsl:variable name="ms1Ancestors" select="$ms1/ancestor::*"/> +<xsl:variable name="ms2Ancestors" select="$ms2/ancestor::*"/> + +<!-- all elements between the milestone elements and also the ancestor elements (for filling the node path to the root) --> +<xsl:template match="element()[local-name() != $ms1Name and local-name() != $ms2Name]"> + <xsl:choose> + <xsl:when test="(. >> $ms1 or fn:contains($ms1Ancestors, .)) and ($ms2 >> . or fn:contains($ms2Ancestors, .))"> + <xsl:element name="{local-name(.)}"><xsl:copy-of select="@*"></xsl:copy-of><xsl:apply-templates/></xsl:element> + </xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- the two milestone elements --> +<xsl:template match="element()[local-name() = $ms1Name or local-name() = $ms2Name]"> + <xsl:choose> + <xsl:when test=". is $ms1"> + <xsl:element name="{local-name(.)}"><xsl:copy-of select="@*"></xsl:copy-of></xsl:element> + </xsl:when> + <xsl:when test=". is $ms2"> + <xsl:element name="{local-name(.)}"><xsl:copy-of select="@*"></xsl:copy-of></xsl:element> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="attribute()|text()|comment()|processing-instruction()"> + <xsl:choose> + <xsl:when test=". >> $ms1 and $ms2 >> ."> + <xsl:copy><xsl:apply-templates/></xsl:copy> + </xsl:when> + <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> + </xsl:choose> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/bin/de/mpg/mpiwg/berlin/mpdl/xml/transform/getFragmentSaxonFast.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,70 @@ +<?xml version="1.0"?> +<!-- Delivers the fragment between two milestones. Takes no care about namespaces. --> + +<xsl:stylesheet version="2.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform" xmlns:fn="local-function"> + +<xsl:function name="fn:contains"> + <xsl:param name="sequence" as="node()*"/> + <xsl:param name="node" as="node()?"/> + <xsl:sequence select="some $nodeInSequence in $sequence satisfies $nodeInSequence is $node"/> +</xsl:function> + +<xsl:output method="xml" encoding="utf-8"/> + +<!-- ms1Name and ms2Name have to be given without namespace: e.g. "pb" --> +<xsl:param name="xmlFileName"></xsl:param> +<xsl:param name="ms1Name"></xsl:param> +<xsl:param name="ms1Position"></xsl:param> +<xsl:param name="ms2Name"></xsl:param> +<xsl:param name="ms2Position"></xsl:param> + +<!-- the doc method is ca. 25% faster than over the source document --> +<xsl:variable name="xmlDocExpr" select="saxon:expression('doc($p1)')" xmlns:saxon="http://saxon.sf.net/"/> +<xsl:variable name ="xmlDoc" select="saxon:eval($xmlDocExpr, $xmlFileName)" xmlns:saxon="http://saxon.sf.net/"/> + +<xsl:variable name="ms1AllExpr" select="saxon:expression('$p1//*[name() = $p2]')" xmlns:saxon="http://saxon.sf.net/"/> +<xsl:variable name ="ms1All" select="saxon:eval($ms1AllExpr, $xmlDoc, $ms1Name)" xmlns:saxon="http://saxon.sf.net/"/> +<xsl:variable name="ms1" select="subsequence($ms1All, $ms1Position, 1)"/> +<xsl:variable name="ms2AllExpr" select="saxon:expression('$p1//*[name() = $p2]')" xmlns:saxon="http://saxon.sf.net/"/> +<xsl:variable name ="ms2All" select="saxon:eval($ms2AllExpr, $xmlDoc, $ms2Name)" xmlns:saxon="http://saxon.sf.net/"/> +<xsl:variable name="ms2" select="subsequence($ms2All, $ms2Position, 1)"/> + +<xsl:variable name="ms1Ancestors" select="$ms1/ancestor::*"/> +<xsl:variable name="ms2Ancestors" select="$ms2/ancestor::*"/> + +<xsl:template match="/"> + <xsl:apply-templates select="$xmlDoc" mode="doc"/> +</xsl:template> + +<xsl:template match="element()[local-name() != $ms1Name and local-name() != $ms2Name]" mode="doc"> + <xsl:choose> + <xsl:when test="(. >> $ms1 or fn:contains($ms1Ancestors, .)) and ($ms2 >> . or fn:contains($ms2Ancestors, .))"> + <xsl:element name="{local-name(.)}"><xsl:apply-templates mode="doc"/></xsl:element> + </xsl:when> + <xsl:otherwise><xsl:apply-templates mode="doc"/></xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="attribute()|text()|comment()|processing-instruction()" mode="doc"> + <xsl:choose> + <xsl:when test=". >> $ms1 and $ms2 >> ."> + <xsl:copy><xsl:apply-templates mode="doc"/></xsl:copy> + </xsl:when> + <xsl:otherwise><xsl:apply-templates mode="doc"/></xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="element()[local-name() = $ms1Name or local-name() = $ms2Name]" mode="doc"> + <xsl:choose> + <xsl:when test=". is $ms1"> + <xsl:element name="{local-name(.)}"><xsl:copy-of select="@*"></xsl:copy-of></xsl:element> + </xsl:when> + <xsl:when test=". is $ms2"> + <xsl:element name="{local-name(.)}"><xsl:copy-of select="@*"></xsl:copy-of></xsl:element> + <!-- <xsl:message terminate="yes">Terminate</xsl:message> --> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> +</xsl:template> + +</xsl:stylesheet>
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--- a/software/mpdl-services/mpiwg-mpdl-xml/build/build.xml Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-xml/build/build.xml Tue Nov 27 12:35:19 2012 +0100 @@ -31,6 +31,7 @@ <target name="dist" depends="compile" description="generate the distribution"> <jar jarfile="${dist}/mpiwg-mpdl-xml.jar" basedir="${build}"/> <copy file="${dist}/mpiwg-mpdl-xml.jar" todir="/Users/jwillenborg/java/mpiwg-mpdl-xml-web/WebContent/WEB-INF/lib"/> + <copy file="${dist}/mpiwg-mpdl-xml.jar" todir="/Users/jwillenborg/java/mpiwg-mpdl-cms-web/WebContent/WEB-INF/lib"/> <copy file="${dist}/mpiwg-mpdl-xml.jar" todir="/Users/jwillenborg/java/mpiwg-mpdl-lt/lib"/> </target>
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Binary file software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/test/TestLocal.class has changed
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Binary file software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/transform/MpdlTransformer.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/transform/PageEchoTransformer.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/transform/PageTransformer.class has changed
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Binary file software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/transform/WordContentHandler.class has changed
--- a/software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/transform/getFragment.xsl Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/transform/getFragment.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -12,7 +12,6 @@ <xsl:output method="xml" encoding="utf-8"/> <!-- ms1Name and ms2Name have to be given without namespace: e.g. "pb" --> -<xsl:param name="xmlFileName"></xsl:param> <xsl:param name="ms1Name"></xsl:param> <xsl:param name="ms1Position"></xsl:param> <xsl:param name="ms2Name"></xsl:param> @@ -25,15 +24,29 @@ <xsl:variable name="ms1Ancestors" select="$ms1/ancestor::*"/> <xsl:variable name="ms2Ancestors" select="$ms2/ancestor::*"/> +<!-- all elements between the milestone elements and also the ancestor elements (for filling the node path to the root) --> <xsl:template match="element()[local-name() != $ms1Name and local-name() != $ms2Name]"> <xsl:choose> <xsl:when test="(. >> $ms1 or fn:contains($ms1Ancestors, .)) and ($ms2 >> . or fn:contains($ms2Ancestors, .))"> - <xsl:element name="{local-name(.)}"><xsl:apply-templates/></xsl:element> + <xsl:element name="{local-name(.)}"><xsl:copy-of select="@*"></xsl:copy-of><xsl:apply-templates/></xsl:element> </xsl:when> <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> </xsl:choose> </xsl:template> +<!-- the two milestone elements --> +<xsl:template match="element()[local-name() = $ms1Name or local-name() = $ms2Name]"> + <xsl:choose> + <xsl:when test=". is $ms1"> + <xsl:element name="{local-name(.)}"><xsl:copy-of select="@*"></xsl:copy-of></xsl:element> + </xsl:when> + <xsl:when test=". is $ms2"> + <xsl:element name="{local-name(.)}"><xsl:copy-of select="@*"></xsl:copy-of></xsl:element> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> +</xsl:template> + <xsl:template match="attribute()|text()|comment()|processing-instruction()"> <xsl:choose> <xsl:when test=". >> $ms1 and $ms2 >> ."> @@ -43,16 +56,4 @@ </xsl:choose> </xsl:template> -<xsl:template match="element()[local-name() = $ms1Name or local-name() = $ms2Name]"> - <xsl:choose> - <xsl:when test=". is $ms1"> - <xsl:element name="{local-name(.)}"><xsl:copy-of select="@*"></xsl:copy-of></xsl:element> - </xsl:when> - <xsl:when test=". is $ms2"> - <xsl:element name="{local-name(.)}"><xsl:copy-of select="@*"></xsl:copy-of></xsl:element> - </xsl:when> - <xsl:otherwise></xsl:otherwise> - </xsl:choose> -</xsl:template> - </xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/transform/pageArchimedes.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,820 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xs="http://www.w3.org/2001/XMLSchema" + xmlns:functx="http://www.functx.com" + xmlns:saxon="http://saxon.sf.net/" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:mml="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml"> + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:variable name="collectionName" select="'echo'"/> +<xsl:variable name="polluxServiceName" select="'http://mpdl-service.mpiwg-berlin.mpg.de/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries'"/> + +<xsl:template match="/"> + <xsl:apply-templates mode="text"/> +</xsl:template> + +<xsl:template match="text" mode="text"> + <xsl:apply-templates mode="text"/> + <!-- TEI Dictionary --> + <xsl:if test="not(empty(//entry))"> + <xsl:for-each select="//entry"> + <xsl:sort select="form/orth"/> + <xsl:variable name="position" select="position()"/> + <span class="entry"> + <xsl:apply-templates mode="text" select="form"/> + <xsl:apply-templates mode="text" select="sense"/> + <xsl:if test="not(empty(figure))"> + <span class="entryDiv"> + <span class="bf"><xsl:value-of select="'Figures: '"/></span> + <span class="entryDiv"> + <xsl:for-each select="figure"> + <xsl:variable name="href" select="graphic/@url"/> + <xsl:variable name="head" select="head"/> + <xsl:variable name="figDesc" select="string(figDesc)"/> + <xsl:if test="$href != ''"> + <div class="figure" style="margin-left:10px;"> + <a href="{$href}"><img alt="Figure: {$figDesc}" src="{$href}" width="200" height="200"/></a> + <br/> + <xsl:value-of select="'[Figure]: '"/><xsl:value-of select="$head"/> + </div> + </xsl:if> + </xsl:for-each> + </span> + </span> + </xsl:if> + <xsl:if test="not(empty(xr))"> + <span class="entryDiv"> + <span class="bf"><xsl:value-of select="'References: '"/></span> + <span class="entryDiv"> + <xsl:for-each select="xr/ref"> + <li><xsl:apply-templates mode="text" select="."/></li> + </xsl:for-each> + </span> + </span> + </xsl:if> + </span> + </xsl:for-each> + </xsl:if> + <!-- Notes --> + <!-- + <xsl:variable name="countEmptyPlaces" select="count($notes/note[empty(@place)])"/> + <xsl:variable name="countBottomPlaces"> + <xsl:choose> + <xsl:when test="$collectionName = 'archimedes' or $collectionName = 'tei'"> + <xsl:value-of select="count($notes/note[contains(@place, 'bottom')])"/> + </xsl:when> + <xsl:otherwise> + <xsl:value-of select="count($notes/note[contains(@position, 'bottom') or empty(string(@position))])"/> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:if test="($collectionName = 'archimedes' or $collectionName = 'tei') and count($notes/*) > 0"> + <div> + <xsl:choose> + <xsl:when test="$countBottomPlaces > 0 or $countEmptyPlaces > 0"> + <hr class="notesBottom"/> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> + <xsl:for-each select="$notes/note"> + <xsl:variable name="notePos" select="position()"/> + <xsl:variable name="place" select="@place"/> + <xsl:variable name="label" select="$notePos"/> + <xsl:variable name="uid" select="@uid"/> + <xsl:variable name="modificationDate" select="@modificationDate"/> + <xsl:variable name="noteWithoutNamespace" select="mpdl-util:copyWithoutNamespace(.)"/> + <xsl:choose> + <xsl:when test="$place = '' or empty($place) or $place = 'bottom'"> + <p> + <a> + <xsl:attribute name="name"><xsl:value-of select="concat('note-', $pageNumber, '-', $label)"/></xsl:attribute> + <xsl:attribute name="href"><xsl:value-of select="concat(urlBase, '#', 'note-', $pageNumber, '-', $label, 'ref')"/></xsl:attribute> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <xsl:value-of select="concat('note-', $pageNumber, '-', $label)"/> + </a> + <xsl:value-of select="': '"/> + <xsl:choose> + <xsl:when test="$uid = '' or empty($uid)"> + <span class="note"><xsl:apply-templates select="$noteWithoutNamespace/node()" mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span class="note"><xsl:apply-templates select="$noteWithoutNamespace/node()" mode="text"/></span> + <xsl:value-of select="concat(' [external note, ', $uid, ', ', $modificationDate, ']')"/> + </xsl:otherwise> + </xsl:choose> + </p> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> + </xsl:for-each> + </div> + </xsl:if> + <xsl:if test="$collectionName = 'echo' and count($notes/*) > 0"> + <div> + <xsl:choose> + <xsl:when test="$countBottomPlaces > 0 or $countEmptyPlaces > 0"> + <hr class="notesBottom"/> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> + <xsl:for-each select="$notes/echo:note"> + <xsl:variable name="label" select="string(@xlink:label)"/> + <xsl:variable name="place" select="@position"/> + <xsl:variable name="uid" select="@uid"/> + <xsl:variable name="modificationDate" select="@modificationDate"/> + <xsl:variable name="noteWithoutNamespace" select="mpdl-util:copyWithoutNamespace(.)"/> + <xsl:choose> + <xsl:when test="$place = '' or empty($place) or $place = 'bottom'"> + <p> + <a> + <xsl:attribute name="name"><xsl:value-of select="$label"/></xsl:attribute> + <xsl:attribute name="href"><xsl:value-of select="concat(urlBase, '#', $label, 'ref')"/></xsl:attribute> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <xsl:value-of select="$label"/> + </a> + <xsl:value-of select="': '"/> + <xsl:choose> + <xsl:when test="$uid = '' or empty($uid)"> + <span class="note"><xsl:apply-templates select="$noteWithoutNamespace/node()" mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span class="note"><xsl:apply-templates select="$noteWithoutNamespace/node()" mode="text"/></span> + <xsl:value-of select="concat(' [external note, ', $uid, ', ', $modificationDate, ']')"/> + </xsl:otherwise> + </xsl:choose> + </p> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> + </xsl:for-each> + </div> + </xsl:if> + --> +</xsl:template> + +<xsl:template match="head" mode="text"> + <xsl:variable name="class"> + <xsl:value-of select="'head bf'"/> + </xsl:variable> + <p> + <span> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </p> +</xsl:template> + +<!-- TEI: segmentation --> +<xsl:template match="seg" mode="text"> + <xsl:choose> + <xsl:when test="@rend = 'highlight'"> + <span class="seg highlight"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="@rend = 'highlightPoint'"> + <span class="seg highlightPoint"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span class="seg"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- TEI: choice --> +<xsl:template match="choice" mode="text"> + <xsl:if test="not(empty(orig))"> + <xsl:apply-templates select="orig" mode="text"/> + </xsl:if> + <xsl:if test="not(empty(abbr))"> + <xsl:apply-templates select="abbr" mode="text"/> + </xsl:if> + <xsl:if test="not(empty(am))"> + <xsl:apply-templates select="am" mode="text"/> + </xsl:if> + <xsl:if test="not(empty(sic))"> + <xsl:apply-templates select="sic" mode="text"/> + </xsl:if> +</xsl:template> + +<!-- TEI: hi (highlighted) --> +<xsl:template match="hi" mode="text"> + <xsl:choose> + <xsl:when test="@rend = 'initial'"> + <span class="dc-unmodified"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="@rend = 'bold'"> + <span class="bf"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:attribute name="class"><xsl:value-of select="@rend"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- TEI: name (of type: place, person, ...) --> +<xsl:template match="name" mode="text"> + <xsl:choose> + <xsl:when test="@type = 'place'"> + <span class="place" title="Place"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="@type = 'person'"> + <span class="person" title="Person"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="@type = 'org'"> + <span class="organization" title="Organization"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:attribute name="class"><xsl:value-of select="@type"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- TEI: place, person, ...) --> +<xsl:template match="placeName" mode="text"> + <span class="place" title="Place: {@type}"><xsl:apply-templates mode="text"/></span> +</xsl:template> +<xsl:template match="persName" mode="text"> + <span class="person" title="Person"><xsl:apply-templates mode="text"/></span> +</xsl:template> + +<!-- TEI: term --> +<xsl:template match="term" mode="text"> + <span class="term" title="Terminology entry"><xsl:apply-templates mode="text"/></span> +</xsl:template> + +<!-- TEI: line --> +<xsl:template match="lg" mode="text"> + <div class="lg"><xsl:apply-templates mode="text"/></div> +</xsl:template> + +<xsl:template match="l" mode="text"> + <span class="l"><xsl:apply-templates mode="text"/></span><br/> +</xsl:template> + +<!-- TEI: reference --> +<xsl:template match="ref" mode="text"> + <span class="ref"> + <xsl:if test="not(empty(@target))"> + <a> + <xsl:attribute name="href"><xsl:value-of select="@target"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </a> + </xsl:if> + <xsl:if test="empty(@target)"> + <xsl:apply-templates mode="text"/> + </xsl:if> + </span> +</xsl:template> + +<!-- TEI: table --> +<xsl:template match="table" mode="text"> + <table> + <xsl:if test="not(empty(head))"> + <caption align="top"><xsl:apply-templates mode="text" select="head"/></caption> + </xsl:if> + <xsl:apply-templates mode="text" select="row"/> + </table> +</xsl:template> + +<xsl:template match="row" mode="text"> + <xsl:choose> + <xsl:when test="@role = 'label'"> + <tr style="font-weight:bold;"><xsl:apply-templates mode="text"/></tr> + </xsl:when> + <xsl:when test="@role = 'data' or empty(@role)"> + <tr><xsl:apply-templates mode="text"/></tr> + </xsl:when> + <xsl:otherwise> + <tr><xsl:apply-templates mode="text"/></tr> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="cell" mode="text"> + <xsl:choose> + <xsl:when test="@role = 'label' and empty(@cols)"> + <td style="font-weight:bold;"><xsl:apply-templates mode="text"/></td> + </xsl:when> + <xsl:when test="@role = 'label' and not(empty(@cols))"> + <td colspan="{@cols}" style="font-weight:bold;"><xsl:apply-templates mode="text"/></td> + </xsl:when> + <xsl:when test="(@role = 'data' or empty(@role)) and empty(@cols)"> + <td><xsl:apply-templates mode="text"/></td> + </xsl:when> + <xsl:when test="(@role = 'data' or empty(@role)) and not(empty(@cols))"> + <td colspan="{@cols}"><xsl:apply-templates mode="text"/></td> + </xsl:when> + <xsl:otherwise> + <td><xsl:apply-templates mode="text"/></td> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- TEI: dictionary --> +<xsl:template match="entry" mode="text"> + <!-- empty: handled in text tag --> +</xsl:template> + +<xsl:template match="form" mode="text"> + <span class="form"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="orth" mode="text"> + <span class="orth"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="sense" mode="text"> + <span class="entryDiv"><xsl:apply-templates mode="text" select="def"/></span> + <span class="entryDiv"><xsl:apply-templates mode="text" select="etym"/></span> +</xsl:template> + +<xsl:template match="def" mode="text"> + <span class="def"> + <span class="bf"><xsl:value-of select="'Definition: '"/></span> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="mentioned" mode="text"> + <span class="mentioned"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="etym" mode="text"> + <span class="etym"> + <span class="bf"><xsl:value-of select="'Etymology: '"/></span> + <span class="entryDiv"> + <xsl:for-each select="cit"> + <li><xsl:value-of select="quote"/><xsl:value-of select="' ('"/><xsl:value-of select="def"/><xsl:value-of select="')'"/></li> + </xsl:for-each> + </span> + </span> +</xsl:template> + +<xsl:template match="cit" mode="text"> + <span class="cit"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<!-- TEI: MathML --> +<xsl:template match="math:*" mode="text"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> +</xsl:template> + +<!-- TEI: SVG --> +<xsl:template match="svg:*" mode="text"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> +</xsl:template> + +<xsl:template match="div" mode="text"> + <xsl:variable name="type" select="@type"/> + <xsl:variable name="level" select="@level"/> + <xsl:variable name="style" select="@style"/> + <xsl:variable name="border" select="@border"/> + <xsl:variable name="width" select="@width"/> + <xsl:variable name="note" select="note"/> + <xsl:variable name="figure" select="figure"/> + <xsl:variable name="floatClassValue"> + <xsl:choose> + <xsl:when test="not(empty($note))"><xsl:value-of select="''"/></xsl:when> + <xsl:when test="not(empty($figure))"><xsl:value-of select="''"/></xsl:when> + <xsl:when test="$type = 'float'"><xsl:value-of select="'float left'"/></xsl:when> + <xsl:otherwise><xsl:value-of select="'float left'"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="levelClassValue"> + <xsl:choose> + <xsl:when test="not(empty($note))"><xsl:value-of select="''"/></xsl:when> + <xsl:when test="$level = '1'"><xsl:value-of select="'level one'"/></xsl:when> + <xsl:when test="$level = '2'"><xsl:value-of select="'level two'"/></xsl:when> + <xsl:when test="$level = '3'"><xsl:value-of select="'level three'"/></xsl:when> + <xsl:when test="$level = '4'"><xsl:value-of select="'level four'"/></xsl:when> + <xsl:when test="$level = '5'"><xsl:value-of select="'level five'"/></xsl:when> + <xsl:when test="$level = '6'"><xsl:value-of select="'level six'"/></xsl:when> + <xsl:when test="$level = '7'"><xsl:value-of select="'level seven'"/></xsl:when> + <xsl:when test="$level = '8'"><xsl:value-of select="'level eight'"/></xsl:when> + <xsl:when test="$level = '9'"><xsl:value-of select="'level nine'"/></xsl:when> + <xsl:when test="empty($level)"><xsl:value-of select="''"/></xsl:when> + <xsl:otherwise><xsl:value-of select="'level n'"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <div> + <xsl:if test="not(empty($style))"><xsl:attribute name="style"><xsl:value-of select="$style"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat($floatClassValue, ' ', $levelClassValue)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> +</xsl:template> + +<xsl:template match="p" mode="text"> + <xsl:variable name="style" select="@style"/> + <xsl:variable name="class"> + <xsl:value-of select="'p'"/> + </xsl:variable> + <xsl:choose> + <xsl:when test="not(empty($style))"> + <div> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </div> + </xsl:when> + <xsl:otherwise> + <div> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="lb|br" mode="text"> + <br></br><xsl:apply-templates mode="text"/> +</xsl:template> + +<xsl:template match="cb" mode="text"> + <br/><xsl:apply-templates mode="text"/> +</xsl:template> + +<xsl:template match="expan" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('expan ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <span class="expan"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="note" mode="text"> + <!-- + <xsl:variable name="hasLabel" select="string(@xlink:label) != ''"/> + <xsl:variable name="place"> + <xsl:choose> + <xsl:when test="$collectionName = 'tei'"><xsl:value-of select="@place"/></xsl:when> + <xsl:otherwise><xsl:value-of select="@position"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="notePos" select="count(preceding::note[. >> $topPB]) + 1"/> + <xsl:variable name="href" select="concat('note-', $pageNumber, '-', $notePos)"/> + <xsl:choose> + <xsl:when test="$collectionName = 'archimedes'"> + <a> + <xsl:attribute name="name"><xsl:value-of select="concat($href, 'ref')"/></xsl:attribute> + <xsl:attribute name="href"><xsl:value-of select="concat(urlBase, '#', $href)"/></xsl:attribute> + <xsl:attribute name="class"><xsl:value-of select="'noteRef super'"/></xsl:attribute> + <xsl:value-of select="$href"/> + </a> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and not($hasLabel) and not(empty($place))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('note margin ', $place)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and not($hasLabel)"> + <p> + <xsl:value-of select="'[Note]: '"/> + <span class="note"><xsl:apply-templates mode="text"/></span> + </p> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and $hasLabel"></xsl:when> + <xsl:when test="$collectionName = 'tei'"> + <xsl:choose> + <xsl:when test="contains($place, 'margin')"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('note ', $place)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <a> + <xsl:attribute name="name"><xsl:value-of select="concat($href, 'ref')"/></xsl:attribute> + <xsl:attribute name="href"><xsl:value-of select="concat(urlBase, '#', $href)"/></xsl:attribute> + <xsl:attribute name="class"><xsl:value-of select="'noteRef super'"/></xsl:attribute> + <xsl:value-of select="$href"/> + </a> + </xsl:otherwise> + </xsl:choose> + </xsl:when> + <xsl:otherwise> + <span class="note"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> + --> +</xsl:template> + +<xsl:template match="emph" mode="text"> + <xsl:variable name="style" select="@style"/> + <xsl:variable name="styleWithoutSC" select="replace($style, 'sc ', '')"/> + <xsl:variable name="text" select="string-join(., '')"/> + <xsl:variable name="length" select="string-length($text)"/> + <xsl:variable name="firstChar" select="substring($text, 1, 1)"/> + <xsl:variable name="first2Chars" select="substring($text, 1, 2)"/> + <xsl:variable name="restChars" select="substring($text, 2, $length)"/> + <xsl:variable name="first2CharsAreUppercase" select="upper-case($first2Chars) = $first2Chars"/> + <!-- an emph as first element in p and also an emph in s in p is recognized --> + <xsl:variable name="isFirstElementInP" select="(not(empty(./parent::*/parent::p)) and (empty(./parent::*/preceding-sibling::node()) and empty(./preceding-sibling::node()))) or (not(empty(./parent::p)) and empty(./preceding-sibling::node()))"/> + <xsl:variable name="rest"> + <xsl:choose> + <xsl:when test="$length < 2 or empty($length)"><xsl:value-of select="''"/></xsl:when> + <xsl:otherwise> + <xsl:choose> + <xsl:when test="not(empty(w))"> + <a class="textPollux" href="{$polluxServiceName}?query={w/@form}&queryDisplay={$text}&language={w/@lang}&outputFormat=html&outputType=morphCompact&outputType=dictFull"><xsl:value-of select="$restChars"/></a> + </xsl:when> + <xsl:otherwise><xsl:value-of select="$restChars"/></xsl:otherwise> + </xsl:choose> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:choose> + <xsl:when test="$collectionName = 'echo' and not(contains($style, 'sc'))"> + <span class="{$style}"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and $style = 'sc' and $length = 1"> + <span class="sc"><xsl:value-of select="$firstChar"/></span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and $style = 'sc' and not($isFirstElementInP)"> + <span class="sc"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <!-- special case (see text() node below): first char dc, rest sc --> + <xsl:when test="$collectionName = 'echo' and $style = 'dc sc logic' and $first2CharsAreUppercase"> + <span class="dc"><xsl:value-of select="$firstChar"/></span><span class="sc"><xsl:sequence select="$rest"/></span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and $style = 'sc' and $first2CharsAreUppercase"> + <span class="dc"><xsl:value-of select="$firstChar"/></span><span class="sc"><xsl:sequence select="$rest"/></span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and contains($style, 'sc') and $first2CharsAreUppercase"> + <span class="dc"><span class="{$styleWithoutSC}"><xsl:value-of select="$firstChar"/></span></span><span class="sc"><span class="{$styleWithoutSC}"><xsl:sequence select="$rest"/></span></span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and $style = 'sc' and not($first2CharsAreUppercase)"> + <span class="sc"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and contains($style, 'sc') and not($first2CharsAreUppercase)"> + <span class="sc"><span class="{$styleWithoutSC}"><xsl:apply-templates mode="text"/></span></span> + </xsl:when> + <xsl:when test="$collectionName = 'archimedes' and $style != ''"> + <span class="{$style}"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="$collectionName = 'archimedes' and $style = ''"> + <span class="emph"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="$style != ''"> + <span class="{$style}"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span class="emph"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="foreign" mode="text"> + <xsl:variable name="lang" select="@lang"/> + <xsl:variable name="xmllang" select="@xml:lang"/> + <xsl:variable name="language"> + <xsl:choose> + <xsl:when test="not(empty($xmllang))"><xsl:value-of select="$xmllang"/></xsl:when> + <xsl:when test="not(empty($lang))"><xsl:value-of select="$lang"/></xsl:when> + <xsl:otherwise><xsl:value-of select="''"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('foreign ', $language)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="q" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('q ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <span class="q"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="quote" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('quote ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> + </xsl:when> + <xsl:otherwise> + <div class="quote"><xsl:apply-templates mode="text"/></div> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="blockquote" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('blockquote ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> + </xsl:when> + <xsl:otherwise> + <div class="blockquote"><xsl:apply-templates mode="text"/></div> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="set-off" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('set-off ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> + </xsl:when> + <xsl:otherwise> + <div class="set-off"><xsl:apply-templates mode="text"/></div> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="reg" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('reg ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <span class="reg"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="var" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('reg ', @type, ' ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <span class="var"> + <xsl:attribute name="class"><xsl:value-of select="concat('var ', @type)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="num" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="'num'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="@style"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </span> + </xsl:when> + <xsl:otherwise> + <span class="num"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="gap" mode="text"> + <xsl:variable name="extent" select="@extent"/> + <xsl:variable name="count"> + <xsl:choose> + <xsl:when test="empty($extent)"><xsl:value-of select="number(3)"/></xsl:when> + <xsl:otherwise><xsl:value-of select="number($extent)"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="gapChars" select="'...'"/> + <xsl:value-of select="concat('[', $gapChars, ']')"/><xsl:apply-templates mode="text"/> +</xsl:template> + +<xsl:template match="anchor" mode="text"> + <a><xsl:attribute name="href"><xsl:value-of select="@xlink:href"/></xsl:attribute><xsl:apply-templates mode="text"/></a> +</xsl:template> + +<!-- GIS Elements --> +<xsl:template match="place" mode="text"> + <span class="place"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="time" mode="text"> + <span class="time"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="person" mode="text"> + <span class="person"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="event" mode="text"> + <span class="event"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<!-- XHTML: remove the xhtml namespace --> +<xsl:template match="xhtml:*" mode="text"> + <xsl:variable name="hasLabel" select="string(@xlink:label) != ''"/> + <xsl:variable name="isTable" select="name() = 'table'"/> + <xsl:choose> + <xsl:when test="(not($hasLabel)) or ($isTable and $hasLabel)"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="figure" mode="text"> + <span class="figure"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="caption" mode="text"> + <span class="figureCaption"><xsl:apply-templates mode="text"/></span> +</xsl:template> + +<xsl:template match="description" mode="text"> + <span class="figureDescription"><xsl:apply-templates mode="text"/></span> +</xsl:template> + +<xsl:template match="handwritten" mode="text"> + <span class="handwritten"><xsl:apply-templates mode="text"/></span> +</xsl:template> + + +<!-- words --> +<xsl:template match="w" mode="text"> + <xsl:variable name="wordLanguage" select="@lang"/> + <xsl:variable name="form" select="encode-for-uri(@form)"/> + <xsl:variable name="displayWord"><xsl:apply-templates mode="text"/></xsl:variable> + <xsl:variable name="displayWordUrlEncoded" select="encode-for-uri($displayWord)"/> + <a class="textPollux"> + <xsl:attribute name="href"><xsl:value-of select="concat($polluxServiceName, '?query=', $form, '&queryDisplay=', $displayWordUrlEncoded, '&language=', $wordLanguage, '&outputFormat=html', '&outputType=morphCompact&outputType=dictFull')"/></xsl:attribute> + <xsl:sequence select="$displayWord"/> + </a> +</xsl:template> + +<xsl:template match="s" mode="text"> + <span class="s"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="text()" mode="text"> + <xsl:value-of select="."/> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/transform/pageEcho.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,485 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xs="http://www.w3.org/2001/XMLSchema" + xmlns:functx="http://www.functx.com" + xmlns:saxon="http://saxon.sf.net/" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:mml="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xs functx saxon dc dcterms echo math mml svg xhtml" + > + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="mode"></xsl:param> +<xsl:param name="normalization"></xsl:param> + +<xsl:variable name="dictionaryServiceName" select="'http://mpdl-service.mpiwg-berlin.mpg.de/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries'"/> + +<xsl:template match="*:echo"> + <span class="page"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="*:anchor|*:figure|*:caption|*:description|*:handwritten|*:place|*:person" mode="text"> + <xsl:element name="span"> + <xsl:attribute name="class"><xsl:value-of select="name()"/></xsl:attribute> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:text" mode="text"> + <xsl:apply-templates mode="text"/> + <!-- Notes --> + <!-- + <xsl:variable name="countEmptyPlaces" select="count($notes/note[empty(@place)])"/> + <xsl:variable name="countBottomPlaces"> + <xsl:value-of select="count($notes/note[contains(@position, 'bottom') or empty(string(@position))])"/> + </xsl:variable> + <xsl:if test="count($notes/*) > 0"> + <div> + <xsl:choose> + <xsl:when test="$countBottomPlaces > 0 or $countEmptyPlaces > 0"> + <hr class="notesBottom"/> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> + <xsl:for-each select="$notes/echo:note"> + <xsl:variable name="label" select="string(@xlink:label)"/> + <xsl:variable name="place" select="@position"/> + <xsl:variable name="uid" select="@uid"/> + <xsl:variable name="modificationDate" select="@modificationDate"/> + <xsl:variable name="noteWithoutNamespace" select="mpdl-util:copyWithoutNamespace(.)"/> + <xsl:choose> + <xsl:when test="$place = '' or empty($place) or $place = 'bottom'"> + <p> + <a> + <xsl:attribute name="name"><xsl:value-of select="$label"/></xsl:attribute> + <xsl:attribute name="href"><xsl:value-of select="concat(urlBase, '#', $label, 'ref')"/></xsl:attribute> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <xsl:value-of select="$label"/> + </a> + <xsl:value-of select="': '"/> + <xsl:choose> + <xsl:when test="$uid = '' or empty($uid)"> + <span class="note"><xsl:apply-templates select="$noteWithoutNamespace/node()" mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span class="note"><xsl:apply-templates select="$noteWithoutNamespace/node()" mode="text"/></span> + <xsl:value-of select="concat(' [external note, ', $uid, ', ', $modificationDate, ']')"/> + </xsl:otherwise> + </xsl:choose> + </p> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> + </xsl:for-each> + </div> + </xsl:if> + --> +</xsl:template> + +<xsl:template match="*:head" mode="text"> + <xsl:variable name="class"> + <xsl:value-of select="'head bf'"/> + </xsl:variable> + <p> + <span> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </p> +</xsl:template> + +<!-- MathML --> +<xsl:template match="math:*" mode="text"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> +</xsl:template> + +<!-- SVG --> +<xsl:template match="svg:*" mode="text"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:div" mode="text"> + <xsl:variable name="type" select="@type"/> + <xsl:variable name="level" select="@level"/> + <xsl:variable name="style" select="@style"/> + <xsl:variable name="border" select="@border"/> + <xsl:variable name="width" select="@width"/> + <xsl:variable name="note" select="note"/> + <xsl:variable name="figure" select="figure"/> + <xsl:variable name="floatClassValue"> + <xsl:choose> + <xsl:when test="not(empty($note))"><xsl:value-of select="''"/></xsl:when> + <xsl:when test="not(empty($figure))"><xsl:value-of select="''"/></xsl:when> + <xsl:when test="$type = 'float'"><xsl:value-of select="'float left'"/></xsl:when> + <xsl:otherwise><xsl:value-of select="'float left'"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="levelClassValue"> + <xsl:choose> + <xsl:when test="not(empty($note))"><xsl:value-of select="''"/></xsl:when> + <xsl:when test="$level = '1'"><xsl:value-of select="'level one'"/></xsl:when> + <xsl:when test="$level = '2'"><xsl:value-of select="'level two'"/></xsl:when> + <xsl:when test="$level = '3'"><xsl:value-of select="'level three'"/></xsl:when> + <xsl:when test="$level = '4'"><xsl:value-of select="'level four'"/></xsl:when> + <xsl:when test="$level = '5'"><xsl:value-of select="'level five'"/></xsl:when> + <xsl:when test="$level = '6'"><xsl:value-of select="'level six'"/></xsl:when> + <xsl:when test="$level = '7'"><xsl:value-of select="'level seven'"/></xsl:when> + <xsl:when test="$level = '8'"><xsl:value-of select="'level eight'"/></xsl:when> + <xsl:when test="$level = '9'"><xsl:value-of select="'level nine'"/></xsl:when> + <xsl:when test="empty($level)"><xsl:value-of select="''"/></xsl:when> + <xsl:otherwise><xsl:value-of select="'level n'"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <div> + <xsl:if test="not(empty($style))"><xsl:attribute name="style"><xsl:value-of select="$style"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat($floatClassValue, ' ', $levelClassValue)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> +</xsl:template> + +<xsl:template match="*:p" mode="text"> + <xsl:variable name="style" select="@style"/> + <xsl:variable name="class"> + <xsl:value-of select="'p'"/> + </xsl:variable> + <xsl:choose> + <xsl:when test="not(empty($style))"> + <div> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </div> + </xsl:when> + <xsl:otherwise> + <div> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="*:pb" mode="text"> + <xsl:element name="span"> + <xsl:attribute name="class"><xsl:value-of select="name()"/></xsl:attribute> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:lb" mode="text"> + <br/><xsl:apply-templates mode="text"/> +</xsl:template> + +<xsl:template match="*:cb" mode="text"> + <br/><xsl:apply-templates mode="text"/> +</xsl:template> + +<xsl:template match="*:expan" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('expan ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <span class="expan"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="*:note" mode="text"> + <!-- + <xsl:variable name="hasLabel" select="string(@xlink:label) != ''"/> + <xsl:variable name="place"><xsl:value-of select="@position"/></xsl:variable> + <xsl:variable name="notePos" select="count(preceding::note[. >> $topPB]) + 1"/> + <xsl:variable name="href" select="concat('note-', $pageNumber, '-', $notePos)"/> + <xsl:choose> + <xsl:when test="not($hasLabel) and not(empty($place))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('note margin ', $place)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:when test="not($hasLabel)"> + <p> + <xsl:value-of select="'[Note]: '"/> + <span class="note"><xsl:apply-templates mode="text"/></span> + </p> + </xsl:when> + <xsl:when test="$hasLabel"></xsl:when> + <xsl:otherwise> + <span class="note"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> + --> +</xsl:template> + +<xsl:template match="*:emph" mode="text"> + <xsl:variable name="style" select="@style"/> + <xsl:variable name="styleWithoutSC" select="replace($style, 'sc ', '')"/> + <xsl:variable name="text" select="string-join(., '')"/> + <xsl:variable name="length" select="string-length($text)"/> + <xsl:variable name="firstChar" select="substring($text, 1, 1)"/> + <xsl:variable name="first2Chars" select="substring($text, 1, 2)"/> + <xsl:variable name="restChars" select="substring($text, 2, $length)"/> + <xsl:variable name="first2CharsAreUppercase" select="upper-case($first2Chars) = $first2Chars"/> + <!-- an emph as first element in p and also an emph in s in p is recognized --> + <xsl:variable name="isFirstElementInP" select="(not(empty(./parent::*/parent::p)) and (empty(./parent::*/preceding-sibling::node()) and empty(./preceding-sibling::node()))) or (not(empty(./parent::p)) and empty(./preceding-sibling::node()))"/> + <xsl:variable name="rest"> + <xsl:choose> + <xsl:when test="$length < 2 or empty($length)"><xsl:value-of select="''"/></xsl:when> + <xsl:otherwise> + <xsl:choose> + <xsl:when test="not(empty(w))"> + <a class="dictionary" href="{$dictionaryServiceName}?query={w/@form}&queryDisplay={$text}&language={w/@lang}&outputFormat=html&outputType=morphCompact&outputType=dictFull"><xsl:value-of select="$restChars"/></a> + </xsl:when> + <xsl:otherwise><xsl:value-of select="$restChars"/></xsl:otherwise> + </xsl:choose> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:choose> + <xsl:when test="not(contains($style, 'sc'))"> + <span class="{$style}"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="$style = 'sc' and $length = 1"> + <span class="sc"><xsl:value-of select="$firstChar"/></span> + </xsl:when> + <xsl:when test="$style = 'sc' and not($isFirstElementInP)"> + <span class="sc"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <!-- special case (see text() node below): first char dc, rest sc --> + <xsl:when test="$style = 'dc sc logic' and $first2CharsAreUppercase"> + <span class="dc"><xsl:value-of select="$firstChar"/></span><span class="sc"><xsl:sequence select="$rest"/></span> + </xsl:when> + <xsl:when test="$style = 'sc' and $first2CharsAreUppercase"> + <span class="dc"><xsl:value-of select="$firstChar"/></span><span class="sc"><xsl:sequence select="$rest"/></span> + </xsl:when> + <xsl:when test="contains($style, 'sc') and $first2CharsAreUppercase"> + <span class="dc"><span class="{$styleWithoutSC}"><xsl:value-of select="$firstChar"/></span></span><span class="sc"><span class="{$styleWithoutSC}"><xsl:sequence select="$rest"/></span></span> + </xsl:when> + <xsl:when test="$style = 'sc' and not($first2CharsAreUppercase)"> + <span class="sc"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="contains($style, 'sc') and not($first2CharsAreUppercase)"> + <span class="sc"><span class="{$styleWithoutSC}"><xsl:apply-templates mode="text"/></span></span> + </xsl:when> + <xsl:when test="$style != ''"> + <span class="{$style}"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span class="emph"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="*:foreign" mode="text"> + <xsl:variable name="lang" select="@lang"/> + <xsl:variable name="xmllang" select="@xml:lang"/> + <xsl:variable name="language"> + <xsl:choose> + <xsl:when test="not(empty($xmllang))"><xsl:value-of select="$xmllang"/></xsl:when> + <xsl:when test="not(empty($lang))"><xsl:value-of select="$lang"/></xsl:when> + <xsl:otherwise><xsl:value-of select="''"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('foreign ', $language)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="*:q" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('q ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <span class="q"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="*:quote" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('quote ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> + </xsl:when> + <xsl:otherwise> + <div class="quote"><xsl:apply-templates mode="text"/></div> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="*:blockquote" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('blockquote ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> + </xsl:when> + <xsl:otherwise> + <div class="blockquote"><xsl:apply-templates mode="text"/></div> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="*:set-off" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('set-off ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> + </xsl:when> + <xsl:otherwise> + <div class="set-off"><xsl:apply-templates mode="text"/></div> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="*:reg" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('reg ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <span class="reg"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="*:var" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('reg ', @type, ' ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <span class="var"> + <xsl:attribute name="class"><xsl:value-of select="concat('var ', @type)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="*:num" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="'num'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="@style"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </span> + </xsl:when> + <xsl:otherwise> + <span class="num"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="*:gap" mode="text"> + <xsl:variable name="extent" select="@extent"/> + <xsl:variable name="count"> + <xsl:choose> + <xsl:when test="empty($extent)"><xsl:value-of select="number(3)"/></xsl:when> + <xsl:otherwise><xsl:value-of select="number($extent)"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="gapChars" select="'...'"/> + <xsl:value-of select="concat('[', $gapChars, ']')"/><xsl:apply-templates mode="text"/> +</xsl:template> + +<!-- XHTML: remove the xhtml namespace --> +<xsl:template match="xhtml:*" mode="text"> + <xsl:variable name="hasLabel" select="string(@xlink:label) != ''"/> + <xsl:variable name="isTable" select="name() = 'table'"/> + <xsl:choose> + <xsl:when test="(not($hasLabel)) or ($isTable and $hasLabel)"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> +</xsl:template> + + +<!-- words --> +<xsl:template match="*:w" mode="text"> + <xsl:variable name="wordLanguage" select="@lang"/> + <xsl:variable name="form" select="encode-for-uri(string(@form))"/> + <xsl:variable name="formNotUrlEncoded" select="string(@form)"/> + <xsl:variable name="formRegularized" select="string(@formRegularized)"/> + <xsl:variable name="formNormalized" select="string(@formNormalized)"/> + <xsl:variable name="lemmas" select="string(@lemmas)"/> + <xsl:variable name="dictionary" select="string(@dictionary)"/> + <xsl:variable name="displayWord"> + <xsl:choose> + <xsl:when test="$normalization = 'orig'"><xsl:apply-templates mode="text"/></xsl:when> + <xsl:when test="$normalization = 'reg' and $formRegularized = ''"><xsl:apply-templates mode="text"/></xsl:when> + <xsl:when test="$normalization = 'reg' and $formRegularized != ''"><xsl:sequence select="$formRegularized"/></xsl:when> + <xsl:when test="$normalization = 'norm'"><xsl:apply-templates mode="text"/></xsl:when> + <xsl:otherwise><xsl:apply-templates mode="text"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordUrlEncoded" select="encode-for-uri($displayWord)"/> + <xsl:choose> + <xsl:when test="$dictionary = 'true' and $mode = 'tokenized'"> + <a class="dictionary"> + <xsl:attribute name="href"><xsl:value-of select="concat($dictionaryServiceName, '?query=', $form, '&queryDisplay=', $displayWordUrlEncoded, '&language=', $wordLanguage, '&outputFormat=html', '&outputType=morphCompact&outputType=dictFull')"/></xsl:attribute> + <xsl:sequence select="$displayWord"/> + </a> + </xsl:when> + <xsl:otherwise><xsl:sequence select="$displayWord"/></xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="*:s" mode="text"> + <span class="s"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="text()" mode="text"> + <xsl:value-of select="."/> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/transform/pageEchoSpan.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,112 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xs="http://www.w3.org/2001/XMLSchema" + xmlns:functx="http://www.functx.com" + xmlns:saxon="http://saxon.sf.net/" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:mml="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml" + exclude-result-prefixes="xsl xlink xs functx saxon dc dcterms echo math mml svg xhtml" + > + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="mode"></xsl:param> +<xsl:param name="normalization"></xsl:param> + +<xsl:variable name="dictionaryServiceName" select="'http://mpdl-service.mpiwg-berlin.mpg.de/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries'"/> + +<xsl:template match="*:echo"> + <span class="page"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="*:head|*:div|*:p|*:s|*:pb|*:cb|*:expan|*:emph|*:quote|*:q|*:blockquote|*:set-off|*:reg|*:var|*:num|*:gap|*:anchor|*:note|*:figure|*:caption|*:description|*:handwritten|*:place|*:person" mode="text"> + <xsl:element name="span"> + <xsl:attribute name="class"><xsl:value-of select="name()"/></xsl:attribute> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> +</xsl:template> + +<!-- MathML --> +<xsl:template match="math:*" mode="text"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> +</xsl:template> + +<!-- SVG --> +<xsl:template match="svg:*" mode="text"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> +</xsl:template> + +<xsl:template match="*:lb" mode="text"> + <br/><xsl:apply-templates mode="text"/> +</xsl:template> + +<!-- XHTML: remove the xhtml namespace --> +<xsl:template match="xhtml:*" mode="text"> + <xsl:variable name="hasLabel" select="string(@xlink:label) != ''"/> + <xsl:variable name="isTable" select="name() = 'table'"/> + <xsl:choose> + <xsl:when test="(not($hasLabel)) or ($isTable and $hasLabel)"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> +</xsl:template> + + +<!-- words --> +<xsl:template match="*:w" mode="text"> + <xsl:variable name="wordLanguage" select="@lang"/> + <xsl:variable name="form" select="encode-for-uri(string(@form))"/> + <xsl:variable name="formNotUrlEncoded" select="string(@form)"/> + <xsl:variable name="formRegularized" select="string(@formRegularized)"/> + <xsl:variable name="formNormalized" select="string(@formNormalized)"/> + <xsl:variable name="lemmas" select="string(@lemmas)"/> + <xsl:variable name="dictionary" select="string(@dictionary)"/> + <xsl:variable name="displayWord"> + <xsl:choose> + <xsl:when test="$normalization = 'orig'"><xsl:apply-templates mode="text"/></xsl:when> + <xsl:when test="$normalization = 'reg' and $formRegularized = ''"><xsl:apply-templates mode="text"/></xsl:when> + <xsl:when test="$normalization = 'reg' and $formRegularized != ''"><xsl:sequence select="$formRegularized"/></xsl:when> + <xsl:when test="$normalization = 'norm'"><xsl:apply-templates mode="text"/></xsl:when> + <xsl:otherwise><xsl:apply-templates mode="text"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="displayWordUrlEncoded" select="encode-for-uri($displayWord)"/> + <xsl:choose> + <xsl:when test="$dictionary = 'true' and $mode = 'tokenized'"> + <span class="w"> + <xsl:copy-of select="@*"/> + <xsl:attribute name="href"><xsl:value-of select="concat($dictionaryServiceName, '?query=', $form, '&queryDisplay=', $displayWordUrlEncoded, '&language=', $wordLanguage, '&outputFormat=html', '&outputType=morphCompact&outputType=dictFull')"/></xsl:attribute> + <xsl:sequence select="$displayWord"/> + </span> + </xsl:when> + <xsl:otherwise> + <xsl:sequence select="$displayWord"/> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="text()" mode="text"> + <xsl:value-of select="."/> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/transform/pageTei.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,820 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xs="http://www.w3.org/2001/XMLSchema" + xmlns:functx="http://www.functx.com" + xmlns:saxon="http://saxon.sf.net/" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:mml="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml"> + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:variable name="collectionName" select="'echo'"/> +<xsl:variable name="polluxServiceName" select="'http://mpdl-service.mpiwg-berlin.mpg.de/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries'"/> + +<xsl:template match="/"> + <xsl:apply-templates mode="text"/> +</xsl:template> + +<xsl:template match="text" mode="text"> + <xsl:apply-templates mode="text"/> + <!-- TEI Dictionary --> + <xsl:if test="not(empty(//entry))"> + <xsl:for-each select="//entry"> + <xsl:sort select="form/orth"/> + <xsl:variable name="position" select="position()"/> + <span class="entry"> + <xsl:apply-templates mode="text" select="form"/> + <xsl:apply-templates mode="text" select="sense"/> + <xsl:if test="not(empty(figure))"> + <span class="entryDiv"> + <span class="bf"><xsl:value-of select="'Figures: '"/></span> + <span class="entryDiv"> + <xsl:for-each select="figure"> + <xsl:variable name="href" select="graphic/@url"/> + <xsl:variable name="head" select="head"/> + <xsl:variable name="figDesc" select="string(figDesc)"/> + <xsl:if test="$href != ''"> + <div class="figure" style="margin-left:10px;"> + <a href="{$href}"><img alt="Figure: {$figDesc}" src="{$href}" width="200" height="200"/></a> + <br/> + <xsl:value-of select="'[Figure]: '"/><xsl:value-of select="$head"/> + </div> + </xsl:if> + </xsl:for-each> + </span> + </span> + </xsl:if> + <xsl:if test="not(empty(xr))"> + <span class="entryDiv"> + <span class="bf"><xsl:value-of select="'References: '"/></span> + <span class="entryDiv"> + <xsl:for-each select="xr/ref"> + <li><xsl:apply-templates mode="text" select="."/></li> + </xsl:for-each> + </span> + </span> + </xsl:if> + </span> + </xsl:for-each> + </xsl:if> + <!-- Notes --> + <!-- + <xsl:variable name="countEmptyPlaces" select="count($notes/note[empty(@place)])"/> + <xsl:variable name="countBottomPlaces"> + <xsl:choose> + <xsl:when test="$collectionName = 'archimedes' or $collectionName = 'tei'"> + <xsl:value-of select="count($notes/note[contains(@place, 'bottom')])"/> + </xsl:when> + <xsl:otherwise> + <xsl:value-of select="count($notes/note[contains(@position, 'bottom') or empty(string(@position))])"/> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:if test="($collectionName = 'archimedes' or $collectionName = 'tei') and count($notes/*) > 0"> + <div> + <xsl:choose> + <xsl:when test="$countBottomPlaces > 0 or $countEmptyPlaces > 0"> + <hr class="notesBottom"/> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> + <xsl:for-each select="$notes/note"> + <xsl:variable name="notePos" select="position()"/> + <xsl:variable name="place" select="@place"/> + <xsl:variable name="label" select="$notePos"/> + <xsl:variable name="uid" select="@uid"/> + <xsl:variable name="modificationDate" select="@modificationDate"/> + <xsl:variable name="noteWithoutNamespace" select="mpdl-util:copyWithoutNamespace(.)"/> + <xsl:choose> + <xsl:when test="$place = '' or empty($place) or $place = 'bottom'"> + <p> + <a> + <xsl:attribute name="name"><xsl:value-of select="concat('note-', $pageNumber, '-', $label)"/></xsl:attribute> + <xsl:attribute name="href"><xsl:value-of select="concat(urlBase, '#', 'note-', $pageNumber, '-', $label, 'ref')"/></xsl:attribute> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <xsl:value-of select="concat('note-', $pageNumber, '-', $label)"/> + </a> + <xsl:value-of select="': '"/> + <xsl:choose> + <xsl:when test="$uid = '' or empty($uid)"> + <span class="note"><xsl:apply-templates select="$noteWithoutNamespace/node()" mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span class="note"><xsl:apply-templates select="$noteWithoutNamespace/node()" mode="text"/></span> + <xsl:value-of select="concat(' [external note, ', $uid, ', ', $modificationDate, ']')"/> + </xsl:otherwise> + </xsl:choose> + </p> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> + </xsl:for-each> + </div> + </xsl:if> + <xsl:if test="$collectionName = 'echo' and count($notes/*) > 0"> + <div> + <xsl:choose> + <xsl:when test="$countBottomPlaces > 0 or $countEmptyPlaces > 0"> + <hr class="notesBottom"/> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> + <xsl:for-each select="$notes/echo:note"> + <xsl:variable name="label" select="string(@xlink:label)"/> + <xsl:variable name="place" select="@position"/> + <xsl:variable name="uid" select="@uid"/> + <xsl:variable name="modificationDate" select="@modificationDate"/> + <xsl:variable name="noteWithoutNamespace" select="mpdl-util:copyWithoutNamespace(.)"/> + <xsl:choose> + <xsl:when test="$place = '' or empty($place) or $place = 'bottom'"> + <p> + <a> + <xsl:attribute name="name"><xsl:value-of select="$label"/></xsl:attribute> + <xsl:attribute name="href"><xsl:value-of select="concat(urlBase, '#', $label, 'ref')"/></xsl:attribute> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <xsl:value-of select="$label"/> + </a> + <xsl:value-of select="': '"/> + <xsl:choose> + <xsl:when test="$uid = '' or empty($uid)"> + <span class="note"><xsl:apply-templates select="$noteWithoutNamespace/node()" mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span class="note"><xsl:apply-templates select="$noteWithoutNamespace/node()" mode="text"/></span> + <xsl:value-of select="concat(' [external note, ', $uid, ', ', $modificationDate, ']')"/> + </xsl:otherwise> + </xsl:choose> + </p> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> + </xsl:for-each> + </div> + </xsl:if> + --> +</xsl:template> + +<xsl:template match="head" mode="text"> + <xsl:variable name="class"> + <xsl:value-of select="'head bf'"/> + </xsl:variable> + <p> + <span> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </p> +</xsl:template> + +<!-- TEI: segmentation --> +<xsl:template match="seg" mode="text"> + <xsl:choose> + <xsl:when test="@rend = 'highlight'"> + <span class="seg highlight"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="@rend = 'highlightPoint'"> + <span class="seg highlightPoint"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span class="seg"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- TEI: choice --> +<xsl:template match="choice" mode="text"> + <xsl:if test="not(empty(orig))"> + <xsl:apply-templates select="orig" mode="text"/> + </xsl:if> + <xsl:if test="not(empty(abbr))"> + <xsl:apply-templates select="abbr" mode="text"/> + </xsl:if> + <xsl:if test="not(empty(am))"> + <xsl:apply-templates select="am" mode="text"/> + </xsl:if> + <xsl:if test="not(empty(sic))"> + <xsl:apply-templates select="sic" mode="text"/> + </xsl:if> +</xsl:template> + +<!-- TEI: hi (highlighted) --> +<xsl:template match="hi" mode="text"> + <xsl:choose> + <xsl:when test="@rend = 'initial'"> + <span class="dc-unmodified"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="@rend = 'bold'"> + <span class="bf"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:attribute name="class"><xsl:value-of select="@rend"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- TEI: name (of type: place, person, ...) --> +<xsl:template match="name" mode="text"> + <xsl:choose> + <xsl:when test="@type = 'place'"> + <span class="place" title="Place"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="@type = 'person'"> + <span class="person" title="Person"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="@type = 'org'"> + <span class="organization" title="Organization"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:attribute name="class"><xsl:value-of select="@type"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- TEI: place, person, ...) --> +<xsl:template match="placeName" mode="text"> + <span class="place" title="Place: {@type}"><xsl:apply-templates mode="text"/></span> +</xsl:template> +<xsl:template match="persName" mode="text"> + <span class="person" title="Person"><xsl:apply-templates mode="text"/></span> +</xsl:template> + +<!-- TEI: term --> +<xsl:template match="term" mode="text"> + <span class="term" title="Terminology entry"><xsl:apply-templates mode="text"/></span> +</xsl:template> + +<!-- TEI: line --> +<xsl:template match="lg" mode="text"> + <div class="lg"><xsl:apply-templates mode="text"/></div> +</xsl:template> + +<xsl:template match="l" mode="text"> + <span class="l"><xsl:apply-templates mode="text"/></span><br/> +</xsl:template> + +<!-- TEI: reference --> +<xsl:template match="ref" mode="text"> + <span class="ref"> + <xsl:if test="not(empty(@target))"> + <a> + <xsl:attribute name="href"><xsl:value-of select="@target"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </a> + </xsl:if> + <xsl:if test="empty(@target)"> + <xsl:apply-templates mode="text"/> + </xsl:if> + </span> +</xsl:template> + +<!-- TEI: table --> +<xsl:template match="table" mode="text"> + <table> + <xsl:if test="not(empty(head))"> + <caption align="top"><xsl:apply-templates mode="text" select="head"/></caption> + </xsl:if> + <xsl:apply-templates mode="text" select="row"/> + </table> +</xsl:template> + +<xsl:template match="row" mode="text"> + <xsl:choose> + <xsl:when test="@role = 'label'"> + <tr style="font-weight:bold;"><xsl:apply-templates mode="text"/></tr> + </xsl:when> + <xsl:when test="@role = 'data' or empty(@role)"> + <tr><xsl:apply-templates mode="text"/></tr> + </xsl:when> + <xsl:otherwise> + <tr><xsl:apply-templates mode="text"/></tr> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="cell" mode="text"> + <xsl:choose> + <xsl:when test="@role = 'label' and empty(@cols)"> + <td style="font-weight:bold;"><xsl:apply-templates mode="text"/></td> + </xsl:when> + <xsl:when test="@role = 'label' and not(empty(@cols))"> + <td colspan="{@cols}" style="font-weight:bold;"><xsl:apply-templates mode="text"/></td> + </xsl:when> + <xsl:when test="(@role = 'data' or empty(@role)) and empty(@cols)"> + <td><xsl:apply-templates mode="text"/></td> + </xsl:when> + <xsl:when test="(@role = 'data' or empty(@role)) and not(empty(@cols))"> + <td colspan="{@cols}"><xsl:apply-templates mode="text"/></td> + </xsl:when> + <xsl:otherwise> + <td><xsl:apply-templates mode="text"/></td> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- TEI: dictionary --> +<xsl:template match="entry" mode="text"> + <!-- empty: handled in text tag --> +</xsl:template> + +<xsl:template match="form" mode="text"> + <span class="form"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="orth" mode="text"> + <span class="orth"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="sense" mode="text"> + <span class="entryDiv"><xsl:apply-templates mode="text" select="def"/></span> + <span class="entryDiv"><xsl:apply-templates mode="text" select="etym"/></span> +</xsl:template> + +<xsl:template match="def" mode="text"> + <span class="def"> + <span class="bf"><xsl:value-of select="'Definition: '"/></span> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="mentioned" mode="text"> + <span class="mentioned"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="etym" mode="text"> + <span class="etym"> + <span class="bf"><xsl:value-of select="'Etymology: '"/></span> + <span class="entryDiv"> + <xsl:for-each select="cit"> + <li><xsl:value-of select="quote"/><xsl:value-of select="' ('"/><xsl:value-of select="def"/><xsl:value-of select="')'"/></li> + </xsl:for-each> + </span> + </span> +</xsl:template> + +<xsl:template match="cit" mode="text"> + <span class="cit"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<!-- TEI: MathML --> +<xsl:template match="math:*" mode="text"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> +</xsl:template> + +<!-- TEI: SVG --> +<xsl:template match="svg:*" mode="text"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> +</xsl:template> + +<xsl:template match="div" mode="text"> + <xsl:variable name="type" select="@type"/> + <xsl:variable name="level" select="@level"/> + <xsl:variable name="style" select="@style"/> + <xsl:variable name="border" select="@border"/> + <xsl:variable name="width" select="@width"/> + <xsl:variable name="note" select="note"/> + <xsl:variable name="figure" select="figure"/> + <xsl:variable name="floatClassValue"> + <xsl:choose> + <xsl:when test="not(empty($note))"><xsl:value-of select="''"/></xsl:when> + <xsl:when test="not(empty($figure))"><xsl:value-of select="''"/></xsl:when> + <xsl:when test="$type = 'float'"><xsl:value-of select="'float left'"/></xsl:when> + <xsl:otherwise><xsl:value-of select="'float left'"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="levelClassValue"> + <xsl:choose> + <xsl:when test="not(empty($note))"><xsl:value-of select="''"/></xsl:when> + <xsl:when test="$level = '1'"><xsl:value-of select="'level one'"/></xsl:when> + <xsl:when test="$level = '2'"><xsl:value-of select="'level two'"/></xsl:when> + <xsl:when test="$level = '3'"><xsl:value-of select="'level three'"/></xsl:when> + <xsl:when test="$level = '4'"><xsl:value-of select="'level four'"/></xsl:when> + <xsl:when test="$level = '5'"><xsl:value-of select="'level five'"/></xsl:when> + <xsl:when test="$level = '6'"><xsl:value-of select="'level six'"/></xsl:when> + <xsl:when test="$level = '7'"><xsl:value-of select="'level seven'"/></xsl:when> + <xsl:when test="$level = '8'"><xsl:value-of select="'level eight'"/></xsl:when> + <xsl:when test="$level = '9'"><xsl:value-of select="'level nine'"/></xsl:when> + <xsl:when test="empty($level)"><xsl:value-of select="''"/></xsl:when> + <xsl:otherwise><xsl:value-of select="'level n'"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <div> + <xsl:if test="not(empty($style))"><xsl:attribute name="style"><xsl:value-of select="$style"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat($floatClassValue, ' ', $levelClassValue)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> +</xsl:template> + +<xsl:template match="p" mode="text"> + <xsl:variable name="style" select="@style"/> + <xsl:variable name="class"> + <xsl:value-of select="'p'"/> + </xsl:variable> + <xsl:choose> + <xsl:when test="not(empty($style))"> + <div> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </div> + </xsl:when> + <xsl:otherwise> + <div> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="lb|br" mode="text"> + <br></br><xsl:apply-templates mode="text"/> +</xsl:template> + +<xsl:template match="cb" mode="text"> + <br/><xsl:apply-templates mode="text"/> +</xsl:template> + +<xsl:template match="expan" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('expan ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <span class="expan"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="note" mode="text"> + <!-- + <xsl:variable name="hasLabel" select="string(@xlink:label) != ''"/> + <xsl:variable name="place"> + <xsl:choose> + <xsl:when test="$collectionName = 'tei'"><xsl:value-of select="@place"/></xsl:when> + <xsl:otherwise><xsl:value-of select="@position"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="notePos" select="count(preceding::note[. >> $topPB]) + 1"/> + <xsl:variable name="href" select="concat('note-', $pageNumber, '-', $notePos)"/> + <xsl:choose> + <xsl:when test="$collectionName = 'archimedes'"> + <a> + <xsl:attribute name="name"><xsl:value-of select="concat($href, 'ref')"/></xsl:attribute> + <xsl:attribute name="href"><xsl:value-of select="concat(urlBase, '#', $href)"/></xsl:attribute> + <xsl:attribute name="class"><xsl:value-of select="'noteRef super'"/></xsl:attribute> + <xsl:value-of select="$href"/> + </a> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and not($hasLabel) and not(empty($place))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('note margin ', $place)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and not($hasLabel)"> + <p> + <xsl:value-of select="'[Note]: '"/> + <span class="note"><xsl:apply-templates mode="text"/></span> + </p> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and $hasLabel"></xsl:when> + <xsl:when test="$collectionName = 'tei'"> + <xsl:choose> + <xsl:when test="contains($place, 'margin')"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('note ', $place)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <a> + <xsl:attribute name="name"><xsl:value-of select="concat($href, 'ref')"/></xsl:attribute> + <xsl:attribute name="href"><xsl:value-of select="concat(urlBase, '#', $href)"/></xsl:attribute> + <xsl:attribute name="class"><xsl:value-of select="'noteRef super'"/></xsl:attribute> + <xsl:value-of select="$href"/> + </a> + </xsl:otherwise> + </xsl:choose> + </xsl:when> + <xsl:otherwise> + <span class="note"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> + --> +</xsl:template> + +<xsl:template match="emph" mode="text"> + <xsl:variable name="style" select="@style"/> + <xsl:variable name="styleWithoutSC" select="replace($style, 'sc ', '')"/> + <xsl:variable name="text" select="string-join(., '')"/> + <xsl:variable name="length" select="string-length($text)"/> + <xsl:variable name="firstChar" select="substring($text, 1, 1)"/> + <xsl:variable name="first2Chars" select="substring($text, 1, 2)"/> + <xsl:variable name="restChars" select="substring($text, 2, $length)"/> + <xsl:variable name="first2CharsAreUppercase" select="upper-case($first2Chars) = $first2Chars"/> + <!-- an emph as first element in p and also an emph in s in p is recognized --> + <xsl:variable name="isFirstElementInP" select="(not(empty(./parent::*/parent::p)) and (empty(./parent::*/preceding-sibling::node()) and empty(./preceding-sibling::node()))) or (not(empty(./parent::p)) and empty(./preceding-sibling::node()))"/> + <xsl:variable name="rest"> + <xsl:choose> + <xsl:when test="$length < 2 or empty($length)"><xsl:value-of select="''"/></xsl:when> + <xsl:otherwise> + <xsl:choose> + <xsl:when test="not(empty(w))"> + <a class="textPollux" href="{$polluxServiceName}?query={w/@form}&queryDisplay={$text}&language={w/@lang}&outputFormat=html&outputType=morphCompact&outputType=dictFull"><xsl:value-of select="$restChars"/></a> + </xsl:when> + <xsl:otherwise><xsl:value-of select="$restChars"/></xsl:otherwise> + </xsl:choose> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:choose> + <xsl:when test="$collectionName = 'echo' and not(contains($style, 'sc'))"> + <span class="{$style}"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and $style = 'sc' and $length = 1"> + <span class="sc"><xsl:value-of select="$firstChar"/></span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and $style = 'sc' and not($isFirstElementInP)"> + <span class="sc"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <!-- special case (see text() node below): first char dc, rest sc --> + <xsl:when test="$collectionName = 'echo' and $style = 'dc sc logic' and $first2CharsAreUppercase"> + <span class="dc"><xsl:value-of select="$firstChar"/></span><span class="sc"><xsl:sequence select="$rest"/></span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and $style = 'sc' and $first2CharsAreUppercase"> + <span class="dc"><xsl:value-of select="$firstChar"/></span><span class="sc"><xsl:sequence select="$rest"/></span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and contains($style, 'sc') and $first2CharsAreUppercase"> + <span class="dc"><span class="{$styleWithoutSC}"><xsl:value-of select="$firstChar"/></span></span><span class="sc"><span class="{$styleWithoutSC}"><xsl:sequence select="$rest"/></span></span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and $style = 'sc' and not($first2CharsAreUppercase)"> + <span class="sc"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and contains($style, 'sc') and not($first2CharsAreUppercase)"> + <span class="sc"><span class="{$styleWithoutSC}"><xsl:apply-templates mode="text"/></span></span> + </xsl:when> + <xsl:when test="$collectionName = 'archimedes' and $style != ''"> + <span class="{$style}"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="$collectionName = 'archimedes' and $style = ''"> + <span class="emph"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="$style != ''"> + <span class="{$style}"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span class="emph"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="foreign" mode="text"> + <xsl:variable name="lang" select="@lang"/> + <xsl:variable name="xmllang" select="@xml:lang"/> + <xsl:variable name="language"> + <xsl:choose> + <xsl:when test="not(empty($xmllang))"><xsl:value-of select="$xmllang"/></xsl:when> + <xsl:when test="not(empty($lang))"><xsl:value-of select="$lang"/></xsl:when> + <xsl:otherwise><xsl:value-of select="''"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('foreign ', $language)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="q" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('q ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <span class="q"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="quote" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('quote ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> + </xsl:when> + <xsl:otherwise> + <div class="quote"><xsl:apply-templates mode="text"/></div> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="blockquote" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('blockquote ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> + </xsl:when> + <xsl:otherwise> + <div class="blockquote"><xsl:apply-templates mode="text"/></div> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="set-off" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('set-off ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> + </xsl:when> + <xsl:otherwise> + <div class="set-off"><xsl:apply-templates mode="text"/></div> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="reg" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('reg ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <span class="reg"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="var" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('reg ', @type, ' ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <span class="var"> + <xsl:attribute name="class"><xsl:value-of select="concat('var ', @type)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="num" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="'num'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="@style"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </span> + </xsl:when> + <xsl:otherwise> + <span class="num"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="gap" mode="text"> + <xsl:variable name="extent" select="@extent"/> + <xsl:variable name="count"> + <xsl:choose> + <xsl:when test="empty($extent)"><xsl:value-of select="number(3)"/></xsl:when> + <xsl:otherwise><xsl:value-of select="number($extent)"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="gapChars" select="'...'"/> + <xsl:value-of select="concat('[', $gapChars, ']')"/><xsl:apply-templates mode="text"/> +</xsl:template> + +<xsl:template match="anchor" mode="text"> + <a><xsl:attribute name="href"><xsl:value-of select="@xlink:href"/></xsl:attribute><xsl:apply-templates mode="text"/></a> +</xsl:template> + +<!-- GIS Elements --> +<xsl:template match="place" mode="text"> + <span class="place"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="time" mode="text"> + <span class="time"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="person" mode="text"> + <span class="person"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="event" mode="text"> + <span class="event"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<!-- XHTML: remove the xhtml namespace --> +<xsl:template match="xhtml:*" mode="text"> + <xsl:variable name="hasLabel" select="string(@xlink:label) != ''"/> + <xsl:variable name="isTable" select="name() = 'table'"/> + <xsl:choose> + <xsl:when test="(not($hasLabel)) or ($isTable and $hasLabel)"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="figure" mode="text"> + <span class="figure"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="caption" mode="text"> + <span class="figureCaption"><xsl:apply-templates mode="text"/></span> +</xsl:template> + +<xsl:template match="description" mode="text"> + <span class="figureDescription"><xsl:apply-templates mode="text"/></span> +</xsl:template> + +<xsl:template match="handwritten" mode="text"> + <span class="handwritten"><xsl:apply-templates mode="text"/></span> +</xsl:template> + + +<!-- words --> +<xsl:template match="w" mode="text"> + <xsl:variable name="wordLanguage" select="@lang"/> + <xsl:variable name="form" select="encode-for-uri(@form)"/> + <xsl:variable name="displayWord"><xsl:apply-templates mode="text"/></xsl:variable> + <xsl:variable name="displayWordUrlEncoded" select="encode-for-uri($displayWord)"/> + <a class="textPollux"> + <xsl:attribute name="href"><xsl:value-of select="concat($polluxServiceName, '?query=', $form, '&queryDisplay=', $displayWordUrlEncoded, '&language=', $wordLanguage, '&outputFormat=html', '&outputType=morphCompact&outputType=dictFull')"/></xsl:attribute> + <xsl:sequence select="$displayWord"/> + </a> +</xsl:template> + +<xsl:template match="s" mode="text"> + <span class="s"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="text()" mode="text"> + <xsl:value-of select="."/> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/transform/pageXhtml.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,820 @@ +<?xml version="1.0"?> +<xsl:stylesheet version="2.0" + xmlns:xsl="http://www.w3.org/1999/XSL/Transform" + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:xs="http://www.w3.org/2001/XMLSchema" + xmlns:functx="http://www.functx.com" + xmlns:saxon="http://saxon.sf.net/" + xmlns:dc="http://purl.org/dc/elements/1.1/" + xmlns:dcterms="http://purl.org/dc/terms" + xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" + xmlns:math="http://www.w3.org/1998/Math/MathML" + xmlns:mml="http://www.w3.org/1998/Math/MathML" + xmlns:svg="http://www.w3.org/2000/svg" + xmlns:xhtml="http://www.w3.org/1999/xhtml"> + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:variable name="collectionName" select="'echo'"/> +<xsl:variable name="polluxServiceName" select="'http://mpdl-service.mpiwg-berlin.mpg.de/mpiwg-mpdl-lt-web/lt/GetDictionaryEntries'"/> + +<xsl:template match="/"> + <xsl:apply-templates mode="text"/> +</xsl:template> + +<xsl:template match="text" mode="text"> + <xsl:apply-templates mode="text"/> + <!-- TEI Dictionary --> + <xsl:if test="not(empty(//entry))"> + <xsl:for-each select="//entry"> + <xsl:sort select="form/orth"/> + <xsl:variable name="position" select="position()"/> + <span class="entry"> + <xsl:apply-templates mode="text" select="form"/> + <xsl:apply-templates mode="text" select="sense"/> + <xsl:if test="not(empty(figure))"> + <span class="entryDiv"> + <span class="bf"><xsl:value-of select="'Figures: '"/></span> + <span class="entryDiv"> + <xsl:for-each select="figure"> + <xsl:variable name="href" select="graphic/@url"/> + <xsl:variable name="head" select="head"/> + <xsl:variable name="figDesc" select="string(figDesc)"/> + <xsl:if test="$href != ''"> + <div class="figure" style="margin-left:10px;"> + <a href="{$href}"><img alt="Figure: {$figDesc}" src="{$href}" width="200" height="200"/></a> + <br/> + <xsl:value-of select="'[Figure]: '"/><xsl:value-of select="$head"/> + </div> + </xsl:if> + </xsl:for-each> + </span> + </span> + </xsl:if> + <xsl:if test="not(empty(xr))"> + <span class="entryDiv"> + <span class="bf"><xsl:value-of select="'References: '"/></span> + <span class="entryDiv"> + <xsl:for-each select="xr/ref"> + <li><xsl:apply-templates mode="text" select="."/></li> + </xsl:for-each> + </span> + </span> + </xsl:if> + </span> + </xsl:for-each> + </xsl:if> + <!-- Notes --> + <!-- + <xsl:variable name="countEmptyPlaces" select="count($notes/note[empty(@place)])"/> + <xsl:variable name="countBottomPlaces"> + <xsl:choose> + <xsl:when test="$collectionName = 'archimedes' or $collectionName = 'tei'"> + <xsl:value-of select="count($notes/note[contains(@place, 'bottom')])"/> + </xsl:when> + <xsl:otherwise> + <xsl:value-of select="count($notes/note[contains(@position, 'bottom') or empty(string(@position))])"/> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:if test="($collectionName = 'archimedes' or $collectionName = 'tei') and count($notes/*) > 0"> + <div> + <xsl:choose> + <xsl:when test="$countBottomPlaces > 0 or $countEmptyPlaces > 0"> + <hr class="notesBottom"/> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> + <xsl:for-each select="$notes/note"> + <xsl:variable name="notePos" select="position()"/> + <xsl:variable name="place" select="@place"/> + <xsl:variable name="label" select="$notePos"/> + <xsl:variable name="uid" select="@uid"/> + <xsl:variable name="modificationDate" select="@modificationDate"/> + <xsl:variable name="noteWithoutNamespace" select="mpdl-util:copyWithoutNamespace(.)"/> + <xsl:choose> + <xsl:when test="$place = '' or empty($place) or $place = 'bottom'"> + <p> + <a> + <xsl:attribute name="name"><xsl:value-of select="concat('note-', $pageNumber, '-', $label)"/></xsl:attribute> + <xsl:attribute name="href"><xsl:value-of select="concat(urlBase, '#', 'note-', $pageNumber, '-', $label, 'ref')"/></xsl:attribute> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <xsl:value-of select="concat('note-', $pageNumber, '-', $label)"/> + </a> + <xsl:value-of select="': '"/> + <xsl:choose> + <xsl:when test="$uid = '' or empty($uid)"> + <span class="note"><xsl:apply-templates select="$noteWithoutNamespace/node()" mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span class="note"><xsl:apply-templates select="$noteWithoutNamespace/node()" mode="text"/></span> + <xsl:value-of select="concat(' [external note, ', $uid, ', ', $modificationDate, ']')"/> + </xsl:otherwise> + </xsl:choose> + </p> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> + </xsl:for-each> + </div> + </xsl:if> + <xsl:if test="$collectionName = 'echo' and count($notes/*) > 0"> + <div> + <xsl:choose> + <xsl:when test="$countBottomPlaces > 0 or $countEmptyPlaces > 0"> + <hr class="notesBottom"/> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> + <xsl:for-each select="$notes/echo:note"> + <xsl:variable name="label" select="string(@xlink:label)"/> + <xsl:variable name="place" select="@position"/> + <xsl:variable name="uid" select="@uid"/> + <xsl:variable name="modificationDate" select="@modificationDate"/> + <xsl:variable name="noteWithoutNamespace" select="mpdl-util:copyWithoutNamespace(.)"/> + <xsl:choose> + <xsl:when test="$place = '' or empty($place) or $place = 'bottom'"> + <p> + <a> + <xsl:attribute name="name"><xsl:value-of select="$label"/></xsl:attribute> + <xsl:attribute name="href"><xsl:value-of select="concat(urlBase, '#', $label, 'ref')"/></xsl:attribute> + <xsl:attribute name="class"><xsl:value-of select="'note'"/></xsl:attribute> + <xsl:value-of select="$label"/> + </a> + <xsl:value-of select="': '"/> + <xsl:choose> + <xsl:when test="$uid = '' or empty($uid)"> + <span class="note"><xsl:apply-templates select="$noteWithoutNamespace/node()" mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span class="note"><xsl:apply-templates select="$noteWithoutNamespace/node()" mode="text"/></span> + <xsl:value-of select="concat(' [external note, ', $uid, ', ', $modificationDate, ']')"/> + </xsl:otherwise> + </xsl:choose> + </p> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> + </xsl:for-each> + </div> + </xsl:if> + --> +</xsl:template> + +<xsl:template match="head" mode="text"> + <xsl:variable name="class"> + <xsl:value-of select="'head bf'"/> + </xsl:variable> + <p> + <span> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </p> +</xsl:template> + +<!-- TEI: segmentation --> +<xsl:template match="seg" mode="text"> + <xsl:choose> + <xsl:when test="@rend = 'highlight'"> + <span class="seg highlight"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="@rend = 'highlightPoint'"> + <span class="seg highlightPoint"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span class="seg"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- TEI: choice --> +<xsl:template match="choice" mode="text"> + <xsl:if test="not(empty(orig))"> + <xsl:apply-templates select="orig" mode="text"/> + </xsl:if> + <xsl:if test="not(empty(abbr))"> + <xsl:apply-templates select="abbr" mode="text"/> + </xsl:if> + <xsl:if test="not(empty(am))"> + <xsl:apply-templates select="am" mode="text"/> + </xsl:if> + <xsl:if test="not(empty(sic))"> + <xsl:apply-templates select="sic" mode="text"/> + </xsl:if> +</xsl:template> + +<!-- TEI: hi (highlighted) --> +<xsl:template match="hi" mode="text"> + <xsl:choose> + <xsl:when test="@rend = 'initial'"> + <span class="dc-unmodified"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="@rend = 'bold'"> + <span class="bf"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:attribute name="class"><xsl:value-of select="@rend"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- TEI: name (of type: place, person, ...) --> +<xsl:template match="name" mode="text"> + <xsl:choose> + <xsl:when test="@type = 'place'"> + <span class="place" title="Place"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="@type = 'person'"> + <span class="person" title="Person"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="@type = 'org'"> + <span class="organization" title="Organization"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span> + <xsl:attribute name="class"><xsl:value-of select="@type"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- TEI: place, person, ...) --> +<xsl:template match="placeName" mode="text"> + <span class="place" title="Place: {@type}"><xsl:apply-templates mode="text"/></span> +</xsl:template> +<xsl:template match="persName" mode="text"> + <span class="person" title="Person"><xsl:apply-templates mode="text"/></span> +</xsl:template> + +<!-- TEI: term --> +<xsl:template match="term" mode="text"> + <span class="term" title="Terminology entry"><xsl:apply-templates mode="text"/></span> +</xsl:template> + +<!-- TEI: line --> +<xsl:template match="lg" mode="text"> + <div class="lg"><xsl:apply-templates mode="text"/></div> +</xsl:template> + +<xsl:template match="l" mode="text"> + <span class="l"><xsl:apply-templates mode="text"/></span><br/> +</xsl:template> + +<!-- TEI: reference --> +<xsl:template match="ref" mode="text"> + <span class="ref"> + <xsl:if test="not(empty(@target))"> + <a> + <xsl:attribute name="href"><xsl:value-of select="@target"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </a> + </xsl:if> + <xsl:if test="empty(@target)"> + <xsl:apply-templates mode="text"/> + </xsl:if> + </span> +</xsl:template> + +<!-- TEI: table --> +<xsl:template match="table" mode="text"> + <table> + <xsl:if test="not(empty(head))"> + <caption align="top"><xsl:apply-templates mode="text" select="head"/></caption> + </xsl:if> + <xsl:apply-templates mode="text" select="row"/> + </table> +</xsl:template> + +<xsl:template match="row" mode="text"> + <xsl:choose> + <xsl:when test="@role = 'label'"> + <tr style="font-weight:bold;"><xsl:apply-templates mode="text"/></tr> + </xsl:when> + <xsl:when test="@role = 'data' or empty(@role)"> + <tr><xsl:apply-templates mode="text"/></tr> + </xsl:when> + <xsl:otherwise> + <tr><xsl:apply-templates mode="text"/></tr> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="cell" mode="text"> + <xsl:choose> + <xsl:when test="@role = 'label' and empty(@cols)"> + <td style="font-weight:bold;"><xsl:apply-templates mode="text"/></td> + </xsl:when> + <xsl:when test="@role = 'label' and not(empty(@cols))"> + <td colspan="{@cols}" style="font-weight:bold;"><xsl:apply-templates mode="text"/></td> + </xsl:when> + <xsl:when test="(@role = 'data' or empty(@role)) and empty(@cols)"> + <td><xsl:apply-templates mode="text"/></td> + </xsl:when> + <xsl:when test="(@role = 'data' or empty(@role)) and not(empty(@cols))"> + <td colspan="{@cols}"><xsl:apply-templates mode="text"/></td> + </xsl:when> + <xsl:otherwise> + <td><xsl:apply-templates mode="text"/></td> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<!-- TEI: dictionary --> +<xsl:template match="entry" mode="text"> + <!-- empty: handled in text tag --> +</xsl:template> + +<xsl:template match="form" mode="text"> + <span class="form"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="orth" mode="text"> + <span class="orth"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="sense" mode="text"> + <span class="entryDiv"><xsl:apply-templates mode="text" select="def"/></span> + <span class="entryDiv"><xsl:apply-templates mode="text" select="etym"/></span> +</xsl:template> + +<xsl:template match="def" mode="text"> + <span class="def"> + <span class="bf"><xsl:value-of select="'Definition: '"/></span> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="mentioned" mode="text"> + <span class="mentioned"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="etym" mode="text"> + <span class="etym"> + <span class="bf"><xsl:value-of select="'Etymology: '"/></span> + <span class="entryDiv"> + <xsl:for-each select="cit"> + <li><xsl:value-of select="quote"/><xsl:value-of select="' ('"/><xsl:value-of select="def"/><xsl:value-of select="')'"/></li> + </xsl:for-each> + </span> + </span> +</xsl:template> + +<xsl:template match="cit" mode="text"> + <span class="cit"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<!-- TEI: MathML --> +<xsl:template match="math:*" mode="text"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> +</xsl:template> + +<!-- TEI: SVG --> +<xsl:template match="svg:*" mode="text"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> +</xsl:template> + +<xsl:template match="div" mode="text"> + <xsl:variable name="type" select="@type"/> + <xsl:variable name="level" select="@level"/> + <xsl:variable name="style" select="@style"/> + <xsl:variable name="border" select="@border"/> + <xsl:variable name="width" select="@width"/> + <xsl:variable name="note" select="note"/> + <xsl:variable name="figure" select="figure"/> + <xsl:variable name="floatClassValue"> + <xsl:choose> + <xsl:when test="not(empty($note))"><xsl:value-of select="''"/></xsl:when> + <xsl:when test="not(empty($figure))"><xsl:value-of select="''"/></xsl:when> + <xsl:when test="$type = 'float'"><xsl:value-of select="'float left'"/></xsl:when> + <xsl:otherwise><xsl:value-of select="'float left'"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="levelClassValue"> + <xsl:choose> + <xsl:when test="not(empty($note))"><xsl:value-of select="''"/></xsl:when> + <xsl:when test="$level = '1'"><xsl:value-of select="'level one'"/></xsl:when> + <xsl:when test="$level = '2'"><xsl:value-of select="'level two'"/></xsl:when> + <xsl:when test="$level = '3'"><xsl:value-of select="'level three'"/></xsl:when> + <xsl:when test="$level = '4'"><xsl:value-of select="'level four'"/></xsl:when> + <xsl:when test="$level = '5'"><xsl:value-of select="'level five'"/></xsl:when> + <xsl:when test="$level = '6'"><xsl:value-of select="'level six'"/></xsl:when> + <xsl:when test="$level = '7'"><xsl:value-of select="'level seven'"/></xsl:when> + <xsl:when test="$level = '8'"><xsl:value-of select="'level eight'"/></xsl:when> + <xsl:when test="$level = '9'"><xsl:value-of select="'level nine'"/></xsl:when> + <xsl:when test="empty($level)"><xsl:value-of select="''"/></xsl:when> + <xsl:otherwise><xsl:value-of select="'level n'"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <div> + <xsl:if test="not(empty($style))"><xsl:attribute name="style"><xsl:value-of select="$style"/></xsl:attribute></xsl:if> + <xsl:attribute name="class"><xsl:value-of select="concat($floatClassValue, ' ', $levelClassValue)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> +</xsl:template> + +<xsl:template match="p" mode="text"> + <xsl:variable name="style" select="@style"/> + <xsl:variable name="class"> + <xsl:value-of select="'p'"/> + </xsl:variable> + <xsl:choose> + <xsl:when test="not(empty($style))"> + <div> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </div> + </xsl:when> + <xsl:otherwise> + <div> + <xsl:attribute name="class"><xsl:value-of select="$class"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="lb|br" mode="text"> + <br></br><xsl:apply-templates mode="text"/> +</xsl:template> + +<xsl:template match="cb" mode="text"> + <br/><xsl:apply-templates mode="text"/> +</xsl:template> + +<xsl:template match="expan" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('expan ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <span class="expan"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="note" mode="text"> + <!-- + <xsl:variable name="hasLabel" select="string(@xlink:label) != ''"/> + <xsl:variable name="place"> + <xsl:choose> + <xsl:when test="$collectionName = 'tei'"><xsl:value-of select="@place"/></xsl:when> + <xsl:otherwise><xsl:value-of select="@position"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="notePos" select="count(preceding::note[. >> $topPB]) + 1"/> + <xsl:variable name="href" select="concat('note-', $pageNumber, '-', $notePos)"/> + <xsl:choose> + <xsl:when test="$collectionName = 'archimedes'"> + <a> + <xsl:attribute name="name"><xsl:value-of select="concat($href, 'ref')"/></xsl:attribute> + <xsl:attribute name="href"><xsl:value-of select="concat(urlBase, '#', $href)"/></xsl:attribute> + <xsl:attribute name="class"><xsl:value-of select="'noteRef super'"/></xsl:attribute> + <xsl:value-of select="$href"/> + </a> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and not($hasLabel) and not(empty($place))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('note margin ', $place)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and not($hasLabel)"> + <p> + <xsl:value-of select="'[Note]: '"/> + <span class="note"><xsl:apply-templates mode="text"/></span> + </p> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and $hasLabel"></xsl:when> + <xsl:when test="$collectionName = 'tei'"> + <xsl:choose> + <xsl:when test="contains($place, 'margin')"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('note ', $place)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <a> + <xsl:attribute name="name"><xsl:value-of select="concat($href, 'ref')"/></xsl:attribute> + <xsl:attribute name="href"><xsl:value-of select="concat(urlBase, '#', $href)"/></xsl:attribute> + <xsl:attribute name="class"><xsl:value-of select="'noteRef super'"/></xsl:attribute> + <xsl:value-of select="$href"/> + </a> + </xsl:otherwise> + </xsl:choose> + </xsl:when> + <xsl:otherwise> + <span class="note"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> + --> +</xsl:template> + +<xsl:template match="emph" mode="text"> + <xsl:variable name="style" select="@style"/> + <xsl:variable name="styleWithoutSC" select="replace($style, 'sc ', '')"/> + <xsl:variable name="text" select="string-join(., '')"/> + <xsl:variable name="length" select="string-length($text)"/> + <xsl:variable name="firstChar" select="substring($text, 1, 1)"/> + <xsl:variable name="first2Chars" select="substring($text, 1, 2)"/> + <xsl:variable name="restChars" select="substring($text, 2, $length)"/> + <xsl:variable name="first2CharsAreUppercase" select="upper-case($first2Chars) = $first2Chars"/> + <!-- an emph as first element in p and also an emph in s in p is recognized --> + <xsl:variable name="isFirstElementInP" select="(not(empty(./parent::*/parent::p)) and (empty(./parent::*/preceding-sibling::node()) and empty(./preceding-sibling::node()))) or (not(empty(./parent::p)) and empty(./preceding-sibling::node()))"/> + <xsl:variable name="rest"> + <xsl:choose> + <xsl:when test="$length < 2 or empty($length)"><xsl:value-of select="''"/></xsl:when> + <xsl:otherwise> + <xsl:choose> + <xsl:when test="not(empty(w))"> + <a class="textPollux" href="{$polluxServiceName}?query={w/@form}&queryDisplay={$text}&language={w/@lang}&outputFormat=html&outputType=morphCompact&outputType=dictFull"><xsl:value-of select="$restChars"/></a> + </xsl:when> + <xsl:otherwise><xsl:value-of select="$restChars"/></xsl:otherwise> + </xsl:choose> + </xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:choose> + <xsl:when test="$collectionName = 'echo' and not(contains($style, 'sc'))"> + <span class="{$style}"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and $style = 'sc' and $length = 1"> + <span class="sc"><xsl:value-of select="$firstChar"/></span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and $style = 'sc' and not($isFirstElementInP)"> + <span class="sc"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <!-- special case (see text() node below): first char dc, rest sc --> + <xsl:when test="$collectionName = 'echo' and $style = 'dc sc logic' and $first2CharsAreUppercase"> + <span class="dc"><xsl:value-of select="$firstChar"/></span><span class="sc"><xsl:sequence select="$rest"/></span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and $style = 'sc' and $first2CharsAreUppercase"> + <span class="dc"><xsl:value-of select="$firstChar"/></span><span class="sc"><xsl:sequence select="$rest"/></span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and contains($style, 'sc') and $first2CharsAreUppercase"> + <span class="dc"><span class="{$styleWithoutSC}"><xsl:value-of select="$firstChar"/></span></span><span class="sc"><span class="{$styleWithoutSC}"><xsl:sequence select="$rest"/></span></span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and $style = 'sc' and not($first2CharsAreUppercase)"> + <span class="sc"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="$collectionName = 'echo' and contains($style, 'sc') and not($first2CharsAreUppercase)"> + <span class="sc"><span class="{$styleWithoutSC}"><xsl:apply-templates mode="text"/></span></span> + </xsl:when> + <xsl:when test="$collectionName = 'archimedes' and $style != ''"> + <span class="{$style}"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="$collectionName = 'archimedes' and $style = ''"> + <span class="emph"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:when test="$style != ''"> + <span class="{$style}"><xsl:apply-templates mode="text"/></span> + </xsl:when> + <xsl:otherwise> + <span class="emph"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="foreign" mode="text"> + <xsl:variable name="lang" select="@lang"/> + <xsl:variable name="xmllang" select="@xml:lang"/> + <xsl:variable name="language"> + <xsl:choose> + <xsl:when test="not(empty($xmllang))"><xsl:value-of select="$xmllang"/></xsl:when> + <xsl:when test="not(empty($lang))"><xsl:value-of select="$lang"/></xsl:when> + <xsl:otherwise><xsl:value-of select="''"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('foreign ', $language)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="q" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('q ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <span class="q"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="quote" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('quote ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> + </xsl:when> + <xsl:otherwise> + <div class="quote"><xsl:apply-templates mode="text"/></div> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="blockquote" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('blockquote ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> + </xsl:when> + <xsl:otherwise> + <div class="blockquote"><xsl:apply-templates mode="text"/></div> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="set-off" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <div> + <xsl:attribute name="class"><xsl:value-of select="concat('set-off ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </div> + </xsl:when> + <xsl:otherwise> + <div class="set-off"><xsl:apply-templates mode="text"/></div> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="reg" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('reg ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <span class="reg"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="var" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="concat('reg ', @type, ' ', @style)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:when> + <xsl:otherwise> + <span class="var"> + <xsl:attribute name="class"><xsl:value-of select="concat('var ', @type)"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="num" mode="text"> + <xsl:choose> + <xsl:when test="not(empty(@style))"> + <span> + <xsl:attribute name="class"><xsl:value-of select="'num'"/></xsl:attribute> + <span> + <xsl:attribute name="class"><xsl:value-of select="@style"/></xsl:attribute> + <xsl:apply-templates mode="text"/> + </span> + </span> + </xsl:when> + <xsl:otherwise> + <span class="num"><xsl:apply-templates mode="text"/></span> + </xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="gap" mode="text"> + <xsl:variable name="extent" select="@extent"/> + <xsl:variable name="count"> + <xsl:choose> + <xsl:when test="empty($extent)"><xsl:value-of select="number(3)"/></xsl:when> + <xsl:otherwise><xsl:value-of select="number($extent)"/></xsl:otherwise> + </xsl:choose> + </xsl:variable> + <xsl:variable name="gapChars" select="'...'"/> + <xsl:value-of select="concat('[', $gapChars, ']')"/><xsl:apply-templates mode="text"/> +</xsl:template> + +<xsl:template match="anchor" mode="text"> + <a><xsl:attribute name="href"><xsl:value-of select="@xlink:href"/></xsl:attribute><xsl:apply-templates mode="text"/></a> +</xsl:template> + +<!-- GIS Elements --> +<xsl:template match="place" mode="text"> + <span class="place"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="time" mode="text"> + <span class="time"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="person" mode="text"> + <span class="person"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="event" mode="text"> + <span class="event"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<!-- XHTML: remove the xhtml namespace --> +<xsl:template match="xhtml:*" mode="text"> + <xsl:variable name="hasLabel" select="string(@xlink:label) != ''"/> + <xsl:variable name="isTable" select="name() = 'table'"/> + <xsl:choose> + <xsl:when test="(not($hasLabel)) or ($isTable and $hasLabel)"> + <xsl:element name="{name()}" namespace=""> + <xsl:copy-of select="@*"/> + <xsl:apply-templates mode="text"/> + </xsl:element> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> +</xsl:template> + +<xsl:template match="figure" mode="text"> + <span class="figure"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="caption" mode="text"> + <span class="figureCaption"><xsl:apply-templates mode="text"/></span> +</xsl:template> + +<xsl:template match="description" mode="text"> + <span class="figureDescription"><xsl:apply-templates mode="text"/></span> +</xsl:template> + +<xsl:template match="handwritten" mode="text"> + <span class="handwritten"><xsl:apply-templates mode="text"/></span> +</xsl:template> + + +<!-- words --> +<xsl:template match="w" mode="text"> + <xsl:variable name="wordLanguage" select="@lang"/> + <xsl:variable name="form" select="encode-for-uri(@form)"/> + <xsl:variable name="displayWord"><xsl:apply-templates mode="text"/></xsl:variable> + <xsl:variable name="displayWordUrlEncoded" select="encode-for-uri($displayWord)"/> + <a class="textPollux"> + <xsl:attribute name="href"><xsl:value-of select="concat($polluxServiceName, '?query=', $form, '&queryDisplay=', $displayWordUrlEncoded, '&language=', $wordLanguage, '&outputFormat=html', '&outputType=morphCompact&outputType=dictFull')"/></xsl:attribute> + <xsl:sequence select="$displayWord"/> + </a> +</xsl:template> + +<xsl:template match="s" mode="text"> + <span class="s"> + <xsl:apply-templates mode="text"/> + </span> +</xsl:template> + +<xsl:template match="text()" mode="text"> + <xsl:value-of select="."/> +</xsl:template> + +</xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/transform/queryDocument.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,93 @@ +<?xml version="1.0"?> + +<xsl:stylesheet version="2.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform"> + +<xsl:output method="xhtml" encoding="utf-8"/> + +<xsl:param name="query"></xsl:param> +<xsl:param name="flags"></xsl:param> +<xsl:param name="outputFormat"></xsl:param> + +<xsl:variable name="apos">'</xsl:variable> +<xsl:variable name="xpathQuery"> + <xsl:choose> + <xsl:when test="$flags = ''"><xsl:value-of select="concat('//*:s[matches(.,', $apos, $query, $apos, ')]', '|', '//*:head[matches(.,', $apos, $query, $apos, ')]')"/></xsl:when> + <xsl:otherwise><xsl:value-of select="concat('//*:s[matches(.,', $apos, $query, $apos, ',', $apos, $flags, $apos, ')]', '|', '//*:head[matches(.,', $apos, $query, $apos, ',', $apos, $flags, $apos, ')]')"/></xsl:otherwise> + </xsl:choose> +</xsl:variable> +<xsl:variable name="queryResult" select="saxon:evaluate($xpathQuery)" xmlns:saxon="http://saxon.sf.net/"/> +<xsl:variable name="queryResultSize"><xsl:value-of select="count($queryResult)"/></xsl:variable> +<xsl:variable name="queryResultPageSize" select="20"/> +<xsl:variable name="queryResultPages"> + <xsl:choose> + <xsl:when test="$queryResultSize = 0"><xsl:value-of select="count($queryResult)"/></xsl:when> + <xsl:otherwise><xsl:value-of select="$queryResultSize idiv $queryResultPageSize + 1"/></xsl:otherwise> + </xsl:choose> +</xsl:variable> +<xsl:variable name="pageBreaks" select="saxon:evaluate('//*:pb')" xmlns:saxon="http://saxon.sf.net/"/> +<xsl:variable name="queryResultHtml"> +<xsl:choose> + <xsl:when test="$outputFormat = 'html'"> + <table> + <thead> + <tr> + <th align="left" valign="top"> + <button name="order-by" value="author" style="padding:0px;font-weight:bold;font-size:14px;background:none;border:none;">No.</button> + </th> + <th align="left" valign="top"> + <button name="order-by" value="author" style="padding:0px;font-weight:bold;font-size:14px;background:none;border:none;">Path</button> + </th> + <th align="left" valign="top"> + <button name="order-by" value="author" style="padding:0px;font-weight:bold;font-size:14px;background:none;border:none;">Hit</button> + </th> + <th align="left" valign="top"> + <button name="order-by" value="author" style="padding:0px;font-weight:bold;font-size:14px;background:none;border:none;">Page</button> + </th> + </tr> + </thead> + <tbody> + <xsl:for-each select="$queryResult"> + <xsl:variable name="hit" select="."/> + <tr> + <td align="left" valign="top"><xsl:value-of select="position()"/></td> + <td align="left" valign="top"><xsl:value-of select="saxon:path(.)" xmlns:saxon="http://saxon.sf.net/"/></td> + <td align="left" valign="top"><xsl:value-of select="."/></td> + <td align="left" valign="top"><xsl:value-of select="count(./preceding::*:pb)"/></td> <!-- better performance: count($pageBreaks[. << $hit]) --> + </tr> + </xsl:for-each> + </tbody> + </table> + </xsl:when> + <xsl:otherwise> + <result> + <query><xsl:value-of select="$query"/></query> + <flags><xsl:value-of select="$flags"/></flags> + <size><xsl:value-of select="$queryResultSize"/></size> + <page-size><xsl:value-of select="$queryResultPageSize"/></page-size> + <pages><xsl:value-of select="$queryResultPages"/></pages> + <pn>1</pn> + <hits> + <xsl:for-each select="$queryResult"> + <xsl:variable name="hit" select="."/> + <hit> + <hitType><xsl:value-of select="'s'"/></hitType> + <pos><xsl:value-of select="position()"/></pos> + <pn><xsl:value-of select="count(./preceding::*:pb)"/></pn> + <hitId>xmlId</hitId> + <hitPos>4711</hitPos> + <hitString><xsl:value-of select="."/></hitString> + <hitSurroundsPB>false</hitSurroundsPB> + </hit> + </xsl:for-each> + </hits> + <query-forms></query-forms> + <query-regularizations></query-regularizations> + </result> + </xsl:otherwise> +</xsl:choose> +</xsl:variable> +<xsl:template match="/"> + <xsl:sequence select="$queryResultHtml"/> +</xsl:template> + +</xsl:stylesheet>
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Binary file software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/xquery/Hits.class has changed
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--- a/software/mpdl-services/mpiwg-mpdl-xml/examples/Aristoteles_1585_XSY685ZD-Ids.xml Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1299 +0,0 @@ -<?xml version="1.0" encoding="utf-8"?> -<echo xmlns="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" - xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" - xmlns:dcterms="http://purl.org/dc/terms" - xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" - xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" - xmlns:xhtml="http://www.w3.org/1999/xhtml" - xmlns:mml="http://www.w3.org/1998/Math/MathML" - xmlns:xlink="http://www.w3.org/1999/xlink" - version="1.0RC"> - <metadata> - <dcterms:identifier>ECHO:XSY685ZD.xml</dcterms:identifier> - <dcterms:creator identifier="GND:118650130">Aristoteles</dcterms:creator> - <dcterms:contributor identifier="GND:102377138">Leonicus Thomaeus, Nicolaus</dcterms:contributor> - <dcterms:title xml:lang="la">Quaestiones Mechanicae</dcterms:title> - <dcterms:date xsi:type="dcterms:W3CDTF">1585</dcterms:date> - <dcterms:language xsi:type="dcterms:ISO639-3">lat</dcterms:language> - <dcterms:rights>open access</dcterms:rights> - <dcterms:license>http://echo.mpiwg-berlin.mpg.de/policy/oa_basics/declaration</dcterms:license> - <dcterms:accessRights>free</dcterms:accessRights> - </metadata> - <text xml:lang="la" type="free"> - <div type="section" level="1" n="1"> - <pb o="504" file="0504" n="1"/> - </div> - <div type="section" level="1" n="2"> - <head xml:id="head1" xml:space="preserve">ARISTOTELIS</head> - <head xml:id="head2" xml:space="preserve">STAGIRITAE, <lb/>PERIPATETICORVM</head> - <head xml:id="head3" xml:space="preserve">PRINCIPIS,</head> - <head xml:id="head4" xml:space="preserve">Quæſtiones Mechanicæ, <lb/>Nicolao Leonico Thomæo <lb/>interprete.</head> - <head xml:id="head5" style="it" xml:space="preserve">Quæ Å¿it artis Mechanicæ facultas.</head> - <p> - <s xml:id="s1" xml:space="preserve">MIraculo Å¿une ea quidem, quæ natura <lb/>contingunt, quorum ignorantur cau <lb/>ſæ: </s> - <s xml:id="s2" xml:space="preserve">illa uerò, quæ prÄ™ter naturam, <lb/>quæcunque ad hominum vtilitatem <lb/>arte fiunt. </s> - <s xml:id="s3" xml:space="preserve">In multis. </s> - <s xml:id="s4" xml:space="preserve">n. </s> - <s xml:id="s5" xml:space="preserve">natura ei. </s> - <s xml:id="s6" xml:space="preserve">qÄ‘ <lb/>nobis uÅ¿ui eſſe poteÅ¿t, contrarium facit: </s> - <s xml:id="s7" xml:space="preserve">natura <lb/>etenim eundem Å¿emper habet modum, & </s> - <s xml:id="s8" xml:space="preserve">Å¿impli <lb/>citer: </s> - <s xml:id="s9" xml:space="preserve">vtile autem multifariam commutat̃. </s> - <s xml:id="s10" xml:space="preserve">Quan <lb/>doigitur quippiam præter naturam oportuerit <lb/>facere, difficultate Å¿ua hæſitationem praÅ¿tat, ar-<lb/>teqÌ indiget: </s> - <s xml:id="s11" xml:space="preserve">quam ob rem eam artis partẽ, quæ <lb/>huiuÅ¿modi Å¿uccurrit difficulcatibus, Mechanicã <lb/>appellamus. </s> - <s xml:id="s12" xml:space="preserve">Quemad modum enim Antipho Å¿cri <lb/>bit poeta, Å¿ic Å¿e res habet: </s> - <s xml:id="s13" xml:space="preserve">arte enim Å¿uperamus <lb/>da, à quibus natura uincimur. </s> - <s xml:id="s14" xml:space="preserve">HuiuÅ¿modi autem <lb/>Å¿unt, in quibus & </s> - <s xml:id="s15" xml:space="preserve">minora Å¿uperant maiora: </s> - <s xml:id="s16" xml:space="preserve">& </s> - <s xml:id="s17" xml:space="preserve"><lb/>quæcunq; </s> - <s xml:id="s18" xml:space="preserve">momentum paruum habẽtia, magna -<pb o="505" file="0505" n="2" rhead="Mechanicæ."/> -mouent pondera: </s> - <s xml:id="s19" xml:space="preserve">& </s> - <s xml:id="s20" xml:space="preserve">omnia ferè illa, quæ mecha <lb/>nica nuncupamus, & </s> - <s xml:id="s21" xml:space="preserve">problemata. </s> - <s xml:id="s22" xml:space="preserve">Sunt autem, <lb/>hæc neque naturalibus omnino quæſtionibus <lb/>eadem, neque Å¿eiugata valde: </s> - <s xml:id="s23" xml:space="preserve">verùm mathema-<lb/>ticarÅ© contemplationÅ© naturaliumqÌ; </s> - <s xml:id="s24" xml:space="preserve">cõmunia.</s> - <s xml:id="s25" xml:space="preserve"/> - </p> - </div> - <div type="section" level="1" n="3"> - <head xml:id="head6" style="it" xml:space="preserve">De dignitatiòus, admirandis̈ Circuli <lb/>proprietatibus.</head> - <p> - <s xml:id="s26" xml:space="preserve">DEnumeto autem eorum, quæ hoc in gene-<lb/>re dubitantur, illa eſſe dicuntur, quæ circæ <lb/>vectem fiunt. </s> - <s xml:id="s27" xml:space="preserve">AbÅ¿urdum enim eſſe videtur, ma-<lb/>gnum moueri pondus ab exigua virtute, cÅ© plu-<lb/>ri præſertim pondere. </s> - <s xml:id="s28" xml:space="preserve">quod. </s> - <s xml:id="s29" xml:space="preserve">n. </s> - <s xml:id="s30" xml:space="preserve">una vecte quiÅ¿piã <lb/>mouere nõ poteÅ¿t, idipÅ¿um ponderis citius mo-<lb/>uet, uectis ad illud pondus adiungens. </s> - <s xml:id="s31" xml:space="preserve">OÄ©um ãt <lb/>huiuÅ¿modi cauſæ principiÅ© habet circulus, IÅ¿tud <lb/>verò ratione contingit; </s> - <s xml:id="s32" xml:space="preserve">ex ad mirabili etemÌ„ mi-<lb/>randum accidere quippiam, non eÅ¿t ab Å¿urdum.</s> - <s xml:id="s33" xml:space="preserve"/> - </p> - <p> - <s xml:id="s34" xml:space="preserve">Maximè autem eÅ¿t admirãdum Å¿imul contra-<lb/>ria fieri. </s> - <s xml:id="s35" xml:space="preserve">Circulus vero ex huiuÅ¿modi eÅ¿t conÅ¿titu <lb/>tus: </s> - <s xml:id="s36" xml:space="preserve">Å¿tatim enim ex commoto effectus eÅ¿t, & </s> - <s xml:id="s37" xml:space="preserve">mæ <lb/>nente, quorÅ© natura ad Å¿e inuicem eÅ¿t cõtraria. <lb/></s> - <s xml:id="s38" xml:space="preserve">quamobrẽ iÅ¿thæc cernentes minus admirari cõ-<lb/>uenit contingentes in illo contrarietas. </s> - <s xml:id="s39" xml:space="preserve">In pri-<lb/>mis enim lineæ illi, quæ circuli orbem amplecti <lb/>tur, nullam habenti latitudinem, contraria quo-<lb/>dammodo ineſſe apparent, concauum Å¿cilicet, <lb/>& </s> - <s xml:id="s40" xml:space="preserve">curuum. </s> - <s xml:id="s41" xml:space="preserve">hæc aÅ©t eo à ſeinuicem diÅ¿tant mõ, <lb/>quo magnum & </s> - <s xml:id="s42" xml:space="preserve">paruum, illorÅ© etenim medium <lb/>eÅ¿t æquale: </s> - <s xml:id="s43" xml:space="preserve">horum verò rectÅ© quapropter cùm, <lb/>ad Å¿einuicem cõmutantur, illa quidem prius æ-<lb/>qualia fieri neceſſe eÅ¿t, quà m extremorum vtrÅ©-<lb/>libet: </s> - <s xml:id="s44" xml:space="preserve">lineam verò rectam, qñ eÅ¿t curua concaua, <lb/>ut ex huiu Å¿moà <unsure/>i rurÅ¿um curua fit, & </s> - <s xml:id="s45" xml:space="preserve">circularis -<pb o="506" file="0506" n="3" rhead="AriÅ¿t. Quæſtiones"/> -Vnum quidem igitur iÅ¿tuc abÅ¿urdum ineÅ¿t circti<unsure/> <lb/>lo. </s> - <s xml:id="s46" xml:space="preserve">Alterum autem, ê™ Å¿imul contrarijs mouetur <lb/>motionibus: </s> - <s xml:id="s47" xml:space="preserve">Å¿imul enim ad anteriorem mouetur <lb/>locum, & </s> - <s xml:id="s48" xml:space="preserve">ad poÅ¿ter<unsure/>iorem. </s> - <s xml:id="s49" xml:space="preserve">Et ea, quæ circulÅ© de-<lb/>Å¿cribit, linea eodem Å¿e habet modo: </s> - <s xml:id="s50" xml:space="preserve">ex quo. </s> - <s xml:id="s51" xml:space="preserve">n. </s> - <s xml:id="s52" xml:space="preserve">in <lb/>cipit loco illius extremum, ad eundem rurlus re <lb/>dit:</s> - <s xml:id="s53" xml:space="preserve"><unsure/> illa. </s> - <s xml:id="s54" xml:space="preserve">n. </s> - <s xml:id="s55" xml:space="preserve">continuò commota, extremum rurÅ¿us <lb/>efficitur primum. </s> - <s xml:id="s56" xml:space="preserve">quamobrem manifeÅ¿tum, ê™ in <lb/>de mutatnm eÅ¿t. </s> - <s xml:id="s57" xml:space="preserve">quapropter (vt dictum eÅ¿t prius) <lb/>non eÅ¿t incõueniens ipÅ¿<unsure/>u<unsure/>m miracolorum oÄ©um <lb/>eſſe principlum. </s> - <s xml:id="s58" xml:space="preserve">Ea igi ur, quæ circa libram fiÅ©t, <lb/>ad circulum referuntur: </s> - <s xml:id="s59" xml:space="preserve">quæ vero circa vectem, <lb/>ad ipÅ¿am librã: </s> - <s xml:id="s60" xml:space="preserve">alia autem ferè omn<unsure/>ia, quæ circa <lb/>mÄ™chanicas Å¿unt motiones, ad vectern<unsure/>. </s> - <s xml:id="s61" xml:space="preserve">Prætereà <lb/>etiã, quoniam vnica exiÅ¿tente, quæ ex cehtro eÅ¿t, <lb/>linea, nullum aliud alij, quæ in illa Å¿unt puncto-<lb/>rum æqua velocitate feratur; </s> - <s xml:id="s62" xml:space="preserve">Å¿ed cirius Å¿emper <lb/>quod à manente termino eÅ¿t remo ius, pleraque <lb/>miraculorum accidÅ©t in circuli motionibus: </s> - <s xml:id="s63" xml:space="preserve">de <lb/>quibus in ijs, quæ poÅ¿thac addacentur, quęſtioni <lb/>bus erit manifeÅ¿tum. </s> - <s xml:id="s64" xml:space="preserve">Quoniam autem Å¿ecundÅ© <lb/>contrarias fimul motiones mouetur circulus: </s> - <s xml:id="s65" xml:space="preserve">& </s> - <s xml:id="s66" xml:space="preserve"><lb/>alterÅ© quidem diametri extremum vbi A, in an-<lb/>tè mouetur, alterum verò vbi B. </s> - <s xml:id="s67" xml:space="preserve">ad retro; </s> - <s xml:id="s68" xml:space="preserve">effi-<lb/>ciunt nonnulli, vt ab vnica motione multi cõtra <lb/>rio Å¿imul moueantur circuli: </s> - <s xml:id="s69" xml:space="preserve">quemadmodum, <lb/>Å¿unt illi, quos in locis proponÅ©t Å¿acris, æneos, & </s> - <s xml:id="s70" xml:space="preserve"><lb/>ferreos, fabricantes orbiculos. </s> - <s xml:id="s71" xml:space="preserve">Si enim A B circu <lb/>lum alter contigerit circulus in quo CD. </s> - <s xml:id="s72" xml:space="preserve">mota <lb/>circuli in quo A B, diametro in antè, mouebitur <lb/>C D ad retro, diametro circuli vbi eÅ¿t A, circa idẽ <lb/>mota. </s> - <s xml:id="s73" xml:space="preserve">In contrarium igitur mouebitur circulus <lb/>vbi CD, ad illum ubi AB. </s> - <s xml:id="s74" xml:space="preserve">& </s> - <s xml:id="s75" xml:space="preserve">rurÅ¿us ipÅ¿e conti-<lb/>guum ubi EF, in contrarium Å¿ibijpÅ¿i mouebitur <lb/>propter eandem cauÅ¿am. </s> - <s xml:id="s76" xml:space="preserve">Eodem ẽt modo Å¿i plu- -<pb o="507" file="0507" n="4" rhead="Mechanicæ."/> -res fuerint, idẽ facient, uno Å¿olo commoto. </s> - <s xml:id="s77" xml:space="preserve">Hane <lb/>igitur in circulo exiÅ¿tentem animaduertens na-<lb/>turam architecti, in Å¿trumentÅ© fabricant, celãtes <lb/>principium, vt machinæ Å¿olum manifeftum Å¿it il <lb/>lud, quod admirationẽ pręſtat, cauÅ¿a verò lateat.</s> - <s xml:id="s78" xml:space="preserve"/> - </p> - <p> - <s xml:id="s79" xml:space="preserve">In primis igitur, quæ accidunt circa librã, du-<lb/>bitare faciunt, quamnà m ob cauÅ¿am exactiores <lb/>minoribus maiores Å¿unt libræ. </s> - <s xml:id="s80" xml:space="preserve">huius autem rei <lb/>principium eÅ¿t, quamobrem in ipÅ¿o circulo, quæ <lb/>plus à cent<unsure/>ro diÅ¿ter linea, eadẽ ui cõmota cit<unsure/>ius <lb/>fertur, quà m illa, quæ minus diÅ¿tar. </s> - <s xml:id="s81" xml:space="preserve">Citius enim <lb/>bifariam dicitur: </s> - <s xml:id="s82" xml:space="preserve">Å¿iue enim in minori tempore <lb/>zqualem pertranÅ¿i@ locÅ©, citius feciſſe dicimus: <lb/></s> - <s xml:id="s83" xml:space="preserve">Å¿eu in æquali maiorem. </s> - <s xml:id="s84" xml:space="preserve">Maior autem in æquali <lb/>tempore maiorem deÅ¿cribit circulum: </s> - <s xml:id="s85" xml:space="preserve">qui enim <lb/>extra eÅ¿t, maior eo qui intus eft. </s> - <s xml:id="s86" xml:space="preserve">Horum autẽ cau <lb/>fa, qmÌ„ duas fertur lationesea, quæ circulum de-<lb/>Å¿cribit, linea. </s> - <s xml:id="s87" xml:space="preserve">Quandoquidem igitur in propor-<lb/>tione fertur aliqua id, quod fertur, Å¿uper rectam <lb/>ferri neceſſe: </s> - <s xml:id="s88" xml:space="preserve">& </s> - <s xml:id="s89" xml:space="preserve">hæc diameter efficitur figuræ, <lb/>quam faciunt illæ, quæ in huiuÅ¿modi proportio <lb/>ne coaptant̃, lineÄ™. </s> - <s xml:id="s90" xml:space="preserve">Sit enim proportio Å¿ecÅ©dum <lb/>quam latum fertur, quam habet A B ad AC: </s> - <s xml:id="s91" xml:space="preserve">& </s> - <s xml:id="s92" xml:space="preserve">A <lb/>quidem fertur verÅ¿us B, A B uerò Å¿ubterferatur <lb/>verÅ¿us M C: </s> - <s xml:id="s93" xml:space="preserve">latum aut<unsure/>em Å¿it A quidem ad D. </s> - <s xml:id="s94" xml:space="preserve">ubi <lb/>autem eÅ¿t A B, uerÅ¿us E: </s> - <s xml:id="s95" xml:space="preserve">quoniã igitur la@ionis <lb/>erat proportio, quam A B habet ad A C, neceſſe <lb/>eÅ¿t & </s> - <s xml:id="s96" xml:space="preserve">A D ad A E hanc habere proportionem. </s> - <s xml:id="s97" xml:space="preserve">Si-<lb/>mile igitur eÅ¿t proportione paruum quadrilate-<lb/>rum maiori; </s> - <s xml:id="s98" xml:space="preserve">quamobrẽ & </s> - <s xml:id="s99" xml:space="preserve">eadẽ illorum eÅ¿t dia-<lb/>meter: </s> - <s xml:id="s100" xml:space="preserve">& </s> - <s xml:id="s101" xml:space="preserve">A erit ad F. </s> - <s xml:id="s102" xml:space="preserve">Eodem etiam oÅ¿tendet̃ mo-<lb/>do, vbicu<unsure/>nq; </s> - <s xml:id="s103" xml:space="preserve">latio deprehendatur; </s> - <s xml:id="s104" xml:space="preserve">Å¿emper enim <lb/>Å¿upra diametrum erit. </s> - <s xml:id="s105" xml:space="preserve">ManifeÅ¿tum igitur, quòd <lb/>id, quod Å¿ecundum diametrum duabus fertur la <lb/>tionibus, neceſſariò Å¿ecundum laterum propor- -<pb o="508" file="0508" n="5" rhead="AriÅ¿t. Quæſtiones."/> -tionem fertur. </s> - <s xml:id="s106" xml:space="preserve">Sienim Å¿ecundum aliam quam-<lb/>piam, nõ fertur ẜ m dia metrum. </s> - <s xml:id="s107" xml:space="preserve">Si autem in nul <lb/>la fertur proportione Å¿ecundum duas lationes <lb/>nullo in tempore, rectã eſſe lationem eÅ¿t impoÅ¿-<lb/>Å¿ibile. </s> - <s xml:id="s108" xml:space="preserve">Sit enim recta. </s> - <s xml:id="s109" xml:space="preserve">PoÅ¿ita igitur hac pro diame <lb/>tro, & </s> - <s xml:id="s110" xml:space="preserve">circum reple is lateribus, illud quod fer-<lb/>tur, Å¿ecundum laterum pro portionem ferri n<unsure/>e-<lb/>ceſſe eÅ¿t: </s> - <s xml:id="s111" xml:space="preserve">hoc enim demonÅ¿tratum eÅ¿t prius. </s> - <s xml:id="s112" xml:space="preserve">non <lb/>igitur rectam efficiet id, quod Å¿ecundùm nullam <lb/>proportionem in nullo fertur tempore. </s> - <s xml:id="s113" xml:space="preserve">Siautẽ <lb/>ẜm quam piam feratur proportionem, & </s> - <s xml:id="s114" xml:space="preserve">in tem <lb/>pore quopiam hoc neceſſe eÅ¿t tem pus rectam eÅ¿ <lb/>Å¿e lationem, per ea, quæ retro Å¿unt dicta. </s> - <s xml:id="s115" xml:space="preserve">Quam-<lb/>obrem circulare eÅ¿tid, quod Å¿ecundÅ© nullam, <lb/>proportionem nullo in tempore duas fertur la-<lb/>tiones Quòd quidem igitur ea, quæ circulum do<unsure/> <lb/>Å¿cribir, duas Å¿imul feratur lationes, manifeÅ¿tum <lb/>eÅ¿t cùm ex iÅ¿tis, tum quia ẜ m rectum lata, aÅ¿t ê‘. <lb/></s> - <s xml:id="s116" xml:space="preserve">pendiculum peruenit, vt Å¿it rurÅ¿us ipÅ¿a à centro <lb/>perpendiculum. </s> - <s xml:id="s117" xml:space="preserve">Sit circulas A B C D: </s> - <s xml:id="s118" xml:space="preserve">extremum <lb/>autem vbi eÅ¿t B, feratur ad ipÅ¿um D: </s> - <s xml:id="s119" xml:space="preserve">peruenit Å¿a <lb/>hè aliquando ad ipÅ¿um C. </s> - <s xml:id="s120" xml:space="preserve">Si quidẽ igit<unsure/>ur in pro <lb/>portione feratur, quam haber BE, EC fertur vti <lb/>que Å¿ecundùm diametrum vbi BC. </s> - <s xml:id="s121" xml:space="preserve">nunc autem <lb/>quoniam in nulla propor:</s> - <s xml:id="s122" xml:space="preserve"><unsure/> ione, in circunferen-<lb/>tia certè fertur vbi B E C. </s> - <s xml:id="s123" xml:space="preserve">Si autem duobus ab <lb/>eadem potemtia latis, hoc quidem plus ræpella-<lb/>tur, illud verò minus, rationi conÅ¿entaneum eÅ¿t <lb/>tardius moueri id, quod plus repellitur, eo qùod <lb/>repellitur minus: </s> - <s xml:id="s124" xml:space="preserve">quod videtur accidere maiori <lb/>& </s> - <s xml:id="s125" xml:space="preserve">minori illarum, qÌuæ ex centro circulos deÅ¿cri <lb/>bunt. </s> - <s xml:id="s126" xml:space="preserve">quoniam enim proprius eÅ¿t manenti, eius <lb/>quæ minor eÅ¿t, extrem um, quà m id quod eÅ¿t ma <lb/>oris, veluti retractum in contrariÅ© ad medium, <lb/>ardius fertur minoris extremÅ©. </s> - <s xml:id="s127" xml:space="preserve">Omni quidem -<pb o="509" file="0509" n="6" rhead="Mechanicæ."/> -igitur circulum deÅ¿cribenti iÅ¿tuc accidit: </s> - <s xml:id="s128" xml:space="preserve">fertur-<lb/>qÌue eam quæ ẜm naturam eÅ¿t lationem, Å¿ecun-<lb/>dum circunferentiam: </s> - <s xml:id="s129" xml:space="preserve">illam verò quÄ™ prÄ™ter na <lb/>turam, in tranÅ¿uerÅ¿um, & </s> - <s xml:id="s130" xml:space="preserve">Å¿ecundùm centrum. <lb/></s> - <s xml:id="s131" xml:space="preserve">maiorem aut<unsure/>ẽ Å¿emper eam, quæ præter naturã <lb/>eÅ¿t, ipÅ¿a minor fertur: </s> - <s xml:id="s132" xml:space="preserve">quia enim centro eÅ¿t vici-<lb/>nior, quod trahit, vincitur magis<unsure/>: </s> - <s xml:id="s133" xml:space="preserve">Quòd autem-<unsure/> <lb/>magis quod præter natui<unsure/>ã eÅ¿t mouetur ipÅ¿a mi-<lb/>nor, quà m maior illarÅ©, quæ ex centro circulos <lb/>deÅ¿cribÅ©t ex ijs eÅ¿t manifeÅ¿tum. </s> - <s xml:id="s134" xml:space="preserve">Sit circulus vbi <lb/>B C D E & </s> - <s xml:id="s135" xml:space="preserve">alter in hoc minorvbi M N O P, circa <lb/>idem centrum A & </s> - <s xml:id="s136" xml:space="preserve">proijciantur diametri in ma <lb/>gno quidem, in quibus CD, BE, in minori ve-<lb/>10 ipſæ MO, N P: </s> - <s xml:id="s137" xml:space="preserve">& </s> - <s xml:id="s138" xml:space="preserve">altera parie longius quadra <lb/>tum Å¿uppleatur DKRC: </s> - <s xml:id="s139" xml:space="preserve">Å¿i quidem AB circu-<lb/>lum deÅ¿cribẽs ad id perueniet, vnde eÅ¿t egreſſa; </s> - <s xml:id="s140" xml:space="preserve"><lb/>manifeÅ¿tum eÅ¿t quod ad ipÅ¿am fertur A B. </s> - <s xml:id="s141" xml:space="preserve">Simi <lb/>literetiam A M ad ipÅ¿am A M perueniet. </s> - <s xml:id="s142" xml:space="preserve">Tar-<lb/>dius autem fertur A M, quà ru A B quemadmo-<lb/>dum dictum eÅ¿t: </s> - <s xml:id="s143" xml:space="preserve">quia maior fit repulÅ¿io & </s> - <s xml:id="s144" xml:space="preserve">magis <lb/>rerrahitur A M. </s> - <s xml:id="s145" xml:space="preserve">Ducatur igitur ipÅ¿a A L F, & </s> - <s xml:id="s146" xml:space="preserve">ab <lb/>ipÅ¿o L perpendiculum ad ip Å¿am A B, ipÅ¿a L Q in <lb/>minore circulo: </s> - <s xml:id="s147" xml:space="preserve">& </s> - <s xml:id="s148" xml:space="preserve">rurÅ¿um ab L ducatur iuxta A <lb/>B L S, & </s> - <s xml:id="s149" xml:space="preserve">S T ad ipÅ¿am A B perpendiculum, & </s> - <s xml:id="s150" xml:space="preserve">ip. </s> - <s xml:id="s151" xml:space="preserve"><lb/>fa F X: </s> - <s xml:id="s152" xml:space="preserve">ipſæ igitur ubi Å¿unt S T, & </s> - <s xml:id="s153" xml:space="preserve">L Q, æquales: </s> - <s xml:id="s154" xml:space="preserve"><lb/>ipÅ¿a ergo B T minor eÅ¿t, quà m M Q. </s> - <s xml:id="s155" xml:space="preserve">æquales <lb/>enim rectæ lineæ in æqualibus coniectæ circulis <lb/>perpẽdiculares a diametro, minorẽ diametri re-<lb/>fecant Å¿ectionẽ in maioribus circulis, eÅ¿t autem <lb/>ipÅ¿a S T æqualis ipÅ¿i L Q. </s> - <s xml:id="s156" xml:space="preserve">In quanto autem tema <lb/>pore ipÅ¿a A L ipÅ¿am M L lata eÅ¿t, in tanto tem@ <lb/>poris Å¿patio in maiori circulo maiorem, quà m <lb/>fit B S, latum erit extremum ipÅ¿ius B A. </s> - <s xml:id="s157" xml:space="preserve">Latio <lb/>quidem igitur Å¿ecundùm naturam æqualis: </s> - <s xml:id="s158" xml:space="preserve">ea <lb/>a<unsure/>utem quæ præter naturam eÅ¿t minor, videlices -<pb o="510" file="0510" n="7" rhead="AriÅ¿t. Quæſtiones."/> -B T, quà m M Q. </s> - <s xml:id="s159" xml:space="preserve">Oportet autem proportionabi-<lb/>liter eſſe, Å¿icut quod eÅ¿t Å¿ecundÅ© naturã, ita quod <lb/>eÅ¿t prÄ™ter naturam, ad id quod eÅ¿t prÄ™ter naturã; <lb/></s> - <s xml:id="s160" xml:space="preserve">maiorem igituÄ© circunferentiam pertranÅ¿iuit, ê—̃ <lb/>Å¿it ipÅ¿a S B. </s> - <s xml:id="s161" xml:space="preserve">Neceſſe autem eÅ¿t ipÅ¿am F B in hoc <lb/>rẽpore pertran Å¿iſſe: </s> - <s xml:id="s162" xml:space="preserve">hic. </s> - <s xml:id="s163" xml:space="preserve">n. </s> - <s xml:id="s164" xml:space="preserve">erit, quando proportio <lb/>nabiliter vtrinq; </s> - <s xml:id="s165" xml:space="preserve">accidit, quod eÅ¿t præter naturã, <lb/>ad id quod eÅ¿t Å¿ecundùm naturam. </s> - <s xml:id="s166" xml:space="preserve">Si igitur ma-<lb/>ius eÅ¿t, quod Å¿ecundum naturam in maiori, & </s> - <s xml:id="s167" xml:space="preserve"><lb/>quod eÅ¿t præter naturam. </s> - <s xml:id="s168" xml:space="preserve">magis vtique hic coin <lb/>cidit vno modo: </s> - <s xml:id="s169" xml:space="preserve">ita quod B Å¿it latum per ipÅ¿am <lb/>B F in tanto tempore, in quo M punctum per <lb/>ipÅ¿am M L. </s> - <s xml:id="s170" xml:space="preserve">hic enim Å¿ecundùm naturam quidẽ <lb/>figno B fit X F: </s> - <s xml:id="s171" xml:space="preserve">eÅ¿t enim ab ipÅ¿o F perpendiculÅ©: </s> - <s xml:id="s172" xml:space="preserve"><lb/>præter naturam verò ad ipÅ¿am X B. </s> - <s xml:id="s173" xml:space="preserve">EÅ¿t autem <lb/>quemadmodum F X ad X B, Å¿ic L Q. </s> - <s xml:id="s174" xml:space="preserve">ad M Q. </s> - <s xml:id="s175" xml:space="preserve">ma <lb/>nifeÅ¿tum autem, Å¿i coniunguntur abipÅ¿a B M, <lb/>ad F L. </s> - <s xml:id="s176" xml:space="preserve">Si autem minor, aut maior quà m Å¿it F B, <lb/>erit illa quam latum eÅ¿t B, non Å¿imiliter erit, ne-<lb/>que proportionale in vtriÅ¿que quod eÅ¿t Å¿ecundÅ© <lb/>naturam, ad id quod eÅ¿t præter naturam. </s> - <s xml:id="s177" xml:space="preserve">Quam <lb/>igicur ob cauÅ¿am ab eadem potentia celerius fe@ <lb/>tur id quod plus à centro diÅ¿tat punctum, ex ijs, <lb/>quæ dicta Å¿unt, eÅ¿t manifeÅ¿tum.</s> - <s xml:id="s178" xml:space="preserve"/> - </p> - </div> - <div type="section" level="1" n="4"> - <head xml:id="head7" style="it" xml:space="preserve">Quæſtiones.</head> - <p> - <s xml:id="s179" xml:space="preserve">C V R aÅ©t maiores libræ exactiores <lb/> -<anchor type="note" xlink:label="note-0510-01a" xlink:href="note-0510-01"/> -Å¿unt minotib<emph style="sub">9</emph>. </s> - <s xml:id="s180" xml:space="preserve">palã eÅ¿t ex ijs. </s> - <s xml:id="s181" xml:space="preserve">Spar <lb/>tum. </s> - <s xml:id="s182" xml:space="preserve">n fit centrÅ©: </s> - <s xml:id="s183" xml:space="preserve">id pãq; </s> - <s xml:id="s184" xml:space="preserve">manet. <lb/></s> - <s xml:id="s185" xml:space="preserve">quod aÅ©t libræ vtrinq; </s> - <s xml:id="s186" xml:space="preserve">eÅ¿t, exeun <lb/>tes à centro. </s> - <s xml:id="s187" xml:space="preserve">Ab eodẽ igit̃ põdere <lb/>citius moueri neceſſe eÅ¿t extre-<lb/>mÅ© libræ, quo plus à ſparto diÅ¿ceſſerit. </s> - <s xml:id="s188" xml:space="preserve">Et nõnul <lb/>la quidẽ in paruis libris impoÅ¿ita nõ manifeÅ¿ta -<pb o="511" file="0511" n="8" rhead="Mechanicæ."/> -fenÅ¿ui Å¿unt pondera: </s> - <s xml:id="s189" xml:space="preserve">in magis aÅ©t maniÅ¿eÅ¿ta: </s> - <s xml:id="s190" xml:space="preserve">ni <lb/>hil enim prohibet minorem moueri magnitudi <lb/>nem, ê—̃ vt viÅ¿ioni Å¿it maniÅ¿eÅ¿ta. </s> - <s xml:id="s191" xml:space="preserve">In magna aÅ©t li-<lb/>bra idem pondus viÅ¿ibile efficit magnitudo. </s> - <s xml:id="s192" xml:space="preserve">QuÄ™ <lb/>dam verò manifeÅ¿ta quidem Å¿unt in vtriÅ¿q;</s> - <s xml:id="s193" xml:space="preserve">, Å¿ed <lb/>multo magis in maioribus, qmÌ„ multo maior in <lb/>clinationis fit magnitudo ab eodem ê‘ondere <lb/>in maioribus. </s> - <s xml:id="s194" xml:space="preserve">Quamobrem machinanturij, qui <lb/>purpuram vendunt, vt pendendo defraudent, tÅ© <lb/>ad medium Å¿partum non ponentes: </s> - <s xml:id="s195" xml:space="preserve">tum plum-<lb/>bum in alteraruram libræ partem infundentes: <lb/></s> - <s xml:id="s196" xml:space="preserve">aut ligni gÄ‘ ad radicẽ vergebat, in eam quam de <lb/>ferri vola<unsure/>nt, partem conÅ¿tituentes; </s> - <s xml:id="s197" xml:space="preserve">aut Å¿i nodum <lb/>habuerit: </s> - <s xml:id="s198" xml:space="preserve">ligni enim grauior illa eÅ¿t pars, in qua <lb/>eÅ¿t radix, nodus vero radix quædam eÅ¿t.</s> - <s xml:id="s199" xml:space="preserve"/> - </p> - <div type="float" level="2" n="1"> - <note position="left" xlink:label="note-0510-01" xlink:href="note-0510-01a" - xml:space="preserve">1</note> - </div> - <p> - <s xml:id="s200" xml:space="preserve">Cut Å¿iquidẽ Å¿urÅ¿um Å¿uerit Å¿partÅ©, qñ deorsÅ© lato <lb/> -<anchor type="note" xlink:label="note-0511-01a" xlink:href="note-0511-01"/> -põdere, quiÅ¿piã id admouet, rurÅ¿um aÅ¿cendit li-<lb/>bra: </s> - <s xml:id="s201" xml:space="preserve">Å¿i autem deorÅ¿um conÅ¿titutum Å¿uerit, non <lb/>aÅ¿cendit, Å¿ed manet? </s> - <s xml:id="s202" xml:space="preserve">An quia Å¿urÅ¿um quidẽ Å¿par <lb/>to exiÅ¿tente, plus libræ extra perpendiculum fit: <lb/></s> - <s xml:id="s203" xml:space="preserve">Å¿partum enim eÅ¿t perpendiculum: </s> - <s xml:id="s204" xml:space="preserve">quare neceſſe <lb/>eÅ¿t deorÅ¿um ferri id quod plus eÅ¿t, donec aÅ¿cen-<lb/>dat quæ bifariam libram diuidit, ad ipÅ¿um per-<lb/>pendiculÅ©, cùm onus incumbat ad libræ partem <lb/>furÅ¿um raptam. </s> - <s xml:id="s205" xml:space="preserve">Sit libra recta vbi B C, Å¿partum <lb/>autem A D. </s> - <s xml:id="s206" xml:space="preserve">hoc igitur deorÅ¿um proiecto perpen <lb/>diculum erit vbi A D M. </s> - <s xml:id="s207" xml:space="preserve">Si igitur in ipÅ¿o B pona <lb/>tur onus, B quidem erit vbi E, C, autem vbi H. </s> - <s xml:id="s208" xml:space="preserve"><lb/>quamobrem ea quæ bifariam libram Å¿ecat, pri-<lb/>mo quidem erit D M ipÅ¿ius perpendiculi: </s> - <s xml:id="s209" xml:space="preserve">incum <lb/>bente autẽ onere erit D G. </s> - <s xml:id="s210" xml:space="preserve">quare libra ipÅ¿ius vbi <lb/>E H, quod extra perpendiculum eÅ¿t A M, vbi eÅ¿t <lb/>P Q, maius eÅ¿t dimidio. </s> - <s xml:id="s211" xml:space="preserve">Si igitur amoueatur <lb/>onus abipÅ¿o E, neceſſe eÅ¿t deorÅ¿um ferri H, mi-<lb/>@@s enim eÅ¿t ipÅ¿um E. </s> - <s xml:id="s212" xml:space="preserve">Si quidem igitur Å¿urÅ¿um -<pb o="512" file="0512" n="9" rhead="AriÅ¿t. Quæſtiones"/> -habuerit Å¿partum, rurÅ¿um propter hoc aÅ¿cendit<unsure/> <lb/>libta. </s> - <s xml:id="s213" xml:space="preserve">Si autẽ deorÅ¿um fuerit, id, quod Å¿ubÅ¿tat, cõ <lb/>trarium facit: </s> - <s xml:id="s214" xml:space="preserve">plus enim dimidio fit libræ, quæ <lb/>deorÅ¿um eÅ¿t pars, quà m quod perpendiculum Å¿e <lb/>cet, quapropter nõ aÅ¿cẽdit: </s> - <s xml:id="s215" xml:space="preserve">eleuata enim pars le <lb/>uior eÅ¿t. </s> - <s xml:id="s216" xml:space="preserve">Sit libra recta vbi N O, perpendiculum <lb/>autem K L M bifariã igitur Å¿ecatu@ K G. </s> - <s xml:id="s217" xml:space="preserve">Impo-<lb/>Å¿ito autem onere in ipÅ¿o N, erit quidẽ N vbi O, <lb/>ipÅ¿um autem G vbi R, K L autem vbi L P. </s> - <s xml:id="s218" xml:space="preserve">quare <lb/>maius eÅ¿t K O, quà m L R, ipÅ¿o P K L. </s> - <s xml:id="s219" xml:space="preserve">Et ablato <lb/>igitur onere, neceſſe eÅ¿t manere: </s> - <s xml:id="s220" xml:space="preserve">in cumbit enim <lb/>ceu onus exceſſus medietatis eius vbi eÅ¿t F.</s> - <s xml:id="s221" xml:space="preserve"/> - </p> - <div type="float" level="2" n="2"> - <note position="right" xlink:label="note-0511-01" xlink:href="note-0511-01a" - xml:space="preserve">2</note> - </div> - <p> - <s xml:id="s222" xml:space="preserve">Cur exiguÄ™ vires (quem admodum à principio <lb/> -<anchor type="note" xlink:label="note-0512-01a" xlink:href="note-0512-01"/> -dictum eÅ¿t) vecte magna mouent pondera, vecis <lb/>inÅ¿uper onus accipien tis? </s> - <s xml:id="s223" xml:space="preserve">cùm Å¿acilius Å¿it mino-<lb/>grem mouere grauitatem: </s> - <s xml:id="s224" xml:space="preserve">minor autẽ eÅ¿t Å¿ine ve-<lb/>cte. </s> - <s xml:id="s225" xml:space="preserve">An quoniã ipÅ¿e vectis eÅ¿t in cauÅ¿a hbra exi-<lb/>Å¿tens, Å¿partÅ© infernè habens, in inæqualia diui-<lb/>Å¿a. </s> - <s xml:id="s226" xml:space="preserve">hypomochlion enim eÅ¿t Å¿partum. </s> - <s xml:id="s227" xml:space="preserve">a mbo nan <lb/>que Å¿tamt, vt centrum. </s> - <s xml:id="s228" xml:space="preserve">quoniam autem ab Ä™qua-<lb/>Å¿i pondere celerius mouetur maior earum, q̃ à <lb/>centro Å¿unt: </s> - <s xml:id="s229" xml:space="preserve">duo verò pondera, quod mouet, & </s> - <s xml:id="s230" xml:space="preserve"><lb/>quod mouetur: </s> - <s xml:id="s231" xml:space="preserve">quod igitur motum pondus ad-<lb/>mouens, longitudo patitur ad longitudinẽ. </s> - <s xml:id="s232" xml:space="preserve">Å¿em <lb/>per autem quanto ab hy pomochlio diÅ¿tabit ma <lb/>gis, tanto facilius mo<unsure/>uebit. </s> - <s xml:id="s233" xml:space="preserve">CauÅ¿a autem eÅ¿t, quÄ™ <lb/>retro cõmemorata eÅ¿t: </s> - <s xml:id="s234" xml:space="preserve">quoniam quÄ™ plus à cen-<lb/>@ro diÅ¿tat, maiorem deÅ¿cribit circuium: </s> - <s xml:id="s235" xml:space="preserve">quare ab <lb/>eadem potentia plus Å¿eparabitur mouens illud, <lb/>quod plus ab hypomochlio diÅ¿tabit. </s> - <s xml:id="s236" xml:space="preserve">Sit uectis <lb/>vbi A B, pondus vbi C, quod mouet autẽvbi D, <lb/>hy pomochlion vbi E. </s> - <s xml:id="s237" xml:space="preserve">quod autem vbi eÅ¿t D mo <lb/>u<unsure/>ens, vbi F. </s> - <s xml:id="s238" xml:space="preserve">motum autem vbi C. </s> - <s xml:id="s239" xml:space="preserve">pondus, vbi G.</s> - <s xml:id="s240" xml:space="preserve"/> - </p> - <div type="float" level="2" n="3"> - <note position="left" xlink:label="note-0512-01" xlink:href="note-0512-01a" - xml:space="preserve">3</note> - </div> - <p> - <s xml:id="s241" xml:space="preserve">Cur ij, qui in nauis medio Å¿unt remiges, maxi <lb/> -<anchor type="note" xlink:label="note-0512-02a" xlink:href="note-0512-02"/> -me nauem mouent? </s> - <s xml:id="s242" xml:space="preserve">An quia remus vectis eÅ¿t: </s> - <s xml:id="s243" xml:space="preserve">hy -<pb o="513" file="0513" n="10" rhead="Mechanicæ."/> -pomochlion auteni fit Å¿calmus, Å¿tat enim ille ẜ <lb/>pondus verò mare eÅ¿t, quod propellit remus: </s> - <s xml:id="s244" xml:space="preserve">ve-<lb/>ctem autem mouens ipÅ¿e eÅ¿t remex. </s> - <s xml:id="s245" xml:space="preserve">Semper aÅ©t <lb/>plus mouet ponderis, quanto magis ab hypomo <lb/>chlio diÅ¿tabit, quicunque id mouet. </s> - <s xml:id="s246" xml:space="preserve">maior enim <lb/>ita fit, quæ ex centro. </s> - <s xml:id="s247" xml:space="preserve">Å¿calmus autem hypomo-<lb/>chlion exiÅ¿tens, centrum eÅ¿t. </s> - <s xml:id="s248" xml:space="preserve">In medio autem na <lb/>uis plurimum remi intus eÅ¿t: </s> - <s xml:id="s249" xml:space="preserve">ilia enim parte la-<lb/>tiſſima eÅ¿t nauis: </s> - <s xml:id="s250" xml:space="preserve">quare maior vtrinq; </s> - <s xml:id="s251" xml:space="preserve">remi pars <lb/>vtrorunq; </s> - <s xml:id="s252" xml:space="preserve">nauis parietum intrin Å¿ecus eÅ¿t. </s> - <s xml:id="s253" xml:space="preserve">Moue-<lb/>tur autem nauis, quoniam appellente ad mare <lb/>remo, extremum illius quod intus eÅ¿t, in ante <lb/>prom ouctur. </s> - <s xml:id="s254" xml:space="preserve">nauem verò Å¿calmo alligatam Å¿i-<lb/>mul promoueri contingit, quo remi extremum. <lb/></s> - <s xml:id="s255" xml:space="preserve">vbi enim plurimum maris diuidit remus, eò ma-<lb/>ximè propelli neceſſe eÅ¿t. </s> - <s xml:id="s256" xml:space="preserve">plurimum autem di@i <lb/>dit, vbi pars plurima remia Å¿calmo eÅ¿t. </s> - <s xml:id="s257" xml:space="preserve">Et eam <lb/>cauÅ¿am remiges, qui in media Å¿unt naui, mouent <lb/>illam maximè. </s> - <s xml:id="s258" xml:space="preserve">maxima enim remi pars à ſcalmo <lb/>in nauis medio intus eÅ¿t.</s> - <s xml:id="s259" xml:space="preserve"/> - </p> - <div type="float" level="2" n="4"> - <note position="left" xlink:label="note-0512-02" xlink:href="note-0512-02a" - xml:space="preserve">4</note> - </div> - <p> - <s xml:id="s260" xml:space="preserve">Cur paruum exiÅ¿tens gubernaculum, & </s> - <s xml:id="s261" xml:space="preserve">in ex-<lb/> -<anchor type="note" xlink:label="note-0513-01a" xlink:href="note-0513-01"/> -tremo nauigio tantas habet vires, vt ab exiguo <lb/>temone, & </s> - <s xml:id="s262" xml:space="preserve">ab hominis vnius viribus alioqùin <lb/>modicè vtentis, magnÄ™ nauigiorum moueantur <lb/>moles? </s> - <s xml:id="s263" xml:space="preserve">An quoniam gubernaculum vectis eÅ¿t: </s> - <s xml:id="s264" xml:space="preserve">o-<lb/>@us autem mare: </s> - <s xml:id="s265" xml:space="preserve">gubernator verò mouens eÅ¿t. <lb/></s> - <s xml:id="s266" xml:space="preserve">Non autem Å¿ecundum latitudinẽ, veluti remus, <lb/>mare accipit gubernaculum: </s> - <s xml:id="s267" xml:space="preserve">non enim in antè <lb/>nauigium mouet, Å¿ed ipÅ¿um commotum mare <lb/>accipiens inclinat obliquè: </s> - <s xml:id="s268" xml:space="preserve">quoniam enim pon-<lb/>dus eÅ¿t mare, contrario innixum modo nauem <lb/>inclinat. </s> - <s xml:id="s269" xml:space="preserve">hypomochlion enim in contrarium <lb/>verÅ¿atur: </s> - <s xml:id="s270" xml:space="preserve">mare vero interius, Å¿i illud exterius. </s> - <s xml:id="s271" xml:space="preserve"><lb/>illud autem Å¿equitur nauis, quoniam illi eÅ¿t ai<unsure/>li-<lb/>@ata. </s> - <s xml:id="s272" xml:space="preserve">Et remus quidem Å¿ecundùm latitudinem -<pb o="514" file="0514" n="11" rhead="AriÅ¿t. Quæſtiones."/> -onus propellens, & </s> - <s xml:id="s273" xml:space="preserve">ab eodem repulÅ¿us, in rectÅ© <lb/>propellit: </s> - <s xml:id="s274" xml:space="preserve">gubernaculum autem, ut obliquum ia <lb/>cet, hinc inde in obliquum motionÄ“ facit. </s> - <s xml:id="s275" xml:space="preserve">In ex <lb/>tremo autÄ“, & </s> - <s xml:id="s276" xml:space="preserve">non in medio iacet, quoniam mo <lb/>uenti facilimÅ« eÅ¿t ab extremo motum mouere. <lb/></s> - <s xml:id="s277" xml:space="preserve">Prim a enim pars celerimè fertur; </s> - <s xml:id="s278" xml:space="preserve">quoniã quem <lb/>adm odum in ijs quæ feruntur, in fine deficit la-<lb/>tio, Å¿ic ipÅ¿ius cÅtinui in fine imbeciliſſima eÅ¿t la-<lb/>tio. </s> - <s xml:id="s279" xml:space="preserve">imbeciliſſima autem ad expellendum eÅ¿t fa <lb/>cilis. </s> - <s xml:id="s280" xml:space="preserve">Propter hæcigitur in puppi gubernaculum <lb/>ponitur. </s> - <s xml:id="s281" xml:space="preserve">Nec minus, quoniÄ parua ibi motione <lb/>Å¿acta, multò maius interuallum fit in ultimo: </s> - <s xml:id="s282" xml:space="preserve">ê—a <lb/>æqualis angulus Å¿emper maiorem Å¿pectat: </s> - <s xml:id="s283" xml:space="preserve">tan-<lb/>toqÌue magis. </s> - <s xml:id="s284" xml:space="preserve">quanto maiores fuerint illæ, quæ <lb/>con@inẽt. </s> - <s xml:id="s285" xml:space="preserve">Ex ijs etiam manifeÅ¿tum eÅ¿t, quam ob <lb/>cauſſam magis in contrarium procedit nauigiÅ©, <lb/>quà m remi ipÅ¿ius palmula. </s> - <s xml:id="s286" xml:space="preserve">eadem enim magni <lb/>tudoijÅ¿dem mota uiribus, in aete plus, quà m in <lb/>aqua progreditur. </s> - <s xml:id="s287" xml:space="preserve">Si enim A B remus, Guerò <lb/>Å¿calmus: </s> - <s xml:id="s288" xml:space="preserve">A autem in nauigio Å¿it remi principiÅ©, <lb/>B uerò in mari palmula. </s> - <s xml:id="s289" xml:space="preserve">Si igitur A ubi D tranÅ¿. </s> - <s xml:id="s290" xml:space="preserve"><lb/>latum eÅ¿t, non erit B ubi E: </s> - <s xml:id="s291" xml:space="preserve">æqualis enim B E ipÅ¿i <lb/>A C. </s> - <s xml:id="s292" xml:space="preserve">æquale igitur tranÅ¿latÅ« erit, Å¿ed minus. </s> - <s xml:id="s293" xml:space="preserve">erit <lb/>igitur ubi eÅ¿t F. </s> - <s xml:id="s294" xml:space="preserve">minor enim C F, ipÅ¿a A D. </s> - <s xml:id="s295" xml:space="preserve">quare <lb/>ipÅ¿a G F, ipÅ¿a D F. </s> - <s xml:id="s296" xml:space="preserve">Å¿imiles enim Å¿unt trianguli. </s> - <s xml:id="s297" xml:space="preserve"><lb/>Stans autem erit medium, ubi eÅ¿t C. </s> - <s xml:id="s298" xml:space="preserve">in contrariÅ© <lb/>enim ipÅ¿i, quod in mari eÅ¿t, extremo uidelicet B <lb/>procedit, ubi extremum in nauigio eÅ¿t A. </s> - <s xml:id="s299" xml:space="preserve">Non <lb/>procedere autÄ“ubi eÅ¿t D. </s> - <s xml:id="s300" xml:space="preserve">niÅ¿i commouere@ur na <lb/>uigium, & </s> - <s xml:id="s301" xml:space="preserve">eo transferretur, ubi remi eÅ¿t princi-<lb/>plum IdipÅ¿nm etiam facit gubernaculam, niÅ¿i ê™ <lb/>(ut dictum eÅ¿t terro) nihil nauigio ad id, quod-<lb/>antè eÅ¿t, confert, Å¿ed Å¿olùm puppim in obliquÅ© <lb/>pell@t, ubicunq; </s> - <s xml:id="s302" xml:space="preserve">fuerit in con@rariÅ« enim eo mo <lb/>do uergit prora. </s> - <s xml:id="s303" xml:space="preserve">Vbi igitur applicatum eÅ¿t gu- -<pb o="515" file="0515" n="12" rhead="Mechanicæ."/> -bernaculum, id oporter rei motæ, ceu quoddam <lb/>intelligere medium: </s> - <s xml:id="s304" xml:space="preserve">& </s> - <s xml:id="s305" xml:space="preserve">quemadmodÅ© Å¿calmus <lb/>remo. </s> - <s xml:id="s306" xml:space="preserve">Medium autem procedit Å¿ecundùm quod <lb/>gubernaculum trãsfertur. </s> - <s xml:id="s307" xml:space="preserve">Å¿i quidÄ“ introrÅ¿us agit, <lb/>& </s> - <s xml:id="s308" xml:space="preserve">puppis eo transfertur. </s> - <s xml:id="s309" xml:space="preserve">prora verò ad contra-<lb/>rium vergit. </s> - <s xml:id="s310" xml:space="preserve">in eodem enim exiÅ¿tente prora, to-<lb/>tum transfertur nauigium.</s> - <s xml:id="s311" xml:space="preserve"/> - </p> - <div type="float" level="2" n="5"> - <note position="right" xlink:label="note-0513-01" xlink:href="note-0513-01a" - xml:space="preserve">5</note> - </div> - <p> - <s xml:id="s312" xml:space="preserve">Cur quando antenna Å¿ublimior fuerit, ijÅ¿dem <lb/> -<anchor type="note" xlink:label="note-0515-01a" xlink:href="note-0515-01"/> -uelis, & </s> - <s xml:id="s313" xml:space="preserve">uento eodẽ celerius ferÅ©tur nauigia? </s> - <s xml:id="s314" xml:space="preserve">An <lb/>quia malus quidem fit uectis: </s> - <s xml:id="s315" xml:space="preserve">hypomochlion ue <lb/>rò mali Å¿edes, in qua collocatur: </s> - <s xml:id="s316" xml:space="preserve">pondus autem, <lb/>quod moueri debet, ipÅ¿um nauigium: </s> - <s xml:id="s317" xml:space="preserve">mou@s ve <lb/>ro is, qui vela tendit, Å¿piri@us. </s> - <s xml:id="s318" xml:space="preserve">Si igitur quanto re <lb/>motius fuerit hypomochion, facilius eadem po-<lb/>tentia, & </s> - <s xml:id="s319" xml:space="preserve">citius idem mouet pondus, altius cer-<lb/>tè Å¿ublata antenna, velum à mali Å¿ede, quæ hypo <lb/>mochlion eÅ¿t, remo@ius faciens, id efficie@.</s> - <s xml:id="s320" xml:space="preserve"/> - </p> - <div type="float" level="2" n="6"> - <note position="right" xlink:label="note-0515-01" xlink:href="note-0515-01a" - xml:space="preserve">6</note> - </div> - <p> - <s xml:id="s321" xml:space="preserve">Cur quando ex puppi nauigare voluerint, non <lb/> -<anchor type="note" xlink:label="note-0515-02a" xlink:href="note-0515-02"/> -flante ex puppi vento, veli quidem partem, quæ <lb/>ad gubernatorem vergit, conÅ¿tringunt: </s> - <s xml:id="s322" xml:space="preserve">illam ve-<lb/>rò quæ proram verÅ¿us eÅ¿t, pedem facientes rela <lb/>xant? </s> - <s xml:id="s323" xml:space="preserve">An quia retrahere quidem multo exiÅ¿ten <lb/>tevento gubernaculum non poteÅ¿t: </s> - <s xml:id="s324" xml:space="preserve">pauco autẽ <lb/>poteÅ¿t, quem conÅ¿tringunt. </s> - <s xml:id="s325" xml:space="preserve">Propellit quidem igi <lb/>tur ipÅ¿e ventus: </s> - <s xml:id="s326" xml:space="preserve">in puppim vero illum cÅÅ¿tituit, <lb/>gubernaculum retrahens, & </s> - <s xml:id="s327" xml:space="preserve">mare compellẽs: </s> - <s xml:id="s328" xml:space="preserve">Å¿i-<lb/>mul & </s> - <s xml:id="s329" xml:space="preserve">nautæ ipÅ¿i cum vento contendunt: </s> - <s xml:id="s330" xml:space="preserve">in cõ-<lb/>trariam enim Å¿e reclinant partem.</s> - <s xml:id="s331" xml:space="preserve"/> - </p> - <div type="float" level="2" n="7"> - <note position="right" xlink:label="note-0515-02" xlink:href="note-0515-02a" - xml:space="preserve">7</note> - </div> - <p> - <s xml:id="s332" xml:space="preserve">Cur ex figurarum genere, q̃cunq; </s> - <s xml:id="s333" xml:space="preserve">rotÅ©dÄ™ Å¿unt <lb/> -<anchor type="note" xlink:label="note-0515-03a" xlink:href="note-0515-03"/> -& </s> - <s xml:id="s334" xml:space="preserve">circinatæ, facilius mouentur? </s> - <s xml:id="s335" xml:space="preserve">Trifariam autẽ <lb/>circulum rotari contingit. </s> - <s xml:id="s336" xml:space="preserve">aut enim @m abÅ¿idem <lb/>centro Å¿imul moto, quemadmodum paluÅ¿triver <lb/>titur rota: </s> - <s xml:id="s337" xml:space="preserve">aut circa manens centrÅ«, veluti tro-<lb/>chleæ Å¿tante centro: </s> - <s xml:id="s338" xml:space="preserve">autin pauimento manente <lb/>@ẽtro, Å¿icut figuli rota conuertitur. </s> - <s xml:id="s339" xml:space="preserve">An celer@ima -<pb o="516" file="0516" n="13" rhead="AriÅ¿t. Quæſtiones"/> -quidem huiuÅ¿modi Å¿unt, quoniam parua Å¿uipa@ <lb/>te planum contingunt, veluti circulas Å¿ecÅ«dum <lb/>punctum: </s> - <s xml:id="s340" xml:space="preserve">& </s> - <s xml:id="s341" xml:space="preserve">quoniam non offenÅ¿ant: </s> - <s xml:id="s342" xml:space="preserve">à terra emÌ„ <lb/>Å¿emotus eÅ¿t angulus. </s> - <s xml:id="s343" xml:space="preserve">PrÄ™terea etiam cui obuiam <lb/>fiunt corpori, id rurÅ¿um Å¿ecundùm puÅ¿illam tan <lb/>gunt. </s> - <s xml:id="s344" xml:space="preserve">Å¿i autem rectilineum eſſet, rectitudine Å¿ua <lb/>multum plani contingeret. </s> - <s xml:id="s345" xml:space="preserve">Ad hæc quò mutat <lb/>pondus, eo motor mouet. </s> - <s xml:id="s346" xml:space="preserve">cùm igitur ad rectum <lb/>Å¿uper plano circuli fuerit diameter, planÅ« Å¿ecun <lb/>dùm punctum, contingÄ“te circulo æquale vtriu-<lb/>que pondus difterminat diameter. </s> - <s xml:id="s347" xml:space="preserve">cùm autÄ“ mo <lb/>uetur, plus illico, ad quòd mouetur, ceui@de nu <lb/>tans. </s> - <s xml:id="s348" xml:space="preserve">ab impellẽre facilius in ante mouetur: </s> - <s xml:id="s349" xml:space="preserve">quò <lb/>enim vnumquodque vergit, mouetur ex facili: <lb/></s> - <s xml:id="s350" xml:space="preserve">Å¿i quidem difficulter ad cÅtrarium nutus Å¿ui mo <lb/>uetur modum. </s> - <s xml:id="s351" xml:space="preserve">Præterea nonnulli autumn@nt ê™ <lb/>circuli linea in perpeti verÅ¿atur motu, quemad-<lb/>modum manentia propter contrarium nixÅ© ma <lb/>nent: </s> - <s xml:id="s352" xml:space="preserve">Å¿icur maioribus contingit circulis ad mino <lb/>res. </s> - <s xml:id="s353" xml:space="preserve">Celerius enim ab æquali mouentur potentia <lb/>maiores circuli, mouentqÌue onera, quoniam cir <lb/>culi maioris angulus ad minoris angulum cir-<lb/>culi nutum habet quendam: </s> - <s xml:id="s354" xml:space="preserve">& </s> - <s xml:id="s355" xml:space="preserve">Å¿icut diameter ad <lb/>diametrum, ita maior circulus ad minorem. </s> - <s xml:id="s356" xml:space="preserve">it fi <lb/>niti autem Å¿unt minores. </s> - <s xml:id="s357" xml:space="preserve">Si autem ad alte@um <lb/>nutum habet circulus, Å¿imiliter eÅ¿t bene mobi-<lb/>lis. </s> - <s xml:id="s358" xml:space="preserve">Et aliam Å¿anè habet inclinationem circulus, <lb/>& </s> - <s xml:id="s359" xml:space="preserve">ea quæ à circulo mouentur, licèt planitiẽ abÅ¿i-<lb/>de non contingat, Å¿ed aut iuxta planitiẽ, aut velu <lb/>tì trocleæ. </s> - <s xml:id="s360" xml:space="preserve">etenim hoc Å¿e habentes modo facilli-<lb/>me mouentur, & </s> - <s xml:id="s361" xml:space="preserve">onera commouent. </s> - <s xml:id="s362" xml:space="preserve">An nõ quia <lb/>parua Å¿ui portione cùm tangit, tum offenÅ¿at cir-<lb/>culus, Å¿ed aliam ob cauÅ¿am. </s> - <s xml:id="s363" xml:space="preserve">ea autem eÅ¿t, quæ di <lb/>cta eÅ¿t prius, quod circulus Å¿cilicet ex duabus ef-<lb/>fectus eÅ¿t lationibus: </s> - <s xml:id="s364" xml:space="preserve">quam ob rem illarum alte- -<pb o="517" file="0517" n="14" rhead="Mechanicæ."/> -ram pro nutu Å¿emper habet, & </s> - <s xml:id="s365" xml:space="preserve">veluti continuo <lb/>motum illum mòuent, quicunq; </s> - <s xml:id="s366" xml:space="preserve">mouent, quan-<lb/>do Å¿ecundùm circun ferentiam illum mouerint: <lb/></s> - <s xml:id="s367" xml:space="preserve">latam enim ipÅ¿am mouent. </s> - <s xml:id="s368" xml:space="preserve">Eam quidem igitur, <lb/>quæ in obliquum eÅ¿t, motionem ipÅ¿um impellit <lb/>mouens: </s> - <s xml:id="s369" xml:space="preserve">Å¿ecundum vero illam, quæ Å¿uper diame <lb/>trum eÅ¿t, Å¿eipÅ¿um mouet circulus.</s> - <s xml:id="s370" xml:space="preserve"/> - </p> - <div type="float" level="2" n="8"> - <note position="right" xlink:label="note-0515-03" xlink:href="note-0515-03a" - xml:space="preserve">8</note> - </div> - <p> - <s xml:id="s371" xml:space="preserve">Cur ea, quæ per maiores circulos tolluntur & </s> - <s xml:id="s372" xml:space="preserve"><lb/> -<anchor type="note" xlink:label="note-0517-01a" xlink:href="note-0517-01"/> -trahuntur, facilius & </s> - <s xml:id="s373" xml:space="preserve">ci@ius moueri contingit, ve <lb/>lu@i maioribus trochleis quà m minoribus, & </s> - <s xml:id="s374" xml:space="preserve"><lb/>fcytalis Å¿imiliter? </s> - <s xml:id="s375" xml:space="preserve">An quoniã, qnanto maior fue-<lb/>ri@@illa quæ à centro eÅ¿t, in Ä™quali tempore maius <lb/>mouetur Å¿patium. </s> - <s xml:id="s376" xml:space="preserve">Quam ob rem æquali inexiÅ¿tẽ <lb/>te onere, idem faciet: </s> - <s xml:id="s377" xml:space="preserve">quemadmodum diximus, <lb/>& </s> - <s xml:id="s378" xml:space="preserve">maiores libras minoribus exactiores eſſe. </s> - <s xml:id="s379" xml:space="preserve">Å¿par <lb/>tum enim in illis centrum eÅ¿t: </s> - <s xml:id="s380" xml:space="preserve">libræ autem vtrin <lb/>que partes, quæ ex centro Å¿unt, exiÅ¿tunt.</s> - <s xml:id="s381" xml:space="preserve"/> - </p> - <div type="float" level="2" n="9"> - <note position="right" xlink:label="note-0517-01" xlink:href="note-0517-01a" - xml:space="preserve">9</note> - </div> - <p> - <s xml:id="s382" xml:space="preserve">Cur facilius quando Å¿ine pondere eÅ¿t, moue-<lb/> -<anchor type="note" xlink:label="note-0517-02a" xlink:href="note-0517-02"/> -tur libra, quà m cum pondus habet? </s> - <s xml:id="s383" xml:space="preserve">SimiliqÌ mo <lb/>do rota, & </s> - <s xml:id="s384" xml:space="preserve">huiuÅ¿modi quippiam, quod grauius <lb/>quidem eÅ¿t, maius autem minore & </s> - <s xml:id="s385" xml:space="preserve">leuiore? </s> - <s xml:id="s386" xml:space="preserve">An <lb/>quia non Å¿olùm in contrarium quod graue eÅ¿t, <lb/>Å¿ed in obliquum etiam difficulter mouetur. </s> - <s xml:id="s387" xml:space="preserve">In <lb/>contrarium enim ei, ad quod vergit onus, moue-<lb/>re difficile eÅ¿t: </s> - <s xml:id="s388" xml:space="preserve">quò autem vergit, eÅ¿t facilè. </s> - <s xml:id="s389" xml:space="preserve">in obli <lb/>quum autem haudquaquam vergit.</s> - <s xml:id="s390" xml:space="preserve"/> - </p> - <div type="float" level="2" n="10"> - <note position="right" xlink:label="note-0517-02" xlink:href="note-0517-02a" - xml:space="preserve">10</note> - </div> - <p> - <s xml:id="s391" xml:space="preserve">Cur Å¿uper Å¿cytalas facilius portantur onera, <lb/> -<anchor type="note" xlink:label="note-0517-03a" xlink:href="note-0517-03"/> -quã Å¿@per currus, cùm tamẽ ij magnas habeant <lb/>rotas, illæ verò puÅ¿illas? </s> - <s xml:id="s392" xml:space="preserve">An qui in Å¿cytalis nulla <lb/>eÅ¿t offen Å¿atio, in curribus autem axis eÅ¿t, ad quẽ <lb/>offenÅ¿ant. </s> - <s xml:id="s393" xml:space="preserve">deÅ¿uper enim illum premunt, & </s> - <s xml:id="s394" xml:space="preserve">late-<lb/>ribus. </s> - <s xml:id="s395" xml:space="preserve">quod autem eÅ¿t in Å¿cycalis, ad iÅ¿thæc duo <lb/>mouetur, & </s> - <s xml:id="s396" xml:space="preserve">inÅ¿ernè Å¿ubÅ¿trato Å¿patio, & </s> - <s xml:id="s397" xml:space="preserve">onere Å¿u <lb/>per impoÅ¿ito. </s> - <s xml:id="s398" xml:space="preserve">in vtriÅ¿que enim ijs reuoluitur lo-<lb/>cis circulus, & </s> - <s xml:id="s399" xml:space="preserve">motus impellitur.</s> - <s xml:id="s400" xml:space="preserve"/> - </p> - <div type="float" level="2" n="11"> - <note position="right" xlink:label="note-0517-03" xlink:href="note-0517-03a" - xml:space="preserve">11</note> - </div> - <pb o="518" file="0518" n="15" rhead="AriÅ¿t. Quæſtiones"/> - <p> - <s xml:id="s401" xml:space="preserve">Curlongius feruntur miſſilia funda, quà m ma <lb/> -<anchor type="note" xlink:label="note-0518-01a" xlink:href="note-0518-01"/> -numiſſa, cùm alioquin proiector mann magis <lb/>pondus comprehendat, quà m cùmillud Å¿uÅ¿pen-<lb/>dit? </s> - <s xml:id="s402" xml:space="preserve">Præterea Å¿ic quidem duo mouet pondera, <lb/>fundæ videlicet, & </s> - <s xml:id="s403" xml:space="preserve">miſſilis: </s> - <s xml:id="s404" xml:space="preserve">illo aÅ©t modo Å¿olum <lb/>miſſile. </s> - <s xml:id="s405" xml:space="preserve">An quia in funda quidem commotum <lb/>miſſile funditur, proijcit: </s> - <s xml:id="s406" xml:space="preserve">fundam enim circulo <lb/>Å¿ubinde rotans, id iaculatur: </s> - <s xml:id="s407" xml:space="preserve">ex manu autem à <lb/>quiete eÅ¿t initium: </s> - <s xml:id="s408" xml:space="preserve">omnia autem cùm in motu <lb/>Å¿unt, quà m cùm quieÅ¿cunt, facilius mouẽtur. </s> - <s xml:id="s409" xml:space="preserve">An <lb/>& </s> - <s xml:id="s410" xml:space="preserve">eam ob cauÅ¿am e@t, Å¿ed nec minus etiam, quia <lb/>in fundæ vÅ¿u manus quidem fit centrum: </s> - <s xml:id="s411" xml:space="preserve">Å¿unda <lb/>vero, quod à centro exit. </s> - <s xml:id="s412" xml:space="preserve">quanto aÅ©t productius <lb/>Å¿ueritid, quod à cÄ“tro eÅ¿t, tanto citius mouetur. <lb/></s> - <s xml:id="s413" xml:space="preserve">iactus aÅ«t qui mÄu fit, fundÄ™ reÅ¿pectu breuis eÅ¿t.</s> - <s xml:id="s414" xml:space="preserve"/> - </p> - <div type="float" level="2" n="12"> - <note position="left" xlink:label="note-0518-01" xlink:href="note-0518-01a" - xml:space="preserve">12</note> - </div> - <p> - <s xml:id="s415" xml:space="preserve">Cur circa idem iugum maiores collopes faci-<lb/> -<anchor type="note" xlink:label="note-0518-02a" xlink:href="note-0518-02"/> -lius, quà m minores mouentur? </s> - <s xml:id="s416" xml:space="preserve">& </s> - <s xml:id="s417" xml:space="preserve">item Å¿uculæ, <lb/>quæ graciliores Å¿unt, ab eadẽ viê—̃ craſſiores? </s> - <s xml:id="s418" xml:space="preserve">An <lb/>quÃa Å¿ecula quidem, & </s> - <s xml:id="s419" xml:space="preserve">iugum cÄ“trum eÅ¿t: </s> - <s xml:id="s420" xml:space="preserve">pro-<lb/>minentes autem longitudines eæ, quæ Å¿unt à cÄ“-<lb/>tro. </s> - <s xml:id="s421" xml:space="preserve">Celerius autem & </s> - <s xml:id="s422" xml:space="preserve">plus mouentur quæ maio <lb/>rum Å¿unt circulorum ab eadem vi, quà m quÄ™ mi <lb/>norum. </s> - <s xml:id="s423" xml:space="preserve">ab eadem enim vi plus transfertur id <lb/>extremum, quod longius à centro diÅ¿tat. </s> - <s xml:id="s424" xml:space="preserve">Quam <lb/>ob rem ad iugum quidem inÅ¿trumenta faciunt <lb/>collopas, quibus facilius verÅ¿ant: </s> - <s xml:id="s425" xml:space="preserve">in gracilibus <lb/>autem Å¿uculis plus fit id, quod extra lignum e@@. <lb/></s> - <s xml:id="s426" xml:space="preserve">hoc autem id efficitur, quod à centro exit.</s> - <s xml:id="s427" xml:space="preserve"/> - </p> - <div type="float" level="2" n="13"> - <note position="left" xlink:label="note-0518-02" xlink:href="note-0518-02a" - xml:space="preserve">13</note> - </div> - <p> - <s xml:id="s428" xml:space="preserve">Cur eiuÅ¿dem magnitudinis lignum facilius <lb/> -<anchor type="note" xlink:label="note-0518-03a" xlink:href="note-0518-03"/> -genus frangitur, Å¿i quiÅ¿piam æquì diductis mani <lb/>bus extrema comprehendens fregerit, quà m Å¿i <lb/>iuxta genu: </s> - <s xml:id="s429" xml:space="preserve">& </s> - <s xml:id="s430" xml:space="preserve">Å¿i terræ illud applicans pede Å¿u-<lb/>perimpoÅ¿ito manu longè diducta confregerit, ê—̃ <lb/>prope? </s> - <s xml:id="s431" xml:space="preserve">An quiaibi quidem genu centrÅ© eÅ¿t, hic <lb/>vero ipÅ¿e pes. </s> - <s xml:id="s432" xml:space="preserve">quanto autem remotius à centro -<pb o="519" file="0519" n="16" rhead="Mechanicæ."/> -Å¿uerit, faciliùs mouetur quodcunq;</s> - <s xml:id="s433" xml:space="preserve">. moueriau-<lb/>tem quod frangitur, neceſſe eÅ¿t.</s> - <s xml:id="s434" xml:space="preserve"/> - </p> - <div type="float" level="2" n="14"> - <note position="left" xlink:label="note-0518-03" xlink:href="note-0518-03a" - xml:space="preserve">14</note> - </div> - <p> - <s xml:id="s435" xml:space="preserve">Cur eæ quæ circa littora appellantur Crocæ, <lb/> -<anchor type="note" xlink:label="note-0519-01a" xlink:href="note-0519-01"/> -rotunda Å¿unt figura, cùm alioquin à principio ex <lb/>magnis Å¿int lapidibus, oÅ¿treiÅ¿ve? </s> - <s xml:id="s436" xml:space="preserve">An quia ea, q̃ <lb/>plus recedunt à medio in motionibus, feruntur <lb/>celeriùs. </s> - <s xml:id="s437" xml:space="preserve">medium enim fit centrum: </s> - <s xml:id="s438" xml:space="preserve">intcruallÅ© <lb/>verò ea, quæ à centro. </s> - <s xml:id="s439" xml:space="preserve">Sem per autem maior ab <lb/>æquali motione maiorem deÅ¿cribit circulum. <lb/></s> - <s xml:id="s440" xml:space="preserve">quod autem maius in æquali pertranÅ¿it tẽpore, <lb/>celerius fertur: </s> - <s xml:id="s441" xml:space="preserve">quæ autem celeriùs ex Ä™ quali fe <lb/>runtur Å¿patio, uehementiùs im petunt: </s> - <s xml:id="s442" xml:space="preserve">quÄ™ autẽ <lb/>magis impetunt impetuntur & </s> - <s xml:id="s443" xml:space="preserve">magis. </s> - <s xml:id="s444" xml:space="preserve">quam ob <lb/>rem ea quæ plus à medio diÅ¿tant, cÅfringi neceÅ¿ <lb/>Å¿e eÅ¿t, id autem cùm patiantur, rotunda fieri eÅ¿t <lb/>neceſſarium. </s> - <s xml:id="s445" xml:space="preserve">Crocis autem propter maris mo-<lb/>tum, quoniam Å¿imul cum illo agitantur, in perpe <lb/>tua eſſe accidit motione, eoqÌ; </s> - <s xml:id="s446" xml:space="preserve">verÅ¿atas mÅ Å¿em <lb/>per offenÅ¿are. </s> - <s xml:id="s447" xml:space="preserve">id autem ipÅ¿is maximè extremis <lb/>contingere partibus eÅ¿t neceſſe.</s> - <s xml:id="s448" xml:space="preserve"/> - </p> - <div type="float" level="2" n="15"> - <note position="right" xlink:label="note-0519-01" xlink:href="note-0519-01a" - xml:space="preserve">15</note> - </div> - <p> - <s xml:id="s449" xml:space="preserve">Cur quanto longiora Å¿untligna, tanto imbe-<lb/> -<anchor type="note" xlink:label="note-0519-02a" xlink:href="note-0519-02"/> -cilliora fiunt: </s> - <s xml:id="s450" xml:space="preserve">& </s> - <s xml:id="s451" xml:space="preserve">Å¿i tollantur, in flectuntur magis: <lb/></s> - <s xml:id="s452" xml:space="preserve">tametÅ¿i quod breue quidem eÅ¿t, ceu bicubitum, <lb/>fuerit tenue: </s> - <s xml:id="s453" xml:space="preserve">quod verò cubitorum centum, craÅ¿ <lb/>Å¿um? </s> - <s xml:id="s454" xml:space="preserve">An ê—a & </s> - <s xml:id="s455" xml:space="preserve">vectis, & </s> - <s xml:id="s456" xml:space="preserve">onus, & </s> - <s xml:id="s457" xml:space="preserve">hypomochlion <lb/>in leuando ipÅ¿a fit ligni proceritas: </s> - <s xml:id="s458" xml:space="preserve">prior nanq; </s> - <s xml:id="s459" xml:space="preserve"><lb/>illius pars ceu hypomochlion fit, quod verò in <lb/>extremo eÅ¿t, pÅdus. </s> - <s xml:id="s460" xml:space="preserve">quam ob rem quanto exten. </s> - <s xml:id="s461" xml:space="preserve"><lb/>Å¿ius fuerit id, quod ab hypocomochlio eÅ¿t, tanto <lb/>inflecti neceſſe ẽ magis. </s> - <s xml:id="s462" xml:space="preserve">quo enim plus ab hypo-<lb/>mochlio diÅ¿tat, eo magis incuruari neceſſe eÅ¿t. </s> - <s xml:id="s463" xml:space="preserve"><lb/>neceſſario igitur extrema vectis eleuantur. </s> - <s xml:id="s464" xml:space="preserve">Siigi <lb/>tur flexilis fuerit vectis, ipÅ¿um inflecti magis <lb/>cùm extollitur neceſſe eÅ¿t: </s> - <s xml:id="s465" xml:space="preserve">quod lõgis accidit li <lb/>gnis. </s> - <s xml:id="s466" xml:space="preserve">in breuibus autem quod vltimÅ© eÅ¿t, quie- -<pb o="520" file="0520" n="17" rhead="AriÅ¿t. Quæſtiones"/> -Å¿centi hy pomochlio deprope Å¿it.</s> - <s xml:id="s467" xml:space="preserve"/> - </p> - <div type="float" level="2" n="16"> - <note position="right" xlink:label="note-0519-02" xlink:href="note-0519-02a" - xml:space="preserve">16</note> - </div> - <p> - <s xml:id="s468" xml:space="preserve">Cur a paruo exift@nte cuneo magna Å¿cindun-<lb/>tur pondera, & </s> - <s xml:id="s469" xml:space="preserve">corporum moles, validaque fit <lb/>impreſſio? </s> - <s xml:id="s470" xml:space="preserve">An quia cuneus duo Å¿unt vectes Å¿ibi <lb/>inuicem contrarij: </s> - <s xml:id="s471" xml:space="preserve">vterque autem & </s> - <s xml:id="s472" xml:space="preserve">pondus ha <lb/>b@t, & </s> - <s xml:id="s473" xml:space="preserve">hypomochlion, quod diuellit, & </s> - <s xml:id="s474" xml:space="preserve">compri-<lb/>mit. </s> - <s xml:id="s475" xml:space="preserve">Plagæ quinetiam ipÅ¿ius latio pondus, quod <lb/>percutit & </s> - <s xml:id="s476" xml:space="preserve">mouet, magnum facit: </s> - <s xml:id="s477" xml:space="preserve">& </s> - <s xml:id="s478" xml:space="preserve">quoniÄ mo <lb/>tum mouet, pÅ¿a celeritate valentius fit. </s> - <s xml:id="s479" xml:space="preserve">paruo au <lb/>tem exiÅ¿tente vecte, magnæ illum conÅ¿equuntur <lb/>vires: </s> - <s xml:id="s480" xml:space="preserve">quam ob rem vltra magnitndinis decentiã <lb/>later mouens, Sit cuneus vbi A B C, quod verò <lb/>cuneo Å¿cinditur, D E F G. </s> - <s xml:id="s481" xml:space="preserve">vectis igitur Å¿it ipÅ¿a <lb/>A B, pondus verò ipÅ¿ius B inferior pars, hypomo <lb/>chlion autem D G, huic autem contrarius vect is <lb/>B C. </s> - <s xml:id="s482" xml:space="preserve">percuſſa igitur A C, vtroque illorum vtitur <lb/>vecte: </s> - <s xml:id="s483" xml:space="preserve">Å¿cindit enim ipÅ¿um B.</s> - <s xml:id="s484" xml:space="preserve"/> - </p> - <p> - <s xml:id="s485" xml:space="preserve">Cur Å¿i quiÅ¿piam trochleas componens duas in <lb/> -<anchor type="note" xlink:label="note-0520-01a" xlink:href="note-0520-01"/> -tignis duobus ad Å¿e inuicem inuctis cõtrario ad <lb/>trochleas modo circulo funem circunduxerit, <lb/>cuius alterum quidem caput tigonrum appenda <lb/>tur alteri, alterum varò troclheis Å¿it innixum, & </s> - <s xml:id="s486" xml:space="preserve"><lb/>à funis initio trahere coeperit, magna trahit pon <lb/>dera, licet imbecillium fuerit virium? </s> - <s xml:id="s487" xml:space="preserve">An quia <lb/>idem põdus à minori potentia Å¿i mouetur, vecte <lb/>medio trãsfertur magis, quà m à manu: </s> - <s xml:id="s488" xml:space="preserve">trochlea <lb/>autem idem vecti facit. </s> - <s xml:id="s489" xml:space="preserve">quam ob rem Å¿i vna faci-<lb/>lius traher, & </s> - <s xml:id="s490" xml:space="preserve">ab vnico tractu multò grauius tra-<lb/>het, quà m facere poſſit manus, idip Å¿um duæ tro-<lb/>chleæ plus quà m in dupla velocitate leuabunt. <lb/></s> - <s xml:id="s491" xml:space="preserve">Minus enim altera trahit, quà m Å¿i ip Å¿a per Å¿eip-<lb/>Å¿am traheret, quando circa alteram iniectus Å¿ue <lb/>rit funis: </s> - <s xml:id="s492" xml:space="preserve">illa nanque minus etiam pondus effe-<lb/>cit. </s> - <s xml:id="s493" xml:space="preserve">PariqÌ; </s> - <s xml:id="s494" xml:space="preserve">modo Å¿i ad plures iniectus fuerit funis <lb/>in paucis trochleis, multa Å¿it differentia. </s> - <s xml:id="s495" xml:space="preserve">quã ob -<pb o="521" file="0521" n="18" rhead="Mechanicæ"/> -rem à prima pondere quat@or minas trahente, <lb/>ab vl im a trahi multò minus. </s> - <s xml:id="s496" xml:space="preserve">Et in re ædifica-<lb/>toria faciliter magn@ a mouen@ pondera, traducÅ©t <lb/>enim ab una trochlea ad aliam, & </s> - <s xml:id="s497" xml:space="preserve">rurÅ¿us ab illa <lb/>ad Å¿uculas, & </s> - <s xml:id="s498" xml:space="preserve">vectes. </s> - <s xml:id="s499" xml:space="preserve">hoc autem idem eÅ¿t, ac Å¿i <lb/>multas facerent trochleas.</s> - <s xml:id="s500" xml:space="preserve"/> - </p> - <div type="float" level="2" n="17"> - <note position="left" xlink:label="note-0520-01" xlink:href="note-0520-01a" - xml:space="preserve">18</note> - </div> - <p> - <s xml:id="s501" xml:space="preserve">Cur Å¿i quis Å¿uper lignum magnã imponat Å¿e-<lb/> -<anchor type="note" xlink:label="note-0521-01a" xlink:href="note-0521-01"/> -carim, deÅ¿uperq; </s> - <s xml:id="s502" xml:space="preserve">illi magnum adijciat pondus, <lb/>ligni quippiam quod curandum Å¿it, non diuidit: <lb/></s> - <s xml:id="s503" xml:space="preserve">Å¿iverò Å¿ecurim extoll@s percutiat, illud Å¿cindit, <lb/>cùm alioquin multò minus habeat ponderis id <lb/>quod percutit, quà m id quod Å¿uperiacet, & </s> - <s xml:id="s504" xml:space="preserve">pre-<lb/>mit? </s> - <s xml:id="s505" xml:space="preserve">An quia omnia cum motu fiunt: </s> - <s xml:id="s506" xml:space="preserve">& </s> - <s xml:id="s507" xml:space="preserve">graue <lb/>ipÅ¿um grauitatis magis aſſumit motum dÅ© mo-<lb/>uetur, ê—̃ dum quieÅ¿cit Incumbensigitur conua-<lb/>tum graui motionem non mouetur, motum ve-<lb/>rò & </s> - <s xml:id="s508" xml:space="preserve">Å¿ecundum hanc mouetur, & </s> - <s xml:id="s509" xml:space="preserve">Å¿ecundùm eã <lb/>quæ eÅ¿t percutientis. </s> - <s xml:id="s510" xml:space="preserve">PrÄ™terea Å¿ecuris ipÅ¿a effici-<lb/>tur cuneus: </s> - <s xml:id="s511" xml:space="preserve">paruus autẽ exiÅ¿tens cuneus magna <lb/>diuidit cum ex duobus Å¿it vectibus contrario ad <lb/>Å¿eÅ¿e inuicem modo conÅ¿titutis</s> - </p> - <div type="float" level="2" n="18"> - <note position="right" xlink:label="note-0521-01" xlink:href="note-0521-01a" - xml:space="preserve">19</note> - </div> - <p> - <s xml:id="s512" xml:space="preserve">Cur Å¿tatera, qua carnes ponderãtur, paruo ap-<lb/> -<anchor type="note" xlink:label="note-0521-02a" xlink:href="note-0521-02"/> -pendiculo magna trutinat onera, cùm alioquin <lb/>tota dimidiata exiftar libra? </s> - <s xml:id="s513" xml:space="preserve">vbi enim onus impo-<lb/>nitur, Å¿olùm Å¿uÅ¿penditur lanx: </s> - <s xml:id="s514" xml:space="preserve">in altera verò par <lb/>te, Å¿ola eÅ¿t Å¿tatera. </s> - <s xml:id="s515" xml:space="preserve">An quia Å¿imul libram & </s> - <s xml:id="s516" xml:space="preserve">vectẽ <lb/>ipÅ¿am contingit eſſe Å¿tateram. </s> - <s xml:id="s517" xml:space="preserve">libram quidem, <lb/>vbi Å¿partorum quodcunque Å¿tateræ fit centrum: <lb/></s> - <s xml:id="s518" xml:space="preserve">in altera enim parte lancem, in altera autem pro <lb/>lance æquipondij appendiculum habet. </s> - <s xml:id="s519" xml:space="preserve">quod li-<lb/>bre incumbit, ceu Å¿i quis alterain apponeret lan-<lb/>cem, & </s> - <s xml:id="s520" xml:space="preserve">ilii pondus imponeret. </s> - <s xml:id="s521" xml:space="preserve">manifeÅ¿tÅ© enim, <lb/>quòd tantundem tradit ponderis ei quod in alte <lb/>raiacet lance. </s> - <s xml:id="s522" xml:space="preserve">Quemadmodum autem Å¿i vna li-<lb/>bra multæ Å¿int libræ, Å¿ic talia inÅ¿unt Å¿parta mul.</s> - <s xml:id="s523" xml:space="preserve"> -<pb o="522" file="0522" n="19" rhead="AriÅ¿t. Quæſtiones"/> -tain eiuÅ¿modi libra: </s> - <s xml:id="s524" xml:space="preserve">quorum vniuÅ¿cuiuÅ¿q; </s> - <s xml:id="s525" xml:space="preserve">quod <lb/>intrinÅ¿ecus eÅ¿t ad appẽdiculum, Å¿tateræ eÅ¿t dimi-<lb/>dium. </s> - <s xml:id="s526" xml:space="preserve">& </s> - <s xml:id="s527" xml:space="preserve">omnino iÅ¿tud libra eÅ¿t, vnam quidẽ ha-<lb/>benslancem, in qua pondus appẽditur: </s> - <s xml:id="s528" xml:space="preserve">alteram <lb/>verò, vbi in Å¿tatera æquipondium. </s> - <s xml:id="s529" xml:space="preserve">Quam ob rem <lb/>appendiculum ad alteram Å¿ui partem eÅ¿t Å¿tatera. <lb/></s> - <s xml:id="s530" xml:space="preserve">huiuÅ¿modi autem exiÅ¿tÄ“s multæ Å¿unt libræ, totqÌ; </s> - <s xml:id="s531" xml:space="preserve"><lb/>quot fuerint Å¿parta. </s> - <s xml:id="s532" xml:space="preserve">Semper autem quod lanci <lb/>propinquius eÅ¿t Å¿partum, appenÅ¿oqÌue oneri, ma-<lb/>ius trahit pÅdus: </s> - <s xml:id="s533" xml:space="preserve">quoniam fit quidem omnis Å¿ta-<lb/>tera inuerÅ¿us vectis: </s> - <s xml:id="s534" xml:space="preserve">hypomochlion nãq; </s> - <s xml:id="s535" xml:space="preserve">vnum-<lb/>quodque Å¿partum Å¿uperne exiÅ¿tens: </s> - <s xml:id="s536" xml:space="preserve">pondus verÅ© <lb/>id quod lanci ineÅ¿t. </s> - <s xml:id="s537" xml:space="preserve">Quanto autem productior ve <lb/>ctis Å¿uerit longitudo ab ipfo hypomochlio, tãto <lb/>ibi quidem faciliùs mouet: </s> - <s xml:id="s538" xml:space="preserve">hic autem æquilibri-<lb/>brium facit, pondusqÌue Å¿tateræ trutinat, quod <lb/>ad æquipondij vergit appendiculum.</s> - <s xml:id="s539" xml:space="preserve"/> - </p> - <div type="float" level="2" n="19"> - <note position="right" xlink:label="note-0521-02" xlink:href="note-0521-02a" - xml:space="preserve">20</note> - </div> - <p> - <s xml:id="s540" xml:space="preserve">Cur medici Å¿acilius dentes extrahÅ«t dentifor-<lb/>cipis onere adiecto, quà m Å¿i Å¿ola vtantur manu? <lb/></s> - <s xml:id="s541" xml:space="preserve">An quia ex manu magis quà m ex dentiforcipe <lb/>lubricus elabitur dẽs? </s> - <s xml:id="s542" xml:space="preserve">An ferro id potius accidit <lb/>quà m digitis, quoniam vndique dentem non cÅ-<lb/>prehendunt: </s> - <s xml:id="s543" xml:space="preserve">quod mollis digitorum facit caro: </s> - <s xml:id="s544" xml:space="preserve"><lb/>adhæret enim, & </s> - <s xml:id="s545" xml:space="preserve">cõplectitur magis. </s> - <s xml:id="s546" xml:space="preserve">An quia den <lb/>@iÅ¿orceps duo contrarij vectes, vnicum habentes <lb/>hypom ochlion, eius Å¿cilicet inÅ¿trumenti conne-<lb/>xionem. </s> - <s xml:id="s547" xml:space="preserve">Hoc igitur ad extractionem vtuntur or-<lb/>gano, vt facilius moueant. </s> - <s xml:id="s548" xml:space="preserve">Sic dentiforcipis alte-<lb/>rum quidem exrremum vbi eÅ¿t A: </s> - <s xml:id="s549" xml:space="preserve">alterum autẽ <lb/>quod extrahit B. </s> - <s xml:id="s550" xml:space="preserve">vectis autem vbi A D F: </s> - <s xml:id="s551" xml:space="preserve">alterve <lb/>rò vectis vbi B C E. </s> - <s xml:id="s552" xml:space="preserve">hypomochlion autẽ C G D: </s> - <s xml:id="s553" xml:space="preserve"><lb/>connexio verò vbi G: </s> - <s xml:id="s554" xml:space="preserve">dẽs autem pondus. </s> - <s xml:id="s555" xml:space="preserve">Vtroq; </s> - <s xml:id="s556" xml:space="preserve"><lb/>igitur B & </s> - <s xml:id="s557" xml:space="preserve">F Å¿imul comprehendentes mouent. </s> - <s xml:id="s558" xml:space="preserve"><lb/>quando autem commotus fuerit, faciliùs manu <lb/>trahitur, quà m inÅ¿trumento.</s> - <s xml:id="s559" xml:space="preserve"/> - </p> - <pb o="523" file="0523" n="20" rhead="Mechanicæ"/> - <p> - <s xml:id="s560" xml:space="preserve">Cur nuces abÅ¿que ictu facilè cõfrigun tinÅ¿tru-<lb/>mẽtis, quæ ad eum fiunt vÅ¿um? </s> - <s xml:id="s561" xml:space="preserve">multumeinim au <lb/>fertur virium, motionis Å¿cilicet, & </s> - <s xml:id="s562" xml:space="preserve">violẽ@iæ. </s> - <s xml:id="s563" xml:space="preserve">PrÄ™-<lb/>terea duro & </s> - <s xml:id="s564" xml:space="preserve">graui comprimens inÅ¿trumento ci <lb/>tiùs conÅ¿ringet, quà m ligneo & </s> - <s xml:id="s565" xml:space="preserve">leui. </s> - <s xml:id="s566" xml:space="preserve">An quia Å¿ic <lb/>vtrinqne à duobus comprimitur vectibus ipÅ¿a <lb/>nux: </s> - <s xml:id="s567" xml:space="preserve">a vecte autem facile diuelluntur onera. </s> - <s xml:id="s568" xml:space="preserve">id <lb/>enim inÅ¿trumẽtum ex duobus componitur vecti <lb/>bus idem habentibus hypomochlion, connexio-<lb/>nem videlicet @pÅ¿am vbi eÅ¿t A. </s> - <s xml:id="s569" xml:space="preserve">Quemadmodum <lb/>igitur fuere diductÄ™ Å¿ecundÅ© extrema motis C D <lb/>ipſæ F E, Å¿ic à parua faciliter potentia conducun <lb/>tur. </s> - <s xml:id="s570" xml:space="preserve">Quod igitur cum percuſſione Å¿eciſſet põdus, <lb/>id valentiores illa E C, & </s> - <s xml:id="s571" xml:space="preserve">F D vectes efficiunt. </s> - <s xml:id="s572" xml:space="preserve">ele <lb/>uatione enim in contrarium ela@i & </s> - <s xml:id="s573" xml:space="preserve">comprimẽ-<lb/>tes, frangunt vbi eÅ¿t K. </s> - <s xml:id="s574" xml:space="preserve">Hanc etiam ob cauÅ¿am <lb/>quanto vicinius fuerit K ipÅ¿um A, confringitur <lb/>celerius. </s> - <s xml:id="s575" xml:space="preserve">quanto enim ab hypomochlio plus di-<lb/>Å¿tat vectis, facilius & </s> - <s xml:id="s576" xml:space="preserve">plus mouet ab eadẽ poten-<lb/>tia. </s> - <s xml:id="s577" xml:space="preserve">EÅ¿t igitur A quidem hypomochlion: </s> - <s xml:id="s578" xml:space="preserve">ipÅ¿a au-<lb/>tem D A F vectis: </s> - <s xml:id="s579" xml:space="preserve">& </s> - <s xml:id="s580" xml:space="preserve">item ipia C A E. </s> - <s xml:id="s581" xml:space="preserve">Quanto igi-<lb/>tur ipÅ¿um K vicinius Å¿uerit angulo ipÅ¿ius A, tãto <lb/>vicinius fit connexioni vbi eÅ¿t A. </s> - <s xml:id="s582" xml:space="preserve">hoc autem eÅ¿t <lb/>hypomochlion. </s> - <s xml:id="s583" xml:space="preserve">Ab eadem igitur potentia appli-<lb/>cante F E plus extolli neceſſe eÅ¿t. </s> - <s xml:id="s584" xml:space="preserve">Quam ob rem <lb/>quoniã ex contrario eÅ¿t eleuatio, neceſſe eÅ¿t ma-<lb/>gis com primi. </s> - <s xml:id="s585" xml:space="preserve">quod autem comprimitur magis, <lb/>ci@ius frangitur.</s> - <s xml:id="s586" xml:space="preserve"/> - </p> - <p> - <s xml:id="s587" xml:space="preserve">Cur Å¿i duo extrema in rhombo puncta dua-<lb/>bus ferantur lationibus, haudquaquam æquale <lb/>vtrunque eorÅ© pertianÅ¿it rectam, Å¿ed multò plus <lb/>alterum ?</s> - <s xml:id="s588" xml:space="preserve">idem autem eÅ¿t Å¿ermo, cur quod Å¿uper <lb/>latus fertur, minus pertranÅ¿it, quà m ipÅ¿um latus. <lb/></s> - <s xml:id="s589" xml:space="preserve">illud enim diametrum minorẽ, hoc verò maius <lb/>latus. </s> - <s xml:id="s590" xml:space="preserve">& </s> - <s xml:id="s591" xml:space="preserve">hoc quidem vnica, illud verò duabus Å¿er -<pb o="524" file="0524" n="21" rhead="AriÅ¿t Quæſtiones."/> -t<unsure/>ur lationibus. </s> - <s xml:id="s592" xml:space="preserve">Feratur enim ex ipÅ¿a A B, A qui-<lb/>dem ad ipÅ¿um B, B vero ad ipÅ¿um D eadem cele <lb/>ritate. </s> - <s xml:id="s593" xml:space="preserve">feratur aurẽ & </s> - <s xml:id="s594" xml:space="preserve">ipÅ¿a A B in ipÅ¿a A C iuxta <lb/>C D eadem celeritate cum illis. </s> - <s xml:id="s595" xml:space="preserve">neceſſe igitur eÅ¿t <lb/>A quidem in ipÅ¿a A D diametro ferri, B verò in <lb/>ipÅ¿a B C. </s> - <s xml:id="s596" xml:space="preserve">& </s> - <s xml:id="s597" xml:space="preserve">vtranq; </s> - <s xml:id="s598" xml:space="preserve">Å¿imul pertranÅ¿iſſe, & </s> - <s xml:id="s599" xml:space="preserve">ipÅ¿a@@ <lb/>A B ipÅ¿um la<unsure/>tus A C. </s> - <s xml:id="s600" xml:space="preserve">latum enim Å¿it ipÅ¿um A ip <lb/>Å¿am A E, A B autem ipÅ¿am A E, & </s> - <s xml:id="s601" xml:space="preserve">proiecta Å¿it <lb/>F G iuxta ipÅ¿um A B, & </s> - <s xml:id="s602" xml:space="preserve">ab ipÅ¿o E Å¿imiliter replea <lb/>tur. </s> - <s xml:id="s603" xml:space="preserve">Simile igitur fit, quod repletum eÅ¿t ipÅ¿i toti. <lb/></s> - <s xml:id="s604" xml:space="preserve">æqualis igirur A F ipÅ¿i A E ipÅ¿a autem A B ipÅ¿am <lb/>A F lata erit. </s> - <s xml:id="s605" xml:space="preserve">in diametro igitur erit Å¿ecÅ«dÅ« K. </s> - <s xml:id="s606" xml:space="preserve">& </s> - <s xml:id="s607" xml:space="preserve"><lb/>fem per neceſſe eÅ¿t ipÅ¿um ferri Å¿ecundùm diame <lb/>trum. </s> - <s xml:id="s608" xml:space="preserve">& </s> - <s xml:id="s609" xml:space="preserve">Å¿imul A B latus pertranfit latus A C, & </s> - <s xml:id="s610" xml:space="preserve"><lb/>ipÅ¿um A diametrum pertranÅ¿it A D. </s> - <s xml:id="s611" xml:space="preserve">Similiter ẽt <lb/>demon Å¿trabitur & </s> - <s xml:id="s612" xml:space="preserve">ipÅ¿um B in ipÅ¿a B C diametro <lb/>latum. </s> - <s xml:id="s613" xml:space="preserve">æqualis enim eÅ¿t ipÅ¿a B E ipÅ¿i B G Reple. </s> - <s xml:id="s614" xml:space="preserve"><lb/>to igitur ab ipÅ¿o G. </s> - <s xml:id="s615" xml:space="preserve">quod intus eÅ¿t, toti eÅ¿t Å¿imi-<lb/>le: </s> - <s xml:id="s616" xml:space="preserve">& </s> - <s xml:id="s617" xml:space="preserve">ipÅ¿um B in ipÅ¿a diame@ro erit Å¿ecundùm la <lb/>terum connexionem. </s> - <s xml:id="s618" xml:space="preserve">Et Å¿imul latus pertranÅ¿it <lb/>latus, & </s> - <s xml:id="s619" xml:space="preserve">B ipÅ¿um B C diametrum. </s> - <s xml:id="s620" xml:space="preserve">Simuligitur A <lb/>multo plus ipÅ¿a A B pertrÄÅ¿it: </s> - <s xml:id="s621" xml:space="preserve">& </s> - <s xml:id="s622" xml:space="preserve">ipÅ¿um latus, mi-<lb/>nus latus eadem lata celeritate: </s> - <s xml:id="s623" xml:space="preserve">& </s> - <s xml:id="s624" xml:space="preserve">ipÅ¿um latus <lb/>maiorem, quà m B, pertranÅ¿iuit vna latum latio-<lb/>ne. </s> - <s xml:id="s625" xml:space="preserve">Quanto enim acutior fuerit rhombus, diame <lb/>ter quidem minor Å¿it. </s> - <s xml:id="s626" xml:space="preserve">A C autẽ maior: </s> - <s xml:id="s627" xml:space="preserve">latus ve-<lb/>ro ipÅ¿ius B C minus. </s> - <s xml:id="s628" xml:space="preserve">AbÅ¿urdÅ« eÅ¿t enim (vt dictÅ©) <lb/>id quod duabus fertur lationibus, aliquãdo ferri <lb/>tardius illo. </s> - <s xml:id="s629" xml:space="preserve">quod fertur vnica: </s> - <s xml:id="s630" xml:space="preserve">& </s> - <s xml:id="s631" xml:space="preserve">vtriÅ¿q; </s> - <s xml:id="s632" xml:space="preserve">poÅ¿itis <lb/>æquali velocitate punctis, alterÅ« pertranÅ¿ire ma <lb/>iorem. </s> - <s xml:id="s633" xml:space="preserve">CauÅ¿a autem eÅ¿t, quoniã ei quod ab obt@ <lb/>Å¿o Å¿ertur angulo ambæ ferè cÅtrariæ fiunt latio-<lb/>n<unsure/>es, & </s> - <s xml:id="s634" xml:space="preserve">illa Å¿ecÅ«dÅ« quÄ ipÅ¿um fertur, & </s> - <s xml:id="s635" xml:space="preserve">illa Å¿ecun <lb/>dÅ© quã ipÅ¿um à latere defertur. </s> - <s xml:id="s636" xml:space="preserve">Ei autẽ quod ab <lb/>@cuto fertur, accidit vt ad idÄ“ feratur, coadiuuat -<pb o="525" file="0525" n="22" rhead="Mechanicæ."/> -@nim quæ ipÅ¿ius eÅ¿t lateris, illã quæ eÅ¿t Å¿uper dia-<lb/>metrum. </s> - <s xml:id="s637" xml:space="preserve">& </s> - <s xml:id="s638" xml:space="preserve">quanto hunc quidem acutiorem fe-<lb/>ceris, illum verò obtuÅ¿um magis: </s> - <s xml:id="s639" xml:space="preserve">hæc quidÄ“ tar-<lb/>dior erit, illa verò celerior. </s> - <s xml:id="s640" xml:space="preserve">Hæ quidẽ igitur ma-<lb/>gis contrariæ fiunt, qmÌ„ obtuÅ¿ior fit angulus: </s> - <s xml:id="s641" xml:space="preserve">illæ <lb/>verò ad idẽ magis, qmÌ„ lineæ coarctantur: </s> - <s xml:id="s642" xml:space="preserve">ipÅ¿um <lb/>enim A ferè ad idem fertur Å¿ecundum ambas la <lb/>tiones. </s> - <s xml:id="s643" xml:space="preserve">Coadiuuatur igitur altera: </s> - <s xml:id="s644" xml:space="preserve">& </s> - <s xml:id="s645" xml:space="preserve">quanto Å¿anè <lb/>acutior fuerit angulus, tanto magis ipÅ¿um A ad <lb/>contrarium: </s> - <s xml:id="s646" xml:space="preserve">ipÅ¿um. </s> - <s xml:id="s647" xml:space="preserve">n. </s> - <s xml:id="s648" xml:space="preserve">ad B fertur: </s> - <s xml:id="s649" xml:space="preserve">latus autẽ de-<lb/>fertipÅ¿um ad D. </s> - <s xml:id="s650" xml:space="preserve">Et quanto Å¿anè obtuÅ¿ior fuerit <lb/>angulus, magis contrariæ fiunt lationes: </s> - <s xml:id="s651" xml:space="preserve">rectior <lb/>enim efficitur linea. </s> - <s xml:id="s652" xml:space="preserve">Å¿i autẽ omnino recta fieret, <lb/>penitus vtique eſſent contrariæ latus verò Å¿ecun <lb/>dùm vnicam latuum lationem à nullo præpedi-<lb/>tur: </s> - <s xml:id="s653" xml:space="preserve">rationabiliter igitur maiorem pertranÅ¿it.</s> - <s xml:id="s654" xml:space="preserve"/> - </p> - <p> - <s xml:id="s655" xml:space="preserve">Dubitatur quam ob cauÅ¿am maior circulus Ä™-<lb/>qualem minori circulo conuoluit<unsure/>ur lineam, qÅ«<unsure/> <lb/>circa idem centrum fuerint poÅ¿iti: </s> - <s xml:id="s656" xml:space="preserve">Å¿eorÅ¿um autẽ <lb/>reuoluti, quemadmodum alterius magnitudo ad <lb/>magnitudiuem Å¿e habet alterius, fic & </s> - <s xml:id="s657" xml:space="preserve">illorÅ© ad <lb/>Å¿einuicem fiunt lineæ. </s> - <s xml:id="s658" xml:space="preserve">Præterea vno et@ã & </s> - <s xml:id="s659" xml:space="preserve">eodÄ“ <lb/>v<unsure/>triÅ¿que exiÅ¿tente centro, aliquando quidem tan <lb/>ta Å¿it linea, quam conuoluuntur, quantam minor <lb/>per Å¿e conuoluitur circulus, quandoq; </s> - <s xml:id="s660" xml:space="preserve">verò quã-<lb/>tam maior. </s> - <s xml:id="s661" xml:space="preserve">Quòd quidem igitur maiorem con <lb/>uoluitur maior, manifeÅ¿tum eÅ¿t. </s> - <s xml:id="s662" xml:space="preserve">angulus enim <lb/>Å¿en Å¿u videtur eſſe cuiuÅ¿que circunferentia pro-<lb/>priæ diametri maioris circuli maior, minoris mi <lb/>nor. </s> - <s xml:id="s663" xml:space="preserve">quam ob rem eandem habebunt propor-<lb/>tionem Å¿ecundùm Å¿enÅ¿um ad Å¿e lineæ. </s> - <s xml:id="s664" xml:space="preserve">Å¿ecun-<lb/>dùm quas fuerint conuoluti. </s> - <s xml:id="s665" xml:space="preserve">verùm enimuerò <lb/>quòd etiam æqualem conuoluuntur, quando cir<unsure/> <lb/>t<unsure/>a idem fuerint poÅ¿iti centrum, manifeÅ¿tÅ© eÅ¿t: </s> - <s xml:id="s666" xml:space="preserve">& </s> - <s xml:id="s667" xml:space="preserve"><lb/>Å¿ic fiunt aliquando quidÄ“ æquales lineæ, Å¿ecun- -<pb o="526" file="0526" n="23" rhead="AriÅ¿t Quæſtiones"/> -dum quam maior conuoluitur circulus, aliquan <lb/>do verò Å¿ecundùm quam minor Sit enim circu-<lb/>lus maior quidem vbi D F C. </s> - <s xml:id="s668" xml:space="preserve">minor verò vbi <lb/>E C B: </s> - <s xml:id="s669" xml:space="preserve">vtriÅ¿qÌ; </s> - <s xml:id="s670" xml:space="preserve">autem centrum A. </s> - <s xml:id="s671" xml:space="preserve">Et quam qui-<lb/>dem per Å¿e magnus conuoluitur, Å¿it vbi F I: </s> - <s xml:id="s672" xml:space="preserve">quã <lb/>vero per Å¿e minor vbi GK, æqualis A. </s> - <s xml:id="s673" xml:space="preserve">Si igitur mi <lb/>norem mouero, idem mouens centÅ« vbi A, ma-<lb/>ior autem Å¿it annexus. </s> - <s xml:id="s674" xml:space="preserve">qñ igitur A B fuerit recta <lb/>ad ipÅ¿am G K, Å¿imul & </s> - <s xml:id="s675" xml:space="preserve">A C fit recta ad ipÅ¿am F I. <lb/></s> - <s xml:id="s676" xml:space="preserve">quam ob rem Ä™qualem Å¿emper tranÅ¿lata erit ip-<lb/>Å¿am quidem G K, vbi eÅ¿t G B circunferentia: </s> - <s xml:id="s677" xml:space="preserve">ip-<lb/>Å¿am verò FL, quæ eÅ¿t vbi F C. </s> - <s xml:id="s678" xml:space="preserve">Si aÅ©t quarta pars <lb/>æqualem conuoluitur, manifeÅ¿tum eÅ¿t, ê™ totus <lb/>circulus toti circulo æqualem conuoluetur. </s> - <s xml:id="s679" xml:space="preserve">qua <lb/>re qñ B C, linea ad ipÅ¿um peruenerit K, & </s> - <s xml:id="s680" xml:space="preserve">ipÅ¿a <lb/>F C circunferentia erit in ipÅ¿a C L: </s> - <s xml:id="s681" xml:space="preserve">& </s> - <s xml:id="s682" xml:space="preserve">vniuerÅ¿us <lb/>crit conuolutus circulus. </s> - <s xml:id="s683" xml:space="preserve">SimiliqÌ; </s> - <s xml:id="s684" xml:space="preserve">modo Å¿i ma-<lb/>gnum mouero, illi paruuÅ« annectens, eodem exi <lb/>ftente cẽtro, Å¿imul cum A C ipÅ¿i A B perpendicu <lb/>lum & </s> - <s xml:id="s685" xml:space="preserve">recta eri@: </s> - <s xml:id="s686" xml:space="preserve">hæc quidem ad ipÅ¿am F I, illa <lb/>verò ad G M. </s> - <s xml:id="s687" xml:space="preserve">Quam ob rem quando hæc quidẽ <lb/>æqualem ipÅ¿i G M pert<unsure/>rãſiuerit, illa verò ipÅ¿i F I, <lb/>& </s> - <s xml:id="s688" xml:space="preserve">rurÅ¿um facta fuerit recta ipÅ¿a F A ad ipÅ¿am <lb/>F L. </s> - <s xml:id="s689" xml:space="preserve">& </s> - <s xml:id="s690" xml:space="preserve">ipÅ¿a A C rurÅ¿um recta, velut à principio <lb/>erant in ipÅ¿is M I. </s> - <s xml:id="s691" xml:space="preserve">Hoc autem neque aliqua in-<lb/>tercedente mora maioris ad minorem, vt. </s> - <s xml:id="s692" xml:space="preserve">Å¿. </s> - <s xml:id="s693" xml:space="preserve">per <lb/>aliquod temporis Å¿patium Å¿taret in eodem pun-<lb/>cto: </s> - <s xml:id="s694" xml:space="preserve">neque tranÅ¿iliente minore aliquod punctÅ«, <lb/>maiorem quidem æqualem minori pertranÅ¿ire, <lb/>hunc a@tem maiori, abÅ¿urdum eÅ¿t. </s> - <s xml:id="s695" xml:space="preserve">Præterea vni-<lb/>ea etiam Å¿emper exiÅ¿tente motione, centrÅ« mo-<lb/>rum interdum quidem magnam, nonnunquà m <lb/>verò minorem conuerti, admirandum eÅ¿t. </s> - <s xml:id="s696" xml:space="preserve">Idem <lb/>enim celeritate eadem latus æqualẽ natum hoc <lb/>eft pertranÅ¿ire: </s> - <s xml:id="s697" xml:space="preserve">eadem autem celeritate vtroque -<pb o="527" file="0527" n="24" rhead="Mechanicæ."/> -modo æqualem licet mouere. </s> - <s xml:id="s698" xml:space="preserve">Principium autem <lb/>Å¿umendum eÅ¿t circa iÅ¿torum cauÅ¿am, ê™ eadem <lb/>potentia & </s> - <s xml:id="s699" xml:space="preserve">æqualis hanc quidem tardiùs mouet <lb/>magnitudinem, illam verò celeriùs. </s> - <s xml:id="s700" xml:space="preserve">Si enim fue-<lb/>rit quippiam, quod à ſeipÅ¿o moueri natÅ© non Å¿it, <lb/>Å¿i Å¿imul & </s> - <s xml:id="s701" xml:space="preserve">illud mouerit, quod natÅ© eÅ¿t moueri. <lb/></s> - <s xml:id="s702" xml:space="preserve">tardius mouebitur, quà m Å¿i ipÅ¿um per Å¿e moue-<lb/>retur. </s> - <s xml:id="s703" xml:space="preserve">Et Å¿i quidem natum fuerit moueri, non Å¿i-<lb/>mul aÅ©t moueatur, Å¿imiliter Å¿e habebit. </s> - <s xml:id="s704" xml:space="preserve">Et im<unsure/>poÅ¿ <lb/>Å¿ibile certè eÅ¿t plus moueri, quà m mouens: </s> - <s xml:id="s705" xml:space="preserve">non <lb/>enim Å¿uam ipÅ¿ius mouetur motionem. </s> - <s xml:id="s706" xml:space="preserve">Si igitur<unsure/> <lb/>circulus maior vbi A, minor autem vbi B, Å¿i mi-<lb/>nor maiorem impellet non reuolutum ex Å¿e, ma<unsure/> <lb/>nifeÅ¿tum eÅ¿t, ê™ tantum ipÅ¿ius rectæ maior pertrã <lb/>Å¿it, quantum eÅ¿t impulÅ¿us: </s> - <s xml:id="s707" xml:space="preserve">tantum aÅ©t eÅ¿t impul-<lb/>Å¿us, quantum paruus eÅ¿t motus: </s> - <s xml:id="s708" xml:space="preserve">æqualem igitur <lb/>ipÅ¿ius rectæ pertranÅ¿iuerunt. </s> - <s xml:id="s709" xml:space="preserve">Neceſſe igitur eÅ¿t <lb/>Å¿i reuolutus minor maiorem impellet, reuol@i Å¿i<unsure/> <lb/>mul cum impulÅ¿ione: </s> - <s xml:id="s710" xml:space="preserve">tantum aÅ«t quantum mi-<lb/>nor reuolutus eÅ¿t, Å¿i nihil ipÅ¿e Å¿ub ipÅ¿ius motio-<lb/>ne mouetur. </s> - <s xml:id="s711" xml:space="preserve">QÅ«o enim, & </s> - <s xml:id="s712" xml:space="preserve">quantum mouit, tan-<lb/>tum motÅ© eſſe neceſſe eÅ¿t, quod mouetur ab ill@ <lb/>Å¿ed profectò paruus circulus tãtum Å¿eipÅ¿um cir-<lb/>culariter mouit, quantum eÅ¿t pedalis quantitas: </s> - <s xml:id="s713" xml:space="preserve"><lb/>(tantum enim Å¿it id quod motus eÅ¿t) & </s> - <s xml:id="s714" xml:space="preserve">magnus <lb/>igitur tantum motus erit. </s> - <s xml:id="s715" xml:space="preserve">SimiliqÌ; </s> - <s xml:id="s716" xml:space="preserve">modo Å¿i ma-<lb/>gnus paruÅ« mouebit, motus erit paruus, quÄ“ad-<lb/>modum maior. </s> - <s xml:id="s717" xml:space="preserve">Per Å¿e aÅ«t motus illorum vtrum-<lb/>libet Å¿iue celeriter, Å¿eu tardè eadem velocitate, <lb/>Å¿tatim quantum maior natus eÅ¿t circunÅ¿erri li-<lb/>neam: </s> - <s xml:id="s718" xml:space="preserve">quod difficultatÄ“ facit, quod non Å¿imiliter <lb/>Å¿aciunt, quando fuerint connexi. </s> - <s xml:id="s719" xml:space="preserve">hoc autem eÅ¿t, <lb/>Å¿i alteri ab altero moueatur, non quã natus eÅ¿t, <lb/>@eque peculiarem motionem. </s> - <s xml:id="s720" xml:space="preserve">Nihil enim referr <lb/>citcÅ©ponere, & </s> - <s xml:id="s721" xml:space="preserve">annectere, aut cÅiungere vtrum- -<pb o="528" file="0528" n="25" rhead="AriÅ¿t QueÅ¿tiones"/> -liber alteri. </s> - <s xml:id="s722" xml:space="preserve">Å¿imiliter enim quando hic quidem <lb/>mouer, ille vero mouetur ab iÅ¿to; </s> - <s xml:id="s723" xml:space="preserve">quantum vtiq; <lb/></s> - <s xml:id="s724" xml:space="preserve">mouerit alter, tantum alter mouebitur. </s> - <s xml:id="s725" xml:space="preserve">Quando <lb/>quidem igitur adiacens mouerit, aut propenÅ¿us, <lb/>non Å¿emper conueluitur: </s> - <s xml:id="s726" xml:space="preserve">quÄdo verò circa idem <lb/>poÅ¿i@i fuerint cẽtrum, alterum ab altero Å¿emper <lb/>conuolui neceſſe eÅ¿t. </s> - <s xml:id="s727" xml:space="preserve">Sed nihilominus non Å¿uam <lb/>ipÅ¿ius motionem mouetur alter, Å¿ed veluti nullã <lb/>haberet motionem: </s> - <s xml:id="s728" xml:space="preserve">& </s> - <s xml:id="s729" xml:space="preserve">Å¿i habuerit, illa autem nõ <lb/>vtatur, tan@undem accidit. </s> - <s xml:id="s730" xml:space="preserve">Quandoquidem igi-<lb/>tur mà gnus moueri Å¿ibi alligatum paruum, par-<lb/>uus mouetur quãtum ille: </s> - <s xml:id="s731" xml:space="preserve">quÄdo autem paruus, <lb/>rurÅ¿us magnus quantum iÅ¿te: </s> - <s xml:id="s732" xml:space="preserve">Å¿eparatus aÅ©t vter-<lb/>que Å¿eipÅ¿um mouet. </s> - <s xml:id="s733" xml:space="preserve">Quod autem eodem exiÅ¿ten <lb/>te centro, & </s> - <s xml:id="s734" xml:space="preserve">mouente eadem velocitate, accidit <lb/>inæqualem illos pertrÄÅ¿ire lineam, paralogiÅ¿mo <lb/>fophiÅ¿ticÄ™ vtitur is, qui dubitat. </s> - <s xml:id="s735" xml:space="preserve">idem enim am-<lb/>bobus eÅ¿t centrÅ©, verùm per accidens, velúti mu <lb/>Å¿icum & </s> - <s xml:id="s736" xml:space="preserve">album. </s> - <s xml:id="s737" xml:space="preserve">eſſe enim vtriuÅ¿que circuli cen-<lb/>tro nÅ eodem vtitur. </s> - <s xml:id="s738" xml:space="preserve">Quandoquidem igitur mo-<lb/>uens fuerit paruus, vt illius centrum & </s> - <s xml:id="s739" xml:space="preserve">princi-<lb/>pium: </s> - <s xml:id="s740" xml:space="preserve">quando verò magnus, vt illius. </s> - <s xml:id="s741" xml:space="preserve">non igitur <lb/>idem Å¿im pliciter mouet, Å¿ed eÅ¿t quomodo.</s> - <s xml:id="s742" xml:space="preserve"/> - </p> - <p> - <s xml:id="s743" xml:space="preserve">Cur lect@lorum Å¿pondas Å¿ecÅ«dum duplam fa-<lb/> -<anchor type="note" xlink:label="note-0528-01a" xlink:href="note-0528-01"/> -ciunt proportionem, hanc quidem Å¿ex pedÅ« vel <lb/>paulo am pliorem, illam vero trium: </s> - <s xml:id="s744" xml:space="preserve">curve non <lb/>Å¿ecundùm diametrum illos reÅ¿tibus extendunt? <lb/></s> - <s xml:id="s745" xml:space="preserve">An tantos quidem magnitudine faciunt, vt cor-<lb/>poribus Å¿int proportionem habẽtes: </s> - <s xml:id="s746" xml:space="preserve">fiunt enim <lb/>Å¿ic Å¿ecundum Å¿pondas dupli, longitudine quidẽ <lb/>cubitorum quatuor, latitudine vero duorÅ©. </s> - <s xml:id="s747" xml:space="preserve">Ex-<lb/>tendunt autem illos nÅ Å¿ecundùm diametrum, <lb/>Å¿ed ex oppoÅ¿ito, vt & </s> - <s xml:id="s748" xml:space="preserve">ligna minus diÅ¿tra hantur. </s> - <s xml:id="s749" xml:space="preserve"><lb/>celertrimè enim Å¿cindÅ©tur Å¿ecundum naturam <lb/>diuiÅ¿a, & </s> - <s xml:id="s750" xml:space="preserve">eodem modo diÅ¿tenta laborant ma- -<pb o="529" file="0529" n="26" rhead="Mechanicæ."/> -@imè. </s> - <s xml:id="s751" xml:space="preserve">Amplius, quoniam opus eÅ¿t, vt reÅ¿tes pon-<lb/>dus ferre poſſint, Å¿ic certè pondere impoÅ¿ito mi-<lb/>nus laborabunt, Å¿i tranÅ¿uerÅ¿im, quà m Å¿i obliquè <lb/>extendantur. </s> - <s xml:id="s752" xml:space="preserve">Præterea hoc etiam modo minus <lb/>abÅ¿umitur reÅ¿tium. </s> - <s xml:id="s753" xml:space="preserve">Sit enim lectulus A F G K, & </s> - <s xml:id="s754" xml:space="preserve"><lb/>bifariam diuidatur ipÅ¿a F G Å¿ecundùm B, æqua-<lb/>lia certè foramina Å¿unt in ipÅ¿a F B, & </s> - <s xml:id="s755" xml:space="preserve">in ipÅ¿a F A. <lb/></s> - <s xml:id="s756" xml:space="preserve">latera enim Å¿unt æqualia. </s> - <s xml:id="s757" xml:space="preserve">nam totum F G duplÅ© <lb/>eÅ¿t. </s> - <s xml:id="s758" xml:space="preserve">Extendunt autem, vt deÅ¿criptum eÅ¿t, ab ipÅ¿o <lb/>A ad ipÅ¿um B ita vbi eÅ¿t C, ita vbi D: </s> - <s xml:id="s759" xml:space="preserve">ita vbi H, <lb/>poÅ¿tea vbi E, & </s> - <s xml:id="s760" xml:space="preserve">eodem Å¿emper modo, donec ad <lb/>angulum peruenerint alium. </s> - <s xml:id="s761" xml:space="preserve">Duo enim anguli <lb/>reÅ¿tis habent capi a. </s> - <s xml:id="s762" xml:space="preserve">æquales autem Å¿unt reÅ¿tes <lb/>Å¿ecundum curuaturas, videlicet A B; </s> - <s xml:id="s763" xml:space="preserve">& </s> - <s xml:id="s764" xml:space="preserve">B C ipÅ¿is <lb/>C D, & </s> - <s xml:id="s765" xml:space="preserve">D H. </s> - <s xml:id="s766" xml:space="preserve">& </s> - <s xml:id="s767" xml:space="preserve">aliæ Å¿imili Å¿e habet modo, quo-<lb/>niam eadem demonÅ¿tratio. </s> - <s xml:id="s768" xml:space="preserve">ipÅ¿a enim AB æqua-<lb/>lis eÅ¿t ipÅ¿i H E: </s> - <s xml:id="s769" xml:space="preserve">æqualia enim Å¿unt latera Å¿patij <lb/>B G, M A, & </s> - <s xml:id="s770" xml:space="preserve">foramina æquè diÅ¿tant, ipÅ¿a autem <lb/>B G æquelis eÅ¿t ipÅ¿i M A: </s> - <s xml:id="s771" xml:space="preserve">angulus enim B æqua-<lb/>lis eÅ¿t angulo G: </s> - <s xml:id="s772" xml:space="preserve">in æqualibus enim hic quidem <lb/>intus, ille vero extra. </s> - <s xml:id="s773" xml:space="preserve">& </s> - <s xml:id="s774" xml:space="preserve">B quidem eÅ¿t Å¿emirectus: </s> - <s xml:id="s775" xml:space="preserve"><lb/>eÅ¿t enim F B æqualis ipÅ¿i F A: </s> - <s xml:id="s776" xml:space="preserve">& </s> - <s xml:id="s777" xml:space="preserve">angulus vbi F, re <lb/>ctus eÅ¿t. </s> - <s xml:id="s778" xml:space="preserve">B autẽ angulus æqualis ei vbi eÅ¿t G, quo-<lb/>niam quadratum altera parte longius duplÅ© eÅ¿t: </s> - <s xml:id="s779" xml:space="preserve"><lb/>& </s> - <s xml:id="s780" xml:space="preserve">ad medium eÅ¿t curuatura. </s> - <s xml:id="s781" xml:space="preserve">quamobrẽ A D ip-<lb/>Å¿i E G eÅ¿t æqualis: </s> - <s xml:id="s782" xml:space="preserve">huic verò ipÅ¿a H M. </s> - <s xml:id="s783" xml:space="preserve">SimiliqÌue <lb/>modo demonÅ¿trantur aliæ, quoniã æquales Å¿unt <lb/>duæ, quæ Å¿ecÅ©dùm curuataras Å¿unt, duabus. </s> - <s xml:id="s784" xml:space="preserve">Qua <lb/>re manifeÅ¿tum eÅ¿t, ê™ tot Å¿unt reÅ¿tes in lectulo, <lb/>quot Å¿unt quatuor, Å¿icut A B. </s> - <s xml:id="s785" xml:space="preserve">Quanta autem fora <lb/>minÅ© eÅ¿t multirudo in ipÅ¿o F G latere, & </s> - <s xml:id="s786" xml:space="preserve">in eius <lb/>dimidio F B eÅ¿t medietas. </s> - <s xml:id="s787" xml:space="preserve">Quamobrem in dimi-<lb/>diato lectulo tantæ reÅ¿tium magnitudines erÅ©t, <lb/>quantum eÅ¿t A B: </s> - <s xml:id="s788" xml:space="preserve">multitudine verò tot, quot in <lb/>B G Å¿unt foramina. </s> - <s xml:id="s789" xml:space="preserve">Hoc autem nihil refert dice -<pb o="530" file="0530" n="27" rhead="AriÅ¿t. Quæſtiones"/> -re, quà m quot Å¿unt in ipÅ¿is A F & </s> - <s xml:id="s790" xml:space="preserve">B F Å¿imul <lb/>Å¿umpris. </s> - <s xml:id="s791" xml:space="preserve">Si autem Å¿ecundum diametrum exten-<lb/>dantur reÅ¿tes, quemad modum Å¿e habet in lectu-<lb/>lo A B C D, dimidia non tot Å¿unt, quot ambo-<lb/>rum latera F A F G. </s> - <s xml:id="s792" xml:space="preserve">æqualia autem, quot in ip <lb/>fis F B, F A, Å¿unt foramina: </s> - <s xml:id="s793" xml:space="preserve">maiores autem Å¿unt</s> - </p> - <div type="float" level="2" n="20"> - <note position="left" xlink:label="note-0528-01" xlink:href="note-0528-01a" - xml:space="preserve">25</note> - </div> - <p> - <s xml:id="s794" xml:space="preserve">ipſæ A F, B F, quæ exiÅ¿tentes, quà m A B, <lb/>quare reÅ¿tis in tantum maior, quantum ambola <lb/>@era diame@ro Å¿unt maiora.</s> - <s xml:id="s795" xml:space="preserve"/> - </p> - <p> - <s xml:id="s796" xml:space="preserve">Cur difficilius eÅ¿t longa ligna ab extremo Å¿u-<lb/> -<anchor type="note" xlink:label="note-0530-01a" xlink:href="note-0530-01"/> -per humeros ferre, quà m Å¿ecundum medium, æ-<lb/>quali exiÅ¿tente pondere? </s> - <s xml:id="s797" xml:space="preserve">An quia vibrato ligno <lb/>ipÅ¿um extremum prohibet ferre vibratione ma-<lb/>gis retrahens lationem? </s> - <s xml:id="s798" xml:space="preserve">An quoniam licèt nihil <lb/>inflectatur, neque multam habeat longitudinẽ, <lb/>difficilius tamen ad ferendum eÅ¿t ab extremo, <lb/>quoniam faciliùs ex medio eleuatur, quà m ab <lb/>extremo, & </s> - <s xml:id="s799" xml:space="preserve">ideo Å¿ic ferre eÅ¿t facilius: </s> - <s xml:id="s800" xml:space="preserve">CauÅ¿a aÅ©t, <lb/>qmÌ„ Å¿ecundùm medium quidem eleuato ligno, <lb/>Å¿emper Å¿eÅ¿e inuicem Å¿uÅ¿pendunt extrema, & </s> - <s xml:id="s801" xml:space="preserve">al-<lb/>tera pars alteram bene Å¿ubleuat medium enim <lb/>veluti centrum fit, vbi habet is, qui eleuat, aut <lb/>Å¿ert. </s> - <s xml:id="s802" xml:space="preserve">ExtremorÅ© igitur v@@rÅ©q; </s> - <s xml:id="s803" xml:space="preserve">deorÅ¿um vergens, <lb/>Å¿urÅ¿um Å¿uÅ¿penditur. </s> - <s xml:id="s804" xml:space="preserve">ê™ Å¿i ab extremo eleuetur, <lb/>aut feratur, non Å¿anè facit: </s> - <s xml:id="s805" xml:space="preserve">Å¿ed vniuerÅ¿um pon-<lb/>dus ad vnum vergit medium, quò eleuatur, aut <lb/>fertur. </s> - <s xml:id="s806" xml:space="preserve">Sit medium vbi A, extrema B C. </s> - <s xml:id="s807" xml:space="preserve">Eleuatio <lb/>igitur aut portato Å¿ecundùm A, ipÅ¿um qÌuidem B <lb/>deorÅ¿um @utans, Å¿urÅ¿um eleuat C: </s> - <s xml:id="s808" xml:space="preserve">ipÅ¿um autem <lb/>C deorÅ¿um nutans, B Å¿urÅ¿um eleuat: </s> - <s xml:id="s809" xml:space="preserve">ambo au-<lb/>tem Å¿urÅ¿um eleuata hoc fac@unt.</s> - <s xml:id="s810" xml:space="preserve"/> - </p> - <div type="float" level="2" n="21"> - <note position="left" xlink:label="note-0530-01" xlink:href="note-0530-01a" - xml:space="preserve">26</note> - </div> - <p> - <s xml:id="s811" xml:space="preserve">Cur Å¿i valde procerum fuerit idẽ pondus diffi-<lb/> -<anchor type="note" xlink:label="note-0530-02a" xlink:href="note-0530-02"/> -cilius Å¿uper humeros geÅ¿tatur, etiam Å¿i medium <lb/>quiÅ¿piam illud ferat, quà m Å¿i breuius Å¿it? </s> - <s xml:id="s812" xml:space="preserve">Quod <lb/>@nim dudum dictum eÅ¿t, cauÅ¿a nõ eÅ¿t: </s> - <s xml:id="s813" xml:space="preserve">Å¿ed vibra- -<pb o="531" file="0531" n="28" rhead="MechanicÄ™."/> -tio nunc eÅ¿t cauÅ¿a. </s> - <s xml:id="s814" xml:space="preserve">qñ. </s> - <s xml:id="s815" xml:space="preserve">n. </s> - <s xml:id="s816" xml:space="preserve">productius fuerit, vibran <lb/>tur extrema: </s> - <s xml:id="s817" xml:space="preserve">quamobrem contingit portantem <lb/>difficilius geÅ¿tare. </s> - <s xml:id="s818" xml:space="preserve">Vibrationis aÅ©t cã eÅ¿t, quoniã <lb/>ab eadẽ motione magis transferuntur extrema, <lb/>quanto procerius fuerit lignÅ©. </s> - <s xml:id="s819" xml:space="preserve">Humerus quidẽ <lb/>fit centrÅ© vbi A: </s> - <s xml:id="s820" xml:space="preserve">manet. </s> - <s xml:id="s821" xml:space="preserve">n. </s> - <s xml:id="s822" xml:space="preserve">is. </s> - <s xml:id="s823" xml:space="preserve">ipſæ aÅ©t AB, AC, quÄ™ <lb/>Å¿unt ex centro. </s> - <s xml:id="s824" xml:space="preserve">quanto aÅ©t maius fuerit id quod <lb/>ex centro eÅ¿t, Å¿iue A B, Å¿eu A C, plus transfertur <lb/>Å¿patij. </s> - <s xml:id="s825" xml:space="preserve">DemonÅ¿tratum autem eÅ¿t hoc prius.</s> - <s xml:id="s826" xml:space="preserve"/> - </p> - <div type="float" level="2" n="22"> - <note position="left" xlink:label="note-0530-02" xlink:href="note-0530-02a" - xml:space="preserve">27</note> - </div> - <p> - <s xml:id="s827" xml:space="preserve">Cur iuxta puteos celonia faciunt eo qui viÅ¿un <lb/> -<anchor type="note" xlink:label="note-0531-01a" xlink:href="note-0531-01"/> -tur modo: </s> - <s xml:id="s828" xml:space="preserve">ligno. </s> - <s xml:id="s829" xml:space="preserve">n. </s> - <s xml:id="s830" xml:space="preserve">plumbi adiÅ©gunt pondus: </s> - <s xml:id="s831" xml:space="preserve">cÅ© <lb/>alioquin vas ipÅ¿um & </s> - <s xml:id="s832" xml:space="preserve">plenum, & </s> - <s xml:id="s833" xml:space="preserve">vacuum pon-<lb/>dus habeat? </s> - <s xml:id="s834" xml:space="preserve">An qmÌ„ duobus tem poribus hau@ien <lb/>di, diuiÅ¿o opere, (intingere. </s> - <s xml:id="s835" xml:space="preserve">n. </s> - <s xml:id="s836" xml:space="preserve">oportet, & </s> - <s xml:id="s837" xml:space="preserve">id Å¿ur-<lb/>Å¿um trahere) continget demittere quidẽ vacuum <lb/>faciliter, trahere vero plenum difficulter. </s> - <s xml:id="s838" xml:space="preserve">CÅmo-<lb/>dum igitur eÅ¿t paulò tardius illud demittere, cÅ« <lb/>multò leuius effectum Å¿uÅ¿tolatur pondus. </s> - <s xml:id="s839" xml:space="preserve">id autẽ <lb/>facit in extremo celonio adiunctum plumbum, <lb/>aut lapis. </s> - <s xml:id="s840" xml:space="preserve">demittenti quidem maius Å¿it pondus, ê— <lb/>fi Å¿olummodo vacuum oportet demittere: </s> - <s xml:id="s841" xml:space="preserve">cum <lb/>verò plenum fuerit, Å¿urÅ¿um id rapit plumbum, <lb/>aut quicquid ponderis inerit. </s> - <s xml:id="s842" xml:space="preserve">Quam ob rem faci <lb/>liora hoc modo ambo Å¿unt, quà m illo.</s> - <s xml:id="s843" xml:space="preserve"/> - </p> - <div type="float" level="2" n="23"> - <note position="right" xlink:label="note-0531-01" xlink:href="note-0531-01a" - xml:space="preserve">28</note> - </div> - <p> - <s xml:id="s844" xml:space="preserve">Cur qñ Å¿uper ligno, aut hmõi quopiam duo <lb/> -<anchor type="note" xlink:label="note-0531-02a" xlink:href="note-0531-02"/> -portauerint homines æquale pondus, non Å¿imili <lb/>ter premuntur, Å¿i ad vnum non declinet pondus, <lb/>Å¿eu magis, quanto vicinius fuerit geÅ¿tãtibus? </s> - <s xml:id="s845" xml:space="preserve">An <lb/>quoniam vectis quidem lignum efficitur, põdus <lb/>verò hypomochlion: </s> - <s xml:id="s846" xml:space="preserve">qui autem propior eÅ¿t pon-<lb/>deri ex ijs, qui illud geÅ¿tant, id quod mouetur: </s> - <s xml:id="s847" xml:space="preserve">al <lb/>ter verò portantiÅ©, quod mouet. </s> - <s xml:id="s848" xml:space="preserve">Quanto igitur <lb/>plus diÅ¿tat à pondere, tanto facilius mouet, & </s> - <s xml:id="s849" xml:space="preserve"><lb/>alerum premit magis inferius, veluti contrani-<lb/>@te pondere impoÅ¿ito, quod hypomochlion -<pb o="532" file="0532" n="29" rhead="AriÅ¿t. Quæſtiones."/> -factum EÅ¿t. </s> - <s xml:id="s850" xml:space="preserve">Si autem in medio inerit pondus, ni-<lb/>hilo magis alter alteri fit pondus, aut mouet: </s> - <s xml:id="s851" xml:space="preserve">Å¿ed <lb/>eodem modo alteri alt<unsure/>er fit pondus.</s> - <s xml:id="s852" xml:space="preserve"/> - </p> - <div type="float" level="2" n="24"> - <note position="right" xlink:label="note-0531-02" xlink:href="note-0531-02a" - xml:space="preserve">29</note> - </div> - <p> - <s xml:id="s853" xml:space="preserve">Cur Å¿urgentes omnes, femori crus ad acutum <lb/> -<anchor type="note" xlink:label="note-0532-01a" xlink:href="note-0532-01"/> -cõſtituentes angulÅ©, & </s> - <s xml:id="s854" xml:space="preserve">thoraci Å¿imiliter femur, <lb/>Å¿urgunt? </s> - <s xml:id="s855" xml:space="preserve">quod Å¿i non, haudquaquam Å¿urgere po-<lb/>terunt. </s> - <s xml:id="s856" xml:space="preserve">An quia id quod æquale eÅ¿t, quietis vbiq; <lb/></s> - <s xml:id="s857" xml:space="preserve">eÅ¿t cauÅ¿a: </s> - <s xml:id="s858" xml:space="preserve">rectus autẽ angulus æqualitatis eÅ¿t, Å¿ta-<lb/>tionemqÌ; </s> - <s xml:id="s859" xml:space="preserve">facit: </s> - <s xml:id="s860" xml:space="preserve">quamobrẽ ad Å¿imiles fertur an-<lb/>gulos ipÅ¿i terræ circumferentiæ: </s> - <s xml:id="s861" xml:space="preserve">non enim quod <lb/>ad rectum eÅ¿t ipÅ¿i pauimento. </s> - <s xml:id="s862" xml:space="preserve">An q̇mÌ„ Å¿urgens fit <lb/>rectus, Å¿tantẽ vero neceſſe eÅ¿t perpẽdiculum eſſe <lb/>ad terram. </s> - <s xml:id="s863" xml:space="preserve">Å¿iquidẽ igitur ad rectÅ© debet eſſe, hoc <lb/>aÅ©t eÅ¿t caput Å¿ecundÅ© pedes habere, & </s> - <s xml:id="s864" xml:space="preserve">fieri opor <lb/>tet, cùm Å¿urgit. </s> - <s xml:id="s865" xml:space="preserve">Quandoquidem igitur fuerit @e <lb/>dens, Å¿ecundum parallelum pedes habet, & </s> - <s xml:id="s866" xml:space="preserve">ca-<lb/>put, & </s> - <s xml:id="s867" xml:space="preserve">non in æquali. </s> - <s xml:id="s868" xml:space="preserve">Caput Å¿it A, thorax A B, fe-<lb/>mur B C, crura C D. </s> - <s xml:id="s869" xml:space="preserve">ad rectum autem Å¿it & </s> - <s xml:id="s870" xml:space="preserve">tho-<lb/>rãx vbi A B, ipÅ¿i femori, & </s> - <s xml:id="s871" xml:space="preserve">cruri Å¿emur Å¿ic feden-<lb/>te. </s> - <s xml:id="s872" xml:space="preserve">quamobrem eo Å¿e habẽtem modo Å¿urgere eÅ¿t <lb/>impoſſibile. </s> - <s xml:id="s873" xml:space="preserve">neceſſe autem eÅ¿t crus reclinare, pe-<lb/>desque conÅ¿tituere Å¿ub capite. </s> - <s xml:id="s874" xml:space="preserve">hoc autem erit, Å¿i <lb/>C D fiet, vbi C F, & </s> - <s xml:id="s875" xml:space="preserve">fimul Å¿urgere continget, & </s> - <s xml:id="s876" xml:space="preserve">in <lb/>eadem Ä™quali habere caput & </s> - <s xml:id="s877" xml:space="preserve">pedes. </s> - <s xml:id="s878" xml:space="preserve">ipÅ¿a autem <lb/>C F a cutum facit angulum ad ipÅ¿am B C.</s> - <s xml:id="s879" xml:space="preserve"/> - </p> - <div type="float" level="2" n="25"> - <note position="left" xlink:label="note-0532-01" xlink:href="note-0532-01a" - xml:space="preserve">30</note> - </div> - <p> - <s xml:id="s880" xml:space="preserve">Cur facilius mouetur cõmotu, quà m manens? <lb/></s> - <s xml:id="s881" xml:space="preserve"> -<anchor type="note" xlink:label="note-0532-02a" xlink:href="note-0532-02"/> -veluti currus citiùs cõmotus agitant, quà m mo-<lb/>ueri in cipientes. </s> - <s xml:id="s882" xml:space="preserve">An quia difficillimÅ© eÅ¿t pondus <lb/>mouere, quod in cõtrariÅ© mouetur. </s> - <s xml:id="s883" xml:space="preserve">aufert enim <lb/>quiddam ex motoris potentia, licèt multo fit ve-<lb/>locior: </s> - <s xml:id="s884" xml:space="preserve">neceſſe nanque eÅ¿t tardiorẽ eſſe impulÅ¿io <lb/>nem illius quod repellitur. </s> - <s xml:id="s885" xml:space="preserve">Secundo autẽloco Å¿i <lb/>quieuerit: </s> - <s xml:id="s886" xml:space="preserve">reÅ¿iÅ¿tit enim ipÅ¿um quieÅ¿cẽs. </s> - <s xml:id="s887" xml:space="preserve">Quod aÅ©t <lb/>mouetur ad id ipÅ¿um ad quod impellitur, impel-<lb/>lenti Å¿imile facit, ceu Å¿i quiÅ¿piam mouẽtis poten -<pb o="533" file="0533" n="30" rhead="De Mundo."/> -tiam, & </s> - <s xml:id="s888" xml:space="preserve">celeritatem augeret. </s> - <s xml:id="s889" xml:space="preserve">quod enim<unsure/> ab illo <lb/>pateretur, vtique ipÅ¿um tacit ex Å¿e commotum.</s> - <s xml:id="s890" xml:space="preserve"/> - </p> - <div type="float" level="2" n="26"> - <note position="left" xlink:label="note-0532-02" xlink:href="note-0532-02a" - xml:space="preserve">31</note> - </div> - <p> - <s xml:id="s891" xml:space="preserve">Cur ea, quæ proijci@@ntur, ceſſant à latione? </s> - <s xml:id="s892" xml:space="preserve">An <lb/> -<anchor type="note" xlink:label="note-0533-01a" xlink:href="note-0533-01"/> -quia impellens deÅ¿init potentia, v@l propter retra <lb/>ctionem, vel propter rei proiectÄ™ inclinationem, <lb/>quando ea valentior fuerit, quam proijcientis vi <lb/>res. </s> - <s xml:id="s893" xml:space="preserve">Aut iÅ¿thæc ambigere, principium relinquen-<lb/>tes, abÅ¿urdum eÅ¿t.</s> - <s xml:id="s894" xml:space="preserve"/> - </p> - <div type="float" level="2" n="27"> - <note position="right" xlink:label="note-0533-01" xlink:href="note-0533-01a" - xml:space="preserve">32</note> - </div> - <p> - <s xml:id="s895" xml:space="preserve">Cur quippiam non pecularem Å¿ibi fertur latio <lb/> -<anchor type="note" xlink:label="note-0533-02a" xlink:href="note-0533-02"/> -nem, impulfore alioquin non conÅ¿equente? </s> - <s xml:id="s896" xml:space="preserve">An <lb/>videlicet quoniam primum id efficit, @@al@erum <lb/>impellat: </s> - <s xml:id="s897" xml:space="preserve">illudque rurÅ¿um, v@ alterum? </s> - <s xml:id="s898" xml:space="preserve">Ceſſat au-<lb/>tem quando non poteÅ¿t amplius facere primum <lb/>impellens id quod fertur, vt impellat, & </s> - <s xml:id="s899" xml:space="preserve">qñ ip-<lb/>Å¿ius lati grauitas nut@ Å¿uo declinat magis, quà m <lb/>impellentis in antè Å¿it potentia,</s> - </p> - <div type="float" level="2" n="28"> - <note position="right" xlink:label="note-0533-02" xlink:href="note-0533-02a" - xml:space="preserve">33</note> - </div> - <p> - <s xml:id="s900" xml:space="preserve">Cur neque parua valde, neque magna longè <lb/> -<anchor type="note" xlink:label="note-0533-03a" xlink:href="note-0533-03"/> -proijci queunt, Å¿ed commẽſurationem quandã <lb/>illa habere oportet ad id quod proijcit? </s> - <s xml:id="s901" xml:space="preserve">An quia <lb/>neceſſe eÅ¿t quod proijcitur, & </s> - <s xml:id="s902" xml:space="preserve">impelli@@r, cõtra-<lb/>ni i ei vnde impellitur? </s> - <s xml:id="s903" xml:space="preserve">quod autem magnitudi-<lb/>ne Å¿ua nihil cedit, aut imbellicitate nihil contra-<lb/>nititur non efficit proiectionem, neque impulÅ¿io <lb/>nem. </s> - <s xml:id="s904" xml:space="preserve">Quod enim multo impellentis excedit vi-<lb/>res, haud quaquã cedit: </s> - <s xml:id="s905" xml:space="preserve">quod verò multo eÅ¿t im-<lb/>berillius. </s> - <s xml:id="s906" xml:space="preserve">nihil contranititur. </s> - <s xml:id="s907" xml:space="preserve">An quia tãtum fer-<lb/>tur id quod fertur, quantÅ© aeris mouerit ad pro-<lb/>fundum: </s> - <s xml:id="s908" xml:space="preserve">quod autem non mouetur, neq; </s> - <s xml:id="s909" xml:space="preserve">moue <lb/>bit quippiam accidit autem illis ambo iÅ¿thæc ha <lb/>bere. </s> - <s xml:id="s910" xml:space="preserve">Valde enim magnum, & </s> - <s xml:id="s911" xml:space="preserve">valde paruum, ceù <lb/>non mota exiÅ¿tunt: </s> - <s xml:id="s912" xml:space="preserve">alterum nanque nihil mouet <lb/>alterum verò nihil mouetur.</s> - <s xml:id="s913" xml:space="preserve"/> - </p> - <div type="float" level="2" n="29"> - <note position="right" xlink:label="note-0533-03" xlink:href="note-0533-03a" - xml:space="preserve">34</note> - </div> - <p> - <s xml:id="s914" xml:space="preserve">Cur ea quæ in vorticoÅ¿is feruntur aquis, ad <lb/> -<anchor type="note" xlink:label="note-0533-04a" xlink:href="note-0533-04"/> -medium tandem aguntur omnia? </s> - <s xml:id="s915" xml:space="preserve">An quia ma-<lb/>gnitudinem habet quocunque ferrur: </s> - <s xml:id="s916" xml:space="preserve">quam -<pb o="534" file="0534" n="31" rhead="AriÅ¿t. Quæſtiones Mech."/> -ob rem illius extrema in duobus Å¿unt circulis, <lb/>hoc quidem minori illo verò maiori: </s> - <s xml:id="s917" xml:space="preserve">quare ma-<lb/>ior diÅ¿trahit: </s> - <s xml:id="s918" xml:space="preserve">quoniã celerius fertur, & </s> - <s xml:id="s919" xml:space="preserve">tranÅ¿uer-<lb/>Å¿um impellit illud ad minorem: </s> - <s xml:id="s920" xml:space="preserve">quoniam autem <lb/>id quod Å¿ertur, latitudinem habet, & </s> - <s xml:id="s921" xml:space="preserve">iÅ¿te rurÅ¿us <lb/>idem eff@cit & </s> - <s xml:id="s922" xml:space="preserve">ad interiorem propellit, donec ad <lb/>medium perueniat. </s> - <s xml:id="s923" xml:space="preserve">An quia quod fertur, Å¿imili <lb/>Å¿e habet modo ad omnes circulos propter me-<lb/>dium. </s> - <s xml:id="s924" xml:space="preserve">medium enim in vnoquoque circulo Ä™qua <lb/>liter diÅ¿tat. </s> - <s xml:id="s925" xml:space="preserve">An quia quorum quidem circÅ©auctæ <lb/>quæ latio non Å¿uperior propter magnitudinem, <lb/>Å¿ed grauitate Å¿ua circuli celeritatẽ excellunt, ea <lb/>neceſſe eÅ¿t relinqui, & </s> - <s xml:id="s926" xml:space="preserve">tardiùs ferri, tardius aÅ©t <lb/>minor circulus fertur. </s> - <s xml:id="s927" xml:space="preserve">non idem enim in tempo-<lb/>re æquali magnus cÅ© paruo reuoluitur circulus, <lb/>quando circa idem Å¿uerint medium. </s> - <s xml:id="s928" xml:space="preserve">quamobrẽ <lb/>in minori circulo relinqui eÅ¿t neceſſe, donec ad <lb/>medium perueniant. </s> - <s xml:id="s929" xml:space="preserve">Quorum cunque autem Å¿u-<lb/>perior à principio fuerit latio, ea finiensidem eÅ¿ <lb/>ficiet. </s> - <s xml:id="s930" xml:space="preserve">oportet enim @ãc quidem Å¿tatim, alterum <lb/>verò celeritate Å¿uperare grauitatem, quam obrẽ <lb/>ad interiorẽ Å¿emper circulum relinquetur quod-<lb/>cunque. </s> - <s xml:id="s931" xml:space="preserve">Neceſſe enim eÅ¿t, quod non Å¿uperatur, <lb/>aut in exteriori, aut in interiori moueri: </s> - <s xml:id="s932" xml:space="preserve">in illo <lb/>autem in quo eÅ¿t, impoſſibile eÅ¿t ferri, quod non <lb/>Å¿uperatur adhuc vero minus in exteriori: </s> - <s xml:id="s933" xml:space="preserve">cele-<lb/>rior enim exterioris circuli eÅ¿t latio: </s> - <s xml:id="s934" xml:space="preserve">reÅ¿tat igitur, <lb/>vtid, quod non Å¿uperatur, ad interiorem tranÅ¿-<lb/>feratur. </s> - <s xml:id="s935" xml:space="preserve">Å¿emper autem vnumquodque proficit, vt <lb/>non Å¿uperetur. </s> - <s xml:id="s936" xml:space="preserve">Quoniam verò peruenire ad me-<lb/>dium, finem quidem efficit, vt quippiam nõ mo-<lb/>ueatur, Å¿tat autem Å¿olummodo ipÅ¿um centrum: <lb/></s> - <s xml:id="s937" xml:space="preserve">ad hoc Å¿anè omnia congregari neceſſe eÅ¿t.</s> - <s xml:id="s938" xml:space="preserve"/> - </p> - <div type="float" level="2" n="30"> - <note position="right" xlink:label="note-0533-04" xlink:href="note-0533-04a" - xml:space="preserve">35</note> - </div> - </div> - <div type="section" level="1" n="5"> - <head xml:id="head8" style="it" xml:space="preserve">Mechanicarum quæſtionum finis.</head> - </div> - </text> -</echo> \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-xml/examples/Aristoteles_1585_XSY685ZD.xml Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1258 +0,0 @@ -<?xml version="1.0" encoding="UTF-8"?><echo xmlns="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" xmlns:dcterms="http://purl.org/dc/terms" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:xhtml="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.0RC"> - <metadata> - <dcterms:identifier>ECHO:XSY685ZD.xml</dcterms:identifier> - <dcterms:creator identifier="GND:118650130">Aristoteles</dcterms:creator> - <dcterms:contributor identifier="GND:102377138">Leonicus Thomaeus, Nicolaus</dcterms:contributor> - <dcterms:title xml:lang="la">Quaestiones Mechanicae</dcterms:title> - <dcterms:date xsi:type="dcterms:W3CDTF">1585</dcterms:date> - <dcterms:language xsi:type="dcterms:ISO639-3">lat</dcterms:language> - <dcterms:rights>open access</dcterms:rights> - <dcterms:license>http://echo.mpiwg-berlin.mpg.de/policy/oa_basics/declaration</dcterms:license> - <dcterms:accessRights>free</dcterms:accessRights> - </metadata> - <text xml:lang="la" type="free"> -<div type="section" level="1" n="1"><pb o="504" file="0504" n="1"/> -</div> -<div type="section" level="1" n="2"> -<head xml:space="preserve">ARISTOTELIS</head> -<head xml:space="preserve">STAGIRITAE, <lb/>PERIPATETICORVM</head> -<head xml:space="preserve">PRINCIPIS,</head> -<head xml:space="preserve">Quæſtiones Mechanicæ, <lb/>Nicolao Leonico Thomæo <lb/>interprete.</head> -<head style="it" xml:space="preserve">Quæ Å¿it artis Mechanicæ facultas.</head> -<p> - <s xml:space="preserve">MIraculo Å¿une ea quidem, quæ natura <lb/>contingunt, quorum ignorantur cau <lb/>ſæ: </s> - <s xml:space="preserve">illa uerò, quæ prÄ™ter naturam, <lb/>quæcunque ad hominum vtilitatem <lb/>arte fiunt. </s> - <s xml:space="preserve">In multis. </s> - <s xml:space="preserve">n. </s> - <s xml:space="preserve">natura ei. </s> - <s xml:space="preserve">qÄ‘ <lb/>nobis uÅ¿ui eſſe poteÅ¿t, contrarium facit: </s> - <s xml:space="preserve">natura <lb/>etenim eundem Å¿emper habet modum, & </s> - <s xml:space="preserve">Å¿impli <lb/>citer: </s> - <s xml:space="preserve">vtile autem multifariam commutat̃. </s> - <s xml:space="preserve">Quan <lb/>doigitur quippiam præter naturam oportuerit <lb/>facere, difficultate Å¿ua hæſitationem praÅ¿tat, ar-<lb/>teqÌ indiget: </s> - <s xml:space="preserve">quam ob rem eam artis partẽ, quæ <lb/>huiuÅ¿modi Å¿uccurrit difficulcatibus, Mechanicã <lb/>appellamus. </s> - <s xml:space="preserve">Quemad modum enim Antipho Å¿cri <lb/>bit poeta, Å¿ic Å¿e res habet: </s> - <s xml:space="preserve">arte enim Å¿uperamus <lb/>da, à quibus natura uincimur. </s> - <s xml:space="preserve">HuiuÅ¿modi autem <lb/>Å¿unt, in quibus & </s> - <s xml:space="preserve">minora Å¿uperant maiora: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve"><lb/>quæcunq; </s> - <s xml:space="preserve">momentum paruum habẽtia, magna -<pb o="505" file="0505" n="2" rhead="Mechanicæ."/> -mouent pondera: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">omnia ferè illa, quæ mecha <lb/>nica nuncupamus, & </s> - <s xml:space="preserve">problemata. </s> - <s xml:space="preserve">Sunt autem, <lb/>hæc neque naturalibus omnino quæſtionibus <lb/>eadem, neque Å¿eiugata valde: </s> - <s xml:space="preserve">verùm mathema-<lb/>ticarÅ© contemplationÅ© naturaliumqÌ; </s> - <s xml:space="preserve">cõmunia.</s> - <s xml:space="preserve"/> -</p> -</div> -<div type="section" level="1" n="3"> -<head style="it" xml:space="preserve">De dignitatiòus, admirandis̈ Circuli <lb/>proprietatibus.</head> -<p> - <s xml:space="preserve">DEnumeto autem eorum, quæ hoc in gene-<lb/>re dubitantur, illa eſſe dicuntur, quæ circæ <lb/>vectem fiunt. </s> - <s xml:space="preserve">AbÅ¿urdum enim eſſe videtur, ma-<lb/>gnum moueri pondus ab exigua virtute, cÅ© plu-<lb/>ri præſertim pondere. </s> - <s xml:space="preserve">quod. </s> - <s xml:space="preserve">n. </s> - <s xml:space="preserve">una vecte quiÅ¿piã <lb/>mouere nõ poteÅ¿t, idipÅ¿um ponderis citius mo-<lb/>uet, uectis ad illud pondus adiungens. </s> - <s xml:space="preserve">OÄ©um ãt <lb/>huiuÅ¿modi cauſæ principiÅ© habet circulus, IÅ¿tud <lb/>verò ratione contingit; </s> - <s xml:space="preserve">ex ad mirabili etemÌ„ mi-<lb/>randum accidere quippiam, non eÅ¿t ab Å¿urdum.</s> - <s xml:space="preserve"/> -</p> -<p> - <s xml:space="preserve">Maximè autem eÅ¿t admirãdum Å¿imul contra-<lb/>ria fieri. </s> - <s xml:space="preserve">Circulus vero ex huiuÅ¿modi eÅ¿t conÅ¿titu <lb/>tus: </s> - <s xml:space="preserve">Å¿tatim enim ex commoto effectus eÅ¿t, & </s> - <s xml:space="preserve">mæ <lb/>nente, quorÅ© natura ad Å¿e inuicem eÅ¿t cõtraria. <lb/></s> - <s xml:space="preserve">quamobrẽ iÅ¿thæc cernentes minus admirari cõ-<lb/>uenit contingentes in illo contrarietas. </s> - <s xml:space="preserve">In pri-<lb/>mis enim lineæ illi, quæ circuli orbem amplecti <lb/>tur, nullam habenti latitudinem, contraria quo-<lb/>dammodo ineſſe apparent, concauum Å¿cilicet, <lb/>& </s> - <s xml:space="preserve">curuum. </s> - <s xml:space="preserve">hæc aÅ©t eo à ſeinuicem diÅ¿tant mõ, <lb/>quo magnum & </s> - <s xml:space="preserve">paruum, illorÅ© etenim medium <lb/>eÅ¿t æquale: </s> - <s xml:space="preserve">horum verò rectÅ© quapropter cùm, <lb/>ad Å¿einuicem cõmutantur, illa quidem prius æ-<lb/>qualia fieri neceſſe eÅ¿t, quà m extremorum vtrÅ©-<lb/>libet: </s> - <s xml:space="preserve">lineam verò rectam, qñ eÅ¿t curua concaua, <lb/>ut ex huiu Å¿moà <unsure/>i rurÅ¿um curua fit, & </s> - <s xml:space="preserve">circularis -<pb o="506" file="0506" n="3" rhead="AriÅ¿t. Quæſtiones"/> -Vnum quidem igitur iÅ¿tuc abÅ¿urdum ineÅ¿t circti<unsure/> <lb/>lo. </s> - <s xml:space="preserve">Alterum autem, ê™ Å¿imul contrarijs mouetur <lb/>motionibus: </s> - <s xml:space="preserve">Å¿imul enim ad anteriorem mouetur <lb/>locum, & </s> - <s xml:space="preserve">ad poÅ¿ter<unsure/>iorem. </s> - <s xml:space="preserve">Et ea, quæ circulÅ© de-<lb/>Å¿cribit, linea eodem Å¿e habet modo: </s> - <s xml:space="preserve">ex quo. </s> - <s xml:space="preserve">n. </s> - <s xml:space="preserve">in <lb/>cipit loco illius extremum, ad eundem rurlus re <lb/>dit:</s> - <s xml:space="preserve"><unsure/> illa. </s> - <s xml:space="preserve">n. </s> - <s xml:space="preserve">continuò commota, extremum rurÅ¿us <lb/>efficitur primum. </s> - <s xml:space="preserve">quamobrem manifeÅ¿tum, ê™ in <lb/>de mutatnm eÅ¿t. </s> - <s xml:space="preserve">quapropter (vt dictum eÅ¿t prius) <lb/>non eÅ¿t incõueniens ipÅ¿<unsure/>u<unsure/>m miracolorum oÄ©um <lb/>eſſe principlum. </s> - <s xml:space="preserve">Ea igi ur, quæ circa libram fiÅ©t, <lb/>ad circulum referuntur: </s> - <s xml:space="preserve">quæ vero circa vectem, <lb/>ad ipÅ¿am librã: </s> - <s xml:space="preserve">alia autem ferè omn<unsure/>ia, quæ circa <lb/>mÄ™chanicas Å¿unt motiones, ad vectern<unsure/>. </s> - <s xml:space="preserve">Prætereà <lb/>etiã, quoniam vnica exiÅ¿tente, quæ ex cehtro eÅ¿t, <lb/>linea, nullum aliud alij, quæ in illa Å¿unt puncto-<lb/>rum æqua velocitate feratur; </s> - <s xml:space="preserve">Å¿ed cirius Å¿emper <lb/>quod à manente termino eÅ¿t remo ius, pleraque <lb/>miraculorum accidÅ©t in circuli motionibus: </s> - <s xml:space="preserve">de <lb/>quibus in ijs, quæ poÅ¿thac addacentur, quęſtioni <lb/>bus erit manifeÅ¿tum. </s> - <s xml:space="preserve">Quoniam autem Å¿ecundÅ© <lb/>contrarias fimul motiones mouetur circulus: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve"><lb/>alterÅ© quidem diametri extremum vbi A, in an-<lb/>tè mouetur, alterum verò vbi B. </s> - <s xml:space="preserve">ad retro; </s> - <s xml:space="preserve">effi-<lb/>ciunt nonnulli, vt ab vnica motione multi cõtra <lb/>rio Å¿imul moueantur circuli: </s> - <s xml:space="preserve">quemadmodum, <lb/>Å¿unt illi, quos in locis proponÅ©t Å¿acris, æneos, & </s> - <s xml:space="preserve"><lb/>ferreos, fabricantes orbiculos. </s> - <s xml:space="preserve">Si enim A B circu <lb/>lum alter contigerit circulus in quo CD. </s> - <s xml:space="preserve">mota <lb/>circuli in quo A B, diametro in antè, mouebitur <lb/>C D ad retro, diametro circuli vbi eÅ¿t A, circa idẽ <lb/>mota. </s> - <s xml:space="preserve">In contrarium igitur mouebitur circulus <lb/>vbi CD, ad illum ubi AB. </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">rurÅ¿us ipÅ¿e conti-<lb/>guum ubi EF, in contrarium Å¿ibijpÅ¿i mouebitur <lb/>propter eandem cauÅ¿am. </s> - <s xml:space="preserve">Eodem ẽt modo Å¿i plu- -<pb o="507" file="0507" n="4" rhead="Mechanicæ."/> -res fuerint, idẽ facient, uno Å¿olo commoto. </s> - <s xml:space="preserve">Hane <lb/>igitur in circulo exiÅ¿tentem animaduertens na-<lb/>turam architecti, in Å¿trumentÅ© fabricant, celãtes <lb/>principium, vt machinæ Å¿olum manifeftum Å¿it il <lb/>lud, quod admirationẽ pręſtat, cauÅ¿a verò lateat.</s> - <s xml:space="preserve"/> -</p> -<p> - <s xml:space="preserve">In primis igitur, quæ accidunt circa librã, du-<lb/>bitare faciunt, quamnà m ob cauÅ¿am exactiores <lb/>minoribus maiores Å¿unt libræ. </s> - <s xml:space="preserve">huius autem rei <lb/>principium eÅ¿t, quamobrem in ipÅ¿o circulo, quæ <lb/>plus à cent<unsure/>ro diÅ¿ter linea, eadẽ ui cõmota cit<unsure/>ius <lb/>fertur, quà m illa, quæ minus diÅ¿tar. </s> - <s xml:space="preserve">Citius enim <lb/>bifariam dicitur: </s> - <s xml:space="preserve">Å¿iue enim in minori tempore <lb/>zqualem pertranÅ¿i@ locÅ©, citius feciſſe dicimus: <lb/></s> - <s xml:space="preserve">Å¿eu in æquali maiorem. </s> - <s xml:space="preserve">Maior autem in æquali <lb/>tempore maiorem deÅ¿cribit circulum: </s> - <s xml:space="preserve">qui enim <lb/>extra eÅ¿t, maior eo qui intus eft. </s> - <s xml:space="preserve">Horum autẽ cau <lb/>fa, qmÌ„ duas fertur lationesea, quæ circulum de-<lb/>Å¿cribit, linea. </s> - <s xml:space="preserve">Quandoquidem igitur in propor-<lb/>tione fertur aliqua id, quod fertur, Å¿uper rectam <lb/>ferri neceſſe: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">hæc diameter efficitur figuræ, <lb/>quam faciunt illæ, quæ in huiuÅ¿modi proportio <lb/>ne coaptant̃, lineÄ™. </s> - <s xml:space="preserve">Sit enim proportio Å¿ecÅ©dum <lb/>quam latum fertur, quam habet A B ad AC: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">A <lb/>quidem fertur verÅ¿us B, A B uerò Å¿ubterferatur <lb/>verÅ¿us M C: </s> - <s xml:space="preserve">latum aut<unsure/>em Å¿it A quidem ad D. </s> - <s xml:space="preserve">ubi <lb/>autem eÅ¿t A B, uerÅ¿us E: </s> - <s xml:space="preserve">quoniã igitur la@ionis <lb/>erat proportio, quam A B habet ad A C, neceſſe <lb/>eÅ¿t & </s> - <s xml:space="preserve">A D ad A E hanc habere proportionem. </s> - <s xml:space="preserve">Si-<lb/>mile igitur eÅ¿t proportione paruum quadrilate-<lb/>rum maiori; </s> - <s xml:space="preserve">quamobrẽ & </s> - <s xml:space="preserve">eadẽ illorum eÅ¿t dia-<lb/>meter: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">A erit ad F. </s> - <s xml:space="preserve">Eodem etiam oÅ¿tendet̃ mo-<lb/>do, vbicu<unsure/>nq; </s> - <s xml:space="preserve">latio deprehendatur; </s> - <s xml:space="preserve">Å¿emper enim <lb/>Å¿upra diametrum erit. </s> - <s xml:space="preserve">ManifeÅ¿tum igitur, quòd <lb/>id, quod Å¿ecundum diametrum duabus fertur la <lb/>tionibus, neceſſariò Å¿ecundum laterum propor- -<pb o="508" file="0508" n="5" rhead="AriÅ¿t. Quæſtiones."/> -tionem fertur. </s> - <s xml:space="preserve">Sienim Å¿ecundum aliam quam-<lb/>piam, nõ fertur ẜ m dia metrum. </s> - <s xml:space="preserve">Si autem in nul <lb/>la fertur proportione Å¿ecundum duas lationes <lb/>nullo in tempore, rectã eſſe lationem eÅ¿t impoÅ¿-<lb/>Å¿ibile. </s> - <s xml:space="preserve">Sit enim recta. </s> - <s xml:space="preserve">PoÅ¿ita igitur hac pro diame <lb/>tro, & </s> - <s xml:space="preserve">circum reple is lateribus, illud quod fer-<lb/>tur, Å¿ecundum laterum pro portionem ferri n<unsure/>e-<lb/>ceſſe eÅ¿t: </s> - <s xml:space="preserve">hoc enim demonÅ¿tratum eÅ¿t prius. </s> - <s xml:space="preserve">non <lb/>igitur rectam efficiet id, quod Å¿ecundùm nullam <lb/>proportionem in nullo fertur tempore. </s> - <s xml:space="preserve">Siautẽ <lb/>ẜm quam piam feratur proportionem, & </s> - <s xml:space="preserve">in tem <lb/>pore quopiam hoc neceſſe eÅ¿t tem pus rectam eÅ¿ <lb/>Å¿e lationem, per ea, quæ retro Å¿unt dicta. </s> - <s xml:space="preserve">Quam-<lb/>obrem circulare eÅ¿tid, quod Å¿ecundÅ© nullam, <lb/>proportionem nullo in tempore duas fertur la-<lb/>tiones Quòd quidem igitur ea, quæ circulum do<unsure/> <lb/>Å¿cribir, duas Å¿imul feratur lationes, manifeÅ¿tum <lb/>eÅ¿t cùm ex iÅ¿tis, tum quia ẜ m rectum lata, aÅ¿t ê‘. <lb/></s> - <s xml:space="preserve">pendiculum peruenit, vt Å¿it rurÅ¿us ipÅ¿a à centro <lb/>perpendiculum. </s> - <s xml:space="preserve">Sit circulas A B C D: </s> - <s xml:space="preserve">extremum <lb/>autem vbi eÅ¿t B, feratur ad ipÅ¿um D: </s> - <s xml:space="preserve">peruenit Å¿a <lb/>hè aliquando ad ipÅ¿um C. </s> - <s xml:space="preserve">Si quidẽ igit<unsure/>ur in pro <lb/>portione feratur, quam haber BE, EC fertur vti <lb/>que Å¿ecundùm diametrum vbi BC. </s> - <s xml:space="preserve">nunc autem <lb/>quoniam in nulla propor:</s> - <s xml:space="preserve"><unsure/> ione, in circunferen-<lb/>tia certè fertur vbi B E C. </s> - <s xml:space="preserve">Si autem duobus ab <lb/>eadem potemtia latis, hoc quidem plus ræpella-<lb/>tur, illud verò minus, rationi conÅ¿entaneum eÅ¿t <lb/>tardius moueri id, quod plus repellitur, eo qùod <lb/>repellitur minus: </s> - <s xml:space="preserve">quod videtur accidere maiori <lb/>& </s> - <s xml:space="preserve">minori illarum, qÌuæ ex centro circulos deÅ¿cri <lb/>bunt. </s> - <s xml:space="preserve">quoniam enim proprius eÅ¿t manenti, eius <lb/>quæ minor eÅ¿t, extrem um, quà m id quod eÅ¿t ma <lb/>oris, veluti retractum in contrariÅ© ad medium, <lb/>ardius fertur minoris extremÅ©. </s> - <s xml:space="preserve">Omni quidem -<pb o="509" file="0509" n="6" rhead="Mechanicæ."/> -igitur circulum deÅ¿cribenti iÅ¿tuc accidit: </s> - <s xml:space="preserve">fertur-<lb/>qÌue eam quæ ẜm naturam eÅ¿t lationem, Å¿ecun-<lb/>dum circunferentiam: </s> - <s xml:space="preserve">illam verò quÄ™ prÄ™ter na <lb/>turam, in tranÅ¿uerÅ¿um, & </s> - <s xml:space="preserve">Å¿ecundùm centrum. <lb/></s> - <s xml:space="preserve">maiorem aut<unsure/>ẽ Å¿emper eam, quæ præter naturã <lb/>eÅ¿t, ipÅ¿a minor fertur: </s> - <s xml:space="preserve">quia enim centro eÅ¿t vici-<lb/>nior, quod trahit, vincitur magis<unsure/>: </s> - <s xml:space="preserve">Quòd autem-<unsure/> <lb/>magis quod præter natui<unsure/>ã eÅ¿t mouetur ipÅ¿a mi-<lb/>nor, quà m maior illarÅ©, quæ ex centro circulos <lb/>deÅ¿cribÅ©t ex ijs eÅ¿t manifeÅ¿tum. </s> - <s xml:space="preserve">Sit circulus vbi <lb/>B C D E & </s> - <s xml:space="preserve">alter in hoc minorvbi M N O P, circa <lb/>idem centrum A & </s> - <s xml:space="preserve">proijciantur diametri in ma <lb/>gno quidem, in quibus CD, BE, in minori ve-<lb/>10 ipſæ MO, N P: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">altera parie longius quadra <lb/>tum Å¿uppleatur DKRC: </s> - <s xml:space="preserve">Å¿i quidem AB circu-<lb/>lum deÅ¿cribẽs ad id perueniet, vnde eÅ¿t egreſſa; </s> - <s xml:space="preserve"><lb/>manifeÅ¿tum eÅ¿t quod ad ipÅ¿am fertur A B. </s> - <s xml:space="preserve">Simi <lb/>literetiam A M ad ipÅ¿am A M perueniet. </s> - <s xml:space="preserve">Tar-<lb/>dius autem fertur A M, quà ru A B quemadmo-<lb/>dum dictum eÅ¿t: </s> - <s xml:space="preserve">quia maior fit repulÅ¿io & </s> - <s xml:space="preserve">magis <lb/>rerrahitur A M. </s> - <s xml:space="preserve">Ducatur igitur ipÅ¿a A L F, & </s> - <s xml:space="preserve">ab <lb/>ipÅ¿o L perpendiculum ad ip Å¿am A B, ipÅ¿a L Q in <lb/>minore circulo: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">rurÅ¿um ab L ducatur iuxta A <lb/>B L S, & </s> - <s xml:space="preserve">S T ad ipÅ¿am A B perpendiculum, & </s> - <s xml:space="preserve">ip. </s> - <s xml:space="preserve"><lb/>fa F X: </s> - <s xml:space="preserve">ipſæ igitur ubi Å¿unt S T, & </s> - <s xml:space="preserve">L Q, æquales: </s> - <s xml:space="preserve"><lb/>ipÅ¿a ergo B T minor eÅ¿t, quà m M Q. </s> - <s xml:space="preserve">æquales <lb/>enim rectæ lineæ in æqualibus coniectæ circulis <lb/>perpẽdiculares a diametro, minorẽ diametri re-<lb/>fecant Å¿ectionẽ in maioribus circulis, eÅ¿t autem <lb/>ipÅ¿a S T æqualis ipÅ¿i L Q. </s> - <s xml:space="preserve">In quanto autem tema <lb/>pore ipÅ¿a A L ipÅ¿am M L lata eÅ¿t, in tanto tem@ <lb/>poris Å¿patio in maiori circulo maiorem, quà m <lb/>fit B S, latum erit extremum ipÅ¿ius B A. </s> - <s xml:space="preserve">Latio <lb/>quidem igitur Å¿ecundùm naturam æqualis: </s> - <s xml:space="preserve">ea <lb/>a<unsure/>utem quæ præter naturam eÅ¿t minor, videlices -<pb o="510" file="0510" n="7" rhead="AriÅ¿t. Quæſtiones."/> -B T, quà m M Q. </s> - <s xml:space="preserve">Oportet autem proportionabi-<lb/>liter eſſe, Å¿icut quod eÅ¿t Å¿ecundÅ© naturã, ita quod <lb/>eÅ¿t prÄ™ter naturam, ad id quod eÅ¿t prÄ™ter naturã; <lb/></s> - <s xml:space="preserve">maiorem igituÄ© circunferentiam pertranÅ¿iuit, ê—̃ <lb/>Å¿it ipÅ¿a S B. </s> - <s xml:space="preserve">Neceſſe autem eÅ¿t ipÅ¿am F B in hoc <lb/>rẽpore pertran Å¿iſſe: </s> - <s xml:space="preserve">hic. </s> - <s xml:space="preserve">n. </s> - <s xml:space="preserve">erit, quando proportio <lb/>nabiliter vtrinq; </s> - <s xml:space="preserve">accidit, quod eÅ¿t præter naturã, <lb/>ad id quod eÅ¿t Å¿ecundùm naturam. </s> - <s xml:space="preserve">Si igitur ma-<lb/>ius eÅ¿t, quod Å¿ecundum naturam in maiori, & </s> - <s xml:space="preserve"><lb/>quod eÅ¿t præter naturam. </s> - <s xml:space="preserve">magis vtique hic coin <lb/>cidit vno modo: </s> - <s xml:space="preserve">ita quod B Å¿it latum per ipÅ¿am <lb/>B F in tanto tempore, in quo M punctum per <lb/>ipÅ¿am M L. </s> - <s xml:space="preserve">hic enim Å¿ecundùm naturam quidẽ <lb/>figno B fit X F: </s> - <s xml:space="preserve">eÅ¿t enim ab ipÅ¿o F perpendiculÅ©: </s> - <s xml:space="preserve"><lb/>præter naturam verò ad ipÅ¿am X B. </s> - <s xml:space="preserve">EÅ¿t autem <lb/>quemadmodum F X ad X B, Å¿ic L Q. </s> - <s xml:space="preserve">ad M Q. </s> - <s xml:space="preserve">ma <lb/>nifeÅ¿tum autem, Å¿i coniunguntur abipÅ¿a B M, <lb/>ad F L. </s> - <s xml:space="preserve">Si autem minor, aut maior quà m Å¿it F B, <lb/>erit illa quam latum eÅ¿t B, non Å¿imiliter erit, ne-<lb/>que proportionale in vtriÅ¿que quod eÅ¿t Å¿ecundÅ© <lb/>naturam, ad id quod eÅ¿t præter naturam. </s> - <s xml:space="preserve">Quam <lb/>igicur ob cauÅ¿am ab eadem potentia celerius fe@ <lb/>tur id quod plus à centro diÅ¿tat punctum, ex ijs, <lb/>quæ dicta Å¿unt, eÅ¿t manifeÅ¿tum.</s> - <s xml:space="preserve"/> -</p> -</div> -<div type="section" level="1" n="4"> -<head style="it" xml:space="preserve">Quæſtiones.</head> -<p> - <s xml:space="preserve">C V R aÅ©t maiores libræ exactiores <lb/> -<anchor type="note" xlink:label="note-0510-01a" xlink:href="note-0510-01"/> -Å¿unt minotib<emph style="sub">9</emph>. </s> - <s xml:space="preserve">palã eÅ¿t ex ijs. </s> - <s xml:space="preserve">Spar <lb/>tum. </s> - <s xml:space="preserve">n fit centrÅ©: </s> - <s xml:space="preserve">id pãq; </s> - <s xml:space="preserve">manet. <lb/></s> - <s xml:space="preserve">quod aÅ©t libræ vtrinq; </s> - <s xml:space="preserve">eÅ¿t, exeun <lb/>tes à centro. </s> - <s xml:space="preserve">Ab eodẽ igit̃ põdere <lb/>citius moueri neceſſe eÅ¿t extre-<lb/>mÅ© libræ, quo plus à ſparto diÅ¿ceſſerit. </s> - <s xml:space="preserve">Et nõnul <lb/>la quidẽ in paruis libris impoÅ¿ita nõ manifeÅ¿ta -<pb o="511" file="0511" n="8" rhead="Mechanicæ."/> -fenÅ¿ui Å¿unt pondera: </s> - <s xml:space="preserve">in magis aÅ©t maniÅ¿eÅ¿ta: </s> - <s xml:space="preserve">ni <lb/>hil enim prohibet minorem moueri magnitudi <lb/>nem, ê—̃ vt viÅ¿ioni Å¿it maniÅ¿eÅ¿ta. </s> - <s xml:space="preserve">In magna aÅ©t li-<lb/>bra idem pondus viÅ¿ibile efficit magnitudo. </s> - <s xml:space="preserve">QuÄ™ <lb/>dam verò manifeÅ¿ta quidem Å¿unt in vtriÅ¿q;</s> - <s xml:space="preserve">, Å¿ed <lb/>multo magis in maioribus, qmÌ„ multo maior in <lb/>clinationis fit magnitudo ab eodem ê‘ondere <lb/>in maioribus. </s> - <s xml:space="preserve">Quamobrem machinanturij, qui <lb/>purpuram vendunt, vt pendendo defraudent, tÅ© <lb/>ad medium Å¿partum non ponentes: </s> - <s xml:space="preserve">tum plum-<lb/>bum in alteraruram libræ partem infundentes: <lb/></s> - <s xml:space="preserve">aut ligni gÄ‘ ad radicẽ vergebat, in eam quam de <lb/>ferri vola<unsure/>nt, partem conÅ¿tituentes; </s> - <s xml:space="preserve">aut Å¿i nodum <lb/>habuerit: </s> - <s xml:space="preserve">ligni enim grauior illa eÅ¿t pars, in qua <lb/>eÅ¿t radix, nodus vero radix quædam eÅ¿t.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="1"> -<note position="left" xlink:label="note-0510-01" xlink:href="note-0510-01a" xml:space="preserve">1</note> -</div> -<p> - <s xml:space="preserve">Cut Å¿iquidẽ Å¿urÅ¿um Å¿uerit Å¿partÅ©, qñ deorsÅ© lato <lb/> -<anchor type="note" xlink:label="note-0511-01a" xlink:href="note-0511-01"/> -põdere, quiÅ¿piã id admouet, rurÅ¿um aÅ¿cendit li-<lb/>bra: </s> - <s xml:space="preserve">Å¿i autem deorÅ¿um conÅ¿titutum Å¿uerit, non <lb/>aÅ¿cendit, Å¿ed manet? </s> - <s xml:space="preserve">An quia Å¿urÅ¿um quidẽ Å¿par <lb/>to exiÅ¿tente, plus libræ extra perpendiculum fit: <lb/></s> - <s xml:space="preserve">Å¿partum enim eÅ¿t perpendiculum: </s> - <s xml:space="preserve">quare neceſſe <lb/>eÅ¿t deorÅ¿um ferri id quod plus eÅ¿t, donec aÅ¿cen-<lb/>dat quæ bifariam libram diuidit, ad ipÅ¿um per-<lb/>pendiculÅ©, cùm onus incumbat ad libræ partem <lb/>furÅ¿um raptam. </s> - <s xml:space="preserve">Sit libra recta vbi B C, Å¿partum <lb/>autem A D. </s> - <s xml:space="preserve">hoc igitur deorÅ¿um proiecto perpen <lb/>diculum erit vbi A D M. </s> - <s xml:space="preserve">Si igitur in ipÅ¿o B pona <lb/>tur onus, B quidem erit vbi E, C, autem vbi H. </s> - <s xml:space="preserve"><lb/>quamobrem ea quæ bifariam libram Å¿ecat, pri-<lb/>mo quidem erit D M ipÅ¿ius perpendiculi: </s> - <s xml:space="preserve">incum <lb/>bente autẽ onere erit D G. </s> - <s xml:space="preserve">quare libra ipÅ¿ius vbi <lb/>E H, quod extra perpendiculum eÅ¿t A M, vbi eÅ¿t <lb/>P Q, maius eÅ¿t dimidio. </s> - <s xml:space="preserve">Si igitur amoueatur <lb/>onus abipÅ¿o E, neceſſe eÅ¿t deorÅ¿um ferri H, mi-<lb/>@@s enim eÅ¿t ipÅ¿um E. </s> - <s xml:space="preserve">Si quidem igitur Å¿urÅ¿um -<pb o="512" file="0512" n="9" rhead="AriÅ¿t. Quæſtiones"/> -habuerit Å¿partum, rurÅ¿um propter hoc aÅ¿cendit<unsure/> <lb/>libta. </s> - <s xml:space="preserve">Si autẽ deorÅ¿um fuerit, id, quod Å¿ubÅ¿tat, cõ <lb/>trarium facit: </s> - <s xml:space="preserve">plus enim dimidio fit libræ, quæ <lb/>deorÅ¿um eÅ¿t pars, quà m quod perpendiculum Å¿e <lb/>cet, quapropter nõ aÅ¿cẽdit: </s> - <s xml:space="preserve">eleuata enim pars le <lb/>uior eÅ¿t. </s> - <s xml:space="preserve">Sit libra recta vbi N O, perpendiculum <lb/>autem K L M bifariã igitur Å¿ecatu@ K G. </s> - <s xml:space="preserve">Impo-<lb/>Å¿ito autem onere in ipÅ¿o N, erit quidẽ N vbi O, <lb/>ipÅ¿um autem G vbi R, K L autem vbi L P. </s> - <s xml:space="preserve">quare <lb/>maius eÅ¿t K O, quà m L R, ipÅ¿o P K L. </s> - <s xml:space="preserve">Et ablato <lb/>igitur onere, neceſſe eÅ¿t manere: </s> - <s xml:space="preserve">in cumbit enim <lb/>ceu onus exceſſus medietatis eius vbi eÅ¿t F.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="2"> -<note position="right" xlink:label="note-0511-01" xlink:href="note-0511-01a" xml:space="preserve">2</note> -</div> -<p> - <s xml:space="preserve">Cur exiguÄ™ vires (quem admodum à principio <lb/> -<anchor type="note" xlink:label="note-0512-01a" xlink:href="note-0512-01"/> -dictum eÅ¿t) vecte magna mouent pondera, vecis <lb/>inÅ¿uper onus accipien tis? </s> - <s xml:space="preserve">cùm Å¿acilius Å¿it mino-<lb/>grem mouere grauitatem: </s> - <s xml:space="preserve">minor autẽ eÅ¿t Å¿ine ve-<lb/>cte. </s> - <s xml:space="preserve">An quoniã ipÅ¿e vectis eÅ¿t in cauÅ¿a hbra exi-<lb/>Å¿tens, Å¿partÅ© infernè habens, in inæqualia diui-<lb/>Å¿a. </s> - <s xml:space="preserve">hypomochlion enim eÅ¿t Å¿partum. </s> - <s xml:space="preserve">a mbo nan <lb/>que Å¿tamt, vt centrum. </s> - <s xml:space="preserve">quoniam autem ab Ä™qua-<lb/>Å¿i pondere celerius mouetur maior earum, q̃ à <lb/>centro Å¿unt: </s> - <s xml:space="preserve">duo verò pondera, quod mouet, & </s> - <s xml:space="preserve"><lb/>quod mouetur: </s> - <s xml:space="preserve">quod igitur motum pondus ad-<lb/>mouens, longitudo patitur ad longitudinẽ. </s> - <s xml:space="preserve">Å¿em <lb/>per autem quanto ab hy pomochlio diÅ¿tabit ma <lb/>gis, tanto facilius mo<unsure/>uebit. </s> - <s xml:space="preserve">CauÅ¿a autem eÅ¿t, quÄ™ <lb/>retro cõmemorata eÅ¿t: </s> - <s xml:space="preserve">quoniam quÄ™ plus à cen-<lb/>@ro diÅ¿tat, maiorem deÅ¿cribit circuium: </s> - <s xml:space="preserve">quare ab <lb/>eadem potentia plus Å¿eparabitur mouens illud, <lb/>quod plus ab hypomochlio diÅ¿tabit. </s> - <s xml:space="preserve">Sit uectis <lb/>vbi A B, pondus vbi C, quod mouet autẽvbi D, <lb/>hy pomochlion vbi E. </s> - <s xml:space="preserve">quod autem vbi eÅ¿t D mo <lb/>u<unsure/>ens, vbi F. </s> - <s xml:space="preserve">motum autem vbi C. </s> - <s xml:space="preserve">pondus, vbi G.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="3"> -<note position="left" xlink:label="note-0512-01" xlink:href="note-0512-01a" xml:space="preserve">3</note> -</div> -<p> - <s xml:space="preserve">Cur ij, qui in nauis medio Å¿unt remiges, maxi <lb/> -<anchor type="note" xlink:label="note-0512-02a" xlink:href="note-0512-02"/> -me nauem mouent? </s> - <s xml:space="preserve">An quia remus vectis eÅ¿t: </s> - <s xml:space="preserve">hy -<pb o="513" file="0513" n="10" rhead="Mechanicæ."/> -pomochlion auteni fit Å¿calmus, Å¿tat enim ille ẜ <lb/>pondus verò mare eÅ¿t, quod propellit remus: </s> - <s xml:space="preserve">ve-<lb/>ctem autem mouens ipÅ¿e eÅ¿t remex. </s> - <s xml:space="preserve">Semper aÅ©t <lb/>plus mouet ponderis, quanto magis ab hypomo <lb/>chlio diÅ¿tabit, quicunque id mouet. </s> - <s xml:space="preserve">maior enim <lb/>ita fit, quæ ex centro. </s> - <s xml:space="preserve">Å¿calmus autem hypomo-<lb/>chlion exiÅ¿tens, centrum eÅ¿t. </s> - <s xml:space="preserve">In medio autem na <lb/>uis plurimum remi intus eÅ¿t: </s> - <s xml:space="preserve">ilia enim parte la-<lb/>tiſſima eÅ¿t nauis: </s> - <s xml:space="preserve">quare maior vtrinq; </s> - <s xml:space="preserve">remi pars <lb/>vtrorunq; </s> - <s xml:space="preserve">nauis parietum intrin Å¿ecus eÅ¿t. </s> - <s xml:space="preserve">Moue-<lb/>tur autem nauis, quoniam appellente ad mare <lb/>remo, extremum illius quod intus eÅ¿t, in ante <lb/>prom ouctur. </s> - <s xml:space="preserve">nauem verò Å¿calmo alligatam Å¿i-<lb/>mul promoueri contingit, quo remi extremum. <lb/></s> - <s xml:space="preserve">vbi enim plurimum maris diuidit remus, eò ma-<lb/>ximè propelli neceſſe eÅ¿t. </s> - <s xml:space="preserve">plurimum autem di@i <lb/>dit, vbi pars plurima remia Å¿calmo eÅ¿t. </s> - <s xml:space="preserve">Et eam <lb/>cauÅ¿am remiges, qui in media Å¿unt naui, mouent <lb/>illam maximè. </s> - <s xml:space="preserve">maxima enim remi pars à ſcalmo <lb/>in nauis medio intus eÅ¿t.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="4"> -<note position="left" xlink:label="note-0512-02" xlink:href="note-0512-02a" xml:space="preserve">4</note> -</div> -<p> - <s xml:space="preserve">Cur paruum exiÅ¿tens gubernaculum, & </s> - <s xml:space="preserve">in ex-<lb/> -<anchor type="note" xlink:label="note-0513-01a" xlink:href="note-0513-01"/> -tremo nauigio tantas habet vires, vt ab exiguo <lb/>temone, & </s> - <s xml:space="preserve">ab hominis vnius viribus alioqùin <lb/>modicè vtentis, magnÄ™ nauigiorum moueantur <lb/>moles? </s> - <s xml:space="preserve">An quoniam gubernaculum vectis eÅ¿t: </s> - <s xml:space="preserve">o-<lb/>@us autem mare: </s> - <s xml:space="preserve">gubernator verò mouens eÅ¿t. <lb/></s> - <s xml:space="preserve">Non autem Å¿ecundum latitudinẽ, veluti remus, <lb/>mare accipit gubernaculum: </s> - <s xml:space="preserve">non enim in antè <lb/>nauigium mouet, Å¿ed ipÅ¿um commotum mare <lb/>accipiens inclinat obliquè: </s> - <s xml:space="preserve">quoniam enim pon-<lb/>dus eÅ¿t mare, contrario innixum modo nauem <lb/>inclinat. </s> - <s xml:space="preserve">hypomochlion enim in contrarium <lb/>verÅ¿atur: </s> - <s xml:space="preserve">mare vero interius, Å¿i illud exterius. </s> - <s xml:space="preserve"><lb/>illud autem Å¿equitur nauis, quoniam illi eÅ¿t ai<unsure/>li-<lb/>@ata. </s> - <s xml:space="preserve">Et remus quidem Å¿ecundùm latitudinem -<pb o="514" file="0514" n="11" rhead="AriÅ¿t. Quæſtiones."/> -onus propellens, & </s> - <s xml:space="preserve">ab eodem repulÅ¿us, in rectÅ© <lb/>propellit: </s> - <s xml:space="preserve">gubernaculum autem, ut obliquum ia <lb/>cet, hinc inde in obliquum motionÄ“ facit. </s> - <s xml:space="preserve">In ex <lb/>tremo autÄ“, & </s> - <s xml:space="preserve">non in medio iacet, quoniam mo <lb/>uenti facilimÅ« eÅ¿t ab extremo motum mouere. <lb/></s> - <s xml:space="preserve">Prim a enim pars celerimè fertur; </s> - <s xml:space="preserve">quoniã quem <lb/>adm odum in ijs quæ feruntur, in fine deficit la-<lb/>tio, Å¿ic ipÅ¿ius cÅtinui in fine imbeciliſſima eÅ¿t la-<lb/>tio. </s> - <s xml:space="preserve">imbeciliſſima autem ad expellendum eÅ¿t fa <lb/>cilis. </s> - <s xml:space="preserve">Propter hæcigitur in puppi gubernaculum <lb/>ponitur. </s> - <s xml:space="preserve">Nec minus, quoniÄ parua ibi motione <lb/>Å¿acta, multò maius interuallum fit in ultimo: </s> - <s xml:space="preserve">ê—a <lb/>æqualis angulus Å¿emper maiorem Å¿pectat: </s> - <s xml:space="preserve">tan-<lb/>toqÌue magis. </s> - <s xml:space="preserve">quanto maiores fuerint illæ, quæ <lb/>con@inẽt. </s> - <s xml:space="preserve">Ex ijs etiam manifeÅ¿tum eÅ¿t, quam ob <lb/>cauſſam magis in contrarium procedit nauigiÅ©, <lb/>quà m remi ipÅ¿ius palmula. </s> - <s xml:space="preserve">eadem enim magni <lb/>tudoijÅ¿dem mota uiribus, in aete plus, quà m in <lb/>aqua progreditur. </s> - <s xml:space="preserve">Si enim A B remus, Guerò <lb/>Å¿calmus: </s> - <s xml:space="preserve">A autem in nauigio Å¿it remi principiÅ©, <lb/>B uerò in mari palmula. </s> - <s xml:space="preserve">Si igitur A ubi D tranÅ¿. </s> - <s xml:space="preserve"><lb/>latum eÅ¿t, non erit B ubi E: </s> - <s xml:space="preserve">æqualis enim B E ipÅ¿i <lb/>A C. </s> - <s xml:space="preserve">æquale igitur tranÅ¿latÅ« erit, Å¿ed minus. </s> - <s xml:space="preserve">erit <lb/>igitur ubi eÅ¿t F. </s> - <s xml:space="preserve">minor enim C F, ipÅ¿a A D. </s> - <s xml:space="preserve">quare <lb/>ipÅ¿a G F, ipÅ¿a D F. </s> - <s xml:space="preserve">Å¿imiles enim Å¿unt trianguli. </s> - <s xml:space="preserve"><lb/>Stans autem erit medium, ubi eÅ¿t C. </s> - <s xml:space="preserve">in contrariÅ© <lb/>enim ipÅ¿i, quod in mari eÅ¿t, extremo uidelicet B <lb/>procedit, ubi extremum in nauigio eÅ¿t A. </s> - <s xml:space="preserve">Non <lb/>procedere autÄ“ubi eÅ¿t D. </s> - <s xml:space="preserve">niÅ¿i commouere@ur na <lb/>uigium, & </s> - <s xml:space="preserve">eo transferretur, ubi remi eÅ¿t princi-<lb/>plum IdipÅ¿nm etiam facit gubernaculam, niÅ¿i ê™ <lb/>(ut dictum eÅ¿t terro) nihil nauigio ad id, quod-<lb/>antè eÅ¿t, confert, Å¿ed Å¿olùm puppim in obliquÅ© <lb/>pell@t, ubicunq; </s> - <s xml:space="preserve">fuerit in con@rariÅ« enim eo mo <lb/>do uergit prora. </s> - <s xml:space="preserve">Vbi igitur applicatum eÅ¿t gu- -<pb o="515" file="0515" n="12" rhead="Mechanicæ."/> -bernaculum, id oporter rei motæ, ceu quoddam <lb/>intelligere medium: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">quemadmodÅ© Å¿calmus <lb/>remo. </s> - <s xml:space="preserve">Medium autem procedit Å¿ecundùm quod <lb/>gubernaculum trãsfertur. </s> - <s xml:space="preserve">Å¿i quidÄ“ introrÅ¿us agit, <lb/>& </s> - <s xml:space="preserve">puppis eo transfertur. </s> - <s xml:space="preserve">prora verò ad contra-<lb/>rium vergit. </s> - <s xml:space="preserve">in eodem enim exiÅ¿tente prora, to-<lb/>tum transfertur nauigium.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="5"> -<note position="right" xlink:label="note-0513-01" xlink:href="note-0513-01a" xml:space="preserve">5</note> -</div> -<p> - <s xml:space="preserve">Cur quando antenna Å¿ublimior fuerit, ijÅ¿dem <lb/> -<anchor type="note" xlink:label="note-0515-01a" xlink:href="note-0515-01"/> -uelis, & </s> - <s xml:space="preserve">uento eodẽ celerius ferÅ©tur nauigia? </s> - <s xml:space="preserve">An <lb/>quia malus quidem fit uectis: </s> - <s xml:space="preserve">hypomochlion ue <lb/>rò mali Å¿edes, in qua collocatur: </s> - <s xml:space="preserve">pondus autem, <lb/>quod moueri debet, ipÅ¿um nauigium: </s> - <s xml:space="preserve">mou@s ve <lb/>ro is, qui vela tendit, Å¿piri@us. </s> - <s xml:space="preserve">Si igitur quanto re <lb/>motius fuerit hypomochion, facilius eadem po-<lb/>tentia, & </s> - <s xml:space="preserve">citius idem mouet pondus, altius cer-<lb/>tè Å¿ublata antenna, velum à mali Å¿ede, quæ hypo <lb/>mochlion eÅ¿t, remo@ius faciens, id efficie@.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="6"> -<note position="right" xlink:label="note-0515-01" xlink:href="note-0515-01a" xml:space="preserve">6</note> -</div> -<p> - <s xml:space="preserve">Cur quando ex puppi nauigare voluerint, non <lb/> -<anchor type="note" xlink:label="note-0515-02a" xlink:href="note-0515-02"/> -flante ex puppi vento, veli quidem partem, quæ <lb/>ad gubernatorem vergit, conÅ¿tringunt: </s> - <s xml:space="preserve">illam ve-<lb/>rò quæ proram verÅ¿us eÅ¿t, pedem facientes rela <lb/>xant? </s> - <s xml:space="preserve">An quia retrahere quidem multo exiÅ¿ten <lb/>tevento gubernaculum non poteÅ¿t: </s> - <s xml:space="preserve">pauco autẽ <lb/>poteÅ¿t, quem conÅ¿tringunt. </s> - <s xml:space="preserve">Propellit quidem igi <lb/>tur ipÅ¿e ventus: </s> - <s xml:space="preserve">in puppim vero illum cÅÅ¿tituit, <lb/>gubernaculum retrahens, & </s> - <s xml:space="preserve">mare compellẽs: </s> - <s xml:space="preserve">Å¿i-<lb/>mul & </s> - <s xml:space="preserve">nautæ ipÅ¿i cum vento contendunt: </s> - <s xml:space="preserve">in cõ-<lb/>trariam enim Å¿e reclinant partem.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="7"> -<note position="right" xlink:label="note-0515-02" xlink:href="note-0515-02a" xml:space="preserve">7</note> -</div> -<p> - <s xml:space="preserve">Cur ex figurarum genere, q̃cunq; </s> - <s xml:space="preserve">rotÅ©dÄ™ Å¿unt <lb/> -<anchor type="note" xlink:label="note-0515-03a" xlink:href="note-0515-03"/> -& </s> - <s xml:space="preserve">circinatæ, facilius mouentur? </s> - <s xml:space="preserve">Trifariam autẽ <lb/>circulum rotari contingit. </s> - <s xml:space="preserve">aut enim @m abÅ¿idem <lb/>centro Å¿imul moto, quemadmodum paluÅ¿triver <lb/>titur rota: </s> - <s xml:space="preserve">aut circa manens centrÅ«, veluti tro-<lb/>chleæ Å¿tante centro: </s> - <s xml:space="preserve">autin pauimento manente <lb/>@ẽtro, Å¿icut figuli rota conuertitur. </s> - <s xml:space="preserve">An celer@ima -<pb o="516" file="0516" n="13" rhead="AriÅ¿t. Quæſtiones"/> -quidem huiuÅ¿modi Å¿unt, quoniam parua Å¿uipa@ <lb/>te planum contingunt, veluti circulas Å¿ecÅ«dum <lb/>punctum: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">quoniam non offenÅ¿ant: </s> - <s xml:space="preserve">à terra emÌ„ <lb/>Å¿emotus eÅ¿t angulus. </s> - <s xml:space="preserve">PrÄ™terea etiam cui obuiam <lb/>fiunt corpori, id rurÅ¿um Å¿ecundùm puÅ¿illam tan <lb/>gunt. </s> - <s xml:space="preserve">Å¿i autem rectilineum eſſet, rectitudine Å¿ua <lb/>multum plani contingeret. </s> - <s xml:space="preserve">Ad hæc quò mutat <lb/>pondus, eo motor mouet. </s> - <s xml:space="preserve">cùm igitur ad rectum <lb/>Å¿uper plano circuli fuerit diameter, planÅ« Å¿ecun <lb/>dùm punctum, contingÄ“te circulo æquale vtriu-<lb/>que pondus difterminat diameter. </s> - <s xml:space="preserve">cùm autÄ“ mo <lb/>uetur, plus illico, ad quòd mouetur, ceui@de nu <lb/>tans. </s> - <s xml:space="preserve">ab impellẽre facilius in ante mouetur: </s> - <s xml:space="preserve">quò <lb/>enim vnumquodque vergit, mouetur ex facili: <lb/></s> - <s xml:space="preserve">Å¿i quidem difficulter ad cÅtrarium nutus Å¿ui mo <lb/>uetur modum. </s> - <s xml:space="preserve">Præterea nonnulli autumn@nt ê™ <lb/>circuli linea in perpeti verÅ¿atur motu, quemad-<lb/>modum manentia propter contrarium nixÅ© ma <lb/>nent: </s> - <s xml:space="preserve">Å¿icur maioribus contingit circulis ad mino <lb/>res. </s> - <s xml:space="preserve">Celerius enim ab æquali mouentur potentia <lb/>maiores circuli, mouentqÌue onera, quoniam cir <lb/>culi maioris angulus ad minoris angulum cir-<lb/>culi nutum habet quendam: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">Å¿icut diameter ad <lb/>diametrum, ita maior circulus ad minorem. </s> - <s xml:space="preserve">it fi <lb/>niti autem Å¿unt minores. </s> - <s xml:space="preserve">Si autem ad alte@um <lb/>nutum habet circulus, Å¿imiliter eÅ¿t bene mobi-<lb/>lis. </s> - <s xml:space="preserve">Et aliam Å¿anè habet inclinationem circulus, <lb/>& </s> - <s xml:space="preserve">ea quæ à circulo mouentur, licèt planitiẽ abÅ¿i-<lb/>de non contingat, Å¿ed aut iuxta planitiẽ, aut velu <lb/>tì trocleæ. </s> - <s xml:space="preserve">etenim hoc Å¿e habentes modo facilli-<lb/>me mouentur, & </s> - <s xml:space="preserve">onera commouent. </s> - <s xml:space="preserve">An nõ quia <lb/>parua Å¿ui portione cùm tangit, tum offenÅ¿at cir-<lb/>culus, Å¿ed aliam ob cauÅ¿am. </s> - <s xml:space="preserve">ea autem eÅ¿t, quæ di <lb/>cta eÅ¿t prius, quod circulus Å¿cilicet ex duabus ef-<lb/>fectus eÅ¿t lationibus: </s> - <s xml:space="preserve">quam ob rem illarum alte- -<pb o="517" file="0517" n="14" rhead="Mechanicæ."/> -ram pro nutu Å¿emper habet, & </s> - <s xml:space="preserve">veluti continuo <lb/>motum illum mòuent, quicunq; </s> - <s xml:space="preserve">mouent, quan-<lb/>do Å¿ecundùm circun ferentiam illum mouerint: <lb/></s> - <s xml:space="preserve">latam enim ipÅ¿am mouent. </s> - <s xml:space="preserve">Eam quidem igitur, <lb/>quæ in obliquum eÅ¿t, motionem ipÅ¿um impellit <lb/>mouens: </s> - <s xml:space="preserve">Å¿ecundum vero illam, quæ Å¿uper diame <lb/>trum eÅ¿t, Å¿eipÅ¿um mouet circulus.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="8"> -<note position="right" xlink:label="note-0515-03" xlink:href="note-0515-03a" xml:space="preserve">8</note> -</div> -<p> - <s xml:space="preserve">Cur ea, quæ per maiores circulos tolluntur & </s> - <s xml:space="preserve"><lb/> -<anchor type="note" xlink:label="note-0517-01a" xlink:href="note-0517-01"/> -trahuntur, facilius & </s> - <s xml:space="preserve">ci@ius moueri contingit, ve <lb/>lu@i maioribus trochleis quà m minoribus, & </s> - <s xml:space="preserve"><lb/>fcytalis Å¿imiliter? </s> - <s xml:space="preserve">An quoniã, qnanto maior fue-<lb/>ri@@illa quæ à centro eÅ¿t, in Ä™quali tempore maius <lb/>mouetur Å¿patium. </s> - <s xml:space="preserve">Quam ob rem æquali inexiÅ¿tẽ <lb/>te onere, idem faciet: </s> - <s xml:space="preserve">quemadmodum diximus, <lb/>& </s> - <s xml:space="preserve">maiores libras minoribus exactiores eſſe. </s> - <s xml:space="preserve">Å¿par <lb/>tum enim in illis centrum eÅ¿t: </s> - <s xml:space="preserve">libræ autem vtrin <lb/>que partes, quæ ex centro Å¿unt, exiÅ¿tunt.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="9"> -<note position="right" xlink:label="note-0517-01" xlink:href="note-0517-01a" xml:space="preserve">9</note> -</div> -<p> - <s xml:space="preserve">Cur facilius quando Å¿ine pondere eÅ¿t, moue-<lb/> -<anchor type="note" xlink:label="note-0517-02a" xlink:href="note-0517-02"/> -tur libra, quà m cum pondus habet? </s> - <s xml:space="preserve">SimiliqÌ mo <lb/>do rota, & </s> - <s xml:space="preserve">huiuÅ¿modi quippiam, quod grauius <lb/>quidem eÅ¿t, maius autem minore & </s> - <s xml:space="preserve">leuiore? </s> - <s xml:space="preserve">An <lb/>quia non Å¿olùm in contrarium quod graue eÅ¿t, <lb/>Å¿ed in obliquum etiam difficulter mouetur. </s> - <s xml:space="preserve">In <lb/>contrarium enim ei, ad quod vergit onus, moue-<lb/>re difficile eÅ¿t: </s> - <s xml:space="preserve">quò autem vergit, eÅ¿t facilè. </s> - <s xml:space="preserve">in obli <lb/>quum autem haudquaquam vergit.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="10"> -<note position="right" xlink:label="note-0517-02" xlink:href="note-0517-02a" xml:space="preserve">10</note> -</div> -<p> - <s xml:space="preserve">Cur Å¿uper Å¿cytalas facilius portantur onera, <lb/> -<anchor type="note" xlink:label="note-0517-03a" xlink:href="note-0517-03"/> -quã Å¿@per currus, cùm tamẽ ij magnas habeant <lb/>rotas, illæ verò puÅ¿illas? </s> - <s xml:space="preserve">An qui in Å¿cytalis nulla <lb/>eÅ¿t offen Å¿atio, in curribus autem axis eÅ¿t, ad quẽ <lb/>offenÅ¿ant. </s> - <s xml:space="preserve">deÅ¿uper enim illum premunt, & </s> - <s xml:space="preserve">late-<lb/>ribus. </s> - <s xml:space="preserve">quod autem eÅ¿t in Å¿cycalis, ad iÅ¿thæc duo <lb/>mouetur, & </s> - <s xml:space="preserve">inÅ¿ernè Å¿ubÅ¿trato Å¿patio, & </s> - <s xml:space="preserve">onere Å¿u <lb/>per impoÅ¿ito. </s> - <s xml:space="preserve">in vtriÅ¿que enim ijs reuoluitur lo-<lb/>cis circulus, & </s> - <s xml:space="preserve">motus impellitur.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="11"> -<note position="right" xlink:label="note-0517-03" xlink:href="note-0517-03a" xml:space="preserve">11</note> -</div> -<pb o="518" file="0518" n="15" rhead="AriÅ¿t. Quæſtiones"/> -<p> - <s xml:space="preserve">Curlongius feruntur miſſilia funda, quà m ma <lb/> -<anchor type="note" xlink:label="note-0518-01a" xlink:href="note-0518-01"/> -numiſſa, cùm alioquin proiector mann magis <lb/>pondus comprehendat, quà m cùmillud Å¿uÅ¿pen-<lb/>dit? </s> - <s xml:space="preserve">Præterea Å¿ic quidem duo mouet pondera, <lb/>fundæ videlicet, & </s> - <s xml:space="preserve">miſſilis: </s> - <s xml:space="preserve">illo aÅ©t modo Å¿olum <lb/>miſſile. </s> - <s xml:space="preserve">An quia in funda quidem commotum <lb/>miſſile funditur, proijcit: </s> - <s xml:space="preserve">fundam enim circulo <lb/>Å¿ubinde rotans, id iaculatur: </s> - <s xml:space="preserve">ex manu autem à <lb/>quiete eÅ¿t initium: </s> - <s xml:space="preserve">omnia autem cùm in motu <lb/>Å¿unt, quà m cùm quieÅ¿cunt, facilius mouẽtur. </s> - <s xml:space="preserve">An <lb/>& </s> - <s xml:space="preserve">eam ob cauÅ¿am e@t, Å¿ed nec minus etiam, quia <lb/>in fundæ vÅ¿u manus quidem fit centrum: </s> - <s xml:space="preserve">Å¿unda <lb/>vero, quod à centro exit. </s> - <s xml:space="preserve">quanto aÅ©t productius <lb/>Å¿ueritid, quod à cÄ“tro eÅ¿t, tanto citius mouetur. <lb/></s> - <s xml:space="preserve">iactus aÅ«t qui mÄu fit, fundÄ™ reÅ¿pectu breuis eÅ¿t.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="12"> -<note position="left" xlink:label="note-0518-01" xlink:href="note-0518-01a" xml:space="preserve">12</note> -</div> -<p> - <s xml:space="preserve">Cur circa idem iugum maiores collopes faci-<lb/> -<anchor type="note" xlink:label="note-0518-02a" xlink:href="note-0518-02"/> -lius, quà m minores mouentur? </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">item Å¿uculæ, <lb/>quæ graciliores Å¿unt, ab eadẽ viê—̃ craſſiores? </s> - <s xml:space="preserve">An <lb/>quÃa Å¿ecula quidem, & </s> - <s xml:space="preserve">iugum cÄ“trum eÅ¿t: </s> - <s xml:space="preserve">pro-<lb/>minentes autem longitudines eæ, quæ Å¿unt à cÄ“-<lb/>tro. </s> - <s xml:space="preserve">Celerius autem & </s> - <s xml:space="preserve">plus mouentur quæ maio <lb/>rum Å¿unt circulorum ab eadem vi, quà m quÄ™ mi <lb/>norum. </s> - <s xml:space="preserve">ab eadem enim vi plus transfertur id <lb/>extremum, quod longius à centro diÅ¿tat. </s> - <s xml:space="preserve">Quam <lb/>ob rem ad iugum quidem inÅ¿trumenta faciunt <lb/>collopas, quibus facilius verÅ¿ant: </s> - <s xml:space="preserve">in gracilibus <lb/>autem Å¿uculis plus fit id, quod extra lignum e@@. <lb/></s> - <s xml:space="preserve">hoc autem id efficitur, quod à centro exit.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="13"> -<note position="left" xlink:label="note-0518-02" xlink:href="note-0518-02a" xml:space="preserve">13</note> -</div> -<p> - <s xml:space="preserve">Cur eiuÅ¿dem magnitudinis lignum facilius <lb/> -<anchor type="note" xlink:label="note-0518-03a" xlink:href="note-0518-03"/> -genus frangitur, Å¿i quiÅ¿piam æquì diductis mani <lb/>bus extrema comprehendens fregerit, quà m Å¿i <lb/>iuxta genu: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">Å¿i terræ illud applicans pede Å¿u-<lb/>perimpoÅ¿ito manu longè diducta confregerit, ê—̃ <lb/>prope? </s> - <s xml:space="preserve">An quiaibi quidem genu centrÅ© eÅ¿t, hic <lb/>vero ipÅ¿e pes. </s> - <s xml:space="preserve">quanto autem remotius à centro -<pb o="519" file="0519" n="16" rhead="Mechanicæ."/> -Å¿uerit, faciliùs mouetur quodcunq;</s> - <s xml:space="preserve">. moueriau-<lb/>tem quod frangitur, neceſſe eÅ¿t.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="14"> -<note position="left" xlink:label="note-0518-03" xlink:href="note-0518-03a" xml:space="preserve">14</note> -</div> -<p> - <s xml:space="preserve">Cur eæ quæ circa littora appellantur Crocæ, <lb/> -<anchor type="note" xlink:label="note-0519-01a" xlink:href="note-0519-01"/> -rotunda Å¿unt figura, cùm alioquin à principio ex <lb/>magnis Å¿int lapidibus, oÅ¿treiÅ¿ve? </s> - <s xml:space="preserve">An quia ea, q̃ <lb/>plus recedunt à medio in motionibus, feruntur <lb/>celeriùs. </s> - <s xml:space="preserve">medium enim fit centrum: </s> - <s xml:space="preserve">intcruallÅ© <lb/>verò ea, quæ à centro. </s> - <s xml:space="preserve">Sem per autem maior ab <lb/>æquali motione maiorem deÅ¿cribit circulum. <lb/></s> - <s xml:space="preserve">quod autem maius in æquali pertranÅ¿it tẽpore, <lb/>celerius fertur: </s> - <s xml:space="preserve">quæ autem celeriùs ex Ä™ quali fe <lb/>runtur Å¿patio, uehementiùs im petunt: </s> - <s xml:space="preserve">quÄ™ autẽ <lb/>magis impetunt impetuntur & </s> - <s xml:space="preserve">magis. </s> - <s xml:space="preserve">quam ob <lb/>rem ea quæ plus à medio diÅ¿tant, cÅfringi neceÅ¿ <lb/>Å¿e eÅ¿t, id autem cùm patiantur, rotunda fieri eÅ¿t <lb/>neceſſarium. </s> - <s xml:space="preserve">Crocis autem propter maris mo-<lb/>tum, quoniam Å¿imul cum illo agitantur, in perpe <lb/>tua eſſe accidit motione, eoqÌ; </s> - <s xml:space="preserve">verÅ¿atas mÅ Å¿em <lb/>per offenÅ¿are. </s> - <s xml:space="preserve">id autem ipÅ¿is maximè extremis <lb/>contingere partibus eÅ¿t neceſſe.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="15"> -<note position="right" xlink:label="note-0519-01" xlink:href="note-0519-01a" xml:space="preserve">15</note> -</div> -<p> - <s xml:space="preserve">Cur quanto longiora Å¿untligna, tanto imbe-<lb/> -<anchor type="note" xlink:label="note-0519-02a" xlink:href="note-0519-02"/> -cilliora fiunt: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">Å¿i tollantur, in flectuntur magis: <lb/></s> - <s xml:space="preserve">tametÅ¿i quod breue quidem eÅ¿t, ceu bicubitum, <lb/>fuerit tenue: </s> - <s xml:space="preserve">quod verò cubitorum centum, craÅ¿ <lb/>Å¿um? </s> - <s xml:space="preserve">An ê—a & </s> - <s xml:space="preserve">vectis, & </s> - <s xml:space="preserve">onus, & </s> - <s xml:space="preserve">hypomochlion <lb/>in leuando ipÅ¿a fit ligni proceritas: </s> - <s xml:space="preserve">prior nanq; </s> - <s xml:space="preserve"><lb/>illius pars ceu hypomochlion fit, quod verò in <lb/>extremo eÅ¿t, pÅdus. </s> - <s xml:space="preserve">quam ob rem quanto exten. </s> - <s xml:space="preserve"><lb/>Å¿ius fuerit id, quod ab hypocomochlio eÅ¿t, tanto <lb/>inflecti neceſſe ẽ magis. </s> - <s xml:space="preserve">quo enim plus ab hypo-<lb/>mochlio diÅ¿tat, eo magis incuruari neceſſe eÅ¿t. </s> - <s xml:space="preserve"><lb/>neceſſario igitur extrema vectis eleuantur. </s> - <s xml:space="preserve">Siigi <lb/>tur flexilis fuerit vectis, ipÅ¿um inflecti magis <lb/>cùm extollitur neceſſe eÅ¿t: </s> - <s xml:space="preserve">quod lõgis accidit li <lb/>gnis. </s> - <s xml:space="preserve">in breuibus autem quod vltimÅ© eÅ¿t, quie- -<pb o="520" file="0520" n="17" rhead="AriÅ¿t. Quæſtiones"/> -Å¿centi hy pomochlio deprope Å¿it.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="16"> -<note position="right" xlink:label="note-0519-02" xlink:href="note-0519-02a" xml:space="preserve">16</note> -</div> -<p> - <s xml:space="preserve">Cur a paruo exift@nte cuneo magna Å¿cindun-<lb/>tur pondera, & </s> - <s xml:space="preserve">corporum moles, validaque fit <lb/>impreſſio? </s> - <s xml:space="preserve">An quia cuneus duo Å¿unt vectes Å¿ibi <lb/>inuicem contrarij: </s> - <s xml:space="preserve">vterque autem & </s> - <s xml:space="preserve">pondus ha <lb/>b@t, & </s> - <s xml:space="preserve">hypomochlion, quod diuellit, & </s> - <s xml:space="preserve">compri-<lb/>mit. </s> - <s xml:space="preserve">Plagæ quinetiam ipÅ¿ius latio pondus, quod <lb/>percutit & </s> - <s xml:space="preserve">mouet, magnum facit: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">quoniÄ mo <lb/>tum mouet, pÅ¿a celeritate valentius fit. </s> - <s xml:space="preserve">paruo au <lb/>tem exiÅ¿tente vecte, magnæ illum conÅ¿equuntur <lb/>vires: </s> - <s xml:space="preserve">quam ob rem vltra magnitndinis decentiã <lb/>later mouens, Sit cuneus vbi A B C, quod verò <lb/>cuneo Å¿cinditur, D E F G. </s> - <s xml:space="preserve">vectis igitur Å¿it ipÅ¿a <lb/>A B, pondus verò ipÅ¿ius B inferior pars, hypomo <lb/>chlion autem D G, huic autem contrarius vect is <lb/>B C. </s> - <s xml:space="preserve">percuſſa igitur A C, vtroque illorum vtitur <lb/>vecte: </s> - <s xml:space="preserve">Å¿cindit enim ipÅ¿um B.</s> - <s xml:space="preserve"/> -</p> -<p> - <s xml:space="preserve">Cur Å¿i quiÅ¿piam trochleas componens duas in <lb/> -<anchor type="note" xlink:label="note-0520-01a" xlink:href="note-0520-01"/> -tignis duobus ad Å¿e inuicem inuctis cõtrario ad <lb/>trochleas modo circulo funem circunduxerit, <lb/>cuius alterum quidem caput tigonrum appenda <lb/>tur alteri, alterum varò troclheis Å¿it innixum, & </s> - <s xml:space="preserve"><lb/>à funis initio trahere coeperit, magna trahit pon <lb/>dera, licet imbecillium fuerit virium? </s> - <s xml:space="preserve">An quia <lb/>idem põdus à minori potentia Å¿i mouetur, vecte <lb/>medio trãsfertur magis, quà m à manu: </s> - <s xml:space="preserve">trochlea <lb/>autem idem vecti facit. </s> - <s xml:space="preserve">quam ob rem Å¿i vna faci-<lb/>lius traher, & </s> - <s xml:space="preserve">ab vnico tractu multò grauius tra-<lb/>het, quà m facere poſſit manus, idip Å¿um duæ tro-<lb/>chleæ plus quà m in dupla velocitate leuabunt. <lb/></s> - <s xml:space="preserve">Minus enim altera trahit, quà m Å¿i ip Å¿a per Å¿eip-<lb/>Å¿am traheret, quando circa alteram iniectus Å¿ue <lb/>rit funis: </s> - <s xml:space="preserve">illa nanque minus etiam pondus effe-<lb/>cit. </s> - <s xml:space="preserve">PariqÌ; </s> - <s xml:space="preserve">modo Å¿i ad plures iniectus fuerit funis <lb/>in paucis trochleis, multa Å¿it differentia. </s> - <s xml:space="preserve">quã ob -<pb o="521" file="0521" n="18" rhead="Mechanicæ"/> -rem à prima pondere quat@or minas trahente, <lb/>ab vl im a trahi multò minus. </s> - <s xml:space="preserve">Et in re ædifica-<lb/>toria faciliter magn@ a mouen@ pondera, traducÅ©t <lb/>enim ab una trochlea ad aliam, & </s> - <s xml:space="preserve">rurÅ¿us ab illa <lb/>ad Å¿uculas, & </s> - <s xml:space="preserve">vectes. </s> - <s xml:space="preserve">hoc autem idem eÅ¿t, ac Å¿i <lb/>multas facerent trochleas.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="17"> -<note position="left" xlink:label="note-0520-01" xlink:href="note-0520-01a" xml:space="preserve">18</note> -</div> -<p> - <s xml:space="preserve">Cur Å¿i quis Å¿uper lignum magnã imponat Å¿e-<lb/> -<anchor type="note" xlink:label="note-0521-01a" xlink:href="note-0521-01"/> -carim, deÅ¿uperq; </s> - <s xml:space="preserve">illi magnum adijciat pondus, <lb/>ligni quippiam quod curandum Å¿it, non diuidit: <lb/></s> - <s xml:space="preserve">Å¿iverò Å¿ecurim extoll@s percutiat, illud Å¿cindit, <lb/>cùm alioquin multò minus habeat ponderis id <lb/>quod percutit, quà m id quod Å¿uperiacet, & </s> - <s xml:space="preserve">pre-<lb/>mit? </s> - <s xml:space="preserve">An quia omnia cum motu fiunt: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">graue <lb/>ipÅ¿um grauitatis magis aſſumit motum dÅ© mo-<lb/>uetur, ê—̃ dum quieÅ¿cit Incumbensigitur conua-<lb/>tum graui motionem non mouetur, motum ve-<lb/>rò & </s> - <s xml:space="preserve">Å¿ecundum hanc mouetur, & </s> - <s xml:space="preserve">Å¿ecundùm eã <lb/>quæ eÅ¿t percutientis. </s> - <s xml:space="preserve">PrÄ™terea Å¿ecuris ipÅ¿a effici-<lb/>tur cuneus: </s> - <s xml:space="preserve">paruus autẽ exiÅ¿tens cuneus magna <lb/>diuidit cum ex duobus Å¿it vectibus contrario ad <lb/>Å¿eÅ¿e inuicem modo conÅ¿titutis</s> -</p> -<div type="float" level="2" n="18"> -<note position="right" xlink:label="note-0521-01" xlink:href="note-0521-01a" xml:space="preserve">19</note> -</div> -<p> - <s xml:space="preserve">Cur Å¿tatera, qua carnes ponderãtur, paruo ap-<lb/> -<anchor type="note" xlink:label="note-0521-02a" xlink:href="note-0521-02"/> -pendiculo magna trutinat onera, cùm alioquin <lb/>tota dimidiata exiftar libra? </s> - <s xml:space="preserve">vbi enim onus impo-<lb/>nitur, Å¿olùm Å¿uÅ¿penditur lanx: </s> - <s xml:space="preserve">in altera verò par <lb/>te, Å¿ola eÅ¿t Å¿tatera. </s> - <s xml:space="preserve">An quia Å¿imul libram & </s> - <s xml:space="preserve">vectẽ <lb/>ipÅ¿am contingit eſſe Å¿tateram. </s> - <s xml:space="preserve">libram quidem, <lb/>vbi Å¿partorum quodcunque Å¿tateræ fit centrum: <lb/></s> - <s xml:space="preserve">in altera enim parte lancem, in altera autem pro <lb/>lance æquipondij appendiculum habet. </s> - <s xml:space="preserve">quod li-<lb/>bre incumbit, ceu Å¿i quis alterain apponeret lan-<lb/>cem, & </s> - <s xml:space="preserve">ilii pondus imponeret. </s> - <s xml:space="preserve">manifeÅ¿tÅ© enim, <lb/>quòd tantundem tradit ponderis ei quod in alte <lb/>raiacet lance. </s> - <s xml:space="preserve">Quemadmodum autem Å¿i vna li-<lb/>bra multæ Å¿int libræ, Å¿ic talia inÅ¿unt Å¿parta mul.</s> - <s xml:space="preserve"> -<pb o="522" file="0522" n="19" rhead="AriÅ¿t. Quæſtiones"/> -tain eiuÅ¿modi libra: </s> - <s xml:space="preserve">quorum vniuÅ¿cuiuÅ¿q; </s> - <s xml:space="preserve">quod <lb/>intrinÅ¿ecus eÅ¿t ad appẽdiculum, Å¿tateræ eÅ¿t dimi-<lb/>dium. </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">omnino iÅ¿tud libra eÅ¿t, vnam quidẽ ha-<lb/>benslancem, in qua pondus appẽditur: </s> - <s xml:space="preserve">alteram <lb/>verò, vbi in Å¿tatera æquipondium. </s> - <s xml:space="preserve">Quam ob rem <lb/>appendiculum ad alteram Å¿ui partem eÅ¿t Å¿tatera. <lb/></s> - <s xml:space="preserve">huiuÅ¿modi autem exiÅ¿tÄ“s multæ Å¿unt libræ, totqÌ; </s> - <s xml:space="preserve"><lb/>quot fuerint Å¿parta. </s> - <s xml:space="preserve">Semper autem quod lanci <lb/>propinquius eÅ¿t Å¿partum, appenÅ¿oqÌue oneri, ma-<lb/>ius trahit pÅdus: </s> - <s xml:space="preserve">quoniam fit quidem omnis Å¿ta-<lb/>tera inuerÅ¿us vectis: </s> - <s xml:space="preserve">hypomochlion nãq; </s> - <s xml:space="preserve">vnum-<lb/>quodque Å¿partum Å¿uperne exiÅ¿tens: </s> - <s xml:space="preserve">pondus verÅ© <lb/>id quod lanci ineÅ¿t. </s> - <s xml:space="preserve">Quanto autem productior ve <lb/>ctis Å¿uerit longitudo ab ipfo hypomochlio, tãto <lb/>ibi quidem faciliùs mouet: </s> - <s xml:space="preserve">hic autem æquilibri-<lb/>brium facit, pondusqÌue Å¿tateræ trutinat, quod <lb/>ad æquipondij vergit appendiculum.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="19"> -<note position="right" xlink:label="note-0521-02" xlink:href="note-0521-02a" xml:space="preserve">20</note> -</div> -<p> - <s xml:space="preserve">Cur medici Å¿acilius dentes extrahÅ«t dentifor-<lb/>cipis onere adiecto, quà m Å¿i Å¿ola vtantur manu? <lb/></s> - <s xml:space="preserve">An quia ex manu magis quà m ex dentiforcipe <lb/>lubricus elabitur dẽs? </s> - <s xml:space="preserve">An ferro id potius accidit <lb/>quà m digitis, quoniam vndique dentem non cÅ-<lb/>prehendunt: </s> - <s xml:space="preserve">quod mollis digitorum facit caro: </s> - <s xml:space="preserve"><lb/>adhæret enim, & </s> - <s xml:space="preserve">cõplectitur magis. </s> - <s xml:space="preserve">An quia den <lb/>@iÅ¿orceps duo contrarij vectes, vnicum habentes <lb/>hypom ochlion, eius Å¿cilicet inÅ¿trumenti conne-<lb/>xionem. </s> - <s xml:space="preserve">Hoc igitur ad extractionem vtuntur or-<lb/>gano, vt facilius moueant. </s> - <s xml:space="preserve">Sic dentiforcipis alte-<lb/>rum quidem exrremum vbi eÅ¿t A: </s> - <s xml:space="preserve">alterum autẽ <lb/>quod extrahit B. </s> - <s xml:space="preserve">vectis autem vbi A D F: </s> - <s xml:space="preserve">alterve <lb/>rò vectis vbi B C E. </s> - <s xml:space="preserve">hypomochlion autẽ C G D: </s> - <s xml:space="preserve"><lb/>connexio verò vbi G: </s> - <s xml:space="preserve">dẽs autem pondus. </s> - <s xml:space="preserve">Vtroq; </s> - <s xml:space="preserve"><lb/>igitur B & </s> - <s xml:space="preserve">F Å¿imul comprehendentes mouent. </s> - <s xml:space="preserve"><lb/>quando autem commotus fuerit, faciliùs manu <lb/>trahitur, quà m inÅ¿trumento.</s> - <s xml:space="preserve"/> -</p> -<pb o="523" file="0523" n="20" rhead="Mechanicæ"/> -<p> - <s xml:space="preserve">Cur nuces abÅ¿que ictu facilè cõfrigun tinÅ¿tru-<lb/>mẽtis, quæ ad eum fiunt vÅ¿um? </s> - <s xml:space="preserve">multumeinim au <lb/>fertur virium, motionis Å¿cilicet, & </s> - <s xml:space="preserve">violẽ@iæ. </s> - <s xml:space="preserve">PrÄ™-<lb/>terea duro & </s> - <s xml:space="preserve">graui comprimens inÅ¿trumento ci <lb/>tiùs conÅ¿ringet, quà m ligneo & </s> - <s xml:space="preserve">leui. </s> - <s xml:space="preserve">An quia Å¿ic <lb/>vtrinqne à duobus comprimitur vectibus ipÅ¿a <lb/>nux: </s> - <s xml:space="preserve">a vecte autem facile diuelluntur onera. </s> - <s xml:space="preserve">id <lb/>enim inÅ¿trumẽtum ex duobus componitur vecti <lb/>bus idem habentibus hypomochlion, connexio-<lb/>nem videlicet @pÅ¿am vbi eÅ¿t A. </s> - <s xml:space="preserve">Quemadmodum <lb/>igitur fuere diductÄ™ Å¿ecundÅ© extrema motis C D <lb/>ipſæ F E, Å¿ic à parua faciliter potentia conducun <lb/>tur. </s> - <s xml:space="preserve">Quod igitur cum percuſſione Å¿eciſſet põdus, <lb/>id valentiores illa E C, & </s> - <s xml:space="preserve">F D vectes efficiunt. </s> - <s xml:space="preserve">ele <lb/>uatione enim in contrarium ela@i & </s> - <s xml:space="preserve">comprimẽ-<lb/>tes, frangunt vbi eÅ¿t K. </s> - <s xml:space="preserve">Hanc etiam ob cauÅ¿am <lb/>quanto vicinius fuerit K ipÅ¿um A, confringitur <lb/>celerius. </s> - <s xml:space="preserve">quanto enim ab hypomochlio plus di-<lb/>Å¿tat vectis, facilius & </s> - <s xml:space="preserve">plus mouet ab eadẽ poten-<lb/>tia. </s> - <s xml:space="preserve">EÅ¿t igitur A quidem hypomochlion: </s> - <s xml:space="preserve">ipÅ¿a au-<lb/>tem D A F vectis: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">item ipia C A E. </s> - <s xml:space="preserve">Quanto igi-<lb/>tur ipÅ¿um K vicinius Å¿uerit angulo ipÅ¿ius A, tãto <lb/>vicinius fit connexioni vbi eÅ¿t A. </s> - <s xml:space="preserve">hoc autem eÅ¿t <lb/>hypomochlion. </s> - <s xml:space="preserve">Ab eadem igitur potentia appli-<lb/>cante F E plus extolli neceſſe eÅ¿t. </s> - <s xml:space="preserve">Quam ob rem <lb/>quoniã ex contrario eÅ¿t eleuatio, neceſſe eÅ¿t ma-<lb/>gis com primi. </s> - <s xml:space="preserve">quod autem comprimitur magis, <lb/>ci@ius frangitur.</s> - <s xml:space="preserve"/> -</p> -<p> - <s xml:space="preserve">Cur Å¿i duo extrema in rhombo puncta dua-<lb/>bus ferantur lationibus, haudquaquam æquale <lb/>vtrunque eorÅ© pertianÅ¿it rectam, Å¿ed multò plus <lb/>alterum ?</s> - <s xml:space="preserve">idem autem eÅ¿t Å¿ermo, cur quod Å¿uper <lb/>latus fertur, minus pertranÅ¿it, quà m ipÅ¿um latus. <lb/></s> - <s xml:space="preserve">illud enim diametrum minorẽ, hoc verò maius <lb/>latus. </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">hoc quidem vnica, illud verò duabus Å¿er -<pb o="524" file="0524" n="21" rhead="AriÅ¿t Quæſtiones."/> -t<unsure/>ur lationibus. </s> - <s xml:space="preserve">Feratur enim ex ipÅ¿a A B, A qui-<lb/>dem ad ipÅ¿um B, B vero ad ipÅ¿um D eadem cele <lb/>ritate. </s> - <s xml:space="preserve">feratur aurẽ & </s> - <s xml:space="preserve">ipÅ¿a A B in ipÅ¿a A C iuxta <lb/>C D eadem celeritate cum illis. </s> - <s xml:space="preserve">neceſſe igitur eÅ¿t <lb/>A quidem in ipÅ¿a A D diametro ferri, B verò in <lb/>ipÅ¿a B C. </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">vtranq; </s> - <s xml:space="preserve">Å¿imul pertranÅ¿iſſe, & </s> - <s xml:space="preserve">ipÅ¿a@@ <lb/>A B ipÅ¿um la<unsure/>tus A C. </s> - <s xml:space="preserve">latum enim Å¿it ipÅ¿um A ip <lb/>Å¿am A E, A B autem ipÅ¿am A E, & </s> - <s xml:space="preserve">proiecta Å¿it <lb/>F G iuxta ipÅ¿um A B, & </s> - <s xml:space="preserve">ab ipÅ¿o E Å¿imiliter replea <lb/>tur. </s> - <s xml:space="preserve">Simile igitur fit, quod repletum eÅ¿t ipÅ¿i toti. <lb/></s> - <s xml:space="preserve">æqualis igirur A F ipÅ¿i A E ipÅ¿a autem A B ipÅ¿am <lb/>A F lata erit. </s> - <s xml:space="preserve">in diametro igitur erit Å¿ecÅ«dÅ« K. </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve"><lb/>fem per neceſſe eÅ¿t ipÅ¿um ferri Å¿ecundùm diame <lb/>trum. </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">Å¿imul A B latus pertranfit latus A C, & </s> - <s xml:space="preserve"><lb/>ipÅ¿um A diametrum pertranÅ¿it A D. </s> - <s xml:space="preserve">Similiter ẽt <lb/>demon Å¿trabitur & </s> - <s xml:space="preserve">ipÅ¿um B in ipÅ¿a B C diametro <lb/>latum. </s> - <s xml:space="preserve">æqualis enim eÅ¿t ipÅ¿a B E ipÅ¿i B G Reple. </s> - <s xml:space="preserve"><lb/>to igitur ab ipÅ¿o G. </s> - <s xml:space="preserve">quod intus eÅ¿t, toti eÅ¿t Å¿imi-<lb/>le: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">ipÅ¿um B in ipÅ¿a diame@ro erit Å¿ecundùm la <lb/>terum connexionem. </s> - <s xml:space="preserve">Et Å¿imul latus pertranÅ¿it <lb/>latus, & </s> - <s xml:space="preserve">B ipÅ¿um B C diametrum. </s> - <s xml:space="preserve">Simuligitur A <lb/>multo plus ipÅ¿a A B pertrÄÅ¿it: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">ipÅ¿um latus, mi-<lb/>nus latus eadem lata celeritate: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">ipÅ¿um latus <lb/>maiorem, quà m B, pertranÅ¿iuit vna latum latio-<lb/>ne. </s> - <s xml:space="preserve">Quanto enim acutior fuerit rhombus, diame <lb/>ter quidem minor Å¿it. </s> - <s xml:space="preserve">A C autẽ maior: </s> - <s xml:space="preserve">latus ve-<lb/>ro ipÅ¿ius B C minus. </s> - <s xml:space="preserve">AbÅ¿urdÅ« eÅ¿t enim (vt dictÅ©) <lb/>id quod duabus fertur lationibus, aliquãdo ferri <lb/>tardius illo. </s> - <s xml:space="preserve">quod fertur vnica: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">vtriÅ¿q; </s> - <s xml:space="preserve">poÅ¿itis <lb/>æquali velocitate punctis, alterÅ« pertranÅ¿ire ma <lb/>iorem. </s> - <s xml:space="preserve">CauÅ¿a autem eÅ¿t, quoniã ei quod ab obt@ <lb/>Å¿o Å¿ertur angulo ambæ ferè cÅtrariæ fiunt latio-<lb/>n<unsure/>es, & </s> - <s xml:space="preserve">illa Å¿ecÅ«dÅ« quÄ ipÅ¿um fertur, & </s> - <s xml:space="preserve">illa Å¿ecun <lb/>dÅ© quã ipÅ¿um à latere defertur. </s> - <s xml:space="preserve">Ei autẽ quod ab <lb/>@cuto fertur, accidit vt ad idÄ“ feratur, coadiuuat -<pb o="525" file="0525" n="22" rhead="Mechanicæ."/> -@nim quæ ipÅ¿ius eÅ¿t lateris, illã quæ eÅ¿t Å¿uper dia-<lb/>metrum. </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">quanto hunc quidem acutiorem fe-<lb/>ceris, illum verò obtuÅ¿um magis: </s> - <s xml:space="preserve">hæc quidÄ“ tar-<lb/>dior erit, illa verò celerior. </s> - <s xml:space="preserve">Hæ quidẽ igitur ma-<lb/>gis contrariæ fiunt, qmÌ„ obtuÅ¿ior fit angulus: </s> - <s xml:space="preserve">illæ <lb/>verò ad idẽ magis, qmÌ„ lineæ coarctantur: </s> - <s xml:space="preserve">ipÅ¿um <lb/>enim A ferè ad idem fertur Å¿ecundum ambas la <lb/>tiones. </s> - <s xml:space="preserve">Coadiuuatur igitur altera: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">quanto Å¿anè <lb/>acutior fuerit angulus, tanto magis ipÅ¿um A ad <lb/>contrarium: </s> - <s xml:space="preserve">ipÅ¿um. </s> - <s xml:space="preserve">n. </s> - <s xml:space="preserve">ad B fertur: </s> - <s xml:space="preserve">latus autẽ de-<lb/>fertipÅ¿um ad D. </s> - <s xml:space="preserve">Et quanto Å¿anè obtuÅ¿ior fuerit <lb/>angulus, magis contrariæ fiunt lationes: </s> - <s xml:space="preserve">rectior <lb/>enim efficitur linea. </s> - <s xml:space="preserve">Å¿i autẽ omnino recta fieret, <lb/>penitus vtique eſſent contrariæ latus verò Å¿ecun <lb/>dùm vnicam latuum lationem à nullo præpedi-<lb/>tur: </s> - <s xml:space="preserve">rationabiliter igitur maiorem pertranÅ¿it.</s> - <s xml:space="preserve"/> -</p> -<p> - <s xml:space="preserve">Dubitatur quam ob cauÅ¿am maior circulus Ä™-<lb/>qualem minori circulo conuoluit<unsure/>ur lineam, qÅ«<unsure/> <lb/>circa idem centrum fuerint poÅ¿iti: </s> - <s xml:space="preserve">Å¿eorÅ¿um autẽ <lb/>reuoluti, quemadmodum alterius magnitudo ad <lb/>magnitudiuem Å¿e habet alterius, fic & </s> - <s xml:space="preserve">illorÅ© ad <lb/>Å¿einuicem fiunt lineæ. </s> - <s xml:space="preserve">Præterea vno et@ã & </s> - <s xml:space="preserve">eodÄ“ <lb/>v<unsure/>triÅ¿que exiÅ¿tente centro, aliquando quidem tan <lb/>ta Å¿it linea, quam conuoluuntur, quantam minor <lb/>per Å¿e conuoluitur circulus, quandoq; </s> - <s xml:space="preserve">verò quã-<lb/>tam maior. </s> - <s xml:space="preserve">Quòd quidem igitur maiorem con <lb/>uoluitur maior, manifeÅ¿tum eÅ¿t. </s> - <s xml:space="preserve">angulus enim <lb/>Å¿en Å¿u videtur eſſe cuiuÅ¿que circunferentia pro-<lb/>priæ diametri maioris circuli maior, minoris mi <lb/>nor. </s> - <s xml:space="preserve">quam ob rem eandem habebunt propor-<lb/>tionem Å¿ecundùm Å¿enÅ¿um ad Å¿e lineæ. </s> - <s xml:space="preserve">Å¿ecun-<lb/>dùm quas fuerint conuoluti. </s> - <s xml:space="preserve">verùm enimuerò <lb/>quòd etiam æqualem conuoluuntur, quando cir<unsure/> <lb/>t<unsure/>a idem fuerint poÅ¿iti centrum, manifeÅ¿tÅ© eÅ¿t: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve"><lb/>Å¿ic fiunt aliquando quidÄ“ æquales lineæ, Å¿ecun- -<pb o="526" file="0526" n="23" rhead="AriÅ¿t Quæſtiones"/> -dum quam maior conuoluitur circulus, aliquan <lb/>do verò Å¿ecundùm quam minor Sit enim circu-<lb/>lus maior quidem vbi D F C. </s> - <s xml:space="preserve">minor verò vbi <lb/>E C B: </s> - <s xml:space="preserve">vtriÅ¿qÌ; </s> - <s xml:space="preserve">autem centrum A. </s> - <s xml:space="preserve">Et quam qui-<lb/>dem per Å¿e magnus conuoluitur, Å¿it vbi F I: </s> - <s xml:space="preserve">quã <lb/>vero per Å¿e minor vbi GK, æqualis A. </s> - <s xml:space="preserve">Si igitur mi <lb/>norem mouero, idem mouens centÅ« vbi A, ma-<lb/>ior autem Å¿it annexus. </s> - <s xml:space="preserve">qñ igitur A B fuerit recta <lb/>ad ipÅ¿am G K, Å¿imul & </s> - <s xml:space="preserve">A C fit recta ad ipÅ¿am F I. <lb/></s> - <s xml:space="preserve">quam ob rem Ä™qualem Å¿emper tranÅ¿lata erit ip-<lb/>Å¿am quidem G K, vbi eÅ¿t G B circunferentia: </s> - <s xml:space="preserve">ip-<lb/>Å¿am verò FL, quæ eÅ¿t vbi F C. </s> - <s xml:space="preserve">Si aÅ©t quarta pars <lb/>æqualem conuoluitur, manifeÅ¿tum eÅ¿t, ê™ totus <lb/>circulus toti circulo æqualem conuoluetur. </s> - <s xml:space="preserve">qua <lb/>re qñ B C, linea ad ipÅ¿um peruenerit K, & </s> - <s xml:space="preserve">ipÅ¿a <lb/>F C circunferentia erit in ipÅ¿a C L: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">vniuerÅ¿us <lb/>crit conuolutus circulus. </s> - <s xml:space="preserve">SimiliqÌ; </s> - <s xml:space="preserve">modo Å¿i ma-<lb/>gnum mouero, illi paruuÅ« annectens, eodem exi <lb/>ftente cẽtro, Å¿imul cum A C ipÅ¿i A B perpendicu <lb/>lum & </s> - <s xml:space="preserve">recta eri@: </s> - <s xml:space="preserve">hæc quidem ad ipÅ¿am F I, illa <lb/>verò ad G M. </s> - <s xml:space="preserve">Quam ob rem quando hæc quidẽ <lb/>æqualem ipÅ¿i G M pert<unsure/>rãſiuerit, illa verò ipÅ¿i F I, <lb/>& </s> - <s xml:space="preserve">rurÅ¿um facta fuerit recta ipÅ¿a F A ad ipÅ¿am <lb/>F L. </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">ipÅ¿a A C rurÅ¿um recta, velut à principio <lb/>erant in ipÅ¿is M I. </s> - <s xml:space="preserve">Hoc autem neque aliqua in-<lb/>tercedente mora maioris ad minorem, vt. </s> - <s xml:space="preserve">Å¿. </s> - <s xml:space="preserve">per <lb/>aliquod temporis Å¿patium Å¿taret in eodem pun-<lb/>cto: </s> - <s xml:space="preserve">neque tranÅ¿iliente minore aliquod punctÅ«, <lb/>maiorem quidem æqualem minori pertranÅ¿ire, <lb/>hunc a@tem maiori, abÅ¿urdum eÅ¿t. </s> - <s xml:space="preserve">Præterea vni-<lb/>ea etiam Å¿emper exiÅ¿tente motione, centrÅ« mo-<lb/>rum interdum quidem magnam, nonnunquà m <lb/>verò minorem conuerti, admirandum eÅ¿t. </s> - <s xml:space="preserve">Idem <lb/>enim celeritate eadem latus æqualẽ natum hoc <lb/>eft pertranÅ¿ire: </s> - <s xml:space="preserve">eadem autem celeritate vtroque -<pb o="527" file="0527" n="24" rhead="Mechanicæ."/> -modo æqualem licet mouere. </s> - <s xml:space="preserve">Principium autem <lb/>Å¿umendum eÅ¿t circa iÅ¿torum cauÅ¿am, ê™ eadem <lb/>potentia & </s> - <s xml:space="preserve">æqualis hanc quidem tardiùs mouet <lb/>magnitudinem, illam verò celeriùs. </s> - <s xml:space="preserve">Si enim fue-<lb/>rit quippiam, quod à ſeipÅ¿o moueri natÅ© non Å¿it, <lb/>Å¿i Å¿imul & </s> - <s xml:space="preserve">illud mouerit, quod natÅ© eÅ¿t moueri. <lb/></s> - <s xml:space="preserve">tardius mouebitur, quà m Å¿i ipÅ¿um per Å¿e moue-<lb/>retur. </s> - <s xml:space="preserve">Et Å¿i quidem natum fuerit moueri, non Å¿i-<lb/>mul aÅ©t moueatur, Å¿imiliter Å¿e habebit. </s> - <s xml:space="preserve">Et im<unsure/>poÅ¿ <lb/>Å¿ibile certè eÅ¿t plus moueri, quà m mouens: </s> - <s xml:space="preserve">non <lb/>enim Å¿uam ipÅ¿ius mouetur motionem. </s> - <s xml:space="preserve">Si igitur<unsure/> <lb/>circulus maior vbi A, minor autem vbi B, Å¿i mi-<lb/>nor maiorem impellet non reuolutum ex Å¿e, ma<unsure/> <lb/>nifeÅ¿tum eÅ¿t, ê™ tantum ipÅ¿ius rectæ maior pertrã <lb/>Å¿it, quantum eÅ¿t impulÅ¿us: </s> - <s xml:space="preserve">tantum aÅ©t eÅ¿t impul-<lb/>Å¿us, quantum paruus eÅ¿t motus: </s> - <s xml:space="preserve">æqualem igitur <lb/>ipÅ¿ius rectæ pertranÅ¿iuerunt. </s> - <s xml:space="preserve">Neceſſe igitur eÅ¿t <lb/>Å¿i reuolutus minor maiorem impellet, reuol@i Å¿i<unsure/> <lb/>mul cum impulÅ¿ione: </s> - <s xml:space="preserve">tantum aÅ«t quantum mi-<lb/>nor reuolutus eÅ¿t, Å¿i nihil ipÅ¿e Å¿ub ipÅ¿ius motio-<lb/>ne mouetur. </s> - <s xml:space="preserve">QÅ«o enim, & </s> - <s xml:space="preserve">quantum mouit, tan-<lb/>tum motÅ© eſſe neceſſe eÅ¿t, quod mouetur ab ill@ <lb/>Å¿ed profectò paruus circulus tãtum Å¿eipÅ¿um cir-<lb/>culariter mouit, quantum eÅ¿t pedalis quantitas: </s> - <s xml:space="preserve"><lb/>(tantum enim Å¿it id quod motus eÅ¿t) & </s> - <s xml:space="preserve">magnus <lb/>igitur tantum motus erit. </s> - <s xml:space="preserve">SimiliqÌ; </s> - <s xml:space="preserve">modo Å¿i ma-<lb/>gnus paruÅ« mouebit, motus erit paruus, quÄ“ad-<lb/>modum maior. </s> - <s xml:space="preserve">Per Å¿e aÅ«t motus illorum vtrum-<lb/>libet Å¿iue celeriter, Å¿eu tardè eadem velocitate, <lb/>Å¿tatim quantum maior natus eÅ¿t circunÅ¿erri li-<lb/>neam: </s> - <s xml:space="preserve">quod difficultatÄ“ facit, quod non Å¿imiliter <lb/>Å¿aciunt, quando fuerint connexi. </s> - <s xml:space="preserve">hoc autem eÅ¿t, <lb/>Å¿i alteri ab altero moueatur, non quã natus eÅ¿t, <lb/>@eque peculiarem motionem. </s> - <s xml:space="preserve">Nihil enim referr <lb/>citcÅ©ponere, & </s> - <s xml:space="preserve">annectere, aut cÅiungere vtrum- -<pb o="528" file="0528" n="25" rhead="AriÅ¿t QueÅ¿tiones"/> -liber alteri. </s> - <s xml:space="preserve">Å¿imiliter enim quando hic quidem <lb/>mouer, ille vero mouetur ab iÅ¿to; </s> - <s xml:space="preserve">quantum vtiq; <lb/></s> - <s xml:space="preserve">mouerit alter, tantum alter mouebitur. </s> - <s xml:space="preserve">Quando <lb/>quidem igitur adiacens mouerit, aut propenÅ¿us, <lb/>non Å¿emper conueluitur: </s> - <s xml:space="preserve">quÄdo verò circa idem <lb/>poÅ¿i@i fuerint cẽtrum, alterum ab altero Å¿emper <lb/>conuolui neceſſe eÅ¿t. </s> - <s xml:space="preserve">Sed nihilominus non Å¿uam <lb/>ipÅ¿ius motionem mouetur alter, Å¿ed veluti nullã <lb/>haberet motionem: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">Å¿i habuerit, illa autem nõ <lb/>vtatur, tan@undem accidit. </s> - <s xml:space="preserve">Quandoquidem igi-<lb/>tur mà gnus moueri Å¿ibi alligatum paruum, par-<lb/>uus mouetur quãtum ille: </s> - <s xml:space="preserve">quÄdo autem paruus, <lb/>rurÅ¿us magnus quantum iÅ¿te: </s> - <s xml:space="preserve">Å¿eparatus aÅ©t vter-<lb/>que Å¿eipÅ¿um mouet. </s> - <s xml:space="preserve">Quod autem eodem exiÅ¿ten <lb/>te centro, & </s> - <s xml:space="preserve">mouente eadem velocitate, accidit <lb/>inæqualem illos pertrÄÅ¿ire lineam, paralogiÅ¿mo <lb/>fophiÅ¿ticÄ™ vtitur is, qui dubitat. </s> - <s xml:space="preserve">idem enim am-<lb/>bobus eÅ¿t centrÅ©, verùm per accidens, velúti mu <lb/>Å¿icum & </s> - <s xml:space="preserve">album. </s> - <s xml:space="preserve">eſſe enim vtriuÅ¿que circuli cen-<lb/>tro nÅ eodem vtitur. </s> - <s xml:space="preserve">Quandoquidem igitur mo-<lb/>uens fuerit paruus, vt illius centrum & </s> - <s xml:space="preserve">princi-<lb/>pium: </s> - <s xml:space="preserve">quando verò magnus, vt illius. </s> - <s xml:space="preserve">non igitur <lb/>idem Å¿im pliciter mouet, Å¿ed eÅ¿t quomodo.</s> - <s xml:space="preserve"/> -</p> -<p> - <s xml:space="preserve">Cur lect@lorum Å¿pondas Å¿ecÅ«dum duplam fa-<lb/> -<anchor type="note" xlink:label="note-0528-01a" xlink:href="note-0528-01"/> -ciunt proportionem, hanc quidem Å¿ex pedÅ« vel <lb/>paulo am pliorem, illam vero trium: </s> - <s xml:space="preserve">curve non <lb/>Å¿ecundùm diametrum illos reÅ¿tibus extendunt? <lb/></s> - <s xml:space="preserve">An tantos quidem magnitudine faciunt, vt cor-<lb/>poribus Å¿int proportionem habẽtes: </s> - <s xml:space="preserve">fiunt enim <lb/>Å¿ic Å¿ecundum Å¿pondas dupli, longitudine quidẽ <lb/>cubitorum quatuor, latitudine vero duorÅ©. </s> - <s xml:space="preserve">Ex-<lb/>tendunt autem illos nÅ Å¿ecundùm diametrum, <lb/>Å¿ed ex oppoÅ¿ito, vt & </s> - <s xml:space="preserve">ligna minus diÅ¿tra hantur. </s> - <s xml:space="preserve"><lb/>celertrimè enim Å¿cindÅ©tur Å¿ecundum naturam <lb/>diuiÅ¿a, & </s> - <s xml:space="preserve">eodem modo diÅ¿tenta laborant ma- -<pb o="529" file="0529" n="26" rhead="Mechanicæ."/> -@imè. </s> - <s xml:space="preserve">Amplius, quoniam opus eÅ¿t, vt reÅ¿tes pon-<lb/>dus ferre poſſint, Å¿ic certè pondere impoÅ¿ito mi-<lb/>nus laborabunt, Å¿i tranÅ¿uerÅ¿im, quà m Å¿i obliquè <lb/>extendantur. </s> - <s xml:space="preserve">Præterea hoc etiam modo minus <lb/>abÅ¿umitur reÅ¿tium. </s> - <s xml:space="preserve">Sit enim lectulus A F G K, & </s> - <s xml:space="preserve"><lb/>bifariam diuidatur ipÅ¿a F G Å¿ecundùm B, æqua-<lb/>lia certè foramina Å¿unt in ipÅ¿a F B, & </s> - <s xml:space="preserve">in ipÅ¿a F A. <lb/></s> - <s xml:space="preserve">latera enim Å¿unt æqualia. </s> - <s xml:space="preserve">nam totum F G duplÅ© <lb/>eÅ¿t. </s> - <s xml:space="preserve">Extendunt autem, vt deÅ¿criptum eÅ¿t, ab ipÅ¿o <lb/>A ad ipÅ¿um B ita vbi eÅ¿t C, ita vbi D: </s> - <s xml:space="preserve">ita vbi H, <lb/>poÅ¿tea vbi E, & </s> - <s xml:space="preserve">eodem Å¿emper modo, donec ad <lb/>angulum peruenerint alium. </s> - <s xml:space="preserve">Duo enim anguli <lb/>reÅ¿tis habent capi a. </s> - <s xml:space="preserve">æquales autem Å¿unt reÅ¿tes <lb/>Å¿ecundum curuaturas, videlicet A B; </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">B C ipÅ¿is <lb/>C D, & </s> - <s xml:space="preserve">D H. </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">aliæ Å¿imili Å¿e habet modo, quo-<lb/>niam eadem demonÅ¿tratio. </s> - <s xml:space="preserve">ipÅ¿a enim AB æqua-<lb/>lis eÅ¿t ipÅ¿i H E: </s> - <s xml:space="preserve">æqualia enim Å¿unt latera Å¿patij <lb/>B G, M A, & </s> - <s xml:space="preserve">foramina æquè diÅ¿tant, ipÅ¿a autem <lb/>B G æquelis eÅ¿t ipÅ¿i M A: </s> - <s xml:space="preserve">angulus enim B æqua-<lb/>lis eÅ¿t angulo G: </s> - <s xml:space="preserve">in æqualibus enim hic quidem <lb/>intus, ille vero extra. </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">B quidem eÅ¿t Å¿emirectus: </s> - <s xml:space="preserve"><lb/>eÅ¿t enim F B æqualis ipÅ¿i F A: </s> - <s xml:space="preserve">& </s> - <s xml:space="preserve">angulus vbi F, re <lb/>ctus eÅ¿t. </s> - <s xml:space="preserve">B autẽ angulus æqualis ei vbi eÅ¿t G, quo-<lb/>niam quadratum altera parte longius duplÅ© eÅ¿t: </s> - <s xml:space="preserve"><lb/>& </s> - <s xml:space="preserve">ad medium eÅ¿t curuatura. </s> - <s xml:space="preserve">quamobrẽ A D ip-<lb/>Å¿i E G eÅ¿t æqualis: </s> - <s xml:space="preserve">huic verò ipÅ¿a H M. </s> - <s xml:space="preserve">SimiliqÌue <lb/>modo demonÅ¿trantur aliæ, quoniã æquales Å¿unt <lb/>duæ, quæ Å¿ecÅ©dùm curuataras Å¿unt, duabus. </s> - <s xml:space="preserve">Qua <lb/>re manifeÅ¿tum eÅ¿t, ê™ tot Å¿unt reÅ¿tes in lectulo, <lb/>quot Å¿unt quatuor, Å¿icut A B. </s> - <s xml:space="preserve">Quanta autem fora <lb/>minÅ© eÅ¿t multirudo in ipÅ¿o F G latere, & </s> - <s xml:space="preserve">in eius <lb/>dimidio F B eÅ¿t medietas. </s> - <s xml:space="preserve">Quamobrem in dimi-<lb/>diato lectulo tantæ reÅ¿tium magnitudines erÅ©t, <lb/>quantum eÅ¿t A B: </s> - <s xml:space="preserve">multitudine verò tot, quot in <lb/>B G Å¿unt foramina. </s> - <s xml:space="preserve">Hoc autem nihil refert dice -<pb o="530" file="0530" n="27" rhead="AriÅ¿t. Quæſtiones"/> -re, quà m quot Å¿unt in ipÅ¿is A F & </s> - <s xml:space="preserve">B F Å¿imul <lb/>Å¿umpris. </s> - <s xml:space="preserve">Si autem Å¿ecundum diametrum exten-<lb/>dantur reÅ¿tes, quemad modum Å¿e habet in lectu-<lb/>lo A B C D, dimidia non tot Å¿unt, quot ambo-<lb/>rum latera F A F G. </s> - <s xml:space="preserve">æqualia autem, quot in ip <lb/>fis F B, F A, Å¿unt foramina: </s> - <s xml:space="preserve">maiores autem Å¿unt</s> -</p> -<div type="float" level="2" n="20"> -<note position="left" xlink:label="note-0528-01" xlink:href="note-0528-01a" xml:space="preserve">25</note> -</div> -<p> - <s xml:space="preserve">ipſæ A F, B F, quæ exiÅ¿tentes, quà m A B, <lb/>quare reÅ¿tis in tantum maior, quantum ambola <lb/>@era diame@ro Å¿unt maiora.</s> - <s xml:space="preserve"/> -</p> -<p> - <s xml:space="preserve">Cur difficilius eÅ¿t longa ligna ab extremo Å¿u-<lb/> -<anchor type="note" xlink:label="note-0530-01a" xlink:href="note-0530-01"/> -per humeros ferre, quà m Å¿ecundum medium, æ-<lb/>quali exiÅ¿tente pondere? </s> - <s xml:space="preserve">An quia vibrato ligno <lb/>ipÅ¿um extremum prohibet ferre vibratione ma-<lb/>gis retrahens lationem? </s> - <s xml:space="preserve">An quoniam licèt nihil <lb/>inflectatur, neque multam habeat longitudinẽ, <lb/>difficilius tamen ad ferendum eÅ¿t ab extremo, <lb/>quoniam faciliùs ex medio eleuatur, quà m ab <lb/>extremo, & </s> - <s xml:space="preserve">ideo Å¿ic ferre eÅ¿t facilius: </s> - <s xml:space="preserve">CauÅ¿a aÅ©t, <lb/>qmÌ„ Å¿ecundùm medium quidem eleuato ligno, <lb/>Å¿emper Å¿eÅ¿e inuicem Å¿uÅ¿pendunt extrema, & </s> - <s xml:space="preserve">al-<lb/>tera pars alteram bene Å¿ubleuat medium enim <lb/>veluti centrum fit, vbi habet is, qui eleuat, aut <lb/>Å¿ert. </s> - <s xml:space="preserve">ExtremorÅ© igitur v@@rÅ©q; </s> - <s xml:space="preserve">deorÅ¿um vergens, <lb/>Å¿urÅ¿um Å¿uÅ¿penditur. </s> - <s xml:space="preserve">ê™ Å¿i ab extremo eleuetur, <lb/>aut feratur, non Å¿anè facit: </s> - <s xml:space="preserve">Å¿ed vniuerÅ¿um pon-<lb/>dus ad vnum vergit medium, quò eleuatur, aut <lb/>fertur. </s> - <s xml:space="preserve">Sit medium vbi A, extrema B C. </s> - <s xml:space="preserve">Eleuatio <lb/>igitur aut portato Å¿ecundùm A, ipÅ¿um qÌuidem B <lb/>deorÅ¿um @utans, Å¿urÅ¿um eleuat C: </s> - <s xml:space="preserve">ipÅ¿um autem <lb/>C deorÅ¿um nutans, B Å¿urÅ¿um eleuat: </s> - <s xml:space="preserve">ambo au-<lb/>tem Å¿urÅ¿um eleuata hoc fac@unt.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="21"> -<note position="left" xlink:label="note-0530-01" xlink:href="note-0530-01a" xml:space="preserve">26</note> -</div> -<p> - <s xml:space="preserve">Cur Å¿i valde procerum fuerit idẽ pondus diffi-<lb/> -<anchor type="note" xlink:label="note-0530-02a" xlink:href="note-0530-02"/> -cilius Å¿uper humeros geÅ¿tatur, etiam Å¿i medium <lb/>quiÅ¿piam illud ferat, quà m Å¿i breuius Å¿it? </s> - <s xml:space="preserve">Quod <lb/>@nim dudum dictum eÅ¿t, cauÅ¿a nõ eÅ¿t: </s> - <s xml:space="preserve">Å¿ed vibra- -<pb o="531" file="0531" n="28" rhead="MechanicÄ™."/> -tio nunc eÅ¿t cauÅ¿a. </s> - <s xml:space="preserve">qñ. </s> - <s xml:space="preserve">n. </s> - <s xml:space="preserve">productius fuerit, vibran <lb/>tur extrema: </s> - <s xml:space="preserve">quamobrem contingit portantem <lb/>difficilius geÅ¿tare. </s> - <s xml:space="preserve">Vibrationis aÅ©t cã eÅ¿t, quoniã <lb/>ab eadẽ motione magis transferuntur extrema, <lb/>quanto procerius fuerit lignÅ©. </s> - <s xml:space="preserve">Humerus quidẽ <lb/>fit centrÅ© vbi A: </s> - <s xml:space="preserve">manet. </s> - <s xml:space="preserve">n. </s> - <s xml:space="preserve">is. </s> - <s xml:space="preserve">ipſæ aÅ©t AB, AC, quÄ™ <lb/>Å¿unt ex centro. </s> - <s xml:space="preserve">quanto aÅ©t maius fuerit id quod <lb/>ex centro eÅ¿t, Å¿iue A B, Å¿eu A C, plus transfertur <lb/>Å¿patij. </s> - <s xml:space="preserve">DemonÅ¿tratum autem eÅ¿t hoc prius.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="22"> -<note position="left" xlink:label="note-0530-02" xlink:href="note-0530-02a" xml:space="preserve">27</note> -</div> -<p> - <s xml:space="preserve">Cur iuxta puteos celonia faciunt eo qui viÅ¿un <lb/> -<anchor type="note" xlink:label="note-0531-01a" xlink:href="note-0531-01"/> -tur modo: </s> - <s xml:space="preserve">ligno. </s> - <s xml:space="preserve">n. </s> - <s xml:space="preserve">plumbi adiÅ©gunt pondus: </s> - <s xml:space="preserve">cÅ© <lb/>alioquin vas ipÅ¿um & </s> - <s xml:space="preserve">plenum, & </s> - <s xml:space="preserve">vacuum pon-<lb/>dus habeat? </s> - <s xml:space="preserve">An qmÌ„ duobus tem poribus hau@ien <lb/>di, diuiÅ¿o opere, (intingere. </s> - <s xml:space="preserve">n. </s> - <s xml:space="preserve">oportet, & </s> - <s xml:space="preserve">id Å¿ur-<lb/>Å¿um trahere) continget demittere quidẽ vacuum <lb/>faciliter, trahere vero plenum difficulter. </s> - <s xml:space="preserve">CÅmo-<lb/>dum igitur eÅ¿t paulò tardius illud demittere, cÅ« <lb/>multò leuius effectum Å¿uÅ¿tolatur pondus. </s> - <s xml:space="preserve">id autẽ <lb/>facit in extremo celonio adiunctum plumbum, <lb/>aut lapis. </s> - <s xml:space="preserve">demittenti quidem maius Å¿it pondus, ê— <lb/>fi Å¿olummodo vacuum oportet demittere: </s> - <s xml:space="preserve">cum <lb/>verò plenum fuerit, Å¿urÅ¿um id rapit plumbum, <lb/>aut quicquid ponderis inerit. </s> - <s xml:space="preserve">Quam ob rem faci <lb/>liora hoc modo ambo Å¿unt, quà m illo.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="23"> -<note position="right" xlink:label="note-0531-01" xlink:href="note-0531-01a" xml:space="preserve">28</note> -</div> -<p> - <s xml:space="preserve">Cur qñ Å¿uper ligno, aut hmõi quopiam duo <lb/> -<anchor type="note" xlink:label="note-0531-02a" xlink:href="note-0531-02"/> -portauerint homines æquale pondus, non Å¿imili <lb/>ter premuntur, Å¿i ad vnum non declinet pondus, <lb/>Å¿eu magis, quanto vicinius fuerit geÅ¿tãtibus? </s> - <s xml:space="preserve">An <lb/>quoniam vectis quidem lignum efficitur, põdus <lb/>verò hypomochlion: </s> - <s xml:space="preserve">qui autem propior eÅ¿t pon-<lb/>deri ex ijs, qui illud geÅ¿tant, id quod mouetur: </s> - <s xml:space="preserve">al <lb/>ter verò portantiÅ©, quod mouet. </s> - <s xml:space="preserve">Quanto igitur <lb/>plus diÅ¿tat à pondere, tanto facilius mouet, & </s> - <s xml:space="preserve"><lb/>alerum premit magis inferius, veluti contrani-<lb/>@te pondere impoÅ¿ito, quod hypomochlion -<pb o="532" file="0532" n="29" rhead="AriÅ¿t. Quæſtiones."/> -factum EÅ¿t. </s> - <s xml:space="preserve">Si autem in medio inerit pondus, ni-<lb/>hilo magis alter alteri fit pondus, aut mouet: </s> - <s xml:space="preserve">Å¿ed <lb/>eodem modo alteri alt<unsure/>er fit pondus.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="24"> -<note position="right" xlink:label="note-0531-02" xlink:href="note-0531-02a" xml:space="preserve">29</note> -</div> -<p> - <s xml:space="preserve">Cur Å¿urgentes omnes, femori crus ad acutum <lb/> -<anchor type="note" xlink:label="note-0532-01a" xlink:href="note-0532-01"/> -cõſtituentes angulÅ©, & </s> - <s xml:space="preserve">thoraci Å¿imiliter femur, <lb/>Å¿urgunt? </s> - <s xml:space="preserve">quod Å¿i non, haudquaquam Å¿urgere po-<lb/>terunt. </s> - <s xml:space="preserve">An quia id quod æquale eÅ¿t, quietis vbiq; <lb/></s> - <s xml:space="preserve">eÅ¿t cauÅ¿a: </s> - <s xml:space="preserve">rectus autẽ angulus æqualitatis eÅ¿t, Å¿ta-<lb/>tionemqÌ; </s> - <s xml:space="preserve">facit: </s> - <s xml:space="preserve">quamobrẽ ad Å¿imiles fertur an-<lb/>gulos ipÅ¿i terræ circumferentiæ: </s> - <s xml:space="preserve">non enim quod <lb/>ad rectum eÅ¿t ipÅ¿i pauimento. </s> - <s xml:space="preserve">An q̇mÌ„ Å¿urgens fit <lb/>rectus, Å¿tantẽ vero neceſſe eÅ¿t perpẽdiculum eſſe <lb/>ad terram. </s> - <s xml:space="preserve">Å¿iquidẽ igitur ad rectÅ© debet eſſe, hoc <lb/>aÅ©t eÅ¿t caput Å¿ecundÅ© pedes habere, & </s> - <s xml:space="preserve">fieri opor <lb/>tet, cùm Å¿urgit. </s> - <s xml:space="preserve">Quandoquidem igitur fuerit @e <lb/>dens, Å¿ecundum parallelum pedes habet, & </s> - <s xml:space="preserve">ca-<lb/>put, & </s> - <s xml:space="preserve">non in æquali. </s> - <s xml:space="preserve">Caput Å¿it A, thorax A B, fe-<lb/>mur B C, crura C D. </s> - <s xml:space="preserve">ad rectum autem Å¿it & </s> - <s xml:space="preserve">tho-<lb/>rãx vbi A B, ipÅ¿i femori, & </s> - <s xml:space="preserve">cruri Å¿emur Å¿ic feden-<lb/>te. </s> - <s xml:space="preserve">quamobrem eo Å¿e habẽtem modo Å¿urgere eÅ¿t <lb/>impoſſibile. </s> - <s xml:space="preserve">neceſſe autem eÅ¿t crus reclinare, pe-<lb/>desque conÅ¿tituere Å¿ub capite. </s> - <s xml:space="preserve">hoc autem erit, Å¿i <lb/>C D fiet, vbi C F, & </s> - <s xml:space="preserve">fimul Å¿urgere continget, & </s> - <s xml:space="preserve">in <lb/>eadem Ä™quali habere caput & </s> - <s xml:space="preserve">pedes. </s> - <s xml:space="preserve">ipÅ¿a autem <lb/>C F a cutum facit angulum ad ipÅ¿am B C.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="25"> -<note position="left" xlink:label="note-0532-01" xlink:href="note-0532-01a" xml:space="preserve">30</note> -</div> -<p> - <s xml:space="preserve">Cur facilius mouetur cõmotu, quà m manens? <lb/></s> - <s xml:space="preserve"> -<anchor type="note" xlink:label="note-0532-02a" xlink:href="note-0532-02"/> -veluti currus citiùs cõmotus agitant, quà m mo-<lb/>ueri in cipientes. </s> - <s xml:space="preserve">An quia difficillimÅ© eÅ¿t pondus <lb/>mouere, quod in cõtrariÅ© mouetur. </s> - <s xml:space="preserve">aufert enim <lb/>quiddam ex motoris potentia, licèt multo fit ve-<lb/>locior: </s> - <s xml:space="preserve">neceſſe nanque eÅ¿t tardiorẽ eſſe impulÅ¿io <lb/>nem illius quod repellitur. </s> - <s xml:space="preserve">Secundo autẽloco Å¿i <lb/>quieuerit: </s> - <s xml:space="preserve">reÅ¿iÅ¿tit enim ipÅ¿um quieÅ¿cẽs. </s> - <s xml:space="preserve">Quod aÅ©t <lb/>mouetur ad id ipÅ¿um ad quod impellitur, impel-<lb/>lenti Å¿imile facit, ceu Å¿i quiÅ¿piam mouẽtis poten -<pb o="533" file="0533" n="30" rhead="De Mundo."/> -tiam, & </s> - <s xml:space="preserve">celeritatem augeret. </s> - <s xml:space="preserve">quod enim<unsure/> ab illo <lb/>pateretur, vtique ipÅ¿um tacit ex Å¿e commotum.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="26"> -<note position="left" xlink:label="note-0532-02" xlink:href="note-0532-02a" xml:space="preserve">31</note> -</div> -<p> - <s xml:space="preserve">Cur ea, quæ proijci@@ntur, ceſſant à latione? </s> - <s xml:space="preserve">An <lb/> -<anchor type="note" xlink:label="note-0533-01a" xlink:href="note-0533-01"/> -quia impellens deÅ¿init potentia, v@l propter retra <lb/>ctionem, vel propter rei proiectÄ™ inclinationem, <lb/>quando ea valentior fuerit, quam proijcientis vi <lb/>res. </s> - <s xml:space="preserve">Aut iÅ¿thæc ambigere, principium relinquen-<lb/>tes, abÅ¿urdum eÅ¿t.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="27"> -<note position="right" xlink:label="note-0533-01" xlink:href="note-0533-01a" xml:space="preserve">32</note> -</div> -<p> - <s xml:space="preserve">Cur quippiam non pecularem Å¿ibi fertur latio <lb/> -<anchor type="note" xlink:label="note-0533-02a" xlink:href="note-0533-02"/> -nem, impulfore alioquin non conÅ¿equente? </s> - <s xml:space="preserve">An <lb/>videlicet quoniam primum id efficit, @@al@erum <lb/>impellat: </s> - <s xml:space="preserve">illudque rurÅ¿um, v@ alterum? </s> - <s xml:space="preserve">Ceſſat au-<lb/>tem quando non poteÅ¿t amplius facere primum <lb/>impellens id quod fertur, vt impellat, & </s> - <s xml:space="preserve">qñ ip-<lb/>Å¿ius lati grauitas nut@ Å¿uo declinat magis, quà m <lb/>impellentis in antè Å¿it potentia,</s> -</p> -<div type="float" level="2" n="28"> -<note position="right" xlink:label="note-0533-02" xlink:href="note-0533-02a" xml:space="preserve">33</note> -</div> -<p> - <s xml:space="preserve">Cur neque parua valde, neque magna longè <lb/> -<anchor type="note" xlink:label="note-0533-03a" xlink:href="note-0533-03"/> -proijci queunt, Å¿ed commẽſurationem quandã <lb/>illa habere oportet ad id quod proijcit? </s> - <s xml:space="preserve">An quia <lb/>neceſſe eÅ¿t quod proijcitur, & </s> - <s xml:space="preserve">impelli@@r, cõtra-<lb/>ni i ei vnde impellitur? </s> - <s xml:space="preserve">quod autem magnitudi-<lb/>ne Å¿ua nihil cedit, aut imbellicitate nihil contra-<lb/>nititur non efficit proiectionem, neque impulÅ¿io <lb/>nem. </s> - <s xml:space="preserve">Quod enim multo impellentis excedit vi-<lb/>res, haud quaquã cedit: </s> - <s xml:space="preserve">quod verò multo eÅ¿t im-<lb/>berillius. </s> - <s xml:space="preserve">nihil contranititur. </s> - <s xml:space="preserve">An quia tãtum fer-<lb/>tur id quod fertur, quantÅ© aeris mouerit ad pro-<lb/>fundum: </s> - <s xml:space="preserve">quod autem non mouetur, neq; </s> - <s xml:space="preserve">moue <lb/>bit quippiam accidit autem illis ambo iÅ¿thæc ha <lb/>bere. </s> - <s xml:space="preserve">Valde enim magnum, & </s> - <s xml:space="preserve">valde paruum, ceù <lb/>non mota exiÅ¿tunt: </s> - <s xml:space="preserve">alterum nanque nihil mouet <lb/>alterum verò nihil mouetur.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="29"> -<note position="right" xlink:label="note-0533-03" xlink:href="note-0533-03a" xml:space="preserve">34</note> -</div> -<p> - <s xml:space="preserve">Cur ea quæ in vorticoÅ¿is feruntur aquis, ad <lb/> -<anchor type="note" xlink:label="note-0533-04a" xlink:href="note-0533-04"/> -medium tandem aguntur omnia? </s> - <s xml:space="preserve">An quia ma-<lb/>gnitudinem habet quocunque ferrur: </s> - <s xml:space="preserve">quam -<pb o="534" file="0534" n="31" rhead="AriÅ¿t. Quæſtiones Mech."/> -ob rem illius extrema in duobus Å¿unt circulis, <lb/>hoc quidem minori illo verò maiori: </s> - <s xml:space="preserve">quare ma-<lb/>ior diÅ¿trahit: </s> - <s xml:space="preserve">quoniã celerius fertur, & </s> - <s xml:space="preserve">tranÅ¿uer-<lb/>Å¿um impellit illud ad minorem: </s> - <s xml:space="preserve">quoniam autem <lb/>id quod Å¿ertur, latitudinem habet, & </s> - <s xml:space="preserve">iÅ¿te rurÅ¿us <lb/>idem eff@cit & </s> - <s xml:space="preserve">ad interiorem propellit, donec ad <lb/>medium perueniat. </s> - <s xml:space="preserve">An quia quod fertur, Å¿imili <lb/>Å¿e habet modo ad omnes circulos propter me-<lb/>dium. </s> - <s xml:space="preserve">medium enim in vnoquoque circulo Ä™qua <lb/>liter diÅ¿tat. </s> - <s xml:space="preserve">An quia quorum quidem circÅ©auctæ <lb/>quæ latio non Å¿uperior propter magnitudinem, <lb/>Å¿ed grauitate Å¿ua circuli celeritatẽ excellunt, ea <lb/>neceſſe eÅ¿t relinqui, & </s> - <s xml:space="preserve">tardiùs ferri, tardius aÅ©t <lb/>minor circulus fertur. </s> - <s xml:space="preserve">non idem enim in tempo-<lb/>re æquali magnus cÅ© paruo reuoluitur circulus, <lb/>quando circa idem Å¿uerint medium. </s> - <s xml:space="preserve">quamobrẽ <lb/>in minori circulo relinqui eÅ¿t neceſſe, donec ad <lb/>medium perueniant. </s> - <s xml:space="preserve">Quorum cunque autem Å¿u-<lb/>perior à principio fuerit latio, ea finiensidem eÅ¿ <lb/>ficiet. </s> - <s xml:space="preserve">oportet enim @ãc quidem Å¿tatim, alterum <lb/>verò celeritate Å¿uperare grauitatem, quam obrẽ <lb/>ad interiorẽ Å¿emper circulum relinquetur quod-<lb/>cunque. </s> - <s xml:space="preserve">Neceſſe enim eÅ¿t, quod non Å¿uperatur, <lb/>aut in exteriori, aut in interiori moueri: </s> - <s xml:space="preserve">in illo <lb/>autem in quo eÅ¿t, impoſſibile eÅ¿t ferri, quod non <lb/>Å¿uperatur adhuc vero minus in exteriori: </s> - <s xml:space="preserve">cele-<lb/>rior enim exterioris circuli eÅ¿t latio: </s> - <s xml:space="preserve">reÅ¿tat igitur, <lb/>vtid, quod non Å¿uperatur, ad interiorem tranÅ¿-<lb/>feratur. </s> - <s xml:space="preserve">Å¿emper autem vnumquodque proficit, vt <lb/>non Å¿uperetur. </s> - <s xml:space="preserve">Quoniam verò peruenire ad me-<lb/>dium, finem quidem efficit, vt quippiam nõ mo-<lb/>ueatur, Å¿tat autem Å¿olummodo ipÅ¿um centrum: <lb/></s> - <s xml:space="preserve">ad hoc Å¿anè omnia congregari neceſſe eÅ¿t.</s> - <s xml:space="preserve"/> -</p> -<div type="float" level="2" n="30"> -<note position="right" xlink:label="note-0533-04" xlink:href="note-0533-04a" xml:space="preserve">35</note> -</div> -</div> -<div type="section" level="1" n="5"> -<head style="it" xml:space="preserve">Mechanicarum quæſtionum finis.</head> - </div></text> -</echo> \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-xml/examples/Benedetti_1585-Ids.xml Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,15419 +0,0 @@ -<?xml version="1.0" encoding="utf-8"?><echo xmlns="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" xmlns:dcterms="http://purl.org/dc/terms" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:xhtml="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.0"> - <metadata> - <dcterms:identifier>ECHO:163127KK.xml</dcterms:identifier> - <dcterms:creator identifier="GND:118859676 LCCN:nr89-8408 VIAF:88720955">Benedetti, Giovanni Battista de</dcterms:creator> - <dcterms:title xml:lang="la">Diversarum Speculationum mathematicum, & physicarum liber</dcterms:title> - <dcterms:language xsi:type="dcterms:ISO639-3">lat</dcterms:language> - <dcterms:date xsi:type="dcterms:W3CDTF">1585</dcterms:date> - <dcterms:rights>open access</dcterms:rights> - <dcterms:license>http://echo.mpiwg-berlin.mpg.de/policy/oa_basics/declaration</dcterms:license> - <dcterms:accessRights>free</dcterms:accessRights> - <echolink>echo.mpiwg-berlin.mpg.de?titleID=163127KK</echolink> - </metadata> - <text type="book" xml:lang="la"> - <div type="front" level="1" n="1"> - <div type="cover" level="2" n="1"> - <pb n="1" file="0001" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0001"/> - <pb n="2" file="0002" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0002"/> - <pb n="3" file="0003" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0003"/> - <pb n="4" file="0004" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0004"/> - <pb n="5" file="0005" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0005"/> - </div> - <div type="title" level="2" n="1"> - <head xml:id="head1" xml:space="preserve">IO. BAPTISTAE <lb/>BENEDICTI <lb/>Patritij Veneti PhiloÅ¿ophi.</head> - <head xml:id="head2" xml:space="preserve"><emph style="it">DIVERSARVM SPECVLATIONVM <lb/>Mathematicarum, & PhyÅ¿icarum</emph> <lb/>Liber.</head> - <head xml:id="head3" xml:space="preserve">Quarum Å¿eriem Å¿equens pagina indicabit.</head> - <head xml:id="head4" xml:space="preserve">AD SERENISSIMVM CAROLVM EMANVELEM <lb/>ALLOBROGVM, ET SVBALPINORVM <lb/>DVCEM INVICTISSIMVM.</head> - <figure position="here"> - <image file="0005-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0005-01"/> - </figure> - <head xml:id="head5" xml:space="preserve"><emph style="sc">Tavrini</emph>, Apud Hæredem Nicolai Beuilaquæ, <num value="1585" style="sc">mdlxxxv</num>. <lb/><emph style="it">Superioribus permiſſum</emph>.</head> - <pb n="6" file="0006" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0006"/> - </div> - <div type="toc" level="2" n="1"> - <head xml:id="head6" style="it" xml:space="preserve">TRACTATVS QVI IN HOC <lb/>volumine continentur.</head> - <head xml:id="head7" xml:space="preserve">Theoremata Arithmetica.</head> - <head xml:id="head8" xml:space="preserve">Derationibus operationum perÅ¿pectiuæ.</head> - <head xml:id="head9" xml:space="preserve">De Mechanicis.</head> - <head xml:id="head10" xml:space="preserve">DiÅ¿putationes de quibuÅ¿dam placitis AriÅ¿t.</head> - <head xml:id="head11" xml:space="preserve">In quintum Euclidis librum.</head> - <head xml:id="head12" xml:space="preserve">PhyÅ¿ica, & Mathematica reÅ¿ponÅ¿a per EpiÅ¿tolas.</head> - <pb n="7" file="0007" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0007"/> - </div> - <div type="dedication" level="2" n="1"> - <head xml:id="head13" xml:space="preserve">SERENISSIMO <lb/>CAROLO EMANVELI <lb/>Sabaudiæ Duci, <reg norm="&c." type="unresolved">&c.</reg></head> - <p style="it"> - <s xml:id="s1" xml:space="preserve"><emph style="sc it">AGitvr</emph> nonusdecimus annus ex quo litte-<lb/>ris Serenißimi patris tuæ CelÅ¿itudinis, ac-<lb/>cerÅ¿itus ex vrbe ParmenÅ¿i in banc me ciui-<lb/>tatem contuli. </s> - <s xml:id="s2" xml:space="preserve">Is aduenientem tam bumanè <lb/>excepit, tanta deinde liberalitate fuit com-<lb/>plexus ego vicißim ei deÅ¿eruiendi, tam vebe-<lb/>menti cupiditate fui accenÅ¿us, vt Å¿ub eius ditione quodÅ¿uper-<lb/>eßet vitæ agere conÅ¿tituerem. </s> - <s xml:id="s3" xml:space="preserve">Cuius in me benignitas, mea <lb/>in illum obÅ¿eruantia mirum in modum mutuo vÅ¿u, & conÅ¿ue-<lb/>tudine eÅ¿t adaucta, vt idem Dux me Å¿ecum dum ruÅ¿ticaretur <lb/>eße vellet, ſæpè etiam Å¿ecum pernoctare; </s> - <s xml:id="s4" xml:space="preserve">quo quidem tempo-<lb/>re de Matbematicis Å¿cientijs mecum agebat, in quibus perdi-<lb/>Å¿cendis mea opera vtebatur, quæſtiones, Arithmeticam, GeoÂ<lb/>metriam, Opticen, MuÅ¿icam, aut Astrologiam Å¿pectantes <lb/>proponens. </s> - <s xml:id="s5" xml:space="preserve">Cui vt quod in me eßet Å¿atisfacerem, acrius <lb/>quà m anteainea studia (adquætamen Å¿emper fui propenÅ¿ißi-<lb/>mus) incubui. </s> - <s xml:id="s6" xml:space="preserve"><reg norm="Illiusque" type="simple">IlliusqÌê«</reg> imitatione (vt ferècæteri Principum <lb/>studiaimitantur) non pauci aut præſentes, aut per litter as me <lb/>de his, atque illis Mathematicis quæstionibus conÅ¿uluerunt. <lb/></s> - <s xml:id="s7" xml:space="preserve">Cùmque ego nunquam laborem amicorum cauÅ¿a defugerim, <lb/>euenit vt post tot annorum curricula, mea Å¿crinia Å¿crutatus, <lb/>inuenerim tot abÅ¿olutas quæſtiones, vt ex eis corpus mediocre <lb/>effici poſſe videretur. </s> - <s xml:id="s8" xml:space="preserve">Quas, cùm rationibus in epiÅ¿tola Å¿ub-<lb/>Å¿equenti allatis edere constituiſſem, non Å¿ub cuiuÅ¿que alte-<lb/>rius nomine, & auÅ¿picijs quam tuæ CelÅ¿itudinis volui apparere; <lb/></s> - <s xml:id="s9" xml:space="preserve">tum quòd patri debitum libellum filio reddere par erat, tum - <pb n="8" file="0008" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0008"/> - quòd in tuæ CelÅ¿itudine paternam in me fouendo, & augendo<unsure/> <lb/>benignit atem ineße Å¿emper Å¿um expertus, tum quòd tuæ CelÅ¿i-<lb/>tudinis interrog ationibus excitatus non pauca quæ hoc volu-<lb/>mine continentur, elucubraui. </s> - <s xml:id="s10" xml:space="preserve">Acceßit, quod ego Å¿emper in <lb/>his dedic ationibus Å¿pectandum put aui, tuam CelÅ¿itudinem t<unsure/>an-<lb/>tos progreßus in Mathematicis feciſſe, vt vel idonea æſtima-<lb/>trix mearum vigiliarum eſſe poßit. </s> - <s xml:id="s11" xml:space="preserve">Quare, & veterum Per-<lb/>Å¿arum Regum gloriam æquauit, & nos veluti in Å¿pem certam <lb/>fælicitatis buius ſæculi induxit, Å¿i verum eÅ¿t Platonis va-<lb/>ticinium, beat am eam futuram Rempublic am in qua <lb/>Principes PhiloÅ¿ophentur. </s> - <s xml:id="s12" xml:space="preserve">Tua igitur celÅ¿i-<lb/>tudo libellum tot ei nominibus debitum, <lb/>ea qua Å¿olet bumanitate accipe-<lb/>re nè grauetur. </s> - <s xml:id="s13" xml:space="preserve">Deus tuas <lb/>omnes cogitationes, <lb/>& conatus ad <lb/>fÅ“licißi-<lb/>mos <lb/>Å¿emper exitus perducat, <lb/><reg norm="teque" type="simple">teqÌê«</reg> diutißimè Å¿er-<lb/>uet incolu-<lb/>mem.</s> - </p> - <pb n="9" file="0009" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0009"/> - </div> - <div type="preface" level="2" n="1"> - <head xml:id="head14" xml:space="preserve">AD LECTOREM</head> - <p> - <s xml:id="s14" xml:space="preserve"><emph style="sc">CVm</emph> Varijs temporibus permulta in diuerÅ¿is <lb/>diÅ¿ciplinis contemplatus Å¿im, partim à præ-<lb/>Å¿tantibus viris patronis ac amicis meis exci-<lb/>tatus, quiÅ¿uper eis Å¿ententiam meam exquire-<lb/>bant, partim, abingenito mihi deÅ¿iderio, ra-<lb/>tionem, & cauÅ¿am eorum percipiendi, com-<lb/>mittendum non putaui, quin qualiacunque <lb/>meaÅ¿cripta in illis Å¿cientijs, Å¿tudioÅ¿is impartirer, <lb/>non dubitans quin illis aliquid commodi atque vtilitatis allatura Å¿int, prÄ™ <lb/>Å¿ertim cum in eiuÅ¿modi quæſtionibus inueÅ¿tigandis atque perpendendis, <lb/>nemo ( quod Å¿ciam ) hactenus elaborauerit. </s> - <s xml:id="s15" xml:space="preserve">Nihil enim his libris à me <lb/>traditum eÅ¿t, quod aut legiſſe, aut ab alijs audiuiſſe meminerim, nam Å¿i <lb/>aliena attigi, ea, aut cum aliqua differentia demonÅ¿trationis, aut diluci-<lb/>dius Å¿cripÅ¿i, quod Å¿i forte alius eadem tradidit, aut eius lucubrationes ad <lb/>me non peruenerunt, aut earum perlectionis memoria excidit. </s> - <s xml:id="s16" xml:space="preserve">Vtenim <lb/>etiam AriÅ¿toteles ipÅ¿e Å¿enÅ¿it facilè fieri poteÅ¿t, vt pluribus, eædem opinio-<lb/>nes in mentem veniant. </s> - <s xml:id="s17" xml:space="preserve">Immo multa Å¿cribenti euenire poteÅ¿t, vt cum <lb/>iamdiu aliquid Å¿cripÅ¿erit, iam oblitus, idem repetat, quod mihi etiam <lb/>nonnunquam accidit. </s> - <s xml:id="s18" xml:space="preserve">In his autemlibris non Å¿uÅ¿cepi munus integræ ali <lb/>cuius Å¿cientiæ tradendæ, ne, quæ abalijs iam tradita Å¿unt, ipÅ¿e inutiliter re <lb/>peterem, mihiqÌue viderer exalienis laboribus laudem voluiſſe comparare. <lb/></s> - <s xml:id="s19" xml:space="preserve">Singularum enim Å¿cientiarum volumina, iam ab alijs collecta, at-<lb/>que in ordinemÅ¿unt digeÅ¿ta, & Å¿i pauciſſimi Å¿int libri quorum omnes <lb/>Å¿ententiæ, omniaqÌue inuenta vnius Å¿int authoris, excipio Archime-<lb/>dis volumina. </s> - <s xml:id="s20" xml:space="preserve">Cumque multi Å¿int, qui vel vnam rem à ſe inuentam <lb/>in publicum proferre non dubitent, multo magis mihi qui multa ex-<lb/>cogitaui, & Å¿i inter Å¿e hætereogenea, atque vtcunque expreſſa, idem <lb/>licere Å¿um arbitratus. </s> - <s xml:id="s21" xml:space="preserve">In his autem meditandis, ex Arithmeticis autho-<lb/>ribus quos inÅ¿pexi, præcipuus fuit Nicolaus Tartalea, quippe quem fe-<lb/>rè omnia ab alijs Å¿cripta collegiſſe conÅ¿tat, nec alios ex præcipuis, quos le-<lb/>gere potui omittendos duxi, inter quos Å¿unt Hieronymus Cardanus, Mi-<lb/>chael Stifelius, Gemma FriÅ¿ius, Ioannes Nouiomagus, Cuthebertus <lb/>TonÅ¿tallus, <reg norm="cæterique" type="simple">cæteriqÌ;</reg> huiuÅ¿inodi. </s> - <s xml:id="s22" xml:space="preserve">Quorundam tamen volumina illorum <lb/>qui à Tartalea citantur, vt Leonardi PiÅ¿ani, ProÅ¿docimi, Ioannis Infor-<lb/>tunati, Fratris Lucæ, Petri Borgi, aliorumqÌue aliquot inÅ¿piciendorum, - <pb n="10" file="0010" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0010"/> - facultas mihi non fuit. </s> - <s xml:id="s23" xml:space="preserve">Præterea, licet in his libris nonnullÄ™ inueniantur <lb/>propoÅ¿itiones, quæ diÅ¿iunctam ab alijs habeant rationem, eæ non Å¿per-<lb/>nendæ tamen Å¿unt, viam fortaſſe alicui aperient vlterius progrediendi. <lb/></s> - <s xml:id="s24" xml:space="preserve">Quemadmodum enim, exempli gratia, ex Å¿ub contraria coni Å¿ectione, <lb/>Å¿umpta poÅ¿tea fuit diuina illa Planisferijdelineation, quæ Å¿ub Ptolomæi no-<lb/>mine legitur, & Å¿icuti ex <ref id="ref-0001">penultima primi Euclidis</ref>, quam Pythagoras <lb/>excogitauit propè innumeræ pulchræ conÅ¿equentiæ in AÅ¿tronomia, in <lb/>Architectura, in <reg norm="multisque" type="simple">multisqÌ;</reg> alijs Å¿cientijs deÅ¿umptæ Å¿unt, immo quemad-<lb/>modum ex Å¿ingulis propoÅ¿itionibus à noÅ¿tris maioribus excogitatis mul-<lb/>ta egregia Å¿unt deducta, ita fortaſſe continget, vt ex mearum muentio-<lb/>num aliqua, <reg norm="nonnihil" type="context">nõnihil</reg> in poÅ¿terum vtilitatis deÅ¿umatur. </s> - <s xml:id="s25" xml:space="preserve">Si quid verò, hic in-<lb/>ueneris, quod tuo genio non arrideat, illa prudentiſſimi hominis Å¿en-<lb/>tentia in mentem veniat. </s> - <s xml:id="s26" xml:space="preserve"><emph style="it">Quot capita, tot Å¿ententiæ</emph>, ac per raro con-<lb/>tingere, vt idem omnibus probari, atque placere queat, & perdifficulter <lb/>inueniri hominem cui placeant omnia quæ alteri Å¿atisfaciunt. </s> - <s xml:id="s27" xml:space="preserve">Nec te mo <lb/>ueat, quodhÄ™c Theoremata Å¿iue excogitationes non videas ordine illo di-<lb/>Å¿poÅ¿itas, quo collocari debere exiÅ¿timaueris, tum in Arithmeticis, tum in <lb/>cæteris. </s> - <s xml:id="s28" xml:space="preserve">Cum enim in huiuÅ¿modi rebus ordo non Å¿it neceſſarr<unsure/>us, vi-<lb/>Å¿um eÅ¿t mihi poſſe me, Å¿ine repræhenÅ¿ione, illum negligere, cum Å¿pe-<lb/>culationi, Å¿iue inuentioni preÄ™cipuè adeo mihi incumbendum decreuerim <lb/>vtin collocatione operam ponere, & tempus abÅ¿umere operæpretium <lb/>non duxerim, quod idem in epiÅ¿tolarum collocatione feci, in quibus per-<lb/>Å¿onarum ad quas Å¿cribo nullus ferè graduum ordo Å¿eruatus eÅ¿t, nec tem-<lb/>poris, quo Å¿unt Å¿criptæ, quæſitorum tantummodo ratione habita. </s> - <s xml:id="s29" xml:space="preserve">Nec <lb/>admirari quenquam velim, quod in Å¿peculandis numerorum paſſioni-<lb/>bus, figuris vtar geometricis, ita enim in <ref id="ref-0002">.2. libr.</ref> fecit Euclides, qui mo-<lb/>dus, eo magis mihi arridet, quo minus eÅ¿t abÅ¿tractus, <emph style="it">quoniam oportet in-<lb/>telligentem phantaÅ¿mata Å¿peculari</emph>, cum prÄ™terea perÅ¿picuum Å¿it, diÅ¿cretum <lb/>omne, ex continui diuiÅ¿ione aliquo modo oriri, Å¿iue actu, Å¿iue potentia. <lb/></s> - <s xml:id="s30" xml:space="preserve">Deinde Å¿i forte meis in deinonÅ¿trationibus tibi videbor aliquando bre-<lb/>uior, illud in cauÅ¿a fuiſſe Å¿cias, quod ibi ad viros Å¿cribebam in his diÅ¿cipli-<lb/>nis exercitatos, quibus Å¿atis fuit rem Å¿ignificare. </s> - <s xml:id="s31" xml:space="preserve">Libuit autem mihi om-<lb/>nes voluminis Arithmetici propoÅ¿itiones potius vocabulo theorema-<lb/>tum appellare, quam problematum, quia pars earum Å¿peculatiua tan-<lb/>tum mea eÅ¿t, & Å¿i ex varijs eiuÅ¿modi propoÅ¿itionibus etiam operatiuam <lb/>adinuenerim. </s> - <s xml:id="s32" xml:space="preserve">Quoniam verò multis in locis accidit, vt veritatis iudi-<lb/>candæ cauÅ¿a neceſſe mihi fuerit quorundam Å¿ententijs aduerÅ¿ari nolim te - <pb n="11" file="0011" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0011"/> - hoc mihi vitio tribuere, <reg norm="meque" type="simple">meqÌ;</reg> hoc nomine carptorem <reg norm="maledicumque" type="simple">maledicumqÌ;</reg> ha-<lb/>bere quod alienos errores aperiam, cum potius habenda Å¿it mihi gratia, <lb/>quod in ijs interdum laborans (quÄ™ AntiÅ¿thenes in diÅ¿ciplinis magis ne-<lb/>ceſſaria eſſe dixit, <emph style="it">vt mala Å¿cilicet prius dediÅ¿cantur<sic comment="closing parenthesis in italics">)</sic></emph> falÅ¿as opiniones euel-<lb/>lere Å¿tudeam, <reg norm="veritatemque" type="simple">veritatemqÌ;</reg> oÅ¿tendere, quam omnis philoÅ¿ophus, AriÅ¿to-<lb/>telis exemplo, pluris quam cuiuÅ¿uis hominis authoritatem, aut gratiam <lb/>facere debet. </s> - <s xml:id="s33" xml:space="preserve">CumqÌue in hoc volumine aliquid eiuÅ¿modi legeris <lb/>te oratum volo, vt in iudicando, affectum omnem exuas, <lb/>SalluÅ¿tianum illud præ oculis habens. </s> - <s xml:id="s34" style="it" xml:space="preserve">Omnes qui dere-<lb/>bus <reg norm="dubijs" type="lig">dubijs</reg> conÅ¿ultant, ab odio amicitia, ira, atque <lb/>miÅ¿ericordia vacuos eſſe decet. </s> - <s xml:id="s35" xml:space="preserve">Hinc fiet, vt <lb/>non perÅ¿onæ (vt multiÅ¿olent) Å¿ed <lb/>veritati, quÄ™ Å¿ummo Å¿tudio di-<lb/>gniſſima eÅ¿t, Å¿emper po <lb/>tius faueas. </s> - <s xml:id="s36" xml:space="preserve">Vale <lb/>noÅ¿trisqÌue <lb/>labo-<lb/>ribus vtere, Å¿i quem inde fructum, <lb/>Å¿icuti Å¿pero tuleris, illi præ-<lb/>cipuè habeas gratiam à <lb/>quo omnes fluunt <lb/>Å¿cientiæ.</s> - </p> - <pb n="12" file="0012" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0012"/> - <pb n="13" file="0013" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0013"/> - </div> - </div> - <div type="body" level="1" n="1"> - <div type="chapter" level="2" n="1"> - <head xml:id="head15" xml:space="preserve">IO. BAPTISTAE <lb/>BENEDICTI <lb/>PATRITII VENETI <lb/>SERENISS. CAR. EM. <lb/>ALLOBROGVM DVCIS <lb/>PHILOSOPHI.</head> - <head xml:id="head16" style="it" xml:space="preserve">Theoremata Arithmetica.</head> - <p> - <s xml:id="s37" xml:space="preserve"><emph style="sc">PRaeclare</emph> multa veteres mathematici philoÅ¿ophi de nuÂ<lb/>meris eorumqÌue effectibus excogitata poÅ¿teris tradide-<lb/>runt, quorum cum vix vllam rationem reddiderint, aut <lb/>certè per exiguam, occaÅ¿ione diuerÅ¿orum problematum <lb/>mihi à Sereniſſimo Sabaudiæ Duce propoÅ¿itorum præbi-<lb/>ta, de ijs quæ ab antiquis propoÅ¿ita fuerunt contemplanda <lb/>nonnulla occurrerunt, quæ poÅ¿teritati comendare non <lb/>inutile arbitratus fum, ne hæ meæ cogitationes intercide-<lb/>rent, & occaÅ¿ionem præberem quamplurimis abÅ¿truÅ¿a hÄ™c <lb/>indagandi, quæ problematibus & thæorematibus inuoluta, vix aliquem qui euol-<lb/>ueret nacta funt.</s> - </p> - <p> - <s xml:id="s38" xml:space="preserve">Inter cætera vero à me queÅ¿ita, hoc fuit theorema.</s> - </p> - <div type="math:theorem" level="3" n="1"> - <head xml:id="head17" xml:space="preserve">THEOREMA PRIMVM.</head> - <p> - <s xml:id="s39" xml:space="preserve"><emph style="sc">INterrogavit</emph> me Sereniſſimus Dux Sabaudiæ, qua ratione cognoÅ¿ci poÅ¿-<lb/>Å¿et Å¿cientificè & Å¿peculatiue (vt dicitur) productum ex duobus fractis numeris, <lb/>quolibet producentium minus eſſe. </s> - <s xml:id="s40" xml:space="preserve">Cui reÅ¿pondi, mente & cogitatione conci-<lb/>piendum eſſe fractos producentes cum fractis productis, non vnius eiuÅ¿demqÌue na-<lb/>turæ eſſe, imò longè diuerfæ.</s> - </p> - <p> - <s xml:id="s41" xml:space="preserve">Exempli gratia, fractis numeris propofitis <var>.a.i.</var> et <var>.a.c.</var> quorum integri Å¿int <var>.a.<lb/>b.</var> et <var>.a.d.</var> qui tanquam lineæ cogitentur, apertum fanè eſſet productum <var>.c.i.</var> fu-<lb/>perficiale futurum, quod nomen caperet à producto Å¿uperficiali <var>.d.b.</var> generato ex <lb/>vno in aliud totorum linearium, nam Å¿i conÅ¿titueretur <var>.a.i.</var> octauum ipÅ¿ius <var>.a.b.</var> et <var>.a.<lb/>c.</var> dimidium <var>.a.d.</var> multiplicato <var>.a.i.</var> cum <var>.a.c.</var> produceretur fextumdecimum ipÅ¿ius <var>.<lb/>d.b</var>. </s> - <s xml:id="s42" xml:space="preserve">Quare <var>.d.b.</var> eſſet totum <reg norm="relatiuum" type="context">relatiuÅ©</reg> ipÅ¿ius <var>.c.i.</var> non aliquod totum producentium. <lb/></s> - <s xml:id="s43" xml:space="preserve">Mirum itaque non eÅ¿t Å¿i productum <var>.c.i.</var> minus videatur fuis producentibus, cum <lb/>toto, diuerſæ naturæ à primis conferatur, fractum fiquidem ab integro eiuÅ¿dem <lb/>naturæ, linearis, Å¿uperficialis, aut corporeæ denominatur.</s> - </p> - <p> - <s xml:id="s44" xml:space="preserve">Quòd Å¿i amplioris cognitionis gratia ex Å¿cientiæ præceptis Å¿peculari voluerit a@ - <pb o="2" rhead="IO. BAPT. BENED." n="14" file="0014" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0014"/> - quis, qua ratione fractus numerus <var>.c.i.</var> minor Å¿it in Å¿uo integro <var>.d.b.</var> fracto <var>.a.i.</var> in <lb/>Å¿uo integro <var>.a.b.</var> aut fracto <var>.a.c.</var> in Å¿uo integro <var>.a.d.</var> conÅ¿ideret is quo pacto pro-<lb/>portio <var>.c.i.</var> ad <var>.d.b.</var> minor Å¿it proportione <var>.a.i.</var> ad <var>.a.b.</var> et <var>.a.c.</var> ad <var>.a.d.</var> hac ratione. </s> - <s xml:id="s45" xml:space="preserve">Ma-<lb/>nifeÅ¿tum eÅ¿t ex <ref id="ref-0003">prima Å¿exti de quantitate <lb/>continua</ref>, aut <ref id="ref-0004">.18. Å¿eptimi Euclidis</ref> de diÅ¿cre <lb/> - <anchor type="figure" xlink:label="fig-0014-01a" xlink:href="fig-0014-01"/> - ta, proportionem ipÅ¿ius <var>.d.i.</var> ad <var>.d.b.</var> eſſe Å¿i-<lb/>cut <var>.a.i.</var> ad <var>.a.b.</var> & cum <var>.c.i.</var> minor Å¿it <var>.d.i.</var> <lb/>velut pars Å¿uo toto, proportio, <var>c.i.</var> ad <var>.d.b.</var> <lb/>minor erit proportione <var>.d.i.</var> ad <var>.d.b.</var> ex .8. <lb/>quinti, </s> - <s xml:id="s46" xml:space="preserve">quare minor erit pariter proportio-<lb/>ne <var>.a.i.</var> ad <var>.a.b.</var> ex <ref id="ref-0005">.12. <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg></ref> vnà etiam pro-<lb/>portio <var>.c.i.</var> ad <var>.d.b.</var> minor erit <var>.a.c.</var> ad <var>.a.d.</var> <lb/>ex eiÅ¿dem cauÅ¿is, medio <var>.c.b</var>. </s> - <s xml:id="s47" xml:space="preserve">Ex quibus pa-<lb/>tet ratio, cur fracti diuerÅ¿arum denomina-<lb/>tionum ad vnicam reducantur. </s> - <s xml:id="s48" xml:space="preserve">Cur etiam <lb/>numeros integros in partes fractis Å¿imiles <lb/>frangere liceat, quæ omnia ex Å¿ubÅ¿equenti <lb/>figura facilè cognoÅ¿ci poſſunt.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0014-01" xlink:href="fig-0014-01a"> - <image file="0014-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0014-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="2"> - <head xml:id="head18" xml:space="preserve">THEOREMA <num value="2">II</num>.</head> - <p> - <s xml:id="s49" xml:space="preserve"><emph style="sc">QVae</emph> Å¿it ratio, cur hi, qui numeros, fractos diuerÅ¿arum denominationum col-<lb/>ligere volunt, & in Å¿ummam redigere, multiplicent vnum ex numerantibus <lb/>per denominatorem alterius, & poÅ¿tmodum denominatores adinuicem, quorum <lb/>vltimum productum, commune eÅ¿t denominans duorum priorum productorum, <lb/>quæ collecta in Å¿ummam efficiunt quod quærebatur.</s> - </p> - <p> - <s xml:id="s50" xml:space="preserve">Qua in re Å¿ciendum eÅ¿t, denominantes conÅ¿iderari tanquam partes vnius <reg norm="eiuÅ¿dem- qÌue" type="context">eiuÅ¿dẽ-<lb/>qÌue</reg> magnitudinis quantitatis continuæ, linearum (verbigratia) <var>a.b.</var> et <var>.a.d.</var> <reg norm="æqualium" type="context">æqualiÅ©</reg> <lb/>in longitudine, <reg norm="quarum" type="context">quarÅ©</reg> <var>.a.b.</var> in quatuor partes diuidatur, et <var>.a.d.</var> in tres. </s> - <s xml:id="s51" xml:space="preserve">Quare Å¿i colli-<lb/>gere voluerimus duo tertia cum tribus quartis, multiplicabimus <var>.a.c.</var> duo tertia, <lb/>cum <var>.a.b.</var> diuiÅ¿a in 4. partes, produceturqÌue <var>.c.b.</var> octo partium Å¿uperficialium, de-<lb/>hinc multiplicando <var>.a.i.</var> tres quartas cum <var>.a.d.</var> diuiÅ¿a in .3. partes producetur <var>.i.d.</var> pri <lb/>mis Å¿ingulis æqualis, nouem partium Å¿uper <lb/>ficialium, multiplicata deinde <var>a.b.</var> diui-<lb/> - <anchor type="figure" xlink:label="fig-0014-02a" xlink:href="fig-0014-02"/> - Å¿a in .4. partes per <var>.a.d.</var> in .3. diuiÅ¿a, produ-<lb/>cetur quadratum <var>.d.b.</var> in continuo, in 12. <lb/>partes diuiÅ¿um, quod erit totum commune <lb/>Å¿ingulis productis, quorum primum erat <var>.c.<lb/>b</var>. </s> - <s xml:id="s52" xml:space="preserve">Quare <var>.c.b.</var> ita Å¿e habet ad totum <var>.d.b.</var> Å¿i-<lb/>cut <var>.a.c.</var> ad <var>.a.d.</var> ex prima Å¿exti in continuis, <lb/>aut .18. Å¿eptimi in diÅ¿cretis quantitatibus, <lb/>et <var>.d.i.</var> ad <var>.d.b.</var> Å¿icut <var>.a.i.</var> ad <var>.a.b.</var> ex eiÅ¿dem <lb/>propoÅ¿itionibus. </s> - <s xml:id="s53" xml:space="preserve">Collectis deinde parti-<lb/>bus producti <var>.c.b.</var> cum partibus producti <var>.<lb/>d.i.</var> manifeÅ¿tè depræhendetur eiuÅ¿modi <lb/>Å¿ummam componi ex partibus vnius totius <lb/>communis Å¿ingulis earum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0014-02" xlink:href="fig-0014-02a"> - <image file="0014-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0014-02"/> - </figure> - </div> - <pb o="3" rhead="THEOR. ARITH." n="15" file="0015" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0015"/> - </div> - <div type="math:theorem" level="3" n="3"> - <head xml:id="head19" xml:space="preserve">THEOREMA <num value="3">III</num>.</head> - <p> - <s xml:id="s54" xml:space="preserve"><emph style="sc">CVr</emph> reperturi qualis Å¿it fractus aliquis numerus reÅ¿pectu alterius; </s> - <s xml:id="s55" xml:space="preserve">multiplicare <lb/>debeant numeratores adinuicem & ita etiam denominatores, ex quo produ-<lb/>ctum ex numeratoribus nomen capiat à producto denominatorum.</s> - </p> - <p> - <s xml:id="s56" xml:space="preserve">Huius Å¿i cauÅ¿am noÅ¿ce vis, Å¿ume <var>.o.i.</var> & <var>.o.u.</var> pro totis denominatoribus, tum <var>.o.e.</var> <lb/>& <var>.o.a.</var> pro numeratoribus (exempli cauÅ¿a) Å¿it <var>.o.i.</var> Å¿enarius <var>.o.u.</var> quaternarius <var>.o.e.</var> <lb/>quinarius <var>.o.a.</var> ternarius. </s> - <s xml:id="s57" xml:space="preserve">Si noÅ¿ce vis quæ Å¿int tres quartÄ™ partes quinque Å¿extarum, <lb/>patet ex regulis practicis oriri quindecim vigeÅ¿imaÅ¿quartas. </s> - <s xml:id="s58" xml:space="preserve">Id quomodo fiat, ex <lb/>Å¿ubÅ¿cripta Å¿igura depræhendetur, memores tamen eſſe oportet, quodlibet <reg norm="productum" type="context">productÅ©</reg> <lb/>conÅ¿iderari <reg norm="tanquam" type="context">tanquã</reg> Å¿uperficiem, producentia <reg norm="autem" type="context">autẽ</reg> tan-<lb/>quam lineas. </s> - <s xml:id="s59" xml:space="preserve">In hac igitur Å¿igura productum ex totis <lb/> - <anchor type="figure" xlink:label="fig-0015-01a" xlink:href="fig-0015-01"/> - linearibus eÅ¿t <var>.u.i.</var> aggregatum ex .24. partibus, & <var>.u.e.</var> <lb/>productum aggregatum ex .20. </s> - <s xml:id="s60" xml:space="preserve">Quodita Å¿e habebit <lb/>ad productum totale <var>.u.i.</var> Å¿icut <var>.o.e.</var> ad <var>o.i.</var> ex prima <lb/>Å¿exti aut .18. Å¿eptimi, ita <var>.u.e.</var> erunt quinque Å¿extæ par <lb/>tes <var>.u.i.</var> quarum in propoÅ¿ito exemplo, tres quartæ <lb/><reg norm="quæruntur" type="context">quærÅ©tur</reg>. </s> - <s xml:id="s61" xml:space="preserve">Si <reg norm="itaque" type="simple">itaq;</reg> multiplicabitur <var>.o.e.</var> <reg norm="cum" type="context">cÅ©</reg> <var>.o.a.</var> orietur <lb/>productum <var>.a.e.</var> ita <reg norm="proportionatum" type="context">proportionatÅ©</reg> ad <var>.u.e.</var> Å¿icut <var>.o.a.</var> ad <lb/><var>o.u.</var> reperitur, ex prædictis rationibus. </s> - <s xml:id="s62" xml:space="preserve">Quòd Å¿i <reg norm="Å¿tatutum" type="context">Å¿tatutÅ©</reg> <lb/>eÅ¿t <var>.o.a.</var> tres quartas partes eſſe ipÅ¿ius <var>.u.o.</var> <reg norm="etiam" type="context">etiã</reg> <var>.a.e.</var> tres <lb/>quartæ partes <reg norm="erunt" type="context">erÅ©t</reg> <var>.u.e.</var> Å¿ed <var>.u.e.</var> quinque Å¿extæ Å¿unt ip-<lb/>Å¿ius <var>.u.i.</var> ex quo Å¿equitur bonum eſſe huiuÅ¿modi opus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0015-01" xlink:href="fig-0015-01a"> - <image file="0015-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0015-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="4"> - <head xml:id="head20" xml:space="preserve">THEOREMA <num value="4">IIII</num>.</head> - <p> - <s xml:id="s63" xml:space="preserve"><emph style="sc">CVr</emph> multiplicaturi fractos cum integris, rectè multiplicent numerantem fra-<lb/>cti per numerum integrorum, partianturqÌue productum per <reg norm="denominantem" type="context">denominantẽ</reg> <lb/>fracti, ex quo numerus quæſitus colligitur.</s> - </p> - <p> - <s xml:id="s64" xml:space="preserve">Propter quod mente concipiamus in Å¿ubÅ¿equenti figura, numerum integrorum <lb/>tanquam lineam <var>.a.e.</var> qui, verbigratia, Å¿it denarius, quorum vnuÅ¿quiÅ¿que Å¿it æqualis <lb/><var>a.i.</var> cogiteturqÌue productum ipÅ¿ius <var>.a.e.</var> in <var>.a.i.</var> Å¿itqÌue <var>.u.e.</var> quod quidem erit dena-<lb/>rius Å¿uperficialis, conÅ¿tituta prius <var>.a.u.</var> æqualis <var>.a.i.</var> & <var>.a.o.</var> Å¿int duæ tertiæ <var>.a.u.</var> <reg norm="quarum" type="context">quarÅ©</reg> <lb/>duarum tertiarum productum in numerum <var>.a.e.</var> Å¿it <var>.o.e.</var> pariter <var>.u.i.</var> vnitas Å¿it Å¿uper-<lb/>ficialis prout <var>.a.i.</var> vnitas eÅ¿t linearis, quam <var>.u.i.</var> reÅ¿picere debet productum <var>.o.e.</var> ex <lb/>quo integer Å¿uperficialis <var>.u.i.</var> erit tanquam ternarius, & productum <var>.o.i.</var> tanquam bi <lb/>narius, & quia quælibet pars è viginti ipÅ¿ius <var>.o.e.</var> æqualis eÅ¿t tertiæ parti <var>.u.i.</var> vnita-<lb/>tis Å¿uperficialis; </s> - <s xml:id="s65" xml:space="preserve">Å¿i cupiamus Å¿cire quot integræ vnitates Å¿int in partibus <var>.o.e.</var> conÅ¿ul-<lb/>tum eÅ¿t eaÅ¿dem diuidere per denominantem <var>.u.i.</var> compoÅ¿itum ex tribus partibus Å¿u <lb/>perficialibus, & cum tam linea <var>u.a.</var> quam Å¿uperficies <var>.u.i.</var> diuidatur in 3. partes <reg norm="aequa les" type="simple">Ä™quaÂ<lb/>les</reg> noÅ¿ce peroportunum eÅ¿t eiuÅ¿modi partitionem numeri <var>.o.e.</var> fieri per numerum <lb/>ipÅ¿ius <var>.u.i.</var> non <var>.u.a.</var> ex prædictis cauÅ¿is.</s> - </p> - <figure position="here"> - <image file="0015-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0015-02"/> - </figure> - <pb o="4" rhead="IO. BAPT. BENED." n="16" file="0016" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0016"/> - </div> - <div type="math:theorem" level="3" n="5"> - <head xml:id="head21" xml:space="preserve">THEOREMA <num value="5">V</num>.</head> - <p> - <s xml:id="s66" xml:space="preserve"><emph style="sc">ALia</emph> quoque via prædicti effe <lb/> - <anchor type="figure" xlink:label="fig-0016-01a" xlink:href="fig-0016-01"/> - ctus cauÅ¿a, Å¿peculando inno-<lb/>teÅ¿cere poteÅ¿t, cuius rei gratia for-<lb/>metur Å¿equens figura <var>.e.o.a.u.n.</var> <lb/>eiuÅ¿modi, vt <var>a.e.</var> Å¿it numerus li-<lb/>linearis integrorum, & <var>o.e.</var> produ-<lb/>ctum numerantis ipÅ¿orum <reg norm="fractorum" type="context">fractorÅ©</reg> <lb/>in integris, ex quo <var>.a.o.</var> erunt duæ <lb/>tertiæ, verbigratia, <var>a.i.</var> aut <var>a.u.</var> qua-<lb/>rum <reg norm="linearum" type="context">linearÅ©</reg> Å¿ingulÄ™ Å¿tatuuntur æqua <lb/>les vnitati lineari, Å¿uperficies autem <lb/>parallelogramma <var>.u.n.</var> conÅ¿tituatur <lb/>æqualis magnitudinis Å¿uperficiei <var>.o.<lb/>e.</var> ex quo <var>.u.n.</var> erit nobis cognita Å¿u-<lb/>perficies. </s> - <s xml:id="s67" xml:space="preserve">CognoÅ¿cetur pariter quan <lb/>titas partium <var>.a.u.</var> quam in propoÅ¿i-<lb/>to exemplo diximus eſſe trium par-<lb/>tium. </s> - <s xml:id="s68" xml:space="preserve">ex regula igitur de tribus, di-<lb/>cemus Å¿i <var>.u.a.</var> dat <var>.a.e.</var> Å¿ine dubio <var>.o.<lb/>a.</var> dabit <var>.a.n.</var> numerum linearem. <lb/>quæ regula ex 15. Å¿exti in continuis, <lb/>& ex 20. Å¿eptimi in diÅ¿cretis, depro-<lb/>mitur. </s> - <s xml:id="s69" xml:space="preserve">rectè igitur <reg norm="multiplicantur" type="context">multiplicãtur</reg> fra-<lb/>cti numerantes cum integris, & productum diuiditur per <reg norm="denominantem" type="context">denominantẽ</reg> fractorum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0016-01" xlink:href="fig-0016-01a"> - <image file="0016-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0016-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="6"> - <head xml:id="head22" xml:space="preserve">THEOREMA <num value="6">VI</num>.</head> - <p> - <s xml:id="s70" xml:space="preserve"><emph style="sc">ITem</emph> & alia Å¿peculatione cognoÅ¿ci poteÅ¿t hoc rectè fieri, mul-<lb/>tiplicantes enim has duas tertias per decem, debemus conÅ¿ide-<lb/> - <anchor type="figure" xlink:label="fig-0016-02a" xlink:href="fig-0016-02"/> - rare quantitatem duarum <reg norm="tertiarum" type="context">tertiarÅ©</reg> decies produci, ex quo oriuntur <lb/>20. tertia, quandoquidem Å¿ingulæ vnitates, </s> - <s xml:id="s71" xml:space="preserve">tunc pro duobus ter-<lb/>tijs Å¿umuntur, Å¿ed c<unsure/>um quilibet integer tria fragmenta contineat, <lb/>ideo ex ratione partiendi quoties ternarius ingrediatur viginti, <lb/>Å¿tatim cognoÅ¿cemus quod optabamus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0016-02" xlink:href="fig-0016-02a"> - <image file="0016-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0016-02"/> - </figure> - </div> - <p> - <s xml:id="s72" xml:space="preserve">Id ipÅ¿um accideret Å¿i integri in eiuÅ¿modi Å¿pecie fractorum diui-<lb/>derentur. quo facto hi multiplicandi eſſent cum numerante propo <lb/>Å¿ito, & <reg norm="partiendum" type="context">partiendÅ©</reg> productum per quadratum denominantis.</s> - </p> - <p> - <s xml:id="s73" xml:space="preserve">Cuius rei hæc eÅ¿t Å¿peculatio. </s> - <s xml:id="s74" xml:space="preserve">Sit linea <var>.a.e.</var> conÅ¿tans ex <reg norm="quinque" type="simple">quinq;</reg> <lb/>integris numeris, quorum <reg norm="vnuÅ¿quiÅ¿que" type="simple">vnuÅ¿quiÅ¿q;</reg> æqualis Å¿it <var>.a.u.</var> vel <var>.a.i.</var> & <var>.a.o.</var> <lb/>Å¿int duo tertia vnitatis integræ linearis. </s> - <s xml:id="s75" xml:space="preserve">cogitemus nunc hos <reg norm="quinque" type="simple">quinq;</reg> <lb/>integros diuidi in Å¿ua <reg norm="fragmenta" type="context">fragmẽta</reg> linearia, quÄ™ in propoÅ¿ito exemplo <lb/>erunt 15. multiplicatis iam 15. cum propoÅ¿itis, videlicet <var>a.o.</var> orie-<lb/>tur productum <var>.o.e.</var> triginta fragmentorum Å¿uperficialium, <reg norm="quorum" type="context">quorÅ©</reg> <lb/>in Å¿ingulos integros Å¿uperficiales <reg norm="cadunt" type="context">cadÅ©t</reg> <reg norm="nouem" type="context">nouẽ</reg> in hoc <reg norm="exemplo" type="context">exẽplo</reg>, & cum <lb/>notauerimus quoties <reg norm="nouem" type="context">nouẽ</reg> ingrediatur triginta, propoÅ¿itum con-<lb/>Å¿equemur.</s> - </p> - <pb o="5" rhead="THEOR. ARITH." n="17" file="0017" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0017"/> - </div> - <div type="math:theorem" level="3" n="7"> - <head xml:id="head23" xml:space="preserve">THEOREMA <num value="7">VII</num>.</head> - <p> - <s xml:id="s76" xml:space="preserve"><emph style="sc">CVr</emph> multiplicaturi integros numeros & fractos, cum integris & fractis, de-<lb/>beant integros reducere ad Å¿pecies fractorum, eos colligendo cum fractis: <lb/></s> - <s xml:id="s77" xml:space="preserve">deinde multiplicare hos vltimos numerantes adinuicem & productum partiri <lb/>per productum denominantium.</s> - </p> - <p> - <s xml:id="s78" xml:space="preserve">Vt (exempli cauÅ¿a) Å¿i volumus multiplicare vnum & duo tertia, per duo & tria <lb/>quarta, reducentur omnia in fractos, ex quo vna ex parte eſſent quinque ter-<lb/>tia, multiplicanda cum vndecim quartis ex altera, quo facto oriretur productum <lb/>quinquagintaquinque fractorum, quod diuiÅ¿um per <lb/>productum ternarijin quaternarium, videlicet per duode <lb/>cim, quatuor integri proferentur cum Å¿eptem duodeci-<lb/>mis fractis vnius integri.</s> - </p> - <figure position="here"> - <image file="0017-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0017-01"/> - </figure> - <p> - <s xml:id="s79" xml:space="preserve">Detur Å¿ubÅ¿equens figura in qua linea <var>a.i.</var> æqualis Å¿it li-<lb/>neæ <var>.u.a.</var> quarum <reg norm="vnaquæque" type="simple">vnaquæq;</reg> <reg norm="conÅ¿ideretur" type="context">cõſideretur</reg> pro integro nume <lb/>ro: </s> - <s xml:id="s80" xml:space="preserve"><reg norm="cogiteturque" type="simple">cogiteturqÌ;</reg> <var>.a.i.</var> valere quatuor in pręſenti <reg norm="exemplo" type="context">exẽplo</reg>, & <var>.a.<lb/>u.</var> tria: </s> - <s xml:id="s81" xml:space="preserve">detur deinde linea <var>.a.o.</var> æquipollens vni integro <reg norm="cum" type="context">cÅ©</reg> <lb/>duobus tertijs, & <var>a.e.</var> æquipollens duobus integris & tri-<lb/>bus quartis. </s> - <s xml:id="s82" xml:space="preserve">Iam Å¿i hæ duæ lineæ in Å¿uos fractos redu-<lb/>cantur, multiplicata (vt in Å¿equenti figura apparet.) <var>a.o.</var> <reg norm="cum" type="context">cÅ©</reg> <lb/><var>a.e.</var> orietur productum <var>o.e.</var> fractorum Å¿uperficialium <lb/><reg norm="quinquagintaquinque" type="simple">quinquagintaquinq;</reg>, quorum integer Å¿uperficialis va-<lb/>let duodecim, Å¿cilicet <var>.u.i.</var> vt cuique manifeÅ¿tum eÅ¿t, ex <lb/>quo, quærenti media partitione, quoties duodecim in-<lb/>grediatur quinquagintaquinque, citra errorem, quæſitum <lb/>occurret.</s> - </p> - </div> - <div type="math:theorem" level="3" n="8"> - <head xml:id="head24" xml:space="preserve">THEOREMA <num value="8">VIII</num>.</head> - <p> - <s xml:id="s83" xml:space="preserve"><emph style="sc">ID ipsvm</emph> accideret Å¿i fractiad vnam eandemqÌue denominationem reduceren-<lb/>tur, qui poÅ¿tmodum Å¿imul multiplicarentur, productumqÌue partiremur per qua-<lb/>dratum denominantis communis.</s> - </p> - <p> - <s xml:id="s84" xml:space="preserve">Exempli cauÅ¿a, Å¿int eadem quinque tertia, & vndecim quarta adinuicem multi-<lb/>plicanda, quæ Å¿i reducantur ad vnam & eandem denominationem quinarius <lb/>numerans vnius, multiplicabitur cum quaternario deno-<lb/>minante alterius, & vndenarius Å¿ecundi cum ternario de-<lb/>nominante primi. ex quo vna ex parte eſſent viginti, ex <lb/> - <anchor type="figure" xlink:label="fig-0017-02a" xlink:href="fig-0017-02"/> - altera 33. numerantia vnius <reg norm="communis" type="context">cõmunis</reg> denominantis, quod <lb/>eſſet productum ternarij in quaternarium, videlicet duo-<lb/>decim, vt ex veteri regula patet. </s> - <s xml:id="s85" xml:space="preserve">Iam Å¿i multiplicentur vi <lb/>ginti cum trigintatribus, dabuntur 660. fracti, quorum in-<lb/>teger erit quadratum duodenarij, nempe 144. quibus qui-<lb/>dem 660. diuiÅ¿is per 144. proferentur quatuor integri & <lb/>Å¿eptem duodecimi.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0017-02" xlink:href="fig-0017-02a"> - <image file="0017-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0017-02"/> - </figure> - </div> - <p> - <s xml:id="s86" xml:space="preserve">Cuius rei gratia Å¿it in Å¿ubÅ¿cripta figura linea <var>.a.i.</var> & ei <lb/>æqualis <var>.a.u.</var> pro integro lineari, quæ <var>.a.i.</var> diuidatur in qua-<lb/>tuor partes, & <var>.a.u.</var> in tres, & linea <var>.a.e.</var> Å¿it vndecim <reg norm="partium" type="context">partiÅ©</reg> <lb/>talium qualium <var>.a.i.</var> eÅ¿t quatuor, & <var>.a.o.</var> Å¿it quinque pro-<lb/>ut <var>.a.u.</var> eÅ¿t trium. </s> - <s xml:id="s87" xml:space="preserve">nunc multiplicato <var>.a.o.</var> & <var>.a.i.</var> orietur pro-<lb/>ductum <var>.o.i.</var> viginti partium Å¿uperficialium. </s> - <s xml:id="s88" xml:space="preserve">tum multipli- - <pb o="6" rhead="IO. BAPT. BENED." n="18" file="0018" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0018"/> - cato <var>.a.e.</var> per <var>.a.u.</var> dabitur productum <var>.u.e.</var> <reg norm="trigintatrium" type="context">trigintatriÅ©</reg> <lb/> - <anchor type="figure" xlink:label="fig-0018-01a" xlink:href="fig-0018-01"/> - partium. </s> - <s xml:id="s89" xml:space="preserve">ad hæc quadratum <var>.u.i.</var> conÅ¿tabit ex duode-<lb/>cim partibus eiuÅ¿dem rationis cum reliquis duobus <lb/>productis, quod quadratum <var>.u.i.</var> vnitas eÅ¿t Å¿uperficia-<lb/>lis, & communis denominans duorum productorum. <lb/></s> - <s xml:id="s90" xml:space="preserve">quod Å¿i in præſentiarum cogitabimus lineam <var>.c.d.</var> tri-<lb/>gintatrium partium æqualium, et <var>.c.t.</var> duodecim Å¿imi-<lb/>lium, et <var>.c.f.</var> viginti <var>.c.n.</var> duodecim, multiplicato <var>.c.<lb/>d.</var> cum <var>.c.f.</var> dabitur Å¿uperficies <var>.f.d.</var> 660. fractorum <lb/>Å¿uperficialium, quorum vnitas integra Å¿uperficialis <lb/>erit quadratum <var>.n.t.</var> 144. partium cuiuÅ¿modi <var>.f.d.</var> <lb/>partes habet .660. diuiÅ¿o itaque <var>.f.d.</var> per <var>.n.t.</var> pro-<lb/>poÅ¿itum conÅ¿equetur. </s> - <s xml:id="s91" xml:space="preserve">eo quòd eadem proportio erit <lb/> - <anchor type="figure" xlink:label="fig-0018-02a" xlink:href="fig-0018-02"/> - producti <var>.f.d.</var> ad <var>.n.t.</var> quæ producti eius quòd fit ex <var>.<lb/>a.e.</var> in <var>.a.o.</var> ad <var>.u.i.</var> nam proportio <var>.c.d.</var> ad <var>.c.t.</var> ea-<lb/>dem eÅ¿t quæ <var>.a.e.</var> ad <var>.a.i.</var> & <var>c.f.</var> ad <var>.c.n.</var> vt <var>.a.o.</var> ad <var>.a.<lb/>u.</var> ex prima Å¿exti vel 18. Å¿eptimi, Å¿ed vt <var>.f.d.</var> ad id <reg norm="quod" type="simple">ê™</reg> <lb/>fit ex <var>.f.c.</var> in <var>.c.t.</var> eÅ¿t vt <var>.c.d.</var> ad <var>.c.t.</var> & vt eius <reg norm="quod" type="simple">ê™</reg> fit ex <lb/><var>f.c.</var> in <var>.c.t.</var> ad <var>.n.t.</var> eÅ¿t vt <var>.f.c.</var> ad <var>.c.n.</var> ex dictis pro-<lb/>poÅ¿itionibus </s> - <s xml:id="s92" xml:space="preserve">quare ex æqua proportionalitate, eodem <lb/>modo diÅ¿currendo in figura <var>.o.a.e.</var> ita Å¿e habebit <var>.f.d.</var> <lb/>ad <var>.n.t.</var> vt <var>.o.e.</var> ad <var>.u.i</var>. </s> - <s xml:id="s93" xml:space="preserve">Porrò ex ijs, quæ hactenus de <lb/>fractorum multiplicatione conÅ¿iderata fuerunt, apertè <lb/>ratio deprehenditur, cur productum, Å¿ingulis producen <lb/>tibus Å¿emper minus Å¿it, cum producta Å¿int Å¿uperficialia <lb/>producentia verò Å¿emper linearia, omiſſis productis <lb/>corporeis, quæ omnia ad Å¿uperficialia reducuntur.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0018-01" xlink:href="fig-0018-01a"> - <image file="0018-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0018-01"/> - </figure> - <figure xlink:label="fig-0018-02" xlink:href="fig-0018-02a"> - <image file="0018-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0018-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="9"> - <head xml:id="head25" xml:space="preserve">THEOREMA <num value="9">IX</num>.</head> - <p> - <s xml:id="s94" xml:space="preserve"><emph style="sc">IN Ipsa</emph> fractorum diuiÅ¿ione, animaduertendum eÅ¿t, denominantes numeros <lb/>Å¿emper æquales inuicem eſſe debere, vnius Å¿cilicet Å¿peciei, quòd Å¿i æquales non <lb/>fuerint, neceſſe eÅ¿t via multiplicationis ipÅ¿orum denominantium adinuicem effice-<lb/>re æquales vt Å¿int, ex quo productum oritur eiuÅ¿modi, vt aptum Å¿it habere partes <lb/>fractorum, quæ deÅ¿iderabantur.</s> - </p> - <p> - <s xml:id="s95" xml:space="preserve">Exempli gratia, Å¿i proponerentur diuidenda Å¿eptem octaua per tria quarta præ-<lb/>cipit antiquorum regula, vt ad vnam tantum denominationem reducantur. </s> - <s xml:id="s96" xml:space="preserve">quare <lb/>multiplicant denominantes inuicem. </s> - <s xml:id="s97" xml:space="preserve">ex quo productum in materia propoÅ¿ita ori-<lb/>tur triginta duarum partium commune denominans, cuius duo numerantes Å¿unt vi-<lb/>gintiquatuor & vigintiocto, producti ex multiplicatione vnius numerantis in deno <lb/>minantem alterius, ex quo dantur vigintiquatuor tamquam tria quarta trigintaduo <lb/>rum, & vigintiocto tanquam Å¿eptem octaua particularum vniformium, prout ope <lb/>primæ Å¿exti aut decimæoctauæ Å¿eptimi in Å¿ubÅ¿cripta figura cognoÅ¿ci poteÅ¿t.</s> - </p> - <pb o="7" rhead="THEOR. ARITH." n="19" file="0019" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0019"/> - <p> - <s xml:id="s98" xml:space="preserve">Sit itaque linea <var>.a.i.</var> diuifa in partes octo, & ei æqualis in longitudine <var>.a.u.</var> in qua-<lb/>tuor, productum verò vnius in alteram <lb/> - <anchor type="figure" xlink:label="fig-0019-01a" xlink:href="fig-0019-01"/> - Å¿it <var>.u.i.</var> trigintaduarum particularum <lb/>fuperficialium fimilium & <reg norm="æqualium" type="context">æqualiÅ©</reg> ad-<lb/>inuicem. </s> - <s xml:id="s99" xml:space="preserve">fit deinde <var>.a.e.</var> Å¿eptem <reg norm="partium" type="context">partiÅ©</reg> <lb/>lineæ <var>.a.i.</var> & <var>.a.o.</var> trium partium <var>.a.u</var>. <lb/></s> - <s xml:id="s100" xml:space="preserve">tunc productum <var>.a.e.</var> in <var>.a.u.</var> erit <var>.u.e.</var> <lb/>particularum Å¿uperficialium vigintiocto <lb/>& productum <var>.a.o.</var> in <var>.a.i.</var> erit <var>.o.i.</var> par <lb/>ticularum <reg norm="Å¿uperficialium" type="context">Å¿uperficialiÅ©</reg> vigintiquatuor <lb/>eiuÅ¿dem naturæ cum partibus triginta-<lb/>duabus totius denominantis communis. <lb/></s> - <s xml:id="s101" xml:space="preserve">vnde diuifo numerante vigintiocto per-<lb/>numerantem vigintiquatuor, dabitur <lb/>vnum cum fexta parte illius vnius.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0019-01" xlink:href="fig-0019-01a"> - <image file="0019-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0019-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="10"> - <head xml:id="head26" xml:space="preserve">THEOREMA <num value="10">X</num>.</head> - <p> - <s xml:id="s102" xml:space="preserve"><emph style="sc">PArtiri</emph> Å¿eu diuidere vno numero alium numerum, eÅ¿t etiam quodammodo <lb/>eiuÅ¿modi partem numeri diuifibilis inuenire refpectu totius numeri diuifibilis, <lb/>cuiuÅ¿modi eÅ¿t vnitas in diuidente refpectu totius diuidentis, partem inquam numeri <lb/>diuiÅ¿ibilis Å¿ic Å¿e habentem ad totum numerum diuiÅ¿ibilem Å¿icut vnitas ad totum di-<lb/>uidentem, quod Å¿imiliter ex regula de tribus præſtamus dicentes, Å¿i tantus numerus <lb/>diuidens dat <reg norm="vnitatem" type="context">vnitatẽ</reg>, quid dabit numerus diuifibilis, quemadmodum ex <ref id="ref-0006">.15. Å¿exti</ref> <lb/>Å¿eu <ref id="ref-0007">.20. Å¿eptimi</ref> licet Å¿peculari, Idcircò quotieÅ¿cunque minorem numerum per <lb/>maiorem diuidimus, Å¿emper qui prouenit fractus eÅ¿t.</s> - </p> - <p> - <s xml:id="s103" xml:space="preserve">Exempli gratia, Å¿i cogitaremus lineam <var>.a.e.</var> diuiÅ¿am in octo partes æquales, qua <lb/>rum vna Å¿cilicet vnitas effet <var>.a.i.</var> & cupere-<lb/>mus eam diuidere in nouem partes, ac Å¿cire <lb/> - <anchor type="figure" xlink:label="fig-0019-02a" xlink:href="fig-0019-02"/> - quan a Å¿it nona illius pars; </s> - <s xml:id="s104" xml:space="preserve">manifeÅ¿tum eſſet, <lb/>nonam partem ipÅ¿ius <var>.a.e.</var> minorem futuram <lb/>ipÅ¿a <var>.a.i.</var> cum <var>.a.i.</var> diminui debeat à ſua inte-<lb/>gritate eadem proportione, qua <var>.a.e.</var> minor <lb/>reperitur vna linea nouem partium æqualium <lb/>fingularum <var>.a.i</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0019-02" xlink:href="fig-0019-02a"> - <image file="0019-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0019-02"/> - </figure> - </div> - <p> - <s xml:id="s105" xml:space="preserve">Quod vt dilucidè cuiuis innoteÅ¿cat, hoc <lb/>etiam modo licebit videre Å¿itlinea <var>.n.c.</var> no-<lb/>nupla ad <var>.a.i.</var> & parallela ad <var>.a.e.</var> dubium non <lb/>eÅ¿t quin <var>.n.c.</var> maior futura Å¿it ipÅ¿a <var>.a.e.</var> iam Å¿i <lb/>earum extrema congiungantur medijs duabus <lb/>lineis <var>.n.a.</var> et <var>.c.e.</var> quæ Å¿imul concurrant in <lb/>puncto <var>.o.</var> (quod eÅ¿t probatu facillimum) da-<lb/>buntur certe duo trianguli fimiles <var>.a.o.e.</var> et <var>.n.o.c</var>. </s> - <s xml:id="s106" xml:space="preserve">Sit deinde <var>.n.t.</var> vna è partibus <lb/>ipÅ¿ius <var>.n.c.</var> quæ <var>.n.t.</var> æqualis erit <var>.a.i.</var> ex præſuppoÅ¿ito. </s> - <s xml:id="s107" xml:space="preserve">ducatur deinde <var>.o.t.</var> quÄ™ <lb/>interÅ¿ecet <var>.a.e.</var> in puncto <var>.x.</var> dico <var>.a.x.</var> tanto minorem futuram <var>.a.i.</var> quanto <var>.a.e.</var> <lb/>minor eÅ¿t <var>.n.c.</var> neque enim dubium eſſe poteÅ¿t quin proportiones <var>.n.t.</var> ad <var>.a.x.</var> et <var>. - <pb o="8" rhead="IO. BAPT. BENED." n="20" file="0020" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0020"/> - n.c.</var> ad <var>.a.e.</var> Å¿int æquales inuicem quandoqui-<lb/> - <anchor type="figure" xlink:label="fig-0020-01a" xlink:href="fig-0020-01"/> - dem vnaquæque earum ex triangulorum Å¿imi <lb/>litudine æqualis eÅ¿t proportioni <var>.o.n.</var> ad <var>.o.a</var>. <lb/></s> - <s xml:id="s108" xml:space="preserve">itaque <var>.n.t.</var> hoc eÅ¿t <var>.a.i.</var> tanto maior erit <var>.a.x.</var> <lb/>quanto <var>.n.c.</var> maior eÅ¿t <var>.a.e.</var> vnde ficut <var>.a.e.</var> con-<lb/>Å¿tat octo nonis ipÅ¿ius <var>.n.c.</var> ita pars <var>.a.x.</var> ipÅ¿ius <var>.<lb/>a.e.</var> octo nonis conÅ¿tabit ipÅ¿ius <var>.a.i</var>.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0020-01" xlink:href="fig-0020-01a"> - <image file="0020-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0020-01"/> - </figure> - </div> - <p> - <s xml:id="s109" xml:space="preserve">Hinc patet ratio cur partituri numerum mino <lb/>rem per maiorem collocent minorem fupra <lb/>virgulam & maiorem infra & zerum ad <reg norm="læuam" type="context">læuã</reg>.</s> - </p> - <p> - <s xml:id="s110" xml:space="preserve">Sciendum eÅ¿t præterea diuidere numerum <lb/>per numerum: </s> - <s xml:id="s111" xml:space="preserve">eſſe inuenire <reg norm="alterum" type="context">alterÅ©</reg> latus à quo <lb/>producitur, Å¿uppoÅ¿ito Å¿emper quòd numerus <lb/>diuifibilis Å¿uperÅ¿icialis Å¿it, & rectangulus.</s> - </p> - <p> - <s xml:id="s112" xml:space="preserve">Exempli gratia, Å¿i proponantur triginta diuidenda per quinarium, nihil aliud erit <lb/>hæc diuiÅ¿io, quam inuentio alterius numeri, qui multiplicatus per quinarium produ-<lb/>cat triginta Å¿uperficies rectangulas, huiuÅ¿modi verò eÅ¿t Å¿enarius, cuius Å¿ingulæ vnita-<lb/>tes Å¿uperficiales erunt.</s> - </p> - <p> - <s xml:id="s113" xml:space="preserve">Cuius rei gratia Å¿it Å¿ubÅ¿criptum rectangulum <var>.a.e.</var> triginta vnitatum <reg norm="Å¿uperÅ¿icialium" type="context">Å¿uperÅ¿icialiÅ©</reg>, <lb/>cuius latus <var>.e.n.</var> Å¿it quinque vnitatum. </s> - <s xml:id="s114" xml:space="preserve">hinc latus <var>.a.n.</var> erit Å¿ex vnitatum; </s> - <s xml:id="s115" xml:space="preserve">ita diuiden-<lb/>tes rectangulum <var>.e.a.</var> nihil a iud faciemus, quam vt inue-<lb/> - <anchor type="figure" xlink:label="fig-0020-02a" xlink:href="fig-0020-02"/> - nia mus quantum valeat latus <var>.a.n.</var> quod erit Å¿ex vnitatum. <lb/></s> - <s xml:id="s116" xml:space="preserve">Sin verò diuiÅ¿erimus per latus <var>.a.n.</var> quæremus latus <var>.e.n.</var> <lb/>quinque vnitatum. </s> - <s xml:id="s117" xml:space="preserve">ex quo, proportio totius numeri diuifi-<lb/>bilis ad numerum qui oritur, erit Å¿icut diuidentis ad vnita-<lb/>tem, ex prima Å¿exti, aut .18. vel .19. Å¿eptimi, & permutatim <lb/>ita Å¿e habebit diuiÅ¿ibile ad diuidentem, Å¿icut numerus qui <lb/>oritur ad vnitatem.</s> - </p> - <div type="float" level="4" n="3"> - <figure xlink:label="fig-0020-02" xlink:href="fig-0020-02a"> - <image file="0020-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0020-02"/> - </figure> - </div> - <p> - <s xml:id="s118" xml:space="preserve">Partiri igitur nihil aliud eÅ¿t, quam inuenire latus rectanguli, quod productum in <lb/>diuidente, numerum diuiÅ¿ibilem compl at, ex quo numerus diuiÅ¿ibilis Å¿uperficialis <lb/>eÅ¿t, diuidens autem, & qui oritur, numeri lineares & latera producentia huiuÅ¿cemodi <lb/>numerum diuiÅ¿ibilem. </s> - <s xml:id="s119" xml:space="preserve">nam multiplicare & diuidere opponuntur inuicem, cum au-<lb/>tem ex multiplicatione laterum Å¿iue linearum generatur Å¿uperficies, ex diuiÅ¿ione po-<lb/>Å¿tea ipÅ¿ius Å¿uperficiei inuenitur alterum latus. </s> - <s xml:id="s120" xml:space="preserve">quare mirum non eÅ¿t, Å¿i proueniens ex <lb/>vna diuiÅ¿ione (via fractorum) Å¿it Å¿emper maius numero diuiÅ¿ibili.</s> - </p> - <p> - <s xml:id="s121" xml:space="preserve">Exempli gratia, diuidendo dimidium per tertiam partem, reÅ¿ultat vnus integer nu <lb/>merus cum dimidio pro numero qui oritur. </s> - <s xml:id="s122" xml:space="preserve">Sit itaque dimidium Å¿uperÅ¿iciale diuiÅ¿i-<lb/>bile <var>.b.c.</var> cuius totum Å¿it <var>.b.p.</var> quadratum. </s> - <s xml:id="s123" xml:space="preserve">tertium verò lineare diuidens, <var>b.n.</var> cuius to-<lb/>tum lineare Å¿it <var>.b.d.</var> quærendum nobis eÅ¿t latus <var>.b.s.</var> quod cum latere <var>.b.n.</var> producat re <lb/>ctangulum <var>.n.s.</var> æquale dimidio Å¿uperÅ¿iciali propoÅ¿ito <var>.b.c.</var> quod Å¿i Å¿iat, ex .15. Å¿exti, <lb/>aut .20. Å¿eptimi. erit eadem proportio <var>.b.n.</var> ad <var>.b.q.</var> quæ eÅ¿t <var>.q.c.</var> ad <var>.b.s.</var> dicemus itaque <lb/>Å¿i <var>.n.b.</var> dat <var>.b.q.</var> quid dabit <var>.q.c</var>? </s> - <s xml:id="s124" xml:space="preserve">certè <var>.b.s.</var> Å¿ed <var>.n.b.</var> eÅ¿t tertium lineare et <var>.b.q.</var> lineare <reg norm="in- tegrum" type="context">in-<lb/>tegrÅ©</reg>, & <var>b.s.</var> proueniens lineare. </s> - <s xml:id="s125" xml:space="preserve">& quia <var>.b.c.</var> dimidium Å¿uperficiale, producitur à <var>.q.c.</var> <lb/>dimidio lineari in <var>.q.b.</var> integro lineari. </s> - <s xml:id="s126" xml:space="preserve">quare cum <var>.n.s.</var> Å¿it Ä™qualis <var>.b.c.</var> & productum ex <var>.<lb/>b.n.</var> minori <var>.q.c.</var> neceſſe eÅ¿t, vt producatur in <var>.b.s.</var> maiore <var>.q.b.</var> quod <var>.q.b.</var> maius eÅ¿t <var>.q.c.</var> <lb/>quod quidem <var>.q.c.</var> ita appellatur Å¿icut <var>.b.c</var>. </s> - <s xml:id="s127" xml:space="preserve">quare mirum non eÅ¿t Å¿i proueniens per fra-<lb/>ctos numeros ex diuiÅ¿ione, maior Å¿it numero diuiÅ¿ibili.</s> - </p> - <pb o="9" rhead="THEOR. ARITH." n="21" file="0021" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0021"/> - <p> - <s xml:id="s128" xml:space="preserve">Hinc manifeÅ¿te patet quamlibet <reg norm="diuiÅ¿ionem" type="context">diuiÅ¿ionẽ</reg> aut partitionem oriri ex regula de tri-<lb/>bus, quandoquidem Å¿inguli diuidentes æquipollent vni integro, & loco illius Å¿u-<lb/>muntur. </s> - <s xml:id="s129" xml:space="preserve">Perinde enim eÅ¿t diuidere centum per viginti, ac <reg norm="regulam" type="context">regulã</reg> obÅ¿eruare de tri-<lb/>bus <reg norm="dicentes" type="context">dicẽtes</reg>, Å¿i viginti æquipollent vni, quibus <reg norm="Ä™quiualebunt" type="context">Ä™quiualebÅ©t</reg> <reg norm="centum" type="context">cẽtum</reg>? </s> - <s xml:id="s130" xml:space="preserve">Hoc autem ex Å¿ub <lb/>Å¿equenti figura facile deprehendetur, in qua linea <var>.a.b.</var> Å¿ignificat viginti, et <var>.a.o.</var> <reg norm="vni- tatem" type="context">vni-<lb/>tatẽ</reg> <reg norm="linearem" type="context">linearẽ</reg>, et <var>.a.c.</var> vnitates lineares <reg norm="centum" type="context">centÅ©</reg>: </s> - <s xml:id="s131" xml:space="preserve">o.c. verò centum vnitates Å¿uperficiales, <lb/>et <var>.a.d.</var> <reg norm="quinque" type="simple">quinq;</reg> vnitates lineares, et <var>.d.b.</var> centum vnitates Å¿uperficiales, ex quo manife-<lb/>ftè deprehenditur quòd quemadmodum multiplicare, nihil aliud eÅ¿t, quam inueni <lb/>re <reg norm="productum" type="context">productÅ©</reg> ex duobus lateribus propoÅ¿itis, it a partiri nihil aliud eÅ¿t, quam da-<lb/>to vno latere inuenire aliud latus producti propoÅ¿iti.</s> - </p> - <figure position="here"> - <image file="0021-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0021-01"/> - </figure> - <figure position="here"> - <image file="0021-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0021-02"/> - </figure> - <p> - <s xml:id="s132" xml:space="preserve">Nam <reg norm="quotieÅ¿cunque" type="simple">quotieÅ¿cunq;</reg> <reg norm="ratiocinantes" type="context">ratiocinãtes</reg> dicimus tantundem numeri, immediate produci <lb/>mus Å¿uperficiem, <reg norm="mediante" type="context">mediãte</reg> vnitate in huiuÅ¿modi numero, qui numerus <reg norm="antequam" type="context">antequã</reg> pro-<lb/>ducatur in vnitatem, mente concipiendus eÅ¿t tanqua m linearis, tanquam linea in-<lb/>quam diuiÅ¿a in totidem particulas lineares, Å¿ingulas continuas & æquales vnitati <lb/>propoÅ¿itæ. </s> - <s xml:id="s133" xml:space="preserve"><reg norm="Cum" type="context">CÅ©</reg> verò productus fuerit numerus in vnitate Å¿uperficialis, erit ac Å¿i tot eÅ¿-<lb/>Å¿ent vnitates quadratæ, quod Å¿i ita non eſſet, nulla mentio facienda eſſet quo-<lb/>rumuis <reg norm="fractorum" type="context">fractorÅ©</reg>. </s> - <s xml:id="s134" xml:space="preserve">Ex <reg norm="eadem" type="context">eadẽ</reg> regula de tribus reduci poteÅ¿tad praxim <reg norm="tertium" type="context">tertiÅ©</reg> theorema.</s> - </p> - <p> - <s xml:id="s135" xml:space="preserve">Quare cupientes Å¿cire quæ Å¿int illæ partes, quæ Å¿unt tres quartÄ™, ipÅ¿arum quin-<lb/>que Å¿extarum, dicemus Å¿i quatuor dant tria, quid dabunt <reg norm="quinque" type="simple">quinq;</reg> Å¿extæ? </s> - <s xml:id="s136" xml:space="preserve">dabunt .15. <lb/>vigeÅ¿imas quartas, quæ quindecim Å¿unt tres quartæ ipÅ¿ius .20. viginti <reg norm="autem" type="context">autẽ</reg> <reg norm="quinque" type="simple">quinq;</reg> Å¿ex <lb/>tæ vigintiquatuor, quandoquidem nos numerum quÄ™rimus, cui ita proportionentur <lb/><reg norm="quinque" type="simple">quinq;</reg> Å¿extæ alterius numeri, Å¿icut quatuor ad tria, vnde Å¿ic Å¿e habent .20. ad .15. Å¿i-<lb/>cut .4. ad .3. ipÅ¿e autem .20. <reg norm="quinque" type="simple">quinq;</reg> Å¿extÄ™ partes Å¿unt vigintiquatuor, vt per Å¿e <reg norm="notum" type="context">notÅ©</reg> eÅ¿t.</s> - </p> - <p> - <s xml:id="s137" xml:space="preserve">Ex eadem regula de tribus, huiuÅ¿modi quęſito reÅ¿ponderi poteÅ¿t, Å¿i conÅ¿tituamus <lb/>prædictas <reg norm="quinque" type="simple">quinq;</reg> Å¿extas eſſe numerum, cuius tres quartæ quÄ™rantur, dicentes, Å¿i vnus <lb/>integer dat tres quartas, quid dabunt <reg norm="quinque" type="simple">quinq;</reg> Å¿extæ? </s> - <s xml:id="s138" xml:space="preserve">quare Å¿equentes regulam de <lb/>tribus, dabuntur quindecim vigeÅ¿imæ quartæ. </s> - <s xml:id="s139" xml:space="preserve">Valet eadem regula de tribus; </s> - <s xml:id="s140" xml:space="preserve">vt quis <lb/>Å¿cire poſſit, quæ pars aut partes numeri propoÅ¿iti Å¿it aliquis numerus.</s> - </p> - <p> - <s xml:id="s141" xml:space="preserve">Exempli gratia, Å¿cire cupienti, quæ pars aut partes ipÅ¿ius vigintiquatuor Å¿int Å¿ex-<lb/>decim, conÅ¿tituentur .24. tanquam vnum totum, cuius pars aut partes Å¿int Å¿exdecim, <lb/>dicemus igitur Å¿i .24. dant Å¿exdecim, quid dabit vnum? </s> - <s xml:id="s142" xml:space="preserve">Å¿exdecim videlicet vigeÅ¿i-<lb/>masquartas, quæ cum ad primos numeros reductæ fuerint, erunt duæ tertiæ. <lb/></s> - <s xml:id="s143" xml:space="preserve">Eadem ratione qui Å¿cire uellet, quæ partes aut pars eſſent tres quartæ, octo no-<lb/>narum, diceret, Å¿i octo nonæ danttres quartas, quid dabit vnum? </s> - <s xml:id="s144" xml:space="preserve">prouenient .27. <lb/>trigeÅ¿imęſecundæ.</s> - </p> - <p> - <s xml:id="s145" xml:space="preserve">SubÅ¿eruit pariter ad <reg norm="Å¿ciendum" type="context">Å¿ciendÅ©</reg> <reg norm="naturam" type="context">naturã</reg> <reg norm="partium" type="context">partiÅ©</reg> numeri propoÅ¿iti. </s> - <s xml:id="s146" xml:space="preserve">Exempli cauÅ¿a, Å¿i quis <lb/>quærat, cuius numeri, duodecim Å¿int duæ tertiæ partes. </s> - <s xml:id="s147" xml:space="preserve">Dicet Å¿i duo dant tria, quid - <pb o="10" rhead="IO. BAPT. BENED." n="22" file="0022" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0022"/> - dabunt duodecim? </s> - <s xml:id="s148" xml:space="preserve">nempe dabunt decemocto, numerum quæſitum Å¿cilicet, <lb/></s> - <s xml:id="s149" xml:space="preserve">Tunc autem nil aliud pręſtamus quam quòd quærimus numerum ad quem ita Å¿e <lb/>habeant duodecim, Å¿icut duo ad tria. </s> - <s xml:id="s150" xml:space="preserve">Ita etiam Å¿i quis quærat, cuius numeri duo <lb/>tertia Å¿int tres quintÄ™, dicet, Å¿i tria dant <reg norm="quinque" type="simple">quinq;</reg>, quid dabunt duo tertia? </s> - <s xml:id="s151" xml:space="preserve">nempe da-<lb/>bunt integrum cum fracto nono. </s> - <s xml:id="s152" xml:space="preserve">Hoc erit <reg norm="itaque" type="simple">itaq;</reg> quÄ™rere numerum ad quem Å¿ic Å¿e <lb/>habeant duo tertia Å¿icut tria ad <reg norm="quinque" type="simple">quinq;</reg>, quod manifeÅ¿tum eÅ¿t per Å¿e.</s> - </p> - <p> - <s xml:id="s153" xml:space="preserve">Eadem ratione qui Å¿cire vellet, cuius numeri duæ Å¿eptimæ, eſſent octo integra-<lb/>rum cum duabus quintis, diceret, Å¿i duo dant Å¿eptem quid dabunt octo integra cum <lb/>duabus quintis? </s> - <s xml:id="s154" xml:space="preserve">nempe dabunt .29. integra cum duabus quintis numerum quæſi-<lb/>tum. </s> - <s xml:id="s155" xml:space="preserve">Sic etiam qui transferre uellet fractum numerum in fractum, id perficeret <lb/>ex regula de tribus.</s> - </p> - <p> - <s xml:id="s156" xml:space="preserve">Exempli gratia Å¿i proponerentur vnde cim tertiædecimæ vnius totius, toto diui-<lb/>Å¿o in .13. partes, <reg norm="deÅ¿ideraremusque" type="simple">deÅ¿ideraremusqÌ;</reg> Å¿cire, quot partes totius <reg norm="eÅ¿sent" type="context">eÅ¿sẽt</reg> vndecim <reg norm="tertiaedeci- mæ" type="simple">tertiÄ™deci-<lb/>mæ</reg>, toto in .4. partes diuiÅ¿o, diceremus Å¿i .13. dant .11. quid dabunt quatuor? </s> - <s xml:id="s157" xml:space="preserve">nem <lb/>pe <reg norm="dabunt" type="context">dabÅ©t</reg> tres quartas <reg norm="cum" type="context">cÅ©</reg> <reg norm="quinque" type="simple">quinq;</reg> tertijsdecimis unius quartæ, hoc verò nihil aliud eÅ¿t <lb/>quam querere numerum, ad quem Å¿ic Å¿e habeat totum in 4. partes diuiÅ¿um, Å¿icut <lb/>idem totum diuiÅ¿um in tredecim Å¿e habet ad undecim tertiasdecimas, Porrò ad <lb/>alia etiam multa hæc regula accommodata eÅ¿t.</s> - </p> - <p> - <s xml:id="s158" xml:space="preserve">Hæc enim <reg norm="non" type="context">nõ</reg> Å¿ine propoÅ¿ito dicta Å¿unt, Å¿ed ut <reg norm="quiÅ¿que" type="simple">quiÅ¿q;</reg> videat cauÅ¿am Å¿imilium ope-<lb/>rationum, quæ à practicis circa fractos numeros Å¿criptæ Å¿unt, omnem à diuina illa <lb/>regula de tribus originem trahere ut etiam in Å¿equentibus videbimus.</s> - </p> - </div> - <div type="math:theorem" level="3" n="11"> - <head xml:id="head27" xml:space="preserve">THEOREMA <num value="11">XI</num>.</head> - <p> - <s xml:id="s159" xml:space="preserve"><emph style="sc">CVr</emph> productum ex eo quod oritur in diuidente, Å¿emper æquale eÅ¿t numero <lb/>diuiÅ¿ibili Å¿i queras ita accipe.</s> - </p> - <p> - <s xml:id="s160" xml:space="preserve">Sit numerus diuiÅ¿ibilis <var>.b.</var> quod oritur Å¿it <var>.c.</var> diuidens <var>.d.</var> & vnitas diuidentis <var>.t.</var> cum <lb/>igitur, vt in præcedenti theoremate dictum <lb/>fuit, eadem Å¿it proportio <var>.b.</var> ad <var>.c.</var> quæ eÅ¿t <var>.d.</var> <lb/> - <anchor type="figure" xlink:label="fig-0022-01a" xlink:href="fig-0022-01"/> - ad <var>.t.</var> manifeÅ¿te deprehenditur ex .20. Å¿epti <lb/>mi, productum ex <var>.b.</var> in <var>.t.</var> æquale eſſe pro-<lb/>ducto <var>.c.</var> in <var>d</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0022-01" xlink:href="fig-0022-01a"> - <image file="0022-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0022-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="12"> - <head xml:id="head28" xml:space="preserve">THEOREMA <num value="12">XII</num>.</head> - <p> - <s xml:id="s161" xml:space="preserve">ID ipÅ¿um alia ratione contemplari licet.</s> - </p> - <p> - <s xml:id="s162" xml:space="preserve">Numerus diuiÅ¿ibilis Å¿ignificetur per lineam <var>.n.e.</var> diuidens verò per lineam <var>.a.e.</var> <lb/>quod oritur linea <var>.u.e.</var> vnitas diuidentis <var>.o.e.</var> <reg norm="quam" type="context">quã</reg> cogitamus eſſe vnitatem linearem; <lb/></s> - <s xml:id="s163" xml:space="preserve">ad hæc productum ex <var>.u.e.</var> in <var>.a.e.</var> Å¿it Å¿uperficies <var>.u.a</var>. </s> - <s xml:id="s164" xml:space="preserve">Dico Å¿uperficiem <var>.u.a.</var> componi <lb/>ex tot vnitatibus Å¿uperficialibus quot linearibus conÅ¿tat linea <var>.n.e.</var> nam ex ijs quæ <lb/>diuidendi ratione notauimus, <reg norm="conÅ¿tituitur" type="context">cõſtituitur</reg> <lb/>eandem proportionem eſſe <var>.n.e.</var> ad <var>.u.e.</var> <lb/> - <anchor type="figure" xlink:label="fig-0022-02a" xlink:href="fig-0022-02"/> - quÄ™ eÅ¿t <var>.a.e.</var> ad <var>.o.e</var>. </s> - <s xml:id="s165" xml:space="preserve">At ex prima Å¿exti aut <lb/>18. Å¿eptimi Å¿ic Å¿e habet totale <reg norm="productum" type="context">productÅ©</reg> <var>.<lb/>u.a.</var> ad partiale <var>.u.o.</var> Å¿icut <var>.a.e.</var> ad <var>.o.e</var>. <lb/></s> - <s xml:id="s166" xml:space="preserve">quare Å¿ic Å¿e habebit <var>.u.a.</var> ad <var>.u.o.</var> Å¿icut <var>.n.<lb/>e.</var> ad <var>.u.e.</var> Å¿ed <var>.u.e.</var> et <var>.u.o.</var> numero non differunt, cum Å¿int vnius & eiuÅ¿dem Å¿peciei, (ta-<lb/>met Å¿i numerus <var>.u.o.</var> Å¿it Å¿uperficialis et <var>.u.e.</var> linearis). </s> - <s xml:id="s167" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> ex nona quinti numerus <var>.<lb/>u.a.</var> æqualis erit numero <var>.n.e</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0022-02" xlink:href="fig-0022-02a"> - <image file="0022-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0022-02"/> - </figure> - </div> - <pb o="11" rhead="THEOREM. ARITH." n="23" file="0023" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0023"/> - </div> - <div type="math:theorem" level="3" n="13"> - <head xml:id="head29" xml:space="preserve">THEOREMA. <num value="13">XIII</num>.</head> - <p> - <s xml:id="s168" xml:space="preserve"><emph style="sc">CVr</emph> diuidentibus numerum diuiÅ¿ibilem per proueniens, oritur numerus diui-<lb/>dens?</s> - </p> - <p> - <s xml:id="s169" xml:space="preserve">Sit Å¿ubÅ¿criptus rectangulus <var>.o.e.</var> numerus diuiÅ¿i <lb/> - <anchor type="figure" xlink:label="fig-0023-01a" xlink:href="fig-0023-01"/> - bilis, qui producitur, tam ex <var>.a.o.</var> in <var>.a.e.</var> quám ex <var>.a.<lb/>e.</var> in <var>.a.o</var>. </s> - <s xml:id="s170" xml:space="preserve">quare Å¿i <var>.a.o.</var> diuidens fuerit <var>.a.e.</var> proue-<lb/>niens erit, Å¿i veró <var>.a.e.</var> diuidens extiterit, <var>a.o.</var> pro-<lb/>ueniens erit futurum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0023-01" xlink:href="fig-0023-01a"> - <image file="0023-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0023-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="14"> - <head xml:id="head30" xml:space="preserve">THEOREMA. <num value="14">XIIII</num>.</head> - <p> - <s xml:id="s171" xml:space="preserve">HOcipÅ¿um, alia <reg norm="quoque" type="simple">quoq;</reg> uia licebit Å¿peculari.</s> - </p> - <p> - <s xml:id="s172" xml:space="preserve">Sit linea <var>.a.</var> <reg norm="denotans" type="context">denotãs</reg> numerum diuiÅ¿ibilem, et <var>.o.</var> primi prouenientis linea <var>.e.</var> pri <lb/>mi diuidentis <var>.u.</var> Å¿ecundi prouenientis ideÅ¿t cum <var>.o.</var> pro diuidente Å¿umetur. </s> - <s xml:id="s173" xml:space="preserve">Iam ex <lb/>indicata definitione diuiÅ¿ionis nono theoremate huius libri, dabitur proportio <var>.a.</var> <lb/>ad <var>.o.</var> prout datur <var>.e.</var> ad vnitatem Å¿ignificatam li-<lb/>nea <var>.i.</var> & permutatim <var>.a.</var> ad <var>.e.</var> Å¿icut <var>.o.</var> ad <var>.i.</var> Å¿ed <var>.a.</var> <lb/> - <anchor type="figure" xlink:label="fig-0023-02a" xlink:href="fig-0023-02"/> - ad <var>.u.</var> Å¿ic Å¿e habet prout <var>.o.</var> ad <var>.i.</var> ex eadem definitio-<lb/>ne diuiÅ¿ionis, <reg norm="itaque" type="simple">itaq;</reg> Å¿ic Å¿e habebit <var>.a.</var> ad <var>.u.</var> Å¿icut <var>.a.</var> ad <var>.<lb/>e.</var> vnde <var>.u.</var> æqualis erit <var>.e.</var> ex .9. quinti.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0023-02" xlink:href="fig-0023-02a"> - <image file="0023-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0023-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="15"> - <head xml:id="head31" xml:space="preserve">THEOREMA. <num value="15">XV</num>.</head> - <p> - <s xml:id="s174" xml:space="preserve">VNde prouenit, vt qui velit cognoÅ¿cere cuius numeri quatuor quintæ par-<lb/>tes, Å¿int duæ tertiÄ™, aut quid Å¿imile, <reg norm="conÅ¿ultiſſime" type="context">cõſultiſſime</reg> faciat, Å¿i ad unam <reg norm="eandemque" type="simple">eandemq;</reg> <lb/>denominationem reduxerit.</s> - </p> - <p> - <s xml:id="s175" xml:space="preserve">Prout in propoÅ¿ito exemplo, <reg norm="cum" type="context">cÅ©</reg> <reg norm="denominans" type="context">denominãs</reg> <reg norm="communis" type="context">cõmunis</reg> Å¿it quindecim, cuius duæ ter <lb/>tiæ Å¿unt <reg norm="decem" type="context">decẽ</reg>, & quatuor quintæ duodecim, <reg norm="communis" type="context">cõmunis</reg> <reg norm="autem" type="context">autẽ</reg> denominans .15. multipli <lb/>candus Å¿it per quatuor quintas, Å¿cilicet duodecim, & productum diuidendum per <lb/>duas tertias, hoc eÅ¿t decem, ex quo oriantur decemocto quęſitus numerus?</s> - </p> - <p> - <s xml:id="s176" xml:space="preserve">Quod ad <reg norm="reductionem" type="context">reductionẽ</reg> <reg norm="numeratorum" type="context">numeratorÅ©</reg> ad vnam & eandem denominationem attinet, <lb/>ea de cauÅ¿a fit quo uti poſſimus regula de tribus, quæ tribus tantummodo notis ter-<lb/>minis indiget, quo quartus à prÄ™dictis dependens, inueniri poſſit, quandoquidem <lb/>bini illi reÅ¿pectus, tribus terminis comprehendi <reg norm="poÅ¿sunt" type="context">poÅ¿sÅ©t</reg>. </s> - <s xml:id="s177" xml:space="preserve">At quod ad multiplicatio-<lb/>nem Å¿pectat denominantis <reg norm="communis" type="context">cõmunis</reg> <reg norm="cum" type="context">cÅ©</reg> numerante denominantis in cogniti & diui-<lb/>Å¿ionem producti per numerantem <reg norm="cognitum" type="context">cognitÅ©</reg> illÄ™ nihil aliud Å¿unt, quam <reg norm="quartum" type="context">quartÅ©</reg> <reg norm="terminum" type="context">terminÅ©</reg> <lb/>inuenire, ita proportionatum tertio, vt Å¿ecundus primo.</s> - </p> - <p> - <s xml:id="s178" xml:space="preserve">Excmpli gratia, Å¿it <var>.a.</var> <reg norm="denotans" type="context">denotãs</reg> nume-<lb/>rantem denominantis cogniti, qui Å¿igni <lb/> - <anchor type="figure" xlink:label="fig-0023-03a" xlink:href="fig-0023-03"/> - ficetur linea <var>.o.</var> et <var>.e.</var> Å¿it denominantis in-<lb/>cogniti numerans, denotati linea <var>.u.</var> imò <lb/>verò & cogniti <var>.o.</var> nempe quatuor <lb/>quintæ, Iam Å¿i <var>.o.</var> cum <var>.e.</var> multiplicemus, & productum per <var>.a.</var> diuidemus dabitur <var>.u.</var> <lb/>Å¿ic Å¿e habens ad <var>.e.</var> Å¿icut <var>.o.</var> ad <var>.a.</var> ex .20. Å¿eptimi.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0023-03" xlink:href="fig-0023-03a"> - <image file="0023-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0023-03"/> - </figure> - </div> - <pb o="12" rhead="IO. BAPT. BENED." n="24" file="0024" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0024"/> - </div> - <div type="math:theorem" level="3" n="16"> - <head xml:id="head32" xml:space="preserve">THEOREMA <num value="16">XVI</num>.</head> - <p> - <s xml:id="s179" xml:space="preserve">INuenire autem cupienti cuius numeri, duæ tertiæ, Å¿int quatuor quintÄ™ partes, mulÂ<lb/>tiplicandÄ™ eſſent duæ tertiæ per denominantem communem, & productum diui-<lb/>dendum per quatuor quintas ipÅ¿ius de-<lb/>nominantis. </s> - <s xml:id="s180" xml:space="preserve">Ac Å¿i quis diceret Å¿i <var>.e.</var> dat <var>.<lb/> - <anchor type="figure" xlink:label="fig-0024-01a" xlink:href="fig-0024-01"/> - o.</var> quid dabit <var>.a</var>? </s> - <s xml:id="s181" xml:space="preserve">nempe dabit <var>.u.</var> nam in <lb/>propoÅ¿ito exemplo, terminus <var>.a.</var> loco <var>.e.</var> <lb/>duos Å¿ortietur denominantes, cognitum <lb/>videlicet <var>.o.</var> et <var>.u.</var> incognitum quod po-<lb/>Å¿tea cognitum oritur ex regula de tribus, vt dictum eÅ¿t.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0024-01" xlink:href="fig-0024-01a"> - <image file="0024-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0024-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="17"> - <head xml:id="head33" xml:space="preserve">THEOREMA <num value="17">XVII</num>.</head> - <p> - <s xml:id="s182" xml:space="preserve">QVA ratione cognoÅ¿ci poterit proportionem quantitatis cenÅ¿icæ cenÅ¿icæ ad <lb/>Å¿imilem quantitatem quadruplam eſſe ad eam, quæ eÅ¿t Å¿uarum radicum; </s> - <s xml:id="s183" xml:space="preserve">pro-<lb/>portionem <reg norm="autem" type="context">autẽ</reg> primarum relatarum eſſe quintuplam, <reg norm="atque" type="simple">atq;</reg> ita deinceps?</s> - </p> - <p> - <s xml:id="s184" xml:space="preserve">Cuiusrei gratia, <reg norm="Å¿ciendus" type="context">Å¿ciẽdus</reg> eÅ¿t modus <reg norm="productionis" type="simple">ê“ductionis</reg> <reg norm="harum" type="context">harÅ©</reg> <reg norm="dignitatum" type="context">dignitatÅ©</reg> qui oritur ex produ-<lb/>ctione primæ radicis in Å¿eipÅ¿am, prout qui <reg norm="cubum" type="context">cubÅ©</reg> requirit, ducat radicé in Å¿uo quadra-<lb/>to, & orietur cubus, hæc poÅ¿tea ducta in cubum, <reg norm="quantitatem" type="context">quantitatẽ</reg> cenÅ¿icam <reg norm="cenÅ¿icam" type="context">cenÅ¿icã</reg>, et in <lb/>hanc, prædictam radicem, dabit quantitatem primam relatam. </s> - <s xml:id="s185" xml:space="preserve">Quod vbi Å¿ciueri-<lb/>mus, meminiſſe oportet Euclidem decimaoctaua Å¿exti aut .11. octaui docere, pro-<lb/>portionem quadrati ad <reg norm="quadratum" type="context">quadratÅ©</reg>, duplam eſſe proportioni Å¿uarum radicum, & .36. <lb/>vndecimi aut .11. octaui, cubi ad <reg norm="cubum" type="context">cubÅ©</reg> triplam eſſe, ego verò nunc aſſero, cenÅ¿ici cen <lb/>Å¿ici ad radicum proportionem quadruplam eſſe, primi verò relati ad primum re-<lb/>latum quintuplam <reg norm="atque" type="simple">atq;</reg> ita gradatim.</s> - </p> - <p> - <s xml:id="s186" xml:space="preserve">Cuius Å¿peculationis gratia, detur linea <var>.d.</var> quæ cubum maiorem Å¿ignificet. et <var>.b.</var> <lb/>minorem <var>.c.</var> verò Å¿it radixipÅ¿ius <var>.d.</var> et <var>.e.</var> ipÅ¿ius <var>.b.</var> ita ordinate adinuicem, vt in Å¿ub-<lb/>Å¿cripta figura cernitur. </s> - <s xml:id="s187" xml:space="preserve">Iam <var>.c.</var> cum <var>.d.</var> producatur <reg norm="proueniatque" type="simple">proueniatqÌ;</reg> <var>.q.</var> cenÅ¿icum cenÅ¿i-<lb/>cum, tum producatur <var>.e.</var> cum <var>.b.</var> et dabitur <var>.p.</var> alterum cenÅ¿icum cenÅ¿icum. </s> - <s xml:id="s188" xml:space="preserve">Dico <lb/>igitur proportionem <var>.q.</var> ad <var>.p.</var> quadruplam eſſe proportioni <var>.c.</var> ad <var>.e.</var> hac de <lb/>cauÅ¿a quòd proportio <var>.q.</var> ad <var>.p.</var> compo-<lb/>natur ex proportione <var>.d.</var> ad <var>.b.</var> et <var>.c.</var> ad <var>.e.</var> <lb/> - <anchor type="figure" xlink:label="fig-0024-02a" xlink:href="fig-0024-02"/> - prout facile ex .24. Å¿exti, aut quinta octaui <lb/>deprÄ™henditur. </s> - <s xml:id="s189" xml:space="preserve">Quare <reg norm="cum" type="context">cÅ©</reg> proportio <var>.d.</var> ad <var>.<lb/>b.</var> proportioni <var>.c.</var> ad <var>.e.</var> tripla Å¿it, patet pro-<lb/>portionem <var>.q.</var> ad <var>.p.</var> quadruplam eſſe pro-<lb/>portioni <var>.c.</var> ad <var>.e</var>. </s> - <s xml:id="s190" xml:space="preserve">Idem de cæteris dignitati <lb/>bus dico, Å¿umptis Å¿emper <var>.d</var> et <var>.b.</var> pro duo-<lb/>bus cenÅ¿ibus cenÅ¿uum, aut duobus primis relatis, aut alio quouis axiomate.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0024-02" xlink:href="fig-0024-02a"> - <image file="0024-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0024-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="18"> - <head xml:id="head34" xml:space="preserve">THEOREMA. <num value="18">XVIII</num>.</head> - <p> - <s xml:id="s191" xml:space="preserve">CVR diuidentibus nobis dignitatem, per dignitatem, radix prouenientis: </s> - <s xml:id="s192" xml:space="preserve">pro <lb/>ueniens Å¿it diuiÅ¿ionis vnius radicis per alteram?</s> - </p> - <p> - <s xml:id="s193" xml:space="preserve">Sint exempli gratia duÄ™ lineæ <var>.b.q.</var> et <var>.f.g.</var> quæ Å¿ignificent duas radices cuiuÅ¿uis <lb/>dignitatis; </s> - <s xml:id="s194" xml:space="preserve"><reg norm="demusque" type="simple">demusqÌ;</reg> eſſe radices duorum quadratorum, <reg norm="quadratumque" type="simple">quadratumqÌ;</reg> ipÅ¿ius <var>b.q.</var> <lb/>per quadratum ipÅ¿ius <var>.f.g.</var> diuidatur; </s> - <s xml:id="s195" xml:space="preserve">quadrataqÌue radix prouenientis Å¿it <var>.d.q.</var> <lb/>vnitas verò linearis Å¿it <var>.i.g</var>. </s> - <s xml:id="s196" xml:space="preserve">Dico ipÅ¿am <var>.d.q.</var> eſſe proueniens ex diuiÅ¿ione <var>.b.q.</var> <lb/>per <var>.f.g</var>. </s> - <s xml:id="s197" xml:space="preserve">Patet enim ex definitione diuiÅ¿ionis nono theoremate tradita quadra- - <pb o="13" rhead="THEOR. ARITH." n="25" file="0025" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0025"/> - tum ipÅ¿ius <var>.d.q.</var> talem eſſe partem quadrati ipÅ¿ius <var>.b.q.</var> qualis quadratum ipÅ¿ius <var>.g.i.</var> <lb/>eÅ¿t quadrati ipÅ¿ius <var>.f.g</var>. </s> - <s xml:id="s198" xml:space="preserve">Scimus prÄ™terea ex .19. Å¿exti, aut vndecima octaui, propor-<lb/>tioné quadrati ipÅ¿ius <var>.b.q.</var> ad <reg norm="quadratum" type="context">quadratÅ©</reg> ipÅ¿ius <var>.d.q.</var> duplam eſſe proportioni <var>.b.q.</var> ad <var>.<lb/>d.q.</var> Å¿uarum radicum (cuborum enim tripla eſſet & cenÅ¿uum cenÅ¿uum, quadrupla, <lb/><reg norm="atque" type="simple">atq;</reg> ita deinceps ex præcedenti theoremate) Id ipÅ¿um dico de dignitatibus ipÅ¿ius <var>.<lb/>f.g.</var> et <var>.i.g.</var> reÅ¿pectu radicum <var>.f.g.</var> et <var>.i.g</var>. </s> - <s xml:id="s199" xml:space="preserve">Vnde <lb/>cum proportio dignitatis ipÅ¿ius <var>.b.q.</var> ad il-<lb/>lam <var>.d.q.</var> Ä™qualis Å¿it proportioni dignitatis <lb/> - <anchor type="figure" xlink:label="fig-0025-01a" xlink:href="fig-0025-01"/> - ipÅ¿ius <var>.f.g.</var> ad illam <var>.g.i.</var> ex communi Å¿cien-<lb/>tia apertè cognoÅ¿cemus Å¿implices propor-<lb/>tiones eſſe interÅ¿e æquales, nempe eam quÄ™ <lb/>eÅ¿t <var>.b.q.</var> ad <var>.d.q.</var> æqualem eſſe ei, quæ eÅ¿t <var>.f.<lb/>g.</var> ad <var>.i.g.</var> <reg norm="itaque" type="simple">itaq;</reg> Å¿equitur ex definitione diuiÅ¿ionis <var>.d.q.</var> eſſe proueniens ex diuiÅ¿ione <var>.<lb/>b.q.</var> per <var>.f.g</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0025-01" xlink:href="fig-0025-01a"> - <image file="0025-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0025-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="19"> - <head xml:id="head35" xml:space="preserve">THEOREMA <num value="19">XVIIII</num>.</head> - <p> - <s xml:id="s200" xml:space="preserve">CVR productum ex duabus radicibus quadratis, eÅ¿t quadrata radix, producti <lb/>Å¿uorum quadratorum Å¿imul?</s> - </p> - <p> - <s xml:id="s201" xml:space="preserve">In cuius rei gratiam, Å¿int duo quadrata <var>.d.a.</var> et <var>n.o.</var> coniuncta Å¿imul, prout in Å¿ub-<lb/>Å¿cripta figura apparet, ita tamen vtangulus <var>.a.n.u.</var> Å¿itre <lb/>ctus, </s> - <s xml:id="s202" xml:space="preserve">quare ex quartadecima primi, duo latera <var>.n.c.</var> et <var>.<lb/> - <anchor type="figure" xlink:label="fig-0025-02a" xlink:href="fig-0025-02"/> - n.a.</var> directe <reg norm="coniungentur" type="context">coniÅ©gentur</reg> adinuicem, prout etiam reli-<lb/>qua duo latera <var>.n.u.</var> et <var>.n.d</var>. </s> - <s xml:id="s203" xml:space="preserve">Cogitato deinde <var>.a.u.</var> pro <lb/>ducto ipÅ¿ius <var>.a.n.</var> in <var>.n.u.</var> duarum videlicet radicum <lb/>quadratarum Å¿imul, dabitur ex prima Å¿exti, aut de-<lb/>cimaottaua Å¿eptimi, productum <var>.a.u.</var> medium propor <lb/>tionale inter quadratum <var>.a.d.</var> et <var>.u.c.</var> quod Å¿i cogi-<lb/>temus has tres Å¿uperficies, tres numeros eſſe, pate-<lb/>bit ex vigeÅ¿imaprima Å¿eptimi productum <var>.a.u.</var> in Å¿e-<lb/>ipÅ¿um, quadratum Å¿cilicet <var>.a.u.</var> æquale eſſe producto <var>.<lb/>a.d.</var> in <var>.u.c.</var> ex quo propoÅ¿iti euidentia conÅ¿equetur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0025-02" xlink:href="fig-0025-02a"> - <image file="0025-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0025-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="20"> - <head xml:id="head36" xml:space="preserve">THEOREMA <num value="20">XX</num>.</head> - <p> - <s xml:id="s204" xml:space="preserve">QVA ratione id ipÅ¿um in cubis cognoÅ¿ci poterit. <lb/></s> - <s xml:id="s205" xml:space="preserve">Sit cubus <var>.l.b.</var> & cubus <var>.o.p.</var> quorum productum Å¿it <var>.u.g.</var> quod aſſero eÅ¿le <lb/> - <anchor type="figure" xlink:label="fig-0025-03a" xlink:href="fig-0025-03"/> - cubum, quamuis Eucli. idem probet <lb/>in <ref id="ref-0008">.4. noni.</ref> cuius radicem demonÅ¿tra-<lb/>bo eſſe numeri æqualis numero <var>.m.q.</var> <lb/>qui <var>.m.q.</var> productum eÅ¿t ipÅ¿ius <var>.m.e.</var> in <var>.e.<lb/>q.</var> radicum propoÅ¿itorum cuborum. </s> - <s xml:id="s206" xml:space="preserve">Pa-<lb/>tet enim ex præcedenti theoremate <var>.m.<lb/> - <anchor type="figure" xlink:label="fig-0025-04a" xlink:href="fig-0025-04"/> - <pb o="14" rhead="IO. BAPT. BENED." n="26" file="0026" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0026"/> - q.</var> radicem eſſe quadratam producti <var>.l.e.</var> in <var>.e.p.</var> quod <reg norm="productum" type="context">productÅ©</reg> Å¿it quadratuni<unsure/> <lb/>corporeum <var>.c.g.</var> cogitemus pariter duo quadrata <var>.l.e.</var> et <var>.e.p.</var> eſſe pariter corpo-<lb/>rea, tantÄ™ profunditatis, quantam, vnitas linearis radicum <var>.m.e.</var> et <var>.e.q.</var> requirit. <lb/></s> - <s xml:id="s207" xml:space="preserve">Hæc duo corpora producentur à ſuperficie in vnitatem, <reg norm="vocenturque" type="simple">vocenturqÌ;</reg> <var>.l.x.</var> et <var>.x.p.</var> quo <lb/>facto, cogitemus corpus <var>.a.g.</var> tamquam productum cubi <var>.l.b.</var> in quadratum <var>.e.p</var>. </s> - <s xml:id="s208" xml:space="preserve">Vn-<lb/>de ex decimaoctaua, aut decimanona Å¿eptimi, eadem erit proportio <var>.a.g.</var> ad <var>.c.g.</var> <lb/>quæ eÅ¿t <var>.l.b.</var> ad <var>.l.x.</var> corporeum, Å¿ed ex .25. vndecimi & prima Å¿exti, ita Å¿e habet <var>.a.K.</var> <lb/>ad <var>.K.c.</var> vnitatem linearé Å¿icut <var>.a.g.</var> ad <var>.c.g.</var> & ex <reg norm="eiÅ¿dem" type="context">eiÅ¿dẽ</reg> ita Å¿e habebit <var>.b.e.</var> ad <var>.e.x.</var> vnita-<lb/>tem linearem, Å¿icut <var>.l.b.</var> ad quadratum <var>.l.x.</var> corporeum. </s> - <s xml:id="s209" xml:space="preserve">Itaque Å¿ic Å¿e habebit <var>.b.e.</var> ad <lb/>vnitatem linearem <var>.e.x.</var> videlicet <var>.K.c.</var> Å¿icut <var>.a.K.</var> ad ipÅ¿am <var>.K.c</var>. </s> - <s xml:id="s210" xml:space="preserve">Vnde ex nona quinti <var>.<lb/>a.K.</var> æqualis erit <var>.e.b.</var> & conÅ¿equenter æqualis <var>.m.e</var>. </s> - <s xml:id="s211" xml:space="preserve">Iam verò Å¿it <var>.u.g.</var> productum <var>.l.b.</var> <lb/>cubi, in cubum <var>.o.p.</var> vt Å¿upra dictum eÅ¿t, Hinc patebit ex quauis duarum propoÅ¿itio-<lb/>num, decimaoctaua, aut decimanona Å¿eptimi, eandem futuram proportionem <var>.u.g.</var> <lb/>ad <var>.a.g.</var> quæ eÅ¿t <var>.o.p.</var> ad <var>.x.p.</var> quadratum corporeum. </s> - <s xml:id="s212" xml:space="preserve">Quare ex poÅ¿tremis, dictis ratio-<lb/>nibus, eadem erit proportio <var>.u.K.</var> ad <var>.a.K.</var> quæ eÅ¿t <var>.o.e.</var> ad vnitatem linearem <var>.e.x.</var> at <lb/>ex dictis decimaoctaua & decimanona Å¿eptimi, ita Å¿e habet <reg norm="numerus" type="simple">numerê°</reg> <var>.m.q.</var> ad <reg norm="numerum" type="context">numerÅ©</reg> <lb/><reg norm="Å¿uperficialem" type="context">Å¿uperficialẽ</reg> <var>.m.e.</var> qui <reg norm="producitur" type="simple">ê“ducitur</reg> à lineari <var>.m.e.</var> in vnitaté <reg norm="linearem" type="context">linearẽ</reg> ipÅ¿ius <var>.e.q.</var> Å¿icut nume <lb/>rus <var>.q.e.</var> ad Å¿uam vnitaté, Å¿ed <reg norm="cum" type="context">cÅ©</reg> numerus <var>.a.K.</var> æqualis Å¿it numero <var>.m.e.</var> vt <reg norm="probatum" type="context">probatÅ©</reg> eÅ¿t <lb/>erit ergo ex vndecima & nona quinti, numerus <var>.u.K.</var> æqualis numero <var>.m.q</var>. </s> - <s xml:id="s213" xml:space="preserve">At <var>.f.g.</var> <lb/>pariter æqualis eÅ¿t numero <var>.m.q.</var> ex præcedenti theoremate, vnde <var>.K.u.</var> pariter æqua <lb/>lis erit <var>.f.g</var>. </s> - <s xml:id="s214" xml:space="preserve">Itaque Å¿equitur <var>.u.g.</var> cubum eſſe, & <var>f.g.</var> radicem ipÅ¿ius, æqualem numero <var>.<lb/>m.q.</var> quod quærebatur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0025-03" xlink:href="fig-0025-03a"> - <image file="0025-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0025-03"/> - </figure> - <figure xlink:label="fig-0025-04" xlink:href="fig-0025-04a"> - <image file="0025-04" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0025-04"/> - </figure> - </div> - <figure position="here"> - <image file="0026-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0026-01"/> - </figure> - <figure position="here"> - <image file="0026-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0026-02"/> - </figure> - </div> - <div type="math:theorem" level="3" n="21"> - <head xml:id="head37" xml:space="preserve">THEOREMA <num value="21">XXI</num>.</head> - <p> - <s xml:id="s215" xml:space="preserve">VT autem in uniuerÅ¿um Å¿ciri poſſit totum <reg norm="infinitum" type="context">infinitÅ©</reg> dignitatum, hoc eÅ¿t radicem <lb/>producti duarum dignitatum Å¿imilium, productum eſſe duarum radicum ea-<lb/>rundem dignitatum.</s> - </p> - <p> - <s xml:id="s216" xml:space="preserve">Ponamus, exempli gratia, duas radices quadratas <var>.q.p.</var> et <var>.g.K.</var> incognitas, quas <lb/>qui velit adinuicem multiplicare, cogatur earum quadrata cognita <var>.n.</var> cum <var>.i.</var> multi-<lb/>plicare, quorum productum Å¿it quadratum <var>.m.</var> radix cuius Å¿it <var>.b.d.</var> quam dico æqualé - <pb o="15" rhead="THEOREM. ARIT." n="27" file="0027" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0027"/> - eſſe <reg norm="producto" type="simple">ê“ducto</reg> <var>.q.p.</var> in <var>.g.k.</var> <reg norm="quod" type="wordlist">qÄ‘</reg> <reg norm="autem" type="context">autẽ</reg> Å¿it <var>.o</var>. </s> - <s xml:id="s217" xml:space="preserve">Patet enim <reg norm="proportionem" type="context">proportionẽ</reg> <var>.o.</var> ad <var>.q.p.</var> <reg norm="eandem" type="context">eandẽ</reg> eſſe <lb/>cum proportione <var>.g.k.</var> ad Å¿uam vnitatem linearem, ex decimaoctaua, aut decima-<lb/>nona Å¿eptimi, hæc vero vnitas linearis Å¿it <var>.t.</var> cuius Å¿uperficialis Å¿it <var>.u.</var> vnitas Å¿cilicet to-<lb/>ties in Å¿eipÅ¿am multiplicata quoties propoÅ¿ita dignitas patitur, tametÅ¿i in præſen <lb/>ti exemplo quadrata dignitas Å¿umatur. </s> - <s xml:id="s218" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> ex eiÅ¿dem propoÅ¿itionibus decimaocta <lb/>ua aut decimanona, Å¿ic Å¿e habet <var>.m.</var> ad <var>.n.</var> Å¿icut <var>.i.</var> ad <var>.u</var>. </s> - <s xml:id="s219" xml:space="preserve">Scimus prÄ™terea <reg norm="proportionem" type="context">proportionẽ</reg> <var>.<lb/>m.</var> ad <var>.n.</var> (eo quod in propoÅ¿ito exemplo Å¿int quadrata) duplam eſſe proportioni <var>.b.<lb/>d.</var> ad <var>.q.p.</var> et ipÅ¿ius <var>.i.</var> ad <var>.u.</var> pariter duplam proportioni <var>.g.k.</var> ad <var>.t.</var> iam autem dictum <lb/>fuit Å¿ic Å¿e habere <var>.m.</var> ad <var>.n.</var> Å¿icut <var>.i.</var> ad <var>.u</var>. </s> - <s xml:id="s220" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> <var>.<lb/>b.d.</var> Å¿ic Å¿e habebit ad <var>.q.p.</var> Å¿icut <var>.g.k.</var> ad <var>.t.</var> <lb/> - <anchor type="figure" xlink:label="fig-0027-01a" xlink:href="fig-0027-01"/> - quandoquidem Å¿ic Å¿e habeattotum ad <reg norm="to- tum" type="context">to-<lb/>tÅ©</reg>, Å¿icut pars ad <reg norm="partem" type="context">partẽ</reg>, <reg norm="dum" type="context">dÅ©</reg> Å¿imiles Å¿int, proba <lb/><reg norm="tum" type="context">tÅ©</reg> <reg norm="autem" type="context">autẽ</reg> eÅ¿t Å¿uperius ita Å¿e habere <var>.o.</var> ad <var>.q.p.</var> <lb/>Å¿icut <var>.g.k.</var> ad <var>.t.</var> <reg norm="itaque" type="simple">itaq;</reg> <var>.o.</var> Å¿ic Å¿e habebit ad <var>.q.p.</var> <lb/>Å¿icut <var>.b.d.</var> ad <var>.q.p.</var> vnde <var>.o.</var> æqualis erit <var>.b.d.</var> <lb/>HocipÅ¿um cÄ™teris dignitatibus conueniet, <lb/>mutatis tantummodo proportionibus <var>.m.<lb/>n.</var> ad proportionem <var>.b.d</var>: <var>q.p.</var> Å¿ic propor-<lb/>tionibus duarum dignitatum <var>.i.u.</var> ad pro-<lb/>portionem Å¿uarum radicum <var>.g.k.t</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0027-01" xlink:href="fig-0027-01a"> - <image file="0027-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0027-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="22"> - <head xml:id="head38" xml:space="preserve">THEOREMA <num value="22">XXII</num>.</head> - <p> - <s xml:id="s221" xml:space="preserve"><emph style="sc">DOcent</emph> veteres, quòd Å¿i quilibet numerus in duas partes inæquales diuiÅ¿us <lb/>fuerit, <reg norm="totumque" type="simple">totumqÌ</reg> diuiÅ¿um per <reg norm="vnam" type="context">vnã</reg> partium, & per eandem pars altera diuiÅ¿a fue-<lb/>rit: </s> - <s xml:id="s222" xml:space="preserve">differentia prouenientium Å¿emper vnitas erit. </s> - <s xml:id="s223" xml:space="preserve">quodquidem veriſſimum eÅ¿t.</s> - </p> - <p> - <s xml:id="s224" xml:space="preserve">Detur enim <var>.b.d.</var> propoÅ¿itus numerus in duas partes inæquales diuiÅ¿us <var>.b.c.</var> et <var>.c.d.</var> <lb/>& in primis <reg norm="totum" type="context">totÅ©</reg> <var>.b.d.</var> per <var>.c.d.</var> diuidatur, ex quo oriatur <var>e.o.</var> vnitas autem <reg norm="per" type="punctuation simple">.ê‘</reg> <var>.i.o.</var> Å¿igni-<lb/>ficetur, tum pars ipÅ¿a <var>.b.c.</var> <reg norm="per" type="simple punctuation">ê‘.</reg> <reg norm="eandem" type="context">eãdem</reg> <var>.c.d.</var> diuidatur, <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <reg norm="proueniens" type="context">proueniẽs</reg> <var>.a</var>. </s> - <s xml:id="s225" xml:space="preserve">Sanè ex defini-<lb/>tione diuiÅ¿ionis, eadem erit proportio <var>.b.d.</var> ad <var>.e.o.</var> quæ eÅ¿t <var>.c.d.</var> ad <var>.i.o.</var> et ita <var>.b.c.</var> ad <var>.a.</var> <lb/>Å¿icut <var>.c.d.</var> ad <var>.i.o</var>. </s> - <s xml:id="s226" xml:space="preserve">Ex <ref id="ref-0009">.19. autem quinti</ref>, ita Å¿e habet <var>.b.c.</var> ad <var>.e.i.</var> Å¿icut <var>.b.d.</var> ad <var>.e.o.</var> at <var>.b.d.</var> <lb/>ad <var>.e.o.</var> Å¿ic Å¿e habet Å¿icut <var>.c.d.</var> ad <var>.i.o.</var> hoc eÅ¿t Å¿icut <var>.b.c.</var> ad <var>.a</var>. </s> - <s xml:id="s227" xml:space="preserve">Quare ex .11. quinti Å¿ic Å¿e <lb/>habebit <var>.b.c.</var> ad <var>.e.i.</var> Å¿icut .ad <var>.a.</var> ex quo ex .9. <reg norm="praedi cti" type="simple">prÄ™diÂ<lb/>cti</reg> <var>.a.</var> æqualis erit <var>.e.i.</var> Å¿ed <var>.e.i.</var> minor eÅ¿t <var>.e.o.</var> <lb/> - <anchor type="figure" xlink:label="fig-0027-02a" xlink:href="fig-0027-02"/> - per <var>.i.o</var>. </s> - <s xml:id="s228" xml:space="preserve">Quare Å¿equitur propoÅ¿itum verum eÅ¿Â<lb/>Å¿e. </s> - <s xml:id="s229" xml:space="preserve">Quod ipÅ¿um pauciſſimis verbis Å¿ic definiri <lb/>poteÅ¿t, Å¿i dixerimus, eiuÅ¿modi diuidens .in par-<lb/>te diuiÅ¿ibili, <reg norm="quam" type="context">quã</reg> in toto, Å¿emel minus ingredi, <lb/>quandoquidem altera pars eÅ¿t, ex qua totum integrum perficitur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0027-02" xlink:href="fig-0027-02a"> - <image file="0027-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0027-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="23"> - <head xml:id="head39" xml:space="preserve">THEOREMA <num value="23">XXIII</num>.</head> - <p> - <s xml:id="s230" xml:space="preserve">HOcipÅ¿um alia ratione contemplari poÂ<lb/> - <anchor type="figure" xlink:label="fig-0027-03a" xlink:href="fig-0027-03"/> - terimus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0027-03" xlink:href="fig-0027-03a"> - <image file="0027-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0027-03"/> - </figure> - </div> - <p> - <s xml:id="s231" xml:space="preserve">Significetur enim totalis numerus per <var>.a.e.</var> <lb/>in duas partes diuiÅ¿us <var>.a.u.</var> et <var>.u.e.</var> totius autem diuidens Å¿it <var>.u.e.</var> & partis alterius <var>.a.u.</var> <lb/>totius verò <reg norm="proueniens" type="context">proueniẽs</reg> Å¿it <var>.a.c.</var> partis <reg norm="autem" type="context">autẽ</reg>, Å¿it <reg norm="proueniens" type="context">proueniẽs</reg> <var>.a.n.</var> tum differentia Å¿it <var>.n.c.</var> vni - <pb o="16" rhead="IO. BAPT. BENED." n="28" file="0028" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0028"/> - tas vero cui <reg norm="differentiam" type="context">differentiã</reg> <var>.n.c.</var> æquari dico, Å¿it <var>.a.i</var>. </s> - <s xml:id="s232" xml:space="preserve">Patet enim in primis, eandem propor <lb/>tionem eſſe <var>.a.e.</var> ad <var>.a.c.</var> quæ eÅ¿t <var>.u.e.</var> ad <var>.a.i.</var> ex definitione diuiÅ¿ionis, et eandem <lb/>eſſe <var>.a.u.</var> ad <var>.a.n.</var> quæ eÅ¿t <var>.u.e.</var> ad <var>.a.i.</var> vnde ex .<lb/>11. quinti Å¿ic Å¿e habebit <var>.a.e.</var> ad <var>.a.c.</var> Å¿icut <var>.a.<lb/> - <anchor type="figure" xlink:label="fig-0028-01a" xlink:href="fig-0028-01"/> - u.</var> ad <var>.a.n.</var> et ex .19. eiuÅ¿dem Å¿ic Å¿e habe-<lb/>bit <var>.u.e.</var> ad <var>.n.c.</var> Å¿icut <var>.a.e.</var> ad <var>.a.c.</var> Å¿ed. Å¿ic Å¿e <lb/>habebat <var>.u.e.</var> ad <var>.a.i</var>. </s> - <s xml:id="s233" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> ex prædicta .11. quinti, Å¿ic Å¿e habebit <var>.u.e.</var> ad <var>.n.c.</var> Å¿icut ad <var>.a.<lb/>i</var>. </s> - <s xml:id="s234" xml:space="preserve">Quare ex .9. eiuÅ¿dem <var>.n.c.</var> æqualis erit <var>.a.i.</var> etidcirco <var>.n.c.</var> pariter vnitas erit.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0028-01" xlink:href="fig-0028-01a"> - <image file="0028-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0028-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="24"> - <head xml:id="head40" xml:space="preserve">THEOREMA <num value="24">XXIIII</num>.</head> - <p> - <s xml:id="s235" xml:space="preserve"><emph style="sc">CVr</emph> quibuslibet duobus numeris diuiÅ¿is adinuicem, <reg norm="multiplicatisque" type="simple">multiplicatisqÌ</reg> prouenien <lb/>tibus Å¿imul, productum, Å¿emper eÅ¿t vnitas Å¿uperficialis? </s> - <s xml:id="s236" xml:space="preserve">Nempe ex .20. Å¿eptimi, <lb/>quoniam vnitas linearis Å¿emper media proportionalis eÅ¿t inter bina prouenientia. <lb/></s> - <s xml:id="s237" xml:space="preserve">Quodita Å¿pecularilicet.</s> - </p> - <p> - <s xml:id="s238" xml:space="preserve"><reg norm="Significentur" type="context">Significẽtur</reg> duo propoÅ¿iti numeri per <var>.b.p.</var> et <var>.b.d.</var> mutuo diuiÅ¿i, proueniens au-<lb/>tem <var>.b.p.</var> per <var>.b.d.</var> diuiÅ¿um Å¿it <var>.b.n.</var> tum proueniens <var>.b.d.</var> diuiÅ¿um per <var>.b.p.</var> Å¿it <var>.b.a.</var> <lb/>et <var>.b.t.</var> Å¿it vnitas <var>.b.p.</var> et <var>.b.e.</var> vnitas <var>.b.d.</var> ex quo <var>.b.t.</var> æqualis erit <var>.b.e</var>.</s> - </p> - <p> - <s xml:id="s239" xml:space="preserve">Iam ex definitio ne diuiÅ¿ionis, dabitur eadem proportio <var>.b.p.</var> ad <var>.b.n.</var> quæ eÅ¿t <var>.b.d.</var> <lb/>ad <var>.b.e.</var> et proportio <var>.b.d.</var> ad <var>.b.a.</var> quæ eÅ¿t <var>.b.p.</var> ad <var>.b.t</var>. </s> - <s xml:id="s240" xml:space="preserve">Sed cum Å¿ic Å¿e habeat <var>.b.<lb/>p.</var> ad <var>.b.n.</var> Å¿icut <var>.b.d.</var> ad <var>.b.e.</var> permutando Å¿ic Å¿e habebit <var>.b.p.</var> ad <var>.b.d.</var> Å¿icut <var>.b.n.</var> ad <var>.b.<lb/>e.</var> hoc eÅ¿t ad <var>.b.t.</var> et cum Å¿ic Å¿e habeat <var>.b.d.</var> ad <var>.b.a.</var> Å¿icut <var>.b.p.</var> ad <var>.b.t</var>: permutando Å¿ic Å¿e <lb/>habebit <var>.b.d.</var> ad <var>.b.p.</var> Å¿icut <var>.b.a.</var> ad <var>.b.t</var>. <lb/></s> - <s xml:id="s241" xml:space="preserve">Quare euerÅ¿im Å¿ic Å¿e habebit <var>.b.p.</var> ad <var>.<lb/> - <anchor type="figure" xlink:label="fig-0028-02a" xlink:href="fig-0028-02"/> - <lb/>b.d.</var> Å¿icut <var>.b.t.</var> ad <var>.b.a.</var> Å¿ed <var>.b.n.</var> ad <var>.b.t.</var> Å¿ic <lb/>Å¿e habebat vt <var>.b.p.</var> ad <var>.b.d</var>. </s> - <s xml:id="s242" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> ex .11. <lb/>quintiÅ¿ic Å¿e habebit <var>.b.n.</var> ad <var>.b.t.</var> Å¿icut <var>.b.<lb/> - <anchor type="figure" xlink:label="fig-0028-03a" xlink:href="fig-0028-03"/> - e.</var> ad <var>.b.a</var>. </s> - <s xml:id="s243" xml:space="preserve">Dictum autem eÅ¿t <var>.b.e.</var> et <var>.b.t.</var> idem omnino eſſe. </s> - <s xml:id="s244" xml:space="preserve">Quare ex .20. Å¿eptimi pro-<lb/>poÅ¿iti veritas innoteÅ¿cet.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0028-02" xlink:href="fig-0028-02a"> - <image file="0028-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0028-02"/> - </figure> - <figure xlink:label="fig-0028-03" xlink:href="fig-0028-03a"> - <image file="0028-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0028-03"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="25"> - <head xml:id="head41" xml:space="preserve">THEOREMA <num value="25">XXV</num>.</head> - <p> - <s xml:id="s245" xml:space="preserve">IDipÅ¿um & hac altera uia patebit.</s> - </p> - <p> - <s xml:id="s246" xml:space="preserve">Duo illi numeri per <var>.o.</var> et <var>.u.</var> Å¿ignificentur mutuo diuiÅ¿i, proueniens <reg norm="autem" type="context">autẽ</reg> <var>.o.</var> per <var>.<lb/>u.</var> Å¿it <var>.e.</var> et proueniens <var>.u.</var> per <var>.o.</var> Å¿it <var>.x.</var> vnitas uerò per <var>.i.</var> Å¿ignificetur, quas tamen quanti-<lb/>tates Å¿ubÅ¿cripto modo ad inuicem diÅ¿poni-<lb/>to. </s> - <s xml:id="s247" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> ex definitione diuiÅ¿ionis, eadem erit <lb/> - <anchor type="figure" xlink:label="fig-0028-04a" xlink:href="fig-0028-04"/> - proportio <var>.o.</var> ad <var>.e.</var> quÄ™ eÅ¿t <var>.u.</var> ad <var>.i.</var> et <var>.o.</var> ad <var>.i.</var> quÄ™ <lb/>eÅ¿t <var>.u.</var> ad <var>.x</var>. </s> - <s xml:id="s248" xml:space="preserve">Quare ex æqualitate <reg norm="proportionum" type="context">proportionÅ©</reg> <var>.<lb/>c.</var> ad <var>.i.</var> Å¿ic Å¿e habebit Å¿icut <var>.i.</var> ad <var>.x.</var> erit enim <var>.i.</var> <lb/>media proportionalis inter <var>.e.</var> et <var>.x.</var> ex .20. <reg norm="autem" type="context">autẽ</reg> <lb/>Å¿eptimi propoÅ¿itum concludetur. </s> - <s xml:id="s249" xml:space="preserve">HuiuÅ¿modi rei cauÅ¿a etiam eÅ¿t, quod proueniens <lb/>diuiÅ¿ionis vnius eÅ¿t numerator æqualis denominatori diuiÅ¿ionis alterius.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0028-04" xlink:href="fig-0028-04a"> - <image file="0028-04" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0028-04"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="26"> - <head xml:id="head42" xml:space="preserve">THEOREMA <num value="26">XXVI</num>.</head> - <p> - <s xml:id="s250" xml:space="preserve"><emph style="sc">CVr</emph> duobus numeris mutuo diuiÅ¿is, <reg norm="sumptis" type="context">sÅ©ptis</reg> deinde prouenientibus Å¿imul et adinui <lb/>cem, & per hanc Å¿ummam, diuiÅ¿a Å¿umma quadratorum dictorum <reg norm="propoÅ¿itorum" type="context">propoÅ¿itorÅ©</reg> - <pb o="17" rhead="THEOREM. ARITH." n="29" file="0029" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0029"/> - numerorum, proueniat numerus æqualis numero producti duorum primorum nu-<lb/>m erorum Å¿imul.</s> - </p> - <p> - <s xml:id="s251" xml:space="preserve">Sint exempli gratia propoÅ¿iti numeri .2. et .8. qui mutuo diuiÅ¿i in primis dent pro <lb/>uenientia quatuor integra, tum quartam partem pro altero proueniente, hæc colle-<lb/>cta dabunt Å¿ummam quatuor integrorum et quartæ partis vnius, Å¿umma autem qua <lb/>dratorum binarij & octonarij erit .68. qui quidem numerus per quatuor & quar <lb/>tam partem vnius diuiÅ¿us dabit .16. pro proueniente, quæ .16. æqualia erunt pro <lb/>ducto binarii in octonarium.</s> - </p> - <p> - <s xml:id="s252" xml:space="preserve">Cuius rei hæc erit Å¿peculatio, Å¿int duæ lineæ <var>.o.e.</var> et <var>.o.n.</var> quæ duos numeros pro-<lb/>poÅ¿itos Å¿ignificent, inuicem ad angulum rectum <var>.o.</var> coniunctæ, quarum quadrata <lb/>Å¿int <var>.o.a.</var> et <var>.o.p.</var> ipÅ¿orum productum Å¿it <var>.n.e.</var> tum <var>.o.t.</var> Å¿it proueniens ex diuiÅ¿ione <var>.o.e.</var> <lb/>per <var>.o.n</var>. </s> - <s xml:id="s253" xml:space="preserve">HÄ™c Å¿ingulatim conÅ¿ideremus (<reg norm="nam" type="context">nã</reg> Å¿i in partibus Å¿implicibus quod dicimus ac <lb/>ciderit, id ipÅ¿um in compoÅ¿itis conÅ¿equenter eueniet) quamobrem ex definitione di <lb/>uiÅ¿ionis dabitur eadem proportio <var>.o.e.</var> ad <var>.o.t.</var> quæ eft <var>.o.n.</var> ad vnitatem, quæ Å¿it <var>.o.<lb/>x</var>. </s> - <s xml:id="s254" xml:space="preserve">Nunc cogitemus <reg norm="Å¿uperficiem" type="context">Å¿uperficiẽ</reg> <reg norm="rectangulam" type="context">rectangulã</reg> <var>.o.c.</var> <reg norm="æqualem" type="context">æqualẽ</reg> quadrato <var>.o.a</var>. </s> - <s xml:id="s255" xml:space="preserve">tunc numerus <var>.<lb/>c.t.</var> proueniens erit, ut patet, ex diuiÅ¿ione numeri quadrati <var>.o.a.</var> per <reg norm="numerum" type="context">numerÅ©</reg> <var>.o.t.</var> <reg norm="eritque" type="simple">eritqÌ</reg> <lb/><reg norm="eadem" type="context">eadẽ</reg> proportio <var>.c.t.</var> ad <var>.o.e.</var> quæ eÅ¿t <var>.o.e.</var> ad <var>.o.t.</var> ex Å¿ecunda parte quintæ decimæ Å¿exti, <lb/>aut .20. Å¿eptimi. </s> - <s xml:id="s256" xml:space="preserve"><reg norm="Iam" type="context">Iã</reg> <reg norm="autem" type="context">autẽ</reg> dictum eÅ¿t <var>.o.e.</var> ad <var>.o.t.</var> Å¿ic Å¿e habere Å¿icut <var>.o.n.</var> ad <var>.o.x</var>. </s> - <s xml:id="s257" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> ex .<lb/>11. quinti Å¿ic Å¿e habebit <var>.c.t.</var> ad <var>.o.e.</var> Å¿icut <var>.o.n.</var> ad <var>.o.x</var>. </s> - <s xml:id="s258" xml:space="preserve">Sed ex prima Å¿exti, aut .18. vel .<lb/>19. Å¿eptimi, Å¿ic Å¿e habet <reg norm="productum" type="simple">ê“ductum</reg> <var>.n.e.</var> ad <var>.e.x.</var> Å¿icut <var>.o.n.</var> ad <var>.o.x</var>. </s> - <s xml:id="s259" xml:space="preserve">quare denuo Å¿ic Å¿e ha-<lb/>bebit numerus <var>.c.t.</var> ad numerum <var>.o.e.</var> Å¿icut nume-<lb/>rus <var>.n.e.</var> ad numerum <var>.x.e</var>. </s> - <s xml:id="s260" xml:space="preserve">Sed numerus <var>.o.e.</var> cum <lb/> - <anchor type="figure" xlink:label="fig-0029-01a" xlink:href="fig-0029-01"/> - numero <var>.x.e.</var> Å¿pecie idem eÅ¿t, igitur ex .9. quinti nu <lb/>merus <var>.c.t.</var> numero <var>.n.e.</var> æqualis erit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0029-01" xlink:href="fig-0029-01a"> - <image file="0029-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0029-01"/> - </figure> - </div> - <p> - <s xml:id="s261" xml:space="preserve">Id ipÅ¿um de quadrato ipÅ¿ius <var>.o.n.</var> videlicet <var>.p.o.</var> <lb/>dico. </s> - <s xml:id="s262" xml:space="preserve">Nam Å¿i proueniens <var>.o.n.</var> diuiÅ¿o per <var>.o.e.</var> ideÅ¿t <var>.<lb/>o.i.</var> proportionale reÅ¿pondens ad <var>.o.t.</var> cum <var>.o.t.</var> <lb/><reg norm="coniunctum" type="context">coniunctÅ©</reg> fuerit, et per <reg norm="hanc" type="context">hãc</reg> Å¿ummam diuiÅ¿a Å¿umma <lb/>quadratorum <var>.o.a.</var> et <var>.o.p.</var> patet per Å¿e proueniens <lb/>futurum eiuÅ¿dem numeri <var>.c.t.</var> <reg norm="ipÅ¿umque" type="simple">ipÅ¿umqÌ</reg> <var>.c.t.</var> proue-<lb/>niens Å¿emper Å¿uturum.</s> - </p> - <p> - <s xml:id="s263" xml:space="preserve">Quo autem lucidius res hæc innoteÅ¿cat. </s> - <s xml:id="s264" xml:space="preserve">Cogi <lb/>temus proueniens quadrati <var>.o.p.</var> diuiÅ¿i ab <var>.o.i.</var> re-<lb/><reg norm="Å¿pondentisque" type="simple">Å¿pondentisq;</reg> <var>.o.t.</var> eſſe <var>.i.u.</var> quod via prædicta inue-<lb/>nitur æqualis eſſe numero <var>.n.e.</var> ex quo conÅ¿e-<lb/>quenter æquale <var>.c.t</var>: cogitato deinde rectangu-<lb/>lo <var>.o.u.</var> æquali <var>.o.p.</var> coniuncto <var>.o.c</var>:totum <var>.t.u.</var> æqua-<lb/>le erit compoÅ¿ito duorum quadratorum <var>.o.a.</var> et <var>.o.<lb/>p.</var> cum in nullo numerus <var>.c.t.</var> mutetur, tam ex com-<lb/>poÅ¿ito <var>.t.u.</var> <reg norm="quam" type="context">quã</reg> ex Å¿implici <var>.o.c.</var> ex quo propoÅ¿iti Å¿e <lb/>Å¿e ueritas profert.</s> - </p> - </div> - <div type="math:theorem" level="3" n="27"> - <head xml:id="head43" xml:space="preserve">THEOREMA <num value="27">XXVII</num>.</head> - <p> - <s xml:id="s265" xml:space="preserve"><emph style="sc">PRoposvervnt</emph> veteres nobile quidem problema, Å¿ed quod tamen citra al-<lb/>gebraticam effectionem, aut neÅ¿cierunt, aut noluerunt diſſoluere, quod nihi-<lb/>lominus facillimum eÅ¿t.</s> - </p> - <pb o="18" rhead="IO. BAPT. BENED." n="30" file="0030" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0030"/> - <p> - <s xml:id="s266" xml:space="preserve">Proponunt hi numerum in binas eiuÅ¿modi partes diuidendum, vt Å¿umma qua-<lb/>dratorum dictarum partium, alteri numero poÅ¿sibili propoÅ¿ito æqualis Å¿it, poſſi-<lb/>bili inquam, etenim Å¿i eiuÅ¿modi numerus propoÅ¿itus, minor eſſet producto totius <lb/>primi in Å¿uum dimidium, eſſet huiuÅ¿modi factum impoſſibile. </s> - <s xml:id="s267" xml:space="preserve">Quod nos exequi <lb/>cupientes, Å¿umamus primum <reg norm="numerum" type="context">numerÅ©</reg> propoÅ¿itum, quem in Å¿e ipÅ¿um multiplice-<lb/>mus. </s> - <s xml:id="s268" xml:space="preserve">ab hoc quadrato deducamus Å¿ecundum numerum propoÅ¿itum, tum quod re-<lb/>manÅ¿erit duplicemus, quod duplum denuo iubeo ex eodem primo quadrato detra-<lb/>hi, accepta poÅ¿tea radice quadrata reÅ¿idui & dempta ex priori numero propoÅ¿ito, <lb/></s> - <s xml:id="s269" xml:space="preserve">tunc dimidium reÅ¿idui vna pars erit ex duabus primi numeri quæſita.</s> - </p> - <p> - <s xml:id="s270" xml:space="preserve">Exempli gratia proponantur .20. diuidenda in duas eiuÅ¿modi partes, vt Å¿umma <lb/>quadratorum ipÅ¿arum partium æqualis Å¿it .272. qui numerus maior eÅ¿t .200. maior <lb/>inquam dimidio quadrati .400. ipÅ¿orum .20. hic autem numerus .272. è quadra-<lb/>to .400. deducatur, <reg norm="remanebunt" type="context">remanebÅ©t</reg> enim .128. quod duplicari iubeo, <reg norm="producentur" type="context">producẽtur</reg> <reg norm="Å¿iquidem" type="context">Å¿iquidẽ</reg> <num value="256">.<lb/>256.</num> quæ pariter deducta è quadrato totali, remanebunt .144. cuius radicem Å¿umi <lb/>volo, quæ erit .12. & dempta ex .20. priori numero dato remanebit .8. cuius di-<lb/>midium erit .4: pars vna ex quæſitis, quæ ex primo numero propoÅ¿ito .20. detra-<lb/>hetur, <reg norm="remanebitque" type="simple">remanebitqÌ</reg> .16. pro altera parte.</s> - </p> - <p> - <s xml:id="s271" xml:space="preserve">Cuius demonÅ¿trationis cauÅ¿a, in primis cogitemus quadratum <var>.a.c.</var> cognitum nu-<lb/>meri <var>.a.b.</var> primò propoÅ¿iti, qui cogitetur diuiÅ¿us in duo quadrata <var>.d.e.</var> et <var>.e.b.</var> <reg norm="duo- que" type="simple">duo-<lb/>qÌ</reg> Å¿upplementa <var>.a.e.</var> et <var>.e.c.</var> numerus autem Å¿ummæ duorum quadratorum <var>.d.e.<lb/>b.</var> pro Å¿ecundo propoÅ¿ito datur; </s> - <s xml:id="s272" xml:space="preserve">ex quo, Å¿umma duorum Å¿upplementorum <var>.a.e.c.</var> <lb/>conÅ¿equenter erit cognita, quÄ™ cum duplicata fuerit, & quatuor hæc Å¿upplementa<unsure/> <lb/>cogitatione accommodata, prout in <lb/>quadrato <var>.f.g.</var> apparet (<reg norm="quanuis" type="context">quãuis</reg> idipÅ¿um <lb/> - <anchor type="figure" xlink:label="fig-0030-01a" xlink:href="fig-0030-01"/> - proueniret Å¿i modo Eucl. octaua <reg norm="Å¿ecundi" type="context">Å¿ecÅ©di</reg> <lb/>aptaretur) æquali quadrato <var>.a.c.</var> ita vt <lb/>cogitatis quatuor Å¿upplementis numeri <lb/>cogniti in quadrato <var>.f.g.</var> ex conÅ¿equen-<lb/>ti cognoÅ¿cetur numerus quadrati partia <lb/>lis <var>.h.i.</var> & vna etiam eius radix qua de-<lb/>tracta ex numero <var>.a.b.</var> aut <var>.f.n.</var> (quod <lb/>idem eÅ¿t) primo propoÅ¿iti, relinquetur numerus cognitus duplum <var>.x.k.n.</var> aut <var>.t.b.</var> <lb/>pars vna totius <var>.a.b.</var> ex quo uerum erit hoc meum problema.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0030-01" xlink:href="fig-0030-01a"> - <image file="0030-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0030-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="28"> - <head xml:id="head44" xml:space="preserve">THEOREMA <num value="28">XXVIII</num>.</head> - <p> - <s xml:id="s273" xml:space="preserve">SI quis & aliam rationem perficiendæ <lb/> - <anchor type="figure" xlink:label="fig-0030-02a" xlink:href="fig-0030-02"/> - huius rei quærat, hoc præſtet inuen-<lb/>to numero huius Å¿upplementi, cum in <lb/>præcedenti theoremate dictum fuerit, <lb/>qua ratione manifeÅ¿tetur duplum Å¿upple-<lb/>menti ipÅ¿ius.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0030-02" xlink:href="fig-0030-02a"> - <image file="0030-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0030-02"/> - </figure> - </div> - <p> - <s xml:id="s274" xml:space="preserve">Cogitemus in Å¿ubÅ¿cripta figura lineam <var>.<lb/>a.b.</var> tanquam primum numerum propoÅ¿i-<lb/>tum, & productum <var>.a.e.</var> Å¿upplemento <var>.a.e.</var> primæ præcedentis figuræ æquale Å¿it, <lb/>ac deinde ordine ab antiquis tradito procedatur, ad quadratum reducto dimidio <var>.<lb/>a.b.</var> videlicet <var>.b.c.</var> quod erit <var>.b.d.</var> ex quo detrahatur deinde <var>.a.e</var>. </s> - <s xml:id="s275" xml:space="preserve">quare remane- - <pb o="19" rhead="THEOREM. ARIT." n="31" file="0031" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0031"/> - bit quadratum <var>.e.d.</var> cognitum, cuius radix æqualis erit <var>.c.t.</var> qua coniuncta dimi-<lb/>dio <var>.c.a.</var> ex quinta Å¿ecundi Eucli. dabit quod propoÅ¿itum erat.</s> - </p> - </div> - <div type="math:theorem" level="3" n="29"> - <head xml:id="head45" xml:space="preserve">THEOREMA <num value="29">XXIX</num>.</head> - <p> - <s xml:id="s276" xml:space="preserve"><emph style="sc">QVid</emph> cauſæ eÅ¿t, cur Å¿ubtracto duplo producti duorum numerorum ad inui-<lb/>cem <reg norm="multiplicatorum" type="context">multiplicatorÅ©</reg> ex Å¿umma Å¿uorum quadratorum, Å¿emper quod Å¿uper <lb/>eÅ¿t duorum numerorum quadratum differentiæ Å¿it?</s> - </p> - <p> - <s xml:id="s277" xml:space="preserve">Exempli gratia Å¿i proponerentur duo numeri .16. et .4. duplum producti eorum <lb/>eſſet .128. quò detracto ex Å¿umma Å¿uorum quadratorum, nempè ex .272. rema-<lb/>neret .144. cuius quadrati radix eſſet .12. tanquam differentia inter .4. et .16.</s> - </p> - <p> - <s xml:id="s278" xml:space="preserve">Id vtÅ¿ciamus, duo numeri propoÅ¿iti, duabus lineis Å¿ignificentur, maiore <var>.q.g.</var> <lb/>et minore <var>.g.p.</var> directè coniunctis, Å¿uper quas, totale quadratum extruatur <var>.a.p.</var> <lb/>in quo cogitetur diameter <var>.a.p.</var> et à puncto <var>.g.</var> ducatur parallela <var>.g.n.c.</var> et à pun-<lb/>cto <var>.n.</var> parallela <var>.n.s.r.</var> ex quo duo producta <reg norm="dabuntur" type="context">dabÅ©tur</reg> <var>.q.n.</var> et <var>.n.u.</var> Å¿ingula æqualia pro-<lb/>ducto <var>.q.g.</var> in <var>g.p.</var> et <var>.a.n.</var> et <var>.n.p.</var> duo quadrata dictorum numerorum propoÅ¿i-<lb/>torum, quod Å¿atis <reg norm="Å¿uperque" type="simple">Å¿uperqÌ</reg> , probatur quarta Å¿ecundi Eucli. </s> - <s xml:id="s279" xml:space="preserve">Cogitemus deinde <var>.n.<lb/>o.</var> æqualem <var>.n.p.</var> et à puncto <var>.o.</var> ducatur <var>.o.m.t.</var> parallela <var>.r.s.</var> et <var>.o.e.</var> ad <var>.n.<lb/>c</var>. </s> - <s xml:id="s280" xml:space="preserve">quare ex allatis ab Eucli. octaua Å¿ecundi, dabi-<lb/>tur quantitas <var>.m.n.</var> æqualis <var>.q.n.</var> producto <var>.q.g.</var> in <lb/> - <anchor type="figure" xlink:label="fig-0031-01a" xlink:href="fig-0031-01"/> - <var>g.p.</var> et quantitas <var>.o.c.</var> minor ipÅ¿o producto, ex <lb/>quantitate quadrati <var>.n.p.</var> ex quo quantitas <var>.m.n.e.</var> <lb/>vna cum quadrato <var>.n.p.</var> æqualis erit duplo produ-<lb/>cti <var>.q.g.</var> in <var>.g.p.</var> Å¿ed hæ duæ quantitates, Å¿unt par-<lb/>tes duorum quadratorum dictorum, & quæ Å¿uper <lb/>eÅ¿t <var>.m.e.</var> quadratum differentiæ vnius numeri pro-<lb/>poÅ¿iti ab altero, prout in Å¿ubÅ¿cripta figura licebit cui <lb/>libet conÅ¿iderare. </s> - <s xml:id="s281" xml:space="preserve">Itaque veritas hæc manifeÅ¿ta <lb/>erit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0031-01" xlink:href="fig-0031-01a"> - <image file="0031-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0031-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="30"> - <head xml:id="head46" xml:space="preserve">THEOREMA <num value="30">XXX</num>.</head> - <p> - <s xml:id="s282" xml:space="preserve"><emph style="sc">CVr</emph> ij qui ex duobus numeris propoÅ¿itis maiorem per minorem diuidunt, Å¿i <lb/>proueniens per maiorem numerum multiplicauerint, productum æquale <lb/>erit prouenienti ex diuiÅ¿ione quadrati maioris numeri per minorem?</s> - </p> - <p> - <s xml:id="s283" xml:space="preserve">Exempli gratia Å¿i proponantur duo numeri .20. et .4. <reg norm="ipÅ¿eque" type="simple">ipÅ¿eqÌ</reg> .20. per .4. diui-<lb/>datur, dabit quinque, tum .400. quadrato .20. diuiÅ¿o per prioré .4. dabit .100. <lb/>quod proueniens, producto ex .20. in .5. primo prouenienti adæquatur.</s> - </p> - <p> - <s xml:id="s284" xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, Å¿int duo numeri, qui lineis <var>.x.u.</var> et <var>.x.s.</var> maiore <reg norm="atque" type="simple">atq;</reg> mi-<lb/>nore Å¿ignificétur, tum <var>.u.x.</var> numerus per <var>.s.x.</var> di-<lb/>uidatur, Å¿itqÌue proueniens <var>.x.n.</var> poÅ¿tmodum qua-<lb/> - <anchor type="figure" xlink:label="fig-0031-02a" xlink:href="fig-0031-02"/> - dratum <var>.u.x.</var> Å¿it <var>.x.o.</var> et productum ex <var>.n.x.</var> in <var>.u.<lb/>x.</var> Å¿it <var>.x.e.</var> quod æquale eſſe dico prouenienti ex <lb/>diuiÅ¿ione quadrati <var>.o.x.</var> per <var>.s.x.</var> quod Å¿it <var>.m</var>. </s> - <s xml:id="s285" xml:space="preserve">Patet <lb/>enim ex definitione diuiÅ¿ionis, talem futuram pro-<lb/>portionem <var>.u.x.</var> ad <var>.n.x.</var> qualis eÅ¿t <var>.s.x.</var> ad vnitatem, <lb/>& quadratum <var>.o.x.</var> ad rectangulum <var>.e.x.</var> ita Å¿e ha- - <pb o="20" rhead="I O. BAPT. BENED." n="32" file="0032" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0032"/> - biturum, Å¿icut <var>.u.x.</var> ad <var>.n.x.</var> ex prima Å¿exti aut .18. vel .19. Å¿eptimi, </s> - <s xml:id="s286" xml:space="preserve">quare ex 11. <lb/>quinti ita Å¿e habebit <var>.o.x.</var> ad <var>.e.x.</var> Å¿icut <var>.s.x.</var> ad vnitatem; </s> - <s xml:id="s287" xml:space="preserve">Å¿ed Å¿icut Å¿e habet <var>.s.x.</var> ad. <lb/>vnitatem, ita Å¿e habet pariter <var>.o.x.</var> ad <var>.m</var>. </s> - <s xml:id="s288" xml:space="preserve">vnde ex .11. prædicta ita Å¿e habebit <var>.o.<lb/>x.</var> ad <var>.m.</var> Å¿icut idipÅ¿um <var>.o.x.</var> ad <var>.e.x.</var> itaqÌue ex .9. prædicti quinti <var>.m.</var> æqualis erit <var>.o.x</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0031-02" xlink:href="fig-0031-02a"> - <image file="0031-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0031-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="31"> - <head xml:id="head47" xml:space="preserve">THEOREMA <num value="31">XXXI</num>.</head> - <p> - <s xml:id="s289" xml:space="preserve">CVR propoÅ¿ito aliquo numero in duas partes inæquales diuiÅ¿o, Å¿i rurÅ¿us per <lb/>quamlibet ipÅ¿arum diuidatur, prouenientia tantumdem coniuncta quantum <lb/>multiplicata efficiant.</s> - </p> - <p> - <s xml:id="s290" xml:space="preserve">Exempli gratia, Å¿it denarius prop oÅ¿itus numerus, per binarium & octonarium <lb/>diuiÅ¿us, prouenientia erunt quinque & vnum cum quarta parte, quæ coniuncta <lb/>crunt .6. cum quarta parte lineari, quæ Å¿i mul multiplicata, pariter erunt .6. cum <lb/>quarta parte Å¿uperficiali.</s> - </p> - <p> - <s xml:id="s291" xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, totalis numerns, linea <var>.q.p.</var> Å¿ignificetur, eius duæ <lb/>partes, per <var>.k.</var> maiorem et <var>.u.</var> minorem, ipÅ¿a vnitas per .t: proueniens ex diuiÅ¿io-<lb/>ne <var>.q.p.</var> per <var>.k.</var> Å¿it <var>.q.i.</var> proueniens autem ipÅ¿ius <var>.q.p.</var> per <var>.u.</var> Å¿it <var>.q.f.</var> </s> - <s xml:id="s292" xml:space="preserve">quare ex defini-<lb/>tione diuiÅ¿ionis ita Å¿e habebit <var>.q.p.</var> ad <var>.q.i.</var> Å¿icut <var>.k.</var> ad <var>.t.</var> et <var>.q.p.</var> ad <var>.q.f.</var> Å¿icut <var>.u.</var> ad <var>.t.</var> <lb/>hoc eÅ¿t <var>.q.f.</var> ad <var>.q.p.</var> Å¿icut <var>.t.</var> ad <var>.u.</var> vnde ex æqualitate <reg norm="proportionum" type="context">proportionÅ©</reg> Å¿ic Å¿e habebit <var>.q.f.</var> <lb/>ad <var>.q.i.</var> Å¿icut <var>.k.</var> ad <var>.u.</var> et conuerÅ¿im. </s> - <s xml:id="s293" xml:space="preserve">Ad hæc in linea <var>.q.p.</var> vnitas, per lineam <var>.q.o.</var> Å¿igni-<lb/>ficetur, quo facto, dicamus, Å¿i <var>.q.p.</var> ad <var>.q.i.</var> Å¿ic Å¿e habet vt <var>.k.</var> ad <var>.q.o.</var> itaque permu-<lb/>tando, Å¿ic Å¿e habebit <var>.q.p.</var> ad <var>.k.</var> Å¿icut <var>.q.i.</var> ad <var>.q.o.</var> hoc eÅ¿t <var>.k.u.</var> ad <var>.k.</var> Å¿icut <var>.i.q.f.</var> ad <var>.<lb/>q.f.</var> (nam <var>.k.u.</var> partes Å¿unt integrales totius <var>.q.p.</var> et <var>.k.u.</var> ad <var>.k.</var> eÅ¿t Å¿icut <var>.i.q.f.</var> ad <var>.q.f.</var> <lb/>ex .18. quinti) </s> - <s xml:id="s294" xml:space="preserve">Quare ita erit <var>.i.q.f.</var> ad <var>.q.f.</var> Å¿icut <var>.q.i.</var> ad vnitatem <var>.q.o.</var> ex .11. quinti <lb/>Addatur deinde <var>.q.i.</var> ad <var>.q.f.</var> et <var>.q.i.</var> per <var>.<lb/>q.f.</var> multiplicetur, cuius multiplicatio-<lb/> - <anchor type="figure" xlink:label="fig-0032-01a" xlink:href="fig-0032-01"/> - nis productum, Å¿it <var>.x.f.</var> quod probabo <lb/>æquale eſſe Å¿ummæ <var>.f.q.</var> cum <var>.q.i</var>. </s> - <s xml:id="s295" xml:space="preserve">Sece-<lb/>tur enim linea <var>.q.x.</var> in puncto <var>.s.</var> ita. vt <var>.<lb/>q.s.</var> æqualis Å¿it <var>.q.o.</var> Å¿igneturqÌue pro-<lb/>ductum <var>.s.f.</var> </s> - <s xml:id="s296" xml:space="preserve">quare <reg norm="eadem" type="context">eadẽ</reg> erit propor-<lb/>tio quantitatis <var>.x.f.</var> ad <var>.s.f.</var> quæ eÅ¿t <var>.q.x.</var> <lb/>ad <var>.q.s.</var> ex prima Å¿exti, aut .18. vel 19. <lb/>Å¿eptimi, hoc eÅ¿t, Å¿icut <var>.q.i.</var> ad <var>.q.o.</var> et <lb/>ex .11. quinti (vt dictum eÅ¿t) Å¿icut <var>.i.q.<lb/>f.</var> ad <var>.q.f.</var> Å¿ed numerus <var>.s.f.</var> fuperficia-<lb/>lis tantus eÅ¿t, quantus linearis <var>.q.f</var>. <lb/></s> - <s xml:id="s297" xml:space="preserve">quare ex .9. quinti tantus erit (Å¿u-<lb/>perficialiter) numerus <var>.x.f.</var> quantus <lb/>(lineariter). <var>f.q.i.</var> quod erat pro-<lb/>poÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0032-01" xlink:href="fig-0032-01a"> - <image file="0032-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0032-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="32"> - <head xml:id="head48" xml:space="preserve">THEOREMA. <num value="32">XXXII</num>.</head> - <p> - <s xml:id="s298" xml:space="preserve">CVR numero aliquo in duas partes inæquales diuiÅ¿o, Å¿i rurÅ¿us diuidatur per <lb/>Å¿ingulas partes, Å¿umma duorum prouenientium per binarium, Å¿emper ma-<lb/>ior Å¿it Å¿umma prouenientium ex diuiÅ¿ione vnius partis per alteram.</s> - </p> - <p> - <s xml:id="s299" xml:space="preserve"><reg norm="Exempli" type="context">Exẽpli</reg> gratia, Å¿i proponeretur numerus .24. qui in duas partes inæquales diuide - <pb o="21" rhead="THEOREM. ARIT." n="33" file="0033" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0033"/> - retur .20. Å¿cilicet et .4. certè .24. perÅ¿ingulas partes diuiÅ¿o, daretur vnum proue-<lb/>niens Å¿ex integra, & alterum vnum & quinta pars, quorum Å¿umma eſſet Å¿eptem in-<lb/>tegra cum quinta parte, tum altera parte per alteram diuiÅ¿a, daretur vnum proue-<lb/>niens quinque integrorum & alterum vnius quinti tantum, quorum Å¿umma eſſet <lb/>quinque integra, & vna quinta pars, minor prima reliquorum duorum prouenien-<lb/>tium per binarium.</s> - </p> - <p> - <s xml:id="s300" xml:space="preserve">Cuius conÅ¿iderationis cauÅ¿a, propoÅ¿itus numerus linea <var>.q.p.</var> Å¿ignificetur, eius duÄ™ <lb/>partes lineis <var>.q.x.</var> et <var>.x.p.</var> <reg norm="tum" type="context">tÅ©</reg> <var>.q.f.</var> Å¿it proueniens ex diuiÅ¿ione totius <var>.q.p.</var> per <var>.x.p.</var> et <var>.<lb/>q.i.</var> Å¿it proueniens ex diuiÅ¿ione eiuÅ¿dem <var>.q.p.</var> per <var>.q.x.</var> adhæc <var>.h.m.</var> Å¿it proueniens, <lb/>ex diuiÅ¿ione <var>.q.x.</var> per <var>x.p.</var> et <var>.h.k.</var> proue-<lb/>niensex diuiÅ¿ione <var>.p.x.</var> per <var>.q.x.</var> patet igi-<lb/> - <anchor type="figure" xlink:label="fig-0033-01a" xlink:href="fig-0033-01"/> - tur ex .22. theoremate huiuslibri proue-<lb/>niés.h.m. minus eſſe proueniente <var>.q.f.</var> per <lb/>vnitaté, & proueniens <var>.h.k.</var> minus proue-<lb/>niente <var>.q.i.</var> per alteram vnitatem. </s> - <s xml:id="s301" xml:space="preserve">Itaque <var>.<lb/>f.q.i.</var> maior erit <var>.m.h.k.</var> per numerum binarium, quoderat propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0033-01" xlink:href="fig-0033-01a"> - <image file="0033-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0033-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="33"> - <head xml:id="head49" xml:space="preserve">THEOREMA. <num value="33">XXXIII</num>.</head> - <p> - <s xml:id="s302" xml:space="preserve"><emph style="sc">QVilibet</emph> numerus, medius eÅ¿t <lb/>proportionalis inter numerum <lb/> - <anchor type="figure" xlink:label="fig-0033-02a" xlink:href="fig-0033-02"/> - Å¿ui quadrati & vnitatem.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0033-02" xlink:href="fig-0033-02a"> - <image file="0033-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0033-02"/> - </figure> - </div> - <p> - <s xml:id="s303" xml:space="preserve">Detur enim numerus propoÅ¿itus, <lb/>qui linea <var>.a.u.</var> Å¿ignificetur, cuiusqua-<lb/>dratum Å¿it <var>.u.n.</var> vnitas linearis Å¿it <var>.i.a.</var> <lb/>et Å¿uperficialis <var>.o.</var> patebit ex .18. Å¿exti <lb/>aut 11. octaui proportionem <var>.u.n.</var> ad <var>.<lb/>o.</var> futuram duplam proportioni <var>.u.a.</var> <lb/>ad <var>.i.a.</var> Å¿ed <var>.i.a.</var> e<unsure/>t.o. eadem (Å¿pecie) <lb/>res <reg norm="sunt" type="context">sÅ©t</reg>, tanta Å¿cilicet <var>.a.i.</var> quanta <var>.o.</var> vni <lb/> - <anchor type="figure" xlink:label="fig-0033-03a" xlink:href="fig-0033-03"/> - tas eÅ¿t, Itaque proportio numeri <var>.u.n.</var> <lb/>ad <var>.u.a.</var> æqualis erit proportioni <var>.u.a.</var> <lb/>ad <var>.i.a</var>. </s> - <s xml:id="s304" xml:space="preserve">Quare numerus <var>.u.a.</var> inter nu-<lb/>merum <var>.u.n.</var> & vnitatem, medius erit <lb/>proportionalis.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0033-03" xlink:href="fig-0033-03a"> - <image file="0033-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0033-03"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="34"> - <head xml:id="head50" xml:space="preserve">THEOREMA <num value="34">XXXIIII</num>.</head> - <p> - <s xml:id="s305" xml:space="preserve"><emph style="sc">HOc</emph> ipÅ¿um quod diximus & alia ratione Å¿peculari licebit.</s> - </p> - <p> - <s xml:id="s306" xml:space="preserve">PropoÅ¿itus numerus, nunc etiam per <var>.a.u.</var> Å¿ignificetur, eius quadratum per <var>.<lb/>u.n.</var> vnitas linearis per <var>.a.i.</var> <reg norm="productumque" type="simple">productumqÌ;</reg> <var>.a.u.</var> in <var>.a.i.</var> terminetur, <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.n.i</var>. </s> - <s xml:id="s307" xml:space="preserve">quare <lb/><var>n.i.</var> conÅ¿tabit numero Ãuperficiali æquali numero lineari <var>.a.u.</var> & ex prima fexti aut .<lb/>18. vel .19. Å¿eptimi, eadem erit proportio <var>.u.n.</var> ad <var>.i.n.</var> quæ eÅ¿t <var>.a.u.</var> ad <var>.a.i.</var> Å¿ed nu-<lb/>merus <var>.a.u.</var> cum numero <var>.n.i.</var> idem Å¿pecie eÅ¿t. </s> - <s xml:id="s308" xml:space="preserve">Itaque medius eÅ¿t proportiona-<lb/>lis inter <var>.u.n.</var> & vnitatem.</s> - </p> - <pb o="22" rhead="IO. BAPT. BENED." n="34" file="0034" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0034"/> - </div> - <div type="math:theorem" level="3" n="35"> - <head xml:id="head51" xml:space="preserve">THEOREMA <num value="35">XXXV</num>.</head> - <p> - <s xml:id="s309" xml:space="preserve"><emph style="sc">QVivis</emph> numerus per alterum multiplicatus, & diuiÅ¿us, medius eÅ¿t propor-<lb/>tionalis inter productum multiplicationis, & proueniens diaiÅ¿ionis.</s> - </p> - <p> - <s xml:id="s310" xml:space="preserve">Exempli gratia, Å¿i .20. <reg norm="multiplicentur" type="context">multiplicẽtur</reg> per quinque & inde per quinque diuidantur <lb/>productum erit .100. proueniens .4. inter quos numeros .20. medius eÅ¿t propor-<lb/>tionalis.</s> - </p> - <p> - <s xml:id="s311" xml:space="preserve">Hoc vt Å¿peculemur, proponatur numerus multiplicandus & diuidendus, qui Å¿i-<lb/>gnificetur linea <var>.u.e.</var> multiplicans autem & diuidens linea <var>.a.u.</var> multiplicationis <lb/>productum Å¿it <var>.e.a.</var> proueniens ex diuiÅ¿ione Å¿it <var>.o.e</var>. </s> - <s xml:id="s312" xml:space="preserve">Nunc proueniens <var>.e.o.</var> per <reg norm="nu- merum" type="context">nu-<lb/>merÅ©</reg> <var>.a.u.</var> diuidentem multiplicetur, cuius multiplicationis productum Å¿it <var>.e.i.</var> <lb/>quare, eadem erit proportio numeri <var>.a.e.</var> <lb/>ad numerum <var>.e.i.</var> quæ eÅ¿t numeri <var>.u.e.</var> ad <lb/> - <anchor type="figure" xlink:label="fig-0034-01a" xlink:href="fig-0034-01"/> - numerum <var>.e.o.</var> ex prima Å¿extiaut .18. vel <lb/>19. Å¿eptimi. </s> - <s xml:id="s313" xml:space="preserve">Sed cum numerus <var>.u.e.</var> ex <ref id="ref-0010">.11. theoremate præſentis libri</ref>, numero <var>.e.<lb/>i.</var> æqualis Å¿it. </s> - <s xml:id="s314" xml:space="preserve">verum eſſe, quod propoÅ¿i-<lb/>tum fuit conÅ¿equetur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0034-01" xlink:href="fig-0034-01a"> - <image file="0034-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0034-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="36"> - <head xml:id="head52" xml:space="preserve">THEOREMA <num value="36">XXXVI</num>.</head> - <p> - <s xml:id="s315" xml:space="preserve">CVR ij, qui propoÅ¿itum numerum ita multiplicare & diuidere cupiunt, vt pro <lb/>ductum multiplicationis, tam Å¿it multiplex prouenienti ex diuiÅ¿ione, quam <lb/>quæritur, rectè Å¿umant aliquem numerum pro multiplicante & diuidente, qui Å¿it ra <lb/>dix quadrata denominantis quęſitÄ™ multiplicitatis.</s> - </p> - <p> - <s xml:id="s316" xml:space="preserve">Exempli gratia, proponuntur .20. multiplicanda atque diuidenda, ita vt pro-<lb/>ductum multiplicationis nonuplum Å¿it prouenienti ex diuiÅ¿ione, nempè, vt pro-<lb/>ueniens, nona pars Å¿it eiuÅ¿modi producti, </s> - <s xml:id="s317" xml:space="preserve">quare quadratam radicem ipÅ¿orum no-<lb/>uem, ideÅ¿t denominantis Å¿umunt, tria Å¿cilicet, multiplicant igitur & diuidunt <lb/>data .20. ex quo productum erit .60. proueniens autem .6. cum duabus tertijs. </s> - <s xml:id="s318" xml:space="preserve">& <lb/>propoÅ¿itum Å¿equitur.</s> - </p> - <p> - <s xml:id="s319" xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, Å¿ignificetur numerus propoÅ¿itus linea <var>.u.e.</var> multipli-<lb/>cans autem & diuidens linea <var>.u.a.</var> productum Å¿it <var>.e.a.</var> proueniens <var>.e.o.</var> quadratum <lb/>verò <var>.a.u.</var> Å¿it <var>.x.a.</var> erit igitur proportio <var>.a.e.</var> ad <var>.e.o.</var> dupla proportioni <var>.a.e.</var> ad nume <lb/>rum <var>.u.e.</var> ex præcedenti theoremate: </s> - <s xml:id="s320" xml:space="preserve">Adhæc, cogitemus in linea <var>.u.a.</var> vnitatem <var>.<lb/>u.i.</var> <reg norm="terminenturque" type="simple">terminenturqÌ;</reg> duo producta <var>.e.i.</var> et <var>.x.i.</var> </s> - <s xml:id="s321" xml:space="preserve">quare eadem erit proportio <var>.a.e.</var> ad <var>.e.i.</var> <lb/>quæ eÅ¿t <var>.a.e.</var> ad <var>.u.e.</var> numerus enim <var>.e.i.</var> (quamuis Å¿uperficialis) idem eÅ¿t cum nume-<lb/>ro lineari <var>.u.e.</var> Å¿ed <var>.a.e.</var> ad <var>.e.i.</var> Å¿ic Å¿e habet Å¿icut <var>.a.u.</var> ad <var>.u.i.</var> ex prima Å¿exti aut .18. <lb/>vel .19. Å¿eptimi, (quod ipÅ¿um dico de <var>.a.x.</var> ad <var>.x.i.</var>) </s> - <s xml:id="s322" xml:space="preserve">quare proportio <var>.a.x.</var> ad <var>.x.i.</var> hoc <lb/>eÅ¿t <var>.x.u.</var> Ä™qualis erit <reg norm="proportioni" type="simple">ê“portioni</reg> <var>.a.e.</var> ad <var>.u.e.</var> at trigeÅ¿imotertio & trigeÅ¿imoquarto theo <lb/>remate probatum eÅ¿t proportionem numeri <var>.a.x.</var> ad vnitatem, duplam eſſe propor-<lb/>tioni eiuÅ¿dem numeri <var>.a.x.</var> ad <var>.u.x.</var> Å¿equitur <lb/>igitur cum dimidia Å¿int æqualia, tota etiam <lb/>æqualia eſſe: </s> - <s xml:id="s323" xml:space="preserve">hoc eÅ¿t proportionem numeri <var>.<lb/> - <anchor type="figure" xlink:label="fig-0034-02a" xlink:href="fig-0034-02"/> - a.e.</var> ad numerum <var>.e.o.</var> æqualem eſſe propor <lb/>tioni numeri <var>.a.x.</var> ad vnitatem. </s> - <s xml:id="s324" xml:space="preserve">Itaque rectè <lb/>Å¿umitur numerus <var>.a.u.</var> eiuÅ¿modi vt <reg norm="quadratum" type="context">quadratÅ©</reg> - <pb o="23" rhead="THEOR. ARITH." n="35" file="0035" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0035"/> - ipÅ¿ius <var>.a.x.</var> tam Å¿it multiplex ad vnitatem, quam cupimus numerum <var>.a.e.</var> numero <var>.<lb/>e.o.</var> multiplicem eſſe.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0034-02" xlink:href="fig-0034-02a"> - <image file="0034-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0034-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="37"> - <head xml:id="head53" xml:space="preserve">THEOREMA <num value="37">XXXVII</num>.</head> - <p> - <s xml:id="s325" xml:space="preserve">CVR inuenire cupientes duos numeros, quorum quadrata in Å¿ummam colle-<lb/>cta, æqualia Å¿int numero propoÅ¿ito, & ijÅ¿dem numeris multiplicatis ad-<lb/>inuicem, productum alteri numero propoÅ¿ito Å¿it æquale, rectè Å¿umant dimidium <lb/>primi numeri propoÅ¿iti, cui Å¿umma quadratorum æquari debet, <reg norm="hocque" type="simple">hocqÌ;</reg> dimidium <lb/>in Å¿eipÅ¿um multiplicent, vnà etiam alterum numerum propoÅ¿itum in Å¿eipÅ¿um <lb/>multiplicent, quod quadratum detrahunt de primo, & reÅ¿idui quadratam radicem, <lb/>dimidio primi numeri propoÅ¿iti coniungunt, ex qua Å¿umma, quadratam radicem <lb/><reg norm="eruunt" type="context">eruÅ©t</reg>, quæ duobus quæſitis numeris maior erit, cuius quadrato de primo numero <lb/>detracto, & exreliquo erutaradice quadrata, detur minor numerus, duorum <reg norm="quae- Å¿itorum" type="simple">quÄ™-<lb/>Å¿itorum</reg>.</s> - </p> - <p> - <s xml:id="s326" xml:space="preserve">Exempli gratia, Å¿i proponerentur .34. pro primo numero cui æquari de-<lb/>beret Å¿umma duorum quadratorum, quorum radicum productum æquale eſſe de-<lb/>beret alteri numero, verbi gratia .15. iubet antiquorum regula, dimidium primi <lb/>numeri in Å¿eipÅ¿um multiplicari, cuius dimidij quadratum erit .289. è quo Å¿i detra-<lb/>has quadratum Å¿ecundi numeri, nempe .225. remanebit .64. <reg norm="atque" type="simple">atq;</reg> huius Å¿i quadra-<lb/>tam radicem Å¿umas nempe .8. quam dimidio primi numeri, nempe .17. coniun-<lb/>gas, dabitur duorum quadratorum numerorum quęſitorum maior numerus .25. hac <lb/>deinde radice è dimidio detracta, minus quadratum dabitur .9. Å¿cilicet, quorum <lb/>radices .5. et .3. eſſent ij numeri, qui quæruntur.</s> - </p> - <p> - <s xml:id="s327" xml:space="preserve">Cuius Å¿peculationis gratia, cogitemus primum numerum, cui quadratorum fum <lb/>ma æquari debet, Å¿ignificari linea <var>.a.n.</var> tum concipiamus quæſita quadrata Å¿ignifi-<lb/>cari, <reg norm="coniungique" type="simple">coniungiqÌ</reg> modo Å¿ubÅ¿cripto <var>.t.b.k.</var> Å¿ecundum porrò numerum propoÅ¿itum, <lb/>Å¿ignificari producto <var>.d.b</var>. </s> - <s xml:id="s328" xml:space="preserve">Iam nil Å¿upereÅ¿t aliud quam vt quantitates <var>.d.p.</var> et <var>.b.p.</var> <lb/>quæramus.</s> - </p> - <p> - <s xml:id="s329" xml:space="preserve">Itaque cum in linea <var>.a.n.</var> Å¿ummæ quadratorum numerus detur, quadratum di-<lb/>midij <var>.o.a.</var> Å¿it <var>.s.a.</var> quod nobis erit cognitum; </s> - <s xml:id="s330" xml:space="preserve">Å¿it etiam <var>.a.u.</var> numerus quadrati ma <lb/>ioris, et <var>.u.n.</var> minoris, et <var>.a.z.</var> productum vnius in alterum; </s> - <s xml:id="s331" xml:space="preserve">qui quidem numerus <var>.a.<lb/>z.</var> æqualis erit <lb/>quadrato nume <lb/> - <anchor type="figure" xlink:label="fig-0035-01a" xlink:href="fig-0035-01"/> - ri <var>.d.b.</var> ex .19. <lb/>theoremate hu-<lb/>ius libri. </s> - <s xml:id="s332" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> <lb/><var>a.z.</var> cognitum <lb/>erit, cum eius <lb/>radix <var>.d.b.</var> Å¿it <reg norm="Å¿e- cundus" type="context">Å¿e-<lb/>cÅ©dus</reg> numerus <lb/>propoÅ¿itus, quæ <lb/>minor erit <var>.a.s.</var> ex quinta Å¿ecundi, aut Å¿eptima conÅ¿equentia poÅ¿t .16. noni Eucli-<lb/>dis. </s> - <s xml:id="s333" xml:space="preserve">Iam Å¿ubtracta quantitate <var>.z.a.</var> è quadrato <var>.a.s.</var> cognoÅ¿cetur quadratum <var>.t.x.</var> <lb/>cuius radix æqualis erit <var>.o.u.</var> ex poÅ¿tremo adductis, Itaque cognoÅ¿cemus <var>.o.u.</var> qui <lb/>numerus coniunctus dimidio <var>.o.a.</var> cognito, dabit quadratum <var>.a.u.</var> cognitum, at-<lb/>queita <var>.u.n.</var> pariter cognoÅ¿cetur, & eorum radices conÅ¿equenter.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0035-01" xlink:href="fig-0035-01a"> - <image file="0035-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0035-01"/> - </figure> - </div> - <pb o="24" rhead="IO. BAPT. BENED." n="36" file="0036" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0036"/> - <p> - <s xml:id="s334" xml:space="preserve">Hoc ipÅ¿um & alia ratione perfici poteÅ¿t, nempe, iuncta Å¿umma <var>.k.b</var>: <var>b.d</var>: ec<unsure/> <var>.<lb/>b.t.</var> alteri rectangulo æquali <var>.b.d.</var> quod Å¿it <var>.b.c.</var> ex quo totum quadratum lineæ <var>.d.k.</var> <lb/>cognitum erit, <reg norm="atque" type="simple">atq;</reg> ita etiam conÅ¿equenter eius radicem <var>.d.k.</var> cognoÅ¿cemus, cuius <lb/>ope ac producti <var>.d.b.</var> cognoÅ¿cemus <var>.d.p.</var> et <var>.p.k.</var> prout ex theoremate quadrageÅ¿i-<lb/>moquinto huius libri patebit.</s> - </p> - <p> - <s xml:id="s335" xml:space="preserve">Michael Stifelius, vndecimo cap. tertij libri, problema eiuÅ¿modi proponit, <lb/>quod tamen ipÅ¿e via algebræ diÅ¿soluit.</s> - </p> - <figure position="here"> - <image file="0036-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0036-01"/> - </figure> - </div> - <div type="math:theorem" level="3" n="38"> - <head xml:id="head54" xml:space="preserve">THEOREMA <num value="38">XXXVIII</num>.</head> - <p> - <s xml:id="s336" xml:space="preserve">CVR ij, qui duos numeros inuenire volunt, quorum productum alicui nu-<lb/>mero propoÅ¿ito æquetur, & quadratorum eorundem differentia alteri nu-<lb/>mero propoÅ¿ito æqualis Å¿ir. </s> - <s xml:id="s337" xml:space="preserve">Rectè dimidium Å¿ecundi numeri propoÅ¿iti in Å¿eipÅ¿um <lb/>multiplicent, cui quidem numero differentia quadratorum æquari debet; </s> - <s xml:id="s338" xml:space="preserve">porrò <lb/>huic quadrato primi propoÅ¿iti numeri, cui æquandum eÅ¿t productum numerorum <lb/>quæſitorum, quadratum adiungant; </s> - <s xml:id="s339" xml:space="preserve">tum radicem quadratam huius Å¿ummæ co-<lb/>pulet dimidio Å¿ecundi numeri propoÅ¿iti, ei inquam, cui differentia quadratorum <lb/>æqualis eſſe debet, ex quo quadratum maius conÅ¿urgit, à quo, detracto Å¿ecundo <lb/>numero, Å¿upereÅ¿t quadratum minus.</s> - </p> - <p> - <s xml:id="s340" xml:space="preserve">Exempli gratia, Å¿i proponeretur primo loco numerus .8. cui æquandum eÅ¿t <lb/>productum numerorum quæſitorum, tum proponeretur numerus .12. cui, detra-<lb/>cto minore à maiore, differentia quadratorum vtriuÅ¿que quæſiti numeri æqualis <lb/>eſſe debet, oportet huius vltimi numeri .12. dimidium in Å¿eipÅ¿um multiplicare, <reg norm="fient- qÌue" type="context">fiẽt-<lb/>qÌue</reg> .36. quadratum dimidij, vnde in Å¿ummam colligeremus quadratum primi <lb/>numeri .8. quod eſſet .64. quæ cum .36. efficerent .100. cuius centenarij radice, nem <lb/>pe .10. collecta in Å¿ummam cum dimidio Å¿ecundi numeri, nempe .6. daretur qua-<lb/>dratum maius, nempe .16. ex quo, detracto Å¿ecundo numero, nempe .12. rema-<lb/>neret quadratum minus .4.</s> - </p> - <p> - <s xml:id="s341" xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, maius quadratum <lb/> - <anchor type="figure" xlink:label="fig-0036-02a" xlink:href="fig-0036-02"/> - incognitum Å¿ignificetur linea <var>.q.g.</var> minus verò <lb/>pariter incognitum linea <var>.g.i.</var> </s> - <s xml:id="s342" xml:space="preserve">quare <var>.q.i.</var> eorum <lb/>differentia, tanquam data remanebit cognita, <lb/>vnà etiam <var>.b.i.</var> et <var>.q.b.</var> Å¿ua dimidia; </s> - <s xml:id="s343" xml:space="preserve">tunc cogite-<lb/>tur quadratum <var>.y.g.</var> Å¿uper <var>.b.g.</var> et <reg norm="parallelogram- mum" type="context">parallelogrã-<lb/>mum</reg> rectangu<unsure/>lum <var>.g.r.</var> deÅ¿ignatum, & ita etiam <lb/>gnomon <var>.u.g.t.</var> prout Å¿exta Å¿ecundi Euclidis pro <lb/>ponitur, ex quo quadratum <var>.b.i.</var> nempe <var>.u.t.</var> co-<lb/>gnitum erit, Å¿ed gnomon æqualis eÅ¿t rectangulo <var>.g.r.</var> ex prædicta, aut ex .8. poÅ¿t .16. - <pb o="25" rhead="THEOREM. ARIT." n="37" file="0037" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0037"/> - noni, <reg norm="hocque" type="simple">hocqÌ;</reg> rectangulum <var>.g.r.</var> quadratum eÅ¿t primi numeri propoÅ¿iti ex .19. theo-<lb/>remate huius libri, <reg norm="itaque" type="simple">itaq;</reg> cognitum erit. </s> - <s xml:id="s344" xml:space="preserve">vnà etiam gnomon <var>.u.g.t.</var> cognoÅ¿cetur, <lb/>quare totum quadratum <var>.g.y.</var> <reg norm="eiusque" type="simple">eiusqÌ;</reg> radix <var>.b.g.</var> manifęſta erit, cui coniuncta <var>.q.b.</var> <lb/>data, maius quadratum <var>.q.g.</var> cognoÅ¿cetur, ex qua <var>.b.g.</var> detracta <var>.b.i.</var> data, cogno-<lb/>Å¿cetur <var>.i.g.</var> quadratum minus conÅ¿equenter, etiam eorum radices notæ erunt.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0036-02" xlink:href="fig-0036-02a"> - <image file="0036-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0036-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="39"> - <head xml:id="head55" xml:space="preserve">THEOREMA <num value="39">XXXIX</num>.</head> - <p> - <s xml:id="s345" xml:space="preserve"><emph style="sc">ALia</emph> etiam ratione idipÅ¿um definiri poteÅ¿t, prætermiſſa antiquorum via, <lb/>nempe multiplicatis in Å¿emetipÅ¿is primo & Å¿ecundo, numeris propoÅ¿itis, qua-<lb/><reg norm="druplicatoque" type="simple">druplicatoqÌ;</reg> quadrato primi, qua Å¿umma coniuncta cum quadrato Å¿ecundi nume-<lb/>ri, & ex hac altera Å¿umma eruta radice quadrata, ex qua detracto Å¿ecundo nume-<lb/>ro, & è reliquo Å¿umpto dimidio, quod erit <reg norm="quadratum" type="context">quadratÅ©</reg> minus, quo detracto ex radi-<lb/>ce poÅ¿tremo iuncta, Å¿upererit quadrarum maius.</s> - </p> - <p> - <s xml:id="s346" xml:space="preserve">Exempli gratia, Å¿i proponeretur numerus .8. cui productum duorum numerorum <lb/>quæſitorum æquandum eÅ¿t, proponeretur idem .12. cui differentia quadratorum <lb/>duorum numerorum æqualis eſſe debet. </s> - <s xml:id="s347" xml:space="preserve">Iubeo primum numerum, nempe .8. in Å¿e <lb/>ipÅ¿um multiplicari, ex quo exurget .64. pro numero Å¿ui quadrati, quod quadru-<lb/>plicari volo, <reg norm="eritque" type="simple">eritqÌ;</reg> productum .256. quod cenÅ¿eo <reg norm="coniungendum" type="context">coniÅ©gendum</reg> cum quadrato Å¿e-<lb/>cundi numeri propoÅ¿iti, nempe .144. <reg norm="eritque" type="simple">eritqÌ;</reg> Å¿umma .400. ex quaÅ¿umetur radix, Å¿ci <lb/>licet .20. & ex hac detrahetur Å¿ecundus numerus .12. <reg norm="reÅ¿iduique" type="simple">reÅ¿iduiqÌ;</reg> dimidium, nempe <num value="4">.<lb/>4.</num> pro quadrato minore, quo in Å¿ummam collecto cum, 12. dabit quadratum <lb/>maius .16.</s> - </p> - <p> - <s xml:id="s348" xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, quadratum maius per lineam <var>.q.g.</var> minus per <var>.g.p.</var> Å¿i-<lb/>gnificetur: </s> - <s xml:id="s349" xml:space="preserve">Å¿uper integram autem <var>.q.p.</var> erigatur quadratum integrum <var>.d.p.</var> diuiÅ¿um, <lb/>vt quadratum <var>.f.g.</var> vigeÅ¿imiÅ¿eptimi theorematis huius libri, (idipÅ¿um accideret di-<lb/>uiÅ¿o quadrato modo octauæ Å¿ecundi Euclidis) quæ quidem diuiÅ¿io, eÅ¿t via quatuor <lb/>productorum <var>.q.g.</var> in <var>.g.p.</var> è quibus vnum Å¿it <var>.g.r.</var> quod erit cognitum ex .19. theore <lb/>mate cum Å¿it <reg norm="quadratum" type="context">quadratÅ©</reg> primi numeri ppoÅ¿iti, ex quo illa quatuor cognita <reg norm="erunt" type="context">erÅ©t</reg>. </s> - <s xml:id="s350" xml:space="preserve">Iam <lb/>verò Å¿i cogitemus <var>.q.p.</var> Å¿ectam in puncto <var>.t.</var> ita vt <var>.q.t.</var> æqualis Å¿it <var>.p.g.</var> dabitur differen <lb/>tia <var>.t.g.</var> cognita, vt radix quadrati <var>.e.o.</var> cum ex præſup-<lb/>poÅ¿ito <var>.r.n.</var> æqualis Å¿it <var>.q.g.</var> et <var>.r.e</var>: <var>g.p.</var> ex quo etiam <var>.q.t.</var> <lb/> - <anchor type="figure" xlink:label="fig-0037-01a" xlink:href="fig-0037-01"/> - ita pariter <var>.e.n.t.g.</var> æqualis erit. </s> - <s xml:id="s351" xml:space="preserve">Collecto <reg norm="itaque" type="simple">itaq;</reg> quadra <lb/>to <var>.e.o.</var> ipÅ¿ius <var>.t.g.</var> cum quadruplo <var>.g.r</var>: cognitum erit <lb/>quadratum <var>.d.p.</var> ipÅ¿ius <var>.q.p.</var> </s> - <s xml:id="s352" xml:space="preserve">quare cognoÅ¿cetur <var>.q.p.</var> de <lb/>quo numero detracta differétia quadratorum cognita <var>.<lb/>t.g.</var> Å¿upererit aggregatum <var>.p.g.</var> et <var>.q.t.</var> cognitum. </s> - <s xml:id="s353" xml:space="preserve">Qua-<lb/>re ex conÅ¿equenti, dimidium aggregati, nempe <var>.g.p.</var> <lb/>cognoÅ¿cetur, tanquam minus duorum quadratorum. <lb/></s> - <s xml:id="s354" xml:space="preserve">cui iuncta <var>.g.t.</var> aut detracta <var>.p.g.</var> ex <var>.p.q.</var> quadratum <var>.q.<lb/>g.</var> maius cognitum remanebit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0037-01" xlink:href="fig-0037-01a"> - <image file="0037-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0037-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="40"> - <head xml:id="head56" xml:space="preserve">THEOREMA <num value="40">XL</num>.</head> - <p> - <s xml:id="s355" xml:space="preserve">CVR ijs, qui volunt duos eiuÅ¿modi numeros inuenire, vt eorum maior mi-<lb/>norem, numero propoÅ¿ito Å¿uperet, & productum vnius in alterum, alteri nu-<lb/>mero propoÅ¿ito adÄ™quetur, conÅ¿ultiÅ¿simum Å¿it dimidium primi numeri propoÅ¿iti, - <pb o="26" rhead="IO. BAPT. BENED." n="38" file="0038" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0038"/> - numerum inquam, cui differentia duorum quæſitorum æquanda eÅ¿t, in Å¿eipÅ¿um <lb/>multiplicare, atque huic quadrato, Å¿ecundum numerum propoÅ¿itum iungere, cui, <lb/>productum numerorum quæſitorum æquale eſſe debet, & ex hac Å¿umma eruere qua <lb/>dratam radicem, quæ coniuncta dimidio primi numeri propoÅ¿iti, dabit maiorem <lb/>duorum numerorum & ex eadem radice detracto dimidio primi numeri, minorem <lb/>numerum duorum quæſitorum.</s> - </p> - <p> - <s xml:id="s356" xml:space="preserve">Exempli gratia, Å¿i proponeretur .12. cui differentia vnius numeri ab altero æqua-<lb/>ri deberet, tum proponeretur .64. cui productum multiplicationis duorum quæſi-<lb/>torum Å¿imul <reg norm="æquandum" type="context">æquãdum</reg> eſſet. </s> - <s xml:id="s357" xml:space="preserve">Dimidium primi numeri in Å¿eipÅ¿um multiplicaremus, <lb/><reg norm="proueniretque" type="simple">proueniretqÌ;</reg> <reg norm="quadratum" type="context">quadratÅ©</reg> .36. cui coniuncto Å¿ecundo, nempe .64. totum eſſet .100. <lb/>ex quo detracta quadrata radice .10. etipÅ¿i coniuncto Å¿enario, dimidio primi nume <lb/>ri, & ex eadem detracto eodem dimidio .6. pro maiore numero proueniret .16. & <lb/>pro minore .4.</s> - </p> - <p> - <s xml:id="s358" xml:space="preserve">Cuius rei Å¿peculatio hæc eÅ¿t. </s> - <s xml:id="s359" xml:space="preserve">Sit <var>.e.o.</var> differentia cognita duorum incognitorum <lb/>numerorum <var>.a.o.</var> et <var>.a.e.</var> quorum productum datum Å¿iue cognitum Å¿it <var>.a.s</var>: conÅ¿ide-<lb/>remus nunc <var>.e.i.</var> dimidium <var>.e.o.</var> datæ differentiæ, & ex compoÅ¿ito <var>.a.i.</var> imaginetur <lb/>quadratum <var>.a.x.</var> in quo protracta Å¿it <var>.t.u.</var> æquidiÅ¿tans lateri <var>.a.i.</var> & tam ab ipÅ¿a <var>.a.i.</var> re <lb/>mota, quam <var>.x.i.</var> ab <var>.s.e.</var> vnde <var>.t.e.</var> quadratum erit <var>.e.i.</var> <lb/>dimidiæ Å¿cilicet differentiæ datæ <var>.e.o.</var> et <var>.t.n.</var> rectan-<lb/> - <anchor type="figure" xlink:label="fig-0038-01a" xlink:href="fig-0038-01"/> - gulum æquale erit rectangulo <var>.n.c.</var> vt cuilibet licet <lb/>per Å¿e conÅ¿iderare, vnde Å¿equitur gnomonem <var>.e.r.t.</var> <lb/>æqualem eſſe producto <var>.a.s.</var> ideo cognitus, qui <reg norm="quidem" type="context">quidẽ</reg> <lb/>gnomon, Å¿i coniunctus fuerit quadrato <var>.e.t.</var> cognito <lb/>ex radice <var>.e.i.</var> cognita (vt dimidia toralis differentiÄ™ <var>.<lb/>e.o.</var> datæ) habebimus quadratum totale <var>.a.x.</var> cogni-<lb/>tum, & ita eius radicem <var>.a.i.</var> cognitam & reliqua om <lb/>nia conÅ¿equenter quæ quidem Å¿peculatio eadem eÅ¿t <lb/>quæ .6. Å¿ecundi Å¿eu .8. noni Euclidis.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0038-01" xlink:href="fig-0038-01a"> - <image file="0038-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0038-01"/> - </figure> - </div> - <p> - <s xml:id="s360" xml:space="preserve">Poteris tamen ex modo & rationibus præceden-<lb/>ti theoremate allatis, hocipÅ¿um concludere.</s> - </p> - </div> - <div type="math:theorem" level="3" n="41"> - <head xml:id="head57" xml:space="preserve">THEOREMA <num value="41">XLI</num>.</head> - <p> - <s xml:id="s361" xml:space="preserve">CVR ij, qui aliquo propoÅ¿ito numero, inuenturi Å¿unt duos numeros inter Å¿e <lb/>differentes, quorum quadratorum Å¿umma altero numero propoÅ¿ito æqualis <lb/>Å¿it, rectè primum numerum propoÅ¿itum in Å¿eipÅ¿um multiplicant, quod quadratum <lb/>exÅ¿ecundo numero <reg norm="detrahunt" type="context">detrahÅ©t</reg>, & dimidium reÅ¿idui Å¿umunt, quod productum erit <lb/>multiplicationis duorum numerorum interÅ¿e, in reliquis præcedentis theorematis <lb/>ordinem Å¿equuntur.</s> - </p> - <p> - <s xml:id="s362" xml:space="preserve">Exempli gratia, Å¿i proponeretur .12. tanquam numerus, cui differentia duorum <lb/>numerorum quæſitorum æquanda eÅ¿t, proponerentur præterea .272. quibus Å¿um-<lb/>ma quadratorum duorum numerorum quæſitorum æquari deberet, oporteret Å¿anè <lb/>primum numerum, nempe .12. in Å¿eipÅ¿um multiplicare, cuius <reg norm="quadratum" type="context">quadratÅ©</reg> hoc loco <lb/>eſſet .144. atque hoc detrahere ex Å¿ecundo numero, Å¿upereſſet .128. Å¿umpto <lb/>deinde dimidio huiuÅ¿ce numeri, népe .64. producto in quam duorum numerorum <lb/><reg norm="quæſitorum" type="context">quæſitorÅ©</reg>. </s> - <s xml:id="s363" xml:space="preserve">Cum hoc .64. proÅ¿tea et duodenario primo propoÅ¿ito numero, præceden <lb/>tis theorematis ordinem Å¿equeremur.</s> - </p> - <pb o="27" rhead="THEOREM. ARIT." n="39" file="0039" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0039"/> - <p> - <s xml:id="s364" xml:space="preserve">Quod vt Å¿peculemus, conÅ¿ideremus Å¿ubÅ¿criptam figuram, vigefiminoni theore-<lb/>matis figuræ Å¿imilem, in qua numeri quæſiti duabus <lb/>lineis directè coniunctis <var>.q.g.</var> et <var>.g.p.</var> fignificentur, ho <lb/> - <anchor type="figure" xlink:label="fig-0039-01a" xlink:href="fig-0039-01"/> - <reg norm="rum" type="context">rÅ©</reg> quadrata <reg norm="erunt" type="context">erÅ©t</reg> <var>.r.c.</var> et <var>.g.s.</var> <reg norm="quorum" type="context">quorÅ©</reg> <reg norm="summa" type="context">sÅ©ma</reg> <reg norm="iterum" type="context">iterÅ©</reg> propo <lb/>nitur, quare etiam cognita. </s> - <s xml:id="s365" xml:space="preserve"><reg norm="Differentia" type="context">Differẽtia</reg> autem <reg norm="duorum" type="context">duorÅ©</reg> <lb/>numerorum primo propofita fit <var>.q.i.</var> eius verò qua-<lb/>dratum <var>.m.e.</var> quod cognitum eÅ¿t ex Å¿ua radice <var>.q.i</var>. <lb/></s> - <s xml:id="s366" xml:space="preserve">quare gnomon <var>.e.n.m.</var> Å¿imul cum quadrato minori <var>.<lb/>g.s.</var> cognitus erit, quæ Å¿umma æqualis eÅ¿t duplo <var>.g.r.</var> <lb/>producto datorum numerorum. </s> - <s xml:id="s367" xml:space="preserve">Itaque & ipÅ¿a <var>.g.<lb/>r.</var> cognoÅ¿cetur, nunc Å¿i præcedentis theorematis Å¿pe-<lb/>culationem in reliquis conÅ¿uluerimus propoÅ¿itum <lb/>conÅ¿equemur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0039-01" xlink:href="fig-0039-01a"> - <image file="0039-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0039-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="42"> - <head xml:id="head58" xml:space="preserve">THEOREMA <num value="42">XLII</num>.</head> - <p> - <s xml:id="s368" xml:space="preserve">ADhuc etiam & alia ratione idipÅ¿um conÅ¿equi poſſemus, non conÅ¿ulto qua-<lb/>drageÅ¿imo theoremate. </s> - <s xml:id="s369" xml:space="preserve">Nam Å¿ubtracto quadrato differentiæ, numeri primi <lb/>(<reg norm="inquam" type="context">inquã</reg>) propoÅ¿iti, ex <reg norm="summa" type="context">sÅ©ma</reg> duorum quadratorum, nempe ex Å¿ecundo numero pro-<lb/>poÅ¿ito colligendum eſſet reÅ¿iduum in Å¿ummam cum prædicto Å¿ecundo numero, & <lb/>ex Å¿umma hac deÅ¿umenda quadrata radix, quæ duorum numerorum Å¿umma erit, <lb/>de qua detracto primo numero, remanebit duplum minoris numeri quæſiti, cuius <lb/>dimidio addito primo numero propoÅ¿ito, aut detracto minore inuento ex radice <lb/>poÅ¿tremo inuenta, dabitur numerus maior, qui quæritur.</s> - </p> - <p> - <s xml:id="s370" xml:space="preserve">Exempli gratia, cum Å¿uperfuerint .128. hæc Å¿i cum Å¿ecundo numero <reg norm="nempe" type="context">nẽpe</reg> .272. <lb/>iunxerimus, dabunt .400. quorum radix erit .20. de quo numero detracto primo <lb/>propoÅ¿ito, nempe .12. Å¿upererunt .8. quorum <reg norm="dimidium" type="context">dimidiÅ©</reg> erit .4. quo ex .20. detracto <lb/>aut coniuncto .12. maior numerus orietur.</s> - </p> - <p> - <s xml:id="s371" xml:space="preserve">Cuius rei contemplatio, præcedenti figura aperitur. </s> - <s xml:id="s372" xml:space="preserve">Nam reÅ¿iduum detractionis <lb/>quadrati <var>.m.e.</var> ex Å¿umma <reg norm="duorum" type="context">duorÅ©</reg> quadratorum <var>.r.c.</var> et <var>.g.s.</var> numerum præbet æqua-<lb/>lem duobus Å¿upplementis <var>.q.n.</var> et <var>.n.u.</var> ex .8. Å¿ecundi Euclidis. qui coniunctus duo-<lb/>bus quadratis (quorum Å¿umma Å¿ecundo propoÅ¿ita fuit) cognitionem profert qua-<lb/>drati <var>.q.u.</var> & eius radicis <var>.q.p.</var> de qua, detracto primo dato numero, Å¿cilicet <var>.q.i.</var> Å¿u-<lb/>pereÅ¿t <var>.i.p.</var> cuius dimidium nempe <var>.g.p.</var> minor eÅ¿t numerus qui quÄ™ritur; </s> - <s xml:id="s373" xml:space="preserve">reÅ¿iduum <lb/>verò totius <var>.g.q.</var> maior Å¿cilicet.</s> - </p> - </div> - <div type="math:theorem" level="3" n="43"> - <head xml:id="head59" xml:space="preserve">THEOREMA <num value="43">XLIII</num>.</head> - <p> - <s xml:id="s374" xml:space="preserve">CVR ij, qui volunt duos numeros inuenire, quorum Å¿umma æqualis propo-<lb/>fito alicui numero futura Å¿it, & Å¿umma quadratorum maior eorum produ-<lb/>cto per quantitatem alterius propoÅ¿iti numeri, rectè dimidium primi dati numeri in <lb/>Å¿eipÅ¿um multiplicant, quod quadratum ex <reg norm="Å¿ecundo" type="context">Å¿ecÅ©do</reg> dato numero detrahunt, Å¿umuntÂ<lb/>qÌue tertiÄ™ partis refidui quadratam radicem, quam dimidio primi numeri coniun-<lb/>gunt, ex quo maior numerus <reg norm="duorum" type="context">duorÅ©</reg> <reg norm="quæſitorum" type="context">quæſitorÅ©</reg> datur, quo ex toto primo detracto, Å¿u-<lb/>pererit minor.</s> - </p> - <p> - <s xml:id="s375" xml:space="preserve">Exempli gratia, propoÅ¿ito numero .20. cui æquanda eÅ¿t Å¿umma duorum nume-<lb/>rorum quæſitorum, <reg norm="datoque" type="simple">datoqÌ;</reg> Å¿ecundo numero .208. qui Å¿emper maior eſſe debet - <pb o="28" rhead="IO. BAPT. BENED." n="40" file="0040" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0040"/> - quadrato dimidij, prout ex Å¿peculatione huiuÅ¿modi operis cognoÅ¿cetur, <reg norm="cuiæquanda" type="context">cuiæquãda</reg> <lb/>eÅ¿t <reg norm="differentia" type="context">differẽtia</reg> inter <reg norm="Å¿ummam" type="context">Å¿ummã</reg> <reg norm="quadratorum" type="context">quadratorÅ©</reg> <reg norm="duorum" type="context">duorÅ©</reg> qui <reg norm="quæruntur" type="context">quærÅ©tur</reg> <reg norm="numerorum" type="context">numerorÅ©</reg>, Å¿imul <reg norm="cum" type="context">cÅ©</reg> pro <lb/>ducto <reg norm="eorum" type="context">eorÅ©</reg> radicum. </s> - <s xml:id="s376" xml:space="preserve">Dimidium numeri .20. in Å¿eipÅ¿um multiplicandum eſſet, qua-<lb/><reg norm="dratumque" type="simple">dratumqÌ;</reg> detrahendum ex .208. vtremanerent .108. quorum .108. tertiæ partis qua <lb/>drata radix eſſet .6. quæ Å¿i iuncta fuerit dimidio .20. nempe .10. daretur maior nu-<lb/>merus quæſitus .16. quo detracto è .20. darentur .4.</s> - </p> - <p> - <s xml:id="s377" xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, datus primus numerus Å¿ignificetur linea <var>.g.h.</var> in qua <lb/>maior numerus incognitus Å¿it <var>.g.h.</var> minor verò <var>.b.h.</var> quorum quadrata Å¿int <var>.y.t.</var> et <var>.<lb/>b.l.</var> in quadrato maximo <var>.g.p.</var> tum productum <var>.g.b.</var> in <var>.b.h.</var> Å¿it <var>.g.c.</var> <reg norm="cogitenturque" type="simple">cogitenturqÌ;</reg> duo <lb/>diametri <var>.q.h.</var> et <var>.g.p.</var> diuiÅ¿i per medium in puncto <var>.o.</var> per quod duÄ™ lineæ ducan-<lb/>tur <var>.f.d.</var> et <var>.k.m.</var> parallelæ lateribus maximi quadrati. </s> - <s xml:id="s378" xml:space="preserve">Hæ dictum quadratum in <lb/>quatuor quadrata æqualia diuident, quorum <reg norm="vnumquodque" type="simple">vnumquodqÌ;</reg>, æquale erit quadrato <var>.<lb/>g.f.</var> dimidij ipÅ¿ius <var>.g.h.</var> datÄ™, </s> - <s xml:id="s379" xml:space="preserve">quare eorum <reg norm="vnumquodque" type="simple">vnumquodqÌ;</reg> cognitum erit. </s> - <s xml:id="s380" xml:space="preserve">Iterum co <lb/>gitemus <var>.s.x.</var> per <var>.e.</var> <reg norm="parallelam" type="context">parallelã</reg> <var>.g.k.</var> tantum diÅ¿tan-<lb/>tem à <var>.g.k.</var> quantum <var>.y.l.</var> ab <var>.g.h.</var> diÅ¿tare inueni-<lb/> - <anchor type="figure" xlink:label="fig-0040-01a" xlink:href="fig-0040-01"/> - tur. </s> - <s xml:id="s381" xml:space="preserve">Cogitetur pariter <var>.z.i.a.</var> per punctum <var>.i.</var> <lb/>parallela <var>.d.p.</var> </s> - <s xml:id="s382" xml:space="preserve">quare <var>.a.t.</var> æqualis erit <var>.f.c.</var> et <var>.y.x.</var> <lb/>æqualis <var>.f.e.</var> et <var>.y.s</var>: <var>b.l.</var> æqualis. </s> - <s xml:id="s383" xml:space="preserve">Ita Å¿ubtractis è <lb/>duobus quadratis Å¿uperius dictis <var>.a.t.y.x.</var> et <var>.b.l.</var> <lb/>producto <var>.y.b.</var> æqualibus, Å¿upererunt <var>.k.d.</var> et <var>.a.c.<lb/>x.</var> cognita, tanquam æqualia dato Å¿ecundo nu-<lb/>mero, Å¿ed <var>.k.d.</var> quadratum eÅ¿t medietatis <var>.g.f.</var> <lb/>cognitæ, cognoÅ¿cetur igitur reÅ¿iduum <var>.a.c.x.</var> vnà <lb/>etiam Å¿ingulæ tertiæ partes nempe quadrata <var>.o.<lb/>i.o.c.</var> et <var>.o.e.</var> & radix <var>.b.f.</var> vel <var>.f.s.</var> Å¿ingularum, <lb/>qua coniuncta dimidio <var>.g.f.</var> <reg norm="rurfusque" type="simple">rurfusqÌ;</reg> ab <reg norm="eodem" type="context">eodẽ</reg> de-<lb/>tracta, propoÅ¿itum conÅ¿equemur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0040-01" xlink:href="fig-0040-01a"> - <image file="0040-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0040-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="44"> - <head xml:id="head60" xml:space="preserve">THEOREMA <num value="44">XLIIII</num>.</head> - <p> - <s xml:id="s384" xml:space="preserve">CVR Å¿i quis cupiat numerum propoÅ¿itum in duas eiuÅ¿modi partes diuidere, vt <lb/>quadratum maioris, quadratum minoris Å¿uperet quantitate alterius numeri <lb/>propoÅ¿iti, rectè primum numerum in Å¿eipÅ¿um multiplicabit, & ab eodem Å¿ecun-<lb/>dum numerum detrahet, reÅ¿iduum verò per duplum primi diuidet, ex quo proue-<lb/>niens primi pars minor erit, quæ ex illo primo detracta, partem maiorem <lb/>proferet.</s> - </p> - <p> - <s xml:id="s385" xml:space="preserve">Exempli gratia, Å¿i proponantur .20. diuiÅ¿a in duas eiuÅ¿modi partes, vt <reg norm="quadratum" type="context">quadratÅ©</reg> <lb/>maioris Å¿uperet quadratum minoris numero æquali ipÅ¿i .240. oportebit primum <lb/>numerum, qui quadratus cum fuerit, erit .400. in Å¿eipÅ¿um multiplicare, & ex hoc <lb/>quadrato Å¿ecundum numerum nempe .240. detrahere, </s> - <s xml:id="s386" xml:space="preserve">tunc remanebunt .160. quÄ™ <lb/>diuiÅ¿a per .40. <reg norm="numerum" type="context">numerÅ©</reg> <reg norm="duplum" type="context">duplÅ©</reg> primo, dabuntur quatuor pro minori numero, à reÅ¿i-<lb/>duo verò .20. detractis quatuor, erunt .16. pro maiorinumero.</s> - </p> - <p> - <s xml:id="s387" xml:space="preserve">Quod vt exactè conÅ¿ideremus, primus numerus propoÅ¿itus Å¿ignificetur linea <var>.q.<lb/>h.</var> diuidendus in duas partes <var>.q.p.</var> et <var>.p.h.</var> tales quales quærimus. </s> - <s xml:id="s388" xml:space="preserve">PoÅ¿tmodum eriga <lb/><gap extent="2"/>r quadratum <var>.q.e.</var> diuiÅ¿um diametro <var>.f.h.</var> <reg norm="ductisque" type="simple">ductisqÌ;</reg> <var>.p.o.t.</var> et <var>.a.o.c.</var> parallelis lateri-<lb/>bus quadrati, dabuntur imaginaria quadrata <var>.c.t.</var> et <var>.p.a.</var> duarum partium <var>.q.p.</var> et <var>.p.<lb/>h.</var> incognitarum. </s> - <s xml:id="s389" xml:space="preserve">Ad hæc cogitemus quadratum <var>.u.n.</var> æquale quadrato <var>.p.a.</var> è quadra - <pb o="29" rhead="THEOR. ARITH." n="41" file="0041" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0041"/> - to maiore <var>.c.t.</var> extractum quare reÅ¿iduum qua-<lb/> - <anchor type="figure" xlink:label="fig-0041-01a" xlink:href="fig-0041-01"/> - drati <var>.c.p.</var> cognitum erit, quam quantitatem co-<lb/>gnitam, cum Å¿it Å¿ecundo loco data, cogitemus <lb/>detrahi è toto quadrato cognito <var>.q.e.</var> ex quo <lb/>Å¿umma duorum Å¿upplementorum <var>.q.o.</var> et <var>.o.e.</var> <lb/>cognoÅ¿cetur, vnà cum quadratis <var>.u.n.</var> et <var>.p.a.</var> du <lb/>plo Å¿cilicet <var>.q.a.</var> quo diuiÅ¿o per duplum <var>.q.h.</var> aut <lb/>Å¿implex <var>.q.a.</var> per <var>.q.h.</var> Å¿implicem, dabitur <var>.a.h.</var> <lb/>nempe <var>.p.h.</var> minor numerus quæſitus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0041-01" xlink:href="fig-0041-01a"> - <image file="0041-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0041-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="45"> - <head xml:id="head61" xml:space="preserve">THEOREMA <num value="45">XLV</num>.</head> - <p> - <s xml:id="s390" xml:space="preserve">CVR volentes diuidere numerum propoÅ¿itum in duas eiuÅ¿modi partes, vt pro <lb/>ductum vnius in alteram, alteri numero propoÅ¿ito æquetur, rectè dimidium <lb/>primi dati numeri in Å¿eipÅ¿um multiplicant, ex quo quadrato Å¿ecundum datum nu-<lb/>merum detrahunt, <reg norm="reÅ¿iduique" type="simple">reÅ¿iduiqÌ;</reg> radicem Å¿umunt, qua coniuncta vni dimidio primi nu-<lb/>meri, pars maior datur, ex altero verò dimidio detracta, minorem manifeÅ¿tabit.</s> - </p> - <p> - <s xml:id="s391" xml:space="preserve">Exempli gratia, Å¿i numerus partiendus eſſet .34. alter verò numerus eſſet .64. cui <lb/>productum vnius partis in alteram æquale eſſe deberet. </s> - <s xml:id="s392" xml:space="preserve">Dimidium primi numeri, in <lb/>Å¿eipÅ¿um multiplicaremus, cuius quadratum eſſet .289. de quo detracto Å¿ecundo nu-<lb/>mero nempe .64. remaneret .225. cuius quadrata radix nempe .15. coniuncta .17. <lb/>dimidio .34. proferet .32. maiorem partem, <reg norm="detractoque" type="simple">detractoqÌ;</reg> ex .17. Å¿upereſſet .2. pars <lb/>inquam minor.</s> - </p> - <p> - <s xml:id="s393" xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, primus numerus propoÅ¿itus Å¿ignificetur linea <var>.a.d.</var> cu-<lb/>ius dimidium <var>.c.d.</var> cognitum erit, vnà etiam eius quadratum <var>.c.f.</var> quo diuiÅ¿o per dia <lb/>metrum <var>.e.d.</var> Å¿upponantur partes ignotæ <lb/> - <anchor type="figure" xlink:label="fig-0041-02a" xlink:href="fig-0041-02"/> - ipÅ¿ius <var>.a.d.</var> eſſe <var>.a.b.</var> et <var>.b.d.</var> & à puncto <var>.b.</var> <lb/>duci lineam <var>.b.h.g.</var> parallelam <var>.d.f.</var> et <var>.m.<lb/>h.k.</var> parallelam <var>.d.a.</var> extructa figura Å¿imi <lb/>li figuræ quintæ Å¿ecundi Eucli. </s> - <s xml:id="s394" xml:space="preserve">quare da <lb/>bitur <reg norm="gnomon" type="context">gnomõ</reg> <var>.l.d.g.</var> æqualis producto <var>.b.<lb/>k.</var> & proinde cognitus, quo detracto è <lb/>quadrato, <var>c.f.</var> remanebit quadratum <var>.g.l.</var> <lb/>cuius radice æquali <var>.c.b.</var> coniuncta <var>.a.c.</var> <lb/>& detracta ex <var>.c.d.</var> partes <var>.a.b.</var> et <var>.b.d.</var> quæſitæ dabuntur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0041-02" xlink:href="fig-0041-02a"> - <image file="0041-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0041-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="46"> - <head xml:id="head62" xml:space="preserve">THEOREMA <num value="46">XLVI</num>.</head> - <p> - <s xml:id="s395" xml:space="preserve">CVR propoÅ¿itis tribus numeris, quorum prior in duas eiuÅ¿modi partes diui-<lb/>dendus Å¿it, ut mutuò diuiſæ, & per Å¿ummam prouenientium diuiÅ¿o Å¿ecundo <lb/>numero, proueniens vltimum Å¿it æquale tertio numerorum propoÅ¿itorum. </s> - <s xml:id="s396" xml:space="preserve">ConÅ¿ul <lb/>tiÅ¿simum Å¿it Å¿ecundum numerum per tertium diuidere, ex quo proueniens Å¿it Å¿um-<lb/>ma prouenientium è duabus partibus mutuò diuiÅ¿is, quam Å¿ummam Å¿i quis velit di-<lb/>Å¿tinguere, rectè poſſit medio operationis <reg norm="prÄ™cedentis" type="context">prÄ™cedẽtis</reg> theorematis <reg norm="sumpta" type="context">sÅ©pta</reg> vnitate Å¿uper <lb/>ficiali pro Å¿ecundo numero diÅ¿tinctis poÅ¿tmodum prouenientibus, rectè meo iudi-<lb/>cio operabimur per <reg norm="regulam" type="context">regulã</reg> de tribus (quod fuit ab antiquis prætermiſſum) Si dixe- - <pb o="30" rhead="IO. BAPT. BENED." n="42" file="0042" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0042"/> - rimus, Å¿i Å¿umma vnius dictorum prouenientium cum vnitate dat primum numerum, <lb/>quid ipÅ¿a eadem vnitas dabit? </s> - <s xml:id="s397" xml:space="preserve">ex quo propoÅ¿itum oriatur.</s> - </p> - <p> - <s xml:id="s398" xml:space="preserve">Exempli gratia, proponuntur tres numeri, primus .20. Å¿ecundus .34. tertius .8. <lb/>Iam quærimus diuidere primum .20. in duas partes quæ mutuò diuiſæ prÄ™beant duo <lb/>prouenientia, quorum Å¿umma tanta Å¿it vt per eam diuiÅ¿o .34. proueniat numerus <lb/>æqualis tertio numero .8. </s> - <s xml:id="s399" xml:space="preserve">Quod vt præſtemus iubet regula Å¿ecundum .34. per <reg norm="tertium" type="context">tertiÅ©</reg> <num value="8">.<lb/>8.</num> diuidi, vnde proueniet .4. cum vna quarta parte, quod proueniens erit Å¿umma pro <lb/>uenientium ex diuiÅ¿ione duarum partium quæſitarum, quæ Å¿i diÅ¿tinguere volueri-<lb/>mus, præcedentis theorematis methodum Å¿equemur, vnitate Å¿uperficiali pro Å¿ecun <lb/>do numero propoÅ¿ito Å¿umpta, ac Å¿i diceremus, diuidatur .4. cum vna quarta parte <lb/>in duas eiuÅ¿modi partes, vt productum vnius in alteram Å¿it vnitas Å¿uperficialis, cer-<lb/>tè fractis integris cum quarta parte coniungendis, darentur vnitatis decemÅ¿eptem <lb/>quartæ lineares, verum cum neceſſe Å¿it, ex præcedenti theoremate, dimidium in <lb/>Å¿eipÅ¿um multiplicare, <reg norm="eſſetque" type="simple">eſſetqÌ;</reg> dimidium .8. quartarum partium cum octaua, com-<lb/>modius totum conÅ¿tituetur .34. octauarum, quarum dimidium, nempe decemÅ¿ep-<lb/>tem octauæ, in Å¿eipÅ¿um multiplicatum erunt .289. Å¿exageÅ¿imæ quartæ vnius integri <lb/>Å¿uperficialis, quandoquidem <reg norm="integrum" type="context">integrÅ©</reg> Å¿uperficiale, cuius vnitas linearis in .8. partes <lb/>diuiditur eÅ¿t .64. vt ex primo theoremate huius libri depræhendi poteÅ¿t. </s> - <s xml:id="s400" xml:space="preserve">Nunc vni-<lb/>tate hac Å¿uperficiali, nempe .64. ex .289. detracta, Å¿upererit .225. cuius radix qua-<lb/>drata, Å¿cilicet .15. coniuncta dimidio dictorum prouenientium, nempe .17. dabit <lb/>maius proueniens .32. <reg norm="detractaque" type="simple">detractaqÌ;</reg> ex altero dimidio, dabit proueniens minus .2. hoc <lb/>eÅ¿t pro maiore proueniente .32. octauas, & pro minore duas, quatuor Å¿cilicet inte-<lb/>gros pro maiore, & quartam partem vnius integri pro minore. </s> - <s xml:id="s401" xml:space="preserve">Nunc Å¿i ex regula <lb/>de tribus dixerimus, Å¿i .4. iuncta vni, nempe .5. dant .20. primum numerum, quid <lb/>dabunt .4. integra (proueniens inquam maius) <reg norm="dabunt" type="context">dabÅ©t</reg> certè .16. partem maiorem. <lb/></s> - <s xml:id="s402" xml:space="preserve">Tum Å¿i dixerimus, Å¿i quarta pars coniuncta vnitati dat .20: </s> - <s xml:id="s403" xml:space="preserve">quid dabit quarta illa <lb/>pars (hoc eÅ¿t proueniens minus) dabit <reg norm="profectò" type="simple">ê“fectò</reg> quatuor Å¿cilicet <reg norm="minorem" type="context">minorẽ</reg> partem, quod <lb/>ab antiquis certè ignoratum fuit, qui, inuentis prouenientibus quieuerunt, ne-<lb/>Å¿cientes ijs vti ad inueniendas duas primi numeri partes.</s> - </p> - <p> - <s xml:id="s404" xml:space="preserve">Cuius Å¿peculationis gratia, demus primum numerum Å¿ignificari linea <var>.e.u.</var> cuius <lb/>partes <var>.e.a.</var> & <var>a.u.</var> Å¿int quæ quæruntur, alter verò numerus Å¿ignificetur linea <var>.b.<lb/>d.</var> tertius linea <var>.g.f.</var> proueniens <reg norm="autem" type="wordlist">aÅ©t</reg> diuiÅ¿ionis <var>.e.a.</var> per <var>.a.u.</var> Å¿it <var>.n.t.</var> diuiÅ¿ionis <reg norm="autem" type="wordlist">aÅ©t</reg> <var>.a.u.</var> <lb/>per <var>.a.e.</var> Å¿it <var>.t.o.</var> Å¿umma erit <var>.n.t.o.</var> vnitas verò <var>.n.i.</var> et <var>.o.i</var>. </s> - <s xml:id="s405" xml:space="preserve">Iam Å¿i numerus <var>.f.g.</var> tertiò <lb/>propoÅ¿itus ex diuiÅ¿ione Å¿ecundi per <var>.o.t.n.</var> proferri debet. </s> - <s xml:id="s406" xml:space="preserve">Ex .13. theoremate patet, <lb/>quòd Å¿i <var>.b.d.</var> per <var>.g.f.</var> diuiÅ¿erimus, proferetur <var>.o.t.n.</var> qui cum fuerit inuentus, <reg norm="Å¿ummam" type="context">Å¿ummã</reg> <lb/>eſſe oportet <reg norm="duorum" type="context">duorÅ©</reg> <reg norm="prouenientium" type="context">prouenientiÅ©</reg>, ex diuiÅ¿ione mutua <reg norm="duorum" type="context">duorÅ©</reg> numerorum, nempe <var>.<lb/>a.e.</var> per <var>.a.u.</var> et <var>.a.u.</var> per <var>.a.e.</var> deinde manifeÅ¿tum eÅ¿t ex .24. aut .25. theoremate <reg norm="eorum" type="context">eorÅ©</reg> <lb/>productum (multiplicatis prouenientibus adinuicem) vnitatem Å¿uperficialem futu <lb/>ram eſſe. </s> - <s xml:id="s407" xml:space="preserve">Hactenus igitur, totum <var>.o.n.</var> ex doctrina præcedentis theorematis diui-<lb/>ditur in puncto <var>.t.</var> ita vt productum <var>.o.t.</var> in <var>.t.n.</var> <lb/>Å¿olam vnitatem Å¿uperficialem <reg norm="contineat" type="context">cõtineat</reg>, quo <lb/> - <anchor type="figure" xlink:label="fig-0042-01a" xlink:href="fig-0042-01"/> - facto, Å¿i, vt antedictum eÅ¿t, cogitauerimus <var>.n.<lb/>t.</var> <reg norm="proueniens" type="context">proueniẽs</reg> eſſe ex diuiÅ¿ione <var>.e.a.</var> per <var>.a.u.</var> et <var>.<lb/>t.o.</var> proueniens ex diuiÅ¿ione <var>.a.u.</var> per <var>.a.e.</var> pa-<lb/>tebit ex definitione diuiÅ¿ionis, quod eadem <lb/>erit proportio <var>.a.e.</var> ad <var>.n.t.</var> quæ eÅ¿t <var>.a.u.</var> ad vni-<lb/>tatem <var>.n.i.</var> et <var>.a.u.</var> ad <var>.o.t.</var> eadem quæ eÅ¿t <var>.e.a.</var> - <pb o="31" rhead="THEOREM. ARITH." n="43" file="0043" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0043"/> - ad vnitatem <var>.o.i.</var> <reg norm="permutandoque" type="simple">permutandoqÌ;</reg> <var>.e.a.</var> ad <var>.a.u.</var> Å¿icut <var>.t.n.</var> ad <var>.n.i.</var> & componendo <var>.e.a.u.</var> <lb/>ad <var>a.u.</var> Å¿icut <var>.t.n.i.</var> ad <var>.n.i</var>: & euerÅ¿im <var>.e.a.u.</var> ad <var>.e.a.</var> vt <var>.t.n.i.</var> ad <var>.t.n</var>. </s> - <s xml:id="s408" xml:space="preserve">Quare, ex .20. Å¿epti <lb/>mi, recte vtimur regula de tribus. </s> - <s xml:id="s409" xml:space="preserve">Idem & de altera parte dico, quamuis qui vnam <lb/>teneat, alteram quo que habiturus Å¿it. </s> - <s xml:id="s410" xml:space="preserve">Non mirum tamen Å¿i huiuÅ¿modi problema <lb/>ab antiquis definitum non fuerit, qui hanc vltimam partem non cognouerunt.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0042-01" xlink:href="fig-0042-01a"> - <image file="0042-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0042-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="47"> - <head xml:id="head63" xml:space="preserve">THEOREMA <num value="47">XLVII</num>.</head> - <p> - <s xml:id="s411" xml:space="preserve">CVR duobus numeris mutuó diuiÅ¿is, Å¿i per Å¿ummam prouenientium, produ-<lb/>ctum vnius in alterum multiplicetur, vltimum productum, Å¿ummæ quadra-<lb/>tn<gap extent="2"/>m duorum numerorum æquale futurum Å¿it.</s> - </p> - <p> - <s xml:id="s412" xml:space="preserve">Exempli gratia, propoÅ¿itis .16. et .4. mutuò diuiÅ¿is, Å¿umma prouenientium erit <num value="4">.<lb/>4.</num> integrorum cum quarta parte, qua Å¿umma multiplicata cum producto <reg norm="primorum" type="context">primorÅ©</reg> <lb/>numerorum, nempe .64. dabuntur .272. integri Å¿uperficiales, qui Å¿ummæ quadra-<lb/>torum duorum numerorum æquantur.</s> - </p> - <p> - <s xml:id="s413" xml:space="preserve">Hoc vt conÅ¿ideremus, duo numeri partibus <var>.a.e.</var> et <var>.e.i.</var> in linea <var>.a.i.</var> Å¿ignificentur, <lb/>quorum productum Å¿it <var>.e.d.</var> & <reg norm="quadratum" type="context">quadratÅ©</reg> ipÅ¿ius <var>.a.e.</var> Å¿it <var>.e.p</var>: ipÅ¿ius verò <var>.e.i.</var> Å¿it <var>.e.q.</var> pro-<lb/>ueniens <reg norm="autem" type="wordlist">aÅ©t</reg> ex diuiÅ¿ione <var>.e.i.</var> per <var>.a.e.</var> Å¿it <var>.o.u.</var> proueniens <reg norm="autem" type="wordlist">aÅ©t</reg> <var>.a.e.</var> per <var>.e.i.</var> Å¿it <var>.o.t.</var> quo-<lb/>rum Å¿umma Å¿it <var>.o.u.t.</var> tum productum <var>.e.d</var>: linea <var>.u.n.</var> Å¿ignificetur ad angulum <reg norm="rectum" type="context">rectÅ©</reg> <lb/>coniuncta in puncto <var>.u.</var> extremo ipÅ¿ius <var>.o.u.t.</var> productum <reg norm="autem" type="wordlist">aÅ©t</reg> <var>.u.o.t.</var> in <var>.u.n.</var> Å¿it <var>.n.t</var>. </s> - <s xml:id="s414" xml:space="preserve">Iam <lb/>probandum nobis eÅ¿t <var>.n.t.</var> æqualem eſſe Å¿ummæ duorum quadratorum <var>.q.e.p</var>. </s> - <s xml:id="s415" xml:space="preserve">Quod <lb/>Å¿ingillatim probo, & aſſero productum <var>.o.n.</var> æquale eſſe quadrato <var>.q.e.</var> & <reg norm="productum" type="context">productÅ©</reg> <var>.<lb/>s.t.</var> quadrato <var>.e.p</var>. </s> - <s xml:id="s416" xml:space="preserve">Nam ex .35. theoremate patet numerum <var>.e.i.</var> medium eſſe <reg norm="pro- portionalem" type="context">pro-<lb/>portionalẽ</reg> inter <var>.e.d.</var> et <var>.o.u</var>: cum numerus <var>.e.i.</var> ex præſuppoÅ¿ito ab <var>.e.a.</var> multiplicetur <lb/>& diuidatur, cuius multiplicationis produ-<lb/>ctum eÅ¿t <var>.d.e</var>: nempe <var>.u.n.</var> & proueniens ex <lb/> - <anchor type="figure" xlink:label="fig-0043-01a" xlink:href="fig-0043-01"/> - diuiÅ¿ione eÅ¿t <var>.o.u</var>: </s> - <s xml:id="s417" xml:space="preserve">quare ex dicto theorema-<lb/>te <var>.e.i.</var> media proportionalis eÅ¿t inter <var>.u.n.</var> et <var>.<lb/>u.o</var>. </s> - <s xml:id="s418" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> productum <var>.o.n.</var> æquale eÅ¿t qua-<lb/>drato <var>.e.q.</var> ex .16. Å¿exti vel .20. Å¿eptimi. </s> - <s xml:id="s419" xml:space="preserve">Idem <lb/>dico de producto <var>.s.t.</var> <reg norm="nempe" type="context">nẽpe</reg> æquale eſſe qua-<lb/>drato <var>.e.p.</var> quandoquidem numerus <var>.a.e.</var> ab <lb/><var>e.i.</var> multiplicatur ac diuiditur, cuius multi-<lb/>plicationis productum eÅ¿t <var>.d.e.</var> nempe <var>o.s.</var> & <lb/>proueniens ex diuiÅ¿ione <var>.o.t</var>: </s> - <s xml:id="s420" xml:space="preserve">inter quæ ex .<lb/>35. theoremate <var>.a.e.</var> media proportionalis <lb/>eÅ¿t. </s> - <s xml:id="s421" xml:space="preserve">Quare ex allatis propoÅ¿itionibus <reg norm="productum" type="context">productÅ©</reg> <var>.s.t.</var> æquale eÅ¿t quadrato <var>.e.p.</var> Å¿ed <reg norm="totum" type="context">totÅ©</reg> <lb/>productum <var>.n.t.</var> Å¿umma eÅ¿t duorum productorum <var>.o.n.</var> et <var>.s.t.</var> ex prima Å¿ecundi Eucli. <lb/></s> - <s xml:id="s422" xml:space="preserve">Itaque verum eſſe quod dictum eÅ¿t, conÅ¿equitur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0043-01" xlink:href="fig-0043-01a"> - <image file="0043-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0043-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="48"> - <head xml:id="head64" xml:space="preserve">THEOREMA <num value="48">XLVIII</num>.</head> - <p> - <s xml:id="s423" xml:space="preserve">CVR Å¿i quis maiorem duorum numerorum Å¿ola vnitate inter Å¿e differentium, <lb/>per minorem diuidat, <reg norm="maioremque" type="simple">maioremqÌ;</reg> per proueniens multiplicet, productum, <lb/><reg norm="summæ" type="context">sÅ©mæ</reg> ipÅ¿ius maioris cum eodem proueniente æquale erit.</s> - </p> - <p> - <s xml:id="s424" xml:space="preserve">Exempli gratia .10 per .9. diuiÅ¿o, datur vnum cum nona parte, quo multiplica-<lb/>to per proueniens, ipÅ¿o nempe .10: </s> - <s xml:id="s425" xml:space="preserve">datur productum .11. cum nona parte, tantum Å¿ci - <pb o="32" rhead="I.O. BAPT. BENED." n="44" file="0044" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0044"/> - licet quanta Å¿umma eÅ¿t maioris cum proueniente.</s> - </p> - <p> - <s xml:id="s426" xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, maior numerus Å¿ignificetur <var>.a.i.</var> et minor linea <var>.a.o.</var> ex <lb/>quo ex præſupoÅ¿ito <var>.o.i.</var> vnitas erit. </s> - <s xml:id="s427" xml:space="preserve">Sit autem proueniens ex diuiÅ¿ione <var>.a.i.</var> per <var>.a.o.<lb/>a.e</var>: </s> - <s xml:id="s428" xml:space="preserve">quod <var>.e.a.</var> directè coniungatur ipÅ¿i <var>.a.i.</var> et productum <var>.a.i.</var> in <var>.a.e.</var> Å¿it <var>.u.i</var>. </s> - <s xml:id="s429" xml:space="preserve">Probabo <lb/>numerum Å¿uperficialem <var>.u.i.</var> æqualem eſſe lineari <var>.i.a.e</var>. </s> - <s xml:id="s430" xml:space="preserve">quare meminiſſe oportet, <lb/>decimotertio theoremate probatum fuiſſe, quod Å¿i numerus diuiÅ¿ibilis per pro-<lb/>ueniens diuidatur, proueniens futurus Å¿it numerus diuidens, </s> - <s xml:id="s431" xml:space="preserve">quare <var>.a.o.</var> erit pro-<lb/>ueniens ex diuiÅ¿ione <var>.a.i.</var> per <var>.a.e.</var> & ex deÅ¿initione diuiÅ¿ionis ita Å¿e habebit <var>.e.a.</var> ad <var>.<lb/>a.i.</var> Å¿icut <var>.o.i.</var> ad <var>.o.a.</var> & componondo ita <var>.e.i.</var> ad <var>.a.i.</var> Å¿icut <var>.i.a.</var> ad <var>.o.a.</var> </s> - <s xml:id="s432" xml:space="preserve">quare <var>.a.i.</var> erit me-<lb/>dia pportionalis inter <var>.e.i.</var> et <var>.a.o.</var> Å¿ed <var>.a.i.</var> non modò diuiÅ¿a <reg norm="nunc" type="context">nÅ©c</reg> cogitatur ab <var>.e.a.</var> ex <lb/>quo Å¿it proueniens <var>.a.o.</var> Å¿ed etiam per eandem <var>.e.a.</var> multiplicata, ex quo produ-<lb/>ctum oriatur <var>.u.i</var>. </s> - <s xml:id="s433" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> ex .25. theobema-<lb/>te <var>.a.i.</var> media eÅ¿t proportionalis inter <var>.u.</var> <lb/> - <anchor type="figure" xlink:label="fig-0044-01a" xlink:href="fig-0044-01"/> - i. et <var>.a.o</var>. </s> - <s xml:id="s434" xml:space="preserve">Quare. ex .11. quinti. eadem erit <lb/>proportio <var>.u.i.</var> ad <var>.a.i.</var> Å¿icut <var>.e.i.</var> ad eandem <var>.<lb/>a.i</var>. </s> - <s xml:id="s435" xml:space="preserve">Igitur ex .9. prædicti numerus <var>.u.i.</var> <lb/>æqualis erit numero <var>.e.i.</var> quod erat propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0044-01" xlink:href="fig-0044-01a"> - <image file="0044-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0044-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="49"> - <head xml:id="head65" xml:space="preserve">THEOREMA <num value="49">XLIX</num>.</head> - <p> - <s xml:id="s436" xml:space="preserve">IDipÅ¿tim etiam alia ratione conÅ¿iderari poteÅ¿t.</s> - </p> - <p> - <s xml:id="s437" xml:space="preserve">Linea <var>.u.a.</var> Å¿ecetur in puncto <var>.t.</var> ita vt <var>.a.t.</var> æqualis Å¿it vnitati <var>.o.i.</var> & media paral <lb/>lela <var>.t.n.</var> terminetur productum <var>.t.i.</var> quod conÅ¿tabit æquali numero, quamuis Å¿uperfi-<lb/>ciali, numero <var>.a.i.</var> tametÅ¿i lineari. </s> - <s xml:id="s438" xml:space="preserve">Tumparallela ducatur à puncto <var>.o.</var> ipÅ¿i <var>.a.u.</var> termi <lb/><reg norm="neturque" type="simple">neturqÌ;</reg> productum <var>.o.u.</var> ex quo bina producta dabuntur <var>.u.o.</var> et <var>.t.i.</var> inter Å¿e æqualia <lb/>ex .15. Å¿exti aut .20. Å¿eptimi cum ita Å¿e habeat <var>.a.i.</var> ad <var>.a.u.</var> Å¿icut <var>.a.o.</var> ad <var>.a.t.</var> Å¿ed <var>.a.i.</var> ad <var>.<lb/>a.o.</var> permutando Å¿ic Å¿e habet Å¿icut <var>.a.u.</var> ad <var>.a.t.</var> & ex prima Å¿exti aut .18. vel .19. Å¿epti-<lb/>mi Å¿ic Å¿e habet <var>.u.i.</var> ad <var>.u.o.</var> Å¿icut <var>.a.i.</var> ad <var>.a.</var> <lb/> - <anchor type="figure" xlink:label="fig-0044-02a" xlink:href="fig-0044-02"/> - o. hoc eÅ¿t <var>.u.i.</var> ad <var>.t.i.</var> ope .11. quinti. </s> - <s xml:id="s439" xml:space="preserve">Iam <lb/>ex definitione diuiÅ¿ionis ita Å¿e habet <var>.a.e.</var> <lb/>ad <var>.a.i.</var> Å¿icut <var>.o.i.</var> ad <var>.o.a.</var> & componendo <var>.<lb/>e.i.</var> ad <var>.a.i.</var> Å¿icut <var>.i.a.</var> ad <var>.o.a</var>. </s> - <s xml:id="s440" xml:space="preserve">Itaque ex præ-<lb/>dicta .11. Å¿ic Å¿e habebit <var>.e.i.</var> ad <var>.i.a.</var> Å¿icut <var>.u.<lb/>i.</var> ad <var>.t.i.</var> Å¿ed <var>.t.i.</var> numero conÅ¿tat æquali <var>.a.<lb/>i</var>. </s> - <s xml:id="s441" xml:space="preserve">quare ex .9. quinti numerus <var>.u.i.</var> numero <var>.e.i.</var> æqualis erit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0044-02" xlink:href="fig-0044-02a"> - <image file="0044-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0044-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="50"> - <head xml:id="head66" xml:space="preserve">THEOREMA <num value="50">L</num>.</head> - <p> - <s xml:id="s442" xml:space="preserve">CVR diuidentes numerum propoÅ¿itum in duas eiuÅ¿modi partes, vt <reg norm="productum" type="context">productÅ©</reg> <lb/>vnius in alteram cum i pÅ¿arum differentia in Å¿ummam collectum, æquale Å¿it <lb/>alicui alteri numero maiori primo. </s> - <s xml:id="s443" xml:space="preserve">Rectè primum ex Å¿ecundo detrahunt, reÅ¿iduum <lb/>verò conÅ¿eruant, tum ex primo Å¿emper binarium deÅ¿umunt, <reg norm="dimidiumque" type="simple">dimidiumqÌ;</reg> conÅ¿er-<lb/>uant, alterum verò dimidium in Å¿eipÅ¿o multiplicant, & ex quadrato numerum con <lb/>Å¿eruatum eruunt, <reg norm="reÅ¿iduique" type="simple">reÅ¿iduiqÌ;</reg> radicem ex dimidio conÅ¿eruato, quod vltimum reÅ¿i-<lb/>duum propoÅ¿iti numeri quæſita pars minor eÅ¿t.</s> - </p> - <p> - <s xml:id="s444" xml:space="preserve">Exempli gratia, Å¿i proponatur numerus .20. ita <reg norm="diuidendus" type="context">diuidẽdus</reg>, vt <reg norm="productum" type="context">productÅ©</reg> vnius partis <lb/>in alteram, cum partium differentia collectum in Å¿ummam, æquale Å¿it propoÅ¿ito - <pb o="33" rhead="THEOR. ARITH." n="45" file="0045" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0045"/> - numero, verbi gratia .92. præcepit regula detrahi primum numerum ex Å¿ecundo, <lb/>nempe .20. ex .92. cuius reÅ¿iduum, Å¿cilicet .72. conÅ¿eruetur, tum detrahi iubet bi <lb/>narium ex primo, Å¿ic in propoÅ¿ito exemplo remanebunt .18. huius autem .18. dimi <lb/>dium in Å¿eipÅ¿um multiplicari iubet, quod cum Å¿it .9. datur numerus .81. ex quo .81. <lb/>primum numerum conÅ¿eruatum, nempe .72. vult regula detrahi, Å¿ic remanebit .9. <lb/>tum huius .9. quadrata radix detrahenda eÅ¿t ex dimidio ipÅ¿ius .18. quod fuit ante qua <lb/>dratum, Å¿ic Å¿upererit .6. hoc eÅ¿t .9. excepta radice quadrata, qui .6. erit minor pars <lb/>quæſita, maior verò .14. quarum productum .84. coniunctum cum partium differen <lb/>tia præbet exactè .92.</s> - </p> - <p> - <s xml:id="s445" xml:space="preserve">Cuius rei hæc eÅ¿t Å¿peculatio. </s> - <s xml:id="s446" xml:space="preserve">Primus numerus minor, qui proponitur diuiÅ¿ibilis <lb/>Å¿ignificetur linea <var>.q.g.</var> maior vero linea <var>.x.</var> tum cogitemus <var>.q.g.</var> diuiÅ¿am, cuius maior <lb/>pars Å¿it <var>.q.o.</var> minor <var>.o.g.</var> differentia <var>.q.p.</var> ex quo <var>.p.o.</var> æqualis erit <var>.o.g.</var> Å¿it autem produ-<lb/>ctum <var>.b.o</var>. </s> - <s xml:id="s447" xml:space="preserve">Oportet igitur, ut <var>.b.o.</var> Å¿imul cum differentia <var>.q.p.</var> æquale Å¿it numero <var>.x.</var> Å¿e-<lb/>cundò propoÅ¿ito, qui notus eÅ¿t, </s> - <s xml:id="s448" xml:space="preserve">quare etiam Å¿umma producti <var>.b.o.</var> cum differentia <lb/><var>q.p.</var> cognita erit, ex qua detracto primo numero <var>.q.g.</var> reÅ¿iduum cognitum erit, nunc <lb/>igitur quodnam erit hoc reÅ¿iduum? </s> - <s xml:id="s449" xml:space="preserve">attendamus qua ratione ex Å¿umma <var>.b.o.</var> et <var>.q.p.</var> <lb/>detrahenda Å¿it <var>.q.g</var>. </s> - <s xml:id="s450" xml:space="preserve">In primis Å¿i Å¿ubtraxerimus ex dicta Å¿umma <var>.q.p.</var> quÄ™ pars eÅ¿t <var>.q.g.</var> <lb/>Å¿upererit detrahenda <var>.p.g.</var> ex <var>.b.o.</var> pars inquam ipÅ¿ius <var>.q.g.</var> quod fiet quotieÅ¿cunque <lb/>cogitauerimus <var>.q.o.</var> duabus vnitatibus diminutam, et per <var>.o.g.</var> multiplicatam, Å¿it au-<lb/>tem productum <var>.b.e.</var> nam cum <var>.o.g.</var> toties <var>.b.o.</var> ingrediatur, quot Å¿unt in <var>.q.o.</var> vnitates <lb/>ex prima Å¿exti aut .18. vel .19. Å¿eptimi, <reg norm="detrahendaque" type="simple">detrahendaqÌ;</reg> Å¿it <var>.p.g.</var> ex <var>.b.o.</var> quæ <var>.p.g.</var> dupla <lb/>eÅ¿t <var>.o.g.</var> patebit <var>.o.c.</var> æqualem eſſe <var>.p.g.</var> fu-<lb/>pererit ita que <var>.b.e.</var> productum <var>.q.e.</var> in <var>.e.</var> <lb/> - <anchor type="figure" xlink:label="fig-0045-01a" xlink:href="fig-0045-01"/> - i. cognitum, erutis autem ex <var>.q.g.</var> ijÅ¿dem <lb/>duabus vnitatibus, remanebit <var>.q.i.</var> nobis <lb/>nota, ex quo <var>.e.i.</var> æqualis erit <var>.e.c</var>. </s> - <s xml:id="s451" xml:space="preserve">Cum <lb/>igitur productum <var>.q.e.</var> in <var>.e.i.</var> cognoÅ¿camus <lb/>Å¿imul cum <var>.q.i</var>: Sivoluerimus partes <var>.q.e.</var> <lb/>et <var>.e.i.</var> cognoÅ¿cere, vtemur .45. theorema-<lb/>te huius libri, & propoÅ¿itum obtinebimus, nam cognoÅ¿cemus <var>.e.i.</var> & ex conÅ¿equen-<lb/>ti <var>.o.g.</var> eius æqualem.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0045-01" xlink:href="fig-0045-01a"> - <image file="0045-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0045-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="51"> - <head xml:id="head67" xml:space="preserve">THEOREMA <num value="51">LI</num>.</head> - <p> - <s xml:id="s452" xml:space="preserve"><emph style="sc">DIvidere</emph> numerum in duas eiuÅ¿modi partes, quæ pro medio proportionali <lb/>alterum numerum propoÅ¿itum recipiant, primi dimidio minorem, aliud ni <lb/>hil eÅ¿t, quà m binas primi numeri partes inuenire, quæ inter Å¿e multiplicatæ quadra <lb/>to Å¿ecundi numeri numerum æqualem proferant, ex .16. Å¿exti aut .20. Å¿eptimi, quod <lb/>tamen .45. theoremate fuit à nobis Å¿peculatum.</s> - </p> - </div> - <div type="math:theorem" level="3" n="52"> - <head xml:id="head68" xml:space="preserve">THEOREMA <num value="52">LII</num>.</head> - <p> - <s xml:id="s453" xml:space="preserve">CVR pro poÅ¿itis tribus numeris quibuÅ¿cunque, Å¿i productum primi in Å¿ecun-<lb/>dum per tertium multiplicetur, atque Å¿ecundum hoc productum <reg norm="corporeum" type="context">corporeÅ©</reg>, <lb/>per primum numerum diuidatur, proueniens erit numerus æqualis producto Å¿e-<lb/>cundi in tertium.</s> - </p> - <p> - <s xml:id="s454" xml:space="preserve">Exempli cauÅ¿a, proponantur hi tres numeri .10. 11. 12. <reg norm="multiplicenturque" type="simple">multiplicenturqÌ;</reg> .10. <reg norm="cum" type="context">cÅ©</reg>.</s> - <pb o="34" rhead="IO. BAPT. BENED." n="46" file="0046" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0046"/> - <s xml:id="s455" xml:space="preserve">11. dabuntur .110. quo producto multiplicato cum .12. dabuntur .1320. hoc pro <lb/>ueniens per primum nempe .10. diuiÅ¿um dabit .132. numerum æqualem producto <lb/>Å¿ecundi in tertium numerorum propoÅ¿itorum, Å¿cilicet .132.</s> - </p> - <p> - <s xml:id="s456" xml:space="preserve">Hoc vt Å¿peculemur, primus numerus Å¿ignificetur line <var>a.o.u.</var> Å¿ecundus <var>.e.o.</var> tertius <var>.<lb/>e.a.</var> productum verò <var>.o.u.</var> in <var>.o.e.</var> Å¿it <var>.o.i.</var> ipÅ¿ius ve <lb/>rò <var>.o.i.</var> per <var>.e.a.</var> <reg norm="productum" type="context">productÅ©</reg> <reg norm="corporeum" type="context">corporeÅ©</reg> Å¿it <var>.i.c.</var> tum <lb/> - <anchor type="figure" xlink:label="fig-0046-01a" xlink:href="fig-0046-01"/> - <reg norm="productum" type="context">productÅ©</reg> <var>.e.o.</var> in <var>.e.a.</var> Å¿it <var>.e.c</var>. </s> - <s xml:id="s457" xml:space="preserve">Dico <reg norm="nunc" type="context">nÅ©c</reg> quod di-<lb/>uiÅ¿o numero corporeo <var>.i.c.</var> per <reg norm="primum" type="context">primÅ©</reg> <var>.o.u.</var> <reg norm="proue" type="simple">ê“ue</reg> <lb/>niens æquale erit numero producti <var>.e.c</var>. </s> - <s xml:id="s458" xml:space="preserve">Qua-<lb/>re in primis cogitandum eÅ¿t, quod cum produ-<lb/>ctum <var>.i.c.</var> ortum fuerit ex multiplicatione <var>.o.i.</var> <lb/>in <var>.e.a</var>: dictum <var>.o.i.</var> toties ingredietur <var>.i.c.</var> quo-<lb/>ties vnitas reperitur in <var>.e.a.</var> eadem ratione, to-<lb/>ties <var>.e.c.</var> in <var>.i.c.</var> quot vnitates erunt in <var>.o.u</var>. </s> - <s xml:id="s459" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> <lb/>Å¿equitur quòd diuiÅ¿o <var>.i.c.</var> per <var>o.u.</var> proueniens Å¿it <lb/><var>e.c.</var> corporeum, æquale nihilominus producto <var>.e.c.</var> Å¿uperficiali.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0046-01" xlink:href="fig-0046-01a"> - <image file="0046-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0046-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="53"> - <head xml:id="head69" xml:space="preserve">THEOREMA <num value="53">LIII</num>.</head> - <p> - <s xml:id="s460" xml:space="preserve">CVR diuidens propoÅ¿itum numerum in tres partes Å¿ic Å¿e habentes vt produ-<lb/>ctum primi in Å¿ecundam, in tertia <reg norm="multiplicatum" type="context">multiplicatÅ©</reg>, præbeat numerum alteri nu-<lb/>mero propoÅ¿ito æqualem. </s> - <s xml:id="s461" xml:space="preserve">Rectè Å¿ecundum numerum per quemcunque alium mino <lb/>rem primo diuidit, qui diuidens vna erit ex tribus partibus quæſitis, proueniens <lb/>autem erit productum vnius in alteram reliquarum duarum, quarum Å¿umma cogni <lb/>ta erit, detracto numero diuidente ex primo dato, quam quidem Å¿i diÅ¿tinguere <lb/>quis voluerit, vtetur theoremate .45.</s> - </p> - <p> - <s xml:id="s462" xml:space="preserve">Exempli gratia, proponitur numerus .20. in tres partes diuidendus, quæ Å¿ic Å¿e <lb/>habeant, ut productum primæ in Å¿ecundam in tertia multiplicatum det .90. itaque <lb/>Å¿umenda erit pro prima vna pars ipÅ¿ius .20. quæcunque illa Å¿it, verbi gratia .2. qua <lb/>Å¿ecundus numerus, nempe .90. diuidatur, dabitur igitur .45. quod erit productum <lb/>cæterarum partium inter Å¿e, quarum Å¿umma eÅ¿t .18. quam Å¿ummam Å¿i diÅ¿tinguere <lb/>volueris in cÄ™teris duabus partibus Å¿eparatis, vteris .45. theoremate, vt quà m citiÅ¿-<lb/>Å¿imè quod cupis exequaris, erunt autem partes .3. et .15.</s> - </p> - <p> - <s xml:id="s463" xml:space="preserve">In cuius Å¿peculationis gratiam nihil aliud occurrit, quà m quod præcedenti theo-<lb/>remate, & Å¿uperiore .45. allatum eÅ¿t.</s> - </p> - </div> - <div type="math:theorem" level="3" n="54"> - <head xml:id="head70" xml:space="preserve">THEOREMA <num value="54">LIIII</num>.</head> - <p> - <s xml:id="s464" xml:space="preserve"><emph style="sc">DIvidere</emph> numerum in .3. eiuÅ¿modi partes, vt quadratum vnius Å¿it æquale <lb/>producto reliquarum duarum inter Å¿e, idem omnino eÅ¿t cum 51. theoremate. <lb/></s> - <s xml:id="s465" xml:space="preserve">Nam qui Å¿umet quamlibet partem propoÅ¿iti numeri, quæ tertia parte maior tamen <lb/>non Å¿it, <reg norm="reÅ¿iduumque" type="simple">reÅ¿iduumqÌ</reg> in duas tales partes diuiÅ¿erit, vt prima Å¿umpta, media proportio <lb/>nalis Å¿it ex probatione .51. theoremate allata, propoÅ¿itum conÅ¿equetur.</s> - </p> - </div> - <div type="math:theorem" level="3" n="55"> - <head xml:id="head71" xml:space="preserve">THEOREMA <num value="55">LV</num>.</head> - <p> - <s xml:id="s466" xml:space="preserve">ID ipÅ¿um alia ratione ab ea diuerÅ¿a <reg norm="quam" type="context">quã</reg> .51. theoremate adduximus, <reg norm="profici" type="simple">ê“fici</reg> poteÅ¿t.</s> - </p> - <pb o="35" rhead="THEOREM. ARIT." n="47" file="0047" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0047"/> - <p> - <s xml:id="s467" xml:space="preserve">Sumantur enimtres numeri continui proportionales, cuiuÅ¿cunque denique pro <lb/>portionalitatis, qui in Å¿ummam colligantur, ac poÅ¿tmodum, regula de trib. dica-<lb/>mus. </s> - <s xml:id="s468" xml:space="preserve">Si Å¿umma hæc primo numero propoÅ¿ito in tres partes diuidendo reÅ¿pondet, <lb/>cuireÅ¿pondebit vna ex tribus partibus huiuÅ¿cÄ™ <reg norm="summæ" type="context">sÅ©mæ</reg>? </s> - <s xml:id="s469" xml:space="preserve">idem dereliquis duabus pa<unsure/>rti <lb/>bus dico.</s> - </p> - <p> - <s xml:id="s470" xml:space="preserve">Exempli gratia, Å¿i proponatur numerus .57. diuidendus in tres continuas partes <lb/>proportionales proportione Å¿eÅ¿quialtera, tres numeros in eiuÅ¿modi proportio-<lb/>nalitate diÅ¿tinctos Å¿umemus, vt potè .4. 6. 9. qui in Å¿ummam collecti dabunt <reg norm="Å¿um- mam" type="context">Å¿um-<lb/>mã</reg> .19. <reg norm="dicemusque" type="simple">dicemusqÌ;</reg> Å¿i .19. dant .4. quid <reg norm="dabunt" type="context">dabÅ©t</reg> .57? </s> - <s xml:id="s471" xml:space="preserve">vnde proueniens vnius partis erit <num value="12">.<lb/>12</num>. </s> - <s xml:id="s472" xml:space="preserve">Tum Å¿i dicamus, Å¿i .19. dat .6. quid dabit .57? </s> - <s xml:id="s473" xml:space="preserve">nempe dabit .18. </s> - <s xml:id="s474" xml:space="preserve">PoÅ¿tremò, Å¿i <num value="19">.<lb/>19.</num> dat .9. quid dabit .57? </s> - <s xml:id="s475" xml:space="preserve">nempe .26. atque ita dabitur .18. cuius quadratum æqua-<lb/>bitur producto reliquarum duarum partium inter Å¿e.</s> - </p> - <p> - <s xml:id="s476" xml:space="preserve">Quod vt Å¿ciamus, numerus propoÅ¿itus in tres quaÅ¿libet partes diuidendus Å¿i-<lb/>gnificetur linea <var>.a.d.</var> tres autem numeri dictæ proportionalitatis, lineis <var>.e.f</var>: <var>f.g.</var> <lb/>et <var>.g.h.</var> directè inter Å¿e coniunctis denotentur. </s> - <s xml:id="s477" xml:space="preserve">Cogitemus pariter lineam <var>.d.a.</var> in <lb/>tres partes diuiÅ¿am <var>.a.b</var>: <var>b.c.</var> et <var>.c.d.</var> eadem cum cæteris proportionalitate, </s> - <s xml:id="s478" xml:space="preserve">tunc ea-<lb/>dem erit proportio <var>.a.d.</var> ad quamlibet Å¿uarum partium, quæ eÅ¿t <var>.e.h.</var> ad reÅ¿ponden <lb/>tem ipÅ¿ius in <var>.a.d</var>: Verbi gratia reÅ¿pondentem <var>.a.b.</var> ipÅ¿i <var>.e.f.</var> et <var>.b.c</var>: <var>f.g.</var> et <var>.c.d</var>: <var>g.h</var>. </s> - <s xml:id="s479" xml:space="preserve">Di <lb/>co enim quòd ita Å¿e habebit <var>.a.d.</var> ad <var>.c.d.</var> Å¿icut <var>.e.h.</var> ad <var>.g.h</var>. </s> - <s xml:id="s480" xml:space="preserve">Nam cum Å¿ic Å¿e habeat <var>.a.<lb/>b.</var> ad <var>.b.c.</var> Å¿icut <var>.e.f.</var> ad <var>.f.g.</var> ex præſuppoÅ¿ito, permutando Å¿ic Å¿e habebit <var>.a.b.</var> ad <var>.e.f.</var> Å¿i-<lb/>cut <var>.b.c.</var> ad <var>.f.g.</var> & eadem ratione Å¿ic Å¿e habe-<lb/>bit <var>.c.d.</var> ad <var>.g.h.</var> Å¿icut <var>.b.c.</var> ad <var>.f.g.</var> & <reg norm="conÅ¿equen- ter" type="context">cõſequen-<lb/> - <anchor type="figure" xlink:label="fig-0047-01a" xlink:href="fig-0047-01"/> - ter</reg> Å¿icut <var>.a.b.</var> ad <var>.e.f.</var> ex quo ex .13. quinti Å¿ic <lb/>Å¿e habebit tota <var>.a.d.</var> ad totam <var>.e.h.</var> Å¿icut <var>.c.d.</var> <lb/>ad <var>.g.h.</var> aut <var>.b.c.</var> ad <var>.f.g.</var> aut <var>.a.b.</var> ad <var>.e.f.</var> per-<lb/>mutando itaque propoÅ¿itum manifeÅ¿tum erit, ipÅ¿um autem productum <var>.a.b.</var> in <var>.c.b.</var> <lb/>æquale erit quadrato <var>.b.c.</var> ex .15. fexti aut .20. Å¿eptimi.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0047-01" xlink:href="fig-0047-01a"> - <image file="0047-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0047-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="56"> - <head xml:id="head72" xml:space="preserve">THEOREMA <num value="56">LVI</num>.</head> - <p> - <s xml:id="s481" xml:space="preserve"><emph style="sc">VEteres</emph> aliud quoque problema indeterminatum propoÅ¿uerunt, quod ex <lb/>more ratione à me definietur, eÅ¿t autem eiuÅ¿modi.</s> - </p> - <p> - <s xml:id="s482" xml:space="preserve">Quomodo propoÅ¿itus numerus in tres eiuÅ¿modi partes diuidatur, vt <reg norm="quadratum" type="context">quadratÅ©</reg> <lb/>vnius æquale fit fummæ quadratorum reliquarum duarum partium.</s> - </p> - <p> - <s xml:id="s483" xml:space="preserve">Hoc vt efficiamus tria quadrata Å¿eparata Å¿umamus, <reg norm="quorum" type="context">quorÅ©</reg> <reg norm="vnum" type="context">vnÅ©</reg> æquale Å¿it reliquis <lb/>duobus; </s> - <s xml:id="s484" xml:space="preserve"><reg norm="eorum" type="context">eorÅ©</reg> <reg norm="autem" type="context">autẽ</reg> radices in Å¿ummam Å¿imul colligantur, tum regulam de tribus Å¿e <lb/>quemur, ratione præcedenti theoremate demonÅ¿trata, & rectè vt infra docebimus, <lb/>quod autem dico de quadratis, etiam de cubis, & quibuÅ¿uis dignitatibus aſſero.</s> - </p> - <p> - <s xml:id="s485" xml:space="preserve">Exempli gratia, Å¿i numerus diuiÅ¿ibilis proponatur .30. in tres eiuÅ¿modi partes di <lb/>uidendus, vt quadratum vnius æquale Å¿it Å¿ummæ quadratorum reliquarum duarum <lb/>partium, in primis radices trium quadratorum Å¿umemus, Å¿ic quomodocunque Å¿e <lb/>habentes, vt maius ipÅ¿orum æquale Å¿it Å¿ummæ reliquorum duorum, verbi gratia .25. <lb/>16. et .9. nempe .5. 4. et .3. quæ Å¿i colligantur in Å¿ummam efficiunt .12. </s> - <s xml:id="s486" xml:space="preserve">Tum ex regu-<lb/>la de tribus dicemus, Å¿i .12. reÅ¿pondet .30: </s> - <s xml:id="s487" xml:space="preserve">cui, 5. radix maior reÅ¿pondebit? </s> - <s xml:id="s488" xml:space="preserve">nem-<lb/>pe .12. cum dimidio.</s> - </p> - <p> - <s xml:id="s489" xml:space="preserve">Deinde Å¿i dixerimus Å¿i .12. valet .30. quid valebit .4. radix media? </s> - <s xml:id="s490" xml:space="preserve">nempe vale-<lb/>bit .10. tertia autem minor .7. cum dimidio. </s> - <s xml:id="s491" xml:space="preserve">Itaquetota Å¿umma erit .30. & quadra- - <pb o="36" rhead="IO. BAPT. BENED." n="48" file="0048" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0048"/> - tum .12. cum dimidio erit .155. quod æquale erit Å¿ummæ quadratorum duarum par <lb/>tium, nempe .100. cum .55.</s> - </p> - <p> - <s xml:id="s492" xml:space="preserve">Hoc vt <reg norm="demonÅ¿tremus" type="context">demõſtremus</reg>, numerus diuiÅ¿ibilis propoÅ¿itus Å¿ignificetur linea <var>.a.d.</var> & Å¿um <lb/>ma radicum, noÅ¿tro modo Å¿umptarum, linea <var>.e.h.</var> quarum prima & maior Å¿it <var>.e.f.</var> Å¿e-<lb/>cunda <var>.f.g.</var> tertia <var>.g.h.</var> cogitemus etiam lineam <var>.a.d.</var> ea ratione diuiÅ¿am eſſe qua <var>.e.h.</var> <lb/>patebit cnim ex modo præcedentis theorematis vnamquanque partium <var>.a.d.</var> ita Å¿e <lb/>habituram ad Å¿uum totum Å¿icut Å¿e habent Å¿ingulæ <var>.e.h.</var> ad Å¿uum. </s> - <s xml:id="s493" xml:space="preserve">Quod ideo dico, vt <lb/>intelligamus rectè nos dicere. </s> - <s xml:id="s494" xml:space="preserve">Si <var>.e.h.</var> dat <var>.a.d.</var> ergo <var>.e.f.</var> dabit <var>.a.b.</var> <reg norm="atque" type="simple">atq;</reg> ita de cæteris. <lb/></s> - <s xml:id="s495" xml:space="preserve">Quare permutando Å¿ic Å¿e habebit <var>.a.b.</var> ad <var>.b.c.</var> Å¿icut <var>.e.f.</var> ad <var>.f.g.</var> idem dico de reliquis. <lb/></s> - <s xml:id="s496" xml:space="preserve">Igitur ex .18. Å¿exti aut .11. octaui, eadem erit proportio quadrati <var>.a.b.</var> ad <reg norm="quadratum" type="context">quadratÅ©</reg> <var>.<lb/>b.c.</var> quæ quadrati <var>.e.f.</var> ad quadratum <var>.f.g.</var> tota enim Å¿unt æqualia, cum eorum partes <lb/>Å¿imiles inter Å¿e Å¿unt æquales. </s> - <s xml:id="s497" xml:space="preserve">Idem dico de proportione qu@drati <var>.a.b.</var> nempe ita <lb/>Å¿e habere ad <var>.c.d.</var> Å¿icut quadratum <var>.e.f.</var> ad quadratum <var>.g.h.</var> ex quo ex .24. quinti pro-<lb/>portio quadrati <var>.a.b.</var> ad Å¿ummam quadratorum duarum partium <var>.b.c.</var> et <var>.c.d.</var> Å¿ic Å¿e ha <lb/>bebit ut quadrati <var>.e.f.</var> ad Å¿ummam quadra-<lb/>torum <var>.f.g.</var> et <var>.g.h</var>. </s> - <s xml:id="s498" xml:space="preserve">At quadratum <var>.e.f.</var> æquale <lb/> - <anchor type="figure" xlink:label="fig-0048-01a" xlink:href="fig-0048-01"/> - eÅ¿t Å¿ummæ quadratorum <var>.f.g.</var> et <var>.g.h.</var> igitur <lb/>Å¿ic etiam Å¿e habebit quadratum <var>.a.b.</var> nempe <lb/>æquale quadratis <var>.b.c.</var> et <var>.c.g</var>. </s> - <s xml:id="s499" xml:space="preserve">IdipÅ¿um de cæ <lb/>teris dignitatibus dices, <reg norm="vterisque" type="simple">vterisqÌ;</reg> .21. theoremate huius libri.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0048-01" xlink:href="fig-0048-01a"> - <image file="0048-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0048-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="57"> - <head xml:id="head73" xml:space="preserve">THEOREMA <num value="57">LVII</num>.</head> - <p> - <s xml:id="s500" xml:space="preserve"><emph style="sc">SImile</emph> quoque problema ab antiquis indeterminatum proponitur, quod eiuÅ¿-<lb/>modi eÅ¿t.</s> - </p> - <p> - <s xml:id="s501" xml:space="preserve">An numerus aliquis in tres eiuÅ¿modi partes di@idi poſſit, vt quadratum vnius æ-<lb/>quale Å¿it Å¿ummæ quadratorum cæterarum duarum partium Å¿imul cum producto <lb/>vnius in alteram.</s> - </p> - <p> - <s xml:id="s502" xml:space="preserve">Exempli gratia, Å¿i proponatur numerus .50. vt iam dictum eÅ¿t diuidendus, repe <lb/>riendus erit alius quilibet numerus, qui tamen Å¿umma Å¿it trium radicum Å¿ic Å¿e ha-<lb/>bentium, vt quadratum vnius æquale Å¿it Å¿ummæ quadratorum duarum partium Å¿i-<lb/>mul cum producto vnius in alteram, eum autem qui primò occurrit Å¿umamus, utpo <lb/>tè .30. qui Å¿umma eÅ¿t numerorum .6. 10. 14. partium Å¿ic Å¿e habentium, vt quadratum <lb/>ipÅ¿ius .14. æquale Å¿it Å¿ummæ quadratorum cæterarum partium Å¿imul cum produ-<lb/>cto vnius in alteram, agamusqÌue regula de tribus, ac dicamus, Å¿i .30. valet <num value="50">.<lb/>50.</num> quid valebit .14. nempe .23. cum tertia parte. </s> - <s xml:id="s503" xml:space="preserve">Idem efficiemus in cæte-<lb/>ris partibus, quarum vna erit .16. cum duabus tertijs, altera verò .10. abÅ¿que @ractis, <lb/>ex quo quadratum primæ erit .544. cum .4. nonis, Å¿ecundæ .277. cum Å¿eptem nonis, <lb/>tertiæ .100. & productum Å¿ecundæ in tertiam .166. cum .6. nonis, quod productum, <lb/>cum quadratis Å¿ecundæ & tertiæ collectum erit .544. cum .4. nonis.</s> - </p> - <p> - <s xml:id="s504" xml:space="preserve">Huius rei Å¿peculatio eadem eÅ¿t, quÄ™ fuit præcedentis theorematis vÅ¿quequo no-<lb/>ueris eandem proportionem eſſe quadrati <var>.a.b.</var> ad Å¿ummam quadratorum <var>.b.c.</var> et <var>.c.<lb/>d.</var> quæ quadrati <var>.e.f.</var> ad Å¿ummam quadratorum <var>.f.g.</var> et <var>.g.h</var>. </s> - <s xml:id="s505" xml:space="preserve">Sed cum hic non demus <lb/>quadratum <var>.e.f.</var> æquale Å¿ummæ quadratorum <var>.f.g.</var> et <var>.g.h.</var> fed maius ex producto <var>.g.h.</var> <lb/>in <var>.f.g.</var> aut quod idem eÅ¿t, è contrario, Å¿ubÅ¿equentes figuræ cogitandæ erunt, qua-<lb/>rum <var>.i.</var> Å¿it quadratum <var>.a.b</var>: l. Å¿it quadratum <var>.e.f</var>: x. quadratum <var>.b.c</var>: y. quadratum <var>.f.g</var>: p. <lb/>quadratum <var>.c.d</var>: q. quadratum <var>.g.h</var>: k. Å¿it productum <var>.b.c.</var> in <var>.c.d</var>: m. Å¿it productum <var>.f.</var> - <pb o="37" rhead="THEOREM. ARITH." n="49" file="0049" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0049"/> - g. in <var>.g.h</var>. </s> - <s xml:id="s506" xml:space="preserve">Nunc ex Å¿peculatione præcedentis theorematis, eadem erit proportio <var>.n.<lb/>t.</var> ad <var>.o.u.</var> quæ eÅ¿t <var>.n.s.</var> ad <var>.o.r.</var> </s> - <s xml:id="s507" xml:space="preserve">quare pro-<lb/>ductum <var>.k.</var> ex definitione Å¿imile erit <lb/> - <anchor type="figure" xlink:label="fig-0049-01a" xlink:href="fig-0049-01"/> - producto <var>.m.</var> cum vtraque Å¿int rectan-<lb/>gula, vnde proportio <var>.k.</var> ad <var>.m.</var> ad pro-<lb/>portionem <var>.n.t.</var> ad <var>.o.u.</var> ex .18. Å¿exti du-<lb/>pla erit. </s> - <s xml:id="s508" xml:space="preserve">Igitur proportio <var>.k.</var> ad <var>.m.</var> æ-<lb/>qualis erit proportioni <var>.x.</var> ad <var>.y.</var> et <var>.p.</var> <lb/>ad <var>.q.</var> et <var>.i.</var> ad <var>.l.</var> & permutando Å¿ic Å¿e ha-<lb/>bebit <var>.k.</var> ad <var>.i.</var> Å¿icut <var>.m.</var> ad <var>.l.</var> Å¿ed <var>.x.p.</var> ad <var>.i.</var> <lb/>Å¿icÅ¿e habere probatum eÅ¿t vt <var>.y.q.</var> ad <var>.l</var>. <lb/></s> - <s xml:id="s509" xml:space="preserve">Quare ex eadem .24. quinti Å¿ic Å¿e habe <lb/>bit <var>.x.p.k.</var> ad <var>.i.</var> Å¿icut <var>.y.q.m.</var> ad <var>.l.</var> Å¿ed <var>.y.q.<lb/>m.</var> æqualis eÅ¿t <var>.l</var>. </s> - <s xml:id="s510" xml:space="preserve">Itaque <var>.x.p.k.</var> pariter <var>.i.</var> <lb/>æqualis erit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0049-01" xlink:href="fig-0049-01a"> - <image file="0049-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0049-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="58"> - <head xml:id="head74" xml:space="preserve">THEOREMA <num value="58">LVIII</num>.</head> - <p> - <s xml:id="s511" xml:space="preserve">ALIVD quoque problema, nec tamen definitum, veteres propoÅ¿uerunt, <lb/>nempe an aliquis numerus in .4. eiuÅ¿modi partes diuidi poſſit, vt Å¿umma qua-<lb/>dratorum duarum partium dupla Å¿it Å¿ummæ quadratorum reliquarum duarum.</s> - </p> - <p> - <s xml:id="s512" xml:space="preserve">Verum huius effectio & Å¿peculatio non erit difficilis, <reg norm="cum" type="context">cÅ©</reg> Å¿it eadem quæ præmiÅ¿sis <lb/>proximè duobus theorematibus allata fuit, Å¿umpta nempe Å¿umma radicum quarun <lb/>cunque Å¿ic Å¿e habentium, prout dictum fuit. </s> - <s xml:id="s513" xml:space="preserve">Verbigratia .44. cuius partes erunt. <lb/>16. 12. 14. 2. <reg norm="tunc" type="context">tÅ©c</reg> progrediemur regula de tribus dicentes. </s> - <s xml:id="s514" xml:space="preserve">Si .44 numerum propoÅ¿i-<lb/>tum valet, quid .16. pars maior? </s> - <s xml:id="s515" xml:space="preserve">nempe valebit partem maiorem numeri propoÅ¿i-<lb/>ti reÅ¿pondentem .16. idem de cæteris dico.</s> - </p> - <p> - <s xml:id="s516" xml:space="preserve">Porrò Å¿peculatio eadem eÅ¿t cum Å¿uperioribus.</s> - </p> - </div> - <div type="math:theorem" level="3" n="59"> - <head xml:id="head75" xml:space="preserve">THEOREMA <num value="59">LIX</num>.</head> - <p> - <s xml:id="s517" xml:space="preserve">CVR diuidens propoÅ¿itum numerum in duas eiuÅ¿modi partes, vt productum <lb/>radicum quadratarum ipÅ¿arum partium æquale Å¿it alteri numero propoÅ¿ito, <lb/>cuius <reg norm="tamen" type="context">tamẽ</reg> quadratum maius <reg norm="non" type="context">nõ</reg> Å¿it quadrato dimidij primi numeri propoÅ¿iti. </s> - <s xml:id="s518" xml:space="preserve">Rectè <lb/>Å¿ecundum numerum propoÅ¿itum in Å¿eipÅ¿um multiplicat, & <reg norm="eundem" type="context">eundẽ</reg> ex quadrato di-<lb/>midij primi detrahit, <reg norm="reÅ¿iduique" type="simple">reÅ¿iduiqÌ;</reg> quadratam radicem Å¿ubtrahit ex dimidio ipÅ¿ius pri-<lb/>mi, ex quo datur minor pars quæſita, quaipÅ¿i dimidio coniuncta, maior pars ha-<lb/>betur.</s> - </p> - <p> - <s xml:id="s519" xml:space="preserve">Exempli gratia, Å¿i proponatur numerus, 20. propoÅ¿ito modo, in duas partes <lb/>eiuÅ¿modi diuidendus, vt productum radicum æquale Å¿it (verbigratia) 8. </s> - <s xml:id="s520" xml:space="preserve">Dimi-<lb/>dium priminumeri in Å¿eipÅ¿um multiplicabimus, cuius quadratum erit .100. ex <lb/>quo quadratum Å¿ecundi numeri, nempe .64. detrahemus, <reg norm="remanebitque" type="simple">remanebitqÌ;</reg> .36. cuius radi <lb/>ce quadrata coniuncta .10. dimidio inquam primi numeri propoÅ¿iti, dabitur nume <lb/>rus .16. pars maior, & Å¿ubtracta à dimidio, dabitur minor pars, nempe .4.</s> - </p> - <pb o="38" rhead="IO. BAPT. BENED." n="50" file="0050" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0050"/> - <p> - <s xml:id="s521" xml:space="preserve">Hoc vt demonÅ¿tremus, primus nu-<lb/> - <anchor type="figure" xlink:label="fig-0050-01a" xlink:href="fig-0050-01"/> - merus linea <var>.a.b.</var> Å¿ignificetur, quam di-<lb/>uiÅ¿am cogitemus in puncto <var>.c.</var> in partes <lb/>quæſitas, ex quo præſupponitur duas li-<lb/>neas <var>.a.c.</var> et <var>.c.b.</var> duo quadrata eſſe, quæ <lb/>in altera figura Å¿ignificetur per <var>.d.</var> et <var>.e.</var> <lb/>productum autem radicum cognitum <var>.<lb/>f.</var> quandoquidem datum eÅ¿t, cuius qua-<lb/>dratum æquale erit producto quadra-<lb/>torum <var>.d.e.</var> adinuicem, nempe <var>.b.c.</var> in <var>.a.c.</var> ex .19. theoremate huius. </s> - <s xml:id="s522" xml:space="preserve">Quod verbi <lb/>gratia Å¿it <var>.x.</var> <reg norm="itaque" type="simple">itaq;</reg> cognitum, quo facto, doctrinam .45. theorematis libri huius Å¿ecuti, <lb/>propoÅ¿itum conÅ¿equemur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0050-01" xlink:href="fig-0050-01a"> - <image file="0050-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0050-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="60"> - <head xml:id="head76" xml:space="preserve">THEOREMA <num value="60">LX</num>.</head> - <p> - <s xml:id="s523" xml:space="preserve">CVR productum differentiæ duarum radicum in Å¿ummam ipÅ¿arum, Å¿emper <lb/>differentia Å¿it quadratorum ipÅ¿arum radicum.</s> - </p> - <p> - <s xml:id="s524" xml:space="preserve"><reg norm="Exempli" type="context">Exẽpli</reg> gratia, quoslibet duos numeros pro radicibus Å¿umpÅ¿erimus, vt potè .3. et <num value="5">.<lb/>5.</num> quorum differentia eÅ¿t .2. certè Å¿i differentiam hanc per Å¿ummam radicum Å¿cili-<lb/>cet .8. multiplicauerimus, dabitur numerus .16. quod productum differentia eÅ¿t <lb/>Å¿uorum quadratorum, nempeinter .9. et .25.</s> - </p> - <p> - <s xml:id="s525" xml:space="preserve">Hoc vt Å¿peculemur, duæ radices in linea <var>.n.i.</var> Å¿ignificentur, quarum vna Å¿it <var>.n.c.</var> & <lb/>altera <var>.c.i.</var> ipÅ¿arum autem differentia <var>.n.t.</var> ex quo <var>.t.<lb/>c.</var> æqualis erit <var>.c.i</var>. </s> - <s xml:id="s526" xml:space="preserve">Tum cogitato toto quadrato <var>.d.i.</var> <lb/> - <anchor type="figure" xlink:label="fig-0050-02a" xlink:href="fig-0050-02"/> - cum diametro <var>.d.i.</var> <reg norm="ductaque" type="simple">ductaqÌ</reg> parallela lateri <var>.n.d.</var> à <lb/>puncto <var>.c.</var> & altera à puncto <var>.t.</var> & à puncto <var>.o.</var> tertia <lb/>ipÅ¿i <var>.n.i.</var> & à puncto <var>.a.</var> quarta <var>.x.a.e.</var> parallela ipÅ¿i <var>.<lb/>o.</var> inueniemus <var>.b.n.</var> productum eſſe differentiæ <var>.n.<lb/>t.</var> in Å¿umma radicum <var>.n.i.</var> & cum <var>.d.o.</var> et <var>.a.o.</var> Å¿int <lb/>quadrata radicum prædictarum: </s> - <s xml:id="s527" xml:space="preserve">b.e. æquale erit <var>.<lb/>n.u.</var> cum vtrunque horum productorum æquale Å¿it <var>.<lb/>x.u.</var> ex quo gnomon <var>.e.d.u.</var> æqualis erit producto <var>.<lb/>b.n.</var> quod Å¿cire cupiebamus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0050-02" xlink:href="fig-0050-02a"> - <image file="0050-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0050-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="61"> - <head xml:id="head77" xml:space="preserve">THEOREMA <num value="61">LXI</num>.</head> - <p> - <s xml:id="s528" xml:space="preserve">CVR propoÅ¿itum aliquem numerum diuiÅ¿uri in duas eiuÅ¿modi partes, vt diffe-<lb/>rentia radicum quadratarum æqualis Å¿it alteri numero propoÅ¿ito, cuius ta-<lb/>men quadratum dimidij primi quadratum non excedat. </s> - <s xml:id="s529" xml:space="preserve">Rectè Å¿ecundum numerum <lb/>in Å¿eipÅ¿um multiplicant, productum verò ex primo numero detrahunt, <reg norm="rurÅ¿usque" type="simple">rurÅ¿usqÌ;</reg> di <lb/>midium reÅ¿idui quadrant, & quadratum hoc ex quadrato dimidij primi Å¿ubtrahunt, <lb/>atque ita radice quadrata reÅ¿idui, dimidio primi coniuncta, pars maior datur, qua <lb/>ex ipÅ¿o dimidio detracta, pars minor relinquitur.</s> - </p> - <p> - <s xml:id="s530" xml:space="preserve">Exempli gratia, propoÅ¿ito numero .20. ita ut propoÅ¿itum eÅ¿t, diuidendo, nem-<lb/>pe vt differentia radicum quadratarum dictarum partium æqualis Å¿it binario, bina-<lb/>rium hocin Å¿eipÅ¿um multiplicabimus, cuius quadratum .4. è primo numero .20. de - <pb o="39" rhead="THEOREM. AR IT." n="51" file="0051" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0051"/> - trahemus, <reg norm="Å¿upereritque" type="simple">Å¿upereritqÌ;</reg> numerus .16. cuius dimidium Å¿cilicet .8. in Å¿eipÅ¿um multipli-<lb/>cabimus, <reg norm="dabiturque" type="simple">dabiturqÌ;</reg> numerus .64. qui cum ex quadrato dimidij primi detractus fue-<lb/>rit, nempe ex .100. & reÅ¿iduo .36. radix quadrata nempe .6. coniuncta denario, di-<lb/>midio primi, dabit .16. partem maiorem, & ex denario detracta, partem minorem.</s> - </p> - <p> - <s xml:id="s531" xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, primus numerus <lb/>propoÅ¿itus Å¿igniÅ¿icetur linea <var>.x.y.</var> pro voto diui-<lb/> - <anchor type="figure" xlink:label="fig-0051-01a" xlink:href="fig-0051-01"/> - Å¿a in puncto <var>.c.</var> et <var>.x.t.</var> productum Å¿it ipÅ¿ius <var>.x.<lb/>c.</var> in <var>.c.y.</var> pariter etiam <var>.q.p.</var> Å¿it Å¿umma radicum <lb/>quadratarum, nempe <var>.q.g.</var> ipÅ¿ius <var>.t.c.</var> et <var>.g.p.</var> ip-<lb/>Å¿ius <var>.c.y</var>. </s> - <s xml:id="s532" xml:space="preserve">Tum Å¿uper <var>.q.p.</var> extruatur & diuidatur <lb/>quadratum <var>.q.u.</var> ea ratione qua .41. theoremate <lb/>aut .29. diuiÅ¿imus, in quo Å¿anè quadrato, quadra <lb/>tum ipÅ¿ius <var>.q.i.</var> cernemus datæ differentiæ, & in <lb/>eo collocata quadrata <var>.x.c.</var> et <var>.c.y.</var> ita etiam & <lb/>rationem, qua cognoÅ¿cimus productum <var>.g.r.</var> (vÅ¿i <lb/>modo .29. theorematis) cuius quidem <var>.g.r.</var> qua-<lb/>dratum, ex .19. theoremate æquale erit produ-<lb/>cto <var>.x.t.</var> ideo etiam <reg norm="cognitum" type="context">cognitÅ©</reg>, ac proinde cum no <lb/>uerimus <var>.x.y.</var> Å¿i rationem Å¿equemur .45. theore <lb/>mate cognoÅ¿cemus non Å¿olum ratione .41. theoremate allata hocrectè perfici, Å¿ed <lb/>hac etiam alia ratione.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0051-01" xlink:href="fig-0051-01a"> - <image file="0051-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0051-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="62"> - <head xml:id="head78" xml:space="preserve">THEOREMA <num value="62">LXII</num>.</head> - <p> - <s xml:id="s533" xml:space="preserve">CVR propoÅ¿itum numerum diuiÅ¿uri in duas eiuÅ¿modi partes, vt differentia <lb/><reg norm="Å¿uarum" type="context">Å¿uarÅ©</reg> <reg norm="radicum" type="context">radicÅ©</reg> <reg norm="quadratarum" type="context">quadratarÅ©</reg> æqualis Å¿it alteri numero propoÅ¿ito. </s> - <s xml:id="s534" xml:space="preserve">Cuius <reg norm="tamen" type="wordlist">tamẽ</reg> <reg norm="qua- dratum" type="context">qua-<lb/>dratÅ©</reg> maius non Å¿it quadrato medietatis ipÅ¿ius primi propoÅ¿iti numeri. </s> - <s xml:id="s535" xml:space="preserve">Rectè <reg norm="etiam" type="context">etiã</reg> <lb/><reg norm="quadratum" type="context">quadratÅ©</reg> dimidij Å¿ecundi numeri ex dimidio primi <reg norm="detrahunt" type="context">detrahÅ©t</reg>, <reg norm="reÅ¿iduique" type="simple">reÅ¿iduiqÌ;</reg> radicem per <lb/>Å¿ecundum multiplicant, & productum ex dimidio primi detrahunt, vt reÅ¿iduum <lb/>pars quæſita minor Å¿it, & illud alterum totius reÅ¿iduum, pars maior.</s> - </p> - <p> - <s xml:id="s536" xml:space="preserve">Exempli gratia, Å¿i numerus .50. in <lb/>prædictas duas partes diuidendus pro-<lb/> - <anchor type="figure" xlink:label="fig-0051-02a" xlink:href="fig-0051-02"/> - poneretur, & alter etiam .6. quadratum <lb/>dimidij Å¿ecundi numeri eſſet .9. eo detra <lb/>cto ex dimidio primi, remaneret .16. cu <lb/>ius radix .4. Å¿cilicet per totum Å¿ecundum <lb/>nempe .6. multiplicata, proferet .24. <lb/>quo producto ex dimidio primi detra-<lb/>cto, nempe .25. dabitur .1. pars minor, <lb/>maior <reg norm="autem" type="context">autẽ</reg> erit <reg norm="reÅ¿idum" type="context">reÅ¿idÅ©</reg> .50. hoc eÅ¿t .49. <lb/>radices autem erunt .1. et .7. differentes <lb/>inter Å¿e, numero Å¿enario.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0051-02" xlink:href="fig-0051-02a"> - <image file="0051-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0051-02"/> - </figure> - </div> - <p> - <s xml:id="s537" xml:space="preserve">Hocvt Å¿ciamus, duo numeri lineis <reg norm="Å¿i- gnificentur" type="context">Å¿i-<lb/>gnificẽtur</reg>, primus linea .b: <reg norm="Å¿ecundus" type="context">Å¿ecÅ©dus</reg> linea <var>.<lb/>c.</var> duæ autem partes <var>.b.</var> duobus quadra-<lb/>tis <var>.q.i.</var> et <var>.i.d.</var> notentur, eorum verò radi-<lb/>ces lineis <var>.a.g.</var> et <var>.g.d.</var> differentia porrò ip <lb/>Å¿i <var>.c.</var> æqualis & co gnita Å¿it <var>.a.h.</var> ex quo <var>.h.</var> - <pb o="40" rhead="IO. BAPT. BENED." n="52" file="0052" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0052"/> - g. æqualis erit <var>.g.d.</var> tum productum <var>.a.g.</var> in <var>.g.d.</var> Å¿it <var>.a.i.</var> et <var>.t.i.</var> æqualis <var>.a.i.</var> et <var>.l.i.</var> pariter <lb/>Å¿ecetur æqualis <var>.t.i.</var> quæ omnia ex diametro <var>.q.d.</var> cogitari poſſunt: </s> - <s xml:id="s538" xml:space="preserve">erit igitur <var>.u.i.</var> æ-<lb/>qualis <var>.i.d.</var> <reg norm="Å¿upereritque" type="simple">Å¿upereritqÌ;</reg> quadratum <var>.q.u.</var> differentiæ <var>.a.h.</var> cognitum, hoc verò cogi-<lb/>temus diuiÅ¿um eſſe in .4. partes æquales medijs diametris <var>.p.r.</var> et <var>.n.e.</var> </s> - <s xml:id="s539" xml:space="preserve">quare <reg norm="vnaquæque" type="simple">vnaquæq;</reg> <lb/>partium cognoÅ¿cetur, & <reg norm="quadratum" type="context">quadratÅ©</reg> erit ipÅ¿ius <var>.a.K.</var> aut ipÅ¿ius <var>.K.h.</var> dimidij <var>.a.h</var>. </s> - <s xml:id="s540" xml:space="preserve">Quòd <lb/>Å¿i aliquod iÅ¿torum quadratorum detrahere voluerimus, nempe <var>.n.r.</var> ex dimidio Å¿um <lb/>mæ <var>.b.</var> duorum quadratorum <var>.q.i.</var> et <var>.i.d.</var> cognitæ, hac via procedemus, primum con <lb/>Å¿iderabimus <var>.t.r.</var> coniunctam <var>.t.i.</var> quæ quantitates erunt Å¿umma dimidij <reg norm="duorum" type="context">duorÅ©</reg> qua-<lb/>dratorum <var>.q.i.</var> et <var>.i.d.</var> quando quidem <var>.t.r.</var> <lb/><reg norm="dimidium" type="context">dimidiÅ©</reg> eÅ¿t quadrati <var>.t.l.</var> et <var>.t.i.</var> <reg norm="dimidium" type="context">dimidiÅ©</reg> <lb/> - <anchor type="figure" xlink:label="fig-0052-01a" xlink:href="fig-0052-01"/> - gnomonis <var>.t.i.l.</var> coniunctum dimidio <lb/>quadrati <var>.i.d.</var> ex quo <var>.i.t.r.</var> dimidium erit <var>.<lb/>b.</var> ex qua quantitate <var>.i.t.r.</var> cogitare debe <lb/>mus detrahi quadratum ipÅ¿ius <var>.K.h.</var> nem <lb/>pe <var>.n.r</var>: </s> - <s xml:id="s541" xml:space="preserve">quare quod Å¿upereÅ¿t cognitum <lb/>erit nempe <var>.y.s.</var> cum <var>.n.i.</var> Å¿ed <var>.y.m.</var> æqualis <lb/>eÅ¿t <var>.n.i.</var> et <var>.y.m.</var> cum <var>.y.s.</var> conÅ¿tituunt qua-<lb/>dratum <var>.p.m</var>. </s> - <s xml:id="s542" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> <var>.p.m.</var> quadratum & <lb/>conÅ¿equenter <var>.p.s.</var> eius radix cognoÅ¿ce-<lb/>tur, ita etiam & productum huius <var>.p.s.</var> in <var>.<lb/>s.x.</var> æqualis <var>.c.</var> nempe <var>.p.x</var>: <reg norm="eÅ¿tque" type="simple">eÅ¿tqÌ;</reg> produ-<lb/>ctum huiuÅ¿modi Å¿emper minus quantita <lb/>te <var>.r.t.i</var>: per <var>.u.i.</var> æquale quadrato minori <var>.<lb/>i.d</var>. </s> - <s xml:id="s543" xml:space="preserve">quare <var>.i.d.</var> cognoÅ¿cetur, conÅ¿equen-<lb/>ter <var>.i.</var> @q. tanquam reÅ¿iduum ex <var>.b.</var> & eo-<lb/>rum radices quadratæ cognoÅ¿centur <var>.a.<lb/>g.</var> et <var>.g.d</var>.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0052-01" xlink:href="fig-0052-01a"> - <image file="0052-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0052-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="63"> - <head xml:id="head79" xml:space="preserve">THEOREMA <num value="63">LXIII</num>.</head> - <p> - <s xml:id="s544" xml:space="preserve">IDEM præſtari hac alia via, meo iudicio poteÅ¿t. </s> - <s xml:id="s545" xml:space="preserve">Secundus numerus in <reg norm="Å¿uum" type="context">Å¿uÅ©</reg> dimi <lb/><reg norm="dium" type="context">diÅ©</reg> multiplicetur, <reg norm="productum" type="context">productÅ©</reg> autem ex dimidio primi detrahatur, ex quo re-<lb/>manens erit productum vnius quadratæ radicis in alteram partium primi numeri <lb/>quæſitarum, deinde productum hoc duplicetur, & primo numero dato coniunga-<lb/>tur, <reg norm="Å¿icque" type="simple">Å¿icqÌ;</reg> huius Å¿ummæ quadrata radix erit Å¿umma radicum quadratarum dictarum <lb/>partium, cui iuncto producto ex quadrageÅ¿imoquinto theoremate Å¿ingulæ radices <lb/>proferentur.</s> - </p> - <p> - <s xml:id="s546" xml:space="preserve">Exempli gratia, primus numerus diuiÅ¿ibilis erat .50. alter verò .6. </s> - <s xml:id="s547" xml:space="preserve">Iam Å¿i multi-<lb/>plicemus .6. per .3. nempe dimidium proferetur numerus .18. quo ex dimidio pri-<lb/>mi, nempe .25. detracto, Å¿upererit .7. productum vnius radicis in alteram, quod du <lb/>plicatum dabit .14. quo coniuncto cum primo numero .50. dabitur numerus .64. <lb/>cuius quadrata radix Å¿cilicet .8. erit Å¿umma radicum duarum partium quæſitarum, <lb/>qua & producto .7. ex quadrag eÅ¿imoquinto theoremate dictæ radices diÅ¿tinguen, <lb/>tur, quarum vna erit .7. & altera <var>.I</var>.</s> - </p> - <p> - <s xml:id="s548" xml:space="preserve">Vtautem hocÅ¿peculemur, præcedenti figura vti poterimus, in qua patet <var>.t.r.</var> pro <lb/>ductum eſſe Å¿ecundi numeri <var>.c.</var> nempe <var>.a.h.</var> hoc eÅ¿t <var>.t.u.</var> in dimidio <var>.a.e.</var> Å¿cilicet <var>.p.t.</var> re-<lb/>Å¿iduum autem dimidij primi <var>.b.</var> eſſe <var>.t.i.</var> nempe <var>.a.i.</var> productum radicum, quod Å¿upple - <pb o="41" rhead="THEOREM. ARITH." n="53" file="0053" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0053"/> - mentum eÅ¿t quadrati <var>.q.d.</var> totalis. </s> - <s xml:id="s549" xml:space="preserve">Quare duplicato <var>.a.i.</var> & coniuncto <var>.b.</var> cognoÅ¿ci-<lb/>mustotum <var>.q.d.</var> & conÅ¿equenter <var>.a.d.</var> Å¿uam radicem, hoc eÅ¿t Å¿ummam duarum radi <lb/>cum <var>.a.g.</var> et <var>.g.d.</var> quæ medio <var>.a.i.</var> cognito, & quadrageÅ¿imoquinto theoremate Å¿ingu-<lb/>læ cognoÅ¿cuntur.</s> - </p> - </div> - <div type="math:theorem" level="3" n="64"> - <head xml:id="head80" xml:space="preserve">THEOREMA <num value="64">LXIIII</num>.</head> - <p> - <s xml:id="s550" xml:space="preserve">CVR propoÅ¿itum aliquem num erum in duas eiuÅ¿modi partes diuiÅ¿uri, vt Å¿um-<lb/>ma radicum dictarum partium æqualis Å¿it alteri numero propoÅ¿ito. </s> - <s xml:id="s551" xml:space="preserve">Rectè Å¿e-<lb/>cundum numerum in Å¿eipÅ¿um multiplicant, ex quo quadrato, primum datum nu-<lb/>merum detrahunt, <reg norm="rurÅ¿usque" type="simple">rurÅ¿usqÌ;</reg> reÅ¿iduum in Å¿eipÅ¿um multiplicant, & ex eo quadrato <lb/>quartam partem deÅ¿umunt, <reg norm="quam" type="context">quã</reg> ex quadrato dimidij primi numeri detrahunt, radi-<lb/>cemqÌue qua dratam reÅ¿idui cum iunxerint, & ex dimidio primi numeri detraxerint, <lb/>partes quæſitæ proferuntur.</s> - </p> - <p> - <s xml:id="s552" xml:space="preserve">Exempli gratia, Å¿i proponeretur primus numerus .20. diuidendus et .6. Å¿ecundus <lb/>pro Å¿umma radicum, hunc Å¿ecundum .6. in Å¿eipÅ¿um multiplicabimus, <reg norm="dabiturque" type="simple">dabiturqÌ;</reg> nu-<lb/>merus .36. ex quo quadrato primus numerus detrahetur, <reg norm="Å¿upereritque" type="simple">Å¿upereritqÌ;</reg> numerus .16. <lb/>qui quadratus dabit .256. cuius numeri quarta pars Å¿umetur, nempe .64. quæ ex qua <lb/>drato dimidij primi numeri detrahetur, nempe .100. <reg norm="Å¿upereritque" type="simple">Å¿upereritqÌ;</reg> .36. cuius radix qua <lb/>drata .6. coniuncta & detracta ex .10. dabit .16. partem maiorem et .4. minorem.</s> - </p> - <p> - <s xml:id="s553" xml:space="preserve">Cuius rei hæc Å¿peculatio, primus numerus diuiÅ¿ibilis Å¿ignificetur linea <var>.a.b.</var> diui-<lb/>Å¿a in puncto <var>.e.</var> in partes adhuc incognitas, et <var>.a.c.</var> Å¿it productum <var>.a.e.</var> in <var>.e.b.</var> item <var>.q.<lb/>p.</var> Å¿ecundum numerum Å¿ignificet, æqualem Å¿ummæ radicum, quæ puncto <var>.n.</var> diÅ¿tin-<lb/>guantur. </s> - <s xml:id="s554" xml:space="preserve">PoÅ¿tmodum totum quadratum <var>.p.d.</var> erigatur (quod nobis eÅ¿t cognitum), <lb/>in duo quadrata diuiÅ¿um <var>.o.p.</var> et <var>.o.d.</var> quorum Å¿umma <var>.a.b.</var> cum detur, cognita rema-<lb/>net Å¿umma <reg norm="duorum" type="context">duorÅ©</reg> <reg norm="Å¿upplementorum" type="context">Å¿upplementorÅ©</reg> <var>.o.u.</var> et <var>.o.q.</var> qua quadrata <reg norm="cum" type="context">cÅ©</reg> fuerit dabit quadru <lb/><reg norm="plum" type="context">plÅ©</reg> quadrati <reg norm="Å¿upplementi" type="context">Å¿upplemẽti</reg> <var>.o.q.</var> <reg norm="nempe" type="context">nẽpe</reg> <reg norm="quadruplum" type="context">quadruplÅ©</reg> producti <var>.a.c.</var> etenim <var>.a.c.</var> ex .19. theo <lb/>remate huius libri quadratum eft ipÅ¿ius <var>.q.o.</var> <reg norm="Å¿icque" type="simple">Å¿icqÌ;</reg> poterant etiam veteres quadrare <lb/>dimidium differentiæ <var>.a.b.</var> ab <var>.p.d.</var> nempe quadrato tantummodo Å¿upplemento <var>.q.<lb/>o</var>. </s> - <s xml:id="s555" xml:space="preserve">Tunc habito <var>.a.c.</var> eius ope tanquam producti <var>.a.e.</var> in <var>.e.b.</var> ex .45. theoremate Å¿ingu <lb/>læ partes cognoÅ¿centur.</s> - </p> - <p> - <s xml:id="s556" xml:space="preserve">Quod alia etiam ratione præſtari poterat, nempe cognito Å¿upplemento <var>.<lb/>q.o.</var> diÅ¿tinguendæ radices <var>q.n.</var> et <var>.n.p.</var> ex .45. theoremate, quibus cognitis, eorum <lb/>etiam quadrata cognoÅ¿cuntur.</s> - </p> - <figure position="here"> - <image file="0053-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0053-01"/> - </figure> - <figure position="here"> - <image file="0053-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0053-02"/> - </figure> - <pb o="42" rhead="IO. BAPT. BENED." n="54" file="0054" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0054"/> - </div> - <div type="math:theorem" level="3" n="65"> - <head xml:id="head81" xml:space="preserve">THEOREMA <num value="65">LXV</num>.</head> - <p> - <s xml:id="s557" xml:space="preserve">CVR propoÅ¿ito numero in tres qualeÅ¿cunque partes diuiÅ¿o, Å¿i prima in <lb/>tertiam multiplicetur, & huic producto, Å¿ecundæ in primam productum <lb/>coniungatur, <reg norm="itemque" type="simple">itemqÌ;</reg> Å¿ecundæ in tertiam, hæc Å¿umma duplicata æqualis Å¿it Å¿ummæ <lb/>productorum Å¿ingularum in cæteras duas.</s> - </p> - <p> - <s xml:id="s558" xml:space="preserve">Exempli gratia, Å¿i proponatur .20. diuiÅ¿us in tres partes nempe .12. 5. 3. multipli-<lb/>cato primo .12. per .3. tertiam partem dabitur .36. Å¿ecunda verò multiplicata per re <lb/>liquas duas, hoc eÅ¿t .5. per .12. et .3. in primis dabitur .60. poÅ¿tea .15. <reg norm="quorum" type="context">quorÅ©</reg> <reg norm="trium" type="context">triÅ©</reg> pro <lb/>ductorum Å¿umma erit .111. quæ duplicata dabit .222. qui numerus æqualis eſſe di-<lb/>citur Å¿ummæ productorum Å¿ingularum partium in reliquas duas, nempe Å¿ummæ .60. <lb/>36. 60. 15. 36. 15. hoc eÅ¿t ipÅ¿is .222.</s> - </p> - <p> - <s xml:id="s559" xml:space="preserve">Cuius rei per Å¿e patet Å¿peculatio, cum in his Å¿ex vltimis productis, Å¿ingula tria <lb/>prima duplicentur.</s> - </p> - </div> - <div type="math:theorem" level="3" n="66"> - <head xml:id="head82" xml:space="preserve">THEOREMA <num value="66">LXVI</num>.</head> - <p> - <s xml:id="s560" xml:space="preserve">CVR propoÅ¿ito numero in .3. qualeÅ¿cunque partes diuiÅ¿o, Å¿i in reliquas duas Å¿in-<lb/>gulæ multiplicentur, & hæc producta cum Å¿umma Å¿uorum quadratorum con-<lb/>iungantur, tota Å¿umma hæc vltima æqualis erit quadrato totali propoÅ¿iti numeri.</s> - </p> - <p> - <s xml:id="s561" xml:space="preserve">Exempli gratia, Å¿i fuerit idem numerus .20. in .3. partes diuiÅ¿us .12. 5. 3. </s> - <s xml:id="s562" xml:space="preserve">Si .12. in <lb/>5. et .3. producatur, Å¿umma productorum erit .96. at .5. in .12. et .3. erit .75. poÅ¿tmo-<lb/>dum .3. in .12. et .5. erit .51. nempe in vniuerÅ¿um .222. quadratorum porrò Å¿umma <lb/>erit .178 quæ coniuncta .222. dabit .400. quadratum ipÅ¿ius .20.</s> - </p> - <p> - <s xml:id="s563" xml:space="preserve">Erit autem huiuÅ¿ce rei facillima Å¿peculatio, Å¿i Å¿equentem figuram mente conce-<lb/>perimus, in qua <var>.a.b.</var> propoÅ¿itum numerum Å¿ignificet, cuius partes diÅ¿tinctæ Å¿int me-<lb/>dio <var>.e.</var> et <var>.c</var>. </s> - <s xml:id="s564" xml:space="preserve">Ip Å¿um autem <var>.q.b.</var> Å¿it quadratum <lb/>totale parallelis <var>.e.s.</var> et <var>.c.x.</var> diuiÅ¿um, quæ qua <lb/> - <anchor type="figure" xlink:label="fig-0054-01a" xlink:href="fig-0054-01"/> - dratum in triarectangula diuident, quorum <lb/>primum erit <var>.q.e.</var> compoÅ¿itum ex producto <var>.a.<lb/>e.</var> in Å¿emetipÅ¿am, nempe quadratum <var>.o.e.</var> & <lb/>ex producto eiuÅ¿dem <var>.a.e.</var> in <var>.e.b.</var> quod erit re <lb/>ctangulum <var>.o.s.</var> ex quo tria rectangula <var>.o.s.</var> et <var>.<lb/>n.x.</var> et <var>.t.u.</var> tria producta erunt Å¿ingularum par <lb/>tium in cæteras duas, et <var>.e.o</var>: <var>c.n</var>: <var>b.t.</var> tria qua-<lb/>drata erunt: </s> - <s xml:id="s565" xml:space="preserve">quibus Å¿ex quantitatibus quadra <lb/>tum totale <var>.q.b.</var> completur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0054-01" xlink:href="fig-0054-01a"> - <image file="0054-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0054-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="67"> - <head xml:id="head83" xml:space="preserve">THEOREMA <num value="67">LXVII</num>.</head> - <p> - <s xml:id="s566" xml:space="preserve"><emph style="sc">VEteres</emph> aliud quoque problema indefinitum propoÅ¿uerunt, quod tamen à <lb/>nobis determinabitur.</s> - </p> - <p> - <s xml:id="s567" xml:space="preserve">Cur diuiÅ¿uri propoÅ¿itum numerum in duas eiuÅ¿modi partes, vt mutuò diuiÅ¿is, & <lb/>per Å¿ummam prouenientium diuiÅ¿a Å¿umma qua dratorum partium, oriatur proue-<lb/>niens alter numerus propoÅ¿itus.</s> - </p> - <p> - <s xml:id="s568" xml:space="preserve">PropoÅ¿ito deinde tertio quolibet numero diuidendo per Å¿ingulas partes primi, - <pb o="43" rhead="THEOREM. ARIT." n="55" file="0055" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0055"/> - ita vt Å¿imul prouenientibus in Å¿ummam collectis huius fummæ ad primum nume-<lb/>rum propoÅ¿itum proportio futura Å¿it ea quæ eÅ¿t tertij ad Å¿ecundum. </s> - <s xml:id="s569" xml:space="preserve">Rectè dimidium <lb/>primi numeri in Å¿eipÅ¿um multiplicant, ex quo quadrato Å¿ecundum numerum detra <lb/>hunt, tum reÅ¿idui radicem Å¿umunt, quam iungentes, & detrahentes ex dimidio <lb/>primi, partes quæſitas habent, cætera ex neceÅ¿sitate Å¿ubÅ¿equuntur, prout nunc a <lb/>me docebitur.</s> - </p> - <p> - <s xml:id="s570" xml:space="preserve">Exempli gratia, proponitur numerus .20. in duas partes diuidendus, quibus po <lb/>Å¿tea mutuò diuiÅ¿is, & per Å¿ummam prouenientium diuiÅ¿a Å¿umma quadratorum, <lb/>dent <reg norm="Å¿ecundum" type="context">Å¿ecundÅ©</reg> numerum propoÅ¿itum .36. nam reliqua conÅ¿equuntur. </s> - <s xml:id="s571" xml:space="preserve">Itaque .10. <lb/>dimidium primi in Å¿eipÅ¿um multiplicatur, & ex quadrato .100. eruitur numerus .36. <lb/>nempe Å¿ecundus propoÅ¿itus reÅ¿idui porrò .64. quadrata radix .8. fumitur, quam con <lb/>iungimus & detrahimus ex dimidio primi Å¿cilicet .10. ex quo partes quæſitæ dabun <lb/>tur .18. et .2. quæ mutuo diuiſæ dabunt Å¿uorum prouenientium Å¿ummam .9. cum no-<lb/>na parte, per quam diuidentes .328. Å¿ummam quadratorum ipÅ¿arum partium, <lb/>exactè dabitur numerus .36. qui fuit Å¿ecundò propoÅ¿itus. </s> - <s xml:id="s572" xml:space="preserve">Tum Å¿i per Å¿ingu-<lb/>las iam inuentas partes quilibet numerus diuiÅ¿us fuerit, verbi gratia .72. Å¿umma pro <lb/>uenientium erit .40. qui num@rus eandem proportionem cum primo nempe .20. Å¿er <lb/>uabit, quam tertius propoÅ¿itus .72. cum Å¿ecundo .36.</s> - </p> - <p> - <s xml:id="s573" xml:space="preserve">Quod vt Å¿peculemur, primus numerus Å¿ignificetur linea <var>.n.e.</var> ita diuidendus à <lb/>puncto <var>.o.</var> vt diuiÅ¿a parte <var>.n.o.</var> per <var>.o.e.</var> et <var>.o.e.</var> per <var>.n.o.</var> & per Å¿ummam prouenien-<lb/>tium diuiÅ¿a Å¿umma quadratorum <var>.n.o.</var> et <var>.o.e.</var> detur Å¿ecundus numerus notatus linea <var type="line">.<lb/>q.K</var>. </s> - <s xml:id="s574" xml:space="preserve">Porrò meminiſſe oportet quòd .26. theoremate probatum fuit vltimum hoc <lb/>proueniens æquale producto partium inter Å¿e futurum, nempe producto <var>.n.o.</var> in <var>.o.<lb/>e.</var> quod Å¿ignificetur rectangulo <var>.n.e</var>. </s> - <s xml:id="s575" xml:space="preserve">Itaque datis <var>.n.e.</var> et <var>.q.K.</var> Å¿i .45. theorema conÅ¿u-<lb/>luerimus, partes <var>.n.o.</var> et <var>.o.e.</var> cognoÅ¿cemus.</s> - </p> - <p> - <s xml:id="s576" xml:space="preserve">Proponitur deinde tertius quilibetnumerus, verbi gratia <var>.x.</var> diuidendus per <var>.o.e.</var> <lb/>et <var>.o.n.</var> qui Å¿i diuidatur per <var>.o.e.</var> dabit pro <lb/>ueniens <var>.b.o</var>. </s> - <s xml:id="s577" xml:space="preserve">Si verò per <var>.n.o.</var> proueniens <lb/> - <anchor type="figure" xlink:label="fig-0055-01a" xlink:href="fig-0055-01"/> - erit <var>.d.n.</var> nunc aſſerimus <reg norm="Å¿ummam" type="context">Å¿ummã</reg> duorum <lb/>horum prouenientium, Å¿ic primo nume-<lb/>ro <var>.n.e.</var> dato proportionatam eſſe, Å¿icut <lb/>tertius <var>.x.</var> <reg norm="Å¿ecundo" type="context">Å¿ecÅ©do</reg> <var>.q.K</var>. </s> - <s xml:id="s578" xml:space="preserve">Producatur enim li-<lb/>nea <var>.d.n.</var> donec <var>.n.q.</var> æqualis Å¿it <var>.o.b.</var> ex <lb/>quo <var>.q.d.</var> erit Å¿umma vltimò prouenien-<lb/>tium: </s> - <s xml:id="s579" xml:space="preserve">item producatur <var>.e.n.</var> donec <var>.n.u.</var> æ-<lb/>qualis Å¿it <var>.o.e.</var> <reg norm="termineturque" type="simple">termineturqÌ</reg> rectangulum <var>.<lb/>q.u.</var> quod tertio numero propoÅ¿ito <var>.x.</var> vt <lb/>patet, æquale erit, </s> - <s xml:id="s580" xml:space="preserve">quare ex .15. Å¿exti aut .<lb/>20. Å¿eptimi eadem erit proportio <var>.d.n.</var> ad <lb/><var>n.q.</var> quæ <var>.u.n.</var> nempe <var>.o.e.</var> ad <var>.o.n.</var> & com-<lb/>ponendo <var>.d.q.</var> ad <var>.q.n.</var> Å¿icut <var>.e.n.</var> ad <var>.n.o.</var> & <lb/>permutando <var>.d.q.</var> ad <var>.e.n.</var> quæ <var>.q.n.</var> hoc eÅ¿t <var>.<lb/>b.o.</var> ad <var>.o.n.</var> nempe Å¿icut <var>.b.e.</var> ad <var>.e.n.</var> Å¿uperficialem, ex prima Å¿exti aut .18. vel .19. <lb/>Å¿eptimi, Å¿ed rectangulum <var>.e.n.</var> conÅ¿titutum fuit æquale numero <var>.q.K</var>. </s> - <s xml:id="s581" xml:space="preserve">itaque verum <lb/>eÅ¿t propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0055-01" xlink:href="fig-0055-01a"> - <image file="0055-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0055-01"/> - </figure> - </div> - <pb o="44" rhead="IO. BAPT. BENED." n="56" file="0056" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0056"/> - </div> - <div type="math:theorem" level="3" n="68"> - <head xml:id="head84" xml:space="preserve">THEOREMA <num value="68">LXVIII</num>.</head> - <p> - <s xml:id="s582" xml:space="preserve">CVR numero per numerum diuiÅ¿o, <reg norm="productoque" type="simple">productoqÌ;</reg> duorum numerorum per pro-<lb/>ueniens multiplicato, quod vltimò productum eÅ¿t, diuiÅ¿i numeri Å¿emper qua <lb/>dratum exiÅ¿tat.</s> - </p> - <p> - <s xml:id="s583" xml:space="preserve">Exempli gratia, Å¿i diuidamus .10. per .2. proueniens erit .5. quo producto ex duo <lb/>bus numeris multiplicato, nempe .20. habe <lb/>bimus .100. quadratum numeri diuiÅ¿i.</s> - </p> - <figure position="here"> - <image file="0056-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0056-01"/> - </figure> - <p> - <s xml:id="s584" xml:space="preserve">Cuius gratia duo numeri Å¿int <var>.a.</var> et <var>.e.</var> por <lb/>rò <var>.a.</var> per <var>.e.</var> diuiÅ¿o detur <var>.u.</var> tum <var>.o.</var> produ-<lb/>ctum <var>.a.</var> in <var>.e.</var> eſſe conÅ¿tituatur, quo per <var>.u.</var> <lb/>multiplicato dabitur <var>.x.</var> quadratum <var>.a.</var> pro-<lb/>ptereà quòd <var>.a.</var> medium eÅ¿t proportionale <lb/>inter <var>.o.</var> et <var>.u.</var> ex .35. theoremate. </s> - <s xml:id="s585" xml:space="preserve">itaque <lb/>ex .16. Å¿exti aut .20. Å¿eptimi, propoÅ¿iti veri-<lb/>tas eluceÅ¿cet.</s> - </p> - </div> - <div type="math:theorem" level="3" n="69"> - <head xml:id="head85" xml:space="preserve">THEOREMA <num value="69">LXIX</num>.</head> - <p> - <s xml:id="s586" xml:space="preserve">CVR numero aliquo per duos alios multiplicato & diuiÅ¿o, Å¿i per horum duo-<lb/>rum productum, Å¿umma duorum primorum productorum diuiÅ¿a fuerit, vl-<lb/>timum proueniens, Å¿ummæ duorum primorum prouenientium æquale Å¿it.</s> - </p> - <p> - <s xml:id="s587" xml:space="preserve">Exempli gratia, proponitur numerus .24. per .8. et .6. multiplicandus & diuiden <lb/>dus Å¿umma productorum crit .336. prouenientium autem .7. Å¿i igitur Å¿ummam .336. <lb/>productorum per productum duorum Å¿ecundorum numerorum nempe .48. diuiÅ¿e-<lb/>rimus, proueniens pariter erit .7.</s> - </p> - <p> - <s xml:id="s588" xml:space="preserve">In cuius <reg norm="gratiam" type="context">gratiã</reg> primus numerus Å¿ignificetur linea <var>.q.b.</var> multiplicandus & diuiden-<lb/>dus numeris deÅ¿ignatis per <var>.k.m.</var> et <var>.y.m.</var> productorum Å¿umma Å¿it <var>.k.z.</var> prouenien-<lb/>tium autem <var>.a.e</var>: et <var>.a.o.</var> ex <var>.k.m.</var> et <var>.o.e.</var> ex <var>.y.m</var>: tum productum <var>.k.m.</var> in <var>.m.y.</var> Å¿it <var>.f.<lb/>m</var>. </s> - <s xml:id="s589" xml:space="preserve">Dico quòd Å¿i <var>.k.z.</var> per <var>.f.m.</var> diuiÅ¿erimus proueni et <var>.a.e</var>. </s> - <s xml:id="s590" xml:space="preserve">Quod cum Å¿ic fuerit, erit <lb/>quoque verum quòd diuiÅ¿a <var>.k.z.</var> per <var>.a.e.</var> proueniet <var>.f.m.</var> numerus Å¿cilicet æqualis <lb/>numero <var>.f.m.</var> ex .13. theoremate huius. </s> - <s xml:id="s591" xml:space="preserve">Itaque quotieÅ¿cunque probauero quòd di-<lb/>uiÅ¿a <var>.k.z.</var> per <var>.a.e.</var> proueniat numerus æqualis ipÅ¿i <var>.f.m.</var> propoÅ¿itum verum eſſe con <lb/>Å¿equetur. ex .13. theoremate. </s> - <s xml:id="s592" xml:space="preserve">Quòd Å¿i proueniens ex diuiÅ¿ione <var>.k.z.</var> per <var>.a.e.</var> æqua <lb/>le fuerit <var>.f.m.</var> patet ex .7. quinti quòd <reg norm="eadem" type="context">eadẽ</reg> erit proportio numeri <var>.k.m.y.</var> ad ipÅ¿um <lb/>proueniens, quæ ad numerum <var>.f.m</var>. </s> - <s xml:id="s593" xml:space="preserve">Cogitemus <reg norm="itaque" type="simple">itaq;</reg> <var>.k.u.</var> æqualem <var>.a.e.</var> Å¿uper quam <lb/>mente concipiamus rectangulum <var>.u.p.</var> æqualem <var>.k.z.</var> ex quo eadem erit proportio <var>.<lb/>k.p.</var> ad <var>.k.y.</var> quæ <var>.g.k.</var> ad <var>.k.u.</var> ex .15. Å¿exti, aut, 20. Å¿eptimi, numerus autem <var>.k.p.</var> erit <lb/>proueniens, quod probandum eÅ¿t æquale eſſe <var>.f.m</var>.</s> - </p> - <p> - <s xml:id="s594" xml:space="preserve">Probabitur autem Å¿ic, ex .9. quinti, nempe demonÅ¿trato quòd numerus <var>.k.p.</var> ean <lb/>dem proportionem habeat ad numerum <var>.k.y.</var> quam habet numerus <var>.f.m.</var> ad eundem <lb/><var>k.y</var>. </s> - <s xml:id="s595" xml:space="preserve">Sed probatum eÅ¿t Å¿ic Å¿e habere <var>.k.g.</var> ad <var>.k.u.</var> Å¿icut <var>.k.p.</var> ad <var>.k.y.</var> Å¿ufficiet igitur pro-<lb/>bare Å¿ic Å¿e habere <var>.k.g.</var> ad <var>.k.u.</var> Å¿icut <var>.f.m.</var> ad <var>.k.y</var>. </s> - <s xml:id="s596" xml:space="preserve">Sed <var>.k.g.</var> dicitur æqualis eſſe <var>.q.b</var>: et <var>.k.</var> <lb/>u; </s> - <s xml:id="s597" xml:space="preserve">a.e. Å¿atis erit igitur probare ita Å¿e habere <var>.q.b.</var> ad <var>.a.e.</var> Å¿icut <var>.f.m.</var> ad <var>.k.y</var>. </s> - <s xml:id="s598" xml:space="preserve">Scimus au-<lb/>tem quòd eadem eÅ¿t proportio <var>.q.b.</var> ad <var>.a.o.</var> quæ <var>.m.k.</var> ad vnitatem, quæ Å¿it <var>.x.</var> & quod <lb/>proportio <var>.o.e.</var> ad <var>.q.b.</var> eadem eÅ¿t, quæ <var>.x.</var> ad <var>.m.y.</var> ex definitione diuiÅ¿ionis. </s> - <s xml:id="s599" xml:space="preserve">Quare <lb/>ex æqualitate proportionum eadem erit proportio <var>.k.m.</var> ad <var>.m.y.</var> quæ <var>.e.o.</var> ad <var>.o.a.</var> & - <pb o="45" rhead="THEOREM. ARIT." n="57" file="0057" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0057"/> - componendo Å¿ic Å¿e habebit <var>.k.y.</var> ad <var>.m.y.</var> Å¿icut <var>.e.a.</var> ad <var>.o.a.</var> & permutando <var>.k.y.</var> ad <var>.e.<lb/>a.</var> Å¿icut <var>.m.y.</var> ad <var>.o.a.</var> & ex .19. quinti ita <var>.k.m.</var> ad <var>.e.o.</var> Å¿icut <var>.k.y.</var> ad <var>.e.a.</var> & permutando <var>.<lb/>k.m.</var> ad <var>.k.y.</var> Å¿icut <var>.e.o.</var> ad <var>.e.a</var>. </s> - <s xml:id="s600" xml:space="preserve">Nunc producatur <var>.f.t.</var> donec <var>.t.i.</var> æqualis Å¿it <var>.k.y.</var> <reg norm="produ- ctaque" type="simple">produ-<lb/>ctaqÌ;</reg> <var>.m.t.</var> done <var>c.t.s.</var> æqualis Å¿it vnitati <var>.x.</var> <reg norm="termineturque" type="simple">termineturqÌ;</reg> rectangulum <var>.s.i.</var> ex quo da-<lb/>bitur proportio numeri <var>.f.m.</var> ad numerum <var>.s.i.</var> compoÅ¿ita ex <var>.m.t.</var> ad <var>.t.s.</var> et <var>.f.t.</var> ad <var>.t.i.</var> <lb/>ex .24. Å¿exti, aut quinta octaui, Å¿ed ita etiam proportio <var>.q.b.</var> ad <var>.a.e.</var> componitur ex <lb/>eiÅ¿dem proportionibus, nempe ex <var>.q.b.</var> ad <var>.o.e.</var> æquali <var>.m.t.</var> ad <var>.t.s.</var> & ex proportione <var>.<lb/>o.e.</var> ad <var>.a.e.</var> æquali <var>.f.t.</var> ad <var>.t.i.</var> ita que proportio numeri <var>.f.m.</var> ad <var>.s.i.</var> hoc eÅ¿t ad <reg norm="numerum" type="context">numerÅ©</reg> <lb/>ipÅ¿ius <var>.k.y.</var> Ä™qualis eÅ¿t proportioni numeri <var>.q.b.</var> ad <var>.a.e.</var> <reg norm="nempe" type="context">nẽpe</reg> <var>.k.g.</var> ad <var>.k.u.</var> hoc eÅ¿t <var>.k.p.</var> ad <lb/><var>x.y.</var> ex quo Å¿equitur <var>.k.p.</var> conÅ¿tare numero Ä™quali <var>.f.m.</var> proueniens igitur ex diuiÅ¿ione <lb/>numeri <var>.k.z.</var> per <var>.f.m.</var> æquale eÅ¿t numero ipÅ¿ius <var>.a.e</var>.</s> - </p> - <figure position="here"> - <image file="0057-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0057-01"/> - </figure> - </div> - <div type="math:theorem" level="3" n="70"> - <head xml:id="head86" xml:space="preserve">THEOREMA <num value="70">LXX</num>.</head> - <p> - <s xml:id="s601" xml:space="preserve">HAEC porrò concluÅ¿io alia etiam via demonÅ¿trari poteÅ¿t.</s> - </p> - <p> - <s xml:id="s602" xml:space="preserve">Significetur numerus diuidendus atque multiplicandus linea <var>.b.a</var>. </s> - <s xml:id="s603" xml:space="preserve">Deinde <lb/>diuidentes & <reg norm="multiplicantes" type="context">multiplicãtes</reg> Å¿int <var>.k.m.</var> et <var>.m.y.</var> prouenientia ex diuiÅ¿ione Å¿int <var>.a.o.</var> et <var>.o.<lb/>e.</var> atque <var>.a.o.</var> ex <var>.m.y</var>: <var>o.e.</var> verò ex <var>.k.m.</var> proueniat, quorum Å¿umma Å¿it <var>.a.e</var>: productum <lb/>autem <var>.b.a.</var> in <var>.k.m.</var> Å¿it <var>.b.p.</var> et <var>.p.s.</var> productum <var>.b.a.</var> in <var>.m.y.</var> ad hæc rectangulum <var>.k.y.</var> Å¿it <lb/>productum <var>.k.m.</var> in <var>.m.y</var>: quo to-<lb/>tum productum <var>.a.s.</var> diuidatur, pro <lb/> - <anchor type="figure" xlink:label="fig-0057-02a" xlink:href="fig-0057-02"/> - <reg norm="ueniensque" type="simple">ueniensqÌ;</reg> Å¿it <var>.a.c.</var> cui, <var>a.c</var>: <reg norm="productum" type="context">productÅ©</reg> <var>.<lb/>a.s.</var> <reg norm="eandem" type="context context">eãdẽ</reg> <reg norm="proportionem" type="context">proportionẽ</reg> Å¿eruabit, <reg norm="quam" type="context">quã</reg> <lb/><var>k.y.</var> rectangulum ad vnitatem ex <lb/>definitione diuiÅ¿ionis, hoc autem <lb/>proueniens <var>.a.c.</var> <reg norm="conÅ¿tare" type="context">cõſtare</reg> numero æ-<lb/>quali aſſero Å¿ummæ <var>.a.e</var>. </s> - <s xml:id="s604" xml:space="preserve">Primum <lb/>enim ex dicta definitione diuiÅ¿io-<lb/>nis habemus eandem eſſe propor-<lb/>tionem <var>.b.a.</var> ad <var>.a.o.</var> quæ <var>.m.y.</var> ad <lb/>vnitatem, & quod Å¿ic Å¿e habet <var>.b.a.</var> <lb/>ad <var>.o.e.</var> Å¿icut <var>.k.m.</var> ad eandem vnita <lb/>tem. </s> - <s xml:id="s605" xml:space="preserve">Itaque vnitas hæc linearis Å¿i-<lb/>gnificetur per <var>.m.x.</var> in Å¿ingulis late-<lb/>ribus <var>.k.m.</var> et <var>.m.y.</var> producentibus rectangulum <var>.k.y</var>: Å¿uperficialis autem vnitas Å¿it. - <pb o="46" rhead="IO. BAPT. BENED." n="58" file="0058" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0058"/> - <var>g.m.</var> <reg norm="cogiteturque" type="simple">cogiteturqÌ;</reg> rectangulum <var>.y.x.</var> & rectangulum <var>.k.x</var>. </s> - <s xml:id="s606" xml:space="preserve">Itaque dabitur eadem pro <lb/>portio <var>.k.m.</var> ad <var>.m.x.</var> nempe <var>.k.x.</var> rectanguli ad <var>.m.g.</var> quæ eÅ¿t <var>.b.a.</var> ad <var>.o.e.</var> et <var>.y.x.</var> ad <var>.m.<lb/>g.</var> quæ <var>.b.a.</var> ad <var>.a.o.</var> Å¿ed ex prima Å¿exti aut .18. vel .19. Å¿eptimi, Å¿ic Å¿e habet rectangu-<lb/>lum <var>.k.y.</var> ad <var>.x.y.</var> Å¿icut <var>.k.m.</var> ad <var>.m.x.</var> </s> - <s xml:id="s607" xml:space="preserve">quare Å¿icut <var>.b.a.</var> ad <var>.o.e.</var> ex .11. quinti, & eiuÅ¿dem <lb/>rectanguli <var>.k.y.</var> ad rectangulum <var>.k.x.</var> Å¿icut <var>.y.m.</var> ad <var>.x.m.</var> nempe <var>.b.a.</var> ad <var>.a.o</var>. </s> - <s xml:id="s608" xml:space="preserve">Quare <lb/>ex communi Å¿cientia, Å¿ic Å¿e habebit duplum rectanguli <var>.k.y.</var> ad Å¿ummam <var>.y.x.</var> cum <var>.<lb/>k.x.</var> rectangulorum, Å¿icut duplum <var>.b.a.</var> ad Å¿ummam <var>.a.o.e.</var> et proportio Å¿ummæ re-<lb/>ctangulorum <var>.y.x.</var> et <var>.k.x.</var> duplo <var>.g.m.</var> Å¿icut duplum <var>.b.a.</var> ad <var>.a.o.e</var>. </s> - <s xml:id="s609" xml:space="preserve">Igitur Å¿umma duo-<lb/>rum rectangulorum <var>.y.x.</var> et <var>.x.k.</var> media proportionalis erit inter duplum rectanguli <var>.<lb/>k.y.</var> & duplum vnitatis Å¿uperÅ¿icialis <var>.g.m</var>. </s> - <s xml:id="s610" xml:space="preserve">Nunc terminetur rectangulum <var>.a.r.</var> ex quo <lb/>dabitur eadem proportio dupli <var>.a.s.</var> ad <var>.a.r.</var> Å¿icut dupli <var>.b.a.</var> ad <var>.a.e.</var> ex propoÅ¿itioni-<lb/>bus notatis, Å¿exti aut Å¿eptimi. </s> - <s xml:id="s611" xml:space="preserve">Quare etiam Å¿icut dupli rectanguli <var>.k.y.</var> ad <reg norm="Å¿ummam" type="context">Å¿ummã</reg> <lb/>rectangulorum <var>.y.x.</var> et <var>.k.x</var>. </s> - <s xml:id="s612" xml:space="preserve">Iam verò Å¿i conÅ¿tituatur <var>.e.c.</var> pro vnitate lineari ipÅ¿ius <var>.<lb/>e.r.</var> certi erimus numerum <var>.a.c.</var> æqualem eſſe <var>.a.e.</var> & proportionem <var>.r.e.</var> ad <var>.e.c.</var> hoc <lb/>eÅ¿t <var>.a.r.</var> ad <var>.a.c.</var> eandem quæ <var>.y.x.</var> et <var>.x.k.</var> rectangulorum ad <var>.m.g.</var> ex prædictis rationi-<lb/>bus, & ex hypotheÅ¿i, nempe quòd <var>.<lb/>e.r.</var> æqualis Å¿it numero <var>.k.m.y.</var> <lb/> - <anchor type="figure" xlink:label="fig-0058-01a" xlink:href="fig-0058-01"/> - hoc eÅ¿t rectangulorum <var>.y.x.</var> et <var>.x.<lb/>k</var>. </s> - <s xml:id="s613" xml:space="preserve">Quamobrem <var>.a.r.</var> ex communi <lb/>Å¿cientia <reg norm="medium" type="context">mediÅ©</reg> proportionale erit <lb/>inter duplum <var>.a.s.</var> & duplum <var>.a.c.</var> <reg norm="ea demque" type="context simple">eaÂ<lb/>dẽqÌ;</reg> <reg norm="proportio" type="simple">ê“portio</reg> dupli prÄ™dicti <var>.a.s.</var> ad <lb/>duplum <var>.a.c.</var> ex æqualitate propor-<lb/>tionum Å¿imul collectarum, eadem <lb/>erit qùæ proportio dupli rectangu-<lb/>li <var>.k.y.</var> ad duplum <var>.m.g.</var> hoc eÅ¿t <var>.a.s.</var> <lb/>Å¿implicis ad Å¿implicem <var>.a.c.</var> quæ Å¿im <lb/>plicis rectanguli <var>.k.y.</var> ad Å¿implicem <lb/>vnitatem <var>.g.m.</var> Å¿ic enim Å¿e habet Å¿im <lb/>plex ad Å¿implex, Å¿icut duplum ad <lb/>duplum. </s> - <s xml:id="s614" xml:space="preserve">Sed pariter ita Å¿e habet <var>.a.s.</var> ad <var>.a.</var> c<unsure/>. cogitato <var>.a.c.</var> tamquam proueniente <lb/>ex diuiÅ¿ione <var>.a.s.</var> per rectangulum <var>.k.y.</var> vt conÅ¿titutum eÅ¿t, Å¿icut <var>.k.y.</var> ad <var>.m.g.</var> ex defi-<lb/>nitione diuiÅ¿ionis vt iam dictum eÅ¿t, </s> - <s xml:id="s615" xml:space="preserve">quare numerus <var>.a.c.</var> æqualis erit numero <var>.a.o.e</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0057-02" xlink:href="fig-0057-02a"> - <image file="0057-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0057-02"/> - </figure> - <figure xlink:label="fig-0058-01" xlink:href="fig-0058-01a"> - <image file="0058-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0058-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="71"> - <head xml:id="head87" xml:space="preserve">THEOREMA <num value="71">LXXI</num>.</head> - <p> - <s xml:id="s616" xml:space="preserve">CVR propoÅ¿itis .4. numeris, duobus nempe diuidentibus ac duobus diuiden-<lb/>dis, Å¿i <reg norm="adinuicem" type="context">adinuicẽ</reg> diuiÅ¿i fuerint, <reg norm="duoque" type="simple">duoqÌ;</reg> <reg norm="prouenientia" type="context">proueniẽtia</reg> <reg norm="inuicem" type="context">inuicẽ</reg> multiplicata <reg norm="quenuis" type="context">quẽuis</reg> nu <lb/>merum producant, qui Å¿eruetur, Å¿i deinde ijdem numeri verÅ¿a vice mutuo diuiÅ¿i fue <lb/>rint, & inter Å¿e multiplicata prouenientia, <reg norm="productum" type="context">productÅ©</reg> hoc, primo Å¿eruato numero <lb/>æquale erit.</s> - </p> - <p> - <s xml:id="s617" xml:space="preserve">Exempli gratia propoÅ¿itis his .4. numeris .20. 30. 5. 10. duo autem .20. Å¿cilicet <lb/>et .30. Å¿int numeri diuidendi, porrò .5. et .10. numeri diuidentes, <reg norm="nempe" type="context">nẽpe</reg> vt primo .20 <lb/>per .5. diuidatur, tum .30. per .10. producetur .4. et .3. qui Å¿imul multiplicati <reg norm="proferent" type="context">proferẽt</reg> <num value="12">.<lb/>12.</num> tum .20. per .10. d iuiÅ¿o et .30. per .5. prouenientia erunt .2. 6. quæ inter Å¿e multi-<lb/>plicata producent etiam .12.</s> - </p> - <pb o="47" rhead="THEOR. ARITH." n="59" file="0059" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0059"/> - <p> - <s xml:id="s618" xml:space="preserve">Cuius rationem Å¿i quæris, Å¿ignificentur .4. numeri lineis, <var>a.e.o.u.</var> <reg norm="diuidaturque" type="simple">diuidaturqÌ;</reg> .2. <lb/>per <var>.o.</var> & <reg norm="oriatur" type="simple">oriat̃</reg>. s. & per <var>.u.</var> <reg norm="oriatur" type="simple">oriat̃</reg> <var>.y.</var> et <var>.<lb/> - <anchor type="figure" xlink:label="fig-0059-01a" xlink:href="fig-0059-01"/> - e.</var> diuiÅ¿o per <var>.o.</var> oriatur <var>.z.</var> & per <var>.u.</var> <lb/>proueniat <var>.f.</var> tum <var>.n.</var> Å¿it productum <var>.z.</var> <lb/>in <var>.y.</var> et <var>.m.</var> productum <var>.s.</var> in <var>.f</var>. </s> - <s xml:id="s619" xml:space="preserve">Dico <lb/>n. futurum æquale <var>.m</var>. </s> - <s xml:id="s620" xml:space="preserve">Sit deinde <var>.<lb/>x.</var> vnitas, quare ex definitione diui-<lb/>Å¿ionis eadem erit proportio <var>.s.</var> ad <var>.a.</var> <lb/>et <var>.z.</var> ad <var>.e.</var> quæ <var>.x.</var> ad <var>.o</var>. </s> - <s xml:id="s621" xml:space="preserve">Sed ita Å¿e ha-<lb/>bet <var>.a.</var> ad <var>.y.</var> et <var>.e.</var> ad <var>.f.</var> Å¿icut <var>.u.</var> ad <var>.x.</var> ex <lb/>quo Å¿ic Å¿e habebit <var>.s.</var> ad <var>.a.</var> Å¿icut <var>.z.</var> ad <lb/>e. et <var>.a.</var> ad. y, Å¿icut <var>.e.</var> ad <var>.f</var>. </s> - <s xml:id="s622" xml:space="preserve">Itaque ex <lb/>æqualitate proportionum Å¿ic Å¿e ha-<lb/>bebit s. ad <var>.y.</var> Å¿icut <var>.z.</var> ad <var>.f</var>. </s> - <s xml:id="s623" xml:space="preserve">Igitur ex <lb/>15. Å¿exti aut .20. Å¿eptimi productum <var>.<lb/>n.</var> producto <var>.m.</var> æquale erit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0059-01" xlink:href="fig-0059-01a"> - <image file="0059-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0059-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="72"> - <head xml:id="head88" xml:space="preserve">THEOREMA <num value="72">LXXII</num>.</head> - <p> - <s xml:id="s624" xml:space="preserve">ALIVD quoque problema à me inuentum eÅ¿t, nempe vt proponantur .4. <lb/>numeri qualeÅ¿cunque tandem, quorum duo diuiÅ¿ibiles Å¿int, tertius diuiÅ¿or <lb/>vnius è duobus pro libito, <reg norm="quæramusque" type="simple">quæramusqÌ;</reg> alterius diuidentem, qui Å¿ic Å¿e habeat vt pro <lb/>ductum duorum prouenientium quarto numero propoÅ¿ito Å¿it æquale.</s> - </p> - <p> - <s xml:id="s625" xml:space="preserve">Exempli gratia, proponuntur .4. numeri .20. 48. 5. 12. porrò .20. et .48. numeri <lb/>Å¿int diuiÅ¿ibiles et .5. <reg norm="diuidens" type="context">diuidẽs</reg> vnius, ut potè .20. </s> - <s xml:id="s626" xml:space="preserve"><reg norm="Quærendus" type="context">Quærẽdus</reg> nunc erit diuidens alterius <lb/>nempe .48. eiuÅ¿modi vt productum prouenientium æquale Å¿it .12. </s> - <s xml:id="s627" xml:space="preserve">Diuidam itaque <num value="20">.<lb/>20.</num> per .5. <reg norm="prouenietque" type="simple">prouenietqÌ;</reg> 4. quem per .48. multiplicabo, nempe per alterum diuiÅ¿ibi-<lb/>lem, <reg norm="Å¿icque" type="simple">Å¿icqÌ;</reg> proueniet .192. quod productum per quartum numerum nempe .12. diui-<lb/>fum dabit .16. qui erit diuidens quæſitus, quo diuiÅ¿o .48. proueniet .3. Å¿ecundum Å¿ci <lb/>licet proueniens, quo per alterum hoc eÅ¿t .4. multiplicato producetur quartus nu-<lb/>merus .12.</s> - </p> - <p> - <s xml:id="s628" xml:space="preserve">Quod vt Å¿ciamus, primus nume-<lb/>rus diuiÅ¿ibilis Å¿ignificetur <reg norm="rectangulo" type="context">rectãgulo</reg> <var>.<lb/> - <anchor type="figure" xlink:label="fig-0059-02a" xlink:href="fig-0059-02"/> - a.i.</var> Å¿ecundus rectangulo <var>.o.u.</var> primus <lb/>diuidens latere <var>.a.e.</var> quartum nume-<lb/>rum rectangulo <var>.i.o.</var> primum proue-<lb/>niens latere <var>.e.i.</var> Å¿ecundus diuidens la <lb/>tere <var>.e.u.</var> (hic autem eÅ¿t quem quæri-<lb/>mus) tum alterum proueniens Å¿igni <lb/>ficetur latere <var>.e.o</var>. </s> - <s xml:id="s629" xml:space="preserve">Iam <reg norm="eadem" type="context">eadẽ</reg> erit pro-<lb/>portio <var>.e.i.</var> ad <var>.e.u.</var> quæ <var>.o.i.</var> ad <var>.o.u.</var> <lb/>Sed cum cognitæ Å¿int tres quantita-<lb/>tes <var>.e.i</var>: <var>i.o</var>: et <var>.o.u.</var> quarta quoque. e<unsure/> <var>.u.</var> exregula de tribus immediatè cognoÅ¿cetur, <lb/>cætera in Å¿ubÅ¿cripta figura facillimè patebunt.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0059-02" xlink:href="fig-0059-02a"> - <image file="0059-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0059-02"/> - </figure> - </div> - <pb o="48" rhead="IO. BAPT. BENED." n="60" file="0060" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0060"/> - </div> - <div type="math:theorem" level="3" n="73"> - <head xml:id="head89" xml:space="preserve">THEOREMA <num value="73">LXXIII</num>.</head> - <p> - <s xml:id="s630" xml:space="preserve">HOC etiam problema à me inuentum eÅ¿t, nempe Å¿i duæ radices quadratæ in <lb/>Å¿ummam collectæ fuerint, & ex dimidio eiuÅ¿modi Å¿ummæ detracta fuerit mi <lb/>nor radix, <reg norm="reÅ¿iduique" type="simple">reÅ¿iduiqÌ;</reg> quadratum duplicatum <reg norm="eique" type="simple">eiqÌ;</reg> Å¿ummæ coniungatur du-<lb/>plum producti ipÅ¿ius reÅ¿idui in dimidium Å¿ummæ radicum, atque huic Å¿ummæ du-<lb/>plum producti eiuÅ¿dem reÅ¿idui in radicem minorem coniunctum fuerit; </s> - <s xml:id="s631" xml:space="preserve">vltima hæc <lb/>Å¿umma differentia erit duorum quadratorum propoÅ¿itorum.</s> - </p> - <p> - <s xml:id="s632" xml:space="preserve">Exempli gratia duæ radices quadraræ Å¿int .5. et .11. harum Å¿umma erit .16. & dimi <lb/>dium .8. differentia minoris ab ipÅ¿o dimidio erit .3: duplum quadrati huius differen <lb/>tiæ erit .18: </s> - <s xml:id="s633" xml:space="preserve">duplum producti huius differentiÄ™ in dimidium Å¿ummÄ™ radicum erit .48. <lb/>item & huius differentiæ duplum in minorem radicem erit .30. quarum omnium <lb/>Å¿umma erit .96. tantaqÌue erit differentia Å¿uorum quadratorum, quorum vnum <lb/>erit .25. alterum verò .121.</s> - </p> - <p> - <s xml:id="s634" xml:space="preserve">Pro cuius rei Å¿cientia, duæ quadratæ radices Å¿int <var>.h.o.</var> et <var>.o.d.</var> directæ inter Å¿e con-<lb/>iunctæ, quæ Å¿umma per medium in puncto <var>.e.</var> diuidatur, tum cogitetur <var>.e.b.</var> æqualis <lb/><var>o.e.</var> perpendicularis <var>.h.d.</var> <reg norm="ducanturque" type="simple">ducanturqÌ;</reg> lineæ <var>.b.h</var>: <var>b.o.</var> et <var>.b.d</var>. </s> - <s xml:id="s635" xml:space="preserve">Iam ex .4. primi <var>.b.h.</var> æqua <lb/>lis erit <var>.b.d.</var> & quadratum <var>.b.h.</var> æquale quadrato <var>.h.o.</var> & quadrato <var>.o.b.</var> Å¿imul cum du <lb/>plo producti <var>.o.e.</var> in <var>.o.h.</var> ex .12. Å¿ecundi Eucli. </s> - <s xml:id="s636" xml:space="preserve">Sed ex .13. <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> quadratum <var>.b.d.</var> <lb/>minus eÅ¿t quadrato <var>.o.d.</var> cum quadrato <var>.o.b.</var> ex duplo producti <var>.o.e.</var> in <var>.o.d.</var> at duplum <lb/>eiuÅ¿modi producti æquale eÅ¿t duplo qua-<lb/>drati <var>.o.e.</var> & duplo producti <var>.o.e.</var> in <var>.e.d.</var> ex <lb/> - <anchor type="figure" xlink:label="fig-0060-01a" xlink:href="fig-0060-01"/> - tertia eiuÅ¿dem, itaque duo quadrata Å¿cili-<lb/>cet <var>.o.b.</var> et <var>.o.d.</var> maiora erunt duobus qua-<lb/>dratis, nempe <var>.o.b.</var> et <var>.o.h.</var> collectis cum du <lb/>plo producti <var>.o.e.</var> in <var>.o.h.</var> ex duplo quadrati <lb/><var>o.e.</var> vna <reg norm="cum" type="context">cÅ©</reg> duplo producti <var>.o.e.</var> in <var>.e.d</var>. </s> - <s xml:id="s637" xml:space="preserve">Qua <lb/>re <reg norm="differentia" type="context">differẽtia</reg> Å¿ummæ duorum quadratorum <lb/><var>o.b.</var> et <var>.o.d.</var> à ſumma duorum <var>o.b.</var> et <var>.o.h.</var> du <lb/>plum erit quadrati <var>.o.e.</var> cum duplo produ-<lb/>cti <var>.o.e.</var> in <var>.e.d.</var> & duplo producti <var>.o.e.</var> in <var>.o.h.</var> <lb/>Quòd Å¿i ex Å¿ingulis duabus Å¿ummis quadratorum demptum fuerit quadratum <var>.o.b.</var> <lb/>eadem producta & quadrata ipÅ¿ius <var>.o.e.</var> remanebunt, tanquam differentia duorum <lb/>quadratorum <var>.o.u.</var> et <var>.h.c</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0060-01" xlink:href="fig-0060-01a"> - <image file="0060-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0060-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="74"> - <head xml:id="head90" xml:space="preserve">THEOREMA <num value="74">LXXIIII</num>.</head> - <p> - <s xml:id="s638" xml:space="preserve">CVR Å¿umma duorum <reg norm="extremorum" type="context">extremorÅ©</reg> quatuor terminorum <reg norm="proportionalium" type="context">proportionaliÅ©</reg> arith-<lb/>meticè, æqualis eÅ¿t Å¿ummæ duorum mediorum, vbi nota hac in re neceſſa-<lb/>rium non eſſe proportionalitatem continuam exiÅ¿tere.</s> - </p> - <p> - <s xml:id="s639" xml:space="preserve">Exempli gratia, Å¿i darentur hi quatuor termini .20. 17. 9. 6. quorum proportio ea <lb/>dem eſſet primi ad Å¿ecundum quæ tertij ad quartum, Å¿umma primi cum quarto eſſet <lb/>26. <reg norm="tantaque" type="simple">tantaqÌ;</reg> Å¿ecundi cum tertio.</s> - </p> - <p> - <s xml:id="s640" xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, primus <reg norm="maiorque" type="simple">maiorqÌ;</reg> numerus Å¿ignificetur linea <var>.e.o.</var> Å¿ecun-<lb/>dus <var>.s.q.</var> tertius <var>.u.c.</var> quartus <var>.g.t.</var> differentia porrò inter <var>.e.o.</var> et <var>.s.q.</var> Å¿it <var>.i.o.</var> quæ æqualis <lb/>erit differentiæ <var>.r.c.</var> qua quartus à tertio Å¿uperatur ex hypotheÅ¿i. </s> - <s xml:id="s641" xml:space="preserve">Itaque aſſero Å¿um <lb/>mam <var>.e.o.</var> cum <var>.g.t.</var> nempe <var>.a.o.</var> æqualem eſſe Å¿ummÄ™ <var>.q.s.</var> et <var>.u.c.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.q.p</var>. </s> - <s xml:id="s642" xml:space="preserve">Nam in <var>.a.o.</var> - <pb o="49" rhead="THEOREM. ARIT." n="61" file="0061" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0061"/> - Secundus tertiusqÌue terminus reperiuntur, eÅ¿t <lb/> - <anchor type="figure" xlink:label="fig-0061-01a" xlink:href="fig-0061-01"/> - enim Å¿ecundus <var>.e.i.</var> tertius <var>.i.o.</var> et <var>.e.a.</var> quando-<lb/>quidem ex præſuppoÅ¿ito <var>.e.i.</var> æqualis eÅ¿t <var>.s.q.</var> et <lb/><var>i.o.</var> æqualis <var>.r.c.</var> et <var>.a.e.</var> cum Å¿it æqualis <var>.g.t.</var> cui <lb/>pariter æqualis eÅ¿t <var>.r.u.</var> ex quo <var>.a.e.</var> æqualis <lb/>eÅ¿t <var>.u.r</var>. </s> - <s xml:id="s643" xml:space="preserve">Itaque illud Å¿equitur <var>.a.o.</var> ipÅ¿i <var>.q.p.</var> <lb/>æqualem eſſe.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0061-01" xlink:href="fig-0061-01a"> - <image file="0061-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0061-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="75"> - <head xml:id="head91" xml:space="preserve">THEOREMA <num value="75">LXXV</num>.</head> - <p> - <s xml:id="s644" xml:space="preserve">CVR Å¿umma duorum terminorum extremorum imparium arithmeticæ pro-<lb/>portionalitatis Å¿emper duplo medij termini æqualis eÅ¿t.</s> - </p> - <p> - <s xml:id="s645" xml:space="preserve">Exempli gratia, Å¿unt hitres termini proportionalitatis arithmeticæ .20. 15. 10 <lb/>Å¿umma duorum extremorum erit .30. quæ duplum eÅ¿t medij termini .15.</s> - </p> - <p> - <s xml:id="s646" xml:space="preserve">Quod vt Å¿peculemur, tres termini, tribus lineis <var>.b.d</var>: <var>n.u.</var> et <var>.q.p.</var> <reg norm="Å¿ignificentur" type="context">Å¿ignificẽtur</reg>. </s> - <s xml:id="s647" xml:space="preserve">Di-<lb/>co nunc quòd Å¿umma <var>.b.d.</var> cum <var>.q.p.</var> nempe <var>.<lb/>h.d.</var> Å¿emper duplo <var>.n.u.</var> Å¿cilicet <var>.g.u.</var> æqualis <lb/> - <anchor type="figure" xlink:label="fig-0061-02a" xlink:href="fig-0061-02"/> - erit. </s> - <s xml:id="s648" xml:space="preserve">Tum differentia <var>.b.d.</var> ad <var>.n.u.</var> Å¿it <var>.c.d.</var> quæ <lb/>æqualis erit <var>.e.u.</var> differentiæ inter <var>n.u.</var> et <var>.q.p.</var> <lb/>patet enim in linea <var>.h.d</var>: <var>b.c.</var> æqualem eſſe <var>.n.<lb/>u.</var> Å¿ed <var>.n.u.</var> ex <var>.n.e.</var> componitur æquali <var>.q.p.</var> et <lb/>ex <var>.e.u.</var> æquali <var>.c.d.</var> cum <reg norm="itaque" type="simple">itaq;</reg> in <var>.h.d.</var> partem <var>.<lb/>h.b.</var> reperiamus æqualem <var>.n.e.</var> gratia <var>.q.p.</var> & <lb/>partem <var>.c.d.</var> æquale <var>m.e.u.</var> manifeÅ¿tum erit <lb/><var>h.d.</var> æqualem eſſe <var>.g.u</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0061-02" xlink:href="fig-0061-02a"> - <image file="0061-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0061-02"/> - </figure> - </div> - <head xml:id="head92" style="it" xml:space="preserve">BINA PROBLEMAT A EX DVOBVS PRAEDICTIS <lb/>THEOREMATIBVS DEPENDENTIA.</head> - <p> - <s xml:id="s649" xml:space="preserve">EX duobus prædictis theorematibus duo problemata oriuntur, <reg norm="quorum" type="context">quorÅ©</reg> primum <lb/>eÅ¿t. </s> - <s xml:id="s650" xml:space="preserve">Datis tribus quantitatibus cognitis, Å¿i quis quartam inuenire voluerit, <lb/>quæ eiuÅ¿modi Å¿it reÅ¿pectu tertiæ, qualis eÅ¿t Å¿ecunda reÅ¿pectu primæ, Å¿ecunda cum <lb/>tertia in Å¿ummam colligenda erit, ex qua detracta prima, Å¿upererit quarta.</s> - </p> - <p> - <s xml:id="s651" xml:space="preserve">Exempli gratia, cognitis tribus quantitatibus .20. 17. 9. Å¿i quartam inuenire vo <lb/>luerimus eiuÅ¿modi proportionem cum tertia arithmeticè Å¿eruantem, quam Å¿ecunda <lb/>cum prima, Å¿ecundam cum tertia in Å¿ummam colligemus, <reg norm="dabiturque" type="simple">dabiturqÌ;</reg> Å¿umma .26. ex <lb/>qua detracta prima quantitate, quarta relinquetur nempe .6. quod ex .74. theore-<lb/>mate dependet.</s> - </p> - <p> - <s xml:id="s652" xml:space="preserve">IdipÅ¿um tamen proueniret Å¿i quis ex tertio termino differentiam primi atque Å¿e-<lb/>cundi detraheret; </s> - <s xml:id="s653" xml:space="preserve">hæc tamen via non tam vniuerÅ¿alis eÅ¿tqu à m illa. </s> - <s xml:id="s654" xml:space="preserve">N Å¿i quartus ter <lb/>minus incognitus tertio maior eſſe deberet, dictam differentiam cum tertio termi-<lb/>mino in Å¿ummam colligere oporteret.</s> - </p> - <p> - <s xml:id="s655" xml:space="preserve">Alterum problema eÅ¿t, quòd inuentis duobus terminis, Å¿i tertius requiratur, Å¿e-<lb/>cundus duplicandus erit, ex qua Å¿umma detracto primo, Å¿tatim tertius proferetur, <lb/>quod problema ex præcedenti theoremate dependet.</s> - </p> - <pb o="50" rhead="IO. BAPT. BENED." n="62" file="0062" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0062"/> - <p> - <s xml:id="s656" xml:space="preserve">Progredi nihilominus etiam hac in re poſſemus per differentiam primi & Å¿ecun-<lb/>di termini, eam detrahendo aut in Å¿ummam cum Å¿ecunda colligendo, attamen prior <lb/>ratio magis latè patet, ideÅ¿t vniuerÅ¿alior eÅ¿t.</s> - </p> - </div> - <div type="math:theorem" level="3" n="76"> - <head xml:id="head93" xml:space="preserve">THEOREMA <num value="76">LXXVI</num>.</head> - <p> - <s xml:id="s657" xml:space="preserve">CVR Å¿i quis cupiat Å¿ecundum terminum inuenire, quatuor terminorum arith-<lb/>meticè proportionalis continuæ, quorum nobis duo extrema proponantur. <lb/></s> - <s xml:id="s658" xml:space="preserve">Rectè primum duplicabit <reg norm="coniungetque" type="simple">coniungetqÌ;</reg> vltimo termino, nempe quarto, ex qua Å¿um-<lb/>ma tertiam partem deÅ¿umet, quæ erit Å¿ecundus terminus quęſitus.</s> - </p> - <p> - <s xml:id="s659" xml:space="preserve">Exempli gratia, Å¿i horum quatuor terminorum .12. 9. 6. 3. duo nobis extrema <lb/>proponantur. </s> - <s xml:id="s660" xml:space="preserve">nempe .12. et .3. quorum Å¿ecundus inueniendus Å¿it, Å¿umpto quolibet <lb/>pro primo, Å¿it autem .3. primus numerus, quartus verò .12. </s> - <s xml:id="s661" xml:space="preserve">quare duplicato 3. vtpo <lb/>tè primo, & coniuncto .12. quarto, Å¿umma erit .18. cuius eÅ¿t tertia pars .6. Å¿ecundus <lb/>numerus Å¿cilicet Å¿umpto principio à minimo. </s> - <s xml:id="s662" xml:space="preserve">IdipÅ¿um euenit Å¿umpto principio à <lb/>maximo. </s> - <s xml:id="s663" xml:space="preserve">Nam Å¿i datur Å¿ecundus à minimo aut à maximo, illico tertius datur diffe-<lb/>rentia inter hunc & primum, Å¿ecundo coniuncta, aut ex eodem detracta.</s> - </p> - <p> - <s xml:id="s664" xml:space="preserve">Cuius ratio Å¿ic demonÅ¿tratur, quatuor termini quatuor lineis <var>.m.g</var>: <var>q.p</var>: <var>u.n</var>: <var>c.t.</var> <lb/>Å¿ignificentur, quorum <var>.m.g.</var> et <var>.c.t.</var> tantummodo cognoÅ¿cantur. </s> - <s xml:id="s665" xml:space="preserve"><reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.m.g.</var> primus ac <lb/>maior terminus: </s> - <s xml:id="s666" xml:space="preserve">k.g. verò Å¿it duplum primi <var>.m.g</var>: cui coniungatur <var>.b.k.</var> æqualis <var>.c.t.</var> <lb/>Dico tertiam partem <var>.b.g.</var> quæ Å¿umma totalis eÅ¿t, æqualem eſſe <var>.q.p</var>. </s> - <s xml:id="s667" xml:space="preserve">In primis enim <lb/>certi Å¿umus <var>.m.f.</var> in <var>.m.g.</var> reperiri æqualem <var>.q.p.</var> <reg norm="Å¿upereÅ¿tque" type="simple">Å¿upereÅ¿tqÌ;</reg> <var>.f.g.</var> differentia inter <var>.m.g.</var> <lb/>et <var>.q.p.</var> æqualis <var>.e.p.</var> differentiæ inter <var>.q.p.</var> et <var>.u.n.</var> & æqualis <var>.o.n.</var> differen-<lb/>tiæ inter <var>.u.n.</var> et <var>.c.t</var>: Å¿imul etiam in <var>.k.m.</var> habemus <var>.d.m.</var> æqualem <var>.m.f.</var> </s> - <s xml:id="s668" xml:space="preserve">quare etiam <var>.q.<lb/>p.</var> et <var>.k.d.</var> æqualem <var>.f.g.</var> nempe <var>.e.p.</var> aut <var>.o.n</var>: Hactenus in <var>.k.g.</var> reperimus duplum <var>.q.<lb/>p.</var> Å¿imul cum <var>.f.g.</var> et <var>.k.d.</var> æqualibus <var>.e.p.</var> et <var>.o.n.</var> & quia <var>.b.K.</var> æqualis <var>.c.t.</var> fuit coniuncta. <lb/></s> - <s xml:id="s669" xml:space="preserve">conÅ¿iderandum eÅ¿t an hætres quantitates <var>.f.g</var>: <var>K.d.</var> et <var>.b.K.</var> Å¿imul æquales Å¿int <var>.q.p.</var> <lb/>quod tamen per Å¿e manifeÅ¿tum eÅ¿t. </s> - <s xml:id="s670" xml:space="preserve">nam <var>.q.p.</var> Å¿uperat <var>.u.n.</var> per <var>.e.p.</var> et <var>.u.n.</var> ex-<lb/>cedit <var>.c.t.</var> per <var>.o.n.</var> æqualem <var>.e.p.</var> </s> - <s xml:id="s671" xml:space="preserve">quare <var>.q.p.</var> per duplum differentiÄ™ <var>.f.g.</var> Å¿uperat <var>.c.t.</var> ita <lb/>que <var>.f.g</var>: <var>k.d.</var> et <var>.K.b.</var> ipÅ¿i <var>.q.p.</var> Å¿unt <reg norm="ae- quales" type="simple">Ä™-<lb/>quales</reg>, ex quo Å¿equitur <var>.q.p.</var> <reg norm="tertiam" type="context">tertiã</reg> <lb/> - <anchor type="figure" xlink:label="fig-0062-01a" xlink:href="fig-0062-01"/> - partem eſſe <var>.b.g.</var> Hæc quæ hacte-<lb/>nus dicta fuerunt, in genere maio-<lb/>ris inæqualitatis probata fuerunt. <lb/></s> - <s xml:id="s672" xml:space="preserve">At in genere minoris, Å¿umpto or-<lb/>dinis principio à minimo termino <lb/>rum, duplicetur <var>.c.t.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> duplum <lb/>hoc <var>.K.t.</var> cui <var>.k.b.</var> æqualis <var>.m.g.</var> con-<lb/>iungatur, quæſumma Å¿it <var>.b.t</var>. </s> - <s xml:id="s673" xml:space="preserve">Di-<lb/>co <var>.u.n.</var> tertiam eſſe partem ipÅ¿ius. <lb/></s> - <s xml:id="s674" xml:space="preserve">Nam in primis in <var>.b.t.</var> datur termi <lb/>nus <var>.b.K.</var> æqualis vltimo <var>.m.g.</var> in <lb/>quo Å¿emel reperitur <var>.u.n.</var> vnà cum <lb/>duabus differentijs, nempe <var>.i.g.</var> in <lb/>ipÅ¿a autem <var>.b.t</var>: <var>u.n.</var> Å¿ignificetur pri <lb/>mo loco per <var>.r.K.</var> ex quo Å¿upererit <var>.b.r.</var> duabus differentijs prædictis æqualis, Å¿ed ex <lb/>præſuppoÅ¿ito <var>.u.n.</var> componitur ex <var>.o.u.</var> æquali <var>.c.t.</var> et <var>.o.n.</var> Ä™quali vni differentiæ. </s> - <s xml:id="s675" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> - <pb o="51" rhead="THEOREM. ARIT." n="63" file="0063" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0063"/> - cum in <var>.b.t.</var> præter <var>.r.K.</var> bis detur <var>.c.t.K.t.</var> et <var>.b.r.</var> duabus differentijs æquipol-<lb/>lens, illud efficitur <var>.u.n.</var> pariter ipÅ¿ius <var>.b.t.</var> eſſe tertiam partem, quod erat <reg norm="propoÅ¿itum" type="context">propoÅ¿itÅ©</reg>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0062-01" xlink:href="fig-0062-01a"> - <image file="0062-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0062-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="77"> - <head xml:id="head94" xml:space="preserve">THEOREMA <num value="77">LXXVII</num>.</head> - <p> - <s xml:id="s676" xml:space="preserve">CVR Å¿i quis velit Å¿ecundum quinque continuorum proportionalium termi-<lb/>num inuenire, Å¿olis extremis cognitis. </s> - <s xml:id="s677" xml:space="preserve">Rectè <reg norm="vltimum" type="context">vltimÅ©</reg> triplo primi coniunget, <lb/>ex qua Å¿umma quartam partem detraher, quæ erit Å¿ecundus terminus quæſitus. <lb/></s> - <s xml:id="s678" xml:space="preserve">Quod ipÅ¿um faciet qui inuenire vult Å¿ecundum terminum Å¿enarij Å¿eptenarij, octo-<lb/>narij aut alterius cuiuÅ¿cunque, creÅ¿cente tamen multiplicatione primi, <reg norm="vltimoque" type="simple">vltimoqÌ;</reg> <lb/>coniuncto.</s> - </p> - <p> - <s xml:id="s679" xml:space="preserve">Exempli gratia, dantur duo extremi termini, horum quinque numerorum .18. <lb/>16. 14. 12. 10. nempe .18. et .10. Å¿i .18. primus erit, hoc eÅ¿t, Å¿i à genere maioris inæ-<lb/>qualitatis progrediemur, triplicabimus terminum .18. <reg norm="dabunturque" type="simple">dabunturqÌ;</reg> .54. cui numero <lb/>coniuncto quinto termino .10. dabitur numerus .64. cuius quarta pars erit .16. vtpo <lb/>tè Å¿ecundus terminus gratia, aut Å¿ecundi Å¿ex terminorum, quadruplicandus eſſet pri <lb/>mus .18. deinde adiuncto vltimo, quinta pars Å¿ummæ eſſet Å¿ecundus terminus, <reg norm="atque" type="simple">atq;</reg> <lb/>ita deinceps.</s> - </p> - <p> - <s xml:id="s680" xml:space="preserve">Cuius Å¿peculationis gratia, dicti termini lineis <var>.z.h</var>: <var>f.s</var>: <var>u.p</var>: <var>e.g.</var> et <var>.r.x.</var> <reg norm="Å¿igniÅ¿icentur" type="context">Å¿igniÅ¿icẽtur</reg>. <lb/></s> - <s xml:id="s681" xml:space="preserve">In primis ex genere maioris inæqualitatis, triplicabimus <var>.z.h.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> triplum hoc <var>.k.<lb/>h.</var> <reg norm="cuiconiungatur" type="context">cuicõiungatur</reg> <var>.b.k.</var> Ä™qualis vltimo termino <var>.r.x</var>. </s> - <s xml:id="s682" xml:space="preserve">Dico <var>.f.s.</var> <reg norm="quartam" type="context">quartã</reg> partem eſſe Å¿um-<lb/>mÄ™ <var>.b.h</var>. </s> - <s xml:id="s683" xml:space="preserve">Nam in <var>.k.h.</var> Å¿ecundus terminus <var>.f.s.</var> ter cum tribus differentijs æqualibus <var>.n.h.</var> <lb/>reperitur. </s> - <s xml:id="s684" xml:space="preserve">Probandum nunc eÅ¿t tres has differentias <var>.n.h</var>: <var>a.c.</var> et <var>.d.k.</var> Å¿imul cum <var>.b.<lb/>K.</var> Ä™quales eſſe <var>.f.s.</var> <lb/> - <anchor type="figure" xlink:label="fig-0063-01a" xlink:href="fig-0063-01"/> - quod in <reg norm="dubium" type="context">dubiÅ©</reg> re <lb/>uocari <reg norm="non" type="context">nõ</reg> poteÅ¿t, <lb/>cum <var>.f.s.</var> Å¿uperet <var>.<lb/>r.x.</var> per <var>.o.s</var>: <var>t.p.</var> et <var>.<lb/>i.g</var>. </s> - <s xml:id="s685" xml:space="preserve">At in genere <lb/>minoris inæquali <lb/>tatis, triplum <var>.r.x.</var> <lb/>Å¿it <var>.x.a.</var> et <var>.a.b.</var> Å¿it <lb/>æqualis <var>.z.h.</var> & <reg norm="cum" type="context">cÅ©</reg> <lb/><var>z.h.</var> tribus <reg norm="differem" type="context">differẽ</reg> <lb/>tijs <var>.n.h</var>: <var>o.s</var>: <var>t.p.</var> Å¿u-<lb/>peret <var>.e.g.</var> quæ in <var>.<lb/>a.b.</var> Å¿int <var>.b.K</var>: <var>K.d</var>: <lb/><var>d.c.</var> ex quo <var>.a.c.</var> <lb/>æqualis erit <var>.e.g.</var> <lb/>et <var>.a.x.</var> cum <var>.b.c.</var> tripla <var>.e.g</var>. </s> - <s xml:id="s686" xml:space="preserve">Itaque tota Å¿umma <var>.b.x.</var> qua drupla erit <var>.e.g</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0063-01" xlink:href="fig-0063-01a"> - <image file="0063-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0063-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="78"> - <head xml:id="head95" xml:space="preserve">THEOREMA <num value="78">LXXVIII</num>.</head> - <p> - <s xml:id="s687" xml:space="preserve">QVantitates quæ fuerint inuicem in proportionalitate arithmetica proportio-<lb/>nales, permutan do quoque proportionales erunt.</s> - </p> - <pb o="52" rhead="IO. BAPT. BENED." n="64" file="0064" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0064"/> - <p> - <s xml:id="s688" xml:space="preserve">Sint exempli gratia .4. quantitates <var>.a.b</var>: <var>c.d</var>: <var>e.f</var>: et <var>.g.h</var>: inuicem proportionales in <lb/>proportionalitate arithmetica. </s> - <s xml:id="s689" xml:space="preserve">Hoc eÅ¿t vt quæ proportio (licet impropriè dicta) <lb/>eÅ¿t ipÅ¿ius <var>.a.b.</var> ad <var>.c.d.</var> <reg norm="eadem" type="context">eadẽ</reg> Å¿it ipÅ¿ius <var>.e.f.</var> ad <var>.g.h</var>. </s> - <s xml:id="s690" xml:space="preserve">Tunc permutando dico eandem pro <lb/>portionem fore ipÅ¿ius <var>.a.b.</var> ad <var>.e.f.</var> quæ ipÅ¿ius <var>.c.d.</var> ad <var>.g.h</var>.</s> - </p> - <p> - <s xml:id="s691" xml:space="preserve">Nam, ex hypotheÅ¿i, differentia qua <var>.a.b.</var> Å¿uperat <var>.c.d.</var> (quæ Å¿it <var>.m.b.</var>) æqualis eÅ¿t <lb/>differentiæ qua <var>.e.f.</var> Å¿uperat <var>.g.h.</var> (quæ Å¿it <var>.i.f.</var>) vnde <var>.a.m.</var> reÅ¿iduum ex <var>.a.b.</var> æquale erit <lb/><var>c.d.</var> & reÅ¿iduum <var>.e.i.</var> æquale <var>.g.h</var>. </s> - <s xml:id="s692" xml:space="preserve">Sit igitur exempli gratia <var>.c.d.</var> maior <var>.g.h.</var> per <var>.c.n.</var> <lb/>vnde <var>.n.d.</var> æqualis erit <var>.g.h.</var> </s> - <s xml:id="s693" xml:space="preserve">quare <var>.a.m.</var> maior erit <var>.e.i.</var> per <var>.a.K.</var> æqualem <var>.c.n.</var> ex com-<lb/>muni Å¿cientia. </s> - <s xml:id="s694" xml:space="preserve">Vnde <var>.K.m.</var> æqualis erit <var>.n.d.</var> hoc eÅ¿t ipÅ¿i <var>.g.h.</var> hoc eÅ¿t ipÅ¿i <var>e.i</var>. </s> - <s xml:id="s695" xml:space="preserve">Quare ex <lb/>communi conceptu <var>.b.K.</var> æqualis erit ipÅ¿i <var>.f.e.</var> Å¿ed <var>.n.d.</var> æqualis eÅ¿t <var>.g.h.</var> vt dictum eÅ¿t. <lb/></s> - <s xml:id="s696" xml:space="preserve">Cum ergo <var>.b.K.</var> æqualis Å¿it <var>.e.f.</var> et <var>.d.n.</var> ipÅ¿i <var>.g.h.</var> et <var>.a.b.</var> maior Å¿it ipÅ¿a <var>.K.b.</var> per <var>.a.K.</var> æqua-<lb/>lem ipÅ¿i <var>.c.n.</var> per quam <var>c.n</var>: <var>d.c.</var> maior eÅ¿t ipÅ¿a <var>.d.n.</var> Å¿equitur verum eſſe <reg norm="propoÅ¿itum" type="context">propoÅ¿itÅ©</reg> hoc <lb/>eÅ¿t, quod eadem proportio Å¿it ipÅ¿ius <var>.a.b.</var> ad <var>.e.f.</var> quæ <var>.c.d.</var> ad <var>.g.h.</var> arithmetice Å¿cilicet.</s> - </p> - <figure position="here"> - <image file="0064-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0064-01"/> - </figure> - </div> - <div type="math:theorem" level="3" n="79"> - <head xml:id="head96" xml:space="preserve">THEOREMA <num value="79">LXXIX</num>.</head> - <p> - <s xml:id="s697" xml:space="preserve">CVR prouenientia duorum numerorum diuidentium eiuÅ¿dem numeri diuiÅ¿i-<lb/>bilis, geometricè <reg norm="eandem" type="context">eandẽ</reg> inter Å¿e <reg norm="proportionem" type="context">proportionẽ</reg> Å¿eruant, <reg norm="quam" type="context">quã</reg> ipÅ¿imet <reg norm="diuidentes" type="context">diuidẽtes</reg>.</s> - </p> - <p> - <s xml:id="s698" xml:space="preserve">Exempli gratia Å¿i per Å¿enarium & octonarium numerus vigintiquatuor diuida-<lb/>tur, prouenientia erunt .4. et .3. eadem proportione, qua diuidentes.</s> - </p> - <p> - <s xml:id="s699" xml:space="preserve">Cuius eÅ¿t ratio numerus diuiÅ¿ibilis Å¿ignificetur rectangulis <var>.u.x.</var> et <var>.n.e.</var> diuidentes <lb/>autem Å¿int <var>.u.o.</var> et <var>.e.o.</var> </s> - <s xml:id="s700" xml:space="preserve">quare ex ijs, quæ .10. <lb/> - <anchor type="figure" xlink:label="fig-0064-02a" xlink:href="fig-0064-02"/> - theoremate dicta fuerunt <var>.u.x.</var> per <var>.u.o.</var> diui-<lb/>Å¿o dabit <var>.x.o.</var> & diuiÅ¿o <var>.n.e.</var> per <var>.e.o.</var> dabit <var>.o.<lb/>n</var>. </s> - <s xml:id="s701" xml:space="preserve">Dicimus itaque <reg norm="eandem" type="context">eandẽ</reg> eſſe <reg norm="proportionem" type="context">proportionẽ</reg> <lb/><var>o.x.</var> ad <var>.o.n.</var> quæ <var>.e.o.</var> ad <var>.o.u.</var> quod patet Å¿ub <lb/>Å¿criptam figuram conÅ¿iderantibus, in qua, <lb/>ex .15. Å¿exti aut .20. Å¿eptimi, eadem propor-<lb/>tio cernitur <var>.o.x.</var> ad <var>.o.n.</var> quæ <var>.o.e.</var> ad <var>.o.u</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0064-02" xlink:href="fig-0064-02a"> - <image file="0064-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0064-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="80"> - <head xml:id="head97" xml:space="preserve">THEOREMA <num value="80">LXXX</num>.</head> - <p> - <s xml:id="s702" xml:space="preserve">CVR quauis quantitate, tribus <lb/> - <anchor type="figure" xlink:label="fig-0064-03a" xlink:href="fig-0064-03"/> - aut quatuor aut etiam pro libi-<lb/>to pluribus diuidentibus numeris di-<lb/>uifa, prouenientia eandem prorÅ¿us <lb/>inter Å¿e proportionem Å¿eruabunt, <lb/>quam ipÅ¿i diuidentes habere compe <lb/>riuntur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0064-03" xlink:href="fig-0064-03a"> - <image file="0064-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0064-03"/> - </figure> - </div> - <p> - <s xml:id="s703" xml:space="preserve">Exempli gratia, proponitur nu-<lb/>merus .60. quinque numeris diuiden <lb/>dus, vtpotè .30. 20. 15. 12. 10. pro-<lb/>uenientia erunt .2. 3. 4. 5. 6. eadem - <pb o="53" rhead="THEOREM. ARITH." n="65" file="0065" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0065"/> - proportione diuidentium, quamuis ex aduerÅ¿o.</s> - </p> - <p> - <s xml:id="s704" xml:space="preserve">Cuius ratio ex .15. Å¿exti aut .20. Å¿eptimi dependet. </s> - <s xml:id="s705" xml:space="preserve">prout in Å¿ubÅ¿cripto ordine fa-<lb/>cillimè deprehendi poteÅ¿t.</s> - </p> - </div> - <div type="math:theorem" level="3" n="81"> - <head xml:id="head98" xml:space="preserve">THEOREMA <num value="81">LXXXI</num>.</head> - <p> - <s xml:id="s706" xml:space="preserve">CVR quantitate in tres continuas partes proportionales Å¿ecta, & per Å¿ingulas <lb/>ipÅ¿arum diuiÅ¿a, Å¿umma trium prouenientium quadrato medij prouenientis <lb/>æqualis eÅ¿t.</s> - </p> - <p> - <s xml:id="s707" xml:space="preserve">Exempli gratia, proponitur .14. diuidendus in tres continuas partes proportio-<lb/>nales, nempe .8. 4. 2. <reg norm="ipÅ¿eque" type="simple">ipÅ¿eqÌ;</reg> numerus .14. per Å¿ingulas diuiditur, ex quo tria proue-<lb/>nientia oriuntur, nempe ex prima parte .8. <reg norm="proueniens" type="context">proueniẽs</reg> erit .1. cum tribus quartis par <lb/>tibus ex Å¿ecunda .4. datur proueniens .3. cum dimidio vnius, & ex tertia .2. proue-<lb/>nient .7. integri, qui in Å¿ummam collecti dant .12. integros & vnam quartam par-<lb/>tem tantumdem, videlicet quantum quadratum prouenientis medij, nempe .3. <lb/>cum dimidio.</s> - </p> - <p> - <s xml:id="s708" xml:space="preserve">Cuius Å¿peculationis gratia, totalis numerus Å¿ignificetur linea <var>.n.c.</var> qui in tres par-<lb/>tes diuidatur <var>.n.a</var>: <var>a.e.</var> et <var>.e.c.</var> quæ Å¿int continuæ proportionales, quarum Å¿ingulis, <lb/>numerum <var>.n.c.</var> diuiÅ¿um eſſe cogitemus, proueniens autem ex diuiÅ¿ione <var>.n.c.</var> per <var>.n.<lb/>a.</var> Å¿it <var>.i.d.</var> quod verò prouenit ex diuiÅ¿ione <var>.n.c.</var> per <var>.a.e.</var> Å¿it <var>.d.u.</var> proueniens quoque ex <lb/>diuiÅ¿ione <var>.n.c.</var> per <var>.e.c.</var> Å¿it <var>.u.o.</var> quorum Å¿umma Å¿it <var>.i.o.</var> quæ aſſeritur eſſe numeri æqua-<lb/>lis numero quadrati <var>.d.u</var>. </s> - <s xml:id="s709" xml:space="preserve">Quod hac ratione probabo, producatur linea <var>.i.o.</var> donec <var>.<lb/>o.p.</var> æqualis Å¿it <var>.o.u.</var> <reg norm="erigaturque" type="simple">erigaturqÌ;</reg> <var>.o.m.</var> æqualis <var>.d.i.</var> perpendiculariter <var>.o.p.</var> in puncto <var>.o.</var> <lb/>quæ producatur donec <var>.o.q.</var> vnitati Å¿it æqualis, <reg norm="terminenturque" type="simple">terminenturqÌ;</reg> duo rectangula <var>.m.p.</var> <lb/>et <var>.q.i.</var> ex quo habebimus rectangulum, aut productum <var>.m.p.</var> æquale quadrato <var>.d.u.</var> <lb/>ex .16 Å¿exti aut .20. Å¿eptimi, quandoquidem tria prouenientia <var>.o.u</var>: <var>u.d.</var> et <var>.d.i.</var> ex <lb/>prÄ™cedenti theoremate Å¿unt inter Å¿e continua proportionalia, proportionalitate qua <lb/>partes <var>.n.c</var>. </s> - <s xml:id="s710" xml:space="preserve">Iam verò Å¿i probauero <var>.q.i.</var> productum, producto <var>.m.p.</var> æquale eſſe, pro-<lb/>poÅ¿itum quoque probatum erit. </s> - <s xml:id="s711" xml:space="preserve">Numerus enim producti <var>.q.i.</var> æqualis eÅ¿t numero. <lb/></s> - <s xml:id="s712" xml:space="preserve">Å¿ummæ <var>.i.o</var>. </s> - <s xml:id="s713" xml:space="preserve">Habemus autem ex definitione diuiÅ¿ionis ita Å¿e habere <var>.n.c.</var> ad <var>.i.d.</var> Å¿icut <var>.<lb/>n.a.</var> ad <var>.o.q</var>. </s> - <s xml:id="s714" xml:space="preserve">Itaque permutando Å¿ic Å¿e habebit <var>.n.c.</var> ad <var>.n.a.</var> Å¿icut <var>.d.i.</var> hoc eÅ¿t <var>.m.o.</var> ad <var>.<lb/>o.q.</var> Å¿ed Å¿icut Å¿e habet <var>.n.c.</var> ad <var>.n.a.</var> ita pariter Å¿e habet <var>.i.o.</var> ad <var>.o.u.</var> hoc eÅ¿t ad <var>.o.p</var>. </s> - <s xml:id="s715" xml:space="preserve">Ita-<lb/>que <var>.i.o.</var> ad <var>.o.p.</var> Å¿ic Å¿e habebit Å¿icut <var>.m.o.</var> ad <var>.o.q.</var> ex quo ex .15. Å¿exti aut .20. Å¿eptimi <var>.<lb/>q.i.</var> æqualis erit <var>.m.p.</var> & conÅ¿equenter quadrato <var>.d.u</var>. </s> - <s xml:id="s716" xml:space="preserve">Vt autem lector minori labo-<lb/>re cognoÅ¿cere queat <var>.i.o.</var> ad <var>.o.u.</var> Å¿ic Å¿e habere, vt <var>.n.c.</var> ad <var>.n.a.</var> Å¿ciendum eÅ¿t quòd, Å¿ic <lb/>Å¿e habet <var>.i.d.</var> ad <var>.d.u.</var> ut <var>.c.e.</var> ad <var>.e.a.</var> ex quo componendo Å¿ic Å¿e habebit <var>.i.u.</var> ad <var>.d.u.</var> Å¿i-<lb/>cut <var>.c.a.</var> ad <var>.a.e.</var> & permutando ita <var>.i.u.</var> <lb/> - <anchor type="figure" xlink:label="fig-0065-01a" xlink:href="fig-0065-01"/> - ad <var>.c.a.</var> vt <var>.d.u.</var> ad <var>.e.a.</var> Å¿ed cum ex <reg norm="præ- cedenti" type="context">præ-<lb/>cedẽti</reg> theoremate Å¿ic Å¿e habeat <var>.d.u.</var> <lb/>ad <var>.u.o.</var> Å¿icut <var>.e.a.</var> ad <var>.a.n.</var> permutando <lb/>Å¿ic Å¿e habebit <var>.d.u.</var> ad <var>.a.e.</var> Å¿icut <var>.u.o.</var> ad <lb/><var>a.n.</var> ex quo ex .11. quinti Å¿ic Å¿e habe-<lb/>bit <var>.i.u.</var> ad <var>.c.a.</var> prout <var>.o.u.</var> ad <var>.a.n.</var> per-<lb/>mutandoqÌue <var>.i.u.</var> ad <var>.u.o.</var> vt <var>.c.a.</var> ad <var>.a.n.</var> & componendo, ita <var>.i.o.</var> ad <var>.u.o.</var> Å¿icut <var>.c.n.</var> <lb/>ad <var>.a.n</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0065-01" xlink:href="fig-0065-01a"> - <image file="0065-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0065-01"/> - </figure> - </div> - <pb o="54" rhead="IO. BAPT. BENED." n="66" file="0066" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0066"/> - </div> - <div type="math:theorem" level="3" n="82"> - <head xml:id="head99" xml:space="preserve">THEOREMA <num value="82">LXXXII</num>.</head> - <p> - <s xml:id="s717" xml:space="preserve">CVR quantitate aliqua in quatuor partes <reg norm="continuas" type="context">cõtinuas</reg> proportionales Å¿ecta per-<lb/>qÌue Å¿ingulas diuiÅ¿a, Å¿umma quatuor prouenientium æqualis Å¿it producto Å¿e-<lb/>cundi in tertium.</s> - </p> - <p> - <s xml:id="s718" xml:space="preserve">Exempli gratia, Å¿i triginta in quatuor partes proportionales Å¿ecetur, hoc eÅ¿t. <lb/>16. 8. 4. 2. <reg norm="perque" type="simple">perqÌ;</reg> harum Å¿ingulas idem numerus .30. diuidatur, primum proueniens <lb/>erit .1. cum Å¿eptem octauis partibus. </s> - <s xml:id="s719" xml:space="preserve">Secundum .3. cum tribus quartis, tertium .7. <lb/>cum dimidio, quartum .15. integri, quorum Å¿umma erit .28. cum octaua parte, tan <lb/><reg norm="tumque" type="simple">tumqÌ;</reg> erit productum Å¿ecundi prouenientis in tertium.</s> - </p> - <p> - <s xml:id="s720" xml:space="preserve">Quod vt Å¿ciamus, quantitas <var>.n.c.</var> in partes continuas proportionales quatuor Å¿e-<lb/>cetur <var>.n.a</var>: <var>a.t</var>: <var>t.e.</var> et <var>.e.c.</var> <reg norm="rurÅ¿usque" type="simple">rurÅ¿usqÌ;</reg> per Å¿ingulas partes illa ipÅ¿a diuiÅ¿a, prouenientia <lb/>Å¿int <var>.i.d</var>: <var>d.x</var>: <var>x.u</var>: <var>u.o.</var> <reg norm="quorum" type="context">quorÅ©</reg> Å¿umma Å¿it <var>.i.o.</var> hanc <reg norm="Å¿ummam" type="context">Å¿ummã</reg> dicimus æqualem eſſe nume-<lb/>ro producti <var>.d.x.</var> in <var>.x.u</var>.</s> - </p> - <p> - <s xml:id="s721" xml:space="preserve">Quod hac ratione probo, cogito productam eſſe lineam <var>.i.o.</var> <reg norm="quousque" type="simple">quousqÌ;</reg> <var>.o.p.</var> æqua <lb/>lis Å¿it <var>.o.u.</var> <reg norm="erectamque" type="simple">erectamqÌ;</reg> <var>.m.o.</var> æqualem <var>.i.d.</var> perpendiculariter <var>.o.p.</var> & productam donec <var>.<lb/>o.q.</var> vnitati Å¿it æqualis. </s> - <s xml:id="s722" xml:space="preserve">Iam terminatis rectangulis <var>.m.p.</var> et <var>.i.q.</var> patebit ex .15. Å¿exti <lb/>aut .20. Å¿eptimi, productum <var>.m.p.</var> producto <var>.d.x.</var> in <var>.x.u.</var> æquale eſſe. </s> - <s xml:id="s723" xml:space="preserve">Ita quòd Å¿i pro-<lb/>bauero productum <var>.i.q.</var> producto <var>.m.p.</var> æquale eſſe, facile patebit propoÅ¿itum. </s> - <s xml:id="s724" xml:space="preserve">Cuius <lb/>gratia, Å¿equuti præcedentis theorematis ordinem, primum ex <reg norm="definitionem" type="context">definitionẽ</reg> diuiÅ¿ionis, <lb/>eadem proportio erit <var>.n.c.</var> ad <var>.i.d.</var> quæ <var>.n.a.</var> ad <var>.o.q.</var> ex quo permutando <var>.n.c.</var> ad <var>.n.a.</var> Å¿ic <lb/>Å¿e habebit vt <var>.i.d.</var> hoc eÅ¿t <var>.m.o.</var> ad <var>.o.q.</var> & Å¿i progrediamur eodem ordine, quo præ-<lb/>cedenti theoremate, Å¿umpto principio ab <var>.i.d.</var> et <var>.e.c.</var> verÅ¿us <var>.d.x.</var> et <var>.e.t.</var> gradatimqÌue <lb/>permutando ac coniungendo, inue-<lb/> - <anchor type="figure" xlink:label="fig-0066-01a" xlink:href="fig-0066-01"/> - niemus eandem proportionem eſſe <lb/><var>c.n.</var> ad <var>.n.a.</var> quæ <var>.i.o.</var> ad <var>.o.u.</var> nempe <var>.<lb/>o.p.</var> ex quo ex .11 quinti, ita Å¿e habe <lb/>bit <var>.i.o.</var> ad <var>.o.p.</var> vt <var>.m.o.</var> ad <var>.o.q.</var> </s> - <s xml:id="s725" xml:space="preserve">quare <lb/>ex .15. Å¿extiaut .20. Å¿eptimi <reg norm="produ- ctum" type="context">produ-<lb/>ctÅ©</reg> <var>.i.q.</var> erit producto<unsure/> <var>.m.p.</var> æquale, <lb/>ex quo etiam æquale erit producto <var>.<lb/>d.x.</var> in <var>.x.u</var>. </s> - <s xml:id="s726" xml:space="preserve">Idem ordo in qualibet <lb/>quantitate in quantaÅ¿uis partes diuiÅ¿a Å¿eruari poterit, cum huiuÅ¿modi <reg norm="Å¿cientia" type="context">Å¿ciẽtia</reg> in vni <lb/>uerÅ¿um pateat.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0066-01" xlink:href="fig-0066-01a"> - <image file="0066-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0066-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="83"> - <head xml:id="head100" xml:space="preserve">THEOREMA <num value="83">LXXXIII</num>.</head> - <p> - <s xml:id="s727" xml:space="preserve">CVR termini medij cubus, trium continuè proportionalium, Å¿emper producto <lb/>rectanguli compræhenÅ¿i à maximo & medio in minimo termino æqualis Å¿it.</s> - </p> - <p> - <s xml:id="s728" xml:space="preserve">Exempli gratia, datis his tribus terminis continuis proportionalibus .9. 6. 4. Å¿i <lb/>Å¿umpÅ¿erimus productum maximi in medium nempe .54. quod per <reg norm="minimum" type="context">minimÅ©</reg> .4. multi-<lb/>plicemus, dabitur numerus .216. cubo medij .6. æqualis.</s> - </p> - <p> - <s xml:id="s729" xml:space="preserve">In cuius gratiam tres numeri continui proportionales tribus lineis <var>.a.e.i.</var> <reg norm="Å¿ignifi- centur" type="context">Å¿ignifi-<lb/>cẽtur</reg>, cubus autem <var>.e.</var> Å¿ignificetur figura <var>.d.n.</var> <reg norm="productumque" type="simple">productumqÌ</reg> <var>.a.</var> in <var>.e.</var> Å¿it <var>.b.n.</var> ipÅ¿ius <reg norm="au- temmet" type="context">au-<lb/>tẽmet</reg> in <var>.i.</var> Å¿it <var>.p.o.</var> ita quod <var>.q.p.</var> aut <var>.b.o.</var> cum Å¿int <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> Å¿peciei, æqualis erit .a: et <var>.o.n.</var> - <pb o="55" rhead="THEOR. ARITH." n="67" file="0067" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0067"/> - æqualis .e: et <var>.q.n.</var> æqualis <var>.i</var>. </s> - <s xml:id="s730" xml:space="preserve">Nunc co-<lb/> - <anchor type="figure" xlink:label="fig-0067-01a" xlink:href="fig-0067-01"/> - gitemus abÅ¿olui corpus <var>.n.h.</var> ita ut <var>.b.<lb/>o.c.</var>Å¿it vnica recta linea, ex quo ex .25. <lb/>vndecimi proportio <var>.n.h.</var> ad <var>.n.k.</var> ea-<lb/>dem eÅ¿t quæ <var>.o.h.</var> ad <var>o.k.</var> Å¿ed Å¿ic Å¿e ha-<lb/>bet <var>.o.h.</var> ad <var>.o.k.</var> vt <var>.h.b.</var> ad <var>.b.k.</var> <lb/>ex prima Å¿exti aut .18. vel .19. Å¿e-<lb/>ptimi itaque <var>.n.h.</var> ad <var>.n.k.</var> ex .11. <lb/>quinti Å¿ic Å¿e habebit. vt <var>.h.b.</var> ad <var>.b.k.</var> <lb/>Å¿ed <var>.n.h.</var> ad <var>.n.d.</var> ex eiÅ¿dem Å¿ic Å¿e habet <lb/>ut <var>.h.u.</var> ad <var>.d.u.</var> et <var>.h.u.</var> ad <var>.u.d.</var> ita ut <var>.h.<lb/>b.</var> ad <var>.b.k.</var> ex præſuppoÅ¿ito. </s> - <s xml:id="s731" xml:space="preserve">Itaque ex <lb/>11. prædicta <var>.n.h.</var> ad <var>.n.k.</var> eadem erit <lb/>proportio quæ <var>.n.h.</var> ad <var>.n.d</var>. </s> - <s xml:id="s732" xml:space="preserve">Quare <lb/>ex .9. quinti <var>.n.k.</var> æqualis erit <var>.n.d.</var> <lb/>Quod erat propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0067-01" xlink:href="fig-0067-01a"> - <image file="0067-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0067-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="84"> - <head xml:id="head101" xml:space="preserve">THEOREMA <num value="84">LXXXIIII</num>.</head> - <p> - <s xml:id="s733" xml:space="preserve">CVR quadrato vnius quantitatis radice proportionalis, per Å¿ingulos tres termi <lb/>nos diuiÅ¿o, prouenientia, Å¿ingulis dictis terminis Å¿int æqualia.</s> - </p> - <p> - <s xml:id="s734" xml:space="preserve"><reg norm="Exempli" type="context">Exẽpli</reg> gratia, datis tribus terminis continuis proportionalibus .9. 6. 4. qua <lb/>dratum medij erit .36. quod per .9. diuiÅ¿um dabit .4: per .6: 6. per .4: 9.</s> - </p> - <p> - <s xml:id="s735" xml:space="preserve">Cuius gratia, Å¿int tres termini <reg norm="continui" type="context">cõtinui</reg> <reg norm="proportionales" type="simple">ê“portionales</reg> <var>.a.o</var>: <var>o.c.</var> et <var>.c.q.</var> <reg norm="quadratum" type="context">quadratÅ©</reg> <reg norm="autem" type="context">autẽ</reg> <lb/>medij Å¿it <var>.e.c</var>. </s> - <s xml:id="s736" xml:space="preserve">Iam Å¿i applicetur <reg norm="rectangulum" type="context">rectangulÅ©</reg> <var>.a.d.</var> æquale quadrato <var>.e.c.</var> ipÅ¿i <var>.a.o.</var> & re-<lb/>ctangulum <var>.q.p.</var> æquale eidem quadrato <var>.e.c.</var> ipÅ¿i <var>.c.q.</var> Å¿i quadratum <var>.e.c.</var> per <var>.a.o.</var> diui <lb/>Å¿erimus, proueniens erit <var>.o.d.</var> <reg norm="diuiÅ¿oque" type="simple">diuiÅ¿oqÌ</reg> per <var>.c.q.</var> proueniens erit <var>.c.p.</var> quod Å¿i per Å¿uam <lb/>radicem <var>.o.c.</var> diuidatur, proueniens erit <var>.o.</var> <lb/> - <anchor type="figure" xlink:label="fig-0067-02a" xlink:href="fig-0067-02"/> - e. quod Å¿ine dubio æquale eÅ¿t <var>.o.c.</var> Å¿ed dico <var>.<lb/>o.d.</var> æqualem eſſe <var>.c.q</var>. </s> - <s xml:id="s737" xml:space="preserve">Nam ex .16. Å¿exti aut <lb/>20. Å¿eptimi eadem eÅ¿t proportio <var>.a.o.</var> ad <var>.o.<lb/>c.</var> quÄ™ <var>.o.e.</var> ad <var>.o.d.</var> nempe <var>.o.c.</var> ad <var>.o.d.</var> itaque <lb/><var>o.d.</var> ex .9. quinti æqualis eÅ¿t <var>.c.q.</var> quandoqui <lb/>dem ex .11. Å¿ic Å¿e habet <var>.o.c.</var> ad <var>.o.d.</var> Å¿icut <var>.o.<lb/>c.</var> ad <var>.c.q</var>. </s> - <s xml:id="s738" xml:space="preserve">Applicatis ijÅ¿dem rationibus ipÅ¿i <var>.<lb/>p.c.</var> probabimus <var>.c.p.</var> æqualem eſſe <var>.a.o.</var> cum <lb/><var>o.c.</var> media Å¿it proportionalis, <reg norm="tam" type="context">tã</reg> inter <var>.c.p.</var> et <lb/><var>c.q.</var> quam inter <var>.a.o.</var> et <var>.c.q.</var> itaque <var>.c.p.</var> æqua-<lb/>lis eÅ¿t <var>.a.o</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0067-02" xlink:href="fig-0067-02a"> - <image file="0067-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0067-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="85"> - <head xml:id="head102" xml:space="preserve">THEOREMA <num value="85">LXXXV</num>.</head> - <p> - <s xml:id="s739" xml:space="preserve">CVR propoÅ¿itis tribus quantitatibus continuis proportionalibus proportione <lb/>aliarum duarum nobis datarum, multiplicata maiori poÅ¿tremarum dua-<lb/>rum in Å¿ummam mediæ cum minima trium primarum, productum æqua-<lb/>le Å¿it producto minoris duarum in Å¿ummam maximæ cum media trium.</s> - </p> - <p> - <s xml:id="s740" xml:space="preserve">Exempli gratia proponuntur quantitates .9. 6. 4. proportione numerorum pro- - <pb o="56" rhead="IO. BAPT. BENED." n="68" file="0068" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0068"/> - poÅ¿itorum .3. et .2. multiplicato .3. per .10. <reg norm="Å¿ummam" type="context">Å¿ummã</reg> .6. cum .4. dantur .30. quod pro-<lb/>ductum æquale erit producto .2. per .15. nempe per Å¿ummam 9. et .6.</s> - </p> - <p> - <s xml:id="s741" xml:space="preserve">Quod vt cognoÅ¿camus, tres quan <lb/> - <anchor type="figure" xlink:label="fig-0068-01a" xlink:href="fig-0068-01"/> - titates continuæ proportionales Å¿int <lb/><var>b.a.p.</var> proportione <var>.d.q.</var> productum <lb/>autem <var>.d.</var> in Å¿ummam <var>.a.</var> cum <var>.p.</var> Å¿it <var>.f.t.</var> <lb/>& productum <var>.q.</var> in Å¿ummam <var>.b.a.</var> Å¿it <var>.<lb/>K.h.</var> et <var>.K.n.</var> Å¿it æqualis <var>.b.</var> et <var>.n.o.</var> æqua <lb/>lis <var>.a.</var> & ita etiam <var>.o.u.</var> eidem <var>.a.</var> et <var>.u.t.</var> <lb/>æqualis <var>.p.</var> et <var>.h.o.</var> ipÅ¿i <var>.q.</var> et <var>.f.o.</var> ipÅ¿i <var>.d</var>. <lb/></s> - <s xml:id="s742" xml:space="preserve">quare ita Å¿e habebit <var>.K.n.</var> ad <var>.n.o.</var> Å¿icut <lb/><var>o.u.</var> ad <var>.u.t.</var> & componendo <var>.K.o.</var> ad <var>.<lb/>n.o.</var> vt <var>.o.t.</var> ad <var>.u.t.</var> & permutando <var>.K.<lb/>o.</var> ad <var>.o.t.</var> vt <var>.n.o.</var> hoc eÅ¿t <var>.o.u.</var> ad <var>.u.t.</var> & <lb/>pariter <var>.f.o.</var> ad <var>.o.h.</var> vt <var>.o.u.</var> ad <var>.u.t</var>. </s> - <s xml:id="s743" xml:space="preserve">Ita-<lb/>que Å¿icut <var>.k.o.</var> ad <var>.o.t.</var> ex quo ex .15. Å¿exti aut .20. Å¿eptimi <var>.K.h.</var> æqualis erit <var>.f.t</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0068-01" xlink:href="fig-0068-01a"> - <image file="0068-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0068-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="86"> - <head xml:id="head103" xml:space="preserve">THEOREMA <num value="86">LXXXVI</num>.</head> - <p> - <s xml:id="s744" xml:space="preserve">CVR multiplicatis Å¿ingulis tribus quantitatibus continuis proportionalibus in <lb/>reliquas duas, Å¿ex producta æqualia Å¿int producto dupli Å¿ummæ ipÅ¿arum trium <lb/>in mediam proportionalem.</s> - </p> - <p> - <s xml:id="s745" xml:space="preserve">Exempli gratia, proponuntur hitres termini continui proportionales .9. 6. 4. pro <lb/>ductum .9. in .6. erit .54. at .9. in .4. erit .36. et .6. in .9: 54. et .6. in .4: 24. et .4. in .9: 36. et <num value="4">.<lb/>4.</num> in .6: 24. quæ producta Å¿imul collecta efficiunt numerum .228 Å¿ed <reg norm="tantum" type="context">tantÅ©</reg> eÅ¿t pro-<lb/>ductum dupli Å¿ummæ trium terminorum in Å¿ecundum nempe .38 in .6.</s> - </p> - <p> - <s xml:id="s746" xml:space="preserve">Cuius <reg norm="intelligentiæ" type="context">intelligẽtiæ</reg> cauÅ¿a, tres termini <reg norm="continui" type="context">cõtinui</reg> proportionales Å¿ignificentur linea <var>.<lb/>b.e.</var> nempe <var>.b.d</var>: <var>d.c</var>: <var>c.e.</var> cuius duplum Å¿it <var>.u.e.</var> et <var>.b.f.</var> æqualis Å¿it <var>.b.d.</var> et <var>.f.n</var>: <var>d.c.</var> et <var>.n.u</var>: <lb/>c. e productum verò <var>.u.e.</var> in <var>.d.c.</var>Å¿it <var>.u.s.</var> cui dico æqualem eſſe Å¿ummam productorum <lb/>Å¿ingulorum trium terminorum in reliquos duos. </s> - <s xml:id="s747" xml:space="preserve">Quamobrem ducantur perpendi-<lb/>culares <var>.c.g</var>: <var>d.o</var>: <var>b.i</var>: <var>f.a.</var> et <var>.n.p.</var> inter <var>.u.e.</var> et <var>.q.s.</var> ex quo pro producto <var>.c.e.</var> in <var>.c.d.</var> ha-<lb/>bebimus rectangulum <var>.c.s.</var> & rectan-<lb/> - <anchor type="figure" xlink:label="fig-0068-02a" xlink:href="fig-0068-02"/> - gulum <var>.d.g.</var> pro producto <var>.c.e.</var> in <var>.d.b.</var> <lb/>ex .16. Å¿exti aut .20. Å¿eptimi itemqÌue <lb/>rectangulum <var>.q.n.</var> pro producto <var>.d.c.</var> <lb/>in <var>.c.e.</var> & rectangulum <var>.b.o.</var> ex <var>.d.c.</var> in <var>.<lb/>b.d.</var> & rectangulum <var>.b.a.</var> ex <var>.b.d.</var> in <var>.d.<lb/>c.</var> et <var>.p.f.</var> ex <var>.d.b.</var> in <var>.c.e.</var> ex .16. aut .20. <lb/>prÄ™dictas. </s> - <s xml:id="s748" xml:space="preserve">Quare Å¿ex producta æquantur inter Å¿e, <reg norm="replentque" type="simple">replentqÌ</reg> productum <var>.u.s.</var> ex quo <lb/>verum eÅ¿t propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0068-02" xlink:href="fig-0068-02a"> - <image file="0068-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0068-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="87"> - <head xml:id="head104" xml:space="preserve">THEOREMA <num value="87">LXXXVII</num>.</head> - <p> - <s xml:id="s749" xml:space="preserve">QVA ratione cognoÅ¿ci poſſit <reg norm="verum" type="context">verÅ©</reg> eſſe proportionem Å¿ummæ quatuor quan-<lb/>titatum continuarum proportionalium ad Å¿ummam Å¿ecundæ & tertiæ, ean-<lb/>dem eſſe, quæ Å¿ummæ primæ & tertiæ ad Å¿ecundam Å¿implicem.</s> - </p> - <p> - <s xml:id="s750" xml:space="preserve">Exempli gratia, Å¿i inue nirentur hæ quatuor quantitates continuæ proportiona-<lb/>es .16. 8. 4. 2. earum Å¿umma erit .30. Å¿unima verò Å¿ecundæ & tertiæ .12. tum Å¿umma - <pb o="57" rhead="THEOREM. ARITH." n="69" file="0069" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0069"/> - primæ cum tertia .20. ex quo Å¿icÅ¿e habet .20. ad .8. nempe ad Å¿ecundam, vt .30. <lb/>ad .12.</s> - </p> - <p> - <s xml:id="s751" xml:space="preserve">Quod vt Å¿ciamus, quatuor prædictæ quantitates Å¿ignificentur linea <var>.a.e.i.o.</var> pro-<lb/>babo ita Å¿e habere <var>.a.e.i.o.</var> ad <var>.e.i.</var> vt <var>.a.i.</var> ad <var>.e</var>. </s> - <s xml:id="s752" xml:space="preserve">Nam cum Å¿ic Å¿e habeat <var>.a.</var> ad <var>.e.</var> ut <var>.e.</var> <lb/>ad <var>.i.</var> & vt <var>.i.</var> ad .o: ex æqualitate proportionum vel permutando ita Å¿e habebit <var>.a.</var> ad <var>.i.</var> <lb/>vt <var>.e.</var> ad <var>.o.</var> & è conuerÅ¿o ita <var>.o.</var> ad <var>.e.</var> vt <var>.i.</var> ad <var>.a.</var> & <reg norm="componendo" type="context">cõponendo</reg> ita <var>.o.e.</var> ad e. vt <var>.i.a.</var> ad <var>.a.</var> <lb/><reg norm="permutandoque" type="simple">permutandoqÌ</reg> <var>.o.e.</var> ad <var>.i.a.</var> vt <var>.e.</var> ad <var>.a.</var> nempe <var>.i.</var> ad <var>.e.</var> & componendo ita <var>.o.i.e.a.</var> ad <var>.<lb/>i.a.</var> vt <var>.i.e.</var> ad <var>.e.</var> & permutando ita <var>.o.i.e.a.</var> ad <var>.i.e.</var> vt <var>.i.a.</var> ad <var>.e.</var> quod erat propoÅ¿itum. <lb/></s> - <s xml:id="s753" xml:space="preserve">Ex quo patet error antiquorum quiidipÅ¿um, accidere arbitrati Å¿unt in quantitatibus <lb/>diÅ¿cretæ proportionalitatis, quod tamen falÅ¿um eÅ¿t.</s> - </p> - <p> - <s xml:id="s754" xml:space="preserve">Exempli gratia, Å¿i proponantur .12. 6. 4. 2. proportio .12. ad .6. eadem eÅ¿t quæ .4. <lb/>ad .2. </s> - <s xml:id="s755" xml:space="preserve">Sed à proportione .6. ad .4. frangitur, cum non Å¿it eadem quæ .12. ad .6. harum <lb/>autem Å¿umma erit .24. & Å¿umma Å¿ecundæ cum tertia .10. Å¿ed primæ cum tertia erit <lb/>16. ex quo .16. ad .6. non Å¿ic Å¿e habebit vt .24. ad .10. <lb/>At in Å¿peculatione quatuor quantitatum <var>.a.</var> <lb/> - <anchor type="figure" xlink:label="fig-0069-01a" xlink:href="fig-0069-01"/> - <var>e.i.o.</var> Å¿i proportio <var>.e.</var> ad <var>.i.</var> non eſſet eadem <lb/>quæ <var>.a.</var> ad <var>.e.</var> minimè licuiſſet dicere ita Å¿e <lb/>habere <var>.i.</var> ad <var>.e.</var> vt <var>.e.</var> ad <var>.a</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0069-01" xlink:href="fig-0069-01a"> - <image file="0069-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0069-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="88"> - <head xml:id="head105" xml:space="preserve">THEOREMA <num value="88">LXXXVIII</num>.</head> - <p> - <s xml:id="s756" xml:space="preserve">CVR extribus quantitatibus quibuÅ¿libet, productum duarum in tertiam, vna <lb/>Å¿emper <reg norm="eademque" type="simple">eademqÌ;</reg> Å¿it quantitas.</s> - </p> - <p> - <s xml:id="s757" xml:space="preserve">Exempli gratia, proponuntur .15. 8. 2. Å¿i multiplicauerimus .15. per .8. tum produ <lb/>ctum per .2. tantum erit quantum Å¿i quis multiplicaret .8. per .2. & hoc per .15. et .15. <lb/>per .2. <reg norm="rurÅ¿usque" type="simple">rurÅ¿usqÌ;</reg> per .8.</s> - </p> - <p> - <s xml:id="s758" xml:space="preserve">Quod ut pateat, tres quantitates tri-<lb/> - <anchor type="figure" xlink:label="fig-0069-02a" xlink:href="fig-0069-02"/> - bus lineis Å¿ignificentur <var>.m.f</var>: a. et <var>.o</var>. </s> - <s xml:id="s759" xml:space="preserve">Dico <lb/>productum <var>.m.f.</var> in <var>.a.</var> multiplicatum. <lb/></s> - <s xml:id="s760" xml:space="preserve">per <var>.o.</var> æquale eſſe producto <var>.a.</var> in <var>.o.</var> mul-<lb/>tiplicato per <var>.m.f.</var> aut producto <var>.m.f.</var> in <var>.<lb/>o.</var> multiplicato per <var>.a</var>. </s> - <s xml:id="s761" xml:space="preserve">Sit enim corpus <var>.d.<lb/>u.</var> <reg norm="rectangulum" type="context">rectãgulum</reg>, cuius latus <var>.n.u.</var> Å¿it æquale <lb/><var>m.f.</var> et <var>.u.t</var>: a: et <var>.u.c</var>: o. patebit manifeÅ¿tè <lb/><var>n.t.</var> eſſe productum <var>.m.f.</var> in <var>.a.</var> quod <var>.n.t.</var> <lb/>multiplicatum in <var>.u.c.</var> æquali <var>.o.</var> producit <lb/>corpus <var>.d.u.</var> Å¿ed idipÅ¿um corpus <var>.d.u.</var> ex <lb/>multiplicatione producti <var>.c.t.</var> in latus <var>.n.<lb/>u.</var> æquale <var>.m.f.</var> oritur, & idipÅ¿um <var>.d.u.</var> ex <lb/>multiplicatione <var>.n.c.</var> in latus <var>.u.t.</var> æquale <var>.a.</var> profertur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0069-02" xlink:href="fig-0069-02a"> - <image file="0069-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0069-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="89"> - <head xml:id="head106" xml:space="preserve">THEOREMA <num value="89">LXXXIX</num>.</head> - <p> - <s xml:id="s762" xml:space="preserve">CVR quarumcunque quatuor quantitatum, Å¿i prima in Å¿ecundam multiplice-<lb/>tur & hoc productum in tertiam, <reg norm="rurÅ¿usque" type="simple">rurÅ¿usqÌ</reg> hoc alterum in quartam, vltimum <lb/>productum æquale Å¿it producto producti Å¿ecundæ in tertiam, in productum primæ <lb/>in quartam.</s> - </p> - <pb o="58" rhead="IO. BAPT. BENED." n="70" file="0070" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0070"/> - <p> - <s xml:id="s763" xml:space="preserve">Exempli gratia, caÅ¿u Å¿eÅ¿e offerunt hi quatuor numeri .8. 5. 3. 2. multiplicato .8. <lb/>per .5. & hoc .40. per .3. rurÅ¿us hoc .120. per .2. vltimum productum eſſet .240. æqua <lb/>le producto .15. (quod ex .5. in .3. oritur) in productum .16. quod ex .8. in .2. pro-<lb/>fertur.</s> - </p> - <p> - <s xml:id="s764" xml:space="preserve">Cuius Å¿peculationis gratia, cogitemus quatuor numeros quatuor lineis <var>.a.e.i.o.</var> <lb/>Å¿ignifi cari, productum autem <var>.e.</var> in <var>.i.</var> eſſe <var>.m.f.</var> et <var>.r.s.</var> Å¿imiliter & productum <var>.a.</var> in <var>.o.</var> eÅ¿-<lb/>Å¿e <var>.m.z</var>: et <var>.z.f.</var> productum eſſe <var>.m.f.</var> in <var>.m.z.</var> cui productum <var>.a.</var> in <var>.e.</var> multiplicatum per <lb/>i. & hoc tandem per <var>.o.</var> æquari debet.</s> - </p> - <p> - <s xml:id="s765" xml:space="preserve">Sit itaque <var>.u.y.</var> productum <var>.a.</var> in <var>.e.</var> quod <var>.u.y.</var> per <var>.i.</var> multiplicatum proferat <var>.u.s.</var> <lb/>hocqÌue <var>.u.s.</var> multiplicatum per <var>.o</var>. </s> - <s xml:id="s766" xml:space="preserve">Dico quod dabit numerum æqualem numero <var>.f.z.</var> <lb/>Quamobrem <var>.r.s.</var> aut <var>.m.f.</var> quod idem eÅ¿t, in figura præcedentis theore matis Å¿igni-<lb/>ficetur linea <var>.n.u.</var> & linea <var>.r.u.</var> hu-<lb/>ius, nempe <var>.a.</var> Å¿ignificetur per <var>.u.t.</var> <lb/> - <anchor type="figure" xlink:label="fig-0070-01a" xlink:href="fig-0070-01"/> - præcedentis, ex quo numerus pro <lb/>ducti <var>.u.s.</var> præſentis, in præcedenti <lb/>Å¿ignificabitur producto <var>.n.t.</var> quod <lb/><reg norm="productum" type="simple context">ê“ductÅ©</reg> <var>.u.s.</var> <reg norm="prÄ™sens" type="context">prÄ™sẽs</reg> <reg norm="per" type="simple">ê‘</reg> <reg norm="præsens" type="context">præsẽs</reg> <var>.o.</var> mulÂ<lb/>tiplicatum, quod erat in præceden <lb/>ti <var>.u.c.</var> Å¿ignificabitur per <var>.d.u.</var> præce <lb/>dentis, quod non modo ex multi-<lb/>plicatione <var>.n.t.</var> præcedentis, nempe <var>.u.s.</var> præſentis. in <var>.u.c.</var> præcedentis æquali <var>.o.</var> præ-<lb/>Å¿entis oritur, Å¿ed etiam ex <var>.c.t.</var> præcedentis æquali <var>.m.z.</var> præſentis in <var>.n.u.</var> præceden <lb/>tis æquali <var>.m.f.</var> præſentis. </s> - <s xml:id="s767" xml:space="preserve">Itaque verum eÅ¿t propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0070-01" xlink:href="fig-0070-01a"> - <image file="0070-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0070-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="90"> - <head xml:id="head107" xml:space="preserve">THEOREMA <num value="90">XC</num>.</head> - <p> - <s xml:id="s768" xml:space="preserve">CVR quibuÅ¿libet & quantiÅ¿uis numeris in Å¿ummam collectis, Å¿i ab vnitate in Å¿e-<lb/>cunda Å¿pecie progreſſionis arithmeticÄ™ imparium numerorum progreſſi fue-<lb/>rimus, eiuÅ¿modi Å¿umma Å¿emper eÅ¿t quadratus numerus.</s> - </p> - <p> - <s xml:id="s769" xml:space="preserve">Exempli gratia, Å¿i horum quatuor diÅ¿parium numerorum <reg norm="Å¿ummam" type="context">Å¿ummã</reg>, in dicta pro-<lb/>greſſione arithmetica quis Å¿umat, principio ab vnitate Å¿umpto, nempe .1. 3. 5. 7. Å¿um-<lb/>ma erit .16. numerus quadratus inquam. </s> - <s xml:id="s770" xml:space="preserve">Idem de cæteris.</s> - </p> - <p> - <s xml:id="s771" xml:space="preserve">Quamobrem animaduertendum eÅ¿t, vnitatem, tam Å¿umi pro Å¿ui ipÅ¿ius radicem, <lb/>quam pro quadrato, cubo, cenÅ¿o cenÅ¿i, primo relato, & alia quauis dignitate. <lb/></s> - <s xml:id="s772" xml:space="preserve">Nunc autem pro quadrato Å¿umamus per <var>.o.</var> Å¿ignificato, <reg norm="cogitemusque" type="simple">cogitemusqÌ</reg> quadratum <var>.o.</var> <lb/>includi quadrato vnitatem Å¿equenti, quod, vt patet, eÅ¿t quatuor vnitatum, ac pro-<lb/>priè primum quadratum numerorum, ex quo etiam nomen accepit, vnde ex Å¿imi-<lb/>litudine quam cætera quadrata cum hoc primo retinent, ex quaternario denomina-<lb/>tionem acceperunt. </s> - <s xml:id="s773" xml:space="preserve"><reg norm="Hocitaque" type="simple">Hocitaq;</reg> Å¿it <var>.o.u.c.e.</var> ita ex communi Å¿cientia quadrato <var>.o.</var> iun-<lb/>gitur gnomon <var>.e.c.u.</var> conÅ¿tans tribus vnitatibus, quare primus gnomon, numero im-<lb/>pari conÅ¿tat. </s> - <s xml:id="s774" xml:space="preserve">Scimus etiam ex additione numeri binarij ad imparem, numeris di-<lb/>Å¿paribus Å¿ummam excreÅ¿cere, cum propius accedere <reg norm="quam" type="context">quã</reg> binario nequeant, ex quo <lb/>medio binario, Å¿ibi inuicem Å¿uccedunt. </s> - <s xml:id="s775" xml:space="preserve">Dico igitur quòd quinario ternarium Å¿ub <lb/>Å¿equente, coniuncto quadrato <var>.o.u.c.e.</var> profertur quadratum, quod in numeris, bi-<lb/>narij quadratum Å¿equitur, <reg norm="eritque" type="simple">eritqÌ;</reg> ternarij, <reg norm="quodque" type="simple">quodqÌ;</reg> Å¿ignificetur per <var>.o.f.</var> patet enim pri <lb/>mo non differre ab <var>.o.c.</var> præter quam gnomone <var>.b.f.d.</var> qui coniungitur quadrato <var>.o.<lb/>c.</var> quique duabus vnitatibus maior eÅ¿t <var>.e.c.u</var>. </s> - <s xml:id="s776" xml:space="preserve"><reg norm="Iam" type="context">Iã</reg> Å¿cimus gnomonem <var>.e.o.u.</var> æqualem - <pb o="59" rhead="THEOREM. ARIT." n="71" file="0071" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0071"/> - eſſe gnomoni <var>.e.c.u.</var> <reg norm="itemque" type="simple">itemqÌ;</reg> gnomonem <var>.b.f.d.</var> æqualem gnomoni <var>.b.o.d.</var> at hic gno-<lb/>mon <var>.b.o.d.</var> ex præſuppoÅ¿ito, maior eÅ¿t gnomone <var>.e.o.u.</var> duabus vnitatibus <var>.b.</var> et <var>.d.</var> <lb/>Itaque etiam gnomon <var>.b.f.d.</var> duabus vnitatibus gnomonem <var>.e.c.u.</var> Å¿uperabit. </s> - <s xml:id="s777" xml:space="preserve">Qua-<lb/>re <var>.b.f.d.</var> erit impar immediatè Å¿equens ternarium, qui coniunctus quadrato <var>.o.c.</var> <lb/>quadratum Å¿ubÅ¿equens componet. </s> - <s xml:id="s778" xml:space="preserve">Eadem ratione probabitur de quadrato <var>.o.n.</var> Å¿e <lb/>quenti <var>.o.f.</var> & gnomone <var>.i.n.a.</var> cum hic ordo Å¿peculationis Å¿it vniuerÅ¿alis. </s> - <s xml:id="s779" xml:space="preserve">In <lb/>quo cernitur quemlibet gnomonem Å¿ibi <reg norm="contiguum" type="context">contiguÅ©</reg> inferiorem Å¿emper duabus vni-<lb/>tat ibus excedere, cumque quadrata non niÅ¿i gnomonibus Å¿ibi inuicem Å¿uccedant. <lb/></s> - <s xml:id="s780" xml:space="preserve">Sed <reg norm="cum" type="context">cÅ©</reg> primus <var>.e.c.u.</var> diÅ¿par fuerit, <reg norm="proculdubio" type="simple">ê“culdubio</reg> <reg norm="etiam" type="context">etiã</reg> <reg norm="neceſſarioque" type="simple">neceſſarioqÌ;</reg> cæteri diÅ¿pares <reg norm="erunt" type="context">erÅ©t</reg>. <lb/></s> - <s xml:id="s781" xml:space="preserve">Ex qua Å¿peculatione, oritur regula ab antiquis tradita <lb/>inueniendi vltimi numeri diÅ¿paris <reg norm="concurrentis" type="context">cõcurrentis</reg> ad <reg norm="compo ſitionem" type="context">cõpoÂ<lb/> - <anchor type="figure" xlink:label="fig-0071-01a" xlink:href="fig-0071-01"/> - Å¿itionem</reg> alicuius quadrati. </s> - <s xml:id="s782" xml:space="preserve">Vt Å¿i quis Å¿eire deÅ¿ideret nu-<lb/>merum vltimum diÅ¿parem, quo mediante quadratum <var>.<lb/>o.n.</var> conÅ¿titutum fuit, quod aliud non eÅ¿t quam Å¿cire <lb/>quantus Å¿it numerus vltimi gnomonis <var>.i.n.a.</var> æqualis gno <lb/>moni <var>.i.o.a</var>. </s> - <s xml:id="s783" xml:space="preserve">Itaque vt Å¿ciamus hunc gnomonem <var>.i.o.a.</var> <lb/>patet duplicandam eſſe radicem <var>.o.e.b.i.</var> <reg norm="dabiturque" type="simple punctuation">dabiturqÌ,</reg> <var>.o.e.<lb/>b.i.</var> et <var>.o.u.d.a.</var> vbi bis reperitur <var>.o.</var> nos autem tantummo <lb/>do quærimus Å¿cire gnomonem .i.b.e.o.u.d.a. </s> - <s xml:id="s784" xml:space="preserve">Itaque <lb/>minor eÅ¿t vnitate duplo radicis, cum unitas <var>.o.</var> bis repe-<lb/>tatur, quæ tamen in gnomone Å¿emel tantum Å¿umebatur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0071-01" xlink:href="fig-0071-01a"> - <image file="0071-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0071-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="91"> - <head xml:id="head108" xml:space="preserve">THEOREMA <num value="91">XCI</num>.</head> - <p> - <s xml:id="s785" xml:space="preserve">CVR Å¿umma quadratorum, quorum radices Å¿unt in proportione Å¿eÅ¿quitertia <lb/>nempe .4. ad .3. quadrata Å¿it.</s> - </p> - <p> - <s xml:id="s786" xml:space="preserve">Exempli gratia, Å¿umemus quadratum .3. Å¿cilicet 9. quod in Å¿ummam cum qua-<lb/>drato .4. colligemus, nempè .16. <reg norm="eritque" type="simple">eritqÌ;</reg> quadratum .25. & ita quadratum .6. hoc eÅ¿t <num value="36">.<lb/>36.</num> collectum cum quadrato .8. nempè .64. efficiet quadratum .100. ita etiam qua-<lb/>dratum .9. hoceÅ¿t .81. coniunctum quadrato .12. nempè .144. producet quadra-<lb/>tum .225.</s> - </p> - <p> - <s xml:id="s787" xml:space="preserve">In cuius gratiam Å¿int duo quadrata Å¿ubÅ¿cripta <var>.q.o.</var> et <var>.q.a.</var> quorum radices Å¿int <var>.q.</var> <lb/> - <anchor type="figure" xlink:label="fig-0071-02a" xlink:href="fig-0071-02"/> - g. et <var>.q.p.</var> hoc eÅ¿t <var>.q.g.</var> quatuor vnitatum, et <var>.q.<lb/>p.</var> trium, ex quo <var>.q.a.</var> erit .16. vnitatum et <var>.q.o.</var> <lb/>nouem. </s> - <s xml:id="s788" xml:space="preserve">Ad hæc cogitemus applicari quadra-<lb/>to <var>.q.a.</var> gnomonem <var>.f.s.h.</var> tam amplum Å¿iue la-<lb/>tum <reg norm="quam" type="context">quã</reg> gnomon <var>.b.a.g.</var> nempè vt <var>.h.</var> Å¿it æqua <lb/>lis .g: g. verò differentia Å¿it qua <var>.q.g.</var> maior eÅ¿t <var>.<lb/>q.p.</var> <reg norm="huncque" type="simple">huncqÌ;</reg> gnomonem <var>.f.s.h.</var> dico Ä™qualem eÅ¿ <lb/>Å¿e quadrato <var>.q.o.</var> nam ex preÅ¿uppoÅ¿ito <var>.g.</var> terra <lb/>dicem <var>.q.p.</var> ingreditur, & quater <var>.q.g.</var> ex quo, <lb/>tres partes <var>.q.k.p.</var> inter Å¿e æquales Å¿unt vnde <lb/>etiam quadratum <var>.q.o.</var> nouem partibus Å¿uper-<lb/>ficialibus quadratis conÅ¿tabit, quarum Å¿ingula <lb/>rum radix æqualis erit <var>.g.</var> cumque præcedenti <lb/>theoremate didicerimus quemlibet gnomo-<lb/>nem quadrati immediatè Å¿equentis æquÄ™ amplitudinis cum gnomone præcedentis, - <pb o="60" rhead="IO. BAPT. BENED." n="72" file="0072" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0072"/> - per duab. vnitatibus Å¿uperficialibus creÅ¿cere, <reg norm="quarum" type="context">quarÅ©</reg> <reg norm="Å¿ingularum" type="context">Å¿ingularÅ©</reg> radix æqualis eÅ¿t <var>.g.</var> ne <lb/>ceſſariò Å¿equitur gnomonem <var>.b.a.g.</var> duabus partibus aut vnitatibus gnomonem <var>.d.<lb/>o.p.</var> Å¿uperare, ita vt gnomon <var>.b.a.g.</var> Å¿eptem vnitatibus, aut partibus Å¿uperficialibus <lb/>quadratis conÅ¿tet. </s> - <s xml:id="s789" xml:space="preserve">Quare eadem ratione gnomon <var>.f.s.h.</var> conÅ¿tabit nouem Å¿imilibus. <lb/></s> - <s xml:id="s790" xml:space="preserve">Itaque æqualis erit quadrato <var>.q.o</var>. </s> - <s xml:id="s791" xml:space="preserve">Quamobrem verum eÅ¿t, quòd quadrato <var>.q.o.</var> <lb/>coniuncto quadrato <var>.q.a.</var> proueniet quadratum <var>.q.s.</var> cuius radix ita differet à <var>.q.g.</var> vt <var>.<lb/>q.g.</var> à <var>.q.p</var>: ex quo tres radices arithmeticè inter Å¿e continuæ proportionales erunt. <lb/></s> - <s xml:id="s792" xml:space="preserve">IdipÅ¿um dico Å¿i <var>.q.p.</var> fuerit .6. et <var>.q.g</var>: 8: </s> - <s xml:id="s793" xml:space="preserve">tunc enim Å¿ingulæ partes <var>.q.k.p.g.h.</var> æquipol <lb/>lebunt duabus vnitatibus, quæ cogitabuntur <lb/> - <anchor type="figure" xlink:label="fig-0072-01a" xlink:href="fig-0072-01"/> - in Å¿ummam collectæ, ut cum patribus <var>.q.k.p.<lb/>g.h.</var> integris contemplari liceat. </s> - <s xml:id="s794" xml:space="preserve">Idem acci-<lb/>det fi <var>.q.p.</var> erit .9. et <var>.q.g.</var> 12. fingulæ enim par-<lb/>tes <var>.q.K.p.g.h.</var> tripartitæ erunt. </s> - <s xml:id="s795" xml:space="preserve">Idcircò dixi <lb/>gnomonem <var>.f.s.h.</var> tam amplum cogitari de-<lb/>bere, quam gnomon <var>.b.a.g.</var> nempè ut <var>.h.</var> æqua <lb/>lis Å¿it <var>.g</var>. </s> - <s xml:id="s796" xml:space="preserve">Idem occurret Å¿i <var>.q.g.</var> erit .12. et <var>.q.p.</var> <lb/>quinque, quod cum fuerit patebitex præce-<lb/>dentis theorematis Å¿peculatione, gnomonem <lb/><var>f.s.h</var>: 25. vnitatibus conÅ¿tare, cogitatum am-<lb/>plitudinis Å¿implicis vnitatis denominatæ in <var>.q.<lb/>p.</var> aut <var>.q.g.</var> non amplitudinis gnomonis <var>.b.a.g.</var> <lb/>qui Å¿eptem vnitatibus latus eſſet. </s> - <s xml:id="s797" xml:space="preserve">Cum igitur <var>.<lb/>q.p.</var> quinque vnitatibus linearibus conÅ¿tet Å¿cimus <var>.q.o</var>: 25. Å¿uperficialibus conÅ¿tare, <lb/>collecto itaque in Å¿ummam quadrato <var>.q.o.</var> cum quadrato <var>.q.a.</var> cognoÅ¿cetur quadra-<lb/>tum <var>.q.s.</var> vnà etiam eius radix. </s> - <s xml:id="s798" xml:space="preserve">Eadem ratione, alia multa quadrata Å¿imilia contem-<lb/>plari licebit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0071-02" xlink:href="fig-0071-02a"> - <image file="0071-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0071-02"/> - </figure> - <figure xlink:label="fig-0072-01" xlink:href="fig-0072-01a"> - <image file="0072-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0072-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="92"> - <head xml:id="head109" xml:space="preserve">THEOREMA <num value="92">XCII</num>.</head> - <p> - <s xml:id="s799" xml:space="preserve">CVR propoÅ¿ito numero pari maiori binario, qui detrahi & in Å¿ummam colli-<lb/>gi debeat ex altero numero quærendo, vt tam reÅ¿iduum quam Å¿umma Å¿int <lb/>quadrata numerorum integrornm. </s> - <s xml:id="s800" xml:space="preserve">Rectè dimidium propoÅ¿iti numeri in Å¿eipÅ¿um <lb/>multiplicamus, & quadrato huic addimus vnitatem, <reg norm="eritque" type="simple">eritqÌ;</reg> numerus quæfitus.</s> - </p> - <p> - <s xml:id="s801" xml:space="preserve">Exempli gratia proponitur .12. numerus detrahendus, & coniungendus nume-<lb/>ro inueÅ¿tigando, ut reÅ¿iduum detractionis, & Å¿umma Å¿int quadrati numeri. </s> - <s xml:id="s802" xml:space="preserve">Addi-<lb/>ta vnitate ipÅ¿i .36. quadrato dimidij, dabitur .37. numerus quæſitus.</s> - </p> - <p> - <s xml:id="s803" xml:space="preserve">Cuius Å¿peculationis gratia, Å¿ubÅ¿cripta quatuor quadrata cogitemus <var>.g.p</var>: <var>u.i</var>: <var>t.c</var>: <var>n.<lb/>K.</var> <reg norm="cogitemusque" type="simple">cogitemusqÌ;</reg> quadratum <var>.g.p.</var> eſſe quadratum Å¿ummæ, <var>K.n.</var> verò reÅ¿idui Å¿ubtractio-<lb/>nis: <var>u.i.</var> <reg norm="autem" type="wordlist">aÅ©t</reg> numerum <reg norm="inueÅ¿tigandum" type="context context">inueÅ¿tigãdÅ©</reg>, ex quo gnomon <var>.u.d.i.</var> cognoÅ¿cetur ita etiam et <var>.n.<lb/>o.K.</var> qui inter Å¿e Å¿unt æquales. </s> - <s xml:id="s804" xml:space="preserve">Iam certi erimus <var>.e.i.</var> eſſe plus quam dimidium gno-<lb/>monis <var>.n.o.K</var>. </s> - <s xml:id="s805" xml:space="preserve">Itaque cogitemus rectangulum <var>.r.c.</var> exactum <reg norm="dimidium" type="context">dimidiÅ©</reg> eſſe gnomonis <var>.<lb/>n.o.K.</var> ex unitatibus Å¿uperficialibus quarum una erit <var>.m.a</var>.</s> - </p> - <p> - <s xml:id="s806" xml:space="preserve">Cuius numeri quadratum Å¿it <var>.t.c.</var> vnde etiam cognitum & cum <var>.K.c.</var> ex communi <lb/>Å¿cientia Å¿it vnitas linearis, </s> - <s xml:id="s807" xml:space="preserve">propterea quod <var>.m.a.</var> eÅ¿t Å¿uperficialis hoc eÅ¿t quadrata, <lb/>quæ detracta ex <var>.q.c.</var> dimidio gnomonis <var>.n.o.K.</var> (quamuis lineari) Å¿upererit <var>.K.q.</var> co <lb/>gnita, numerorum integrorum (nota <var>q.K.i.</var> Å¿emper minor erit duabus vnitatibus li-<lb/>nearibus & maior vna ex dictis vnitatibus, ut ex te ipÅ¿o contemplari potes) </s> - <s xml:id="s808" xml:space="preserve">quare <var>. - <pb o="61" rhead="THEOR. ARITH." n="73" file="0073" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0073"/> - n.k.</var> ipÅ¿ius quadratum numerorum integrorum cognoÅ¿cetur, cui addito gnomone <var type="gnomon">.<lb/>n.o.K.</var> cognoÅ¿cemus numerum <var>.u.i.</var> quæſitum.</s> - </p> - <p> - <s xml:id="s809" xml:space="preserve">Sed cum nobis hæc via, tenenda propoÅ¿itum non fuit, hoc eÅ¿t primo loco inue <lb/>niendi quadrati minoris <var>.n.K.</var> ideo Å¿upereÅ¿t probandum gnomonem <var>.t.o.c.</var> vnitati <reg norm="ae- qualem" type="simple">Ä™-<lb/>qualem</reg> eſſe, nempe quadratulo <var>.m.a.</var> quod patebit, Å¿i conÅ¿ideremus nos Å¿umpÅ¿iſſe <lb/>rectangulum <var>.r.c.</var> pro dimidio gnomonis <var>.n.o.K</var>. </s> - <s xml:id="s810" xml:space="preserve">etenim Å¿i Å¿upplemento etiam <var>.n.r.</var> qua <lb/>dratulum æquale <var>.m.a.</var> adderetur, pateret gnomonem <var>.n.a.K.</var> cum dicto quadratulo <lb/>collectum, æqualem eſſe gnomoni <var>.n.o.K</var>: cum duo Å¿upplementa <var>.m.t.</var> et <var>.m.c.</var> inter Å¿e <lb/>fint æqualia. </s> - <s xml:id="s811" xml:space="preserve">Quamobrem inuento quadrato <var>.t.c.</var> ex dimidio gnomonis cognito, <lb/>additur vnitas, gnomon Å¿cilicet <var>.t.o.c.</var> ex quo cognoÅ¿citur numerus <var>.u.i.</var> quæſitus. <lb/></s> - <s xml:id="s812" xml:space="preserve">Quod autem quadratum <var>.g.p.</var> numeris integris conÅ¿tet, hac ratione probatur viÅ¿um <lb/>enim fuit Å¿upra quadratum <var>.n.K.</var> verè quadratum eſſe, & numeris integris conÅ¿tare, <lb/>pariter etiam <var>.t.c.</var> <reg norm="Å¿eque" type="simple">Å¿eqÌ;</reg> mutuo conÅ¿equi (nam <var>.K.c.</var> eÅ¿t vnitas linearis) ex quo gnomon <lb/><var>n.a.K.</var> numero diÅ¿pari conÅ¿tabit, ex ijs quæ .90. theoremate probata fuerunt. </s> - <s xml:id="s813" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> <lb/>ex eodem theoremate neceſſe eÅ¿t gnomonem <var>.t.d.c.</var> etiam numero diÅ¿pari conÅ¿tare, <lb/>ita vt à numero <var>.n.a.K.</var> non niÅ¿i duabus vnitatibus differat, nempe vt <var>.c.p.</var> Å¿it vnitas li-<lb/>nearis, Å¿ed ita reuera eÅ¿t, numerus enim <var>.u.d.i.</var> ex præſuppoÅ¿ito par eÅ¿t, </s> - <s xml:id="s814" xml:space="preserve">quare nume <lb/>rus <var>.t.d.c.</var> diÅ¿par erit, cum alterum vnitate Å¿uperet, videlicet gnomone <var>.t.o.c.</var> vnita <lb/>ri æquali, tum <var>.n.a.K.</var> minor eÅ¿t <var>.n.o.K.</var> ex eodem gnomone <var>.t.o.c.</var> unitati æquali. </s> - <s xml:id="s815" xml:space="preserve">Ita <lb/>que <var>.n.a.K.</var> minor erit <var>.u.d.i.</var> per vnitatem, & minor <var>.t.d.c.</var> per duas unitates, ex quo Å¿e-<lb/>quitur <var>.g.p.</var> eſſe quadratum <reg norm="integrorum" type="context">integrorÅ©</reg> ex dicto theoremate ac con Å¿equens quadrato <lb/><var>t.c</var>. </s> - <s xml:id="s816" xml:space="preserve">quare <var>.c.p.</var> vnitas erit, & radices <var>.q.K.</var> et <var>.q.p.</var> horum quadratorum numero bina-<lb/>rio inter Å¿e different. </s> - <s xml:id="s817" xml:space="preserve">Vnà etiam Å¿cienda eÅ¿t cauÅ¿a, cur numerus propoÅ¿itus neceſſa <lb/> - <anchor type="figure" xlink:label="fig-0073-01a" xlink:href="fig-0073-01"/> - riò binario maior eſſe debeat. </s> - <s xml:id="s818" xml:space="preserve">Etenim <reg norm="cum" type="context">cÅ©</reg> ipÅ¿e <lb/>Å¿it futurus gnomon <var>.n.o.K.</var> nec poſſit minor eſſe <lb/>numero ternario, vt patet ex .90. theoremate, <lb/>idcirco Å¿equitur neceſſariò maiorem eſſe bina-<lb/>rio debere. </s> - <s xml:id="s819" xml:space="preserve">Quòd Å¿i diÅ¿par numerus propone-<lb/>retur, nec forma operis nec Å¿peculationis <reg norm="mutan- da" type="context">mutã-<lb/>da</reg> eſſet. </s> - <s xml:id="s820" xml:space="preserve">Non erit tamen neceſſarium vt ipÅ¿a <lb/>quadrata <var>.n.K.</var> et <var>.g.p.</var> numeris integris conÅ¿ta-<lb/>rent. </s> - <s xml:id="s821" xml:space="preserve">Sæpius enim fractis <reg norm="componerentur" type="context">cõponerentur</reg>, quod <lb/>ex .90. theoremate facile erit Å¿peculari nihilo-<lb/>minus fractis integris, <reg norm="ipÅ¿isque" type="simple">ipÅ¿isqÌ;</reg> collectis cum Å¿uis <lb/>fractis Å¿ummæ eſſent quadratæ.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0073-01" xlink:href="fig-0073-01a"> - <image file="0073-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0073-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="93"> - <head xml:id="head110" xml:space="preserve">THEOREMA <num value="93">XCIII</num>.</head> - <p> - <s xml:id="s822" xml:space="preserve">CVR propoÅ¿itis duobus numeris altero pari, altero verò diÅ¿pari, duplo primi <lb/>minore per vnitatem, Å¿i alium inuenire numerum voluerimus, cui alterum iÅ¿to <lb/>rum coniunctum proferat quadratum, & altero detracto, quadratum Å¿uperÅ¿it. </s> - <s xml:id="s823" xml:space="preserve">Re-<lb/>ctè datos numeros in Å¿ummam colligemus, quam Å¿ummam in duas quam maximas <lb/>poterimus partes diuidemus, quarum vna pari, altera diÅ¿pari conÅ¿tet, tum vtran-<lb/>que in Å¿eipÅ¿am multiplicabimus, & quadrato minori, duorum numerorum propo-<lb/>Å¿itorum quemuis ademus, ex quo cupimus nobis quadratum minus Å¿upereſſe, & pro <lb/>ueniet nobis numerum quæſitum.</s> - </p> - <p> - <s xml:id="s824" xml:space="preserve">Exempli gtatia, proponuntur numeri .11. et .6. quorum alter alicui numero ad- - <pb o="62" rhead="IO. BAPT. BENED." n="74" file="0074" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0074"/> - dendus, alter ex eodem detrahendus Å¿it, ex quo proferri debeant bina qua-<lb/>drata. </s> - <s xml:id="s825" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> numeri illi in Å¿ummam collecti dabunt .17. differentiam minoris quadra <lb/>ti & maioris. </s> - <s xml:id="s826" xml:space="preserve">I am Å¿i ex hoc .17. binas partes fecerimus, altera erit .8. altera .9. qui <lb/>bus in Å¿eipÅ¿is multiplicatis alterum quadratum erit .64. alterum .81. addito <reg norm="itaque" type="simple">itaq;</reg> ipÅ¿i. <lb/>64. 11. aut .6. pro libito, propoÅ¿itum numerum conÅ¿equemur. </s> - <s xml:id="s827" xml:space="preserve">cui addito .6. vel .11. <lb/>dabit nobis .81. vel ex ipÅ¿o detracto .11. vel .6. relinquet nobis 64. in pręſenti autem <lb/>exemplo talis numerus erit, aut .70. vel .75. </s> - <s xml:id="s828" xml:space="preserve">Huius autem theorematis Å¿peculatio <lb/>ex .90. dependet, quo demonÅ¿tratum fuit gnomonem proximè quadratum Å¿equen <lb/>tem, vnitate duplo radicis minorem eſſe.</s> - </p> - </div> - <div type="math:theorem" level="3" n="94"> - <head xml:id="head111" xml:space="preserve">THEOREMA <num value="94">XCIIII</num>.</head> - <p> - <s xml:id="s829" xml:space="preserve">CVR Å¿i quis cupiat Å¿ummam progreſſionis arithmeticæ quam citiſſimè cogno <lb/>Å¿cere. </s> - <s xml:id="s830" xml:space="preserve">Rectè coniunget vltimo termino vnitatem primum terminum, huius <lb/>poÅ¿tea vltimi termini dimidium cum numero terminorum multiplicabit, ex <lb/>quo multiplicationis productum, erit omnium propoÅ¿itorum terminorum Å¿umma, <lb/>aut eundem vltimum terminum iunctum primo, per dimidium numeri terminorum <lb/>multiplicabit. </s> - <s xml:id="s831" xml:space="preserve">Nam idipÅ¿um eueniet.</s> - </p> - <p> - <s xml:id="s832" xml:space="preserve">Exempli gratia, Å¿i proponerentur .17. termini in prima progreſſione arithmeti-<lb/>ca naturali, vltimus eſſet .17. cui coniuncta vnitate primo termino Å¿umma erit .18. <lb/>cuius dimidium cum numero terminorum, nempe .17. multiplicatum cum fuerit, <lb/>oritur productum .153. </s> - <s xml:id="s833" xml:space="preserve">IdpÅ¿um eueniet, multiplicato dimidio numeri <reg norm="terminorum" type="context">terminorÅ©</reg> <lb/>per vltimum coniunctum vnitati primo termino.</s> - </p> - <p> - <s xml:id="s834" xml:space="preserve">Quod vt Å¿ciamus, cogitemus terminos progreſſionis collocari, vt in figura Å¿ub-<lb/>Å¿cripta <var>.a.o.n.</var> collocantur, <reg norm="tanquam" type="context">tanquã</reg> per gradus, Å¿umpto principio ab vnitate <var>.n.</var> tum <var>.<lb/>u.t.</var> atque ita gradatim. </s> - <s xml:id="s835" xml:space="preserve">Sic cogitato abÅ¿oluto parallelogrammo <var>.q.o.</var> Å¿ciemus aper-<lb/>tè Å¿ummam progreſſionis tanto maiorem eſſe dimidio totius <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>, quan <lb/>tum dimidium numeri diametri <var>.a.e.i.c.u.n.</var> requirit. </s> - <s xml:id="s836" xml:space="preserve">Nam cum parallelogram-<lb/>mum diuidatur à dl<unsure/>ametro in tres partes, diameter vnam occupat, reliquæ verò duÄ™ <lb/>ambientes diametrum inter Å¿e Å¿unt æquales. </s> - <s xml:id="s837" xml:space="preserve">Sumpto <reg norm="itaque" type="simple">itaq;</reg> diametro cum altera di <lb/>ctarum duarum partium, patet Å¿umi pluÅ¿quam <reg norm="dimidium" type="context">dimidiÅ©</reg> totius <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>. </s> - <s xml:id="s838" xml:space="preserve">pro <lb/>tanta portione, quantum eÅ¿t dimidiam occupatam à diametro, qui <reg norm="cam" type="context">cã</reg><unsure/> ex diÅ¿cretis <lb/>reÅ¿pondentibus numero terminorum componatur, conÅ¿tat numero æquali eſſe di-<lb/>cto numero terminorum <var>.o.n</var>. </s> - <s xml:id="s839" xml:space="preserve">Iam Å¿i quis multiplicet <var>.a.o.</var> per dimidium <var>.o.n.</var> procul <lb/>dubio, ex prima Å¿exti aut .18. Å¿eptimi, orietur <reg norm="dimidium" type="context">dimidiÅ©</reg> numeri <reg norm="parallelogrammi" type="context">parallelogrãmi</reg> <var>.q.o.</var> <lb/>quod minus erit Å¿umma progreſſionis dimidio numeri diametri, aut quod idem eÅ¿t <lb/>dimidio <var>.o.n.</var> Å¿ed hoc dimidium <var>.o.n.</var> æquale eÅ¿t producto dimidij vnitatis <var>.n.</var> in <var>.o.n.</var> <lb/>ex .20. Å¿eptimi, cum dimidium <var>.o.n.</var> Å¿it eius productum in <reg norm="vnitatem" type="context">vnitatẽ</reg>. </s> - <s xml:id="s840" xml:space="preserve">Itaque multipli-<lb/>cato <var>.n.o.</var> per dimidium <var>.o.a.</var> coniunctum dimidio vnitatis <var>.n.</var> oritur Å¿umma quæſita <lb/>propoÅ¿itæ progreſſionis. </s> - <s xml:id="s841" xml:space="preserve">IdipÅ¿um accidet multiplicata Å¿umma <var>.o.a.</var> & vnitate <var>.n.</var> <reg norm="per" type="simple">ê‘</reg> <lb/>dimidium <var>.o.n.</var> ex .20. Å¿eptimi, cum proportio totius ad totum eadem Å¿it, quæ dimi <lb/>dijad dimidium, ex cauÅ¿a permutationalitatis. </s> - <s xml:id="s842" xml:space="preserve">Patet etiam in progreſſionibus, <lb/>quæ ab vnitate initium ducunt, Å¿i fiat aÅ¿cenÅ¿us per binarium Å¿umma vltimi termini <lb/>cum primo Å¿emper duplam futuram eſſe numero terminorum, quod Å¿equentes figu - <pb o="63" rhead="THEOREM. ARIT." n="75" file="0075" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0075"/> - ras confideranti Å¿peculari licebit, Diametros harum figurarum notaui literis Å¿iue <lb/>characteribus <var>.a.e.i.c.u.n</var>.</s> - </p> - <figure position="here"> - <image file="0075-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0075-01"/> - </figure> - <figure position="here"> - <image file="0075-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0075-02"/> - </figure> - </div> - <div type="math:theorem" level="3" n="95"> - <head xml:id="head112" xml:space="preserve">THEOREMA <num value="95">XCV</num>.</head> - <p> - <s xml:id="s843" xml:space="preserve">IN progreſſionibus, quæ ab alio termino quam vnitate incohantur, idipÅ¿um vt <lb/>monuimus accidit, hoc tamen notato, quòd ex conÅ¿equenti quælibet pars dia-<lb/>metri <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>, minimo termino æqualis erit, prout in progreſſionibus quæ <lb/>ab vnitate originem ducunt, Å¿ingulæ partes diametri, vnitati Å¿ui primi termini æ-<lb/>quales Å¿unt. </s> - <s xml:id="s844" xml:space="preserve">At in reliquis progreſſionibus, vt in figura patet, eadem eÅ¿t propor-<lb/>tio totius diametri ad <var>.o.n.</var> quæ minimi termini ad vnitatem ex .13. quinti, nempe <var>.<lb/>a.o.</var> ad <var>.o.n.</var> vt <var>.n.n.n.n.</var> ad <var>.n</var>. </s> - <s xml:id="s845" xml:space="preserve">In eiuÅ¿modi progreſſionibus accidit quoque <reg norm="parallelo- grammum" type="context">parallelo-<lb/>grãmum</reg> à diametro in tres partes diuidi, quarum vnam ipÅ¿e occupat, reliquæ ve-<lb/>ro inter Å¿e æquales ipÅ¿um ambiunt. </s> - <s xml:id="s846" xml:space="preserve">Ex quo illud etiam Å¿equitur, productum <var>.a.o.</var> in <lb/>dimidium <var>.o.n.</var> æquale eſſe dimidio <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>, quod minus eÅ¿t Å¿umma progreÅ¿-<lb/>Å¿ionis dimidio diametri, quod dimidum Å¿i inuenire voluerimus, minimum <reg norm="terminum" type="context">terminÅ©</reg> <var>.<lb/>n.n.n.n.</var> per dimidium <var>.o.n.</var> multiplicabimus, & ex .18. aut .19. Å¿eptimi ipÅ¿um habe-<lb/>bimus, <reg norm="quandoquidem" type="context">quandoquidẽ</reg> minimo termino per totum <var>.o.n.</var> multiplicato profertur integer <lb/>diameter ex .20. prædicti. </s> - <s xml:id="s847" xml:space="preserve">Etenim vt diximus, eadem eÅ¿t proportio totius diame-<lb/>tri ad <var>.o.n.</var> quæ minimi termini ad vnitatem. </s> - <s xml:id="s848" xml:space="preserve">Ita etiam dico ex dicta .20. Å¿eptimi. <lb/></s> - <s xml:id="s849" xml:space="preserve">idem dimidium diametri oriri, Å¿i quis dimidium minimi termini nempè <var>.n.n.</var> per to <lb/>tum <var>.o.n.</var> multiplicauerit. </s> - <s xml:id="s850" xml:space="preserve">Quamobrem qui Å¿tatim Å¿ummam propoÅ¿itæ progreſſionis <lb/>cognoÅ¿cere voluerit, <lb/> - <anchor type="figure" xlink:label="fig-0075-03a" xlink:href="fig-0075-03"/> - Å¿emper primum termi <lb/>num <var>.n.n.n.n.</var> cum <var>.a.o.</var> <lb/>coniunget, qua Å¿umma <lb/>per <reg norm="dimidium" type="context">dimidiÅ©</reg> <var>.o.n.</var> mul-<lb/>tiplicata, aut <var>.o.n.</var> per <lb/>dimidium dictæ Å¿um-<lb/>mæ, ex prædictis rationibus propofitum conÅ¿equemur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0075-03" xlink:href="fig-0075-03a"> - <image file="0075-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0075-03"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="96"> - <head xml:id="head113" xml:space="preserve">THEOREMA <num value="96">XCVI</num>.</head> - <p> - <s xml:id="s851" xml:space="preserve">CVR Å¿i quis numerum terminorum inuenire velit, cognitis tantummodo pri <lb/>mo atque vltimo, rectè vltimum per primum diuidet, ex quo proueniens - <pb o="64" rhead="IO. BAPT. BENED." n="76" file="0076" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0076"/> - numerus quæſitus erit.</s> - </p> - <p> - <s xml:id="s852" xml:space="preserve">Quod intelligendum eÅ¿ttamen quoties primus terminus differentia <reg norm="terminorum" type="context">terminorÅ©</reg> <lb/>eÅ¿t, nempe aÅ¿cendens ipÅ¿orum ter minorum.</s> - </p> - <p> - <s xml:id="s853" xml:space="preserve">Cuius ratio manifeÅ¿tè Å¿peculari poteÅ¿t in figura præcedentis theorematis. </s> - <s xml:id="s854" xml:space="preserve">Nam <lb/>diuiÅ¿a <var>.a.o.</var> per <var>.n.n.n.n.</var> eadem proportio erit <var>.a.o.</var> ad proueniens, quæ. n <var>.n.n.<lb/>n.</var> ad vnitatem <var>.n.</var> ex definitione diuiÅ¿ionis. </s> - <s xml:id="s855" xml:space="preserve">At Å¿uperius dictum fuit ita Å¿e ha bere <var>.a.<lb/>o.</var> ad <var>.o.n.</var> vt <var>.n.n.n.n.</var> ad <var>.n.</var> ex quo Å¿equitur ex .11. et .9. quinti pr oueniens eſſe nume-<lb/>rum quæſitum <var>.o.n</var>.</s> - </p> - </div> - <div type="math:theorem" level="3" n="97"> - <head xml:id="head114" xml:space="preserve">THEOREMA <num value="97">XCVII</num>.</head> - <p> - <s xml:id="s856" xml:space="preserve">VBI verò primus terminus, reliquorum non erit differentia. </s> - <s xml:id="s857" xml:space="preserve">Hac de caufa ne-<lb/>ceſſe eÅ¿t detrahere primum ex vltimo, <reg norm="reÅ¿iduumque" type="simple">reÅ¿iduumqÌ;</reg> per numerum aÅ¿cenden-<lb/>tem differentiam Å¿cilicet, partiri, <reg norm="proueniensque" type="simple">proueniensqÌ;</reg> vnitati coniungere, quò numerum <lb/>terminorum habere poſſimus. </s> - <s xml:id="s858" xml:space="preserve">Scimus etenim tam multas vnitates eſſe in vltimo <lb/>terminorum quot in omnibus interuallis aut differentijs in Å¿ummam collectis Å¿imul <lb/>cum vnitatibus primi termini, <reg norm="totque" type="simple">totqÌ;</reg> funt termini, quot interualla Å¿imul cum pri-<lb/>motermino. </s> - <s xml:id="s859" xml:space="preserve">Quare fi minimus terminus interuallo æqualis fuerit. </s> - <s xml:id="s860" xml:space="preserve">Vltimo per pri-<lb/>mum diuiÅ¿o, ex a dductis præcedenti theoremate propofitum confequemur. </s> - <s xml:id="s861" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> <lb/>primo termino ex vltimo detracto <reg norm="refiduoque" type="simple">refiduoqÌ;</reg> per interuallum, hoc eft numerum dif-<lb/>ferentiæ diuifo, proueniens erit numerus terminorum abÅ¿que primo quod vnus eft, <lb/>coni uncto quoque dicto prouenienti propoÅ¿itum conÅ¿equemur.</s> - </p> - </div> - <div type="math:theorem" level="3" n="98"> - <head xml:id="head115" xml:space="preserve">THEOREMA <num value="98">XCVIII</num>.</head> - <p> - <s xml:id="s862" xml:space="preserve">CVR fi quis arithmeticæ progreſſionis dato primo & vltimo fimul cum nume <lb/>ro terminorum, afcendentem numerum cognofcere voluerit. </s> - <s xml:id="s863" xml:space="preserve">Rectè primuin <lb/>ex vltimo detrahet, <reg norm="refiduumque" type="simple">refiduumqÌ;</reg> per numerum terminorum excepto vno diuidet.</s> - </p> - <p> - <s xml:id="s864" xml:space="preserve">Huius theorematis Å¿peculatio ex .13. theoremate manifeÅ¿ta crit, nam in præce-<lb/>denti cap. numerus terminorum erat proueniens diuiÅ¿ionis reÅ¿idui Å¿ubtractionis pri-<lb/>mi termini ex vltimo.</s> - </p> - </div> - <div type="math:theorem" level="3" n="99"> - <head xml:id="head116" xml:space="preserve">THEOREMA <num value="99">XCIX</num>.</head> - <p> - <s xml:id="s865" xml:space="preserve">CVR Å¿i quis maximum omnium terminorum dictæ progreffionis cognofcere <lb/>voluerit, dato primo vnà cum numero aÅ¿cendenti, <reg norm="numeroque" type="simple">numeroqÌ;</reg> terminorum. </s> - <s xml:id="s866" xml:space="preserve">Re-<lb/>ctè numerum afcendentem cum numero terminorum excepto vno multiplicabit, <lb/><reg norm="productoque" type="simple">productoqÌ;</reg> primum terminum coniunget.</s> - </p> - <p> - <s xml:id="s867" xml:space="preserve">Cuius quidem theorematis tum ex vndecimo, tum ex ijs quæ præcedentibus ca-<lb/>pitibus dicta fuerunt, aperta eÅ¿t ratio.</s> - </p> - </div> - <div type="math:theorem" level="3" n="100"> - <head xml:id="head117" xml:space="preserve">THEOREMA <num value="100">C</num>.</head> - <p> - <s xml:id="s868" xml:space="preserve">CVR veteres cupientes obtinere Å¿ummam pr<unsure/>ogreffionis continuæ naturalis, <lb/>quæab vnitate initium ducit, dato vltimo termino tantummodo. </s> - <s xml:id="s869" xml:space="preserve">Dimidium <lb/>vltimi-termini <reg norm="cum" type="context">cÅ©</reg> toto fequente multiplicabant, <reg norm="productumque" type="simple">productumqÌ;</reg> Å¿umma quæſita erat.</s> - </p> - <p> - <s xml:id="s870" xml:space="preserve">Exempli gratia, Å¿i vltimus terminus eiuÅ¿modi progreſſionis fuerit .7. multiplica- - <pb o="65" rhead="THEOR. ARITH." n="77" file="0077" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0077"/> - to dimidio ipÅ¿ius nempe .3. & dimidio, cum numero ipÅ¿um terminum <reg norm="Å¿equenti" type="context">Å¿equẽti</reg>, nem <lb/>pè .8. Å¿umma dictorum terminorum erit .28.</s> - </p> - <p> - <s xml:id="s871" xml:space="preserve">Huius autem Å¿peculatio ex .94. theoremate dependet, in quo facilè depræhen-<lb/>dere licet ex figura continuæ progreſſionis naturalis, numerum terminorum maxi-<lb/>mo termino Å¿emper æqualem eſſe; </s> - <s xml:id="s872" xml:space="preserve">ex quo <reg norm="tantum" type="context">tãtum</reg> eÅ¿t dimidium numeriterminorum, <lb/>quantum maximi dimidium, <reg norm="tantusque" type="simple">tantusqÌ;</reg> eÅ¿t vltimus terminus vnitati coniunctus, quan <lb/>tus numerus is, qui vltimum terminum conÅ¿equitur.</s> - </p> - </div> - <div type="math:theorem" level="3" n="101"> - <head xml:id="head118" xml:space="preserve">THEOREMA <num value="101">CI</num>.</head> - <p> - <s xml:id="s873" xml:space="preserve">CVR antiqui idip fum, quod iam dictum eft, in ea progreſſione, cuius vltimus ter <lb/>minus diÅ¿par eÅ¿t Å¿cire cupientes, numerum integrorum proximè dimidium <lb/>maximi Å¿equentem Å¿umebant, quem per maximum multiplicabant, ex quo <lb/>Å¿umma quæſita oriebatur.</s> - </p> - <p> - <s xml:id="s874" xml:space="preserve">Exempli gratia, Å¿i dimidium maximi fuiſſet .3. cum dimidio, fumebant quatuor, <lb/>& per maximum .7. multiplicabant, ex quo pariter proferebatur Å¿umma .28.</s> - </p> - <p> - <s xml:id="s875" xml:space="preserve">Cuius ratio ex .20. Å¿eptimi Euclidis oritur, cum eadem Å¿it proportio numeri fe-<lb/>quentis ma ximum ad numerum dimidium maximi Å¿equentem; </s> - <s xml:id="s876" xml:space="preserve">quæ maximi ad <reg norm="fuum" type="context">fuÅ©</reg> <lb/>dimidium, eÅ¿t enim dupla.</s> - </p> - </div> - <div type="math:theorem" level="3" n="102"> - <head xml:id="head119" xml:space="preserve">THEOREMA <num value="102">CII</num>.</head> - <p> - <s xml:id="s877" xml:space="preserve">TRaditum eÅ¿t à nonnullis, à veteribus obÅ¿eruatam fuiſſe hancregulam, qua Å¿ci-<lb/>re poſſent Å¿ummam alicuius progreſſionis arithmeticæ diÅ¿continuæ aut inter <lb/>cifæ, quæ numero pari terminetur. </s> - <s xml:id="s878" xml:space="preserve"><reg norm="Multiplicabant" type="context">Multiplicabãt</reg> enim <reg norm="dimidium" type="context">dimidiÅ©</reg> vltimi termini per <lb/>pro ximum numerum dimidio dicto maiorem, ex quo <reg norm="inquiebant" type="context">inquiebãt</reg> Å¿emper productum <lb/>Å¿ummæ quæſitæ æquale eſſe, <reg norm="Å¿ubijciuntque" type="simple">Å¿ubijciuntqÌ;</reg> exemplum progreſſionis, quæ à binario in-<lb/>choata crefcit per binarium. </s> - <s xml:id="s879" xml:space="preserve">In qua quidem progreſſione non per fe, fed per acci-<lb/>dens regula vera eft. </s> - <s xml:id="s880" xml:space="preserve">Hoc eÅ¿t, non quia ex Å¿e vnus ex producentibus numeris dimi-<lb/>dium termini maioris futurus Å¿it, alter uerò proximè Å¿equens dimidium, fed quia <lb/>vt dictum eÅ¿t .95. theoremate, eadem eÅ¿t proportio maximi termini ad numerum <lb/>terminorum, quæ minimi ad vnitatem. </s> - <s xml:id="s881" xml:space="preserve">CumqÌue in præfenti exemplo minimum <lb/>Å¿it duplum vnitati in eiuÅ¿modi caÅ¿u, numerus terminorum, dimidio maximi termini <lb/>æqualis eÅ¿t, qui terminorum numerus ex Å¿e, vt patet, vnus eÅ¿t ex producentibus, al-<lb/>ter verò producens numerus, eÅ¿t proximè dimidium Å¿equens, non exÅ¿e, fed quia nu <lb/>merus Å¿equens, dimidium eÅ¿t Å¿ummæ maximi, & minimi, quæ per fe alter eſſe de-<lb/>bet producens numerus. </s> - <s xml:id="s882" xml:space="preserve">In cæteris enim progreſſionibus, quæ binario non creÅ¿cút <lb/>regulafal<unsure/>fa eÅ¿t, prout facilè patere poteÅ¿t ei, qui ex Å¿cientiæ legibus ope Å¿peculatio-<lb/>nis .95. theorematis Å¿peculatus fuerit.</s> - </p> - </div> - <div type="math:theorem" level="3" n="103"> - <head xml:id="head120" xml:space="preserve">THEOREMA <num value="103">CIII</num>.</head> - <p> - <s xml:id="s883" xml:space="preserve">ALIAM quoque tradunt regulam, qua veteres vÅ¿os fuiſſe dicunt, quo Å¿um-<lb/>mam Å¿cire poſſent progreſſionis diÅ¿continuæ, quænumero diÅ¿pari abÅ¿olui-<lb/>tur. </s> - <s xml:id="s884" xml:space="preserve">Ea autem eÅ¿t eiuÅ¿modi. </s> - <s xml:id="s885" xml:space="preserve">Vltimum terminum in duas quam maximè poterant ma-<lb/>ximas partes diuidebant, quarum vna Å¿emper altera maior erat, banc autem maio-<lb/>rem in Å¿eipÅ¿am multiplicabant, at que quadratum hoc, Å¿ummam progreffionis effe - <pb o="66" rhead="IO. BAPT. BENED." n="78" file="0078" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0078"/> - affirmabant. </s> - <s xml:id="s886" xml:space="preserve">Quæ Å¿anè regula, non Å¿emper, etÅ¿i interdum vera Å¿it.</s> - </p> - <p> - <s xml:id="s887" xml:space="preserve">Sumebant hi exemplum progreſſionis, quæ ab vnitate incohata creÅ¿cit per bina <lb/>rium, in qua per accidens euenit vt numerus dimidium vltimi termini proximè Å¿e-<lb/>quens, nempe è duabus partibus vltimi termini maior, æqualis Å¿it numero termino <lb/>rum, qui per Å¿e vnus è producentibus, ex ijs que .94. theoremate diximus, eſſe debet; <lb/></s> - <s xml:id="s888" xml:space="preserve">alter vero producens, qui per Å¿e dimidium Å¿ummæ primi & vltimi eſſe debet, per <lb/>accidens pars maior eÅ¿t duarum vltimi termini, & alteri producenti æqualis.</s> - </p> - <p> - <s xml:id="s889" xml:space="preserve">Aut alio modo ratiocinemur, dicentes, in huiuÅ¿modi progreſſione dimidium <lb/>Å¿ummæ vltimi termini cum primo, Å¿emper medium proportionale eÅ¿t inter eam <lb/>Å¿ummam & dimidium numeri terminorum, etenim huiuÅ¿modi Å¿umma numero ter-<lb/>minorum Å¿emper dupla eÅ¿t, prout .94. theoremate tradimus. </s> - <s xml:id="s890" xml:space="preserve">Itaque ex .20. Å¿eptimi, <lb/>quadratum partis maioris, producto Å¿ummæ dictæ in numerum dimidij <reg norm="terminorum" type="context">terminorÅ©</reg> <lb/>æquale erit, quod productum per Å¿e Å¿ummæ progreſſionis eÅ¿t æquale. </s> - <s xml:id="s891" xml:space="preserve">At in cæte-<lb/>ris eiuÅ¿modi progreſſionibus fallit regula, vt ex Å¿upradictis facilè demonÅ¿tratur.</s> - </p> - </div> - <div type="math:theorem" level="3" n="104"> - <head xml:id="head121" xml:space="preserve">THEOREMA <num value="104">CIIII</num>.</head> - <p> - <s xml:id="s892" xml:space="preserve">PErmultis terminis ad libitum propoÅ¿itis, diÅ¿poÅ¿itis nihilominus progreſſio-<lb/>ne, aut proportionalitate geometrica continua, Å¿i minimus ex maximo & exfe-<lb/>quenti minimum detrahatur, reÅ¿iduum maximi, eam proportionem ad fum-<lb/>mam reliquorum omnium terminorum retinebit, quam reÅ¿iduum Å¿ecundi ad pri-<lb/>mum.</s> - </p> - <p> - <s xml:id="s893" xml:space="preserve">Proponuntur, exempli gratia, quatuor termini .3. 12. 48. 192. continui geome-<lb/>tricè proportionales, Å¿i primum, hoc eÅ¿t minimum, ex Å¿ecundo, & maximo detra <lb/>has, exÅ¿ecundo Å¿upererit .9. ex maximo .189. quod Å¿i minimum per reÅ¿iduum maxi <lb/>mi multiplicaueris, hoc eÅ¿t .189. orietur .567. tum Å¿i huiuÅ¿modi productum per .9. <lb/>( refiduum Å¿ecundi ) diuiÅ¿eris, proueniet .63. quod proueniens æquale erit Å¿ummæ <lb/>reliquorum omnium terminorum, maximo excepto. </s> - <s xml:id="s894" xml:space="preserve">Ex quo inferre licet ex .20. Å¿e <lb/>ptimi eandem proportionem eſſe .189. ad .63. quæ .9. ad .3. aut Å¿i reÅ¿iduum Å¿ecundi <lb/>per Å¿ummam dictorum terminorum multiplicaueris produceturidem .567. </s> - <s xml:id="s895" xml:space="preserve">quare <lb/>ex .20. Å¿eptimi & cætera.</s> - </p> - <p> - <s xml:id="s896" xml:space="preserve">Quod vt <reg norm="Å¿cientificè" type="context">Å¿ciẽtificè</reg> poſſimus, & in vniuerÅ¿um Å¿peculari. </s> - <s xml:id="s897" xml:space="preserve">Quatuor termini propo-<lb/>Å¿iti, quatuor Å¿ubÅ¿criptis lineis <reg norm="Å¿ignificentur" type="context">Å¿ignificẽtur</reg> <var>.b.i</var>: <var>c.a</var>: <var>f.r</var>: <var>m.s.</var> (quod <reg norm="autem" type="wordlist">aÅ©t</reg> de his quatuor di <lb/>co de <reg norm="centummillibus" type="context">centÅ©millibus</reg>, & eo amplius dicere poſſum.) </s> - <s xml:id="s898" xml:space="preserve">Nunc minimus terminus <var>.m.s.</var> ex <lb/>maximo <var>.b.i.</var> detrahatur, <reg norm="Å¿uperÅ¿itque" type="simple">Å¿uperÅ¿itqÌ;</reg> <var>.n.i.</var> <reg norm="idemque" type="simple">idemqÌ;</reg> <var>.m.s.</var> ex Å¿ecundo termino <var>.f.r.</var> Å¿ubtra-<lb/>hatur, <reg norm="Å¿uperÅ¿itque" type="simple">Å¿uperÅ¿itqÌ;</reg> <var>.o.r</var>. </s> - <s xml:id="s899" xml:space="preserve">Dico proportionem <var>.n.i.</var> ad Å¿ummam reliquorum omnium ter-<lb/>minorum <var>.c.a</var>: <var>f.r</var>: <var>m.s.</var> eandem effe, quæ <var>.o.r.</var> ad <var>.m.s</var>. </s> - <s xml:id="s900" xml:space="preserve">Quamobrem ex tertio & quar-<lb/>to Å¿ecundus <var>.f.r.</var> <reg norm="detrahatur" type="simple">detrahat̃</reg>, ex <reg norm="tertioque" type="simple">tertioqÌ;</reg> Å¿uperÅ¿it <var>.t.a.</var> & ex quarto <var>.e.i.</var> ita etiam tertius <var>.<lb/>c.a.</var> ex quarto <var>.b.i.</var> <reg norm="Å¿uperÅ¿itque" type="simple">Å¿uperÅ¿itqÌ;</reg> <var>.d.i.</var> Å¿anè <lb/> - <anchor type="figure" xlink:label="fig-0078-01a" xlink:href="fig-0078-01"/> - Å¿ic Å¿e habebit <var>.c.a.</var> ad <var>.f.r.</var> vt <var>.c.t.</var> ad <var>.f.o.</var> <lb/>vt <reg norm="quisque" type="simple">quisq;</reg> per Å¿e Å¿cire poteÅ¿t. </s> - <s xml:id="s901" xml:space="preserve">Quare ex <lb/>19. quinti Å¿ic Å¿e habebit <var>.a.t.</var> ad <var>.r.o.</var> vt <var>.<lb/>c.a.</var> ad <var>.f.r.</var> & permutando ita <var>.a.t.</var> ad <var>.a.<lb/>c.</var> vt <var>.o.r.</var> ad <var>.r.f.</var> & Å¿eparando Å¿ic <var>.a.t.</var> ad <var>.<lb/>a.c.</var> (hoc eÅ¿t <var>.f.r.</var>) vt <var>.r.o.</var> ad <var>.o.f.</var> vide-<lb/>licet <var>.m.s</var>. </s> - <s xml:id="s902" xml:space="preserve"><reg norm="Idem" type="context">Idẽ</reg> dico de <var>.d.i.</var> ad <var>.a.c.</var> nem-<lb/>pe Å¿ic Å¿e habebit <var>.d.i.</var> ad <var>.a.c.</var> vt <var>.a.t.</var> ad <var>. - <pb o="67" rhead="THEOREM. ARIT." n="79" file="0079" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0079"/> - r.f.</var> hoc eÅ¿t <var>.o.r.</var> ad <var>.m.s.</var> ex .11. quinti. </s> - <s xml:id="s903" xml:space="preserve">Itaque ex communi Å¿cientia Å¿ic Å¿e habe-<lb/>bit <var>.d.i.</var> ad <var>.d.b.</var> vt <var>.e.d.</var> ad <var>.e.b</var>: cum <var>.e.d.</var> æqualis Å¿it <var>.t.a</var>. </s> - <s xml:id="s904" xml:space="preserve">Ita etiam vt <var>.e.n.</var> ad <var>.n.b</var>: cum <var>.n.<lb/>e.</var> æqualis Å¿it <var>.o.r</var>. </s> - <s xml:id="s905" xml:space="preserve">Iam Å¿i Å¿ic Å¿e habeat <var>.d.i.</var> ad <var>.d.b.</var> vt <var>.d.e.</var> ad <var>.e.b.</var> permutando <reg norm="quoque" type="simple">quoq;</reg> Å¿ic <lb/>Å¿e habebit <var>.d.i.</var> ad <var>.d.e.</var> vt <var>.d.b.</var> ad <var>.b.e.</var> & compon endo ita <var>.i.d.e.</var> ad <var>.e.d.</var> vt <var>.d.b.e.</var> ad <var>.e.<lb/>b.</var> & permutando Å¿ic <var>.i.d.e.</var> ad <var>.d.b.e.</var> vt. de <var>.a.d.e.b.</var> nempe vt <var>.e.n.</var> ad <var>.n.b.</var> & permutan <lb/>do ita <var>.i.d.e.</var> ad <var>.e.n.</var> vt <var>.d.b.e.</var> ad <var>.b.n.</var> & componendo ita <var>.i.d.e.n.</var> ad <var>.n.e.</var> vt <var>.d.b.e.</var> et <var>.b.<lb/>n.</var> ad <var>.b.n.</var> & permutando Å¿ic <var>.i.d.e.n.</var> ad <var>.d.b.e.</var> et <var>.b.n.</var> nempe ad <var>.a.c</var>: <var>f.r</var>: <var>m.s</var>: vt <var>.e.n.</var> ad <var>.<lb/>n.b.</var> hoc eÅ¿t. ut <var>.o.r.</var> ad <var>.m.s.</var> quod erat propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0078-01" xlink:href="fig-0078-01a"> - <image file="0078-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0078-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="105"> - <head xml:id="head122" xml:space="preserve">THEOREMA <num value="105">CV</num>.</head> - <p> - <s xml:id="s906" xml:space="preserve">CVR deÅ¿ideranti Å¿ummam quorumcunque terminorum progreſſionis conti-<lb/>nuæ geometricæ cognoÅ¿cere. </s> - <s xml:id="s907" xml:space="preserve">Rectè minimus terminus ex maximo detrahen <lb/>dus eÅ¿t, <reg norm="reÅ¿iduumque" type="simple">reÅ¿iduumqÌ;</reg> per denominantem progreſſionis dempta vnitate diuidendum, <lb/><reg norm="prouenientique" type="simple">prouenientiqÌ;</reg> maximum terminum addendum, ex quo oritur Å¿umma quæſita.</s> - </p> - <p> - <s xml:id="s908" xml:space="preserve">Exempli gratia, Å¿i darentur quatuor termini continui proportionales .8. 12. 18. <lb/>27. primum hoc eÅ¿t minimum .8. ex vltimo .27. detraheremus: </s> - <s xml:id="s909" xml:space="preserve"><reg norm="remaneretque" type="simple">remaneretqÌ;</reg> .19. qui <lb/>per denominantem progreſſionis, dempta vnitate, diuideretur. </s> - <s xml:id="s910" xml:space="preserve">Quo loco animad <lb/>uertendum eÅ¿t, quamlibet <reg norm="denominationem" type="context">denominationẽ</reg> cuiuÅ¿cunque proportionis numerorum <lb/>Å¿upra vnitatem fieri, nam de proportionibus multiplicibus dubitandum non eÅ¿t, & <lb/>idipÅ¿um de Å¿uperparticularibus, & Å¿uperpartientibus eÅ¿t intelligendum, vt in præ-<lb/>Å¿enti proportio Å¿eÅ¿quialtera inter duos terminos cogitanda eÅ¿t, nempe inter vnum <lb/>& dimidium, atque vnum. </s> - <s xml:id="s911" xml:space="preserve">SeÅ¿quitertia autem inter vnum & tertiam partem, <lb/>& vnum. </s> - <s xml:id="s912" xml:space="preserve">SeÅ¿quiquinta inter vnum cum quinta parte, & vnum. </s> - <s xml:id="s913" xml:space="preserve">De Å¿uperpartien <lb/>tibus idem aſſero quod de proportione <reg norm="Å¿uperbipartiente" type="context">Å¿uperbipartiẽte</reg> tertias appellata, vt .5. <lb/>ad .3. quæ cogitanda eſſet inter vnum duas tertias, & vnum, Å¿uperbipartiens quar-<lb/>tas inter vnum tres quartas, & vnum, ita vt minor terminus, numerans Å¿cilicet, Å¿em <lb/>per Å¿it vnitas, alter verò denominans. </s> - <s xml:id="s914" xml:space="preserve">Idem de cæteris. </s> - <s xml:id="s915" xml:space="preserve">Quare in præſenti exem <lb/>plo, detracta vnitate ex denominante progreſſionis, Å¿upererit tantummodo dimi-<lb/>dium, quo diuiÅ¿o .19. proueniet .38. qui numerus æqualis erit Å¿ummæ <reg norm="reliquorum" type="context">reliquorÅ©</reg> <lb/>omnium terminorum, cui coniuncto vltimo termino .27. dabitur Å¿umma quæſita .65.</s> - </p> - <p> - <s xml:id="s916" xml:space="preserve">Pro cuius Å¿peculatione, quatuor termini Å¿ignificentur, quatuor lineis <var>.m.s</var>: <var>f.r</var>: <var>c.a.<lb/>b.i.</var> primus autem terminus <var>.m.s.</var> ex vltimo <var>.b.i.</var> detrahatur, <reg norm="reÅ¿iduumque" type="simple">reÅ¿iduumqÌ;</reg> Å¿it <var>.n.i.</var> & ex <lb/>Å¿ecundo <var>.f.r.</var> cuius reÅ¿iduum Å¿it <var>.o.r.</var> proportio verò progreſſionis ea Å¿it, quæ <var>.g.h.</var> ad <var>.<lb/>y.</var> quo vnitas repræſentatur (ex quo Å¿ic Å¿e habebit <var>.g.h.</var> ad <var>.y.</var> vt <var>.f.r.</var> ad <var>.m.s.</var>) qua <var>.y.</var> de <lb/>tracta ex <var>.g.h.</var> Å¿uperÅ¿it <var>.h</var>. </s> - <s xml:id="s917" xml:space="preserve">Tum erecta <lb/>cogitetur linea <var>.n.u.x.</var> indefinita per <lb/> - <anchor type="figure" xlink:label="fig-0079-01a" xlink:href="fig-0079-01"/> - pendicularis <var>.b.i.</var> à puncto <var>.n.</var> quÄ™ diui <lb/>datur in puncto <var>.x.</var> ita vt <var>.n.x.</var> æqualis <lb/>Å¿it vnitati <var>.y.</var> & in puncto <var>.u.</var> ita. vt <var>.n.<lb/>u.</var> æqualis Å¿it <var>.h.</var> ex quo eadem erit <lb/>proportio <var>.n.u.</var> ad <var>.n.x.</var> vt <var>.h.</var> ad <var>.y.</var> <reg norm="nem- pe" type="context">nẽ-<lb/>pe</reg> <var>.o.r.</var> ad <var>.m.s</var>. </s> - <s xml:id="s918" xml:space="preserve">Nam cú Å¿ic Å¿e habeat <var>.<lb/>f.r.</var> ad <var>.m.s.</var> hoc eÅ¿t ad <var>.f.o.</var> vt <var>.g.h.</var> ad <var>.y</var> <lb/>hoc eÅ¿t ad <var>.g.</var> permutando <reg norm="quoque" type="simple">quoq;</reg> Å¿ic <lb/>Å¿e habebit <var>.f.r.</var> ad <var>.g.h.</var> vt <var>.f.o.</var> ad <var>.g</var>. </s> - <s xml:id="s919" xml:space="preserve">Ita <lb/>que ex .19. quinti <var>.o.r.</var> ad <var>.h.</var> vt <var>.f.r.</var> ad <var>.g.h.</var> ex quo ex .11. eiuÅ¿dem <var>.o.r.</var> ad <var>.h.</var> vt <var>.f.o.</var> ad - <pb o="70" rhead="IO. BAPT. BENED." n="80" file="0080" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0080"/> - g. & permutando <var>.o.r.</var> ad <var>.f.o.</var> hoc eÅ¿t ad <var>.m.s.</var> vt <var>.h.</var> ad <var>.g.</var> hoc eÅ¿t <var>.y</var>. </s> - <s xml:id="s920" xml:space="preserve">Quamobrem ea-<lb/>dem erit proportio <var>.o.r.</var> ad <var>.m.s.</var> quæ <var>.n.u.</var> ad <var>.n.x</var>. </s> - <s xml:id="s921" xml:space="preserve">Abfoluantur itaque duo rectangu <lb/>la <var>.x.i.</var> et <var>.u.z.</var> ita tamen vt <reg norm="rectangulum" type="context">rectangulÅ©</reg> <var>.u.z.</var> cogitetur Ä™quale rectangulo <var>.x.i.</var> cuius <var>.x.i.</var> <lb/>Å¿uperficialis numerus ex communi conceptione lineari <var>.n.i.</var> æqualis erit, </s> - <s xml:id="s922" xml:space="preserve">quare ex <lb/><reg norm="eadem" type="context">eadẽ</reg> communi conceptione, numerus Å¿uperficialis <var>.u.z.</var> lineari <var>.n.i.</var> æqualis erit, qui <lb/>quidem numerus in figura rectangu-<lb/>la Å¿uperficialis cognitandus erit, cum <lb/> - <anchor type="figure" xlink:label="fig-0080-01a" xlink:href="fig-0080-01"/> - diuidendus Å¿it per <var>.h.</var> hoc eÅ¿t per <var>.n.u.</var> <lb/>ex quo proueniens ex huiuÅ¿modi di <lb/>uiÅ¿ione erit numerus <var>.n.z.</var> ex ijs <lb/>quæ .10. theoremate dicta fuerunt. <lb/></s> - <s xml:id="s923" xml:space="preserve">Sed ex .15. fexti aut .20. Å¿epti-<lb/>mi eadem eÅ¿t proportio <var>.n.i.</var> ad <var>.n.z.</var> <lb/>quæ <var>.n.u.</var> ad <var>.n.x.</var> hoc eÅ¿t <var>.o.r.</var> ad <var>.m.s.</var> <lb/>videlicet vt <var>.n.i.</var> ad aggregatum reli-<lb/>quorum omnium terminorum <var>.c.a</var>: <var>f.<lb/>r</var>: </s> - <s xml:id="s924" xml:space="preserve">m.s. ex præcedenti theoremate, & ex .11. quinti Euclidis. </s> - <s xml:id="s925" xml:space="preserve">Itaque ex .9. eiuf-<lb/>dem numerus <var>.n.z.</var> æqualis erit Å¿ummæ trium terminorum <var>.c.a</var>: f. num <var>.s.</var> cui coniuncto <lb/>quarto termino <var>.b.i.</var> propoÅ¿itum obtinetur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0079-01" xlink:href="fig-0079-01a"> - <image file="0079-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0079-01"/> - </figure> - <figure xlink:label="fig-0080-01" xlink:href="fig-0080-01a"> - <image file="0080-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0080-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="106"> - <head xml:id="head123" xml:space="preserve">THEOREMA <num value="106">CVI</num>.</head> - <p> - <s xml:id="s926" xml:space="preserve">PRopoÅ¿uere veteres quæſita nonnulla de itineribus <reg norm="interque" type="simple">interqÌ;</reg> hoc vnum fuit. </s> - <s xml:id="s927" xml:space="preserve">Po-<lb/>namus duos iter agere per eandem viam quorum alter quatuor milliaria Å¿in-<lb/>gulis diebus conficiat, alter verò prima die milliare vnum, Å¿ecunda duo, tertia tria, <lb/>atque ita Å¿ingulis diebus milliare addit; </s> - <s xml:id="s928" xml:space="preserve">quærimus quot dierum Å¿pacio Å¿ocium con <lb/>Å¿equetur.</s> - </p> - <p> - <s xml:id="s929" xml:space="preserve">Quamobrem numerus milliarium primi viatoris duplicatur, Å¿ic Å¿unt .8. milliaria. <lb/></s> - <s xml:id="s930" xml:space="preserve">ex quo Å¿emper vnitas detrahicur, quæ in præſenti exemplo erit .7. <reg norm="totque" type="simple">totqÌ;</reg> dies erunt <lb/>quibus Å¿ocius Å¿ocium conÅ¿equetur, & milliarium numerum æqualem abÅ¿oluerit.</s> - </p> - <p> - <s xml:id="s931" xml:space="preserve">Cuius rei facilis erit Å¿peculatio, Å¿i Å¿ubÅ¿cripta figura diligenter conÅ¿ideretur, in <lb/>qua primus viator, die prima, quatuor milliaria linea <var>.q.d.</var> Å¿ignificata conficit, at-<lb/>que illa ipÅ¿a die alter vnum tantum defignatum per <var>.d.</var> perficit, ita vt primus via-<lb/>tor tribus milliaribus Å¿ocium anteceſſerit, altera verò die fecundus uiator cum duo <lb/>milliaria <reg norm="conficiat" type="context">cõficiat</reg>, excedètur à primo duobus milliaribus tantummodo, quÄ™ cum tri-<lb/>bus primæ diei quinque erunt; </s> - <s xml:id="s932" xml:space="preserve">tertia die ijÅ¿dem de cauÅ¿is primus Å¿ex tantum millia-<lb/>ribus à ſecundo diÅ¿tabit, cum verò quarta die tot Å¿ecundus quot primus milliaria <lb/>conficiat, primus à ſecundo amplius quam antea non diÅ¿tabit; </s> - <s xml:id="s933" xml:space="preserve">quinta verò cum Å¿e <lb/>cundus vnum milliare amplius quam primus conficiat. </s> - <s xml:id="s934" xml:space="preserve">propius accedit ad primum <lb/>vno ex Å¿ex milliaribus, quibus anteà diÅ¿tabat, tum Å¿exta cum duobus primum Å¿upe-<lb/>ret, detrahet ex Å¿ex milliaribus præteritæ diÅ¿tantiæ tria, Å¿eptima tandem illa Å¿ex <lb/>detraxerit. </s> - <s xml:id="s935" xml:space="preserve">In quo conÅ¿iderandum eÅ¿t Å¿ecundum viatorem iter agere progreſſio-<lb/>ne arithmetica continua naturali <var>.d.c.f.</var> primum autem per rectangulum <var>.q.f.</var> <reg norm="quarum" type="context">quarÅ©</reg> <lb/>duarum figurarum <var>.d.o.p.f.</var> pars <reg norm="communis" type="context">cõmunis</reg> eſſe reperitur, quæ quantitates Å¿i inuicem <lb/>æquales eſſe debent, neceſſe eÅ¿t Å¿eparatas partes <var>.u.q.n.</var> et <var>.t.i.c.</var> inter Å¿e æquales eſſe, <lb/>& quoniam quarta die (hoc eÅ¿t die Å¿ic diÅ¿tante à primo, nempè numero milliarium - <pb o="71" rhead="THEOREM. ARITH." n="81" file="0081" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0081"/> - primi viatoris) tot milliaria abÅ¿oluat vnus <lb/> - <anchor type="figure" xlink:label="fig-0081-01a" xlink:href="fig-0081-01"/> - quot alter abÅ¿que vlla differentia, quæ Å¿igni-<lb/>ficetur per <var>.o.s.</var> neceſſe eÅ¿titaque ex communi <lb/>conceptione tot dies eſſe poÅ¿t <var>.o.s.</var> quotante-<lb/>ceſſerant, vt exceſſus æqualis Å¿it defectui, qui <lb/>Å¿imul collecti, iuncta etiam <var>.o.s.</var> duplum erunt <lb/><var>d.s.</var> dempta vnitate, prout facilè in Å¿ubÅ¿cripta <lb/>figura qui Å¿que per Å¿e Å¿cientificè poterit Å¿pecu <lb/>lari. </s> - <s xml:id="s936" xml:space="preserve">Quamobrem conÅ¿ultum erit duplicare <lb/>numerum <var>.o.s.</var> & exduplo vnitatem detrahe-<lb/>re, quandoquidem dies Å¿upra <reg norm="infraque" type="simple">infraqÌ;</reg> <var>.o.s.</var> cum <lb/>die <var>.o.s.</var> minores Å¿unt duplo numeri <var>.d.s.</var> aut <var>.o.<lb/>s.</var> (quodidem eÅ¿t) vnitate.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0081-01" xlink:href="fig-0081-01a"> - <image file="0081-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0081-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="107"> - <head xml:id="head124" xml:space="preserve">THEOREMA <num value="107">CVII</num>.</head> - <p> - <s xml:id="s937" xml:space="preserve"><emph style="sc">QVod</emph> <reg norm="Å¿iÅ¿ecundus" type="context">Å¿iÅ¿ecÅ©dus</reg> viator <reg norm="ordinem" type="context">ordinẽ</reg> <reg norm="Å¿ecundæ" type="context">Å¿ecÅ©dæ</reg> progreſſionis arithmeticæ <reg norm="Å¿eruansiter" type="context">Å¿eruãsiter</reg> <lb/>agat, nempe ea quæ ab vno per binarium aÅ¿cendit, Å¿emper numerus dierum <lb/>æqualis erit numero milliarium diurnorum primi viatoris.</s> - </p> - <p> - <s xml:id="s938" xml:space="preserve">In cuius gratiam animaduertendum eÅ¿t numerus ne milliarium diurnorum pri-<lb/>mi viatoris par an impar Å¿it. </s> - <s xml:id="s939" xml:space="preserve">Etenim Å¿i par eÅ¿t, primus viator in fine Å¿ingulorum die-<lb/>rum primæ medietatis numeri omnium dierum Å¿ecundum antecedet numero diÅ¿pa <lb/>ri milliarium; </s> - <s xml:id="s940" xml:space="preserve">altero verò dimidio numero dierum, à ſecundo numero etiam diÅ¿pa <lb/>ri præteribitur, vt in Å¿equenti figura patet. </s> - <s xml:id="s941" xml:space="preserve">Nam prima die, Å¿ecundus ex primo <lb/>milliare vnum ex numero pari, qui à primo conficitur detrahit; </s> - <s xml:id="s942" xml:space="preserve">Å¿ecunda verò die <lb/>idem Å¿ecundus, duo Å¿ubtrahit milliaria exdiÅ¿pari, qui primo reliquus fuerat, <reg norm="Å¿icque" type="simple">Å¿icqÌ;</reg> <lb/>perpetuò diÅ¿par remanet vÅ¿que ad vnitatem, ad quam cum peruenerint, nempe ad <lb/>illius diei exitum, quo primus Å¿ecundum vnitate tantummodò Å¿uperat, manifeÅ¿tè <lb/>depræhendetur Å¿ubÅ¿equente dieÅ¿ecundum vnitate primum Å¿uperaturum, altera ve <lb/>rò tribus vnitatibus, prout penultima die Å¿ecundus à primo tribus vnitatibus Å¿upera <lb/>batur. </s> - <s xml:id="s943" xml:space="preserve">Quare neceſſe erit, tot diebus Å¿ecundum cum primo iter agere, inchoan-<lb/>do ab ea die, qua fecundus primam Å¿uperabit, quot egerat dum à primo Å¿uperare-<lb/>tur, vt ex communi conceptione, media figura <var>.A.</var> depræhendi poteÅ¿t. </s> - <s xml:id="s944" xml:space="preserve">Quod au-<lb/>tem Å¿ingula dimidia dierum, dimi-<lb/> - <anchor type="figure" xlink:label="fig-0081-02a" xlink:href="fig-0081-02"/> - dia Å¿int numeri milliarium diurno-<lb/>rum primi; </s> - <s xml:id="s945" xml:space="preserve">patebit exÅ¿equenti fi-<lb/>gura, cogitato termino <var>.u.n.</var> vlti-<lb/>mo progreſſionis Å¿uperatÄ™ à primo <lb/>vÅ¿que ad vnitatem <var>.e.</var> quiterminus <lb/><var>u.n.</var> coniunctus primo <var>.o.</var> nempe <var>.e.</var> <lb/>Å¿emper <reg norm="duplum" type="context">duplÅ©</reg><unsure/> eÅ¿t numeritermino-<lb/>rum <var>.o.n.</var> vt .94. theoremate circa <lb/>finem dictum fuit. </s> - <s xml:id="s946" xml:space="preserve">Sed <var>.u.n.</var> cum <var>.e.</var> <lb/>numero æquali conÅ¿tat numero <lb/>milliarium diurnorum primi viatoris, ex quo Å¿equitur totum numerum dierum, quo <lb/>rum <var>.o.n.</var> dimidium eÅ¿t, æqualem eſſe numero milliarium diuruorum primi via-<lb/>toris.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0081-02" xlink:href="fig-0081-02a"> - <image file="0081-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0081-02"/> - </figure> - </div> - <pb o="70" rhead="IO. BAPT. BENED." n="82" file="0082" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0082"/> - </div> - <div type="math:theorem" level="3" n="108"> - <head xml:id="head125" xml:space="preserve">THEOREMA <num value="108">CVIII</num>.</head> - <p> - <s xml:id="s947" xml:space="preserve">AT Å¿i numerus milliarium primi viatoris diÅ¿par fuerit, Å¿ecundum numero pari <lb/>Å¿emper Å¿uperabit, vt facile erit Å¿equentem figuram conÅ¿ideranti intelligere, <lb/>ex quo illud Å¿equetur, futuram quandam diem, qua paria milliaria conficient. </s> - <s xml:id="s948" xml:space="preserve">Sit-<lb/>qÌue illa dies <var>.u.n.</var> Å¿equitur <reg norm="etiam" type="context">etiã</reg> tranÅ¿acta ea die, tot diebus vtrique ambulandum eſſe <lb/>quot iter egere anteaquam ad diem <var>.u.n.</var> <lb/>peruenirent, vt tanto numero primus à ſe-<lb/> - <anchor type="figure" xlink:label="fig-0082-01a" xlink:href="fig-0082-01"/> - cundo Å¿uperetur, <reg norm="quanto" type="context">quãto</reg> Å¿ecundum primus <lb/>Å¿uperauerat, vnde totalis numerus <var>.o.f.</var> mi <lb/>nor erit duplo <var>.o.n.</var> vnitate ex communi <lb/>conceptione, Å¿ed ita etiam Å¿e habet termi-<lb/>nus <var>.u.n.</var> hoc eÅ¿t minor duplo <var>.o.n.</var> per <var>.o.</var> vt <lb/>94. theoremate dictum fuit, itaque <var>.o.f.</var> æ-<lb/>qualis erit <var>.u.n.</var> quoderat propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0082-01" xlink:href="fig-0082-01a"> - <image file="0082-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0082-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="109"> - <head xml:id="head126" xml:space="preserve">THEOREMA <num value="109">CIX</num>.</head> - <p> - <s xml:id="s949" xml:space="preserve">SIN verò progreſſio Å¿ecundi viatoris, non ab vnitate Å¿ed à binario inchoata, <lb/>per binarium quoque aÅ¿cenderet, <reg norm="numerusque" type="simple">numerusqÌ;</reg> milliarium diurnorum primi via <lb/>toris par eſſet, abÅ¿que dubio quadam die paria milliaria <reg norm="vterque" type="simple">vterq;</reg> conficeret, quÄ™ Å¿igni <lb/>ficetur <var>.u.n.</var> qua tranÅ¿acta, tot diebus vtrique <reg norm="ambulandum" type="context">ambulãdum</reg> erit, quot <reg norm="fuerunt" type="context">fuerÅ©t</reg> <reg norm="dum" type="context">dÅ©</reg> primus <lb/>Å¿ecundum Å¿uperaret, vt totidem alijs pri-<lb/>mus à ſecundo Å¿uperetur, in qua tamen <lb/> - <anchor type="figure" xlink:label="fig-0082-02a" xlink:href="fig-0082-02"/> - progreſſione terminus <var>.u.n.</var> Å¿emper duplus <lb/>eÅ¿t numero terminorum <var>.o.n.</var> ex .95. theo-<lb/>remate, <reg norm="totque" type="simple">totqÌ;</reg> Å¿unt infra <var>.u.n.</var> termini vÅ¿que <lb/>ad <var>.f.</var> quot Å¿upra. </s> - <s xml:id="s950" xml:space="preserve">ex quo illud Å¿equitur om <lb/>nesterminosaut dies <var>.o.n.f.</var> pauciores eſſe <lb/><var>u.n.</var> vnitate, atque ita præcipit, regula de-<lb/>trahendam eſſe vnitatem ex numero mil-<lb/>liarium diurnorum primi viatoris, Å¿i dierum numerum habere voluerimus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0082-02" xlink:href="fig-0082-02a"> - <image file="0082-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0082-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="110"> - <head xml:id="head127" xml:space="preserve">THEOREMA <num value="110">CX</num>.</head> - <p> - <s xml:id="s951" xml:space="preserve">SED Å¿i in eiuÅ¿modi progreſſione numerus milliarium diurnorum primi viato-<lb/>ris diÅ¿par fuerit, patet quòd primus Å¿ecundum numero diÅ¿pari Å¿uperabit, do-<lb/>necad vnitatem perueniatur <reg norm="vi- ciſſimque" type="simple">vi-<lb/> - <anchor type="figure" xlink:label="fig-0082-03a" xlink:href="fig-0082-03"/> - ciſſimqÌ;</reg> primum Å¿ecundus, in-<lb/>choando ab vnitate, </s> - <s xml:id="s952" xml:space="preserve">quare nul-<lb/>la <reg norm="vnquam" type="context">vnquã</reg> die paria milliaria vter-<lb/>que conficiet, Å¿it itaque vltima <lb/>dies, qua primus Å¿ecundum vnita <lb/>tem antecedit <var>.u.n.</var> qui terminus <lb/>duplus eÅ¿t numero terminorum <lb/><var>o.n.</var> & cum illa die primus Å¿ecun-<lb/>dum milliario antecedat, Å¿equen - <pb o="71" rhead="THEOREM. ARIT." n="83" file="0083" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0083"/> - te verò à ſecundo milliario vno primus antecedatur, ex communi Å¿cientia neceſſe <lb/>eÅ¿t Å¿ecundum tot diebus <reg norm="cum" type="context">cÅ©</reg> primo iter agere quot Å¿unt <var>.o.n.</var> qui Å¿imul æquales erunt <var>.<lb/>u.n.</var> Å¿ed <var>.u.n.</var> minor eft numero milliarium diurnorum primi vnitate <var>.e</var>. </s> - <s xml:id="s953" xml:space="preserve">Itaque rectè <lb/>Å¿equemur regulam, quæ iubet ex numero milliarium vnitatem demere, quo nu <lb/>merum dierum habere poſſimus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0082-03" xlink:href="fig-0082-03a"> - <image file="0082-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0082-03"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="111"> - <head xml:id="head128" xml:space="preserve">THEOREMA <num value="111">CXI</num>.</head> - <p> - <s xml:id="s954" xml:space="preserve">SI verò Å¿ecundi viatoris progreſſio per ternarium aÅ¿cenderet, Å¿umpto initio ab <lb/>ipÅ¿o ternario, animaduertendum eÅ¿t an numerus milliarium diurnorum primi, <lb/>ternario menÅ¿uretur necne, etenim Å¿i menÅ¿uretur, tandem aliquando paria millia-<lb/>ria conficient, quæ dies Å¿it <var>.u.n.</var> </s> - <s xml:id="s955" xml:space="preserve">quare Å¿ub <lb/> - <anchor type="figure" xlink:label="fig-0083-01a" xlink:href="fig-0083-01"/> - <var>u.n.</var> totidem quot Å¿upra termini <reg norm="erunt" type="context">erÅ©t</reg>, <lb/>& <reg norm="cum" type="context">cÅ©</reg> <var>.o.n.</var> tertia Å¿it pars <var>.u.n.</var> ex .95. theo-<lb/>remate. </s> - <s xml:id="s956" xml:space="preserve">Itaque tota <var>.o.f.</var> minor erit <lb/>duabus tertijs <var>.u.n.</var> vnitate, vtiam re-<lb/>ctè Å¿umendæ Å¿int duæ tertiæ partes <var>.u.n.</var> <lb/>ex quibus vnitas detrahatur Å¿uperÅ¿itqÌue <lb/>numerus <var>.o.f.</var> dierum quæſitorum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0083-01" xlink:href="fig-0083-01a"> - <image file="0083-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0083-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="112"> - <head xml:id="head129" xml:space="preserve">THEOREMA <num value="112">CXII</num>.</head> - <p> - <s xml:id="s957" xml:space="preserve">CVM verò milliarium numerus p rimi viatoris metirinon poterit à numero <lb/>aÅ¿cendente Å¿ecundi, patet nullam futuram diem qua pari milliaria conficient, <lb/></s> - <s xml:id="s958" xml:space="preserve">quare illa vltima qua primus Å¿ecundum antecedet, vno aut duobus milliaribus an-<lb/>tecedet in præſenti caÅ¿u. </s> - <s xml:id="s959" xml:space="preserve">Antecedat itaque duobus milliaribus, <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> dies <var>.u.n.</var> & alte <lb/>ra <var>.t.i.</var> Å¿ecundus primum vno milliari Å¿uperabit, ita quod Å¿ub <var>.t.i.</var> non poterunt plu-<lb/>res integros dies iter agere, quam ambulauerunt ante diem <var>.u.n.</var> hoc eÅ¿t vÅ¿quequo <lb/>Å¿ecundtis iunctus Å¿it primo, qui numerus dierum, tertia parte <var>.o.n.</var> ipÅ¿ius <var>.u.n.</var> vnitate <lb/>minor erit, cum ex .95. theoremate <var>.o.n.</var> Å¿it tertia pars <var>.u.n.</var> ex quo numerus <var>.o.f.</var> ter-<lb/>minorum aut dierum intergrorum cognitus erit, qui Å¿i cum numero alcendente <lb/>cognoÅ¿cetur, Å¿tatim ex .99. theoremate deueniemus in cognitionem vltimi diei in <lb/>tegri <var>.s.f.</var> atque ita etiam totius Å¿ummæ progreſſionis ex .95. theoremate. </s> - <s xml:id="s960" xml:space="preserve">Iam verò <lb/>cognito numero milliarium diurnorum primi, Å¿imul cum numero terminorum, aut <lb/>dierum conÅ¿equenter nouerimus rectanguli Å¿ummam, hoc eÅ¿t productum à primo <lb/>viatore formatum, quarum duarum Å¿ummarum in præſenti caÅ¿u Å¿emper ea, quæ <lb/>huiuÅ¿modi producti eÅ¿t, maior erit, cum conÅ¿titutum fuerit Å¿ecundum viatorem à <lb/>primo Å¿uperari ipÅ¿a die <var>.u.n.</var> vno milliari amplius quam Å¿equente die <var>.t.i.</var> primus à ſe <lb/>cundo Å¿uperatur, tum pari gradu iter egerunt Å¿ub <var>.t.i.</var> quo Å¿upra <var>.u.n.</var> ambulauerant. <lb/></s> - <s xml:id="s961" xml:space="preserve">Hoc animaduertendo, quòd Å¿i Å¿umma progreſſionis maior eſſet rectangulo, ex ea <lb/>Å¿umma neceſſe eſſet <reg norm="numerum" type="context">numerÅ©</reg> mil <lb/>liarium vltimi termini in Å¿umma <lb/> - <anchor type="figure" xlink:label="fig-0083-02a" xlink:href="fig-0083-02"/> - incluÅ¿i detrahere, & reÅ¿iduo ope-<lb/>rari. </s> - <s xml:id="s962" xml:space="preserve">Nunc verò Å¿ummam pro-<lb/>greſſionis exÅ¿umma rectanguli à <lb/>primo viatore facti Å¿ubtrahi de-<lb/>bet, <reg norm="reÅ¿iduumque" type="simple">reÅ¿iduumqÌ;</reg> Å¿eruari <reg norm="voceturque" type="simple">voceturqÌ;</reg> - <pb o="72" rhead="IO. BAPT. BENED." n="84" file="0084" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0084"/> - <reg norm="primum" type="context">primÅ©</reg> <reg norm="reÅ¿iduum" type="context">reÅ¿iduÅ©</reg>. </s> - <s xml:id="s963" xml:space="preserve">Ad hæc <reg norm="numerum" type="context">numerÅ©</reg> <reg norm="milliariorum" type="context">milliariorÅ©</reg>, quæ <reg norm="Å¿ecundus" type="context">Å¿ecÅ©dus</reg> viator die <reg norm="Å¿eqennti" type="context">Å¿eqẽnti</reg> <var>.s.f.</var> confi <lb/>ciet <reg norm="Å¿umatur" type="simple">Å¿umat̃</reg>, ex quo numerus <reg norm="milliarium" type="context">milliariÅ©</reg> <reg norm="diurnorum" type="context">diurnorÅ©</reg> primi detrahatur, <reg norm="residuumque" type="context simple">reÅ¿iduÅ©qÌ;</reg> pariter re <lb/>Å¿eruetur, <reg norm="voceturque" type="simple">voceturqÌ;</reg> Å¿ecundum reÅ¿iduum, poÅ¿tmodum numerum milliarium primi <lb/>vnius diei multiplicetur per primum reÅ¿iduum Å¿eruatum, <reg norm="productumque" type="simple">productumqÌ;</reg> per <reg norm="Å¿ecundum" type="context">Å¿ecundÅ©</reg> <lb/>reÅ¿iduum diuidatur <var>.a.c.</var> <reg norm="proueniens" type="context">proueniẽs</reg> (<reg norm="quod" type="simple">ê™</reg> erit iter primi in Å¿equenti die) iungatur reÅ¿i-<lb/>Å¿iduo primo, tot enim erunt milliaria conficienda a Å¿ecundo Å¿equenti die, vt <lb/>Å¿eÅ¿e conÅ¿equantur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0083-02" xlink:href="fig-0083-02a"> - <image file="0083-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0083-02"/> - </figure> - </div> - <p> - <s xml:id="s964" xml:space="preserve">Vtautem Å¿ciamus quantam partem diei <reg norm="Å¿eqenntis" type="context">Å¿eqẽntis</reg>, Å¿ingulos itinere agere oporteat, <lb/><reg norm="proueniens" type="context">proueniẽs</reg> per .24. horas multiplicetur (Å¿uppoÅ¿ito quod <reg norm="ambulantes" type="context">ambulãtes</reg> <reg norm="nullam" type="context">nullã</reg> <reg norm="requiem" type="context">requiẽ</reg> nec <lb/>die nec nocte <reg norm="capiant" type="context">capiãt</reg>), <reg norm="productumque" type="simple context simple">ê“ductÅ©qÌ;</reg> <reg norm="per" type="simple">ê‘</reg> <reg norm="numerum" type="context">numerÅ©</reg> milliariorum vuius diei primi viatoris di <lb/>uidatur, ex quo dabitur <reg norm="quantitas" type="context">quãtitas</reg> <reg norm="horarum" type="context">horarÅ©</reg>, & pars horæ, qua <reg norm="cuique" type="simple">cuiq;</reg> illa die <reg norm="ambulandum" type="context">ambulandÅ©</reg> <lb/>eÅ¿t. </s> - <s xml:id="s965" xml:space="preserve">Idem accideret Å¿i primum reÅ¿iduum reÅ¿eruatum cum proueniente in Å¿ummam <lb/>colligeretur, <reg norm="eaque" type="simple">eaqÌ;</reg> Å¿umma per .24. horas multiplicaretur, <reg norm="productumque" type="simple">productumqÌ;</reg> per <reg norm="nume- rum" type="context">nume-<lb/>rÅ©</reg> milliariorum Å¿equenti die à ſecundo conficiendorum diuideretur. </s> - <s xml:id="s966" xml:space="preserve">I dipÅ¿um quo-<lb/>que eueniret multiplicato primo reÅ¿iduo per .24. & producto per Å¿ecundum reÅ¿i-<lb/>duum diuiÅ¿o.</s> - </p> - <p> - <s xml:id="s967" xml:space="preserve">Exempli gratia, primus viator diurna milliaria vndecim conficit, Å¿ecundus, pri-<lb/>ma die tria, Å¿ecunda .6. tertia .9. <reg norm="atque" type="simple">atq;</reg> ita deinceps, diuidatur ergo .11. per .3. vnde <lb/>pro numero <var>.o.n.</var> dabitur .3. <reg norm="Å¿upereritque" type="simple">Å¿upereritqÌ;</reg> .2. </s> - <s xml:id="s968" xml:space="preserve">quare: <var>u.n.</var> ab <var>.e.n.</var> duobus milliaribus Å¿u-<lb/>perabitur, et <var>.i.t.</var> dictum <var>.e.n.</var> vno milliario, ex quo ante diem <var>.e.u.n.</var> duobus diebus <lb/>iter egerunt, <reg norm="totque" type="simple">totqÌ;</reg> diebus ambulandum erit poÅ¿t <var>.t.i.</var> hoceÅ¿t .6. in vniuerÅ¿um inte-<lb/>gris. </s> - <s xml:id="s969" xml:space="preserve">Ad hÄ™c multiplicato <var>.o.f.</var> hoc eÅ¿t .6. per <var>.x.o.</var> hoc eÅ¿t .3. habebimus <var>.s.f.</var> <reg norm="milliariorum" type="context">milliariorÅ©</reg> <lb/>18. <reg norm="tum" type="context">tÅ©</reg> <reg norm="coniuncto" type="context context">cõiÅ©cto</reg> <var>.x.o.</var> primo termino hoc eÅ¿t .3. <reg norm="cum" type="context">cÅ©</reg> <var>.s.f.</var> hoc eÅ¿t .18. vltimo termino, habe <lb/>bimus .21. quo multiplicato <reg norm="cum" type="context">cÅ©</reg> dimidio <var>.o.f.</var> hoc eÅ¿t .3. habebimus totam Å¿ummam <lb/>progreſſionis .63. Å¿ex dierum integrorum ex .94. theoremate, tum multiplicato .11. <lb/>nempe numero <reg norm="milliariorum" type="context">milliariorÅ©</reg> diurnorum primi cum .6. hoc eÅ¿t cum <var>.o.f.</var> habebimus pa <lb/>rallelogrammum à primo Å¿ex diebus integris confectum milliariorum .66. ex quo <lb/>detracta .63. Å¿umma inquam progreſſionis, Å¿upererit pro primo reÅ¿iduo .3. Å¿umptis <lb/>poÅ¿tea milliaribus .21. pro itinere, quod Å¿ecundus die Å¿equenti <var>.s.f.</var> conficeret, & ex <lb/>ijs detracto numero <reg norm="milliariorum" type="context">milliariorÅ©</reg> diurnorum primi, nempe .11. Å¿ecundum reÅ¿iduum <lb/>erit 10. quod pro diuidenti Å¿eruabitur. </s> - <s xml:id="s970" xml:space="preserve">Iam multiplicato .11. cum primo reÅ¿iduo <num value="3">.<lb/>3.</num> dabitur .33. qui diuiÅ¿us per .10. Å¿ecundum reÅ¿iduum profert .3. cum tribus decimis, <lb/><reg norm="eritque" type="simple">eritqÌ</reg> iter à primo viatore Å¿equenti die conficiendum, hoc etiam ipÅ¿um proueniens <lb/>cum primo reÅ¿iduo .3. coniunctum, dat .6. cum tribus decimis, quod eÅ¿t iter Å¿ecundi <lb/>viatoris illa Å¿equenti die. </s> - <s xml:id="s971" xml:space="preserve">Ad inueniendam autem quantitatem diei, qua vtrique <lb/>ambulandum eÅ¿t, perinde erit multiplicare proueniens .3. & tres decimas per .24. ho <lb/>ras, & productum per .11. dimidium iter primi viatoris partiri, ac multiplicare Å¿um <lb/>mam .6. & tres decimas cum .24. horis, <reg norm="productumque" type="simple">productumqÌ</reg> diuidere per .21. hoc eÅ¿t periter <lb/>Å¿ecundi viatoris Å¿equentis diei, vtrinque enim Å¿emper Å¿eptem horæcum .12. minu <lb/>tis prouenient. </s> - <s xml:id="s972" xml:space="preserve">IdipÅ¿um accidet multiplicato per .24. horas primo reÅ¿iduo .3. pro-<lb/><reg norm="ductoque" type="simple">ductoqÌ;</reg> diuiÅ¿o per Å¿ecundum reÅ¿iduum .10.</s> - </p> - <p> - <s xml:id="s973" xml:space="preserve">Quarum Å¿peculationum gratia, totum iter parallelogrammi primi viatoris die-<lb/>rum integrorum fignificetur linea <var>.n.e.</var> Å¿umma verò progreſſionis Å¿ecundi linea <var>.f.m.</var> <lb/>parallela <var>.n.e.</var> <reg norm="eritque" type="simple">eritqÌ;</reg> <var>.f.m.</var> minor <var>.n.e</var>. </s> - <s xml:id="s974" xml:space="preserve">ConÅ¿tituamus deinde à termino <var>.f.n.</var> (majoris <lb/><reg norm="intelligentiÄ™" type="context">intelligẽtiÄ™</reg> gratia) vtranque <reg norm="perpendiculariter" type="context">perpẽdiculariter</reg> duci, <reg norm="producatur" type="simple">ê“ducatur</reg> deinde <var>.n.e.</var> donec <var>.e.<lb/>d.</var> æqualis Å¿it itineri diurno primi viatoris, item etiam producatur <var>.f.m.</var> donec <var>.m.K.</var> <lb/>æqualis Å¿it itineri à ſecundo confecto Å¿equenti die vltimum integrum progreſſio- - <pb o="73" rhead="THEOREM. ARITH." n="85" file="0085" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0085"/> - nis, ex quo <var>.m.k.</var> prolixior erit <var>.e.d.</var> ex præſup poÅ¿ito. </s> - <s xml:id="s975" xml:space="preserve">PoÅ¿tmodum <var>.m.e.</var> et <var>.k.d.</var> dua-<lb/>bus lineis rectis coniungantur, quæ productæ concurrentin puncto <var>.b.</var> ducatur pari-<lb/>ter <var>.e.g.</var> à puncto <var>.e.</var> parallela <var>.b.k.</var> et <var>.m.a</var>: <var>e.h.</var> et <var>.b.q.</var> parallelæ <var>.f.n.</var> ex quo <var>.f.m.</var> æqua-<lb/>lis erit <var>.n.a.</var> et <var>.m.h</var>: <var>a.e.</var> et <var>.h.q</var>: <var>e.o.</var> et <var>.g.k</var>: <var>e.d.</var> et <var>.f.q</var>: <var>n.o.</var> ex .34. primi Eucli. </s> - <s xml:id="s976" xml:space="preserve">vnde pro <lb/>portio <var>.m.h.</var> ad <var>.h.q.</var> erit vt <var>.m.g.</var> ad <var>.g.k.</var> quandoquidem vtraque æqualis eÅ¿t propor-<lb/>tioni <var>.m.e.</var> ad <var>.e.b.</var> ex .2. Å¿exti, Å¿ed cum <var>.m.k.</var> et <var>.g.k.</var> notæ Å¿int, pariter cognoÅ¿cetur <var>.m.<lb/>g.</var> Å¿ecundum reÅ¿iduum, cum etiam notæ Å¿int <var>.n.e.</var> et <var>.n.a</var>. </s> - <s xml:id="s977" xml:space="preserve">Itaque cognoÅ¿cemus <var>.a.e.</var> hoc <lb/>eÅ¿t <var>.m.h.</var> cognitis verò <var>.m.g</var>: <var>g.k.</var> et <var>.m.h.</var> ex .15. Å¿exti aut .20. Å¿eptimi cognoÅ¿cetur <var>.h.<lb/>q.</var> erit igitur <var>.a.e.</var> aut quod idem eÅ¿t <var>.m.</var> hprimum reÅ¿iduum, et <var>.m.g.</var> Å¿ecundum, et <var>.h.<lb/>q.</var> aut <var>.e.o.</var> proueniens, et <var>.n.o.</var> et <var>.f.q.</var> itinera vtriuÅ¿que viatoris inter Å¿e æqualia. <lb/></s> - <s xml:id="s978" xml:space="preserve">Nec verò prætermittenda eÅ¿t Å¿peculatio vltimæ rationis inueniendæ quantitatis <lb/>diei, quæ conÅ¿tat ope diuiÅ¿ionis producti <var>.m.h.</var> in .24. per <var>.m.g</var>. </s> - <s xml:id="s979" xml:space="preserve">Ea autem eiuÅ¿modi <lb/>eÅ¿t. </s> - <s xml:id="s980" xml:space="preserve">Probatum fuit Å¿ic Å¿e habere <var>.m.h.</var> ad <var>.h.q.</var> ut <var>.m.g.</var> ad <var>.g.k</var>. </s> - <s xml:id="s981" xml:space="preserve">Itaque componendo <lb/>Å¿ic Å¿e habebit <var>.m.q.</var> ad <var>.h.q.</var> vt <var>.m.k.</var> ad <var>.g.k.</var> & permutando <var>.m.q.</var> ad <var>.m.k.</var> vt <var>.h.q.</var> ad <var>.g.<lb/>k</var>. </s> - <s xml:id="s982" xml:space="preserve">Sed cum Å¿ic Å¿e habeat <var>.m.h.</var> ad <var>.h.q.</var> vt <var>.m.g.</var> ad <var>.g.k.</var> permutando Å¿ic Å¿e habebit <var>.m.<lb/>h.</var> ad <var>.m.g.</var> vt <var>.h.q.</var> ad <var>.g.k.</var> itaque <lb/>ex .11. quinti ita <var>.m.h.</var> ad <var>.m.g.</var> vt <var>.<lb/> - <anchor type="figure" xlink:label="fig-0085-01a" xlink:href="fig-0085-01"/> - m.q.</var> ad <var>.m.k.</var> ex quo permutando <lb/><var>m.h.</var> ad <var>.m.q.</var> vt <var>.m.g.</var> ad <var>.m.k.</var> Å¿ed <lb/><reg norm="cum" type="context">cÅ©</reg> <var>.m.k.</var> Å¿it motus toti diei reÅ¿pon <lb/>dens, Å¿ecurè dicere poterimus, Å¿i <lb/><var>m.g.</var> talis eÅ¿t reÅ¿pectu horarum <num value="24">.<lb/>24.</num> Å¿ignificatarum per <var>.m.k.</var> qualis <lb/> - <anchor type="figure" xlink:label="fig-0085-02a" xlink:href="fig-0085-02"/> - erit <var>.m.h.</var> & quo <lb/>tæ parti dieire-<lb/>Å¿pondens: </s> - <s xml:id="s983" xml:space="preserve">quæ <lb/><reg norm="poÅ¿tmodum" type="context">poÅ¿tmodÅ©</reg> erit <var>.<lb/>m.q.</var> quæ, vt <reg norm="di- ctum" type="context">di-<lb/>ctÅ©</reg> fuit, talis eÅ¿t <lb/>reÅ¿pectu <var>.m.k.</var> <lb/>qualis <var>.m.h.</var> re-<lb/>Å¿pectu <var>.m.g</var>. </s> - <s xml:id="s984" xml:space="preserve">Reli <lb/>quÄ™ duæ Å¿pecula <lb/>tiones priorum <lb/><reg norm="modorum" type="context">modorÅ©</reg>, vna & <lb/>eadem eÅ¿t, <reg norm="facilisque" type="simple">facilisqÌ;</reg> per Å¿e mediocriter intelligenti. </s> - <s xml:id="s985" xml:space="preserve">Eodem modo reliquæ omnes <lb/>progreſſiones Å¿ecundi viatoris <reg norm="cum" type="context">cÅ©</reg> rectangulo primi conferri ex hoc theoremate <lb/>poterunt.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0085-01" xlink:href="fig-0085-01a"> - <image file="0085-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0085-01"/> - </figure> - <figure xlink:label="fig-0085-02" xlink:href="fig-0085-02a"> - <image file="0085-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0085-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="113"> - <head xml:id="head130" xml:space="preserve">THEOREMA <num value="113">CXIII</num>.</head> - <p> - <s xml:id="s986" xml:space="preserve">PRoponitur & aliud, primum Å¿cilicet viatorem iter incipere diebus aliquot an-<lb/>tè Å¿ecundum, primum tamen lentius, quà m Å¿ecundum ambulare, & utrunque <lb/>eorum certa quædam milliaria conficere. </s> - <s xml:id="s987" xml:space="preserve">Iam Å¿iÅ¿cire voluerimus in quot diebus <lb/>Å¿eÅ¿e conÅ¿equentur, uulgaris regula iubet, inÅ¿pici quot milliaria primus Å¿olus iter a-<lb/>gens confecerit, tum animaduerti differentiam diurnam motus vnius ab altero, <reg norm="atque" type="simple">atq;</reg> <lb/>milliarium numerum primi viatoris Å¿oli abundantis per hanc <reg norm="differentiam" type="context">differentiã</reg> diuidi, pro <lb/>ueniens autem erit numerus dierum quæſitus.</s> - </p> - <pb o="74" rhead="IO. BAPT. BENED." n="86" file="0086" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0086"/> - <p> - <s xml:id="s988" xml:space="preserve">Exempli gratia, Å¿i primus octo diebus antequam Å¿ecundus iter arripuiſſet, con-<lb/><reg norm="feciſſetque" type="simple">feciſſetqÌ;</reg> fingulis diebus .20. milliaria, tum Å¿ecundus .25. quotidie perfeciſſet, mul <lb/>tiplicandus eſſet numerus .8. cum .20. ex quo darentur .160. milliaria à primo Å¿olo <lb/><reg norm="ambulante" type="context">ambulãte</reg> confecta, quibus diuiÅ¿is per .5. differentiam motuum diurnorum, daretur <num value="32">.<lb/>32.</num> numerus quæſitus dierum.</s> - </p> - <p> - <s xml:id="s989" xml:space="preserve">Cuius ratio apertiſſima eÅ¿t. </s> - <s xml:id="s990" xml:space="preserve">Sint enim duo rectanguli <var>.a.n.</var> et <var>.u.i.</var> æquales inter <lb/>Å¿e, quibus motus itinerarium Å¿ignificentur, quorum <var>.a.n.</var> Å¿it primi, et <var>.u.i.</var> Å¿ecundi, præ <lb/>tereà . <var>a.c.</var> numerum milliarium diurnorum primi .et <var>.u.e.</var> Å¿ecundi, ex quo <var>.a.c.</var> minor <lb/>erit <var>.u.e.</var> per <var>.o.e.</var> atque ita <var>.o.e.</var> co-<lb/>gnoÅ¿cetur. </s> - <s xml:id="s991" xml:space="preserve">Tum <var>.c.o.</var> numerum <reg norm="dierum" type="context">dierÅ©</reg> <lb/> - <anchor type="figure" xlink:label="fig-0086-01a" xlink:href="fig-0086-01"/> - primi Å¿oli iter agentis denotet, <reg norm="cumque" type="context simple">cÅ©q;</reg> <lb/>conÅ¿tituamus <var>.a.n.</var> æqualem eſſe <var>.u.i.<lb/>o.i.</var> Ä™qualis erit <var>.o.a.</var> atque <var>.o.a.</var> cogni <lb/>tus ex Å¿uis producentibus <var>.a.c.</var> et <var>.c.o.</var> <lb/>itaque <var>.o.i.</var> etiam cognitus, qui diui-<lb/>Å¿us per latus cognitum <var>.o.e.</var> dabit <var>.e.<lb/>i.</var> cognitum numerum Å¿cilicet dierum, quibus Å¿ecundo ambulandum eÅ¿t, vt primum <lb/>conÅ¿equatur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0086-01" xlink:href="fig-0086-01a"> - <image file="0086-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0086-01"/> - </figure> - </div> - <head xml:id="head131" xml:space="preserve">APPENDIX THEOREM. CXIII.</head> - <p> - <s xml:id="s992" xml:space="preserve">AB hoc theoremate Å¿umpÅ¿i ordinem illius operationis, numeris mediantibus, ad <lb/>inueniendam exactam temporis quantitatem, Å¿eu interuallum, <reg norm="quod" type="simple">ê™</reg> tranÅ¿it, vel in <lb/>tercedit inter vnam mediocrem coniunctionem & aliam proximam <reg norm="Å¿equentem" type="context">Å¿equentẽ</reg> duo <lb/>rum planetarum, vt patet in epiÅ¿tola noÅ¿tra ad IlluÅ¿trem Bernardum Trottum con-<lb/>tra Benedictum Altauillam repræhenÅ¿orem Ephemeridum. </s> - <s xml:id="s993" xml:space="preserve">Verum tamen eÅ¿t <reg norm="quod" type="simple">ê™</reg> <lb/>cum praxis huiuÅ¿modi theorematis Å¿it multiplex, viÅ¿um fuit vnam proponere, quÄ™ <lb/>non ita perÅ¿picua Å¿it, Å¿ed Å¿ubobÅ¿cura, non quòd aliquid voluerim latere illum ami <lb/>cum mihi dilectiſſimum, cui priuatim omnes modos prius oÅ¿tenderam, Å¿ed vt cere-<lb/>brum illius mei aduerÅ¿arij in laberintum conijcerem inextricabilem vt feci, quam-<lb/>uis modus ille egregius etiam Å¿it, vt nunc oÅ¿tendam.</s> - </p> - <p> - <s xml:id="s994" xml:space="preserve">In dicta epiÅ¿tola igitur mente cogitaui medium motum tardioris planetæ, pu-<lb/>ta Å¿aturni, illius temporis quo velocior planeta, Å¿cilicet Iupiter, percurrit Å¿uo medio <lb/>motu totum zodiacum, incipiendo ambo eodem temporis puncto, nec non ab vna <lb/>eorum media coniunctione, hoc eÅ¿t ab eodem zodiaci puncto, in quo coniunctÄ™ fue <lb/>runt eorum lineæ mediorum motuum, vbi inueni vi regulæ de tribus, quòd Satur <lb/>nus Å¿pacio dierum vnius mediocris reuolutionis Iouis, qui Å¿unt .4328. progreditur <lb/>medio motu gra .145. min .4. hoc eÅ¿t min .8704. pofito quòd ipÅ¿e Saturnus perficiat <lb/>vnam mediam reuolutionem Å¿pacio dierum .10740. vt dixi. </s> - <s xml:id="s995" xml:space="preserve">Incipiendo igitur ite <lb/>rum Iupiter aliam reuolutionem percurrere, reperto Saturno per min .8704. ante <lb/>ipÅ¿um Å¿pacio .4328. dierum, certus eram hos dies Å¿ignificatos eſſe à linea <var>.a.u.</var> vel <var>.c.<lb/>o.</var> (æquales enim inuicem Å¿unt) in figura huiuÅ¿modi theorematis, & quòd rectangu <lb/>lum <var>.a.o.</var> præbebat Å¿ummam graduum .145. min .4. hoc eÅ¿t min .8704. et quòd <var>.a.c.</var> <lb/>vel <var>.o.u.</var> Å¿ignificabat iter vnius diei ipÅ¿ius Saturni, et <var>.u.e.</var> iter vnius diei Iouis. </s> - <s xml:id="s996" xml:space="preserve">Cogi-<lb/>temus nunc <var>.u.x.</var> Å¿ignificari dies .30. & à puncto <var>.x.</var> productam eſſe <var>.x.f.</var> parallelam ipÅ¿i <lb/><var>u.o.e.</var> vnde certi erimus rectangulum <var>.e.x.</var> Å¿ignificare iter Ionis Å¿pacio temporis die-<lb/>rum .30. rectangulum verò <var>.o.x.</var> iter Saturni eodem temporis interuallo, vnde rectan - <pb o="75" rhead="THEOREM. ARIT." n="87" file="0087" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0087"/> - gulum <var>.e.x.</var> erit minutorum .149. & Å¿ecundorum .43. et <var>.o.x.</var> minutorum .60. & Å¿ecun. <lb/>20. vt in dicta epiÅ¿tola, vnde rectangulum <var>.o.f.</var> erit min .89. & Å¿ecun .23. & quia re-<lb/>ctangulum <var>.o.i.</var> æquale eÅ¿t rectangulo <var>.a.o.</var> ergo <var>.o.i.</var> Å¿imiliter continebit min .8704. <lb/>Nunc quia <var>.a.c.</var> vel <var>.o.u.</var> denotat iter vnius diei Saturni et <var>.u.e.</var> vnius diei Iouis vt di-<lb/>ximus ergo <var>.u.o.</var> erit minutorum .2. Å¿ecun <var>.o.</var> & tertiarum .40. videlicet tertiarum <num value="7240">.<lb/>7240.</num> Å¿uppoÅ¿ito periodo totaliipÅ¿ius Saturni dierum .10740. et <var>.u.e.</var> erit <reg norm="minutorum" type="context">minutorÅ©</reg>. <lb/>4. Å¿ecun .59. & ter .27. vel circa hoc eÅ¿t tertiarum .17967. vnde <var>.o.e.</var> erit tertiarum <num value="10727">.<lb/>10727.</num> </s> - <s xml:id="s997" xml:space="preserve">NuncÅ¿i dixerimus cum <var>.o.e.</var> tertiarum .10727. dat <var>.o.u.</var> vel <var>.a.c.</var> (nam tam <lb/>vna quam altera eÅ¿t tertiarum .7240.) quid dabit <var>.a.u.</var> vel <var>.o.c.</var> (quia tam vna quam <lb/>altera eÅ¿t partium .4328.) clarum erit quòd dabit <var>.o.n.</var> vel <var>.u.t.</var> uel <var>.e.i.</var> quia tam vna <lb/>quam altera erit partium .2921. quæ partes coniunctæ cum fuerint cum partibus ip-<lb/>Å¿ius <var>.a.u.</var> dabunt totam <var>.a.t.</var> <reg norm="partium" type="context">partiÅ©</reg> .7249. quæ erunt tot dies, hoc eÅ¿t periodus quæſita.</s> - </p> - <p> - <s xml:id="s998" xml:space="preserve">Alia methodo Å¿imiliter poſſumus idem cognoÅ¿cere, Å¿cilicet dicendo Å¿i rectangu <lb/>lum <var>.f.o.</var> quod eÅ¿t minutorum .89. & Å¿ecun .23. hoc eÅ¿t Å¿ecundorum .5363. dat rectan <lb/>gulum <var>.o.x.</var> minutorum .60. & Å¿ecun .20. hoc eÅ¿t Å¿ecun .3620. quid dabit <var>.a.u.</var> partium <lb/>4328. vnde veniet <var>.u.t.</var> partium .2921. Å¿imiliter, eo quod eadem proportio eÅ¿t rectan <lb/>guli <var>.f.o.</var> ad <var>.o.x.</var> quæ <var>.e.o.</var> ad <var>.o.u.</var> ex prima Å¿exti, vel .18. 19. Å¿eptimi Å¿eu .15. quinti.</s> - </p> - <p> - <s xml:id="s999" xml:space="preserve">Poſſet etiam aliquis dicere Å¿i <var>.f.o.</var> dat <var>.o.x.</var> quid dabit <var>.o.a.</var> vnde veniet <var>.o.t.</var> quo <lb/>diuiÅ¿o per <var>.o.u.</var> daret <var>.u.<lb/>t.</var> quia ita Å¿e habet <var>.a.o.</var> <lb/> - <anchor type="figure" xlink:label="fig-0087-01a" xlink:href="fig-0087-01"/> - ad <var>.o.t.</var> vt <var>.a.u.</var> ad <var>.u.t.</var> ex <lb/>Å¿upra hic iam citatis.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0087-01" xlink:href="fig-0087-01a"> - <image file="0087-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0087-01"/> - </figure> - </div> - <p> - <s xml:id="s1000" xml:space="preserve">Sed ego, in dicta epi-<lb/>Å¿tola, aliam methodum <lb/>obÅ¿eruaui, quæ eÅ¿t multi <lb/>plicando minuta .8704. <lb/>per .30. <reg norm="productumque" type="simple">productumqÌ;</reg> di <lb/>uiÅ¿i per min .5363. quaÅ¿i <lb/>dicens. </s> - <s xml:id="s1001" xml:space="preserve">Si <var>.o.f.</var> dat <var>.o.i.</var> <lb/>quid dabit <var>.e.f</var>. </s> - <s xml:id="s1002" xml:space="preserve">Vnde exiam Å¿upradictis propoÅ¿itionibus veniet <var>.e.i.</var> & quia permu-<lb/>tando ita Å¿e habet <var>.o.f.</var> ad <var>.e.f.</var> vt <var>.o.i.</var> ad <var>.e.i.</var> ideo dixi, Å¿i min .89. cum Å¿ecun .23. dat .30 <lb/>quid dabit min .8704.</s> - </p> - </div> - <div type="math:theorem" level="3" n="114"> - <head xml:id="head132" xml:space="preserve">THEOREMA <num value="114">CXIIII</num>.</head> - <p> - <s xml:id="s1003" xml:space="preserve">PRoponunt veteres & quærunt aliud, nempe Å¿i duo iter agentes, eodem in-<lb/>Å¿tanti diuerÅ¿is è locis proficiÅ¿cantur, ita vt vnus locum vnde alter profectus <lb/>eÅ¿t petat, <reg norm="alterque" type="simple">alterqÌ;</reg> altero velocior Å¿it, quo loco <reg norm="quanue" type="context">quãue</reg> die Å¿ibi inuicem occurrent.</s> - </p> - <p> - <s xml:id="s1004" xml:space="preserve">Exempli gratia, Patauio profectus quidam Taurinum petit, eodem inÅ¿tanti al-<lb/>ter Taurino Patauium, <reg norm="eÅ¿tque" type="simple">eÅ¿tqÌ;</reg> iter .400. milliarium, ille tamen vndecim diebus, hic <lb/>9. motu regulari & vniformi appellit. </s> - <s xml:id="s1005" xml:space="preserve">Quærimus quot milliaria quiÅ¿que confece-<lb/>rit, <reg norm="quotque" type="simple">quotqÌ;</reg> diebus iter egerit, priuÅ¿quam Å¿ibi occurrant.</s> - </p> - <p> - <s xml:id="s1006" xml:space="preserve">Iubent nos veteres dies vtriuÅ¿que inuicem inter Å¿e multiplicare, <reg norm="eritque" type="simple">eritqÌ;</reg> produ-<lb/>ctum .99. item etiam in Å¿ummam colligere, <reg norm="eritque" type="simple">eritqÌ;</reg> Å¿umma .20. per quam <reg norm="productum" type="context">productÅ©</reg>. <lb/>99.@diuiÅ¿erimus dabuntur dies .4. cum .19. vigeÅ¿imis vnius diei. </s> - <s xml:id="s1007" xml:space="preserve">At pro milliaribus <lb/>vtriuÅ¿que, pro eo qui .11. diebus iter conficit, multiplicatis .400. per .4. et .19. vigeÅ¿i <lb/>mis, tum diuiÅ¿o per .11. dabitur numerus .180. à Patauio Taurinum & è contra, qui - <pb o="76" rhead="IO. BAPT. BENED." n="88" file="0088" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0088"/> - Taurino Patauium .220. quæ quiÅ¿que confecerit.</s> - </p> - <p> - <s xml:id="s1008" xml:space="preserve">Dum autem hæc Å¿pecularer attentius, occurrit alius Å¿oluendi modus, quamuis pro <lb/>lixior. </s> - <s xml:id="s1009" xml:space="preserve">Is <reg norm="autem" type="wordlist">aÅ©t</reg> eÅ¿t eiuÅ¿modi. </s> - <s xml:id="s1010" xml:space="preserve"><reg norm="Accipiatur" type="simple">Accipiat̃</reg> medietas minoris numeri <reg norm="dierum" type="context">dierÅ©</reg>, <reg norm="nempe" type="context">nẽpe</reg> .4. <reg norm="cum" type="context">cÅ©</reg> dimi <lb/>dio, & per .400. multiplicetur, <reg norm="productumque" type="context simple">productÅ©qÌ;</reg> per <reg norm="maiorem" type="context">maiorẽ</reg> numerum diuidemus Å¿cilicet <lb/>11. ex quo dabuntur .163. cum .7. vndecimis, quo proueniente è dimidio <reg norm="millia- riorum" type="context">millia-<lb/>riorÅ©</reg> itineris .200. detracto, & <reg norm="preÅ¿iduum" type="context">preÅ¿iduÅ©</reg> <reg norm="nempe" type="context">nẽpe</reg> .36. <reg norm="cum" type="context">cÅ©</reg> .4. vndecimis multiplicato pro <lb/><reg norm="ductoque" type="simple">ductoqÌ;</reg> diuiÅ¿o <reg norm="per" type="simple">ê‘</reg> <reg norm="Å¿ummam" type="context">Å¿ummã</reg> dimidij itineris .200. <reg norm="cum" type="context">cÅ©</reg> primo <reg norm="prouentu" type="context">prouẽtu</reg> .163. et .7. vndecimis <lb/><reg norm="nempe" type="context">nẽpe</reg> <reg norm="per" type="simple">ê‘</reg> .363. ct .7. vndecimas partes <reg norm="proueniet" type="simple">ê“ueniet</reg> .16. <reg norm="cum" type="context">cÅ©</reg> .4. vndecimis, quo <reg norm="coniuncto" type="context">cõiuncto</reg> pri <lb/>mo <reg norm="prouenienti" type="simple context">ê“ueniẽti</reg>, primus .180. milliaria <reg norm="confecerit" type="context">cõfecerit</reg>, quæ è .400. detracta Å¿upererunt .220. <lb/>pro itinere Å¿ecundi, qui .9. diebus iter abÅ¿oluit. </s> - <s xml:id="s1011" xml:space="preserve">Ad hæc Å¿i tempus Å¿cire velimus <lb/>eius, qui .11. diebus appellit, multiplicabimus .11. cum .180. <reg norm="productumque" type="simple">productumqÌ;</reg> per .400. <lb/>partiemur, <reg norm="prouenientque" type="simple">prouenientqÌ;</reg> paulominus, quam quinque dies, nempe .4. cum .22. horis <lb/>et .48. minutis, quod tempus vtrique viatori inÅ¿eruiet, quandoquidem idipÅ¿um pro <lb/>uenit multiplicato .220. per .9. <reg norm="productoque" type="simple">productoqÌ;</reg> per .400. diuiÅ¿o.</s> - </p> - <p> - <s xml:id="s1012" xml:space="preserve">Huius autem, qui à me pręſcribitur modi, Å¿peculatio talis eÅ¿t. </s> - <s xml:id="s1013" xml:space="preserve">Duo termini duabus <lb/>rectis lineis æqualibus, & parallelis inter Å¿e <var>.b.p.</var> et <var>.d.q.</var> Å¿ignificentur, quæ alijs dua-<lb/>bus <var>.b.d.</var> et <var>.q.p.</var> <reg norm="coniungantur" type="simple">coniungant̃</reg>, quÄ™ parallelæ & æquales erunt ex .33. primi, quibus Å¿igni <lb/>ficentur duo itinera. </s> - <s xml:id="s1014" xml:space="preserve">Viator primus quidem lentior à . b in <var>.d.</var> velocior à <var>.q.</var> in <var>.p</var>. </s> - <s xml:id="s1015" xml:space="preserve">Iam <lb/>Å¿umatur <reg norm="punctum" type="context">punctÅ©</reg> medium <var>.q.p.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.k.</var> & ab ipÅ¿o ad <var>.b.d.</var> ducatur <var>.k.i.</var> parallela <var>.d.q.</var> aut <lb/><var>b.p.</var> quod idem eÅ¿t, ex quo <var>.b.i.</var> æqualis erit <var>.p.k.</var> ex .34. primi, hoc eÅ¿t <var>.q.k.</var> <reg norm="certique" type="simple">certiqÌ;</reg> eri-<lb/>mus primum viatorem <var>.q.p.</var> in dimidio itineris <var>.q.k.</var> occurrere non potuiſſe viatori ip <lb/>Å¿ius <var>.b.i.</var> quandoquidem eo tempore, quo is, qui ipÅ¿ius <var>.q.p.</var> mouetur per <var>.q.k.</var> (cum Å¿it <lb/>altero velocior) qui per <var>.b.d.</var> nondum peruenerit ad .i: Sit itaque punctum <var>.c.</var> in quo <lb/>lentior reperitur, dum velocior eÅ¿t in <var>.k.</var> ex quo certi erimus eos inter <var>.c.</var> et <var>.i.</var> Å¿ibi in-<lb/>uicem obuiaturos eſſe. </s> - <s xml:id="s1016" xml:space="preserve">Cogito deinde rectam lineam ductam <var>.k.c.</var> & ut Å¿e habet <var>.i.<lb/>c.</var> ad <var>.c.b.</var> ita cogito Å¿e habere. u<unsure/> <var>.k.</var> ad <var>.k.q.</var> & à puncto <var>.u.</var> ad <var>.i.</var> duco <var>.u.i.</var> quæ, vt manife <lb/>Å¿tum eÅ¿t, lineam <var>.k.c.</var> in puncto <var>.e.</var> interÅ¿ecabit, à quo cum fuerit ducta <var>.e.o.n.</var> parallela <lb/><var>k.i.</var> habebimus <var>.o.n.</var> ea Å¿cilicet puncta, quibus occurrunt Å¿ibijpÅ¿is, nam cum Å¿ic Å¿e ha <lb/>beat <var>.q.k.</var> ad <var>.k.u.</var> vt <var>.b.c.</var> ad <var>.c.i.</var> et <var>.k.u.</var> ad <var>.k.n.</var> vt <var>.c.i.</var> ad <var>.c.o.</var> ex Å¿imilitudine manifeÅ¿ta <lb/>triangulorum, ex æqualitate proportionum Å¿ic Å¿e habebit <var>.q.k.</var> ad <var>.k.n.</var> vt <var>.b.c.</var> ad <var>.c.o.</var> <lb/>& permutando ita <var>.k.q.</var> ad <var>.b.c.</var> vt <var>.k.n.</var> ad <var>.c.o.</var> & cum <var>.q.k.</var> et <var>.b.c.</var> Å¿patia Å¿int tempori-<lb/>bus æqualibus confecta, itaque Å¿patia <var>.k.n.</var> et <var>.c.o.</var> ex communi Å¿cientia temporibus <lb/>æqualibus conficientur.</s> - </p> - <p> - <s xml:id="s1017" xml:space="preserve">Quare rectè dicimus, Å¿i tot diebus à <var>.b.</var> in <var>.d.</var> aliquis peruenit, quot milliaria in di <lb/>midio temporis alterius viatoris idem conficiet? </s> - <s xml:id="s1018" xml:space="preserve">ex quo ex regula de tribus quam <lb/>primum iter <var>.b.c.</var> cognoÅ¿citur, quo ex dimidio itineris detracto, remanet <var>.c.i.</var> cogni <lb/>tus, Å¿ed cum probauerimus <var>.q.k.</var> ad <var>.k.n.</var> hoc eÅ¿t <var>.i.o.</var> (cum Å¿int æquales inter Å¿e, ex .34 <lb/>primi) ita Å¿e habere. vt <var>.b.c.</var> ad <var>.c.o.</var> permutando Å¿ic Å¿e habebit <var>.q.k.</var> ad <var>.b.c.</var> vt <var>.i.o.</var> ad <var>.<lb/>o.c.</var> & <reg norm="componendo" type="context">cõponendo</reg> <var>.q.k.</var> et <var>.b.c.</var> ad <var>.b.c.</var> vt <var>.i.c.</var> ad <var>.c.o.</var> </s> - <s xml:id="s1019" xml:space="preserve">quare rectè dicimus Å¿i Å¿umma <var>.q.<lb/>k.</var> cum <var>.b.c.</var> dat <var>.b.c.</var> quid dabit <var>.i.c</var>? </s> - <s xml:id="s1020" xml:space="preserve">nempe dabit <var>.c.o.</var> quo coniuncto cum <var>.b.c.</var> cogno-<lb/>Å¿citur <var>.b.o.</var> quo <var>.b.o.</var> detracto ex <var>.b.d.</var> remanet cognitus <var>.o.d.</var> nempe <var>.q.n.</var> illi æqualis <lb/>ex .34. prædicta. </s> - <s xml:id="s1021" xml:space="preserve">Gratia verò <reg norm="temporis" type="context">tẽporis</reg> patet nos rectè dicere Å¿i <var>.b.d.</var> tot diebus abÅ¿olui <lb/>tur, aut etiam <var>.q.p</var>: quo <var>.b.o.</var> aut <var>.q.n.</var> abÅ¿oluetur.</s> - </p> - <p> - <s xml:id="s1022" xml:space="preserve">Vt autem ad Å¿peculationem regulæ antiquorum deueniamus, cogitemus pri-<lb/>mum viatorem ipÅ¿ius <var>.q.p.</var> velociorem eo, qui per <var>.b.d.</var> iter agit, tanto tempore præ <lb/>tergredi <var>.p.</var> quanto alter <var>.b.d.</var> abÅ¿oluit. </s> - <s xml:id="s1023" xml:space="preserve">Is autem ad <var>.g.</var> pertingat, ex quo eadem pro-<lb/>portio Å¿pacij <var>.q.g.</var> ad <var>.q.p.</var> hoc eÅ¿t <var>.b.d.</var> dabitur, quæ temporis quo <var>.b.d.</var> abÅ¿oluitur ab - <pb o="77" rhead="THEOREM. ARIT." n="89" file="0089" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0089"/> - eo qui per <var>.b.d.</var> ad tempus quo <var>.q.p.</var> Å¿olum, qui per <var>.q.p.</var> mouetur (mo-<lb/>tus enim continui regulares & vniformes conÅ¿tituuntur) eadem ratione ita-<lb/>que ea erit proportio <var>.q.k.</var> ad <var>.b.c.</var> quæ <var>.q.g.</var> ad <var>.q.p.</var> & cum probatum <lb/>fuerit ita Å¿e habere <var>.k.n.</var> ad <var>.c.o.</var> vt <var>.q.k.</var> ad <var>.b.c.</var> itaque Å¿ic Å¿e habebit <var>.k.n.</var> ad <var>.c.<lb/>o.</var> ut <var>.q.g.</var> ad <var>.q.p.</var> probatum etiam fuit ita Å¿e habere <var>.q.k.</var> ad <var>.k.n.</var> vt <var>.b.c.</var> ad <var>.c.o.</var> ex quo <lb/>componendo Å¿ic Å¿e habebit <var>.q.n.</var> ad <var>.n.k.</var> vt <var>.b.o.</var> ad <var>.o.c.</var> & permutando ita <var>.q.n.</var> ad <var>.b.<lb/>o.</var> vt <var>.k.n.</var> ad <var>.c.o.</var> hoc eÅ¿t <var>.q.g.</var> ad <var>.q.p.</var> nempe vt tempus lenti ad tempus velocis itine-<lb/>rantis, & componendo ita <var>.q.n.</var> cum <var>.o.b.</var> hoc eÅ¿t <var>.b.d.</var> ad <var>.b.o.</var> vt Å¿umma <reg norm="dierum" type="context">dierÅ©</reg> vnius & <lb/>alterius viatoris ad <reg norm="minorem" type="context">minorẽ</reg> <reg norm="numerum" type="context">numerÅ©</reg> <reg norm="dierum" type="context">dierÅ©</reg> velocioris. </s> - <s xml:id="s1024" xml:space="preserve">Breuiter <reg norm="itaque" type="simple">itaq;</reg> obtineremus in <lb/><reg norm="tentum" type="context">tentÅ©</reg> <reg norm="quando" type="wordlist">qñ</reg> diceremus Å¿i Å¿umma dierum, quibus iter agitur à viatoribus talis eÅ¿t (20) re-<lb/>Å¿pectu numeri dierum velocioris(9) qualis & cui <reg norm="reÅ¿pondebit" type="context">reÅ¿põdebit</reg> totum Å¿pacium <var>.b.d</var>? </s> - <s xml:id="s1025" xml:space="preserve">vn-<lb/>de dabitur Å¿pacium <var>.b.o.</var> vnde reliqua omnia nobis cognita emergent.</s> - </p> - <p> - <s xml:id="s1026" xml:space="preserve">Cum autem antiquorum regula iubeat numerum dierum vnius, cum numero die-<lb/>rum alterius multiplicari, ac poÅ¿tmodum diuidi productum per Å¿ummam omnium <lb/>dierum, rectèid quidem fit. </s> - <s xml:id="s1027" xml:space="preserve">Nam cum Å¿ic Å¿e habeat <var>.b.d.</var> ad <var>.b.o.</var> vt Å¿umma omnium <lb/>dierum ad minorem quantitatem dierum velocioris Å¿cilicet. </s> - <s xml:id="s1028" xml:space="preserve">Ideo temporis propor <lb/>tio à mobili per <var>.b.d.</var> abÅ¿umpti ad tempus mobilis per <var>.b.o.</var> eadem erit, quæ Å¿ummæ <lb/>omnium dierum ad namerum dierum velocioris. </s> - <s xml:id="s1029" xml:space="preserve">Quarerectè dicemus, Å¿i eiuÅ¿modi <lb/>Å¿umma talem reÅ¿pectum habet ad minorem numerum dierum, quem numerum re-<lb/>Å¿piciet dies ipÅ¿ius <var>.b.d</var>? </s> - <s xml:id="s1030" xml:space="preserve">ex quo proferentur dies reÅ¿pondentes ipÅ¿i <var>.b.o.</var> cætera iam <lb/>dicta fuerunt.</s> - </p> - <p> - <s xml:id="s1031" xml:space="preserve">HuiuÅ¿modi verò Å¿peculationis am-<lb/> - <anchor type="figure" xlink:label="fig-0089-01a" xlink:href="fig-0089-01"/> - plitudo ad pauciſſima verba reduci <lb/>poteft, in cuius <reg norm="gratiam" type="context">gratiã</reg> Å¿it Å¿ubÅ¿cripta <lb/>figura pars <reg norm="inquam" type="context">inquã</reg> prÄ™cedentis, in qua <lb/><reg norm="conÅ¿tituamus" type="context simple">cõſtituamê°</reg> <var>.o.n.</var> <reg norm="locum" type="context">locÅ©</reg> <reg norm="eum" type="context">eÅ©</reg> eſſe quo Å¿ibi <lb/>viatores obuient, ex quo Å¿pacium <var>.q.<lb/>n.</var> à ſuo viatore conficietur, eo ipÅ¿o <lb/>tempore, quo à ſuo Å¿pacium <var>.b.o.</var> ita <lb/>que eadem erit proportio <var>.q.n.</var> ad <var>.b.</var> <lb/> - <anchor type="figure" xlink:label="fig-0089-02a" xlink:href="fig-0089-02"/> - o. quæ <var>.q.g.</var> ad <var>.b.d.</var> eadem erit <reg norm="inquam" type="context">inquã</reg> <lb/>proportio <var>.d.o.</var> ad <var>.o.b.</var> quæ numeri <lb/>dierum eius, qui à <var>.b.</var> pergit in <var>.d.</var> ad <lb/>numerum dierum alterius qui à <var>.q.</var> in <lb/>p. proficiÅ¿citur, & componendo <reg norm="eadem" type="context">eadẽ</reg> <lb/>erit proportio <var>.d.b.</var> ad <var>.b.o.</var> quæ Å¿um-<lb/>mæ dierum ad minorem numerum ipÅ¿orum, & eadem quæ dierum <var>.b.d.</var> ad dies <lb/>ipÅ¿ius <var>.b.o</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0089-01" xlink:href="fig-0089-01a"> - <image file="0089-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0089-01"/> - </figure> - <figure xlink:label="fig-0089-02" xlink:href="fig-0089-02a"> - <image file="0089-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0089-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="115"> - <head xml:id="head133" xml:space="preserve">THEOREMA <num value="115">CXV</num>.</head> - <p> - <s xml:id="s1032" xml:space="preserve">CIRCA hæc ipÅ¿a itinera aliud quæritur peruenuÅ¿tè, in quo quæſito illud conÂ<lb/>Å¿tituitur cognitum eſſe, nempe interuallum inter duo diuerÅ¿a loca, è quibus <lb/>duo viatores eodem inÅ¿tanti vt Å¿ibi occurrant proficiÅ¿cuntur, <reg norm="certaque" type="simple">certaqÌ;</reg> milliaria Å¿in-<lb/>gulis diebus conficiant, ita tamen, ut unus ordinatè plura altero ambulet, quæritur <lb/>deinde quoto die Å¿ibi occurrent. </s> - <s xml:id="s1033" xml:space="preserve">Hoc autem fit diuiÅ¿o toto interuallo locorum per <lb/>Å¿ummam milliariorum quam vterque quotidie abÅ¿oluit.</s> - </p> - <pb o="78" rhead="IO. BAPT. BENED." n="90" file="0090" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0090"/> - <p> - <s xml:id="s1034" xml:space="preserve">Exempli gratia, diÅ¿tant loca .100. milliaribus à ſe inuicem; </s> - <s xml:id="s1035" xml:space="preserve">vnus autem viator <lb/>Å¿ingulis diebus .15 milliaria, alter .10. conficit Å¿i ita que .15. cum .10. <reg norm="coniungamus" type="context">coniÅ©gamus</reg>, <lb/>Å¿umma erit .25. per <reg norm="quam" type="context">quã</reg> diuiÅ¿is milliaribus .100. totius interualli proferetur .4. nume <lb/>rus quæſitus dierum quo viatoribus iter agendum erit prius quam Å¿ibi obuient.</s> - </p> - <p> - <s xml:id="s1036" xml:space="preserve">In cuius Å¿peculationis gratiam totum iter Å¿ignificetur linea <var>.a.u</var>: primi autem via-<lb/>toris iter diurnum fit <var>.a.e.</var> & alterius <var>.u.o</var>: terminus verò <var>.i.</var> Å¿it occurÅ¿us ita vt eodem <lb/>tempore, alter Å¿pacium <var>.a.i.</var> alter <var>.u.i.</var> confecerit, Å¿pacij autem <var>.a.e.</var> tempus <lb/>per <var>.b.</var> Å¿ignificetur & tempus Å¿pacij <var>.u.o.</var> per <var>.c.</var> quæ tempora erunt inter Å¿e <lb/>æqualia, porrò Å¿pacij <var>.a.i.</var> tempus per <var>.d.</var> & Å¿pacij <var>.u.i.</var> per <var>.f.</var> denotetur, æquali <lb/>bus inquam, ex quo eadem proportio erit <var>.a.e.</var> ad <var>.a.i.</var> quæ <var>.b.</var> ad <var>.d.</var> et <var>.o.u.</var> ad <var>.u.i.</var> quæ <lb/>c. ad <var>.f.</var> vnde permutando eadem erit proportio itineris ipÅ¿ius <var>.b.</var> ad iter ipÅ¿ius <var>.c.</var> quæ <lb/>itineris <var>.d.</var> ad iter ipÅ¿ius <var>.f.</var> & componendo itinerum ipÅ¿ius <var>.b.c.</var> ad iter <var>.c.</var> vt itinerum <var>.<lb/>d.f.</var> ad iter <var>.f.</var> & permutando itinerum <lb/><var>b.c.</var> ad itinera <var>.d.f.</var> vt itineris <var>.c.</var> ad. iter <lb/> - <anchor type="figure" xlink:label="fig-0090-01a" xlink:href="fig-0090-01"/> - ipÅ¿ius <var>.f.</var> meritò itaque quÄ™ritur Å¿i itine <lb/>ra <var>.b.c.</var> dat itinera <var>.d.f.</var> quid dabit tem-<lb/>pus <var>.c.</var> nempe dabit tempus <var>.f.</var> Å¿ed <var>.c.</var> <lb/>Å¿ignatum eÅ¿t pro vna die, </s> - <s xml:id="s1037" xml:space="preserve">quare in pro <lb/>poÅ¿ito exemplo <var>.f.</var> Å¿ignificabit 4: dies.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0090-01" xlink:href="fig-0090-01a"> - <image file="0090-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0090-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="116"> - <head xml:id="head134" xml:space="preserve">THEOREMA <num value="116">CXVI</num>.</head> - <p> - <s xml:id="s1038" xml:space="preserve">ANtiquorum monumentis traditum motum reperimus diuinandi numeri quem <lb/>quis mente conceperit, quo iubemus eum qui numerum cogitauerit, dimi-<lb/>dium cogitari numeri addere cogitato, atque huic Å¿ummæ, rurÅ¿us eiuÅ¿dem Å¿ummÄ™ <lb/>dimidium adiungere, tum quærimus, quoties noueratius totam eam Å¿ummam ingre <lb/>diatur patefactis fractis Å¿i qui occurrant.</s> - </p> - <p> - <s xml:id="s1039" xml:space="preserve">Exempli gratia, Å¿i quis cogitaſſet numerum .12. iubebant huic dimidium addi, <lb/>nempe .6. ex quo Å¿umma erat .18. iubebant, præterea dimidium huius Å¿ummæ nem-<lb/>pe .9. toti Å¿ummæ adiungi, quæ fuiſſet .27. adhæc quærebant Å¿ibi patefieri quoties <num value="9">.<lb/>9.</num> Å¿ummam prædictam ingrederetur, & Å¿i in prima aut Å¿ecunda diuiÅ¿ione aut <reg norm="etiam" type="context">etiã</reg> <lb/>vtraque, fracti reperirentur, ac quoties nouem vltimam Å¿ummam ingrediebatur, <lb/>toties .4. multiplicabant. </s> - <s xml:id="s1040" xml:space="preserve">Quod Å¿i in prima diuiÅ¿ione fracti erant, vltimo produ-<lb/>cto addebant vnitatem; </s> - <s xml:id="s1041" xml:space="preserve">Å¿in verò in Å¿ecunda, binarium adiungebant, ex quo exa-<lb/>ctus numerus quæſitus proferebatur.</s> - </p> - <p> - <s xml:id="s1042" xml:space="preserve">Pro cuius rei ratione Å¿it <var>.a.</var> numerus cogitatione compræhenÅ¿us et <var>.e.</var> ipÅ¿ius <var>.a.</var> cum <lb/>eiuÅ¿dem medietate Å¿umma et <var>.i.</var> ipÅ¿ius <var>.e.</var> cum eiuÅ¿dem medietate itidem Å¿umma, vn <lb/>de <var>.i.e.a.</var> tres numeri continui proportionales, in Å¿eÅ¿quialtera proportione euadent. <lb/></s> - <s xml:id="s1043" xml:space="preserve">Sumantur nunc tres numeri .4. 6. 9. in eadem proportionalitate. </s> - <s xml:id="s1044" xml:space="preserve">Vnde ratione Ä™qua <lb/>litatis proportionum ita Å¿e habebit <var>.i.</var> ad <var>.a.</var> <reg norm="quemadmodum" type="wordlist">quẽadmodum</reg> .9. ad .4. & permutando <var>.i.</var> <lb/>ad .9. quemadmodum <var>.a.</var> ad .4. & ob id .4. toties ingredietur <var>.a.</var> quoties .9. ipÅ¿am <var>.i</var>. </s> - <s xml:id="s1045" xml:space="preserve">Sed <lb/>quia ſępe contingit, vt in Å¿ecunda diuiÅ¿ione, aut in ambabus etiam diuiÅ¿ionibus re <lb/>periantur numeri fracti, anima duertendum eÅ¿t numerum animo compræhenÅ¿um <var>.a.</var> <lb/>Å¿cilicet aut parem aut imparem Å¿emper futurum. </s> - <s xml:id="s1046" xml:space="preserve">Si par eÅ¿t, aut multiplex erit ad <num value="4">.<lb/>4.</num> aut non. </s> - <s xml:id="s1047" xml:space="preserve">Si priori modo Å¿e habebit in duabus diuiÅ¿ionibus, nullus numerus fra-<lb/>ctus admittetur, Å¿ed Å¿i ad .4. multiplex non erit, à multiplicibus per duo Å¿emper dif <lb/>feret, & Å¿i per medium diuidatur, eiuÅ¿dem medietas impar Å¿emper erit, vnde prior - <pb o="79" rhead="THEOR. ARITH." n="91" file="0091" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0091"/> - quoque Å¿umma par nunquam exiÅ¿tet, cuius medietatem aliquod medium Å¿emper <lb/>ingredietur, & hanc ob cauÅ¿am poÅ¿terior Å¿umma cum fracto Å¿emper erit, & nume-<lb/>rum deÅ¿umptum maiorem eſſe multiplici ad quatuor per duo Å¿ignificabit.</s> - </p> - <p> - <s xml:id="s1048" xml:space="preserve">At verò Å¿i inter impares reponatur, aut eorum erit qui Å¿uperant multiplicem <lb/>ipÅ¿ius quatuor per vnum, Å¿eu per tria, quod hinc innoteÅ¿cet, nempe, quia Å¿i eorum <lb/>erit qui dictum multiplicem per vnum tantum vincunt, Å¿ua medietate ipÅ¿i numero <lb/>addita, & præter hanc medietatem medio etiam integro adiuncto, tota hæc prior <lb/>Å¿umma in numerum parem Å¿emper euadet, vnde in poÅ¿teriori Å¿umma nullus nume-<lb/>rus fractus conÅ¿picietur, & hanc ob <reg norm="causam" type="context">causã</reg> multiplici ipÅ¿ius .4. vnitas Å¿emper addetur.</s> - </p> - <p> - <s xml:id="s1049" xml:space="preserve">Sed Å¿i numerus deÅ¿umptus, in Å¿erie eorum, qui multiplicem ipÅ¿ius .4. pertria Å¿u-<lb/>perant, collocabitur, hinc compræhendetur, quia primæ Å¿ummæ numerus cum <lb/>media vnitate Å¿emper impar erit, vnde Å¿ecunda Å¿umma præter integras cum me-<lb/>dia vnitate nobis Å¿emper occur ret.</s> - </p> - <p> - <s xml:id="s1050" xml:space="preserve">Quod autem nobis prodere faciamus an in prima diuiÅ¿ione, & Å¿ecunda numerus <lb/>aliquis fractus conÅ¿iÅ¿tat, eò tantum nobis inÅ¿eruit, quò deueniamus in cognitionem <lb/>an numerus animo conceptus multiplicem ipÅ¿ius .4. per vnum, per duo, aut tria Å¿upe <lb/>ret. </s> - <s xml:id="s1051" xml:space="preserve">Quòd etiam medias eas vnitates ad integros reducere faciamus, eò tantum re <lb/>fertur, vt minori labore eum, qui numerum imaginatione compræhendit, onere-<lb/>mus, quia reuera numerus impar nunquam mente concipi poteÅ¿t, quin aliquis fra-<lb/>ctus in prima diuiÅ¿ione, aut in Å¿ecunda Å¿equatur: </s> - <s xml:id="s1052" xml:space="preserve">vnde à numeris imparibus, qui mul <lb/>tiplicem ipÅ¿ius .4. unitatis tantum exceſſu <reg norm="Å¿uperant" type="context">Å¿uperãt</reg>, poÅ¿terior Å¿umma <reg norm="cum" type="context">cÅ©</reg> quarta parte <lb/>vnitatis, præter integros numeros, & ab imparibus qui dictum multiplicem ipÅ¿ius <num value="4">.<lb/>4.</num> per tria vincunt, cum tribus quartis vnius integri præter integras vnitates ; </s> - <s xml:id="s1053" xml:space="preserve">& à <lb/>numeris paribus, qui multiplicem ipÅ¿ius .4. per duo cum medietate vnitatis præter <lb/>integros Å¿emper procedit. </s> - <s xml:id="s1054" xml:space="preserve">Ita cum is qui numerum Å¿ecum conÅ¿iderat, Å¿i in nume-<lb/> - <anchor type="figure" xlink:label="fig-0091-01a" xlink:href="fig-0091-01"/> - ris fractis verÅ¿atus eſſet, qui eum in-<lb/>terrogat prudenter Å¿e gereret, Å¿i Å¿ibi <lb/>declarari curaret, quis nam ex fractis <lb/>Å¿u per integros <reg norm="Å¿ecundæ" type="context">Å¿ecÅ©dæ</reg> <reg norm="summæ" type="context">sÅ©mæ</reg> remane <lb/>ret, quia <reg norm="per" type="simple">ê‘</reg> quot quarta integros <reg norm="Å¿ecun- dæ" type="context">Å¿ecÅ©-<lb/>dæ</reg> Å¿ummæ Å¿uperaret, per <reg norm="totidem" type="context">totidẽ</reg> inte <lb/>gros numerus mente conceptus multiplicem ipÅ¿ius .4. Å¿uperaret.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0091-01" xlink:href="fig-0091-01a"> - <image file="0091-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0091-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="117"> - <head xml:id="head135" xml:space="preserve">THEOREMA <num value="117">CXVII</num>.</head> - <p> - <s xml:id="s1055" xml:space="preserve">VNDE fiat, vt Å¿i ali quis quemuis numerum animo compræhendat, eique <lb/>numero alium etiam quemlibet numerum propoÅ¿itum addat, & à tertia par <lb/>te huius Å¿ummæ tertiam partem numeri imaginati detrah et, reÅ¿iduum Å¿ecundi nu-<lb/>meri adiuncti, ideÅ¿t propoÅ¿iti, tertia pars erit.</s> - </p> - <p> - <s xml:id="s1056" xml:space="preserve">Vt exempli gratia, Å¿i aliquis de numero denario cogitaſſet, <reg norm="huicque" type="simple">huicqÌ;</reg> .24. adderet, <lb/>vnde triginta quatuor efficerent, detra hendo nunc tertiam partem numeri de na-<lb/>rij cogitatione concepti, ideÅ¿t .3. cum tertia parte vnius, à tertia parte huius Å¿um mæ <lb/>ideÅ¿t ab vndecim & vna tertia parte remanerent .8. ideÅ¿t tertia pars numeri additi. <lb/></s> - <s xml:id="s1057" xml:space="preserve">Id quod mihi inter iocos in honeÅ¿torum hominum cætu in mentem venit.</s> - </p> - <p> - <s xml:id="s1058" xml:space="preserve">Pro cuius ratione, prior numerus ima <lb/> - <anchor type="figure" xlink:label="fig-0091-02a" xlink:href="fig-0091-02"/> - ginatus mediante linea <var>.a.b.</var> et is, qui ad-<lb/>ditus eÅ¿t <reg norm="intercedente" type="context">intercedẽte</reg> linea <var>.b.d.</var> è directo - <pb o="80" rhead="IO. BAPT. BENED." n="92" file="0092" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0092"/> - coniunctis denotetur, et <var>.b.e.</var> Å¿it tertia pars ipÅ¿ius <var>.a.b.</var> prioris numeri im aginati, et. b <lb/>c. tertia pars ipÅ¿ius, <var>b.d.</var> Å¿ecundi numeri propoÅ¿iti, vnde coniunctum vnius harum ter <lb/>tiarum <reg norm="partium" type="context">partiÅ©</reg> <reg norm="cum" type="context">cÅ©</reg> alia Å¿it <var>.e.c.</var> quod quidem <var>.e.c.</var> eſſe tertiam partem Å¿ummæ duorum <lb/>primorum ideÅ¿t <var>.a.d.</var> aſſero. </s> - <s xml:id="s1059" xml:space="preserve">Iam manifeÅ¿tum eÅ¿t ipÅ¿ius <var>.d.b.</var> ad <var>.b.c.</var> eſſe quemadmo <lb/>dum ipÅ¿ius <var>.a.b.</var> ad <var>.b.e.</var> vnde viciſſim ipÅ¿ius <var>.d.b.</var> ad <var>.b.a.</var> erit quemadmodum ipÅ¿ius <var>.b.<lb/>c.</var> ad <var>.b.e.</var> & coniunctim ipÅ¿ius <var>.d.a.</var> ad <var>.a.b.</var> quemadmodum ipÅ¿ius <var>.c.e.</var> ad <var>.e.b.</var> & viciÅ¿-<lb/>Å¿im ipÅ¿ius <var>.d.a.</var> ad <var>.c.e.</var> quemadmodum ipÅ¿ius <var>.b.a.</var> ad <var>.b.e.</var> Å¿ed proportio ipÅ¿ius <var>.b.a.</var> ad <var>.<lb/>b.e.</var> eÅ¿t tripla, ergo ea quæ eÅ¿t ipÅ¿ius <var>.a.d.</var> ad <var>.e.c.</var> erit quoque tripla; </s> - <s xml:id="s1060" xml:space="preserve">vnde Å¿umendo <var>.e.<lb/>c.</var> pro tertia parte ipÅ¿ius <var>.a.d.</var> & ab ipÅ¿a <var>.e.c.</var> Å¿ubtrahendo tertiam partem ipÅ¿ius <var>.a.b.</var> <lb/>tertia pars ipÅ¿ius <var>.b.d.</var> remanebit <var>.b.c</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0091-02" xlink:href="fig-0091-02a"> - <image file="0091-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0091-02"/> - </figure> - </div> - <p> - <s xml:id="s1061" xml:space="preserve">Aut alio hoc modo, Å¿upponendo <var>.e.c.</var> tertiam partem ipÅ¿ius <var>.a.d.</var> et <var>.e.b.</var> ipÅ¿ius <var>.a.<lb/>b.</var> exiÅ¿ter. </s> - <s xml:id="s1062" xml:space="preserve">Dico <var>.b.c.</var> tertiam partem ipÅ¿ius <var>.b.d.</var> futuram:</s> - <s xml:id="s1063" xml:space="preserve">quia Å¿i totius <var>.a.d.</var> ad totum <lb/><var>e.c.</var> ita Å¿e habet, quemadmodum <var>.a.b.</var> à toto <var>.a.d.</var> diffecti atque diuulÅ¿i ad <var>.e.b.</var> à toto <var>.<lb/>e.c.</var> diÅ¿ractum, ergo ex <ref id="ref-0011">.19. lib. quinti Eu-<lb/> - <anchor type="figure" xlink:label="fig-0092-01a" xlink:href="fig-0092-01"/> - clid.</ref> reÅ¿idui <var>.b.d.</var> totius <var>.a.d.</var> ad reÅ¿iduum <var>.b.c.</var> <lb/>totius <var>.e.c.</var> erit, vt totius <var>.a.d.</var> ad <reg norm="totum" type="context">totÅ©</reg> <var>.e.c.</var> at-<lb/>que hic quidem modus rem <reg norm="propoÅ¿itam" type="context">propoÅ¿itã</reg> Å¿pe-<lb/>culandi mihi aptior & commodior eſſe videtur.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0092-01" xlink:href="fig-0092-01a"> - <image file="0092-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0092-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="118"> - <head xml:id="head136" xml:space="preserve">THEOREMA <num value="118">CXVIII</num>.</head> - <p> - <s xml:id="s1064" xml:space="preserve">PErmulta ac varia problemata inuenerunt antiqui, longioribus verò vijs reÅ¿olu-<lb/>ta, proptereà quòd <reg norm="non" type="context">nõ</reg> Å¿emper nobis Å¿uccurrit breuiſſima in vnaquaque re ex-<lb/>plicatio. </s> - <s xml:id="s1065" xml:space="preserve">Vt exempli gratia, proponitur numerus .50. diuidendus in tres tales par-<lb/>tes, quod Å¿ecunda dupla Å¿it primÄ™, & adhuc eam Å¿uperet tribus vnitatibus, tertia ve <lb/>rò æqualis Å¿it aggregato primæ cum Å¿ecunda, & amplius ipÅ¿um aggregatum Å¿uperet <lb/>quinque vnitatibus.</s> - </p> - <p> - <s xml:id="s1066" xml:space="preserve">Ad hoc autem quæſitum Å¿oluendum antiqui vtebantur regula falÅ¿i, quod reuera <lb/>breuiori modo poteÅ¿t Å¿olui, videlicet detra hendo illud Å¿ecundum exceſſum, quin-<lb/>que Å¿cilicet ex .50. ita vt nobis .45. remaneret, cui medietati hoc eÅ¿t .22. cum dimidia <lb/>vnitate, Å¿i addiderimus illud quinque habebimus .27. cum dimidia vnitate pro ter-<lb/>tia parte quæſita ipÅ¿ius numeri .50. deinde Å¿i ab eodem numero .22. cum dimidia <lb/>vnitate detractum fuerit illud .3. primus exceſſus datus, remanebit .19. cum dimi-<lb/>dia vnitate, cuius tertia pars, hoc eÅ¿t .6. cum dimidia vnitate, prima pars, ex tri-<lb/>bus quæſita erit, quæ quidem Å¿i detraxerimus ex .19. cum dimidia vnitate, reli-<lb/>quum erit .13. cui <reg norm="cum" type="context">cÅ©</reg> additus fuerit primus exceſſus ideÅ¿t .3. </s> - <s xml:id="s1067" xml:space="preserve">Iam propoÅ¿itum re-<lb/>Å¿ultabit nobis .16. pro Å¿ecunda parte quæſita.</s> - </p> - <p> - <s xml:id="s1068" xml:space="preserve">Ratio verò huiuÅ¿modi operationis talis eÅ¿t, Å¿it verbi gratia totalis numerus pro-<lb/>poÅ¿itus Å¿ignificatus per lineam <var>.a.b.</var> cuius Å¿ecundæ partis numerus datus Å¿ignificetur <lb/>per lineam <var>.g.</var> & numerus tertiæ partis propoÅ¿itus per lineam <var>.h</var>. </s> - <s xml:id="s1069" xml:space="preserve">Nunc dempta <var>.h.</var> ex <lb/><var>a.b.</var> nobis cognita, remanebit <var>.f.a.</var> qua <reg norm="quidem" type="context">quidẽ</reg> per æqualia imaginatione diuiÅ¿a in pun <lb/>cto <var>.e.</var> & ipÅ¿i <var>.e.f.</var> addita <var>.f.b.</var> tota <var>.e.b.</var> nobis cognita erit, quæ quidem tertia pars <lb/>quæſita ipÅ¿ius <var>.a.b.</var> erit, proptereà quòd <var>.a.e.</var> (quæ æqualis eÅ¿t ipÅ¿i <var>.e.f.</var>) erit Å¿umma <lb/>primæ, & Å¿ecundæ partis. </s> - <s xml:id="s1070" xml:space="preserve">Detrahatur poÅ¿teà . g. ex <var>.e.a.</var> & remanebit <var>.d.a.</var> cuius ter <lb/>tia pars Å¿it <var>.a.c.</var> quæ quidem prima pars quæſita erit, & nunc cognita, & ita <var>.c.d.</var> <lb/>cognita, cui cum addita fuerit <var>.d.e.</var> habebimus Å¿ecundam partem quæſitam, quæ <lb/>compo- - <pb o="81" rhead="THEOREM. ARIT." n="93" file="0093" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0093"/> - componitur ex <var>.d.c.</var> dupla .ad <var>.a.c.</var> pri-<lb/> - <anchor type="figure" xlink:label="fig-0093-01a" xlink:href="fig-0093-01"/> - mam partem, & ex <var>.d.e.</var> numero dato. <lb/></s> - <s xml:id="s1071" xml:space="preserve">tertia verò pars <var>.e.b.</var> compoÅ¿ita eÅ¿t ex <var>.<lb/>e.f.</var> æquali <var>.a.e.</var> hoc eÅ¿t æquali compoÅ¿i-<lb/>to ex prima, & Å¿e cunda parte, & ex <var>.f.<lb/>b.</var> numero dato vt proponebatur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0093-01" xlink:href="fig-0093-01a"> - <image file="0093-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0093-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="119"> - <head xml:id="head137" xml:space="preserve">THEOREMA <num value="119">CXIX</num>.</head> - <p> - <s xml:id="s1072" xml:space="preserve">INter alia problemata ab antiquis inuenta, hoc etiam ponitur. </s> - <s xml:id="s1073" xml:space="preserve">Aliquis inter-<lb/>rogat quot Å¿int horæ, alius verò reÅ¿pondit tot eſſe, quot duæ tertiæ præteriti <lb/>temporis Å¿imul iuncta cum tribus quintis futuri temporis totius dieri naturalis effi-<lb/>ciunt. </s> - <s xml:id="s1074" xml:space="preserve">Nunc quæritur quot Å¿int horÄ™.</s> - </p> - <p> - <s xml:id="s1075" xml:space="preserve">Antiqui, hoc etiam problema Å¿oluebant mediante regula falÅ¿i, Å¿ed mihi alio mo <lb/>do Å¿oluendum eſſe dictum problema videtur. </s> - <s xml:id="s1076" xml:space="preserve">Accipio enim ex quinque, tres vni-<lb/>tates, pro parte futuri temporis, quas quidem in tres vnitates præteriti temporis <lb/>duco, vnde proueniunt mihi nouem vnitates, quod productum coniungo <reg norm="cum" type="context">cÅ©</reg> quin-<lb/>que futuri temporis, vnde veniunt .14. vnitates, ex regula poftea de tribus ita dico <lb/>Å¿i ex .14. mihi prouenit .9. quid reÅ¿ultabit ex .24. & prouenient mihi horæ .15. cum <lb/>tribus Å¿eptimis vnius horæ, hoc eÅ¿t minuta ferè .26.</s> - </p> - <p> - <s xml:id="s1077" xml:space="preserve">Pro cuius ratione, quinque vnitates, feu partes temporis futuri Å¿ignificentur à <lb/>linea <var>.e.u.</var> quarum trium Å¿igniÅ¿icentur a linea <var>.e.i.</var> Å¿umpta deinde Å¿it linea <var>.e.o.</var> æqualis <lb/>lineæ <var>.e.i.</var> et <var>.e.a.</var> tripla Å¿it ad <var>.o.e.</var> vel ad <var>.e.i.</var> quod idem eÅ¿t, vnde <var>.a.e.</var> compoÅ¿ita erit <lb/>ex <var>.a.o.</var> (hoc eÅ¿t ex duabus tertijs ip Å¿ius <var>.a.e.</var>) & ex <var>o.e.</var> (hoc eÅ¿t ex. tribus quintis ip-<lb/>Å¿ius <var>.e.u.</var>) vnde <var>.a.u.</var> ad <var>.a.e.</var> eandem rationem obtinebit, quæ .14. ad .9. </s> - <s xml:id="s1078" xml:space="preserve">propterea igi <lb/>tur poſſumus recte ratiotinari <lb/> - <anchor type="figure" xlink:label="fig-0093-02a" xlink:href="fig-0093-02"/> - fi .14. nobis dat .9. quid dabit .24. <lb/>qui quidem .24. nobis dabit .15. <lb/>cum min .26. quod rectè factum <lb/>erit ex <ref id="ref-0012">.20. Å¿eptimi Euclidis</ref>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0093-02" xlink:href="fig-0093-02a"> - <image file="0093-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0093-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="120"> - <head xml:id="head138" xml:space="preserve">THEOREMA <num value="120">CXX</num>.</head> - <p> - <s xml:id="s1079" xml:space="preserve">SVpponunt etiam antiqui tres Å¿ocios nummos habere, quorum Å¿umma primi & <lb/>Å¿ecundi cognita Å¿it, item Å¿umma primi & tertij cognita & Å¿umma Å¿ecundi & <lb/>tertij item cognita, at que ex huiuÅ¿modi tribus aggregatis veniunt in cognitionem <lb/>particularem vniuÅ¿cuiuÅ¿que illorum.</s> - </p> - <p> - <s xml:id="s1080" xml:space="preserve">GemafriÅ¿ius Å¿oluit hoc problema ex regula Å¿alÅ¿i. </s> - <s xml:id="s1081" xml:space="preserve">At ego tali ordine progredior. <lb/></s> - <s xml:id="s1082" xml:space="preserve">Sit verbi gratia, Å¿umma primi cum Å¿ecundo .50. & Å¿ecundi cum tertio .70. & primi <lb/>cum tertio .60. harum trium Å¿ummarum accipiantur duæ quæuis, vt puta .50. & .70 <lb/>quæ coniunctæ Å¿imul dabunt .120. à qua Å¿umma detrahatur reliqua, ideÅ¿t .60. & <lb/>reÅ¿tabit nobis .60. cuius medietas erit .30. hoc eÅ¿t numerus nummorum Å¿ecundi <lb/>Å¿ocij quo numero detracto à .70. hoc eÅ¿t à ſumma Å¿ecundi cum tertio remanebit .40. <lb/>hoc eÅ¿t numerus tertij Å¿ocij, & hic numerus deÅ¿umptus à .60. reÅ¿iduus erit nume-<lb/>rus primi Å¿ocij.</s> - </p> - <pb o="82" rhead="IO. BAPT. BENED." n="94" file="0094" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0094"/> - <p> - <s xml:id="s1083" xml:space="preserve">Pro cuius ratione conÅ¿ideremus triangulum hic Å¿ubnotatum <var>.a.b.c.</var> cuius <lb/>unumquodque latus Å¿ignificet Å¿ummam duorum Å¿ociorum, vtputa latus <var>.a.b.</var> Å¿ignifi-<lb/>cet Å¿ummam primi cum Å¿ecundo, latus verò <var>.b.c.</var> Å¿ummam Å¿ecundi cum tertio, la-<lb/>rus autem <var>.a.c.</var> Å¿ummam primi cum tertio, et <var>.a.e.</var> Å¿eu <var>.a.o.</var> Å¿it numerus primi Å¿ocij, et <var>.<lb/>e.b.</var> vel <var>.b.u.</var> Å¿it Å¿ecundi Å¿ocij, et <var>.c.u.</var> Å¿eu <var>.c.o.</var> Å¿it tertij, cum autem <var>.a.e.</var> æqualis Å¿it <var>.a.o.</var> <lb/> - <anchor type="figure" xlink:label="fig-0094-01a" xlink:href="fig-0094-01"/> - et <var>.b.e</var>:æ qualis <var>.b.u.</var> et <var>.c.u.</var> æqualis <var>.c.o.</var> <lb/>ex Å¿uppoÅ¿ito Å¿i <reg norm="dempta" type="context">dẽpta</reg> fuerit Å¿umma Å¿eu <lb/>latus <var>.a.c.</var> datum ex aggregato laterum <var>.<lb/>a.b.</var> cum <var>.b.c.</var> reliquarum Å¿ummarum, re <lb/>linquet nobis cognitum aggregatum <lb/>ex <var>.b.e.</var> cum <var>.b.u</var>. </s> - <s xml:id="s1084" xml:space="preserve">Quare & eius medic-<lb/>tas <var>.b.e.</var> Å¿iue <var>.b.u.</var> nobis cognita erit, qua <lb/>detracta exÅ¿umma <var>.b.a.</var> relinquetur no <lb/>bis cognitus numerus <var>.a.e.</var> detracto ve-<lb/>ro numero <var>.a.e.</var> hoc eÅ¿t <var>.a.o.</var> ex <var>.a.c.</var> Å¿um-<lb/>ma, Å¿eu latus, aut <var>.b.u.</var> ex <var>.b.c.</var> remanebit <lb/><var>o.c.</var> Å¿eu <var>.c.u.</var> cognitus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0094-01" xlink:href="fig-0094-01a"> - <image file="0094-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0094-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="121"> - <head xml:id="head139" xml:space="preserve">THEOREMA <num value="121">CXXI</num>.</head> - <p> - <s xml:id="s1085" xml:space="preserve">HAC etiam methodo hoc facere poſſumus non <reg norm="Å¿olum" type="context">Å¿olÅ©</reg> de tribus Å¿ocijs, Å¿ed <reg norm="etiam" type="context">etiã</reg> <lb/>de omnibus quotquot volueris, vt exempli gratia, <lb/> - <anchor type="figure" xlink:label="fig-0094-02a" xlink:href="fig-0094-02"/> - Å¿int Å¿ex Å¿ocij <var>.a.b.c.d.e.f.</var> quorum Å¿umma per binos co-<lb/>gnita, vtputà ſumma numeri <var>.a.</var> cum <var>.b.</var> cognita nobis Å¿it, <lb/>& Å¿umma numeri <var>.b.</var> cum <var>.c.</var> & Å¿umma <var>.c.</var> cum <var>.d.</var> & Å¿um-<lb/>ma <var>.d.</var> cum <var>.e.</var> & Å¿umma <var>.e.</var> cum <var>.f.</var> neceÅ¿le eft etiam Å¿cire <lb/>Å¿ummam duorum vno relicto, vtputa Å¿ummam <var>.a.</var> cum <lb/>c. vt poſſimus triangulum <var>.a.b.c.</var> conÅ¿tituere. </s> - <s xml:id="s1086" xml:space="preserve">Vnde ex <lb/>præmiffa, cognitus numerus nobis erit vniuÅ¿cuiuÅ¿que <var>.a.<lb/>b.c</var>. </s> - <s xml:id="s1087" xml:space="preserve">Quapropter dempto numero <var>.c.</var> ex Å¿umma <var>.c.</var> cum <lb/>d. & numero <var>.d.</var> ex Å¿umma <var>.d.</var> cum <var>.e.</var> & numero <var>.e.</var> ex Å¿um <lb/>ma <var>.e.</var> cum <var>.f.</var> habebimus intentum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0094-02" xlink:href="fig-0094-02a"> - <image file="0094-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0094-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="122"> - <head xml:id="head140" xml:space="preserve">THEOREMA <num value="122">CXXII</num>,</head> - <p> - <s xml:id="s1088" xml:space="preserve">CVM aliquando, illud quod Archimedes inuenit, vt furtum Regiab aurifa-<lb/>bro in regia corona factum, quemadmodum Å¿cribit Vitruuius, proderet, con-<lb/>templarer, mihi etiam viÅ¿um eÅ¿t, vt aliquem modum Å¿cientiÅ¿icum inueÅ¿tigarem, quo <lb/>proportio auri ad argentum, quod in aliquo propoÅ¿ito corpore exipÅ¿is miÅ¿to cogni <lb/>ti ponderis cognoÅ¿ci poſſet. </s> - <s xml:id="s1089" xml:space="preserve">Et cum multos diuerÅ¿is temporibus excogitarim offi-<lb/>cio meo deeſſe nolui in ijÅ¿dem literarum monumentis mandandis, quorum hic <lb/>vnus erit: </s> - <s xml:id="s1090" xml:space="preserve">propoÅ¿ita nobis Å¿int tria corpora <var>.A.M.V.</var> æqualia inter Å¿e, Å¿ed diuer-<lb/>Å¿arum Å¿pecierum materiei, vtputa quod <var>.A.</var> Å¿it argenteum, & omogeneum <var>.V.</var> ve-<lb/>rò aureum omogeneum, & M. mixtum exauro, & argento, ideÅ¿t heterogeneum, <lb/>cupimusergo Å¿cire <reg norm="iuÅ¿tam" type="context">iuÅ¿tã</reg> quantitatem auri & argenti, quæ eÅ¿t in ipÅ¿o corpore <var>.M.</var> <lb/>miÅ¿to. </s> - <s xml:id="s1091" xml:space="preserve">Ita igitur faciamus. </s> - <s xml:id="s1092" xml:space="preserve">Videamus primum quantum Å¿it pondus vniuÅ¿cuiuÅ¿que <lb/>ipÅ¿orum corporum, ponamus autem pondus corporis <var>.V.</var> auri eſſe vt .234. pondus - <pb o="89" rhead="THEOR. ARITH." n="95" file="0095" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0095"/> - autem corporis <var>.M.</var> miÅ¿ti <reg norm="vt" type="punctuation">.vt</reg> .216. argentei verò <var>.A.</var> vt .156. detrahatur nunc pon-<lb/>dus <var>.A.</var> ex pondere <var>.V</var>. </s> - <s xml:id="s1093" xml:space="preserve">Reliquum erit .78. quod vocetur prima differentia Å¿eruan-<lb/>da, dematur etiam pondus <var>.M.</var> ex pondere <var>.V.</var> reliquum erit .18. pro Å¿ecunda diffe-<lb/>rentia, etiam Å¿eruanda, multiplicetur poÅ¿teà pondus <var>.A.</var> per Å¿ecundam differen-<lb/>tiam, productum verò diuidatur per primam differentiam. </s> - <s xml:id="s1094" xml:space="preserve">Vnde in præſenti exem <lb/>plo proueniet nobis .36. quiquidem prouentus erit quantitas argenti ipÅ¿ius corpo-<lb/>ris miÅ¿ti <var>.M.</var> quo etiam detracto ex pondere totali ipÅ¿ius <var>.M.</var> reliquum erit quanti-<lb/>tas auri eius corporis, hoc eÅ¿t .180.</s> - </p> - <p> - <s xml:id="s1095" xml:space="preserve">In cuius operationis Å¿peculatione, aliquid natura Å¿ua prius cognitum præcedere <lb/>oportet hoc eÅ¿t, quod omnia corpora omogenea eandem proportionem obtinent <lb/>inter quantitates, quam inter pondera. </s> - <s xml:id="s1096" xml:space="preserve">Quo Å¿uppoÅ¿ito denotetur corpus <var>.A.</var> li-<lb/>nea <var>.o.a.</var> corpus autem <var>.V.</var> linea <var>.o.c.</var> & corpus <var>.M.</var> linea <var>.e.u</var>: Å¿ed <var>.e.o.</var> Å¿ignificet par-<lb/>tem argenti, et <var>.o.u.</var> partem auri in corpore miÅ¿to <var>.M.</var> vnde ex communi conceptu <lb/>habebimus <var>.o.e.</var> æqualem <var>.u.c.</var> cum ex hypotheÅ¿i <var>.e.u.</var> æqualis Å¿it <var>.o.c.</var> et <var>.a.o.</var> Å¿imiliter. <lb/></s> - <s xml:id="s1097" xml:space="preserve">Significetur poÅ¿teà pondus <var>.a.o.</var> ab <var>.f.</var> & pondus <var>.e.u.</var> ab <var>.b.x.</var> & pondus <var>.o.c.</var> ab <var>.f.g.</var> pon <lb/>dus verò <var>.o.e.</var> ab <var>.b.</var> pondus autem <var>.o.u.</var> ab <var>.x.</var> pondus enim <var>.u.c.</var> ab <var>.b.d.</var> et <var>.g.</var> Å¿it diffe-<lb/>rentia, qua <var>.f.g.</var> maior eÅ¿t .f: et <var>.d.</var> <lb/> - <anchor type="figure" xlink:label="fig-0095-01a" xlink:href="fig-0095-01"/> - differentia qua <var>.b.d.</var> maior eÅ¿t <var>.b</var>. <lb/></s> - <s xml:id="s1098" xml:space="preserve">Vnde ex ratione omogeneitatis ea <lb/>dem proportio erit <var>.a.o.</var> ad <var>.e.o.</var> vt <var>.<lb/>f.</var> ad <var>.b.</var> et <var>.o.c.</var> ad <var>.u.c.</var> quæ <var>.x.b.d.</var> Å¿eu <lb/><var>f.g.</var> (quodidem eÅ¿t) ad <var>.b.d</var>. </s> - <s xml:id="s1099" xml:space="preserve">Quare <lb/>ex <ref id="ref-0013">.11. quinti <reg norm="eadem" type="context">eadẽ</reg></ref> erit proportio <var>.<lb/>f.</var> ad <var>.b.</var> vt <var>.f.g.</var> ad <var>.b.d.</var> & permutan-<lb/>do ita erit <var>.f.</var> ad <var>.f.g.</var> vt <var>.b.</var> ad <var>.b.d.</var> & <lb/>Å¿eparando ita <var>.f.</var> ad <var>.g.</var> vt <var>.b.</var> ad <var>.d</var>. </s> - <s xml:id="s1100" xml:space="preserve">Sed <var>.g.</var> cognita nobis eÅ¿t, vt differentia in <lb/>ter <var>.f.</var> g, et <var>.f</var>: cognita nobis eÅ¿t etiam <var>.f</var>: cognoÅ¿cimus itidem <var>.d.</var> vt differentiam inter <var>.<lb/>x.b.d.</var> et <var>.b.x.</var> quapropter cognoÅ¿cemus <var>.b.</var> ex .20. Å¿eptimi Eucli. & Å¿ic <var>.x.</var> reÅ¿iduum. <lb/></s> - <s xml:id="s1101" xml:space="preserve">ex <var>.b.x</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0095-01" xlink:href="fig-0095-01a"> - <image file="0095-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0095-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="123"> - <head xml:id="head141" xml:space="preserve">THEOREMA <num value="123">CXXIII</num>.</head> - <p> - <s xml:id="s1102" xml:space="preserve">NVNC ex methodo præcedentis propoÅ¿iti deuenire poſſumus in cognitio-<lb/>nem veræ quantitatis auri, & argenti confuÅ¿i in corona Hieronis conÅ¿tituen-<lb/>do primum duo corpora Å¿implicia æqualia inter Å¿e, & coronæ hoc modo videlicet, <lb/>immergendo coronam, Å¿eu corpus miÅ¿tum in aliquod vas aqua plenum, & diligen-<lb/>ter colligere aquam, quæ ex eo effundetur, poÅ¿teà verò oportet, aliud vas inuenire <lb/>præciſæ capax illius a quæ collectæ, in quod demum infundatur tantum auri, & po-<lb/>Å¿teà tantum argenti, quantum Å¿ieri poteÅ¿t, vnde vnumquodque horum duorum cor <lb/>porum Å¿implicium æquale erit mixto, Å¿eu coronæ, & Å¿ic quod dictum eÅ¿t in præce-<lb/>cedenti theoremate exequemur.</s> - </p> - </div> - <div type="math:theorem" level="3" n="124"> - <head xml:id="head142" xml:space="preserve">THEOREMA <num value="124">CXXIIII</num>.</head> - <p> - <s xml:id="s1103" xml:space="preserve">SED vt breuiori methodo idem præſtemus, quod in antecedenti propoÅ¿ito di-<lb/>ctum eÅ¿t, quædam theoremata præmittenda Å¿unt, videlicet quòd quotÃeÅ¿cunque <lb/>fuerint tria corpora, quorum duo inuicem æqualia Å¿int in quantitate, Å¿ed diuerÅ¿a- - <pb o="84" rhead="IO. BAPT. BENED." n="96" file="0096" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0096"/> - rum Å¿pecierum materiæ, tertium verò corpus maius, vel minus Å¿it in quantitate vtro-<lb/>que illorum, Å¿ed eiuÅ¿dem materiæ vnius quod vis illorum, ponderis verò alterius, <lb/><reg norm="semper" type="context">sẽper</reg> eadem proportio erit inter pondera æqualium corporum, quæ inter <reg norm="quantita- tem" type="context">quãtita-<lb/>tem</reg> corporis inæqualis, & eam quæ vnius cuiuÅ¿uis æqualium.</s> - </p> - <p> - <s xml:id="s1104" xml:space="preserve">Exempli gratia, Å¿it <var>.b.</var> corpus aliquod aureum æquale corpori <var>.u.</var> argenteo, Å¿it <lb/>etiam corpus <var>.a.</var> argenteum maius corpore <var>.b.</var> vel <var>.u.</var> Å¿ed ponderis eiuÅ¿dem, quod au-<lb/>ri <var>.b</var>. </s> - <s xml:id="s1105" xml:space="preserve">Tunc dico eandem eſſe proportionem ponde-<lb/> - <anchor type="figure" xlink:label="fig-0096-01a" xlink:href="fig-0096-01"/> - ris <var>.b.</var> ad pondus <var>.u.</var> quæ eÅ¿t magnitudinis <var>.a.</var> ad ma-<lb/>gnitudinem <var>.u</var>. </s> - <s xml:id="s1106" xml:space="preserve">Quod ratiocinemur hoc modo, nam <lb/>cum proportio corporeitatis <var>.a.</var> ad corporeitatem <var>.u.</var> <lb/>eadem Å¿it, quæ ponderis <var>.a.</var> ad pondus <var>.u.</var> ex ratione <lb/>omogeneitatis, ponderis verò <var>.b.</var> ad pondus <var>.u.</var> ex .7. <lb/>quinti, eadem quæ ponderis <var>.a.</var> ad pondus <var>.u.</var> ideo ex <lb/>11. eiuÅ¿dem proportio ponderis <var>.b.</var> ad pondus <var>.u.</var> eadem erit, quæ corporeitatis <var>.a.</var> <lb/>ad corporeitatem <var>.u.</var> vel ad corporeitatem <var>.b.</var> quæ æqualis eÅ¿t alteri.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0096-01" xlink:href="fig-0096-01a"> - <image file="0096-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0096-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="125"> - <head xml:id="head143" xml:space="preserve">THEOREMA <num value="125">CXXV</num>.</head> - <p> - <s xml:id="s1107" xml:space="preserve">QVotieÅ¿cunque nobis propoÅ¿ita fuerint duo corpora cuiuÅ¿uis magnitudinis æ-<lb/>que ponderantia, Å¿ed diuerÅ¿arum Å¿pecierum materiæ, cum Å¿cire volueri-<lb/>mus proportionem ponderum illarum Å¿pecierum inter ipÅ¿as hoc modo faciemus.</s> - </p> - <p> - <s xml:id="s1108" xml:space="preserve">Sint exempli gratia, duo nobis propoÅ¿ita corpora <var>.a.</var> et <var>.b.</var> (vt dictum eÅ¿t) quæ Å¿i <lb/>fuerint æqualium magnitudinum inter Å¿e, clarum erit quod quæritur, Å¿ed inæqua-<lb/>lia erunt, immergatur <reg norm="unumquodque" type="simple">unumquodq;</reg> eorum in vas aqua plenum, & collecta Å¿it aqua <lb/>effuÅ¿a ab vnoquoque illorum, </s> - <s xml:id="s1109" xml:space="preserve">tunc <reg norm="vnaquæque" type="simple">vnaquæq;</reg> iÅ¿tarum aquarum æqualis magnitudi-<lb/>nis erit Å¿ui corporis impellentis, & proportio ponderoÅ¿itatis illarum eadem erit, <lb/>quæ earum magnitudinum ex omogeneitate, quapropter Å¿i vnamquamque illarum <lb/>ponderabimus, habebimus propoÅ¿itum ex præcedenti theoremate.</s> - </p> - </div> - <div type="math:theorem" level="3" n="126"> - <head xml:id="head144" xml:space="preserve">THEOREMA <num value="126">CXXVI</num>.</head> - <p> - <s xml:id="s1110" xml:space="preserve">SED cum Å¿cire voluerimus pondus alicuius magnitudinis aquæ æqualis alicui <lb/>corpori ponderoÅ¿o, breuiſſimus modus erit ponderando ipÅ¿um corpus tam in ae-<lb/>re, quà m in aqua, & quia Å¿emper leuius erit in aqua, </s> - <s xml:id="s1111" xml:space="preserve">tunc differentia ponderum ip-<lb/>Å¿ius corporis, erit pondus quæſitum, hoc eÅ¿t vnius corporis aquei æqualis magnitu-<lb/>dinis magnitudini corporis propoÅ¿iti ex <ref id="ref-0014">.7. propoÅ¿itione lib. Archimedis de inÅ¿i-<lb/>dentibus aquæ</ref>. </s> - </p> - <p> - <s xml:id="s1112" xml:space="preserve">Quare ex præmiſſis quotieÅ¿cunque immerÅ¿a fuerint in aquam dicti vaÅ¿is duo cor <lb/>pora æquè ponderantia, Å¿ed diuerÅ¿arum Å¿pecierum, vt dictum eÅ¿t, proportio pon-<lb/>deris aquæ maioris ad pondus aquæ minoris magnitudinis eadem Å¿emper erit, quæ <lb/>ponderis minoris corporis ad pondus alicuius corporis eidem æqualis, Å¿peciei verò <lb/>maioris, vel eadem proportio ponderis alicuius corporis æqualis maiori, Å¿peciei ve <lb/>rò minoris ad pondus ipÅ¿ius maioris.</s> - </p> - <p> - <s xml:id="s1113" xml:space="preserve">Vt puta Å¿it corpus <var>.a.</var> argenteum æqualis ponderis corpori <var>.b.</var> aurei, & corpus <var>.u.</var> <lb/>argenteum æqualis magnitudinis corpori <var>.b.</var> aurei, corpus verò <var>.n.</var> aureum æqualis <lb/>magnitudinis corpori <var>.a.</var> argentei, corpus verò <var>.f.</var> aqueum æqualis magnitudinis cor- - <pb o="85" rhead="THEOREM. ARIT." n="97" file="0097" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0097"/> - pori <var>.a.</var> argentei, corpus autem <var>.e.</var> <reg norm="aqueum" type="context">aqueÅ©</reg> æqualis ma-<lb/> - <anchor type="figure" xlink:label="fig-0097-01a" xlink:href="fig-0097-01"/> - gnitudinis corpori <var>.b.</var> aurei. </s> - <s xml:id="s1114" xml:space="preserve">Tunc dico proportio-<lb/>nem ponderis <var>.f.</var> ad pondus <var>.e.</var> eadem eſſe, quæ pon-<lb/>deris <var>.b.</var> ad pondus <var>.u.</var> vt in præcedenti theoremate <lb/>iam dictum eÅ¿t, vel quæ ponderis <var>.n.</var> ad pondus <var>.a.</var> ex <lb/>11. quinti Euclidis. </s> - <s xml:id="s1115" xml:space="preserve">Proptereà quòe ponderis <var>.<lb/>n.</var> ad pondus <var>.a.</var> eft vt poderis <var>.b.</var> ad pondus <var>.u.</var> eo <lb/>quòd permutando ponderis <var>.n.</var> ad pondus <var>.b.</var> eÅ¿t vt <lb/>ponderis <var>.a.</var> ad pondus <var>.u.</var> ex corporum omogenei-<lb/>tate, & ex æqualitate magnitudinum corporum antecedentium & conÅ¿equentium.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0097-01" xlink:href="fig-0097-01a"> - <image file="0097-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0097-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="127"> - <head xml:id="head145" xml:space="preserve">THEOREMA <num value="127">CXXVII</num>.</head> - <p> - <s xml:id="s1116" xml:space="preserve">SCire etiam nos oportet, quòd quotieÅ¿cumque fuerint duo corpora aquea, quo-<lb/>rum vnum æqualis magnitudinis Å¿it alicui miÅ¿to, quod quidem miÅ¿tum graue <lb/>Å¿it tam in aere, quà m in aqua, alterum verò corpus aquem æqualis Å¿it magnitudi-<lb/>nis alicui corpoli Å¿implici, quod quidem corpus Å¿implex æqualis ponderis Å¿it dicto <lb/>corpori miÅ¿to. </s> - <s xml:id="s1117" xml:space="preserve">Tunc proportio ponderis aquei, cuius magnitudo æquatur magni <lb/>tudini corporis miÅ¿ti, ad pondus corporis aquei, cuius magnitudo æqualis eÅ¿t ma-<lb/>gnitudini corporis Å¿implicis, eadem erit, quæ proportio ponderis alicuius corpo-<lb/>ris Å¿implicis, cuius magnitudo æqualis Å¿it magnitudini corporis miÅ¿ti Å¿uperius dicti, <lb/>Å¿ed Å¿peciei corporis Å¿implicis iam dicti, ad pondus dicti miÅ¿ti.</s> - </p> - <p> - <s xml:id="s1118" xml:space="preserve">Exempli gratia, Å¿it corpus aqueum <var>.e.</var> magnitudinis æqualis corpori <var>.m.</var> mixto, <lb/>corpus verò aqueum <var>.i.</var> æqualis magnitudinis Å¿it corpori Å¿implici <var>.a.</var> quod quidem <lb/>corpus <var>.a.</var> æqualis ponderis Å¿it cum corpore <var>.m.</var> & corpus <var>.u.</var> Å¿it æqualis magnitudinis <lb/>cum corpore <var>.m.</var> Å¿ed Å¿peciei corporis <var>.a</var>. </s> - <s xml:id="s1119" xml:space="preserve">Tunc dico proportionem ponderis <var>.e.</var> ad <lb/>pondus <var>.i.</var> <reg norm="eandem" type="context">eãdem</reg> eſſe, quæ ponderis <var>.u.</var> ad pondus <var>.m.</var> primum nulli dubium eÅ¿t, quin <lb/>eadem proportio Å¿it magnitudinis <var>.e.</var> ad magnitudinem <var>.i.</var> quæ magnitudinis <var>.m.</var> ad <lb/>a. Å¿ed <var>.m.</var> ad <var>.a.</var> eÅ¿t vt <var>.u.</var> ad <var>.a.</var> ex .7. quinti </s> - <s xml:id="s1120" xml:space="preserve">quare <lb/>ex .11. eiuÅ¿dem proportio <var>.e.</var> ad <var>.i.</var> erit vt <var>.u.</var> ad <var>.a.</var> <lb/> - <anchor type="figure" xlink:label="fig-0097-02a" xlink:href="fig-0097-02"/> - de ipÅ¿ius magnitudinibus loquendo, Å¿ed propor-<lb/>tio ponderis <var>.u.</var> ad pondus <var>.a.</var> eadem eÅ¿t, quæ ma <lb/>gnitudinis <var>.u.</var> ad magnitudinem <var>.a.</var> ex omogenei-<lb/>tate. </s> - <s xml:id="s1121" xml:space="preserve">Idem dico de pondere <var>.e.</var> ad pondus .1. </s> - <s xml:id="s1122" xml:space="preserve">Qua-<lb/>re proportio ponderis <var>.e.</var> ad pondus <var>.i.</var> eadem erit <lb/>quæ ponderis <var>.u.</var> ad pondus <var>.a</var>. </s> - <s xml:id="s1123" xml:space="preserve">Sed ponderis <var>.u.</var> <lb/>ad pondus <var>.m.</var> eadem eÅ¿t quæ ponderis <var>.u.</var> ad pondus <var>.a.</var> ex .7. quinti, ergò ex .11. <lb/>eiuÅ¿dem proportio ponderis <var>.e.</var> ad pondus <var>.i.</var> eadem erit, quæ ponderis <var>.u.</var> ad pon-<lb/>dus <var>.m.</var> quod eÅ¿t propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0097-02" xlink:href="fig-0097-02a"> - <image file="0097-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0097-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="128"> - <head xml:id="head146" xml:space="preserve">THEOREMA <num value="128">CXXVIII</num>.</head> - <p> - <s xml:id="s1124" xml:space="preserve">NVNC ad cognoÅ¿cendam proportionem duarum diuerÅ¿arum Å¿pecierum in <lb/>corpore miÅ¿to propoÅ¿ito, tribus corporibus aqueis mediantibus, quæ <reg norm="quidem" type="context">quidẽ</reg> <lb/>corpora æqualium magnitudinum Å¿int alijs tribus corporibus vnius & eiuÅ¿dem pon <lb/>deris, quorum vnum Å¿it mixtum, reliqua verò duo Å¿implicia, Å¿ed Å¿pecierum mixti, <lb/>hoc ordine procedemus.</s> - </p> - <pb o="96" rhead="IO. BAPT. BENED." n="98" file="0098" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0098"/> - <p> - <s xml:id="s1125" xml:space="preserve">Sint exempli gratia, tria corpora æquè ponderantia, & vnumquodque illorum <lb/>Å¿itquinque librarum, quorum vnum Å¿it aureum, aliud argenteum, reliquum verò <lb/>mixtum ex ijs duobus metallis, vnde corpus aureum Å¿implex minus erit, & argen <lb/>teum maius corpore mixto, quod nulli dubium eÅ¿t, Å¿it nunc pondus corporis aquei <lb/>Ä™qualis corpori aureo, <reg norm="librarum" type="context">librarÅ©</reg> .3. aquei verò Ä™qualis miÅ¿to, Å¿it <reg norm="librarum" type="context">librarÅ©</reg> 3. <reg norm="cum" type="context">cÅ©</reg> quarta par <lb/>te, aquei demum æqualisargenteo, librarum .4. cum dimidia, vnde exijs, quæ in præ <lb/>cedenti theoremate, & in .126. theoremate diximus, Å¿i imaginatione concipiemus <lb/>alia duo corpora Å¿implicia, auri, & argenti, Å¿ed æqualium magnitudinum mixto, <lb/>habebimus proportionem ponderis aurei ad pondus corporis mixti vt <reg norm="trium" type="context">triÅ©</reg> librarum <lb/>cum quarta vnius ad .3. libras, & proportio ponderis mixti ad pondus argentei erit, <lb/>vt proportio librarum .4. cum dimidia ad tres libras cum quarta parte vnius libræ, <lb/>& proportio ponderis aurei ad pondus argentei vt librarum .4. cum dimidia ad li-<lb/>bras .3: hoc eÅ¿t aurei ad mixtum, vt .13. ad .12. & mixti ad argenteum, vt .18. ad .13. <lb/>& aurei ad argenteum, vt .3. ad .2. ideÅ¿t, vt .18. ad .12.</s> - </p> - <p> - <s xml:id="s1126" xml:space="preserve">Nunc inueniantur duo numeri ita inter Å¿e proportionati, vt .3. ad .2. habentes ta-<lb/>men inter ipÅ¿os numerum ita proportionatum ad maximum, vt .12. Å¿e habet ad <num value="13">.<lb/>13.</num> & ita proportionatum ad minimum, vt Å¿e habet .18. ad .13. quod hoc modo in-<lb/>ueniemus, multiplicabimus .18. per .12. & proueniet nobis .216. pro numero me-<lb/>dio, poÅ¿teà multiplicabimus .18. per .13. & proueniet .234. pro maximo, <reg norm="demum" type="context">demÅ©</reg> multi <lb/>plicando .12. per .13. proueniet .156. pro minimo, ita quod .234. correÅ¿pondebit <lb/>ponderi corporis aurei: </s> - <s xml:id="s1127" xml:space="preserve">216. verò ponderi mixti, et .156. ponderi argentei æqua-<lb/>lium magnitudinum.</s> - </p> - <p> - <s xml:id="s1128" xml:space="preserve">Cum autem proportiones horum trium corporum inuenerimus, Å¿i ordinem theo <lb/>rematis .122. Å¿equemur, habebimus quod quærebamus, & inueniemus in præſenti <lb/>exemplo proportionem ponderis auri ad pondus argenti in corpore mixto eſſe, vt <num value="180">.<lb/>180.</num> ad .36. Å¿ed quia Å¿uppoÅ¿itum fuit corpus mixtum eſſe quinque librarum, propte-<lb/>reà dicemus. </s> - <s xml:id="s1129" xml:space="preserve">Si .216. hoc eÅ¿t toti corpori mixto correÅ¿pondent quinque libræ tunc <lb/>parti .180. hoc eÅ¿t auro in ipÅ¿o corpore mixto, correÅ¿pondent libræ .4. cum duabus <lb/>vncijs, ex regula detribus, reÅ¿iduum verò quinque librarum, ideÅ¿t vnciæ decem, <lb/>correÅ¿pondent parti .36. hoc eÅ¿t argento in dicto corpore mixto.</s> - </p> - <p> - <s xml:id="s1130" xml:space="preserve">Sed Å¿i tria corpora dicta fuiſſent inuicem ita proportionata, vt .40. 47. 60. </s> - <s xml:id="s1131" xml:space="preserve">tunc <lb/>proportio auri ad argentum in corpore mixto eſſet vt .13. ad .7. quapropter <reg norm="cum" type="context">cÅ©</reg> pon <lb/>dus mixti fuiſſet .120. librarum, </s> - <s xml:id="s1132" xml:space="preserve">tunc aurum ipÅ¿ius eſſet librarum .78. argentum ve-<lb/>rò librarum .42. ex eadem regula.</s> - </p> - <p> - <s xml:id="s1133" xml:space="preserve">Pro quarum rerum Å¿peculatione nil aliud oportet nunc dicere cum Å¿atis dictum à no <lb/>bis Å¿uperius fuerit, vno excepto, hoc eÅ¿t rationem reddere, qua motus fui ad inue <lb/>niendos illos .3. numeros ita inter Å¿e diÅ¿poÅ¿itos, vt dictum eÅ¿t, quæ quidem ratio fuit, <lb/>vt haberemus .3. numeros ita inter ipÅ¿os ordinatè diÅ¿poÅ¿itos, vt Å¿unt pondera trium <lb/>illorum corporum æqualium magnitudinum. </s> - <s xml:id="s1134" xml:space="preserve">Proptereà quòd quamuis inter pri-<lb/>mos .3. numeros ponderum corporum aqueorum eædem fuerint proportiones pon <lb/>derum corporum metallicorum, nihilominus medius numerus extra proprium lo-<lb/>cum, & inordinatè inueniebatur, reÅ¿pectu extremorum, vnde medius numerus in <lb/>Å¿uo vero Å¿itu inter .18. et .12. fuiſſent .16. <reg norm="cum" type="context">cÅ©</reg> .8. tertijs decimis, Å¿ed vt <reg norm="fractorum" type="context">fractorÅ©</reg> incom <lb/>moditatem euitemus, præcepi, vt multiplicarentur extrema per .13. vnde produ-<lb/>cti fuerunt numeri .234. et .156. in <reg norm="eadem" type="context">eadẽ</reg> proportione, quæ eÅ¿t .18. ad .12. ex .18. Å¿epti <lb/>mi, iuſſi etiam multiplicari .18. per .12. vt nobis prodiret .216. ad quem numerum, <lb/>numerus .234. ita Å¿e haberet, ut .13. ad .12. ex .19. Å¿eptimi, quod autem ita Å¿it propor - <pb o="87" rhead="THEOREM. ARITH." n="99" file="0099" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0099"/> - tionatus .216. ad .156. vt .18. ad .13. maniteÅ¿tum eÅ¿t exijÅ¿dem, nam tam .18. quam <num value="13">.<lb/>13.</num> multiplicatus fuit per .12.</s> - </p> - </div> - <div type="math:theorem" level="3" n="129"> - <head xml:id="head147" xml:space="preserve">THEOREMA <num value="129">CXXIX</num>.</head> - <p> - <s xml:id="s1135" xml:space="preserve">ALIVD proponitur problema hoc modo: </s> - <s xml:id="s1136" xml:space="preserve">Å¿upponitur obÅ¿idio alicuius loci, vbi <lb/>alimento ad nutriendos .10000. homines Å¿ufficiunt pro quinque menÅ¿ibus tan-<lb/>tum, Å¿ed quia eum locum obÅ¿idione non liberari putatur niÅ¿i .18. menÅ¿ibus exactis, <lb/>quæritur, quot homines eo tempore illis alimentis nutriri poſſint, hoc eÅ¿t .18. <lb/>menÅ¿ibus.</s> - </p> - <p> - <s xml:id="s1137" xml:space="preserve">Præcipitregula, vt multiplicetur primus numerus, hoc eÅ¿t hominum .10000. cum <lb/>Å¿ecundo, hoc eÅ¿t menÅ¿ium quinque, productum verò diuidatur per .18. hoc eÅ¿t men-<lb/>Å¿ium, </s> - <s xml:id="s1138" xml:space="preserve">tunc proueniet .2777. cum .7. nonis.</s> - </p> - <p> - <s xml:id="s1139" xml:space="preserve">Cuius operationis ratio eÅ¿t hæc, Å¿int exempli gratia duo hic Å¿ubÅ¿cripta producta <lb/>Å¿uperficialia <var>.a.n.</var> et <var>.o.u.</var> inuicem æqualia, Å¿ed tal@ figura delineata, vt proportio <var>.u.<lb/>x.</var> ad <var>.x.o.</var> Å¿it, vt .10000. ad quinque, & proportio <var>a.x.</var> ad <var>.x.o.</var> Å¿it vt .18. ad quinque, <lb/>ct <var>.x.n.</var> Å¿it nobis ignota, quæ quidem eÅ¿t illa, quæ indagatur, ita <reg norm="quod" type="simple">ê™</reg> vnumquodque <lb/>iÅ¿torum productorum Å¿ignificabit alimentum, et <var>.u.x.</var> Å¿ignificabit numerum homi-<lb/>num .10000. qui quidem homines comederent totum alimentum <var>.u.o.</var> Å¿pacio tem-<lb/>poris <var>.x.o.</var> quinque menÅ¿ium, proptereà quòd <var>u.o.</var> Å¿upponitur productum eſſe ab <var>.<lb/>u.x.</var> in <var>.x.o</var>. </s> - <s xml:id="s1140" xml:space="preserve">Deinde <reg norm="Å¿upponendo" type="context">Å¿upponẽdo</reg> <var>.a.x.</var> tem <lb/> - <anchor type="figure" xlink:label="fig-0099-01a" xlink:href="fig-0099-01"/> - pus eſſe .18. menÅ¿ium, ergo <var>.x.n.</var> Å¿ignifi-<lb/>cabit numerum hominum, qui eo tem-<lb/>poris Å¿pacio ali poſſunt, hoc eÅ¿t <var>.x.a.</var> ali-<lb/>mento <var>.n.a.</var> eo quòd <var>.a.n.</var> producitur ex <var>.<lb/>n.x.</var> in <var>.a.x.</var> vnde ex .15. Å¿exti, Å¿eu ex, 20. <lb/>Å¿eptimi proportio <var>.x.u.</var> ad <var>.x.n.</var> <reg norm="eadem" type="context">eadẽ</reg> erit, <lb/>quÄ™ <var>.a.x.</var> ad <var>.x.o.</var> quapropter rectè factum <lb/>erit accipere <reg norm="productum" type="context">productÅ©</reg> <var>.u.o.</var> quodidem <lb/>eÅ¿t in quantitate, quod productum .2. n. & ipÅ¿um diuidere per <var>.a.x.</var> vnde nobis <lb/>proueniat <var>.n.x</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0099-01" xlink:href="fig-0099-01a"> - <image file="0099-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0099-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="130"> - <head xml:id="head148" xml:space="preserve">THEOREMA <num value="130">CXXX</num>.</head> - <p> - <s xml:id="s1141" xml:space="preserve">QVotieÅ¿cunque nobis propoÅ¿itum fuerit inuenire tertium terminum, trium ter <lb/>minorum continuè proportionalium armonicæ proportionalitatis, quo-<lb/>tum duo nobis cogniti Å¿int, ita agemus.</s> - </p> - <p> - <s xml:id="s1142" xml:space="preserve">Sint, exempli gratia, tres termini <var>.q.p</var>: <var>a.g.</var> et <var>.e.c.</var> continuæ proportionalium at <lb/>monicæ proportionalitatis, quorum <var>.q.p.</var> maior et <var>.a.g.</var> medius Å¿int nobis cogniti, <lb/>cum ergo voluerimus tertium <var>.e.<lb/>c.</var> cognitum nobis eſſe: </s> - <s xml:id="s1143" xml:space="preserve">a.g. detra-<lb/> - <anchor type="figure" xlink:label="fig-0099-02a" xlink:href="fig-0099-02"/> - hatur ex <var>.q.p.</var> differentia verò <var>.d.<lb/>p.</var> addatur <var>.q.p.</var> quorum Å¿umma <lb/>erit <var>.q.o.</var> cognita, qua mediante <lb/>diuidatur productum, quod ex <var>.a.<lb/>g.</var> in <var>.d.p.</var> exurgit, & proueniet no <lb/>bis <var>.n.g.</var> hoc e@t minor differentia, eo quòd productum <var>.q.o.</var> in <var>.n.g.</var> æquale eÅ¿t pro- - <pb o="88" rhead="IO. BAPT. BENED." n="100" file="0100" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0100"/> - ducto .2. g. in <var>.d.p.</var> ex .20. Å¿eptimi, proptereà quòd proportio <var>.q.o.</var> ad <var>.o.p.</var> hoc eÅ¿t ad <var>.<lb/>d.p.</var> eÅ¿t vt <var>.a.g.</var> ad <var>.g.n.</var> coniunctim cum diÅ¿iunctim it a Å¿it <var>.q.p.</var> ad <var>.p.o.</var> vt <var>.a.n.</var> ad <var>.n.g.</var> <lb/><reg norm="permutando" type="context">permutãdo</reg> eo quòd <var>.q.p.</var> ad <var>.a.n.</var> (ideÅ¿t ad <var>.e.c.</var>) ita Å¿e <reg norm="hent" type="context">hẽt</reg> ut <var>.p.o.</var> (hoc eÅ¿t <var>.d.p.</var>) ad <var>.n.g.</var> <lb/>ex <reg norm="conditionibus" type="context">cõditionibus</reg> armonicæ proportio nalitatis. </s> - <s xml:id="s1144" xml:space="preserve">Deinde Å¿i detraxerimus <var>.n.g.</var> ex <var>.a.g.</var> <lb/>remanebit <var>.e.c.</var> minor terminus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0099-02" xlink:href="fig-0099-02a"> - <image file="0099-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0099-02"/> - </figure> - </div> - <p> - <s xml:id="s1145" xml:space="preserve">Sed Å¿i <var>.e.c.</var> tertius terminus nobis propoÅ¿itus eſſet Å¿imul cum <var>.a.g.</var> medio, & volue <lb/>rimus maiorem inuenire <var>.q.p.</var> Å¿cilicet, oportebit <var>.e.c.</var> ex <var>.a.g.</var> detrahere, differentiam <lb/>verò <var>.n.g.</var> Å¿imiliter demeremus <lb/>ex <var>.e.c.</var> unde remaneret nobis <var>.e.t.</var> <lb/> - <anchor type="figure" xlink:label="fig-0100-01a" xlink:href="fig-0100-01"/> - cognitum, quo reÅ¿iduo <var>.c.t.</var> me-<lb/>diante diuidemus productum, <reg norm="quod" type="simple">ê™</reg> <lb/>furgit ex <var>.a.g.</var> in <var>.t.c.</var> & prouentus <var>.<lb/>d.p.</var> erit differentia maior, eo <reg norm="quod" type="simple">ê™</reg> <lb/><reg norm="productum" type="context">productÅ©</reg> quod Å¿it ex <var>.e.t.</var> in <var>.d.p.</var> <lb/>æquale eÅ¿t producto quòd fit ex <var>.a.g.</var> in <var>.t.c.</var> per 20. Å¿eptimi Eucli. eo quòd <var>.a.g.</var> (id-<lb/>eÅ¿t <var>.q.d.</var>) ad <var>.d.p.</var> eÅ¿t ut <var>.e.t.</var> ad <var>.t.c.</var> diÅ¿iunctim, cum coniunctim ita Å¿it <var>.q.p.</var> ad <var>.d.p.</var> vt <var>.e.<lb/>c.</var> ad <var>.t.c.</var> permutando, quia <var>.q.p.</var> ad <var>.e.c.</var> eÅ¿t vt <var>.d.p.</var> ad <var>.t.c.</var> hoc eÅ¿t ad <var>.n.g.</var> ex legibus <lb/>dictis.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0100-01" xlink:href="fig-0100-01a"> - <image file="0100-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0100-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="131"> - <head xml:id="head149" xml:space="preserve">THEOREMA <num value="131">CXXXI</num>.</head> - <p> - <s xml:id="s1146" xml:space="preserve">ALIA etiam methodo hoc perfici poſſe comperi. </s> - <s xml:id="s1147" xml:space="preserve">PropoÅ¿iti enim cum nobis fue <lb/>rint duo termini <var>.c.e.</var> minimus et <var>.g.a.</var> medius, maximus verò quærendus Å¿it, de <lb/>trahatur differentia <var>.g.n.</var> ex <var>.e.c.</var> & per reÅ¿iduum <var>.e.t.</var> diuidatur productum <reg norm="quod" type="simple">ê™</reg> fit ex <var>.a.<lb/>g.</var> in <var>.e.c.</var> prouentus quæ erit <var>.q.p.</var> terminus quæſitus.</s> - </p> - <p> - <s xml:id="s1148" xml:space="preserve">Pro cuius ratione, ponamus in eſſe terminum <var>.q.p.</var> </s> - <s xml:id="s1149" xml:space="preserve">tunc ex forma huius proportio <lb/>nalitatis nulli dubium erit quin <var>.q.p.</var> ad <var>.e.c.</var> fit vt <var>.d.p.</var> ad <var>.n.g.</var> hoc eft ad <var>.t.c.</var> vnde ex <lb/>19. quinti vel .12. Å¿eptimi ita eſſet <var>.q.d.</var> ad <var>.e.t.</var> vt <var>.q.p.</var> ad <var>.e.c.</var> </s> - <s xml:id="s1150" xml:space="preserve">quare ex .20. @cptimi pro <lb/>ductum <reg norm="quod" type="simple">ê™</reg> naÅ¿citur ex <var>.p.d.</var> (hoc eÅ¿t <var>.a.g.</var>) in <var>.e.c.</var> æquale eric producto <var>.e.t.</var> in <var>.q.p.</var> qua-<lb/>propter Å¿i diuiÅ¿erimus id per <var>.e.t.</var> proueniet nobis <var>.q.p</var>.</s> - </p> - <p> - <s xml:id="s1151" xml:space="preserve">Sed <reg norm="cum" type="context">cÅ©</reg> nobis propoÅ¿iti fuerint duo termini <var>.q.p.</var> maximus, et <var>.a.g.</var> medius, Å¿i <reg norm="mini- mum" type="context">mini-<lb/>mÅ©</reg> <var>.e.c.</var> <reg norm="voluerimus" type="simple">voluerimê°</reg> inuenire. </s> - <s xml:id="s1152" xml:space="preserve">Termino <var>.q.p.</var> maximo, <reg norm="iungatur" type="context simple">iÅ©gat̃</reg>. <var>p.o.</var> Ä™qualis, <var>p.d.</var> <reg norm="differentiÄ™" type="context">differẽtiÄ™</reg> <lb/>propoÅ¿itæ, diuidatur poÅ¿tea productum <reg norm="quod" type="simple">ê™</reg> ex <var>.q.p.</var> in <var>.a.g.</var> generatur per <var>.q.o.</var> prouen <lb/>tus autem Å¿it <var>.e.c.</var> qui quidem erit terminus quæſitus.</s> - </p> - <p> - <s xml:id="s1153" xml:space="preserve">Cuius operationis Å¿peculutio hæc erit, Å¿upponatur terminum <var>.e.c.</var> inuentum eſſe <lb/>vnde <var>.n.g.</var> differentia Å¿it inter <var>.e.c.</var> <lb/>et <var>.a.g.</var> ex forma igitur armonicæ <lb/> - <anchor type="figure" xlink:label="fig-0100-02a" xlink:href="fig-0100-02"/> - proportionalitis ita erit <var>.q.p.</var> ad <var>.a.<lb/>n.</var> vt <var>.p.o.</var> ad <var>.n.g.</var> vnde ex .13. quin-<lb/>ti. </s> - <s xml:id="s1154" xml:space="preserve">Ita erit <var>.q.o.</var> ad <var>.a.g.</var> vt <var>.q.p.</var> ad <var>.a.<lb/>n.</var> ergo <reg norm="productum" type="context">productÅ©</reg> quòd fit ex <var>.a.g.</var> <lb/>in <var>.q.p.</var> (ex .20. Å¿eptimi) æquale erit <lb/>producto <var>.q.o.</var> in <var>.a.n</var>. </s> - <s xml:id="s1155" xml:space="preserve">Quare Å¿i diuiÅ¿um fuerit tale productum per <var>.q.o.</var> proueniet no-<lb/>bis <var>.e.c.</var> quòd querebamus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0100-02" xlink:href="fig-0100-02a"> - <image file="0100-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0100-02"/> - </figure> - </div> - <pb o="89" rhead="THEOR. ARITH." n="101" file="0101" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0101"/> - </div> - <div type="math:theorem" level="3" n="132"> - <head xml:id="head150" xml:space="preserve">THEOREMA <num value="132">CXXXII</num>.</head> - <p> - <s xml:id="s1156" xml:space="preserve">SED quia aliquis poſſet in dubium reuocare, an poſſibile Å¿it inuenire tertium <lb/>terminum rationalem, Å¿eu communicantem duobus datis terminis inter Å¿e com <lb/>municantibus in tali proportionalitate, hoc eÅ¿t harmonica. </s> - <s xml:id="s1157" xml:space="preserve">Vthoc oÅ¿tendatur.</s> - </p> - <p> - <s xml:id="s1158" xml:space="preserve">Sint duo termini dati <var>.a.o.</var> et <var>.a.e.</var> inter Å¿e communicantes, tertius verò inuentus <lb/>Å¿it <var>.a.c.</var> qui maximus, primò, Å¿it in ea proportionalitate, quem dico communicantem <lb/>eſſe cum primis datis.</s> - </p> - <p> - <s xml:id="s1159" xml:space="preserve">Nam ex conditionibus huiuÅ¿modi proportionalitatis, habebimus primum ean-<lb/>dem proportionem eſſe <var>.a.c.</var> ad <var>.a.o.</var> quæ eÅ¿t <var>.e.c.</var> ad <var>.e.o.</var> vnde permutando ita erit <var>.a.<lb/>c.</var> ad <var>.e.c.</var> vt <var>.a.o.</var> ad <var>.o.e.</var> & quia ex .9. decimi Euclid <var>.a.o.</var> communicat cum <var>.o.e.</var> </s> - <s xml:id="s1160" xml:space="preserve">quare <lb/>ex .10. eiuÅ¿dem <var>.a.c.</var> communicabit cum <var>.e.c.</var> & per .9. cum <var>.a.e.</var> et per .8. cum <var>.a.o.</var> <lb/>quod<unsure/>eÅ¿t propoÅ¿itum.</s> - </p> - <p> - <s xml:id="s1161" xml:space="preserve">Sed Å¿i datus fuerit maximus <var>.a.c.</var> cum medio <var>.a.e.</var> interÅ¿e communicantes mini-<lb/>mum verò <var>.a.o.</var> probabo <reg norm="communicantem" type="context">cõmunicantem</reg> cum illis eſſe. </s> - <s xml:id="s1162" xml:space="preserve">Cogitemus ergo <var>.c.f.</var> æqua-<lb/>jem eſſe differentiæ <var>.c.e.</var> cognitæ, vnde habebimus proportionem, <var>a.c.</var> ad <var>.c.f.</var> vt <var>.a.o.</var> <lb/>ad <var>.o.e.</var> & componendo <var>.a.f.</var> ad <var>.f.c.</var> vt <var>.a.e.</var> ad <var>.e.o.</var> & quia (ex Å¿uppoÅ¿ito). <var>a.c.</var> commu-<lb/>nicat cum <var>.e.c.</var> hoc eÅ¿t cum <var>.c.f.</var> </s> - <s xml:id="s1163" xml:space="preserve">quare <lb/>ex eadem .9. dicti decimi <var>.a.f.</var> et <var>.f.c.</var> <reg norm="erunt" type="context">erÅ©t</reg> <lb/> - <anchor type="figure" xlink:label="fig-0101-01a" xlink:href="fig-0101-01"/> - inter Å¿e communicantes, & per .10. <var>a.e.</var> <lb/>communicabit cum <var>.o.e.</var> & per .9. <var>a.e.</var> cò <lb/>municabit cum <var>.a.o.</var> vnde per .8. <var>a.o.</var> communicabit cum <var>.a.c.</var> Å¿imiliter.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0101-01" xlink:href="fig-0101-01a"> - <image file="0101-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0101-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="133"> - <head xml:id="head151" xml:space="preserve">THEOREMA <num value="133">CXXXIII</num>.</head> - <p> - <s xml:id="s1164" xml:space="preserve">SED Å¿i nobis duo extremi termini propoÅ¿iti fuerint, & medium inuenire deÅ¿ide <lb/>remus in dicta proportionalitate, ita faciendum erit.</s> - </p> - <p> - <s xml:id="s1165" xml:space="preserve">Sint, exempli gratia, duo termini dati <var>.q.b.</var> et <var>.b.r.</var> minor <var>.b.r.</var> ex maiori <var>.b.q.</var> de-<lb/>trahatur, reÅ¿iduum verò <var>.q.x.</var> multiplicetur per <var>.b.r.</var> productum poÅ¿teà diuidatur per <lb/><var>q.r.</var> vnde proueniet nobis <var>.x.l.</var> pro differentia minori, quæ addita cum <var>.b.x.</var> minimo <lb/>termino, dabit nobis <var>.b.l.</var> mcdium terminum harmonicum.</s> - </p> - <p> - <s xml:id="s1166" xml:space="preserve">Pro cuius ratione cogitemus dictum medium terminum <var>.b.l.</var> iam inuentum eſſe, <lb/>vnde ita erit proportio <var>.q.l.</var> ad <var>.l.x.</var> vt <var>.q.b.</var> ad <var>.b.r.</var> ex forma huius proportionalitatis, <lb/></s> - <s xml:id="s1167" xml:space="preserve">quare coniunctim ita erit <var>.q.r.</var> ad <var>.r.b.</var> vt <lb/><var>q.x.</var> ad <var>.x.l.</var> & proptereà ex .20. Å¿eptimi <lb/> - <anchor type="figure" xlink:label="fig-0101-02a" xlink:href="fig-0101-02"/> - productum, quod fit ex <var>.q.r.</var> in <var>.x.l.</var> æqua-<lb/>le erit producto <var>.q.x.</var> in <var>.b.r</var>. </s> - <s xml:id="s1168" xml:space="preserve">Rectè igitur <lb/>fit cum diuiditur hoc productum per <var>.q.r.</var> vt proueniat nobis <var>.x.l.</var> differentia minor.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0101-02" xlink:href="fig-0101-02a"> - <image file="0101-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0101-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="134"> - <head xml:id="head152" xml:space="preserve">THEOREMA <num value="134">CXXXIIII</num>.</head> - <p> - <s xml:id="s1169" xml:space="preserve">POſſumus etiam harmonicè diuidere vnam datam proportionem abÅ¿que aliqua <lb/>diuiÅ¿ione productorum, ne nobis fractiones proueniant, hoc modo videlicet. <lb/></s> - <s xml:id="s1170" xml:space="preserve">Nobis propoÅ¿itum Å¿it diuidere harmonicè Å¿eÅ¿quialteram <reg norm="proportionem" type="context">proportionẽ</reg> inuenian-<lb/>tur primo minimi termini huius proportionis ut putà .3. et .2. quarum Å¿umma, hoc <lb/>eÅ¿t quinque, multiplicetur per minorem ideÅ¿t .2. vnde proueniet nobis .10. qui qui-<lb/>dem erit minor terminus trium quæſitorum, quorum maximus erit productum Å¿um - <pb o="90" rhead="IO. BAPT. BENED." n="102" file="0102" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0102"/> - mæ iam dictæ in maiorem eorum, hoc eÅ¿t quod fit ex quinque in .3. quod erit .15. </s> - <s xml:id="s1171" xml:space="preserve">Vt <lb/>autem medium terminum harmonicum inter iÅ¿tos habeamus, accipiatur <reg norm="duplum" type="context">duplÅ©</reg> pro-<lb/>ducti, quod fit ex primis minimis terminis, quod erit .12.</s> - </p> - <p> - <s xml:id="s1172" xml:space="preserve">Cuius rei Å¿peculatio eÅ¿t iÅ¿ta: </s> - <s xml:id="s1173" xml:space="preserve">Å¿ignificentur duo termini datæ proportionis ab <var>.q.b.</var> <lb/>et <var>.b.r.</var> quorum Å¿umma erit <var>.q.r.</var> cuius quadratum Å¿it <var>.q.o.</var> Å¿it etiam imaginata <var>.b.e.</var> <lb/>parallela ad <var>.o.r</var>. </s> - <s xml:id="s1174" xml:space="preserve"><reg norm="Sitque" type="simple">SitqÌ;</reg> <var>.b.x.</var> æqualis <var>.b.r.</var> et <var>.q.u.</var> Å¿imiliter, & ducatur <var>.x.y.</var> parallela ad <lb/><var>r.o.</var> et <var>.u.l.</var> ad <var>.q.x</var>. </s> - <s xml:id="s1175" xml:space="preserve">Tunc habebimus <var>.b.o.</var> æquale ei producto, quod fit ex <var>.q.r.</var> in <var>.b.r.</var> <lb/>et <var>.b.y.</var> eidem etiam æquale, et <var>.q.e.</var> pro producto, quod fit ex <var>.q.r.</var> in <var>.q.b.</var> et <var>.q.l.</var> pro <lb/>eo, quod fit ex <var>.q.x.</var> in <var>.b.r</var>. </s> - <s xml:id="s1176" xml:space="preserve">Vnde <var>.q.l.</var> cum <var>.b.y.</var> æquale fiet duplo ei, quod fit ex <var>.q.b.</var> <lb/>in <var>.b.r</var>. </s> - <s xml:id="s1177" xml:space="preserve">Dico nunc <var>.b.o.</var> eſſe minimum terminum eorum, quos quærimus, et <var>.y.b.</var> cum <var>.<lb/>x.u.</var> medium <var>.q.e.</var> verò maximum huiuÅ¿modi proportionalitatis.</s> - </p> - <p> - <s xml:id="s1178" xml:space="preserve">Primum ergo certi Å¿cimus ex prima Å¿exti vel .18. Å¿eptimi eandem exiÅ¿tere pro-<lb/>portionem <var>.q.e.</var> ad <var>.b.o.</var> Å¿eu ad <var>.b.y.</var> quæ <var>.q.b.</var> ad <var>.b.r</var>: Å¿ed <var>.u.y.</var> ad <var>.u.x.</var> eÅ¿t vt <var>.y.l.</var> ad <var>.l.x.</var> <lb/>hoc eÅ¿t vt <var>.q.b.</var> ad <var>.b.r.</var> ideÅ¿t vt <var>.q.e.</var> ad <var>.b.o.</var> & Å¿umma <var>.u.y.</var> cum <var>.u.x.</var> ideÅ¿t <var>.q.y.</var> minor eÅ¿t <lb/>quam <var>.q.e.</var> maximus terminus per <var>.b.y.</var> minimum ter-<lb/>minum. </s> - <s xml:id="s1179" xml:space="preserve">& <reg norm="coniunctim" type="context">cõiunctim</reg> <var>.q.y.</var> ad <var>.q.l.</var> vt <var>.y.x.</var> ad <var>.x.l.</var> hoc eÅ¿t <lb/> - <anchor type="figure" xlink:label="fig-0102-01a" xlink:href="fig-0102-01"/> - vt <var>.q.r.</var> ad <var>.r.b</var>. </s> - <s xml:id="s1180" xml:space="preserve">Vnde ex Å¿peculatione <reg norm="præcedentis" type="context">præcedẽtis</reg> theo <lb/>rematis, Å¿equitur <var>.u.y.</var> eſſe differentiam inter <reg norm="maximum" type="context">maximÅ©</reg> <lb/>& medium terminum, et <var>.u.x.</var> eſſe differentiam inter <lb/>medium & minimum dictæ proportionalitatis. </s> - <s xml:id="s1181" xml:space="preserve">Nam <lb/>eadem proportio eÅ¿t <var>.q.e.</var> maximi termini ad <var>.b.o.</var> mi-<lb/>nimi. quæ <var>.u.y.</var> (differentia inter <var>.q.e.</var> & gnomonem <var>.<lb/>u.b.y.</var>) ad <var>.u.x.</var> (differentia inter dictum <var>.u.b.y.</var> et <var>.b.y.</var> <lb/>minimum terminum, quia Å¿unt ambæ ut <var>.q.b.</var> ad <var>.b.r.</var> <lb/>vt diximus. </s> - <s xml:id="s1182" xml:space="preserve">Quare <var>.b.y.</var> <reg norm="coniunctum" type="context">coniunctÅ©</reg> cum <var>.x.u.</var> medius <lb/>terminus erit, qui quidem (vt dictum eÅ¿t) duplus eÅ¿t ei <lb/>quod fit ex <var>.q.b.</var> in <var>.b.r</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0102-01" xlink:href="fig-0102-01a"> - <image file="0102-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0102-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="135"> - <head xml:id="head153" xml:space="preserve">THEOREMA <num value="135">CXXXV</num>.</head> - <p> - <s xml:id="s1183" xml:space="preserve">ALIVM etiam modum ab antiquis traditum ad hoc problema perficiendum <lb/>inueni, qui talis eÅ¿t. </s> - <s xml:id="s1184" xml:space="preserve">Inueniatur primo inter datos terminos extremos, me-<lb/>dius terminus in arithmetica proportione, per <reg norm="quem" type="context">quẽ</reg> <lb/> - <anchor type="figure" xlink:label="fig-0102-02a" xlink:href="fig-0102-02"/> - multiplicetur vnuÅ¿quiÅ¿que dictorum extremorum, <lb/>deinde multiplicentur ipÅ¿i extremi interÅ¿e, vnde <lb/>habebimus tria producta eadem proportione inui <lb/>cem exiÅ¿tentia, vt quærebatur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0102-02" xlink:href="fig-0102-02a"> - <image file="0102-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0102-02"/> - </figure> - </div> - <p> - <s xml:id="s1185" xml:space="preserve">Exempli gratia, ponamus duos propoÅ¿itos ter-<lb/>minos eſſe .3. et .2. quorum medius arithmeticè <lb/>eſſet .2. cum dimidia vnitate, per quem cum vnum <lb/>quemque priorum multiplicauerimus, <reg norm="emergent" type="context">emergẽt</reg> no-<lb/>bis duo producta, quorum primum ideÅ¿t maius eſſet <lb/>7. cum dimidia vnitate, reliquum verò eſſet <lb/>quinque, productum poÅ¿teà quod ex ipÅ¿is extremis <lb/>prouenit, erit .6. quod quidem eÅ¿t harmonicè collo <lb/>catum inter .7. cum dimidia vnitate, & quinque.</s> - </p> - <p> - <s xml:id="s1186" xml:space="preserve">Cuius rei Å¿peculatio omnis à præcedenti theore-<lb/>mate dependet. </s> - <s xml:id="s1187" xml:space="preserve">Sint exempli gratia, duo termini - <pb o="91" rhead="THEOREM. ARIT." n="103" file="0103" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0103"/> - propoÅ¿iti <var>.a.e.</var> maior, et <var>.e.o.</var> minor, <reg norm="Sitque" type="simple">SitqÌ;</reg> <var>.o.k.</var> medius arithmeticus inter dictos, vn-<lb/>de clarè patebit <var>.o.k.</var> eſſe dimidium Å¿ummæ dictorum terminorum ex .75. theorema <lb/>te huius libri. </s> - <s xml:id="s1188" xml:space="preserve">Sit ergo productum <var>a.t.</var> id quod fit ex <var>.a.e.</var> in <var>.o.k.</var> et <var>.o.t.</var> Å¿it <reg norm="productum" type="context">productÅ©</reg> <lb/>quod fit ex <var>.e.o.</var> in <var>.o.k.</var> et <var>.n.m.</var> Å¿it productum quod Å¿it ex <var>.a.e.</var> in <var>.e.o.</var> quorum vnum-<lb/>quodque erit dimid ium vniuÅ¿cuiuÅ¿que producti præcedentis theorematis, <lb/>ex .18. et .19. Å¿eptimi Eucli. vnumquodque Å¿ui relatiui. </s> - <s xml:id="s1189" xml:space="preserve">Quare argumentando per <lb/>mutando à concluÅ¿ionibus præcedentis theorematis ad has præſentis, habebimus <lb/>productum.</s> - </p> - </div> - <div type="math:theorem" level="3" n="136"> - <head xml:id="head154" xml:space="preserve">THEOREMA <num value="136">CXXXVI</num>.</head> - <p> - <s xml:id="s1190" xml:space="preserve">MEDIVM autem contra <reg norm="harmonicum" type="context">harmonicÅ©</reg> inuenire cum quis voluesit inter duos <lb/>propoÅ¿itos terminos, ita faciendum erit, hoc eÅ¿t per Å¿ummam datorum ex <lb/>tremorum diuidatur productum quod fit ex minimo termino in <reg norm="differentiam" type="context">differẽtiam</reg> dato-<lb/>rum, prouentus poÅ¿tea erit differentia inter maximum & med<unsure/>um quæſitum.</s> - </p> - <p> - <s xml:id="s1191" xml:space="preserve">Vt exempli gratia, Å¿i nobis propoÅ¿iti fuerint hi duo termini .3. et .2. Å¿umma eo-<lb/>rum erit quinque, per quam cum diuiÅ¿erimus productum, quod naÅ¿citur ex mini-<lb/>mo .2. in differentiam eorum, quæ eÅ¿t vnum, quod quidem erit .2. </s> - <s xml:id="s1192" xml:space="preserve">tunc duæ quintæ <lb/>partes prouenient, quæ Å¿i demptæ fuerint ex maximo termino, reliquum erit .2. <reg norm="cum" type="context">cÅ©</reg> <lb/>3. quintis, hoc eÅ¿t medius terminus contta harmonicus.</s> - </p> - <p> - <s xml:id="s1193" xml:space="preserve">Pro cuius ratione cogitemus <var>.u.d.</var> et <var>.x.c.</var> eſſe duosterminosnobis propoÅ¿itos, in-<lb/>ter quos deÅ¿ideremus inuenire <var>.o.s.</var> medium ita illis <reg norm="relatum" type="context">relatÅ©</reg>, vt proportio exceſſus ip-<lb/>Å¿ius Å¿upra <var>.x.c.</var> (qui Å¿it <var>.e.n.</var>) ad exceÅ¿-<lb/>Å¿um <var>.u.d.</var> Å¿upra <var>.o.s.</var> (qui Å¿it <var>.n.d.</var>) ea-<lb/> - <anchor type="figure" xlink:label="fig-0103-01a" xlink:href="fig-0103-01"/> - dem Å¿it quæ <var>.u.d.</var> ad <var>.x.c</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0103-01" xlink:href="fig-0103-01a"> - <image file="0103-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0103-01"/> - </figure> - </div> - <p> - <s xml:id="s1194" xml:space="preserve">Cogitemus igitur <var>.x.c.</var> coniunctum <lb/>eſſe cum <var>.u.d.</var> & hæcÅ¿umma vocetur <var>.<lb/>b.d.</var> vnde habebimus proportionem <var>.<lb/>u.d.</var> ad <var>.u.b.</var> vt <var>.e.n.</var> ad <var>.n.d</var>. </s> - <s xml:id="s1195" xml:space="preserve">Quare <reg norm="com- ponendo" type="context">cõ-<lb/>ponendo</reg> ita erit <var>.d.b.</var> ad <var>.u.b.</var> ut <var>.e.d.</var> 3d.n.d. Å¿ed quia <var>.d.b</var>: <var>u.b.</var> et <var>.e.d.</var> quantitates no-<lb/>bis cognitÄ™ Å¿unt, ideò <var>.d.n.</var> ex .20. Å¿eptimi cognita nobis erit.</s> - </p> - </div> - <div type="math:theorem" level="3" n="137"> - <head xml:id="head155" xml:space="preserve">THEOREMA <num value="137">CXXXVII</num>.</head> - <p> - <s xml:id="s1196" xml:space="preserve">SVpponunt antiqui aliquot mercatores dantes pecunias lucro in diuerÅ¿is vnius <lb/>anni temporibus, </s> - <s xml:id="s1197" xml:space="preserve">tunc in fine anni Å¿umma torius lucri datur cognita, Å¿ed quæ-<lb/>ritur quantuni<unsure/> vnicuique illorum exipÅ¿a Å¿umma debeatur.</s> - </p> - <p> - <s xml:id="s1198" xml:space="preserve">Exempli gratia, primus in principio anni poÅ¿uit .100. aurcos, Å¿ecundus verò .100 <lb/>diebus poÅ¿t primum poÅ¿uit .50. aureos tertius autem .200. diebus poÅ¿t primum po-<lb/>Å¿uit .25. aureos Å¿umma lucri poÅ¿tea in fine anni fuit aureorum .60.</s> - </p> - <p> - <s xml:id="s1199" xml:space="preserve">Nunc vt Å¿ciamus quantum huius Å¿ummæ vniduique illorum proueniat, præcipit <lb/>regula, vt faciamus tria producta, quorum primum Å¿it ex numero dierum totius an-<lb/>ni in numerum aureorum primi, vnde tale productum in præſenti caÅ¿u erit .36500. <lb/>Å¿ecundum verò Å¿it ex numero dierum à primo die in quo ipÅ¿e Å¿ecundus poÅ¿uit uÅ¿que <lb/>ad finem anni, in numerum ipÅ¿orum nummorum, quod erit .13250. tertium autem <lb/>productum ex diebus tertij in numerum Å¿uorum aureorum, quod <reg norm="quidem" type="context">quidẽ</reg> erit .4125. <lb/>quæ producta Å¿imul collecta faciunt .53875. deinde multiplicetur vnumquodque - <pb o="92" rhead="IO. BAPT. BENED." n="104" file="0104" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0104"/> - ipÅ¿orum prochictorum per Å¿ummam lucri hoc eÅ¿t per .60. vnde multiplicatio primi <lb/>producti erit .2190000. multiplicatio verò Å¿ecundi producti erit .795000. tertij po <lb/>Å¿tca erit .247500. quarum multiplicationum vnaquæque diuidatur per Å¿ummam <lb/>53875. productori<unsure/>t, & proueniet ex prima diuiÅ¿ione .40. <reg norm="cum" type="context">cÅ©</reg> fractis .35000. vnius in-<lb/>tegri diuiÅ¿i in partes .53875. quod erit lucrum primi, prouentus autem Å¿ecundæ di-<lb/>uiÅ¿ionis erit .14. cum fractis .41050. vnius integri diuiÅ¿i in partes .53875. lucrum <reg norm="Å¿ecu di." type="context">Å¿ecÅ©Â<lb/>di</reg> </s> - <s xml:id="s1200" xml:space="preserve">prouentus verò quartæ diuiÅ¿ionis erit .4. cum fractis .32000. vnius integri, vt Å¿u <lb/>pra diuiÅ¿i in partes .53875. hoc eÅ¿t lucrum tertij.</s> - </p> - <p> - <s xml:id="s1201" xml:space="preserve">Cuius rei Å¿peculatio ex Å¿e in Å¿ub Å¿cripta figura patet, vbi <var>.a.q.</var> Å¿ignificat numerum <lb/>dierum totius anni pro primo mercatore <var>.q.n.</var> autem Å¿ignificat numerum dierum Å¿e <lb/>cundi mercatoris <var>.e.q.</var> poÅ¿teà ſignificat numerum dierum tertij Å¿it etiam <var>.s.a.</var> pro nu-<lb/>mero denariorum primi, et <var>.o.n.</var> pro numero Å¿ecundi, et <var>.e.t.</var> pro numero <lb/>tertij, productum autem <var>.q.s.</var> Å¿ignificet valorem primi lucri, et <var>.q.o.</var> Å¿ecundi, <lb/>et <var>.q.t.</var> tertij <var>.x.y.</var> autem Å¿ignificet Å¿ummam lucri omnium, et <var>.x.i.</var> Å¿ignificet <lb/>partem primi, et <var>.i.p.</var> Å¿ecundi, et <var>.p.y.</var> tertij. </s> - <s xml:id="s1202" xml:space="preserve">vnde clarè patebit ex communi <lb/>Å¿cientia quòd eadem proportio erit <var>.x.y.</var> ad <var>.x.i.</var> quæ aggregati omnium producto-<lb/>rum <var>.q.s</var>: <var>q.o.</var> et <var>.q.t.</var> ad <var>.q.s.</var> & ita <var>.x.y.</var> ad <var>.i.p.</var> vt aggregati dictiad <var>.q.o.</var> et <var>.x.y.</var> ad <var>.p.y.</var> <lb/>vt dicti aggregati ad <var>.q.t</var>. </s> - <s xml:id="s1203" xml:space="preserve">Rectè igitur ex regula de tribus multiplicatio <var>.q.s.</var> in <var>.x.y.</var> <lb/>diuiditur per aggregatum omnium <lb/> - <anchor type="figure" xlink:label="fig-0104-01a" xlink:href="fig-0104-01"/> - productorum, ita vt Å¿i aliquis dice-<lb/>ret, Å¿i ex dicto aggregato, prouenit <lb/><var>x.y.</var> quid proueniet vnicuique <reg norm="illo- rum" type="context">illo-<lb/>rÅ©</reg> <reg norm="productorum" type="context">productorÅ©</reg>. </s> - <s xml:id="s1204" xml:space="preserve"><reg norm="Nam" type="context">Nã</reg> Å¿i numerus dena-<lb/>riorum <reg norm="Å¿ecundi" type="context">Å¿ecÅ©di</reg> æqualis eſſet numero <lb/><var>a.s.</var> primi vt putà . <var>n.b</var>. </s> - <s xml:id="s1205" xml:space="preserve">tunc eius <reg norm="lucrum" type="context">lucrÅ©</reg> <lb/>Å¿ignificaretur à rectangulo <var>.q.b.</var> & ita <lb/>de tertio dico <reg norm="quod" type="simple">ê™</reg> Å¿ignificaretur à <reg norm="re- ctangulo" type="context">re-<lb/>ctãgulo</reg> <var>.q.c.</var> vel Å¿i Å¿i<unsure/>antibus <reg norm="ijÅ¿dem" type="context">ijÅ¿dẽ</reg> <reg norm="denariorum" type="context">denariorÅ©</reg> quantitatibus <var>.n.o.</var> et <var>.e.t.</var> omnes Å¿uas pe-<lb/>cunias eodem tempore poÅ¿uiſſent, </s> - <s xml:id="s1206" xml:space="preserve">tunc rectangula Å¿ignificantia eorum lucra eÅ¿lent <lb/><var>q.s.q.d.</var> et <var>.q.f.</var> Å¿ed cum nec eodem tempore, nec eandem quantitatem poÅ¿ueruntr<unsure/>e <lb/>ctè eorum lucra Å¿ignificantur à rectangulis <var>.q.s.q.o.</var> et <var>.q.t.</var> <reg norm="quod" type="simple">ê™</reg> ex prima .6. vel .18. aut <num value="19">.<lb/>19.</num> Å¿eptimi ratiocinando clarè patebit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0104-01" xlink:href="fig-0104-01a"> - <image file="0104-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0104-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="138"> - <head xml:id="head156" xml:space="preserve">THEOREMA <num value="138">CXXXVIII</num>.</head> - <p> - <s xml:id="s1207" xml:space="preserve">NIcolaus Tartalea in primo libro vltimæ partis numerorum ad .35. quæſitum <lb/>docet inuenire quantitatem laterum vnius propoÅ¿iti trianguli, cuius la-<lb/>r<unsure/>erum proportio nobis data Å¿it Å¿imul cum area Å¿uperÅ¿iciali ipÅ¿ius trianguli, Å¿ed quia <lb/>ipÅ¿e Tartalea vtiturregula algebræ, mihi viÅ¿um eÅ¿t breuiori methodo hoc idein fa <lb/>cere, & etiam vniuerÅ¿aliori via.</s> - </p> - <p> - <s xml:id="s1208" xml:space="preserve">Supp onamus igitur duo triangula, quorum vnum <var>.u.n.i.</var> Å¿it nobis <reg norm="propoÅ¿itum" type="context">propoÅ¿itÅ©</reg>, & <lb/>cognitæ Å¿uperficiei, proportiones Å¿imiliter laterum <var>.i.n.</var> ad <var>.n.u</var>: et <var>.u.n.</var> ad <var>.u.i.</var> Å¿int no <lb/>bis datæ, <reg norm="alterum" type="context">alterÅ©</reg> verò <reg norm="triangulum" type="context">triangulÅ©</reg> Å¿it <var>.a.o.u.</var> à nobis tamen ita <reg norm="confectum" type="context">confectÅ©</reg>, v<unsure/>latera Å¿int inÂ<lb/>er Å¿e proportionata eodem modo, quo latera prioris trianguli, Å¿ed hæc nobis <reg norm="etiam" type="context">etiã</reg> <lb/>cognita Å¿int, <reg norm="quod" type="simple">ê™</reg> facillimum eÅ¿t. </s> - <s xml:id="s1209" xml:space="preserve">Nunc vero Å¿i <reg norm="demptum" type="context">demptÅ©</reg> fuerit <reg norm="quadratum" type="context">quadratÅ©</reg> <var>.a.o.</var> minimi <lb/>lateris, ex quadrato <var>.o.u.</var> maximi, relinquet nobis duplum producti <var>.o.u.</var> in <var>.u.e.</var> per <lb/><reg norm="penultimam" type="context">penultimã</reg> .2. Eucli. <reg norm="Å¿upponendo" type="context">Å¿upponẽdo</reg> <var>.a.e.</var> perpendicularem ad <var>.o.u.</var> vnde tale productum <lb/>quòd fit ex <var>.o.u.</var> in <var>.u.e.</var> conÅ¿equenter nobis cognitum erit, & quia <var>.o.u.</var> nobis cogni- - <pb o="93" rhead="THEOREM. ARIT." n="105" file="0105" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0105"/> - tum eÅ¿t, ideo cognoÅ¿cemus <var>.e.u.</var> Å¿ed <reg norm="cum" type="context">cÅ©</reg> <var>.e.u.</var> minor Å¿it <var>.a.u.</var> ex .18. & penultima primi, <lb/>Å¿i <reg norm="demptum" type="context">demptÅ©</reg> fuerit quadratum <var>.e.u.</var> ex quadrato <var>.a.u.</var> remanebit nobis <reg norm="cognitum" type="context">cognitÅ©</reg> <reg norm="quadra- tum" type="context">quadra-<lb/>tÅ©</reg> <var>.a.e.</var> & Å¿ic nota erit nobis perpendicularis <var>.a.e.</var> ex penultima primi, quæ quidem <var>.<lb/>a.e.</var> Å¿i multiplicata fuerit in dimidium <var>.o.u.</var> dabit nobis <reg norm="Å¿uperficiem" type="context">Å¿uperficiẽ</reg> trianguli <var>.a.o.u.</var> ex <lb/>41. dicti libri. </s> - <s xml:id="s1210" xml:space="preserve">Et quia proportio trianguli <var>.a.o.u.</var> ad triangulum <var>.u.i.n.</var> (propter Å¿imi <lb/>litudinem) eÅ¿t vt quadrati <var>.o.u.</var> ad quadratum <var>.n.i.</var> ex communi Å¿cientia cum vna-<lb/>quæque iÅ¿tarum proportionum dupla Å¿it proportioni <var>.o.u.</var> ad <var>.n.i.</var> ex .17. et .18. Å¿exti, <lb/>deinde cum nobis cognitæ Å¿int tres iÅ¿tarum quatuor quantitatum hoc eÅ¿t Å¿uperficies <lb/>trianguli <var>.a.o.u.</var> Å¿uperficies trianguli <var>.u.n.i.</var> & quadrati <var>.o.u.</var> </s> - <s xml:id="s1211" xml:space="preserve">quare ex regula de tribus <lb/>cognoÅ¿cemus etiam quadratum <var>.n.i.</var> & Å¿ic <var>.n.i.</var> latus primi trianguli, vnde reliqua la <lb/>tera illicò nobis innoteÅ¿cent exipÅ¿a regula de tribus, cum dixerimus, Å¿i <var>.o.u.</var> dat nobis <lb/><var>u.a</var>. </s> - <s xml:id="s1212" xml:space="preserve">tunc <var>.i.n.</var> dabit <var>.u.n.</var> quòd etiam infero de <var>.u.i</var>.</s> - </p> - <p> - <s xml:id="s1213" xml:space="preserve">Poſſemus etiam ita hoc perficere, <lb/>Å¿cilicet inuenire <var>.x.</var> quantitatem me-<lb/> - <anchor type="figure" xlink:label="fig-0105-01a" xlink:href="fig-0105-01"/> - diam proportionalem inter duas Å¿u-<lb/>perficies triangulorum, vnde Å¿uper-<lb/>ficies trianguli <var>.i.a.u.o.</var> ad <var>.x.</var> Å¿e ha-<lb/>beret ut <var>.o.u.</var> ad <var>.i.n.</var> & ita ex regula <lb/>detribus cognoÅ¿cemus <var>.i.n</var>. </s> - <s xml:id="s1214" xml:space="preserve">Multo <reg norm="tem" type="context">tẽ</reg> <lb/>pore poÅ¿tquà m hoc theorema conÅ¿truxi, ipÅ¿um conÅ¿criptum inueni in decimo <lb/>Å¿ecundi libri Ioannis de monte Regio, Å¿atis tamen obÅ¿curè expreſſum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0105-01" xlink:href="fig-0105-01a"> - <image file="0105-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0105-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="139"> - <head xml:id="head157" xml:space="preserve">THEOREMA <num value="139">CXXXIX</num>.</head> - <p> - <s xml:id="s1215" xml:space="preserve">IN eodem primo libro vltimæ partis numerorum, Tartalea probat, via algebrÄ™ <lb/>quòd quælibet duo latera trianguli orthogonij, angulumrectum continentia, <lb/>Å¿int tertio longiora per diame-<lb/> - <anchor type="figure" xlink:label="fig-0105-02a" xlink:href="fig-0105-02"/> - trum circuli inÅ¿criptibilis in ip-<lb/>Å¿o triangulo. </s> - <s xml:id="s1216" xml:space="preserve">Å¿ed hoc breuius <lb/>geometricè poteÅ¿t <reg norm="demonÅ¿trari" type="context">demõſtrari</reg>, <lb/>quemadmodum in Å¿ubÅ¿cripta <lb/>hic figura videre eÅ¿t, proptereà <lb/>quòd cum anguli <var>.A.o.u.</var> et <var>.n.</var> <lb/>omnes Å¿int recti et <var>.A.u.</var> æqualis <lb/><var>o.n.</var> et <var>.A.n.</var> Ä™qualis <var>.u.o.</var> ipſæ <var>.A.<lb/>u.</var> et <var>.A.n.</var> æquales erunt diame-<lb/>tro ipÅ¿ius circuli. </s> - <s xml:id="s1217" xml:space="preserve">Sed eædem <var>.<lb/>A.u.</var> et <var>.A.n.</var> Å¿unt Å¿uperfluum, quo <var>.A.B.</var> et <var>.A.C.</var> Å¿unt maiores <var>.B.C.</var> cum <var>.B.u.</var> et <var>.C.n.</var> <lb/>Å¿int æquales <var>.B.C.</var> ex penultima tertij Eucli.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0105-02" xlink:href="fig-0105-02a"> - <image file="0105-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0105-02"/> - </figure> - </div> - <head xml:id="head158" xml:space="preserve">THEO. SEQVENS THEO. CXXXIX.</head> - <p> - <s xml:id="s1218" xml:space="preserve">SImiliter in nono capite Å¿ecundi libri nouæ Å¿cientiæ poterat ipÅ¿e Tartalea breuio <lb/>ri methodo abÅ¿que vlla operatione ipÅ¿ius Algebræ inuenire <var>.A.H.</var> reÅ¿pectu <var>.A.<lb/>E.</var> eſſe vt .4. <reg norm="cum" type="context">cÅ©</reg> vno Å¿eptimo ad <reg norm="vnum" type="context">vnÅ©</reg>. </s> - <s xml:id="s1219" xml:space="preserve"><reg norm="Nam" type="context">Nã</reg> ipÅ¿e Å¿upponit <var>.A.E.</var> <reg norm="decimam" type="context">decimã</reg> <reg norm="partem" type="context">partẽ</reg> eſſe ipÅ¿ius - <pb o="94" rhead="IO. BAPT. BENED." n="106" file="0106" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0106"/> - <var>A.I.</var> vnde quadratum lineæ <var>.A.I.</var> erit .100. idem dico de quadrato lineæ <var>.I.L</var>. </s> - <s xml:id="s1220" xml:space="preserve">quare <lb/>ex penultima primi <var>.A.L.</var> erit radix quadrata quadrati .200. ideÅ¿t .14. cum vno Å¿epti-<lb/>mo ferè. </s> - <s xml:id="s1221" xml:space="preserve">quare <var>.A.L.</var> iuncta <var>.A.O.</var> erit .28. cum duobus Å¿eptimis. </s> - <s xml:id="s1222" xml:space="preserve">Å¿ed <var>.L.O.</var> ex Å¿uppoÅ¿i-<lb/>to erit .20. eo quòd <var>.L.I.</var> Ä™quatur ipÅ¿i <var>.A.I.</var> Å¿imiliter et <var>.I.O.</var> vt ipÅ¿e etiam probauit. </s> - <s xml:id="s1223" xml:space="preserve">qua <lb/>dempta ex <var>.L.A.O.</var> relinquetur <var>.H.A.M.</var> (nam <var>.L.H.</var> cum <var>.O.M.</var> æquatur ipÅ¿i <var>.L.O.</var> ex .<lb/>35. tertij ipÅ¿ius Eucli. partium .8. <reg norm="cum" type="context">cÅ©</reg> duabus Å¿eptimis. cuius <reg norm="dimidium" type="context">dimidiÅ©</reg> hoc eÅ¿t <var>.A.H.</var> erit <lb/>4. cum una Å¿eptima, quod eÅ¿t propoÅ¿itum. </s> - <s xml:id="s1224" xml:space="preserve">ReÅ¿pice figuram ipÅ¿ius Tartaleæ.</s> - </p> - </div> - <div type="math:theorem" level="3" n="140"> - <head xml:id="head159" xml:space="preserve">THEOREMA <num value="140">CXL</num>.</head> - <p> - <s xml:id="s1225" xml:space="preserve">QVadrageÅ¿imum nonum quæſitum Å¿imiliter poſſumus alio modo Å¿oluere, vt <lb/>putà cum vnumquodque latus rhombi Å¿imul cum area cognitum, Å¿eu datum <lb/>nobis Å¿it <reg norm="cognitum" type="context">cognitÅ©</reg> Å¿imiliter nobis erit quadratum lateris <var>.a.d.</var> hoc eÅ¿t Å¿umma duorum <lb/>quadratorum <var>.a.o.</var> et <var>.o.d.</var> ex penultima primi Euclid. </s> - <s xml:id="s1226" xml:space="preserve">cúmque nobis cognita etiam <lb/>Å¿it totalis Å¿uperficies rhombi, cognita etiam nobis erit eius medietas, hoc eÅ¿t produ-<lb/>ctum <var>.o.d.</var> in <var>.o.a.</var> vnde ex methodo .37. Theorematis cognoÅ¿cemus .a<unsure/><var>.o.</var> et <var>.o.d.</var> & Å¿ic <lb/>etiam eorum dupla, quod quærebatur.</s> - </p> - <figure position="here"> - <image file="0106-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0106-01"/> - </figure> - </div> - <div type="math:theorem" level="3" n="141"> - <head xml:id="head160" xml:space="preserve">THEOREMA <num value="141">CXLI</num>.</head> - <p> - <s xml:id="s1227" xml:space="preserve">PVlchrum quæſitum fuit id, quod Tartalea ponit pro .18. noni libri in quarto fo-<lb/>lio, quod huiuÅ¿modi eÅ¿t. </s> - <s xml:id="s1228" xml:space="preserve">Aliquis habet dolium mero plenum, ex quo <lb/>duas vrnas extrahit ipÅ¿ius vini, Å¿ed loco ipÅ¿ius vini infundit duas vrnas aquæ. </s> - <s xml:id="s1229" xml:space="preserve">Dein <lb/>de poÅ¿t aliquot dies extrahit iterum alias duas vrnas illius miÅ¿ti, & iterum infundit <lb/>duas vrnas aquæ, & poÅ¿t alios aliquot dies idem facit, & hac vltima tertia vice in-<lb/>uenit aquam tantam eſſe, quantum vinum. </s> - <s xml:id="s1230" xml:space="preserve">Quæritur nunc quot vrnas capiat il-<lb/>lud dolium.</s> - </p> - <p> - <s xml:id="s1231" xml:space="preserve">Solutio ipÅ¿ius Tartaleæ bona eÅ¿t, cum Å¿upponat illas quatuor quantitates vini eſſe <lb/>inuicem continuas proportionales, vt putà primò totum vinum merum, poÅ¿teà re-<lb/>Å¿iduum pro Å¿ecunda quantitate, deinde pro tertia in Å¿ecunda, & pro quarta in ter-<lb/>tia extractione, hoc eÅ¿t quòd proportio totius vini meri ad vinum in prima Å¿it, vt hu <lb/>ius ad vinum in Å¿ecunda, & vt huius ad vinum in tertia miÅ¿tione. </s> - <s xml:id="s1232" xml:space="preserve">Sed quia ipÅ¿e <lb/>non probat hanc continuam proportionalitatem ex methodo Å¿cientifica, mihi <reg norm="visum" type="context">visÅ©</reg> <lb/>eÅ¿t hoc loco illam deÅ¿cribere.</s> - </p> - <p> - <s xml:id="s1233" xml:space="preserve">Cogitemus igitur <var>a.u.</var> pro capacitate dolij, et <var>.a.i.</var> pro quantitate duarum vrna-<lb/>rum. </s> - <s xml:id="s1234" xml:space="preserve">Nunc uerò Å¿upponamus quamlibet partem huius miÅ¿ti omogeneam eſſe Å¿uo <lb/>toto, quapropter Å¿equetur eandem proportionem eſſe vini ad aquam in qualibet <lb/>parte, quæ erit in toto, & ideò imaginemur <var>.e.o.</var> æqualem <var>.a.i</var>. </s> - <s xml:id="s1235" xml:space="preserve">Sed in puncto <var>.i.</var> tali <lb/>modo diuiÅ¿am, vt proportio <var>.i.e.</var> ad <var>.i.o.</var> eadem Å¿it quæ <var>.i.a.</var> ad <var>.i.u</var>. </s> - <s xml:id="s1236" xml:space="preserve">Supponamus <reg norm="etiam" type="context">etiã</reg> - <pb o="95" rhead="THEOREM. ARITH." n="107" file="0107" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0107"/> - <var>e.o.</var> eſſe duas primas vrnas vini miÅ¿ti hoc eÅ¿t primæ miÅ¿tionis, vnde cum eadem pro <lb/>portio Å¿it <var>.a.i.</var> ad <var>.i.u.</var> vt <var>.e.i.</var> ad <var>.i.o.</var> ita erit (ex .19. quinti). <var>a.e.</var> ad <var>.o.u.</var> ut <var>.a.i.</var> ad <var>.i.u.</var> & <lb/><reg norm="componendo" type="context">componẽdo</reg> ita erit <var>.a.e.</var> cum <var>.o.u.</var> hoc eÅ¿t <var>.i.o.u.</var> (proptereà quòd <var>.i.o.</var> æqualis eÅ¿t <var>.a.e.</var> <lb/>vt reÅ¿idua totorum æqualium) ad <var>.o.u.</var> quemadmodum <var>.a.i.u.</var> ad <var>.i.u</var>. </s> - <s xml:id="s1237" xml:space="preserve">Quare <var>.i.u.</var> erit <lb/>media proportionalis inter <var>.a.u.</var> et <var>.o.u.</var> vnde proportio <var>.a.u.</var> ad <var>.o.u.</var> dupla erit pro <lb/>portioni <var>.i.u.</var> ad <var>.o.u</var>. </s> - <s xml:id="s1238" xml:space="preserve">Nunc autem cum extracta fuerit quantitas <var>.e.o.</var> ex primo mi-<lb/>Å¿to, & poÅ¿teà infuÅ¿a aqua vÅ¿que ad plenitudinem dolij, proportio ingredientium <lb/>huius Å¿ecundi miÅ¿ti erit ea, quæ eÅ¿t inter <var>.o.u.</var> et <var>.o.a.</var> eo quòd in prima miÅ¿tione pro-<lb/>proportio ingredientium erat ea, quæ eÅ¿t inter <var>.o.u.</var> et <var>.a.e.</var> vel inter <var>.a.e.</var> et <var>.o.u.</var> <lb/>vt demonÅ¿trauimus. </s> - <s xml:id="s1239" xml:space="preserve">Accipiamus ergo <var>.t.m.</var> huiuÅ¿modi Å¿ecundi mifti, magnitudi-<lb/>nis <var>.a.i.</var> vel <var>.e.o.</var> Å¿ignificantis duas vrnas, & permutemus eum in tantam aquam, <lb/><reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> punctum <var>.o.</var> quod nobis diuidat <var>t.m.</var> in <var>.o.m.</var> et, <var>o.t.</var> partes Å¿implices, tali propor <lb/>tione inuicem relatas, vt Å¿unt <var>.o.u.</var> et <var>.o.a.</var> vnde habebimus ex Å¿upradictis rationibus <lb/>eandem proportionem ipÅ¿ius <var>.a.t.</var> ad <var>.m.u.</var> vt <var>.a.o.</var> ad <var>.o.u.</var> & componendo <var>.a.t.</var> cum <var>.m.<lb/>u.</var> hoc eÅ¿t <var>.i.m.u.</var> (eo quod cum <var>.t.m.</var> æqualis Å¿it <var>.a.i.</var> per conÅ¿equens <var>.i.m.</var> æqualis erit <var>.<lb/>a.t.</var>) ad <var>.m.u.</var> vt <var>.a.o.u.</var> ad <var>.o.u.</var> Å¿ed proportio <var>.a.o.u.</var> ad <var>.o.u.</var> dupla erat proportioni <var>.i.o.<lb/>u.</var> ad <var>.o.u.</var> quemadmodum Å¿upra diximus. </s> - <s xml:id="s1240" xml:space="preserve">Ergo proportio <var>.i.m.u.</var> ad <var>.m.u.</var> erit dupla <lb/>Å¿imiliter proportioni <var>.i.o.u.</var> ad <var>.o.<lb/>u.</var> quapropter <var>.o.u.</var> erit media proÂ<lb/> - <anchor type="figure" xlink:label="fig-0107-01a" xlink:href="fig-0107-01"/> - portionalis inter <var>.i.u.</var> et <var>.m.u</var>. </s> - <s xml:id="s1241" xml:space="preserve">Ec-<lb/>ce igitur quomodo eadem eÅ¿t pro <lb/>portio <var>.a.u.</var> ad <var>.i.u.</var> quæ <var>.i.u.</var> ad <var>.o.u.</var> & quæ <var>.o.u.</var> ad <var>.m.u.</var> qui quidem modus neceſſarius <lb/>eÅ¿t vt intellectus acquieÅ¿cat, id quod experientia non facit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0107-01" xlink:href="fig-0107-01a"> - <image file="0107-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0107-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="142"> - <head xml:id="head161" xml:space="preserve">THEOREMA <num value="142">CXLII</num>.</head> - <p> - <s xml:id="s1242" xml:space="preserve">PRæcedens Tartaleæ quæſitum elegans quidem eÅ¿t, Å¿ed pulchrum etiam vide-<lb/>tur quærere proportionem ingredientium in ultima miÅ¿tione, cum cognita fue <lb/>rit nobis proportio continentiæ dolij ad capacitatis vrnæ Å¿imul <reg norm="cum" type="context">cÅ©</reg> numero vitium <lb/>extractionum & impletionum.</s> - </p> - <p> - <s xml:id="s1243" xml:space="preserve">Exempli gratia, Å¿i proportio <var>.a.u.</var> ad <var>.a.i.</var> cognita nobis fuerit, cognoÅ¿cemus etiam <lb/><var>e.i.</var> ex regula de tribus & per conÅ¿equens etiam <var>.i.o.</var> reÅ¿iduum ex <var>.e.o.</var> & Å¿imiliter ag-<lb/>gregatum <var>.a.i.</var> cum <var>.i.o.</var> & Å¿ic <var>.o.u.</var> reÅ¿iduum totius, et <var>.o.t.</var> Å¿imiliter, eo quòd <var>.a.u.</var> ad <var>.a.<lb/>o.</var> eÅ¿t ut <var>.t.m.</var> ad <var>.o.t.</var> vnde cognoÅ¿cemus etiam <var>.o.m.</var> vt reÅ¿iduum <var>.t.m.</var> & Å¿imiliter ag-<lb/>gregatum <var>.a.o.</var> cum <var>.o.m.</var> hoc eÅ¿t <var>.a.m.</var> & etiam <var>.m.u.</var> reÅ¿iduum totius.</s> - </p> - <p> - <s xml:id="s1244" xml:space="preserve">CognoÅ¿cere autem proportionem totius dolij ad vrnam, vel ècontrà , cum cogni <lb/>ta nobis fuerit proportio ingredientium in vltima miÅ¿tione Å¿imul cum numero vi-<lb/>tium extractionum, & repletionum, quod Å¿cribit Tartalea, hoc etiam modo <lb/>poſſumus.</s> - </p> - <p> - <s xml:id="s1245" xml:space="preserve">Exempli gratia, Å¿i proportio <var>.m.u.</var> ad <var>.m.a.</var> cognita nobis fuerit, illicò Å¿cie-<lb/>mus proportionem <var>.a.u.</var> ad <var>.m.u.</var> & cum Å¿ciuerimus numerum vitium extractionum, <lb/>& impletionum illicò cognoÅ¿ci-<lb/>mus multiplicitatem proportio-<lb/>nis <var>.a.u.</var> ad <var>.m.u.</var> ad proportionem <var>.<lb/> - <anchor type="figure" xlink:label="fig-0107-02a" xlink:href="fig-0107-02"/> - o.u.</var> ad <var>.m.u.</var> quapropter propor-<lb/>tio <var>.o.u.</var> ad <var>.m.u.</var> nobis cognita erit <lb/>hoc eÅ¿t <var>.a.u.</var> ad <var>.i.u.</var> & Å¿imiliter ea, quæ eÅ¿t <var>.a.u.</var> ad <var>.a.i.</var> & è conuerÅ¿o Å¿imiliter.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0107-02" xlink:href="fig-0107-02a"> - <image file="0107-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0107-02"/> - </figure> - </div> - <pb o="96" rhead="IO. BAPT. BENED." n="108" file="0108" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0108"/> - <p> - <s xml:id="s1246" xml:space="preserve">Vnde cum aliquis diceret priori modo, dolium habeo vrnarum .400. vini, & per <lb/>vices .25. extraxi & impleui ipÅ¿um, vt dictum eÅ¿t. </s> - <s xml:id="s1247" xml:space="preserve">Nunc verò velim Å¿cire proportio-<lb/>nem vini ad a quam hac vltima vice. </s> - <s xml:id="s1248" xml:space="preserve">Nunc igitur Å¿i procedemus iuxta doctrinam <lb/>primi exempli huius theorematis, obtinebimus quod quærebamus.</s> - </p> - <p> - <s xml:id="s1249" xml:space="preserve">Sed Å¿i diceret iuxta Tartaleæ quæſitum, hoc eÅ¿t dolium habeo, quod ignoro quot <lb/><reg norm="nam" type="context">nã</reg> urnas contineat, volo tamen per .25. vices extrahere, & implere vt <reg norm="Å¿upradictum" type="context">Å¿upradictÅ©</reg> <lb/>eÅ¿t, ita vt vltima vice proportio vini ad aquam Å¿it Å¿eÅ¿quialtera. </s> - <s xml:id="s1250" xml:space="preserve">Tunc Å¿i iuxta mo-<lb/>dum Å¿ecundi exempli huius theorematis procedemus habebimus quod cupimus.</s> - </p> - <p> - <s xml:id="s1251" xml:space="preserve">Alio etiam modo aliquis quærere poſſet, hoc eÅ¿t, habeo <reg norm="dolium" type="context">doliÅ©</reg> quod capit .400. <lb/>vrnas. </s> - <s xml:id="s1252" xml:space="preserve">Habeo etiam vas trium vrnarum, quo mediante me oportet extrahere, & <lb/>implere. </s> - <s xml:id="s1253" xml:space="preserve">Velim tamen Å¿cire quoties me hoc facere oporteat, ita vt poÅ¿trema vi-<lb/>ce vinum Å¿e habeat ad aquam in proportione Å¿eÅ¿quialtera, vnde multoties accidet <lb/>vltimam extractionem, & impletionem mutilatam, Å¿eu imperfectam, euadere.</s> - </p> - <p> - <s xml:id="s1254" xml:space="preserve">Exempli gratia, Å¿i proportio vini ad aquam in vltima miÅ¿tione deberet eſſe vt <var>.n.<lb/>u.</var> ad <var>.n.a.</var> ita vt extrema vice fuiſſet <var>.t.m.</var> quæ quidem <var>.t.m.</var> excederet terminum per <var>.<lb/>n.m.</var> quæ <var>.n.m.</var> reuera eſſet nobis cognita, eò quòd ex priori modo hic Å¿upra dicto <lb/>proportio <var>.a.m.</var> ad <var>.m.u.</var> nobis in-<lb/>noteÅ¿ceret, & proportio <var>.n.a.</var> ad <var>.<lb/>n.u.</var> nobis data eÅ¿t Å¿imul cum <reg norm="quan titate" type="context">quãÂ<lb/> - <anchor type="figure" xlink:label="fig-0108-01a" xlink:href="fig-0108-01"/> - titate</reg> <var>.a.u.</var> </s> - <s xml:id="s1255" xml:space="preserve">quare quantitas <var>.n.u.</var> & <lb/><var>m.u.</var> nobis cognita, remanebit, et <lb/><var>n.m.</var> eorum differentia Å¿imiliter, etiam, et <var>.t.n.</var> reÅ¿iduum vaÅ¿is, quo metimur, vnde <lb/>neceſſe erit, quo<unsure/>d vltima vice vas contineret Å¿olum <var>.t.n.</var> reliqua uerò per Å¿e patent.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0108-01" xlink:href="fig-0108-01a"> - <image file="0108-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0108-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="143"> - <head xml:id="head162" xml:space="preserve">THEOREMA <num value="143">CXLIII</num>.</head> - <p> - <s xml:id="s1256" xml:space="preserve">HIeronymus Cardanus in <ref id="ref-0015">lib. Å¿uæ arithmeticæ cap .66. quæſtione .56.</ref> quam CarÂ<lb/>danicam vocat, ita inquit.</s> - </p> - <p> - <s xml:id="s1257" xml:space="preserve">Quidam perambulauit prima die certam quantitatem Å¿patij, & Å¿ecunda die, <reg norm="tan tò" type="context">tãÂ<lb/>tò</reg> plus proportionaliter, quantò diameter eÅ¿t maior coÅ¿ta, & tertia die tantò plus <lb/>Å¿ecunda, quantò proportionaliter portio lineæ diuiſæ Å¿ecundum proportionem ha <lb/>bentem medium, & duo extrema excedit minorem portionem, & quarta die in <lb/>proportione ad tertiam vt Å¿ecunda ad primam, & quinta die proportionaliter tan-<lb/>tò plus quarta, quantò in tertia plus Å¿ecunda, & ita alternatis vicibus in diebus no-<lb/>uem peregit nouem milliaria. </s> - <s xml:id="s1258" xml:space="preserve">Quæritur igitur quantum ambulauit die prima.</s> - </p> - <p> - <s xml:id="s1259" xml:space="preserve">Hoc autem nihil aliud eÅ¿t, quà m Å¿i aliquis diceret, propono tibi, exempli gratia, <lb/>lineam <var>.a.l.</var> nouem partibus inuicem non æqualibus ita diuiÅ¿am <var>.a.c</var>: <var>c.d</var>: <var>d.e</var>: & cæte-<lb/>ris, quarum partium proportiones tibi etiam do, vt putà . <var>a.c.</var> ad <var>.c.d.</var> et <var>.c.d.</var> ad <var>.d.e.</var> et <var>.<lb/>d.e.</var> ad <var>.e.f.</var> & Å¿ic de cæteris vÅ¿que ad poÅ¿tremam <var>.k.l.</var> quæ quidem proportiones Å¿int <lb/>etiam inuicem diſſimiles, Å¿eu inæquales, do tibi etiam <reg norm="proportionem" type="context">proportionẽ</reg> totius lineæ <var>.a.l.</var> <lb/>ad <var>.a.b.</var> Å¿uam partem, quæ vt in propoÅ¿ito exemplo nonupla eÅ¿t.</s> - </p> - <p> - <s xml:id="s1260" xml:space="preserve">Quæro nunc quam proportionem habebit <var>.a.c.</var> ad <var>.a.b.</var> & Å¿ic de cæteris partibus <lb/>eiuÅ¿dem ad eandem <var>.a.b</var>.</s> - </p> - <p> - <s xml:id="s1261" xml:space="preserve">Quod quidem facillimum erit Å¿peculari, nec non operari vnicuique, qui omnino <lb/>practicæ numerorum ignarus non fuerit, dum ab ordine Å¿cientifico non diÅ¿cedat.</s> - </p> - <p> - <s xml:id="s1262" xml:space="preserve">Cum enim cognoÅ¿cimus proportionem <var>.a.c.</var> ad <var>.c.d.</var> conÅ¿equenter cognoÅ¿cemus <lb/>ctiam proportion em aggregati <var>.a.c.d.</var> ad <var>.c.d.</var> cum autem cognouerimus proportio- - <pb o="97" rhead="THEOREM. ARIT." n="109" file="0109" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0109"/> - nem <var>.c.d.</var> ad <var>.d.e.</var> Å¿i <var>.c.d.</var> accipiemus, vt medium inter <var>.a.d.</var> et <var>.d.e.</var> cognoÅ¿cemus etiam <lb/>proportionem <var>.a.d.</var> ad <var>.d.e.</var> </s> - <s xml:id="s1263" xml:space="preserve">quare etiam eam quæ <var>.a.e.</var> ad <var>.d.e.</var> collocando poÅ¿teà . <lb/><var>d.e.</var> inter <var>.e.f.</var> et <var>.a.e.</var> innoteÅ¿cet ea, quæ eÅ¿t <var>.a.e.</var> ad <var>.e.f.</var> & ita gradatim accedenrus ad <lb/>perfectam cognitionem proportionis totius <var>.a.l.</var> ad <var>.k.l</var>. </s> - <s xml:id="s1264" xml:space="preserve">Nunc autem mediante <var>.k.l.</var> <lb/>cognoÅ¿cemus proportionem totius <var>.a.l.</var> ad <var>.i.k.</var> & hac mediante, cam cognoÅ¿cemus, <lb/>quæ totius <var>.a.l.</var> ad <var>.g.h.</var> & hac mediante eam quæ totius <var>.a.l.</var> ad <var>.f.g.</var> & Å¿ic gradatim, co <lb/>gnita nobis erit proportio totius <lb/>lineæ <var>.a.l.</var> ad Å¿uam partem <var>.a.c.</var> be-<lb/> - <anchor type="figure" xlink:label="fig-0109-01a" xlink:href="fig-0109-01"/> - neficio poÅ¿teà totius lineæ <var>.a.l.</var> co <lb/>gnoÅ¿cemus proportionem <var>a.c.</var> ad <lb/><var>a.b.</var> & Å¿ic aliarum reÅ¿pectu lineæ <var>.a.b.</var> vt quærebatur, quæ quidem propoÅ¿itio, etÅ¿i car <lb/>danica uocetur leuiſſima tamen eÅ¿t.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0109-01" xlink:href="fig-0109-01a"> - <image file="0109-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0109-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="144"> - <head xml:id="head163" xml:space="preserve">THEOREMA <num value="144">CXLIIII</num>.</head> - <p> - <s xml:id="s1265" xml:space="preserve">QVamuis multi de modo in Å¿umma colligendi, Å¿ubtrahendi, <reg norm="multiplicandi" type="context">multiplicãdi</reg>, & di <lb/>uidendi proportiones Å¿cripÅ¿erint, nullus tamen (quod Å¿ciam) perfectè, ac <lb/>Å¿cientificè Å¿peculatus eÅ¿t has operationes, quapropter hanc rem cum Å¿ilentio tranÅ¿i <lb/>re nolui, quin aliquid de ipÅ¿a conÅ¿cribam à ſumma dictarum proportionum in-<lb/>cohando.</s> - </p> - <p> - <s xml:id="s1266" xml:space="preserve">QuotieÅ¿cunque igitur volunt duas proportiones inuicem aggregare, Å¿imul ea-<lb/>rum antecedentia multiplicant, & Å¿imiliter earum conÅ¿equentia. </s> - <s xml:id="s1267" xml:space="preserve">Tunc proportio <lb/>terminata ab illis productis euadit in Å¿ummam illarum duarum propoÅ¿itarum <lb/>proportionum.</s> - </p> - <p> - <s xml:id="s1268" xml:space="preserve">Vt exempli gratia, Å¿i voluerimus colligere proportionem Å¿eÅ¿quialteram cum Å¿eÅ¿-<lb/>quitertia, multiplicando .3. cum .4. antecedentia Å¿cilicet, pro ductum erit .12. poÅ¿teà <lb/>multiplicando .2. cum .3. conÅ¿equentia, tunc productum erit .6. </s> - <s xml:id="s1269" xml:space="preserve">Proportio igitur, <lb/>quæ inter .12. et .6. reperitur. (quæ dupla eÅ¿t) eÅ¿t Å¿umma propoÅ¿itarum <reg norm="proportionum" type="context">proportionÅ©</reg>.</s> - </p> - <p> - <s xml:id="s1270" xml:space="preserve">Cuius rei Å¿peculatio erit huiuÅ¿modi Å¿int <var>.x.</var> et <var>.u.</var> <lb/>duo antecedentia quarunruis proportionum <var>.t.</var> <lb/> - <anchor type="figure" xlink:label="fig-0109-02a" xlink:href="fig-0109-02"/> - verò et. n Å¿int eorum conÅ¿equentia, productum <lb/>autem antecedentium Å¿it <var>.a.g.</var> illud verò quod <reg norm="con" type="context">cõ</reg> <lb/>Å¿equentium Å¿it <var>.d.a.</var> vnde proportio <var>.a.g.</var> ad <var>.a.d.</var> <lb/>compoÅ¿ita erit ex proportione <var>.x.</var> ad <var>.t.</var> & ex ea, <lb/>quæ eÅ¿t <var>.u.</var> ad <var>.n.</var> per .24. Å¿exti vel quintam octaui. <lb/></s> - <s xml:id="s1271" xml:space="preserve">Patet igitur ratio rectè faciendi, vt Å¿uprà dictum <lb/>eÅ¿t.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0109-02" xlink:href="fig-0109-02a"> - <image file="0109-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0109-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="145"> - <head xml:id="head164" xml:space="preserve">THEOREMA <num value="145">CXLV</num>.</head> - <p> - <s xml:id="s1272" xml:space="preserve">QVotieÅ¿cunque deinde detrahere volunt vnam proportionem ex altera mul-<lb/>tiplicant antecedens vnius cum conÅ¿equenti alterius. </s> - <s xml:id="s1273" xml:space="preserve">Tunc proportio, quÄ™ <lb/>inter talia duo producta incluÅ¿a reperitur, eÅ¿t reÅ¿iduum, Å¿eu differentia illarum dua-<lb/>rum proportionum datarum.</s> - </p> - <p> - <s xml:id="s1274" xml:space="preserve">Vt exempli gratia, Å¿i aliquis vellet ex proportione dupla detrahere Å¿eÅ¿quialte-<lb/>ram, multiplicaret .2. antecedens duplæ cum .2. conÅ¿equenti Å¿eÅ¿quialteræ, quorum <lb/>productum eſſet .4. pro antecedenti reÅ¿iduÄ™ proportionis. </s> - <s xml:id="s1275" xml:space="preserve">Deinde multiplicaret .3 <lb/>antecedens Å¿eÅ¿quialteræ cum .1. conÅ¿equenti duplæ, & productum eſſet .3. pro <reg norm="con- Å¿equenti" type="context">cõ- - <pb o="98" rhead="IO. BAPT. BENED." n="110" file="0110" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0110"/> - Å¿equenti</reg> reÅ¿iduÄ™ proportionis; </s> - <s xml:id="s1276" xml:space="preserve">quæ quidem reÅ¿idua proportio eſſet vt .4. ad .3. hoc <lb/>eÅ¿t Å¿eÅ¿quitertia, & Å¿ic de cæteris.</s> - </p> - <p> - <s xml:id="s1277" xml:space="preserve">Pro cuius ratione, Å¿it proportio <var>.x.</var> ad <var>.n.</var> ea quæ (exempli gratia) maior Å¿it, à <lb/>qua volumus demere proportionem <var>.t.</var> ad <var>.u.</var> minorem Å¿cilicet. </s> - <s xml:id="s1278" xml:space="preserve">Nunc autem <lb/>productum <var>.x.</var> in <var>.u.</var> Å¿it <var>.a.g.</var> illud verò <var>.t.</var> in <var>.<lb/>n.</var> Å¿it <var>.a.d</var>. </s> - <s xml:id="s1279" xml:space="preserve">Tunc dico proportionem <var>.a.g.</var> ad <var>.a.</var> <lb/> - <anchor type="figure" xlink:label="fig-0110-01a" xlink:href="fig-0110-01"/> - d. eſſe reÅ¿iduam quæſitam. </s> - <s xml:id="s1280" xml:space="preserve">Sit <var>.b.a.</var> productum <lb/>u. in <var>.n.</var> vnde eadem proportio erit producti <var>.a.<lb/>g.</var> ad productum <var>.a.b.</var> quÄ™ <var>.x.</var> ad <var>.n.</var> et <var>.a.d.</var> ad <var>a.b.</var> <lb/>quæ <var>.t.</var> ad <var>.u.</var> ex prima Å¿exti, Å¿eu .18. vel .19. Å¿e-<lb/>ptimi, Å¿ed proportio <var>.a.g.</var> ad <var>.a.b.</var> hoc eÅ¿t <var>.x.</var> ad <var>.<lb/>n.</var> componitur ex ea, quæ eÅ¿t <var>.a.g.</var> ad <var>.a.d.</var> & ea, <lb/>quæ eÅ¿t <var>.a.d.</var> ad <var>.a.b.</var> hoc eÅ¿t <var>.t.</var> ad <var>.u.</var> ergò ea, quÄ™ <lb/>eÅ¿t <var>.a.g.</var> ad <var>.a.d.</var> erit quà m quærebamus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0110-01" xlink:href="fig-0110-01a"> - <image file="0110-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0110-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="146"> - <head xml:id="head165" xml:space="preserve">THEOREMA <num value="146">CXLVI</num>.</head> - <p> - <s xml:id="s1281" xml:space="preserve">RATIO verò, quòd rectè fiat, quotieÅ¿cunque aliquam proportionem dupli-<lb/>care volentes, quadramus terminos ipÅ¿ius proportionis, vel Å¿i eam triplicare <lb/>voluerimus, cubamus ipÅ¿os terminos, vel Å¿i eam quadruplicare voluerimus <lb/>inuenimus cenÅ¿icos cenÅ¿icos terminorum ipÅ¿ius proportionis, & Å¿ic de Å¿ingulis, in <ref id="ref-0016">.17 <lb/>Theo. huiuÅ¿modi tractatus</ref> manifeÅ¿ta eÅ¿t.</s> - </p> - </div> - <div type="math:theorem" level="3" n="147"> - <head xml:id="head166" xml:space="preserve">THEOREMA <num value="147">CXLVII</num>.</head> - <p> - <s xml:id="s1282" xml:space="preserve">QVotieÅ¿cunque nobis propoÅ¿iti fuerint duo numeri ad libitum, deÅ¿ideraremusÂ<lb/>qÌue duas proportiones tali relatione inuicem refertas, quali Å¿unt hi duo pro <lb/>poÅ¿iti numeri inter Å¿e, ita faciendum erit.</s> - </p> - <p> - <s xml:id="s1283" xml:space="preserve">Sciendum primo eÅ¿t proportionem maioris numeri propoÅ¿iti ad minorem Å¿em-<lb/>per eſſe alicuius ex quinque generum, hoc eÅ¿t aut erit generis multiplicis, aut Å¿u-<lb/>perparticularis, aut multiplicis Å¿uperparticularis, aut Å¿uper partientis, aut multi-<lb/>plicis Å¿uperpartientis.</s> - </p> - <p> - <s xml:id="s1284" xml:space="preserve">Nunc autem Å¿i erit ex genere multiplici, iam ab antiquis traditus eÅ¿t modus, <reg norm="quem" type="context">quẽ</reg> <lb/>Å¿equi debemus. </s> - <s xml:id="s1285" xml:space="preserve">Cuius Å¿peculatio à me inuenta patet .in .17. Theo. huius libri, vt <lb/>in præcedenti dixi.</s> - </p> - <p> - <s xml:id="s1286" xml:space="preserve">Sed Å¿i talis proportio datorum numerorum erit alicuius aliorum generum, ita <lb/>agemus, Å¿i fuerit Å¿uperparticularis.</s> - </p> - <p> - <s xml:id="s1287" xml:space="preserve">Sit exempli gratia, Å¿eÅ¿quialtera, tunc Å¿umantur duo numeri inuicem inæquales, <lb/>quos à caÅ¿u volueris <var>.o.</var> et <var>.c.</var> qui quidem cubentur, & eorum cubi Å¿int <var>.a.</var> et <var>.e</var>. </s> - <s xml:id="s1288" xml:space="preserve">Inuenia <lb/>tur poÅ¿teà . u. ita proportionatus ad <var>.o.</var> vt <var>.o.</var> eÅ¿t ad <var>.c.</var> ex regula de tribus, hoc eÅ¿t diui-<lb/>dendo quadratum ipÅ¿ius <var>.o.</var> per <var>.c.</var> vnde nobis proueniat <var>.u.</var> & quia proportio <var>.a.</var> ad <var>.e.</var> <lb/>tripla eÅ¿t proportioni <var>.o.</var> ad <var>.c.</var> & proportio <var>.u.</var> ad <var>.c.</var> dupla eÅ¿t <reg norm="eidem" type="context">eidẽ</reg>, quæ <var>.o.</var> ad <var>.c.</var> ideo <lb/>proportio <var>.a.</var> ad <var>.e.</var> Å¿eÅ¿quialtera erit proportioni <var>.u.</var> ad <var>.c</var>.</s> - </p> - <p> - <s xml:id="s1289" xml:space="preserve">Sed Å¿i proportio numerorum propoÅ¿itorum fuerit Å¿eÅ¿quitertia, faciemus <var>.a.</var> et <var>.e.</var> <lb/>eſſe cenÅ¿ica cenÅ¿ica ipÅ¿ius <var>.o.</var> et <var>.c</var>. </s> - <s xml:id="s1290" xml:space="preserve">tunc Å¿umemus <var>.u.</var> conÅ¿equentem ad <var>.o.</var> vt dictum eÅ¿t, <lb/>deinde inueniremus <var>.i.</var> conÅ¿equens ad <var>.u.</var> ita ut <var>.u.</var> conÅ¿equens ipÅ¿ius <var>.o</var>. </s> - <s xml:id="s1291" xml:space="preserve">tunc habebi-<lb/>mus proportionem <var>.i.</var> ad <var>.c.</var> triplam, & eam quæ eÅ¿t <var>.a.</var> ad <var>.e.</var> quadruplam proportio- - <pb o="99" rhead="THEOREM. ARIT." n="111" file="0111" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0111"/> - ni <var>.o.</var> ad <var>.c</var>. </s> - <s xml:id="s1292" xml:space="preserve">Idem dico de reliquis proportionibus Å¿uperparticularibus.</s> - </p> - <p> - <s xml:id="s1293" xml:space="preserve">Sed Å¿i data proportio numerorum fuerit ex Å¿uper partientibus, vt exempli gra-<lb/>tia de quinque ad tria, efficiemus, vt <var>.a.</var> et <var>.e.</var> Å¿int prima relata ipÅ¿ius <var>.o.</var> et <var>.c.</var> vnde <lb/>proportio <var>.a.</var> ad <var>.e.</var> ita Å¿e habe-<lb/>bit ad proportionem <var>.o.</var> ad <var>.c.</var> <lb/> - <anchor type="figure" xlink:label="fig-0111-01a" xlink:href="fig-0111-01"/> - vt quinque ad <reg norm="vnum" type="context">vnÅ©</reg> & propor-<lb/>tio <var>.i.</var> ad <var>.c.</var> ut tria ad <reg norm="vnum" type="context">vnÅ©</reg>. </s> - <s xml:id="s1294" xml:space="preserve">Qua-<lb/>re proportio <var>.a.</var> ad <var>.e.</var> ad pro-<lb/>portionem <var>.i.</var> ad <var>.c.</var> Å¿e habebit, <lb/>vt quinque ad tria, & Å¿ic de reliquis.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0111-01" xlink:href="fig-0111-01a"> - <image file="0111-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0111-01"/> - </figure> - </div> - <p> - <s xml:id="s1295" xml:space="preserve">Pro alijs, eundem ordinem Å¿eruando, obtinebimus quod volumus.</s> - </p> - </div> - <div type="math:theorem" level="3" n="148"> - <head xml:id="head167" xml:space="preserve">THEOREMA <num value="148">CXLVIII</num>.</head> - <p> - <s xml:id="s1296" xml:space="preserve">QVamuis in .16. Å¿exti et .20. Å¿eptimi manifeÅ¿tè pateat ratio, quare rectè fiatac <lb/>cipiendam radicem quadratam illius producti, quod fit ex duobus datis <lb/>terminis, vt medium proportionale geometricè inter ipÅ¿os habeamus: </s> - <s xml:id="s1297" xml:space="preserve">nihilomi-<lb/>nus, quia per aliam methodum hoc idem Å¿cire poſſumus, inconueniens non erit a-<lb/>liquid circa hoc dicere.</s> - </p> - <p> - <s xml:id="s1298" xml:space="preserve">Cogitemus igitur exempli gratia, tres numeros continuè proportionales geo-<lb/>metricè <var>.a.b</var>: <var>c.d.</var> et <var>.e.f.</var> quorum <var>.a.b.</var> et <var>.e.f.</var> tantummodo nobis cogniti Å¿int, imagine-<lb/>mur etiam <var>.g.a.</var> eſſe productum quod fit ex <var>.a.b.</var> in <var>.e.f.</var> et <var>.d.k.</var> quadratum <var>.c.d.</var> et <var>.a.h.</var> <lb/>id quod fit ex <var>.a.b.</var> vnde eandem proportionem habebimus <var>.a.h.</var> ad <var>.a.g.</var> quæ eÅ¿t <var>.h.b.</var> <lb/>ad <var>.b.g.</var> ex prima .6. aut .18. vel .19. Å¿epti-<lb/>mi, Å¿ed per .11. octaui ita eÅ¿t quadrati <var>.a.</var> <lb/> - <anchor type="figure" xlink:label="fig-0111-02a" xlink:href="fig-0111-02"/> - h. ad quadratum <var>.k.d.</var> vt <var>.a.b.</var> ad <var>.e.f.</var> hoc <lb/>eÅ¿t vt <var>.h.b.</var> ad <var>.b.g.</var> ergo per .11. quinti ita <lb/>erit <var>.a.h.</var> ad <var>.a.g.</var> vt ad <var>.k.d.</var> vnde <var>.a.g.</var> æqua <lb/>le erit <var>.k.d.</var> per .9. quinti. </s> - <s xml:id="s1299" xml:space="preserve">Rectè ergo erit <lb/>accipere radicem quadratam <var>.a.g.</var> pro <var>.c.<lb/>d.</var> quod etiam eÅ¿t diuidere vnam datam <lb/><reg norm="proportionem" type="context">proportionẽ</reg> per æqualia, hoc eÅ¿t in duas <lb/>æquales partes, non dubito quin poſſer aliquis dicere non oportere vti poÅ¿teriori-<lb/>bus Theorematibus ad demonÅ¿trandum priora illis, Å¿ed hoc .148. dictum Å¿it luden <lb/>di loco.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0111-02" xlink:href="fig-0111-02a"> - <image file="0111-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0111-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="149"> - <head xml:id="head168" xml:space="preserve">THEOREMA <num value="149">CXLIX</num>.</head> - <p> - <s xml:id="s1300" xml:space="preserve"><emph style="sc">Vnde</emph> fiat <reg norm="quod" type="simple">ê™</reg> Å¿i quis inuenire voluerit Å¿ecundum terminum ex quatuor nume <lb/>ris continuè, & geometricè proportionalibus, quorum duo extremi tantum-<lb/>modo nobis cogniti Å¿int, rectè factum Å¿it quadrare primum eorum, & hoc quadra-<lb/>tum poÅ¿teà per alium terminum cognitum multiplicare, cuius producti demum ac-<lb/>cipere radicem cubam pro Å¿ecundo termino quæſito, hocloco videbimus.</s> - </p> - <p> - <s xml:id="s1301" xml:space="preserve">Imaginemur quatuor terminos continuè proportionales, vt dictum eÅ¿t, eſſe.</s> - <pb o="100" rhead="IO. BAPT. BENED." n="112" file="0112" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0112"/> - <s xml:id="s1302" xml:space="preserve">a.b: <var>c.d</var>: <var>e.f.</var> et <var>.g.h.</var> quorum <var>.a.b.</var> et <var>.g.h.</var> nobis tantummodo cogniti Å¿int, <reg norm="Å¿itque" type="simple">Å¿itqÌ</reg> imagina <lb/>tione deÅ¿criptus cubus <var>.a.q.</var> primi termini, <reg norm="cubusque" type="simple">cubusqÌ</reg> <var>.d.k.</var> Å¿ecundi rermini, conÅ¿idere-<lb/>mus etiam baÅ¿im <var>.a.i.</var> quadratam ipÅ¿ius cubi <var>.a.q.</var> hoc eÅ¿t præcedentem dignitatem ip <lb/>Å¿ius cubi eiuÅ¿dem radicis, quæ quidem baÅ¿is <var>.a.i.</var> multiplicetur per quartum <reg norm="terminum" type="context">terminÅ©</reg> <lb/><var>g.h.</var> productum autem Å¿it <var>.g.a.</var> vnde eadem proportio erit <var>.a.q.</var> ad <var>.a.g.</var> quæ <var>.b.q.</var> ad <var>.b.<lb/>g.</var> per .25. vndecimi, Å¿ed per primam Å¿exti, vel .18. aut .19. Å¿eptimi ita eÅ¿t <var>.q.i.</var> ad <var>.i.g.</var> <lb/>vt <var>.b.q.</var> ad <var>.b.g.</var> </s> - <s xml:id="s1303" xml:space="preserve">quare per .11. quinti <lb/>ita erit <var>.a.q.</var> ad <var>.a.g.</var> vt <var>.q.i.</var> ad <var>.i.g.</var> ideÅ¿t <lb/> - <anchor type="figure" xlink:label="fig-0112-01a" xlink:href="fig-0112-01"/> - vt <var>.a.b.</var> ad <var>.g.h.</var> Å¿ed vt eÅ¿t <var>.a.b.</var> ad <var>.g.h.</var> <lb/>Å¿ic eÅ¿t <var>.a.q.</var> ad <var>.k.d.</var> per .36. vndecimi, <lb/>Å¿eu per .11. octaui, vnde per .11. quin <lb/>ti Å¿ic erit <var>.a.q.</var> ad <var>.a.g.</var> vt ad <var>.k.d</var>. </s> - <s xml:id="s1304" xml:space="preserve">Qua-<lb/>re per .9. eiuÅ¿dem <var>.a.g.</var> Ä™qualis erit <var>.k.<lb/>d</var>. </s> - <s xml:id="s1305" xml:space="preserve">Vnde rectè erit accipere radicem <lb/>cubam <var>.a.g.</var> pro <reg norm="Å¿ecundo" type="context">Å¿ecÅ©do</reg> termino <var>.c.d.</var> <lb/>id, quod nobis inÅ¿eruit ad inueniendam tertiam partem vnius propoÅ¿itæ propor-<lb/>tionis.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0112-01" xlink:href="fig-0112-01a"> - <image file="0112-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0112-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="150"> - <head xml:id="head169" xml:space="preserve">THEOREMA <num value="150">CL</num>.</head> - <p> - <s xml:id="s1306" xml:space="preserve"><emph style="sc">Sed</emph> vt Å¿peculatio iÅ¿ta ita vniuerÅ¿alis fiat vt ad <reg norm="oens" type="context">oẽs</reg> dignitates applicari poſſit; <lb/></s> - <s xml:id="s1307" xml:space="preserve">Supponamus <var>.a.q.</var> et <var>.k.d.</var> eſſe duas dignitates quas volueris vnius, Å¿ed eiuÅ¿dem <lb/>Å¿peciei, et <var>.a.i.</var> dignitas præcedens dignitatem <var>.a.q.a.</var> cuius multiplicatione in <var>.a.b.</var> <lb/>eius radix producitur dignitas <var>.a.q.</var> & ab ipÅ¿ius <var>.a.i.</var> multiplicatione in <var>.g.h.</var> reÅ¿ultet <var>.a.<lb/>g.</var> vnde ex .18. vel .19. Å¿eptimi eadem proportio erit <var>.a.q.</var> ad <var>.a.g.</var> quæ <var>.a.b.</var> ad <var>.g.h.</var> Å¿ed <lb/>eadem etiam eÅ¿t <var>.a.q.</var> ad <var>.k.d.</var> ex ijs, quæ in .17. theoremare dixi, vnde ex .11. quinti, <lb/>ita erit <var>.a.q.</var> ad <var>.a.g.</var> vt ad <var>.k.d</var>. </s> - <s xml:id="s1308" xml:space="preserve">Quapropter <var>.a.g.</var> æqualis erit <var>.k.d.</var> & ideo cum inuenta <lb/>fuerit radix huiuÅ¿modi dignitatis ex quantitate <var>.a.g.</var> habebimus <var>.c.d.</var> Å¿ecundum ter-<lb/>minum quæſitum.</s> - </p> - </div> - <div type="math:theorem" level="3" n="151"> - <head xml:id="head170" xml:space="preserve">THEOREMA <num value="151">CLI</num>.</head> - <p> - <s xml:id="s1309" xml:space="preserve"><emph style="sc">Vnde</emph> verò fiat, quòd cum quis voluerit dimidium alicuius datæ proportio-<lb/>nis inuenire, rectè faciat, Å¿i accipiat radices quadratas illorum datorum rer-<lb/>minorum, etÅ¿i voluerit tertiam partem, accipiat radices cubas: </s> - <s xml:id="s1310" xml:space="preserve">Å¿i autem quartam, <lb/>accipereradices cenÅ¿icas cenÅ¿icas ipÅ¿orum, & Å¿ic de Å¿ingulis in .17. </s> - <s xml:id="s1311" xml:space="preserve">Theoremate om-<lb/>nia patent.</s> - </p> - </div> - <div type="math:theorem" level="3" n="152"> - <head xml:id="head171" xml:space="preserve">THEOREMA <num value="152">CLII</num>.</head> - <p> - <s xml:id="s1312" xml:space="preserve"><emph style="sc">Vnde</emph> autem fiat, vt cum quis voluerit multiplicare aliquam proportionem <lb/>per fractos, rectè faciat prius multiplicando eam per numeratorem, dein-<lb/>de productum diuiÅ¿erit per denominationem ipÅ¿orum fractorum.</s> - </p> - <p> - <s xml:id="s1313" xml:space="preserve">Vt exempli gratia, cum aliquis voluerit multiplicare proportionem Å¿eÅ¿quiquar-<lb/>tam per duo tertia, multiplicabit prius ipÅ¿am proportionem per numeratorem .2. <lb/>& productum, erit proportio .25. ad .16. qua poÅ¿tea diuiÅ¿a per .3. denominatorem, <lb/>prouentus erit proportio radicis cubæ .25. ad radicem cubam .16. vel vt proportio.</s> - <pb o="101" rhead="THEOREM. ARIT." n="113" file="0113" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0113"/> - <s xml:id="s1314" xml:space="preserve">25. ad radicem cubam .10000. quæ quidem proportiones æquales inuicem Å¿unt, cu <lb/>tam vna, quà m alia, Å¿it tertia pars totius.</s> - </p> - <p> - <s xml:id="s1315" xml:space="preserve">Pro cuius ratione cogitem is <var>.a.b.</var> eſſe aliquod totum, quod multiplicare cupimus <lb/>per duas tertias, quod <reg norm="quidem" type="context">quidẽ</reg> nihil aliud eÅ¿t, quà m accipere duas tertias partes vnius <lb/>totius Å¿uperficialis, imaginemur igitur hoc totum <var>.a.b.</var> lineare diuiÅ¿um eſſe in tertias <lb/>partes mediantibus <var>.e.</var> et <var>.d</var>. </s> - <s xml:id="s1316" xml:space="preserve">& tunc multiplicando ipÅ¿um per 2. tertias lineares produ-<lb/>ctum erit <var>.a.c.</var> Å¿ex vnitatum Å¿uperficialium, quod quidem productum poÅ¿teà diuiÅ¿um <lb/>per .3. dabit <var>.d.c.</var> hoc eÅ¿t duas tertias Å¿uperficiales (quæ eÅ¿t tertia pars ipÅ¿ius <var>.a.c.</var>) & <lb/>Ä™quales numero <var>.c.b.</var> duabus vnitatibus linearibus, ideÅ¿t duabus tertijs ipÅ¿ius <var>.a.b</var>. </s> - <s xml:id="s1317" xml:space="preserve">No <lb/>tandum etiam eÅ¿t, quòd cum ferè omnia reducantur ad regulam de tribus, proptereà <lb/>etiam multiplicatio alicuius quantitatis per aliam quantitatem, nihil aliud eÅ¿t quà m <lb/>quædam operatio ipÅ¿ius regulæ de tribus, vt eyempli gratia volo multiplicare .25. <lb/>per 20. hoc nihil aliud eÅ¿t niÅ¿i quærere alium numerum ita proportionatum ad .25. <lb/>vt 20. Å¿e habetad vnum, vnde multiplicando .25. cum .20. & productum diuidendo <lb/>per vnum exregula de tribus, prouentus eÅ¿t idem numerus ipÅ¿ius producti, & propte <lb/>rea cum volumus multiplicare aliquem numerum per fractos hoc nihil aliud eÅ¿t <lb/>quà m quærere aliquem numerum ita proportionatum ad ipÅ¿um numerum datum, <lb/>vt Å¿e habet numerator ad denominatorem, exempli gratia Å¿i .24. aliquis voluerit mul <lb/>tiplicare per duo tertia hoc idem eÅ¿t vt Å¿i quæreret numerum ad quem .24. ita Å¿e <lb/>habeat, vt .3. ad .2. & idem dico de proportionibus, hoc eÅ¿t quod aliud non eÅ¿t mulri-<lb/>plicare aliquam proportionem per fractos, quà m aliam proportionem quærere ad <lb/><reg norm="quam" type="context">quã</reg> data Å¿e habeat, vt denominator Å¿e <reg norm="hent" type="context">hẽt</reg> ad <reg norm="numeratorem" type="context">numeratorẽ</reg>; </s> - <s xml:id="s1318" xml:space="preserve">& hoc exregula de tribus <lb/>perficitur, <reg norm="conÅ¿tituendo" type="context context">cõſtituẽdo</reg> <reg norm="denominatorem" type="context">denominatorẽ</reg> in primo loco, quilocus eÅ¿t diuiÅ¿oris, numerato <lb/><reg norm="rem" type="context">rẽ</reg> verò in <reg norm="Å¿ecundo" type="context">Å¿ecÅ©do</reg> loco, <reg norm="multiplicando" type="context">multiplicãdo</reg> poÅ¿teà pro <lb/>portionem per <reg norm="numeratorem" type="context">numeratorẽ</reg>, & <reg norm="productum" type="context">productÅ©</reg> <reg norm="diuidem" type="context">diuidẽ</reg> <lb/> - <anchor type="figure" xlink:label="fig-0113-01a" xlink:href="fig-0113-01"/> - do per denominatorem, prouentus demum erit <lb/>proportio, ad quam data Å¿e habebit, vt denomi-<lb/>nator Å¿e <reg norm="hent" type="context">hẽt</reg> ad numeratorem ex ratione ipÅ¿ius re <lb/>gulÄ™ de tribus. </s> - <s xml:id="s1319" xml:space="preserve">Ratio verò methodi <reg norm="diuidendi" type="context">diuidẽdi</reg> <reg norm="vnam" type="context">vnã</reg> <lb/>datam <reg norm="proportionem" type="context">proportionẽ</reg> per fractos, ex Å¿e Å¿atis patet, <lb/>cum idem Å¿it modus diuidendi quemhbet nume <lb/>rum integrum per fractos. </s> - <s xml:id="s1320" xml:space="preserve">Quare, quæ vnius, <lb/>& alterius eÅ¿t ratio.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0113-01" xlink:href="fig-0113-01a"> - <image file="0113-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0113-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="153"> - <head xml:id="head172" xml:space="preserve">THEOREMA <num value="153">CLIII</num>.</head> - <p> - <s xml:id="s1321" xml:space="preserve">NIcolaus Tartalea in .3. lib. quintæ partis numerorum Å¿oluit .24. quæſitum Å¿i-<lb/>bi propoÅ¿itum à Hieronymo Cardano, via particulari & non generali. </s> - <s xml:id="s1322" xml:space="preserve">Quæ-<lb/>Å¿itum autem tale eÅ¿t quamlibet propoÅ¿itam rectam lineam in duas partes ita diuide <lb/>re via Euclidis, ut cubus totius lineæ ad cubos partium Å¿e habeat in proportione <lb/>tripla.</s> - </p> - <p> - <s xml:id="s1323" xml:space="preserve">Tartalea igitur inquit quòd vt Å¿atisfiat Å¿peculatiuis ingenijs Å¿oluendum Å¿it huiuÅ¿-<lb/>modi quæſitum, Å¿ecando lineam propoÅ¿itam <var>.a.b.</var> in tres æquales partes, quarum vna <lb/>fit <var>.c.b.</var> vnde problema Å¿olutum erit.</s> - </p> - <p> - <s xml:id="s1324" xml:space="preserve">Verum dicit, Å¿ed hæc non eÅ¿t methodus generalis, proptereà , quod cum tale <lb/>problema alterius fuiÅ¿let proportionis quam triplæ, talis methodus nihil valeret.</s> - <pb o="102" rhead="IO. BAPT. BENED." n="114" file="0114" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0114"/> - <s xml:id="s1325" xml:space="preserve">Quapropter non tacebo quod mihi in mentem venit circa hoc problema.</s> - </p> - <p> - <s xml:id="s1326" xml:space="preserve">Sit ergo linea <var>.a.b.</var> diuiÅ¿ibilis in puncto <var>.c.</var> ita vt cubum totius dictæ <var>.a.b.</var> lineæ ad <lb/>Å¿ummam cuborum <reg norm="Å¿uarum" type="context">Å¿uarÅ©</reg> partium <var>.a.c.</var> et <var>.c.b.</var> oporteat eam proportionem <reg norm="habere" type="conjecture">habére</reg>, <lb/>exempli gratia, vt .125. ad .65. vt vitemus fracta pro nunc, notantes talem propor-<lb/>tionem quadrupla nunquam maiorem eſſe poſſe, vt quilibet ex Å¿e contemplari po-<lb/>teÅ¿t, conÅ¿tituendo punctum <var>.c.</var> in medio loco inter <var>.a.</var> et <var>.b.</var> vnde proportio totalis <lb/>cubi ad Å¿ummam partialium eſſet omnium maxima quæ poſſint eſſe, collocando <var>.c.</var> <lb/>vbi volueris in dicta linea <var>.a.b.</var> & hæc eſſet quadrupla.</s> - </p> - <p> - <s xml:id="s1327" xml:space="preserve">Sed vt ad propoÅ¿itum reuertamur, conÅ¿iderabimus cubum totalem ipÅ¿ius <var>.a.b.</var> <lb/>eſſe vt .125. & Å¿ummam partialium vt .65. quam detrahemus ex cubo totali & nobis <lb/>remanebit .60. pro Å¿umma trium Å¿olidorum inuicem æqualium, quorum longitu-<lb/>do vniuÅ¿cuiuÅ¿que erit tota linea <var>.a.b.</var> nobis cognita vt radix dati cubi totalis, quæ erit <lb/>in hoc exemplo quinque partium, latitudo verò vniuÅ¿cuiuÅ¿que dictorum <reg norm="Å¿olidorum" type="context">Å¿olidorÅ©</reg> <lb/>erit <var>.a.c.</var> pars maior ipÅ¿ius <var>.a.b.</var> quæ quidem <var>.a.c.</var> adhuc nobis ignota eÅ¿t, profunditas <lb/>Å¿eu altitudo vniuÅ¿cuiuÅ¿que illorum Å¿olidorum, erit <var>.c.b.</var> pars reliqua ipÅ¿ius <var>.a.b.</var> & <reg norm="etiam" type="context">etiã</reg> <lb/>nobis incognita, Å¿ed quia Å¿umma horum trium Å¿olidorum nobis manifeÅ¿ta Å¿uperius <lb/>fuit, quæ erat .60. propterà nobis cognita erit quantitas vniuÅ¿cuiuÅ¿que illorum Å¿oli-<lb/>dorum, vt tertia pars totius Å¿ummæ ipÅ¿orum quæ erit .20. in propoÅ¿ito <reg norm="exemplo" type="context">exẽplo</reg>, dein <lb/>de cum vnumquodque illorum Å¿olidorum producatur à ſuperficie contenta Å¿eu pro<lb/> ducta ab <var>.c.a.</var> in <var>.c.b.</var> in tota linea <var>.a.b.</var> Å¿equitur quòd Å¿i diuiÅ¿erimus hoc Å¿olidum .20. <lb/>per lineam <var>.a.b.</var> quinque partium proueniet nobis cognita Å¿uperficies producta ab <var>.<lb/>a.c.</var> in <var>.c.b.</var> quatuor partium, Å¿ed cum quadratum totius <var>.a.b.</var> nobis cognitum Å¿it, eo <lb/>quod <var>.a.b.</var> vt eius latus etiam cognitum eÅ¿t. </s> - <s xml:id="s1328" xml:space="preserve">Tunc dictum quadratum erit .25. quod <lb/>quidem æquale eÅ¿t quadruplo illius quod fit ex <var>.a.c.</var> in <var>.c.b.</var> Å¿imul cum quadrato diffe <lb/>rentiæ inter <var>.a.c.</var> et <var>.c.b.</var> per .8. Å¿ecundi Eucli. </s> - <s xml:id="s1329" xml:space="preserve">Vnde quia quadruplum illius quod fit <lb/>ex <var>.a.c.</var> in <var>.c.b.</var> nobis cognitum eÅ¿t, vt <lb/>16. eo quod Å¿implum quod eÅ¿t .4. <reg norm="iam" type="context">iã</reg> <lb/> - <anchor type="figure" xlink:label="fig-0114-01a" xlink:href="fig-0114-01"/> - inuentum fuit, ideo Å¿i hoc quadru-<lb/>plum .16. demptum fuerit ex totali <lb/>quadrato .25. reliquum erit .9. qua <lb/><reg norm="dratum" type="context">dratÅ©</reg> Å¿cilicet vnius partis <var>.a.c.</var> ipÅ¿ius <lb/>hoc eÅ¿t illius partis, quæ differentia <lb/>eÅ¿t inter <var>a.c.</var> et <var>.c.b.</var> quæ quidem erit <num value="3">.<lb/>3.</num> partium quæ differentia cum Å¿ub-<lb/>tracta fuerit ex <var>.a.b.</var> reliquum erit du <lb/>plum ipÅ¿ius <var>.c.b.</var> duo Å¿cilicet. </s> - <s xml:id="s1330" xml:space="preserve">Quare <var>.<lb/>c.b.</var> erit vt <var>.I.</var> et <var>.a.c.</var> vt .4. & productum <var>.a.c.</var> in <var>.c.b.</var> erit .4. vnitatum Å¿uperficialium. <lb/></s> - <s xml:id="s1331" xml:space="preserve">& c.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0114-01" xlink:href="fig-0114-01a"> - <image file="0114-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0114-01"/> - </figure> - </div> - <pb o="103" n="115" file="0115" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0115"/> - </div> - <div type="appendix" level="3" n="1"> - <head xml:id="head173" xml:space="preserve">APPENDIX</head> - <head xml:id="head174" xml:space="preserve">DE SPECVLATIONE <lb/>REGVLAE FALSI.</head> - <p> - <s xml:id="s1332" xml:space="preserve"><emph style="sc">Nvnc</emph> idem ferè mihi accidit, quod & Michaeli Stifelio, à quo <lb/>cum Petreius Tipographus nuper totam Å¿uam Arithmeticam re <lb/>cepiſſet, mox poÅ¿teà per literas petijt <reg norm="explicationem" type="context">explicationẽ</reg> regulæ falÅ¿i.</s> - </p> - <p> - <s xml:id="s1333" xml:space="preserve">Similiter poÅ¿t inciÅ¿as omnes Å¿uperiorum Theorematum figu-<lb/>ras, <reg norm="opereque" type="simple">opereqÌ;</reg> Typographo commiſſo, amicus quidam omnium <lb/>Å¿cientiarum ornatiſſimus maxima neceſſitudine mecum coniun-<lb/>ctus monuit me, vt aliquid de regula falÅ¿i Å¿cribere vellem, cuius <lb/>Å¿uaÅ¿u hæc, quæ Å¿equuntur appendicis vice ponere libuit, nelector, quidpiam quod <lb/>ad hancrem pertinet iure merito à nobis deÅ¿iderare poſſet; </s> - <s xml:id="s1334" xml:space="preserve">vt autem ad ipÅ¿am <reg norm="re- gulam" type="context">re-<lb/>gulã</reg> accedamus Ego Å¿icut, & in alijs multis, ita & in huiuÅ¿cæ regulÄ™ inuentione cum <lb/>ipÅ¿o Stifelio maximè conuenio, putans regulam falÅ¿i, Å¿eu falÅ¿arum poÅ¿itionum in-<lb/>uentam fuiſſe per paruos numeros in quæſtionibus facillimis & cognitis, eodem fer <lb/>mè modo, quo ipÅ¿e monÅ¿trat illis duobus exemplis, quæ quamuis ipÅ¿e appellet theo <lb/>remata, nihilominus the oremata ego illa non vocarem, niÅ¿i adiuncta fuerit Å¿pecu-<lb/>latio ab ipÅ¿o præterita, & non experientia tantummodo, vt ipÅ¿e fecit. </s> - <s xml:id="s1335" xml:space="preserve">Primum eius <lb/>exemplum eÅ¿t, quòd.</s> - </p> - <p> - <s xml:id="s1336" xml:space="preserve">Quorumcumque <reg norm="duorum" type="context">duorÅ©</reg> numerorum differentia, Å¿i fuerit multiplicata in aggre <lb/>gatum eorum, producit ipÅ¿am differentiam, quæ eÅ¿t inter quadrata eorum.</s> - </p> - <p> - <s xml:id="s1337" xml:space="preserve">Secundum verò exemplum eÅ¿t, quod.</s> - </p> - <p> - <s xml:id="s1338" xml:space="preserve">Datis tribus numeris Å¿ecundum progreſſionem arithmeticam diÅ¿poÅ¿itis, facit mul <lb/>tiplicatio medij in Å¿e, <reg norm="quantum" type="context">quãtum</reg> multiplicatio extremorum inter Å¿e cum multiplicatio <lb/>ne differentiarum inter Å¿e.</s> - </p> - <p> - <s xml:id="s1339" xml:space="preserve">Talia enim exempla ipÅ¿e aliter non probat niÅ¿i experientia in aliquibus numeris, <lb/>arbitratus ex eo inuentam eſſe regulam falÅ¿i, experientia tantummodo confirma-<lb/>tam, quod quidem etiam & ego credo. </s> - <s xml:id="s1340" xml:space="preserve">At experientia in philoÅ¿ophia mathema-<lb/>tica, aut <reg norm="nullam" type="context">nullã</reg> prorÅ¿us facit <reg norm="Å¿cientiam" type="context">Å¿cientiã</reg>, aut omnino Å¿uperfluus fuit Euclides in multis <lb/>Å¿uis propoÅ¿itionibus, & præcipuè in eius Å¿ecundo libro, Å¿i Å¿ufficeret experientia. </s> - <s xml:id="s1341" xml:space="preserve">Id-<lb/>circo quo magis ad euidentiam ipÅ¿ius veritatis, quam profiteor, deuenire poſſim, <lb/><reg norm="accipiam" type="context">accipiã</reg> primò primum exemplum <lb/>ipÅ¿ius Stifelij hic Å¿uperius citatum, <lb/>& pro numero maiori, in prima hic <lb/> - <anchor type="figure" xlink:label="fig-0115-01a" xlink:href="fig-0115-01"/> - Å¿ubÅ¿cripta figura .AE. accipio <var>.a.i.</var> <lb/>cuius quadratum Å¿it <var>.a.c</var>: pro minori <lb/>vero numero capio <var>.a.e.</var> <reg norm="partem" type="context">partẽ</reg> ipÅ¿ius <lb/><var>a.i.</var> cuius quadratum fit <var>.a.t.</var> differen <lb/>tia autem horum numerorum erit <var>.<lb/>e.i.</var> reliqua pars ipÅ¿ius <var>.a.i</var>: & differen <lb/>tia ipÅ¿orum quadratorum erit gno-<lb/>mon <var>.e.c.o</var>: Nunc autem protraho <var>.<lb/>i.c.</var> latus quadrati maioris quouÅ¿que <lb/><var>c.n.</var> æqualis Å¿it <var>.a.e.</var> numero minori, <lb/><reg norm="perficioque" type="simple">perficioqÌ;</reg> rectangulum <var>.e.n.</var> quod - <pb o="104" rhead="IO. BAPT. BENED." n="116" file="0116" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0116"/> - producitur ex <var>.i.e.</var> differentia in <var>.i.n.</var> aggregatum amborum numerorum, Å¿ed hoc pro <lb/>ductum excedit productum <var>e.c</var>: partem gnomonis dicti per <var>.u.n.</var> quod quidem <var>.u.<lb/>n.</var> æquatur ipÅ¿i <var>.u.o.</var> reliquæ Å¿cilicet parti ipÅ¿ius gnomonis, <reg norm="nam" type="context">nã</reg> <var>.e.u.</var> æqualis eft <var>.i.c.</var> qua <lb/>re et <var>.a.i.</var> Å¿ed <var>.e.t.</var> Ä™quatur <var>.e.a.</var> vnde <var>.t.u.</var> æqualis erit <var>.e.i.</var> </s> - <s xml:id="s1342" xml:space="preserve">quare et <var>.u.c</var>: at cum <var>.c.n.</var> æqua <lb/>lis Å¿it ipÅ¿i <var>.a.e.</var> erit etiam æqualis ipÅ¿i <var>.<lb/>o.t</var>. </s> - <s xml:id="s1343" xml:space="preserve">quare <var>.u.n.</var> æqualis erit ipÅ¿i <var>.u.o.</var> <lb/>& tunc intellectus quieÅ¿cit, & <reg norm="abÅ¿que" type="simple">abÅ¿q;</reg> <lb/> - <anchor type="figure" xlink:label="fig-0116-01a" xlink:href="fig-0116-01"/> - aliqua alia experientia verè Å¿cientifi <lb/><reg norm="ceque" type="simple">ceqÌ;</reg> dicere poteft, quòd.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0115-01" xlink:href="fig-0115-01a"> - <image file="0115-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0115-01"/> - </figure> - <figure xlink:label="fig-0116-01" xlink:href="fig-0116-01a"> - <image file="0116-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0116-01"/> - </figure> - </div> - <p> - <s xml:id="s1344" xml:space="preserve">Quorumcumque duorum nume-<lb/>rorum differentia, fi fuerit multipli-<lb/>cata in aggregatum eorum, producit <lb/>ipÅ¿am <reg norm="differentiam" type="context">differentiã</reg>, quæ eftinter qua-<lb/>drata eorum.</s> - </p> - <p> - <s xml:id="s1345" xml:space="preserve">Hæcautem propoÅ¿itio à me ipÅ¿o <lb/>etiam in .60. </s> - <s xml:id="s1346" xml:space="preserve">Theoremate huius libri <lb/>aliter demonftrata fuit.</s> - </p> - <p> - <s xml:id="s1347" xml:space="preserve">DE Å¿peculatione autem, etÅ¿cientia Å¿ecundi exempli, in Å¿ecunda hic Å¿ubÅ¿cripta <lb/>figura <var>.ω.</var> cogitemus lineam <var>.u.a.</var> tribusin partibus arithmeticè diuiÅ¿am, qua <lb/>rum maxima Å¿it <var>.u.o.</var> media. Å¿it <var>.o.e.</var> minima verò Å¿it <var>.e.a.</var> multiplicatio autem mediæ <var>.<lb/>o.e.</var> in Å¿e Å¿it quadratum <var>.o.t.</var> abÅ¿cindatur deinde ex <var>.o.e</var>: <var>e.i.</var> æqualis <var>.e.a.</var> </s> - <s xml:id="s1348" xml:space="preserve">tunc <var>.o.i.</var> erit <lb/>differentia inter <var>.o.e.</var> et <var>.e.a.</var> & æqualis differentiæ inter <var>.o.e.</var> et <var>.o.u.</var> ex hypotefi, quæ <lb/>quidem <var>.o.i.</var> in Å¿e ducta procreabit quadratum <var>.o.c.</var> quod erit productum ex differen <lb/>tijs ipÅ¿arum partium, & erit pars quadrati <var>.o.t.</var> Å¿uperius dicti, vt exÅ¿e patet. </s> - <s xml:id="s1349" xml:space="preserve">Nunc <lb/>autem dico gnomonem <var>.i.t.n.</var> æqualem eſſe ei quod fit ex <var>.a.e.</var> in <var>.o.u</var>. </s> - <s xml:id="s1350" xml:space="preserve">Producatur igi <lb/>tur <var>.e.t.</var> quouÅ¿que <var>.t.r.</var> æqualis Å¿it ipÅ¿i <var>.o.i</var>. </s> - <s xml:id="s1351" xml:space="preserve">tunc <var>.e.r.</var> erit æqualis <var>.o.u.</var> quod etiam clarum <lb/>eÅ¿t. </s> - <s xml:id="s1352" xml:space="preserve">Claudatur ergo rectangulum <var>.i.r.</var> quod erit æquale producto ipÅ¿ius <var>.e.a.</var> in <var>.o.u.</var> <lb/>Nam <var>.e.i.</var> Å¿umpta fuit <lb/>æqualis <var>.e.a.</var> Å¿ed ex ra <lb/> - <anchor type="figure" xlink:label="fig-0116-02a" xlink:href="fig-0116-02"/> - tionibus in priori <reg norm="exem" type="context">exẽ</reg> <lb/>plo allatis, <reg norm="productum" type="simple">ê“ductum</reg> <var>.<lb/>i.r.</var> æquale erit gno-<lb/>moni <var>.i.t.n</var>. </s> - <s xml:id="s1353" xml:space="preserve">Nuncau <lb/>tem verè, Å¿cientifice-<lb/>qÌue poſſumus affirma <lb/>re, quòd. </s> - <s xml:id="s1354" xml:space="preserve">Datis tribus <lb/>numeris <reg norm="Å¿ecundum" type="context">Å¿ecundÅ©</reg> pro <lb/>greffionem arithme-<lb/>ticam diÅ¿pofitis, fa-<lb/>cit multiplicatio me-<lb/>dij in Å¿e quantum mul <lb/>tiplicatio extremorum inter Å¿e, cum multiplicatione differentiarum inter Å¿e.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0116-02" xlink:href="fig-0116-02a"> - <image file="0116-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0116-02"/> - </figure> - </div> - <p> - <s xml:id="s1355" xml:space="preserve">Et Å¿ic de alijs huiuÅ¿modi inuentionibus infero.</s> - </p> - <p> - <s xml:id="s1356" xml:space="preserve">DIcturus igitur aliquid circa <reg norm="regulam" type="context">regulã</reg> falÅ¿i, videtur mihi nullam oportere facere <lb/>mentionem de origine huiuÅ¿cæ regulæ, cum in hoc Stifelius Å¿atisfecerit, Å¿ed - <pb o="105" rhead="THEOREM. ARITH." n="117" file="0117" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0117"/> - potius veras rationes <reg norm="propriaque" type="simple">propriaqÌ;</reg> fundamenta huiuÅ¿modi operationis oftendere, fu-<lb/>mendo eadem exempla propoÅ¿ita abipÅ¿is practicis, & maximè à Nicolao Tartalea <lb/>viro accuratiffimo, qui vbicunque potuit Å¿peculatus eÅ¿t cauiÅ¿as <reg norm="ipÅ¿arum" type="context">ipÅ¿arÅ©</reg> operationum, <lb/>etÅ¿i de huiuÅ¿modi falÅ¿i regula circa finem cap .8. lib. 17. promittat poÅ¿tea loqui, nub-<lb/>libi tamen loquutus eft. </s> - <s xml:id="s1357" xml:space="preserve">Monendum etiam cenÅ¿eo, me nihil de rationibus regulæ <lb/>falÅ¿i Å¿implicis dicturum, cum ex Å¿eipÅ¿is Å¿atis appareant, quod non ita eÅ¿t de poÅ¿itio-<lb/>nibus duplis. </s> - <s xml:id="s1358" xml:space="preserve">Incipiam ergo à primo problemate lib. 17. ipÅ¿ius Tartaleæ, quo <reg norm="etiam" type="context">etiã</reg> <lb/>ipÅ¿e vtitur pro exemplo docendi gratia, ipÅ¿am regulam duplæ poÅ¿itionis, quod qui <lb/>dem problema aliter à me <reg norm="Å¿olutum" type="context">Å¿olutÅ©</reg> fuit in .118. </s> - <s xml:id="s1359" xml:space="preserve">Theoremate huius mei lib. quod Å¿imi <lb/>liter ob hanc demum occaÅ¿ionem mihi oblatam, alia etiam via, Å¿peculatus Å¿umidem <lb/>poſſe fieri, quæ quidem via Å¿eu methodus generalis erit, & ita Å¿e habet.</s> - </p> - <p> - <s xml:id="s1360" xml:space="preserve">Accipio enim propoÅ¿itum numerum diuiÅ¿ibilem, à quo detraho Å¿ummam <lb/>datorum numerorum, primo duplicato, eo quòd tam in Å¿ecunda quam in <lb/>tertia parte reperitur, vt in propofito exemplo, datus numerus eft, 50. à <lb/>quo detraho Å¿ummam dictorum numerorum, quæ eÅ¿t .11. nam tres, & tres, & <lb/>quinque Å¿unt vndecim, eo quòd primus ingreditur in Å¿ecunda, & in tertia parte, <lb/>dempto igitur hoc numero .11. ex .50. remanet .39. qui quidem numerus intelligen-<lb/>dus eÅ¿t pro Å¿umma trium partium Å¿implicium adhuc incognitarum, à quo extrahen <lb/>da eÅ¿t prima, eo modo quo nunc proponam exregula de tribus, hoc eÅ¿t aggregan <lb/>do dictas partes Å¿implices Å¿ine aliqua additione vtcunque volueris (Å¿ed commodius <lb/>erit in minimis numeris) iuxta propoÅ¿itum, quod quidem propoÅ¿itum eÅ¿t, vt Å¿ecun <lb/>da pars dupla Å¿it primæ, tertia verò æqualis fit primæ & Å¿ecundæ, quæ partes in di-<lb/>ctis minimis numeris, ita diÅ¿poÅ¿itæ erunt .1. 2. 3. quarum Å¿umma erit .6. </s> - <s xml:id="s1361" xml:space="preserve">Nunc Å¿i ex <lb/>regula de tribus dixerimus, cum hæc Å¿umma proueniat nobis ab vno, à quo proue-<lb/>niet .39. et veniet nobis .6. cum dimidio pro prima parte quæfita in propoÅ¿ito nume-<lb/>ro .39. cum ergo habuerimus primam <reg norm="partem" type="context">partẽ</reg>, reliquas poÅ¿teà illicò cognoÅ¿cemus.</s> - </p> - <p> - <s xml:id="s1362" xml:space="preserve">HuiuÅ¿modi verò operationis ratio ex Å¿e manifeÅ¿ta patet, eo quòd proportio Å¿um <lb/>mæ partium in minimis numeris ad primam eorum partem eadem eſſe debet, quæ <lb/>ipÅ¿ius .39. ad primam partem quæſitam huiuÅ¿modi aggregati partium <reg norm="Å¿implicium" type="context">Å¿impliciÅ©</reg>, Å¿ed <lb/>quia nemo adhuc, quod Å¿ciam, Å¿atis animaduertit rationem modorum, qui ab anti-<lb/>quis obÅ¿eruati Å¿unt, qui quidem modi duo Å¿unt circa hoc Helcataym duplæ falſæ <lb/>pofitionis, igitur non prætermittam aliquid de hacreÅ¿peculari, & primo de pri-<lb/>mo modo.</s> - </p> - <p> - <s xml:id="s1363" xml:space="preserve">In primis igitur <reg norm="Å¿ciendum" type="context">Å¿ciendÅ©</reg> eft, <reg norm="quod" type="simple">ê™</reg> <lb/> - <anchor type="figure" xlink:label="fig-0117-01a" xlink:href="fig-0117-01"/> - veritas ita inueniri poterit eo-<lb/>rum modo, me diantibus Å¿impliÂ<lb/>cibus partibus, vt <reg norm="etiam" type="context">etiã</reg> <reg norm="median- tibus" type="simple">median-<lb/>tibê°</reg> <reg norm="compoÅ¿itis" type="context">cõpoÅ¿itis</reg>, ut in pręſenti <reg norm="exem" type="context">exẽ</reg> <lb/>plo pro primis pofitionibus ac-<lb/>ceperunt .10. et .8. pro Å¿ecundis <lb/>verò compoÅ¿itis <reg norm="cum" type="context">cÅ©</reg> numero .3. <lb/><reg norm="inuenerunt" type="context">inuenerÅ©t</reg> .23. et .19. pro tertijs <lb/><reg norm="autem" type="wordlist">aÅ©t</reg> <reg norm="compoÅ¿itis" type="context">cõpoÅ¿itis</reg> <reg norm="cum" type="context">cÅ©</reg> <reg norm="quinque" type="simple">quinq;</reg>, notaue <lb/>runt .38. et .32. vnde prima Å¿um <lb/>marefultauit .71. Å¿ecunda verò <lb/>59. ita <reg norm="quod" type="simple">ê™</reg> <reg norm="primus" type="simple">primê°</reg> error remanebat <lb/>21. <reg norm="Å¿ecundus" type="context simple">Å¿ecÅ©dê°</reg> <reg norm="autem" type="wordlist">aÅ©t</reg> .9. vt in figura <var>.A</var>.</s> - </p> - <div type="float" level="4" n="3"> - <figure xlink:label="fig-0117-01" xlink:href="fig-0117-01a"> - <image file="0117-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0117-01"/> - <caption xml:space="preserve">Compositorum</caption> - </figure> - </div> - <pb o="106" rhead="IO. BAPT. BENED." n="118" file="0118" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0118"/> - <p> - <s xml:id="s1364" xml:space="preserve">SEDijdem errores proueniunt exÅ¿ummis partium Å¿implicium.</s> - </p> - <p> - <s xml:id="s1365" xml:space="preserve">Vtexempli gratia, in figura <var>.B.</var> Å¿umma propoÅ¿ita partium Å¿implicium eÅ¿t .39. <lb/>vt diximus, eo quòd ab ipÅ¿o .50. detraxerimus .11. Å¿umma Å¿cilicet numerorum adij <lb/>ciendorum ad efficiendas partes compofitas, Å¿umma poÅ¿teà fimplicium partium <lb/>primæ poÅ¿itionis, erit .60. eo quòd prima pars erat .10. Å¿ecunda autem Å¿implex 20. <lb/>tertia verò fimplex .30. iuxta ordinem propoÅ¿iti. </s> - <s xml:id="s1366" xml:space="preserve">Summa deinde Å¿implicium <reg norm="partium" type="context">partiÅ©</reg> <lb/>fecundæ poÅ¿itionis effet .48. quia prima eius pars erat .8. Å¿ecunda verò Å¿implex .16. <lb/>tertia autem Å¿implex .24. vnde prima Å¿umma excederet datam .39. per .21. differen-<lb/>tiæ, Å¿ecunda verò per .9. vt Å¿upra vidimus de Å¿ummis compoÅ¿itis à dato .50. compo-<lb/>fito, & hoc quidem mirandum non eft, quod Å¿cilicet tres Å¿ummæ fimplicium par-<lb/>tium Å¿intinuicem inæqua-<lb/>les, ijÅ¿dem differentijs me-<lb/> - <anchor type="figure" xlink:label="fig-0118-01a" xlink:href="fig-0118-01"/> - diantibus, quibus <reg norm="differunt" type="context">differÅ©t</reg> <lb/>dictæ tres Å¿ummæ compofi <lb/>tæ, cum ab vnaquaque <reg norm="con" type="context">cõ</reg> <lb/><reg norm="poÅ¿itarum" type="context">poÅ¿itarÅ©</reg> ablatus fit nume-<lb/>rus .11. æqualiter, vnde ex <lb/>neceſſitate, permutando, <lb/><reg norm="earum" type="context">earÅ©</reg> differentiæ <reg norm="relinquem" type="context">relinquẽ</reg> <lb/>dæ erant æquales inuicem <lb/>ex <ref id="ref-0017">.78. theoremate hu-<lb/>ius noÅ¿tri lib.</ref> Å¿ummæ enim <lb/>compofitæ erant .71. 59. et <lb/>50. fimplices verò .60. 48. <lb/>et .39. differentes à primis <lb/>per .11. vt dictum eft, qua <lb/>re veritas ita manabit à compofitis, quemadmodum à fimplicibus, Å¿ed à fimplici-<lb/>bus per Å¿e, & a compofitis per accidens vtiam iam videbimus.</s> - </p> - <div type="float" level="4" n="4"> - <figure xlink:label="fig-0118-01" xlink:href="fig-0118-01a"> - <image file="0118-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0118-01"/> - <caption xml:space="preserve">Simpricium</caption> - </figure> - </div> - <p> - <s xml:id="s1367" xml:space="preserve">ANtiquorumigitur primus m odus vtitur regula detribus, hocordine, multi-<lb/>plicando Å¿cilicet Å¿ecundum errorem, qui eft .9. cum differentia primarum par <lb/>tium pofitarum, quæ eft .2. & productum diuidendo per differentiam errorum, quæ <lb/>eft .12. proueniens poftea quod eft .1. cum dimidio additur hoc loco primæ parti Å¿e-<lb/>cundæ poÅ¿itionis. <reg norm="&c." type="unresolved">&c.</reg> quòd benè Å¿e habet. </s> - <s xml:id="s1368" xml:space="preserve">Vbi animaduertendum eÅ¿t, quod ille <lb/>numerus .12. non eft accipiendus per Å¿e vt differentia errorum hoc eft .21. et .9. nifi <lb/>perà ccidens, fed benè perfe, vt <reg norm="differentia" type="context">differẽtia</reg> inter .60. er .48. Å¿implices Å¿ummas, quem <lb/>admodum .9. in hoc propoÅ¿ito eft differentia per Å¿e inter .48. et .39 per accidens ve-<lb/>ro inter .59. et .50.</s> - </p> - <p> - <s xml:id="s1369" xml:space="preserve">CognoÅ¿cendum igitur eft mediante .24. quinti Eucli. quod eadem proportio <lb/>eft primæ Å¿ummæ (Å¿implicium dico) ad Å¿uam primam partem, quæ Å¿ecundæ Å¿um-<lb/>mæ ad Å¿uam, & tertiæ Å¿ummæ ad fuam fimiliter (vbi rectè etiam feciffent hoc in lo-<lb/>co antiqui Å¿i multiplicauiffent tertiam Å¿ummam fim plicem cum prima parte prioris <lb/>fummæ fimplicis, & productum diuififfent per primam Å¿ummam, vnde prima pars <lb/>quæſita tertiæ Å¿ummæ orta fuiffet, abÅ¿que ullo negotio ipfius plus velminus) </s> - <s xml:id="s1370" xml:space="preserve">Quare <lb/>habebimus tres terminos antecedentes ab vna parte, & tres terminos conÅ¿equen-<lb/>tesab alia parte continentes vnam <reg norm="eandemque" type="simple">eandemqÌ;</reg> proportionem, vnde ex .19. quinti, <lb/>vel .12. Å¿eptimi eorum differentiæ proportionales erunt, hoc eft, <reg norm="quod" type="simple">ê™</reg> eadem propor - <pb o="107" rhead="THEOREM. ARIT." n="119" file="0119" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0119"/> - tio erit eius differentiæ, quæ eÅ¿t inter primam & fecundam Å¿ummam, ad differen-<lb/>tiam quæ eÅ¿t inter primas earum partes, quæ illius differentiæ, quæ eÅ¿t inter Å¿ecun-<lb/>dam & tertiam Å¿ummam, ad differentiam, quæ eft inter primas illarum partes, Å¿ed <lb/>harum .4. differentiarum, tres nobis cognitæ Å¿unt, ideft .12. 2. et .9. ergo ex regula de <lb/>tribus ab Eucli. in .20. Å¿eptì<unsure/>mi Å¿peculata inueniebatur quarta differentia, quæ eft .1. <lb/>cum dimidio.</s> - </p> - <p> - <s xml:id="s1371" xml:space="preserve">A compofitis Å¿ummis idem etiam proueniet, Å¿ed non vt ex proprijs caufis, & per <lb/>Å¿e, Å¿edper accidens. </s> - <s xml:id="s1372" xml:space="preserve">Nam quamuis eadem differentia fit inter 71. et .59. quæ in-<lb/>ter .60. et .48. & <reg norm="eadem" type="context">eadẽ</reg> inter .59. et .50. quæ inter .48. et .39. </s> - <s xml:id="s1373" xml:space="preserve">Nihilominus non eft <reg norm="eadem" type="context">eadẽ</reg> <lb/>proportio (propriè) ipÅ¿ius .71. ad .59. quæ ipÅ¿ius .60. ad .48. nec ea quæ ipÅ¿ius .59. ad <num value="50">.<lb/>50.</num> eft quæ ipÅ¿ius .48. ad .39: </s> - <s xml:id="s1374" xml:space="preserve">Vnde non erit eadem proportio ipÅ¿ius .71. ad .59. quæ <lb/>ipfius .10. ad .8. ne@ea quæ eft ipfius .59. ad .50. quæ ipÅ¿ius .8. ad .6. cum dimidio. </s> - <s xml:id="s1375" xml:space="preserve">Sed <lb/>minores illis. </s> - <s xml:id="s1376" xml:space="preserve">Nam ex æqualibus additamentis diminuuntur proportiones maio-<lb/>ris inÄ™qualitatis.</s> - </p> - <p> - <s xml:id="s1377" xml:space="preserve">A fimplicibus igitur Å¿ummis pendet ratio huiuÅ¿modi effectus.</s> - </p> - <p> - <s xml:id="s1378" xml:space="preserve">Si vero prima pars fecundæ poÅ¿itionis effet .4. tunc Å¿ecunda eius pars effet .8. & ter-<lb/>tia .12. quarum Å¿umma effet .24. (harum fimplicium partium Å¿eilicet) & minor vera <lb/>(39.) per .15. & differens à ſumma primarum. (60.) per .36. & differentia primarum <lb/>partium effet .6. differentia vero primæpartis Å¿ecundæ poÅ¿itionis, a prima parte quÄ™ <lb/>fita effet .2. cum dimidio. </s> - <s xml:id="s1379" xml:space="preserve">Vnde in huiuÅ¿modi exemplo videre eft quare colligan-<lb/>tur errores inuicem, quando alter eorum eccedit, reliquus vero deficit à numero pro <lb/>pofito. </s> - <s xml:id="s1380" xml:space="preserve">Quod quidem ob aliam caufam non fit, nifi vt cognoÅ¿catur differentia .36. <lb/>differentia Å¿cilicet Å¿implicium Å¿ummarum ipÅ¿arum poÅ¿itionum.</s> - </p> - <p> - <s xml:id="s1381" xml:space="preserve">Secundus autem modus ab antiquis magis exercitatus eÅ¿t, quod multiplicabant <lb/>diametraliter errores cum primis partibus, hoc eÅ¿t primum errorem cum prima par <lb/>te, hoc eÅ¿t cum numero Å¿ecundæ poÅ¿itionis, Å¿ecundum vero errorem cum prima <lb/>parte, hoc eÅ¿t cum numero primæ poÅ¿itionis, differentiam poÅ¿teà vel aggregatum <lb/>horum duorum productorum diuidebant per differentiam vel aggregatum dicto-<lb/>rum errorum, proueniens poÅ¿teà erat prima pars quæſita numeri propoÅ¿iti. </s> - <s xml:id="s1382" xml:space="preserve">Vn-<lb/>de oriebantur tria producta, quorum <reg norm="tertium" type="context">tertiÅ©</reg>, hoc eÅ¿t differentia, Å¿eu aggregatum il-<lb/>lorum conÅ¿tituebatur ex differentia feuaggregato errorum, & ex numero quæ-<lb/>fito.</s> - </p> - <p> - <s xml:id="s1383" xml:space="preserve">Vtin præfenti exemplo, primus error eÅ¿t .21. qui multiplicatus cum prima par-<lb/>te Å¿ecundæ poÅ¿itionis, quæ eÅ¿t .8. producit .168. <reg norm="Å¿ecundus" type="context">Å¿ecÅ©dus</reg> verò error eÅ¿t .9. qui multi-<lb/>plicatus cum prima parte primÄ™ poÅ¿itionis producit .90. differentia autem horum <lb/>productorum eÅ¿t .78. quæ diuifa per differentiam errorum, quæ eÅ¿t 12. dabit .6. <reg norm="cum" type="context">cÅ©</reg> di <lb/>midio, pro prima parte quæſita dati numeri diuiÅ¿ibilis, qui erat .50.</s> - </p> - <p> - <s xml:id="s1384" xml:space="preserve">Hæc omnia rectè Å¿e habent. </s> - <s xml:id="s1385" xml:space="preserve">Sed, vt Å¿upra dixi diuiÅ¿or non eft per Å¿e differentia <lb/>errorum, neque etiam differentia per Å¿e Å¿ummarum compoÅ¿itarum, fed bene fim-<lb/>plicium.</s> - </p> - <p> - <s xml:id="s1386" xml:space="preserve">Pro cuius rei Å¿peculatione, accipiendæ Å¿unt Å¿ummæ Å¿implices, quarum differen-<lb/>tiæ per Å¿e vtiles Å¿unt in huiuÅ¿modi operatione; </s> - <s xml:id="s1387" xml:space="preserve">& quia etiam rationes veritatis ex <lb/>iÅ¿tis, & non ex illis fluunt; </s> - <s xml:id="s1388" xml:space="preserve">quamuis tam vnæ, quam aliæ Å¿int eædem in quantitate, <lb/>ideÅ¿t æquales.</s> - </p> - <pb o="108" rhead="IO. BAPT. BENED." n="120" file="0120" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0120"/> - <p> - <s xml:id="s1389" xml:space="preserve">DiÅ¿ponantur igitur huiuÅ¿mo-<lb/> - <anchor type="figure" xlink:label="fig-0120-01a" xlink:href="fig-0120-01"/> - di numeri tali ordine, vt fim-<lb/>plex Å¿umma, quæ ab vna reli-<lb/>quarum Å¿uperatur, & aliam Å¿upe-<lb/>rat, medium locum teneat; </s> - <s xml:id="s1390" xml:space="preserve">@t <lb/>in propoÅ¿ito exemplo Å¿umma <lb/>mediocris eft .48. quÄ™ à ſumma <num value="60">.<lb/>60.</num> Å¿uperatur, & Å¿uperat Å¿um-<lb/>mam .39. locata igitur fit hÄ™c .48. <lb/>inter illas, Å¿uæ verò primæ partes <lb/>fimiliter conftitutæ Å¿int Å¿upra di-<lb/>ctas Å¿ummas, cum Å¿uis <reg norm="differentijs" type="context">differẽtijs</reg>, <lb/>& tria producta iam dicta, vt in fi <lb/>guris <var>.C.</var> et <var>.D.</var> arithmeticis <lb/>clarè patet: </s> - <s xml:id="s1391" xml:space="preserve">figura enim <var>.C.</var> eft <lb/>pro exemplo ipÅ¿ius plus Å¿impli-<lb/>citer: </s> - <s xml:id="s1392" xml:space="preserve">figura verò <var>.D.</var> pro exem-<lb/>plo ipÅ¿ius plus, & minus. </s> - <s xml:id="s1393" xml:space="preserve">Et fic <lb/> - <anchor type="figure" xlink:label="fig-0120-02a" xlink:href="fig-0120-02"/> - in figura <var>.C.</var> habebimus tres <lb/>numeros confequentes .60. 48. <lb/>39. & tres antecedentes .10. 8. <lb/>6. cum dimidio, vnam, & ean-<lb/>dem proportionem terminantes, <lb/>ex .24. quinti, vt diximus; </s> - <s xml:id="s1394" xml:space="preserve">qua-<lb/>re eorum differentiæ fimiliter <lb/>proportionales erunt, quod etiam <lb/>vidimus. </s> - <s xml:id="s1395" xml:space="preserve">Supponamus nunc nos <lb/>ignorare æqualitatem maximi <lb/>producti cum reliquis duobus, <lb/>accipiendo Å¿olum pro hypoteÅ¿i, <lb/>quòd dicta producta oriantur <lb/>ex lateribus iam dictis.</s> - </p> - <div type="float" level="4" n="5"> - <figure xlink:label="fig-0120-01" xlink:href="fig-0120-01a"> - <image file="0120-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0120-01"/> - </figure> - <figure xlink:label="fig-0120-02" xlink:href="fig-0120-02a"> - <image file="0120-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0120-02"/> - </figure> - </div> - <p> - <s xml:id="s1396" xml:space="preserve">DemonÅ¿trandum nobis nunc relinquetur, maximum productum æquale effere-<lb/>liquis duobus; </s> - <s xml:id="s1397" xml:space="preserve">hoc eÅ¿t productum .168. æquale effe productis .90. et .78. quorum <lb/>duorum productorum alterum .90. Å¿cilicet, generatur à differentia .9. quæ eft Å¿e-<lb/>cundÄ™, & tertiÄ™ Å¿ummæ, in primum numerum antecedentem, qui eÅ¿t .10. alterum vc-<lb/>ro productum .78. Å¿cilicet, generatur à differentia .12. quÄ™ eÅ¿t primÄ™, & Å¿ecundÄ™, Å¿um <lb/>mÄ™ in tertium numerum antecedentem, qui eÅ¿t .6. cum dimidio, maximum vero <lb/>productum .168. Å¿cilicet generatur à differentia maxima .21. quÄ™ eft primÄ™, & tertiÄ™ <lb/>Å¿ummÄ™ (& Å¿emper Ä™qualis prioribus duabus differentijs .12. et .9.) in Å¿ecundum nu-<lb/>merum antecedentem, qui eÅ¿t .8.</s> - </p> - <p> - <s xml:id="s1398" xml:space="preserve">ConÅ¿tituantur igitur duo producta fimul iuncta Ä™qualia duobus .90. et .78. <lb/>lateralibus Å¿upra vnam aliquam rectam lineam <var>.q.p.</var> <reg norm="fitque" type="simple">fitqÌ;</reg> productum <var>.f.g.</var> Ä™quale <num value="90">.<lb/>90.</num> productum verò <var>.g.n.</var> Ä™quale .78. fit etiam baÅ¿is <var>.g.p.</var> vt .9. et <var>.g.q.</var> vt .12. vnde <var>.g.i.</var> <lb/>vel <var>.q.n.</var> erit vt .6. cum dimidio .et <var>.g.d.</var> vel <var>.p.f.</var> vt .10. & ideo <var>.i.d.</var> differentia erit .3. - <pb o="109" rhead="THEOREM. ARIT." n="121" file="0121" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0121"/> - cum dimidio, ut in figura <var>.C.</var> geometrica hic Å¿ubÅ¿cripta videre licet, et <var>.q.p.</var> erit .21. <lb/>Cogitemus nunc differentiam <var>.d.i.</var> diuiÅ¿am eſſe in puncto <var>.e.</var> ita vt eadem proportio <lb/>Å¿it ipÅ¿ius <var>.d.e.</var> ad <var>.e.i.</var> quæ ipÅ¿ius <var>.q.g.</var> ad <var>.g.p.</var> hoc eÅ¿t vt .1 2. ad .9. quapropter <var>.d.e.</var> erit <num value="2">.<lb/>2.</num> et <var>.e.i.</var> erit .1. cum dimidio, vt in dicta figura <var>.C.</var> arithmetica reperiuntur eſſe dif-<lb/>ferentiæ ipÅ¿orum antecedentium numerorum, deinde à puncto <var>.e.</var> ducatur imagina-<lb/>tione <var>.u.e.o.</var> æ quidiÅ¿tans ipÅ¿i <var>.q.p.</var> & producatur <var>.q.n.</var> vÅ¿que ad <var>.u.</var> vnde ita Å¿e habebit <lb/><var>u.e.</var> ad <var>.e.o.</var> ut <var>.q.g.</var> ad <var>g.p</var>. </s> - <s xml:id="s1399" xml:space="preserve">quare vt <var>.d.e.</var> ad <var>.e.i.</var> ideo ex .15. Å¿exti vel .20. Å¿eptimi <var>.n.e.</var> <lb/>rectangulum æquale crit ipÅ¿i <var>.e.f.</var> qua propter rectang ulum <var>.q.o.</var> æquale erit duobus<lb/> rectangulis <var>.f.g.</var> et <var>.g.n</var>: Å¿ed cum <var>.g.i.</var> Å¿it vt .6. cum dimidio, et <var>.i.e.</var> vt .1. cum dimidio, er <lb/>go <var>.g.e.</var> erit ut .8. qui quidem numerus multiplicatus cum <var>.q.p.</var> 21. producit .168. ve <lb/>rum eÅ¿t igitur quod dictum fuit, hoc eÅ¿t <reg norm="quod" type="simple">ê™</reg> maximum productum Ä™quale Å¿it reliquis <lb/>duobus.</s> - </p> - <figure position="here"> - <image file="0121-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0121-01"/> - </figure> - <figure position="here"> - <image file="0121-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0121-02"/> - </figure> - <pb o="110" rhead="IO. BAPT. BENED." n="122" file="0122" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0122"/> - <p> - <s xml:id="s1400" xml:space="preserve">DEmpto poÅ¿teà quo volueris horum altero productorum ex maximo, <reg norm="diuiÅ¿oque" type="simple">diuiÅ¿oqÌ;</reg> <lb/>reliquo per differentiam conÅ¿equentium, ipÅ¿i diametraliter oppoÅ¿itam, pro <lb/>ueniet tibi numerus antecedens <reg norm="correÅ¿pondensque" type="simple">correÅ¿pondensqÌ;</reg> illi.</s> - </p> - <p> - <s xml:id="s1401" xml:space="preserve">Animaduertendum tamen eÅ¿t, quòd Å¿i in figura à me ita ordinata, Å¿umma Å¿im-<lb/>plex propoÅ¿ita medium locum occuparet, vt in figura <var>.D.</var> arithmetica videri poteÅ¿t; <lb/></s> - <s xml:id="s1402" xml:space="preserve">tunc vt habeatur eius productum, addenda Å¿imul erunt circunÅ¿tantia producta .eo <lb/><reg norm="quod" type="simple">ê™</reg> eius Å¿ecundum latus eſſet antecedens medio loco conÅ¿titutum, & prima pars <reg norm="quae- Å¿ita" type="simple">quÄ™-<lb/>Å¿ita</reg> numeri propoÅ¿iti: </s> - <s xml:id="s1403" xml:space="preserve">in qua figura <var>.D.</var> manifeÅ¿tè patet ratio, quare colligendi Å¿int <lb/>tam errores, quam producta, dum eorum alterum eÅ¿t plus, reliquum verò minus.</s> - </p> - <p> - <s xml:id="s1404" xml:space="preserve">Speculatio figurÄ™ <var>.D.</var> arithmeticÄ™ videbitur in figura <var>.D.</var> geometrica, eodem fe <lb/>rè modo quo fecimus in figuris <var>.C.</var> mutatis mutandis, reÅ¿pectu ipÅ¿ius plus, & minus.</s> - </p> - <p> - <s xml:id="s1405" xml:space="preserve">Collectio namque <reg norm="errorum" type="context">errorÅ©</reg> Å¿imiliter accidentalis eÅ¿t, eo quod eſſentialis numerus <lb/>diuiÅ¿or per Å¿e, eÅ¿t maxima differentia Å¿ummarum Å¿implicium, vt in dicta figura <var>.D.</var> <lb/>cerni poteÅ¿t.</s> - </p> - <p> - <s xml:id="s1406" xml:space="preserve">Sed vt Å¿uperius dixi, nunc etiam repeto, quòd rectè hoc loco multiplicabatur <lb/>Å¿umma Å¿implex propoÅ¿ita, cum prima par <lb/>te primÄ™ poÅ¿itionis, vt productum diuide <lb/>retur per primam Å¿implicem Å¿ummam, <lb/> - <anchor type="figure" xlink:label="fig-0122-01a" xlink:href="fig-0122-01"/> - vnde proueniret nobis pars prima <reg norm="quaeÅ¿i- ta" type="simple">quęſi-<lb/>ta</reg> noÅ¿tri numeri propoÅ¿iti, ex regula de <lb/>tribus, vnica poÅ¿itione.</s> - </p> - <div type="float" level="4" n="6"> - <figure xlink:label="fig-0122-01" xlink:href="fig-0122-01a"> - <image file="0122-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0122-01"/> - </figure> - </div> - <p> - <s xml:id="s1407" xml:space="preserve">Vt exempli gratia, datus numerus diui <lb/>dendus Å¿it .100. in quinque partes, tales <lb/>verò, <reg norm="quod" type="simple">ê™</reg> Å¿ecunda duplo maior Å¿it prima <lb/>cum .2. Å¿imul, tertia autem æqualis Å¿it pri-<lb/>mæ & Å¿ecundæ cum .3. vnitatibus iunctis, <lb/>quarta poÅ¿teà maior Å¿it prima Å¿ecunda, & <lb/>tertia per .4. vnitates, quinta demum Å¿u-<lb/>peret reliquas omnes per quinque vnita <lb/>tes, vt in figura <var>.E.</var> videre eÅ¿t, quæ quidem <lb/>partes compoÅ¿itæ (Å¿umpta vnitate pro <lb/>prima) ita diÅ¿poÅ¿itæ erunt .1. 4. 8. 17. 35. <lb/>quarum Å¿umma erit .65. Å¿implices autem <lb/>cum diÅ¿poÅ¿itæ fuerint erunt .1. 2. 3. 6. 12. <lb/>quarum Å¿umma erit .24. dempta igitur <lb/>cum fuerit hæc Å¿implex Å¿umma .24. à com <lb/>poÅ¿ita .65. reÅ¿iduum erit .41. hoc eÅ¿t Å¿um-<lb/>ma numerorum propoÅ¿itorum cum Å¿uis <lb/>iterationibus in ipÅ¿is partibus, quod cum <lb/>per Å¿e clariſſimum Å¿it, Å¿uperÅ¿luum eÅ¿t <reg norm="ipsam" type="context">ipsã</reg> <lb/>Å¿ummam annatomizare per Å¿ingulas par-<lb/>tes, niÅ¿i quis habuerit eius cerebrum à fi-<lb/>gura Omega <reg norm="terminatum" type="context">terminatÅ©</reg>, cui tamen poÅ¿-<lb/>Å¿emus dicere dictam Å¿ummam .41. in .4. <lb/>partes diuidi, cuius prima eſſet .2. pro ad <lb/>ditione ad <reg norm="Å¿ecundam" type="context">Å¿ecÅ©dam</reg> partem Å¿implicium, - <pb o="111" rhead="THEOREM. ARIT." n="123" file="0123" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0123"/> - Å¿ecunda verò eſſet .5. pro additione ad tertiam partem Å¿implicium, tertia autem eÅ¿-<lb/>Å¿et .11. pro additione ad quartam partem Å¿implicium, quarta demum eſſet .23. pro <lb/>additione quintæ partis Å¿implicium, quarum partium .2. 5. 11. 23. Å¿umma eÅ¿t .41. vt <lb/>diximus. </s> - <s xml:id="s1408" xml:space="preserve">Hæc igitur Å¿umma .41. Å¿ubducenda eÅ¿t à numero .100. propoÅ¿ito, vnde re-<lb/>linquetur .59. pro Å¿umma partium Å¿implicium numeri propoÅ¿iti, quarum prima erit <lb/>2. cum vndecim vigeÅ¿imisquartis ex diuiÅ¿ione huiuÅ¿modi .59. per .24. Å¿ummam par-<lb/>tium Å¿implicium ex viregulæ de tribus, dicendo Å¿i .24. prouenit nobis ab .1. prima <lb/>partium Å¿implicium, à quo proueniet nobis .59? </s> - <s xml:id="s1409" xml:space="preserve">vnde proueniet à .2. cum vndecim <lb/>vigeÅ¿imisquartis pro prima parte quæſita, Å¿ecunda verò iuxta propoÅ¿itum, erit .6. <lb/>cum .22. vigeÅ¿imisquartis, tertia autem .12. cum nouem vigeÅ¿imisquartis, quarta poÂ<lb/>Å¿teà .25. cum .18. vigeÅ¿imisquartis, quinta demum erit .52. cum .12. vigeÅ¿imisquartis, <lb/>quarum omnium Å¿umma erit .100.</s> - </p> - <p> - <s xml:id="s1410" xml:space="preserve">STifelius in primo exemplo regulæ falÅ¿i, ita inquit.</s> - </p> - <p> - <s xml:id="s1411" xml:space="preserve">Quæratur numerus, à cuius dimidio Å¿ubtractæ partes tertia, & quarta relin-<lb/>quatur .300.</s> - </p> - <p> - <s xml:id="s1412" xml:space="preserve">IpÅ¿e enim Å¿upponit .300. pro reÅ¿iduo cognito alterius numeri incogniti, deinde <lb/>accipit .24. pro prima poÅ¿itione numeri cogniti, à cuius medietate abÅ¿cindit tertiam <lb/>& quartam partem ipÅ¿ius medietatis, vnde remanet .5. qui quidem numerus .5. ex <num value="22">.<lb/>22.</num> quinti vel .15. Å¿eptimiÅ¿e ha-<lb/>bebit ad .24. vt .300. ad <reg norm="numerum" type="context">numerÅ©</reg> <lb/> - <anchor type="figure" xlink:label="fig-0123-01a" xlink:href="fig-0123-01"/> - quæſitum, </s> - <s xml:id="s1413" xml:space="preserve">quare cum quis multi <lb/>plicauerit .300. per .24. & produ-<lb/>ctum diuiÅ¿erit per .5. proueniet <num value="1440">.<lb/>1440.</num> numerus quæſitus, ex vi <lb/>regulæ de tribus.</s> - </p> - <div type="float" level="4" n="7"> - <figure xlink:label="fig-0123-01" xlink:href="fig-0123-01a"> - <image file="0123-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0123-01"/> - </figure> - </div> - <p> - <s xml:id="s1414" xml:space="preserve">ConÅ¿ideremus igitur <reg norm="meam" type="context">meã</reg> di-<lb/>Å¿poÅ¿itionem numerorum huiuÅ¿-<lb/>modi exempli, in figura hic Å¿up-<lb/>poÅ¿ita <var>.F.</var> in qua videre licebit <lb/>quo pacto ipÅ¿e etiam Stifelius ac <lb/>cipiat diuiÅ¿orem .5. vt <reg norm="differentiam" type="context">differentiã</reg> <lb/>errorum & non ut differentiam <lb/>duorum conÅ¿equentium .5. et .10 <lb/>Å¿icuti eÅ¿t re uera, ut diuiÅ¿or dico, <lb/>ex rationibus à me hic Å¿upra ad-<lb/>ductis, quamuis vna & eadem Å¿it <lb/>quantitas neceſſariò ut patet.</s> - </p> - <p> - <s xml:id="s1415" xml:space="preserve">ACcipiamus adhuc aliud exemplum à Tartalea propoÅ¿itione .9. <reg norm="datum" type="context">datÅ©</reg>, & <reg norm="oppoÅ¿itum" type="context">oppoÅ¿itÅ©</reg> <lb/>priori; </s> - <s xml:id="s1416" xml:space="preserve">nam Å¿icut in illo numerus Å¿implex habebatur per Å¿ubtractionem Å¿um-<lb/>mæ numerorum adijciendorum, in hoc fitèconuerÅ¿o, hoc eÅ¿t per additionem nu-<lb/>merorum Å¿ubtrahendorum.</s> - </p> - <p> - <s xml:id="s1417" xml:space="preserve">Problema igitur ita Å¿e habet. </s> - <s xml:id="s1418" xml:space="preserve">Fuit quidam mercator qui habebat aliquot au-<lb/>reos, cuius quantitas poÅ¿teà quærenda erit, hic enim fecit duo itinera, ut aliquod <lb/>dictis aureis mediantibus lucrum faceret, in primo autem itinere duplicauit nume-<lb/>rum Å¿uorum aureorum, ex quibus poÅ¿teà conÅ¿umpÅ¿it .4. pro aliquibus expenÅ¿is, in - <pb o="112" rhead="IO. BAPT. BENED." n="124" file="0124" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0124"/> - Å¿ecundo itinere iterum duplicauit Å¿uos aureos, ex quibus etiam poÅ¿tea conÅ¿umpÅ¿it <num value="8">.<lb/>8.</num> numeratis poÅ¿teà pecunijs reperit tantummodo .24. aureos in eius marÅ¿upio, <reg norm="quae ritur" type="simple">quÄ™Â<lb/>ritur</reg> nunc quot habebat aureos in principio primi itineris.</s> - </p> - <p> - <s xml:id="s1419" xml:space="preserve">Intali caÅ¿u, cum ipÅ¿e quolibet itinere duplicabat eius pecuniam, nulli dubium <lb/>eÅ¿t quòd in fine Å¿ecundi itineris ipÅ¿e habuiſſet pecuniam Å¿uam quadruplicatam, Å¿i <lb/>ex ipÅ¿a nihil detractum fuiſſet, Å¿ed quia in fine primi itineris conÅ¿umpÅ¿it .4. aureos, <lb/>quibus alios .4. lucratus eſſet in Å¿ecundo itinere, poÅ¿teà conÅ¿umpÅ¿it iterum <num value="8">.<lb/>8.</num> aureos, ita <reg norm="quod" type="simple">ê™</reg> ex quadruplo Å¿uæ primæ pecuniæ, rectè dici poteÅ¿t, quod conÅ¿um-<lb/>pÅ¿erit .16. aureos; </s> - <s xml:id="s1420" xml:space="preserve">qui quidem numerus ex communi conceptu erit differentia in-<lb/>ter .24. & quadruplum prioris pecuniæ, cum qua profectus fuit in principio eius iti-<lb/>neris; </s> - <s xml:id="s1421" xml:space="preserve">quapropter Å¿i addiderimus .16. ipÅ¿i .24. habebimus .40. pro quadruplo eius <lb/>prioris pecuniæ. </s> - <s xml:id="s1422" xml:space="preserve">Rectè igitur dici poteÅ¿t, Å¿i .4. prouenit ab vno, à quo numero pro <lb/>ueniet .40.</s> - </p> - <p> - <s xml:id="s1423" xml:space="preserve">Videamus igitur nunc quo pacto hoc reÅ¿pondeat cum methodo antiquorum. <lb/></s> - <s xml:id="s1424" xml:space="preserve">Ego enim inueni duas poÅ¿itiones Å¿criptas à Tartalea pro prima pecunia hoc eÅ¿t .12. <lb/>et .14. Å¿ed à .12. pro primo errore reperi .8. more antiquo à .14. verò pro Å¿ecundo er-<lb/>rore proueniebat .16. producta autem horum numerorum diametraliter, Å¿unt .112. <lb/>et .192. quorum differentia eÅ¿t .80. pro tertio producto, quo diuiÅ¿o per differen-<lb/>tiam <reg norm="errorum" type="context">errorÅ©</reg> .8. Å¿cilicet, præbetnobis .10. pro pecunia quæſita, vt etiam ego inueni.</s> - </p> - <p> - <s xml:id="s1425" xml:space="preserve">Sed hoc mihi viÅ¿um eÅ¿t Å¿ubtilius examinare mea methodo mediante, vtin figu-<lb/>ra <var>.G.</var> videre eÅ¿t, prius enim Å¿uo loco poÅ¿uitria producta dicta, deinde duas poÅ¿itio <lb/>nes .12. et .14. & quia Å¿ciebam productum .112. oriri à multiplicatione .14. cum .8. <lb/>ideo poÅ¿ui talem numerum .8. Å¿uo loco diametraliter oppoÅ¿ito ei producto .112. <lb/>& quia Å¿ciebam etiam productum .192. naÅ¿ci ex .12. et .16. ideo Å¿uo loco poÅ¿ui hunc <lb/>numerum .16. qui eÅ¿t maxima differentia inter duos conÅ¿equentes ( ita à me Å¿upra <lb/>nominatos) à qua differentia dempta priori .8. iam inuenta, reliqua .8. mihi daba-<lb/>tur, <reg norm="quam" type="context">quã</reg> Å¿uo loco notaui, Å¿uo <reg norm="etiam" type="context">etiã</reg> <lb/>loco Å¿cripÅ¿i .2. differentiam inter <lb/>12. et .14. antecedentium. </s> - <s xml:id="s1426" xml:space="preserve">Å¿ed <lb/> - <anchor type="figure" xlink:label="fig-0124-01a" xlink:href="fig-0124-01"/> - quia Å¿ciebam eandem proportio <lb/>nem eſſe inter hanc differentiam <lb/>& <reg norm="differentiam" type="context">differentiã</reg> .8. huic <reg norm="Å¿uppoÅ¿itam" type="context">Å¿uppoÅ¿itã</reg>, <lb/>quæ reperitur inter .12. <reg norm="antecedem" type="context">antecedẽ</reg> <lb/>tem, & <reg norm="Å¿uum" type="context">Å¿uÅ©</reg> conÅ¿equentem; </s> - <s xml:id="s1427" xml:space="preserve">ideo <lb/>poÅ¿ui .48. pro dicto conÅ¿equenti, <lb/>diuiÅ¿i poÅ¿tea productum .80. per <num value="8">.<lb/>8.</num> differentiam ei diametraliter <lb/>oppoÅ¿itam, vnde prouenit mihi <lb/>10. cui ita proportionatus eÅ¿t <reg norm="Å¿uus" type="simple">Å¿uê°</reg> <lb/>numerus conÅ¿equens .40. vt .48. <lb/>ad .12. et .56. ad .14. exijÅ¿dem ra-<lb/>tionibus à meÅ¿upra dictis. <lb/></s> - <s xml:id="s1428" xml:space="preserve">In tali igitur figura videntur nu-<lb/>merinaturaliter <reg norm="correÅ¿pondentes" type="context">correÅ¿pondẽtes</reg> <lb/>ipÅ¿is poÅ¿itionibus, & hac metho-<lb/>do poſſumus inuenire tales numeros conÅ¿equentes in omnibus alijs exemplis à no-<lb/>Å¿tris maioribus Å¿criptis.</s> - </p> - <div type="float" level="4" n="8"> - <figure xlink:label="fig-0124-01" xlink:href="fig-0124-01a"> - <image file="0124-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0124-01"/> - </figure> - </div> - <pb o="113" rhead="THEOREM. ARIT." n="125" file="0125" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0125"/> - <p> - <s xml:id="s1429" xml:space="preserve">PRoponitur etiam quoddam vas, cuius pes Å¿it quarta pars totius vaÅ¿is cum oper <lb/>culo, pars autem media Å¿ine operculo, Å¿it quinta pars ipÅ¿ius pedis, operculum <lb/>verò .18. libras pendeat. </s> - <s xml:id="s1430" xml:space="preserve">quæritur nunc quantitas dicti pedis.</s> - </p> - <p> - <s xml:id="s1431" xml:space="preserve">Ex methodo enim antiquorum inuentus eÅ¿t pes .4. cum .14. decimisnonis ta-<lb/>lium partium, Å¿eu librarum, qualium operculus eÅ¿t .18. </s> - <s xml:id="s1432" xml:space="preserve">Videamus igitur & nos ex <lb/>noÅ¿tra figura, quo pacto hoc reÅ¿pondeat veritati.</s> - </p> - <p> - <s xml:id="s1433" xml:space="preserve">Inuenta enim Å¿unt tria producta, <reg norm="iam" type="context">iã</reg> orta ex dicta methodo .10. 100. 90. quæ Å¿uis <lb/>locis notaui, vt in Å¿igura <var>.H.</var> Å¿ubÅ¿cripÅ¿i etiam duas illorum poÅ¿itiones .5. et .10. cum <lb/>Å¿ua differentia .5. & cum productum .10. oriretur ab vno latere .10. reliquum erat <num value="1">.<lb/>1.</num> quod Å¿uo loco notaui, Å¿imiliter quia .100. productum, pro vno eius laterum erat <num value="5">.<lb/>5.</num> reliquum autem .20. Å¿uo loco poÅ¿ui, & quia differentia inter .20. et .1. duo latera, <lb/>quÄ™ eÅ¿t .19. æqualis eÅ¿t ei, quæ inter duo conÅ¿equentia duarum poÅ¿itionum, etiam <lb/>Å¿uo loco ipÅ¿am conÅ¿titui, Å¿ed quia hæc differentia eÅ¿t vnum laterum producti .90. er <lb/>go reliquum latus <reg norm="quæſitum" type="context">quæſitÅ©</reg> erit .4. cum .14. decimisnonis, rectè igitur operatur. <lb/></s> - <s xml:id="s1434" xml:space="preserve">Å¿ed cum eadem proportio Å¿it inter differentiam .5. Å¿uperiorem, et .19. inferiorem, <lb/>quæ eÅ¿t vnius <reg norm="antecedentis" type="context">antecedẽtis</reg> ad Å¿uum <lb/>conÅ¿equens, </s> - <s xml:id="s1435" xml:space="preserve">quare .10. <reg norm="antecedens" type="context">antecedẽs</reg> <lb/>habebit pro Å¿uo conÅ¿equenti .38. <lb/> - <anchor type="figure" xlink:label="fig-0125-01a" xlink:href="fig-0125-01"/> - et .5. habebit .19. et .4. cum .14. de-<lb/>cimisnonis habebit .18. rectè <reg norm="igitur" type="simple">igit̃</reg> <lb/>dictum fuiſſet Å¿i .19. prouenit .à .5. <lb/>à quo proueniet .18?</s> - </p> - <div type="float" level="4" n="9"> - <figure xlink:label="fig-0125-01" xlink:href="fig-0125-01a"> - <image file="0125-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0125-01"/> - </figure> - </div> - <p> - <s xml:id="s1436" xml:space="preserve">HuiuÅ¿modi autem rei ratio ita <lb/>Å¿e <reg norm="hent" type="context">hẽt</reg>, eÅ¿to linea <var>.a.e.u.</var> cuius pars <lb/>a. Å¿it quarta reliquarum <var>.e.u.</var> iuncta <lb/>rum, Å¿ed <var>.e.</var> Å¿it quinta ipÅ¿ius <var>.a</var>. </s> - <s xml:id="s1437" xml:space="preserve">Tunc <lb/>clarum erit quod <var>.e.</var> erit vigeÅ¿ima <lb/>dictarum <var>.e.u.</var> </s> - <s xml:id="s1438" xml:space="preserve">quare erit decima-<lb/>nona ipÅ¿ius <var>.u.</var> Å¿ed <reg norm="cum" type="context">cÅ©</reg> u. <reg norm="sumpta" type="context">sÅ©pta</reg> Å¿it vt <num value="18">.<lb/>18.</num> rectè igitur dici poteÅ¿t, Å¿i .u: ut <num value="19">.<lb/>19.</num> prouenit ab <var>.a.</var> ut quinque, à <lb/>quot ipÅ¿ius <var>.a.</var> proueniet <var>.u.</var> ut .18.</s> - </p> - <p> - <s xml:id="s1439" xml:space="preserve">Quis enim non uidet quod diui <lb/>Å¿a cum fuerit <var>.u.</var> in partes .19. quod <lb/>quinque illarum æquabuntur ipÅ¿i <var>.<lb/>a.</var> cum quælibet fuerit æqualis <var>.e.</var> <lb/>quintæ parti ipÅ¿ius <var>.a</var>.</s> - </p> - <p> - <s xml:id="s1440" xml:space="preserve">HAc igitur mea numerorum diÅ¿poÅ¿itione mediante reperiuntur ipÅ¿i numeri in <lb/>feriores naturaliter conÅ¿equentes, correÅ¿pondentesqÌue ipÅ¿is Å¿uperioribus an <lb/>tecedentibus; </s> - <s xml:id="s1441" xml:space="preserve">quamuis multoties <reg norm="contingere" type="context">cõtingere</reg> poſſit, ut generationes Å¿eu com-<lb/>poÅ¿itiones ipÅ¿orum ignorentur: </s> - <s xml:id="s1442" xml:space="preserve">& quia tam à differentijs errorum, quam ab illis, <lb/>quÄ™ Å¿unt inter ueros conÅ¿equentes numeros ( propter eorum æqualitatem ) elicitur <lb/>ipÅ¿a ueritas, proptereà rectè antiqui illis vÅ¿i Å¿unt, quamuis Å¿int potius Å¿enÅ¿um <lb/>Å¿equuti, uel experientiam, quam rationem: </s> - <s xml:id="s1443" xml:space="preserve">quæ quidem ratio pendet ab ipÅ¿is na-<lb/>turalibus numeris conÅ¿equentibus ( ut Å¿upra uidimus ) etÅ¿i incognitis ut plurimum, <lb/>quod Å¿i ipÅ¿os inuenire primò nobis datum fuiſſet, unica <reg norm="tantummodo" type="context">tantÅ©modo</reg> poÅ¿itio Å¿uffice- - <pb o="114" rhead="IO. BAPT. BENED." n="126" file="0126" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0126"/> - ret, mediante ipÅ¿a regula de tribus, vt <reg norm="iam" type="context">iã</reg> ſępius <reg norm="dictum" type="context">dictÅ©</reg> eÅ¿t, quod <reg norm="etiam" type="context">etiã</reg> clarè patet ex di-<lb/>uerÅ¿is problematibus .17. lib. ipÅ¿ius Tartaleæ, vt ex primo, quod aſſumpÅ¿imus pro <lb/>noÅ¿tro etiam primo exemplo, ex .9. 15. 16. 17. 18. 19. 20. 27. 28. 29. 30. 33. & ex <lb/>alijs multis, vbi facillimè inue nitur conÅ¿equens ipÅ¿ius poÅ¿itionis, qui quidem nume-<lb/>rus eÅ¿t diuiÅ¿or producti ipÅ¿ius numeri propoÅ¿iti in numerum poÅ¿itionis, vnde poÅ¿teà <lb/>prouenit <reg norm="Å¿ecundum" type="context">Å¿ecundÅ©</reg> latus huiuÅ¿modi producti, hoc eÅ¿t numerus quæſitus, per <reg norm="regulam" type="context">regulã</reg> <lb/>de tribus, vt dixi.</s> - </p> - <p> - <s xml:id="s1444" xml:space="preserve">Alia verò multa problemata inueniuntur, pro quorum re@olutione poſſumus ali <lb/>qua methodo vti, in qua manifeÅ¿tè pateant <reg norm="eorum" type="context">eorÅ©</reg> rationes abÅ¿que regula falÅ¿i, cuius <lb/>regulæ rationes non ita promptè ipÅ¿i intellectui Å¿e offerunt, vt Å¿upra vidimus.</s> - </p> - <p> - <s xml:id="s1445" xml:space="preserve">Accipiamus pro exemplo .21. problema ipÅ¿ius Tartalæ in dicto .17. libr. vbi Å¿up-<lb/>ponit vnum hædum diuiÅ¿um in .4. partes, quarum quælibet vendebatur eodem pre <lb/>cio, interiora vero .6. denarijs minus quam quælibet dictarum partium, Å¿umma <lb/>autem omnium iÅ¿torum denariorum fuit .127. quæritur nunc precium cuiuÅ¿que <lb/>partis.</s> - </p> - <p> - <s xml:id="s1446" xml:space="preserve">Tale enim problema hoc etiam alio breuiori modo poteÅ¿t Å¿olui, vt rationes ma-<lb/>gis pateant, quam ex regula falÅ¿i.</s> - </p> - <p> - <s xml:id="s1447" xml:space="preserve">Nam Å¿i illi numero .127. denariorum, additus fuerit numerus .6. Å¿umma erit .133. <lb/>qua diuiÅ¿a per quinque, illico proueniet .26. cum tribusquintis pro precio vniuÅ¿cu-<lb/>iuÅ¿que quatuor partium, à quo .26. cum tribusquintis dempto .6. remanebit .20. cum <lb/>tribusquintis pro precio interiorum.</s> - </p> - <p> - <s xml:id="s1448" xml:space="preserve">Simili modo in .24. problemate inquit.</s> - </p> - <p> - <s xml:id="s1449" xml:space="preserve">Duodecim pyra cum .28. pomis venduntur .36. denarijs, et .20. pyra. cum .200 po <lb/>mis <reg norm="venduntur" type="context">vẽduntur</reg> .44. denarijs, <reg norm="quæritur" type="simple">quærit̃</reg> nunc, quod <reg norm="nam" type="context">nã</reg> fuerit <reg norm="precium" type="context">preciÅ©</reg> <reg norm="vniuÅ¿cuiuÅ¿que" type="simple">vniuÅ¿cuiuÅ¿q;</reg> illorum.</s> - </p> - <p> - <s xml:id="s1450" xml:space="preserve">Hoc etiam problema, hac alia methodo Å¿olui poteÅ¿t, dicendo exregula de tribus, <lb/>Å¿i ex .20. vtrorunque qui ea vendit, vult .44. quid volet ex .12? </s> - <s xml:id="s1451" xml:space="preserve"><reg norm="manifeÅ¿tum" type="context">manifeÅ¿tÅ©</reg> erit quod <lb/>volet .26. cum duobus quintis, </s> - <s xml:id="s1452" xml:space="preserve">quare .12. pyra cum .12. pomis valebunt .26. cum duo <lb/>bus quintis, Å¿ed 12. cum .28. pomis valebant .36. ergo .16. poma Å¿ola valebunt .9. <lb/>cum tribus quintis, hoc enim clarè ex Å¿e patet; </s> - <s xml:id="s1453" xml:space="preserve">quare cum dixerimus, Å¿i .16. poma Å¿o <lb/>la valent .9. cum tribusquintis, vnum valebit <var>.o.</var> cum tribusquintis, Å¿ed quemadmo-<lb/>dum .20. pyra cum .20. pomis valent .44. vnum pyrum, cum vno pomo valebunt .2. <lb/>cum quinta parte, à quo numero detractus cum fuerit <var>.o.</var> cum tribus quintis, precio <lb/>Å¿cilicet vnius pomi, reliquum .1. cum tribusquintis, erit precium vnius pyri.</s> - </p> - <p> - <s xml:id="s1454" xml:space="preserve">Idem etiam dico de .28. problemate, vbi Å¿upponit quod quidam comparaſſet <lb/>quatuor petias, vt vulgo dicitur, panni pro ducatis .96. quarum primæ precium ob-<lb/>litus Å¿it, Å¿ed memoria tenet pro Å¿ecunda Å¿oluiſſe .6. plusquam pro prima, & pro ter-<lb/>tia Å¿oluiſſe .8. plus quam pro Å¿ecunda, & pro quarta Å¿oluiſſe .10. plus quam pro ter-<lb/>tia, quæritur nunc quantum fuerit precium vniuÅ¿cuiuÅ¿que illarum.</s> - </p> - <p> - <s xml:id="s1455" xml:space="preserve">Quod <reg norm="quidem" type="context">quidẽ</reg> problema <lb/> - <anchor type="figure" xlink:label="fig-0126-01a" xlink:href="fig-0126-01"/> - breuius eſſetita Å¿olui, vt in <lb/>Å¿ubÅ¿cripta figura <var>.I.</var> videri <lb/>poteÅ¿t, <reg norm="addendo" type="context">addẽdo</reg> Å¿imul omnes <lb/>exceſſus. </s> - <s xml:id="s1456" xml:space="preserve">Nam exceſſus <reg norm="Å¿ecum" type="context">Å¿ecÅ©</reg> <lb/>dæ Å¿upra primam eÅ¿t .6. Å¿ed <lb/>cum exceſſus tertiæ Å¿upra Å¿e <lb/>cundam Å¿it .8. ergo exceſſus <lb/>tertiæ Å¿upra primam erit .14 - <pb o="115" rhead="THEOREM. ARIT." n="127" file="0127" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0127"/> - Å¿ed exceſſus quartæ Å¿upra tertiam eÅ¿t .10. vnde Å¿upra Å¿ecundam erit .18. & Å¿upra pri-<lb/>mam erit .24. quæ omnia Å¿imul addita erunt .44. & in qualibet harum trium remane-<lb/>bit una pars æqualis primæ quantitati, </s> - <s xml:id="s1457" xml:space="preserve">quare Å¿i ex .96. detractus fuerit numerus .44. <lb/>reliquus 52. erit quadruplus primæ, </s> - <s xml:id="s1458" xml:space="preserve">quare prima pars valebit .13. Å¿ecunda .19. ter-<lb/>tia .27. & quarta .37. quarum omnium Å¿umma eÅ¿t .96.</s> - </p> - <div type="float" level="4" n="10"> - <figure xlink:label="fig-0126-01" xlink:href="fig-0126-01a"> - <image file="0126-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0126-01"/> - </figure> - </div> - <p> - <s xml:id="s1459" xml:space="preserve">EX poÅ¿itionibus autem Tartaleæ in noÅ¿tra figura <var>.K.</var> digeÅ¿tis, videre poſſumus <lb/>quo pacto <reg norm="colligantur" type="context">colligãtur</reg> huiuÅ¿ <lb/>modi <reg norm="conÅ¿equentes" type="context">conÅ¿equẽtes</reg> numeri Å¿impli-<lb/>ces .36. et .52. more figuræ <var>.E.</var> quia <lb/> - <anchor type="figure" xlink:label="fig-0127-01a" xlink:href="fig-0127-01"/> - colliguntur primò partes compoÅ¿i <lb/>tæ .9. 15. 23. 33. ex quarum Å¿umma <lb/>80. Å¿ubtrahitur .36. Å¿umma Å¿im-<lb/>plex ex Å¿implicibus partibus .9. 9. <lb/>9. 9. & <reg norm="reÅ¿iduum" type="context">reÅ¿iduÅ©</reg> quod eÅ¿t .44. Å¿ubdu <lb/>citur ex .96. Å¿umma compoÅ¿ita & <lb/>propoÅ¿ita, vnde remanet .52. pro <lb/>Å¿umma Å¿implici, ex numero dato, <lb/>cuius proportio ad .13. eadem eÅ¿t <lb/>quæ .36 ad .9. & proptereà ſuper-<lb/>flua eÅ¿t Å¿ecunda poÅ¿itio, <reg norm="quando" type="context">quãdo</reg> Å¿ci <lb/>mus inuenire tales duos numeros <lb/>conÅ¿equentes, vt in hoc exemplo <lb/>Å¿unt .36. et .52. quia ex regula de <lb/>tribus poÅ¿teà elicitur veritas quæ-<lb/>Å¿ita. </s> - <s xml:id="s1460" xml:space="preserve"><reg norm="Idem" type="context">Idẽ</reg> dico de 33. problemate.</s> - </p> - <div type="float" level="4" n="11"> - <figure xlink:label="fig-0127-01" xlink:href="fig-0127-01a"> - <image file="0127-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0127-01"/> - </figure> - </div> - <p> - <s xml:id="s1461" xml:space="preserve">PRO quo .33. problemate acci <lb/>piantur poÅ¿itiones primi <reg norm="exem" type="context">exẽ</reg> <lb/>pli TonÅ¿talli hoc eÅ¿t .33. et .31. vt in figuris hic Å¿ubiectis <var>.P.Q.</var> facile quis poteÅ¿t vi-<lb/>dere, vbi in figura P. videbit nume-<lb/>ros compoÅ¿itos, in figura verò <var>.Q.</var> cer <lb/> - <anchor type="figure" xlink:label="fig-0127-02a" xlink:href="fig-0127-02"/> - net numeros Å¿implices, à quibus pro <lb/>ueniunt rationes per Å¿e huiaÅ¿modi <lb/>operationis, in figura autem <var>.R.</var> vide <lb/>bitur meus ordo, & iÅ¿tæ tres figuræ Å¿i <lb/>miles <reg norm="erunt" type="context">erÅ©t</reg> tribus illis primis <var>.A.B.C.</var> <lb/>ita quòd cum quis illas intellexerit, il <lb/>lico etiam iÅ¿tas cognoÅ¿cet, vbi <reg norm="etiam" type="context">etiã</reg> <lb/>videbit quam confusè <reg norm="ratiocinentur" type="context">ratiocinẽtur</reg> ij <lb/>qui ignorant hunc meum ordinem <lb/>Å¿implicium <reg norm="numerorum" type="context">numerorÅ©</reg>, à quibus fluit <lb/>tota ratio (vt Å¿upra dixi) huiuÅ¿cemo <lb/>di operationis.</s> - </p> - <div type="float" level="4" n="12"> - <figure xlink:label="fig-0127-02" xlink:href="fig-0127-02a"> - <image file="0127-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0127-02"/> - </figure> - </div> - <pb o="116" rhead="IO. BAPT. BENED." n="128" file="0128" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0128"/> - <figure position="here"> - <image file="0128-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0128-01"/> - </figure> - <figure position="here"> - <image file="0128-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0128-02"/> - </figure> - <p> - <s xml:id="s1462" xml:space="preserve">I Dem etiam poteÅ¿t dici de .15. problemate (Å¿icut de alijs multis) vbi ponit tres <lb/>homines habentes .40. aureos quorum primus habet duas quintas partes Å¿ecun-<lb/>di, <reg norm="Å¿ecundus" type="context">Å¿ecÅ©dus</reg> verò <reg norm="quinque" type="simple">quinq;</reg> octauas tertij, <reg norm="quæritur" type="simple">quærit̃</reg> <reg norm="nunc" type="context">nÅ©c</reg> quot ducatos habeat vnuÅ¿quiÅ¿que.</s> - </p> - <p> - <s xml:id="s1463" xml:space="preserve">Quis non videt quæſo, <reg norm="quod" type="simple">ê™</reg> omnes partes erunt .15. quare cum dixerimus Å¿i .15. dat <lb/>nobis .2. (pro prima portione primi hominis) quid dabit .40? </s> - <s xml:id="s1464" xml:space="preserve">vnde nobis proueniet <lb/>5. cum tertia parte.</s> - </p> - <p> - <s xml:id="s1465" xml:space="preserve">Et de .29. Å¿imiliter aſſero, vbi ponit <reg norm="alium" type="context">aliÅ©</reg> emiſſe tria fruÅ¿ta panni pro ducatis .48. <lb/>quarum Å¿ecundam habuit pro dimidio precio primæ, tertiam autem pro quarta <lb/>parte ipÅ¿ius Å¿ecundæ, </s> - <s xml:id="s1466" xml:space="preserve">quare omnes partes erunt .13. quapropter precium tertiæ pe-<lb/>tiæ erit tertiadecima pars ipÅ¿ius .48. hoc eÅ¿t .3. cum .9. tertijs decimis.</s> - </p> - <p> - <s xml:id="s1467" xml:space="preserve">Adhuc duo exempla videtur mihi proponere, quorum primum eÅ¿t .38. eiuÅ¿dem <lb/>lib. vbi Å¿upponitur operarium quendam velle perficere opus quoddam Å¿pacio die-<lb/>rum .36. tali pacto, quod qualibet die, in qua ipÅ¿e operaturus Å¿it lucretur Å¿olidos <num value="16">.<lb/>16.</num> qualibet verò die, in qua nihil agat perdat Å¿olidos .24. </s> - <s xml:id="s1468" xml:space="preserve">Tunc accidit, vt exacto <lb/>termino <reg norm="perfectoque" type="simple">perfectoqÌ;</reg> opere, tantum lucratus Å¿it, quantum perdiderit. </s> - <s xml:id="s1469" xml:space="preserve">Quæritur <reg norm="nunc" type="context">nÅ©c</reg> <lb/>quot fuerint dies lucri, quotúe perditionis.</s> - </p> - <p> - <s xml:id="s1470" xml:space="preserve">HuiuÅ¿modi problematis operatio breuiſſima abÅ¿que vlla falÅ¿a poÅ¿itione ita erit, <lb/>hoc eÅ¿t diuidendo productum .36. in .24. per .40. ideÅ¿t per aggregatum ipÅ¿ius .24. <reg norm="cum" type="context">cÅ©</reg> <lb/>16. & prouentus erit .21. cum tribus quintis pro diebus lucri, vnde reliquum ex .36. <lb/>erit .14. cum duabus quintis pro diebus perditionis.</s> - </p> - <p> - <s xml:id="s1471" xml:space="preserve">Cuius operationis ratio ex Å¿e Å¿atis patet, cum duo producta, vnum lucri, alterum <lb/>vero perditio-<lb/>nis æqualia eſſe <lb/> - <anchor type="figure" xlink:label="fig-0128-03a" xlink:href="fig-0128-03"/> - debeant, vnde <lb/>ex duodecima, <lb/>& vigeÅ¿imaÅ¿epti <lb/>mi ex regula de <lb/>tribus reperiun <lb/>tur partes ipÅ¿ius <lb/>36. eodem mo- - <pb o="117" rhead="THEOR. ARITH." n="129" file="0129" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0129"/> - do Å¿e inuicem habentes, vt .24. et .16. quæ Å¿unt .21. cum tribus quintis, et .14. <reg norm="cum" type="context">cÅ©</reg> dua <lb/>bus quintis, ex quo Å¿equitur, vt quod fit ex .21. cum tribus quintis, in 16. Ä™quale Å¿it ei <lb/>quod fit ex .14 cum duabus quintis, in .24. & ita reperiuntur duo producta æqualia, <lb/>vnum lucri, reliquum vero perditionis, vt in figura <var>.M.</var> clarè videtur.</s> - </p> - <div type="float" level="4" n="13"> - <figure xlink:label="fig-0128-03" xlink:href="fig-0128-03a"> - <image file="0128-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0128-03"/> - </figure> - </div> - <p> - <s xml:id="s1472" xml:space="preserve">A Liud verò exemplum eÅ¿t .39. quod quidem à ſuperiori non differt, niÅ¿i quod <lb/>in fine operationis, operarius dictus lucratus eÅ¿t Å¿olidos .60: </s> - <s xml:id="s1473" xml:space="preserve">quÄ™ritur <reg norm="nunc" type="context">nÅ©c</reg> vt Å¿u-<lb/>pra, quot fuerunt dies lucri, & quot perditionis.</s> - </p> - <p> - <s xml:id="s1474" xml:space="preserve">Hoc etiam abÅ¿que vlla falÅ¿a poÅ¿itione dicto citius poteÅ¿t Å¿olui, hoc modo, <reg norm="diuidem" type="context">diuidẽ</reg> <lb/>do Å¿cilicet illos .60 Å¿olidos per .40. ideÅ¿t per aggregatum .24. cum .16. proueniens <lb/>autem, quod erit .1. cum dimidio, adde ad latus Å¿uperius inuentum, hoc eÅ¿t .21. cum <lb/>tribus quintis, & Å¿unima erit .23. cum decima parte pro numero dierum lucri, dein-<lb/>de idem prouentum deme ex alio latere Å¿uperius reperto .14. cum duabus quintis, & <lb/>refiduum erit .12. cum nouem decimis, vnde habebis numerum dierum perdi-<lb/>tionis.</s> - </p> - <p> - <s xml:id="s1475" xml:space="preserve">Pro cuius rei Å¿peculatione cogitemus in figura <var>.N.</var> duo dicta producta inuicem <lb/>æqualia <var>.o.b.</var> et <var>.n.c.</var> exiÅ¿tente latere <var>.u.c.</var> vt .24. <var>u.o.</var> ut .16: </s> - <s xml:id="s1476" xml:space="preserve">b.u. vt .21. cum tribus quin <lb/>tis, et <var>.u.n.</var> vt .14. cum duabus quintis. </s> - <s xml:id="s1477" xml:space="preserve">Nunc verò Å¿i mente concepta fuerit recta <var>.e.<lb/>a.t.</var> æquidiÅ¿tans <var>.o.c.</var> ita vt rectangulum <var>.o.e.</var> Å¿it .60. </s> - <s xml:id="s1478" xml:space="preserve">tunc rectangulum, Å¿eu productum <lb/><var>b.t.</var> Å¿uperabit rectangulum Å¿eu productum <var>.n.e.</var> per idem .60. ex communi conceptu, <lb/>eo quòd ex producto <var>.n.c.</var> Å¿ublatum eÅ¿t productum <var>.a.c.</var> 24. & producto <var>.o.b.</var> additum <lb/>eÅ¿t productum <var>e.a.</var> 16. rectè igitur feci cum diuiÅ¿erim .60 per .40. vnde prouenit mi <lb/>hi <var>.u.a.</var> ideÅ¿t .1. cum dimidio, quod additum ipÅ¿i <var>.b.u.</var> compoÅ¿uit <var>.b.a.</var> & dempto ex <var>.u.<lb/>n.</var> relinquit <var>.a.n.</var> pro lateribus duorum productorum <var>.b.t.</var> et <var>.n.e</var>.</s> - </p> - <p> - <s xml:id="s1479" xml:space="preserve">Sed Å¿i idem operator perdidiſſet .60. tunc cogitaremus parallelam dictam <var>.e.a.</var> t <lb/>Å¿uperius ductam eſſe ita vt Å¿ecaret <var>.b.u.</var> & non <var>.u.n.</var> vnde adderet .24. ipÅ¿i producto <var>.n.<lb/>c.</var> & d@meret .16. à producto <var>.b.o</var>.</s> - </p> - <figure position="here"> - <image file="0129-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0129-01"/> - </figure> - <p> - <s xml:id="s1480" xml:space="preserve">CIRCA verò talia quæſita videtur mihi non inutile fore Å¿i aliquid notatu di-<lb/>gnum aduerterim, hoc eÅ¿t quod ſæpe accidere poterit ut caÅ¿us impoſſibiles <lb/>proponantur. </s> - <s xml:id="s1481" xml:space="preserve">Quemadmodum Å¿i aliquis diceret, cupio mihi ueÅ¿timentum con-<lb/>ficere ex duobus pannis colore & pretio differentibus, quorum unus exempli gra- - <pb o="118" rhead="IO. BAPT. BENED." n="130" file="0130" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0130"/> - tia Å¿it albus, rubeus uerò alter, deinde albus Å¿it pretij .40. Å¿olidorum uniuÅ¿cuiuÅ¿-<lb/>que cubiti, rubeusuerò precij .50. uellemqÌue omnes cubitos eſſe .8. nec plus <lb/>nec minus. </s> - <s xml:id="s1482" xml:space="preserve">Vellem etiam Å¿oluere Å¿olidos 450. neque minus.</s> - </p> - <p> - <s xml:id="s1483" xml:space="preserve">Hic igitur caÅ¿us impoſſibilis eÅ¿t, eo quòd .8. cubiti totius rubei eſſent precij Å¿o-<lb/>lidorum .400. tantummodo, unde ex alio panno albo minoris precij Å¿umere ali-<lb/>quid non poſſumus.</s> - </p> - <p> - <s xml:id="s1484" xml:space="preserve">Idem etiam eueniret Å¿i uoluiſſet Å¿oluere Å¿olidos .320. neque plus, eo quòd .8. cu-<lb/>biti illius minoris precij, hoc eÅ¿t .40. Å¿olidorum, eÅ¿sent ualoris .320. Å¿olidorum tan <lb/>tummodo, quare pro alio panno nullus eÅ¿set locus. </s> - <s xml:id="s1485" xml:space="preserve">Animaduertendum igitur erit <lb/>quod numerus poſſibilis ad Å¿oluendum tale quæſitum erit inter .400. et .320. & non <lb/>extra iÅ¿tos terminos, vt vnicuique patere poteÅ¿t.</s> - </p> - <p> - <s xml:id="s1486" xml:space="preserve">Similiter idem in hoc alio caÅ¿u accidere poterit, ut Å¿i quis diceret.</s> - </p> - <p> - <s xml:id="s1487" xml:space="preserve">Emi quinque petias panni pro aureis .55. pretium tamen primæ oblitus Å¿um, Å¿ed <lb/>memoria teneo, quòd Å¿ecunda altioris pretij erat quam ipÅ¿a prima per .4. & ter-<lb/>tia precioÅ¿ior Å¿ecunda per .7. et quarta carior tertia per .9. quinta verò Å¿uperabat <lb/>quartam per .2.</s> - </p> - <p> - <s xml:id="s1488" xml:space="preserve">Hic etiam reperitur impoſſibilitas quædam, eo quòd aggregatum omnium ha-<lb/>rum rerum, dato etiam quòd pro prima nihil Å¿olutum eſſet, Å¿uperat aureos .55 quòd <lb/>quidem nullo pacto fieri poteÅ¿t, vt veri Å¿int Å¿upra dicti exceÅ¿sus, Å¿i verus eÅ¿t numerus <lb/>totalis aureorum .55. </s> - <s xml:id="s1489" xml:space="preserve">Nam .4. cum .7. faciunt .11. qui quidem .11. cum .9. efficiunt <num value="20">.<lb/>20.</num> & hic cum .2. facit .22. Å¿ed .22. cum .20. et .11. et .4. dant .57. qui numerus maior <lb/>eÅ¿t quam .55.</s> - </p> - <head xml:id="head175" type="footer" xml:space="preserve">FINIS THEOREM. ARIT.</head> - <pb o="119" n="131" file="0131" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0131"/> - </div> - </div> - <div type="chapter" level="2" n="2"> - <head xml:id="head176" xml:space="preserve">DE RATIONIBVS <lb/>OPERATIONVM <lb/>PERSPECTIVAE.</head> - <div type="section" level="3" n="1"> - <head xml:id="head177" xml:space="preserve">CAP.I.</head> - <p> - <s xml:id="s1490" xml:space="preserve">CVM nullus adhuc (quod Å¿ciam) veras <reg norm="internasque" type="simple">internasqÌ</reg> cauÅ¿as <lb/>operationis perÅ¿pectiuæ perÅ¿ectè docuerit, operæpre-<lb/>cium exiÅ¿timaui <reg norm="aliquam" type="context">aliquã</reg> de ijs diÅ¿putationem Å¿uÅ¿cipere.</s> - </p> - <p> - <s xml:id="s1491" xml:space="preserve">Multi enim <reg norm="eorum" type="context">eorÅ©</reg>, qui huiuÅ¿modi operationis regulas <lb/>præſcribunt, cum eius effectuum veras cauÅ¿as igno-<lb/>rent, varios <reg norm="diuerÅ¿osque" type="simple">diuerÅ¿osqÌ;</reg> errores committunt, vt exempli <lb/>gratia in Å¿ubÅ¿cripta figura Å¿uperficiali <var>.A.</var> volentes degra <lb/>dare (vt dicunt) rectangulum <var>.q.a.</var> in triangulo <var>.i.d.q.</var> du-<lb/>cunt <reg norm="parallelam" type="context">parallelã</reg> ipÅ¿i <var>.q.d.</var> à puncto <var>.B.</var> interÅ¿ecationis lineæ-<lb/>o.l. cum latere <var>.i.d.</var> trianguli, & (idem) indifferenter, ean-<lb/>dem quoque à puncto <var>.Z.</var> interÅ¿ecationis ipÅ¿ius <var>.o.l.</var> cum perpendiculari <var>.x.i.</var> ducunt. <lb/></s> - <s xml:id="s1492" xml:space="preserve">neÅ¿cientes hunc Å¿olum eſſe verum modum, n onitem alium, quia Å¿i alius, talis eſſet, <lb/>hic, verus non exiÅ¿teret, nam Å¿i vellent Å¿eÅ¿e excuÅ¿ are, quòd ducendo dictam paralle-<lb/>lam à puncto <var>.B.</var> hoc fiat præſupponendo planum ipÅ¿ius <var>.i.d.q.</var> verÅ¿us rectangulum <var>.<lb/>q.a.</var> orizontale inclinatum, Å¿ecundum angulum <var>.i.d.q.</var> hæc excuÅ¿atio accipien-<lb/>da non eſſet, quia horum conÅ¿enÅ¿u, præſupponendo planum <var>.i.d.q.</var> inclinatum, <lb/>anguli inferiores rectanguli degradati, non tam acuti, quam Å¿unt duo <var>.i.d.q.</var> et <var>.i.q.<lb/>d.</var> eſſe deberent, quod facilè eorum ratione innoteÅ¿cet, quæ de figura corporea <var>.<lb/>A.</var> hîc Å¿ubÅ¿cripta mox proponam, præter id, quòd volentes deinde aÅ¿picere qua-<lb/>dratum degradatum, oporteret huiuÅ¿modi planum reÅ¿pectu oculi ita collocare, <lb/>quemadmodum Å¿e habet linea <var>.i.d.</var> reÅ¿pectu <var>.o.</var> quod factu nimis arduum eſſet.</s> - </p> - <p> - <s xml:id="s1493" xml:space="preserve">Vera igitur ratio erit ducere parallelam <var>.e.r.</var> ad <var>.q.d.</var> à puncto <var>.Z.</var> communi ip-<lb/>Å¿is <var>.o.l.</var> et <var>.x.i.</var> perpendiculari ipÅ¿i <var>.l.p</var>.</s> - </p> - <figure position="here"> - <image file="0131-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0131-01"/> - </figure> - <pb o="120" rhead="IO. BAPT. BENED." n="132" file="0132" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0132"/> - <p> - <s xml:id="s1494" xml:space="preserve">Pro cuius rei Å¿peculatione imaginemur in figura corporea .A: <var>q.a.</var> eſſe figuram re-<lb/>ctangulam <reg norm="orizontalemque" type="simple">orizontalemqÌ;</reg> ad degradandam Å¿uper aliquod planum perpendiculare <lb/>orizonti, & cum eo primum coniunctam in linea <var>.q.d.</var> cuius plani triangulum <var>.i.q.d.</var> <lb/>pars erit, Å¿it autem oculus reÅ¿picientis <var>.o.</var> cuius altitudo <var>.o.p.</var> ab orizonte, qui <reg norm="quidem" type="context">quidẽ</reg> <lb/>conÅ¿picit rectangulum dictum orizontale <var>.q.a.</var> in pyramide <var>.o.q</var>: <var>o.u</var>: <var>o.a.</var> et <var>.o.d.</var> <lb/>terminata quatuor triangulis <var>.o.q.u</var>: <var>o.u.a</var>: <var>o.a.d.</var> et <var>.o.d.q.</var> Å¿it verò primum ita <lb/>collocatus pes <var>.p.</var> eius qui reÅ¿picit, vt linea <var>.p.l.</var> perpendicularis ipÅ¿i <var>.u.a.</var> lateri re-<lb/>ctanguli, medio loco poÅ¿ita Å¿it, inter <var>.a.n.</var> et <var>.u.s</var>. </s> - <s xml:id="s1495" xml:space="preserve"><reg norm="Idque" type="simple">IdqÌ;</reg> primum nobis erit exem-<lb/>plum.</s> - </p> - <p> - <s xml:id="s1496" xml:space="preserve">Imaginemur nunc lineas <var>.u.q.</var> et <var>.a.d.</var> indefinitè productas eſſe, quæ in Å¿uperficie-<lb/>bus duorum triangulorum <var>.o.u.q.</var> et <var>.o.a.d.</var> & rectanguli orizontalis <var>.q.a.</var> ex <ref id="ref-0018">prima <lb/>vndecimi Euclid.</ref> poÅ¿itæ erunt. </s> - <s xml:id="s1497" xml:space="preserve">Imaginemur etiam lineam <var>.p.s.n.</var> perpendicula-<lb/>rem ipÅ¿i <var>.p.l.</var> quæ etiam cum duabus <var>.u.q.s.</var> et <var>.a.d.n.</var> ex .34. primi Euclid. angulos <lb/>rectos conÅ¿tituet, cum ex .28. duæ <var>.u.q.s.</var> et <var>.a.d.n.</var> Å¿int parallelæ ipÅ¿i <var>.p.l.</var> et <var>.s.n.</var> ipÅ¿i <var>.u.<lb/>a.</var> & quia Å¿upponitur <var>.o.p.</var> perpendicularis plano orizontali, Angulus ergò <var>.o.p.l.</var> re-<lb/>ctus erit ex Å¿ecunda definitione .11. Euclid. </s> - <s xml:id="s1498" xml:space="preserve">Imaginemur quoque ductas eſſe <lb/>duas <var>.o.s.</var> et <var>.o.n.</var> vnde <var>.l.p.</var> ei Å¿uperficiei, in qua Å¿unt duæ lineæ <var>.o.p.</var> et <var>.s.n.</var> ex .4. <lb/>11. perpendicularis erit, & Å¿uperficies orizontalis <var>.a.s.</var> perpendicularis erit cum dicta <lb/><var>o.s.n.</var> ex .18. eiuÅ¿dem lib. vnde ex dicta definitione <var>.o.s.u.</var> et <var>.o.n.a.</var> erunt anguli recti <lb/>et <var>.o.s.</var> et <var>.o.n.</var> ex communi Å¿cientia, in Å¿uperficiebus duorum triangulorum <var>.o.u.q.</var> et <var>.<lb/>o.a.d.</var> erunt, Å¿i noluerimus cogere aduerÅ¿arium ad confitendum duas lineas rectas in-<lb/>cludere Å¿uperficiem, quemadmodum cogere-<lb/> - <anchor type="figure" xlink:label="fig-0132-01a" xlink:href="fig-0132-01"/> - tur facere, Å¿i opinaretur duas alias rectas per <lb/>eadem puncta <var>.o.s.n.</var> tranÅ¿ire, quæſunt in di-<lb/>ctis Å¿uperficiebus. </s> - <s xml:id="s1499" xml:space="preserve">Vnde <var>.o.s.</var> et <var>.o.n.</var> communes <lb/>erunt Å¿ectiones duarum dictarum <reg norm="Å¿uperficierum" type="context">Å¿uperficierÅ©</reg> <lb/>cum Å¿uperficie <var>.o.s.n</var>. </s> - <s xml:id="s1500" xml:space="preserve">Imaginemur nunc has <lb/>duas Å¿uperficies <var>.o.u.</var> et <var>.o.a.</var> quarum commu-<lb/>nis Å¿ectio Å¿it <var>.o.t.</var> (quæ erit linea recta ex .3. lib. <lb/>II.) quæ erunt perpendiculares Å¿uperficiei <var>.o.s.<lb/>n.</var> ex .4. et .14. iam dictis. </s> - <s xml:id="s1501" xml:space="preserve">& ex .19. eiuÅ¿dem <lb/><var>o.t.</var> perpendicularis eidem Å¿uperficiei <var>.o.s.n.</var> <lb/>erit, & ex .6. eiuÅ¿dem hæc linea <var>.o.t.</var> duabus <var>.u.<lb/>q.s.</var> et <var>.a.d.n.</var> parallela exiÅ¿ter, & ex .9. eiuÅ¿dem <lb/>hæc linea <var>.o.t.</var> duabus <var>.u.q.s.</var> et <var>.a.d.n.</var> parallela <lb/>exiÅ¿tet, & ex eadem .9. erit parallela ipÅ¿i <var>.p.l.</var> <lb/>Imaginemur nunc planum, Å¿uper quod deÅ¿ide <lb/>remus videre quadrangulum orizontale, quod <lb/>planum, exempli gratia, Å¿it primo, vt iam dixi-<lb/>mus, locatum in linea <var>.q.d.</var> ad angulos rectos <lb/>cum plano orizontali, cuius communes Å¿ectio <lb/>nes cum Å¿uperficiebus <var>.s.t.</var> et <var>.n.t.</var> viÅ¿ionis la-<lb/>terum <var>.u.q.</var> et <var>.a.d.</var> Å¿int <var>.i.q.</var> et <var>.i.d.</var> & com-<lb/>munis Å¿ectio trianguli <var>.o.u.a.</var> ideÅ¿t viÅ¿ionis <lb/>lateris <var>.a.u.</var> cum dicto plano, Å¿it <var>.r.e</var>. </s> - <s xml:id="s1502" xml:space="preserve">Vnde ex <lb/>communi Å¿cientia rectangulum orizontale, <lb/>oculo <var>.o.</var> Å¿eipÅ¿um patefaciet in plano <var>.i.q.d.</var> Å¿e- - <pb o="121" rhead="DE PERSPECT." n="133" file="0133" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0133"/> - cundum figuram quadrilateram <var>.q.d.r.e</var>. </s> - <s xml:id="s1503" xml:space="preserve">Communis autem Å¿ectio Å¿uperficiei <var>.p.t.</var> <lb/>cum dicto plano, Å¿it <var>.i.x.</var> quæ <var>.i.x.</var> perpendicularis erit <var>.s.a.</var> Å¿uperficiei orizontali ex <lb/>19. lib. 11. quia <var>.p.t.</var> eÅ¿t etiam orizonti perpendicularis ex .18. eiuÅ¿dem, cum <var>.o.p.</var> ei-<lb/>dem perpendicularis exiÅ¿tat. </s> - <s xml:id="s1504" xml:space="preserve">Vnde <var>.i.x.</var> erit altitudo trianguli <var>.i.q.d.</var> & æqualis ipÅ¿i <var>.<lb/>o.p.</var> ex <ref id="ref-0019">.34. primi</ref>. </s> - <s xml:id="s1505" xml:space="preserve">Sit deinde <var>.o.l.</var> <reg norm="communis" type="context">cõmunis</reg> Å¿ectio Å¿uperficiei triangularis <var>.o.a.u.</var> <reg norm="cum" type="context">cÅ©</reg> <lb/>Å¿uperficie <var>.p.t.</var> quæ <var>.o.l.</var> Å¿ecando lineam <var>.e.r.</var> in puncto <var>.Z.</var> nobis oÅ¿tendet quantum di-<lb/>Å¿tare Å¿eu eminens eſſe debeat latus <var>.e.r.</var> in plano ab <var>.q.d.</var> medio ipÅ¿ius <var>.z.x</var>. </s> - <s xml:id="s1506" xml:space="preserve">Et quia <lb/>præſuppoÅ¿uimus <var>.p.l.</var> in eodem medio, inter <var>.u.s.</var> et <var>.a.n.</var> ideo <var>.x.q.</var> Ä™qualis erit <var>.x.d.</var> <lb/>& ex .4. lib. primi <var>.i.q.</var> ipÅ¿i <var>.i.d.</var> et <var>.e.r.</var> parallela ipÅ¿i <var>.q.d.</var> ex .6. lib. 11. cum ipÅ¿a quoque <lb/>Å¿it perpendicularis Å¿uperficiei <var>.p.t.</var> ex <ref id="ref-0020">.19. eiuÅ¿dem</ref>. </s> - <s xml:id="s1507" xml:space="preserve">HucuÅ¿que igitur in figura cor-<lb/>porea <var>.A.</var> prodeunt in lucem omnes cauſæ effectuum figuræ Å¿uperficialis <var>.A.</var> ideÅ¿t vn <lb/>de fiat, vt in ipÅ¿a figura Å¿uperficiali, triangulum <var>.o.p.l.</var> tale conÅ¿urgat, & quid Å¿ignifi-<lb/>cet <var>.o.</var> et <var>.o.p.</var> et <var>.p.l.</var> et <var>.o.l.</var> & quam ob cauÅ¿am tale quoque formetur triangulum <var>.i.<lb/>q.d.</var> atque in tantam altitudinem, quantam obtinet <var>.o.p.</var> & quid Å¿int latera <var>.i.q.</var> et <var>.i.<lb/>d.</var> & quare erigatur <var>.x.i.</var> parallela ipÅ¿i <var>.p.o.</var> ab eadem <var>.p.o.</var> tanto Å¿patio diÅ¿tans, & qua <lb/>ratione producatur à puncto <var>.Z.</var> ipÅ¿a <var>.Z.r.e.</var> parallela ipÅ¿i <var>.q.d</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0132-01" xlink:href="fig-0132-01a"> - <image file="0132-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0132-01"/> - </figure> - </div> - <p> - <s xml:id="s1508" xml:space="preserve">Nunc obÅ¿eruandum eÅ¿t, quòd Å¿i planum ipÅ¿ius <var>.i.q.d.</var> in figura corporea aliquan-<lb/>tulum inclinatum eſſet orizontem verÅ¿us, anguli <var>.i.q.d.</var> et <var>.i.d.q.</var> maiores exiÅ¿terent, <lb/>quà m cum idem eÅ¿t ipÅ¿i orizonti perpendiculare, quemadmodum clarè demonÅ¿tra-<lb/>tum fuit in .39. primi Vitelionis.</s> - </p> - <p> - <s xml:id="s1509" xml:space="preserve">Non igitur rectè fit Å¿i in figura Å¿uperficiali ducatur à puncto <var>.B.</var> parallela ipÅ¿i <var>.q.d.</var> <lb/>abÅ¿que maiori apertura angulorum <var>.i.q.d.</var> et <var>.i.d.q</var>.</s> - </p> - <figure position="here"> - <image file="0133-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0133-01"/> - <caption xml:space="preserve">SVPERFICIALIS.</caption> - </figure> - </div> - <div type="section" level="3" n="2"> - <head xml:id="head178" xml:space="preserve">CAP. II.</head> - <p> - <s xml:id="s1510" xml:space="preserve">CVM verò duæ præcedentes figuræ intellectæ erunt, facilè quoque erit intel-<lb/>ligere duas Å¿ubÅ¿equentes <var>.B.B.</var> in corporea quarum <var>.p.l.</var> extra lineas <var>.u.s.</var> et <var>.a.n.</var> <lb/>reperitur, vbi enim aduertendum erit oportere Å¿umere Å¿emper <var>.p.x.</var> figuræ Å¿uperfi-<lb/>cialis æqualem ei, quæ eÅ¿t corporeæ, & eidem Å¿uperficiali, adiungere <var>.x.d.</var> æqualem <lb/>ei, quæ eÅ¿t corporeæ, & compoÅ¿ito <var>.p.d.</var> ex dictis duabus lineis, in figura Å¿uperficiali, <lb/>addere <var>.d.q.</var> æqualem ei, quæ eÅ¿t figuræ corporeæ, deinde accipere punctum <var>.l.</var> in fu-<lb/>perficiali - <pb o="122" rhead="IO. BAPT. BENED." n="134" file="0134" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0134"/> - perficiali, ita diÅ¿tans à <var>.p.</var> vt in corporea reperitur. </s> - <s xml:id="s1511" xml:space="preserve">Po-<lb/> - <anchor type="figure" xlink:label="fig-0134-01a" xlink:href="fig-0134-01"/> - Å¿teà . <var>x.i.</var> erigetur æqualis lineæ <var>.p.o.</var> & Å¿uis terminis <lb/>concludetur triangulum <var>.i.q.d.</var> & id quod remanet. <lb/></s> - <s xml:id="s1512" xml:space="preserve">Vnde Å¿i longius diÅ¿erendo progrediare, patebit ex <num value="4">.<lb/>4.</num> primi <var>.i.d.</var> Å¿uperficialem, futuram æqualem <var>.i.d.</var> <lb/>corporeæ. </s> - <s xml:id="s1513" xml:space="preserve">Idem dico de <var>.i.q.</var> & de reliquis.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0134-01" xlink:href="fig-0134-01a"> - <image file="0134-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0134-01"/> - <caption xml:space="preserve">CORPOREA.</caption> - </figure> - </div> - <figure position="here"> - <image file="0134-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0134-02"/> - <caption xml:space="preserve">SVPERFICIALIS.</caption> - </figure> - </div> - <div type="section" level="3" n="3"> - <head xml:id="head179" xml:space="preserve">CAP. III.</head> - <p> - <s xml:id="s1514" xml:space="preserve"><emph style="sc">MOdvs</emph> hic, proprius eÅ¿t, & vniuerÅ¿alis, licet in figura Å¿uperficiali <var>.A.</var> Å¿upe-<lb/>rius poÅ¿ita, Å¿ecundum communem antiquorum conÅ¿uetudinem exemplum <lb/>dederim, effectus enim idem eÅ¿t. </s> - <s xml:id="s1515" xml:space="preserve">Sed Å¿i quis vellet conÅ¿iderare dictam figuram Å¿u-<lb/>perficialem <var>.A.</var> Å¿ecundum eum modum, quem de figura <var>.B.</var> Å¿uperficiali pręſcripÅ¿i, id <lb/>poterit in Å¿ubÅ¿cripta figura <var>.C.</var> Å¿peculari in hunc modum. </s> - <s xml:id="s1516" xml:space="preserve">Accipiet enim <var>.p.x.</var> Å¿uper <lb/>ficialem, æqualem corporeæ, & quia in ipÅ¿a corporea <var>.A.</var> Å¿uppoÅ¿ita fuit linea <var>.p.l.</var> <lb/>ideÅ¿t punctum <var>.x.</var> inter duas <var>.u.s.</var> et <var>.a.n.</var> Å¿ecabimus <var>.p.x.</var> Å¿uperficialem in puncto <var>.d.</var> ita, <lb/>vt <var>.d.x.</var> Å¿uperficialis, æqualis Å¿it corporeæ, & ipÅ¿i Å¿uperficiali <var>.p.x.</var> addetur <var>.x.q.</var> æqualis <lb/>corporeæ. </s> - <s xml:id="s1517" xml:space="preserve">vnde <var>.q.d.</var> Å¿uperfi-<lb/>cialis æqualis erit corporeÄ™,<lb/> - <anchor type="figure" xlink:label="fig-0134-03a" xlink:href="fig-0134-03"/> - et <var>.p.x.</var> Å¿uperficiali addetur <var>.<lb/>x.l.</var> æqualis corporeæ. </s> - <s xml:id="s1518" xml:space="preserve">De <lb/>ijs poÅ¿tea quæ dicenda Å¿uper-<lb/>Å¿unt, iam Å¿atis Å¿uperqÌue di-<lb/>ximus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0134-03" xlink:href="fig-0134-03a"> - <image file="0134-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0134-03"/> - <caption xml:space="preserve">SVPERFICIALIS.</caption> - </figure> - </div> - <p> - <s xml:id="s1519" xml:space="preserve">Quamobrem, punctum <var>.<lb/>x.</var> aut intra, aut extra lineam <var>.<lb/>q.d.</var> veniat, hunc modum Å¿e-<lb/>quentes, in errorem non in-<lb/>cidemus, imò efficietur qua -<lb/>drilaterum <var>.q.r.</var> Å¿uperficiale, <lb/>Å¿imile, & æquale corporeo.</s> - </p> - </div> - <div type="section" level="3" n="4"> - <head xml:id="head180" xml:space="preserve">CAP. IIII.</head> - <p> - <s xml:id="s1520" xml:space="preserve"><emph style="sc">PVnctvm</emph> verò <var>.i.</var> (quod verum eÅ¿t punctum perÅ¿pectiuæ, vt practici dicere Å¿o <lb/>lent) quid Å¿it, hac via & ratione Å¿ub noÅ¿tram cognitionem cadit: </s> - <s xml:id="s1521" xml:space="preserve">quòd nihil - <pb o="123" rhead="DE PERSPECT." n="135" file="0135" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0135"/> - aliud eÅ¿t, quà m punctum, varijs Å¿ectionibus commune, & huiuÅ¿modi punctum, ocu <lb/>lus non eÅ¿t, quemadmodum multi Pictores, Sculptores, Architecti, & PerÅ¿pectiui <lb/>ignari, ipÅ¿um punctum, oculum appellando, falsò crediderunt, quaÅ¿i <reg norm="punctum" type="context">punctÅ©</reg> <var>.i.</var> per-<lb/>Å¿pectiuæ oculus eſſet.</s> - </p> - <p> - <s xml:id="s1522" xml:space="preserve">In Å¿upradictis igitur figuris manifeÅ¿te eluceÅ¿cit cauÅ¿a diminutionis obiectorum, <lb/>& altitudinis trianguli æqualis ei, quæ eÅ¿t oculi à plano orizontali, vt etiam diÅ¿tantiÄ™ <var>.<lb/>p.l.p.x.</var> & cuiuÅ¿uis tandem rei. </s> - <s xml:id="s1523" xml:space="preserve">Sed vt huius effectus Å¿cientia magis in vniuerÅ¿um pa-<lb/>retur. </s> - <s xml:id="s1524" xml:space="preserve">Volo duas hic Å¿ubÅ¿criptas figuras <var>.D.</var> corpoream, & <var>.D.</var> Å¿uperficialem à vo-<lb/>bis conÅ¿iderari, in quarum corporea, linea <var>.p.l.</var> Å¿it extra duas <var>.u.s.</var> et <var>.a.n.</var> vt in figu-<lb/>ra <var>.B.</var> locata, ita tamen vt planum trianguli <var>.i.q.d.</var> diÅ¿iunctum Å¿it à rectangulo Å¿uper-<lb/>ficiali, ideÅ¿t, vt Å¿eparatum exiÅ¿tat à linea <var>.q.d.</var> latere ipÅ¿ius rectanguli, & Å¿it etiam obli <lb/>quum, reÅ¿pectu ipÅ¿ius rectanguli, ideÅ¿t vt communis Å¿ectio dicti plani cum Å¿uperficie <lb/><var>a.s.</var> orizontalis ipÅ¿i <var>.u.a.</var> parallela non Å¿it, Å¿ed Å¿it obliqua, Å¿i tamen idem planuni per-<lb/>pendiculare dictæ Å¿uperficiei orizontali <var>.a.s.</var> erit: </s> - <s xml:id="s1525" xml:space="preserve">& dicta communis Å¿ectio exprimaÂ<lb/>tur characteribus .q.ω.α.d.x. nunc in figura corporea habebimus figuram <var>.e.r.c.m.</var> <lb/>in plano, quod viÅ¿ualem pyramidem Å¿ecat, medio cuius figuræ <var>.e.r.c.m.</var> oculus po-<lb/>Å¿itus in <var>.o.</var> rectangulum orizontale conÅ¿picit. </s> - <s xml:id="s1526" xml:space="preserve">Volentes vero nunc in figura <var>.D.</var> Å¿uper-<lb/>ficiali eam deÅ¿cribere, faciem us <var>.p.x.</var> Å¿uperficialem, æqualem corporeæ, eiqÌue <lb/>addemus <var>.x.l.</var> æqualem corporeæ, aut Å¿umemus <var>.p.l.</var> eidem corporeæ <reg norm="aequa- lem" type="simple">Ä™qua-<lb/>lem</reg>, quam Å¿ecabimus in puncto <var>.x.</var> eodem planè modo, quo corporea reperi-<lb/>tur diuiÅ¿a; </s> - <s xml:id="s1527" xml:space="preserve">erigemus deinde <var>.p.o.</var> et <var>.x.i.</var> æquales corporeis. </s> - <s xml:id="s1528" xml:space="preserve">Secabimus deinde <var>.x.q.</var> <lb/>æqualem corporeæ, & ducemus <var>.q.i.</var> et <var>.l.o.</var> vnde habebimus triangulos <var>.o.p.l.</var> et <var>.i.x.<lb/>q.</var> Å¿imiles & æquales corporeis ex .4. primi Eucli. </s> - <s xml:id="s1529" xml:space="preserve">Secabimus deinde <var>.q.x.</var> in pun-<lb/>cto <var>.d.</var> eadem ratione, qua Å¿ecta fuit corporea, & ducemus lineam <var>.d.i.</var> vnde habebi-<lb/>mus triangulos <var>.i.d.q.</var> et <var>.i.d.x.</var> Å¿imiles corporeis. </s> - <s xml:id="s1530" xml:space="preserve">& mediante triangulo <var>.i.q.d.</var> hu-<lb/> - <anchor type="figure" xlink:label="fig-0135-01a" xlink:href="fig-0135-01"/> - cuÅ¿que habebimus Å¿itus duorum laterum figurÄ™ <lb/>rectanguli degradati, ideÅ¿t Å¿itus ipÅ¿ius <var>.e.m.</var> et <var>.r.<lb/>c.</var> etiam Å¿i adhuc neÅ¿ciatur in qua parte ipÅ¿ius <var>.i.<lb/>q.</var> & ipÅ¿ius <var>.i.d.</var> eſſe <reg norm="debeant" type="context">debeãt</reg>. </s> - <s xml:id="s1531" xml:space="preserve">Quod Å¿i Å¿cire volue <lb/>rimus <reg norm="Å¿ecabitur" type="simple">Å¿ecabit̃</reg>. <var>p.l.</var> in <reg norm="puncto" type="context">pÅ©cto</reg> <var>.g.</var> Å¿imilis corporeæ, <lb/>Å¿i in ipÅ¿a tamen corporea prius protraxerimus <lb/>lineam <var>.q.d.</var> latus rectanguli vÅ¿que ad <var>.p.l.</var> in pun <lb/>cto <var>.g</var>. </s> - <s xml:id="s1532" xml:space="preserve">Ducetur deinde linea <var>.o.g.</var> Å¿uperficialis, <lb/>quæ Å¿ecabit lineam <var>.i.x.</var> in puncto <var>.f.</var> linea vero <var>.<lb/>o.l.</var> in puncto <var>.z.</var> punctis Å¿itis in <var>.i.x.</var> Å¿uperficiali, <lb/>prÄ™cisè vt in corporea, <reg norm="quemadmodum" type="context">quemadmodÅ©</reg> quilibet <lb/>ex Å¿e facilè cognoÅ¿cere poteÅ¿t. </s> - <s xml:id="s1533" xml:space="preserve">Deinde in cor <lb/>porea, in Å¿uperficie orizontali ducatur <var>.p.q.</var> et <lb/> - <anchor type="figure" xlink:label="fig-0135-02a" xlink:href="fig-0135-02"/> - <pb o="124" rhead="IO. BAPT. BENED." n="136" file="0136" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0136"/> - <var>.p.u.</var> & imaginemur <var>.o.q.</var> in Å¿uperficie <var>.t.s.</var> vnde trianguli <var>.o.p.q.</var> et <var>.o.p.u.</var> erunt perpen <lb/>diculares orizonti ex <ref id="ref-0021">.18. lib. 11.</ref> et <var>.ω.m.</var> et. α. e. communes Å¿ectiones dictorum <reg norm="duorum" type="context">duorÅ©</reg> <lb/>triangulorum cum plano trianguli <var>.i.q.x.</var> ipÅ¿i quoque plano ex .19. eiuÅ¿dem lib. erunt <lb/>perpendiculares. </s> - <s xml:id="s1534" xml:space="preserve">Nuncautem Å¿ecetur <var>.q.x.</var> Å¿uperficialis in punctis <var>.ω.</var> et .α. eadem ra-<lb/>tione; </s> - <s xml:id="s1535" xml:space="preserve">qua corporea Å¿ecta Å¿uit à duabus <var>.p.q.</var> et <var>.p.u.</var> à quibus punctis <var>.ω.</var> et. α. Å¿uperficia <lb/>libus ductæ Å¿int duæ <var>ω.m.</var> et .α.e. perpendiculares vÅ¿que ad latus <var>.i.q.</var> in punctis <var>.m.</var> et <var>.<lb/>e.</var> quÄ™ Å¿itum habebunt in <var>.i.q.</var> Å¿uperficiali prÄ™cisè, vt in corporea, ex .26. primi, du-<lb/>cendo deinde in Å¿uperficiali duas <var>.m.f.</var> et <var>.e.Z.</var> eæ æquales erunt corporeis ex .4. pri-<lb/>mi, & Å¿ic anguli <var>.i.e.z.</var> et <var>.i.m.f.</var> & eÄ™ duÄ™ lineÄ™ <var>.e.z.</var> et <var>.m.f.</var> fectÄ™ erunt à linea <var>.i.d.</var> in duo<lb/> - <anchor type="figure" xlink:label="fig-0136-01a" xlink:href="fig-0136-01"/> - bus punctis <var>.r.</var> et <var>.c.</var> vnde <var>.e.r.</var> et <var>.m.c.</var> æquales <lb/>erunt corporeis ex .26. primi, Å¿ed ita quo-<lb/>que Å¿e habent duÄ™ <var>.e.m.</var> et <var>.r.c.</var> Å¿i verum eÅ¿t <reg norm="quod" type="simple">ê™</reg> dif <lb/>ferentiÄ™ rerum æqualium Å¿int adinuicem etiam <lb/>æquales. </s> - <s xml:id="s1536" xml:space="preserve">Hac ratione igitur habebimus figu-<lb/>ram quadrilateram <var>.m.e.r.c.</var> Å¿uperficialem om <lb/>ninò Å¿inlilem, & Ä™qualem corporeæ. </s> - <s xml:id="s1537" xml:space="preserve">Is tamen <lb/>modus prolixus eÅ¿t, & arduus, quam ob cau-<lb/>Å¿am neque ego vnquam <reg norm="eum" type="context">eÅ©</reg> viui accommo-<lb/>darem, neque alijs, vt eodem vterentur Å¿ua-<lb/>derem.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0135-01" xlink:href="fig-0135-01a"> - <image file="0135-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0135-01"/> - <caption xml:space="preserve">CORPOREA.</caption> - </figure> - <figure xlink:label="fig-0135-02" xlink:href="fig-0135-02a"> - <image file="0135-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0135-02"/> - <caption xml:space="preserve">SVPERFICIALIS.</caption> - </figure> - <figure xlink:label="fig-0136-01" xlink:href="fig-0136-01a"> - <image file="0136-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0136-01"/> - <caption xml:space="preserve">CORPOREA.</caption> - </figure> - </div> - <figure position="here"> - <image file="0136-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0136-02"/> - <caption xml:space="preserve">SVPERFICIALIS.</caption> - </figure> - </div> - <div type="section" level="3" n="5"> - <head xml:id="head181" xml:space="preserve">CAP.V.</head> - <p> - <s xml:id="s1538" xml:space="preserve"><emph style="sc">ESt</emph> igitur Å¿ciendum, quòd qui Å¿ciuerit vnum Å¿olum punctum locare in perÅ¿pe <lb/>ctiua, eo modo quem nunc proponam, facilè quoque Å¿ciet Å¿upra quoduis <reg norm="planum" type="context">planÅ©</reg> <lb/>(quod tamen Å¿it perpendiculare orizonti) quamlibet rem locare. </s> - <s xml:id="s1539" xml:space="preserve">Quam ob cauÅ¿am <lb/>imaginemur hic Å¿ubÅ¿criptas duas figuras <var>.E.</var> <reg norm="corpoream" type="context">corporeã</reg>, & E. Å¿uperficialem, & in qua-<lb/>drilatero rectangulo orizontali <var>.a.u.q.d.</var> imaginemur eſſe punctum <var>.b.</var> quodlibet col-<lb/>locandum in aliquo plano perpendiculari orizonti locato, quemadmodum Å¿uppo-<lb/>nebatur in figura <var>.A.</var> corporea. </s> - <s xml:id="s1540" xml:space="preserve">Imaginemur ergo in ipÅ¿a figura <var>.E.</var> corporea radi-<lb/>um viÅ¿ualem <var>.o.b.</var> qui Å¿ectus Å¿it à noÅ¿tro plano in <var>.k.</var> quod quidem <var>.k.</var> quærendum eÅ¿t <lb/>in triangulo <var>.i.q.d.</var> ipÅ¿ius plani. </s> - <s xml:id="s1541" xml:space="preserve">Volo ob hanc igitur rem, vt à puncto <var>.b.</var> in figura <var>.E.</var> <lb/>Å¿uperficiali ducatur <var>.b.c.</var> ad rectos <reg norm="cum" type="context">cÅ©</reg> <var>.q.d.</var> & à puncto <var>.c.</var> ad <var>.i.</var> ducatur linea <var>.c.i.</var> et <var>.b.m.</var> <lb/>parallela ipÅ¿i <var>.q.d.</var> quÄ™ ab ipÅ¿a <var>.x.l.</var> in puncto <var>.m.</var> erit diuiÅ¿a, & hÄ™c <var>.x.m.</var> è directo con-<lb/>iuncta cum <var>.p.x.</var> ducatur <var>.o.m.</var> quæ ab <var>.i.x.</var> Å¿ecta erit in puncto <var>.f.</var> à quo ducendo dein-<lb/>de <var>.f.g.h.</var> parallela <var>.q.d.</var> ab <var>.i.c.</var> in puncto <var>.K.</var> erit diuiÅ¿a. </s> - <s xml:id="s1542" xml:space="preserve">Atque id erit quod nobis <lb/>inquirendum propoÅ¿ueramus.</s> - </p> - <pb o="125" rhead="DE PERSPECT." n="137" file="0137" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0137"/> - <p> - <s xml:id="s1543" xml:space="preserve">Ad cuius rei <reg norm="Å¿peculationem" type="context">Å¿peculationẽ</reg>, imaginatione con<lb/> - <anchor type="figure" xlink:label="fig-0137-01a" xlink:href="fig-0137-01"/> - cipiamus lineam <var>.b.c.</var> corpoream, protractam eÅ¿ <lb/>Å¿e vÅ¿que ad <var>.y.</var> lineæ <var>.s.n.</var> & imaginatione Å¿it com <lb/><reg norm="præhenÅ¿a" type="context">præhẽſa</reg> linea <var>.y.o.</var> et <var>.b.</var> R<unsure/>. parallela eidem, ideo <lb/>ob rationes iam dictas de figura <var>.A.</var> hæ tres li-<lb/>neæ <var>.o.y</var>: <var>i.c</var>: et. </s> - <s xml:id="s1544" xml:space="preserve">R<unsure/> <var>.b.</var> Å¿imul cum linea <var>.o.b.</var> erunt <lb/>in vna eademqÌue Å¿uperficie plana, quam cha-<lb/>racteribus <var>.y.</var> R<unsure/>. notemus .et <var>.i.c.</var> eius erit Å¿e-<lb/>ctio communis cum plano, in quo quæritur <reg norm="pun- ctum" type="context">pÅ©-<lb/>ctum</reg>, et <var>.f.k.</var> ipÅ¿ius plani cum triangulo <var>.o.b.m.</var> <lb/>erit Å¿ectio communis, & parallela ipÅ¿i <var>.q.d.</var> ex <ref id="ref-0022">.<lb/>6. lib. 11.</ref> quia <var>.k.f.</var> perpendicularis eÅ¿t Å¿uperfi-<lb/>ciei <var>.p.t.</var> ex .19. eiuÅ¿dem cum triangulus <var>.o.<lb/>b.m.</var> eidem Å¿uperficiei <var>.p.t.</var> ex .18. eiuÅ¿dem <lb/>perpendicularis exiÅ¿tat. </s> - <s xml:id="s1545" xml:space="preserve">Vnde perÅ¿picuè pa-<lb/>tet ratio quare protracta<unsure/> Å¿it parallela <var>.b.c.</var> et <lb/>quare ducta Å¿it <var>.i.c.</var> et coniuncta <var>.x.m.</var> cum <var>.x.<lb/>p.</var> directè, & quare ducta Å¿it <var>.o.m.</var> et <var>.f.k</var>. </s> - <s xml:id="s1546" xml:space="preserve">Lau-<lb/>do igitur vt Å¿emper præſupponatur <var>.p.x.</var> perpen <lb/>dicularis baÅ¿i ipÅ¿ius plani & præſupponatur, (vt <lb/>rem totam vnò verbo complectar) Å¿uperficies <var>.<lb/>p.t.</var> perpendicularis plano, & orizonti. </s> - <s xml:id="s1547" xml:space="preserve">Quod <lb/>reliquum eÅ¿t, neceſſariv<unsure/>m non eÅ¿t, niÅ¿i ad Å¿pe-<lb/>culandum. </s> - <s xml:id="s1548" xml:space="preserve">Neceſſariæ ergo non Å¿unt aliæli-<lb/>neæ, quà m.p.x: <var>p.o.x.i</var>: <var>b.c</var>: et <var>.x.m.</var> è dire-<lb/>cto coniuncta cum <var>.p.x.</var> (quæ <var>.x.m.</var> coniuncta <lb/> - <anchor type="figure" xlink:label="fig-0137-02a" xlink:href="fig-0137-02"/> - <pb o="126" rhead="IO. BAPT. BENED." n="138" file="0138" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0138"/> - æqualis Å¿it ipÅ¿i <var>.b.c.</var>) <lb/> - <anchor type="figure" xlink:label="fig-0138-01a" xlink:href="fig-0138-01"/> - <var>o.m.</var> etiam <var>.i.c</var>: et <var>.f.<lb/>k.</var> vt in figura <var>.F.</var> cla <lb/>riſſimè patet. </s> - <s xml:id="s1549" xml:space="preserve">Alias <lb/><reg norm="autem" type="context">autẽ</reg> multas lineas in <lb/>alijs figuris non <reg norm="aliam" type="context">aliã</reg> <lb/>ob <reg norm="camm" type="context">cãm</reg> duxi, <reg norm="quam" type="context">quã</reg> ad <lb/>facilius <reg norm="eruendas" type="context">eruẽdas</reg> è te-<lb/>nebris ignorantiæ, & <lb/>in cognitionis lucem <lb/>proferendas horum <lb/>effectuum cauÅ¿as, vt <lb/>dixi.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0137-01" xlink:href="fig-0137-01a"> - <image file="0137-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0137-01"/> - <caption xml:space="preserve">CORPOREA.</caption> - </figure> - <figure xlink:label="fig-0137-02" xlink:href="fig-0137-02a"> - <image file="0137-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0137-02"/> - <caption xml:space="preserve">SVPERFICIALIS.</caption> - </figure> - <figure xlink:label="fig-0138-01" xlink:href="fig-0138-01a"> - <image file="0138-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0138-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="6"> - <head xml:id="head182" xml:space="preserve">CAP. VI.</head> - <p> - <s xml:id="s1550" xml:space="preserve"><emph style="sc">SEd</emph> vtlocum altitudinis, in noÅ¿tro plano perpendiculari orizonti, & ita <reg norm="locatum" type="context">locatÅ©</reg><unsure/>, <lb/>vt poÅ¿tremo diximus, inueniamus; </s> - <s xml:id="s1551" xml:space="preserve">duas hîc Å¿ubÅ¿criptas figuras conÅ¿iderabimus <var>.<lb/>G.</var> corpoream, & G. Å¿uperficialem, Å¿imiles duabus <var>.E.E.</var> proximè præcedentibus, <lb/>in quarum corporea Å¿it linea <var>.b.M.</var> altitudinis perpendicularis orizonti. </s> - <s xml:id="s1552" xml:space="preserve">Quare Å¿i <lb/>deÅ¿iderabis inuenire in noÅ¿tro plano Å¿itum puncti <var>.M.</var> ideÅ¿t punctum radij <var>.o.M.</var> vi-<lb/>Å¿ualis in quo ipÅ¿e radius à plano eÅ¿t diuiÅ¿us, quod Å¿it <var>.R.</var> quamuis extra <reg norm="triangulum" type="context">triangulÅ©</reg> <lb/><var>i.q.d.</var> tibi imaginatione confige ductam eſſe lineam <var>.p.b.</var> quæ erit Å¿ectio commu-<lb/>nis orizontis cum Å¿uperficie <var>.o.p.b.M.</var> quæ Å¿uperficies erit perpendicularis ipÅ¿i ori-<lb/>zonti ex .18. lib 11. </s> - <s xml:id="s1553" xml:space="preserve">Quòd autemnon minus <var>.o.p.</var> quà m.M.b. Å¿it in vna eademqÌue <lb/>Å¿uperficie dubitandum non eÅ¿t, quia Å¿i imaginabimur ductam eſſe lineam <var>.p.M.</var> ha <lb/>bebimus triangulum <var>.o.p.b.</var> cum triangulo <var>.M.b.p.</var> communibus partibus in vna ea-<lb/>demqÌue Å¿uperficie conÅ¿tantem, vt triangulum quoque <var>.o.p.M.</var> cum triangulo <var>M.b.</var> <lb/>o & triangulum <var>.o.p.b.</var> cum triangulo <var>.o.p.M.</var> & triangulum <var>.M.b.p.</var> cum triangulo <var>.<lb/>M.b.o</var>. </s> - <s xml:id="s1554" xml:space="preserve">Vnde cum quilibet triangulus in vnica tantum Å¿uperficie Å¿it ex .2. lib. 11. Å¿e-<lb/>quetur Å¿uperficiem <var>.o.p.b.M.</var> planam eſſe, & vnicam, cuius communis Å¿ectio cum no-<lb/>Å¿tro plano Å¿it. θ.K.R. quæ perpendicularis orizonti exiÅ¿tet ex .19. lib. 11. eritqÌue pa-<lb/>rallela ipÅ¿i <var>.i.x.</var> ex .6. eiuÅ¿dem. </s> - <s xml:id="s1555" xml:space="preserve">Imaginare nunc erectam eſſe <var>.m.T.</var> æqualem ipÅ¿i <var>.<lb/>b.M.</var> orizonti perpendicularem, quæ extenÅ¿a erit in Å¿uperficie <var>.p.t.</var> quod ex Å¿e ad <lb/>conÅ¿iderandum admodum facilè, clarumqÌue exiÅ¿tit, reducendo ad impoſſibilia <lb/>quemlibet hæc negare volentem. </s> - <s xml:id="s1556" xml:space="preserve">Imaginemur quoque ductam eſſe lineam <var>.M.<lb/>T.</var> quæ <var>.b.m.</var> ex .33. primi erit parallela, quia <var>.m.T.</var> Ä™qualis <var>.b.M.</var> parallela eÅ¿t <lb/>ipÅ¿i <var>.b.M.</var> ex .6. lib. 11. præter hæc <var>.b.m.</var> parallela eÅ¿t ipÅ¿i <var>.q.d.</var> quia Å¿ic fuit ducta <lb/>Å¿uperius, vnde <var>.M.T.</var> parallela erit ipÅ¿i <var>.q.d.</var> ex .9. vndecimi, & obid perpendi-<lb/>cularis erit Å¿uperficiei <var>.b.t.</var> ex .8. eiuÅ¿dem. </s> - <s xml:id="s1557" xml:space="preserve">Nunc Å¿it <var>.R.V.</var> communis Å¿ectio trian-<lb/>guli <var>.o.M.T.</var> cum noÅ¿tro plano, vnde <var>.R.V.</var> perpendicularis erit Å¿uperficiei <var>.p.t.</var> <lb/>ex .19. lib. 11. quam ob cauÅ¿am parallela erit ipÅ¿i <var>.q.d.</var> ex .6. aut ex .9. eiuÅ¿dem <lb/>quia ex .6. dicta, parallela eÅ¿t ipÅ¿i <var>.M.T</var>. </s> - <s xml:id="s1558" xml:space="preserve">AtÅ¿i <var>.R.V.</var> parallela eÅ¿t ipÅ¿i <var>.q.d.</var> <lb/>etiam <var>.f.K.</var> probatum iam fuit parallelam eſſe eidem, ergo <var>.R.V.</var> parallela erit <lb/>ipÅ¿i <var>.K.f.</var> ex .30. primi, </s> - <s xml:id="s1559" xml:space="preserve">Vnde ex .34. æqualis erit ipÅ¿i <var>.K.f</var>. </s> - <s xml:id="s1560" xml:space="preserve">Accedamus nunc <lb/>ad <reg norm="figuram" type="context">figurã</reg> <var>.G.</var> <reg norm="extructam" type="context">extructã</reg> Å¿upra figuram <var>.E.</var> Å¿uperficialem, & erigamus <var>.m.T.</var> perpendi-<lb/>cularem ipÅ¿i <var>.m.p.</var> Å¿ed æqualem perfectæ altitudini, & ducamus <var>.T.o.</var> vt Å¿ecet li-<lb/>neam <var>.i.x.</var> in puncto <var>.V.</var> ab ipÅ¿o ducentes <var>.V.R.</var> parallelam ipÅ¿i <var>.q.d.</var> ducendo de- - <pb o="127" rhead="DE PERSPECT." n="139" file="0139" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0139"/> - inde <var>.k.R.</var> parallelam ipÅ¿i <var>.i.a.</var> habebimus altitudinem <var>.k.R.</var> quam quærebamus in <lb/>noÅ¿tro plano. </s> - <s xml:id="s1561" xml:space="preserve">Quod cum Å¿ui natura clarum euadat, laborem ratiocinandi de eo, <lb/> - <anchor type="figure" xlink:label="fig-0139-01a" xlink:href="fig-0139-01"/> - cuilibet vel mediocriter in præclariſſima hac Å¿cien-<lb/>tia erudito relinquo. </s> - <s xml:id="s1562" xml:space="preserve">ideÅ¿t, vt probetur <var>.k.R.</var> <reg norm="Å¿u- perficialem" type="context">Å¿u-<lb/>perficialẽ</reg>, <reg norm="æqualem" type="context">æqualẽ</reg> eſſe corporeæ. </s> - <s xml:id="s1563" xml:space="preserve">Sed <reg norm="tollendo" type="context">tollẽdo</reg> <reg norm="Å¿uper- fluitatem" type="context">Å¿uper-<lb/>fluitatẽ</reg> linearum, & hoc <reg norm="accommodantes" type="context">accõmodantes</reg> vt in figura <var>.<lb/>F.</var> diligenter conÅ¿ideretur figura <var>.H</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0139-01" xlink:href="fig-0139-01a"> - <image file="0139-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0139-01"/> - <caption xml:space="preserve">CORPOREA.</caption> - </figure> - </div> - <figure position="here"> - <image file="0139-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0139-02"/> - <caption xml:space="preserve">SVPERFICIALIS.</caption> - </figure> - <figure position="here"> - <image file="0139-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0139-03"/> - <caption xml:space="preserve">SVPERFICIALIS</caption> - </figure> - <pb o="128" rhead="IO. BAPT. BENED." n="140" file="0140" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0140"/> - </div> - <div type="section" level="3" n="7"> - <head xml:id="head183" xml:space="preserve">CAP. VII.</head> - <p> - <s xml:id="s1564" xml:space="preserve">ALiarn tamen inueni viam breuiorem vt in figura <var>.H.H.</var> in qua Å¿it punctus <var>.<lb/>b.</var> perfecti, & <var>.k.</var> degradati plani. </s> - <s xml:id="s1565" xml:space="preserve">Nunc ducatur <var>.b.c.s.</var> ad rectos cum <var>.<lb/>p.m.</var> indefinitè, quæ quidem abÅ¿cindatur in puncto <var>.s.</var> ita quòd <var>.c.s.</var> æqualis Å¿it alti <lb/>tudini perfectæ, deinde coniungatur rectà . s. cum <var>.i</var>. </s> - <s xml:id="s1566" xml:space="preserve">Tunc Å¿i ab <var>.k.</var> vÅ¿que ad <reg norm="protractam" type="context">protractã</reg> <lb/><var>i.s.</var> ducta fuerit <var>.k.R.</var> parallela li-<lb/> - <anchor type="figure" xlink:label="fig-0140-01a" xlink:href="fig-0140-01"/> - neÄ™ <var>.c.s.</var> hæc <var>.R.k.</var> erit altitudo <lb/>quæſita Å¿eu degradata.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0140-01" xlink:href="fig-0140-01a"> - <image file="0140-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0140-01"/> - </figure> - </div> - <p> - <s xml:id="s1567" xml:space="preserve">Quod ita probo. </s> - <s xml:id="s1568" xml:space="preserve">Iam nulli du <lb/>bium eÅ¿t quin <var>.f.V.</var> Å¿it æqualis alti-<lb/>tudini quęſitæ Å¿eu degradatÄ™, quo <lb/><reg norm="tieÅ¿cunque" type="simple">tieÅ¿cunq;</reg> ergo <reg norm="probauerimus" type="simple">ê“bauerimus</reg> <var>.k.R.</var> <lb/>æqualem eſſe lineæ <var>.f.V.</var> habebi-<lb/>mus propoÅ¿itum. </s> - <s xml:id="s1569" xml:space="preserve">Quare certum <lb/>nobis erit eandem proportionem <lb/>eſſe lineæ <var>.c.s.</var> ad <var>.k.R.</var> quam <var>.c.i.</var> ad <lb/><var>k.i.</var> et <var>.c.i.</var> ad <var>.k.i.</var> vt <var>.x.i.</var> ad <var>.f.i.</var> et <var>.x.<lb/>i.</var> ad <var>.f.i.</var> vt <var>.m.o.</var> ad <var>.f.o.</var> et <var>.m.o.</var> ad <var>.<lb/>f.o.</var> vt <var>.m.T.</var> ad <var>.f.V.</var> ex Å¿imilitudine <lb/>triangulorum. </s> - <s xml:id="s1570" xml:space="preserve">Ergo <var>.m.T.</var> ad <var>.f.V.</var> <lb/>erit vt <var>.c.s.</var> ad <var>.k.R.</var> ex .11. quinti, <lb/>Å¿ed <var>.c.s.</var> Å¿umpta fuit æqualis <var>.m.T</var>. <lb/></s> - <s xml:id="s1571" xml:space="preserve">quare <var>.c.s.</var> ad <var>.f.V.</var> erit, vt <var>.m.T.</var> ad <lb/><reg norm="eandem" type="context context">eãdẽ</reg> <var>.f.V.</var> ex .7. <reg norm="quinti" type="simple">ê—nti</reg>, & ex .11. eiuÅ¿-<lb/>dem <var>.c.s.</var> ad <var>.f.V.</var> erit vt <var>.c.s.</var> ad <var>.k.R.</var> <lb/>quapropter ex .9. eiuÅ¿dem <var>.k.R.</var> æqualis erit <var>.f.V</var>.</s> - </p> - </div> - <div type="section" level="3" n="8"> - <head xml:id="head184" xml:space="preserve">CAP. VIII.</head> - <p> - <s xml:id="s1572" xml:space="preserve">MOdus ab antiquis philoÅ¿ophis obÅ¿eruatus, eÅ¿t etiam vtilis, <reg norm="compendioÅ¿aque" type="simple">compendioÅ¿aqÌ;</reg> via <lb/>progreditur, cuius Å¿peculationem, in Å¿ubÅ¿cripta figura, quadam ex parte <reg norm="Å¿ecun- dum" type="context">Å¿ecÅ©-<lb/>dum</reg> morem antiquum, quadam etiam ex parte Å¿ecundum ingenij mei vires <reg norm="conÅ¿tru- cta" type="context">cõſtru-<lb/>cta</reg>, cognoÅ¿cemus. </s> - <s xml:id="s1573" xml:space="preserve">In qua ego diuiÅ¿i <var>.x.i.</var> in puncto <var>.s.</var> ab <var>.x.</var> ita eleuato, quanta eÅ¿t <lb/> - <anchor type="figure" xlink:label="fig-0140-02a" xlink:href="fig-0140-02"/> - <pb o="129" rhead="DE PERSPECT." n="141" file="0141" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0141"/> - vera altitudo ipÅ¿ius <var>.M.T.</var> et <var>I.s.</var> duxi Å¿upponendo eſſe <var>.I.</var> <reg norm="punctum" type="context">punctÅ©</reg> pÅ¿pectiuæ <reg norm="Å¿ecundum" type="context">Å¿ecundÅ©</reg> <lb/>antiquos, ideÅ¿t angulum Å¿upremum trianguli antiquorum à punctoqÌue <var>.k.</var> meo duxi <lb/><var>k.f.</var> parallelam ipÅ¿i <var>.c.m.p.</var> vÅ¿que ad <var>.i.x.</var> in puncto <var>.f.</var> & à puncto à communi ipÅ¿is <var>.k.f.</var> <lb/>et <var>.i.x.</var> vÅ¿que ad <var>.I.s.</var> duxi quoque <var>.A.B.</var> parallelam ipÅ¿i <var>.i.x.</var> atque hæc omnia ex more <lb/>antiquo præſtiti.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0140-02" xlink:href="fig-0140-02a"> - <image file="0140-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0140-02"/> - </figure> - </div> - <p> - <s xml:id="s1574" xml:space="preserve">Nunc verò eum conÅ¿iderans modum, quem ego de figuris <var>.G.H.</var> antecedentibus <lb/>præſcripÅ¿i, videndum eÅ¿t, an punctum <var>.B.</var> tribus lineis <var>.A.B.I.s.</var> et <var>.R.V.</var> quarum hÄ™c vl <lb/>tima à me iam ducta fuit, commune exiÅ¿tat, ideÅ¿t vtrum <var>.A.B.</var> æqualis exiÅ¿tat ipÅ¿i <var>.K.<lb/>R.</var> quam Å¿ecundum modum à me adinuentum, reuera Å¿cimus eſſe deÅ¿ideratam altitu <lb/>dinem in perÅ¿pectiua. </s> - <s xml:id="s1575" xml:space="preserve">Quod tunc à nobis probatum erit, quando rationibus clarè <lb/>patebit ipÅ¿am <var>.A.B.</var> æqualem eſſe ipÅ¿i <var>.f.V</var>. </s> - <s xml:id="s1576" xml:space="preserve">Quamobrem ducamus <var>.I.f.</var> vÅ¿que ad <var>.ω.</var> <lb/>lineæ <var>.c.p.</var> vnde ratione Å¿imilitudinis triangulorum manifeÅ¿tè intelligemus, eandem <lb/>proportionem eſſe ipÅ¿ius <var>.m.T.</var> ad <var>.f.V.</var> quæ eÅ¿t <var>.m.o.</var> ad <var>.f.o.</var> & eius, quæ eÅ¿t <var>.m.o.</var> ad <var>.f.<lb/>o.</var> quæ eÅ¿t <var>.ω.I.</var> ad <var>.f.I.</var> & eius, quæ eÅ¿t <var>.ω.I.</var> ad <var>.f.I.</var> quæ eÅ¿t <var>.x.I.</var> ad <var>.A.I.</var> & eius, quæ eÅ¿t <var>.x.<lb/>I.</var> ad <var>.A.I.</var> quæ eÅ¿t <var>.x.s.</var> ad <var>.A.B.</var> ideÅ¿t vt eius, quæ eÅ¿t <var>.m.T.</var> ad <var>.A.B.</var> Å¿ed idem <reg norm="quoque" type="simple">quoq;</reg> erat <lb/>de <var>.m.T.</var> ad <var>.f.V</var>. </s> - <s xml:id="s1577" xml:space="preserve">Vnde Å¿equitur <var>.A.B.</var> æqualem eſſe <var>.f.V.</var> ex .9. quinti Eucli. <reg norm="atque" type="simple">atq;</reg> etiam <lb/>ipÅ¿i <var>.k.R.</var> quod à nobis propoÅ¿itum eÅ¿t inquirendum.</s> - </p> - </div> - <div type="section" level="3" n="9"> - <head xml:id="head185" xml:space="preserve">CAP. IX.</head> - <figure position="here"> - <image file="0141-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0141-01"/> - </figure> - <p> - <s xml:id="s1578" xml:space="preserve"><emph style="sc">INstitvens</emph> etiam Å¿ermonem de figuris Å¿u-<lb/>perficialibus orizontalibus, Å¿eu de plantis, <lb/>pulcherrimum quendam modum, quem ego ad <lb/>locandum quodlibet punctum in perÅ¿pectiua, <lb/>(degradatum cum fuerit <reg norm="parallelogrammum" type="context">parallelogrãmum</reg> quod <lb/>dam rectangulum, in noÅ¿tro plano perpendicula <lb/>ri orizonti, quemadmodum in Å¿uperioribus figu-<lb/>ris <var>.A.</var> demonÅ¿trauimus) conÅ¿ideraui, Å¿ilentio <lb/>haud prætereundum eſſe.</s> - </p> - <p> - <s xml:id="s1579" xml:space="preserve">Sit igitur in Å¿ubÅ¿cripta hîc figura <var>.K.</var> in paralle <lb/><reg norm="logrammo" type="context">logrãmo</reg> perfecto <reg norm="punctum" type="context">pÅ©ctum</reg> <var>.b.</var> quod locari debeat <lb/>in degradato <var>.e.q.d.r</var>. </s> - <s xml:id="s1580" xml:space="preserve">Nunc à duobus quorumli-<lb/>bet quatuor angulorum <var>.q.u.a.d.</var> ducuntur duæ li-<lb/>neæ occultæ <var>.q.g.</var> et <var>.u.f.</var> per punctum <var>.b.</var> vÅ¿que ad <lb/>latera <var>.q.d.</var> et <var>.u.a.</var> ita tamen vt eorum extremita-<lb/>tes <var>.g.</var> et <var>.f.</var> intus cadant inter <var>.q.d.</var> et <var>.u.a.</var> ipÅ¿orum <lb/>laterum, ideÅ¿t vt non Å¿ecent duo latera <var>.q.u.</var> aut <var>.d.<lb/>a</var>. </s> - <s xml:id="s1581" xml:space="preserve">Deinde punctum <var>.f.</var> inter <var>.q.</var> et <var>.d.</var> <reg norm="coniungatur" type="context">cõiungatur</reg> oc-<lb/>cultè cum angulo degradato <var>.e.</var> qui <reg norm="correÅ¿pondet" type="context">correÅ¿põdet</reg> <var>.<lb/>u.</var> perfecti, mediante linea <var>.e.f.</var> quæ erit <var>.u.f.</var> degra <lb/>dita in noÅ¿tro plano. </s> - <s xml:id="s1582" xml:space="preserve">Deinde Å¿umatur punctum <var>.<lb/>n.</var> in linea <var>.q.d.</var> tam diÅ¿tans à <var>.q.</var> quà m.g. diÅ¿tat ab <var>.<lb/>u.</var> ducaturqÌue linea <var>.i.n.</var> quæ lineam <var>.e.r.</var> in puncto <lb/>c. diuidet, quod exijs, quæ Å¿uperius iam diximus <lb/>ad ipÅ¿um <var>.g.</var> referetur. </s> - <s xml:id="s1583" xml:space="preserve">Ducendo poÅ¿tea lineam oc <lb/>cultam <var>.q.c.</var> patebit eam correÅ¿pondere lineæ <var>.q.g.</var> quæ Å¿ecans lineam <var>.e.f.</var> in puncto <var>.<lb/>t.</var> hoc, communi Å¿cientiæ ratione, reÅ¿pondebit ipÅ¿i <var>.b.</var> vt omnes cognoÅ¿cent.</s> - </p> - <pb o="130" rhead="IO. BAPT. BENED." n="142" file="0142" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0142"/> - <p> - <s xml:id="s1584" xml:space="preserve">Sed Å¿i fortè punctum <var>.b.</var> eſſet in aliquo <lb/> - <anchor type="figure" xlink:label="fig-0142-01a" xlink:href="fig-0142-01"/> - laterum, puta <var>.q.u.</var> volo vt in rectangulo per-<lb/>fecto <var>.q.d.a.u.</var> ducta Å¿it vna diagonalis quam <lb/>volueris puta <var>.q.a.</var> deinde à puncto <var>.b.</var> ad reli-<lb/>quum angulum oppoÅ¿iti lateris ducta Å¿it recta <var>.<lb/>b.d.</var> ita quod à diagonali Å¿ecetur in puncto <var>.ω.</var> <lb/>per quod punctum demum à reliquo angulo la <lb/>teris <var>.q.u.</var> ducta Å¿it <var>.u.ω.</var> vÅ¿que ad latus <var>.q.d.</var> in <reg norm="pum" type="context">pÅ©</reg> <lb/>cto <var>.f.</var> quo facto, ita faciendum erit in rectangu <lb/>lo degradato, hoc eÅ¿t ducenda erit diagonalis <var>.<lb/>q.r.</var> quÄ™ correÅ¿pondet diagonali <var>.q.a.</var> perfecti <lb/>deinde <var>.f.e.</var> quæ correÅ¿pondet rectæ <var>.f.u.</var> perfe-<lb/>cti, quæ etiam interÅ¿ecabitur à diagonali <var>.q.r.</var> <lb/>in puncto <var>.o.</var> correÅ¿pondens <var>.ω.</var> perfecti, per <reg norm="quem" type="context">quẽ</reg> <var>.<lb/>o.</var> à puncto <var>.d.</var> cum ducta fuerit <var>.d.o.</var> vÅ¿que ad <var>.t.</var> <lb/>in latere <var>.q.e.</var> hoc <reg norm="punctum" type="context">punctÅ©</reg> <var>.t.</var> correÅ¿pondebit pun <lb/>cto <var>.b.</var> perfecti.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0142-01" xlink:href="fig-0142-01a"> - <image file="0142-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0142-01"/> - </figure> - </div> - <p> - <s xml:id="s1585" xml:space="preserve">Idem eueniet Å¿i loco diametri <var>.q.a.</var> Å¿umpta <lb/>fuerit diameter <var>.u.d.</var> & loco <var>.b.d.</var> protracta fue <lb/>rit <var>.b.a.</var> deinde loco <var>.u.ω.f.</var> ducta fuerit <var>.q.ω.f.</var> vn-<lb/>de punctum correÅ¿pondens ipÅ¿i <var>.f.</var> in figura de-<lb/>gradata erit in latere Å¿upremo <var>.e.r.</var> correÅ¿pon-<lb/>dens lateri <var>.u.a.</var> & ita ducenda erit diameter <var>.d.<lb/>e.</var> correÅ¿pondens diametro <var>.d.u.</var> et <var>.q.f.</var> Å¿urÅ¿um <lb/>verÅ¿us correÅ¿pondens <var>.q.f.</var> imum verÅ¿us deinde <var>.<lb/>r.o.</var> reÅ¿pondens <var>.a.ω.</var> quæ terminabicur ab <reg norm="eodem- met" type="context">eodẽ-<lb/>met</reg> puncto <var>.t.</var> vt prius.</s> - </p> - </div> - <div type="section" level="3" n="10"> - <head xml:id="head186" xml:space="preserve">CAP.X.</head> - <p> - <s xml:id="s1586" xml:space="preserve"><emph style="sc">Ex</emph> mea figura <var>.F.</var> Å¿uperficiali perÅ¿pectiuæ facillimum modum locandi quoduis <lb/>punctum in perÅ¿pectiua elicui. </s> - <s xml:id="s1587" xml:space="preserve">Iuſſi enim vt aptaretur tabula quædam rectan-<lb/>gula exactè plana, triplo aut quadruplo, aut quanto volueris maioris longitu-<lb/>dinis, quà m latitudinis protenÅ¿a, quæ quidem latitudo erat ad duos circiter pedes <lb/>deÅ¿ignata ab <var>.A.B.C.D.</var> cuius duobus lateribus <var>.A.B.</var> et <var>.B.C.</var> iuſſi, vt duæ regulæ affi <lb/>gerentur, quæ Å¿uperficiem eiuÅ¿dem tabulæ excederent, vt <reg norm="vnum" type="context">vnÅ©</reg> ex lateribus alicuius <lb/>anguli recti materialis, qui <reg norm="appellatur" type="simple">appellat̃</reg> norma (vt inferius dicam) ei adherere poſſit, cu-<lb/>raui poÅ¿tea, ut iuxta angulum <var>.D.</var> in puncto <var>.o.</var> fixo mobilis regula <var>.o.Q.</var> affigeretur <lb/>tantæ longitudinis, aut paulò minoris, quantam occupabat latus <var>.D.A.</var> quæ circum <var>.<lb/>o.</var> volueretur, in rectitudine poÅ¿teà . <var>o.i.</var> parallela ipÅ¿i <var>.D.A.</var> in puncto <var>.i.</var> duobus pedi-<lb/>bus longè à latere <var>.A.B.</var> aliam quoque mobilem appendere feci <var>.i.M.</var> in tantam ferè <lb/>longitudinem extenÅ¿am, quanta conÅ¿tat <var>.A.B.</var> <reg norm="conÅ¿titui" type="context">cõſtitui</reg> etiam, vt <reg norm="quoddam" type="context">quoddã</reg> angulum re <lb/>ctum materiale tantæ magnitudinis, quanta nobis vÅ¿ui eſſe poterat Å¿uper eadem ta-<lb/>bula; </s> - <s xml:id="s1588" xml:space="preserve">necnon regula quædam materialis neceſſariæ longitudinis <reg norm="Å¿tatuerentur" type="simple">Å¿tatuerent̃</reg>, <reg norm="atque" type="simple">atq;</reg> hæc <lb/>omnia tenuiſſima, vt fierent curaui. </s> - <s xml:id="s1589" xml:space="preserve">Quandam deinde lineam ad <var>.o.i.</var> parallelam, <lb/>ideÅ¿t <var>.p.E.</var> Å¿uper eadem tabula adeò diÅ¿tantem ab <var>.o.i.</var> vt inter <var>.E.p.</var> et <var>.B.c.</var> perfe-<lb/>ctæ res, quæ degradari debebant, locari poſſent, Å¿ignaui. </s> - <s xml:id="s1590" xml:space="preserve">Hæc autem diÅ¿tantia, quæ - <pb o="131" rhead="DE PERSPECT." n="143" file="0143" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0143"/> - inter <var>.o.i.</var> et <var>.p.E.</var> intercedebat, altitudinem oculi ab orizonte Å¿ignificabat. </s> - <s xml:id="s1591" xml:space="preserve">Signa-<lb/>ui etiam lineam <var>.i.G.</var> perpendicularem lineæ <var>.E.p.</var> cui affigi poſſet non nihil chartæ <lb/>quotieÅ¿cunque volebam in perÅ¿pectiua aliquid delineare. </s> - <s xml:id="s1592" xml:space="preserve">Quod cum facere deÅ¿i-<lb/>derabam, ponebam perfectum optimè affixum in quadrangulo <var>.E.G</var>: & in quadran <lb/>gulo <var>.E.i.</var> aliquod folium papyri affigebam. </s> - <s xml:id="s1593" xml:space="preserve">Ponamus nunc, me voluiſſe conÅ¿titaere <lb/><reg norm="punctum" type="context">punctÅ©</reg> <var>.b.</var> <reg norm="Å¿umebam" type="context">Å¿umebã</reg> <reg norm="angulum" type="context">angulÅ©</reg> <reg norm="rectum" type="context">rectÅ©</reg> <reg norm="materialem" type="context">materialẽ</reg>, Å¿eu <reg norm="normam" type="context">normã</reg>, & eius vnum latus, iuxta latus <var>.<lb/>B.c.</var> ponebam, atque aliud per punctum <var>.b.</var> tranÅ¿ire faciebam, & vbi hoc latus <reg norm="lineam" type="context">lineã</reg> <lb/><var>E.p.</var> diuidebat, punctum .c Å¿ignabam per quod efficiebatur, vt regula <var>.i.M.</var> tranÅ¿iret, <lb/>quieÅ¿ceretqÌue aliquantulum aliquo modo in huiuÅ¿modi Å¿itu, opera deinde circini <lb/>interuallum <var>.b.c.</var> Å¿umebam, & in <var>.p.E.</var> à puncto <var>.x.</var> verÅ¿us <var>.E.</var> punctum <var>.m.</var> <reg norm="Å¿ignabam" type="context">Å¿ignabã</reg>: </s> - <s xml:id="s1594" xml:space="preserve">per <lb/>quod faciebam, vt tranÅ¿iret regula <var>.o.Q.</var> quæ lineam <var>.x.i.</var> in puncto <var>.f.</var> diuidebat. <lb/></s> - <s xml:id="s1595" xml:space="preserve">Angulum deinderectum materialem accipiebam, cuius vnum latus <var>.A.B.</var> ponebam, <lb/>aliud verò per punctum <var>.f.</var> tranÅ¿ibat, quod quidem latus regulam <var>.i.M.</var> in puncto <var>.k.</var> <lb/>(quod Å¿tatim Å¿uper folio papyri Å¿ignabatur) interÅ¿ecabat, atque hoc erat <reg norm="punctum" type="context">punctÅ©</reg>, <lb/>quod quærebam, puncto <var>.b.</var> correÅ¿pondens. </s> - <s xml:id="s1596" xml:space="preserve">HuiuÅ¿modi effectus rationes ab ijs, quæ <lb/>Å¿uperius dixi eliciuntur. </s> - <s xml:id="s1597" xml:space="preserve">Atque hæc ad baÅ¿es rerum, vt in Å¿ubÅ¿cripta figura eluceÅ¿cit, <lb/>Å¿pectabant.</s> - </p> - <figure position="here"> - <image file="0143-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0143-01"/> - </figure> - </div> - <div type="section" level="3" n="11"> - <head xml:id="head187" xml:space="preserve">CAP. XI.</head> - <p> - <s xml:id="s1598" xml:space="preserve"><emph style="sc">Ad</emph> degradandas deinde altitudines, vÅ¿us Å¿um mea figura tam <var>.H.</var> <reg norm="quam" type="context">quã</reg> <reg norm="ent" type="context">ẽt</reg> <var>.H.H.</var> <lb/>vt milii Å¿eÅ¿e offerebat occaÅ¿io. </s> - <s xml:id="s1599" xml:space="preserve">In primis ratione modi figuræ <var>.H.</var> <reg norm="curabam" type="context">curabã</reg>, vt - <pb o="132" rhead="IO. BAPT. BENED." n="144" file="0144" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0144"/> - vnum ex lateribus anguli recti, Å¿eu normæ regulæ <var>.B.C.</var> anniteretur, aliud verò <lb/>per <var>.m.</var> in rectitudine cuius Å¿ignabam <var>m.T.</var> interuallum æquale altitudini perfecti, <lb/>ideÅ¿t punctum <var>.T.</var> æqualiter diÅ¿tans ab <var>.m.</var> tranÅ¿ire faciebam, deinde regulam <var>.o.Q.</var> <lb/>per punctum <var>.T.</var> tranÅ¿ire quoque faciebam: </s> - <s xml:id="s1600" xml:space="preserve">& notabam interÅ¿ectionem ipÅ¿ius cum <lb/>linea <var>.i.x.</var> in puncto <var>.V.</var> efficiebam deinde vt vnum ex lateribus anguli recti, lateri ta-<lb/>bulæ <var>.B.C.</var> anniteretur, aliudqÌue per punctum <var>.k.</var> tranÅ¿ire faciebam, & in huiuÅ¿modi <lb/>rectitudine à puncto k. Å¿ignabam quandam menÅ¿uram æ qualem lineæ <var>.f.V.</var> quÄ™ erat <lb/><var>k.R.</var> pro altitudine degradata.</s> - </p> - <head xml:id="head188" xml:space="preserve">ALITER IDEM.</head> - <p> - <s xml:id="s1601" xml:space="preserve"><emph style="sc">MEdiante</emph> deinde figura <var>.H.H.</var> vnum ex lateribus anguli recti, lateri tabu-<lb/>læ <var>.B.C.</var> vt anniteretur faciebam; </s> - <s xml:id="s1602" xml:space="preserve">aliud verò per punctum <var>.b.</var> perfecti, ideÅ¿t ba <lb/>Å¿is eiuÅ¿dem perfecti tranÅ¿ire faciebam. </s> - <s xml:id="s1603" xml:space="preserve">Et in huiuÅ¿modi rectitudine Å¿ignabam <var>.c.s.</var> <lb/>æquale interuallum altitudini perfecti, ideÅ¿t punctum <var>.s.</var> ita diÅ¿tans à <var>.c.</var> efficiendo de <lb/>inde, vt latus anguli recti, lateri <var>.B.C.</var> tabulæ anniteretur, aliudqÌue per punctum <var>.k.</var> <lb/>tranÅ¿ire faciens Å¿ignabam <var>.k.R.</var> indeterminatam. </s> - <s xml:id="s1604" xml:space="preserve">Faciens deinde tranÅ¿ire regulam <lb/><var>i.M.</var> per punctum <var>.s.</var> notabam punctum <var>.R.</var> interÅ¿ectionis eiuÅ¿dem cum linea <var>.k.R.</var> <reg norm="iam" type="context">iã</reg> <lb/>ducta. </s> - <s xml:id="s1605" xml:space="preserve">Itaque altitudinem <var>.k.R.</var> degradatam habebam. </s> - <s xml:id="s1606" xml:space="preserve">Hæc autem via aliquan-<lb/>tulum breuior, expeditiorqÌue altera.</s> - </p> - <figure position="here"> - <image file="0144-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0144-01"/> - </figure> - <pb o="133" rhead="DE PERSPECT." n="145" file="0145" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0145"/> - </div> - <div type="section" level="3" n="12"> - <head xml:id="head189" style="it" xml:space="preserve">JACOBO SOLDATO MEDIOLANENSI <lb/>Serenißimi Ducis Sabaudiæ Architecto peritißimo.</head> - <head xml:id="head190" xml:space="preserve">CAP. VII.</head> - <p> - <s xml:id="s1607" xml:space="preserve"><emph style="sc">SVperioribvs</emph> diebus non diu poÅ¿tquam de perÅ¿pectiuis inter nos Å¿ermonem <lb/>habuimus, dum animus totus adhuc in his eſſet. </s> - <s xml:id="s1608" xml:space="preserve">Illud in mentem venit quòd exi <lb/>mius ille vir, & profundiſſimæ doctrinæ, nec vnquam Å¿atis laudatus Daniel Barba-<lb/>rus Å¿e accepiſſe profitetur à Ioanne Zamberto patritio Veneto, qui ad verbum om <lb/>nia deÅ¿umpÅ¿erat a Ioanne CuÅ¿ino PariÅ¿ienÅ¿e. </s> - <s xml:id="s1609" xml:space="preserve">Nec parum mirabar peritiſſimum il-<lb/>lum CuÅ¿inum, quod in capite quarto Å¿ecundæ partis perÅ¿pectiuæ, vt quodpiam pla-<lb/>num quadrilaterum in quadratam figuram redigeret, Å¿uper vnam datam <reg norm="lineam" type="context">lineã</reg> qua-<lb/>dratam compoÅ¿uiſſe. </s> - <s xml:id="s1610" xml:space="preserve">Non animaduertens diÅ¿tantiam aut interuallum <var>.b.c.</var> degra-<lb/>datum Ãn linea <var>.b.f.</var> (quod eÅ¿t <var>.b.E.</var>) ita eſſe poſſe latus parallelogrammi rectanguli <lb/>magis longi quam lati, aut magis lati quam longi, vt etiam latus quadrati, quod be-<lb/>neficio Å¿ubÅ¿criptæ hic figuræ facilè depræhendi poteÅ¿t. </s> - <s xml:id="s1611" xml:space="preserve">Vbi <var>.b.c.</var> latitudo eſſe po-<lb/>teÅ¿t, tam perfecti degradati in triangulo <var>.b.n.m.</var> aut in triangulo <var>.b.q.t.</var> quam in trian <lb/>gulo <var>.b.a.c</var>. </s> - <s xml:id="s1612" xml:space="preserve">Sed perfectum degradati in triangulo <var>.b.n.m.</var> magis longum quam <reg norm="latum" type="context">latÅ©</reg> <lb/>& perfectum degradatum in triangulo <var>.b.q.t.</var> magis latum quam longum, & perfe-<lb/>ctum degradati in triangulo <var>.b.a.c.</var> quadratum erit quemadmodum à meis etiam fi-<lb/>guris <var>.A.</var> Å¿cientificè intelligi poteÅ¿t. </s> - <s xml:id="s1613" xml:space="preserve">Hinc, ad inueniendum perfectum alicuius pla-<lb/>ni degradati, non Å¿ufficere degradationem Å¿olum interualli inter duos terminos <lb/>Å¿olos, ideÅ¿t <var>.b.E.</var> aſſignare, apertè patet, quia non omnia parallelogramma. perfecta <lb/>ab vno <reg norm="tantum" type="wordlist">tm̃</reg> interuallo producuntur, eo <reg norm="quod" type="simple">ê™</reg> non Å¿unt omnia quadrata. </s> - <s xml:id="s1614" xml:space="preserve">Ad <reg norm="inquirendum" type="context">inquirendÅ©</reg> <lb/>igitur perfectum alicuius plani <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>, alicuius propoÅ¿iti degradati, oportet <lb/>vniuerÅ¿am degradationem tam latitudinis, <reg norm="quam" type="simple">ê—̃</reg> longitudinis, & <reg norm="non" type="context">nõ</reg> folius longitudinis <lb/>aſſignare; </s> - <s xml:id="s1615" xml:space="preserve">Vt <reg norm="exempli" type="context">exẽpli</reg> gratia, in Å¿ubÅ¿cripta hic figura, <reg norm="volendo" type="context">volẽdo</reg> inuenire perfectum paral <lb/>lelogrammum degradati <var>.b.h.l.m.</var> dando diÅ¿tantiam orizontalem <var>.b.d.</var> à pede <var>.d.</var> ho-<lb/> - <anchor type="figure" xlink:label="fig-0145-01a" xlink:href="fig-0145-01"/> - <pb o="134" rhead="IO. BAPT. BENED." n="146" file="0146" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0146"/> - minis vÅ¿que ad planiÅ¿itum in quo degradatio facta Å¿it: </s> - <s xml:id="s1616" xml:space="preserve">Å¿tatim altitudo <var>.A.</var> oculi à pe <lb/>de, quæ tanta Å¿emper eſſe debet quanta eÅ¿t altitudo trianguli <var>.b.n.m.</var> qui clauditur, <lb/>protrahendo <var>.m.l.</var> et <var>.b.h.</var> vÅ¿que ad concurÅ¿um in <var>.n.</var> in lucem prodibit. </s> - <s xml:id="s1617" xml:space="preserve">Oporter <lb/>deinde erigere lineam <var>.b.o.</var> perpendicularem lineæ <var>.d.b.m.</var> & vÅ¿que ad eandem pro <lb/>ducere lineam <var>.l.h.</var> in puncto <var>.E.</var> et à puncto <var>.A.</var> per <var>.E.</var> vÅ¿que ad <var>.c.</var> ipÅ¿ius <var>.d.b.m.</var> produ <lb/>ctæ ducere <var>.A.E.c.</var> atque deinde protrahere lineam <var>.o.b.</var> vÅ¿que ad <var>.T.</var> ita vt <var>.b.T.</var> Ä™qua <lb/>lis Å¿it ipÅ¿i <var>.b.c.</var> & ad ipÅ¿am à puncto <var>.m.</var> ducere parallelam <var>.m.R.</var> & à puncto <var>.T.</var> ducere <var>.<lb/>T.R.</var> parallelam ipÅ¿i <var>.b.m</var>. </s> - <s xml:id="s1618" xml:space="preserve">Vnde ex .34. primi Eucli <var>.m.R.</var> æqualis erit ipÅ¿i <var>.b.T.</var> et <var>.R.<lb/>T.</var> ipÅ¿i <var>.m.b.</var> & anguli in rectos euadent, atque hoc parallelo grammum rectangulum <lb/>erit verum perfectum degradati <var>.b.m.l.h.</var> obrationes à me circa <reg norm="figuram" type="context">figurã</reg> <var>.A.</var> adductas. <lb/> - <anchor type="figure" xlink:label="fig-0146-01a" xlink:href="fig-0146-01"/> - </s> - <s xml:id="s1619" xml:space="preserve">Sed eÅ¿t hic quod magis nos commoueat, quia cum ex linea <var>.b.c.</var> quadratum <var>.b.g.</var> pro <lb/>duxerit, vult eum poÅ¿tea degradare. </s> - <s xml:id="s1620" xml:space="preserve">Quod vt faciat (hanc figuram videbis in cap <num value="4">.<lb/>4.</num> Å¿ecundæ partis Danielis Barbari) oculum <var>.A.</var> in eadem Å¿uperficie extenÅ¿a, <lb/>quadrati <var>.b.g.</var> collocat. </s> - <s xml:id="s1621" xml:space="preserve">Quod rectè fieti non poteÅ¿t, quia oculum hoc mo-<lb/>do locantes, viÅ¿ualesqÌue radios beneficio vnius plani Å¿ituati in <var>.b.f.</var> Å¿ecantes in ipÅ¿o <lb/>plano, nihil aliud quam <reg norm="dictam" type="context">dictã</reg> <reg norm="lineam" type="context">lineã</reg> <var>.b.f.</var> & <reg norm="nullam" type="context">nullã</reg> <reg norm="degradationem" type="context">degradationẽ</reg> <reg norm="inuenient" type="context">inueniẽt</reg>. </s> - <s xml:id="s1622" xml:space="preserve">Id quod, & Å¿i <lb/>natura Å¿ua Å¿it omnibus notum, ponit <reg norm="tantum" type="wordlist/context">tñ</reg> id ipÅ¿um Vitelio pro quinta propoÅ¿itione <lb/>quarti libri de perÅ¿pectiua. </s> - <s xml:id="s1623" xml:space="preserve">Præter hæc, credit latera <var>.b.d.</var> et <var>.c.e.</var> quadrati degra-<lb/>dati Å¿emper videri mediantibus angulis <var>.b.A.c.</var> et <var>.f.A.g.</var> quod fieri <reg norm="non" type="context">nõ</reg> poteÅ¿t, quem-<lb/>a dmodum ex mea figura corporea <var>.A.</var> facilè cognoÅ¿cere poſſumus, </s> - <s xml:id="s1624" xml:space="preserve">propterea quòd <lb/>latera <var>.d.r.</var> et <var>.q.e.</var> meæ figuræ, mediantibus angulis <var>.d.o.r.</var> et <var>.q.o.e.</var> qui extra Å¿uperfi-<lb/>ciem <var>.s.a.</var> exiÅ¿tunt videntur, vnde Å¿i quis imaginaretur in puncto <var>.p.</var> oculum eſſe, & <lb/>ab ipÅ¿o ad <var>.u.</var> et <var>.q.</var> duas lineas duceret, angulus <var>.q.o.u.</var> nunc maior, nunc minor eſſet <lb/>angulo <var>.q.p.u.</var> aliquando etiam æqualis, quamuis rariſſime; </s> - <s xml:id="s1625" xml:space="preserve">Sub diuerÅ¿is igitur an-<lb/>gulis, pro maiori parte, deteguntur latera, à partibus quadrati tam degradati, quà m <lb/>perfecti, quæ non Å¿unt anguli <var>.b.A.c.</var> et <var>.f.A.G</var>. </s> - <s xml:id="s1626" xml:space="preserve">Quod vero idem poÅ¿tea dicat eam <lb/>proportionem eſſe ab <var>.b.E.</var> ad <var>.f.h.</var> & Å¿imul ad <var>.c.g.</var> quæ eÅ¿t ab <var>.a.g.</var> ad <var>.h.g.</var> id tuo relin - <pb o="135" rhead="DE PERSPECT." n="147" file="0147" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0147"/> - quam iudicio. </s> - <s xml:id="s1627" xml:space="preserve">Tibi quoque conÅ¿iderandum relinquo; </s> - <s xml:id="s1628" xml:space="preserve">cum rationabilis degrada-<lb/>tio eſſe debeat, qua ratione neceſſarium Å¿it, vt diÅ¿tantiæ reÅ¿qÌue, in vna & <reg norm="eadem" type="context">eadẽ</reg> pro-<lb/>portione cum altitudine oculi ad rem degradatam exiÅ¿tant? </s> - <s xml:id="s1629" xml:space="preserve">Cum poÅ¿tea degrada-<lb/>uerit <reg norm="quadratum" type="context">quadratÅ©</reg>, is Å¿criptor, in figura <var>.d.b.c.e.</var> eum bene & ex perÅ¿pectiuæ optimis <lb/>legibus degradatum fuiſſe probare nititur; </s> - <s xml:id="s1630" xml:space="preserve">Å¿olum probans <var>.d.e.</var> æqualem eſſe ipÅ¿i <var>.<lb/>E.h.</var> <reg norm="qua" type="wordlist">q̃</reg> <var>.E.h.</var> <reg norm="Å¿ecundum" type="context">Å¿ecundÅ©</reg> ipÅ¿um eÅ¿t degra datio lateris <var>.c.g.</var> & <reg norm="cum" type="context">cÅ©</reg> Å¿uperius dixerit, Å¿etria <lb/>quadrati plana degradauiſſe, quia <var>.b.E.</var> degradat <var>.b.c.</var> et <var>.E.h.</var> degradat <var>.c.g.</var> et <var>.f.<lb/>h.</var> degradat <var>.f.g.</var> nec quidem de lateribus <var>.b.d.</var> et <var>.c.e.</var> loquitur, quia Å¿i <var>.c.g.</var> <lb/>perfecti, degradatum eÅ¿t in <var>.E.h</var>: et <var>.d.e.</var> rectè protracta exiÅ¿tit, cum Å¿it æqua-<lb/>lis ipÅ¿i <var>.E.h.</var> cum etiam <var>.b.d.</var> et <var>.c.e.</var> rectè protractæ eſſe debeant: </s> - <s xml:id="s1631" xml:space="preserve">qua de cau-<lb/>Å¿a ipÅ¿is <var>.b.E.</var> et <var>.f.h.</var> quæ, ex ipÅ¿o, Å¿unt degradationes <var>.b.c.</var> et <var>.f.g.</var> æquales eſſe non de-<lb/>bent? </s> - <s xml:id="s1632" xml:space="preserve">Poſſet is mihi quidem reÅ¿pondere, <reg norm="quod" type="simple">ê™</reg> hoc pacto nulla Å¿uperficies clauderetur. <lb/></s> - <s xml:id="s1633" xml:space="preserve">Ergo tria latera <var>.b.c</var>: <var>c.g.</var> et <var>.g.f.</var> <reg norm="non" type="context">nõ</reg> benè Å¿unt degradata, <reg norm="eiusque" type="simple">eiusqÌ;</reg> <reg norm="proportionalitates" type="simple">ê“portionalitates</reg> ma <lb/>lè intellectæ nil probant. </s> - <s xml:id="s1634" xml:space="preserve">quia Å¿i dictæ proportionalitates, nobis tutò promitterent <lb/>degradationes, ab eo primum effectas, in linea <var>.b.f.</var> eſſe bonas, ergo duæ <var>.b.d.</var> et <var>.e.c.</var> <lb/>falſæ exiÅ¿terent, quarum quælibet maior eÅ¿t <var>.b.E.</var> et <var>.f.h.</var> ex .18. primi Eucli. </s> - <s xml:id="s1635" xml:space="preserve">Omitta-<lb/>mus etiam quod vbi is Å¿cribit eam eſſe rationem, aut comparationem ab <var>.A.d.</var> ad <var>.b.<lb/>E.</var> quæ eÅ¿t ab <var>.d.c.</var> ad <var>.b.c.</var> eandemqÌue eſſe ab <var>.E.h.</var> ad <var>.c.g.</var> quæ eÅ¿t ab <var>.A.E.</var> ad <var>.A.c.</var> nil <lb/>probet; </s> - <s xml:id="s1636" xml:space="preserve">nec Å¿imilitudinem triangulorum, nec aliquam propoÅ¿itionem Eucli. citans. <lb/></s> - <s xml:id="s1637" xml:space="preserve">In quo excuÅ¿ari non poteÅ¿t, quòd non Å¿oleat Euclidem, aut alium quemuis autorem <lb/>citare, cum vel in ipÅ¿o operis principio capite .3. primæ partis, A pollonium <reg norm="Pergeum" type="context">PergeÅ©</reg> <lb/><reg norm="Euclidemque" type="simple">EuclidemqÌ;</reg>, & Å¿i etiam præter rem, citet. </s> - <s xml:id="s1638" xml:space="preserve">Deinde <reg norm="quum" type="context">quÅ©</reg> idem probare vult <var>.d.e.</var> æqua <lb/>lem eſſe ipÅ¿i <var>.E.h.</var> eandem inquit eſſe proportionem <var>.a.b.</var> ad <var>.a.d.</var> quæ eÅ¿t ipÅ¿ius <var>.A.c.</var> <lb/>ad <var>.A.E.</var> quod & Å¿i verum Å¿it, hic tamen modus ratiocinandi nullo ordine nititur, <lb/>quia rectius dixiſſet pro clariori intelligentia ipÅ¿ius <var>.a.c.</var> ad <var>.a.e.</var> eandem proportio-<lb/>nem eſſe, quæ eÅ¿t <var>.A.c.</var> ad <var>.A.E.</var> propter Å¿imilitudinem, quæ inter duos triangulos <var>.A.<lb/>c.a.</var> et <var>.E.c.e.</var> intercedit, cum <var>.E.e.</var> Å¿upponatur parallela ipÅ¿i <var>.A.a.</var> quod etiam vt de-<lb/>monÅ¿traretur longiori oratione ei opus fuiſſet Å¿i voluiſſet intellectum eorum, qui pa <lb/>rum Å¿unt exercitati, perduci ad <reg norm="cognoÅ¿cendum" type="context">cognoÅ¿cendÅ©</reg> idem planè futurum de <var>.a.c.</var> ad <var>.a.e.</var> vt <lb/>eÅ¿t ipÅ¿ius <var>.A.c.</var> ad <var>.A.E.</var> in hunc modum, ideÅ¿t probando primùm duos triangulos <var>.A.<lb/>c.a.</var> et <var>.E.c.e.</var> æquiangulos eſſe, mediante .29. primi Eucli. cum <var>.A.a.</var> et <var>.E.e.</var> inuicem <lb/>Å¿int parallelæ. </s> - <s xml:id="s1639" xml:space="preserve">Vnde ex .4. Å¿exti. idem extitiſſet de <var>.A.c.</var> ad <var>.E.c.</var> vt <var>.a.c.</var> ad <var>.e.c.</var> et. ex <lb/>16. quinti idem de <var>.A.c.</var> ad <var>.a.c.</var> vt ipÅ¿ius <var>.E.c.</var> ad <var>.e.c.</var> & ex .19. eiuÅ¿dem de <var>.A.E.</var> ad <var>.a.<lb/>e.</var> vt ipÅ¿ius <var>.A.c.</var> ad <var>.a.c.</var> & ex .16. iam dicta de <var>.A.E.</var> ad <var>.A.c.</var> vt ipÅ¿ius <var>.a.e.</var> ad <var>.a.c.</var> ideÅ¿t <lb/>ipÅ¿ius <var>.A.c.</var> ad <var>.A.E.</var> vt eÅ¿t ipÅ¿ius <var>.a.c.</var> ad <var>.a.e</var>: Aut hoc alio modo, qui breuior eÅ¿t pro-<lb/>cedendum, incipiendo Å¿cilicet à ſecunda Å¿exti Eucli. dicendo <reg norm="quod" type="simple">ê™</reg> exiÅ¿tente <var>.E.e.</var> paral <lb/>lela ipÅ¿i <var>.A.a</var>: ex dicta .2. lib. 6. erit idem de <var>.c.E.</var> ad <var>.E.A.</var> vt de <var>.c.e.</var> ad <var>.e.a.</var> vnde ex <num value="18">.<lb/>18.</num> quinti innotuiſſet Å¿tatim quod de <var>.c.A.</var> ad <var>.E.A.</var> vt de <var>.c.a.</var> ad <var>.e.a.</var> extitiſſet. </s> - <s xml:id="s1640" xml:space="preserve">Nunc <lb/>mediantibus Å¿upradictis duabus propoÅ¿itionibus ideÅ¿t .29. primi, & 4. Å¿exti, cogno-<lb/>Å¿citur idem planè eſſe de <var>.b.c.</var> ad <var>.d.e.</var> quod ipÅ¿ius <var>.a.c.</var> ad <var>.a.e.</var> & ex eiÅ¿dem idem eſſe <lb/>de <var>.c.g.</var> ad <var>.E.h.</var> quod ipÅ¿ius <var>.A.c.</var> ad <var>.A.E.</var> vnde ex .11. quinti bis repetita idem erit de <lb/><var>b.c.</var> ad <var>.d.e.</var> quod de <var>.c.g.</var> ad <var>.E.h.</var> Å¿ed cum ex Å¿uppoÅ¿ito <var>.c.g.</var> Å¿it æqualis ipÅ¿i <var>.c.b.</var> idem <lb/>erit de <var>.c.g.</var> ad <var>.e.d.</var> quod ipÅ¿ius <var>.c.b.</var> ad eandem ex .7. quinti, vnde ex .11. idem erit de <lb/><var>c.g.</var> ad <var>.E.h.</var> quod eiuÅ¿dem <var>.c.g.</var> ad <var>.e.d.</var> ex .9. igitur eiuÅ¿dem <var>.d.e.</var> æqualis erit ipÅ¿i <var>.E.<lb/>h.</var> atque hic verus eÅ¿t modus ducendi intellectum parum exercitatum in cognicio-<lb/>nis campum. </s> - <s xml:id="s1641" xml:space="preserve">quem quidem mihi obÅ¿eruandum proponerem Å¿i onus Å¿cribendi Å¿u-<lb/>Å¿ciperem ijs, qui in Å¿cientijs parum verÅ¿ati Å¿unt, quos tanquam puerulos manu du- - <pb o="136" rhead="IO. BAPT. BENED." n="148" file="0148" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0148"/> - cere oportet. </s> - <s xml:id="s1642" xml:space="preserve">Ratio verò ab ipÅ¿o adducta propter quam <var>.E.</var> repreÅ¿entatur oculo al-<lb/>tius quam <var>.b.</var> nempe eo quod <var>.A.</var> Å¿uperÅ¿tet ipÅ¿i <var>.E.</var> nihil valet, quia Å¿i inferius eſſet, <lb/>idem contingeret, Å¿ed hoc euenit eo quod <var>.E.</var> altius eÅ¿t ipÅ¿o <var>.b</var>. </s> - <s xml:id="s1643" xml:space="preserve">Idem dico de <var>.h.</var> <lb/>vbi Å¿imiliter decipitur. </s> - <s xml:id="s1644" xml:space="preserve">Idem etiam in .7. cap. fallitur in Å¿ecundo modo, quem oÅ¿ten <lb/>dit pro Å¿ecundo quadrato aliquo degradato à parallelogrammo degradato magis <lb/>longo quà m lato, cum ducat parallelam <var>.l.m.</var> ad <var>.b.c.</var> à puncto <var>.l.</var> interÅ¿ection is ipÅ¿ius <var>.<lb/>o.c.</var> id, quod non rectè efficitur quemadmodum ex rationibus à me allegatis circa <lb/>meas figuras <var>.A.A.</var> facilè innoteÅ¿cit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0145-01" xlink:href="fig-0145-01a"> - <image file="0145-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0145-01"/> - </figure> - <figure xlink:label="fig-0146-01" xlink:href="fig-0146-01a"> - <image file="0146-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0146-01"/> - </figure> - </div> - <p> - <s xml:id="s1645" xml:space="preserve">Nono deinde cap. contrario planè ordine, quam oporteret proceÅ¿sit, quia <reg norm="cum" type="context">cÅ©</reg> <lb/>angulus .2. trianguli perfecti magis diÅ¿tet à plano Å¿uper quod degradari debet <lb/>triangulum, quà m latus .1. 3. oppoÅ¿itum dicto angulo .2. & per confequens longère <lb/>motior Å¿it ab oculo, ipÅ¿e in degradato, <reg norm="eum" type="context">eÅ©</reg> magis propinquum eſſe facit, è con-<lb/>tra eap .10. rectè fecit contra id, quod capite .9. tradiderat.</s> - </p> - <p> - <s xml:id="s1646" xml:space="preserve">Quod autem deinceps in prima parte .11. & vltimi capitis aſſerit eÅ¿t, <reg norm="admittendum" type="context">admittendÅ©</reg>. <lb/></s> - <s xml:id="s1647" xml:space="preserve">Quod verò in Å¿ecunda parte ab eo traditur, ideÅ¿t alius quidam modus quem de <reg norm="tranÅ¿" type="context">trãſ</reg> <lb/>ferendis punctis à perfecto in degradato proponit, non eÅ¿t modus vniuerÅ¿alis; </s> - <s xml:id="s1648" xml:space="preserve">quia <lb/>Å¿i altitudo <var>.T.Q.</var> oculi à plano orizontali, non eſſet æqualis medietati lateris <var>.B.D.</var> <lb/>perfecti, interualla <var>.a.b.c.d.e.</var> lateris <var>B.D.</var> admittenda non eſſent.</s> - </p> - <p> - <s xml:id="s1649" xml:space="preserve">Pro cuius rei intelligentia Å¿it in Å¿ubÅ¿cripta hic figura corporea <var>.ω.</var> parallelogram-<lb/>mum rectangulum <var>A.B.C.D.</var> in plano orizontali, & linea <var>.Q.H.</var> illud per medium <lb/>diuidat, quæ Å¿it parallela duobus lateribus <var>.A.B.</var> et <var>.C.D.</var> in cuius quolibet puncto <var>.<lb/>Q.</var> Å¿it infimus terminus altitudinis oculi, & in <var>.<lb/> - <anchor type="figure" xlink:label="fig-0148-01a" xlink:href="fig-0148-01"/> - T.</var> ad perpendiculum ipÅ¿ius <var>.Q.</var> Å¿it verus Å¿itus <lb/>eiuÅ¿dem, tantum eleuatus à <var>.Q.</var> quanta eÅ¿t <lb/>medietas ipÅ¿ius <var>.D.B.</var> Å¿itqÌue figura corpo-<lb/>rea finita Å¿imilis meæ <var>.A.</var> vnde <var>.Q.T.</var> æqualis <lb/>erit ipÅ¿i <var>.Q.æ.</var> & planum perpendiculare <reg norm="orizon- ti" type="context">orizõ-<lb/>ti</reg>, Å¿uper quod punctum <var>.k.</var> perfecti duci debet <lb/>Å¿it <var>.R.D.B.</var> Å¿intqÌue ductæ per imaginationem <lb/>lineæ <var>.T.K</var>: <var>Q.K.</var> et Å¿it <var>.K.N.</var> perpendicularis la-<lb/>teri <var>.C.D.</var> à quo puncto <var>.N.</var> imaginatione Å¿it <reg norm="con" type="context">cõ</reg> <lb/>præhenÅ¿a linea <var>.N.Q.</var> at que hæ tres lineæ Å¿ectæ <lb/>Å¿int à plano in punctis <var>.c.i.</var> et .2. quorum <reg norm="punctum" type="context">punctÅ©</reg>. <lb/>2. erit quæſitum plani. </s> - <s xml:id="s1650" xml:space="preserve">Imaginemur nunc duos <lb/>triangulos <var>.K.T.Q.</var> et <var>.N.Q.æ.</var> qui Å¿ecti <reg norm="erunt" type="context">erÅ©t</reg> <lb/>à plano <var>.R.B.D.</var> quorum communes Å¿ectiones <lb/>erunt .1. 2. et <var>.D.c.</var> & quia <var>.N.K.D.i.</var> et <var>.æ.Q.</var> <lb/>inuicem Å¿unt parallelæ, Å¿equitur eandem pro-<lb/>portionem futuram ipÅ¿ius <var>.Q.K.</var> ad <var>.K.i.</var> quæ eÅ¿t <lb/>ipÅ¿ius <var>.æ.N.</var> ad <var>.N.D.</var> imaginatione concipien <lb/>do a puncto <var>.K.</var> vÅ¿que ad <var>.æ.Q.</var> quandam paral-<lb/>lelam ipÅ¿i <var>.N.æ.</var> quemadmo dum ex te ipÅ¿o intel <lb/>ligere potes. </s> - <s xml:id="s1651" xml:space="preserve">Sed ratione Å¿imilitudinis trian-<lb/>gulorum ita Å¿e res habet de <var>.æ.Q.</var> ad <var>.D.c.</var> vt de <var>.<lb/>æ.N.</var> ad <var>.N.D.</var> vt quoque de <var>.T.Q.</var> ad .2. 1. quemadmodum ipÅ¿ius <var>.Q.K.</var> ad <var>.K.i.</var> vn-<lb/>de ex .11. quinti, idem erit de <var>.Q.T.</var> ad .1. 2. quod de <var>.Q.æ.</var> ad <var>.c.D.</var> & ex .16. eiuÅ¿dem <lb/>de <var>.Q.T.</var> ad <var>.Q.æ.</var> quod de .1. 2. ad <var>.c.D.</var> & exiÅ¿tente <var>.æ.Q.</var> ex Å¿uppoÅ¿ito æquali ipÅ¿i.</s> - <pb o="137" rhead="DE PERSPECT." n="149" file="0149" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0149"/> - <s xml:id="s1652" xml:space="preserve">1. 2. </s> - <s xml:id="s1653" xml:space="preserve">Vnde huiuÅ¿modi regula tunc bona redditur, quando <var>T.Q.</var> æqualis eÅ¿t ipÅ¿i .Å“.<lb/>Q. ideÅ¿t medietati ipÅ¿ius <var>.D.B.</var> at verò Å¿i æqualis non eſſet hoc minime Å¿equeretur, <lb/>vt facilè patet. </s> - <s xml:id="s1654" xml:space="preserve">Quòd verò .2. <var>R.Z.</var> &. Å¿int benè diÅ¿poÅ¿ita, dubitandum non eÅ¿t, quia <lb/>punctum <var>.i.</var> meæ hic Å¿ubÅ¿criptæ figuræ, quod coreÅ¿pondet K. eius Å¿iguræ adeò diÅ¿tat <lb/>a medio <var>.R.X.</var> trianguli <var>.R.B.D.</var> vt .2. cum .1. 2. dicto medio <var>.R.X.</var> ex .6. </s> - <s xml:id="s1655" xml:space="preserve">Vndecimi fit <lb/>parallela. </s> - <s xml:id="s1656" xml:space="preserve">Idem de reliquis dico. quod manifeÅ¿tè cognoÅ¿ci poteÅ¿t, ab eo, quod in <lb/>Å¿uperius poÅ¿itis figuris corporeis dixi. </s> - <s xml:id="s1657" xml:space="preserve">HuiuÅ¿modi modus ducendi res in perÅ¿pectiua, <lb/>non Å¿olum à Gallis, Å¿ed à Germanis etiam in vÅ¿um reducitur. </s> - <s xml:id="s1658" xml:space="preserve">Sed quia ad hæc <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> <lb/>tempora eiuÅ¿dem perfectionis ratio, quam ego Å¿uperius propoÅ¿ui, <reg norm="nondum" type="context">nõdum</reg> in lucem <lb/>emerÅ¿it, factum fuit, vt <reg norm="errorum" type="context">errorÅ©</reg> laqueis irretirentur, Å¿umentes <var>.T.Q.</var> modo maiorem, <lb/>modo minorem medietate lateris <var>.D.B</var>. </s> - <s xml:id="s1659" xml:space="preserve">Cum hunc igitur modum hic Autor <lb/>vniuerÅ¿alem eſſe putet, labitur in errorem, cum debuiſſet longitudinem ipÅ¿ius <var>.T.Q.</var> <lb/>debere eſſe æqualem medietati ipÅ¿ius <var>.D.B.</var> proferre. </s> - <s xml:id="s1660" xml:space="preserve">Aſſerit deinde diÅ¿tantiam ip-<lb/>Å¿ius <var>.T.Q.</var> à latere <var>.B.D.</var> æ qualem eſſe debere lateri <var>.C.D.</var> quod neceſſarium non eÅ¿t, <lb/>quia in quibuslibet diÅ¿tantijs, iuÅ¿ta operatio fieri poteÅ¿t, quemadmodum in Å¿ubÅ¿cri-<lb/>pta hîc figura facile patet, ideÅ¿t, quòd quibuÅ¿cunque modis <var>.c.D.</var> æqualis remaneat <lb/>ipÅ¿i .1. 2. & Å¿ic interualla, quæ <reg norm="per" type="simple">ê‘</reg> tranÅ¿uerÅ¿um aguntur <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> ad <reg norm="medium" type="context">mediÅ©</reg> trianguli <var>.D.R.B.</var> <lb/>Neque etiam probandus eÅ¿t auctor ille, cum pro oculo, Å¿uum <var>.T.</var> loco <var>.Q.</var> à me poÅ¿i-<unsure/><lb/>ti, ponit, cum is locus Å¿it verus Å¿itus pedis eius quireÅ¿picit, & non oculi. </s> - <s xml:id="s1661" xml:space="preserve">Quòd <reg norm="autem" type="context">autẽ</reg> <lb/>Auctor iÅ¿te, modo vniuerÅ¿ali intelligat, vt iam diximus, <reg norm="conÅ¿ideretur" type="context">cõſideretur</reg> figura tertij mo <lb/>di primi cap. tertiæ partis, in qua Å¿uum oculum (vt ita dicam) ponit in <var>.o.</var> altius Å¿eu <lb/>diÅ¿tans à rectitudine lateris <var>.c.d.</var> plus quam Å¿it totum latus <var>.d.b</var>.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0148-01" xlink:href="fig-0148-01a"> - <image file="0148-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0148-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="13"> - <head xml:id="head191" xml:space="preserve">AD EVNDEM IACOBVM. <lb/>CAP. XIII.</head> - <p> - <s xml:id="s1662" xml:space="preserve"><emph style="sc">TVas</emph> accepiliteras omnis humanitatis & officij plenas, in quibus requiris cau-<lb/>Å¿am, quæ me in alijs meis literis impulit ad <reg norm="dicendum" type="context">dicendÅ©</reg>, <reg norm="angulum" type="context">angulÅ©</reg> <var>.q.o.u.</var> modo ma-<lb/>iorem, modo verò minorem futurum angulo <var>.q.p.u.</var> meæ figuræ corporeæ <var>.A.</var> hanc <lb/>igitur ob cauÅ¿am imagineris in Å¿ubÅ¿cripta hîc figura duo triangula <var>.q.o.u.</var> et <var>.q.p.u.</var> <lb/>quorum <var>.q.p.u.</var> perpendiculariter Å¿it Å¿uper Å¿uperficie trianguli <var>.q.o.p.</var> collocatum, <lb/>præcisè vt in mea figura corporea <var>.A.</var> Å¿uperficies verò trianguli <var>.q.o.p.</var> Å¿it exempli-<lb/>gratia <var>.V.M.</var> & trian-<lb/>guli <var>.u.o.p.</var> Å¿it <var>.V.D.</var> <lb/> - <anchor type="figure" xlink:label="fig-0149-01a" xlink:href="fig-0149-01"/> - quarum <reg norm="communis" type="context">cõmunis</reg> Å¿e-<lb/>ctio Å¿it <var>.V.p.o.x.</var> non <lb/>eÅ¿t enim <reg norm="dubitandum" type="context">dubitãdum</reg> <lb/>quin triangulum <var>.q.<lb/>p.u.</var> Å¿it perpendicula-<lb/>re triangulo <var>.q.o.p.</var> <lb/><reg norm="cum" type="context">cÅ©</reg> hoc ex .18. lib. 11. <lb/>Eucli. perpendicula-<lb/>re Å¿it Å¿uperficiei <var>.a.s.</var> <lb/>in qua reperitur <reg norm="trian- gulum" type="context">triã-<lb/>gulum</reg> <var>.q.p.u.</var> & hoc <lb/>ex linea <var>.o.p.</var> perpendiculari dictæ Å¿uperficiei <var>.a.s</var>. </s> - <s xml:id="s1663" xml:space="preserve">Nunc dico angulum <var>.q.o.u.</var> modo <lb/>maiorem, modo minorem eſſe angulo <var>.q.p.u</var>. </s> - <s xml:id="s1664" xml:space="preserve">Notiſſimum igitur primum nobis - <pb o="138" rhead="IO. BAPT. BENED." n="150" file="0150" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0150"/> - eÅ¿t angulum <var>.p.q.u.</var> obtuÅ¿um eſſe; </s> - <s xml:id="s1665" xml:space="preserve">Imaginemur ergo circa triang ulum <var>.p.q.u.</var> circun-<lb/>Å¿criptum eſſe circulum, cuius portio <var>.p.q.u.</var> minor erit medietate eiuÅ¿dem medij cir-<lb/>culi, vt iam ex <ref id="ref-0023">30. Eucli. lib. tertij</ref> nouiÅ¿ti. </s> - <s xml:id="s1666" xml:space="preserve">nunc imaginemur dictum circulum circum <lb/>lineam <var>.q.u.</var> loco axis verÅ¿us <var>.x.</var> moueri, vnde girus eiuÅ¿dem, per quem tranÅ¿ibat linea <lb/><var>V.x.</var> remouebitur ab eadem linea non nihil cum motus erit à primo Å¿itu vÅ¿quequò <lb/>ad Å¿ecandam dictam lineam <var>.V.x.</var> in alio quodam puncto inter <var>.p.</var> et <var>.x.</var> redibit; </s> - <s xml:id="s1667" xml:space="preserve">quod <lb/>quidem punctum Å¿i <lb/>erit inter <var>.o.</var> et <var>.x.</var> angu <lb/> - <anchor type="figure" xlink:label="fig-0150-01a" xlink:href="fig-0150-01"/> - lus <var>.q.o.u.</var> maior erit <lb/>angulo <var>.q.p.u</var>. </s> - <s xml:id="s1668" xml:space="preserve">Sed Å¿i <lb/>idem <reg norm="punctum" type="context">punctÅ©</reg> erit in-<lb/>ter <var>.p.</var> et <var>.o.</var> dictus an-<lb/>gulus <var>.q.o.u.</var> minor <lb/>erit <var>.q.p.u.</var> de qua <reg norm="qui- dem" type="simple context">ê—-<lb/>dẽ</reg> re tu ipÅ¿e median-<lb/>te .20. lib. 3. et .16. lib. <lb/>primi certior fieri po-<lb/>tes. </s> - <s xml:id="s1669" xml:space="preserve">Valde miror <reg norm="quod" type="simple">ê™</reg> <lb/>hæc Ioannis CuÅ¿ini di <lb/>cta ad hæc vÅ¿que tempora tanto in prætio Å¿int habita, vt ab excellentibus Å¿criptori-<lb/>bus quaÅ¿i Å¿i proprij eorum ingenij partus eſſent, de verboad verbum vt theÅ¿auros, in <lb/>fuis <reg norm="ipÅ¿orummet" type="context">ipÅ¿orÅ©met</reg> libris reÅ¿cripta fuerint, quemadmodum iam omnes admonui in mea <lb/>gnomonica Orontium, MunÅ¿terum, <reg norm="aliosque" type="simple">aliosqÌ;</reg> permultos feciſſe.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0149-01" xlink:href="fig-0149-01a"> - <image file="0149-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0149-01"/> - </figure> - <figure xlink:label="fig-0150-01" xlink:href="fig-0150-01a"> - <image file="0150-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0150-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="14"> - <head xml:id="head192" xml:space="preserve">CAP. XIIII.</head> - <p> - <s xml:id="s1670" xml:space="preserve"><emph style="sc">Ex</emph> ijs, qu æ de nonnullis effectibus ducendo in perÅ¿pectiua tertÃum corpus regu <lb/>lare, <reg norm="quod" type="simple">ê™</reg> octo triangulis æquilateribus eÅ¿t term inatum, Å¿cire deÅ¿ideras, hoc <reg norm="vnum" type="context">vnÅ©</reg> <lb/>eÅ¿t caput: </s> - <s xml:id="s1671" xml:space="preserve">vnde fiat, aut quomodo probetur quaÅ¿libet duas facies oppoÅ¿itas eiuÅ¿-<lb/>dem corporis octoaedri <reg norm="inuicem" type="context">inuicẽ</reg> æquidiÅ¿tantes eſſe. </s> - <s xml:id="s1672" xml:space="preserve">Quamobrem Å¿it hîc <reg norm="Å¿ubÅ¿criptum" type="context">Å¿ubÅ¿criptÅ©</reg> <lb/><reg norm="octoaedrum" type="context">octoaedrÅ©</reg>, cuius diameter vna Å¿it <var>.b.q.</var> et <var>.b.p.<lb/>l.</var> vna ex faciebus, cui opponatur facies <var>.q.k.</var> <lb/> - <anchor type="figure" xlink:label="fig-0150-02a" xlink:href="fig-0150-02"/> - d. quas <reg norm="adinuicem" type="context">adinuicẽ</reg> æquidiÅ¿tantes eſſe contendo <lb/>Å¿int aliæ duæ facies, quæ inter has ponuntur <var>.<lb/>b.d.k.</var> et <var>.q.p.l.</var> & à punctis extremis <var>.b.q.</var> dia-<lb/>metri. ductæ Å¿int quatuor lineæ <var>.b.a</var>: <var>b.u</var>: <var>q.a</var>: <var>q.<lb/>u.</var> ad puncta <var>.a.</var> et <var>.u.</var> diuidentia <var>.k.d.</var> et <var>.l.p.</var> per <lb/>medium, vnde ex 4. primi Eucli. quatuor hæ <lb/>lineæ adinuicem Ä™quales erunt <reg norm="Å¿umendo" type="context">Å¿umẽdo</reg> eas vt <lb/>baÅ¿es <reg norm="triangulorum" type="context">triangulorÅ©</reg> <var>.a.d.b</var>: <var>u.l.b</var>: <var>a.d.q.</var> et <var>.u.l.q.</var> <lb/><reg norm="adinuicem" type="context">adinuicẽ</reg> <reg norm="quoque" type="simple">quoq;</reg> <reg norm="æquidiÅ¿tabunt" type="simple context">æê—diÅ¿tabÅ©t</reg> <var>.a.b.</var> ab <var>.u.q.</var> et <var>.b.<lb/>u.</var> ab <var>.q.a.</var> ex .27. primi; </s> - <s xml:id="s1673" xml:space="preserve"><reg norm="quia" type="simple">ê—a</reg> Å¿i imaginabimur dia <lb/>metrum <var>.b.q.</var> tunc ex .4. aut ex .8. eiuÅ¿dem lib. <lb/>habebimus angulos <var>.a.b.q.</var> et <var>.u.q.b.</var> æquales <lb/>inuicem; </s> - <s xml:id="s1674" xml:space="preserve">Å¿ed ob eaÅ¿dem rationes <var>.p.l.</var> paralle-<lb/>la eÅ¿t ipÅ¿i <var>.d.k.</var> vnde ex 15. lib. 11. facies <var>.b.p.l.</var> <lb/>parallela fit, aut æquidiÅ¿tans ipÅ¿i <var>.q.d.k.</var> ideÅ¿t <lb/>primum propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0150-02" xlink:href="fig-0150-02a"> - <image file="0150-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0150-02"/> - </figure> - </div> - <pb o="139" rhead="DE PERSPECT." n="151" file="0151" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0151"/> - <p> - <s xml:id="s1675" xml:space="preserve">Ad habendam deinde quantitatem diÅ¿tantiæ, aut interualli Å¿imul cum Å¿itu, in fa-<lb/>cie <var>.q.d.k.</var> quem latus <var>.p.l.</var> perpendiculariter reÅ¿picit. </s> - <s xml:id="s1676" xml:space="preserve">Imaginemur à puncto <var>.u.</var> Å¿uper <lb/><var>q.a.</var> cad ere lineam perpendicularem <var>.u.o.</var> quæ illico reperitur cum triangulum <var>.a.<lb/>u.q.</var> ex lateribus datis & cognitis conÅ¿tet, <reg norm="quodquidem" type="context">quodquidẽ</reg> triangulum, medietas eÅ¿t qua-<lb/>drilateri, Å¿eu. rumbi <var>.q.a.b.u.</var> cui vnaquæque dictarum quatuor facierum perpendi-<lb/>cularis exiÅ¿tit ex .4. ct .18. lib. 11. & ob id linea <var>.u.o.</var> extenÅ¿a in Å¿uperficie dicti quadri-<lb/>lateri, & perpendicularis lineæ <var>.q.a.</var> perpendicularis erit faciei <var>.q.d.k.</var> & ex .29. <lb/>primi, angulus <var>.b.u.o.</var> rectus erit, ut <reg norm="etiam" type="context">etiã</reg> angulus <var>.o.u.l.</var> ex .2. definitione lib. 11. vnde <lb/>ex .4. eiuÅ¿dem lib <var>.o.u.</var> perpendicularis erit faciei <var>.b.p.l</var>. </s> - <s xml:id="s1677" xml:space="preserve">Ha bebimus ergo Å¿itum in fa-<lb/>cie <var>.q.d.k.</var> qui reÅ¿picietur ad angulos rectos à linea <var>.p.l.</var> quiquidem erit in perpendi-<lb/>culari à puncto <var>.o.</var> ad <var>.q.a.</var> ducta.</s> - </p> - <p> - <s xml:id="s1678" xml:space="preserve">Quòd autem <var>.a.o.</var> Å¿it latus exagoni æquilateris circumÅ¿crip tibilis ab eodem circu <lb/>lo, qui vnam ex faciebus triangularibus æquilateribus propoÅ¿iti corporis circunÅ¿cri-<lb/>bere pot eÅ¿t, ita oÅ¿tenditur. Å¿it <reg norm="comprehenÅ¿um" type="context">cõprehenÅ¿um</reg> imaginatione, triangulum <var>.a.q.u.</var> Å¿epara <lb/>tim, cuius latus <var>.a.u.</var> æquale eÅ¿t vni ex lateribus <reg norm="triangulorum" type="context">triangulorÅ©</reg> eiuÅ¿dem corporis ex .33. <lb/>primi, quo dlibet verò aliorum duorum æquale perpendicularibus dictorum trian-<lb/>gulorum, in quo triangulo <var>.a.u.q.</var> ducta Å¿it perpendicularis <var>.u.o.</var> ab vna <reg norm="extremitatum" type="context">extremitatÅ©</reg> <lb/>lateris maioris, ad vnum ex minoribus lateribus, quÄ™ perpendicularis intra triangu-<lb/>lum cadet, quia dictum triangulum oxigonium eÅ¿t. </s> - <s xml:id="s1679" xml:space="preserve">quod autem attinet ad duos angu <lb/>los <var>.a.</var> et <var>.u.</var> cum æquales Å¿int ex quinta lib. primi; </s> - <s xml:id="s1680" xml:space="preserve">17. nos certiores facit; </s> - <s xml:id="s1681" xml:space="preserve">quod verò anÂ<lb/>gulus <var>.q.</var> Å¿it <reg norm="etiam" type="context">etiã</reg> acutus: </s> - <s xml:id="s1682" xml:space="preserve">30. lib. tertii nos cer-<lb/>tos reddit, <reg norm="quia" type="simple">ê—a</reg><var>.a.u.</var> minor eÅ¿t diametro <reg norm="Å¿phae ræ" type="simple">Å¿phÄ™Â<lb/> - <anchor type="figure" xlink:label="fig-0151-01a" xlink:href="fig-0151-01"/> - ræ</reg> datum corpus circumÅ¿cribentis, cum <var>.q.</var> <lb/>dictæ Å¿phÄ™rÄ™ Å¿uperficiem tangat.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0151-01" xlink:href="fig-0151-01a"> - <image file="0151-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0151-01"/> - </figure> - </div> - <p> - <s xml:id="s1683" xml:space="preserve">Ad probandum <var>.a.o.</var> Ä™qualem eſſe lateri <lb/>exagoni dicti, Å¿atis erit probare <var>.a.q.</var> Å¿eÅ¿qui <lb/>alteram eſſe ad <var>.a.o.</var> quia Å¿i in Å¿ubÅ¿cripto <lb/>hîc circulo ducemus duas Å¿emidiametros <var>.<lb/>n.p.</var> et <var>.n.l.</var> ad. angulos <reg norm="trianguli" type="context">triãguli</reg> Ä™quilateri <var>.p.</var> <lb/>et <var>.l.</var> & cum quodlibet laterum ipÅ¿ius exago <lb/>ni, Ä™quale Å¿it Å¿emidiametro circuli ex .15. <lb/>lib. 4. habebimus ex .8. primi, angulum <var>.n.<lb/>p.l.</var> æqualem angulo <var>.q.p.l</var>. </s> - <s xml:id="s1684" xml:space="preserve">Vnde ex .4. eiuÅ¿ <lb/>dem <var>.o.n.</var> Ä™qualiserit ipÅ¿i <var>.o.q.</var> ideÅ¿t <var>.q.a.</var> Å¿eÅ¿ <lb/>quialtera erit ad <var>.a.o</var>.</s> - </p> - <p> - <s xml:id="s1685" xml:space="preserve">Ad probandum nunc in triangulo <var>.a.q.<lb/>u</var>: <var>a.q.</var> Å¿eÅ¿quialteram eſſe ad <var>.a.o.</var> eÅ¿t <reg norm="quoque" type="simple">quoq;</reg> <lb/>Å¿ciendum primò omne latus trianguli Ä™quilateri in potentia Å¿eÅ¿quitertium eſſe ad <lb/>perpendicularem eiuÅ¿dem trianguli, quod vndecima lib. 14. Eucli. breuiter demon <lb/>Å¿tratum eÅ¿t.</s> - </p> - <pb o="140" rhead="IO. BAPT. BENED." n="152" file="0152" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0152"/> - <p> - <s xml:id="s1686" xml:space="preserve">Ponamus nunc quadratum lateris <var>.a.u.</var> eſſe .12. clarum erit quodlibet quadratum <lb/>aliorum duorum laterum <var>.a.q.</var> et <var>.u.q.</var> futurum nouem, ex ijs quæ poÅ¿teriore loco dixi <lb/>mus, & quia quadratum ipÅ¿ius <var>.q.a.</var> eÅ¿t tantò minus aliorum duorum quadratorum <lb/>Å¿umma, quantum eÅ¿t duplum producti ipÅ¿ius <var>.q.a.</var> in <var>.a.o.</var> ex .13. Å¿ecundi, Å¿ed alia duo <lb/>quadrata Å¿imul collecta faciunt .21. à quo numero Å¿ubtrahendo quadratum ipÅ¿ius <var>.a.<lb/>q.</var> ideÅ¿t nouem, remanebit numerus .12. pro duplo producti ipÅ¿ius <var>.q.a.</var> in <var>.a.o.</var> cuius <lb/>dupli me-<lb/> - <anchor type="figure" xlink:label="fig-0152-01a" xlink:href="fig-0152-01"/> - dia pars, id-<lb/>eÅ¿t Å¿implex <lb/>productum <lb/>ipÅ¿ius <var>.q.a.</var> <lb/><reg norm="in" type="wordlist">Ä©</reg> <var>a.o.</var> erit 6. <lb/>Sed <reg norm="quia" type="simple">ê—a</reg> qua <lb/>dratum ip-<lb/>Å¿ius <var>.q.a.</var> eÅ¿t <lb/>nouem, <lb/>eius radix <var>.<lb/>q.a.</var> crit .3. <lb/>per <reg norm="quam" type="context">quã</reg> di-<lb/>uidendo .6. <lb/>productum <lb/>ipÅ¿ius <var>.q.a.</var> <lb/>in <var>.a.o.</var> pro <lb/>latere <var>.a.o.</var> <lb/>conÅ¿urgent <lb/>duo, cum er <lb/>go <var>.a.o.</var> Å¿int <lb/>duo tertia <lb/>ipÅ¿ius <var>.a.q.</var> <lb/>certi <reg norm="erimus" type="simple">erimê°</reg> <lb/><var>a.o.</var> eſſe latus dicti exagoni.</s> - </p> - <div type="float" level="4" n="3"> - <figure xlink:label="fig-0152-01" xlink:href="fig-0152-01a"> - <image file="0152-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0152-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="15"> - <head xml:id="head193" xml:space="preserve">CAP. XV.</head> - <p> - <s xml:id="s1687" xml:space="preserve"><reg norm="DEÅ¿iderantes" type="context">DEÅ¿iderãtes</reg> Å¿cire deinde <var>.l.k.</var> in figura <var>.M.</var> quar <lb/> - <anchor type="figure" xlink:label="fig-0152-02a" xlink:href="fig-0152-02"/> - ti cap. tertiæ partis perÅ¿pectiuÄ™ Danielis <lb/>Barbari, Å¿eu Zamberti, eſſe veram <reg norm="altitudinem" type="context">altitudinẽ</reg> cor-<lb/>poris octoaedri, <reg norm="primum" type="context">primÅ©</reg> Å¿cire debemus <reg norm="quod" type="simple">ê™</reg> <reg norm="exiÅ¿tente" type="context">exiÅ¿tẽte</reg> <var>.b.<lb/>h.</var> vt <reg norm="etiam" type="context">etiã</reg> <var>.b.l.</var> tripla ad <var>.b.k.</var> vt ex ijs, quÄ™ Å¿uperius <reg norm="iam" type="context">iã</reg> <lb/>diximus, facile percipi poteÅ¿t; </s> - <s xml:id="s1688" xml:space="preserve">ex penultima primi <var>.<lb/>b.l.</var> in potentia, Å¿eÅ¿quioctaua erit ad <var>.k.l.</var> ipÅ¿a et <var>.k.<lb/>l.</var> dupla <reg norm="inpotentia" type="context">inpotẽtia</reg> ad <var>.h.k.</var> & ob id ducta <reg norm="cum" type="context">cÅ©</reg> eſſet <var>.h.<lb/>l.</var> exiÅ¿teret in potentia tripla ad <var>.h.k.</var> & Å¿eÅ¿quialtera <lb/>ad <var>.l.k.</var> & Å¿eÅ¿quitertia ad <var>.l.b.</var> & Å¿ic ad <var>.h.b.</var> vnde <var>.l.h.</var> <lb/>æqualis eſſet vni ex lateribus <reg norm="trianguli" type="context">triãguli</reg> Ä™quilateri di-<lb/>cti corporis. </s> - <s xml:id="s1689" xml:space="preserve">Ex rationibus igitur Å¿uperius hîc poÅ¿i-<lb/>tis <var>.l.k.</var> erit altitudo dicta, id eÅ¿t diÅ¿tantia inter duas <lb/>facies inuicem oppoÅ¿itas, octoaedri.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0152-02" xlink:href="fig-0152-02a"> - <image file="0152-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0152-02"/> - </figure> - </div> - <p> - <s xml:id="s1690" xml:space="preserve"><reg norm="Neque" type="simple">Neq;</reg> volo te ignorare <reg norm="alium" type="context">aliÅ©</reg> <reg norm="non" type="context">nõ</reg> <reg norm="paruum" type="context">paruÅ©</reg> fuiſſe <reg norm="errorem" type="context">errorẽ</reg> <lb/>illius Zamberti: </s> - <s xml:id="s1691" xml:space="preserve">cum <reg norm="eodem" type="context">eodẽ</reg> capite affirmet angulos <lb/>octoacdri rectos eſſe <reg norm="cum" type="context">cÅ©</reg> Å¿int acuti, <reg norm="nam" type="context">nã</reg> <reg norm="vnuÅ¿quiÅ¿que" type="simple">vnuÅ¿quiÅ¿q;</reg> minor eÅ¿t angulo cubi Å¿olido.</s> - </p> - <pb o="141" n="153" file="0153" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0153"/> - </div> - </div> - <div type="chapter" level="2" n="3"> - <head xml:id="head194" xml:space="preserve">DE MECHANICIS.</head> - <p style="it"> - <s xml:id="s1692" xml:space="preserve"><emph style="sc">SCripservnt</emph> multi multa, & quidem Å¿citißimè, de mechn<unsure/>-<lb/>nicis, at cum natura <reg norm="vÅ¿usque" type="simple">vÅ¿usq;</reg> aliquid Å¿emper vel nouum, vel <lb/>Latens in apertum emittere Å¿oleant, nec ingenui aut grati Å¿it <lb/>animi, posteris inuidere, Å¿i quid ei contigerit comperuiße prius <lb/>tenebris inuolutum: </s> - <s xml:id="s1693" xml:space="preserve">cum tam multa ipÅ¿e ex aliorum diligentia <lb/>Å¿it conÅ¿equut us. </s> - <s xml:id="s1694" xml:space="preserve">Paucula <reg norm="quædam" type="context">quædã</reg> futùra, vt reor, non ingrata his <lb/>qui in biÅ¿ce mechanicis verÅ¿antur, nuÅ¿quam ante bac tentata, <lb/>aut Å¿atis exastè explicata in medium proferre volui: </s> - <s xml:id="s1695" xml:space="preserve">quo vel iuuandi deÅ¿iderium, vel <lb/>Å¿altem non ocioÅ¿i ingenioli argumentum aliquod exbiberem: </s> - <s xml:id="s1696" xml:space="preserve">at que vel boc vno modo me <lb/>inter bumanos vixiſſe testatum relinquerem.</s> - </p> - <div type="section" level="3" n="1"> - <head xml:id="head195" style="it" xml:space="preserve">De differentia Å¿itus brachiorum libra.</head> - <head xml:id="head196" xml:space="preserve">CAP.I.</head> - <p> - <s xml:id="s1697" xml:space="preserve"><emph style="sc">OMne</emph> pondus poÅ¿itum in extremitate alicuius brachij libræ maiorem, aut mi-<lb/> - <anchor type="handwritten" xlink:label="hd-0153-01a" xlink:href="hd-0153-01"/> - norem grauitatem habet, pro diuerÅ¿a ratione Å¿itus ipÅ¿ius brachij. </s> - <s xml:id="s1698" xml:space="preserve">Å¿it exe<unsure/>mpli <lb/>gratia <var>.B.</var> centrum, aut, quod diuidit brachia alicuius libræ, & <var>.A.B.Q.</var> vertica-<lb/>lis linea, aut, vt rectius dicam, axis orizontis, & <var>.B.C.</var> vnum brachium dictæ li-<lb/>bræ, & in <var>.C.</var> Å¿it pondus, & <var>.C.O.</var> linea inclinationis, Å¿euicineris<unsure/> <var>.C.</var> verÅ¿us cen-<lb/>trum mundi, cum qua <var>.B.C.</var> angulum rectum conÅ¿tituat in puncto <var>.C</var>. </s> - <s xml:id="s1699" xml:space="preserve">ExiÅ¿tente <lb/>igitur in huiuÅ¿modi Å¿itu brachio <var>.B.C.</var> dico pondus <var>.C.</var> grauius futurum, quam <lb/>in alio quolibet Å¿itu. </s> - <s xml:id="s1700" xml:space="preserve">quia Å¿upra centrum <var>.B.</var> omninò non quieÅ¿cet, quemadmodum <lb/>in quouis alio Å¿itu faceret. </s> - <s xml:id="s1701" xml:space="preserve">Ad quod intelligendum, Å¿it dictum brachium, in Å¿itu <var>.B.<lb/>F.</var> cum eodem pondere in puncto <var>.F.</var> & linea itineris Å¿eu inclinationis dicti ponderis <lb/>Å¿it <var>.F.u.M.</var> per quam lineam dictum pondus progredi non poteÅ¿t, niÅ¿i brachium <var>.B.F.</var> <lb/>breuius redderetur. </s> - <s xml:id="s1702" xml:space="preserve">Vnde clarum erit <lb/> - <anchor type="figure" xlink:label="fig-0153-01a" xlink:href="fig-0153-01"/> - quòd pondus <var>.F.</var> aliquantulum Å¿upra cen <lb/>trum <var>.B.</var> mediante brachio <var>.B.F.</var> nititur. <lb/></s> - <s xml:id="s1703" xml:space="preserve">EÅ¿t quidem verum, quòd pondus <var>.C.</var> nec <lb/>ipÅ¿um etiam per lineam <var>.C.O.</var> proficiÅ¿ce-<lb/>tur, quia iter extremitatis brachij eÅ¿t cir-<lb/>cularis, & <var>.C.O.</var> in vno <reg norm="quodam" type="context">quodã</reg> puncto eÅ¿t <lb/>contingens. </s> - <s xml:id="s1704" xml:space="preserve">Sit hociter <var>.A.C.Q</var>. </s> - <s xml:id="s1705" xml:space="preserve">Opor-<lb/>tet nunc præſupponere pondus extremi-<lb/>tatis brachij deberetanto magis <reg norm="centro" type="context">cẽtro</reg> <var>.B.</var> <lb/>inniti, quanto magis linea Å¿uæ inclinatio-<lb/>nis (ponamus <var>.F.u.M.</var>) propinqua erit di <lb/>cto centro <var>.B.</var> quod Å¿equenti cap. proba-<lb/>bo, vt exempli gratia, Å¿it <var>.F.</var> Å¿uper <var>.u.</var> pun-<lb/>ctum medij ex æquo inter <var>.C.</var> et <var>.B.</var> qua-<lb/>propter <var>.u.B.</var> æqualis erit <var>.u.C.</var> vndeÅ¿e- - <pb o="142" rhead="IO. BAPT. BENED." n="154" file="0154" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0154"/> - quetur dictum pondus grauius futurum pro parte <var>.F.C.</var> quam pro ea, quæ eÅ¿t <var>.A.F.</var> & <lb/>minus Å¿upra centrum <var>.B.</var> pro dicta parte <var>.F.C.</var> quam pro parte <var>.A.F.</var> quieturum; </s> - <s xml:id="s1706" xml:space="preserve">& <lb/>dictum brachium quanto magis orizontale erit à ſitu <var>.B.F.</var> tantò minus-Å¿upra dictum <lb/>centrum <var>.B.</var> quieÅ¿cet, & hac ratione grauius quoque erit, & quanto magis vicinum <lb/>erit ipÅ¿i <var>.A.</var> à dicto <var>.F.</var> tantò magis Å¿uper centrum <var>.B.</var> quoque quieÅ¿cet, vnde <reg norm="tantò" type="context">tãtò</reg> quo-<lb/>que leuius exiÅ¿tet. </s> - <s xml:id="s1707" xml:space="preserve">Idem dico de omni Å¿itu brachij per girum inferiorem <var>.C.Q.</var> vbi <lb/>pondus pendebit à centro <var>.B.</var> dictum centrum attrahendo, quemadmodum Å¿uperius <lb/>illud impellebat. </s> - <s xml:id="s1708" xml:space="preserve">Hæc verò omnia cap. Å¿equenti melius percipientur.</s> - </p> - <div type="float" level="4" n="1"> - <handwritten xlink:label="hd-0153-01" xlink:href="hd-0153-01a"/> - <figure xlink:label="fig-0153-01" xlink:href="fig-0153-01a"> - <image file="0153-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0153-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="2"> - <head xml:id="head197" style="it" xml:space="preserve">De proportione ponderis extremitatis brachij libr & <lb/>in diuerÅ¿o Å¿itu ab orizontali.</head> - <head xml:id="head198" xml:space="preserve">CAP. II.</head> - <p> - <s xml:id="s1709" xml:space="preserve"><emph style="sc">PRoportio</emph> ponderis in <var>.C.</var> ad idem pondus in F. erit quemadmodum totius <lb/>brachij <var>.B.C.</var> ad partem <var>.B.u.</var> poÅ¿itam inter centrum & lineam <var>.F.u.M.</var> inclinatio-<lb/>nis, quam pondus ab extremitate <var>.F.</var> liberum verÅ¿us mundi <reg norm="centrum" type="context">centrÅ©</reg> conficeret. </s> - <s xml:id="s1710" xml:space="preserve">Quod <lb/>vt facilius intelligamus imaginemur <reg norm="alterum" type="context">alterÅ©</reg> brachium libræ <var>.B.D.</var> & in extremo <var>.D.</var> <lb/>locatum aliquod pondus minus pondere <var>.C.</var> vt <var>.B.u.</var> pars <var>.B.C.m.</var> nor eÅ¿t <var>.B.D.</var> cla-<lb/>rè cognoÅ¿cetur ex .6. lib. primi de ponderibus Archimedis, quòd Å¿i in puncto <var>.u.</var> col-<lb/>locatum erit pondus ipÅ¿ius <var>.C.</var> libra nihil penitus à ſitu orizontali dimouebitur. </s> - <s xml:id="s1711" xml:space="preserve">Sed <lb/>perinde eÅ¿t quòd pondus <var>.F.</var> æquale <var>.C.</var> Å¿it in extremo <var>.F.</var> in Å¿itu brachij <var>.B.F.</var> <reg norm="quam" type="context">quã</reg> vt Å¿it <lb/>in puncto <var>.u.</var> in Å¿itu ipÅ¿ius <var>.B.u.</var> orizontali. </s> - <s xml:id="s1712" xml:space="preserve">Ad cuius rei euidentiam imaginemur <reg norm="filum" type="context">filÅ©</reg> <var>.<lb/>F.u.</var> perpendiculare, & in cuius extremo <var>.u.</var> pendere pondus, quod erat in <var>.F.</var> vnde cla <lb/>rum erit quòd eundem effectum gignet, ac Å¿i fuiſſet in <var>.F.</var> quod, vt iam diximus re-<lb/>manens affixum puncto <var>.u.</var> brachij <var>.B.u.</var> tantò minus graue eÅ¿t Å¿itu ipÅ¿ius <var>.C.</var> quantò <var>.u.<lb/>B.</var> minus eÅ¿t ipÅ¿o <var>.B.C</var>. </s> - <s xml:id="s1713" xml:space="preserve">Idem aſſero Å¿i brachium eſſet in Å¿itu <var>.e.B.</var> quod facilè cogno-<lb/>Å¿cere poterimus, Å¿i imaginemur filum appenÅ¿um ipÅ¿i <var>.u.</var> brachij <var>.B.C.</var> & vÅ¿que ad <var>.e.</var> <lb/><reg norm="perpendicularem" type="context">perpendicularẽ</reg>, in quo extremo <reg norm="appensum" type="context">appensÅ©</reg> eſſet pondus æquale ponderi <var>.C.</var> & <reg norm="liberum" type="context">liberÅ©</reg> <lb/>ab <var>.e.</var> brachij <var>.B.e.</var> vnde libra orizontalis manebit. </s> - <s xml:id="s1714" xml:space="preserve">Sed Å¿i brachium <var>.B.e.</var> conÅ¿olida-<lb/>tum fuiſſet in tali Å¿itu cum orizontali <var>.B.D.</var> <lb/> - <anchor type="figure" xlink:label="fig-0154-01a" xlink:href="fig-0154-01"/> - & <reg norm="appenÅ¿o" type="context">appẽſo</reg> <reg norm="pondere" type="context">põdere</reg> <var>.C.</var> in <var>.e.</var> libero à filo, nec <lb/><reg norm="aÅ¿cenderet" type="context">aÅ¿cẽderet</reg>, <reg norm="neque" type="simple">neq;</reg> deÅ¿cenderet. </s> - <s xml:id="s1715" xml:space="preserve">quia tantum <lb/>eÅ¿t quod ipÅ¿um Å¿it appenÅ¿um filo, <reg norm="quod" type="simple">ê™</reg> pendet <lb/>ab <var>.u.</var> quantum quòd ab ipÅ¿o liberum <reg norm="appem" type="context">appẽ</reg> <lb/>nÅ¿um fuiſſet <var>.e.</var> brachij <var>.B.e.</var> & hoc procede <lb/>ret ab eo quòd partim pendereta centro <var>.<lb/>B.</var> & Å¿i <reg norm="brachium" type="context">brachiÅ©</reg> eſſet in Å¿itu <var>.B.Q.</var> totum <reg norm="pon" type="context">põ</reg> <lb/>dus centro <var>.B.</var> remaneret appenÅ¿um, <reg norm="quem- admodum" type="context">quem-<lb/>admodÅ©</reg> in Å¿itu <var>.B.A.</var> <reg norm="totum" type="context">totÅ©</reg> dicto centro an-<lb/>niteretur. </s> - <s xml:id="s1716" xml:space="preserve">vnde fit vt hoc modo pondus <lb/>magis aut minus Å¿it graue, quò magis <lb/>aut minus à centro pendet, aut eidem niti-<lb/>tur: </s> - <s xml:id="s1717" xml:space="preserve"><reg norm="atque" type="simple">atq;</reg> hæc eÅ¿t cauÅ¿a proxima, & per Å¿e, <lb/> - <anchor type="handwritten" xlink:label="hd-0154-01a" xlink:href="hd-0154-01"/> - qua fit vt vnum <reg norm="idemque" type="simple">idemq;</reg> pondus in vno eo-<lb/><reg norm="demque" type="simple">demqÌ;</reg> medio magis aut minus graue exi- - <pb o="143" rhead="DE MECHAN." n="155" file="0155" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0155"/> - Å¿tat. </s> - <s xml:id="s1718" xml:space="preserve">Et quamuis appellem latus <var>.B.C.</var> orizontale, Å¿upponens illud angulum rectum <lb/>cum <var>.C.O.</var> facere, vnde angulus <var>.C.B.Q.</var> fit vt minor Å¿it recto, ob quantitatem vnius <lb/>anguli Ä™qualis ei, quem duæ <var>.C.O.</var> et <var>.B.Q.</var> in centro regionis <reg norm="elementaris" type="context">elemẽtaris</reg> <reg norm="conÅ¿tituunt" type="context">conÅ¿tituÅ©t</reg>, <lb/>hoc tamen nihil refert, cum dictus angulus inÅ¿enÅ¿ibilis Å¿it magnitudinis. </s> - <s xml:id="s1719" xml:space="preserve">Ab iÅ¿tis au-<lb/>tem rationibus elicere poſſumus, quod Å¿i punctus <var>.u.</var> erit ex æquo medius inter cen-<lb/>trum <var>.B.</var> & extremum <var>.C.</var> pondus <var>.F.</var> aut <var>.M.</var> pendebit, aut nitetur pro medietate dicto <lb/>centro <var>.B.</var> & Å¿i dictum <var>.u.</var> erit propius <var>.B.</var> quam puncto <var>.C.</var> pendebit ab ipÅ¿o, aut nitetur <lb/>ipÅ¿i amplius <reg norm="quam" type="context">quã</reg> exmedietate, & Å¿i magis verÅ¿us <var>.C.</var> minus <reg norm="quam" type="context">quã</reg> ex medietate <reg norm="nitetur" type="simple">nitet̃</reg>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0154-01" xlink:href="fig-0154-01a"> - <image file="0154-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0154-01"/> - </figure> - <handwritten xlink:label="hd-0154-01" xlink:href="hd-0154-01a"/> - </div> - </div> - <div type="section" level="3" n="3"> - <head xml:id="head199" style="it" xml:space="preserve">Quòd quantit as cuiuÅ¿libet ponderis, aut uirtus mouens re-<lb/>Å¿pectu alterius quantitatis cognoÅ¿catur beneficio <lb/>perpendicularium ductarum à centro <lb/>libr & ad line am inclinationis.</head> - <head xml:id="head200" xml:space="preserve">CAP. III.</head> - <p> - <s xml:id="s1720" xml:space="preserve">EX ijs, quæ à nobis hucuÅ¿que Å¿unt dicta, facilè intelligi poteÅ¿t, <reg norm="quod" type="simple">ê™</reg> quantitas <var>.B.u.</var> <lb/>quæ ferè perpendicularis eÅ¿t à centro <var>.B.</var> ad lineam <var>.F.u.</var> inclinationis, ea eÅ¿t, <lb/> - <anchor type="handwritten" xlink:label="hd-0155-01a" xlink:href="hd-0155-01"/> - quæ nos ducit in cognitionem quantitatis virtutis ipÅ¿ius <var>.F.</var> in huiuÅ¿modi Å¿itu, conÅ¿ti <lb/>tuens videlicet linea <var>.F.u.</var> cum brachio <var>.F.B.</var> angulum acutum <var>.B.F.u</var>. </s> - <s xml:id="s1721" xml:space="preserve">Vt hoc tamen <lb/>melius intelligamus, imaginemur libram <var>.b.o.a.</var> fixam in centro <var>.o.</var> ad. cuius etrema <lb/>Å¿int appenÅ¿a duo pondera, aut duæ virtutes mouentes <var>.e.</var> et <var>.c.</var> ita tamen <reg norm="quod" type="simple">ê™</reg> linea incli-<lb/>nationis <var>.e.</var> ideÅ¿t <var>.b.e.</var> faciat angulum rectum cum <var>.o.b.</var> in puncto <var>.b.</var> linea verò inclina <lb/>tionis <var>.c.</var> ideÅ¿t <var>.a.c.</var> faciat angulum acutum, aut obtuÅ¿um cum <var>.o.a.</var> in puncto <var>.a</var>. </s> - <s xml:id="s1722" xml:space="preserve">Imagi-<lb/>nemur ergo lineam <var>.o.t.</var> perpendicularem lineæ <var>.c.a.</var> inclinationis, vnde <var>.o.t.</var> minor <lb/>erit <var>.o.a.</var> ex .18. primi Euclidis. Å¿ecetur deinde imaginatione <var>o.a.</var> in puncto <var>.i.</var> ita ut <lb/><var>o.i.</var> æqualis. </s> - <s xml:id="s1723" xml:space="preserve">Å¿it <var>.o.t.</var> & puncto <var>.i.</var> appenÅ¿um Å¿it pondus æquale ipÅ¿i <var>.c.</var> cuius inclinationis <lb/>linea parallela Å¿it lineæ inclinationis ponderis <var>.e.</var> Å¿upponendo tamen pondus aut vir <lb/>tutem <var>.c.</var> ea ratione maiorem eſſe ea, quæ eÅ¿t <var>.e.</var> qua <var>.b.o.</var> maior eÅ¿t <var>.o.t.</var> abÅ¿que dubio <lb/>ex .6. lib. primi Archi. de ponderibus <var>.b.o.i.</var> non mouebitur Å¿itu, Å¿ed Å¿i loco <var>.o.i.</var> imagi <lb/>nabimur <var>.o.t.</var> conÅ¿olidatam cum <var>.o.b.</var> & per lineam <var>.t.c.</var> attractam virtute <var>.c.</var> Å¿imiliter <lb/>quoque continget ut <var>b.o.</var> t; </s> - <s xml:id="s1724" xml:space="preserve">communi quadam Å¿cientia, non moueatur Å¿i tu. </s> - <s xml:id="s1725" xml:space="preserve">EÅ¿t ergo <lb/> - <anchor type="handwritten" xlink:label="hd-0155-02a" xlink:href="hd-0155-02"/> - quod propoÅ¿uimus verum quantitatem alicuius ponderis reÅ¿pectu ad eam, quæ eÅ¿t <lb/>alterius debere depræhendi à perpendicularibus, quæ à centro libræ ad lineas incli <lb/>nationis exiliunt. </s> - <s xml:id="s1726" xml:space="preserve">Hinc autem innoteÅ¿cit facillimè, quantum vigoris, & vis pondus, <lb/>aut virtus <var>.c.</var> ad angulum rectum cum <var>.o.a.</var> minimè trahens, amitttat. </s> - <s xml:id="s1727" xml:space="preserve">Hinc quoque co <lb/>rollarium quoddam Å¿equetur, quò d quantò propinquius erit centrum <var>.o.</var> libræ cen-<lb/>tro regionis elementaris, tantò quo que minus erit graue.</s> - </p> - <div type="float" level="4" n="1"> - <handwritten xlink:label="hd-0155-01" xlink:href="hd-0155-01a"/> - <handwritten xlink:label="hd-0155-02" xlink:href="hd-0155-02a"/> - </div> - <figure position="here"> - <image file="0155-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0155-01"/> - </figure> - <pb o="144" rhead="IO. BAPT. BENED." n="156" file="0156" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0156"/> - </div> - <div type="section" level="3" n="4"> - <head xml:id="head201" style="it" xml:space="preserve">Quemadmodum exÅ¿upradictis cauÅ¿is omnes staterarum & <lb/>uectium cauſæ dependeant.</head> - <head xml:id="head202" xml:space="preserve">CAP. IIII.</head> - <p> - <s xml:id="s1728" xml:space="preserve">VIs brachij longioris alicuius Å¿tateræ, aut vectis, maior breuioris, ab ijs, quæ in Å¿u <lb/>perioribus capitibus diximus, ideÅ¿t <reg norm="quod" type="simple">ê™</reg> nitatur <reg norm="pendeatuem" type="context">pendeatuẽ</reg> magis aut minus à <lb/>centro pondus in extremitate brachij maioris poÅ¿itum, oboritur. </s> - <s xml:id="s1729" xml:space="preserve">Quamobrem illud <lb/>à nobis primò eÅ¿t cognoÅ¿cendum, Å¿tateras, aut vectes, puras mathematicas li-<lb/>neas non eſſe, Å¿ed naturales, hincque exiÅ¿tere corpora cum materia coniuncta. </s> - <s xml:id="s1730" xml:space="preserve">Nunc <lb/>igitur imaginemur <var>.n.s.</var> eam Å¿uperficiem eſſe, quæ Å¿ecundum longitudinem axem Å¿ta <lb/>teræ Å¿cindit. </s> - <s xml:id="s1731" xml:space="preserve">& Å¿upponamus ipÅ¿ius centrum eſſe primum in <var>.i.</var> & maius brachium eſſe <lb/> <var>.i.u</var>: minus autem <var>.i.n.</var> & lineam verticalem <var>.i.o.</var> quæ tanta Å¿it, quanta eÅ¿t Å¿piſſitu-<lb/>do, aut craſſities ipÅ¿ius Å¿tateræ à ſuperiori latere ad inferius, ad faciliorem intelligen-<lb/>tiam, Å¿upponendo <var>.n.s.</var> <reg norm="parallelogrammam" type="context">parallelogrãmam</reg>. </s> - <s xml:id="s1732" xml:space="preserve">PoÅ¿itis igitur duobus ponderibus æquali-<lb/> - <anchor type="handwritten" xlink:label="hd-0156-01a" xlink:href="hd-0156-01"/> - bus in extremitatibus brachiorum, experientia innoteÅ¿cit, <reg norm="quod" type="simple">ê™</reg> pondus ad <var>.u.s.</var> appen-<lb/>Å¿um, viol entiam faciet ponderi appenÅ¿o ad <var>.n.x.</var> Å¿ed nos volumus inueÅ¿tigare <reg norm="causam" type="context">causã</reg> <lb/>huius effectus, quæ à nemine vnquam literarum monumentis, <reg norm="quod" type="simple">ê™</reg> Å¿ciam, conÅ¿ignata <lb/> - <anchor type="handwritten" xlink:label="hd-0156-02a" xlink:href="hd-0156-02"/> - fuit. </s> - <s xml:id="s1733" xml:space="preserve">Iam diximus Å¿tateram, aut vectem materialem eſſe & <var>.n.s.</var> eius Å¿uperficiem me-<lb/>diam, Å¿upponendo <var>.i.</var> eſſe centrum quo nititur dicta Å¿tatera aut vectis; </s> - <s xml:id="s1734" xml:space="preserve">Cum hocer-<lb/>go ita Å¿e habeat, Å¿int <var>.u.s.</var> et <var>.n.x.</var> lineæ inclinationum ponderum, & imaginemur, <reg norm="quod" type="simple">ê™</reg> <lb/>dicta pondera pendeant à punctis <var>.u.</var> et <var>.n.</var> vt reuera pendent, etiam Å¿i appenÅ¿a eſſent <lb/>Å¿ub <var>.s.</var> et <var>.x.</var> quia punctum <var>.u.</var> & punctum <var>.n.</var> ita coniuncta Å¿unt cum <var>.s.</var> et <var>.x.</var> ut qui <reg norm="vnum" type="context">vnÅ©</reg> <lb/>trahit alterum quoque trahat. </s> - <s xml:id="s1735" xml:space="preserve">Imaginemur quoque duas lineas <var>.i.u</var>: <var>i.n.</var> et <var>.i.e.</var> quÄ™ <lb/><var>i.e.</var> faciat angulum <var>.o.i.e.</var> æqualem angulo <var>.o.i.n</var>. </s> - <s xml:id="s1736" xml:space="preserve">Hinc clarè nobis patebit, Å¿i quis ipÅ¿i <lb/>e. pondus ipÅ¿ius <var>.u.</var> (<reg norm="quod" type="simple">ê™</reg> æquale eÅ¿t ponderi <var>.n.</var>) appenderet, id eandem planè vim habe <lb/>ret, quam pondus ipÅ¿ius <var>.n.</var> habet, & Å¿tateram neque Å¿urÅ¿um, neque deorÅ¿um moue-<lb/>ret, quia ambo pondera ad centrum <var>.i.</var> mediantibus lineis <var>.e.i.</var> et <var>.n.i.</var> exÄ™quo annite-<lb/>rentur, Å¿ed dicto pondere poÅ¿ito in .u: linea <var>.u.i.</var> per quam pondus centro annititur, <lb/>magis orizontalis quam <var>.e.i.</var> fit, & linea <var>.u.s.</var> inclinationis longius diÅ¿tans à centro <var>.i.</var> <lb/> - <anchor type="handwritten" xlink:label="hd-0156-03a" xlink:href="hd-0156-03"/> - quà m linea <var>.e.t.</var> vnde huiuÅ¿modi pondus magis quoque liberum à centro <var>.i.</var> reÅ¿ultat. <lb/></s> - <s xml:id="s1737" xml:space="preserve">magisque ponderoÅ¿um, quam cum erat in <var>.e.</var> ratione eorum, quæ primo & Å¿ecundo <lb/>capitibus diximus, & ob hanc cauÅ¿am Å¿uperat pondus poÅ¿itum in <var>.n</var>. </s> - <s xml:id="s1738" xml:space="preserve">Sed Å¿i centrum <lb/>fuerit .in <var>.o.</var> imaginabimur duas lineas <var>.o.s.</var> et <var>.o.x.</var> & Å¿upponemus quòd pondera po-<lb/>Å¿ita Å¿int in <var>.s.</var> et <var>.x.</var> vnde exiÅ¿tente magis orizontali linea <var>.o.s.</var> quam erit <var>.o.x.</var> & linea <lb/><var>u.s.</var> inclinationis longius diÅ¿tante à centro <var>.o.</var> quà m linea <var>.e.t.</var> eius pondus erit <reg norm="quoque" type="simple">quoq;</reg> <lb/> - <anchor type="figure" xlink:label="fig-0156-01a" xlink:href="fig-0156-01"/> - <anchor type="figure" xlink:label="fig-0156-02a" xlink:href="fig-0156-02"/> - <pb o="145" rhead="DE MECHAN." n="157" file="0157" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0157"/> - grauius, quia tantò minus pendebit à centro <var>.o.</var> & ratiocinando, vt Å¿uperius dixi-<lb/>mus, inueniemus eundem effectum verum eſſe. </s> - <s xml:id="s1739" xml:space="preserve">In Å¿tateris, rectè & propriè appella <lb/>ri poteÅ¿t <var>.x.i.s.</var> aut <var>.n.o.u.</var> orizontalis, Å¿ed in omni vectium Å¿pecie, hoc <reg norm="tantum" type="context">tãtum</reg> per quan <lb/>dam Å¿imilitudinem dicetur. </s> - <s xml:id="s1740" xml:space="preserve">Idem contemplari licet Å¿upponendo centrum in medio <lb/>inter <var>.o.</var> et <var>.i.</var> quod vnuÅ¿quiÅ¿que ex Å¿e abÅ¿que alterius auxilio facile præſtare poterit.</s> - </p> - <div type="float" level="4" n="1"> - <handwritten xlink:label="hd-0156-01" xlink:href="hd-0156-01a"/> - <handwritten xlink:label="hd-0156-02" xlink:href="hd-0156-02a"/> - <figure xlink:label="fig-0156-01" xlink:href="fig-0156-01a"> - <image file="0156-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0156-01"/> - </figure> - <handwritten xlink:label="hd-0156-03" xlink:href="hd-0156-03a"/> - <figure xlink:label="fig-0156-02" xlink:href="fig-0156-02a"> - <image file="0156-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0156-02"/> - </figure> - </div> - </div> - <div type="section" level="3" n="5"> - <head xml:id="head203" style="it" xml:space="preserve">De quibuÅ¿dam rebus animaduerÅ¿ione dignis.</head> - <head xml:id="head204" xml:space="preserve">CAP.V.</head> - <p> - <s xml:id="s1741" xml:space="preserve">NOn omittenda mihi <reg norm="videntur" type="context">vidẽtur</reg> quædam, quæ ad <reg norm="tractationem" type="context">tractationẽ</reg> vectium admodum <lb/>Å¿unt neceſſaria. </s> - <s xml:id="s1742" xml:space="preserve">Quod autem quærimus, in eo conÅ¿iÅ¿tit, quòd aliqui vectes <lb/>adhibeantur ad opus, quorum centrum, quod Græci <reg norm="hypomochlion" type="context">hypomochliõ</reg> appellant vnum <lb/>eÅ¿t ex extremis ipÅ¿ius vectis, & pondus, quod Å¿urÅ¿um eleuari debet, inter ipÅ¿a-<lb/>met extrema iacet, propinquum tamen hypomochlio, vt exempli gratia, Å¿i vectis <lb/>eſſet infraÅ¿cripta figura <var>.o.s.u.x.</var> cuius hypomochlion eſſet in puncto <var>.o.</var> & pondus in <lb/>puncto <var>.n.</var> clarum erit, <reg norm="quod" type="simple">ê™</reg> cum eleuari debeat <var>.n.</var> oportebit quoque opera manus ele-<lb/>uari <var>.u</var>. </s> - <s xml:id="s1743" xml:space="preserve">Nunc conÅ¿iderandum eÅ¿t quomodo pondus <var>.n.</var> annitatur ad <var>.u</var>. </s> - <s xml:id="s1744" xml:space="preserve">Hanc ob cau <lb/>Å¿am imaginabimur rectas lineas <var>.n.o</var>: <var>n.i</var>: <var>n.e</var>: <var>n.t.</var> et <var>.n.u.</var> quarum <var>.n.i.</var> verÅ¿us mundi cen <lb/>trum Å¿it poÅ¿ita, et <var>.n.t.</var> faciat angulum <var>.i.n.t.</var> æqualem angulo <var>.i.n.o</var>. </s> - <s xml:id="s1745" xml:space="preserve">Nunc ponendo ali <lb/>quam virtutem in <var>.i.</var> æquali inclinatione ad Å¿uperius conÅ¿tante, vt <var>.n.</var> ad inferius (re-<lb/>mota tamen grauitate materiæ vectis) huiuÅ¿modi virtus, totum pondus ipÅ¿ius <var>.n.</var> com <lb/>muni quadam Å¿cientiæ notione Å¿uÅ¿tinebit. </s> - <s xml:id="s1746" xml:space="preserve">& Å¿i <reg norm="pondus" type="context">põdus</reg> ipÅ¿ius <var>.n.</var> eſſet in <var>.x.</var> è directo Å¿u-<lb/>per <var>.o.</var> totum pondus Å¿uper hypomochlio Å¿e haberet, & tanta virtus ipÅ¿ius hypomo-<lb/>chlij Å¿ufficeret ad reÅ¿iÅ¿tendum pro Å¿uÅ¿tinendo, quanta eÅ¿t grauitas ipÅ¿ius ponderis, <lb/>Å¿ed ipÅ¿um iterum ponamus in <var>.n.</var> ibi clarum erit, quòd Å¿i alia virtus à parte inÅ¿eriori <lb/>ad Å¿uperiorem vectis non opponitur, excepto tamen hypomochlio, oportebit virtu <lb/>te cuiuÅ¿dam partis ponderis <var>.n.</var> (abÅ¿que conÅ¿ideratione tamen, vt iam dixi, ponderis <lb/>materiæ vectis) vt vectis à parte <var>.s.u.</var> deprimatur, & dixi vnius cuiuÅ¿dam partis pon-<lb/>deris <var>.n.</var> quia alia <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> ponderis pars annititur ipÅ¿i hypomochlio <var>.o.</var> <reg norm="mediante" type="context">mediãte</reg> linea <lb/><var>o.n.</var> quæ angulos rectos cum <var>.o.x.</var> non facit. </s> - <s xml:id="s1747" xml:space="preserve">Si autem à puncto <var>.t.</var> opponet Å¿eÅ¿e huiuÅ¿-<lb/>modi reÅ¿iÅ¿tentia, vt vectis non deprimatur, clarum erit communi Å¿cientia, <reg norm="quod" type="simple">ê™</reg> virtus <lb/>ponderis <var>.n.</var> diuiÅ¿a erit per medium æqualiter, cuius vna medietas Å¿uper <var>.o.</var> quieÅ¿cet, <lb/>& alia Å¿uper <var>.t.</var> mediantibus duabus lineis <var>.n.o.</var> et <var>.n.t</var>. </s> - <s xml:id="s1748" xml:space="preserve">Imaginemur nunc reÅ¿iÅ¿tentiam <lb/>t. ablatam eſſe, <reg norm="poÅ¿itamque" type="simple">poÅ¿itamq;</reg> in <var>.e.</var> clarum quoque erit, <reg norm="quod" type="simple">ê™</reg> maior pars ponderis <var>.n.</var> ipÅ¿i <var>.e.</var> <lb/>annitetur beneficio lineæ <var>.n.e.</var> quà m ipÅ¿i <var>.o.</var> cum linea <var>.n.i.</var> inclinationis ipÅ¿i <var>.e.</var> Å¿it pro <lb/>pinquior quam <var>.o.</var> quia omnis reÅ¿iÅ¿tentia aut in <var>.i.</var> aut in <var>.e.</var> aut in <var>.t.</var> aut in <var>.u.</var> eÅ¿t loco <lb/>centri, quemadmodum eÅ¿t <var>.o.</var> & alter alterius opera iuuatur. </s> - <s xml:id="s1749" xml:space="preserve">Si verò eadem reÅ¿iÅ¿ten <lb/>tia poÅ¿ita erit in <var>.u.</var> clarum quoque erit, <reg norm="quod" type="simple">ê™</reg> minor pars ponderis <var>.n.</var> annitetur ipÅ¿i <var>.u.</var> <reg norm="quam" type="context">quã</reg> <lb/>ipÅ¿i <var>.o.</var> cum dicta <var>.n.i.</var> à centro <var>.u.</var> longius quam à centro <var>.o.</var> diÅ¿ter, & proportio partis <lb/>ponderis <var>.n.</var> in <var>.o.</var> ad propor-<lb/>tionem partis ponderis <var>.n.</var> in <lb/> - <anchor type="figure" xlink:label="fig-0157-01a" xlink:href="fig-0157-01"/> - u. non erit <reg norm="Å¿ecundum" type="context">Å¿ecÅ©dum</reg> propor <lb/>tionem angulorum <var>.u.n.i.</var> et <lb/><var>o.n.i.</var> Å¿ed Å¿ecundum propor <lb/>tionem <var>.u.i.</var> ad <var>.i.o.</var> quod cla <lb/>rè compræhendi poteÅ¿t ab - <pb o="146" rhead="IO. BAPT. BENED." n="158" file="0158" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0158"/> - huius effectus conuerÅ¿o, ideÅ¿t, vt quemadmodum nunc Å¿upponuntur <var>.o.</var> et <var>.u.</var> eſſe duo <lb/>centra quibus <reg norm="Å¿uÅ¿tinetur" type="simple">Å¿uÅ¿tinet̃</reg> pondus <var>.e.</var> ipÅ¿ius <var>.n.</var> imaginemur <var>.n.</var> eſſe quoddam centrum à <lb/>quo pendeant duo pondera <var>.o.</var> et <var>.u.</var> Å¿ic inuicem proportionata, ut Å¿unt <var>.u.i.</var> et <var>.i.o.</var> <lb/>certe horum ponderum cauÅ¿a Å¿tatera <var>.o.s.</var> quam vectem appellabamus à nulla parte <lb/>inclinabitur. </s> - <s xml:id="s1750" xml:space="preserve">Redeuntes nunc ad propoÅ¿itum, dicemus <reg norm="quod" type="simple">ê™</reg> annitente pondere ipÅ¿ius <var>.<lb/>n.</var> minus ad <var>.u.</var> quam ad <var>.o.</var> ideÅ¿t ad <var>.t.</var> minori vi opus erit in <var>.u.</var> quà m in <var>.t.</var> ad attollen-<lb/>dum pondus ipÅ¿ius <var>.n.</var> & Å¿ic per conÅ¿equens quantò longius erit punctum <var>.u.</var> ab <var>.t.</var> tan <lb/>tò minori quoque vi egebit, & conÅ¿equenter quando vis, aut reÅ¿iÅ¿tentia in <var>.u.</var> ita pro <lb/>portionata erit illi, quæ eÅ¿t ipÅ¿ius <var>.o.</var> vt eÅ¿t <var>.o.i.</var> ad <var>.i.u.</var> vectis non mouebitur. </s> - <s xml:id="s1751" xml:space="preserve">Sed quan <lb/>do erit proportio maior, reÅ¿iÅ¿tentiæ ipÅ¿ius <var>.u.</var> ad eam, quæ eÅ¿t ipÅ¿ius <var>.o.</var> ea, quæ eÅ¿t <var>.o.<lb/>i.</var> ad <var>.i.u.</var> </s> - <s xml:id="s1752" xml:space="preserve">tunc vectis à par-<lb/>teipÅ¿ius <var>.u.s.</var> eleuabitur, Å¿i <lb/> - <anchor type="figure" xlink:label="fig-0158-01a" xlink:href="fig-0158-01"/> - vero proportio minor eſſet <lb/>quà m.o.i. ad <var>.i.u.</var> </s> - <s xml:id="s1753" xml:space="preserve">tunc ve-<lb/>ctis ab eadem parte depri-<lb/>metur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0157-01" xlink:href="fig-0157-01a"> - <image file="0157-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0157-01"/> - </figure> - <figure xlink:label="fig-0158-01" xlink:href="fig-0158-01a"> - <image file="0158-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0158-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="6"> - <head xml:id="head205" style="it" xml:space="preserve">De ratione cuiuÅ¿dam uis adauctæ.</head> - <head xml:id="head206" xml:space="preserve">CAP. VI.</head> - <p> - <s xml:id="s1754" xml:space="preserve">QVibuÅ¿dam in locis vtuntur <reg norm="quidam" type="context">quidã</reg> <reg norm="quodam" type="context">quodã</reg> <reg norm="inÅ¿trumento" type="context">inÅ¿trumẽto</reg> piÅ¿torio ad <reg norm="Å¿ubigendam" type="context context">Å¿ubigẽdã</reg> pa-<lb/>Å¿tam, vnius tantum hominis ui adhibita, quæ quidem machina cum mihi di-<lb/>gna contemplatione eſſe videatur, eius aliquam rationem proponere volui, pro cu-<lb/>ius deÅ¿criptione imaginemur planum, in quo Å¿edet ille, qui voluit paÅ¿tam, & in quo <lb/>ipÅ¿a paÅ¿ta eÅ¿t repoÅ¿ita <var>.T.S.D.</var> & triangulum <var>.T.A.S.</var> immobile perpendiculare-<lb/>qÌue Å¿uperficiei dicti plani, angulo autem <var>.A.</var> coniunctum lignum <var>.A.E.</var> vt Å¿emidiame <lb/>trum mobilem, & æqualem perpendiculari ipÅ¿ius trianguli, </s> - <s xml:id="s1755" xml:space="preserve">unde <var>.A.</var> loco centri erit <lb/>et <var>.D.O.</var> Å¿it Å¿emidiameter, qui paÅ¿tam contundit, & ab eius extremo <var>.O.</var> (quod <var>.O.</var> <lb/>quando <var>.D.O.</var> orizontalis eÅ¿t, in baÅ¿i dicti trianguli reperitur) veniat lignum <var>.O.V.</var> <lb/>quod cum <var>.A.V.</var> Å¿it æquale perpendiculari imaginatæ ab angulo <var>.A.</var> baÅ¿i <var>.T.S.</var> <reg norm="deno- datum" type="context">deno-<lb/>datÅ©</reg> <reg norm="tantum" type="wordlist/context">tñ</reg> utvulgo <reg norm="dicitur" type="simple">dicit̃</reg> Å¿eu flexile in <var>.O.</var> & in <var>.V.</var> vt elleuare <reg norm="atque" type="simple">atq;</reg> deprimere Å¿emidiame <lb/>trum <var>.D.O.</var> poſſit, et <var>.V.O.</var> Å¿it æqualis <var>.A.V.</var> et <var>.V.</var> medium Å¿it inter <var>.A.</var> et <var>.E.</var> vnde <var>.A.V.</var> <lb/>cum <var>.O.V.</var> æquales erunt <var>.A.E.</var> Å¿unt deinde duo ligna <reg norm="perpendicularia" type="context">perpẽdicularia</reg> ab <var>.A.</var> ad baÅ¿im <lb/>fixa, & immobilia inter Å¿e adeò diÅ¿tantia, vt inter ipÅ¿a <reg norm="pertranÅ¿eant" type="context context">pertrãſeãt</reg> <var>.O.V.</var> et <var>.D.O.</var> Å¿upra <lb/>& infra, ne deuiet Å¿emidiametrum <var>.D.O</var>. </s> - <s xml:id="s1756" xml:space="preserve">In extremitate deinde ipÅ¿ius <var>.E.</var> Å¿it lignum <lb/>quoddam tenue, vt digitus polex, ad angulos rectos cum <var>.A.E.</var> quod ab aliquo, qui <lb/>antedictam machinam Å¿tet, manibus teneatur, qui quidem homo idipÅ¿um lignum, <lb/>ideÅ¿t Å¿emidiametrum <var>.A.E.</var> à ſuperficie trianguli dicti, ad Å¿e trahendo, & deinde ver <lb/>Å¿us eundem triangulum impellendo, vim quandam maximam mediante Å¿emidia <lb/>metro <var>.D.O.</var> Å¿uper paÅ¿tam excitat.</s> - </p> - <p> - <s xml:id="s1757" xml:space="preserve">Pro cuius rei contemplatione volo vt Å¿ecundam hanc Å¿ubÅ¿criptam figuram <var>.b.a.<lb/>u.x.</var> imaginemur, in qua <var>.u.</var> exprimat <var>.A.</var> primæ figuræ, & <var>.a.</var> denotet <var>.O.</var> & <var>.o.V.</var> & <var>.x.<lb/>E.</var> imaginemur etiam <var>.u.a.</var> baÅ¿em trianguli <var>.a.u.o.</var> cui <var>.o.t.</var> perpendicularis dictæ baÅ¿i <var>.<lb/>u.a.</var> addatur. </s> - <s xml:id="s1758" xml:space="preserve"><reg norm="HucuÅ¿que" type="simple">HucuÅ¿q;</reg> igitur <var>.u.o.</var> æqualis erit <var>.o.x.</var> & ipÅ¿i <var>.o.a.</var> imaginemur etiam <var>.a.o.</var> <lb/>vÅ¿que ad <var>.b.</var> ita productam vt <var>.o.b.</var> æqualis Å¿it <var>.o.a.</var> ponamus etiam pondus in <var>.a.</var> impel- - <pb o="147" rhead="DE MECHAN." n="159" file="0159" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0159"/> - lere verÅ¿us <var>.u.</var> vnde linea eius inclinationis Å¿it Å¿emper <var>.a.u.</var> Å¿upponamus etiam <var>.a.o.b.</var> <lb/>eſſe <reg norm="libram" type="context">librã</reg>, aut Å¿tateram, aut vectem, & <var>.o.</var> eius centrum, vnde vis, aut virtus ipÅ¿ius <var>.a.</var> <lb/>proportionalis erit ipÅ¿i <var>.o.t.</var> reÅ¿pectu virtutis, aut vis imaginatæ in <var>.b.</var> inclinationis <lb/>perpendicularis ipÅ¿i <var>.b.a.</var> quæ quidem virtus, aut vis in <var>.b.</var> proportionalis erit ipÅ¿i <var>.b.<lb/>o.</var> ex tertio capite huius tractatus; </s> - <s xml:id="s1759" xml:space="preserve">Si ergo fuiſſet poÅ¿ita in <var>.b.</var> virtus quædarn ad an-<lb/>gulum rectum, trahens lineam <var>.b.o.</var> tam proportionatam virtuti perpendiculari ip-<lb/>Å¿ius <var>.a.</var> quam eÅ¿t <var>.o.t.</var> proportionata ipÅ¿i <var>.o.b.</var> Å¿tatera <var>.b.o.a.</var> non moueretur, Å¿ed quæuis <lb/>portio maior in <var>.b.</var> Å¿uperaret <var>.a.</var> cum autem fuerit <var>.o.x.</var> æqualis ipÅ¿i <var>.o.b.</var> <reg norm="idem" type="context">idẽ</reg> planè eue-<lb/> - <anchor type="figure" xlink:label="fig-0159-01a" xlink:href="fig-0159-01"/> - <anchor type="figure" xlink:label="fig-0159-02a" xlink:href="fig-0159-02"/> - niet, communi quadam Å¿cientia, ponen-<lb/>do virtutem <var>.b.</var> in <var>.x</var>. </s> - <s xml:id="s1760" xml:space="preserve">Quantitas ergo virtu <lb/>tis in <var>.x.</var> quæ Å¿uperare debet reÅ¿iÅ¿tentiam <lb/>in <var>.a.</var> quæ ipÅ¿i <var>.u.</var> contraponitur, debet ha-<lb/>bere aliquantulum maioris proportionis <lb/>ad reÅ¿iÅ¿tentiam, quæ in <var>.a.</var> angulum re-<lb/>ctum efficeret cum <var>.a.o.</var> ea, quæ eÅ¿t <var>.o.t.</var> <lb/>ad <var>.o.x</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0159-01" xlink:href="fig-0159-01a"> - <image file="0159-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0159-01"/> - </figure> - <figure xlink:label="fig-0159-02" xlink:href="fig-0159-02a"> - <image file="0159-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0159-02"/> - </figure> - </div> - <pb o="148" rhead="IO. BABPT. BENED." n="160" file="0160" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0160"/> - </div> - <div type="section" level="3" n="7"> - <head xml:id="head207" style="it" xml:space="preserve">De quibuÅ¿dam erroribus Nicolai Tartaleæ circa pondera <lb/>corporum & eorum motus, quorum aliqui deÅ¿umpti <lb/>fuerunt à fordano Å¿criptore quodam antiquo.</head> - <head xml:id="head208" xml:space="preserve">CAP. VII.</head> - <p> - <s xml:id="s1761" xml:space="preserve">CVm magis amici veritatis eſſe debeamus quà m cuiuÅ¿quam hominis, quemad-<lb/>modum AriÅ¿to. Å¿cribit, detegam hoc loco quoÅ¿dam errores Nicolai TartaleÄ™ <lb/>de ponderibus corporum, & velocitatibus motuum localium. </s> - <s xml:id="s1762" xml:space="preserve">Et primum decipitur <lb/>is in .8. lib. Å¿uarum diuerÅ¿arum inuentionum in Å¿ecunda propoÅ¿itione, cum non ani-<lb/>maduerterit quanti momenti Å¿int extrinÅ¿ecæ reÅ¿iÅ¿tentiæ.</s> - </p> - <p> - <s xml:id="s1763" xml:space="preserve">Subiectum quoque tertiæ propoÅ¿itionis eÅ¿t malè demonÅ¿tratum, quia idem pla-<lb/>nè ex eius demonÅ¿tratione iam dicta corporibus hætereogeneis, aut figura diuerÅ¿is <lb/>contingeret, quod ad velocitates attinet.</s> - </p> - <p> - <s xml:id="s1764" xml:space="preserve">In quarta propoÅ¿itione, quod ad <reg norm="diÅ¿putandum" type="context context">diÅ¿putãdÅ©</reg> proponit <reg norm="non" type="context">nõ</reg> concludit melius. </s> - <s xml:id="s1765" xml:space="preserve"><reg norm="autem" type="context">autẽ</reg> id <lb/>ab eo <reg norm="Å¿equitur" type="simple">Å¿equit̃</reg>, quod Archimedes in .6. propoÅ¿itione lib. primi de <reg norm="ponderibus" type="context">põderibus</reg> <reg norm="probauit" type="simple">ê“bauit</reg>.</s> - </p> - <handwritten/> - <p> - <s xml:id="s1766" xml:space="preserve">Sed in Å¿ecunda parte quintÄ™ propoÅ¿itionis non uidet <reg norm="quod" type="simple">ê™</reg> uigore Å¿itus eo modo, quo <lb/>ipÅ¿e diÅ¿putat, nulla elicitur ponderis differentia. </s> - <s xml:id="s1767" xml:space="preserve">quia Å¿i corpus <var>.B.</var> deÅ¿cendere debet <lb/>per arcum <var>.i.l.</var> corpus <var>.A.</var> aÅ¿cendere debet per arcum <var>.u.s.</var> æqualem, & Å¿imilem. eadem <lb/>quoque ratione Å¿ituatum, vt eÅ¿t arcus <var>.i.l.</var> vnde vt eÅ¿t facilè corpori <var>.B.</var> deÅ¿cendere <lb/>per arcum <var>.i.l.</var> difficile ita erit corpori <var>.A.</var> aÅ¿cendere per arcum <var>.u.s</var>. </s> - <s xml:id="s1768" xml:space="preserve">HÄ™c autem qnin <lb/>ta propoÅ¿itio Tartaleæ eÅ¿t Å¿ecuuda quæſtio à Iordano propoÅ¿ita.</s> - </p> - <p> - <s xml:id="s1769" xml:space="preserve">Quòd autem ad primum corollarium dictæ propoÅ¿itionis attinet, verum ille qui <lb/>dem Å¿cribit, eius tamen effectus cauÅ¿a & à Iordano prius, & ab ipÅ¿o poÅ¿tea citata, na-<lb/>tura Å¿ua vera non eÅ¿t. </s> - <s xml:id="s1770" xml:space="preserve">quia vera cauÅ¿a per Å¿e ab eo oritur, <reg norm="quod" type="simple">ê™</reg> à centro libræ dependeat <lb/>vt primo cap. huius tractatus oÅ¿tendi. </s> - <s xml:id="s1771" xml:space="preserve">Secundum verò corollarium falÅ¿um eſſe, ijs ra <lb/>tionibus quas nunc Å¿ubiungam, patebit. </s> - <s xml:id="s1772" xml:space="preserve">Imaginemur <var>.u.</var> pro centro regionis ele-<lb/>mentaris, & libram <var>.b.o.a.</var> obliquam reÅ¿pectu ad <var>.u.</var> & brachijs æqualibus <reg norm="conÅ¿tantem" type="context">conÅ¿tãtem</reg>, <lb/>& pondera in <var>.a.</var> et in <var>.b.</var> etiam æqualia. </s> - <s xml:id="s1773" xml:space="preserve">lineæ autem inclinationum Å¿int <var>.a.u.</var> et <var>.b.u.</var> <lb/>imaginemur etiam lineam <var>.o.u.</var> & à centro <var>.o.</var> libræ duas <var>.o.t.</var> et <var>.o.e.</var> perpendiculares <lb/>inclinationum lineis; </s> - <s xml:id="s1774" xml:space="preserve">vnde pondus ipÅ¿ius <var>.a.</var> in huiuÅ¿modi Å¿itu tam erit proportiona <lb/>tum ponderi <var>.b.</var> quam proportionata erit linea <var>.o.t.</var> lineæ <var>.o.e.</var> ex eo <reg norm="quod" type="simple">ê™</reg> tertio cap. hu-<lb/>iustractatus probaui, Å¿ed linea <var>.o.t.</var> maior eÅ¿t linea <var>.o.e.</var> quod Å¿ic probo. </s> - <s xml:id="s1775" xml:space="preserve">Imaginemur <lb/>triangulum <var>.u.a.b.</var> circunÅ¿criptum eſſe à circulo <var>.u.a.n.b.</var> cuius <var>.c.</var> Å¿it centrum, <reg norm="quod" type="simple">ê™</reg> erit <lb/>extra lineam <var>.u.o.</var> cum Å¿upponatur <var>.a.o.b.</var> obliquam eſſe reÅ¿pectu ad <var>.u.o</var>. </s> - <s xml:id="s1776" xml:space="preserve">Imagine-<lb/>mur deinde à centro <var>.c.</var> lineam <var>.c.o.s.</var> vÅ¿que ad circunferentiam, quæ perpendicula-<lb/>ris erit ipÅ¿i <var>.a.b.</var> ex tertia lib. 3. Eucli. </s> - <s xml:id="s1777" xml:space="preserve">Å¿i poÅ¿teà imaginemur duas lineas <var>.c.a.</var> et <var>.c.b.</var> ha <lb/>bebimus ex .8. lib. primi, angulum <var>.a.c.o.</var> æqualem angulo <var>.b.c.o</var>. </s> - <s xml:id="s1778" xml:space="preserve">Vnde ex .25. lib. 3. <lb/>arcus <var>.a.s.</var> æqualis erit arcui <var>.b.s.</var> Å¿ed Å¿i imaginabimur <var>.u.o.</var> ad circunferentiam vÅ¿que <lb/>productam, clarum erit <reg norm="quod" type="simple">ê™</reg> arcum <var>.s.b.</var> Å¿ecaret in puncto <var>.n.</var> vnde arcus <var>.n.b.</var> minor erit <lb/>arcu <var>.n.a.</var> & Å¿ic etiam angulus <var>.n.u.b.</var> minor erit angulo <var>.n.u.a.</var> ex <ref id="ref-0024">ultima lib. 6.</ref> </s> - <s xml:id="s1779" xml:space="preserve">Imagi-<lb/>nemur nunc alium quendam circulum, cuius <var>.o.u.</var> Å¿it diameter, cuius circunferentia <lb/>per duo puncta <var>.e.</var> et <var>.t.</var> <reg norm="prætergradiatur" type="simple">prætergradiat̃</reg>, cum in ipÅ¿is Å¿int angulirecti, quod quilibet ex <lb/>Å¿eratio cinando colligere poteÅ¿t, Å¿i .30. lib. 3. in mentem reuocauerit. </s> - <s xml:id="s1780" xml:space="preserve">Sed cum angu-<lb/>lus <var>.o.u.t.</var> Å¿it maior angulo <var>.o.u.e.</var> arcus <var>.o.t.</var> maior erit arcu <var>.o.e.</var> ex vltima .6. vnde cor <lb/>da <var>.o.t.</var> maior erit corda ipÅ¿ius <var>.o.e.</var> ex conuerÅ¿o .27. lib. 3. quod eÅ¿t propoÅ¿itum. </s> - <s xml:id="s1781" xml:space="preserve">Pon-<lb/> - <anchor type="handwritten" xlink:label="hd-0160-01a" xlink:href="hd-0160-01"/> - dusigitur ipÅ¿ius <var>.a.</var> in huiuÅ¿modi Å¿itu, pondere ipÅ¿ius <var>.b.</var> grauius erit. </s> - <s xml:id="s1782" xml:space="preserve">Quod è directo ijs <lb/>repugnat quæ Tartalea in 2. parte quinræ propoÅ¿itionis ediÅ¿erit, & per conÅ¿equens <lb/>2. corollarij falÅ¿itatem oÅ¿tendit, vt eam quoque, quæ in 6. propoÅ¿itione latet. </s> - <s xml:id="s1783" xml:space="preserve">quia <reg norm="cum" type="context">cÅ©</reg> - <pb o="149" rhead="DE MECHAN." n="161" file="0161" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0161"/> - proportio <reg norm="ponderis" type="context">põderis</reg> <var>.a.</var> ad pon <lb/>dusipÅ¿ius <var>.b.</var> eadem Å¿it cum <lb/>ea quÄ™ eÅ¿t <var>.o.t.</var> ad <var>.o.e.</var> Å¿ub co <lb/><reg norm="gnitionem" type="context">gnitionẽ</reg> noÅ¿tram cadere po <lb/>teÅ¿t, primum cognoÅ¿cendo <lb/>angulos obliquitatis librÄ™, <lb/>ideÅ¿t angulos <var>.b.o.u.</var> et <var>.a.o.<lb/>u.</var> quia oportet Å¿emper Å¿up-<lb/> - <anchor type="figure" xlink:label="fig-0161-01a" xlink:href="fig-0161-01"/> - ponere Å¿itum aliquem no-<lb/>tum. </s> - <s xml:id="s1784" xml:space="preserve">Si nobis deinde co-<lb/>gnita erit proportio ipÅ¿ius <var>.<lb/>o.u.</var> ad <var>.o.b.</var> et. ad <var>.o.a.</var> aſſe-<lb/>quemur cognitionem angu <lb/>li <var>.b.</var> et <var>.o.a.u.</var> & per conÅ¿e-<lb/>quens ipÅ¿ius <var>.o.a.t.</var> eius reÅ¿i-<lb/>dui, vnde poÅ¿tea beneficio <lb/>angulorum <var>.e.</var> et <var>.t.</var> rectorum <lb/>& laterum <var>.o.b.</var> et <var>.o.a.</var> cogni <lb/>torum in cognitionem <var>.o.t.</var> <lb/>et <var>.o.e.</var> facile deueniemus.</s> - </p> - <div type="float" level="4" n="1"> - <handwritten xlink:label="hd-0160-01" xlink:href="hd-0160-01a"/> - <figure xlink:label="fig-0161-01" xlink:href="fig-0161-01a"> - <image file="0161-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0161-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="8"> - <head xml:id="head209" xml:space="preserve">CAP. VIII.</head> - <p> - <s xml:id="s1785" xml:space="preserve">QVod autem idem Tartalea in .6. propoÅ¿itione, & Iordanus in Å¿ecunda parte. <lb/></s> - <s xml:id="s1786" xml:space="preserve">Å¿ecundæ propoÅ¿itionis Å¿cribunt, maximum quoque errorem inÅ¿e continet. <lb/></s> - <s xml:id="s1787" xml:space="preserve">Dicunt enim <reg norm="angulum" type="context">angulÅ©</reg> <lb/><var>h.a.f.</var> differentem ab <lb/>angulo <var>.d.b.f.</var> alia ra-<lb/>tione non eſſe quà m <lb/>per angulum conta-<lb/>ctus <reg norm="duorum" type="context">duorÅ©</reg> <reg norm="circulorum" type="context">circulorÅ©</reg>, <lb/>vt in Å¿ua figura Å¿cribit <lb/>Tartalea; </s> - <s xml:id="s1788" xml:space="preserve">id quod fal-<lb/>Å¿iſſimum eÅ¿t. </s> - <s xml:id="s1789" xml:space="preserve"><reg norm="Quam" type="context">Quã</reg> ob <lb/>cauÅ¿am in Å¿ubÅ¿cripta <lb/>figura Å¿it libra <var>.B.A.</var> <lb/> - <anchor type="figure" xlink:label="fig-0161-02a" xlink:href="fig-0161-02"/> - & eius centrum. </s> - <s xml:id="s1790" xml:space="preserve">C et <var>.<lb/>u.</var> <reg norm="centrum" type="context">centrÅ©</reg> regionis ele <lb/>mentaris, et <var>.A.u.</var> et <var>.B.<lb/>u.</var> lineæ <reg norm="inclinationum" type="context">inclinationÅ©</reg>. <lb/></s> - <s xml:id="s1791" xml:space="preserve">Imaginemur deinde <lb/>lineam <var>.B.K.</var> <reg norm="parallelam" type="context">parallelã</reg> <lb/>ipÅ¿i <var>.A.u.</var> quæ gyrum <var>.<lb/>B.F.A.</var> in puncto <var>.K.</var> <lb/>communi Å¿cientiæ <reg norm="prae cepto" type="simple">prÄ™Â<lb/>cepto</reg> Å¿cindet, & habe <lb/>bimus angulum <var>.K.B.<lb/>Z.</var> æqualem angulo <var>.<lb/>H.A.F.</var> ideÅ¿t <var>.u.A.F.</var> <lb/>(quia <var>.H.u.</var> et <var>.D.</var> <reg norm="unum" type="context">unÅ©</reg> <lb/>Å¿unt) cum ex .29. libr. <lb/>primi Euclidis angu- - <pb o="150" rhead="IO. BAPT. BENED." n="162" file="0162" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0162"/> - lus <var>u.A.C.</var> æqualis Å¿it <lb/>angulo <var>.K.B.T.</var> & an-<lb/>gulus <var>.C.A.F.</var> æqua-<lb/>lis angulo <var>.T.B.Z.</var> <reg norm="nunc" type="context">nÅ©c</reg> <lb/>comparatio eÅ¿t inter <lb/>angulum <var>.D.B.F.</var> & an <lb/>gulum <var>.K.B.Z.</var> miÅ¿tili-<lb/>neos, qui quidem duo <lb/>anguli, <reg norm="communem" type="context">cõmunem</reg> ha-<lb/>bent angulum miÅ¿tili <lb/>neum <var>.K.B.F.</var> quapro-<lb/> - <anchor type="figure" xlink:label="fig-0162-01a" xlink:href="fig-0162-01"/> - pter Å¿i angulus <var>.K.B.Z.</var> <lb/>miÅ¿tilineus maior eÅ¿t <lb/>angulo <var>.D.B.F.</var> miÅ¿ti-<lb/>lineo per angulum <var>.<lb/>K.B.Z.</var> contingentiæ, <lb/>circulorum ergo angu <lb/>lus miÅ¿tilineus com-<lb/>munis <var>.K.B.F.</var> æqualis <lb/>erit miÅ¿tilineo, angu-<lb/>lo <var>.D.B.F.</var> pars vide-<lb/>licet Å¿ui toto. </s> - <s xml:id="s1792" xml:space="preserve">Omnis <lb/>autem error in quem <lb/>Tartalea, <reg norm="Iordanusque" type="simple">Iordanusq;</reg> <lb/>lapÅ¿i fuerunt ab eo, <reg norm="quod" type="simple">ê™</reg> <lb/> - <anchor type="handwritten" xlink:label="hd-0162-0a" xlink:href="hd-0162-01"/> - lineas inclinationum <lb/>pro parallelis viciſſim <lb/>Å¿umpÅ¿erunt, emana-<lb/>uit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0161-02" xlink:href="fig-0161-02a"> - <image file="0161-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0161-02"/> - </figure> - <figure xlink:label="fig-0162-01" xlink:href="fig-0162-01a"> - <image file="0162-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0162-01"/> - </figure> - <handwritten xlink:label="hd-0162-01" xlink:href="hd-0162-01a"/> - </div> - <p> - <s xml:id="s1793" xml:space="preserve">Septima propoÅ¿itio Tartaleæ, quæ eÅ¿t <reg norm="quinta" type="simple">ê—nta</reg> quæſtio Iordani mihi <reg norm="videtur" type="simple">videt̃</reg> excipien-<lb/>da riÅ¿u, cum pondus ipÅ¿ius <var>.A.</var> ponderi ipÅ¿ius <var>.B.</var> exiÅ¿tens æquale, grauius Å¿it pondere <lb/>eiuÅ¿dem <var>.B.</var> ratione minoris aperturæ anguli contingentiæ in <var>.A.</var> quam in <var>.B.</var> in quo <lb/>idem error committitur, qui in præcedenti committebatur, cum Å¿cilicet ipÅ¿e putet <lb/>lineas <var>.A.E.</var> et <var>.B.D.</var> figuræ ab eo confictæ Å¿ibi inuicem eſſe parallelas, quæ etiam Å¿i <lb/>æquidiÅ¿tantes eſſent (vnde angulus <var>.E.A.G.</var> minor eſſet angulo <var>.D.B.F.</var>) non eam ta <lb/>men ob cauÅ¿am huiuÅ¿modiangulorum differentia cauÅ¿a eſſet differentiæ <reg norm="grauitatum" type="context">grauitatÅ©</reg> <lb/>ipÅ¿orum <var>.A.</var> et <var>.B.</var> ob ea quæ cap .4. huius tractatus poÅ¿ui.</s> - </p> - <p> - <s xml:id="s1794" xml:space="preserve">Octaua autem propoÅ¿itio, quæ eÅ¿t .6. quæſtio Iordani Iongè melius demonÅ¿tratur <lb/>ab Archi. in .6. lib. primi de ponderibus, cum nec à Iordano, nec à Tartalæa probata <lb/>fuerit, cum ijdem non probauerint præcedentes, quas in dicta .8. </s> - <s xml:id="s1795" xml:space="preserve">TartalÄ™a citat, qui <lb/>neque etiam probat nonam .10. 11. 12. et .13. cum ad prÄ™cedentes probandas mini <lb/>mè acceſſerit.</s> - </p> - <p> - <s xml:id="s1796" xml:space="preserve">Quartadecima verò, quæ eÅ¿t .10. quęſtio Iordani, duas ob cauÅ¿as eÅ¿t falÅ¿a, quarum <lb/>vna eÅ¿t, <reg norm="quod" type="simple">ê™</reg> (Å¿upponendo <var>.A.D.E.G.B.</var> eſſe vnum brachium librÄ™ et <var>.A.</var> punctum <reg norm="centri" type="context">cẽtri</reg> <lb/>eiuÅ¿dem, et <var>.D.</var> pondus Ä™quale ponderi <var>.E.</var> & lineas inclinationum <var>.D.K.</var> et <var>.E.M.</var>) an <lb/>guli <var>.K.D.E.</var> et <var>.M.E.G.</var> Å¿ibi <reg norm="inuicem" type="context">inuicẽ</reg> <reg norm="non" type="context">nõ</reg> Å¿unt Ä™quales; </s> - <s xml:id="s1797" xml:space="preserve"><reg norm="cum" type="context">cÅ©</reg> ille angulus Å¿it intrinÅ¿ecus, hic <lb/>verò extrinÅ¿ecus & oppoÅ¿itus dicto intrinÅ¿eco <reg norm="vnius" type="simple">vniê°</reg> <reg norm="trianguli" type="context">triãguli</reg> terminati à . <var>D.E.</var> à <var>.D.K.</var> - <pb o="151" rhead="DE MECHAN." n="163" file="0163" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0163"/> - et <var>.E.M.</var> lineis productis vÅ¿que ad centrum regionis elementaris, vnde dictus angu-<lb/>lus <var>.M.E.G.</var> maior eÅ¿t alio, ex .16 lib. primi Eucli. </s> - <s xml:id="s1798" xml:space="preserve">Qua ratione fit, vt hanc ob cauÅ¿am <lb/>E. grauius Å¿it ipÅ¿o <var>.D.</var> cum minus dependeat à centro <var>.A.</var> vt primo cap. huius tractatus <lb/>iam dixi. </s> - <s xml:id="s1799" xml:space="preserve">Alia quoque eÅ¿tratio, qua dictum <var>.E.</var> grauius fit ipÅ¿o <var>.D.</var> quÄ™ quidem eÅ¿t <lb/>maior diÅ¿tantia à centro <var>.A.</var> libræ, per Å¿imiles rationes capit .4. huius tractatus ci-<lb/>tatas.</s> - </p> - <p> - <s xml:id="s1800" xml:space="preserve">Decimaquinta <reg norm="quoque" type="simple">quoq;</reg> nil penitus valet, quÄ™ eÅ¿t .11. quęſtio Iordani, cuius Autho-<lb/>ris opuÅ¿culum opera Traiani BibliopolÄ™ Venetijs è tenebris in lucem emerÅ¿it.</s> - </p> - </div> - <div type="section" level="3" n="9"> - <head xml:id="head210" style="it" xml:space="preserve">QuòdÅ¿ummaratione Å¿tateræper æqualia interualla <lb/>Å¿int diuiſæ.</head> - <head xml:id="head211" xml:space="preserve">CAP. IX.</head> - <p> - <s xml:id="s1801" xml:space="preserve">MAgna cum ratione <reg norm="diuiduntur" type="context">diuidÅ©tur</reg> Å¿tateræ per interualla Ä™qualia, in libras, aut in <lb/>vncias, aut quoquo alio modo. </s> - <s xml:id="s1802" xml:space="preserve">Nam Å¿it Å¿tatera exempli gratia <var>.a.b.</var> <lb/>& punctum, <reg norm="quod" type="simple">ê™</reg> eam Å¿uÅ¿tinet Å¿it <var>.c.</var> & vas illud, <reg norm="quod" type="simple">ê™</reg> continetid, quod ponderari debet <lb/>f. </s> - <s xml:id="s1803" xml:space="preserve">Imaginemur nunc quod pondus brachij <var>.c.b.</var> ab una parte, & pondus brachij <var>.c.a.</var> <reg norm="cum" type="context">cÅ©</reg> <lb/>eo, <reg norm="quod" type="simple">ê™</reg> eÅ¿t dicti vaÅ¿is <var>.f.</var> ab altera parte, Å¿int cauſę, quibus Å¿tatera <var>.a.b.c.</var> Å¿tet orizonta-<lb/>lis. cui Å¿ic orizontali manenti imaginemur ad punctum <var>.a.</var> adiunctum eſſe pondus, <lb/>veluti vnius librÄ™. & ad punctum <var>.d.</var> tam diÅ¿tanti à <var>.c.</var> ut eÅ¿t <var>.a.</var> ab ipÅ¿o <var>.c.</var> aliud quoque <lb/>pondus vnius libræ <reg norm="additum" type="context">additÅ©</reg> eſſe, vnde <reg norm="coni" type="context">cõi</reg> <reg norm="quadam" type="context">quadã</reg> Å¿cientia Å¿tatera, non mouebitur Å¿itu. <lb/></s> - <s xml:id="s1804" xml:space="preserve"><reg norm="quia" type="simple">ê—a</reg> exiÅ¿tentibus duobus hiÅ¿ce ponderibus æqualibus, altero in <var>.d.</var> & altero in <var>.a.</var> remo <lb/>ta cum eſſent <var>.d.b.</var> et <var>.f.</var> abÅ¿que dubio <var>.a.d.</var> non mutaret Å¿itum, Å¿ed <var>.d.b.</var> et, f. in Å¿itu, in <lb/>quo reperiuntur, à centro paribus viribus prÄ™dita Å¿unt. </s> - <s xml:id="s1805" xml:space="preserve">Addendo igitur <var>.d.b.</var> ipÅ¿i <var>.d.</var> <lb/>et <var>.f.</var> ipÅ¿i .a: Å¿umma earum, æqualibus quoque viribus conÅ¿tabunt. ex communi Å¿en-<lb/>tentia, quæ habet Å¿i Ä™qualibus addas Ä™qualia, tota quoque fient Ä™qualia. </s> - <s xml:id="s1806" xml:space="preserve">Si verò <lb/>ponderi ipÅ¿ius <var>.a.</var> aliud adderetur eidem Ä™quale, haberemus in <var>.a.</var> duplum pon-<lb/>dus ei <reg norm="quod" type="simple">ê™</reg> eÅ¿t ipÅ¿ius <var>.d.</var> Å¿ed volentes vt Å¿olum cum pondere ipÅ¿ius <var>.d.</var> Å¿tatera Å¿tet orizon <lb/>talis, Å¿i dictum pondus ipÅ¿ius <var>.d.</var> longè diÅ¿tabit à centro <var>.c.</var> per duplum ipÅ¿ius <var>.c.a.</var> ideÅ¿t <lb/>ipÅ¿ius <var>.c.d.</var> id <reg norm="quod" type="simple">ê™</reg> volumus aſſeque-<lb/>mur, beneficio Å¿upradictarum ra <lb/> - <anchor type="figure" xlink:label="fig-0163-01a" xlink:href="fig-0163-01"/> - tionum, adiuti opera Å¿extÄ™ lib. priÂ<lb/>mi de <reg norm="ponderibus" type="context">põderibus</reg> Archimedis. </s> - <s xml:id="s1807" xml:space="preserve">Et <lb/>Å¿i quis aliud <reg norm="quoque" type="simple">quoq;</reg> pondus adiun <lb/>geret ipÅ¿i <var>.a.</var> æquale illi priori, ad <lb/><reg norm="efficiendum" type="context">efficiẽdum</reg>, vt Å¿tatera Å¿emper ori <lb/>zontalis maneret, oporteret, vt <reg norm="pondus" type="context">põdus</reg> ipÅ¿ius <var>.d.</var> ab <var>.c.</var> longè diÅ¿taret, ita vt huiuÅ¿modi <lb/>diÅ¿tantia tripla eſſet primæ, & Å¿ic per quoÅ¿dam quaÅ¿i gradus interualla redderentur <lb/>æqualia.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0163-01" xlink:href="fig-0163-01a"> - <image file="0163-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0163-01"/> - </figure> - </div> - <pb o="152" rhead="IO. BAPT. BENED." n="164" file="0164" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0164"/> - </div> - <div type="section" level="3" n="10"> - <head xml:id="head212" style="it" xml:space="preserve">Quòd line a circularis non habe at concauum cum con-<lb/>uexo coniunctum, & quod Aristo. cir caproportio <lb/>nes motuum aberrauerit.</head> - <head xml:id="head213" xml:space="preserve">CAP.X.</head> - <p> - <s xml:id="s1808" xml:space="preserve">ARiÅ¿toteles in principio quæſtionum Mechanicarum ait lineam, quæ terminat <lb/> - <anchor type="handwritten" xlink:label="hd-0164-01a" xlink:href="hd-0164-01"/> - circulum videtur conuexum habere coniunctum cum concauo, quod falÅ¿um <lb/>eÅ¿t: </s> - <s xml:id="s1809" xml:space="preserve">quia huiuÅ¿modi linea partes nullas Å¿ecundum latitudinem habet, (vt ipÅ¿e etiam <lb/>confirmat) Å¿ed eÅ¿t idem conuexum circuli: </s> - <s xml:id="s1810" xml:space="preserve">linea verò quæ terminus eÅ¿t Å¿uperficiei <lb/>ambientis, & amplectentis circulum eÅ¿t eadem concauitas dictæ Å¿uperficiei eun-<lb/>dem circulum ambientis, quæ nullam conuexitatem habet. </s> - <s xml:id="s1811" xml:space="preserve">& hæ duæ Å¿unt lineæ, <lb/>quarum vna diuerÅ¿a eÅ¿t ab alia, neque altera alterius, quod ad conuexum, & ad con-<lb/>cauum attinet.</s> - </p> - <div type="float" level="4" n="1"> - <handwritten xlink:label="hd-0164-01" xlink:href="hd-0164-01a"/> - </div> - <p> - <s xml:id="s1812" xml:space="preserve">Sed illud, quod AriÅ¿toteles Å¿cribit de duplici reÅ¿pectu motus vnius puncti Å¿ecun <lb/>dum vnam datam pro portionem, non Å¿ufficit, ille enim Å¿ic ait.</s> - </p> - <p> - <s xml:id="s1813" xml:space="preserve">Sit proportio Å¿ecundum quam latum fertur, quam habet <var>.A.B.</var> ad <var>.A.C.</var> et <var>.A.</var> qui <lb/>dem feratur verÅ¿us .B: <var>A.B.</var> verò Å¿ubterferatur verÅ¿us <var>.M.C.</var> latum autem Å¿it <var>.A.</var> <reg norm="quidem" type="context">quidẽ</reg> <lb/>ad <var>.D.</var> vbi autem eÅ¿t <var>.A.B.</var> verÅ¿us <var>.E</var>. </s> - <s xml:id="s1814" xml:space="preserve">Quoniam igitur lationis erat proportio, quam <var>.<lb/>A.B.</var> habet ad <var>.A.C.</var> neceſſe eÅ¿t & <var>.A.D.</var> ad <var>.A.E.</var> hanc habere rationem. </s> - <s xml:id="s1815" xml:space="preserve">Simile igi <lb/> - <anchor type="handwritten" xlink:label="hd-0164-02a" xlink:href="hd-0164-02"/> - tur eÅ¿t pro portione paruum quadr ilaterum maiori. </s> - <s xml:id="s1816" xml:space="preserve">Quamobrem etc.</s> - </p> - <div type="float" level="4" n="2"> - <handwritten xlink:label="hd-0164-02" xlink:href="hd-0164-02a"/> - </div> - <p> - <s xml:id="s1817" xml:space="preserve">Cui reÅ¿pondeo, punctum <var>.A.</var> quod mouetur in linea <var>.A.M.</var> ab <var>.A.</var> verÅ¿us <var>.M.</var> vÅ¿que <lb/>ad <var>.F.</var> non moueriab aliqua proportione determinata magis quà m ab alia: </s> - <s xml:id="s1818" xml:space="preserve">vnde <reg norm="non" type="context">nõ</reg> <lb/>Å¿olum poſſumus imaginari dictum punctum <var>.A.</var> moueri ab <var>.A.</var> vÅ¿que ad <var>.F.</var> eiuÅ¿dem <lb/>velocitatis Å¿ub alia quadam proportione, Å¿ed etiam Å¿ub alia, quæ iam datæ contraria <lb/>Å¿it, vt eÅ¿t proportio ipÅ¿ius <var>.A.C.</var> ad <var>.A.B.</var> <reg norm="imaginantes" type="context">imaginãtes</reg> moueri <var>.A.</var> verÅ¿us <var>.C.</var> et <var>.A.C.</var> ver <lb/>Å¿us <var>.B.M.</var> delatam. </s> - <s xml:id="s1819" xml:space="preserve">Dico etiam idem <var>.A.</var> moueri vÅ¿que ad <var>.F.</var> Å¿ecundum proportio-<lb/>nem ipÅ¿ius <var>.A.O.</var> ad <var>.A.N</var>. </s> - <s xml:id="s1820" xml:space="preserve">Quamobrem imaginemur à puncto <var>.F.</var> lineam <var>.F.H.</var> cum <lb/>linea <var>.F.A.</var> efficere angu-<lb/>lum æqualem angulo <var>.O.<lb/>P.A.</var> & à puncto <var>.A.</var> <reg norm="lineam" type="context">lineã</reg> <lb/> - <anchor type="figure" xlink:label="fig-0164-01a" xlink:href="fig-0164-01"/> - <var>A.H.</var> <reg norm="cum" type="context">cÅ©</reg> linea <var>.A.F.</var> face-<lb/>re <reg norm="angulum" type="context">angulÅ©</reg> <reg norm="æqualem" type="context">æqualẽ</reg> angulo <lb/><var>O.A.P.</var> unde angulus <var>.H.</var> <lb/>æqualis erit angulo <var>.O.</var> <lb/>ex .32. libr. primi Eucl. <lb/></s> - <s xml:id="s1821" xml:space="preserve">& <reg norm="triangulum" type="context">triangulÅ©</reg> <var>.A.H.F.</var> Ä™qui <lb/>angulum erit triangulo <var>.<lb/>A.O.P</var>. </s> - <s xml:id="s1822" xml:space="preserve">Quam ob <reg norm="causam" type="context">causã</reg> <lb/><reg norm="eadem" type="context">eadẽ</reg> proportio erit <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <lb/><var>A.H.</var> ad <var>.F.H.</var> quÄ™ <reg norm="enipÅ¿ius" type="context">ẽipÅ¿ius</reg> <lb/><var>A.O.</var> ad <var>.O.P.</var> punctum <lb/>igitur <var>.A.</var> vÅ¿que ad <var>.F.</var> mouetur Å¿ecundum proportionem etiam ipÅ¿ius <var>.A.O.</var> ad <var>.O.P.</var> <lb/>HuiuÅ¿modi igitur conÅ¿ideratio, ab AriÅ¿totele facta, nullius eÅ¿t momenti.</s> - </p> - <div type="float" level="4" n="3"> - <figure xlink:label="fig-0164-01" xlink:href="fig-0164-01a"> - <image file="0164-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0164-01"/> - </figure> - </div> - <pb o="153" rhead="DE MECHAN." n="165" file="0165" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0165"/> - </div> - <div type="section" level="3" n="11"> - <head xml:id="head214" style="it" xml:space="preserve">Quod Aristo. in prima mechanicarum quæstionum eius quod <lb/>inquir it, uer am cauÅ¿am non attulerit.</head> - <head xml:id="head215" xml:space="preserve">CAP. XI.</head> - <p> - <s xml:id="s1823" xml:space="preserve">QVærens AriÅ¿toteles vnde fiat, vt eæ libræ, quæ brachia habent alijs longiora, <lb/>Å¿int exactiores cæteris, ait hoc euenire ratione maioris velocitatis extremo <lb/>rum earundem. </s> - <s xml:id="s1824" xml:space="preserve">Quod verum non eÅ¿t; </s> - <s xml:id="s1825" xml:space="preserve">quia hîc effectus nil aliud eÅ¿t, quam clarius pro <lb/>ponere ob omnium oculos obliquitatem brachiorum à linea orizontali, & oÅ¿tende-<lb/>re etiam facilius à dicto orizontali Å¿itu exire brachia iam dicta. </s> - <s xml:id="s1826" xml:space="preserve">Quæ quidem per Å¿e <lb/>neque à velocitate, neque à tarditate motus, Å¿ed à ratione vectis, & à ma-<lb/>iori interuallo inter Å¿ecundum Å¿itum extremorum à primo proficiÅ¿cuntur. </s> - <s xml:id="s1827" xml:space="preserve">Vt exem-<lb/>pli gratia, imaginemur magnam libram <var>.A.B.</var> orizontalem, cuius centrum Å¿it <var>.E.</var> et <lb/>pondus <var>.B.</var> maius Å¿it pondere ipÅ¿ius <var>.A.</var> vnde conceditur, quòd ob hanc rationem di-<lb/>cta libra Å¿itum mutabit, qui Å¿ecundus Å¿itus Å¿it in <var>.H.F</var>. </s> - <s xml:id="s1828" xml:space="preserve">Imaginemur etiam <reg norm="paruam" type="context">paruã</reg> <reg norm="quan- dam" type="context">quã-<lb/>dam</reg> libram <var>.a.e.b.</var> orizontalem, quæ pondera habeat <var>.a.</var> et <var>.b.</var> æqualia duobus ponde <lb/>ribus alterius libræ & Å¿ecundus Å¿itus Å¿it in <var>.h.f.</var> ita tamen vt anguli circa <var>.e.</var> æquales <lb/>Å¿int ijs, qui Å¿unt circa <var>.E.</var> ideÅ¿t <var>.b.e.f.</var> Å¿it Ä™qualis <var>.B.E.F</var>. </s> - <s xml:id="s1829" xml:space="preserve">Nunc dico Å¿itum <var>.H.F.</var> <reg norm="exa- ctiorem" type="context">exa-<lb/>ctiorẽ</reg> futurum & clariorem Å¿itu <var>.h.e.f.</var> ratione interualli <var>.B.F.</var> maioris, interuallo <var>.<lb/>b.f.</var> quod <var>.B.F.</var> in eadem proportione maior eÅ¿t ipÅ¿o <var>.b.f.</var> in qua <var>.B.E.</var> maius eÅ¿t <var>.b.e.</var> <lb/>quod autem interuallum <var>.B.F.</var> breuiori, aut longiori temporis Å¿pacio quam <var>.b.f.</var> Å¿it fa <lb/>ctum, nil planè refert. </s> - <s xml:id="s1830" xml:space="preserve">Ratione vectis deinde, dico <reg norm="quod" type="simple">ê™</reg> Å¿i Å¿upponemus duas libras pa-<lb/>res <reg norm="æqualesque" type="simple">æqualesqÌ;</reg> in omni alio reÅ¿pectu, præter quà m in brachiorum longitudine, pon-<lb/>dus <var>.B.</var> maiorem vim habebit ad deprimendum brachium <var>.E.B.</var> quà m pondus <var>.b.</var> quia <lb/>libræ materiales, cum Å¿uÅ¿tineantur ab <var>.E.e.</var> & non à puncto mathematico, Å¿ed <lb/>à linea, aut Å¿uperficie naturali in materia exiÅ¿tente. </s> - <s xml:id="s1831" xml:space="preserve">vnde aliqua reÅ¿iÅ¿tentia ipÅ¿i mo-<lb/>tui brachiorum oritur, & hanc ob cauÅ¿am, Å¿upponendo hanc reÅ¿iÅ¿tentiam æqualem <lb/>tam in <var>.E.</var> quà m in <var>.e.</var> clarum erit ob ea, quæ in cap .4. huius tractatus oÅ¿tendi <var>.B.</var> cum <lb/>minus dependeat ab <var>.E.</var> aut minus quoque eidem <var>.E.</var> annitatur, ponderoÅ¿um magis <lb/>futurum, quam <var>.b.</var> & hac de cauÅ¿a mouebit ad partem inferiorem, maiori cum agilita <lb/>te, brachium <var>.E.B.</var> multo magis etiam illud ipÅ¿um deprimet, ideÅ¿t maiorem etiam an <lb/>gulum <var>.B.E.F.</var> quà m erit angulus <var>.b.e.f.</var> faciet.</s> - </p> - <figure position="here"> - <image file="0165-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0165-01"/> - </figure> - <pb o="154" rhead="IO. BAPT. BENED." n="166" file="0166" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0166"/> - </div> - <div type="section" level="3" n="12"> - <head xml:id="head216" style="it" xml:space="preserve">De uer a cauÅ¿a Å¿ecundæ, & tertiæ quæstionis mechanicæ <lb/>ab AriÅ¿totele nonperÅ¿pecta.</head> - <head xml:id="head217" xml:space="preserve">CAP. XII.</head> - <p> - <s xml:id="s1832" xml:space="preserve">ARiÅ¿toteles in Å¿ecunda quæſtionum mechanicarum quærens illius rationem Å¿ic <lb/>Å¿cribit.</s> - </p> - <p> - <s xml:id="s1833" xml:space="preserve">Cur Å¿iquidem Å¿urÅ¿um fuerit Å¿partum quando deorÅ¿um lato pondere quiÅ¿piam id <lb/>admouet rurÅ¿us aÅ¿cendit libra: </s> - <s xml:id="s1834" xml:space="preserve">Å¿i autem deorÅ¿um conÅ¿titutum fuerit non aÅ¿cendit, <lb/>Å¿ed manet? </s> - <s xml:id="s1835" xml:space="preserve">an quia Å¿urÅ¿um quidem Å¿parto exiÅ¿tente plus libræ extra perpendiculum <lb/>Å¿i (Å¿partum enim eÅ¿t perpendiculum) quare neceſſe eÅ¿t deorÅ¿um ferriid, quod plus <lb/>eÅ¿t, quare & cætera.</s> - </p> - <p> - <s xml:id="s1836" xml:space="preserve">Sed vera cauÅ¿a, vnde fiat, vt Å¿i Å¿partum fuerit Å¿urÅ¿um, & brachium vnum <lb/>ipfius libræ deprimendo, & idem liberum deinde permittendo, ad Å¿itum ori-<lb/>zontalem redeat, non Å¿olum eÅ¿t maior quantitas ponderis brachiorum quæ iam præ <lb/>tergreſſa eÅ¿t vltra verticalem lineam, Å¿ed etiam eÅ¿t longitudo brachij eleuati, quæ vl <lb/>tra verticalem lineam reperitur, vnde eius extremi pondus redditur grauius in pro-<lb/>portione, quam in hoc exemplo proponam, Å¿it <var>.A.B.</var> libra in Å¿itu orizontali, cuius <lb/>Å¿partum Å¿it <var>.E.</var> Å¿uper ipÅ¿am. </s> - <s xml:id="s1837" xml:space="preserve">& deprimentes brachium ipÅ¿ius <var>.A.</var> vÅ¿que ad <var>.F.</var> eius Å¿itus <lb/>Å¿it in <var>.F.H.</var> vnde medium <reg norm="punctum" type="context">pÅ©ctum</reg> <var>.G.</var> prætergreſſum erit lineam verticalem <var>.V.Z.</var> ver <lb/>Å¿us <var>.B.</var> quæ <var>.V.Z.</var> Å¿ecabit brachium <var>.F.G.</var> in puncto <var>.D.</var> vnde <var>.D.H.</var> longius erit ipÅ¿o <var>.<lb/>F.D</var>. </s> - <s xml:id="s1838" xml:space="preserve">Nunc nobis Å¿upponendum eÅ¿t id, <lb/>quod veriſſimum exiÅ¿tit, dictam Å¿cilicet li <lb/> - <anchor type="figure" xlink:label="fig-0166-01a" xlink:href="fig-0166-01"/> - bram in Å¿itu <var>.F.H.</var> <reg norm="etiam" type="context">etiã</reg> Å¿i Å¿uÅ¿tineatur à pun-<lb/>cto <var>.E.</var> idem tamen futurum ac Å¿i Å¿uÅ¿tenta-<lb/>retur in puncto <var>.D.</var> vnde Å¿equitur, quod <lb/>pondus appenÅ¿um ex ipÅ¿a <var>.H.</var> ita grauius <lb/>reddatur, ipÅ¿o <var>.F.</var> in eadem propor-<lb/>tione, quæ maioreÅ¿t <var>.D.H.</var> ipÅ¿o <var>.D.F.</var> ob <lb/>rationes quas in primis huius tractatus ca-<lb/>pitibus poÅ¿ui, vt etiam Å¿i <var>.D.H.</var> quodmate <lb/>riale eſſe Å¿upponitur, nullam planÅ¿e4; </s> - <s xml:id="s1839" xml:space="preserve">graui-<lb/>tatem haberet, <reg norm="Å¿olustamem" type="context">Å¿olustamẽ</reg> exceſſus vis pon <lb/>deris in <var>.H.</var> poÅ¿iti, longè maior pondere in <lb/>F. collocato pro maiorilongitudine ipÅ¿ius <lb/><var>D.H.</var> Å¿ufficiat. ad præſtandum vt libra ad <lb/>Å¿itum orizontalem redeat.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0166-01" xlink:href="fig-0166-01a"> - <image file="0166-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0166-01"/> - </figure> - </div> - <p> - <s xml:id="s1840" xml:space="preserve">In Å¿ecunda deinde huius quęſtionis par <lb/> - <anchor type="handwritten" xlink:label="hd-0166-01a" xlink:href="hd-0166-01"/> - te, in qua Å¿cribit libram in Å¿itu, in quo poÅ¿i <lb/>ta eÅ¿t, firmam manere, toto cÄ™lo aberrat, quia <reg norm="neceſſarium" type="context">neceſſariÅ©</reg> eÅ¿t, vt omninò cadat, eò <reg norm="uÅ¿que" type="simple">uÅ¿q;</reg> <lb/>quò Å¿partum Å¿urÅ¿um remaneat: </s> - <s xml:id="s1841" xml:space="preserve">ablato tamen omni impedimento, quod nulla eget <lb/>probatione, cum natura Å¿ua clariſſimè pateat.</s> - </p> - <div type="float" level="4" n="2"> - <handwritten xlink:label="hd-0166-01" xlink:href="hd-0166-01a"/> - </div> - <p> - <s xml:id="s1842" xml:space="preserve">CauÅ¿a, deinde, vera tertiæ quæſtionis non eÅ¿t ea, quam AriÅ¿toteles ponit, Å¿ed hu-<lb/>iuÅ¿modi effectus ab eo, quod capitibus .4. et .5. huius tractatus propoÅ¿ui originem <lb/>habet.</s> - </p> - <pb o="155" rhead="DE MECHAN." n="167" file="0167" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0167"/> - </div> - <div type="section" level="3" n="13"> - <head xml:id="head218" style="it" xml:space="preserve">Quòd AriÅ¿totelisratio in 6. quæſtione poÅ¿it a non Å¿it admittenda.</head> - <head xml:id="head219" xml:space="preserve">CAP. XIII.</head> - <p> - <s xml:id="s1843" xml:space="preserve">VOlens AriÅ¿toteles rationem proponere, vnde fiat, vt nauis velocius moueatur <lb/>cum antennam altiorem quà m cum depræſſiorem habet, id ad vectis ratio-<lb/>nem refert, quod verum <reg norm="non" type="context">nõ</reg> eÅ¿t. </s> - <s xml:id="s1844" xml:space="preserve">HuiuÅ¿modi enim ratione nauis tardius potius, quà m <lb/>velocius ferri deberet, quia quantò altius eÅ¿t velum, vi venti impulÅ¿um, <reg norm="tantò" type="context">tãtò</reg> magis <lb/>proram ipÅ¿ius nauis in aquam demergit. </s> - <s xml:id="s1845" xml:space="preserve">Sed huiuÅ¿modi effectus à maiori potius <lb/>quantitate venti quam recipit, quà m ab alia aliqua cauÅ¿a oritur, quia ventus liberius <lb/><reg norm="vehementiusque" type="simple">vehementiusqÌ;</reg> in altiore parte, quà m in depræſſione vagatur & perflat.</s> - </p> - </div> - <div type="section" level="3" n="14"> - <head xml:id="head220" style="it" xml:space="preserve">Quòdrationes ab AriÅ¿totele de octaua quæstione confictæ <lb/>Å¿ufficient es non Å¿int.</head> - <head xml:id="head221" xml:space="preserve">CAP. XIIII.</head> - <p> - <s xml:id="s1846" xml:space="preserve">RAtiones etiam ab AriÅ¿totele propoÅ¿itæ pro indaganda octauæ quæſtionis ve-<lb/>ritate, in qua quærit vnde fiat, vt corpora rotundæ figuræ, ad <reg norm="voluendum" type="context">voluendÅ©</reg> Å¿int <lb/>faciliora reliquis, quarum reuolutionum corporum tres Å¿pecies aſſignat, <reg norm="quarum" type="context">quarÅ©</reg> vna <lb/>eÅ¿t, vt rotarum <reg norm="curruum" type="context">curruÅ©</reg>; </s> - <s xml:id="s1847" xml:space="preserve">altera vt rotarum puteorum, aut trochlearum, quibus hauri-<lb/>tur aqua; </s> - <s xml:id="s1848" xml:space="preserve">& tertia, vt paruorum vaÅ¿orum a figulis fabricatorum, <reg norm="Å¿ufficientes" type="context">Å¿ufficiẽtes</reg> <reg norm="non" type="context">nõ</reg> Å¿unt.</s> - </p> - <p> - <s xml:id="s1849" xml:space="preserve">Incipiens autem à prima dico dubium non eſſe, quin tangente corpore aliquo ro <lb/>tundo aliquod planum mediante Å¿olo quodam puncto contingat, quemadmodum <lb/>probat TheodoÅ¿ius in .3. lib. primi & Vitellio in .71. lib. primi, & <reg norm="ducendo" type="context">ducẽdo</reg> per <reg norm="centrum" type="context">centrÅ©</reg> <lb/>Å¿phæræ lineam vÅ¿que ad punctum contactus, ipÅ¿a erit perpendicularis plano contin-<lb/>genti Å¿phÄ™ram dictam, vt probat <reg norm="idem" type="context">idẽ</reg> ThÄ™odoÅ¿ius in .4. lib. primi <reg norm="Alhazem" type="context">Alhazẽ</reg> in .25. quar-<lb/>ti, & Vitellio in .7. primi. </s> - <s xml:id="s1850" xml:space="preserve">Verum etiam eÅ¿t omnem inclinationem ponderoÅ¿am huiuÅ¿ <lb/>modi corporis homogÄ™nei totam hanc lineam æqualiter omni ex parte circundare; <lb/></s> - <s xml:id="s1851" xml:space="preserve">cuius quidem rei exemplum in carta deÅ¿cribere poſſumus mediante figura circulari <lb/>hîc Å¿ubÅ¿cripta <var>.a.n.e.u.</var> contigua lineæ rectæ <var>.b.d.</var> in puncto <var>.a.</var> vnde <var>.e.o.a.</var> perpendicu <lb/>laris erit ipÅ¿i <var>.b.d.</var> ex .17. lib. 3. Eucli. & <reg norm="tantum" type="context">tantÅ©</reg> ponderis habebimus à parte <var>.a.u.e.</var> quan <lb/>tum ab ipÅ¿a <var>.a.n.e</var>. </s> - <s xml:id="s1852" xml:space="preserve">Nuncigitur Å¿i imaginabimur ductum eſſe centrum verÅ¿us <var>.u.</var> per <lb/>lineam <var>.o.u.</var> parallelam ipÅ¿i <var>.a.d.</var> clarum nobis <lb/>erit, <reg norm="quod" type="simple">ê™</reg> <reg norm="abÅ¿que" type="simple">abÅ¿q;</reg> vlla difficultate aut reÅ¿iÅ¿tentia <reg norm="idem" type="context">idẽ</reg> <lb/> - <anchor type="figure" xlink:label="fig-0167-01a" xlink:href="fig-0167-01"/> - ducemus, quia huiuÅ¿modi centrum ab inferiori <lb/>parte ad Å¿uperiorem, nunquam mutabit Å¿itum <lb/>reÅ¿pectu <reg norm="diÅ¿tantiæ" type="context">diÅ¿tãtiæ</reg> Å¿eu interualli, quÄ™ inter ipÅ¿um <lb/>lineamqÌue <var>.a.d.</var> intercedit, <reg norm="quod" type="simple">ê™</reg> quidem centrum <lb/>in Å¿e colligittotum pondus figurÄ™ <var>.a.n.e.u.</var> & be <lb/>neficio lineæ <var>.e.o.a.</var> illud ipÅ¿um puncto <var>.a.</var> in li-<lb/>nea <var>.b.a.d.</var> committit, productum <var>.a.</var> nil refert, <lb/>vt magis, aut minus verÅ¿us ipÅ¿um <var>.d.</var> aut verÅ¿us <lb/>b. <reg norm="dirigatur" type="simple">dirigat̃</reg>; </s> - <s xml:id="s1853" xml:space="preserve">ita vt <reg norm="cum" type="context">cÅ©</reg> non oporteat vt huius figuræ <lb/><reg norm="pondus" type="context">põdus</reg>, vna vice, magis eleuetur, quà m alia, Å¿ed <lb/>Å¿emper Ä™qualiter Å¿uper lineam <var>.b.a.d.</var> quieÅ¿cat.</s> - <pb o="156" rhead="IO. BAPT. BENED." n="168" file="0168" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0168"/> - <s xml:id="s1854" xml:space="preserve"><reg norm="Sitque" type="simple">Sitq;</reg> Å¿emper diuiÅ¿um à linea <var>.a.o.e.</var> per medium, Å¿equitur communi quodam con-<lb/>ceptu, nullam nobis difficultatem oborituram, dictum centrum ad quam volueri-<lb/>mus partem ducendo, quemadmodum à qualibet alia figura, quæ perfectè rotunda <lb/>non eſſet, emergeret; </s> - <s xml:id="s1855" xml:space="preserve">Vt <reg norm="exempli" type="context">exẽpli</reg> gratia, Å¿i imaginabimur pentagonum <var>.K.i.h.f.l.</var> quie <lb/>Å¿cere <reg norm="Å¿uper" type="simple">Å¿uê‘</reg> <reg norm="eandem" type="context">eandẽ</reg> <reg norm="lineam" type="context">lineã</reg> <var>.a.b.K.</var> ita ut <reg norm="primum" type="context">primÅ©</reg> <reg norm="totum" type="context">totÅ©</reg> latus <var>.i.K.</var> in linea <var>.b.K.</var> <reg norm="extendatur" type="context simple">extẽdat̃</reg>, <reg norm="ducen- do" type="context">ducẽ-<lb/>do</reg> poÅ¿teà centrum <var>.o.</var> (ponamus.) verÅ¿us <var>.l.</var> dubium non eÅ¿t, quin oporteat, vt dictum <lb/>centrum <var>.o.</var> à linea <var>.b.d.</var> eleuetur, ab <reg norm="eademque" type="simple">eademq;</reg> magis diÅ¿tet, voluens Å¿e per <reg norm="arcum" type="context">arcÅ©</reg> vnum <lb/>circuli, <reg norm="qui" type="simple">ê—</reg> <reg norm="pro" type="simple">ê“</reg> Å¿uo Å¿emidiametro habeat <var>.o.K.</var> quÄ™ maior eÅ¿t ipÅ¿a <var>.o.a.</var> ex .18. li. primi Eu <lb/>cli. vnde Å¿i à puncto <var>.K.</var> imaginabimur lineam <var>.K.c.</var> reÅ¿picientem centrum regionis <lb/>elementaris, dubium non eÅ¿t, quin Å¿i velimus transferre <reg norm="centrum" type="context">cẽtrum</reg> hoc à priori Å¿itu <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> <lb/>ad dictam lineam, oporteat addere pondus parti ipÅ¿ius <var>.l.</var> quæ à linea <var>.K.c.</var> fuit Å¿ecta, <lb/>aut aliquid de ipÅ¿o pondere partis centri detrahere. </s> - <s xml:id="s1856" xml:space="preserve">quod quibuÅ¿uis modis fiat, ar-<lb/>duum certè eÅ¿t ad efficiendum; </s> - <s xml:id="s1857" xml:space="preserve">neque hoc etiam accidit figuræ perfectè rotundæ, <lb/>cum <reg norm="centrum" type="context">cẽtrum</reg> <reg norm="quod" type="simple">ê™</reg> perfectè in medio ipÅ¿ius ponderis eÅ¿t, reperiatur Å¿emper in linea per-<lb/>pendiculari ipÅ¿i plano, in quo animaduertendum eÅ¿t, <reg norm="quod" type="simple">ê™</reg> etiam Å¿i ipÅ¿um planum ap-<lb/>pellem; </s> - <s xml:id="s1858" xml:space="preserve">pro plano tamen perfecto intelligi nolo, Å¿ed pro Å¿uperficie perfectè <reg norm="Å¿phaeri- ca" type="simple">Å¿phÄ™ri-<lb/>ca</reg> circa centrum à corporibus grauibus expetitum; </s> - <s xml:id="s1859" xml:space="preserve">nam ratione magnæ amplitudi-<lb/>nis huiuÅ¿modi Å¿uperficiei, nullam differentiam notatu dignam à perfecto aliquo pla <lb/>no exigui interualli ad curuitatem eiuÅ¿dem Å¿uperficiei imaginari poterimus. </s> - <s xml:id="s1860" xml:space="preserve">Sed ut <lb/>redeamus ad Å¿ermonem de reuolutione figuræ rotundæ Å¿uÅ¿ceptum, <reg norm="clarum" type="context">clarÅ©</reg> igitur erit <lb/>quamlibet minimam vim (vt ita dicam) quÄ™ trahat, aut impellat centrum <var>.o.</var> verÅ¿us <var>.u.</var> <lb/>huiuÅ¿modi figuram reuoluturam, cuius media pars ad trahendum, aut impellendum <lb/>punctum <var>.e.</var> Å¿ufficiere; </s> - <s xml:id="s1861" xml:space="preserve">Imaginemur autem <reg norm="quod" type="simple">ê™</reg> li <lb/>nea <var>.n.o.u.</var> eſſet libra <reg norm="quÄ™dam" type="context">quÄ™dã</reg> in figura perfectè <lb/> - <anchor type="figure" xlink:label="fig-0168-01a" xlink:href="fig-0168-01"/> - rotunda <var>.a.n.e.u.</var> poÅ¿ita, & vis, quÄ™ trahere cen <lb/>trum deberet, diuiÅ¿a eſſet per medium, cuius <lb/>medietas appenÅ¿a eſſet extremitati <var>.u.</var> diame-<lb/>tri <var>.n.o.u.</var> <reg norm="clarum" type="context">clarÅ©</reg> erit, <reg norm="quod" type="simple">ê™</reg> abÅ¿que vlla difficultate <lb/>reuolueret figuram Å¿uper lineam <var>.b.a.d.</var> verÅ¿us <var>.<lb/>d.</var> quia huius vis, aut pondus <reg norm="nullum" type="context">nullÅ©</reg> contra pon <lb/>dus haberet vltra centrum <var>.o.</var> uerÅ¿us <var>.n.</var> <reg norm="quod" type="simple">ê™</reg> cen-<lb/>trum <var>.o.</var> perpetuo quieÅ¿cit <reg norm="Å¿uper" type="simple">Å¿uê‘</reg>. a. in linea <var>.e.o.<lb/>a.</var> per medium diuidente Å¿emper totum pon-<lb/>dus figurÄ™ Å¿uppoÅ¿itÄ™. </s> - <s xml:id="s1862" xml:space="preserve">Tantò facilius ergo tota <lb/>dicta vis ap <lb/> - <anchor type="figure" xlink:label="fig-0168-02a" xlink:href="fig-0168-02"/> - <anchor type="figure" xlink:label="fig-0168-03a" xlink:href="fig-0168-03"/> - plicata cen <lb/>tro, <reg norm="ipsum" type="context">ipsÅ©</reg> ver <lb/>Å¿us <var>.u.</var> <reg norm="trahens" type="context">trahẽs</reg> <lb/>per lineam <lb/><reg norm="parallelam" type="context">parallelã</reg> ip <lb/>Å¿i <var>.a.d.</var> <reg norm="dictam" type="context">dictã</reg> <lb/>figuram re-<lb/>uolueret. </s> - <s xml:id="s1863" xml:space="preserve">Et <lb/>Å¿i linea qua <lb/>dictum cen <lb/>trum trahi-<lb/>tur ab ipÅ¿o - <pb o="157" rhead="DE MECHAN." n="169" file="0169" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0169"/> - <var>b.a.d.</var> non æquediÅ¿taret, Å¿ed Å¿urÅ¿um traheret Å¿uper <var>.u.</var> aut Å¿ubter, aliquid de Å¿ua vi vir <lb/><reg norm="tuteque" type="simple">tuteqÌ;</reg> amitteret, & tantò plus, quantò inclinata magis eſſet verÅ¿us <var>.a.o.e.</var> & tandem <lb/>cum eſſet vnita cum <var>.a.o.e.</var> aut ad Å¿uperius, aut ad inferius quantalibet ui, etiam Å¿i in-<lb/>finita, figuram extra Å¿itum primæ lineæ <var>.a.o.e.</var> non moueret, Å¿ed Å¿i Å¿urÅ¿um traheret Å¿e <lb/>iungeret eam à linea <var>.b.a.d.</var> non ob id tamen efficeret, ut centrum <var>.o.</var> exiret extra pri <lb/>mam lineam <var>.a.o.e</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0167-01" xlink:href="fig-0167-01a"> - <image file="0167-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0167-01"/> - </figure> - <figure xlink:label="fig-0168-01" xlink:href="fig-0168-01a"> - <image file="0168-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0168-01"/> - </figure> - <figure xlink:label="fig-0168-02" xlink:href="fig-0168-02a"> - <image file="0168-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0168-02"/> - </figure> - <figure xlink:label="fig-0168-03" xlink:href="fig-0168-03a"> - <image file="0168-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0168-03"/> - </figure> - </div> - <p> - <s xml:id="s1864" xml:space="preserve">Secunda verò Å¿pecies, tribus reuolutionum modis, abÅ¿que axis mutatione conÅ¿ta <lb/>re poteÅ¿t, ideÅ¿t modo, quo reuoluuntur trochleæ mediante fune, & quo reuoluuntur <lb/>aliquæ rotæ, in quibus aliquod animal incedit; </s> - <s xml:id="s1865" xml:space="preserve">& quo reuoluuntur illæ, quæ in homi <lb/>nis manu circunuoluuntur medio alicuius manubrij inflexi. </s> - <s xml:id="s1866" xml:space="preserve">Hi omnes modi cum <lb/>circulari figura magis, <reg norm="quam" type="context">quã</reg> cum alia quauis, faciliores euadunt. </s> - <s xml:id="s1867" xml:space="preserve">Et primò Å¿i priorem <lb/>modum conÅ¿iderabimus, vt mediante fune quælibet figura, quæ circularis non Å¿it, <lb/>voluatur, Å¿upponamus exemplo debere reuolui pentagonum æquiangulum <var>.a.e.i.o.<lb/>u.</var> circa centrum <var>.c.</var> mediante fune <var>.q.u.a.e.i.p.</var> neceſſariò occurrent (in hac figura an-<lb/>gulorum, <reg norm="laterumque" type="simple">laterumqÌ;</reg> diÅ¿parium) plures inæqualitates, quæ reuolutionem eiuÅ¿dem fi-<lb/>guræ irregularem efficient; </s> - <s xml:id="s1868" xml:space="preserve">quarum vna erit, quod duæ partes funis, ideÅ¿t <var>.u.q.</var> et <var>.i.p.</var> <lb/>non erunt in vna <reg norm="eademque" type="simple">eademqÌ;</reg> inter Å¿e diÅ¿tantia Å¿emper, quod facile intellectu erit, Å¿i ima <lb/>ginabimur ductas eſſe lineas <var>.a.i</var>: <var>u.i</var>: et <var>.i.c.t.</var> Å¿i funis duo pondera habebit alterum <lb/>altero maius, Å¿uis extremis appenÅ¿a, vnde debeat figura virtute ponderis maioris cir <lb/>cunuolui: </s> - <s xml:id="s1869" xml:space="preserve">dictæ duæ partes <var>.u.q.</var> et <var>.i.p.</var> eiuÅ¿dem funis, <reg norm="mundi" type="context">mÅ©di</reg> centrum, dum firmæ ma <lb/>nebunt, reÅ¿picient; </s> - <s xml:id="s1870" xml:space="preserve">Å¿ed permittentes pondera libera; </s> - <s xml:id="s1871" xml:space="preserve">maius, efficiens vt circunuolua-<lb/>tur figura; </s> - <s xml:id="s1872" xml:space="preserve">efficiet, vt aliquando vnum exlateribus, eiuÅ¿dem figuræ mundi <reg norm="quoque" type="simple">quoq;</reg> cen <lb/>trum reÅ¿piciet, vt in <lb/> - <anchor type="figure" xlink:label="fig-0169-01a" xlink:href="fig-0169-01"/> - figura <var>.A.</var> <reg norm="Å¿icque" type="simple">Å¿icqÌ;</reg> etiam <lb/>linea <var>.i.c.t.</var> (pro <reg norm="exem- plo" type="context">exẽ-<lb/>plo</reg>) erit menÅ¿ura di-<lb/>Å¿tantiæ funium inter <lb/>ipÅ¿as, & deinde <reg norm="circum" type="context">circÅ©</reg> <lb/>uoluendo etiam di-<lb/>Å¿tabuntinter Å¿e per li <lb/><reg norm="neam" type="context">neã</reg> <var>.i.a.</var> aut <var>.i.u.</var> vt in <lb/>figura <var>.B.</var> <reg norm="innotuit" type="context">Ä©notuit</reg> <reg norm="exem" type="context">exẽ</reg> <lb/>plo, & Å¿ic etiam ali-<lb/>quando erunt magis <lb/><reg norm="diÅ¿tantes" type="context">diÅ¿tãtes</reg>, quà m linea <lb/><var>t.i.</var> & minus quà m.i.<lb/>a: </s> - <s xml:id="s1873" xml:space="preserve">nunquam tamen minus quam <var>.t.i.</var> neque magis <reg norm="quam" type="context">quã</reg> <lb/><var>i.a.</var> aut <var>.i.u.</var> quæ Å¿unt æquales; </s> - <s xml:id="s1874" xml:space="preserve">Quæ quidem varietas, <lb/> - <anchor type="figure" xlink:label="fig-0169-02a" xlink:href="fig-0169-02"/> - in hanc, & in illam partem impellet partes penden-<lb/>tes funis, vnde æqualiter non trahent. </s> - <s xml:id="s1875" xml:space="preserve">Idem dico, Å¿i <lb/>extrema <var>.q.</var> et <var>.p.</var> eſſent quoque Å¿emper in vna <reg norm="eademque" type="context simple">eadẽqÌ;</reg> <lb/>diÅ¿tantia; </s> - <s xml:id="s1876" xml:space="preserve">neque à corpore <reg norm="ponderoÅ¿o" type="context">põderoÅ¿o</reg> eſſent attracta, <lb/>quia aliæ partes ipÅ¿ius <var>.u.q.</var> et <var>.i.p.</var> ex Å¿upradictis ratio-<lb/>nibus vnam <reg norm="eademque" type="simple">eademqÌ;</reg> diÅ¿tantiam <reg norm="non" type="context">nõ</reg> Å¿emper <reg norm="Å¿eruarent" type="context">Å¿eruarẽt</reg>. <lb/></s> - <s xml:id="s1877" xml:space="preserve">vnde fieret vt cum diuerÅ¿is angulis tam <var>.i.p.</var> <reg norm="quam" type="context">quã</reg> <var>.u.q.</var> <lb/><reg norm="traherent" type="context">traherẽt</reg> Å¿emidiametros <var>.c.i</var>: <var>c.e</var>: <var>c.a</var>: <var>c.u.</var> et <var>.c.o.</var> quia <reg norm="non" type="context">nõ</reg> <lb/>Å¿emper traherent ope Å¿eu virtute anguli æqualis ipÅ¿i <var>.<lb/>c.i.p</var>. </s> - <s xml:id="s1878" xml:space="preserve">Hæc autem inÄ™ qualitas communis eÅ¿t omnibus - <pb o="158" rhead="IO. BAPT. BENED." n="170" file="0170" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0170"/> - figuris rectilineis tam paris, quà m diÅ¿paris numeri. </s> - <s xml:id="s1879" xml:space="preserve">Sed aliam quandam maiorem <lb/>inÄ™qualitatem habent hæ figuræ numeri diÅ¿paris, quæ eÅ¿t, quòd <reg norm="quando" type="context">quãdo</reg> linea <var>.t.i.</var> tam <lb/> <var>.u.q.</var> quà m ipÅ¿i <var>.i.p.</var> <lb/> - <anchor type="figure" xlink:label="fig-0170-01a" xlink:href="fig-0170-01"/> - <reg norm="perpendicularis" type="simple context">ê‘pẽdicularis</reg> fuerit, <lb/>ideÅ¿t <reg norm="quando" type="context">quãdo</reg> <var>.t.i.</var> cum <lb/>dictis partibus funis <lb/>angulos rectos con-<lb/>Å¿tituerit, <reg norm="tunc" type="context">tÅ©c</reg> ratione <lb/><reg norm="longitudinis" type="context">lõgitudinis</reg> ipÅ¿ius <var>.c.<lb/>i.</var> maioris quam <var>.t.<lb/>c.</var> (quia cum Å¿it <var>.c.i.</var> <reg norm="ae- qualis" type="simple">Ä™-<lb/>qualis</reg> ipÅ¿i <var>.c.a.</var> et <var>.c.a.</var> <lb/>maior ipÅ¿a <var>.c.t</var>: <var>c.i.</var> <lb/>etiam maior Å¿it ipÅ¿a <var>.<lb/>c.t.</var>) pondus aut vis <lb/>ipÅ¿ius <var>.p.</var> Å¿uperabit <reg norm="eam" type="context">eã</reg> <lb/>quæ eÅ¿t ipÅ¿ius <var>.q.</var> Å¿ed <lb/>quando <var>.t.</var> erit in oppoÅ¿ita parte, et <var>.i.</var> in ea, quæ eÅ¿t <lb/>ipÅ¿ius .t: q. <reg norm="eandem" type="context">eãdem</reg> ob cauÅ¿am Å¿uperabit <var>.p.</var> & Å¿ic mo <lb/> - <anchor type="figure" xlink:label="fig-0170-02a" xlink:href="fig-0170-02"/> - tum faciet irregularem, & <reg norm="non" type="context">nõ</reg> vniformem; </s> - <s xml:id="s1880" xml:space="preserve">& obid <lb/>etiam perarduum, præter ictus, quos infligunt an-<lb/>guli in partem pendentem <reg norm="aÅ¿cendentem" type="context">aÅ¿cendẽtem</reg> funis, <reg norm="quan- do" type="context">quã-<lb/>do</reg> vnum exlateribus vnitur cum fune.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0169-01" xlink:href="fig-0169-01a"> - <image file="0169-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0169-01"/> - </figure> - <figure xlink:label="fig-0169-02" xlink:href="fig-0169-02a"> - <image file="0169-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0169-02"/> - </figure> - <figure xlink:label="fig-0170-01" xlink:href="fig-0170-01a"> - <image file="0170-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0170-01"/> - </figure> - <figure xlink:label="fig-0170-02" xlink:href="fig-0170-02a"> - <image file="0170-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0170-02"/> - </figure> - </div> - <p> - <s xml:id="s1881" xml:space="preserve">Aliam inÄ™qualitatem habent figuræ pares, quæ <lb/>etiam in imparibus cernitur, etÅ¿i aliquantulum di-<lb/>uerÅ¿a; </s> - <s xml:id="s1882" xml:space="preserve">quæ ab eo oritur, quod funes Å¿it modò ma-<lb/>gis, modo minus propinquÄ™ centro; </s> - <s xml:id="s1883" xml:space="preserve">quæ inæqualis <lb/>diÅ¿tantia, maiorem <reg norm="minoremque" type="simple">minoremqÌ;</reg> vim Å¿uper dictum <lb/>centrum ob rationes in Å¿ecunda parte cap. decimi <lb/>huius tractatus propoÅ¿itas, gignit. </s> - <s xml:id="s1884" xml:space="preserve">Nulla autem <lb/>ex ijs inæqualitatibus circulari figuræ contingit. </s> - <s xml:id="s1885" xml:space="preserve">Illud verò, quod de pentagonis fi-<lb/>guris dixi, omnibus aliis figuris diÅ¿paribus accommodari poteÅ¿t.</s> - </p> - <p> - <s xml:id="s1886" xml:space="preserve">Secundus modus eÅ¿t earum rotarum, in quibus aliquod animal incedit, quæ Å¿i cir-<lb/>culares non eſſent, tantò difficilius voluerentur, quantò pauciores angulos haberent. <lb/></s> - <s xml:id="s1887" xml:space="preserve">quod cum per Å¿e pateat, non demonÅ¿trabo. </s> - <s xml:id="s1888" xml:space="preserve">Si ergo quantò plures angulos habebit <lb/>dicta figura, tantò ad circunuoluendum hoc modo agilior erit. </s> - <s xml:id="s1889" xml:space="preserve">Circularis igitur fi-<lb/> - <anchor type="handwritten" xlink:label="hd-0170-01a" xlink:href="hd-0170-01"/> - gura, quæ ex infinitis angulis efficitur, omnium agillima erit.</s> - </p> - <div type="float" level="4" n="3"> - <handwritten xlink:label="hd-0170-01" xlink:href="hd-0170-01a"/> - </div> - <p> - <s xml:id="s1890" xml:space="preserve">Tertius modus eÅ¿t earum rotarum, quæ manubrium habent, quæ etiam quantò <lb/>pauciores angulos habebunt, tanto <reg norm="quoque" type="simple">quoq;</reg> difficiliores reddentur, tam ratione inimi <lb/>citiæ: </s> - <s xml:id="s1891" xml:space="preserve">quam exercet cum vacuo natura, quà m <reg norm="violentiÄ™" type="context">violẽtiÄ™</reg>, quam anguli aeri faciunt, eum <lb/>expellendo, vt ipÅ¿i occupent locum, quem ipÅ¿e <reg norm="aer" type="conjecture">aér</reg> implebat. </s> - <s xml:id="s1892" xml:space="preserve">Quod nullo modo po <lb/>teÅ¿t euenire circulari figuræ.</s> - </p> - <p> - <s xml:id="s1893" xml:space="preserve">Nunc nobis ad dicendum reÅ¿tat de Å¿pecie reuolutionis rotarum, quæ parallelæ <lb/>Å¿unt orizonri, quibus accidit poſſe volui primo <reg norm="tertioque" type="simple">tertioqÌ;</reg> modo Å¿ecundÄ™ Å¿peciei, & ob <lb/>id Å¿i circulares non erunt, eadem Å¿ubibunt incommoda, de quibus in Å¿ecunda illa Å¿pe <lb/>cie loquuti Å¿umus. </s> - <s xml:id="s1894" xml:space="preserve">Å¿ed circulares rotæ huius tertiæ Å¿peciei ad reuoluendum erunt re-<lb/>liquis eò faciliores, <reg norm="quod" type="simple">ê™</reg> vno <reg norm="Å¿olum" type="context">Å¿olÅ©</reg> polo nituntur; </s> - <s xml:id="s1895" xml:space="preserve">Quod alijs nequaquam conceditur.</s> - </p> - <pb o="159" rhead="DE MECHAN." n="171" file="0171" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0171"/> - <p> - <s xml:id="s1896" xml:space="preserve">Super hac tertia Å¿pecie formari poteÅ¿t problema, vnde fiat, vt quieÅ¿cens huiuÅ¿-<lb/>modi rota parallela orizonti Å¿uper vnum punctum, & quantò fieri poteÅ¿t exiÅ¿tens <reg norm="ae- qualis" type="simple">Ä™-<lb/>qualis</reg>, Å¿i eam circunuoluamus maiore qua poterimus ui, & <reg norm="eandem" type="context">eãdem</reg> poÅ¿tea dimitten-<lb/>tes non perpetuò circunuoluatur.</s> - </p> - <p> - <s xml:id="s1897" xml:space="preserve">Hoc quidem, quatuor fit ob cauÅ¿as. quarum prima eÅ¿t, quia huiuÅ¿modi motus, eius <lb/>rotæ non Å¿it naturalis. </s> - <s xml:id="s1898" xml:space="preserve">Å¿ecunda eÅ¿t, quia etiamÅ¿i rota Å¿uper punctum mathematicum <lb/>quieÅ¿ceret, oporteret tamen vt Å¿uperius <reg norm="alterum" type="context">alterÅ©</reg> haberet polum, qui ipÅ¿am <reg norm="orizontalem" type="context">orizontalẽ</reg> <lb/>teneret, qui quidem munimento aliquo corporeo indigeret; </s> - <s xml:id="s1899" xml:space="preserve">vnde fricatio quÄ™dam <lb/>conÅ¿equeretur, ex qua reÅ¿iÅ¿tentia prodiret.</s> - </p> - <p> - <s xml:id="s1900" xml:space="preserve">Tertia eÅ¿t, quia aer contiguus eam perpetuò aÅ¿tringit, <reg norm="hocque" type="simple">hocqÌ;</reg> modo eius motui <lb/>reÅ¿iÅ¿tit.</s> - </p> - <p> - <s xml:id="s1901" xml:space="preserve">Quarta eÅ¿t, quia quÄ™libet pars corporea, quÄ™ à ſe mouetur, impetu eidem à quali-<lb/>bet extrinÅ¿eca virtute mouente impręſſo, habet naturalem inclinationem ad rectum <lb/>iter, non autem curuum, vnde Å¿i à dicta rota particula aliqua Å¿uÄ™ circunferentiæ <reg norm="diÅ¿ium" type="context">diÅ¿iÅ©</reg> <lb/>geretur, abÅ¿que dubio per aliquod temporis Å¿patium pars Å¿eparata recto itinere fer <lb/>retur per aerem, vt exemplo à fundis, quibus iaciuntur lapides, Å¿umpto, cognoÅ¿ce <lb/>re poÅ¿sumus, in quibus, impetus motus impręſſus naturali quadam propenÅ¿ione <lb/>rectum iter peragit, cum euibratus lapis, per lineam rectam contiguam giro, quem <lb/>primo faciebat, in puncto, in quo dimiſſus fuit, rectum iter inÅ¿tituat, vt rationi con-<lb/>Å¿entaneum eÅ¿t.</s> - </p> - <p> - <s xml:id="s1902" xml:space="preserve">Eadem, quoque ratione fit, vt quantò maior eÅ¿t aliqua rota, tantò maiorem quo <lb/>que impetum, & impreſſionem motus eius circunferentiæ partesrecipiant, vnde <reg norm="Å¿ae pe" type="simple">ſęÂ<lb/>pe</reg> euenit, vt dum eam Å¿iÅ¿tere volumus, id <reg norm="cum" type="context">cÅ©</reg> labore & cum diſſicultate agamus ; </s> - <s xml:id="s1903" xml:space="preserve">quia <lb/>quantò maior eÅ¿t diameter vnius circuli, tantò minus curua eÅ¿t eiuÅ¿dem circunferen <lb/>tia, & tantò propius accedit angulum eiuÅ¿dem circunferentiæ ad quantitatem duo-<lb/>rum angulorum rectorum rectilineorum, ideÅ¿t circunferentia ad rectitudinem linea <lb/>rem. </s> - <s xml:id="s1904" xml:space="preserve">Vnde earundem partium dictæ circunferentiæ motus ad inclinationem Å¿ibi à <lb/>natura tributam, quæ eÅ¿t incedendi per lineam rectam, magis accedit.</s> - </p> - </div> - <div type="section" level="3" n="15"> - <head xml:id="head222" style="it" xml:space="preserve">Quod Aristotelis ratio none queÅ¿tionis <lb/>admittendanon Å¿it.</head> - <head xml:id="head223" xml:space="preserve">CAP. XV.</head> - <p> - <s xml:id="s1905" xml:space="preserve">VEra ratio nonæ quęſtionis à ſecunda parte decimi cap. huius tractatus, & non <lb/>aliunde, accerÅ¿iri debet.</s> - </p> - </div> - <div type="section" level="3" n="16"> - <head xml:id="head224" style="it" xml:space="preserve">Quod Aristotelis rationes de decima queÅ¿tione <lb/>Å¿int reijciende.</head> - <head xml:id="head225" xml:space="preserve">CAP. XVI.</head> - <p> - <s xml:id="s1906" xml:space="preserve">ARiÅ¿totelis rationes, vnde fiat, vt facilius moueantur libræ vacuæ, quà m plenè <lb/>ad propoÅ¿itam diÅ¿putationem non pertinent; </s> - <s xml:id="s1907" xml:space="preserve">quia Å¿emper ineunda eÅ¿t ratio <lb/>proportionis virtutis mouentis Å¿uper mobile; </s> - <s xml:id="s1908" xml:space="preserve">quod ipÅ¿e non fecit.</s> - </p> - <pb o="160" rhead="IO. BABPT. BENED." n="172" file="0172" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0172"/> - <p> - <s xml:id="s1909" xml:space="preserve">Sit exempli gratia libra <var>.a.i.e.</var> quæ in vtraque extremitate vnciam vnam Å¿olum <lb/>ponderis obtineat, & Å¿it libra <var>.n.i.u.</var> æqualis priori, quæ pro Å¿ingula extremitate <reg norm="vnam" type="context">vnã</reg> <lb/> - <anchor type="handwritten" xlink:label="hd-0172-01a" xlink:href="hd-0172-01"/> - ponderis libram habeat. </s> - <s xml:id="s1910" xml:space="preserve">AriÅ¿toteles admiratur, quòd addendo ipÅ¿i <var>.e.</var> mediam pon <lb/>deris vnciam, brachium <var>.i.e.</var> velocius cadat, quà m <reg norm="adijciendo" type="context">adijciẽdo</reg> <reg norm="ipsam" type="context">ipsã</reg> <reg norm="mediam" type="context">mediã</reg> <reg norm="vnciam" type="context">vnciã</reg> ipÅ¿i <var>.u.</var> <lb/>brachij <var>.i.u</var>. </s> - <s xml:id="s1911" xml:space="preserve">Quod à duabus cauÅ¿is proficiÅ¿citur, quarum prior eÅ¿t, magna differentia <lb/>proportionis vnius libræ ad medietatem vnius vnciæ, ad proportionem vnius vnciæ <lb/>ad ipÅ¿am medietatem, quia Å¿i pondus adiectum extremo <var>.u.</var> dimidiæ eſſet libræ , & <lb/>cum eadem tarditate brachium moueret, optimo iure in admirationem poſſet Ari-<lb/>Å¿toteles duci. </s> - <s xml:id="s1912" xml:space="preserve">Sed hoc fieri non poſſet, quia ipÅ¿um deprimeret cum eadem quaÅ¿i ve <lb/>locitate, qua media vncia brachium <var>.i.e</var>. </s> - <s xml:id="s1913" xml:space="preserve">Dixi autem quaÅ¿i, quia nonnihil diÅ¿crimi-<lb/>nis intercederet, quod proficiÅ¿citur à ſecunda ratione. </s> - <s xml:id="s1914" xml:space="preserve">Et hæc, reÅ¿iÅ¿tentia eÅ¿t , quæ <lb/>oritur à ſparto, quia quantò maius pondus continet libra, tantò magis præmit Å¿par <lb/>tum in loco, in quo Å¿uÅ¿tinetur; </s> - <s xml:id="s1915" xml:space="preserve">vnde maior reÅ¿iÅ¿tentia in circunuolutione <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> Å¿par <lb/>ti, in loco, in quo quieÅ¿cit, exoritur, quia ipÅ¿um eÅ¿t corpus materiale. </s> - <s xml:id="s1916" xml:space="preserve">Si quis autem <lb/>vellet, vt brachium <var>.i.u.</var> eadem agilitate, qua <var>.i.e.</var> deÅ¿cenderet, oporteret, vt propor-<lb/>tio dimidiæ librÄ™ adiectæ ponderi ipÅ¿ius <var>.u.</var> <lb/> - <anchor type="figure" xlink:label="fig-0172-01a" xlink:href="fig-0172-01"/> - quod eÅ¿t vnius libræ, vim Å¿uam haberet, <lb/>quæ excederet reÅ¿iÅ¿tentiam Å¿ui Å¿parti (me-<lb/>dio brachiorum maiorum ijs qui Å¿unt <var>.a.i.<lb/>e.</var>) ita proportionatam, vt proportionata <lb/>eÅ¿t vis dimidiæ vnciæ ipÅ¿i e. iunctæ, reÅ¿iÅ¿ten <lb/>tiæ Å¿ui Å¿parti. </s> - <s xml:id="s1917" xml:space="preserve">HuiuÅ¿modi rationes cum ro-<lb/>tis grauioribus <reg norm="leuioribusque" type="simple">leuioribusq;</reg>, & ijs, quÄ™ à cor <lb/>poribus quibuÅ¿libet grauibus impelluntur, accommodatæ fuerint, titubantem intel <lb/>lectum confirmabunt.</s> - </p> - <div type="float" level="4" n="1"> - <handwritten xlink:label="hd-0172-01" xlink:href="hd-0172-01a"/> - <figure xlink:label="fig-0172-01" xlink:href="fig-0172-01a"> - <image file="0172-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0172-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="17"> - <head xml:id="head226" style="it" xml:space="preserve">De uer a cauÅ¿a .12. questionis mechanice.</head> - <head xml:id="head227" xml:space="preserve">CAP. XVII.</head> - <p> - <s xml:id="s1918" xml:space="preserve">VEra ratio, cur multò longius corpus aliquod graue impellatur funda, quam <lb/>manu, inde oritur, quòd circunuoluendo fundam, maior impræſſio impetus <lb/>motus fit in corpore graui, quà m fieret manu, quod corpus liberatum deinde cum <lb/>fuerit à funda, natura duce, iter <reg norm="fuum" type="context">fuÅ©</reg> à puncto, à quo proÅ¿ilijt, per lineam contiguam <lb/>giro, quem poÅ¿tremò faciebat, Å¿uÅ¿cipit. </s> - <s xml:id="s1919" xml:space="preserve"><reg norm="Dubitandumque" type="simple">DubitandumqÌ</reg> non eÅ¿t, quin dicta funda <lb/>maior impetus motus dicto corpori imprimi poſſit, <reg norm="cum" type="context">cÅ©</reg> ex multis circumactibus, ma-<lb/>ior Å¿emper impetus dicto corpori accedat. </s> - <s xml:id="s1920" xml:space="preserve">Manus autem eiuÅ¿dem corporis motus, <lb/>dum illud ipÅ¿um circunuoluitur (pace AriÅ¿totelis dixerim) centrum non eÅ¿t, neque <lb/>funis eÅ¿t Å¿emidiameter. </s> - <s xml:id="s1921" xml:space="preserve">Immo manus quam maximè fieri poteÅ¿t in orbem cietur; <lb/></s> - <s xml:id="s1922" xml:space="preserve">qui quidem motus in orbem, vt circumagatur etiam ipÅ¿um corpus, cogit, quod qui-<lb/>dem corpus, naturali quadam inclinatione, exiguo quodam impetu iam incÄ™pto, <lb/>vellet recta iter peragere, vt in Å¿ubÅ¿cripta figura patet, in qua <var>.e.</var> Å¿ignificat manum <var>.a.</var> <lb/>corpus <var>.a.b.</var> lineam rectam tangentem girum <var>.a.a.a.a.</var> quando corpus liberum rema-<lb/>net. </s> - <s xml:id="s1923" xml:space="preserve">Verum quidem eÅ¿t, impręſſum illum impetum, continuò paulatim decreÅ¿cere <lb/>vnde Å¿tatim inclinatio grauitatis eiuÅ¿dem corporis Å¿ubingreditur, quæ Å¿eÅ¿e miÅ¿cens <lb/>cum impręſſione facta per vim, non permittit, vt linea <var>.a.b.</var> longo tempore recta per - <pb o="161" rhead="DE MECHAN." n="173" file="0173" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0173"/> - maneat, Å¿ed citò fiat curua, cum dictum corpus <var>.a.</var> duabus virtutibus moueatur, qua-<lb/>rum vna eÅ¿t, violentia impræſſa, & alia natura, contra opinionem Tartaleæ, qui ne-<lb/>gat corpus aliquod motibus violen <lb/> - <anchor type="figure" xlink:label="fig-0173-01a" xlink:href="fig-0173-01"/> - to & naturali Å¿imul & Å¿emel moueri <lb/>poſſe. </s> - <s xml:id="s1924" xml:space="preserve"><reg norm="Neque" type="simple">Neq;</reg> eÅ¿t <reg norm="Å¿ilentio" type="context">Å¿ilẽtio</reg> <reg norm="prætereundus" type="context">prætereÅ©dus</reg> <lb/>hac in re <reg norm="quidam" type="simple context">ê—dã</reg> notatu <reg norm="dignus" type="simple">dignê°</reg> effectus, <lb/>qui eiuÅ¿modi eÅ¿t, <reg norm="quod" type="simple">ê™</reg> quanto magis <lb/>creÅ¿cit impetus in corpore <var>.a.</var> cauÅ¿a <lb/>tus ab augumento velocitatis giri <lb/>ipÅ¿ius <var>.e.</var> <reg norm="tantò" type="context">tãtò</reg> magis oportet, vt Å¿en-<lb/>tiat Å¿e trahi manus à dicto corpore <lb/>a. mediante fune, quia quantò ma-<lb/>ior impetus motus ipÅ¿i <var>.a.</var> eÅ¿t impręſ <lb/>Å¿us, tantò magis dictum corpus <var>.a.</var> <lb/>ad rectum iter peragendum incli-<lb/>natur, vnde vt recta incedat, tantò <lb/>maiore quoque vi trahit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0173-01" xlink:href="fig-0173-01a"> - <image file="0173-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0173-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="18"> - <head xml:id="head228" style="it" xml:space="preserve">De decimatertia questione.</head> - <head xml:id="head229" xml:space="preserve">CAP. XVIII.</head> - <p> - <s xml:id="s1925" xml:space="preserve">DEcimatertia quæſtio ad vectem omnino eÅ¿t referenda. </s> - <s xml:id="s1926" xml:space="preserve">Imaginari debemus <lb/>axem cylindrici iugi, hypomochlion eſſe. </s> - <s xml:id="s1927" xml:space="preserve">Quod reÅ¿tat, illud ipÅ¿um totum de <lb/>pendet à .4. <reg norm="quintoque" type="simple">quintoqÌ;</reg> cap. huius tractatus. </s> - <s xml:id="s1928" xml:space="preserve">Vna tamen differentia inter hanc machi-<lb/>nam, <reg norm="vectemque" type="simple">vectemqÌ;</reg> reperitur, quæ eÅ¿t, <reg norm="quod" type="simple">ê™</reg> iugum aliquam reÅ¿iÅ¿tentiam pro coniunctione <lb/>calcata in loco, in quo voluitur, magis quà m hypomochlion vecti efficiat.</s> - </p> - </div> - <div type="section" level="3" n="19"> - <head xml:id="head230" style="it" xml:space="preserve">De decimaquart a queÅ¿tione.</head> - <head xml:id="head231" xml:space="preserve">CAP. XIX.</head> - <p> - <s xml:id="s1929" xml:space="preserve">RAtiones etiam decimæquartæ quæſtionis dependent ab ijs, quæ Å¿unt vectis, vt <lb/>exempli gratia Å¿it lignum <var>.a.b.c.d.</var> frang endum in medio, annitendo genibus <lb/>in punctum <var>.o.</var> clariſſimè tunc videbimus, <reg norm="quod" type="simple">ê™</reg> tenentes marlus longè à medio, in locis <lb/>a. et <var>.c.</var> facilius <reg norm="minorique" type="simple">minoriqÌ;</reg> cum labore illum frangemus, quà m Å¿i eaÅ¿dem vicinas me-<lb/>dio eiuÅ¿dem ligni in locis <var>.e.</var> et <var>.i.</var> poneremus. </s> - <s xml:id="s1930" xml:space="preserve">Cuius rei rationes <reg norm="eædem" type="context">eædẽ</reg> Å¿unt <reg norm="cum" type="context">cÅ©</reg> ijs, quæ <lb/>primis huius tractatus capitibus propoÅ¿itæ fuerunt. </s> - <s xml:id="s1931" xml:space="preserve">Imaginemur lineas rectas ductas <lb/>à puncto <var>.o.</var> ad loca <var>.a.e.i.</var> et <var>.c.</var> hinc manifeÅ¿tè perÅ¿piciemus eorum, quæ iam diximus <lb/>ratione, <reg norm="quod" type="simple">ê™</reg> loca <var>.e.</var> et <var>.i.</var> mediantibus duabus lineis <var>.e.o.</var> et <var>.i.o.</var> magis annitentur <var>.o.</var> cen <lb/>tro, quà m loco <var>.a.</var> et <var>.c.</var> <reg norm="duarum" type="context">duarÅ©</reg> <reg norm="linearum" type="context">linearÅ©</reg> <var>.a.o.</var> et <var>.c.o.</var> beneficio; </s> - <s xml:id="s1932" xml:space="preserve">vnde vim <reg norm="quoque" type="simple">quoq;</reg> maiorem <lb/><reg norm="habebunt" type="context">habebÅ©t</reg> <lb/> - <anchor type="figure" xlink:label="fig-0173-02a" xlink:href="fig-0173-02"/> - in <var>.a.</var> et <var>.c.</var> <lb/>quà m in <lb/>e. et <var>.i</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0173-02" xlink:href="fig-0173-02a"> - <image file="0173-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0173-02"/> - </figure> - </div> - <pb o="162" rhead="IO. BAPT. BENED." n="174" file="0174" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0174"/> - </div> - <div type="section" level="3" n="20"> - <head xml:id="head232" style="it" xml:space="preserve">De uer a r atione .17. queÅ¿tionis.</head> - <head xml:id="head233" xml:space="preserve">CAP. XX.</head> - <p> - <s xml:id="s1933" xml:space="preserve">DEcimaÅ¿eptima quæſtio ab AriÅ¿totele haud benè percepta fuit, quia is non ac-<lb/>commodat partes vectis Å¿uis locis. </s> - <s xml:id="s1934" xml:space="preserve">Quamobrem imaginemur duos vectes <var>.<lb/>a.o.n.</var> et <var>.o.e.u.</var> quorum centra, quæ hypomochlia appellantur Å¿int <var>.o.</var> & pondera, <lb/>quæ Å¿unt attollenda Å¿int <var>.a.</var> et <var>.e.</var> inter Å¿e æqualia, & diÅ¿tantiÄ™ Å¿int <var>.a.o.</var> et <var>.e.o.</var> Å¿ibi <reg norm="inuicem" type="context">inuicẽ</reg> <lb/>æquales, Å¿ed <var>.o.n.</var> æqualis Å¿it ipÅ¿i <var>.o.u</var>: clarum erit, <reg norm="quod" type="simple">ê™</reg> ad eleuandum <var>.a.</var> oportebit depri <lb/>mere <var>.n.</var> & ad eleuandum <var>.e.</var> oportebit attollere <var>.u</var>. </s> - <s xml:id="s1935" xml:space="preserve">Et quia omnia Å¿upponuntur æ qua <lb/> - <anchor type="handwritten" xlink:label="hd-0174-01a" xlink:href="hd-0174-01"/> - lia, clarum quoque erit, commu-<lb/>ni Å¿cientia, tantam virtutem in <lb/> - <anchor type="figure" xlink:label="fig-0174-01a" xlink:href="fig-0174-01"/> - n. quanta Å¿ufficiet ad <reg norm="attollendum" type="context">attollendÅ©</reg> <lb/>a. in <var>.u.</var> <reg norm="quoque" type="simple">quoq;</reg> Å¿uffecturam ad ele-<lb/>uandum <var>.e.</var> quia <reg norm="cum" type="context">cÅ©</reg> æqualibus an <lb/>gulis ijs, quibus duæ virtutes <var>.a.</var> <lb/>et <var>.n.</var> annituntur <var>.o.</var> centro, ita <var>.e.</var> <lb/>et <var>.u.</var> è contrario Å¿uo centro <var>.o.</var> an <lb/>nituntur. </s> - <s xml:id="s1936" xml:space="preserve">& omnes rationes pro <lb/>vecte <var>.a.o.n.</var> quarto <reg norm="quintoque" type="simple">quintoqÌ;</reg> huius tracta-<lb/> - <anchor type="figure" xlink:label="fig-0174-02a" xlink:href="fig-0174-02"/> - tus capitibus citatæ, vecti <var>.o.e.u.</var> vt Å¿atis Å¿u <lb/><reg norm="perque" type="simple">perqÌ;</reg> dixi in dicto capit .5. conuenire poÅ¿-<lb/>Å¿unt.</s> - </p> - <div type="float" level="4" n="1"> - <handwritten xlink:label="hd-0174-01" xlink:href="hd-0174-01a"/> - <figure xlink:label="fig-0174-01" xlink:href="fig-0174-01a"> - <image file="0174-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0174-01"/> - </figure> - <figure xlink:label="fig-0174-02" xlink:href="fig-0174-02a"> - <image file="0174-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0174-02"/> - </figure> - </div> - <p> - <s xml:id="s1937" xml:space="preserve">Nunc Å¿it aliqua pars ligni cindenda Å¿e-<lb/>cundum venulas Å¿uas <var>.d.e.f.g.</var> & Å¿it cuneus <lb/><var>a.b.c.</var> qui vi mallei <var>.P.</var> vÅ¿que ad <var>.t.x.</var> pene-<lb/>trarit. </s> - <s xml:id="s1938" xml:space="preserve">Hinc clarum erit, quòd apertura <lb/><var>i.m.r.</var> ligni, poÅ¿t quam infigitur cuneus Å¿e <lb/>cundum venas, longÃor erit parte <var>.x.b.t.</var> cu <lb/>nei, quæ ingreſſa eÅ¿t. </s> - <s xml:id="s1939" xml:space="preserve">Oportet nunc ima-<lb/>ginari duos vectes Å¿imiles Å¿upradictæ <var>.u.e.<lb/>o.</var> in hunc modum, vt puncta <var>i.r.</var> lìni Å¿int <lb/>loco <var>.u.</var> extremi <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> vectis, et <var>.t.x.</var> loco vir <lb/>tutis applicatæ ipÅ¿i <var>.u.</var> & reÅ¿iÅ¿tentia circa <lb/>punctum <var>.m.</var> loco ponderis <var>.e.</var> vectis <var>.o.e.u.</var> <lb/>dicti, & pars <var>.K.</var> quaÅ¿i immediata poÅ¿t <var>.m.</var> <lb/>verÅ¿us extremitatem <var>.f.e.</var> ligni, Å¿it loco hy-<lb/>pomochlij <var>.o</var>. </s> - <s xml:id="s1940" xml:space="preserve">Hinc fiet vt quanto longio <lb/>res erunt lineæ <var>.i.m.K.</var> et <var>.r.m.K.</var> tantò quo <lb/>que facilius virtutes <var>.t.x.</var> impellent <var>.i.r</var>.</s> - </p> - <pb o="163" rhead="DE MECHAN." n="175" file="0175" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0175"/> - </div> - <div type="section" level="3" n="21"> - <head xml:id="head234" style="it" xml:space="preserve">De uera & intrinÅ¿eca cauÅ¿a trocble arum.</head> - <head xml:id="head235" xml:space="preserve">CAP. XXI.</head> - <p> - <s xml:id="s1941" xml:space="preserve">PRo intelligenda vera, & intrinÅ¿eca ratione, vnde fiat ut multitudo rotularum in <lb/>trochleis cauÅ¿a Å¿it, ut exigua vis Å¿urÅ¿um moueat, aut attollat <reg norm="pondera" type="context">põdera</reg> magna. </s> - <s xml:id="s1942" xml:space="preserve">Ima <lb/>ginemur duas hîc Å¿ubÅ¿criptas trochlæas explicatas tranÅ¿uerÅ¿aliter in hunc modum, <lb/>ideÅ¿t Å¿it <reg norm="paruum" type="context">paruÅ©</reg> <reg norm="tignum" type="context">tignÅ©</reg> <var>.a.b.</var> fixum & <reg norm="parallelum" type="context">parallelÅ©</reg> orizonti. cui Å¿int rotulæ appenÅ¿e ab infe <lb/>riori parte ad Å¿uperiorem <reg norm="huicque" type="simple">huicqÌ;</reg> è regione <reg norm="oppoÅ¿itus" type="simple">oppoÅ¿itê°</reg> Å¿it aliud <reg norm="tignum" type="context">tignÅ©</reg> <var>.c.d.</var> quod moueri <lb/>poſſit ab imo ad Å¿umum, Å¿uper quod totidem Å¿int rotulæ aut radij, <reg norm="cum" type="context">cÅ©</reg> annexa poÅ¿tea <lb/>fuerit funis puncto <var>.b.</var> fixo, eam faciendo pertranÅ¿ire per rotulas tam à parte Å¿upe-<lb/>riore, quam ab inferiore; </s> - <s xml:id="s1943" xml:space="preserve">& appenÅ¿um deinde cum erit paruo illi tigno <var>.c.d.</var> mobili <lb/>pondus <var>.E.</var> ducendo poÅ¿tmodum extremum <var>.f.</var> funis tranÅ¿euntis per rotulas, idem pla <lb/>nè fiet quod à trochlÄ™is Å¿imul unitis fieri Å¿olet. </s> - <s xml:id="s1944" xml:space="preserve">Cuius quidem effectus ratio Å¿ub no-<lb/>Å¿tram cognitionem cadet facilius in huiuÅ¿modi figura. </s> - <s xml:id="s1945" xml:space="preserve">Imaginemur Å¿eparatim Å¿ta-<lb/>teram <var>.g.h.</var> cuius <reg norm="centrum" type="context">cẽtrum</reg> Å¿it <var>.K.</var> ita Å¿itum, ut brachium <var>.g.k.</var> Å¿it duplum ad brachium <var>.K.<lb/>h.</var> Å¿upponendo igitur in puncto <var>.g.</var> pondus, aut virtutem mouentem unius libræ, & in <lb/>h. duarum librarum, <reg norm="abÅ¿que" type="simple">abÅ¿q;</reg> dubio hæ duæ uirtutes in huiuÅ¿modi diÅ¿tantijs à centro <lb/> - <anchor type="handwritten" xlink:label="hd-0175-01a" xlink:href="hd-0175-01"/> - Ä™quales <reg norm="inuicem" type="context">inuicẽ</reg> <reg norm="erunt" type="context">erÅ©t</reg>, ob rationes prioribus capitibus iam allatas, & Å¿tatera orizontalis <lb/>manebit. </s> - <s xml:id="s1946" xml:space="preserve">Vnde clarum erit, <reg norm="quod" type="simple">ê™</reg> quæuis etiam exigua virtus adiuncta ipÅ¿i <var>.g.</var> mouebit <lb/>Å¿tateram extra orizontalem Å¿itum. </s> - <s xml:id="s1947" xml:space="preserve">Nunc Å¿i puncto <var>.i.</var> ex æquo medio inter <var>.g.</var> et <var>.K.</var> <lb/>applicata erit virtus ipÅ¿ius <var>.h.</var> non amplius conÅ¿iderato brachio <var>.K.h.</var> inclinante uirtu-<lb/>te ipÅ¿ius <var>.i.</var> eandem partem verÅ¿us, in quam inclinabat, quando erat in <var>.h.</var> Å¿ed uirtus ip <lb/>Å¿ius <var>.g.</var> inclinet contrario modo, <reg norm="diuerÅ¿oque" type="simple">diuerÅ¿oqÌ;</reg> ab eo, quo inclinabat prius; </s> - <s xml:id="s1948" xml:space="preserve">clarum <reg norm="quoque" type="simple">quoq;</reg> <lb/>erit, communi conceptu, & ob ea, quæ cap .5. huius tractatus Å¿unt dicta <var>.g.h.</var> Å¿emper <lb/>in eodem Å¿itu abÅ¿que motu manÅ¿uram, <reg norm="hancque" type="simple">hancqÌ;</reg> Å¿tateram appellabimus mobilem, & <lb/>primam. </s> - <s xml:id="s1949" xml:space="preserve">Imaginemur nunc à puncto <var>.e.</var> fixo deÅ¿cendere funem <var>.e.K.</var> quæ fulciat pun <lb/>ctum <var>.K.</var> extremum diametri <var>.g.K.</var> quam intelligo pro diametro vnius ex rotulis infe <lb/>rioribus trochleæ; </s> - <s xml:id="s1950" xml:space="preserve">& Å¿it <var>.n.l.m.</var> diameter vnius ex rotulis Å¿uperioribus alterius parui <lb/>tigni defixi à parte inclinationis ipÅ¿ius <var>.g.</var> & parallela diametro <var>.g.K.</var> cuius diametri <lb/>centrum fixum Å¿it <var>.l.</var> & Å¿it coniunctum <var>.g.</var> punctum, à fune cum puncto <var>.m.</var> quæ <reg norm="tam" type="context">tã</reg> per-<lb/>pendicularis Å¿it primo diametro <var>.g.i.K.</var> quà m Å¿ecundo <var>.n.m.</var> ideÅ¿t ita vt anguli <var>.n.m.g.</var> <lb/> - <anchor type="figure" xlink:label="fig-0175-01a" xlink:href="fig-0175-01"/> - <pb o="164" rhead="IO. BAPT. BENED." n="176" file="0176" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0176"/> - et <var>.m.g.k.</var> Å¿int recti. </s> - <s xml:id="s1951" xml:space="preserve">Imaginemur <reg norm="quoque" type="simple">quoq;</reg> virtutem ipÅ¿ius <var>.g.</var> applicatam eſſe extremo <var>.<lb/>n.</var> cum inclinatione tamen contraria, ideÅ¿t ad inferiorem partem, quæ quidem virtus <lb/>communi quodam conceptu eandem poſſidebit vim Å¿uÅ¿tentandi immobilem diame <lb/>trum <var>.g.i.k.</var> quam habebat, <reg norm="quando" type="wordlist">qñ</reg> erat in <var>.g.</var> cum inclinatione ad Å¿uperiorem partem, <lb/>& Å¿ic etiam diameter <var>.n.l.m.</var> non magis ab una, quà m ab alia parte declinabit, quia <lb/>cum quædam virtus in <var>.n.</var> reperiatur æqualis medietati uirtutis ipÅ¿ius <var>.i.</var> quæ uirtus ip <lb/>Å¿ius <var>.i.</var> uim habet deprimendi ipÅ¿um <var>.g.</var> ideÅ¿t <var>.m.</var> pro dimidia Å¿ui ipÅ¿ius parte, Å¿equitur <var>.<lb/>n.m.</var> debere immobilem permanere. </s> - <s xml:id="s1952" xml:space="preserve">Nunc Å¿i alia diameter rotulæ mobilis erit de-<lb/>Å¿umpta, quæ Å¿it <var>.p.q.o.</var> cuius centrum Å¿it <var>.q.</var> in Å¿itu parallelo ipÅ¿i <var>.n.l.m.</var> & Å¿ic collocata, <lb/>vt coniungendo <var>.o.</var> cum <var>.n.</var> anguli <var>.m.n.o.</var> et <var>.n.o.p.</var> Å¿int recti: </s> - <s xml:id="s1953" xml:space="preserve">Å¿i imaginati fuerimus <reg norm="tranÅ¿" type="context">trãſ</reg> <lb/>latum eſſe <reg norm="pondusipÅ¿ius" type="simple">pondusipÅ¿iê°</reg> <var>.n.</var> in <var>.o.</var> <reg norm="cum" type="context">cÅ©</reg> <reg norm="eadem" type="context">eadẽ</reg> inclinatione ad depræſſiorem partem, illud ip <lb/>Å¿um, ac Å¿i eſſet in <var>.n.</var> communi conceptu, Å¿ine alicuius diametri mutatione præſtabit. <lb/></s> - <s xml:id="s1954" xml:space="preserve">Et Å¿i centrum <var>.q.</var> fixum eſſet, & extremo <var>.p.</var> appoÅ¿itum fuiſſet pondus ipÅ¿ius <var>.o.</var> cum in <lb/>clinatione ad Å¿uperiorem partem, idem etiam planè pręſtaret, etiam Å¿i nullum ullius <lb/>diametri Å¿itum, communi Å¿cientia, mutaret, cum extremum <var>.m.</var> deorÅ¿um Å¿it ductum <lb/>à . g. uirtute dimidiæ partis ipÅ¿ius <var>.i.</var> & ab alia huic Å¿imili <var>.m.</var> quoque deorÅ¿um Å¿it tra-<lb/>ctum ab .o: quod quidem <var>.o.</var> deorÅ¿um eÅ¿t alteratum, ob inclinationem ad Å¿uperius <lb/>à uirtute poÅ¿ita in <var>.p.</var> Å¿upponendo centrum <var>.q.</var> fixum. </s> - <s xml:id="s1955" xml:space="preserve">Sed Å¿i loco centri fixi, imagina <lb/>bimur in <var>.q.</var> pondus aliquod æquale ipÅ¿i <var>.i.</var> quod duplum erit in uirtute ad eam, quæ <lb/>eÅ¿t ipÅ¿ius <var>.p.</var> & ipÅ¿ius quoque .g: Å¿equetur <reg norm="etiam" type="context">etiã</reg> eadem immobilitas horum trium dia-<lb/>metrorum. </s> - <s xml:id="s1956" xml:space="preserve">Quia cum Å¿it huiuÅ¿modi pondus Å¿eu virtus in <var>.q.</var> cum inclinatione con-<lb/>traria virtuti in <var>.p.</var> quæ æquipollet dimidiæ parti ipÅ¿ius <var>.q.</var> & Å¿ic ei quæ eÅ¿t ipÅ¿ius <var>.o.</var> Å¿i-<lb/>militer quia <var>.o.</var> tractum eÅ¿t Å¿upra ab <var>.n.</var> virtute ipÅ¿ius <var>.g.</var> quod <var>.m.</var> deorÅ¿um trudit; </s> - <s xml:id="s1957" xml:space="preserve">idcir <lb/>co quanta erit vis quam habebit virtus in <var>.q.</var> ferendi deorÅ¿um diametrum <var>.p.o.</var> tanta <lb/>quoque virtutes ipÅ¿orum <var>.p.</var> et <var>.o.</var> æquales, & æqualiter diÅ¿tantes à <var>.q.</var> ipÅ¿um ad Å¿upe-<lb/>riorem partem inclinabunt. </s> - <s xml:id="s1958" xml:space="preserve">Quamobrem nec aÅ¿cender, nec deÅ¿cendet, nec locum <lb/>mutabit. </s> - <s xml:id="s1959" xml:space="preserve">Supponamus nunc quartum diametrum rotulæ <var>.s.t.r.</var> quæ Å¿it Å¿ecunda rotu <lb/>larum fixarum, parallela ipÅ¿i <var>.p.o.</var> & in eo Å¿itu, quo coniungendo extrema <var>.r.p.</var> anguli <lb/><var>o.p.r.</var> et <var>.p.r.s.</var> Å¿int recti, & imaginemur virtutem ipÅ¿ius <var>.p.</var> reperiri in <var>.s.</var> cum inclinatio <lb/>ne tamen contraria, ideÅ¿t deorÅ¿um verÅ¿us, ex his <reg norm="idem" type="context">idẽ</reg> quoque planè Å¿equetur, ideÅ¿t <reg norm="quod" type="simple">ê™</reg> <lb/>nulla <reg norm="harum" type="context">harÅ©</reg> quatuor diametrorum mouebitur. </s> - <s xml:id="s1960" xml:space="preserve">quia eundem <reg norm="effectum" type="context">effectÅ©</reg> <reg norm="cum" type="context">cÅ©</reg> inclinatione <lb/>deorÅ¿um verÅ¿us efficeret dicta virtus in <var>.s.</var> quem in <var>.p.</var> cum inclinatione Å¿urÅ¿um verÅ¿us. <lb/></s> - <s xml:id="s1961" xml:space="preserve">et iam dictum eÅ¿t virtutem ipÅ¿ius <var>.g.</var> dimidium virtutis ipÅ¿ius <var>.i.</var> trahere <var>.m.</var> quæ <reg norm="mediam" type="context">mediã</reg> <lb/> - <anchor type="handwritten" xlink:label="hd-0176-01a" xlink:href="hd-0176-01"/> - te <var>.n.</var> attrahit <var>.o.</var> eodem robore, et <var>.s.</var> eadem vi trahit <var>.p.</var> medio ipÅ¿ius <var>.r</var>. </s> - <s xml:id="s1962" xml:space="preserve">HucuÅ¿que <reg norm="Å¿cien- tificè" type="context">Å¿ciẽ-<lb/>tificè</reg> nouimus pondus, aut virtutem ipÅ¿ius <var>.s.</var> quæ eÅ¿t dimidium <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <var>.i.</var> Å¿uÅ¿tinere uim <lb/>ipÅ¿orum <var>.i.</var> et <var>.q.</var> nam quater tantum, quanta ipÅ¿amet virtus ipÅ¿ius <var>.s.</var> eſſe conÅ¿picitur. <lb/></s> - <s xml:id="s1963" xml:space="preserve">Et Å¿i adiunctÄ™ nobis eſſent duæ aliæ diametri cum ijÅ¿dem planè conditionibus <reg norm="ijÅ¿dem" type="context">ijÅ¿dẽ</reg> <lb/>rationibus vtentes, cognoÅ¿ceremus quod eadem medietas ipÅ¿ius <var>.i.</var> Å¿exies tantum <reg norm="pon" type="context">põ</reg> <lb/>deris, quanta ipÅ¿a exiÅ¿teret, Å¿eÅ¿tineret. </s> - <s xml:id="s1964" xml:space="preserve">Vnde <reg norm="manifeÅ¿tum" type="context">manifeÅ¿tÅ©</reg> euadit, <reg norm="quod" type="simple">ê™</reg> eidem medietati <lb/>ipÅ¿ius <var>.i.</var> in <var>.s.</var> nonnihil virtutis addendo, dictæ diametri, illicò <reg norm="mouerentur" type="context">mouerẽtur</reg> Å¿itu. </s> - <s xml:id="s1965" xml:space="preserve">Et quia <lb/>rotulæ in quolibet puncto, aliquam diametrum habent, neceſſariò Å¿equitur <reg norm="quod" type="simple">ê™</reg> infe-<lb/>riores ad Å¿uperiores accedere debeant. </s> - <s xml:id="s1966" xml:space="preserve">Attamen Å¿i forte extremum immobile ip-<lb/>Å¿ius funis non pendet à puncto <var>.e.</var> trochleæ Å¿uperioris, Å¿ed alligatum fuerit ad <reg norm="medium" type="context">mediÅ©</reg> <lb/>inferioris trochleæ ut ad punctum <var>.i.</var> ope unius trochleÄ™ Å¿uperioris immobilis vt in fi <lb/>gura <var>.A.</var> videre licet, clarè patebit <reg norm="quod" type="simple">ê™</reg> à tribus virtutibus æqualibus pondus in <var>.i.</var> <reg norm="poÅ¿itum" type="context">poÅ¿itÅ©</reg> <lb/>Å¿uÅ¿tinebitur: </s> - <s xml:id="s1967" xml:space="preserve">hoc eÅ¿t à <var>.g.</var> ab <var>.i.</var> & ab <var>.k.</var> <reg norm="quarum" type="context">quarÅ©</reg> vnaquÄ™que tertia pars erit ipÅ¿ius <var>.i.</var> in con <lb/>c<unsure/>ontrariam <reg norm="partem" type="context">partẽ</reg>, hoc eÅ¿t tertia pars reÅ¿iÅ¿tentiæ. </s> - <s xml:id="s1968" xml:space="preserve">propterea <reg norm="quod" type="simple">ê™</reg> ex æquo inter Å¿e <reg norm="diÅ¿tant" type="context">diÅ¿tãt</reg>. - <pb o="165" rhead="DE MECHAN." n="177" file="0177" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0177"/> - <var>g.i.</var> et .K: </s> - <s xml:id="s1969" xml:space="preserve">Quà propter augebitur virtus per numeros impares, hoc modo; </s> - <s xml:id="s1970" xml:space="preserve">Nam <var>.g.</var> <lb/>eſſet tertia pars reÅ¿iÅ¿tentiÄ™, quemadmodum prius media erat. </s> - <s xml:id="s1971" xml:space="preserve">Idem infero de <var>.m.n.<lb/>o.p.r.</var> et <var>.s</var>. </s> - <s xml:id="s1972" xml:space="preserve">Sed cum oporteat pondus <var>.q.</var> tantum eſſe vt <reg norm="Å¿uffieiant" type="context">Å¿uffieiãt</reg> reÅ¿iÅ¿tentiæ in <var>.o.</var> et <var>.p.</var> <lb/>ipÅ¿um Å¿uÅ¿tinere, idcirco ipÅ¿um pondus <var>.q.</var> Å¿ubÅ¿eÅ¿quialter erit <reg norm="ponderi" type="context">põderi</reg> in <var>.i.</var> poÅ¿iti. </s> - <s xml:id="s1973" xml:space="preserve">Qua-<lb/>propter <var>.s.</var> quinta pars erit ponderum <var>.i.</var> et <var>.q</var>. </s> - <s xml:id="s1974" xml:space="preserve">Deinde Å¿i adhuc. duo diametri vnus in-<lb/>ferior, alter verò Å¿uperior additi fuerint cum pondere æquali <var>.q.</var> ad medium diame-<lb/>tri inferioris, </s> - <s xml:id="s1975" xml:space="preserve">tunc pondus <var>.s.</var> erit Å¿eptima pars trium ponderum <var>.i.q.</var> & tertij additi, ex <lb/> - <anchor type="figure" xlink:label="fig-0177-01a" xlink:href="fig-0177-01"/> - Å¿upradictis rationibus. </s> - <s xml:id="s1976" xml:space="preserve">Et quia virtus Å¿uÅ¿ti <lb/> - <anchor type="figure" xlink:label="fig-0177-02a" xlink:href="fig-0177-02"/> - nens totale pondus trochleæ inferiori ap-<lb/>penÅ¿um in tot diuiditur partes æquales, <lb/>quot Å¿unt diametri orbiculorum trochleæ <lb/>inferioris, quando extremum immobile fu <lb/>nis alligatum fuerit trochleÄ™ Å¿uperiori, vt <lb/>puta in puncto <var>.e.</var> cum verò alligatum fue-<lb/>rit trochleæ inferiori, virtus primi diame-<lb/>tri <var>.g.i.K.</var> trochleæ inferioris Å¿emper Å¿eÅ¿qui <lb/>altera erit vnicuique aliorum <reg norm="diametrorum" type="context">diametrorÅ©</reg> <lb/>ideò virtus reÅ¿iÅ¿tentiÄ™ alterius extremi mo <lb/>bilis funis, puta <var>.s.</var> Å¿ubmultiplex erit totalis <lb/>ponderis, eo modo quo diximus, cuius vir <lb/>tus, Å¿eu grauitas diuiditur Å¿eu diÅ¿trubuitur <lb/>diametris inferioris trochleæ vt dictum eÅ¿t.</s> - </p> - <div type="float" level="4" n="1"> - <handwritten xlink:label="hd-0175-01" xlink:href="hd-0175-01a"/> - <handwritten xlink:label="hd-0176-01" xlink:href="hd-0176-01a"/> - <figure xlink:label="fig-0175-01" xlink:href="fig-0175-01a"> - <image file="0175-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0175-01"/> - </figure> - <figure xlink:label="fig-0177-01" xlink:href="fig-0177-01a"> - <image file="0177-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0177-01"/> - </figure> - <figure xlink:label="fig-0177-02" xlink:href="fig-0177-02a"> - <image file="0177-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0177-02"/> - </figure> - </div> - </div> - <div type="section" level="3" n="22"> - <head xml:id="head236" style="it" xml:space="preserve">Depropria cauÅ¿a .24. quæſtionis.</head> - <head xml:id="head237" xml:space="preserve">CAP. XXII.</head> - <p> - <s xml:id="s1977" xml:space="preserve">VEra cauÅ¿a effectus, qui vigeÅ¿imaquarta quæſtione exprimitur, adhuc à nemine <lb/>(quod Å¿ciam) animaduerÅ¿a fuit, licet non Å¿it admodum ardua vel obÅ¿cura. </s> - <s xml:id="s1978" xml:space="preserve">Ima <lb/>ginemur ergo duos circulos <var>.c.f.</var> et <var>.b.g.</var> concentricos, <reg norm="itaque" type="simple">itaq;</reg> Å¿imul coniunctos, vt Å¿i ip <lb/>Å¿orum vnus feratur in orbem, alius quoque circumagatur, eo modo, quo curruum ro <lb/>tæ voluuntur. </s> - <s xml:id="s1979" xml:space="preserve">Et imaginemur primò Å¿uper lineam <var>.f.i.</var> reuolui maiorem, & quando <lb/>idem circulus erit in <var>.l.</var> dictam lineam <var>.f.i.</var> tangere circunferentiam eiuÅ¿dem in pun- - <pb o="166" rhead="IO. BAPT. BENED." n="178" file="0178" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0178"/> - cto <var>.c.</var> vnde linea <var>.g.m.</var> mediante <var>.K.</var> continget circunferentiam circuli minoris in pun <lb/>cto .b: et <var>.K.g.</var> ex .34. primi Eucli. æqualis erit ipÅ¿i <var>.f.l.</var> quia ex .17. tertii, anguli <var>.f.</var> et <var>.g.</var> <lb/>Å¿unt æquales, vnde ex .28. primi <var>.f.l.</var> et <var>.g.K.</var> Å¿unt parallelæ. </s> - <s xml:id="s1980" xml:space="preserve">& Å¿ic erunt <var>.k.l.</var> cum <var>.f.g.</var> ex <lb/>eadem Å¿upradicta. </s> - <s xml:id="s1981" xml:space="preserve">Ratio autem, qua arcus <var>.g.b.</var> tranÅ¿ierit lineam <var>.g.K.</var> maiorem ipÅ¿a, <lb/>eÅ¿t, quia dum mouetur, quodlibet punctum ipÅ¿ius <var>.g.b.</var> virtute reuolutionis ipÅ¿ius <var>.f.c.</var> <lb/>omne punctum eiuÅ¿dem arcus <var>.g.b.</var> vlterius verÅ¿us <var>.K.</var> quam Å¿i moueretur virtute re-<lb/>uolutionis ipÅ¿ius <var>.g.b.</var> Å¿uper lineam <var>.g.m.</var> defertur. </s> - <s xml:id="s1982" xml:space="preserve">vt exempli gratia, quando virtute <lb/>reuolutionis maioris circuli, centrum <var>.a.</var> reperitur in Å¿itu lineæ <var>.l.K.</var> punctum <var>.g.</var> confe <lb/>cerit iter <var>.g.u.</var> & punctum <var>.b.</var> iter <var>.b.K.</var> etiam reliqua omnia puncta inter <var>.g.b.</var> magna <lb/>itinera egerint, cum à magno circulo Å¿int ante delata. </s> - <s xml:id="s1983" xml:space="preserve">Imaginemur quoque hos cir <lb/>culos eſſe delatos virtute reuolutionis circuli minoris, & <reg norm="partem" type="context">partẽ</reg> <var>.g.t.</var> rectè <var>.g.m.</var> dimen-<lb/>Å¿am fuiſſe ab arcu <var>.g.b</var>. </s> - <s xml:id="s1984" xml:space="preserve"><reg norm="Quando" type="context">Quãdo</reg> ergo <var>.b.</var> erit in <var>.t.</var> factum erit iter <var>.b.t.</var> ab ipÅ¿o <var>.b.</var> et <var>.g.</var> fa-<lb/>ciet iter <var>.g.n.</var> quÄ™ itinera alijs multò breuiora Å¿unt, quia breuioribus cruribus reuolu-<lb/>ta Å¿unt dicta puncta; </s> - <s xml:id="s1985" xml:space="preserve">& Å¿ic dico de reliquis omnibus punctis inter <var>.g.</var> et <var>.b.</var> & in hoc ca <lb/>Å¿u punctum <var>.f.</var> erit in <var>.q.</var> & punctum <var>.c.</var> erit in <var>.e</var>. </s> - <s xml:id="s1986" xml:space="preserve">Quamobrem omnia puncta <reg norm="contingen- tiæ" type="context">cõtingen-<lb/>tiæ</reg> inter <var>.f.</var> et <var>.c.</var> non Å¿olum non erunt delata anteà , Å¿ed potius à primo Å¿itu retrorÅ¿um <lb/>erunt repulÅ¿a. </s> - <s xml:id="s1987" xml:space="preserve">Vnde non eÅ¿t, quòd in tantam admirationem ducamur Å¿i dum reuol <lb/>uitur circulus maior, arcus <var>.g.b.</var> circuli minoris, totam lineam <var>.g.K.</var> tranÅ¿ire videtur, <lb/>& dum reuoluitur minor, apparet arcum <var>.f.c</var>: maius iter quam ab <var>.f.</var> ad <var>.e.</var> non facere, <lb/>cum maiore Å¿eÅ¿e in orbem ferente, quodlibet punctum arcus <var>.g.b.</var> ad vnam <reg norm="eandemque" type="context simple">eandẽq;</reg> <lb/>partem duos motus obtineat. </s> - <s xml:id="s1988" xml:space="preserve">vt exempli gratia punctum <var>.b.</var> non Å¿olum mouetur ver <lb/>Å¿us <var>.m.</var> quòd circa centrum <var>.a.</var> feratur, cum ipÅ¿um etiam centrum moueatur verÅ¿us <var>.m.</var> <lb/>Å¿ed quia prÄ™ter hoc deferantur quoque à circulo maiori verÅ¿us <var>.m.</var> vÅ¿que ad lineam <var>.<lb/>k.l</var>. </s> - <s xml:id="s1989" xml:space="preserve">Dum verò minor circulus in girum ducitur, habet quodlibet punctum arcus <var>.f.c.</var> <lb/>duos motus contrarios, quorum alter verÅ¿us <var>.i.</var> virtute reuolutionis circuli minoris, <lb/>& alter ex eo, <reg norm="quod" type="simple">ê™</reg> dictus circulus maior circa centrum <var>.a.</var> voluatur, vnde omne <reg norm="punctum" type="context">punctÅ©</reg> <lb/>contactus circuli maioris cum recta <var>.f.i.</var> tetrorÅ¿um pellitur verÅ¿us <var>.x</var>.</s> - </p> - <figure position="here"> - <image file="0178-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0178-01"/> - </figure> - <pb o="167" rhead="DE MECHAN." n="179" file="0179" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0179"/> - </div> - <div type="section" level="3" n="23"> - <head xml:id="head238" style="it" xml:space="preserve">De uer a cauÅ¿a .30. quæstionis.</head> - <head xml:id="head239" xml:space="preserve">CAP. <sic comment="should be XXIII">XXIIII.</sic></head> - <p> - <s xml:id="s1990" xml:space="preserve">VEra ratio, cur homo dum Å¿edet ( non tamen Turcarum more ) Å¿i velit <lb/>Å¿eÅ¿e in pedes erigere, calcaneos retrahit, vt efficiat angulum acutum, cum <reg norm="fae- moribus" type="simple">fÄ™-<lb/>moribus</reg> coxis à parte inferiori, & ventrem inclinat, ad conÅ¿tituendum etiam angu <lb/>lum acutum in Å¿uperiori parte, ea eÅ¿t; </s> - <s xml:id="s1991" xml:space="preserve">vt totius corporis pondus, ex Ä™quo, ideÅ¿t ab <lb/>oppoÅ¿itis partibus circundet lineam rectam, quæ tranÅ¿it per locum, in quo conquie <lb/>Å¿cunt pedes verÅ¿us mundi centrum. </s> - <s xml:id="s1992" xml:space="preserve">ideÅ¿t, ut edatur Ä™quilibrium ponderis ipÅ¿ius cor-<lb/>poris circum lineam illam, quÄ™ Å¿ub pedibus inÅ¿eruit pro Å¿parto. </s> - <s xml:id="s1993" xml:space="preserve">Vnde aperiendo, <lb/>deinde dictos duos angulos circa dictam <reg norm="lineam" type="context">lineã</reg>, abÅ¿que vlla difficultate erigitur cor-<lb/>pus, & abÅ¿que periculo in alterutram partem cadendi.</s> - </p> - </div> - <div type="section" level="3" n="24"> - <head xml:id="head240" style="it" xml:space="preserve">Deratione .35. & ultimæ quæstionis.</head> - <head xml:id="head241" xml:space="preserve">CAP. <sic comment="should be XXIIII">XXV.</sic></head> - <p> - <s xml:id="s1994" xml:space="preserve">VEra ratio, quare, quÄ™ reperiuntur in vorticibus aquarum, Å¿emper verÅ¿us <lb/>medium ipÅ¿arum vertiginum vniuntur, inde promanat, quod media <lb/>vertiginum Å¿emper depreſſiora Å¿unt. </s> - <s xml:id="s1995" xml:space="preserve">vnde quòd dicta corpora ad medium acce-<lb/>dant, nihil aliud eÅ¿t, quà m ipÅ¿a corpora Å¿uo pondere grauitateqÌue deÅ¿cendere, figu <lb/>ra enim vorticibus eÅ¿t quaÅ¿i conica, & concaua cum angulo deorÅ¿um, & gyro baÅ¿is <lb/>Å¿urÅ¿um. </s> - <s xml:id="s1996" xml:space="preserve">Atque hæc vera eÅ¿t huius effectus cauÅ¿a, & non ea quam AriÅ¿toteles ponit, <lb/>à <unsure/> quo aliarum omnium quæſtionum, quas ego omiÅ¿i rationes Å¿unt benè propoÅ¿itæ.</s> - </p> - <pb o="168" n="180" file="0180" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0180"/> - </div> - </div> - <div type="chapter" level="2" n="4"> - <head xml:id="head242" xml:space="preserve">DISPVTATIONES <lb/>DE QVIBVSDAM PLACITIS <lb/>ARISTOTELIS.</head> - <p style="it"> - <s xml:id="s1997" xml:space="preserve">TANTA A eÅ¿t certè AriÅ¿totelis amplitudo at que authoritas, vt dif-<lb/>ficillimum ac periculoÅ¿um Å¿it quidpiam Å¿cribere contra quam <lb/>ipÅ¿e docuerit, & mihi præſertim, cui Å¿emper viÅ¿a est viri <lb/>illius Å¿apientia admirabilis. </s> - <s xml:id="s1998" xml:space="preserve">Veruntamen studio veritatis im-<lb/>pulÅ¿us, cuius ipÅ¿e amore in Å¿eipÅ¿um Å¿iviueret excitaretur, in me <lb/>dium <reg norm="quædam" type="context">quædã</reg> proferre non dubitaui, in quibus me inconcußa mathematicæ philoÅ¿ophiæ <lb/>baÅ¿is, cui Å¿emper inÅ¿iÅ¿to, ab eo dißentire coegit.</s> - </p> - <div type="section" level="3" n="1"> - <head xml:id="head243" style="it" xml:space="preserve">Qualiter & ubi AriÅ¿toteles de uelocitate motuum natura-<lb/>lium localium aliter tractauerit quam nos <lb/>Å¿entiamus.</head> - <head xml:id="head244" xml:space="preserve">CAP.I.</head> - <p> - <s xml:id="s1999" xml:space="preserve">VOlens AriÅ¿toreles probare vacuum non eſſe in rerum natura .8. cap. lib. 4. phyÂ<lb/>Å¿icorum ait, idem corpus per varia <reg norm="diuerÅ¿aque" type="simple">diuerÅ¿aqÌ;</reg> media, vt per <reg norm="aerem" type="context">aerẽ</reg>, & per <reg norm="aquam" type="context">aquã</reg> <lb/>Å¿i moueretur, proportionem velocitatis eiuÅ¿dem corporis per aerem, ei, quæ per <lb/>aquam fit, vnam <reg norm="eandemque" type="simple">eandemqÌ;</reg> futuram cum ea, quæ eÅ¿t Å¿ubtilitatis aereÄ™ ad Å¿ubtilitatem <lb/>aquæ. </s> - <s xml:id="s2000" xml:space="preserve">In poÅ¿trema autem parte eiuÅ¿dem capitis Å¿ic Å¿cribit: </s> - <s xml:id="s2001" xml:space="preserve">Nam cum ea quÄ™ ma-<lb/>iorem vel ponderis velleuitatis pręſtantiam habent, Å¿i Å¿imili Å¿igura Å¿int, <reg norm="Å¿pacium" type="context">Å¿paciÅ©</reg> par, <lb/>& æquale, maiore celeritate conficere cernamus, ea quam magnitudines inter Å¿e ha <lb/>bent, proportione: </s> - <s xml:id="s2002" xml:space="preserve">profectò idem etiam perinane fieret. </s> - <s xml:id="s2003" xml:space="preserve">Aliam quoque rationem <lb/>proponit phyloÅ¿ophus .2. cap. Å¿exti phyÅ¿icorum Å¿cribens eademmet proportione, <lb/>qua tempus diuiditur, magnitudinem etiam diuidi. </s> - <s xml:id="s2004" xml:space="preserve">Sexto autem cap. primi de cÅ“-<lb/>lo Å¿cribit, tempora eandem proportionem habere, quam habentè conuerÅ¿o ponde-<lb/>ra; </s> - <s xml:id="s2005" xml:space="preserve">vt Å¿i media pars vnius ponderis, vnius horæ Å¿patio moueretur, vniuerÅ¿um pondus <lb/>in media hora moueretur. </s> - <s xml:id="s2006" xml:space="preserve">Secundo cap. lib. 3. de cÅ“lo duobus in locis apertè comÂ<lb/>monÅ¿trat velocitatem corporis minoris, maiori corpori comparatam, in eadem exi-<lb/>Å¿tere proportione, in qua dicta corpora adinuicem relata exiÅ¿tunt. </s> - <s xml:id="s2007" xml:space="preserve">Quinto cap. eiuÅ¿Â<lb/>dem lib. idem affirmat, exemplo ab igne deÅ¿umpto. </s> - <s xml:id="s2008" xml:space="preserve">Ex alijs etiam plurimis locis <lb/>cognoÅ¿ci poteÅ¿t, Å¿enÅ¿iſſe AriÅ¿totelem duo corpora eadem Å¿pecie, & figura prædita <lb/>eandem planè proportionem in Å¿uorum motuum velocitatibus, quam in Å¿uis ma-<lb/>gnitudinibus habent, retinere. </s> - <s xml:id="s2009" xml:space="preserve">Alij quoque permulti eandem opinionem retinue <lb/>runt, & <reg norm="omnium" type="context">omniÅ©</reg> poÅ¿tremus Nicolaus Tartalea, Å¿ecunda propoÅ¿itione vigeÅ¿iminoni <lb/>quæſiti octaui libri, vbi profitetur Å¿e demonÅ¿tratiuè probare hanc propoÅ¿itio-<lb/>nem veram exiÅ¿tere; </s> - <s xml:id="s2010" xml:space="preserve"><reg norm="neque" type="simple">neq;</reg> videt quà m magna reÅ¿iÅ¿tentiarum Å¿it differentia, quætam <lb/>ex diuerÅ¿itate figurarum, quà m ex magnitudinum varietate exoriri poteÅ¿t; </s> - <s xml:id="s2011" xml:space="preserve">quas qui <lb/>dem diuerÅ¿itates ne conÅ¿iderat quidem.</s> - </p> - <pb o="169" rhead="DISPVTATIONES." n="181" file="0181" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0181"/> - </div> - <div type="section" level="3" n="2"> - <head xml:id="head245" style="it" xml:space="preserve">Quædam Å¿upponenda ut conÅ¿tet cur circa uelocit atem motuum <lb/>natur alium localium ab AriÅ¿totelis placitis <lb/>recedamus.</head> - <head xml:id="head246" xml:space="preserve">CAP. II.</head> - <p> - <s xml:id="s2012" xml:space="preserve">CVM Å¿uÅ¿ceperimus prouinciam probandi quod AriÅ¿toteles circa motus <lb/>locales naturales deceptus fuerit, Å¿unt quædam primo veriſſima & obie-<lb/>cta intellectus perÅ¿e cognita pręſupponenda, ac primum quælibet duo corpora, <lb/>grauia, aut leuia, area æquali, <reg norm="Å¿imilique" type="simple">Å¿imiliqÌ</reg> figura, Å¿ed ex materia diuerÅ¿a conÅ¿tantia, <reg norm="eodem qÌue" type="context">eodẽÂ<lb/>qÌue</reg> modo Å¿itum habentia, eandem proportionem velocitatis inter Å¿uos motus loca <lb/>les naturales, ut inter Å¿uamet pondera aut leuitates in vno <reg norm="eodemque" type="simple">eodemqÌ;</reg> medio, Å¿eruatu-<lb/>ra. </s> - <s xml:id="s2013" xml:space="preserve">Quod quidem natura Å¿ua notiſſimum eÅ¿t Å¿i conÅ¿iderabimus non aliunde maio-<lb/>rem tarditatem, aut velocitatem gigni, quà m à .4. cauÅ¿is (dummodo medium vnifor <lb/>mè Å¿it & quietum) ideÅ¿t à maiori aut minori pondere aut leuitate; </s> - <s xml:id="s2014" xml:space="preserve">à diuerÅ¿a figura; <lb/></s> - <s xml:id="s2015" xml:space="preserve">à ſitu ciuÅ¿dem figuræ diuerÅ¿o, reÅ¿pectu lineÄ™ directionis, quæ recta inter mundi cen-<lb/>trum, & circunferentiam extenditur; </s> - <s xml:id="s2016" xml:space="preserve">& ab inæquali magnitudine. </s> - <s xml:id="s2017" xml:space="preserve">Vnde patebit, <lb/>quòd figuram non variando, nec in qualitate nec in quantitate, neque eiuÅ¿dem figu-<lb/>ræ Å¿itum, motum fore proportionatum virtuti mouenti, quæ erit pondus aut leuitas. <lb/></s> - <s xml:id="s2018" xml:space="preserve">Quod autem de qualitate, de quantitate & Å¿itu eiuÅ¿dem figuræ dico, reÅ¿pectu reÅ¿i-<lb/>ftentiæ ipÅ¿ius medii dico: </s> - <s xml:id="s2019" xml:space="preserve">Quia diffimilitudo aut inÄ™qualitas figurarum, aut Å¿itus di-<lb/>uerÅ¿us non <reg norm="parum" type="context">parÅ©</reg> alterat dictorum corporum motus, cum figura parua facilius diui-<lb/>dat continuitatem medij, quam magna; </s> - <s xml:id="s2020" xml:space="preserve">vt etiam cÄ™lerius idem facit acuta, quà m ob <lb/>tufa; </s> - <s xml:id="s2021" xml:space="preserve">& illa quæ cum angulo, qui antecedat mouebitur velocius quà m illa, quæ Å¿ecus. <lb/></s> - <s xml:id="s2022" xml:space="preserve">QuotieÅ¿cunque igitur duo corpora vnam <reg norm="eandemque" type="simple">eandemqÌ;</reg> reÅ¿iÅ¿tentiam ipÅ¿orum Å¿uperfi-<lb/>ciebus, aut habebunt aut recipient, eorum motus inter Å¿eipÅ¿os eodem planè modo <lb/>proportionati conÅ¿urgent, quo erunt ipÅ¿orum virtutes mouentes: </s> - <s xml:id="s2023" xml:space="preserve">& è conuerÅ¿o, quo <lb/>tieÅ¿cunque duo corpora vnam <reg norm="eandemque" type="simple">eandemqÌ;</reg> grauitatem, aut leuitatem, & diuerÅ¿as reÅ¿i <lb/>Å¿tentias habebunt, eorum motus inter Å¿eipÅ¿os eandem <reg norm="proportionem" type="context">proportionẽ</reg> Å¿ortientur, <reg norm="quam" type="context">quã</reg> <lb/>habebunteorum reÅ¿iÅ¿tentiæ conuerÅ¿o modo; </s> - <s xml:id="s2024" xml:space="preserve">quæ quidem reÅ¿iÅ¿tentiæ inter Å¿eipÅ¿as, <lb/>eandem proportionem quà m ipÅ¿arum Å¿uperficies habebunt, aut in qualitate Å¿ola fi <lb/>guræ, aut in quantitate Å¿ola, aut in Å¿itu, aut in aliquibus ex dictis rebus, eo tamen mo <lb/>do, quiÅ¿uperius poÅ¿itus fuit, vt Å¿cilicet corpus illud quod alteri comparatum, æqua-<lb/>lis erit ponderis, aut leuitatis, Å¿ed minoris reÅ¿iÅ¿tentiæ, exiÅ¿tet velocius altero, in <reg norm="eadem" type="context">eadẽ</reg> <lb/>proportione, cuius Å¿uperficies reÅ¿iÅ¿tentiam Å¿uÅ¿cipit minorem ea quæ alterius eÅ¿t cor-<lb/>poris, ratione facilioris diuiÅ¿ionis continuitatis aeris, aut aquæ; </s> - <s xml:id="s2025" xml:space="preserve">Vt exempli gratia, <lb/>Å¿i proportio Å¿uperficiei corporis maioris Å¿uperficiei minoris Å¿eÅ¿quitertia eſſet, pro-<lb/>portio velocitatis dicti corporis maioris, velocitati corporis minoris, eſſet Å¿ubÅ¿eÅ¿qui <lb/>tertia; </s> - <s xml:id="s2026" xml:space="preserve">vnde velocitas minoris corporis, maior eſſet velocitate corporis maioris, <reg norm="quem" type="context">quẽ</reg> <lb/>admodum quaternarius numerus ternario maior exiÅ¿tit.</s> - </p> - <p> - <s xml:id="s2027" xml:space="preserve">Aliud quoque Å¿upponendum eÅ¿t, velocitatem Å¿cilicet motus naturalis alicuius <lb/>corporis grauis, in diuerÅ¿is medijs, propor-<lb/>tionatam eſſe ponderi eiuÅ¿dem corporis in <lb/>ijÅ¿dem medijs; </s> - <s xml:id="s2028" xml:space="preserve">Vt exempli gratia, Å¿i pondus <lb/> - <anchor type="figure" xlink:label="fig-0181-01a" xlink:href="fig-0181-01"/> - totale alicuius corporis grauis Å¿ignificatum <lb/>crit ab <var>.a.i.</var> quo corpore poÅ¿ito in aliquo me- - <pb o="170" rhead="IO. BABPT. BENED." n="182" file="0182" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0182"/> - dio minus denÅ¿o, quà m ipÅ¿um Å¿it, (quia in medio Å¿e denÅ¿iore Å¿i poner etur, non graue <lb/>e<gap extent="2"/>et, Å¿ed leue, quemadmodum Archimedes oÅ¿tendit) illud medium Å¿ubtrahat par-<lb/>tem <var>.e.i.</var> vnde pars <var>.a.e.</var> eiuÅ¿dem ponderis libera manear; </s> - <s xml:id="s2029" xml:space="preserve">& poÅ¿ito deinde eodem cor <lb/>pore in aliquo alio medio denÅ¿iore, minus tamen denÅ¿o quam ipÅ¿um Å¿it corpus, <lb/>hoc medium Å¿ubtrahat partem <var>.u.i.</var> dicti ponderis, vnde pars <var>.a.u.</var> ei uÅ¿dem <lb/>ponderis remanebit. </s> - <s xml:id="s2030" xml:space="preserve">Dico proportionem velocitatis eiuÅ¿dem corporis per <reg norm="medium" type="context">mediÅ©</reg> <lb/>minus denÅ¿um, ad velocitatem eiuÅ¿dem per medium magis denÅ¿um futuram vt <var>.a.e.</var> <lb/>ad <var>.a.u.</var> vt eÅ¿t etiam rationi conÅ¿onum, magis quà m Å¿i dicamus huiuÅ¿modi velocitates <lb/>eſſe, vt <var>.u.i.</var> ad <var>.e.i.</var> cum velocitates à virtutibus mouentibus Å¿olum (cum figura vna, <lb/><reg norm="eademque" type="simple">eademqÌ;</reg> in qualitate, quantitate <reg norm="Å¿ituque" type="simple">Å¿ituqÌ</reg> erit) proportionentur. </s> - <s xml:id="s2031" xml:space="preserve">Quæne<unsure/>nc diximus, <lb/>planè Å¿imilia Å¿untijs, quæ Å¿upra Å¿cripÅ¿imus, quia idem eÅ¿t dicere, proportionem velo <lb/>citatum, duorum corporum hetereogeneorum, Å¿ed Å¿imilium figura, & magnitudine <lb/>æ qualium, in vno Å¿olo medio, æqualem eſſe <lb/>proportioni ponderum ipÅ¿orum, vt Å¿i dicam?</s> - <s xml:id="s2032" xml:space="preserve"><unsure/> <lb/>proportionem velocitatum vnius Å¿olum cor-<lb/> - <anchor type="figure" xlink:label="fig-0182-01a" xlink:href="fig-0182-01"/> - poris per diuerÅ¿a media eandem eſſe cum ea. <lb/>quæ eÅ¿t <reg norm="ponderum" type="context">ponderÅ©</reg> dicti corporis in iÅ¿idem medijs.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0181-01" xlink:href="fig-0181-01a"> - <image file="0181-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0181-01"/> - </figure> - <figure xlink:label="fig-0182-01" xlink:href="fig-0182-01a"> - <image file="0182-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0182-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="3"> - <head xml:id="head247" style="it" xml:space="preserve">Poſſe uelocitatem alicuius corporis proportionem contrariam <lb/>in diuerÅ¿is medijs habere cum denÅ¿itate eorum.</head> - <head xml:id="head248" xml:space="preserve">CAP. III.</head> - <p> - <s xml:id="s2033" xml:space="preserve">POſſibile eÅ¿t in rerum natura corpus aliquod huiuÅ¿modi denÅ¿itate præditum re-<lb/>periri, vt velocitas eius motus naturalis per aerem, velocitati per aquamita pro <lb/>portionata exiÅ¿tat, vt eÅ¿t <reg norm="denÅ¿itas" type="context">dẽſitas</reg> aquæ denÅ¿itati aeris. </s> - <s xml:id="s2034" xml:space="preserve">DenÅ¿itas aquæ notetur (exem-<lb/>pli gratia) per <var>.u.i.</var> & ea, quæaeris eÅ¿t per <var>.e.i.</var> & pondus alicuius corporis in aere per <lb/><var>e.a.</var> & pondus eiuÅ¿dem corporis in aqua per <var>.u.a.</var> ita tamen, quod eadem proportio <lb/>Å¿it <var>.e.a.</var> ad <var>.u.a.</var> vt <var>.u.i.</var> ad <var>.e.i.</var> vnde per vltimam Å¿uppoÅ¿itionem <reg norm="præcedentis" type="context">præcedẽtis</reg> capitis, pro <lb/>portio velocitatis prædicti corporis per aerem, <lb/>proportioni eiuÅ¿dem corporis per aquam erit, vt <lb/> - <anchor type="figure" xlink:label="fig-0182-02a" xlink:href="fig-0182-02"/> - <var>e.a.</var> ad <var>.u.a.</var> ergo per .11. quinti, vt <var>.u.i.</var> ad <var>.e.i</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0182-02" xlink:href="fig-0182-02a"> - <image file="0182-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0182-02"/> - </figure> - </div> - </div> - <div type="section" level="3" n="4"> - <head xml:id="head249" style="it" xml:space="preserve">OÅ¿citanter ab AriÅ¿totele nonnibil prolatum cap 8. <lb/>lib. 4 PhyÅ¿icorum.</head> - <head xml:id="head250" xml:space="preserve">CAP. IIII.</head> - <p> - <s xml:id="s2035" xml:space="preserve">EX Å¿upradictis patet in vniuerÅ¿um non eſſe verum quod AriÅ¿to .8. cap .4. lib. phyÂ<lb/>Å¿icorum Å¿cribit, velocitates Å¿cilicet motuum alicuius corporis per diuerÅ¿a me-<lb/>dia, proportionatas eſſe denÅ¿itatibus eorundem mediorum. </s> - <s xml:id="s2036" xml:space="preserve">Quocirca, Å¿it propor-<lb/>tio <var>.u.i.</var> ad <var>.e.i.</var> vt <reg norm="denÅ¿itatis" type="context">dẽſitatis</reg> aquÄ™ ad <reg norm="aeream" type="context">aereã</reg> <reg norm="denÅ¿itatem" type="context">dẽſitatem</reg> .et <var>.e.a.</var> ad <var>.u.a.</var> vt ponderis alicuius <lb/>corporis in aere ad pondus eiuÅ¿dem in aqua, ita tamen vt maior aut minor propor-<lb/>tio Å¿it <var>.e.a.</var> ad <var>.u.a.</var> quam <var>.u.i.</var> ad <var>.e.i.</var> vnde exiÅ¿tente proportione velocitatis per <reg norm="aerem" type="context">aerẽ</reg> - <pb o="171" rhead="DISPVTATIONES." n="183" file="0183" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0183"/> - ad velocitatem per aquam vt <var>.e.a.</var> ad <var>.a.u.</var> non erit ergo vt <var>.u.i.</var> ad <var>.e.i</var>. </s> - <s xml:id="s2037" xml:space="preserve">Ob hanc igitur <lb/>cauÅ¿am nimis diſſentaneum eÅ¿t rationi, opi-<lb/>nari proportionem velocitatis omnium cor <lb/>porum grauium per aerem vnam <reg norm="eandemque" type="simple">eandemqÌ;</reg> <lb/> - <anchor type="figure" xlink:label="fig-0183-01a" xlink:href="fig-0183-01"/> - eſſe cum velocitate eorundem per aquam, <lb/>quemadmodum AriÅ¿toteles Å¿enÅ¿it.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0183-01" xlink:href="fig-0183-01a"> - <image file="0183-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0183-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="5"> - <head xml:id="head251" style="it" xml:space="preserve">Exempla dictorum.</head> - <head xml:id="head252" xml:space="preserve">CAP.V.</head> - <p> - <s xml:id="s2038" xml:space="preserve">POnamus, exempli gratia, aquam eſſe in denÅ¿itate dupla ad aerem, & aliquod <lb/>graue corpus in aqua duplum ad denÅ¿itatem ipÅ¿ius aquæ, vnde dictum corpus in <lb/>denÅ¿itate ad aerem quadruplum erit; </s> - <s xml:id="s2039" xml:space="preserve">quam ob cauÅ¿am, mediam Å¿ui ponderis tota-<lb/>lis partem in aqua, & in aere quartam partem, ex .7. lib. de inÅ¿identibus aquæ ab Ar-<lb/>chimede conÅ¿cripto, amitteret. </s> - <s xml:id="s2040" xml:space="preserve">Moueretur igitur in aqua virtute illius mediæ partis <lb/><reg norm="ponderis" type="context">põderis</reg> Å¿ui, in aere <reg norm="autem" type="wordlist">aÅ©t</reg> uirtute <reg norm="trium" type="context">triÅ©</reg> <reg norm="quartarum" type="context">quartarÅ©</reg>; </s> - <s xml:id="s2041" xml:space="preserve">vnde proportio facultatis <reg norm="mouentis" type="context">mouẽtis</reg> dicti <lb/>corporis in aere ad facultatem mouentem eiuÅ¿de m in aqua Å¿eÅ¿quialtera erit. </s> - <s xml:id="s2042" xml:space="preserve"><reg norm="hocque" type="simple">hocqÌ;</reg> <lb/>corpus appelletur <var>.A</var>. </s> - <s xml:id="s2043" xml:space="preserve">Sit aliud quoque corpus, quod <var>.B.</var> nominetur, Å¿imile figura, & <lb/>magnitudine corporea corpori <var>.A.</var> Å¿ed <reg norm="denÅ¿itate" type="context">dẽſitate</reg>, in proportione Å¿eÅ¿quialtera ad <reg norm="aquam" type="context">aquã</reg>, <lb/>& denÅ¿ius erit aere in proportione tripla. </s> - <s xml:id="s2044" xml:space="preserve">quamobrem corpus <var>.A.</var> grauius erit cor-<lb/>pore <var>.B.</var> in aere in proportione Å¿eÅ¿quialtera, vnde etiam velocius erit ipÅ¿o <var>.B.</var> in aere <lb/>in eadem proportione, Å¿ed corpus <var>.B.</var> in aere, duplo maius pondus habebit, <reg norm="quam" type="context">quã</reg> in <lb/>aqua, cum in aere remaneant ei duæ ponderis tertiæ partes, & in aqua vna tantum, <lb/>ita vt AriÅ¿toteli concedam corpus <var>.B.</var> in aere, quam in aqua velocius futurum in ea-<lb/>dem proportione, in qua, aqua eÅ¿t <reg norm="denÅ¿ior" type="context">dẽſior</reg> aere, ex Euclidis vndecima propoÅ¿itione <lb/>lib. quinti. </s> - <s xml:id="s2045" xml:space="preserve">Sed præter hæc omnia, Å¿i corpus <var>.A.</var> eſſet etiam velocius in aere, <reg norm="quam" type="context">quã</reg> in <lb/>aqua, in eadem proportione, Å¿equeretur ex .16. dicti lib. quinti proportionem velo-<lb/>citatis <var>.A.</var> in aqua ad <reg norm="velocitatem" type="context">velocitatẽ</reg> ipÅ¿ius <var>.B.</var> in aqua etiam Å¿eÅ¿quialteram eſſe. </s> - <s xml:id="s2046" xml:space="preserve">Sed cum <lb/>corpus <var>.A.</var> in denÅ¿itate ad aquam <reg norm="duplum" type="context">duplÅ©</reg> Å¿it, & corpus <var>.B.</var> <reg norm="Å¿eÅ¿quialterum" type="context">Å¿eÅ¿quialterÅ©</reg> ad ipÅ¿am <reg norm="aquam" type="context">aquã</reg>, <lb/>Å¿equetur <reg norm="proportionem" type="context">proportionẽ</reg> ponderis ipÅ¿ius <var>.A.</var> ad <reg norm="pondus" type="context">põdus</reg> ipÅ¿ius <var>.B.</var> in aqua eſſe in propor-<lb/>tione dupla; </s> - <s xml:id="s2047" xml:space="preserve">Vnde ex primo Å¿uppoÅ¿ito capitis Å¿ecundi proportio velocitatis <var>.A.</var> ad <lb/>velocitatem <var>.B.</var> in aqua dupla erit, non Å¿eÅ¿quialtera. </s> - <s xml:id="s2048" xml:space="preserve">Si ergo proportio velocitatis <var>.<lb/>A.</var> ad eam quæ eÅ¿t <var>.B.</var> in aqua dupla eÅ¿t, & ea, quæeÅ¿t <var>.B.</var> in aere, ad eam, quæ eÅ¿t ipÅ¿ius <lb/>per aquam eÅ¿t etiam dupla (vnde ea quÄ™ eÅ¿t <var>.A.</var> per aquam Ä™qualis erit ei, quæ eÅ¿t <var>.B.</var> <lb/>peraerem, ex .9. lib. quinti) & cum ea, quæ eÅ¿t <var>.A.</var> Å¿it ei, quæ eÅ¿t <var>.B.</var> per aerem Å¿eÅ¿qui-<lb/>altera, erit ergo ea, quæ eÅ¿t <var>.A.</var> per aerem, ei, quæ eÅ¿t Å¿uimet ipÅ¿ius per aquam Å¿eÅ¿qui <lb/>altera, non autem dupla, ex .7. eiuÅ¿dem libr. quinti. </s> - <s xml:id="s2049" xml:space="preserve">HiÅ¿ce rationibus accedimus ad <lb/>confirmandam veritatem vltimi Å¿uppoÅ¿iti cap .2. proportionem videlicet velocita <lb/>tis <reg norm="motus" type="simple">motê°</reg> naturalis in diuerÅ¿is medijs <reg norm="alicuius" type="simple">alicuiê°</reg> corporis <reg norm="ponderoÅ¿i" type="context">põderoÅ¿i</reg> in ipÅ¿is medijs eſſe ean <lb/>dem cum ea, quæeÅ¿t inter pondera <lb/> - <anchor type="figure" xlink:label="fig-0183-02a" xlink:href="fig-0183-02"/> - dicti corporis in dictis medijs. de ijs <lb/>tamen medijs intelligendo, quæ un-<lb/>ctuoÅ¿a, aut pinguia non Å¿unt, ut Å¿unt <lb/>oleum, lac, aut huiuÅ¿modialia, quæà <lb/>qualibet minima qualitate frigoris <lb/>aut caloris alterantur, & impermea-<lb/>biles fiunt.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0183-02" xlink:href="fig-0183-02a"> - <image file="0183-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0183-02"/> - </figure> - </div> - <pb o="172" rhead="IO. BAPT. BENED." n="184" file="0184" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0184"/> - </div> - <div type="section" level="3" n="6"> - <head xml:id="head253" style="it" xml:space="preserve">Quod proportiones ponderum eiuÅ¿dem corporis in diuerÅ¿is medijs pro <lb/>portiones eorum mediorum denÅ¿it atum non Å¿eruant. Unde ne-<lb/>ceßariò inæquales proportiones uelocitatum <lb/>producuntur.</head> - <head xml:id="head254" xml:space="preserve">CAP. VI.</head> - <p> - <s xml:id="s2050" xml:space="preserve">OMne corpus graue variat proportionem ponderis per diuerÅ¿a media, vnde <lb/>proportiones velocitatum inæquales exiÅ¿tunt. </s> - <s xml:id="s2051" xml:space="preserve">Vt exempli gratia, Å¿i fue-<lb/>rit corpus <var>.A.</var> cuius pondus totale Å¿it <var>.o.a.</var> quod in aqua diminutum Å¿it ratione partis <var>.<lb/>e.o.</var> ita vt ei Å¿olum relinquatur pondus <var>.a.e.</var> & in aeie adempta Å¿it ei pars <var>.i.o.</var> vnde Å¿o <lb/>lum remaneat pondus <var>.a.i</var>. </s> - <s xml:id="s2052" xml:space="preserve">Supponamus aliud <reg norm="quoque" type="simple">quoq;</reg> medium in eadem proportio-<lb/>ne minus denÅ¿um, quà m aer, quemadmodum aer minus denÅ¿us eÅ¿t, aqua, in quo, cor <lb/>pus <var>.A.</var> ammittat partem <var>.t.o.</var> ponderis Å¿ui, vnde ex .7. lib. de inÅ¿identibus aquæ Ar-<lb/>chimedis, eadem proportio erit <var>.e.o.</var> ad <var>.i.o.</var> quæ eÅ¿t <var>.i.o.</var> ad <var>.t.o</var>. </s> - <s xml:id="s2053" xml:space="preserve">Supponamus <reg norm="quoque" type="simple">quoq;</reg> <lb/>eandem proportionem eſſe <var>.a.i.</var> ad <var>.a.e.</var> eÅ¿t <var>.e.o.</var> ad <var>.i.o.</var> </s> - <s xml:id="s2054" xml:space="preserve">tunc dico non futuram ean-<lb/>dem proportionem <var>.t.a.</var> ad <var>.a.i.</var> quæ eÅ¿t <var>.i.o.</var> ad <var>.t.o</var>. </s> - <s xml:id="s2055" xml:space="preserve">Cum Å¿it ergo proportio <var>.a.i.</var> <lb/>ad <var>.a.e.</var> ut <var>.e.o.</var> ad <var>.i.o.</var> erit diÅ¿iunctim <var>.e.i.</var> ad <var>.e.a.</var> vt <var>.e.i.</var> ad <var>.i.o</var>. </s> - <s xml:id="s2056" xml:space="preserve">Quare ex .9. libr. quinÂ<lb/>ti erit <var>.a.e.</var> æqualis <var>.i.o.</var> Å¿ed cum ita Å¿ehabeat <var>.e.o.</var> ad <var>.i.o.</var> vt <var>.i.o.</var> ad <var>.t.o.</var> ita quoque <lb/>Å¿e habebit, ex vndecima quinti <var>.a.i.</var> ad <var>.e.a.</var> ut <var>.i.o.</var> ad <var>.t.o</var>. </s> - <s xml:id="s2057" xml:space="preserve">Cum autem (vt vidimus). <var>a.e.</var> <lb/>Ä™qualis Å¿it ipÅ¿i <var>.i.o.</var> non poterit eſſe proportio <var>.t.a.</var> ad <var>.i.a.</var> vt eÅ¿t <var>.o.i.</var> ad <var>.t.o.</var> quia Å¿i <lb/>hoc eſſet, eſſet etiam diÅ¿iunctim proportio <var>.i.t.</var> ad <var>.i.a.</var> vt eÅ¿t <var>.i.t.</var> ad <var>.t.o.</var> & ex Å¿upradicta <lb/>9. lib. quinti <var>.a.i.</var> æqualis eſſet <var>.t.o</var>. </s> - <s xml:id="s2058" xml:space="preserve">Maximum autem inconueniens eſſet <var>.t.o.</var> minorem <lb/><var>o.i.</var> ideÅ¿t minorem <var>.a.e.</var> æqualem eſſe <var>.a.i.</var> quæ maior eÅ¿t <var>.a.e</var>. </s> - <s xml:id="s2059" xml:space="preserve">OÅ¿tenÅ¿iuè tamen idem <lb/>hoc modo probari poteÅ¿t, vt exiÅ¿tente <var>.i.o.</var> Ä™quali ipÅ¿i <var>.a.e.</var> per conÅ¿equens <reg norm="quoque" type="simple">quoq;</reg> erit <lb/>minor ipÅ¿a <var>.a.i.</var> cum <var>.a.e.</var> pars Å¿it ipÅ¿ius <var>a.i</var>. </s> - <s xml:id="s2060" xml:space="preserve"><reg norm="Pereandem" type="context">Pereãdem</reg> tamen rationem <var>.o.t.</var> minoreÅ¿t <var>.<lb/>o.i</var>. </s> - <s xml:id="s2061" xml:space="preserve">Tanto magis igitur minor erit <var>.t.o.</var> ipÅ¿a <var>.i.a</var>. </s> - <s xml:id="s2062" xml:space="preserve">Vnde ex .8. libri quinti maiorem pro <lb/>portionem habebit <var>.i.t.</var> <lb/>ad <var>.t.o.</var> quam ad <var>.i.a.</var> & <lb/>ex .28. <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> lib <var>.i.o.</var> ad <lb/><var>t.o.</var> <reg norm="maiorem" type="context">maiorẽ</reg> proportio-<lb/> - <anchor type="figure" xlink:label="fig-0184-01a" xlink:href="fig-0184-01"/> - <reg norm="nem" type="context">nẽ</reg> habebit, quà m.t.a. <lb/>ad <var>.i.a.</var> ex .12. igitur di-<lb/>cti quinti maiorem pro <lb/>portionem habebit <var>.i.a.</var> ad <var>.e.a.</var> quà m.t.a. ad <var>.i.a.</var> ita ergo Å¿e habebunt ipÅ¿orum velo-<lb/>citates.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0184-01" xlink:href="fig-0184-01a"> - <image file="0184-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0184-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="7"> - <head xml:id="head255" style="it" xml:space="preserve">Corpora grauia aut leuia eiuÅ¿dem figur æ et materiæ Å¿ed inæqualis <lb/>magnitudinis, in Å¿uis motibus natur alibus uelocit atis, in eo <lb/>dem medio, proportionem longè diuerÅ¿am Å¿eruatura <lb/>eße quam AristoteliuiÅ¿um fuerit.</head> - <head xml:id="head256" xml:space="preserve">CAP. VII.</head> - <p> - <s xml:id="s2063" xml:space="preserve">ESt mihi nunc probandum <reg norm="quod" type="simple">ê™</reg> in uno <reg norm="eodemque" type="simple">eodemqÌ;</reg> mcdio duo corpora inæqualia, Å¿ed <lb/>Å¿imili figura & materia, mouebuntur naturali motu, diuerÅ¿a tamen ratione ab - <pb o="173" rhead="DISPVTATIONES." n="185" file="0185" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0185"/> - ea, quà m AriÅ¿toteles præſcripÅ¿it.</s> - </p> - <p> - <s xml:id="s2064" xml:space="preserve">Sintigitur corpora <var>.a.</var> et <var>.o.</var> inæqualia, <reg norm="eadentamen" type="context">eadẽtamen</reg> figura & materia prædita, quo-<lb/>rum <var>.a.</var> maius Å¿it, & per conÅ¿equens in eadem quoque proportione grauius ipÅ¿o <var>.o.</var> in <lb/>qua eÅ¿t maius, communi omnium Å¿ententia.</s> - </p> - <p> - <s xml:id="s2065" xml:space="preserve">Scribit ergo AriÅ¿toteles proportionem velocitatis corporis <var>.a.</var> ad eam, quæ eÅ¿t <lb/>corporis <var>.o.</var> (naturaliterÅ¿e vnoquoque mouente) eandem futuram, quæ eÅ¿t magnitu <lb/>dinis, aut grauitatis corporis <var>.a.</var> ad magnitudinem, aut grauitatem corporis <var>.o</var>. </s> - <s xml:id="s2066" xml:space="preserve">Ima-<lb/>ginemur igitur corpus u. eadem magnitudine & figura, qua corpus <var>.a.</var> præditum eÅ¿t, <lb/>Å¿ed eandem grauitatem obtinere, quæ communicata eÅ¿t corpori <var>.o.</var> quod ex quauis <lb/>materia conÅ¿ter. </s> - <s xml:id="s2067" xml:space="preserve">Hinc ex primo Å¿uppoÅ¿ito Å¿ecundi capitis certi erimus proportio-<lb/>nem velocitatis corporis <var>.a.</var> Å¿i comparetur cum velocitate corporis <var>.u.</var> futuram, vt <reg norm="eam" type="context">eã</reg>, <lb/>quæ eÅ¿t ponderis corporis <var>.a.</var> ad pondus ipÅ¿ius corporis <var>.u</var>. </s> - <s xml:id="s2068" xml:space="preserve">Ex .9. igitur lib. quinti Eu-<lb/>cli. cogitur fateri AriÅ¿toteles velocitatem corporis <var>.o.</var> eſſe vnam <reg norm="eandemque" type="simple">eandemqÌ;</reg> in Å¿pe-<lb/>cie, quæ eÅ¿t corporis <var>.u</var>. </s> - <s xml:id="s2069" xml:space="preserve">Quod primo Å¿uppoÅ¿ito cap. Å¿ecundi huius lib. planè repugnaÂ<lb/>ret. </s> - <s xml:id="s2070" xml:space="preserve">Igitur hæc AriÅ¿totelis opinio falÅ¿a eÅ¿t. </s> - <s xml:id="s2071" xml:space="preserve">Idem quoque probaretur mediante cor <lb/>pore <var>.i.</var> æquali magnitudine, <reg norm="Å¿imilique" type="simple">Å¿imiliqÌ;</reg> figura cum corpore <var>.o.</var> prædito, Å¿ed, quòd ad <lb/>quantitatem attinet, æquali corpori <var>.a.</var> vnde ex primo Å¿uppoÅ¿ito cap. Å¿ecundi huius li <lb/>bri in eadem pro <lb/>portione <reg norm="velocius" type="simple">velociê°</reg> <lb/>eſſet corpore <var>.o.</var> <lb/> - <anchor type="figure" xlink:label="fig-0185-01a" xlink:href="fig-0185-01"/> - in qua grauius eÅ¿t. <lb/>ex .9. igitur quin-<lb/>ti cogitur AriÅ¿to-<lb/>teles affirmare <reg norm="tam" type="context">tã</reg> <lb/>velox eſſe corpus <lb/>a. <reg norm="quam" type="context">quã</reg> eÅ¿t corpus <lb/>i. vnde idem pla-<lb/>nè inconueniens emergit ex Å¿ecundo Å¿uppoÅ¿ito cap. Å¿ecundi huius lib..</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0185-01" xlink:href="fig-0185-01a"> - <image file="0185-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0185-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="8"> - <head xml:id="head257" style="it" xml:space="preserve">Quod duo corpor a in æqualia eiuÅ¿dem materia in diuerÅ¿is <lb/>medijs eandem uelocitatis proportionem <lb/>retinebunt.</head> - <head xml:id="head258" xml:space="preserve">CAP. VIII.</head> - <p> - <s xml:id="s2072" xml:space="preserve">QVælibet duo corpora inæqualia Å¿imili tamen figura & eadem materia con-<lb/>Å¿tantia, naturaliter Å¿e per diuerÅ¿a media mouentia, vnam <reg norm="eandemque" type="simple">eandemqÌ;</reg> Å¿em-<lb/>per proportionem velocitatum Å¿eruant.</s> - </p> - <p> - <s xml:id="s2073" xml:space="preserve">Sint duo corpora <var>.A.</var> et <var>.B.</var> Å¿ibi inuicem inæqualia quorum <var>.A.</var> Å¿it maius, Å¿ed Å¿imile <lb/>figura & idem materia, <lb/>cuius pondus totaleÅ¿it <var>.<lb/> - <anchor type="figure" xlink:label="fig-0185-02a" xlink:href="fig-0185-02"/> - x.o.</var> & pondus totaleip <lb/>Å¿ius <var>.B.</var> Å¿it <var>.u.s</var>. </s> - <s xml:id="s2074" xml:space="preserve">Imagine-<lb/>mur quoque corpus <var>.A.</var> <lb/>poÅ¿itum in aqua amitte <lb/>re <reg norm="partem" type="context">partẽ</reg> <var>.o.e.</var> ponderis. - <pb o="174" rhead="IO. BAPT. BENED." n="186" file="0186" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0186"/> - <var>o.x.</var> et <var>.B.</var> quoque in eodem loco amittere <var>.c.s.</var> et <var>.A.</var> in <reg norm="aem" type="context">aẽ</reg><unsure/>re partem <var>.i.o.</var> et <var>.B.</var> partem. <lb/> <var>.t.s</var>. </s> - <s xml:id="s2075" xml:space="preserve">Nunc quia corpus aqueum, cui correÅ¿pondet <var>.e.o.</var> æquale eÅ¿t ipÅ¿i <var>.A.</var> & corpus <lb/>aqueum, cui correÅ¿pondet <var>.c.s.</var> æquale eÅ¿t i pÅ¿i <var>.B.</var> vt eÅ¿t ab Archimede <reg norm="probatum" type="context">probatÅ©</reg>: </s> - <s xml:id="s2076" xml:space="preserve">com <lb/>muni quadam Å¿cientiæ ratione, Å¿equitur eandem proportionem futuram <var>.o.x.</var> ad <var>.e.o.</var> <lb/>quæ eÅ¿t <var>.u.s.</var> ad <var>.c.s.</var> ob <reg norm="eaÅ¿demque" type="simple">eaÅ¿demqÌ;</reg> rationes idem erit de <var>.x.o.</var> ad <var>.i.o.</var> ut <var>.u.s.</var> ad <var>.t.s.</var> & <reg norm="idem" type="context">idẽ</reg> <lb/>etiam erit de <var>.o.x.</var> ad <var>.s.u.</var> vt de <var>.e.o.</var> ad <var>.c.s.</var> vt etiam de <var>.o.i.</var> ad <var>.s.t</var>. </s> - <s xml:id="s2077" xml:space="preserve">Vnde ex .19. lib. <lb/>quintà erit de <var>.x.i.</var> ad <var>.u.t.</var> quemadmodum de <var>.x.o.</var> ad <var>.u.s.</var> idem dico de <var>.x.e.</var> ad <var>.u.c</var>. </s> - <s xml:id="s2078" xml:space="preserve">Ex <lb/>11. igitur dicti lib. erit. de <var>.x.i.</var> ad <var>.u.t.</var> quemadmodum de <var>.x.e.</var> ad <var>.u.c.</var> ex quibus <reg norm="quidem" type="context">quidẽ</reg> <lb/>proportionibus, Å¿i Å¿ubtra <lb/> - <anchor type="figure" xlink:label="fig-0186-01a" xlink:href="fig-0186-01"/> - hantur proportiones @reÅ¿i <lb/><reg norm="Å¿tentiarum" type="context">Å¿tẽtiarum</reg> extrinÅ¿ecus <reg norm="ad- uenentium" type="context">ad-<lb/>uenẽtium</reg>, proportiones <lb/>quæ remanebunt, exter-<lb/>tio communi axiomate <lb/>ab Eucli. in principio priÂ<lb/>mi lib. poÅ¿ito, ad inuicem <lb/>erunt æquales, Å¿ecundum quas eorundem corporum Å¿unt velocitates.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0185-02" xlink:href="fig-0185-02a"> - <image file="0185-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0185-02"/> - </figure> - <figure xlink:label="fig-0186-01" xlink:href="fig-0186-01a"> - <image file="0186-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0186-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="9"> - <head xml:id="head259" style="it" xml:space="preserve">Anrectè Aristoteles diÅ¿eruerit de proportionibus mo-<lb/>tuum in uacuo.</head> - <head xml:id="head260" xml:space="preserve">CAP. IX.</head> - <p> - <s xml:id="s2079" xml:space="preserve">CVm verò AriÅ¿toteles circa finem cap .8. lib. 4. phyÅ¿icorum Å¿ubiungit quod ea-<lb/>dem proportione dicta corpora mouerentur in vacuo, vt in pleno, id pace <reg norm="eius" type="simple">eiê°</reg> <lb/><reg norm="dictum" type="context">dictÅ©</reg> Å¿it planè <reg norm="erroneum" type="context">erroneÅ©</reg> eÅ¿t. </s> - <s xml:id="s2080" xml:space="preserve">quia in pleno dictis corporibus Å¿ubtrahitur proportio reÅ¿i <lb/>Å¿tentiarum extrinÅ¿ecarum à proportione ponderum, vt velocitatum proportio re-<lb/>maneat, quÄ™ nulla eſſet, Å¿i dictarum reÅ¿iÅ¿tentiarum proportio, ponderum propor-<lb/>tioni æqualis eſſet, & hanc ob cauÅ¿am diuerÅ¿am velocitatum proportionem in va-<lb/>cuo haberent ab ea, quæ eÅ¿t in pleno.</s> - </p> - </div> - <div type="section" level="3" n="10"> - <head xml:id="head261" style="it" xml:space="preserve">Quòd in uacuo corpor a eiuÅ¿dem materiæ æquali uelocita-<lb/>te mouerentur.</head> - <head xml:id="head262" xml:space="preserve">CAP.X.</head> - <p> - <s xml:id="s2081" xml:space="preserve">QVòd Å¿upradicta corpora in vacuo naturaliter pari velocitate mouerentur, <lb/>hac ratione aſſero.</s> - </p> - <p> - <s xml:id="s2082" xml:space="preserve">Sint enim duo corpora <var>.o.</var> et <var>.g.</var> omogenea, et <var>.g.</var> Å¿it dimidia pars ipÅ¿ius <var>.o.</var> Å¿int alia <lb/>quoque duo corpora <var>.a.</var> et <var>.e.</var> omogenea primis, quorum quodlibet æquale Å¿it ipÅ¿i <var>.g.</var> <lb/>& imaginatione comprÄ™hendamus ambo poÅ¿ita in extremitatibus alicuius lineæ, cu <lb/>ius medium Å¿it <var>.i.</var> clarum erit, tantum pondus habiturum, punctum <var>.i.</var> quantum <reg norm="centrum" type="context">centrÅ©</reg> <lb/>ipÅ¿ius <var>.o.</var> quod <var>.i.</var> virtute corporis <var>.a.</var> et <var>.e.</var> in vacuo, <lb/> - <anchor type="figure" xlink:label="fig-0186-02a" xlink:href="fig-0186-02"/> - eadem velocitate moueretur, quacentrum ipÅ¿ius .<lb/>o: </s> - <s xml:id="s2083" xml:space="preserve">cum autem difiuncta eſſent dicta corpora <var>.a.</var> et <var>.e.</var> <lb/>à dicta linea, non ideo aliquo modo Å¿uam velocita - <pb o="175" rhead="DISPVTATIONES." n="187" file="0187" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0187"/> - tem mutarent, quorum quodlibet eſſet quoque tam velox, quam eÅ¿t .g: igitur <var>.g.</var> <lb/>tam velox eſſet quam <var>.o</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0186-02" xlink:href="fig-0186-02a"> - <image file="0186-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0186-02"/> - </figure> - </div> - </div> - <div type="section" level="3" n="11"> - <head xml:id="head263" style="it" xml:space="preserve">Corpora licet inæqualia eiuÅ¿dem materiæ & figuræ, Å¿ireÅ¿iÅ¿ten-<lb/>tias habuerint ponderibus proportionales <lb/>æqualiter mouebuntur.</head> - <head xml:id="head264" xml:space="preserve">CAP. XI.</head> - <p> - <s xml:id="s2084" xml:space="preserve">EAdem ratione, quam cap. antecedente præſcripÅ¿imus, poſſet oÅ¿tendi, Å¿i duo cor-<lb/>pora <var>.o.</var> et <var>.g.</var> Å¿uas reÅ¿iÅ¿tentias, ita ad inuicem proportionatas haberent, utÅ¿unt <lb/>eorum pondera, in pleno pari velocitate prædita eſſe, quod in fine capitis noni leui <lb/>ter attigi, quia punctum <var>.i.</var> tam velox eſſet, ut centrum ipÅ¿ius <var>.o.</var> cum à tanto pondere <lb/>i. motum eſſet; </s> - <s xml:id="s2085" xml:space="preserve">quanto centrum ipÅ¿ius <var>.o.</var> atquetan <lb/> - <anchor type="figure" xlink:label="fig-0187-01a" xlink:href="fig-0187-01"/> - tam reÅ¿iÅ¿tentiam duo corpora <var>.a.</var> et <var>.e.</var> <reg norm="quanta" type="context">quãta</reg> ipÅ¿um <lb/>o. Å¿olum haberet ex hypotheÅ¿i, dicta tamen corpo <lb/>ra <var>.a.</var> et <var>.e.</var> tam Å¿eparata, quam coniuncta, eandem <lb/>velocitatem retinerent <var>.g.</var> igitur tam velox eſſet, <lb/>quam <var>.o</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0187-01" xlink:href="fig-0187-01a"> - <image file="0187-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0187-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="12"> - <head xml:id="head265" style="it" xml:space="preserve">Maior hic demonÅ¿ir atur eſſe proportio ponder is corpor is den <lb/>Å¿ioris ad pondus minus denÅ¿i in <reg norm="medijs" type="lig">medijs</reg> <reg norm="denÅ¿ioribus" type="context">dẽſioribus</reg>, quam <lb/>Å¿it eorundem corporum in medio minus denÅ¿o, nec <lb/>corporum ponder a Å¿eruare proportionem <lb/>denÅ¿itatis mediorum.</head> - <head xml:id="head266" xml:space="preserve">CAP. XII.</head> - <p> - <s xml:id="s2086" xml:space="preserve">PRopoÅ¿ita nobis cum fuerint duo corpora <var>.A.</var> et <var>.B.</var> area corporea æqualia, quo-<lb/>rum <var>.A.</var> denÅ¿ius Å¿it ipÅ¿o <var>.B.</var> probabo in medio magis denÅ¿o, maiorem proportio <lb/>nem futuram ponderis ipÅ¿ius <var>.A.</var> ad pondus <var>.B.</var> quà m in medio minus denÅ¿o.</s> - </p> - <p> - <s xml:id="s2087" xml:space="preserve">Sit igitur <var>.p.g.</var> pondus totale ipÅ¿ius corporis <var>.A.</var> et <var>.q.k.</var> ipÅ¿ius corporis <var>.B.</var> vnde <var>.p.g.</var> <lb/>maius erit ipÅ¿o <var>.q.k</var>. </s> - <s xml:id="s2088" xml:space="preserve">Sit quoque <var>.o.g.</var> pondus, quod medium magis denÅ¿um Å¿ubtra-<lb/>hit à pondere <var>.p.g.</var> et <var>.n.k.</var> Å¿it pondus, quod idem medium Å¿ubtrahit à pondere <var>.q.k.</var> et <lb/><var>f.g.</var> Å¿it pondus, quod medium minus denÅ¿um Å¿ubtrahit à <var>.p.g.</var> et <var>.i.k.</var> illud, quodid@m <lb/><reg norm="medium" type="context">mediÅ©</reg> Å¿ubtrahit ab <var>.q.k.</var> vnde <var>.o.g.</var> æquale erit <var>.n.k.</var> et <var>.f.g.</var> ipÅ¿i <var>.i.k.</var> quia quod ad <reg norm="aream" type="context">areã</reg> <lb/>attinet, corpora Å¿upponuntur æqualia, vnde proportio <var>.p.f.</var> ad <var>.q.i.</var> maior erit ea, quæ <lb/>eÅ¿t <var>.o.f.</var> ad <var>.n.i.</var> communi <lb/> - <anchor type="figure" xlink:label="fig-0187-02a" xlink:href="fig-0187-02"/> - Å¿cientiæ notione, quia Å¿i <lb/>Å¿cinderet <reg norm="aliquis" type="simple">aliê—s</reg>.p.f. in pun <lb/>cto <var>.c.</var> ita. vt <var>.c.f.</var> æquale eÅ¿-<lb/>Å¿et ipÅ¿i <var>.q.i.</var> proportio <var>.c.f.</var> <lb/>ad <var>.q.i.</var> eſſet vt ea, quæ eÅ¿t <var>.<lb/>o.f.</var> ad <var>.n.i.</var> (hoc eÅ¿t nulla) - <pb o="176" rhead="IO. BAPT. BENED." n="188" file="0188" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0188"/> - Å¿ed proportio <var>.p.f.</var> ad <var>.q.i.</var> maior eſſet ea, quæ eÅ¿t <var>.c.f.</var> ad <var>.q.i.</var> ex. octaua lib. quinti, vn-<lb/>de ex .12. eiuÅ¿dem lib. maior eſſet <var>.p.f.</var> ad <var>.q.i.</var> quà m.o.f. ad <var>.n.i.</var> ex .33. igitur eiuÅ¿dem, <lb/>maior erit proportio <var>.p.o.</var> ad <var>.q.n.</var> quà m.p.f. ad <var>.q.i</var>. </s> - <s xml:id="s2089" xml:space="preserve">Sic quoque Å¿e habebunt ad inui <lb/>cem velocitates, quod eÅ¿t propoÅ¿itum. </s> - <s xml:id="s2090" xml:space="preserve">Cum autem proportio <var>.p.o.</var> ad <var>.q.n.</var> maior Å¿it, <lb/>quà m.p.f. ad <var>.q.i.</var> permurando igitur maior erit proportio <var>.p.o.</var> ad <var>.p.f.</var> quam <var>.q.n.</var> ad <var>.<lb/>q.i.</var> aut euerÅ¿im maior erit proportio <var>.q.i.</var> ad <var>.q.n.</var> quà m.p.f. ad <var>.p.o.</var> vnde Å¿i proportio <lb/><var>p.f.</var> ad <var>.p.o.</var> eſſet ac ea, quæ eÅ¿t <var>.o.g.</var> ad <var>.f.g.</var> non eſſet <var>.q.i.</var> ad <var>.q.n.</var> ut eÅ¿t <var>.o.g.</var> ad <var>.f.g.</var> aut <lb/>vt <var>.n.k.</var> ad <var>.i.k.</var> quodidem <lb/>eÅ¿t, de quibus quidem re-<lb/> - <anchor type="figure" xlink:label="fig-0188-01a" xlink:href="fig-0188-01"/> - bus, exemplis propoÅ¿itis <lb/>quinto capite <reg norm="mentionem" type="context">mẽtionem</reg> <lb/>feci.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0187-02" xlink:href="fig-0187-02a"> - <image file="0187-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0187-02"/> - </figure> - <figure xlink:label="fig-0188-01" xlink:href="fig-0188-01a"> - <image file="0188-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0188-01"/> - </figure> - </div> - <p> - <s xml:id="s2091" xml:space="preserve">Velocitatibus autem Å¿e-<lb/>quentibus pondera, Å¿equi <lb/>tur proportionem veloci-<lb/>citatum duorum corporum hetereogeneorum eandem non eſſe per diuerÅ¿a media, <lb/>contra id, quod Å¿equeretur Å¿i AriÅ¿totelis opinionem .8. cap. lib. 4. phyÅ¿icorum re-<lb/>ciperemus.</s> - </p> - </div> - <div type="section" level="3" n="13"> - <head xml:id="head267" style="it" xml:space="preserve">Longe aliter ueritatem Å¿e habere quam Aristoteles <lb/>doceat in fine libri Å¿eptimi phyÅ¿icorum.</head> - <head xml:id="head268" xml:space="preserve">CAP. XIII.</head> - <p> - <s xml:id="s2092" xml:space="preserve">NOn tam facile eÅ¿t aſſignare proportionem velocitatum duorum corporum na <lb/>turalium, quam AriÅ¿toteles vltimo cap. lib. 7. phyÅ¿icorum putauit.</s> - </p> - <p> - <s xml:id="s2093" xml:space="preserve">Quamobrem Å¿int duo corpora <var>.B.</var> et <var>.D.</var> materia <reg norm="magnitudineque" type="simple">magnitudineqÌ;</reg> diuerÅ¿a, pondere <lb/>tamen, & figura Å¿imilia, & proportio reÅ¿iÅ¿tentiarum, quas recipiunt à medio <reg norm="dum" type="context">dÅ©</reg> mo-<lb/>uentur, Å¿it. ut <var>.o.i.</var> ad <var>.a.e.</var> denotentur deinde velocitates totales abÅ¿que vlla reÅ¿iÅ¿ten-<lb/>tia ab <var>.a.u.</var> et <var>.o.c.</var> quæ æquales erunt ad inuicem per communem Å¿cientiam ex Å¿up-<lb/>poÅ¿ito, Å¿int alia deinde duo corpora <var>.V.</var> et <var>.M.</var> eodem modo Å¿e habentia ut prima <var>.B.</var> <lb/>et <var>.D.</var> in eodem medio, Å¿ed ex diuerÅ¿a materia ab ea, quæ eÅ¿t illorum duorum corpo <lb/>rum, magnitudine tamen & figura ijÅ¿dem Å¿imilia: </s> - <s xml:id="s2094" xml:space="preserve">Å¿ignificentur quoque eo-<lb/>rundem reÅ¿iÅ¿tentiæ per <var>.t.s.</var> et <var>.n.r.</var> & eorundem velocitates à nulla ex reÅ¿iÅ¿tentijs di-<lb/>minutæ, per <var>.n.x.</var> et <var>.t.g.</var> vnde <var>.n.r.</var> æqualis erit <var>.a.e.</var> et <var>.t.s.</var> ipÅ¿i <var>.o.i.</var> et <var>.n.x.</var> ipÅ¿i <var>.t.g</var>: <var>n.x.</var> ta-<lb/>men et <var>.t.g.</var> non erunt Ä™qualia <var>.a.u.</var> et <var>.o.c</var>. </s> - <s xml:id="s2095" xml:space="preserve">Sed exempli gratia, ponamus ea eſſe mi-<lb/>nora. </s> - <s xml:id="s2096" xml:space="preserve">Supponamus nunc <var>.e.u.</var> velocitatem eſſe quæ remanet ipÅ¿i <var>.B.</var> cum applicata <lb/>erit reÅ¿iÅ¿tentia <var>.a.e.</var> dicto corpori <var>.B.</var> quæ diminutam facit totam <var>.a.u.</var> per <var>.a.e.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.i.c.</var> <lb/>ea, quÄ™ remanet ipÅ¿i <var>.o.c.</var> corporis <var>.D.</var> et <var>.r.x.</var> ea, quæ remanet <var>.n.x.</var> corporis <var>.V.</var> et <var>.s.g.</var> <lb/>ea, quæ eÅ¿t ex <var>.t.g.</var> corporis <var>.M</var>. </s> - <s xml:id="s2097" xml:space="preserve">Vnde communi omnium <reg norm="conÅ¿enÅ¿u" type="context">cõſenÅ¿u</reg> aſſequemur <var>.e.u.</var> ma <lb/>iorem futuram <var>.r.x.</var> et <var>.i.c.</var> ipÅ¿a <var>.s.g</var>. </s> - <s xml:id="s2098" xml:space="preserve">Scindatur deinde <var>.a.m.</var> ad Ä™qualitatem <var>.n.x.</var> et <var>.o.z.</var> <lb/>ipÅ¿ius <var>.t.g.</var> vnde <var>.a.m.</var> ad <var>.o.z.</var> et <var>.m.u.</var> ad <var>.z.c.</var> æquales habebimus, ut quoque <var>.e.m.</var> ad <var>.r.<lb/>x.</var> et <var>.i.z.</var> ad <var>.s.g.</var> quamobrem <var>.e.m.</var> maior erit ipÅ¿a <var>.z.i.</var> maior igitur erit proportio <var>.z.c.</var> <lb/>ad <var>.z.i.</var> quà m.m.u. ad <var>.m.e.</var> (quia <var>.z.c.</var> ad <var>.z.i.</var> ita Å¿e habet vt <var>.m.u.</var> ad <var>.i.z.</var> ex .7. lib. quin-<lb/>ti, Å¿ed <var>.m.u.</var> ad <var>.i.z.</var> maior eÅ¿t quam ad <var>.m.e.</var> ex .8. dicti lib. vnde ex .12. eiuÅ¿dem <var>.z.c.</var> ad <lb/>ad <var>.z.i.</var> maior erit, quà m.m.u. ad <var>.m.e</var>. </s> - <s xml:id="s2099" xml:space="preserve">Ergo ex .28. maior proportio erit <var>.c.i.</var> ad <var>.z.i.</var> - <pb o="177" rhead="DISPVTATIONES." n="189" file="0189" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0189"/> - quam <var>.u.</var> ad <var>.m.e.</var> & ex .27. maior erit proportio <var>.c.i.</var> ad <var>.u.e.</var> quam <var>.z.i.</var> ad <var>.e.m.</var> ideÅ¿t <var>.s.</var> g <lb/>ad <var>.r.x.</var> quod AriÅ¿toteli in mentem non venerat. </s> - <s xml:id="s2100" xml:space="preserve">Alijs quoque modis idem proba-<lb/>ri poteÅ¿t, vt Å¿i diceret aliquis, maiorem proportionem eſſe <var>.e.m.</var> ad <var>.m.u.</var> quam <var>.i.z.</var> ad <lb/><var>z.c.</var> (quia <var>.e.m.</var> ad <var>.m.u.</var> eadem eÅ¿t ratio vt ad <var>.z.c.</var> ex .7. quinti, Å¿ed proportio <var>.e.m.</var> ad <var>.<lb/>z.c.</var> maior eÅ¿t quam <var>.i.z.</var> ad <var>.z.c.</var> ex .8. eiuÅ¿dem, ergo ea, quæ eÅ¿t <var>.e.m.</var> ad <var>.m.u.</var> ex .12. ma <lb/>for erit, quam <var>.i.z.</var> ad <var>.z.c.</var>) vnde componendo, ea quæ eÅ¿t <var>.e.u.</var> ad <var>.m.u.</var> maior erit illa, <lb/>quæ eÅ¿t <var>.i.c.</var> ad <var>.z.c.</var> & <reg norm="permutando" type="context">permutãdo</reg>, quam ea, quæ eÅ¿t <var>.e.u.</var> ad <var>.i.c.</var> ea, quæ eÅ¿t <var>.m.u.</var> ad <var>.z.c.</var> <lb/>& ex .33. quinti, ea, quæ eÅ¿t <var>.e.m.</var> ad <var>.i.z.</var> maior erit ea, quæ eÅ¿t <var>.e.u.</var> ad <var>.i.c</var>.</s> - </p> - <figure position="here"> - <image file="0189-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0189-01"/> - </figure> - </div> - <div type="section" level="3" n="14"> - <head xml:id="head269" style="it" xml:space="preserve">Quid Å¿equatur ex Å¿upradistis.</head> - <head xml:id="head270" xml:space="preserve">CAP. XIIII.</head> - <p> - <s xml:id="s2101" xml:space="preserve">EX præcedenti capite manifeÅ¿tè depræhenditur, in vniuerÅ¿um AriÅ¿totelis opi-<lb/>nionem veram non eſſe in prima parte vltimi capitis. lib. 7. phyÅ¿icorum; </s> - <s xml:id="s2102" xml:space="preserve">quia <lb/>in eo loco Å¿upponens ipÅ¿e corpus <var>.B.</var> prÄ™cedentis capitis eſſe dimidiam partem ipÅ¿ius <lb/>D. quantum ad aream corpoream Å¿pectat (Å¿unt tamen pondere ad inuicem æqualia) <lb/>ait <var>.B.</var> futurum duplo velocius ipÅ¿o <var>.D</var>. </s> - <s xml:id="s2103" xml:space="preserve">Ego verò præcedenti capite accepi <var>.e.u.</var> pro <lb/>velocitate reÅ¿idua corporis <var>.B.</var> (Å¿ubtracta ea tamen parte, quam ei reÅ¿iÅ¿tentia adimit, <lb/>quæ erat <var>.e.a.</var>) et <var>.i.c.</var> pro ea, quæ eÅ¿t corporis <var>.D.</var> et <var>.r.x.</var> pro ea, quæ eÅ¿t corporis <var>.V.</var> et <var>.<lb/>s.g.</var> pro ea, quæ eÅ¿t corporis <var>.M</var>. </s> - <s xml:id="s2104" xml:space="preserve">Dicat nunc AriÅ¿toteles, quÄ™ nam harum duarum pro <lb/>portionum dupla erit? </s> - <s xml:id="s2105" xml:space="preserve">quia Å¿i earum aliqua talis erit, alia nullo modo eſſe poterit, <lb/>vt iam oÅ¿tendi, etiamÅ¿i duo corpora <var>.V.</var> et <var>.M.</var> eaÅ¿dem conditiones habeant, quas <var>.B.</var> <lb/>et <var>.D</var>. </s> - <s xml:id="s2106" xml:space="preserve">Ratio autem, quæ AriÅ¿totelem induxerit ad illud credendum, nulla alia eſſe <lb/>potuit, quà m quod putarit reÅ¿iÅ¿tentias proportionatas eſſe magnitudinibus corpo-<lb/>reis, ideÅ¿t quemadmodum <var>.B.</var> erat corporaliter dimidia pars ipÅ¿ius <var>.D.</var> Å¿ic etiam habe <lb/>ret medietatem eius reÅ¿iÅ¿tentiæ, quam habuiſſet corpus <var>.D</var>. </s> - <s xml:id="s2107" xml:space="preserve">Quod etÅ¿i verum eſſet, <lb/>non tamen Å¿equeretur neceſſariò in quibuÅ¿libet corporibus futuram velocitatum <lb/>proportionem eandem, quæ reÅ¿iÅ¿tentiarum eÅ¿t, vt Å¿uperiore capite oÅ¿tendimus.</s> - </p> - </div> - <div type="section" level="3" n="15"> - <head xml:id="head271" style="it" xml:space="preserve">Numrestè Å¿enÅ¿erit PhiloÅ¿ophus reÅ¿istentias proportionales <lb/>eße cum corporibus mobilibus.</head> - <head xml:id="head272" xml:space="preserve">CAP. XV.</head> - <p> - <s xml:id="s2108" xml:space="preserve">QVòd AriÅ¿toteles crediderit reÅ¿iÅ¿tentias proportionatas eſſe corporibus, erra-<lb/>uit. </s> - <s xml:id="s2109" xml:space="preserve">Si Å¿uperficies ijÅ¿dem proportionatæ eſſent, dubium non eÅ¿t, quin <lb/>reÅ¿iÅ¿tentiæ quoque ipſæ, ijÅ¿dem proportionatæ exiÅ¿terent, Å¿upponendo eas Å¿imiles <lb/>Å¿itu, dum eadem corpora mouerentur. </s> - <s xml:id="s2110" xml:space="preserve">Sed eadem proportio non eÅ¿t inter Å¿uperfi- - <pb o="178" rhead="IO. BAPT. BENED." n="190" file="0190" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0190"/> - cies, & quæ inter corpor a <reg norm="reperitur" type="simple">reperit̃</reg>: </s> - <s xml:id="s2111" xml:space="preserve">AriÅ¿toteles igitur in eo defecit. </s> - <s xml:id="s2112" xml:space="preserve">Quòd <reg norm="autem" type="context">autẽ</reg> inter <lb/>Å¿uperficies non eadem Å¿it proportio, quæ inter corpora extat, Å¿i primo ad Å¿phÄ™ricas <lb/>mentem verterimus, intelligemus proportionem eam, quæ inter duas Å¿phæras repe <lb/>ritur triplam Å¿emper exiÅ¿tere ei, quæ eÅ¿t inter ipÅ¿arum diametros ex vltima .12. libr. <lb/>Euclid. </s> - <s xml:id="s2113" xml:space="preserve">EÅ¿t autem proportio, quæ eÅ¿t inter Å¿uperficies Å¿phÄ™ricas Ä™qualis ei, quæ eÅ¿t <lb/>ipÅ¿orum circulorum maiorum ex .16. lib. quinti, cum ex .31. primi de Å¿phæra & cy-<lb/>lindro Archimedis, omnis Å¿phærica Å¿uperficies quadrupla, Å¿it maiori circulo ipÅ¿ius <lb/>Å¿phÄ™ræ, Å¿ed proportio, quæ eÅ¿t inter dictos circulos, eÅ¿t dupla ei, quæ eÅ¿t inter <reg norm="eorun- dem" type="context context">eorÅ©-<lb/>dẽ</reg> diametros ex .2. lib. 12. Euc. </s> - <s xml:id="s2114" xml:space="preserve">ergo <reg norm="proportio" type="simple">ê“portio</reg>, quæ eÅ¿t inter corpora, Å¿eÅ¿quialtera erit <lb/>ei, quæ eÅ¿t Å¿uperficierum, & non æqualis, ut AriÅ¿toteles putauit. </s> - <s xml:id="s2115" xml:space="preserve">Idem de corporibus <lb/>Å¿imilibus à ſuperficiebus planis terminatis dico, ratiocinando mediante .36. lib. 11. <lb/>et .18. Å¿exti, vnde cognoÅ¿cemus proportionem corporum, proportioni laterum, tri-<lb/>plam futuram, & Å¿uperficierum proportionem, laterum proportioni duplam. </s> - <s xml:id="s2116" xml:space="preserve">Quare <lb/>corporum proportio, ei, quæ Å¿uperficierum eÅ¿t, Å¿eÅ¿quialtera erit, ita ut Å¿i velocitates <lb/>extitiſſent ad inuicem proportionatæ, vt Å¿uperficies, proportio velocitatis corporis <var>.<lb/>B.</var> ei, quæ eÅ¿t corporis <var>.C.</var> fuiſſet Å¿ubÅ¿eÅ¿quialtera proportioni corporum, & non æqua <lb/>lis eidem.</s> - </p> - </div> - <div type="section" level="3" n="16"> - <head xml:id="head273" style="it" xml:space="preserve">FdipÅ¿um aliter demonÅ¿tr atur.</head> - <head xml:id="head274" xml:space="preserve">CAP. XVI.</head> - <p> - <s xml:id="s2117" xml:space="preserve">ALio quoque modo probari poteÅ¿t non eſſe in vniuerÅ¿um verum id, quod Ari-<lb/>Å¿toteles in prima parte capitis vltimi lib. 7. phyÅ¿icorum ait, Å¿ic Å¿cribens.</s> - </p> - <p> - <s xml:id="s2118" xml:space="preserve">Si <var>.A.</var> quidem Å¿it id quod mouet <var>.B.</var> verò id quod mouetur, et <var>.C.</var> Å¿it longitudo per <lb/>quam, et <var>.D.</var> tempus in quo eÅ¿t motum, in tempore nimirum Ä™quali, potentia æqua-<lb/>lis <var>.A.</var> dimidium ipÅ¿ius <var>.B.</var> per duplum mouebit ipÅ¿ius <var>.C.</var> per ipÅ¿um autem <var>.C.</var> in dimi <lb/>dio temporis <var>.D.</var> Å¿ic enim erit rationis Å¿imilitudo.</s> - </p> - <p> - <s xml:id="s2119" xml:space="preserve">Sit ergo corpus <var>.o.</var> Å¿eptimi capitis pondere æquali corpori <var>.u.</var> eiuÅ¿dem capitis, Å¿ed <lb/>area corporea minusipÅ¿o <var>.u.</var> pro medietate. </s> - <s xml:id="s2120" xml:space="preserve">Simile tamen figura. </s> - <s xml:id="s2121" xml:space="preserve">Imaginemur <reg norm="nunc" type="context">nÅ©c</reg> <lb/>tertium aliud corpus omogeneum ipÅ¿i <var>.u.</var> quod Å¿it <var>.i.</var> magnitudine & figura Å¿imile ipÅ¿i <lb/>o. vnde minor erit ipÅ¿o <var>.u.</var> pro media parte, & hanc ob cauÅ¿am ipÅ¿um <var>.u.</var> erit duplo ma <lb/>gis graue, quà m ipÅ¿um <var>.i.</var> & per conÅ¿equens ipÅ¿um quoque <var>.o.</var> duplo grauius erit <reg norm="quam" type="context">quã</reg> <lb/>Å¿it ipÅ¿um <var>.i.</var> ex .7. libr. quinti Euclidis. </s> - <s xml:id="s2122" xml:space="preserve">IpÅ¿um ergo corpus <var>.o.</var> duplo velocius erit, <lb/>quà m ipÅ¿um <var>.i.</var> ex primo Å¿uppoÅ¿ito cap .2. huius lib. </s> - <s xml:id="s2123" xml:space="preserve">Vnde ex .9. quinti, velocitas ipÅ¿ius <lb/>i. æqualis eſſet ei, quæ eÅ¿t ipÅ¿ius u. cum AriÅ¿toteles Å¿cribat <var>.o.</var> quoque futurum duplo <lb/>velocius ipÅ¿o <var>.u.</var> <reg norm="quod" type="simple">ê™</reg> cap .7. huius lib. falÅ¿um eſſe demonÅ¿traui.</s> - </p> - </div> - <div type="section" level="3" n="17"> - <head xml:id="head275" style="it" xml:space="preserve">De alio Aristo. lapÅ¿u.</head> - <head xml:id="head276" xml:space="preserve">CAP. XVII.</head> - <p> - <s xml:id="s2124" xml:space="preserve">SCribit AriÅ¿toteles in ultimo cap. lib. 7. phyÅ¿icorum in hunc modum. <lb/></s> - <s xml:id="s2125" xml:space="preserve">Si duo quædam Å¿eorÅ¿um per tantum Å¿patium tanto tempore duo Å¿eorÅ¿um pon <lb/>dera mouent, & compoÅ¿ita per longitudinem æqualem, <reg norm="Ä™qualiuem" type="context">Ä™qualiuẽ</reg> in tempore, com-<lb/>poÅ¿itum ex ponderibus <reg norm="vtriÅ¿que" type="simple">vtriÅ¿q;</reg> mouebunt, eÅ¿t enim in eis eadem ratio.</s> - </p> - <pb o="179" rhead="DISPVTATIONES." n="191" file="0191" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0191"/> - <p> - <s xml:id="s2126" xml:space="preserve">Quod in vniuerÅ¿um nec etiam poteÅ¿t eſſe verum in pleno, quia cap .14. iam pro-<lb/>baui, non eandem proportionem eſſe inter Å¿uperÅ¿icies corporum, & ipÅ¿a corpora.</s> - </p> - </div> - <div type="section" level="3" n="18"> - <head xml:id="head277" style="it" xml:space="preserve">Quomodo dignoÅ¿catur proportio uelocitatis duorum Å¿imilium <lb/>corporum omogeniorum inaqualium.</head> - <head xml:id="head278" xml:space="preserve">CAP. XVIII.</head> - <p> - <s xml:id="s2127" xml:space="preserve">ETiam Å¿i reperire in qua proportione motus naturaliter moueantur duo corpo-<lb/>ra, figura & materia Å¿imilia, inÄ™qualia tamen ad inuicem, non facile Å¿it, oÅ¿ten-<lb/>dam tamen qua ratione id conÅ¿equi poſſimus.</s> - </p> - <p> - <s xml:id="s2128" xml:space="preserve">Proponantur nobis, exempli gratia, duo corpora <var>.a.</var> et <var>.o.</var> Å¿phÄ™rica, inÄ™qualia inui-<lb/>cem, omogenea tamen materia, quorum <var>.a.</var> maius Å¿it; </s> - <s xml:id="s2129" xml:space="preserve">Å¿i voluerimus inuenire in qua <lb/>nam velocitatis proportione naturaliter mouerentur. </s> - <s xml:id="s2130" xml:space="preserve">Volo vt inquiratur corpus <var>.i.</var> <lb/>Å¿phÄ™ricum, alia tamen & diuerÅ¿a materia conÅ¿tans, Å¿ed pondere Ä™quale corpori <var>.o.</var> & <lb/>Å¿uperÅ¿icie tam proportionata Å¿uperficiei corp oris <var>.a.</var> quà m eÅ¿t ea, quæ eÅ¿t Å¿ui ponde-<lb/>ris ad pondus ipÅ¿ius <var>.a</var>. </s> - <s xml:id="s2131" xml:space="preserve">Hoc facto, indagetur, quænam erit proportio inter Å¿u-<lb/>perficies corporum <var>.i.</var> et <var>.o.</var> quÄ™ Å¿emper dupla eÅ¿t, vel Å¿ubdupla ei quæ eÅ¿t diametro-<lb/>rum; </s> - <s xml:id="s2132" xml:space="preserve">ut iam cap .15. dixi, & hÄ™c proportio Å¿uperficierum Å¿phÄ™ricarum <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <var>.o.</var> et <var>.i.</var> Å¿ub <lb/>trahatur ab æqualitate, quod igitur remanebit, erit proportio <reg norm="velocitatum" type="context">velocitatÅ©</reg> inter duo <lb/>corpora <var>.o.</var> et <var>.i.</var> ideÅ¿t inter <var>.o.</var> et <var>.a.</var> vt exempli gratia, Å¿i proportio Å¿uperficiei <var>.o.</var> Å¿uperfi <lb/>ciei ipÅ¿ius <var>.i.</var> Å¿eÅ¿quitertiα<unsure/> eſſet, Å¿ub <lb/>trahendo eam ab Ä™qualitate, rema-<lb/> - <anchor type="figure" xlink:label="fig-0191-01a" xlink:href="fig-0191-01"/> - neret <reg norm="proportio" type="simple">ê“portio</reg> Å¿ubÅ¿eÅ¿quitertia, vnde <lb/>velocitas corporis maioris ( quod in <lb/>pręſenti loco Å¿upponitur eſſe <var>.o.</var>) ei, <lb/>quÄ™ eÅ¿t corporis minoris, quale eÅ¿t <lb/>corpus <var>.i.</var> Å¿ubÅ¿eÅ¿quitertia eſſet; </s> - <s xml:id="s2133" xml:space="preserve">aut <lb/>dicamus quòd <var>.i.</var> eſſet velocius ipÅ¿o <lb/>o. in proportione Å¿eÅ¿quitertia ex Å¿e <lb/>cundo Å¿uppoÅ¿ito Å¿ecundi capitis huius libri. </s> - <s xml:id="s2134" xml:space="preserve">Sed <var>.i.</var> tam velox eÅ¿t quam ipÅ¿um <var>.a.</var> ex <num value="11">.<lb/>11.</num> cap. ergo proportio velocitatis ipÅ¿ius <var>.a.</var> Å¿eÅ¿quitertia erit ei. quæ eÅ¿t ipÅ¿ius <var>.o</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0191-01" xlink:href="fig-0191-01a"> - <image file="0191-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0191-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="19"> - <head xml:id="head279" style="it" xml:space="preserve">Quam Å¿it inanis ab AriÅ¿totele Å¿uÅ¿cepta demonÅ¿tratio quod <lb/>uacuum non detur.</head> - <head xml:id="head280" xml:space="preserve">CAP. XIX.</head> - <p> - <s xml:id="s2135" xml:space="preserve">EX ijs, quæ Å¿uperius <reg norm="demonÅ¿trauimus" type="context">demõſtrauimus</reg> facilè cognoÅ¿ci poteÅ¿t irritam eſſc eam ratio <lb/>nem, quam AriÅ¿toteles .8. cap. lib. 4. phyÅ¿icorum ad deÅ¿truendum vacuum, <reg norm="con" type="context">cõ</reg> <lb/>finxit. </s> - <s xml:id="s2136" xml:space="preserve">Vtigitur idem facilius oÅ¿tendamus, compræhendamus imaginatione infini-<lb/>ta media corporea, quorum vnum altero rarius Å¿it, in qua placuerit nobis ex propor <lb/>tionibus, incipiendo ab uno, imaginemur etiam corpus <var>.Q.</var> denÅ¿ius primo medio, cu-<lb/>ius corporis, totalis grauitas Å¿it <var>.a.b.</var> & poÅ¿itum in ipÅ¿o medio, amittat partem <var>.e.b.</var> ip-<lb/>Å¿ius grauitatis, & in Å¿ecundo medio amittat <var>.i.b.</var> & Å¿ic per gradus vnde nobis patebie<unsure/> - <pb o="180" rhead="IO. BAPT. BENED." n="192" file="0192" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0192"/> - dicto corpori <var>.Q</var>. </s> - <s xml:id="s2137" xml:space="preserve">Nunquam remanÅ¿uram Å¿uam totalem grauitatem <var>.a.b.</var> in quolibet <lb/>ex-dictis medijs. </s> - <s xml:id="s2138" xml:space="preserve">Nunc Å¿i quærat à me AriÅ¿toteles proportionem velocitatis corpo-<lb/>ris <var>.Q.</var> per vacuum ad velocitatem dicti corporis per plenum, ego ei proponam pro-<lb/>portionem ipÅ¿ius <var>.a.b.</var> ad <var>.a.e.</var> exempli gratia, dicens, <reg norm="quod" type="simple">ê™</reg> <reg norm="quemadmodum" type="wordlist">quẽadmodum</reg> <var>.a.b.</var> maius eÅ¿t <lb/>ip Å¿o <var>.a.e.</var> Å¿ic etiam corpus <var>.Q.</var> velocius erit in vacuo, quà m in pleno, dicti autem ple-<lb/>ni denÅ¿itatem appellabimus <var>.e.b</var>. </s> - <s xml:id="s2139" xml:space="preserve">AriÅ¿toteles dicet nunc, <reg norm="quod" type="simple">ê™</reg> aliud quoddam medium <lb/>in eadem proportione Å¿ubtilius ipÅ¿o <var>.e.b.</var> deÅ¿umatur; </s> - <s xml:id="s2140" xml:space="preserve">quemadmodum <var>.a.e.</var> minus eÅ¿t <lb/>ipÅ¿o <var>.a.b.</var> Å¿it ergo iÅ¿tud <var>.i.b.</var> in quo AriÅ¿toteles credit corpus Q. futurum tam velox ut <lb/>in vacuo, in quo aberrat, <reg norm="quia" type="simple">ê—a</reg> proportio velocitatis corporis <var>.Q.</var> in medio <var>.i.b.</var> ad velo <lb/>citatem eiuÅ¿dem in medio <lb/><var>e.b.</var> ita Å¿e hà bebit, ut <var>.i.a.</var> ad <lb/> - <anchor type="figure" xlink:label="fig-0192-01a" xlink:href="fig-0192-01"/> - <var>e.a.</var> ex ultimo Å¿uppoÅ¿ito ca <lb/>pit .2. huius libr. quæ minor <lb/>eſſet ea, quæ eÅ¿t <var>.a.b.</var> ad <var>.a.e.</var> ex .8. lib. quinti Eucli.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0192-01" xlink:href="fig-0192-01a"> - <image file="0192-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0192-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="20"> - <head xml:id="head281" style="it" xml:space="preserve">Non Å¿atis dilucidè AriÅ¿totelem de loco ratiocinatum fuiße.</head> - <head xml:id="head282" xml:space="preserve">CAP. XX.</head> - <p> - <s xml:id="s2141" xml:space="preserve">QVæ AriÅ¿toteles de loco Å¿cribit multas in Å¿e continent difficultates. </s> - <s xml:id="s2142" xml:space="preserve">Primum, <lb/>cap .4. lib. 4. phyÅ¿icorum ait, omne corpus eſſe in Å¿uo proprio loco, Å¿upponen <lb/>do vnum centrum pro loco grauium, et unam circunferentiam pro loco leuium cor <lb/>porum. </s> - <s xml:id="s2143" xml:space="preserve">Sed quomodo punctum poteÅ¿t eſſe locus ipÅ¿ius corporis, cum omni dimen <lb/>Å¿ione <reg norm="capacitateque" type="simple">capacitateqÌ;</reg> Å¿it denudatum? </s> - <s xml:id="s2144" xml:space="preserve">vnde Å¿i <reg norm="centrum" type="context">centrÅ©</reg> locus eſſet corporum grauium, om <lb/>nia dicta corpora grauia, extra proprium locum exiÅ¿terent, quia nullum ex iis eÅ¿t, <reg norm="quod" type="simple">ê™</reg> <lb/>Å¿it in centro. </s> - <s xml:id="s2145" xml:space="preserve">Adde quod neque hoc cum loci definitione ab ipÅ¿o poÅ¿ita conÅ¿entiret <lb/>cum ipÅ¿e dicat in eodem cap. locum eſſe Å¿uperÅ¿iciem quandam, & non interuallum, <lb/>licet huiuÅ¿modi definitio falÅ¿a appareat primo ex <reg norm="inconuenienti" type="context">incõuenienti</reg> falÅ¿o, quod ipÅ¿e hinc <lb/>Å¿equuturum dicit, ideÅ¿t, quod Å¿i locus interuallum eſſet, infinita loca exiÅ¿terent, quod <lb/>reuera nec ob hanc cauÅ¿am inconueniens exiÅ¿tit, quia eodem planè modo quo ali-<lb/>quod corpus poteÅ¿t eſſe infinita corpora, (quod ipÅ¿e diceret in potentia) Å¿ic etiam in <lb/>teruallum aliquod poſſet eſſe infinita interualla. </s> - <s xml:id="s2146" xml:space="preserve">Cum autem dicat Å¿uperficies cor-<lb/>poris ambientis eſſe locum eius corporis, quod continetur, cogitur dicere lineam, <lb/>quæ circundat Å¿uperficiem, Å¿uperficiei locum eſſe, & puncta ipÅ¿ius lineæ, quod reue <lb/>ra abÅ¿urdum eÅ¿t. </s> - <s xml:id="s2147" xml:space="preserve">Locus corporis eÅ¿t interuallum illud eadem magnitudine & figu-<lb/>ra, qua corpus ipÅ¿um prÄ™ditum eÅ¿t, quod Å¿i non eſſet, Å¿ed eſſet Å¿uperficies, quemad-<lb/>modum AriÅ¿toteles voluit, maximum inconueniens Å¿equeretur, Å¿cilicet æquales lo-<lb/>cos capere inÄ™qualia corpora, aut corpora æqualia, locos inÄ™quales occupare, quod <lb/>Å¿citu facillimum eÅ¿t, cum Theon Å¿uper PtolomÄ™i AlmageÅ¿tum iam probarit Å¿phæ-<lb/>ricam Å¿uperficiem maius interuallum corporeum continere, quà m aliam <reg norm="quanuis" type="context">quãuis</reg> Å¿u-<lb/>perficiem dictæ Å¿phÄ™ricæ æqualem, vnde poſſent facilè reperiri duo loci, quorum al-<lb/>ter millies altero maior eſſet, capaces tamen corporum æqualium, aut reperiri duo <lb/>corpora, quorum alterum millies maius eſſet altero, quÄ™ tamen corpora apta eſſent <lb/>ad occupandos locos Ä™quales, quamuis AriÅ¿toteles dicat, locum, neque maiorem ne <lb/>que minorem eſſe debere locato. </s> - <s xml:id="s2148" xml:space="preserve">Sed interualla corporea Ä™qualia à quauis figura <lb/>terminata, continebunt Å¿emper corpora Ä™qualia. </s> - <s xml:id="s2149" xml:space="preserve">Corporeum igitur interuallum eÅ¿t - <pb o="181" rhead="DISPVTATIONES." n="193" file="0193" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0193"/> - reuera locus corpori adÄ™quatus, cum corpus in interuallum Å¿uperÅ¿iciale non intret, <lb/>quam @is interuallum corporeum ingrediatur. </s> - <s xml:id="s2150" xml:space="preserve">Et hoc modo <reg norm="nullum" type="context">nullÅ©</reg> eÅ¿t corpus, quod <lb/>in m@ do aut extra mundum ( dicat autem AriÅ¿toteles quicquid voluerit ) locum <lb/>Å¿uum non habeat.</s> - </p> - </div> - <div type="section" level="3" n="21"> - <head xml:id="head283" style="it" xml:space="preserve">V<unsure/>trum bene Aristoteles Å¿enÅ¿erit de infinito.</head> - <head xml:id="head284" xml:space="preserve">CAP. XXI.</head> - <p> - <s xml:id="s2151" xml:space="preserve">TRactans AriÅ¿toteles in fine quinti cap. lib. 3. phyÅ¿icorum de infinito ait, impoÅ¿Â<lb/>Å¿ibile cum Å¿it inuenire locum infinitum, & omne corpus in loco cum Å¿it, impoÅ¿ <lb/>Å¿ibile quoque eſſe in rerum natura aliquod: </s> - <s xml:id="s2152" xml:space="preserve">infinitum corpus reperiri. </s> - <s xml:id="s2153" xml:space="preserve">Omittamus <lb/>quòd cum AriÅ¿toteles debuerit beneficio loci deÅ¿truere infinitum, ordine peruerÅ¿o <lb/>de infinito prius, quà m de loco diÅ¿putationem inÅ¿tituat; </s> - <s xml:id="s2154" xml:space="preserve">Å¿ed dicamus ipÅ¿um intelli-<lb/>gere de infinito corporeo, & cum probauerimus corporis locum eſſe corporeum in <lb/>teruallum, non autem Å¿uperficiem, neque opus Å¿it in definitione interualli mentio <lb/>nem aliquam facere terminorum, vnde ipÅ¿um infinitum eſſe poteÅ¿t, neque aliqua ra <lb/>tione de hac re dubitari poteÅ¿t; </s> - <s xml:id="s2155" xml:space="preserve">hoc modo nullum inconueniens Å¿equeretur, quòd <lb/>extra cÄ™lum reperiri poſſit corpus aliquod infinitum, quamuis, id ipÅ¿e nulla euiden-<lb/>ti ratione inductus perneget. </s> - <s xml:id="s2156" xml:space="preserve">SenÅ¿it quoque, abÅ¿que eo, <reg norm="quod" type="simple">ê™</reg> aliquam rationem propo <lb/>nat, aliquid extra cÅ“lum reperiri quemadmodum apparet ex fine cap .9. lib. primi <lb/>de cÅ“lo, cum etiam ait cap .8. lib. 8. phyÅ¿icorum, infinitas partes alicuius continui eÅ¿-<lb/>Å¿e Å¿olum in potentia, non item in actu, hoc non eÅ¿t illico concedendum, quia Å¿i omne <lb/>totum continuum, & re ipÅ¿a exiÅ¿tens, in actu eÅ¿t, omnis quoque eius pars erit in actu, <lb/>quia Å¿tultum eſſet credere, ea quæ actu Å¿unt, ex ijs, quæ potentia exiÅ¿tunt, componi. <lb/></s> - <s xml:id="s2157" xml:space="preserve">Neque etiam dicendum eÅ¿t continuationem earundem partium efficere, vt poten-<lb/>tia Å¿int ipſæ partes, & omni actu priuatæ; </s> - <s xml:id="s2158" xml:space="preserve">Sit exempli gratia linea recta <var>.a.u.</var> continua <lb/>quæ deinde diuidatur in puncto <var>.e.</var> per æqualia, dubium non eÅ¿t, quin ante <reg norm="diuiÅ¿ionem" type="context">diuiÅ¿ionẽ</reg>, <lb/>medietas <var>.a.e.</var> tam in actu (licet coniuncta cum alia <var>.e.u.</var>) reperiretur, quà m totum .2. <lb/>u. licet à ſenÅ¿u diÅ¿tincta non eſſet. </s> - <s xml:id="s2159" xml:space="preserve">Idem affirmo de medietate <var>.a.e.</var> ideÅ¿t de quarta <lb/>parte totius <var>.a.u.</var> & pariter de octaua, de milleÅ¿ima, & de quauis, ita vt eſſentia actua <lb/>lis infiniti hoc modo tutò concedi poſſit, <reg norm="cum" type="context">cÅ©</reg> ita Å¿it in natura. </s> - <s xml:id="s2160" xml:space="preserve">Sed peius etiam Å¿enÅ¿it <lb/>AriÅ¿toteles eodem loco capitis quinti lib. 3. phyÅ¿icorum, negando infinitum poſſe <lb/>connumerari inter quantitates, dicens vnam aliquam quantitatem intelligi vt cubi <lb/>tum, tricubitum, & cætera; </s> - <s xml:id="s2161" xml:space="preserve">vbi non conÅ¿iderat eadem etiam ratione intelligi poſſe <lb/>aliquam quantitatem <reg norm="infinitorum" type="context">infinitorÅ©</reg> cubitorum, & in quantitatis definitione nullam eÅ¿-<lb/>Å¿e neceſſitatem terminorum, vt exempli gratia in definitione numeri, non eÅ¿t neceÅ¿ <lb/>Å¿itas alicuius determinati numeri, quia multitudo, non minus infinita, quà m finita, <lb/>intelligi poteÅ¿t. </s> - <s xml:id="s2162" xml:space="preserve">Vbi poÅ¿teà cap .8. libr .4. phyÅ¿icorum ait nullam eſſe differentiam <lb/>inter infinitum, & vacuum, reuera nihil abÅ¿urdius hoc dicere fingereue poterat.</s> - </p> - <pb o="182" rhead="IO. BAPT. BENED." n="194" file="0194" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0194"/> - </div> - <div type="section" level="3" n="22"> - <head xml:id="head285" style="it" xml:space="preserve">Exagitatur ab AriÅ¿totele adductatemporis definitio.</head> - <head xml:id="head286" xml:space="preserve">CAP. XXII.</head> - <p> - <s xml:id="s2163" xml:space="preserve">CVM Å¿enÅ¿erit AriÅ¿toteles <reg norm="tempus" type="context">tẽpus</reg> abÅ¿que motu eſſe <reg norm="non" type="context">nõ</reg> poſſe, ea tamen ab <reg norm="inui- cem" type="context">inui-<lb/>cẽ</reg> Å¿eparans, <reg norm="volens" type="context">volẽs</reg> definire <reg norm="tempus" type="context">tẽpus</reg> ait, <reg norm="ipsum" type="context">ipsÅ©</reg> eſſe <reg norm="motus" type="simple">motê°</reg> <reg norm="menÅ¿uram" type="context">menÅ¿urã</reg> <reg norm="numerumque" type="context simple">numerÅ©qÌ;</reg>. </s> - <s xml:id="s2164" xml:space="preserve">Quæ <lb/>quidem definitio, natura Å¿ua non eÅ¿t bona, quia tempus, neque numerus eÅ¿t, neque <lb/><reg norm="etiam" type="context">etiã</reg> eÅ¿t <reg norm="menÅ¿ura" type="context">mẽſura</reg> motus <reg norm="per" type="simple">ê‘</reg> Å¿e, Å¿ed <reg norm="tantum" type="wordlist">tm̃</reg> <reg norm="per" type="simple">ê‘</reg> <reg norm="accidens" type="context">accidẽs</reg>, quia nihil eÅ¿t, <reg norm="quod" type="wordlist">qÄ‘</reg> numeret aut menÅ¿uret <lb/>aliud, quod non Å¿it <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> Å¿peciei <reg norm="cum" type="context">cÅ©</reg> illo quod <reg norm="menÅ¿uratur" type="context">mẽſuratur</reg>, aut numero <reg norm="circunÅ¿cribitur" type="simple">circunÅ¿cribit̃</reg>, <lb/>vt <reg norm="exempli" type="context">exẽpli</reg> gratia, nulla <reg norm="vnquam" type="context">vnquã</reg> Å¿uperficies <reg norm="per" type="simple">ê‘</reg> Å¿e numerabit aut <reg norm="menÅ¿urabit" type="context">mẽſurabit</reg> <reg norm="lineam" type="context">lineã</reg>, aut cor-<lb/>pus; </s> - <s xml:id="s2165" xml:space="preserve">neclinea Å¿uperficiem <reg norm="aliquam" type="context">aliquã</reg>, aut corpus: </s> - <s xml:id="s2166" xml:space="preserve">nec corpus <reg norm="lineam" type="context">lineã</reg> <reg norm="aliquam" type="context">aliquã</reg> aut Å¿uperfi-<lb/>ciem; </s> - <s xml:id="s2167" xml:space="preserve">Sed linea lineam menÅ¿urabit; </s> - <s xml:id="s2168" xml:space="preserve">Å¿uperficies Å¿uperficiem; </s> - <s xml:id="s2169" xml:space="preserve">& corpus corpus; </s> - <s xml:id="s2170" xml:space="preserve"><reg norm="etianÅ¿i" type="context">etiãſi</reg> <lb/>tam vna ex iis quantitatibus quà m altera Å¿it continua. </s> - <s xml:id="s2171" xml:space="preserve">Cum verò motus non Å¿it tem <lb/>pus, neque tempus Å¿it motus, Å¿ed inter Å¿e maximè differant, Å¿equetur ex iis, alterum <lb/>nullo modo per Å¿e eſſe menÅ¿uram alterius, niÅ¿i per accidens. </s> - <s xml:id="s2172" xml:space="preserve">Et Å¿i alicui videtur, <reg norm="quod" type="simple">ê™</reg> <lb/>ad Å¿ignificandam aliquam quantitatem motus, dicere huiuÅ¿modi operationem dua-<lb/>rum horarum, aut duorum dierum, aut duorum annorum Å¿patio completam eſſe, Å¿it <lb/>ponere tantum tempus: </s> - <s xml:id="s2173" xml:space="preserve">animaduertere debet hoc Å¿impliciter non eſſe verum, quia <lb/>horarum, dierum. </s> - <s xml:id="s2174" xml:space="preserve">& annorum interualla, imaginatione <reg norm="concipiuntur" type="context">concipiÅ©tur</reg> vt motus corpo-<lb/>rum cÄ™leÅ¿tium, Å¿ine quibus, neque anni, neque dies, neque horÄ™ exiÅ¿terent, <reg norm="etiam" type="context">etiã</reg> Å¿i om <lb/>nis motus Å¿it (vt ita dicam) locatus in tempore, ut corpus in loco, vnde motus motu, <lb/>& tempus tempore, non autem aliud ab alio menÅ¿uratur. </s> - <s xml:id="s2175" xml:space="preserve">Tempus ex neceſſita-<lb/>te (phyloÅ¿ophicè tamen loquendo) res eÅ¿t æterna, motus non item, quia diuerÅ¿is mo <lb/>dis terminari poteÅ¿t & ceſſare, & interim dum ceſſabit quieÅ¿cet corpus, quod <lb/>primo mouebatur. </s> - <s xml:id="s2176" xml:space="preserve">nihilominus tamen, tempus continuabit curÅ¿um Å¿uum. </s> - <s xml:id="s2177" xml:space="preserve">Tempus <lb/>igitur potius locus motus erit dicendum, quà m numerus aut menÅ¿ura eius, & tale eÅ¿t, <lb/>vt conÅ¿umatum uideatur à continuò quodam fluxu vnius inÅ¿tantis, quemadmodum <lb/>iam dixi in .38. capite meÄ™ gnomonicæ, & cum dico ab vno inÅ¿tanti, vnum in Å¿pecie, <lb/>& non in numero intelligo, quod à ſenÅ¿ibus noÅ¿tris percipi non poteÅ¿t, <reg norm="neque" type="simple">neq;</reg> etiam <lb/>notari, quia nouum Å¿emper inÅ¿tans nobis occurrit. </s> - <s xml:id="s2178" xml:space="preserve">& Å¿i aliquis aliquod <reg norm="exemplum" type="context">exemplÅ©</reg> (lar <lb/>go modo) incompræhenÅ¿ibilitatis ipÅ¿ius inÅ¿tantis deÅ¿ideraret, imaginetur rotam ali <lb/>quam albam, in qua Å¿it nigrum aliquod punctum Å¿enÅ¿ibile, aut è contra rotam <reg norm="nigram" type="context">nigrã</reg> <lb/>imaginetur, in qua Å¿it punctum album, quæ rota velociſſimè moueatur; </s> - <s xml:id="s2179" xml:space="preserve">huiuÅ¿modi <lb/>punctum, nullo modo aſſignari poterit, magis ab una parte quà m ab altera; </s> - <s xml:id="s2180" xml:space="preserve">immo Å¿e <lb/>Å¿e nobis offeret Å¿emper in forma lineæ circularis. </s> - <s xml:id="s2181" xml:space="preserve">poſſumus aliquo modo etiam Å¿u-<lb/>mere exemplum à ſono, quia omnis chorda cuiuÅ¿libet inÅ¿trumenti muÅ¿ici, dum Å¿o-<lb/>nus editur, tremit, unde huiuÅ¿modi Å¿onus, appellari poteÅ¿t aggregatum aliquod ex <lb/>innumerabilibus Å¿onis. </s> - <s xml:id="s2182" xml:space="preserve">eodem modo Å¿e habet Å¿onus, quem Ä™dunt campanÄ™, & omnia <lb/>inÅ¿trumenta tam naturalia, quà m artificialia, quæ quantò velocius <reg norm="tremunt" type="context">tremÅ©t</reg>, tanto acu <lb/><reg norm="tiorem" type="context">tiorẽ</reg> generant Å¿onum, & quantò tardius, tantò grauiorem. </s> - <s xml:id="s2183" xml:space="preserve">Neque eÅ¿t quòd in ad-<lb/>mirationem ducamur, quòd Å¿enÅ¿ui unum aliquod continuum appareat id, quod di-<lb/>Å¿cretorum eÅ¿t multitudo ( non putet tamen aliquis me negare continuitatem Å¿ucceÅ¿ <lb/>Å¿iuam ipÅ¿ius temporis) quod clare cognoÅ¿ci poteÅ¿t à niue, aut à chryÅ¿tallo, aut à vi-<lb/>tro, aut à ſaccaro in minutiſſimas partes redacto, quæ continuam aliquam <reg norm="albedinem" type="context">albedinẽ</reg> <lb/>nobis ad inÅ¿piciendum offerunt, quod nihil aliud eÅ¿t, quà m innumerabilis quædam <lb/>multitudo minutorum reflexorum. </s> - <s xml:id="s2184" xml:space="preserve"><reg norm="Idem" type="context">Idẽ</reg> dico de Å¿puto, & qualibet Å¿puma, & quan- - <pb o="183" rhead="DISPVTATIONES." n="195" file="0195" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0195"/> - to minutiora Å¿unt corpuÅ¿cula à quibus vt à ſpeculis reflectitur lumen, tantò magis ag <lb/><reg norm="gregatum" type="context">gregatÅ©</reg> illud <reg norm="album" type="context">albÅ©</reg> apparet. </s> - <s xml:id="s2185" xml:space="preserve">Hæc <reg norm="autem" type="context">autẽ</reg> exempla <reg norm="cum" type="context">cÅ©</reg> Å¿int, nec non largo modo Å¿umpta, <lb/><reg norm="mirum" type="context">mirÅ©</reg> non erit Å¿i claudicare <reg norm="videbuntur" type="context simple">videbÅ©t̃</reg>. </s> - <s xml:id="s2186" xml:space="preserve">Sed ut ad <reg norm="motum" type="context">motÅ©</reg>, & <reg norm="tempus" type="context">tẽpus</reg> reuertamur ( quæ Å¿unt <lb/><reg norm="continua" type="context">cõtinua</reg> Å¿ucceſſiua) AriÅ¿toteles in definiendo tempore, non reduxit in mentem, quod <lb/>Å¿cribit decimo metaphyÅ¿icÄ™ et .4. cap. Å¿ecundo. libr. de cÄ™lo, omnia videlicet, ab eo, <lb/>quod minimum eÅ¿t in Å¿uo genere, menÅ¿urari, & ex Å¿eipÅ¿o in phyÅ¿icorum libris, tem-<lb/>pus non eÅ¿t de genere motus; </s> - <s xml:id="s2187" xml:space="preserve">ergo eius ipÅ¿ius rationum ui, tempus non erit menÅ¿ura <lb/>motus, Å¿ed motus quidem poteÅ¿t menÅ¿urare motum, videlicet velocior minus velo-<lb/>cem, & breuior longiorem; </s> - <s xml:id="s2188" xml:space="preserve">& <reg norm="numerus" type="simple">numerê°</reg> <reg norm="menÅ¿uratur" type="simple">menÅ¿urat̃</reg> numero, & tempus tempore in quan <lb/>tum longum eÅ¿t, aut breue, non in quantum velox, aut tardum; </s> - <s xml:id="s2189" xml:space="preserve">Nullum autem in-<lb/>conueniens Å¿equetur Å¿umendo tempus tam ptoportionale motui, quam locus cor-<lb/>pori, quia motus decem milliarium, quæ aliquis vnius horæ Å¿patio conficiat, erit pro <lb/>portionalis corpori denÅ¿o, & motus vnius milliaris eadem hora peracti, proportiona <lb/>lis erit corpori raro; </s> - <s xml:id="s2190" xml:space="preserve">& quemadmodum corpus denÅ¿um occupat minus interuallum <lb/>loci, contra quam fiat in corpore raro: </s> - <s xml:id="s2191" xml:space="preserve">Å¿ic etiam motus velox breuiori temporis Å¿pa-<lb/>tio peragetur, quam tardus.</s> - </p> - </div> - <div type="section" level="3" n="23"> - <head xml:id="head287" style="it" xml:space="preserve">Motum rectum eſſe continuum, uel dißentiente <lb/>AriÅ¿totele.</head> - <head xml:id="head288" xml:space="preserve">CAP. XXIII.</head> - <p> - <s xml:id="s2192" xml:space="preserve">ARiÅ¿toteles .8. capi .8. phyÅ¿icorum ait impoſſibile eſſe aliquid per <reg norm="lineam" type="context">lineã</reg> rectam <lb/>nunc vno modo, nunc altero, ideÅ¿t eundo, & redeundo per dictam lineam in <lb/>extremis abÅ¿que quiete moueri. </s> - <s xml:id="s2193" xml:space="preserve">Id quod contrà poſſibile eſſe dico. </s> - <s xml:id="s2194" xml:space="preserve">Pro Å¿pecula-<lb/>tione cuius rei imaginemur circulum <var>.u.a.n.</var> motu continuo circa centrum <var>.o.</var> in <reg norm="quam" type="context">quã</reg> <lb/>libet partem, aut <reg norm="dextram" type="context">dextrã</reg>, aut <reg norm="Å¿iniÅ¿tram" type="context">Å¿iniÅ¿trã</reg> ferri; </s> - <s xml:id="s2195" xml:space="preserve">& imaginemur <reg norm="punctum" type="context">pÅ©ctum</reg> <var>.b.</var> extra ipÅ¿um, ubi <lb/>magis nobis videbitur, à quo ducantur duæ lineæ recte <var>.b.u.</var> et <var>.b.n.</var> contiguæ ipÅ¿i cir-<lb/>culo in punctis <var>.u.</var> et <var>.n</var>. </s> - <s xml:id="s2196" xml:space="preserve">Imaginatione quoque inter has duas lineas, alteram quæ Å¿it <var>.<lb/>u.n.</var> aut <var>.c.d.</var> aut <var>.e.f.</var> aut <var>.g.h.</var> conÅ¿tituamus in quali <lb/> - <anchor type="figure" xlink:label="fig-0195-01a" xlink:href="fig-0195-01"/> - bet parte, Å¿umemus etiam punctum <var>.a.</var> circun-<lb/>ferentiæ dicti circuli, à quo vÅ¿que ad <var>.b.</var> lineam <var>.<lb/>b.a.</var> imaginemur <reg norm="fixam" type="context">fixã</reg> in <var>.b.</var> Å¿ed quod remanat mo <lb/>bile, Å¿ecundum quod mouebitur punctum <var>.a.</var> vn-<lb/>de <reg norm="aliquando" type="context">aliquãdo</reg> hæc linea erit eadem cum <var>.b.u.</var> & ali <lb/>quando cum <var>.b.n.</var> & aliquando ab <var>.b.u.</var> verÅ¿us <var>.b.<lb/>n.</var> proficiÅ¿cetur, & aliquando ab <var>.b.n.</var> verÅ¿us <var>.b.u.</var> <lb/>vt accidit lineæ directionis, & retrogradationis <lb/>planetarum, vnde circulus <var>.u.a.n.</var> erit vt epiciclus <lb/>et <var>.b.</var> vt terræ centrum. </s> - <s xml:id="s2197" xml:space="preserve">Clarum nunc erit, quòd <lb/>quando linea <var>.b.a.</var> eadem erit cum <var>.b.u.</var> aut cum <lb/><var>b.n.</var> non quieÅ¿cet, quia in inÅ¿tanti reuertetur, quia <lb/><var>b.u.</var> et <var>.b.n.</var> in puncto, <reg norm="dictum" type="context">dictÅ©</reg> circulum tangunt, & <lb/>dicta <var>.b.a.</var> interÅ¿ecabit Å¿emper aliquam ex dictis <lb/><var>u.n.</var> aut <var>.c.d.</var> aut <var>.e.f.</var> aut <var>.g.h.</var> quod interÅ¿ectionis <lb/>punctum Å¿it <var>.t</var>. </s> - <s xml:id="s2198" xml:space="preserve">Imaginemur nunc quod <reg norm="Å¿ecundum" type="context context">Å¿ecÅ©dÅ©</reg> <lb/>punctum <var>.t.</var> aliquid per aliquam ex dictis lineis - <pb o="184" rhead="IO. BABPT. BENED." n="196" file="0196" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0196"/> - moueatur, clarum erit quod tale aliquid, nunquam quieÅ¿cet, etiam Å¿i Å¿it in quouis ex <lb/>tremo. </s> - <s xml:id="s2199" xml:space="preserve">AriÅ¿totelis igitur opinio, tuta non eÅ¿t.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0195-01" xlink:href="fig-0195-01a"> - <image file="0195-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0195-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="24"> - <head xml:id="head289" style="it" xml:space="preserve">Idem uir grauisÅ¿imus an bene Å¿enÅ¿erit de motibus corporum <lb/>uiolentis & natur alibus.</head> - <head xml:id="head290" xml:space="preserve">CAP. XXIIII.</head> - <p> - <s xml:id="s2200" xml:space="preserve">ARiÅ¿toteles in fine .8. phyÅ¿icorum Å¿entit corpus per vim motum, & Å¿eparatum à <lb/>primo mouente, moueri, aut motum eſſe per aliquod tempus ab aere, aut ab <lb/>aqua, quæ ipÅ¿um <reg norm="Å¿equuntur" type="context">Å¿equÅ©tur</reg>. quod fieri non poteÅ¿t; </s> - <s xml:id="s2201" xml:space="preserve">quia imo aer, qui in locum defer-<lb/>tum à corpore Å¿ubintrat ad fugandum vacuum, non Å¿olum hoc corpus non impellit, <lb/>Å¿ed potius id cohibet à motu, quia aer per vim à corpore ducitur retrò, & diuiÅ¿us à <lb/>parte anteriori à dicto corpore, reÅ¿iÅ¿tit Å¿imiliter, & quantum dictus aer in dicta parte <lb/>condenÅ¿atur, tantum in poÅ¿teriori rarefit, vnde per vim Å¿eÅ¿e rarefaciens non permit-<lb/>tit, vt dictum corpus cum ea velocitate fugiat, cum qua aufugeret, quia omne agens <lb/>in agendo patitur. </s> - <s xml:id="s2202" xml:space="preserve">Quamobrem cum aer à dicto corpore rapiatur, corpus quoque <lb/>ipÅ¿um ab aere rapitur. </s> - <s xml:id="s2203" xml:space="preserve">HuiuÅ¿modi autem rarefactio aeris, naturalis non eÅ¿t, Å¿ed vio <lb/>lenta; </s> - <s xml:id="s2204" xml:space="preserve">& hanc ob cauÅ¿am reÅ¿iÅ¿tit, & ad Å¿e trahit, Å¿ed non Å¿ufferente natura, vt inter <reg norm="vnum" type="context">vnÅ©</reg> <lb/>& aliud ex dictis corporibus reperiatur vacuum; </s> - <s xml:id="s2205" xml:space="preserve">iccirco funt hæc Å¿emper contigua, <lb/>& mobile corpus aerem deÅ¿erere cum nequeat, eius velocitas impeditur. </s> - <s xml:id="s2206" xml:space="preserve">HuiuÅ¿mo <lb/>di igitur corporis Å¿eparatim à primo mouente velo citas oritur à quadam naturali im <lb/>pręſſione, ex impetuofitate recepta à dicto mobili, quæ imprÄ™sÅ¿io & impetuoÅ¿itas, <lb/>in motibus rectis naturalibus continuò creÅ¿cit, cum perpetuò inÅ¿e cauÅ¿am <reg norm="mouentem" type="context context">mouẽtẽ</reg>, <lb/>ideÅ¿t propenÅ¿ionem eundi ad locum ei à natura asÅ¿ignatum habeat. </s> - <s xml:id="s2207" xml:space="preserve">AriÅ¿to .8. cap. <lb/>primi lib. de cÅ“lo, dicere non deberet <reg norm="qui" type="simple">ê—</reg> quantò propius accedit corpus ad <reg norm="terminum" type="context">terminÅ©</reg> <lb/>ad quem, tantò magis Å¿it velox; </s> - <s xml:id="s2208" xml:space="preserve">Å¿ed potius, <reg norm="qui" type="simple">ê—</reg> quantò longius diÅ¿tat à termino à quò <lb/>tantò velocius exiÅ¿tit. </s> - <s xml:id="s2209" xml:space="preserve">quia tantò maior fit femper impræsfio, quantò magis moue-<lb/>tur naturaliter corpus, & continuò nouum impetum recipit, cum in fe motus caufam <lb/>contineat, quæ eÅ¿t inclinatio ad locum Å¿uum eundi, extra quem per vim confiftit. <lb/></s> - <s xml:id="s2210" xml:space="preserve">Neque etiam rectè Å¿eripÅ¿it AriÅ¿to .9. cap. lib. 8. phyficorum et .2. lib. primi de cÅ“lo <lb/>eſſe aliquem motum ex recto & circulari mixtum, <reg norm="qui" type="simple">ê—</reg> omninò imposÅ¿ibile eÅ¿t.</s> - </p> - </div> - <div type="section" level="3" n="25"> - <head xml:id="head291" style="it" xml:space="preserve">Motum rectum & natur alem non eſſe primo & per Å¿e <lb/>quicquid AriÅ¿toteli uiÅ¿um Å¿it.</head> - <head xml:id="head292" xml:space="preserve">CAP. XXV.</head> - <p> - <s xml:id="s2211" xml:space="preserve">MOtus rectus corporum naturalium Å¿urÅ¿um, aut dcorfum, non eft naturalis pri <lb/>mò & per Å¿e, quia motus naturalis perpetuus eÅ¿t, aut vt melius dicam, inceÅ¿-<lb/>fabilis, & alius eſſe non poteft quà m circularis, <reg norm="nullaque" type="simple">nullaqÌ;</reg> pars cum fuo toto coniun-<lb/>cta, alium motum naturalem habere poteft, quà m eum, qui eft totius. </s> - <s xml:id="s2212" xml:space="preserve">fi autem à ſuo <lb/>toto diuulfa atque difiuncta fit, <reg norm="libereque" type="simple">libereqÌ;</reg> vagetur, Å¿pontè, & quà m breuisÅ¿ima poteft <lb/>via, ad locum, <reg norm="Å¿uitotius" type="simple">Å¿uitotiê°</reg> à natura Å¿tatutum proficiÅ¿citur. </s> - <s xml:id="s2213" xml:space="preserve">hic motus primò, & per fe di-<lb/>cti corporis, naturalis non eft, cum à caufa naturæ fuæ contraria fit generatus, ideft, - <pb o="185" rhead="DISPVTATIONES." n="197" file="0197" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0197"/> - ab co quod fit extra Å¿uum locum, vbi contra naturam Å¿uam reperitur. </s> - <s xml:id="s2214" xml:space="preserve">Vnde hu-<lb/>iuÅ¿modi motus, partim & non omninò, naturalis eft. </s> - <s xml:id="s2215" xml:space="preserve">Is autem proprius eÅ¿t & natura <lb/>lis motus, qui dicti corporis eſſentiam conÅ¿eruat. </s> - <s xml:id="s2216" xml:space="preserve">hoc autem non præſtat hic rectus, <lb/>cum deÅ¿truat, ergò hic motus primò & per Å¿e naturalis non eft.</s> - </p> - </div> - <div type="section" level="3" n="26"> - <head xml:id="head293" style="it" xml:space="preserve">Omne corpus eſſe in loco proprio graue, ut Aristoteli placuit, <lb/>non eft admittendum.</head> - <head xml:id="head294" xml:space="preserve">CAP. XXVI.</head> - <p> - <s xml:id="s2217" xml:space="preserve">ARift .4. cap. lib. 4. de cÄ™lo fic Å¿cribit.</s> - </p> - <p> - <s xml:id="s2218" xml:space="preserve">Suo enim in loco grauitatem habent omnia præter ignem, fignum cuius eft <lb/>vtrem inflatum plus ponderis, quam vacuum habere, & c.</s> - </p> - <p> - <s xml:id="s2219" xml:space="preserve">Quo in loco, manifeftè indicat Å¿e caufam nec grauitatis, nec leuitatis corporum <lb/>naturalium nofce, quæ eft denfitas auto raritas corporis grauis, aut leuis, maior denÅ¿i-<lb/>tate, aut raritate medij permeabilis, in quo reperitur.</s> - </p> - <p> - <s xml:id="s2220" xml:space="preserve">Exemplum <reg norm="qui" type="simple">ê—</reg> ipÅ¿e de vtre inflato proponit, debuiſſet Å¿altem ei oculos ad verita-<lb/>tem, quæ clarisÅ¿imè fulget, inÅ¿piciendum aperire. </s> - <s xml:id="s2221" xml:space="preserve">VerisÅ¿imum eÅ¿t, vtrem inflatum <lb/>plus ponderis habere quà m vacuum, aut quando aer in eo non eft per vim inclufus.</s> - </p> - <p> - <s xml:id="s2222" xml:space="preserve">Ratio autem huius rei eft, quia quando inflatus eft, ea quantitas aeris, in eum <lb/>per vim iniecti, minorem occupat locum, quà m Å¿i eidem liberè vagari permit-<lb/>teretur, vnde violenter, quodam modo, con denfata eft, & quia corpus denfum in <lb/>minus denfo, femper deÅ¿cendit, & minus denÅ¿um in magis denÅ¿o aÅ¿cendit. </s> - <s xml:id="s2223" xml:space="preserve">Hanc ob <lb/>caufam vter inflatus plenus corpore magis denÅ¿o, quà m eft medium quod eum cir-<lb/>cundat, deÅ¿cendit, non quia aer inaere, aut aqua in aqua fit grauis.</s> - </p> - </div> - <div type="section" level="3" n="27"> - <head xml:id="head295" style="it" xml:space="preserve">Haud admittendam opinionem Principis Peripateticorum <lb/>de circulo, & Å¿pbæra.</head> - <head xml:id="head296" xml:space="preserve">CAP. XXVII.</head> - <p> - <s xml:id="s2224" xml:space="preserve">CVm Ariftoteles fenÅ¿erit circulum eſſe figurarum Å¿uperficialium <reg norm="primam" type="context">primã</reg>, & <reg norm="Å¿phae ram" type="simple context">Å¿phÄ™Â<lb/>rã</reg> eſſe <reg norm="primam" type="context">primã</reg> <reg norm="corporearum" type="context">corporearÅ©</reg> <reg norm="prope" type="wordlist">ê‘ê‘</reg> <reg norm="earum" type="context">earÅ©</reg> periferias, decipitur. </s> - <s xml:id="s2225" xml:space="preserve">Sunt enim vltimæ, <lb/>non primæ. </s> - <s xml:id="s2226" xml:space="preserve">Sunt quidem (in quò rectè Å¿entit) perfectè, licet rationem huius rei non <lb/>nouerit. </s> - <s xml:id="s2227" xml:space="preserve">Nam <reg norm="centrum" type="context">centrÅ©</reg> cuiuÅ¿libet rei, eiuÅ¿dem rei <reg norm="principium" type="context">principiÅ©</reg> eft, & eÄ™ figurÄ™, quæ ipÅ¿um <lb/>æqualiter circundant, poſſunt appellari perÅ¿ectæ, Å¿iue Å¿int Å¿uperficiales, Å¿iue corpo-<lb/>reæ, & ècontrà illæ, quæ contrario modo Å¿e habent, imperfectæ. </s> - <s xml:id="s2228" xml:space="preserve">Quòd autem per-<lb/>Å¿ectum eÅ¿t, licet natura fit primum, eÅ¿t tamen vltimum generatione. </s> - <s xml:id="s2229" xml:space="preserve">Sed quando <lb/>Ariftoteles duas dictas figuras pronuntiauit primas, vt perfectas, prioritate Å¿cilicet <lb/>ea, quæ oritur à perfectione, verum dixit; </s> - <s xml:id="s2230" xml:space="preserve">fed quando de figuris Å¿uperficialibus lo-<lb/>quens, vult circulum effe primum, quia ab vna <reg norm="tantum" type="context">tãtum</reg> linea terminetur; </s> - <s xml:id="s2231" xml:space="preserve"><reg norm="non" type="context">nõ</reg> minus pro <lb/>circulo, quam pro oxigonia Å¿eu elipÅ¿i, aut cucurbitali, aut aliis multis figuris ab vna <lb/>tantum linea terminatis concludit. </s> - <s xml:id="s2232" xml:space="preserve">Neque etiam hæc ratio perfectionem circuli <reg norm="mon" type="context">mõ</reg> <lb/>Å¿trat, quia aliæ figuræ, à lineis curuis terminatÄ™, eandem conditionem fortiuntur. <lb/></s> - <s xml:id="s2233" xml:space="preserve">Circulus Å¿phÄ™<reg norm="raque" type="simple">raqÌ;</reg>, non ex vno Å¿olo angulo recto conÅ¿tant, vt idem Ariftoteles putat - <pb o="186" rhead="IO. BAPT. BENED." n="198" file="0198" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0198"/> - cap .4. lib. 4. de cÄ™lo, etiam fi triangulus ex duobus angulis rectis conÅ¿urgat, Å¿ed Å¿unt <lb/>figurÄ™ infinitorum angulorum rectorum, & hanc ob cauÅ¿am à me dicuntur vltimæ & <lb/>perfectÄ™, quia infinito nihil addi poteÅ¿t. </s> - <s xml:id="s2234" xml:space="preserve">Numerus angulorum rectorum circuli, eft <lb/>minor duplo infinito per duo infinita angulorum contingentiæ, quæ duo infinita mi <lb/>nora funt quouis angulo acuto rectilineo, & numerus angulorum rectorum <reg norm="folidorum" type="context">folidorÅ©</reg> <lb/>Å¿phÄ™ræ, minor eft quadruplo infinito per .4. infinita angulorum Å¿olidorum <reg norm="contingen- tiæ" type="context">cõtingen-<lb/>tiæ</reg>, quæ .4. infinita, minora Å¿unt quouis angulo Å¿olido acuto terminato à tribus pla-<lb/>nis. </s> - <s xml:id="s2235" xml:space="preserve">Triangulus inter figuras planas Å¿uperÅ¿iciales eft primus, & circulus vltimus; </s> - <s xml:id="s2236" xml:space="preserve">& <lb/>pyramis quadrilatera, inter corpora eft prima, & Å¿phÄ™ra vltima.</s> - </p> - </div> - <div type="section" level="3" n="28"> - <head xml:id="head297" style="it" xml:space="preserve">Occultam fuiße grauisÅ¿imo Stagirit & canÅ¿am Å¿cintilla-<lb/>tionis Å¿tellarum.</head> - <head xml:id="head298" xml:space="preserve">CAP. XXVIII.</head> - <p> - <s xml:id="s2237" xml:space="preserve">VBi AriÅ¿toteles ait Å¿cintillationem Å¿tellarum Å¿ieriratione aÅ¿pectus @oÅ¿tri ob, ma <lb/>ximam diÅ¿tantiam, maximum errorem committit, vt etiam facid quum putat <lb/>vifionem fieri extramittendo, contra id, quod alio loco, immo contra veritatem ip <lb/>Å¿am afferuit. </s> - <s xml:id="s2238" xml:space="preserve">Scintillatio ergo Å¿tellarum, neque aÅ¿pectus noÅ¿tri ratione, neque ali-<lb/>cuius mutationis earundem Å¿tellarum, Å¿ed ab inæqualitate motus corporum diapha <lb/>norum mediorum naÅ¿citur, <reg norm="quemadmodum" type="wordlist">quẽadmodum</reg> clarè cernitur, quòd fi inter aliquod obie <lb/>ctum, & nos, aliquis Å¿umus, qui aÅ¿cendat, intercefferit, videbimus obiectum illud qua <lb/>Å¿i tremere. </s> - <s xml:id="s2239" xml:space="preserve">Hoc autem tantò magis fiet, quantò magis diÅ¿tabit obiectum ab ipÅ¿o fu <lb/>mo; </s> - <s xml:id="s2240" xml:space="preserve">vnde admirationi locus non erit, fi ftellas fixas magis Å¿cintillare, quam errantes <lb/>cernamus. </s> - <s xml:id="s2241" xml:space="preserve">Lumen Å¿tellæ ad oculum noÅ¿trum accedens, perpetuò per diuerfas dia-<lb/>phaneitates penetrat, medio continuorum motuum corporum mediorum, vnde <lb/>continuò eorum lumen variatur, & hoc in <reg norm="longinquis" type="context">lõginquis</reg> magis, quà m in propinquis Å¿tel <lb/>lis apparet, quemadmodum ab exemplo de fumo allato, & etiam ab aliquibus vi-<lb/>tris ex Å¿uperficie non plana, Å¿ed irregulari conÅ¿tantibus, quilibet cognoÅ¿cere poteft.</s> - </p> - </div> - <div type="section" level="3" n="29"> - <head xml:id="head299" style="it" xml:space="preserve">Daricontinuum infinitum motum Å¿uper rectam at que <lb/>finitam lineam.</head> - <head xml:id="head300" xml:space="preserve">CAP. XXIX.</head> - <p> - <s xml:id="s2242" xml:space="preserve">OMnes hactenus Å¿enÅ¿erunt imposfibile eſſe dari per <reg norm="imaginationem" type="context">imaginationẽ</reg> motum con-<lb/>tinuum & <reg norm="perpetuum" type="context">perpetuÅ©</reg> <lb/> - <anchor type="figure" xlink:label="fig-0198-01a" xlink:href="fig-0198-01"/> - Å¿uper vnam lineam rectam <lb/>finit: </s> - <s xml:id="s2243" xml:space="preserve">in quo <reg norm="tantum" type="wordlist/context">tñ</reg> decipiuntur. <lb/></s> - <s xml:id="s2244" xml:space="preserve">Imaginemur <reg norm="ion" type="context">iõ</reg> duas lineas <lb/>parallelas <var>.a.b.</var> et <var>.t.x.</var> <reg norm="quarum" type="context">quarÅ©</reg> <lb/><var>b.a.</var> fit <reg norm="infinita" type="context">Ä©finita</reg> à qualibet par <lb/>te, & in ea imaginemur pun <lb/>ctum <var>.a.</var> moueri continuò ad <lb/>quam voluerimus partem, <lb/>& - <pb o="187" rhead="DISPVTATIONES." n="199" file="0199" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0199"/> - & in linea <var>.t.x.</var> imaginemur punctum fixu@, quod fit <var>.c.</var> imaginemur etiam inter <var>.c.</var> @. <lb/>a. vnam lineam rectam <var>.c.a.</var> & inter duas parallelas dictas <var>.r.x.</var> fixam, & motus punct@i <lb/>fit ab <var>.b.</var> verfus <var>.a.</var> ita ut <var>.c.a.</var> fecet <var>.r.x.</var> in puncto <var>.i.</var> quod interfectionis punctum mo-<lb/>uebitur ab <var>.r.</var> verfus <var>.x.</var> continuò, in tempore infinito, neque vnquam idem erit cum <lb/>puncto <var>.x</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0198-01" xlink:href="fig-0198-01a"> - <image file="0198-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0198-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="30"> - <head xml:id="head301" style="it" xml:space="preserve">Non eſſe Å¿olis calorem à motu localι ipÅ¿ius corporis Å¿olaris, <lb/>ut AriÅ¿toteli placuit.</head> - <head xml:id="head302" xml:space="preserve">CAP. XXX.</head> - <p> - <s xml:id="s2245" xml:space="preserve">JD nullo planè modo eÅ¿t admittendum quod Ariftoteles credidit calorem folis à <lb/>motu locali ipÅ¿iuÅ¿met corporis folaris, & non à lumine, prouenire, quemadmo-<lb/>dum manifeftè aſſerit primo metheororum cap .3. circa finem fic fcribens.</s> - </p> - <p> - <s xml:id="s2246" xml:space="preserve">Vtigitur repor gignatur atque calor, folis latio duntaxat, Å¿atis eÅ¿t eſſicere, & c. Å¿ed <lb/>cap .7. lib. 2. de cælo fic Å¿eribit, Caliditas autem ab ipÅ¿is, <reg norm="lumenque" type="simple">lumenqÌ;</reg> ideo fit, quia aer <lb/>ab illorum motione fricatur.</s> - </p> - <p> - <s xml:id="s2247" xml:space="preserve">Vbi non folum oftendit fe opinari, quòd motus corporum cÅ“leÅ¿tium fit caufa ca <lb/>loris, Å¿ed eriam luminis, paulò autem poÅ¿t dicit, Å¿uperiorum autem corporum vnum <lb/>quodque fertur in Å¿phæra, vt ipÅ¿a quidem non igniantur. </s> - <s xml:id="s2248" xml:space="preserve">Opinio profecto abfur-<lb/>da. </s> - <s xml:id="s2249" xml:space="preserve">Nam cùm corpus Å¿olate fixum fit in Å¿pisfitudine Å¿ui orbis deferentis, fe-<lb/>cundum communem opinionem, non mouetur per fe, Å¿ed accidentaliter, cum Å¿ei-<lb/>licet fertur à dicto Å¿uo orbe, vnde fieri poteÅ¿t, vt in motu fui orbis, nullum ex <lb/>orbibus fuorum deferentium augis fricet, fed fi fricaret, id faceret mediante vno fo <lb/>lo puncto, vt cuilibet, aliquantulum in mathematicis verfato patet. </s> - <s xml:id="s2250" xml:space="preserve">Quam ob cau <lb/>Å¿am, rationi <reg norm="confentaneum" type="context">cõfentaneum</reg> non eſſet credere, quòd tantùm caloris gigneretur. </s> - <s xml:id="s2251" xml:space="preserve">Quod <lb/>tamen fi posÅ¿ibile eſſet, quid ergo fricatio Å¿uperficierum orbis Å¿ui, cum iis, quæ funt <lb/>deferentium augis efficeret? </s> - <s xml:id="s2252" xml:space="preserve"><reg norm="Quando" type="context">Quãdo</reg> tamen hoc fieret, vt Å¿cilicet à fricatione fuper <lb/>ficierum procederet calor, nil planè diferiminis inter hyemen, & æftatem intercede <lb/>ret, nec inter calorem diei, & noctis, nec inter unam horam, aut alteram; </s> - <s xml:id="s2253" xml:space="preserve">fed fecun-<lb/>dum Ariftotelis rationes, Venus, <reg norm="Mercuriusque" type="simple">MercuriusqÌ;</reg>, magis calefacere quam fol de <reg norm="berent" type="context">berẽt</reg>, <lb/>cum ita Å¿int veloces vt ipÅ¿e Sol, & eodem magis propinqua terræ. </s> - <s xml:id="s2254" xml:space="preserve">Verum Ari-<lb/>Å¿totelis <reg norm="temporibus" type="context">tẽporibus</reg>, <reg norm="nullum" type="context">nullÅ©</reg> <reg norm="alium" type="context">aliÅ©</reg> <reg norm="planetam" type="context">planetã</reg> quam folem <reg norm="putabant" type="context">putabãt</reg> philofophi Å¿upra <reg norm="Lunam" type="context">Lunã</reg> eÅ¿-<lb/>Å¿e. </s> - <s xml:id="s2255" xml:space="preserve">Atque etiam <reg norm="contigeret" type="context">cõtigeret</reg> menfe Decembri, quam Iunio, magis inualeÅ¿ceret calor, <lb/>cum huiuÅ¿modi menÅ¿e Å¿olad nos propius accedat, quam menfe Iunii. per differen-<lb/>tiam maiorem diametro regionis elementaris, (nam folaris eccentricitas maior eft <lb/>Å¿emidiametro <reg norm="elementaris" type="context">elemẽtaris</reg> regionis) non confiderans Ariftoteles differentiam ca-<lb/>loris, quæ naÅ¿citur ex eo, <reg norm="qui" type="simple">ê—</reg> Sol aut altius Å¿upra orizontem excurrat, aut infra <reg norm="eundem" type="context">eundẽ</reg> <lb/>feratur; </s> - <s xml:id="s2256" xml:space="preserve">neque eam, quÄ™ à longitudine, aut breuitate diei proficiÅ¿citur. </s> - <s xml:id="s2257" xml:space="preserve">Sed quia Ari <lb/>Å¿toteles eodem cap tertio Metheororum intelligit de motu rapto, ideÅ¿t diurno, Å¿iue <lb/>dicamus vniuerfali, hinc Å¿equi deberet, <reg norm="qui" type="simple">ê—</reg> Sol maiorem caloris uim menÅ¿e Martij & <lb/>Septembris, quà m aliis menfibus, profunderet, quia in iiÅ¿dem temporibus, Å¿ol virtu <lb/>te huiuÅ¿modi motus velocior exiftat, quà m alio quolibet tempore anni, cum tunc <lb/>per æquatorem circuÅ©oluatur. </s> - <s xml:id="s2258" xml:space="preserve">Multa quoque alia incommoda Å¿equerentur Å¿i Ari <lb/>Å¿torelis rationes admitteremus. </s> - <s xml:id="s2259" xml:space="preserve">Sed clarè uidemus, mediante refl exione aut refra-<lb/>ctione radiorum folarium, <reg norm="qui" type="simple">ê—</reg> vniente Å¿eÅ¿e lumine, unitur quoque, & augetur calor, <lb/>atque omnis res ad comburendum apta accenditur, & inflammatur. </s> - <s xml:id="s2260" xml:space="preserve">In lumine igi- - <pb o="188" rhead="IO. BAPT BENED." n="200" file="0200" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0200"/> - tur continetur calor, & non in motu ipÅ¿ius Å¿olis, & ita in lumine Å¿edem habet, ut fi Å¿ol <lb/>quieÅ¿ceret, neque in orbe Å¿uo circumager etur, infÄ™liciſſima eſſet ea regio, in cuiu; <lb/></s> - <s xml:id="s2261" xml:space="preserve">Zenith ipÅ¿e reperiretur.</s> - </p> - </div> - <div type="section" level="3" n="31"> - <head xml:id="head303" style="it" xml:space="preserve">Vnde caloris Å¿olis prode at incrementum & state, et byeme <lb/>decrementum.</head> - <head xml:id="head304" xml:space="preserve">CAP. XXXI.</head> - <p> - <s xml:id="s2262" xml:space="preserve">CVm capite Å¿uperiore oÅ¿tenderim calorem Å¿olis non aliunde, quà m à lumine <lb/>prouenire, oÅ¿tendam nunc ex ordine, ex quot, <reg norm="quibusque" type="simple">quibusqÌ;</reg> cauÅ¿is oriatur magna <lb/>differentia eius caloris æſtatis ad hyemem, quarum nonnullæ ab antiquis obÅ¿erua-<lb/>tæ fuerunt, aliæ autem à nemine, quod Å¿ciam. </s> - <s xml:id="s2263" xml:space="preserve">Sunt autem quinque ad minus eæ cau <lb/>ſæ, quarum vna eft diuturna folis mora, tempore æſtatis Å¿upra orizontem, quæ cau-<lb/>Å¿a ab antiquis pofita, & citata fuit. </s> - <s xml:id="s2264" xml:space="preserve">Aliam quoque huius rei cauÅ¿am iidem antiqui <lb/>dicebant eſſe propinquitatem Å¿olis noftro Zenith, Å¿ed hæc cauÅ¿a immediata non <lb/>eÅ¿t, quia ab ea tres caufæ immediatæ naÅ¿cuntur; </s> - <s xml:id="s2265" xml:space="preserve">quarum vna eft maior unio radij re <lb/>flexi cum radio incidenti; </s> - <s xml:id="s2266" xml:space="preserve">Å¿ecunda maior quantitas luminis in Å¿uperficie terrÄ™; <lb/></s> - <s xml:id="s2267" xml:space="preserve">tertia, minor <reg norm="reÅ¿iÅ¿tentia" type="context">reÅ¿iÅ¿tẽtia</reg> vaporum ipÅ¿i radio luminoÅ¿o facta; </s> - <s xml:id="s2268" xml:space="preserve">quarta verò eft impresfio <lb/>caloris facta in terra, quæ cum aliis caufis coniuncta calorem adauget. quæ quidem <lb/>caufæ nemini adhuc, quod fciam, in <reg norm="mentem" type="context">mentẽ</reg> venerunt. </s> - <s xml:id="s2269" xml:space="preserve">Quòd autem attinet ad ma-<lb/>iorem coniunctionem radii reÅ¿lexi cum incidente, quiÅ¿que, uel Å¿altem mediocriter <lb/>in cathoptricÄ™ cognitione verÅ¿atus hoc verum eſſe cognoÅ¿cet. </s> - <s xml:id="s2270" xml:space="preserve">Vt hoc tamen in-<lb/>noteÅ¿cat facilius. </s> - <s xml:id="s2271" xml:space="preserve">Imaginemur <var>.q.p.</var> et <var>.b.d.</var> eſſe duas particulas Ä™quales Å¿uperficiei <lb/>ipfius terræ, Å¿uper quas cadant duo radii luminofi Å¿olis <var>.e.q.</var> et <var>.n.d.</var> quorum <var>.e.q.</var> fit ad <lb/>modum obliquus, et <var>.n.d.</var> quaÅ¿i perpendicularis, vnde radii reÅ¿lexi <var>.p.a.</var> et <var>.b.u.</var> aÅ¿cen <lb/>dent cum angulis æqualibus eis, qui funt Å¿uorum cadentium, cum omnis angulus re-<lb/>flexionis femper æqualis Å¿it angulo Å¿uæ incidentiÄ™, vt cuilibet in cathoptrica, vel me <lb/>diocriter verfato pater. </s> - <s xml:id="s2272" xml:space="preserve">Mixtio autem primorum obliquorum erit <var>.q.o.p.</var> & ea, quæ <lb/>eft minus obliquorum <var>.b.i.d.</var> quorum duorum triangulorum nullus unquam erit, qui <lb/>dubitare posfit <var>.q.o.p.</var> non eſſe minorem <var>.b.i.d.</var> cum anguli <var>.q.</var> et <var>.p.</var> trianguli <var>.q.o.p.</var> a-<lb/>cutiores Å¿int angulis <var>.b.</var> et <var>.d.</var> trianguli <var>.b.i.d.</var> ex Å¿uppoÅ¿ito. </s> - <s xml:id="s2273" xml:space="preserve">Quòd uero attinet ad ma <lb/>iorem quantitatem luminis Å¿uper terræ Å¿uperÅ¿iciem; </s> - <s xml:id="s2274" xml:space="preserve">Imaginemur radium <var>.a.q.</var> cuius <lb/>reÅ¿pectu etiam imagine mur duos Å¿uperficiei terræ Å¿itus, quorum vnus fit <var>.q.o</var>: cui di-<lb/>ctus radius fit perpendicularis, & alter <var>.q.p.</var> cui radius <var>.a.q.</var> ex obliquo incidat. </s> - <s xml:id="s2275" xml:space="preserve">Ima-<lb/>ginemur ergo triangulum <var>.q.o.p.</var> cuius angulus <var>.o.</var> rectus eÅ¿t ex Å¿uppoÅ¿ito, unde <var>.q.o.</var> <lb/>minor erit <var>.q.p.</var> ex .18. primi Euclidis. </s> - <s xml:id="s2276" xml:space="preserve">hinc fit, vt Å¿uper <var>.q.o.</var> cadat vniuerÅ¿um lumen, <lb/>quod Å¿uper <var>.q.p.</var> diffunditur. </s> - <s xml:id="s2277" xml:space="preserve">Sit <var>.q.u.</var> æqualis <var>.q.o.</var> & fit imaginatione protracta <var>.u.n.</var> <lb/>æquidiftans <var>.p.o.a.</var> vnde <var>.q.u.</var> illuminata erit à radio <var>.n.q.</var> minore radio <var>.a.q.</var> ergo mi-<lb/>nus calida erit Å¿uperficies <var>.q.u.</var> ipÅ¿ius terræ, quà m.q.o. quia maius lumen in Å¿e maio-<lb/>rem calorem includit: </s> - <s xml:id="s2278" xml:space="preserve">quod manifeÅ¿tè apparet in radiorum vnione mediante refle-<lb/>xione, aut refractione. </s> - <s xml:id="s2279" xml:space="preserve">Sed quod attinet ad minorem refiÅ¿tentiam vaporum ad ip-<lb/>Å¿um radium luminoÅ¿um, etfi primo capite meæ Gnomonicæ leuiter id attigerim, ni <lb/>hilominus tamen, & idem ipÅ¿um hoc loco proponam. </s> - <s xml:id="s2280" xml:space="preserve">Denotetur, exempli gratia, <lb/>Å¿uperficies terræ ab <var>.o.g.</var> et ea, quæ eft vaporum ab <var>.n.a.</var> Å¿upponatur etiam Å¿ol in fitu.</s> - <pb o="189" rhead="DISPVTATIONES." n="201" file="0201" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0201"/> - <s xml:id="s2281" xml:space="preserve">q. qui Å¿it Zenith <reg norm="puncti" type="context">pÅ©cti</reg> <var>.o.</var> & <reg norm="etiam" type="context">etiã</reg> in <var>.p.</var> ipÅ¿i <reg norm="orizonti" type="context">orizõti</reg> propinquus, aut extra Zenith, cuius <lb/>duos radios <var>.q.o.</var> et <var>.p.o</var>. </s> - <s xml:id="s2282" xml:space="preserve">Imaginemur, quorum duæ partes <var>.a.o.</var> et <var>.n.o.</var> erunt aliquo <lb/>modo ab ipÅ¿is vaporibus offuÅ¿catæ, Å¿ed <var>.o.n.</var> breuior eÅ¿t <var>.o.a.</var> ex .7. lib. 3. Eucli. </s> - <s xml:id="s2283" xml:space="preserve">mino-<lb/>rem ergo reÅ¿iÅ¿tentiam habebit à vaporibus Å¿ol in Zenith, quà m extra cundem com -<lb/>morans, & quantò longius erit idem ab ipÅ¿o Zenith, tanto maiorem reÅ¿iÅ¿tentiam à <lb/>dictis vaporibus inferri ex eadem .7. lib. 3. Eucli. dicemus.</s> - </p> - <figure position="here"> - <image file="0201-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0201-01"/> - </figure> - <figure position="here"> - <image file="0201-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0201-02"/> - </figure> - </div> - <div type="section" level="3" n="32"> - <head xml:id="head305" style="it" xml:space="preserve">Nullum corpus Å¿enÅ¿us expers à ſono offendi, præterquam <lb/>Aristoteles crediderit.</head> - <head xml:id="head306" xml:space="preserve">CAP. XXXII.</head> - <p> - <s xml:id="s2284" xml:space="preserve">POſſe Å¿onum corpus aliquod, quod Å¿enÅ¿u Å¿it deÅ¿titutum, vt AriÅ¿toteles .9. cap. li-<lb/>br .2. de cælo putauit, offendere, eÅ¿t falÅ¿um.</s> - </p> - <p> - <s xml:id="s2285" xml:space="preserve">Corpus enim non niÅ¿i à corpore poteÅ¿t lædi, non ergo à ſono, cum Å¿onus corpus <lb/>non Å¿it. </s> - <s xml:id="s2286" xml:space="preserve">Sed aer, & ignis, cum è contra Å¿int corpora, hoc facilè præſtare poſſunt im-<lb/>plendo aliquem locum velociter ad excludendum vacuum; </s> - <s xml:id="s2287" xml:space="preserve">vnde generatur Å¿onus. <lb/></s> - <s xml:id="s2288" xml:space="preserve">Quod hucuÅ¿que à nemine animaduerÅ¿um fuiſſe comperio.</s> - </p> - <pb o="190" rhead="IO. BAPT. BENED." n="202" file="0202" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0202"/> - </div> - <div type="section" level="3" n="33"> - <head xml:id="head307" style="it" xml:space="preserve">Pytagoreorum opinionem de Å¿onitu corporum cælestium non <lb/>fuiſſe ab Aristotele Å¿ublatam.</head> - <head xml:id="head308" xml:space="preserve">CAP. XXXIII.</head> - <p> - <s xml:id="s2289" xml:space="preserve">SEnÅ¿erunt Pythagorici orbes cæleÅ¿tes dum circunuoluuntur, non autem corpora <lb/>Å¿tellarum Å¿olum, æd ere Å¿onu. </s> - <s xml:id="s2290" xml:space="preserve">Quibus dum AriÅ¿toteles contradicere cogitat, <lb/>maximè fauet. </s> - <s xml:id="s2291" xml:space="preserve">Eatamen opinio è phyloÅ¿ophorum Å¿cholis eÅ¿t explodenda, quia aut <lb/>orbes Å¿unt Å¿ibi ipÅ¿is contigui, aut inuicem diÅ¿tantes: </s> - <s xml:id="s2292" xml:space="preserve">Å¿i ab inuicem diÅ¿tant (quod <lb/>nemo adhuc conceſſit, quia hac ratione vacuum introduceretur) clarum eÅ¿t, quod <lb/>cum Å¿e minime tangant, Å¿onum edere nequeunt: </s> - <s xml:id="s2293" xml:space="preserve">Si verò eorum vnus alteri Å¿it conti <lb/>guus, <reg norm="neque" type="simple">neq;</reg> etiam ab ipÅ¿is Å¿onus reÅ¿ultare poterit, quia pro certo putandum eÅ¿t, <reg norm="ipÅ¿orum" type="context">ipÅ¿orÅ©</reg> <lb/>Å¿uperficies tam politas eſſe, ac lenas, vt nihil omnino aÅ¿peritatis, aut inæqualitatis <lb/>contineant. </s> - <s xml:id="s2294" xml:space="preserve">Vt exempli gratia, Å¿i aliquis duo Å¿pecula plana inuicem confricaret, nul <lb/>lum planè Å¿onum audiret, Å¿ed Å¿i hoc faceret cum duabus Å¿uperficiebus a Å¿peris, <reg norm="Å¿onum" type="context">Å¿onÅ©</reg> <lb/>perÅ¿entiret, & tanto maiorem; </s> - <s xml:id="s2295" xml:space="preserve">quantò aÅ¿periores eſſent dictæ Å¿uperficies, & qui vult <lb/>vtarcus lirÄ™, ex corda Å¿onum eliciat, colophonia dictum arcum illinet, vt aÅ¿periorem <lb/>reddat. </s> - <s xml:id="s2296" xml:space="preserve">Neceſſarium quoque eÅ¿t vt tremat Å¿iue trepidet corpus, quod <reg norm="Å¿onum" type="context">Å¿onÅ©</reg> edere <lb/>debet; </s> - <s xml:id="s2297" xml:space="preserve">Neque etiam abÅ¿que aere Å¿onus efficipotelt, quia aer Å¿onat ingrediendo <lb/>velociter ad implendum locum, vt non remaneat vacuus. </s> - <s xml:id="s2298" xml:space="preserve">Sed Å¿upponendo in æche <lb/>rea regione neque aerem eſſe, neque corpus aliquod fluidum, clarè patebit orbes <lb/>cÅ“leÅ¿tes ex Å¿eÅ¿e nullum emittere Å¿onum. </s> - <s xml:id="s2299" xml:space="preserve">Idem affirmo de fricatione Å¿uperficiei con <lb/>cauæ infimi orbis lunaris cum conuexa materiæ à dicto orbe contentæ, Å¿uperioribus <lb/>rationibus fultus, vt etiam experientia à corpore aliquo fluido, quod in alio velociÅ¿ <lb/>Å¿imè moueretur deÅ¿umpta fretus, cuius corporis Å¿uperficies tamen lenis eſſet, à quo <lb/>Å¿onus non gigneretur. </s> - <s xml:id="s2300" xml:space="preserve">Et non minus dicere poſſum, corpus fluidum moueri in con-<lb/>tinente loco immobili, quam dictum corpus continens illud eſſe, quod moueretur, <lb/>& non fluidum corpus. </s> - <s xml:id="s2301" xml:space="preserve">Cuius rei poſſumus etiam exemplum habere à quouis <lb/>corpore perfectè rotundo, quod circa Å¿uum axem velociſſimè moueatur, nullum Å¿o-<lb/>num efficiet, quia nullam aeris partem extra Å¿uum <reg norm="locum" type="context">locÅ©</reg> impellit dum mouetur non <lb/>Å¿ecundum totum, Å¿ed Å¿ecundum Å¿uas partes, quarum quælibet abÅ¿que reÅ¿iÅ¿tentia im-<lb/>mediatè Å¿ubintrat locum alterius, abÅ¿que temporis interpoÅ¿itione. </s> - <s xml:id="s2302" xml:space="preserve">nec huiuÅ¿modi <lb/>locum aliquo modo eadem materia dicti corporis, quod circunuoluitur: </s> - <s xml:id="s2303" xml:space="preserve">deÅ¿titutum <lb/>dimittat. </s> - <s xml:id="s2304" xml:space="preserve">Sed Å¿i Pythagorici de alia quadam harmoniæ Å¿pecie ab ea, quæ eÅ¿t Å¿ono-<lb/>rum, vt à diuerÅ¿is velocitatibus motuum, aut à diuerÅ¿is magnitudinibus aut diÅ¿tantiis, <lb/>aut Å¿tellarum influxibus intellexiſſent, rectè Å¿enÅ¿iſſent exparte, non autem omnino, <lb/>quia ea harmoniam efficere nequeunt, quæ ad <reg norm="inuicem" type="context">inuicẽ</reg> Å¿ecundum interualla harmoni-<lb/>ca proportionata non Å¿unt, vt Å¿unt dupla, Å¿eÅ¿quial tera, Å¿eÅ¿quitertia, Å¿eÅ¿quiquarta, Å¿eÅ¿-<lb/>quiquinta, Å¿upertripartientia quintas, <reg norm="Å¿uperbipartientia" type="context">Å¿uperbipartiẽtia</reg> tertias, & quÄ™ ab ijs <reg norm="dependent" type="context">dependẽt</reg> <lb/>ideÅ¿t coniuncta Å¿unt cum duplis; </s> - <s xml:id="s2305" xml:space="preserve">de conÅ¿onantijs loquendo. de diſſonantiis idem di <lb/>co, quæ harmonicis inÅ¿eruiunt modulationibus, vt <reg norm="Å¿eÅ¿quioctauum" type="context">Å¿eÅ¿quioctauÅ©</reg>, <reg norm="Å¿eÅ¿quinonum" type="context">Å¿eÅ¿quinonÅ©</reg>, Å¿eÅ¿qui <lb/><reg norm="quintundecimum" type="context context">quintÅ©decimÅ©</reg>, <reg norm="Å¿equiuigeÅ¿imunquartum" type="context context">Å¿equiuigeÅ¿imÅ©quartÅ©</reg>, <reg norm="Å¿eÅ¿quioctuogeÅ¿imum" type="context">Å¿eÅ¿quioctuogeÅ¿imÅ©</reg>, & Å¿uperbipartiens vigeÅ¿i <lb/>masquintas. </s> - <s xml:id="s2306" xml:space="preserve"><reg norm="Verum" type="context">VerÅ©</reg> quidem eÅ¿t nonnulla harmonica interualla in aÅ¿pectibus <reg norm="comperta" type="context">cõperta</reg> <lb/>fuiſſe, vt Prolomeus oÅ¿tendit, & alii quoque aſſerunt. </s> - <s xml:id="s2307" xml:space="preserve">ineÅ¿t tamen huic rei nonnihil <lb/>difficultatis. </s> - <s xml:id="s2308" xml:space="preserve">vt exempli gratia, Å¿i Å¿ubtrahamus diateſſaron extra diapaÅ¿on, remanet <lb/>diapente, & Å¿i à diapente Å¿ubtrahamus Å¿emiditonum, remanet ditonum (quæ duæ - <pb o="191" rhead="DISPVTATIONES." n="203" file="0203" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0203"/> - conÅ¿onantiæ, eum habent reÅ¿pectum ad inuicem, quem habent diateſſaron, & dia <lb/>pente, quia quemadmodum Å¿emiditonum, & ditonum Å¿imul coniuncta, compo <lb/>nunt diapente, Å¿ic diateſſaron, & diapente Å¿imul vnita componunt diapaÅ¿on; </s> - <s xml:id="s2309" xml:space="preserve">& <reg norm="quem" type="context">quẽ</reg> <lb/>admodum terminus, qui diuidit diapaÅ¿on in <reg norm="diateſſaron" type="context">diateſſarõ</reg>, & diapente, eÅ¿t mediator ha@ <lb/>monicus inter extrema diapaÅ¿on diuiÅ¿i, Å¿ic etiam terminus, qui diuidit <reg norm="diapente" type="context">diapẽte</reg> in Å¿e-<lb/>miditonum, & ditonum, mediator eÅ¿t harmonicus inter extrema ipÅ¿ius diapente diui <lb/>Å¿i) Å¿ubtrahendo deinde à diapaÅ¿on Å¿emiditonum remanet exachordum maius, & ab <lb/>eodem diapaÅ¿on Å¿ubtrahendo ditonum remanet exachordum minus, quÄ™ <reg norm="quidem" type="context">quidẽ</reg> <reg norm="non" type="context">nõ</reg> <lb/>accidunt aÅ¿pectuum circulo, quia Å¿ubtrahendo aÅ¿pectum quadratum ab oppoÅ¿ito, <lb/>remanet aliud quadratum, & Å¿ubtrahendo Å¿extilem à trino remanet quoque alius <lb/>Å¿extilis. </s> - <s xml:id="s2310" xml:space="preserve">Quòd autem attinet ad motus, ad magnitudines, ad diÅ¿tantias, & ad influ-<lb/>xus, nihil eÅ¿t, quod hiÅ¿ce proportionibus conueniat, Å¿ed quia hæc omnia <reg norm="dependent" type="context">depẽdent</reg> <lb/>ab <reg norm="infinita" type="context">Ä©finita</reg>, & diuina <reg norm="prouidentia" type="simple context">ê“uidẽtia</reg> Dei, neceſſariò fit vt iÅ¿tæ velocitates, eæ magnitudines, <lb/>diÅ¿tantiæ, & influxus, talem ordinem, & reÅ¿pectum inter Å¿eipÅ¿a, & vniuerÅ¿um <reg norm="habeant" type="context">habeãt</reg>, <lb/>qualis perfectiſſimus Å¿it.</s> - </p> - </div> - <div type="section" level="3" n="34"> - <head xml:id="head309" style="it" xml:space="preserve">Deraro et denÅ¿o nonnulla, minus diligenter à Peripateticis <lb/>perpenÅ¿a.</head> - <head xml:id="head310" xml:space="preserve">CAP. XXXIIII.</head> - <p> - <s xml:id="s2311" xml:space="preserve">ANtiqui Peripatetici de videndo in hyeme animalium halitu. </s> - <s xml:id="s2312" xml:space="preserve">Id, quod in æſta <lb/>te non euenit, malè diÅ¿putauerunt, quia hoc naÅ¿citurà condenÅ¿atione hali <lb/>tus, quÄ™ ab ambiente frigore fit. </s> - <s xml:id="s2313" xml:space="preserve">quia halitus is abore, aut naÅ¿o animalis <reg norm="exiens" type="context">exiẽs</reg> <lb/>non eÅ¿t purus aer attractus primò, Å¿ed mixtus eÅ¿t cum quodam vapore excrementi-<lb/>tio, & Å¿ubtili, quo Å¿emper ab ea parte <reg norm="euacuatur" type="simple">euacuat̃</reg> corpus, qui Å¿tatim ab aere frigido cir-<lb/>cundatur, & denÅ¿atur, quam ob cauÅ¿am ab ipÅ¿o ea luminis pars reflectitur, quæ eum <lb/>penetrare non poteÅ¿t, quod in hypocauÅ¿tis, <reg norm="huiuÅ¿modique" type="simple">huiuÅ¿modiqÌ;</reg> calidis locis non fit. </s> - <s xml:id="s2314" xml:space="preserve">Idem <lb/>exemplo ab aqua Å¿tatim à ciÅ¿ternis, aut profundis puteis in hyeme extracta compro <lb/>bari poteÅ¿t, quia tunc temporis, huiuÅ¿modi aqua, cum magis calida Å¿it, quà m fri-<lb/>gida, emittit vaporem, qui facillimè videtur, ob rationem iam dictam, quod <lb/>in æſtate non cernitur in aqua, etÅ¿i ea magis calida eſſet, quam ea, quæ in hyeme <lb/>hauritur.</s> - </p> - <p> - <s xml:id="s2315" xml:space="preserve">Ratio autem, quam ab antiperiÅ¿taÅ¿i deÅ¿umptam citarunt iidem ad inquirendum, <lb/>cur aqua Å¿ubterranea magis calida, aut minus frigida, hyberno tempore, quà m ea, <lb/>quæ eÅ¿t Å¿upra terram Å¿it, vana eÅ¿t, quia hoc non aliunde fit, quà m ab eo, <reg norm="quod" type="simple">ê™</reg> terræ por-<lb/>ri à frigoris Å¿iccitate Å¿int clauÅ¿i, vnde vapores & exalationes non tam facilè exire poÅ¿ <lb/>Å¿unt. </s> - <s xml:id="s2316" xml:space="preserve">quamobrem calefiunt Å¿ubterraneæ partes. </s> - <s xml:id="s2317" xml:space="preserve">Fimum, fÅ“num, frumentum hac in <lb/>re Å¿unt nobis exemplo, in quibus ſępiſſimè viÅ¿um eÅ¿t ignem accendi.</s> - </p> - <p> - <s xml:id="s2318" xml:space="preserve">Priore illa quoque ratione de antiperiÅ¿taÅ¿i dicta, volunt philoÅ¿ophi maiorem ca-<lb/>liditatem hyÄ™me, quà m ęſtate in animalium Å¿tomacho contineri, non animaduerten <lb/>tes Å¿iccitatem, frigiditatis partes Å¿uperficiales corporis, <reg norm="reÅ¿tringentem" type="context">reÅ¿tringẽtem</reg>, Å¿anguinem ver <lb/>Å¿us originem Å¿uam impellere, qui in eo loco copioÅ¿ior cum Å¿it, eas partes tunc tem-<lb/>poris calefacit magis.</s> - </p> - <p> - <s xml:id="s2319" xml:space="preserve">Neque etiam ijdem nouerunt cauÅ¿am, vnde fiat, ut in æſtate impleto vaÅ¿e vitreo, <lb/>aut argenteo, aut ex materia non poroÅ¿a conÅ¿tante, aqua frigida, vas Å¿udet, quod - <pb o="192" rhead="IO. BAPT. BENED." n="204" file="0204" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0204"/> - tempore hyemis, non niÅ¿i in calidis locis euenit, quem Å¿udorem, dicebantipÅ¿i, eſſe <lb/>eandem aquam, quæ per porros vaÅ¿is exiret, quod falÅ¿iſſimum eÅ¿t, quia Å¿i per porros <lb/>aqua frigida exiret multò magis exiret calida, cum Å¿ubtilior Å¿it, & ad penetrandum <lb/>aptior. </s> - <s xml:id="s2320" xml:space="preserve">Sed hoc non aliunde oritur, quà m à condenÅ¿atione aeris vas circundantis, <lb/>cauÅ¿ata à frigiditate vaÅ¿is refrigerati ab aqua, quemadmodum tempore hyberno <lb/>clarè videmus mane Å¿uperficies interiores vitri feneÅ¿trarum Å¿udare, quia <reg norm="extrinÅ¿ecum" type="context">extrinÅ¿ecÅ©</reg> <lb/>frigus refrigerando vitrum, intrinÅ¿ecum aerem Å¿ibi contiguum congelat.</s> - </p> - <p> - <s xml:id="s2321" xml:space="preserve">Neque Å¿ilentio inuoluendum eÅ¿t, nec AriÅ¿totelem, neque alium ex Å¿uis fautoribus <lb/>animaduertiſſe denÅ¿um, & rarum eſſe cauÅ¿am ventorum. </s> - <s xml:id="s2322" xml:space="preserve"><reg norm="Rarum" type="context">RarÅ©</reg> autem & denÅ¿um, me <lb/>diante calore & frigore fit, & Å¿i à partibus, in omogeneis, licet <reg norm="argumentari" type="context">argumẽtari</reg>, de toto <lb/>deducat conÅ¿equentiam qui velit, obÅ¿eruans in calidis æſtatis diebus, dum aliqua nu <lb/>becula ad Solem cooperiendum incedit, ibi Å¿tatim agitationem aeris Å¿entiri; </s> - <s xml:id="s2323" xml:space="preserve">ea verò <lb/>nubecula prætergreſſa cum fuerit, & in ea parte, aer ad priÅ¿tinam raritatem cauÅ¿a-<lb/>tam à calore Solis redierit, quieÅ¿cit; </s> - <s xml:id="s2324" xml:space="preserve">huiuÅ¿modi autem aeris agitatio, à nulla certè ex <lb/>halatione proficiÅ¿citur, Å¿ed à motu Å¿olum locali, quem dum condenÅ¿atur, facit. </s> - <s xml:id="s2325" xml:space="preserve">Om <lb/>ne denÅ¿um natura Å¿ua frigidum eÅ¿t; </s> - <s xml:id="s2326" xml:space="preserve">omne rarum calidum, & è conuerÅ¿o. </s> - <s xml:id="s2327" xml:space="preserve">Et frigida <lb/>aura, quæ à flabellis cauÅ¿atur, non Å¿olum à nouo aere qui nos tangit, Å¿ed etiam à den-<lb/>Å¿o, quod in agitatione eiuÅ¿dem aeris fit, naÅ¿citur.</s> - </p> - <p> - <s xml:id="s2328" xml:space="preserve">Cum autem de raritate & denÅ¿itate diÅ¿putationem Å¿uÅ¿ceperim, non Å¿ine ratione <lb/>mihi <reg norm="videtur" type="simple">videt̃</reg> <reg norm="illorum" type="context">illorÅ©</reg> <reg norm="opinionem" type="context">opinionẽ</reg> <reg norm="explodendam" type="context context">explodẽdã</reg> eſſe, qui LunÄ™ maculas <reg norm="non" type="context">nõ</reg> aliud eſſe dixerunt, <lb/>quà m aliquas partes rariores aliis eiuÅ¿dem Lunæ partibus, non obÅ¿eruantes rarum, & <lb/>denÅ¿um, proportionabilia lumini, quod ab huiuÅ¿modi corporibus reflectitur, non eÅ¿Â<lb/>Å¿e. </s> - <s xml:id="s2329" xml:space="preserve">quia corpus aliquod rarum aliquando aptum erit ad reflectendum maius lumen, <lb/>quà m corpus minus rarum ut manifeÅ¿tè apparet à nubibus reflecti lumen: </s> - <s xml:id="s2330" xml:space="preserve">quod <lb/>ab aere non fit. </s> - <s xml:id="s2331" xml:space="preserve">Non defuerunt qui contrarium dixerunt, ideÅ¿t, eas LunÄ™ partes, den <lb/>Å¿iores eſſe; </s> - <s xml:id="s2332" xml:space="preserve">neque unquam aliquis fuit qui de diaphano, aut opaco mentionem fece <lb/>rit, quia melius eÅ¿t credere, eas partes diaphanas, Å¿iue perÅ¿picuas magis eſſe, quà m a-<lb/>lias, quÄ™ per aliquod Å¿patium, Å¿olis radio ingreſſum permittant, & aliÄ™ partes <reg norm="cum" type="context">cÅ©</reg> Å¿int <lb/>opacæ ipÅ¿um à ſuperficie reflectant. </s> - <s xml:id="s2333" xml:space="preserve">diuerÅ¿a tamen ratione à ſpeculo, cum in pleni-<lb/>lunio tota ferè LunÄ™ pars illuminata cernatur, quamuis dictum lumen extenÅ¿iuè & in <lb/>tenÅ¿iuè Å¿it minus eo, quod ipÅ¿a in nouilunio recipit. </s> - <s xml:id="s2334" xml:space="preserve">Indignum autem mihi videtur <lb/>ijs reÅ¿pondere, qui dixerunt huiuÅ¿modi maculas, terræ vmbras exiÅ¿tere, cum craſſiſſi-<lb/>mæ ignorantiæ tenebris Å¿int circunfuÅ¿i, vt <reg norm="etiam" type="context">etiã</reg> fuit Cornelius Agrippa, qui primo de <lb/>occulta philoÅ¿ophia dicens Å¿e noſſe modum <reg norm="quendam" type="context">quendã</reg> naturalem à Pythagora inuen-<lb/>tum, quo in Luna id totum, quod ipÅ¿e Å¿uper Å¿peculum Å¿cripÅ¿iſſet, videretur. </s> - <s xml:id="s2335" xml:space="preserve">oÅ¿tendit <lb/>manifeÅ¿tè Å¿e ignorare luminum <reg norm="vmbrarumque" type="simple">vmbrarumqÌ;</reg> naturam. </s> - <s xml:id="s2336" xml:space="preserve">quia nulla vnquam vmbra ge <lb/>nerari poteÅ¿t à corpore, quod aut opacum non Å¿it, aut officio opaci non fungatur, <lb/>vt nunc dicemus de diaphaneitate aquæ. </s> - <s xml:id="s2337" xml:space="preserve">Neque corpus opacum illuminatum <reg norm="adum- brare" type="context">adÅ©-<lb/>brare</reg> poteÅ¿t, niÅ¿i opacum illud in linea recta Å¿itum obtineat, quæ inter lucidum & il <lb/>luminatum extenditur. </s> - <s xml:id="s2338" xml:space="preserve">Neque etiam reÅ¿pondebimus ijs, qui Å¿entiunt quotieÅ¿cun-<lb/>que nulla eſſet terra, Å¿ed totus hic globus eſſet aqua, toties non futuram eclipÅ¿im lu-<lb/>narem, ratione diaphaneitatis aquæ. </s> - <s xml:id="s2339" xml:space="preserve">Quod <reg norm="falÅ¿iſſimum" type="context">falÅ¿iſſimÅ©</reg> eÅ¿t, quia omne corpus <reg norm="Å¿phae- ricum" type="simple">Å¿phÄ™-<lb/>ricum</reg> quantumuis diaphanum Å¿it, dummodo Å¿it denÅ¿ius aere, luminoÅ¿os radios re-<lb/>frangit, & eos ad inuicem interÅ¿ecare facit, qui deinde vltra interÅ¿ectionis <reg norm="punctum" type="context">punctÅ©</reg> di-<lb/>Å¿gregantur, ita vt amittant illuminationis actum. </s> - <s xml:id="s2340" xml:space="preserve">Adde <reg norm="quod" type="simple">ê™</reg> etÅ¿i huiuÅ¿modi corpus <lb/>aqueum, Å¿phÄ™ricum non eſſet, Å¿ed cubicum, illud Å¿uper <reg norm="aliquam" type="context">aliquã</reg> ex eius Å¿uperficiebus <lb/>ad angulos rectos radius Å¿olaris percuteret, non eum tamen penetraret, quia dictus <lb/>radius perpetuò debilitatur, & eò magis, quo maiorem profunditatem in diaphano - <pb o="193" rhead="DISPVTATIONES." n="205" file="0205" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0205"/> - eius corporis, quod Å¿it denÅ¿ius aere acquirit, nec totus radius vnquam dictum corpus <lb/>ingr editur, cum ab eius Å¿uperficie magna pars reflectatur. </s> - <s xml:id="s2341" xml:space="preserve">ReÅ¿iÅ¿tit ergo huiuÅ¿mo-<lb/>di corpus lumini, & quantò magis Å¿piſſum aut profundum exiÅ¿tit, tantò, validius reÅ¿i <lb/>Å¿tit. </s> - <s xml:id="s2342" xml:space="preserve">Habemus huius rei teÅ¿tes, piÅ¿catores vnionum, in ijs mundi partibus, quæ pau-<lb/>cis ab hinc annis HiÅ¿panorum opera nobis innotuerunt, qui aÅ¿firmanu ad maris <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> <lb/>fundum lumen Solis non peruenire.</s> - </p> - <p> - <s xml:id="s2343" xml:space="preserve">Immediata ratio, cur nebulÄ™ in ijs locis in quibus <reg norm="conÅ¿piciuntur" type="context">cõſpiciuntur</reg> permaneant, & <reg norm="uunc" type="context">uÅ©c</reg> <lb/>altiores, nunc vero depręſſiores cernantur, non ea eÅ¿t, quam AriÅ¿toteles cap .3. lib. 1. <lb/>metheororum proponit, Å¿ed inde oritur, quòd Å¿int eædem denÅ¿iores ea parte aeris, <lb/>quæ ipÅ¿is Å¿upereminet, & rariores e a, quæ ijÅ¿dem Å¿ubiacet. </s> - <s xml:id="s2344" xml:space="preserve">Quòd autem alicuius cor <lb/>poris denÅ¿itas maior ea, quæ eÅ¿t medij, in quo reperitur, cauÅ¿a Å¿it, vt ipÅ¿um corpus de <lb/>Å¿cendat, & maior raritas eiuÅ¿dem corporis, ea, quæ eÅ¿t quoque me dij, efficiat, vt di-<lb/>ctum corpus aÅ¿cendat, iam Archimedes in lib. de inÅ¿identibus aquæ docuit.</s> - </p> - <p> - <s xml:id="s2345" xml:space="preserve"><reg norm="Rectiſſimem" type="context">Rectiſſimẽ</reg> inÅ¿tituit natura, vt corpora denÅ¿iora verÅ¿us loca anguÅ¿tiora, & minora <lb/>(intelligendo ea loca orbicularis figuræ) quæ ad centrum propius accedunt, & rario <lb/>ra ad ampliora loca, & maius Å¿patium occupantia, Å¿eÅ¿e reciperent. </s> - <s xml:id="s2346" xml:space="preserve">tum quia eadem <lb/>quantitas materiæ condenÅ¿atæ, eget minori loco quam rarefacta, <reg norm="cum" type="context">cÅ©</reg> etiani, quia <reg norm="cum" type="context">cÅ©</reg> <lb/>corpus denÅ¿um non ita ad velocitatem motus localis, vt rarum, idoneum Å¿it, ad eas <lb/>partes accedat, quæ motibus tardioribus magis Å¿unt aptæ, corpora <reg norm="autem" type="context">autẽ</reg> rara ad eas, <lb/>quæ velocioribus motibus Å¿unt aptiores Å¿eÅ¿e transferant. </s> - <s xml:id="s2347" xml:space="preserve">præterquam <reg norm="quod" type="simple">ê™</reg> reuera ap-<lb/>pareat pro maiori parte, corpus magis denÅ¿um, minus diaphanum; </s> - <s xml:id="s2348" xml:space="preserve">aut magis <reg norm="opacum" type="context">opacÅ©</reg> <lb/>futurum, quà m rarum, licet ſæpiſſimè videamus contrarium, vt Å¿uperius innuimus. <lb/></s> - <s xml:id="s2349" xml:space="preserve">eÅ¿t tamen naturale <reg norm="proportionatumque" type="simple">proportionatumqÌ;</reg> magis opacum denÅ¿o, & diaphanum raro, <lb/>quà m è contra. </s> - <s xml:id="s2350" xml:space="preserve">Quamobrem Å¿umma ratione inducta natura voluit, vt corpora ma <lb/>gis opaca, aut minus diaphana, magis vicina centro colligantur, vt Å¿patium, quòd re <lb/>manet, abÅ¿que vllo impedimento à radijs Å¿olaribus penetrari poſſit. </s> - <s xml:id="s2351" xml:space="preserve">Tres autem eæ <lb/>cauſæ, quas hoc loco poÅ¿ui, propriæ Å¿unt, immediatæ, & per Å¿e, ex quibus fit, vt corpo <lb/>ra denÅ¿iora deÅ¿cendant, & rariora aÅ¿cendant in mediis minus denÅ¿is, aut minus raris <lb/>dictorum corporum, quæ à nemine, <reg norm="quod" type="simple">ê™</reg> Å¿ciam, hucuÅ¿que propoÅ¿itæ fuerunt.</s> - </p> - <p> - <s xml:id="s2352" xml:space="preserve">Qui autem aſſerunt cucurbitæ, quam apponunt chirurgi, effectum ex eo naÅ¿ci, <reg norm="quod" type="simple">ê™</reg> <lb/>calidi Å¿it attrahere, valdè aberrant à vero quia hoc, non niÅ¿i à raro, & à denÅ¿o imme-<lb/>diatè, à calido & frigido cauÅ¿atis efficitur, quia aer in cucurbita rarefactus à calore <lb/>& per conÅ¿equens dilatatus, Å¿tatim vt à dicto calore deÅ¿eritur, iterum condenÅ¿atur & <lb/>tantò citius, quantò aer ambiens frigidior exiÅ¿tet, & quia eadem materia cum con-<lb/>denÅ¿ata fuerit minorem Å¿emper occupat locum, reÅ¿tringens igitur Å¿eÅ¿e in cucurbi-<lb/>ta aer dum condenÅ¿atur, neceſſariò fit, ne ulla, Å¿cilicet pars vacua remaneat, <reg norm="quod" type="simple">ê™</reg> cum <lb/>alius aer ingredi cucurbitam nequeat aliud corpus ingrediatur. </s> - <s xml:id="s2353" xml:space="preserve">Idem cum amphora <lb/>in qua nullum aliud, quam aèreum Å¿it corpus experiri poſſumus, Å¿i <reg norm="eam" type="context">eã</reg> ad ignem pri-<lb/>mò calefactam, deinde <reg norm="cum" type="context">cÅ©</reg> ore in amplo aliquo cyatho, aut alio vaÅ¿e. </s> - <s xml:id="s2354" xml:space="preserve">vino, aut aqua <lb/>pleno vbi videbimus huiuÅ¿modi liquorem Å¿tatim Å¿urÅ¿um ferri, quia dum calefit am-<lb/>phora, rarefit quoque aer qui in ea continetur, & quia rateÅ¿cit dilaratur, & quia <lb/>dilatatur, eget maiore loco; </s> - <s xml:id="s2355" xml:space="preserve">& ideo magna pars eius foras exit; </s> - <s xml:id="s2356" xml:space="preserve">Cum verò ea aeris <lb/>portio, quæ intus remanÅ¿erit, iterum condenÅ¿atur ob defectum caloris, reÅ¿tringitur, <lb/><reg norm="minorique" type="simple">minoriqÌ;</reg> indiget loco; </s> - <s xml:id="s2357" xml:space="preserve">Quod cum ita Å¿e habeat, neceſſarium eÅ¿t, ne aliquis locus va <lb/>cuus remaneat, vt aliud quoddam corpus ingrediatur, cum ad <reg norm="ingrediendum" type="context">ingrediẽdum</reg> aeri non <lb/>patuerit aditus. </s> - <s xml:id="s2358" xml:space="preserve">quod Å¿i corpus admodum non erit fluxile, aut humidum, ita vt ingre <lb/>di amphoram poſſit ita amphorÄ™ hærebit, vt non cito diuelli poſſit, & eo modo ſępe - <pb o="194" rhead="IO. BAPT. BENED." n="206" file="0206" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0206"/> - <reg norm="cum" type="context">cÅ©</reg> admiratione <reg norm="videmus" type="simple">videmê°</reg> fragile vas <reg norm="vitreum" type="context">vitreÅ©</reg> <reg norm="magnum" type="context">magnÅ©</reg>, & graue <reg norm="lapideum" type="context">lapideÅ©</reg> corpus eleuare. <lb/></s> - <s xml:id="s2359" xml:space="preserve">Sed vt ad denÅ¿um & ad rarum redeamus, mihi videtur frigidum eſſe conÅ¿equentem <lb/>qualitatem denÅ¿i, & calidum rari, quia quæuis res dum calefit, rarefit, & quælibet <lb/>materia dum refrigeratur, Å¿imul condenÅ¿atur. </s> - <s xml:id="s2360" xml:space="preserve">Qua ratione fit, vt terra frigidior <lb/>Å¿it aqua, & ignis calidior Å¿it aere.</s> - </p> - <p> - <s xml:id="s2361" xml:space="preserve">Nec propriè locutus eÅ¿t AriÅ¿toteles .9. & .10. capite primi lib. & Å¿ecundo Å¿ecundi <lb/>metheororum cum dixerit <reg norm="calorem" type="context">calorẽ</reg> Solis eum eſſe, qui Å¿urÅ¿um humores, <reg norm="vaporesque" type="simple">vaporesqÌ;</reg> eue <lb/>hat, quia Sol nil aliud facit, quam calefacere, cuius caloris ratione, ea materia rarefit, <lb/>& ob rarefactionem leuior facta aÅ¿cendit, non quia Å¿urÅ¿um à Sole feratur.</s> - </p> - <p> - <s xml:id="s2362" xml:space="preserve">QuÄ™ Å¿ubÅ¿equuntur, cum raro ac denÅ¿o Å¿imbolum habere videntur. </s> - <s xml:id="s2363" xml:space="preserve">cum raro, Å¿cili-<lb/>cet calidum, humidum, leue, Å¿ublime, diaphanum, lumen, clarum, lux, <reg norm="album" type="context">albÅ©</reg>, dies, mo-<lb/>tus, velox, Å¿implex, diÅ¿gregatum, molle, lene, acutum, Å¿ubtile, coctum, <reg norm="Å¿paciosum" type="context">Å¿paciosÅ©</reg>, <lb/>dulce, voluptas, audacia, lætitia, liberalitas, veritas, induÅ¿tria, amor, miÅ¿ericordia, hu-<lb/>manitas, Å¿anitas, vita, & iis Å¿imilia. </s> - <s xml:id="s2364" xml:space="preserve">Cum denÅ¿o verò frigidum, Å¿iccum, graue, imum, <lb/>opacum, vmbra, obÅ¿curum, tenebræ, nigrum, nox, quies, tardum, mixtum, congrega <lb/>tum, durum, aÅ¿perum, ob tuÅ¿um, craſſum, crudum, anguÅ¿tum, amarum, dolor, cimor, <lb/><reg norm="melancholia" type="context">melãcholia</reg>, auaritia, mendacium, inertia, odium, crudelitas, feritas, infirmitas, mors, <lb/>& ijs Å¿imilia.</s> - </p> - <p> - <s xml:id="s2365" xml:space="preserve">Verum eÅ¿t quod ea ratio, qua AriÅ¿toteles ait aerem humidum eſſe, parui eÅ¿t mo-<lb/>menti, quia Å¿imiliter deigne inferri poſſet, qui facilius à termino alieno, <reg norm="quam" type="context">quã</reg> aer, aut <lb/>aqua terminari poteÅ¿t.</s> - </p> - </div> - <div type="section" level="3" n="35"> - <head xml:id="head311" style="it" xml:space="preserve">Motum rectum curuo poſſe comparari etiam diÅ¿entiente <lb/>AriÅ¿totele.</head> - <head xml:id="head312" xml:space="preserve">CAP. XXXV.</head> - <p> - <s xml:id="s2366" xml:space="preserve">SEd vt ad <reg norm="AriÅ¿totelem" type="context">AriÅ¿totelẽ</reg> redeamus, rectè dicere non poteÅ¿t motum rectum ad <reg norm="curuum" type="context">curuÅ©</reg> <lb/>comparabilem non eſſe .4. cap. lib. 7. phyÅ¿icorum, vbi errat quoque dicens repe <lb/>riri non poſſe lineam aliquam rectam alicuius circuli circunferentiæ æqualem. </s> - <s xml:id="s2367" xml:space="preserve">quia <lb/>Archimedes iam probauit in lib. de quadratura circuli, triangulum illum orthogo-<lb/>nium, cuius vnum ex lateribus circundantibus angulum rectum æquale eſſet Å¿emi-<lb/>piametro alicuius circuli, & aliud circunferentiæ, æqualem futurum dicto circulo. </s> - <s xml:id="s2368" xml:space="preserve">Il <lb/>lud igitur triangulum orthogonium, quod æquale erit alicui circulo, & habebit ali-<lb/>quod ex Å¿uis lateribus circundantibus angulum rectum æquale Å¿emidiametro dicti <lb/>circuli, aliud quoque latus ipÅ¿um angulum rectum circundans, ex neceſſitate, <reg norm="circun- ferentiæ" type="context">circÅ©-<lb/>ferentiæ</reg> dicti circuli habebit æquale. </s> - <s xml:id="s2369" xml:space="preserve">PoteÅ¿t igitur dari vna quædam recta linea <reg norm="ae- qualis" type="simple">Ä™-<lb/>qualis</reg> circulari contra AriÅ¿totelis opinionem, qui non benè reuocauit in mentem, <lb/>quod Å¿cripÅ¿it de relatiuis, cum dixit quadraturam circuli poſſe quidem dari, etÅ¿i <reg norm="tunc" type="context">tÅ©c</reg> <lb/><reg norm="temporis" type="wordlist">tê‘is</reg> de ea <reg norm="non" type="context">nõ</reg> <reg norm="haberetur" type="simple">haberet̃</reg> Å¿cientia. </s> - <s xml:id="s2370" xml:space="preserve">Si <reg norm="igitur" type="simple">igit̃</reg> dicta quadratura dari <reg norm="pont" type="context">põt</reg>, poteÅ¿t <reg norm="etiam" type="context">etiã</reg> dari vna <lb/>recta linea Ä™qualis circunferentiæ <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> circuli, ob rationes <reg norm="iam" type="context">iã</reg> dictas. </s> - <s xml:id="s2371" xml:space="preserve">Sed Å¿i AriÅ¿t. <lb/>dixiſſet, circularem corporum cÄ™leÅ¿tium motum, comparabilem non eſſerecto cor-<lb/>porum elementarium, verum dixiſſet, non quia eorum alter circularis, alter ve-<lb/>rò Å¿it rectus, Å¿ed quia cÅ“leÅ¿tis regularis Å¿it, neque modò tardus, modò velox, <lb/>Å¿ed vnam Å¿emper & eandem velocitatem retinens, <reg norm="motus" type="simple">motê°</reg> <reg norm="autem" type="wordlist">aÅ©t</reg>, qui eÅ¿t <reg norm="corporum" type="context">corporÅ©</reg> elemen - <pb o="195" rhead="DISPVTATIO NES." n="207" file="0207" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0207"/> - tarium è contrà ſe habeat, præter id, <reg norm="quod" type="simple">ê™</reg> nunquam fuit neque Å¿it futurus aliquis <reg norm="horum" type="context">horÅ©</reg> <lb/>rectorum, qui naturales dicuntur, qui tam velociter moueatur, ut motus cÅ“li, quia Å¿i <lb/>voluerimus conÅ¿iderare motum diurnum .24. horarum, Å¿ecundum opinionem com-<lb/>munem, reperiemus calculando, Lunam in quadraturis cum Sole, dum inuenitur in <lb/>æquatore, Å¿ingulis horarum minutis moueri per .500. milliaria Italica vel circa, & in <lb/>coniunctionibus, & oppoÅ¿itionibus ipÅ¿ius Solis .1000. vel circa, & Solem tempore <reg norm="ae- quinoctiorum" type="simple context">Ä™-<lb/>quinoctiorÅ©</reg> .18000. & <reg norm="Saturnum" type="context">SaturnÅ©</reg> circa æquatoris <reg norm="Å¿itum" type="context">Å¿itÅ©</reg> .260000. & <reg norm="amplius" type="simple">ampliê°</reg> de Å¿tellis <reg norm="autem" type="wordlist">aÅ©t</reg> <lb/>fixis circa æquatorem poÅ¿itis quiuis cogitet; </s> - <s xml:id="s2372" xml:space="preserve">quod reuera diffi cillimum quibuÅ¿dam <lb/>videbitur, quod quidem non occurrit <reg norm="Å¿ecundum" type="context">Å¿ecÅ©dum</reg> pulcherrimam AriÅ¿tarchi Å¿amij opi-<lb/>nionem, diuinitus à Nicolao Copernico expreſſam, contra quam nil planè valent <lb/>rationes ab AriÅ¿totele; </s> - <s xml:id="s2373" xml:space="preserve">neque etiam à Ptolomeo propoÅ¿itÄ™. </s> - <s xml:id="s2374" xml:space="preserve">Motu verò proprio, quo <lb/>libet horÄ™ minuto, Sol <reg norm="mouetur" type="simple">mouet̃</reg> per milliaria circa .48. </s> - <s xml:id="s2375" xml:space="preserve">Luna <reg norm="quando" type="context">quãdo</reg> <reg norm="coniuncta" type="context">cõiuncta</reg> eÅ¿t, aut op <lb/>poÅ¿ita reperitur Soli .36. milliaria, & in quadraturis .18. Saturnus .24. Iupiter .40: <lb/>Mars .100: Venus .26: Mercur .5. </s> - <s xml:id="s2376" xml:space="preserve">Sed Saturnus motu rapido, vno horæ minuto <reg norm="mo- uetur" type="simple">mo-<lb/>uet̃</reg> circa .260000. milliaria, vt diximus Iupiter circa .170000. Mars .75000. Venus. <lb/>10000, Mercurius .2000. corpus <reg norm="autem" type="context">autẽ</reg> elementare, & Å¿i <reg norm="moueretur" type="simple">moueret̃</reg> motu recto hoc <reg norm="mon" type="context">mõ</reg>, <lb/>& velocius etiam corpore cÄ™leÅ¿ti, non obÅ¿eruans <reg norm="tamen" type="wordlist">tamẽ</reg> uniformitatem, ut dictum cÅ“ <lb/>leÅ¿te facit, cum eodem nullo modo comparari poſſet, quia rectus dictus naturalis, <lb/>Å¿uam Å¿emper velocitatem adauget, ob continuam impreſſionem, quam recipit à cau <lb/>Å¿a perpetuò coniuncta cum ipÅ¿o corpore, quÄ™ eÅ¿t propenÅ¿io illa naturalis eundi bre-<lb/>uiori quadam via ad locum Å¿uum, ita vt etiam Å¿i dictum corpus elementare à motu <lb/>tardiore ad velociorem, Å¿uperare poſſet <reg norm="motum" type="context">motÅ©</reg> alicuius corporis cÄ™leÅ¿tis, ij duo motus <lb/>interÅ¿ecarent Å¿eÅ¿e in vno Å¿olo <reg norm="puncto" type="context">pÅ©cto</reg>, quod diuidi <reg norm="diÅ¿tribuique" type="simple">diÅ¿tribuiqÌ;</reg> in partes nequiret, ideÅ¿t <lb/>non niÅ¿i in vno Å¿olo temporis inÅ¿tanti redderentur æquales, vt ita dicam. </s> - <s xml:id="s2377" xml:space="preserve"><reg norm="Neque" type="simple">Neq;</reg> <reg norm="Å¿olum" type="context">Å¿olÅ©</reg> <lb/>loquor de circulari cÅ“leÅ¿ti cum recto elementari, Å¿ed de qualibet alia motuum Å¿pe-<lb/>cie, Å¿iue Å¿int ambo recti, Å¿iue ambo curui, quando aliquis eorum irregularis erit.</s> - </p> - </div> - <div type="section" level="3" n="36"> - <head xml:id="head313" style="it" xml:space="preserve">Minus Å¿ufficienter exploÅ¿am fuiſſe ab AriÅ¿totele opinionem cre-<lb/>dentium plures mundos exiÅ¿tere.</head> - <head xml:id="head314" xml:space="preserve">CAP. XXXVI.</head> - <p> - <s xml:id="s2378" xml:space="preserve">MAior ratio, qua AriÅ¿toteles eorum opinionem, qui plures eſſe mundos dixe <lb/>runt, refutare nititur, in eo conÅ¿iÅ¿tit, quod is credat partes terræ, quæ alijs <lb/>mundis aſſignarentur, ad huius mundi centrum inclinationem habere, & Å¿ic ignem <lb/>illorum, propenÅ¿ionem habiturum ad circunferentiam huius.</s> - </p> - <p> - <s xml:id="s2379" xml:space="preserve">Quæ certè ratio tam debilis eÅ¿t, vt per Å¿e cadat, non conÅ¿ideransipÅ¿e, quòd Å¿i <lb/>eſſent dicti mundi, eorum quilibet Å¿uum proprium centrum, <reg norm="Å¿uamque" type="simple">Å¿uamqÌ;</reg> propriam cir-<lb/>cunferentiam haberet, <reg norm="terrasque" type="simple">terrasqÌ;</reg> & ignes haberent inclinationem ad centra circunfe-<lb/><reg norm="rentiasque" type="simple">rentiasqÌ;</reg> Å¿uorum mundorum, abÅ¿que eo, <reg norm="quod" type="simple">ê™</reg> vna terra, alterius centrum appeteret; </s> - <s xml:id="s2380" xml:space="preserve">vt <lb/>exempli gratia, Å¿i doctiſſimi AriÅ¿tarchi opinio eÅ¿t vera, rationi <reg norm="quoque" type="simple">quoq;</reg> conÅ¿entaneum <lb/>erit maximè, vt quod Lunæ contingit, cuilibet <reg norm="etiam" type="context">etiã</reg> ex aliis quinque planetis eue-<lb/>niat, ideÅ¿t, vt quemadmodum Luna Å¿uorum epicyclorum ope <reg norm="circum" type="context">circÅ©</reg> terram voluitur, <lb/>quaÅ¿i per circunferentiam alterius cuiuÅ¿dam epicycli, in quo terra Å¿it inÅ¿tar centri <lb/>naturalis (ideÅ¿t Å¿it in medio) delati ab orbe annuo circa Solem; </s> - <s xml:id="s2381" xml:space="preserve">Sic etiam Saturnus, <lb/>Iupiter, Mars, Venus, atque Mercurius, cir cum aliquod corpus in medio Å¿ui epici- - <pb o="196" rhead="IO. BABPT. BENED." n="208" file="0208" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0208"/> - cli maioris, Å¿itum habens, voluantur; </s> - <s xml:id="s2382" xml:space="preserve">quod quidem corpus, & aliquem quoque ha-<lb/>beat motum circa Å¿uum axem, Å¿it opacum, ijs conditionibus, quæ terræ Å¿unt Å¿imi-<lb/>les, præditum exiÅ¿tat, & in dicto epyciclo Å¿int res Å¿imiles iÅ¿tis lunaribus.</s> - </p> - </div> - <div type="section" level="3" n="37"> - <head xml:id="head315" style="it" xml:space="preserve">Anrectè loquutus Å¿it PhyloÅ¿opbus de extenÅ¿ione luminis <lb/>per uacuum.</head> - <head xml:id="head316" xml:space="preserve">CAP. XXXVII.</head> - <p> - <s xml:id="s2383" xml:space="preserve">ARriÅ¿toteles Å¿ecundo lib. de anima Å¿entit <reg norm="quod" type="simple">ê™</reg> per vacuum non extenderetur <reg norm="lu- mem" type="context">lu-<lb/>mẽ</reg>, quod procederet à corpore lucido. </s> - <s xml:id="s2384" xml:space="preserve">Quod veriÅ¿imile <reg norm="non" type="context">nõ</reg> eÅ¿t; </s> - <s xml:id="s2385" xml:space="preserve"><reg norm="quia" type="simple">ê—a</reg> <reg norm="quemadmo dum" type="wordlist">quẽadmoÂ<lb/>dum</reg> quantò rarius eÅ¿t aliquod corpus, tanto aptius eÅ¿t vt diaphanum exiÅ¿tat; </s> - <s xml:id="s2386" xml:space="preserve">& <reg norm="quan- tò" type="context">quã-<lb/>tò</reg> rarius eÅ¿t dictum corpus, tantò minorem quantitatem materiæ contineat; </s> - <s xml:id="s2387" xml:space="preserve">Å¿ic <reg norm="quam" type="context">quã</reg> <lb/>tò magis diaphanum eÅ¿t, cum ex perexigua materia conÅ¿tet, tantò magis liber tran-<lb/>Å¿itus luminis patet; </s> - <s xml:id="s2388" xml:space="preserve">Vnde quantò minor quantitas materiæ erit in dicto Å¿patio, tan <lb/>tò nitidius pertranÅ¿ibit lumen. </s> - <s xml:id="s2389" xml:space="preserve">Sequitur ergo, quòd vbi nulla eſſet materia, totum <lb/>lumen libere tranÅ¿iret. </s> - <s xml:id="s2390" xml:space="preserve">Color cÄ™ruleus quem videmus in profunditate aquæ, & ae-<lb/>ris, color eÅ¿t a quæ & aeris, qui denotat reÅ¿iÅ¿tentiam factam ab aere & ab aqua ipÅ¿i lu <lb/>mini; </s> - <s xml:id="s2391" xml:space="preserve">Quod quidem lumen ubi corpus aliquod non eſſet, minime reflecteretur, Å¿ed <lb/>abÅ¿que vllo impedimento rectà tranÅ¿iret.</s> - </p> - </div> - <div type="section" level="3" n="38"> - <head xml:id="head317" style="it" xml:space="preserve">An rectè phyloÅ¿ophiÅ“<unsure/> penus AriÅ¿toteles Å¿enÅ¿erit de loco im-<lb/>pellendo à pyramide.</head> - <head xml:id="head318" xml:space="preserve">CAP. XXXVIII.</head> - <p> - <s xml:id="s2392" xml:space="preserve">ARiÅ¿toteles .8. cap. lib. 3. de cÅ“lo, diÅ¿putans contra antiquos de elementorum <lb/>figuris, ait pyramidem implere poſſe locum corporeum. </s> - <s xml:id="s2393" xml:space="preserve">quod verum non <lb/>eÅ¿t. </s> - <s xml:id="s2394" xml:space="preserve">Cubus quidem id facit ab .8. enim cubis perfectè impletur locus, Å¿ed non <lb/>item .12. pyramides, ut AriÅ¿toteles Å¿enÅ¿it (ideÅ¿t Å¿ex Å¿uper aliquam exagonam figu-<lb/>ram Å¿uperficialem & Å¿exÅ¿ub eadem) id præſtant, cum potius maius vacuum rema-<lb/>neatad quamlibet partium Å¿upra, & infra, quam plenum. </s> - <s xml:id="s2395" xml:space="preserve">Rectius AriÅ¿toteles <lb/>egiſſet, Å¿i probaſſet ratione immobilitatis conuenire pyramidem terræ, quam cu-<lb/>bum. </s> - <s xml:id="s2396" xml:space="preserve">quamuis, de horum corporum altero, Å¿it Å¿tultum hoc credere. </s> - <s xml:id="s2397" xml:space="preserve">decepti tamen <lb/>fuerunt antiqui, credentes cubum ad motum minus idoneum eſſe, quam reliqua <lb/>quatuor corpora regularia (loquor autem habita volubilitatis ratione) quia pyra-<lb/>midale eÅ¿t illud, quod ita Å¿e habet, vt multis rationibus probari poteÅ¿t, quarum vna <lb/>hæc nobis Å¿ufficiet. </s> - <s xml:id="s2398" xml:space="preserve">Scimus iam ex communi conceptu corpus Å¿phÄ™ricum eſſe ma-<lb/>gis volubile, <reg norm="inÅ¿tabileque" type="simple">inÅ¿tabileqÌ;</reg>, quà m alia Å¿int. </s> - <s xml:id="s2399" xml:space="preserve">Illud ergo corpus, cuius figura ad Å¿phæri-<lb/>cam magis accedet, ad uoluendum, & ad mouendum facilius erit quouis alio, quod <lb/>æqualis Å¿it quantitatis, & Å¿ibi omogeneum materia, vt exempli gratia corpus .20. ba <lb/>Å¿ium ad voluendum, & ad mouendum promptius erit eo, quod ex .12. conÅ¿tat, & id, <lb/>quod eÅ¿t .12. eo, quod eÅ¿t .8. & id, quod eÅ¿t .8. eo, quod eÅ¿t .6. & id, quod eÅ¿t .6. vt <lb/>cubus eÅ¿t, eo, quod eÅ¿t .4. cuiuÅ¿modi eÅ¿t pyramidale. </s> - <s xml:id="s2400" xml:space="preserve">Huc accedit, quòd pyrami-<lb/>dale corpus aliam conditionem habet, quà m cubicum, cum in quauis facie inalte- - <pb o="197" rhead="DISPVTATIONES." n="209" file="0209" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0209"/> - rabile Å¿it, cubicum autem econtrà ſit alterabile vndequaque, <reg norm="Å¿uaque" type="simple">Å¿uaqÌ;</reg> quadrata in <reg norm="rhum- bos" type="context">rhÅ©-<lb/>bos</reg> mutare poſſit, iiÅ¿dem exiÅ¿tentibus lateribus.</s> - </p> - </div> - <div type="section" level="3" n="39"> - <head xml:id="head319" style="it" xml:space="preserve">Examinatur quam ualida Å¿it ratio Aristotelis de <lb/>inalterabilitate CÅ“li.</head> - <head xml:id="head320" xml:space="preserve">CAP. XXXIX.</head> - <p> - <s xml:id="s2401" xml:space="preserve">ARriÅ¿toteles textu .22. primi lib. de CÅ“lo ita inquit.</s> - </p> - <p> - <s xml:id="s2402" xml:space="preserve">Accidit autem, & hoc per Å¿enÅ¿um Å¿ufficienter, quo ad humanam dixiſſe fi-<lb/>dem, & omni prÄ™terito tempore Å¿ecundum traditam inuicem memoriam, nihil vi-<lb/>detur tranÅ¿mutatum neque Å¿ecundum totum vltimum cÄ™lum, neque Å¿ecundum par-<lb/>tem ipÅ¿ius propriam vllam.</s> - </p> - <p> - <s xml:id="s2403" xml:space="preserve">Hoc autem in loco AriÅ¿to. non conÅ¿iderauit, <reg norm="quod" type="simple">ê™</reg> Å¿imiliter de terra dici poſſet, quan <lb/>do ipÅ¿a ita eminus proÅ¿piceretur, imo abÅ¿que dubio putandum eÅ¿t, <reg norm="quod" type="simple">ê™</reg> Å¿i terra luce So <lb/>lis prædita eſſet, & aliquis ipÅ¿am ab octauo orbe vellet videre, nullo pacto cerne-<lb/>ret, cum Å¿idera illa quæ primæ magnitudinis vocantur, & quæ pluÅ¿quam centies ma <lb/>iora ip<unsure/>Å¿a terra putantur non niÅ¿i vt puncta videantur.</s> - </p> - <pb o="198" n="210" file="0210" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0210"/> - </div> - </div> - <div type="chapter" level="2" n="5"> - <head xml:id="head321" xml:space="preserve">IN QVINTVM <lb/>EVCLIDIS LIBRVM</head> - <p type="title" style="it"> - <s xml:id="s2404" xml:space="preserve">QVamuis omnia libri quinti Euclid. uerißima Å¿int. <lb/></s> - <s xml:id="s2405" xml:space="preserve">Animaduertimus tamen permultos Å¿umma <reg norm="cum" type="context">cÅ©</reg> <lb/>difficultate <reg norm="eorum" type="context">eorÅ©</reg> demonstr ationes percipere. </s> - <s xml:id="s2406" xml:space="preserve">PrÅ“<unsure/>-<lb/>cipuè ubi quint a, aut Å¿eptima deffinitiones eiuÅ¿-<lb/>dem libri neceſſariÅ“<unsure/> Å¿unt. </s> - <s xml:id="s2407" xml:space="preserve">IllÅ“ enim adeo obÅ¿curÅ“ <lb/>uidentur, ut longè facilius admißuri Å¿int hÅ“c no-<lb/>Å¿tra poÅ¿tulat at anquam clarior a. </s> - <s xml:id="s2408" xml:space="preserve">At que etiam tanquam intellectui <lb/>commodiora, quam Å¿it illud quintum <reg norm="idemque" type="simple">idemqÌê«</reg> ultimum postulatum <lb/>eiuÅ¿dem in primo libro poÅ¿itum, de line a duas alias Å¿ecante. </s> - <s xml:id="s2409" xml:space="preserve">Quan-<lb/>doquidem <reg norm="ijs" type="lig">ijs</reg> noÅ¿tris postulatis admißis, Å¿equentia Theoremata per <lb/>facillima reddentur.</s> - </p> - <div type="section" level="3" n="1"> - <div type="section" level="4" n="1"> - <head xml:id="head322" style="it" xml:space="preserve">Horum autem primum est.</head> - <p> - <s xml:id="s2410" xml:space="preserve"><emph style="sc">Qvod</emph> tota compoÅ¿ita ex æquali numero partium æqualium, Å¿unt inuicem <lb/>æqualia.</s> - </p> - <p> - <s xml:id="s2411" xml:space="preserve">VtÅ¿i quis diceret omnes proportiones quæ <reg norm="compoÅ¿itæ" type="context">cõpoÅ¿itæ</reg> Å¿unt ex æquali numero alia-<lb/>rum proportionum inuicem æqualium, Å¿unt etiam inuicem æquales, quod Eucli-<lb/>des conatur demonÅ¿trare in .22. et .23. quinti libri.</s> - </p> - </div> - <div type="section" level="4" n="2"> - <head xml:id="head323" xml:space="preserve">SECVNDVM.</head> - <p> - <s xml:id="s2412" xml:space="preserve"><emph style="sc">Qvod</emph> Å¿i à totis æqualibus detractæ fuerint æquales partes, quæ remanent erunt <lb/>partes inuicem æquales.</s> - </p> - <p> - <s xml:id="s2413" xml:space="preserve">Et è conuerÅ¿o Å¿i æqualibus æqualia addas compoÅ¿ita erunt inuicem æqualia.</s> - </p> - <p> - <s xml:id="s2414" xml:space="preserve">Quod in ipÅ¿is proportionibus hoc loco Å¿emper intelligendum eÅ¿t.</s> - </p> - </div> - <div type="section" level="4" n="3"> - <head xml:id="head324" xml:space="preserve">TERTIVM.</head> - <head xml:id="head325" style="it" xml:space="preserve">QuÄ™ est εuclidis Å¿eptima propoÅ¿itio.</head> - <p> - <s xml:id="s2415" xml:space="preserve"><emph style="sc">Qvod</emph> Å¿i fuerint plures termini æquales inuicem, ratio Å¿eu proportio vnius ip-<lb/>Å¿orum ad alium tertium terminum maiorem, minoremúe, Å¿ed eiuÅ¿dem generis, erit <lb/>cadem quæ cuiuÅ¿uis alterius termini ad eundem tertium. </s> - <s xml:id="s2416" xml:space="preserve">Et è conuerÅ¿o, quæ fuerit <lb/>proportio tertij termini ad vnum prædictorum æqualium, eadem erit, Å¿pecie, cum <lb/>alio eorundem terminorum.</s> - </p> - <pb o="199" rhead="IN QVINT. LIB. EVCLI." n="211" file="0211" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0211"/> - </div> - <div type="section" level="4" n="4"> - <head xml:id="head326" xml:space="preserve">QVARTVM.</head> - <head xml:id="head327" style="it" xml:space="preserve">εuclidis uerò nona propoÅ¿itio.</head> - <p> - <s xml:id="s2417" xml:space="preserve"><emph style="sc">Qvotiescvnqve</emph> proportio vnius plurium terminorum collatorum cum ali <lb/>quo tertio eiuÅ¿dem generis, eadem fuerit cum ea quæ eÅ¿t cuiuÅ¿uis alterius dictorum <lb/>terminorum cum eodem tertio, aut proportio dicti tertij, cum aliquo dictorum, ea-<lb/>dem fuerit cum ea quæ ipÅ¿ius eÅ¿t ad aliquem alium eorundem <reg norm="terminorum" type="context">terminorÅ©</reg>, tunc eiuÅ¿-<lb/>modi termini, æquales erunt inter Å¿e.</s> - </p> - </div> - <div type="section" level="4" n="5"> - <head xml:id="head328" xml:space="preserve">QVINTVM.</head> - <head xml:id="head329" style="it" xml:space="preserve">Euclidis uerò octaua propoÅ¿itio.</head> - <p> - <s xml:id="s2418" xml:space="preserve"><emph style="sc">Qvoties</emph> plures erunt termini, quorum vnus fuerit maior altero, Å¿i compa-<lb/>rentur alicui tertio eiuÅ¿dem generis, proportio maioris adtertium illum, maior erit <lb/>ea, quæ eÅ¿t minoris ad prædictum tertium, & proportio illius tertij ad maiorem, mi-<lb/>nor erit ea quæ eiuÅ¿dem tertij ad minorem terminum comparati.</s> - </p> - </div> - <div type="section" level="4" n="6"> - <head xml:id="head330" xml:space="preserve">SEXTVM.</head> - <head xml:id="head331" style="it" xml:space="preserve">εuclidis uerò decima propoÅ¿itio.</head> - <p> - <s xml:id="s2419" xml:space="preserve"><emph style="sc">Qvoties</emph> proportio vnius, ex pluribus terminis comparatis ad aliquem ter-<lb/>tium, maior fuerit proportione alicuius alterius dictorum cum eodem tertio, primus <lb/>ille terminus, altero maior erit. </s> - <s xml:id="s2420" xml:space="preserve">Et quoties proportio tertij termini ad vnum <reg norm="quam" type="context">quã</reg> <lb/>ad alterum terminum maior fuerit, eiuÅ¿modi terminus altero minor erit.</s> - </p> - </div> - <div type="section" level="4" n="7"> - <head xml:id="head332" xml:space="preserve">SEPTIMVM.</head> - <head xml:id="head333" style="it" xml:space="preserve">Euclidis uerò undecima propoÅ¿itio.</head> - <p> - <s xml:id="s2421" xml:space="preserve"><emph style="sc">Proportiones</emph>, quarum vnaquÄ™que cum aliqua tertia æqualis eÅ¿t, ipſæ quo-<lb/>que inter Å¿e Å¿unt æquales. </s> - <s xml:id="s2422" xml:space="preserve">Vtillud, Quæ vni & eidem Å¿unt æqualia, Å¿ibi inuicem <lb/>Å¿unt æqualia.</s> - </p> - </div> - <div type="section" level="4" n="8"> - <head xml:id="head334" xml:space="preserve">OCTAVVM.</head> - <head xml:id="head335" style="it" xml:space="preserve">εuclidis uerò duodecima propoÅ¿itio.</head> - <p> - <s xml:id="s2423" xml:space="preserve"><emph style="sc">Qvotiescvnqve</emph> proportio vnius ex pluribus antecedentibus cum Å¿uo ex <lb/>pluribus conÅ¿equentibus, æqualis fuerit ei cuiuÅ¿uis alterius dictorum <reg norm="antecedentium" type="context">antecedentiÅ©</reg>, <lb/>cum Å¿uo plurium <reg norm="conÅ¿equentium" type="context">cõſequentium</reg>, proportio totius aggregati antecedentium cum to-<lb/>to aggregato conÅ¿equentium, dictæ primÄ™ proportioni Ä™qualis erit, nempe illius an <lb/>tecedentis ad Å¿uum conÅ¿equens.</s> - </p> - <pb o="200" rhead="IO. BAPT. BENED." n="212" file="0212" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0212"/> - </div> - <div type="section" level="4" n="9"> - <head xml:id="head336" xml:space="preserve">NONVM.</head> - <head xml:id="head337" style="it" xml:space="preserve">Euclidis uero tertiadecima propoÅ¿itio.</head> - <p> - <s xml:id="s2424" xml:space="preserve"><emph style="sc">Qvotiescvnqve</emph> aliqua proportio plurium proportionum inuicem æqua-<lb/>lium, tertia aliqua proportione, maior aut minor fuerit, quælibet prædictarum æqua <lb/>lium inter Å¿e, tertia illa proportione maior aut minor pariter erit.</s> - </p> - </div> - <div type="section" level="4" n="10"> - <head xml:id="head338" xml:space="preserve">DECIMVM.</head> - <p> - <s xml:id="s2425" xml:space="preserve"><emph style="sc">Qvotiescvnqve</emph> fuerint ex vna parte plurestermini (Å¿iue coniuncti Å¿iue di-<lb/>Å¿iuncti Å¿int) æquales Å¿inguli vni tertio termino; </s> - <s xml:id="s2426" xml:space="preserve">ex altera verò parte totidem fuerint <lb/>alteri tertio termino æquales, proportio aggregati priorum terminorum ad <reg norm="Å¿uum" type="context">Å¿uÅ©</reg> ter-<lb/>tium, æqualis erit proportioni aggregati reliquorum terminorum ad Å¿uum tertium, <lb/>& è conuerÅ¿o, ita Å¿e habebit primus tertius terminus ad Å¿uos multos terminos, Å¿icut <lb/>Å¿e habet Å¿ecundus tertius terminus ad Å¿uos Å¿imul Å¿umptos.</s> - </p> - </div> - <div type="section" level="4" n="11"> - <head xml:id="head339" xml:space="preserve">VNDECIMVM.</head> - <p> - <s xml:id="s2427" xml:space="preserve">Aggregatum ex partibus proportiona litatis continuæ, quod inter maximum, & <lb/>minimum terminum omnium terminorum proportionalium compræhenditur, Å¿em <lb/>per multiplex eÅ¿t ad Å¿ingulas partiales proportiones, ex quibus ipÅ¿um componitur.</s> - </p> - </div> - <div type="section" level="4" n="12"> - <head xml:id="head340" xml:space="preserve">DVODECIMVM.</head> - <p> - <s xml:id="s2428" xml:space="preserve">Quæuis proportio quocunque modo diuiÅ¿a fuerit, ex iis partibus componitur, in <lb/>quas diuiditur.</s> - </p> - <p style="it"> - <s xml:id="s2429" xml:space="preserve">Cum enim bæ præpoÅ¿itiones Å¿int ita conÅ¿picuæ ipÅ¿i intellectui, ut <reg norm="abÅ¿que" type="simple">abÅ¿q;</reg> dubio inter obie <lb/>ct a ipÅ¿ius intellectus connumerari poſſint, nullus Å¿anæ mentis eas negabit.</s> - </p> - </div> - </div> - <div type="section" level="3" n="2"> - <div type="section" level="4" n="1"> - <head xml:id="head341" xml:space="preserve">THEOR.I. II. ET III.</head> - <p> - <s xml:id="s2430" xml:space="preserve">PRimum, Å¿ecundum, & tertium theorema quinti Euclidis ab ipÅ¿o Å¿atis exactè de <lb/>monÅ¿tratur, Å¿tudioÅ¿us itaque autorem conÅ¿ulat.</s> - </p> - </div> - <div type="section" level="4" n="2"> - <head xml:id="head342" xml:space="preserve">THEOREM. IIII.</head> - <p> - <s xml:id="s2431" xml:space="preserve">QVartum vero Theorema Eu-<lb/> - <anchor type="figure" xlink:label="fig-0212-01a" xlink:href="fig-0212-01"/> - clidis ego Å¿ic <reg norm="demonÅ¿trarem" type="context">demonÅ¿trarẽ</reg>. <lb/></s> - <s xml:id="s2432" xml:space="preserve">Å¿it, verbi gratia, proportio <var>.a.</var> ad <var>.b.</var> <lb/>quæ eÅ¿t <var>.c.</var> ad <var>.d.</var> Å¿umptis multiplici-<lb/>bus <var>.e.</var> et <var>.f.</var> ad <var>.a.</var> et <var>.c.</var> æqualiter, item <lb/>multiplicibus <var>.g.</var> et <var>.h.</var> ad <var>.b.</var> et <var>.d.</var> dico <lb/>proportionem <var>.e.</var> ad <var>.g.</var> eſſe eandem <lb/>quæ eÅ¿t <var>.f.</var> ad <var>.h</var>. </s> - <s xml:id="s2433" xml:space="preserve">Habemus enim ex .10 <lb/>poÅ¿tulato præmiſſo, eandem futuram <lb/>proportionem <var>.e.</var> ad <var>.a.</var> quæ eÅ¿t <var>.f.</var> ad <var>.c.</var> <lb/>& ita <var>.b.</var> ad <var>.g.</var> quæ eÅ¿t <var>.d.</var> ad <var>.h.</var> ex præ-<lb/>Å¿uppoÅ¿ito verò <reg norm="cum" type="context">cÅ©</reg> Å¿ic Å¿e habeat <var>.a.</var> ad <lb/>b. Å¿icut <var>.c.</var> ad <var>.d.</var> erit ex primo poÅ¿tula-<lb/>to <reg norm="eadem" type="context">eadẽ</reg> proportio <var>.e.</var> ad <var>.g.</var> quæ eÅ¿t <var>.f.</var> <lb/>ad <var>.h</var>. </s> - <s xml:id="s2434" xml:space="preserve">Nam proportio <var>.e.</var> ad <var>.g.</var> compo <lb/>nitur ex eis quæ Å¿unt <var>.e.</var> ad .a: et <var>.a.</var> ad <var>. - <pb o="201" rhead="IN QVINT. LIB. EVCLI." n="213" file="0213" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0213"/> - b.</var> et <var>.b.</var> ad <var>.g.</var> & Å¿imiliter proportio <var>.f.</var> ad <var>.h.</var> <reg norm="componitur" type="context">cõponitur</reg> ex eis quæſunt <var>.f.</var> ad <var>.c.</var> et <var>.c.</var> ad <var>.d.</var> <lb/>et <var>.d.</var> ad <var>.h</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0212-01" xlink:href="fig-0212-01a"> - <image file="0212-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0212-01"/> - </figure> - </div> - </div> - <div type="section" level="4" n="3"> - <head xml:id="head343" xml:space="preserve">THEOR.V. ET VI.</head> - <p> - <s xml:id="s2435" xml:space="preserve"><emph style="sc">Circa</emph> 5. et .6. theorema nihil notandum occurrit.</s> - </p> - </div> - <div type="section" level="4" n="4"> - <head xml:id="head344" xml:space="preserve">THEOR. VII. VIII. IX.X. XI. XII. XIII.</head> - <p> - <s xml:id="s2436" xml:space="preserve">THeoremata à .6. in .13. cum Å¿int de obiectis intelligibilibus, Å¿ine vllo medio, <lb/>ab intellectu cognitis, inter axiomata à me relata fuerunt .7. inquam quinti <lb/>Euclid. fecimus tertium PoÅ¿tulatum, .8. quintum, .9. quartum, .10. Å¿extum, .11. Å¿eptiÂ<lb/>mum, .12. octauum, .13. nonum.</s> - </p> - </div> - <div type="section" level="4" n="5"> - <head xml:id="head345" xml:space="preserve">THEOREM. XIIII.</head> - <p> - <s xml:id="s2437" xml:space="preserve">QVartumdecimum Theorema ex Euclide demonÅ¿trabitur, mutatis tantum <lb/>theorematibus ab interprete notatis, ita vt loco .7. 8. noni, & decimi citetur <lb/>tertium .5. 4. et .6. poÅ¿tulatum à me propoÅ¿itum.</s> - </p> - </div> - <div type="section" level="4" n="6"> - <head xml:id="head346" xml:space="preserve">THEOR. XV.</head> - <p> - <s xml:id="s2438" xml:space="preserve">QVintumdecimum Theorema Å¿ic demonÅ¿trabo; </s> - <s xml:id="s2439" xml:space="preserve">Sit, exempli gratia, a. termi-<lb/>nus antecedens. et <var>.b.</var> conÅ¿equens, qui-<lb/>bus duo multiplices Å¿umantur <var>.c.</var> et <var>.d</var>. </s> - <s xml:id="s2440" xml:space="preserve">Dico <lb/> - <anchor type="figure" xlink:label="fig-0213-01a" xlink:href="fig-0213-01"/> - eandem proportionem habiturum <var>.c.</var> ad <var>.d.</var> <lb/>quam <var>.a.</var> ad <var>.b.</var> habet. </s> - <s xml:id="s2441" xml:space="preserve">In primis enim manife-<lb/>Å¿tè patet quamlibet partem ipÅ¿ius <var>.c.</var> habitu-<lb/>ram eandem proportionem cum qualibet par <lb/>te <var>.d.</var> quam habet <var>.a.</var> ad <var>.b.</var> quare ex .7. et .8. po <lb/>Å¿tulato propoÅ¿itum eluceÅ¿cet.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0213-01" xlink:href="fig-0213-01a"> - <image file="0213-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0213-01"/> - </figure> - </div> - </div> - <div type="section" level="4" n="7"> - <head xml:id="head347" xml:space="preserve">THEOREM. XVI.</head> - <p> - <s xml:id="s2442" xml:space="preserve">SExtumdecimum theorema Å¿ic demonÅ¿trabitur. </s> - <s xml:id="s2443" xml:space="preserve">Sit, exempli cauÅ¿a, eadem pro <lb/>portio <var>.a.</var> ad <var>.b.</var> quæ eÅ¿t <var>.c.</var> ad <var>.d</var>. </s> - <s xml:id="s2444" xml:space="preserve">Dico <reg norm="quod" type="simple">ê™</reg> ita Å¿e habebit <var>.a.</var> ad <var>.c.</var> Å¿icut <var>.b.</var> ad <var>.d</var>. </s> - <s xml:id="s2445" xml:space="preserve">Cogi-<lb/>temus itaque alterum iÅ¿torum terminorum <var>.c.</var> aut <var>.b.</var> medium inter <var>.a.</var> et <var>.d</var>. </s> - <s xml:id="s2446" xml:space="preserve">quare <lb/>primum intelligamus <var>.b.</var> inter <var>.a.</var> et. d proportio ipÅ¿ius <var>.a.</var> ad <var>.d.</var> componetur ex ea quÄ™ <lb/>eÅ¿t <var>.a.</var> ad <var>.b.</var> & ea quæ eÅ¿t <var>.b.</var> ad <var>.d.</var> ex .12. poÅ¿tulato. </s> - <s xml:id="s2447" xml:space="preserve">Et ex eodem, illa ipÅ¿a proportio <var>.<lb/>a.</var> ad <var>.d.</var> pariter componetur ex ea quæ eÅ¿t <var>.a.</var> ad <var>.c.</var> & ea quæ eÅ¿t <var>.c.</var> ad <var>.d.</var> Å¿umpto <var>.c.</var> pro <lb/>medio termino. </s> - <s xml:id="s2448" xml:space="preserve">Ex quo Å¿equitur, aggregatum duarum proportionum, videlicet <var>.a.</var> <lb/>ad <var>.b.</var> et <var>.b.</var> ad <var>.d.</var> æquale eſſe aggregato <var>.a.</var> ad <var>.c.</var> et <var>.c.</var> ad <var>.d.</var> ex quibus aggregatis æqua-<lb/>libus Å¿i duas proportiones æquales Å¿ubtraxerimus, eam videlicet quæ eÅ¿t <var>.a.</var> ad <var>.b.</var> & il <lb/>lam quæ eÅ¿t <var>.c.</var> ad <var>.d.</var> Å¿upererunt duæ proportiones <lb/>inter Å¿e æquales. </s> - <s xml:id="s2449" xml:space="preserve">erit enim proportio <var>.a.</var> ad <var>.c.</var> æqua <lb/> - <anchor type="figure" xlink:label="fig-0213-02a" xlink:href="fig-0213-02"/> - lis proportioni <var>.b.</var> ad <var>.d.</var> ex prima parte Å¿ecundi po <lb/>Å¿tulati diuiÅ¿im.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0213-02" xlink:href="fig-0213-02a"> - <image file="0213-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0213-02"/> - </figure> - </div> - <p> - <s xml:id="s2450" xml:space="preserve">Alia etiam ratione idipÅ¿um <reg norm="demonÅ¿trari" type="context">demõſtrari</reg> poteÅ¿t, <lb/>Å¿umpto <var>.b.</var> pro medio termino inter <var>.a.</var> et .c: et <var>.c.</var> <lb/>pro termino medio inter <var>.b.</var> et <var>.d</var>. </s> - <s xml:id="s2451" xml:space="preserve">quare propor-<lb/>tio <var>.a.</var> ad <var>.c.</var> componetur ex <var>.a.</var> ad <var>.b.</var> et <var>.b.</var> ad <var>.c.</var> illa <lb/>verò quæ eÅ¿t <var>.b.</var> ad <var>.d.</var> ex <var>.b.</var> ad <var>.c.</var> et <var>.c.</var> ad <var>.d.</var> ex .12. - <pb o="202" rhead="IO. BAPT. BENED." n="214" file="0214" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0214"/> - poÅ¿tulato. </s> - <s xml:id="s2452" xml:space="preserve">Sed cum proportio <var>.a.</var> ad <var>.b.</var> Ä™qualis Å¿it <lb/> - <anchor type="figure" xlink:label="fig-0214-01a" xlink:href="fig-0214-01"/> - proportioni <var>.c.</var> ad <var>.d.</var> communis autem <var>.b.c</var>: propor <lb/>tio. </s> - <s xml:id="s2453" xml:space="preserve">itaque <var>.a.</var> ad <var>.c.</var> æqualis erit <var>.b.</var> ad <var>.d.</var> ex Å¿ecunda <lb/>parte .2. poÅ¿tulati compoÅ¿itè, & Å¿ic habebimus pro <lb/>poÅ¿itum, ita quòd quotieÅ¿cunque <reg norm="dabuntur" type="context">dabÅ©tur</reg> .4. <reg norm="quam" type="context">quã</reg> <lb/>titates ex una parte proportionales, illæ ipſæ ex <lb/>altera proportionales erunt.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0214-01" xlink:href="fig-0214-01a"> - <image file="0214-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0214-01"/> - </figure> - </div> - </div> - <div type="section" level="4" n="8"> - <head xml:id="head348" xml:space="preserve">THEOR. XVII.</head> - <p> - <s xml:id="s2454" xml:space="preserve">DEcimiÅ¿eptimi theorematis hæc eÅ¿t demonÅ¿tratio. </s> - <s xml:id="s2455" xml:space="preserve">Ita Å¿e ha beat <var>a.c.b.</var> ad <var>.c.<lb/>b.</var> Å¿icut Å¿e habet <var>.d.f.e.</var> ad <var>.f.e</var>. </s> - <s xml:id="s2456" xml:space="preserve">Probo ita Å¿e habere <var>.a.c.</var> ad <var>.c.b.</var> Å¿icut Å¿e habet <var>.d.<lb/>f.</var> ad <var>.f.e</var>. </s> - <s xml:id="s2457" xml:space="preserve">Cogitemus itaque alterum terminum Å¿cilicet <var>.n.f.</var> qui Å¿ic Å¿e habeat. ad <var>.f.e.</var> <lb/>Å¿icut Å¿e habet <var>.a.c.</var> ad <var>.c.b</var>. </s> - <s xml:id="s2458" xml:space="preserve">Quare ex præcedenti theoremate ita Å¿e habebit <var>.a.c.</var> ad <var>.n.<lb/>f.</var> Å¿icut Å¿e habet <var>.c.b.</var> ad <var>.f.e.</var> & ex .8 poÅ¿tulato ita Å¿e habebit <var>.a.c.b.</var> ad <var>.n.f.e.</var> Å¿icut Å¿e ha-<lb/>bet <var>.c.b.</var> ad <var>.f.e</var>. </s> - <s xml:id="s2459" xml:space="preserve">Sed cum ex præſuppoÅ¿ito ita Å¿e habeat <var>.a.c.b.</var> ad <var>.c.b.</var> Å¿icut Å¿e habet <var>.<lb/>d.f.e.</var> ad <var>.f.e.</var> ideo ex præcedenti theoremate ita Å¿e habebit <var>.a.c.b.</var> ad <var>.d.f.e.</var> Å¿icut Å¿e ha <lb/>bet <var>.c.b.</var> ad <var>.f.e.</var> demonÅ¿tratum autem eÅ¿t ita Å¿e habere <var>.c.b.</var> ad <var>.f.e.</var> Å¿icut Å¿e habet <var>.a.c.b.</var> <lb/>ad <var>.n.f.e</var>. </s> - <s xml:id="s2460" xml:space="preserve">Quare ex .7. poÅ¿tulato proportio <var>.a.c.b.</var> ad <var>.d.f.</var> e, æqualis erit proportioni <var>.a.<lb/>c.b.</var> ad <var>.n.f.e.</var> & ex .4. poÅ¿tulato <var>.d.f.e.</var> æqualis erit <var>.n.f.e</var>. </s> - <s xml:id="s2461" xml:space="preserve">Itaque ex 3. poÅ¿tulato primi <lb/>Euclidis <var>.f.d.</var> æqualis erit <var>.n.f</var>. </s> - <s xml:id="s2462" xml:space="preserve">Quamob <lb/>rem proportio <var>.a.c.</var> ad <var>.d.f.</var> Ä™qualis erit <lb/> - <anchor type="figure" xlink:label="fig-0214-02a" xlink:href="fig-0214-02"/> - proportioni <var>.a.c.</var> ad <var>.n.f.</var> ex Å¿ecunda par-<lb/>te tertij axiomatis præmiſſi. </s> - <s xml:id="s2463" xml:space="preserve">Igitur ita <lb/>Å¿e habebit <var>.a.c.</var> ad <var>.d.f.</var> Å¿icut <var>.c.b.</var> ad <var>.f.e.</var> ex <lb/>7. poÅ¿tulato. </s> - <s xml:id="s2464" xml:space="preserve">& Å¿ic ex præcedenti theo-<lb/>remate ita Å¿e habebit <var>.a.c.</var> ad <var>.c.b.</var> Å¿icut <var>.d.f.</var> ad <var>.f.e.</var> quod erat propoÅ¿itum: </s> - <s xml:id="s2465" xml:space="preserve">QuotieÅ¿-<lb/>cunque igitur dabuntur .4. quantitates coniunctim proportionales, diuiÅ¿im quoque <lb/>proportionales erunt.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0214-02" xlink:href="fig-0214-02a"> - <image file="0214-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0214-02"/> - </figure> - </div> - </div> - <div type="section" level="4" n="9"> - <head xml:id="head349" xml:space="preserve">THEOREM. XVIII.</head> - <p> - <s xml:id="s2466" xml:space="preserve">THeorema .18. hac ratione demonÅ¿trari poteÅ¿t. </s> - <s xml:id="s2467" xml:space="preserve">Detur proportio <var>.a.c.</var> ad <var>.c.b.</var> Å¿i-<lb/>milis ei quæ eÅ¿t <var>.d.f.</var> ad <var>.f.e.</var> probo ita Å¿e habere <var>.a.c.b.</var> ad <var>.c.b.</var> Å¿icut Å¿e habet <var>.d.f.<lb/>e.</var> ad <var>.f.e</var>. </s> - <s xml:id="s2468" xml:space="preserve">In primis notum eÅ¿t ex .16. theoremate ita Å¿e habiturum, <var>a.c.</var> ad <var>.d.f.</var> Å¿i <lb/>cut <var>.c.b.</var> ad <var>.f.e</var>. </s> - <s xml:id="s2469" xml:space="preserve">Quare ex .8. poÅ¿tulato ita <lb/>Å¿e habebit <var>.a.c.b.</var> ad <var>.d.f.e.</var> Å¿icut <var>.c.b.</var> ad <var>.f.e.</var> <lb/> - <anchor type="figure" xlink:label="fig-0214-03a" xlink:href="fig-0214-03"/> - </s> - <s xml:id="s2470" xml:space="preserve">Itaque ex .16. theoremate ita Å¿e habebit <var>.<lb/>a.c.b.</var> ad <var>.c.b.</var> Å¿icut <var>.d.f.e.</var> ad <var>.f.e</var>. </s> - <s xml:id="s2471" xml:space="preserve">Quod erat <lb/>propoÅ¿itum. </s> - <s xml:id="s2472" xml:space="preserve">QuotieÅ¿cunque igitur .4. <lb/>quantitates dabuntur vnius <reg norm="eiuÅ¿demque" type="simple">eiuÅ¿demqÌ;</reg> generis diÅ¿iunctim proportionales, coniun-<lb/>ctim quoque proportionales erunt.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0214-03" xlink:href="fig-0214-03a"> - <image file="0214-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0214-03"/> - </figure> - </div> - </div> - <div type="section" level="4" n="10"> - <head xml:id="head350" xml:space="preserve">THEOREM. XIX.</head> - <p> - <s xml:id="s2473" xml:space="preserve">THeorema .19. Å¿atis quidem apud Euclidem demonÅ¿tratur: </s> - <s xml:id="s2474" xml:space="preserve">eius tamentertia <lb/>pars commodius hac ratione demonÅ¿trari poterit (nempe) quod cum Å¿it pro- - <pb o="203" rhead="IN QVINT. LIB. EVCLI." n="215" file="0215" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0215"/> - portio <var>.a.</var> ad <var>.b.</var> quæ eÅ¿t <var>.c.</var> ad <var>.d.</var> probabo ita Å¿e habituram proportionem <var>.b.</var> ad <var>.a.</var> Å¿icut <lb/>Å¿e habet <var>.d.</var> ad <var>.c.</var> hoc argumento: </s> - <s xml:id="s2475" xml:space="preserve">Å¿i <var>.a.</var> ad <var>.b.</var> ita Å¿e <lb/>habet Å¿icut <var>.c.</var> ad <var>.d.</var> ex .16. theoremate ita Å¿e ha <lb/> - <anchor type="figure" xlink:label="fig-0215-01a" xlink:href="fig-0215-01"/> - bebit <var>.a.</var> ad <var>.c</var>, Å¿icut <var>.b.</var> ad <var>.d</var>. </s> - <s xml:id="s2476" xml:space="preserve">Quare Å¿ic Å¿e habebit <lb/>b. ad <var>.d.</var> Å¿icut <var>.a.</var> ad <var>.c</var>. </s> - <s xml:id="s2477" xml:space="preserve">Itaque ex eodem .16. ita Å¿e <lb/>Å¿e habebit <var>.b.</var> ad <var>.a.</var> Å¿icut <var>.d.</var> ad <var>.c</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0215-01" xlink:href="fig-0215-01a"> - <image file="0215-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0215-01"/> - </figure> - </div> - </div> - <div type="section" level="4" n="11"> - <head xml:id="head351" xml:space="preserve">THEOREM. XX.</head> - <p> - <s xml:id="s2478" xml:space="preserve">QVamuis .20. theorema apud Eucli. perfectè demonÅ¿tratum fuerit, poteÅ¿t ni-<lb/>hilominus & hac via demonÅ¿trari. </s> - <s xml:id="s2479" xml:space="preserve">Sic Å¿e habeat proportio <var>.a.</var> ad <var>.b.</var> Å¿icut Å¿e <lb/>habet <var>.c.</var> ad <var>.d.</var> & proportio <var>.b.</var> ad <var>.e.</var> Å¿icut <var>.d.</var> ad <var>.<lb/>f</var>. </s> - <s xml:id="s2480" xml:space="preserve">Dico <reg norm="quod" type="simple">ê™</reg> Å¿i <var>.a.</var> maius fuerit <var>.e.</var> pariter <var>.c.</var> maius <lb/> - <anchor type="figure" xlink:label="fig-0215-02a" xlink:href="fig-0215-02"/> - erit <var>.f.</var> & Å¿i <var>.a.</var> minus fuerit .e: c. <reg norm="quoque" type="simple">quoq;</reg> minus erit <lb/>f. Å¿in verò Ä™quale, <reg norm="ent" type="context">ẽt</reg> æquale erit. </s> - <s xml:id="s2481" xml:space="preserve">Nam ex pri <lb/>mo poÅ¿tulato certi Å¿umus ita Å¿e habere pro <lb/><reg norm="portionem" type="context">portionẽ</reg> <var>.a.</var> ad <var>.e.</var> Å¿icut Å¿e habet proportio <var>.c.</var> ad <lb/>p. </s> - <s xml:id="s2482" xml:space="preserve">Quare ex .12. theor <reg norm="propoÅ¿itum" type="simple context">ê“poÅ¿itÅ©</reg> <reg norm="manifeÅ¿tum" type="context">manifeÅ¿tÅ©</reg> erit.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0215-02" xlink:href="fig-0215-02a"> - <image file="0215-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0215-02"/> - </figure> - </div> - </div> - <div type="section" level="4" n="12"> - <head xml:id="head352" xml:space="preserve">THEOREM. XXI.</head> - <p> - <s xml:id="s2483" xml:space="preserve">VIgeÅ¿imum primum theorema, Å¿atis apud Eucli. probatum, nihilominus præ-<lb/>Å¿cripto nunc modo demonÅ¿trari poterit.</s> - </p> - </div> - <div type="section" level="4" n="13"> - <head xml:id="head353" xml:space="preserve">THEOREM. XXII. XXIII.</head> - <p> - <s xml:id="s2484" xml:space="preserve">DVO hæc theoremata in primum poÅ¿tulatum collegimus. <lb/></s> - <s xml:id="s2485" xml:space="preserve">Sequentia verò cum exactè apud Eucli. demonÅ¿trentur non eÅ¿t cur nos in <lb/>ijs immoremur.</s> - </p> - <pb o="204" n="216" file="0216" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0216"/> - </div> - </div> - </div> - <div type="chapter" level="2" n="6"> - <head xml:id="head354" xml:space="preserve">PHYSICA, <lb/>ET MATHEMATICA <lb/>RESPONSA.</head> - <head xml:id="head355" style="it" xml:space="preserve">FO. BAPTISTAE BεNεDICTI PATRITII <lb/>Veneti, PhiloÅ¿ophi Mathematici.</head> - <div type="preface" level="3" n="1"> - <head xml:id="head356" xml:space="preserve">Ad Lectorem.</head> - <p> - <s xml:id="s2486" xml:space="preserve">VT Nilmagis virtutis eÅ¿t proprium, quà m <lb/>agitari, & inceßabili motu prodeße. </s> - <s xml:id="s2487" xml:space="preserve">Ac velu <lb/>ti fulgidum Å¿ydus ante oculos <reg norm="Å¿pectantium" type="context">Å¿pectantiÅ©</reg> com <lb/>micare. </s> - <s xml:id="s2488" xml:space="preserve">Ita mihi mathematicis <reg norm="ijsque" type="simple">ijsq;</reg> maxi <lb/>mè philoÅ¿ophicis Å¿peculationibus dedito, Å¿apiÅ¿-<lb/>Å¿imè, ut in principium Å¿ummorum aulis, & <lb/>amplißimis ciuitatibus degenti, ubi multa Å¿em <lb/>per Nobilium mir a curioÅ¿itate, Å¿ciendi deÅ¿iderio, & conferendicu <lb/>piditate referta, uerÅ¿antur ingenia, contigit, modo ab his, modo ab <lb/>illis, aut uerbis tentari, aut literis prouocari ad diſſerendum, de <lb/>his, in quorum Å¿tudijs uerÅ¿amur. </s> - <s xml:id="s2489" xml:space="preserve">Quarum concertationum & re <lb/>Å¿ponÅ¿ionum, quoniam non omnino indigna exiÅ¿timaui, quæmemoriæ<unsure/> <lb/>comendarentur, partem aliquam apud me conÅ¿eruaui. </s> - <s xml:id="s2490" xml:space="preserve">Vbi uerò <lb/>per ocium licuit, collegi, relegi, ac tandem de manu mittere decreui. <lb/></s> - <s xml:id="s2491" xml:space="preserve">Tum ut Å¿cientia ipÅ¿a quo magis diffundetur, creÅ¿cat; </s> - <s xml:id="s2492" xml:space="preserve">& quicquid <lb/>ualeo, Å¿ine inuidia in communem utilitatem conferam. </s> - <s xml:id="s2493" xml:space="preserve">Tum ut ui-<lb/>rorum præctantiβimorum, qui me Å¿uis interrogationibus excitaue <lb/>runt, quantum in me erit, gratitudinis ergo, nomina reddam im-<lb/>mortalia, & eorum exemplo alios, ocio Å¿ordidiore abiecto, quod Å¿olet <lb/>ourialium præcipuè excelÅ¿a ingenia corrumpere, ad Å¿ciÅ¿cit andum <lb/>conferendum, & diſſerendum, derebus <reg norm="Å¿erijs" type="lig">Å¿erijs</reg>, & quæuÅ¿ui aliquan-<lb/>do eße poßint, & <reg norm="quandoque" type="simple">quandoq;</reg> euulgari mereantur, alliciam. </s> - <s xml:id="s2494" xml:space="preserve">Tuinte-<lb/>rim nostris laboribus fruere, & nostram diligentiam boni, & æqui <lb/>conÅ¿ule, & Vale.</s> - </p> - <pb o="205" n="217" file="0217" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0217"/> - </div> - <div type="section" level="3" n="1"> - <div type="letter" level="4" n="1"> - <head xml:id="head357" xml:space="preserve">DETEMPORVM <lb/>EMENDATIONE <lb/>IO. BAPTIST AE BENEDICTI <lb/>Patritij Veneti, PhiloÅ¿ophi <lb/>Mathematici.</head> - <head xml:id="head358" style="it" xml:space="preserve">AD SERENISS. CMANVELEM PHILIB. <lb/>Allobrogum & Å¿ubalpinarum gentium Ducem <lb/>Inuictiβimum.</head> - <head xml:id="head359" xml:space="preserve">EPISTOLA.</head> - <p> - <s xml:id="s2495" xml:space="preserve"><emph style="sc">MIrvm</emph>, Quà m lectione epiÅ¿tolæſeu (vt vocant) Breuis <var>.S.D.<lb/>N</var>. </s> - <s xml:id="s2496" xml:space="preserve">Gregorij XIII. Pont. Max. quod ad me nuper tua CelÅ¿itu <lb/>do miÅ¿it ex Nicea, vt meam de ea re Å¿ententiam proferrem, <lb/>delectatus Å¿im; </s> - <s xml:id="s2497" xml:space="preserve">ex quo, non tantum recta illius mens ac verè <lb/>Å¿ancta cogitatio, Å¿ed etiam aperta <reg norm="maximaque" type="simple">maximaqÌ;</reg>, Å¿i ad exitum per <lb/><reg norm="ducatur" type="simple">ducat̃</reg>, imo Å¿ummè neceſſaria vniuerÅ¿o orbi vtilitas percipi <lb/>poteÅ¿t; </s> - <s xml:id="s2498" xml:space="preserve">qua de re memini cum CelÅ¿itudine tua aliquando Å¿er-<lb/>monem habuiſſe. </s> - <s xml:id="s2499" xml:space="preserve">Vidi præterea cum ipÅ¿o breui tranÅ¿-<lb/>miſſum compendium Domini AloiÅ¿ij Lilij: </s> - <s xml:id="s2500" xml:space="preserve">cuius mihi Å¿ententia perplacet, de corre <lb/>ctione eius diei, qui 134. quoque anno præter, neceſſitatem, gignitur. </s> - <s xml:id="s2501" xml:space="preserve">qui Å¿anè dies <lb/>perpetuæ retrogradationis ingreſſus Solis in Zodiaci Å¿igna, cauÅ¿a fuit. </s> - <s xml:id="s2502" xml:space="preserve">quod ita per-<lb/>Å¿picuè patebit. </s> - <s xml:id="s2503" xml:space="preserve">Cum Numa Pompilius anni curÅ¿um correxit <reg norm="emendauitque" type="simple">emendauitqÌ;</reg>, ea Å¿anè <lb/>mente id videtur præſtitiſſe, vt principium Ianuarij primi menÅ¿is anni, præcisè in ip <lb/>Å¿o hyemalis Å¿oltitij puncto collocaretur. </s> - <s xml:id="s2504" xml:space="preserve">quod hac tempeÅ¿tate, dictam ob cauÅ¿am <lb/>adeò retroceſſit, vt circa vndecimam diem Decembris eſſe reperiatur. </s> - <s xml:id="s2505" xml:space="preserve">quod Å¿i cen-<lb/>teÅ¿imo trigeÅ¿imo quarto quoque anno detractus dies vnus fuiſſet, nihil erroris pror-<lb/>Å¿us accidiſſet. </s> - <s xml:id="s2506" xml:space="preserve"><reg norm="Atque" type="simple">Atq;</reg> dies hic (vt alias CelÅ¿it. tuæ Å¿ignificaui) inde generatur, quod quar <lb/>to quoque anno addentes nos ad quarti anni dies .365. diem horarum .24. ob erro-<lb/>rem annuum horarum quinque minutorum .49. Å¿ecundorum ferè .16. (anni æqualis <lb/>Å¿iue medij) fallimur quarto quoque anno in minutis .42. <reg norm="Å¿ecundis" type="context">Å¿ecÅ©dis</reg> propè .56. amplius <lb/>quà m par Å¿it minutis Å¿cilicet .10. Å¿ecundis ferè .44. Å¿ingulis annis; </s> - <s xml:id="s2507" xml:space="preserve">qui numerus .134. <lb/>multiplicatus, diem penè horarum .24. conÅ¿tituit; </s> - <s xml:id="s2508" xml:space="preserve">penè inquam, quia minutum <reg norm="vnum" type="context">vnÅ©</reg> <lb/>deeſſet <reg norm="tantummodo" type="wordlist">tm̃modo</reg>, & Å¿ecunda .44. Å¿i <reg norm="decem" type="context">decẽ</reg> illa minuta, & .44. Å¿e cunda annua, exquiÅ¿ita eÅ¿ <lb/>Å¿ent atque perfecta; </s> - <s xml:id="s2509" xml:space="preserve">quæ tamen differentia nullius adeo eſſet momenti, aut certè pe-<lb/>rexigui, vt vix exactis .111086. annis, diem vnum afferret. </s> - <s xml:id="s2510" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> planè neceſſaria <lb/>eiuÅ¿modi eſſet emendatio, <reg norm="aptaque" type="simple">aptaqÌ;</reg> eius ratio à D. </s> - <s xml:id="s2511" xml:space="preserve">Lilio oÅ¿tenditur, prout etiam Pe-<lb/>trus Pitatus VeronenÅ¿is tradidit, in eo, quem de vera anni quantitate tractatu con-<lb/>Å¿cripÅ¿it, nempe vt tribus primis centeÅ¿imis annis, centeÅ¿imus quiÅ¿que annus commu-<lb/>nis Å¿it, quartus Å¿ubÅ¿equens centeÅ¿imus intercalaris: </s> - <s xml:id="s2512" xml:space="preserve">quod Å¿anè fierineceſſe eÅ¿t. </s> - <s xml:id="s2513" xml:space="preserve">Nam - <pb o="206" n="218" file="0218" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0218"/> - cùm tribus centeÅ¿imis <reg norm="communibus" type="context">cõmunibus</reg>, tres quartas diei partes plus æquo detraxerimus, <lb/>non enim centeÅ¿imo quoque anno, Å¿ed centeÅ¿imo trigeÅ¿imo quarto, dictus dies de-<lb/>trahi debet, poÅ¿tquam tres integros dies, qui quadringentis detrahendi erant, tre-<lb/>centorum annorum Å¿pacio detraxerimus; </s> - <s xml:id="s2514" xml:space="preserve">Å¿itqÌue 134. penè tertia pars .400. quarto <lb/>annorum centenario, tres quartæ diei partes recuperabuntur; </s> - <s xml:id="s2515" xml:space="preserve">atque ita in fine qua-<lb/>dringentorum annorum omnia exactè Å¿uo loco reÅ¿tituta erunt. </s> - <s xml:id="s2516" xml:space="preserve">Idcirco dictus iam <lb/>quadringenteÅ¿imus annus intercalaris & non communis conÅ¿tituendus erit, non alia <lb/>de cauÅ¿a, quam vt biſſexti ordinem Å¿equamur.</s> - </p> - <p> - <s xml:id="s2517" xml:space="preserve">Is verò modus, qui à D. </s> - <s xml:id="s2518" xml:space="preserve">Lilio traditus eÅ¿t, de ratione inueniendi Å¿ingulis menÅ¿ibus <lb/>Nouilunij diem, interdum fallere nos poſſet vno die; </s> - <s xml:id="s2519" xml:space="preserve">prout Ianuario proximè <lb/>lapÅ¿o accidit; </s> - <s xml:id="s2520" xml:space="preserve">quo ex præſcripto modo nouilunij, dies nonus illius menÅ¿is eſſe debuiÅ¿ <lb/>Å¿et, qui fuit tamen dies Å¿eptimus, Å¿exta decima hora cum dimidia poÅ¿t meridiem. </s> - <s xml:id="s2521" xml:space="preserve">Ne <lb/>que etiam tutum eÅ¿t, via integrorum dierum, nulla habita horarum aut minutorum <lb/>ratione, nec minus ea, quæ à Pitato tradita eÅ¿t, mediorum Å¿eu æqualium <reg norm="motuum" type="context">motuÅ©</reg> pro <lb/>gredi: </s> - <s xml:id="s2522" xml:space="preserve">At cenÅ¿erem potius veros motus Å¿equendos eſſe ex calculis exactarum tabu-<lb/>larum, quales Prutenicas eſſe iudico; </s> - <s xml:id="s2523" xml:space="preserve">Et cum Å¿olius PaÅ¿chæ cauÅ¿a laboremus hac in <lb/>re, pleniluniorum verorum, in multos annos tabulas formarem, quæ æquinoctia ver <lb/>nalia Å¿equuntur, cum aſſignatione diei PaÅ¿chatis præcisè, prout fecit Pitatus; </s> - <s xml:id="s2524" xml:space="preserve">non <lb/>via tamen æqualium pleniluniorum Å¿ed verorum. </s> - <s xml:id="s2525" xml:space="preserve">Porrò quod ad PaÅ¿chatis cele-<lb/>brationem attinet, rationi conÅ¿entaneum eÅ¿t, concilij Niceni decretum ea de re Å¿er <lb/>uari, prima Å¿cilicet dominica die poÅ¿t primum plenilunium, quod æquinoctium ver-<lb/>nale Å¿equitur; </s> - <s xml:id="s2526" xml:space="preserve">hoc tamen animaduerÅ¿o, Å¿i dictum plenilunium primum poÅ¿t æquino-<lb/>ctium contingens, <reg norm="diem" type="context">diẽ</reg> dominicum Å¿ortiretur; </s> - <s xml:id="s2527" xml:space="preserve">nulla ratione tali die PaÅ¿cha celebran-<lb/>dum eſſe; </s> - <s xml:id="s2528" xml:space="preserve">verum Å¿ubÅ¿equenti, ne cum HÄ™breis conÅ¿entiat EccleÅ¿ia ChriÅ¿ti: </s> - <s xml:id="s2529" xml:space="preserve">quæ fuit <lb/>cauÅ¿a, vt in decreto concilij Niceni Å¿tatutum Å¿it, à quartadecima, in vigeÅ¿imam pri-<lb/>mam celebrari debere: </s> - <s xml:id="s2530" xml:space="preserve">Quod mihi Petrus Pitatus non animaduertiſſe videtur, cum <lb/>ex <reg norm="eius" type="simple">eiê°</reg> <reg norm="Å¿ententia" type="context">Å¿entẽtia</reg> in Å¿uis tabulis die PaÅ¿chate declarata, huiuÅ¿ce anni PaÅ¿ca <reg norm="celebrandum" type="context">celebrandÅ©</reg> <lb/>fuerit .23. </s> - <s xml:id="s2531" xml:space="preserve">Martij, ipÅ¿omet de plenilunij non tantum æqualis, Å¿ed veri.</s> - </p> - <p> - <s xml:id="s2532" xml:space="preserve">Dies autem PaÅ¿chatum elapÅ¿orum, quos hactenus examinaui, reperi omnes con <lb/>cordare cum ea regula, quam nonnulli de die carnis priuij tradiderunt. </s> - <s xml:id="s2533" xml:space="preserve">nempe pri-<lb/>mum diem martis poÅ¿t nouilunium Februarij, carnis priuij diem eſſe; </s> - <s xml:id="s2534" xml:space="preserve">non autem <reg norm="cum" type="context">cÅ©</reg> <lb/><reg norm="sanctione" type="context">sãctione</reg> Patrum concilij Niceni, qua Å¿tatuerunt à vigeÅ¿ima prima Martii dirigen-<lb/>dum eſſe PaÅ¿chatis diem, vt potè qui Å¿ibi perÅ¿uaÅ¿erunt, circa eum diem <reg norm="æquinoctium" type="context">æquinoctiÅ©</reg> <lb/>perpetuò eſſe debere; </s> - <s xml:id="s2535" xml:space="preserve">prout tunc temporis erat. </s> - <s xml:id="s2536" xml:space="preserve">Non <reg norm="itaque" type="simple">itaq;</reg> error accidit, quod Pa <lb/>Å¿cha ex huiuÅ¿modi Å¿uppoÅ¿itione concilij, poÅ¿t vigeÅ¿imam primam lunÄ™ celebretur, <lb/>cum Å¿eruata regula concilij non fuerit. </s> - <s xml:id="s2537" xml:space="preserve">Prout <reg norm="manifeÅ¿tum" type="context">manifeÅ¿tÅ©</reg> eÅ¿t de PaÅ¿chate anni .1566. <lb/>celebrato .14. </s> - <s xml:id="s2538" xml:space="preserve">Aprilis (quÄ™ fuit .24. lunÄ™) quod .7. dicti menÅ¿is <reg norm="celebrandum" type="context">celebrãdum</reg> erat. </s> - <s xml:id="s2539" xml:space="preserve">Tum <lb/>anno .1569. 10. </s> - <s xml:id="s2540" xml:space="preserve">Aprilis Å¿olenne fuit PaÅ¿ca, quod tertia eiuÅ¿dem eſſe debuerat. </s> - <s xml:id="s2541" xml:space="preserve">Anno <lb/>deinde 1572. 6. Aprilis, dies fuit PaÅ¿chatis, quæ .30. Martij futura erat, anno vero <lb/>1575. in tertiam Aprilis PaÅ¿cha incidit, caÅ¿urum in .27. Martii.</s> - </p> - <p> - <s xml:id="s2542" xml:space="preserve">Cum igirur (vt ex diplomate ad CelÅ¿it. tuam miſſo patet) <var>S.D.N.</var> mens Å¿it <reg norm="atque" type="simple">atq;</reg> vo <lb/>luntas, ut quiÅ¿que liberè in medium proferat quid hac dere Å¿entiat: </s> - <s xml:id="s2543" xml:space="preserve">quædam mihi <lb/>non omnino præmittenda occurrunt, quæ tantis cÅ“ptis non nihil adiumenti for-<lb/>taſſe adferre queant.</s> - </p> - <p> - <s xml:id="s2544" xml:space="preserve">Atque illud in primis non tantum ut corrigatur Calendarium ob PaÅ¿cha cætera-<lb/>qÌue feÅ¿ta mobilia ab illo manantia, vt decreto concilij Niceni Å¿ancitum eÅ¿t, Å¿cilicet <lb/>vt ipÅ¿um PaÅ¿cha celebretur prima dominica poÅ¿t primum plenilunium, quod æqui- - <pb o="207" n="219" file="0219" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0219"/> - noctium vernale proximè Å¿equitur; </s> - <s xml:id="s2545" xml:space="preserve">verum etiam quò anni principium emendetur, <lb/>Å¿cilicet vt ad Å¿uum verum principium reuocetur annus. </s> - <s xml:id="s2546" xml:space="preserve">Nempè ad diem hyemalis <lb/>Å¿olÅ¿titij, quæ prima Ianuarij dies eſſe debet.</s> - </p> - <p> - <s xml:id="s2547" xml:space="preserve">Deinde, tot dierum menÅ¿es conÅ¿tituantur, quot hac noÅ¿tra tempeÅ¿tate, Å¿ol in ipÅ¿is <lb/>Zodiaci Å¿ignis verÅ¿atur. </s> - <s xml:id="s2548" xml:space="preserve">PoÅ¿tremò, quædam feÅ¿ta immobilia in alios dies <reg norm="transferam" type="context">transferã</reg> <lb/>tur, <reg norm="celebrenturque" type="simple">celebrenturqÌ;</reg> aptis temporibus: </s> - <s xml:id="s2549" xml:space="preserve">quod à <var>.S.D.N.</var> mente diſſentire minimè vide-<lb/>tur. </s> - <s xml:id="s2550" xml:space="preserve">cum non magis de feÅ¿tis mobilibus quam immobilibus agat, imo etiam planè <lb/>æquum Å¿it, vt habeatur vtrorunque ratio, quò Å¿tatutis temporibus celebrentur.</s> - </p> - <p> - <s xml:id="s2551" xml:space="preserve">Vt autem ad primam Ianuarij <reg norm="diem" type="context">diẽ</reg> verum principium anni reuocetur; </s> - <s xml:id="s2552" xml:space="preserve">cenÅ¿erem <lb/>ex eo anno, quem corrigere voluerimus, non modò dies .10. eſſe detrahendos, <reg norm="verum" type="context">verÅ©</reg> <lb/>etiam vnum & uiginti, illo ipÅ¿o anno; </s> - <s xml:id="s2553" xml:space="preserve"><reg norm="idque" type="simple">idqÌ;</reg> duplici via; </s> - <s xml:id="s2554" xml:space="preserve">aut partiendo menÅ¿es, atque <lb/>ex illis demendo eos dies, qui minus ad rem hanc facere videbuntur, ac tum rema-<lb/>neat annus trecentorum quadraginta quatuor dierum ita vt decem menÅ¿es Å¿int die. <lb/></s> - <s xml:id="s2555" xml:space="preserve">rum duorum Å¿patio Å¿olito breuiores, alter menÅ¿is vno deficiat: </s> - <s xml:id="s2556" xml:space="preserve">aut conÅ¿tituto <reg norm="Decem- bri" type="context">Decẽ-<lb/>bri</reg> dicti anni dierum decem, dies autem ille, qui decimum <reg norm="proximem" type="context">proximẽ</reg> Å¿equitur, Å¿it & <lb/>primus Ianuarij, & dies Å¿olÅ¿titij ob quam cauÅ¿am exiÅ¿timarem conÅ¿ultiſſimum eiuÅ¿ <lb/>modi annum eſſe mileÅ¿imum quingenteÅ¿imum Å¿eptuageÅ¿imum nonum. </s> - <s xml:id="s2557" xml:space="preserve">Quo quam <lb/>primum <var>.S.D.N</var>. </s> - <s xml:id="s2558" xml:space="preserve">Pontifex max. Å¿uis temporibus huius correctionis manifeÅ¿tos effe-<lb/>ctus experiri & perpendere, atque diÅ¿poÅ¿itionem anni non Å¿olum principio, Å¿ed <reg norm="cae teris" type="simple">cÄ™Â<lb/>teris</reg> partibus Å¿uis in vniuerÅ¿um tam concinnè <reg norm="apteque" type="simple">apteqÌ;</reg> reÅ¿pondere, & aÅ¿trorum moti-<lb/>bus, & EccleÅ¿iæ Å¿acroÅ¿anctæ Å¿anctionibus, Å¿e authore lætari poſſit.</s> - </p> - <p> - <s xml:id="s2559" xml:space="preserve">Omnino <reg norm="itaque" type="simple">itaq;</reg> iudico detrahendos eſſe vnum & viginti dies elapÅ¿i erroris: </s> - <s xml:id="s2560" xml:space="preserve">non de <lb/>cem tantum, quo hyemmalis conuerÅ¿io ad initium Ianuarij reuocetur; </s> - <s xml:id="s2561" xml:space="preserve">idqÌue ne à <lb/>communi opinione de ipÅ¿o anni principio veritas diÅ¿crepet, quæ principium Ianua-<lb/>rij, anni principium arbitratur. </s> - <s xml:id="s2562" xml:space="preserve">etenim cum credant omnes <reg norm="annum" type="context">annÅ©</reg> à Ianuario inchoa-<lb/>ri, veritas autem ipÅ¿a Å¿ic Å¿e habeat, vt nobis Å¿eptentrionalibus tunc inchoet annus, <lb/>cum ad nos Sol accedere incipit, aut dies augetur; </s> - <s xml:id="s2563" xml:space="preserve">non conuenit principia eiuÅ¿mo-<lb/>di Å¿eparata & diÅ¿crepantia eſſe. </s> - <s xml:id="s2564" xml:space="preserve">Et hanc fuiſſe Numæ Pompilio mentem credibile <lb/>eÅ¿t, qui ad annum Romuli decem menÅ¿ium, Ianuarium & Februarium addidit, vt <lb/>principium Ianuarij principium eſſet anni: </s> - <s xml:id="s2565" xml:space="preserve">cuius rei argumentum eſſe poteÅ¿t, quod <lb/>C. </s> - <s xml:id="s2566" xml:space="preserve">Iulij Cæſaris temporibus (qui multis annis poÅ¿t Numam fuit) <reg norm="atque" type="simple">atq;</reg> vti Pont. </s> - <s xml:id="s2567" xml:space="preserve">Max. <lb/></s> - <s xml:id="s2568" xml:space="preserve">corrigendorum feÅ¿torum curam Å¿uÅ¿cepit hyemale Å¿olÅ¿titium per aliquot dies retro-<lb/>ceſſerat; </s> - <s xml:id="s2569" xml:space="preserve">nec mirum tamen eſſet, Å¿i Numæ temporibus, exactè prima Ianuarij die non <lb/>fuiſſet hyemale Å¿olÅ¿titium, adhuc pubeÅ¿cente in Italia AÅ¿tronomia.</s> - </p> - <p> - <s xml:id="s2570" xml:space="preserve">HuiuÅ¿modi autem correctio dierum .21. poÅ¿t .2300. annos à Numa, quæ Å¿it per-<lb/>petuo Å¿eruitura, media emendatione ea, quæ de tribus centeÅ¿imis annis communi-<lb/>bus, & quarto intercalari, Å¿uperius propoÅ¿ita fuit, non repudianda ei videatur, qui <lb/>Å¿ciet, qua ratione Numæ Pompilij annus corrigeretur, octauo quoque anno, inter-<lb/>calando annum vltimum medijs diebus .90. quo prima dies Ianuarij ad verum prin <lb/>cipium anni, hoc eÅ¿t hyemale Å¿olÅ¿titium, reduceretur.</s> - </p> - <p> - <s xml:id="s2571" xml:space="preserve">Alio item argumento cuique patere poteÅ¿t, priÅ¿cos Romanos Å¿tatuiſſe annum ab <lb/>hyemali Å¿olÅ¿titio initium Å¿umere, vt inquit Ouidius primo FaÅ¿torum.</s> - </p> - <p> - <s xml:id="s2572" xml:space="preserve">Bruma noui prima eÅ¿t, <reg norm="veterisque" type="simple">veterisqÌ;</reg> nouiſſima Solis.</s> - </p> - <p> - <s xml:id="s2573" xml:space="preserve">Principium capiunt PhÄ™bus, & annus idem.</s> - </p> - <p> - <s xml:id="s2574" xml:space="preserve">co quod diem naturalem à medio noctis inchoarent, ab eo puncto Å¿cilicet, quo Sol <lb/>ad noÅ¿trum hemiÅ¿pherium accedere incipit.</s> - </p> - <p> - <s xml:id="s2575" xml:space="preserve">Tribuebant igitur veteres diei, atque anno principium ab eo puncto, quo Sol - <pb o="208" n="220" file="0220" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0220"/> - ad nos accedit: </s> - <s xml:id="s2576" xml:space="preserve">cum punctum Zodiaci, quod tropicum hyemalem Capricorni nobis <lb/>producit, reÅ¿pondeat puncto meridiani Å¿ub terra, in quo Sol Å¿emel in die reperitur: <lb/></s> - <s xml:id="s2577" xml:space="preserve">Quòd apertè norunt hi, qui Å¿ub polo boreali conÅ¿tituti Å¿unt. </s> - <s xml:id="s2578" xml:space="preserve"><reg norm="Atque" type="simple">Atq;</reg> facilè diÅ¿cerne <lb/>re poſſumus, diem Å¿cilicet & annum, quaÅ¿i Å¿ibi ad inuicem medio Å¿uarum partium <lb/>reÅ¿pondere; </s> - <s xml:id="s2579" xml:space="preserve">Å¿olÅ¿titium inquam hyemale, mediæ nocti, æſtiuum meridiei, æquino-<lb/>ctium vernale ortui Solis, autumnale occaÅ¿ui. </s> - <s xml:id="s2580" xml:space="preserve">Quam tamen Å¿imilitudinem, multò <lb/>quam nos manifeÅ¿tius deprehendunt, hi qui (ut diximus) Å¿ub polo borcali verÅ¿antur.</s> - </p> - <p> - <s xml:id="s2581" xml:space="preserve">Quod Å¿i quis dubitet hac ratione correcto anno, quo nam pacto ad calculos coe-<lb/>leÅ¿tes motus medijs tabulis aÅ¿tronomicis hactenus in lucem æditis redigi poſſint, id <lb/>facilimum Å¿anè erit, exempli gratia; </s> - <s xml:id="s2582" xml:space="preserve">aliquis planetÄ™ Å¿itum, aut alicuius Å¿tellÄ™ fixæ, quo <lb/>cunque die menÅ¿is anni correcti inuenire cupit, detrahat ex huiuÅ¿modi <reg norm="tempore" type="context">tẽpore</reg> dies <num value="21">.<lb/>21.</num> ab Aera ChriÅ¿ti, cum reÅ¿iduo Å¿upputet Å¿tellam, cuius Å¿itum Å¿cire deÅ¿iderat; </s> - <s xml:id="s2583" xml:space="preserve">Å¿um-<lb/>pta quacunque tabula, Å¿upputatio erit exacta: </s> - <s xml:id="s2584" xml:space="preserve">Cuius ratio cuilibet manifeÅ¿ta erit, <lb/>qui Å¿ciet annum vt potè .1579. dierum .344. tantummodo conÅ¿titutum fuiſſe. </s> - <s xml:id="s2585" xml:space="preserve">Nam <lb/>in ijÅ¿dem locis cÅ“li prima die Ianuarij correcti, erunt Å¿tellæ quibus eſſe Å¿olebant .11 <lb/>Decembris præcedentis anni ex Å¿upputatione tabularum: </s> - <s xml:id="s2586" xml:space="preserve">atque ita deinceps. </s> - <s xml:id="s2587" xml:space="preserve">Alia <lb/>præterea via idem perfici poſſet inuentione omnium motuum cÄ™leÅ¿tium ipÅ¿o princi <lb/>pio anni .1580. correcti: </s> - <s xml:id="s2588" xml:space="preserve">hoc Å¿tatuto, vt hi motus radices eſſent Aeræ <var>S.D.N.</var> Grego <lb/>rij XIII. quod Å¿i alio <reg norm="tempore" type="context">tẽpore</reg> quiÅ¿piam motus cÄ™leÅ¿tes ad calculos redigere voluerit, <lb/>Å¿upputabit ab Aera huiuÅ¿modi, quæ anno .1580. principium habuerit: </s> - <s xml:id="s2589" xml:space="preserve">Quæ vt nobi <lb/>lius nomen Å¿ortiatur, <reg norm="idque" type="simple">idqÌ;</reg> merito ex nomine Gregorij. XIII. Pont. Max. appelletur; <lb/></s> - <s xml:id="s2590" xml:space="preserve">exemplo antiquarum, quæ ex Principum nominibus Å¿unt appellate: </s> - <s xml:id="s2591" xml:space="preserve">vt tanto Pontifi <lb/>ci, cùm ex alijs multi, tum etiam ex hac non infima re, inter mortales immortale no <lb/>men comparetur. </s> - <s xml:id="s2592" xml:space="preserve">Ei verò Å¿ummæ, quæ ex huiuÅ¿modi Aera Gregoriana ex tabu-<lb/>lis colligetur, ipÅ¿iuÅ¿met Aeræ radices addantur, vt exactus calculus habeatur. </s> - <s xml:id="s2593" xml:space="preserve">Et <lb/>hæc Å¿it primæ Å¿ententiæ noÅ¿træ explicatio.</s> - </p> - <p> - <s xml:id="s2594" xml:space="preserve">Altera erit numerum dierum menÅ¿ium anni alia ratione quam nunc Å¿e habeat, or <lb/>dinandum eſſe: </s> - <s xml:id="s2595" xml:space="preserve">nempe vt Ianuarius, Nouember <reg norm="atque" type="simple">atq;</reg> December dies .29. Å¿inguli con <lb/>tineant, Februarius, Martius, & October .30. </s> - <s xml:id="s2596" xml:space="preserve">Aprilis, Maius, AuguÅ¿tus, & September <lb/>dies .31. Iunius, ac Iulius .32. atque id hac potiſſimum de cauÅ¿a, vt Sol unum <reg norm="quodque" type="simple">quodqÌ;</reg> <lb/>Å¿ignum calendis menÅ¿rum ingredi poſſit. </s> - <s xml:id="s2597" xml:space="preserve">Nam detractis (ut dictum eÅ¿t) diebus .21. <lb/>& reuocato ingreſſu Solis in principium Capricorni ad principium Ianuarij, in quo <lb/>Å¿igno hac noÅ¿tra tempeÅ¿tate, Sol, dies propè .29. & quartam vnam verÅ¿atur: </s> - <s xml:id="s2598" xml:space="preserve">Å¿i Ianua <lb/>rius .29. dies continebit, exactis hiÅ¿ce diebus, ingredietur Aquarium circa princi-<lb/>pium Februarii; </s> - <s xml:id="s2599" xml:space="preserve">hæret autem hoc noÅ¿tro ſæculo in Aquario Sol dies propè .29. cum <lb/>dimidio; </s> - <s xml:id="s2600" xml:space="preserve">quare Å¿i Februarius erit .30. dierum, elapÅ¿is ipÅ¿is diebus, Sol ingredietur pi-<lb/>Å¿ces circa principium Martii: </s> - <s xml:id="s2601" xml:space="preserve">& Å¿ic de cæteris.</s> - </p> - <p> - <s xml:id="s2602" xml:space="preserve">Quamobrem Å¿i generali correctione annus emendandus erit, pulcherrimè acci-<lb/>det, Å¿i menÅ¿es anni cum duodecim partibus cÅ“leÅ¿tibus, itineris annui Solis, concor-<lb/>dauerint; </s> - <s xml:id="s2603" xml:space="preserve"><reg norm="eiÅ¿que" type="simple">eiÅ¿qÌ;</reg> aptè reÅ¿ponderint. </s> - <s xml:id="s2604" xml:space="preserve">Qua ex re, variÄ™ vtilitates promanabunt, pręſertim <lb/>Nautis, Agricolis, Medicis, & alijs qui vera principia, & interualla temporum per <lb/>Å¿pecta habebunt: </s> - <s xml:id="s2605" xml:space="preserve">terminos item & interualla incrementi & diminutionis dierum & <lb/>noctium, & eorundem æqualitatis. </s> - <s xml:id="s2606" xml:space="preserve">Exempli cauÅ¿a, Å¿cient omnes principium Ianua-<lb/>rij, eſſe non modo anni principium, verum etiam hyemis, eſſe minimam anni diem, <lb/>& eius noctem maximam; </s> - <s xml:id="s2607" xml:space="preserve">principium incrementi diei, & diminutionis noctis; </s> - <s xml:id="s2608" xml:space="preserve">atque <lb/>etiam omnia illa, quæ ex huiuÅ¿modi conuerÅ¿ione Solis ad nos dependent. </s> - <s xml:id="s2609" xml:space="preserve">Pariter <lb/>Å¿cient omnes primam diem Iulij, non tantum æqualiter annum diuidere, Å¿ed prin - <pb o="209" rhead="EPISTOLA." n="221" file="0221" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0221"/> - cipìum quoque eſſe ęſtatis, maximam diem, noctem minimam totius anni; </s> - <s xml:id="s2610" xml:space="preserve">princi-<lb/>pium diminutionis diei & incrementi noctis, vnà <reg norm="etiam" type="context">etiã</reg> ea, quÄ™ Solis conuerÅ¿ionem <lb/>ad auÅ¿trales Å¿equuntur.</s> - </p> - <p> - <s xml:id="s2611" xml:space="preserve">Neconon intelliget vnuÅ¿quiÅ¿que primam diem Aprilis, <reg norm="primamque" type="simple">primamqÌ;</reg> Octobr. æqui-<lb/>noctiorum dies eſſe; </s> - <s xml:id="s2612" xml:space="preserve">primam autem diem Aprilis, initium veris; </s> - <s xml:id="s2613" xml:space="preserve">Octobris Autumni; <lb/></s> - <s xml:id="s2614" xml:space="preserve">Item Aprilis diem eſſe eum, quo dies noctis prolixitatem vincere incipit: </s> - <s xml:id="s2615" xml:space="preserve">Octobris, <lb/>quo nox diei longitudinem Å¿uperat, & alia huiuÅ¿modi, quæ ab æquinoctijs <reg norm="dependent" type="context context">depẽdẽt</reg>.</s> - </p> - <p> - <s xml:id="s2616" xml:space="preserve">Si vero quiÅ¿piam obijciat, modum hunc noÅ¿trum & ordinem perpetuum eſſe non <lb/>poſſe, ob motum augis Solis; </s> - <s xml:id="s2617" xml:space="preserve">quod punctum cum fuerit in principio Capricorni, <reg norm="tunc" type="context">tÅ©c</reg> <lb/>Sol hærebit in Å¿igno Sagittarij .32. diebus, totidem in Capricorno, in Geminis vero <lb/>29. totidem in Cancro; </s> - <s xml:id="s2618" xml:space="preserve">ex quo Å¿equetur prioribus <reg norm="contrarius" type="context">cõtrarius</reg> effectus; </s> - <s xml:id="s2619" xml:space="preserve">huic ego reÅ¿pon <lb/>debo, tale <reg norm="quidpiam" type="context">quidpiã</reg> non euenturum, niÅ¿i exactis ab hoc anno annis .240@0. quod Å¿i <lb/>mundus poÅ¿thac totidem annis, quot fuit antehac, perdurauerit, punctus augis non <lb/>amplius à ſitu præſenti, quà m .45. gradibus diÅ¿tabit. </s> - <s xml:id="s2620" xml:space="preserve">Verum demus <reg norm="modum" type="context">modÅ©</reg> noÅ¿trum <lb/>& regulam in annos ter, aut quater mille Å¿ubÅ¿eruire poſſe, nec amplius, certè hoc <lb/>toto tempore nullius momenti penè erit, quæ accidere poterit mutatio, tametÅ¿i ela-<lb/>pÅ¿is quatuor millibus annorum Februarius eſſe debebit .29. dierum. Aprilis & No-<lb/>uember .30. Iunius & October .31. AuguÅ¿tus .32. in aliis verò menÅ¿ibus nihil mutan-<lb/>dum erit. </s> - <s xml:id="s2621" xml:space="preserve">Ecce quam Å¿it nullius momenti mutatio.</s> - </p> - <p> - <s xml:id="s2622" xml:space="preserve">Quæ Å¿i Iulij Cæſaris temporibus fuiſſent animaduerÅ¿a nunquam omiſſa fuiſſent, <lb/>Å¿ed Å¿cientiæ AÅ¿tronomicæ nondum (vt ita dicam) confirmata ætas, cum alibi, <reg norm="tum" type="context">tÅ©</reg> maxi <lb/>mè in Italia, quo minus hæc aut Å¿cirentur aut Å¿tatuerentur impediebat.</s> - </p> - <p> - <s xml:id="s2623" xml:space="preserve">Tertia ratio eÅ¿t, vt non <reg norm="Å¿olum" type="context">Å¿olÅ©</reg> feÅ¿ta mobilia, verum <reg norm="ent" type="context">ẽt</reg> immobilia ad meliorem regu <lb/>lam (ut dictum eÅ¿t) reuocentur, Å¿i Å¿uis temporibus celebranda erunt. </s> - <s xml:id="s2624" xml:space="preserve">Quorum <reg norm="primum" type="context">primÅ©</reg> <lb/>eÅ¿t Natiuitas Domini, & quæ ab ea pendent; </s> - <s xml:id="s2625" xml:space="preserve">nempe Circuncifio, Epiphania, Purifi-<lb/>catio, Annunciatio, & Natiuitas Io. BaptiÅ¿tæ. </s> - <s xml:id="s2626" xml:space="preserve">ita vt dies Natalis Domini celebretur <lb/>prima die anni, cum Dei filius naÅ¿ci voluerit circa verum principium anni, quod à <lb/>Å¿olÅ¿titio hyemali initium ducit, & in ipÅ¿o principio diei naturalis ex <reg norm="Romanorum" type="context">RomanorÅ©</reg> Å¿en <lb/>tentia, media Å¿cilicet nocte, tanquam qui Å¿ummæ lætitiæ principium, poÅ¿t longos & <lb/>graues filiorum Adæ mærores, eſſet allaturus. </s> - <s xml:id="s2627" xml:space="preserve">Nec forÅ¿an Ianuarij nomini, à vete-<lb/>ribus Iano bifronti dicati hæc mutatio non conueniret, cum in ip Å¿o Å¿eruatore, duæ <lb/>veluti frontes & formæ vnitæ Å¿int, duæ Å¿cilicet naturæ diuina & humana. </s> - <s xml:id="s2628" xml:space="preserve">Hac ratio <lb/>ne abuÅ¿us tolletur, natus ex diuerÅ¿is moribus Tabulariorum, quorum alij monumen-<lb/>ta, Å¿eu quæ uocant InÅ¿trumenta, à die Natiuitatis Domini incohant, alij à Circunci <lb/>Å¿ione, alij à Calendis Martij, nonnulli à PaÅ¿chate; </s> - <s xml:id="s2629" xml:space="preserve">quæ varietas innumerabiles lites <lb/>affert & abuÅ¿us propè infinitos, ob dubiam & ancipitem Å¿cripturam. </s> - <s xml:id="s2630" xml:space="preserve">Indictionum <lb/>præterea ordini, hic noÅ¿ter modus nihil officiet; </s> - <s xml:id="s2631" xml:space="preserve">celebrato Natali celebrabitur Cir-<lb/>cunciÅ¿io octaua Ianuarij. </s> - <s xml:id="s2632" xml:space="preserve">Epiphania .13. eiuÅ¿dem. </s> - <s xml:id="s2633" xml:space="preserve">Purificatio .11. Februarij quæ erit <lb/>40: </s> - <s xml:id="s2634" xml:space="preserve">dies à Natiuitate Å¿eruatoris. </s> - <s xml:id="s2635" xml:space="preserve">Prima Aprilis Annunciatio Virginis Å¿olennis erit, <lb/>ipÅ¿o nempè die æquinoctij, natiuitas Diui Io: </s> - <s xml:id="s2636" xml:space="preserve">BaptiÅ¿tÄ™ celebrabitur Prima Iulij die <lb/>quæ erit Å¿olÅ¿titij æſtiui, cum illa diminutionem capit. vtrectè Diuus AuguÅ¿tinus il-<lb/>la verba Io: </s> - <s xml:id="s2637" xml:space="preserve">BaptiÅ¿tÄ™ interpretatus fuerit. </s> - <s xml:id="s2638" xml:space="preserve">Illum opportet creÅ¿cere, me autem minui: <lb/></s> - <s xml:id="s2639" xml:space="preserve">in quibus Å¿ic tantus Doctor philoÅ¿ophatur, vt tempus etiam natiuitatis Å¿erui & do-<lb/>mini præclare notet dicens, natus eÅ¿t Å¿eruus cum decreÅ¿cunt dies, natus eÅ¿t Dominus <lb/>cùm creÅ¿cere incipiunt.</s> - </p> - <p> - <s xml:id="s2640" xml:space="preserve">InÅ¿ignes etiam Theologi admonuerunt habendam rationem eſſe nonnullorum <lb/>feÅ¿torum, vt Diui Antonij, diuorum Fabiani & SebaÅ¿tiani, & aliorum Å¿anctorum, - <pb o="210" rhead="IO. BAPT. BENED." n="222" file="0222" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0222"/> - fi forte in octauam Epiphaniæ inciderint: </s> - <s xml:id="s2641" xml:space="preserve">Verum hÄ™c <var>.S.D.N.</var> curæ erunt, ut in aptiÅ¿ <lb/>Å¿ima tempora transferantur.</s> - </p> - <p> - <s xml:id="s2642" xml:space="preserve">Admonuerunt præterea transferendos eſſe dies feÅ¿tos Beati Stephani, Ioannis, <lb/>& Innocentium, vt quemadmodum factum eÅ¿t hactenus, diem natalis proximè Å¿e <lb/>quantur, ob multorum Doctorum, non recentium modo, Å¿ed etiam antiquorum ob <lb/>Å¿eruantiam; </s> - <s xml:id="s2643" xml:space="preserve">qui Å¿uis omelijs & concionibus multa piè, de myÅ¿teriis Å¿ucceſſionis FeÅ¿to <lb/>rum huiuÅ¿imodi tradiderunt.</s> - </p> - <p> - <s xml:id="s2644" xml:space="preserve">Cuperent etiam præclari Theologi diem Aſſumptionis Beatæ virginis incidere <lb/>in primam Septembris, Natiuitatem autem in .25. vt quemadmodum toto illo men <lb/>Å¿e in Å¿igno Virginis Å¿ol verÅ¿abitur, ita EccleÅ¿ia Der in cÄ™lebrandi tantæ Virginis ma <lb/>tris Deilaudibus occupetur.</s> - </p> - <p> - <s xml:id="s2645" xml:space="preserve">Atque hÄ™c Å¿unt SerenisÅ¿ime Princeps, quÄ™ longa & attenta cogitatione à me exa <lb/>minata, atque perpenÅ¿a fuerunt; </s> - <s xml:id="s2646" xml:space="preserve">quæſitam diligenter & accuratè expendentur ab <lb/>his, quorum intereÅ¿t, quam mihi apta & rationi conÅ¿entanea, ac vera penitus, imo <lb/>(quod me magis afficit) etiam tibi viÅ¿a fuerunt; </s> - <s xml:id="s2647" xml:space="preserve">non dubito quin placitura Å¿int; </s> - <s xml:id="s2648" xml:space="preserve">& vo <lb/>tis Å¿ummi Pont. aliqua ex parte Å¿atisfactura. </s> - <s xml:id="s2649" xml:space="preserve">eò magis quòd te iubente, & cogitata à <lb/>me, & Å¿cripta fuerint. </s> - <s xml:id="s2650" xml:space="preserve">Vale Princeps Sereniſſime, & qua Å¿oles hylaritate cÄ™tera no <lb/>Å¿tra, etiam has breues vigilias Å¿uÅ¿cipe & foue. </s> - <s xml:id="s2651" xml:space="preserve">Dat. AuguÅ¿tæ Taurinorum Kal. <lb/></s> - <s xml:id="s2652" xml:space="preserve">Aprilis. <num value="1578">MDLXXVIII</num>. </s> - </p> - <p> - <s xml:id="s2653" xml:space="preserve">T. CelÅ¿itudinis.</s> - </p> - <p> - <s xml:id="s2654" xml:space="preserve">Deditiſſimus Mathematicus.</s> - </p> - <p> - <s xml:id="s2655" xml:space="preserve">Io. Bap. Benedictus.</s> - </p> - <pb o="211" n="223" file="0223" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0223"/> - </div> - </div> - <div type="section" level="3" n="2"> - <div type="letter" level="4" n="1"> - <head xml:id="head360" xml:space="preserve">DE CIRCVLO <lb/>AMBIENTE QVADRILATERVM.</head> - <head xml:id="head361" xml:space="preserve">AD SERENISS. CAROLVM EMANVELEM <lb/>Pedemontis Principem.</head> - <p> - <s xml:id="s2656" xml:space="preserve">PRoblema quod à celÅ¿itudine tua nobis proponitur non Å¿olum poſſibile eÅ¿t, Å¿ed <lb/>facile etiam ad Å¿oluendum, hoc eÅ¿t quod circulus talis inueniatur, qui poſſit cir <lb/>cunÅ¿cribere, Å¿eu capere quadrilaterum ex quatuor datis rectis lineis terminatum, vel <lb/>Å¿ic, datis quatuor rectis lineis ex quibus quadrilaterum poſſit eftici, tale efficiatur vt <lb/>circa ipÅ¿um, circulus poſſit circunÅ¿cribi.</s> - </p> - <p> - <s xml:id="s2657" xml:space="preserve">Sint igitur .4. lineæ propoÅ¿itæ <var>.b.d</var>: <var>q.b</var>: <var>a.q</var>: et <var>.a.d.</var> ex <reg norm="quibus" type="simple">quibê°</reg> poſſibile Å¿it quadrilate <lb/>rum conÅ¿titui, tale vero conÅ¿tituatur, vt aliquis circulus poſſit ipÅ¿um circunÅ¿cribere. <lb/></s> - <s xml:id="s2658" xml:space="preserve">imaginemur autem hoc factum eſſe, quod quidem quadrilaterum Å¿it <var>.a.d.q.b.</var> cuius <lb/> - <anchor type="figure" xlink:label="fig-0223-01a" xlink:href="fig-0223-01"/> - <anchor type="figure" xlink:label="fig-0223-02a" xlink:href="fig-0223-02"/> - <anchor type="figure" xlink:label="fig-0223-03a" xlink:href="fig-0223-03"/> - <anchor type="figure" xlink:label="fig-0223-04a" xlink:href="fig-0223-04"/> - <anchor type="figure" xlink:label="fig-0223-05a" xlink:href="fig-0223-05"/> - <pb o="212" rhead="IO. BAPT. BENED." n="224" file="0224" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0224"/> - diametri Å¿int <var>.q.d.</var> et <var>.a.b.</var> quæ Å¿e inuicem interÅ¿ecent in puncto <var>.o.</var> vnde cum anguli <lb/>contra Å¿e poÅ¿iti circa <var>.o.</var> æquales inuicem Å¿int ex .15. primi Eucli. </s> - <s xml:id="s2659" xml:space="preserve">& angulus <var>.a.q.d.</var> æ-<lb/>qualis angulo <var>.a.b.d.</var> & angulus <var>.q.b.a.</var> æqualis angulo <var>.q.d.a.</var> et <var>.b.q.d.</var> angulo <var>.b.a.d.</var> <lb/>ex .20. tertij tunc triangulus <var>.a.o.q.</var> Å¿imilis erit triangulo <var>.d.o.b.</var> et <var>.q.o.b.</var> Å¿imilis trian-<lb/>gulo <var>.a.o.d.</var> ex definitione. </s> - <s xml:id="s2660" xml:space="preserve">Vnde eadem proportio erit ipÅ¿ius <var>.q.o.</var> ad <var>.b.o.</var> quæ ipÅ¿ius <lb/><var>q.a.</var> ad <var>.b.d.</var> & ipÅ¿ius <var>.b.o.</var> ad <var>.o.d.</var> eadem quæ <var>.q.b.</var> ad <var>.a.d.</var> & ipÅ¿ius <var>.q.o.</var> ad <var>.o.a.</var> eadem <lb/>quæ <var>.q.b.</var> ad <var>.a.d.</var> proportio igitur <var>.q.o.</var> ad <var>.o.d.</var> cognita nobis erit, vt compoÅ¿ita ex <lb/>ea quæ eÅ¿t <var>.q.o.</var> ad <var>.o.b.</var> ex <var>.o.b.</var> ad <var>.o.d.</var> quæ nobis cognitÄ™ Å¿unt, mediante <lb/>proportione ipÅ¿ius <var>.q.a.</var> ad <var>.b.d.</var> & ipÅ¿ius <var>.q.b.</var> ad <var>.a.d.</var> proportio Å¿imiliter ipÅ¿ius <var>.b.o.</var> <lb/>ad <var>.o.a.</var> nobis cognita erit, vt compoÅ¿ita ex proportione ipÅ¿ius <var>.b.o.</var> ad <var>.o.q.</var> & <lb/>ipÅ¿ius <var>.o.q.</var> ad <var>.o.a.</var> cognitis, mediante proportione ipÅ¿ius <var>.b.d.</var> ad <var>.q.a.</var> & ipÅ¿ius <var>.q.b.</var> ad <lb/><var>a.d.</var> cum <reg norm="autem" type="context">autẽ</reg> proportio ipÅ¿ius <var>.q.o.</var> ad <var>.o.b.</var> nobis cognita Å¿it, </s> - <s xml:id="s2661" xml:space="preserve">tunc nobis cognita erit <lb/>proportio ipÅ¿ius <var>.q.d.</var> ad <var>.a.b</var>. </s> - <s xml:id="s2662" xml:space="preserve">Nam ut <var>.q.o.</var> ad <var>.o.b.</var> eÅ¿t vt <var>.a.o.</var> ad <var>.o.d.</var> ex Å¿imilitudine, <lb/></s> - <s xml:id="s2663" xml:space="preserve">quare proportio compoÅ¿iti ex primo, & quarto terminorum ad compoÅ¿itum ex .2. & <lb/>tertio, cognita erit. </s> - <s xml:id="s2664" xml:space="preserve">Å¿ed quod fit ex <var>.q.d.</var> in <var>.a.b.</var> cognitum nobis eÅ¿t, vt æquale duobus <lb/>productis, hoc eÅ¿t ex <var>.q.a.</var> in <var>.d.b.</var> & ex <var>.q.b.</var> in <var>.d.a.</var> ex Å¿ecunda primi AlmageÅ¿ti. </s> - <s xml:id="s2665" xml:space="preserve">quæ <lb/>producta nobis cognita Å¿unt, cum nobis data Å¿int eorum latera. </s> - <s xml:id="s2666" xml:space="preserve">Quapropter facta <lb/>cum fuerit figura quadrilatera rectangula Å¿imilis alicui alterirectangulæ figuræ pro <lb/>ductæ à duobus lateribus inuicem ita proportionatis, vt Å¿e habet <var>.q.d.</var> ad <var>.a.b.</var> æqua-<lb/>lis tamen duobus productis, hoc eÅ¿t producto ex <var>.q.a.</var> in <var>.d.b.</var> & ex <var>.q.b.</var> in <var>.d.a.</var> ex <lb/>doctrina, 25. Å¿exti Eucli quæ quidem figura, exempli gratia, Å¿it <var>.u.t.</var> eius verò latera <lb/>Å¿int <var>.u.n.</var> et <var>.n.t.</var> Hæc enim dico æqualia eſſe <var>.q.d.</var> et <var>.b.a.</var> hoc eÅ¿t <var>.n.t.</var> maius maio-<lb/>ri <var>.b.a.</var> et <var>.u.n.</var> minus minori <var>.q.d</var>. </s> - <s xml:id="s2667" xml:space="preserve">Quod ita probabo. </s> - <s xml:id="s2668" xml:space="preserve">cogitemus rectangulum <var>.s.r.</var> <lb/>productum eſſe ex duobus lateribus <var>.q.d.</var> et <var>.a.b.</var> Å¿ed, <var>s.n.</var> æqualis Å¿it <var>.q.d.</var> et <var>.n.r.</var> æqua-<lb/>lis <var>.a.b.</var> <reg norm="Å¿intque" type="simple">Å¿intqÌ;</reg> duæ lineæ <var>.s.n.</var> et <var>.n.t.</var> inuicem directè coniunctæ, vnde <var>.u.n.</var> directè <lb/>coniuncta etiam erit cum <var>.n.r.</var> ex quo rectangulum <var>.u.t.</var> æquale erit rectangulo <var>.s.r.</var> ex <lb/>communi conceptu, <reg norm="eademque" type="simple">eademqÌ</reg> proportio erit <var>.u.n.</var> ad <var>.n.t.</var> quÄ™ <var>.s.n.</var> ad <var>.n.r.</var> eo <reg norm="quod" type="simple">ê™</reg> ita fa-<lb/>ctum fuit, cum autem ita Å¿it <var>.u.n.</var> ad <var>.n.t.</var> vt <var>.s.n.</var> ad <var>.n.r.</var> </s> - <s xml:id="s2669" xml:space="preserve">tunc permutando ita erit <var>.n.t.</var> ad <lb/><var>n.r.</var> vt <var>.u.n.</var> ad <var>.n.s.</var> Å¿ed quia ita eÅ¿t <var>.u.n.</var> ad <var>.n.r.</var> vt <var>.s.n.</var> ad <var>.n.t.</var> ex 15. Å¿exti, </s> - <s xml:id="s2670" xml:space="preserve">tunc permutan <lb/>do ita erit <var>.n.r.</var> ad <var>.n.t.</var> vt <var>.n.u.</var> ad <var>.n.s.</var> </s> - <s xml:id="s2671" xml:space="preserve">quare ex 11. quinti ita erit <var>.n.t.</var> ad <var>.n.r.</var> vt <var>.n.r.</var> ad <var>.n.<lb/>t.</var> quapropter ex neceſſitate Å¿equitur <var>.n.t.</var> et <var>.n.r.</var> inuicem æquales eſſe, et <var>.u.n.</var> Å¿imiliter <lb/>cum <var>.n.s</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0223-01" xlink:href="fig-0223-01a"> - <image file="0223-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0223-01"/> - </figure> - <figure xlink:label="fig-0223-02" xlink:href="fig-0223-02a"> - <image file="0223-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0223-02"/> - </figure> - <figure xlink:label="fig-0223-03" xlink:href="fig-0223-03a"> - <image file="0223-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0223-03"/> - </figure> - <figure xlink:label="fig-0223-04" xlink:href="fig-0223-04a"> - <image file="0223-04" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0223-04"/> - </figure> - <figure xlink:label="fig-0223-05" xlink:href="fig-0223-05a"> - <image file="0223-05" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0223-05"/> - </figure> - </div> - <p> - <s xml:id="s2672" xml:space="preserve">Inuentæ nunc cum fuerint duæ diametri <var>.q.d.</var> et <var>.a.b.</var> ipÅ¿ius quadrilateri, difficile <lb/>non erit eius angulos inuenire, eo <reg norm="quod" type="simple">ê™</reg> mediante <var>.a.b.</var> cognita, Å¿imul cum <var>.b.d.</var> et <var>.a.d.</var> da <lb/>tis, faciemus triangulum <var>.a.b.d.</var> vel <reg norm="mediante" type="context">mediãte</reg> <var>.q.d.</var> et <var>.q.a.</var> et <var>.a.d.</var> cognitis faciemus <reg norm="triam" type="context">triã</reg> <lb/>gulum <var>.a.q.d.</var> ex .22. primi. </s> - <s xml:id="s2673" xml:space="preserve">Vnde cum centrum circuli circunÅ¿criptibilis cuiuÅ¿uis di-<lb/>ctorum triangulorum ex quinta quarti inuentum fuerit, triangulum reliquum, ab eo <lb/>dem circulo circunÅ¿criptum erit, ex communi Å¿cientia.</s> - </p> - <p> - <s xml:id="s2674" xml:space="preserve">SEd vt ipÅ¿a operatio facilior fiat, Sint eÄ™dem lineæ <var>.b.d</var>: <var>b.q</var>: <var>a.q.</var> et <var>.a.d.</var> ex quibus <lb/>poſſit <reg norm="quadrilaterum" type="context">quadrilaterÅ©</reg> effici. </s> - <s xml:id="s2675" xml:space="preserve">Videatur deinde primò quas volumus oppoÅ¿itas Å¿ibi <lb/>inuicem eſſe, ponatur ergò ut <var>.q.a.</var> et <var>.b.d.</var> velimus oppoÅ¿itas inuicem facere, et <var>.q.b.</var> <lb/>cum <var>.a.d.</var> Å¿imiliter, accipiemus nunc <var>.K.</var> cuiuÅ¿uis magnitudinis, cui comparetur <var>.e.</var> <lb/>ita proportionata, vt <var>.q.b.</var> eÅ¿t ipÅ¿i <var>.a.d.</var> ex doctrina .10. Å¿exti Eucli. vel accipiatur <var>.a.d.</var> <lb/>vice <var>.K.</var> et <var>.q.b.</var> vice <var>.e.</var> quod idem erit, & expeditius, inuenietur Å¿imiliter <var>.h.</var> ita pro-<lb/>portionata ad <var>.e.</var> et <var>.g.</var> ad <var>.k.</var> vt <var>.b.d.</var> eÅ¿t ad <var>.q.a.</var> vel <var>.g.</var> ad <var>.h.</var> vt <var>.a.d.</var> ipÅ¿i <var>.q.b.</var> quod <reg norm="idem" type="context">idẽ</reg> erit.</s> - </p> - <p> - <s xml:id="s2676" xml:space="preserve">Hoc facto coniungantur inuicem directè <var>.g.</var> et <var>.e.</var> quarum compoÅ¿itum Å¿it <var>.g.e.</var> & <lb/>ita duæ <var>.K.</var> et <var>.h.</var> ex quibus Å¿it <var>.K.h</var>. </s> - <s xml:id="s2677" xml:space="preserve">Nunc ex iÅ¿tis duabus lineis <var>.e.g.</var> et <var>K.h.</var> fiat paral- - <pb o="213" rhead="EPISTOLAE" n="225" file="0225" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0225"/> - lelogrammum <var>.Z.</var> deinde fiant alia duo parallelogramma rectangula quorum vnum <lb/>Å¿it ex <var>.q.a.</var> in <var>.b.d.</var> reliquum verò Å¿it ex <var>.q.b.</var> in <var>.a.d.</var> quæ quidem Å¿int <var>.f.m</var>.</s> - </p> - <p> - <s xml:id="s2678" xml:space="preserve">Quo facto deÅ¿ignetur rectangulum <var>.u.t.</var> ex .25. Å¿exti, quod æquale Å¿it duobus re-<lb/>ctangulis <var>.f.</var> et <var>.m.</var> Å¿imile tamen <var>.Z.</var> cuius rectanguli vnum latus correÅ¿pondet <var>.e.g.</var> reli-<lb/>quum verò <var>.K.h.</var> in proportione, Å¿ed in æqualitate, vnum correÅ¿pondet <var>.q.d.</var> <reg norm="reliquum" type="context">reliquÅ©</reg> <lb/>vero <var>.a.b.</var> diametris ipÅ¿ius quadrilateri.</s> - </p> - <p> - <s xml:id="s2679" xml:space="preserve">Accipiatur nunc latus illud quod correÅ¿pondet <var>.K.h.</var> hoc eÅ¿t ipÅ¿i <var>.a.b.</var> maius Å¿cili-<lb/>cet, & Å¿imul cum <var>.b.d.</var> et <var>.a.d.</var> formetur <reg norm="triangulum" type="context context">triãgulÅ©</reg> <var>.a.b.d.</var> ex .22. primi Eucli. circa quod <lb/>circunÅ¿cribatur circulus ex .5. quarti. </s> - <s xml:id="s2680" xml:space="preserve">& inuentum erit quod quÄ™rebamus.</s> - </p> - <figure position="here"> - <image file="0225-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0225-01"/> - </figure> - <figure position="here"> - <image file="0225-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0225-02"/> - </figure> - <figure position="here"> - <image file="0225-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0225-03"/> - </figure> - <figure position="here"> - <image file="0225-04" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0225-04"/> - </figure> - <figure position="here"> - <image file="0225-05" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0225-05"/> - </figure> - <pb o="214" n="226" file="0226" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0226"/> - </div> - </div> - <div type="section" level="3" n="3"> - <div type="letter" level="4" n="1"> - <head xml:id="head362" xml:space="preserve">PER EVNDEM PARALLELVM <lb/>abÅ¿que correctione Å¿emper nauigari <lb/>non poſſe.</head> - <head xml:id="head363" style="it" xml:space="preserve">Vbi not antur Petri <reg norm="Nonij" type="lig">Nonij</reg> lapÅ¿us in correctione erroris nauis. <lb/>Et alij Petri Medinæ errores.</head> - <head xml:id="head364" xml:space="preserve">ILLVSTRISSIMO ANDREAE PROVANAE <lb/>Leinici Domino, FruzaÅ¿ci comiti, Aequiti Torquato, inthimo Sere-<lb/>niÅ¿simi Sabaudiæ Ducis ConÅ¿iliario, <reg norm="eiuÅ¿que" type="simple">eiuÅ¿qÌ;</reg>, & Å¿acræ religio-<lb/>nis Å¿anctorum Mauritij, & Lazari ClaÅ¿si Præfecto.</head> - <p> - <s xml:id="s2681" xml:space="preserve"><emph style="sc">INter</emph> Eximiastuas virtutes, reinauticæ peritia IlluÅ¿tris emicat <lb/>merito ad te Å¿cribendum duxi, quod ad eam facultatem perti-<lb/>nens excogitaui, Å¿imul cum quibuÅ¿dam alijs inÅ¿trumentis, vt non-<lb/>nihil commodi attuliſſe videar maritimis negotijs, & aliqua ex <lb/>parte animi mei erga te propenÅ¿ionem indicauiſſe.</s> - </p> - <p> - <s xml:id="s2682" xml:space="preserve">PEr vnum <reg norm="eum" type="context">eÅ©</reg><reg norm="demque" type="simple">demqÌ;</reg> parallelum in primis <reg norm="abÅ¿que" type="simple">abÅ¿q;</reg> aliqua corre <lb/>ctione Å¿emper nauigari poſſe, omninò nego. </s> - <s xml:id="s2683" xml:space="preserve"><reg norm="Nam" type="context">Nã</reg>, <reg norm="verum" type="context">verÅ©</reg> eÅ¿tid quod Petrus Nonius in <lb/>initio Å¿ui operis oÅ¿tendit, ideÅ¿t nauim verÅ¿us æquatorem Å¿emper declinare: </s> - <s xml:id="s2684" xml:space="preserve">qui <reg norm="cum" type="context">cÅ©</reg> <lb/>corrigit errorem, fallitur, cum ipÅ¿e, eandem nauim, parallelam æquatori in vno ver <lb/>ticali ipÅ¿i æquatori propinquiori, & non in primo parallelo dirigit, itaque exiÅ¿timat <lb/>in fine itineris, vbi deÅ¿cribit punctum <var>.o.</var> eam in eodem parallelo priori repe-<lb/>riri debere, quod <reg norm="verum" type="context">verÅ©</reg> <reg norm="non" type="context">nõ</reg> eÅ¿t, quia ea correctio efficit, vt motus nauis <reg norm="effectum" type="context">effectÅ©</reg> <reg norm="cuiuÅ¿dam" type="context">cuiuÅ¿dã</reg> <lb/><reg norm="deÅ¿cenÅ¿us" type="context">deÅ¿cẽſus</reg> <reg norm="per" type="simple">ê‘</reg> Å¿caligradum pręſtet, in quo à gradu in <reg norm="gradum" type="context">gradÅ©</reg> fiat deÅ¿cenÅ¿us, Å¿ed Å¿i per gra <lb/>dus <reg norm="tantum" type="wordlist">tm̃</reg> aÅ¿cenderet <reg norm="quantum" type="context context">quãtÅ©</reg> <reg norm="deÅ¿cendit" type="context">deÅ¿cẽdit</reg>, <reg norm="dubium" type="context">dubiÅ©</reg> <reg norm="non" type="context">nõ</reg> eÅ¿t quin in fine ita eſſet Å¿e habitura <reg norm="quem" type="context">quẽ</reg> <lb/>ad modum in medio & in principio, cum verò Å¿emper deÅ¿cendat, abÅ¿que vlla aÅ¿cen-<lb/>Å¿ione, neceſſariò Å¿ic Å¿emper procedens, remota cum eſſent impedimenta terræ, Å¿ub <lb/>æquatore reperiretur, Å¿ub quo perpetuò circuiret globum.</s> - </p> - <p> - <s xml:id="s2685" xml:space="preserve">Idem Å¿ub quolibet meridiano præſtare poteÅ¿t, ideÅ¿t vno <reg norm="eodemque" type="simple">eodemqÌ;</reg> vento circun-<lb/>uerti: </s> - <s xml:id="s2686" xml:space="preserve">Å¿ed per alios circulos quam per hos duos (Å¿iue circulus magnus Å¿iue paruus) id <lb/>nunquam perfectè efficere poteÅ¿t, de parallelis iam manifeÅ¿tum eÅ¿t, cum impetus na <lb/>turalis corporum, quæ mota Å¿unt Å¿int Å¿emper in Å¿uperficiebus circulorum maiorum, <lb/>quorum circunferentiÄ™ cum circunferentijs minorum, præter quam per vnum quod <lb/>dam punctum quando adinuicem contiguæ Å¿unt, aut per duo ideÅ¿t cum Å¿e Å¿e interÅ¿e-<lb/>cant non communicant, ita quod ad efficiendum, vt triremis aliqua, aut nauis, per <lb/>aliquem ex parallelis ad æquatorem moueatur, neceſſario Å¿it futurum, vt ratione <reg norm="con- tiguitatis" type="context">cõ-<lb/>tiguitatis</reg> & non continuitatis eam moueri curemus. </s> - <s xml:id="s2687" xml:space="preserve">quia ratione continuitatis om-<lb/>ninò fieri non <reg norm="pont" type="context">põt</reg>, aut conÅ¿tet virtus mouens remis, aut velis. </s> - <s xml:id="s2688" xml:space="preserve">Sed per quemlibet <reg norm="alium" type="context">aliÅ©</reg> <lb/>circulum maiorem, qui non Å¿it aut æquator aut aliquis ex meridianis, eÅ¿t penitus im <lb/>poſſibile. </s> - <s xml:id="s2689" xml:space="preserve">ideÅ¿t vt vnius venti vi nauis impellatur. </s> - <s xml:id="s2690" xml:space="preserve">Quod vt clarè pateat, Å¿it orizon <var>.<lb/>a.c.b.d.</var> & æquator <var>.c.q.t.d.</var> & vnus meridianorum Å¿it <var>.a.r.n.t.b.</var> in quo <var>.n.</var> Å¿it Zenit Å¿ub <lb/>quo primum nauis reperiatur et <var>.r.</var> Å¿it polus Å¿eptentrionalis. </s> - <s xml:id="s2691" xml:space="preserve">Ponamus etiam quod - <pb o="215" rhead="EPISTOLA." n="227" file="0227" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0227"/> - azimut <var>.f.q.u.n.p.</var> conÅ¿tituat angulum <var>.a.n.p.</var> Å¿eu <var>.f.n.b.</var> cum meridiano graduum .45. <lb/>vnde tot graduum eruntarcus <var>.a.p.</var> et <var>.b.f.</var> orizontis, quapropter punctum <var>.f.</var> commu-<lb/>ne ipÅ¿ius orizontis cum azimut, erit medio in loco inter <var>.b.</var> et <var>.c.</var> & ideo quarta <var>.n.f.</var> ip <lb/>Å¿ius azimut Å¿ecabit quartam <var>.c.t.</var> ipÅ¿ius æquatoris in puncto <var>.q.</var> & habebimus triangu-<lb/>Ium <var>.q.t.n.</var> cuius angulus <var>.t.</var> rectus erit, & angulus <var>.n.</var> cognitus Å¿imul cum latere <var>.n.t.</var> la-<lb/>titudinis loci, quibus rebus mediantibus deueniemus in cognitionem lateris <var>.q.n.</var> la-<lb/>teris <var>.q.t.</var> & anguli <var>.q.</var> ex .4. primi Copernici Å¿i voluerimus.</s> - </p> - <p> - <s xml:id="s2692" xml:space="preserve">Ponamus nunc nauem à puncto <var>.n.</var> diÅ¿cedere Å¿eu iter facere verÅ¿us <var>.u.</var> punctum, & <lb/> - <anchor type="figure" xlink:label="fig-0227-01a" xlink:href="fig-0227-01"/> - in ipÅ¿o <var>.u.</var> reperiri, iam in hoc Å¿itu ha-<lb/>bebimus angulum huius Å¿ecundi me-<lb/>ridiani <var>.r.u.p.</var> qui quidem in hoc caÅ¿u <lb/>minor eſſet angulo <var>.r.n.p.</var> extrinÅ¿eco <lb/>trianguli <var>.r.u.n.</var> ex conuerÅ¿o Å¿ecundæ <lb/>partis .48. <reg norm="propoÅ¿itionis" type="simple">ê“poÅ¿itionis</reg> tertij lib. de <reg norm="trian gulis" type="context">triãÂ<lb/>gulis</reg> Monteregij, Å¿eu ex .13. primi Me <lb/>nelai, cuius anguli <var>.u.</var> arcus orizontalis <lb/>Å¿it <var>.x.e.</var> qui quidem minor erit arcu <var>.a.<lb/>p.</var> vt patet ratione anguli <var>.r.u.e.</var> mino-<lb/>ris, ergo alius ventus nauem impellet à <lb/>puncto .11. verÅ¿us <var>.q.</var> diuerÅ¿us ab illo qui <lb/>prius ab <var>.n.</var> verÅ¿us <var>.u.</var> eam impellebat.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0227-01" xlink:href="fig-0227-01a"> - <image file="0227-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0227-01"/> - </figure> - </div> - <p> - <s xml:id="s2693" xml:space="preserve">Vnde clarè patet verum eſſe quod <lb/>dico, hoc eÅ¿t quod aliquo modo fieri <lb/>non poteÅ¿t, vt nauis ab aliquo loco ad <lb/>alium, breuiſſimo interuallo ire poſſit <lb/>ideÅ¿t per gyrum circuli maioris Å¿phæræ vno tantummodo <reg norm="vento" type="context">vẽto</reg> eam impellente, præ <lb/>ter quam in Ä™quatore, Å¿eu in aliquo quouis meridianorum, nos autem ire per gyrum <lb/>alicuius paralleli dementia eſſet, niÅ¿i neceſſitas cogeret.</s> - </p> - <p> - <s xml:id="s2694" xml:space="preserve">HuiuÅ¿modi demonÅ¿trationis ope, quantum decipiatur Petrus Medina cap .6. lib. <lb/>3. cognoÅ¿citur, vbi Å¿ic Å¿cribit; </s> - <s xml:id="s2695" xml:space="preserve">Vbicunque locorum reperiatur homo, aliquem circu-<lb/>lum qui vniuerÅ¿um ambiat imaginatione Å¿ibi confingens, per totum eum circulum <lb/>vno <reg norm="eodemque" type="simple">eodemqÌ;</reg> vento nauigatio Å¿uÅ¿cipitur. </s> - <s xml:id="s2696" xml:space="preserve">Ex hac etiam demonÅ¿tratione, quà m fal <lb/>Å¿a Å¿it charta maritima patet, cuius beneficio exiÅ¿timant nautÄ™ Å¿e per breuiſſimum iter <lb/>a loco ad locum vehi etiamÅ¿i dicti loci non Å¿int ambo in æquatore, aut in aliquo me <lb/>ridiano, Å¿ed extra dictos circulos vnico tantum vento impellente & Å¿i in paruis æquo <lb/>fibus hic error parum depræhenditur, forte tamen in magno Oceano clarè pateret. <lb/></s> - <s xml:id="s2697" xml:space="preserve">In Å¿uperius igitur dicta demonÅ¿tratione iam oÅ¿tendi, quod Å¿i velimus vehi ab vno lo <lb/>co ad alium beneficio alicuius circuli maioris, præter duos iam dictos, hoc fieri non <lb/>poteÅ¿t vno <reg norm="eodemque" type="simple">eodemqÌ;</reg> <reg norm="vento" type="context">vẽto</reg> impellente. </s> - <s xml:id="s2698" xml:space="preserve">Vnde Å¿equitur, omnia ea interualla quæ vno <lb/><reg norm="eodemque" type="simple">eodemqÌ</reg> vento tranÅ¿ibimus futura longiora, præterquam in duobus dictis circulis <lb/>æquinoctiali & meridiano.</s> - </p> - <p> - <s xml:id="s2699" xml:space="preserve">Cum verò Petrus Medina cap .7. volens probare chartam maritimam bonam eÅ¿-<lb/>Å¿e, planiÅ¿phÄ™rium Prolomei & Iordani citat, non animaduertit quam diuerÅ¿o mo-<lb/>do a charta maritima huiuÅ¿modi inÅ¿trumentum Å¿it fabricatum, cum exceptis orizon <lb/>te recto, & meridiano in dicto inÅ¿trumento quilibet alius circulus Å¿it circulus, Å¿iue Å¿it <lb/>almicantarat, Å¿iue azimur, Å¿iue æquator, Å¿iue tropicus, Å¿iue zodiacus, Å¿iue alius quiuis <lb/>circulus, eum in charta maritima ne vna quidem Å¿it linea, quÄ™ non Å¿it recta, quolibet <lb/>nomine vocetur.</s> - </p> - <pb o="216" rhead="IO. BABPT. BENED." n="228" file="0228" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0228"/> - <p> - <s xml:id="s2700" xml:space="preserve">Superius poÅ¿itæ meæ demonÅ¿trationis ope, deuenimus in cognitionem magnitu <lb/>dinis arcus <var>.n.q.</var> cognoÅ¿cimus etiam angulum <var>.n.q.t.</var> vnde nobis <reg norm="manifeÅ¿tum" type="context">manifeÅ¿tÅ©</reg> eſſet quo <lb/>vento oporteret iter facere. cum à puncto <var>.q.</var> nauis aliqua diÅ¿ceſſura eſſet, in eodem <lb/>azimut propoÅ¿ito. </s> - <s xml:id="s2701" xml:space="preserve">Idem etiam dico de puncto <var>.u.</var> cum cogniti eſſent arcus <var>.n.u.</var> et <var>.n.<lb/>r.</var> vt Å¿upponitur, Å¿imul cum angulo <var>.r.n.u.</var> vnde cognitus eſſet nobis angulus <var>.n.u.r.</var> ex <lb/>11. primi lib. Copernici, ex quo ventus nobis cognitus foret.</s> - </p> - <p> - <s xml:id="s2702" xml:space="preserve">Modus autem quem idem Medina cap .9. lib. tertij ad cognoÅ¿cendam diÅ¿tantiam <lb/>vnius meridiani ab alio præſcribit, in genere eÅ¿t falÅ¿us, etiam Å¿i is ab antiquis eum de <lb/>Å¿umat, qui, hic non viderunt quam magna inter meridianos differentia Å¿it interuallo <lb/>rum eorum quæ Å¿unt vicina polis & eorum quæ Å¿unt circa æquatorem.</s> - </p> - <p> - <s xml:id="s2703" xml:space="preserve">FalÅ¿us eÅ¿t etiam modus ab eo traditus ad cognoÅ¿cendos gradus longitudinis per <lb/>medium itineris cogniti in quouis parallelo extra æquatorem facti, & hoc cap .14. li <lb/>bri tertij eiuÅ¿dem, & primo cap. lib. 4. <reg norm="continetur" type="context">cõtinetur</reg>, vbi .17. leucas cum dimidia cuilibet <lb/>gradui tam paralleli quà m meridiani aſſignat.</s> - </p> - <p> - <s xml:id="s2704" xml:space="preserve">FalÅ¿um eÅ¿t etiam quod ab eo aſſeritur, Solem, cum reperiretur in æquatore, circa <lb/>eos qui Å¿ub ipÅ¿o æquatore habitant, vnius diei <reg norm="noctisque" type="simple">noctisqÌ;</reg> Å¿patio per omnes uentos cir-<lb/>cunuolui. </s> - <s xml:id="s2705" xml:space="preserve">quia illis æquator idem eÅ¿t cum verticali, qui duos tantum rhumbos pro-<lb/>ducit, ideÅ¿t orientis, & occidentis: </s> - <s xml:id="s2706" xml:space="preserve">hic verò error, in Å¿ecundo cap. lib. 6. habetur.</s> - </p> - <p> - <s xml:id="s2707" xml:space="preserve">FalÅ¿um eÅ¿t etiam quod profert Solem ijs qui habitant Å¿phæram obliquam, qua-<lb/>libet hora tertia, regulariter ab vno rhumbo ad alium ex præcipuis ideÅ¿t ab vno azi <lb/>mut ad alium progredi, quemadmodum eadem cap .2. lib. 6. et .7. cap. Å¿eptimi libr. <lb/>Å¿cribit. </s> - <s xml:id="s2708" xml:space="preserve">Huius autem rei falÅ¿itas ita facile depræhendetur, ponamus hemiÅ¿phÄ™rium <lb/>orientale, verbi gratia, cuius meridianus Å¿it <var>.p.z.b.</var> æquator <reg norm="autem" type="wordlist">aÅ©t</reg> <var>.e.m.</var> vnus verò paral <lb/>lelorum Å¿eptentrionalium Å¿it <var>.c.a.</var> in quo Solem exiÅ¿tere ponamus, orizon autem Å¿it <var>.<lb/>b.m.</var> zenit vero <var>.z.</var> polus arcticus <var>.p.</var> Å¿it poÅ¿tea azimut <var>.z.q.</var> à meridiano diÅ¿tans per gra <lb/>dus .45. qui quidem azimut in hoc hemiÅ¿phÄ™rio erit rhumbus illius venti, quem uul-<lb/>go Itali Sirocum dicunt, et <var>.z.m.</var> Å¿it azimut verticalis qui in hoc hemiÅ¿phærio erit <reg norm="ron" type="context">rõ</reg> <lb/>bus venti orientalis, ita <reg norm="quod" type="simple">ê™</reg> Å¿ecundum Medinam à rhumbo <var>.z.m.</var> ad <var>.z.q</var>. </s> - <s xml:id="s2709" xml:space="preserve">SolabÅ¿oluet <lb/>Å¿patium <reg norm="temporis" type="context">tẽporis</reg> trium horarum, & aliud <lb/>æquale temporis Å¿patium abÅ¿oluet à <reg norm="rhum" type="context">rhÅ©</reg> <lb/> - <anchor type="figure" xlink:label="fig-0228-01a" xlink:href="fig-0228-01"/> - bo <var>.z.q.</var> ad <var>.z.b.</var> ex ipÅ¿o Medina, vnde ar-<lb/>cus <var>.a.o.</var> paralleli eſſet graduum .45. & <lb/>item arcus <var>.o.c</var>.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0228-01" xlink:href="fig-0228-01a"> - <image file="0228-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0228-01"/> - </figure> - </div> - <p> - <s xml:id="s2710" xml:space="preserve">Ponamus <reg norm="nunc" type="context">nÅ©c</reg> Solem reperiri in Ä™qua <lb/>tore, vbi per ipÅ¿um Medinam arcus <var>.u.m.</var> <lb/>Å¿imiliter eſſet graduum .45. & Å¿ic <var>.u.e</var>: pro <lb/>tracto ergo arcu <var>.p.o.f.</var> palam erit ar-<lb/>cum <var>.f.e.</var> fore graduum .45. Å¿ed cum <lb/>arcus <var>.e.u.</var> Å¿it graduum .45. ex Å¿uppoÅ¿ito <lb/>ipÅ¿ius Medinæ, Å¿equeretur arcum <var>.e.f.</var> <lb/>æqualem eſſe arcui <var>.e.u.</var> pars igitur æqua <lb/>lis erit Å¿uo toto.</s> - </p> - <p> - <s xml:id="s2711" xml:space="preserve">Id etiam quod Petrus Nonius pagina <lb/>124. et .125. lib. de arte nauigandi con-<lb/>tra nautas de diÅ¿tantijs Solis à meridiano Å¿cribit, hanc opinionem Petrià Medina & <lb/>corum qui idem ei perÅ¿uaÅ¿erunt falÅ¿am eſſe demonÅ¿trat.</s> - </p> - <pb o="217" rhead="EPISTOLAE." n="229" file="0229" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0229"/> - <p> - <s xml:id="s2712" xml:space="preserve">FalÅ¿um eÅ¿t etiamid quod cap .3. lib. 6. pronuntiat, ita dicens.</s> - </p> - <p> - <s xml:id="s2713" xml:space="preserve">Quod cum verum eſſet à parte <reg norm="orientali" type="context">oriẽtali</reg> inÅ¿ularum quæ azore dicuntur, pyxidem <lb/>verÅ¿us eum ventum qui vulgò Græcus dicitur, & ab occidentali verÅ¿us eum qui Ma <lb/>giÅ¿ter dicitur, vergere, huius rei nulla eÅ¿t ratio.</s> - </p> - <p> - <s xml:id="s2714" xml:space="preserve">Ego enim huiuÅ¿modi rationem reperiri poſſe contendo, quæ talis eÅ¿t, quia pars <lb/>roſæ (ut vocant) à magnete tacta, ad aliquod punctum, aut Å¿itum globi terrÄ™, in eo-<lb/>dem meridiano inÅ¿ularum, quæ Azore dicuntur, vltra Å¿itum poli arctici in terra diri-<lb/>geretur, ita vt Å¿itus dicti poli in terra eſſet in dicto meridiano, inter locum qui ab in <lb/>dice roſæ aut pyxidis reÅ¿piceretur, & dictas inÅ¿ulas, id quod Å¿uperius Å¿cripto meridia <lb/>no facile cognoÅ¿ci poteÅ¿t, Å¿umendo pro inÅ¿ulis Å¿itum <var>.e.</var> in meridiano et <var>.z.</var> pro polo, <lb/>et <var>.p.</var> pro loco qui à pyxide Å¿it viÅ¿us, <reg norm="imaginando" type="context">imaginãdo</reg> deinde pyxidem in <var>.f.</var> magis orienta <lb/>li quam eÅ¿t <var>.e.</var> clarum eÅ¿t lineam quæ reÅ¿picit (ponamus) <var>f.p.</var> verÅ¿us Græcum & ab alia <lb/>parte verÅ¿us MagiÅ¿trum declinare.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head365" style="it" xml:space="preserve">De Armilla Nautica.</head> - <head xml:id="head366" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s2715" xml:space="preserve">CVm ſæpe viderim quam in magnis æquoribus nos fallant, <reg norm="atque" type="simple">atq;</reg> decipiant mari <lb/>timæ, Å¿eu nauigatoriæ chartÄ™, quemadmodum aliquoties inter nos Å¿ermonem <lb/>habuimus: </s> - <s xml:id="s2716" xml:space="preserve">in id totus incubui vt aliquam machinam excogitarem, quæ difficilis non <lb/>eſſet, <reg norm="efficeretque" type="simple">efficeretqÌ;</reg> vt nauis Å¿uper aquÄ™ globum, beneficio circulorum maiorum, quam <lb/>optimè poſſet, ideÅ¿t breuiſſimo itinere ab uno loco <reg norm="and" type="context">ãd</reg> alium ferretur. </s> - <s xml:id="s2717" xml:space="preserve">Id <reg norm="quod" type="simple">ê™</reg> mihi ex <lb/>animi voto Å¿ucceſſurum putaui, beneficio <reg norm="quinque" type="simple">quinq;</reg> circulorum circundantium <reg norm="aliquem" type="context">aliquẽ</reg> <lb/>globum <reg norm="terreÅ¿trem" type="context">terreÅ¿trẽ</reg> & maritimum, quales ij Å¿unt qui in inferiori Germania à Gerardo <lb/>Mercatore Å¿truuntur, qui vno pede cum dimidio diametri conÅ¿tet, ideÅ¿tÅ¿eÅ¿quipede.</s> - </p> - <p> - <s xml:id="s2718" xml:space="preserve">Sit ergo, exempli gratia, huiuÅ¿modi globus <var>.a.b.d.</var> circa quem duo circuli, aut cirÂ<lb/> - <anchor type="figure" xlink:label="fig-0229-01a" xlink:href="fig-0229-01"/> - <pb o="218" rhead="IO. BAPT. BENED." n="230" file="0230" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0230"/> - culares lineæ ex aurichalco applicentur inuicem coniuncti per medium ad angulos <lb/>rectos, quorum prior <var>.f.e.g.</var> in Å¿e globi polos mediantibus extremitatibus axis mun-<lb/>di contineat, qui quidem poli à punctis Å¿uarum interÅ¿ectionum per <reg norm="quartam" type="context">quartã</reg> ex æquo <lb/>in punctis <var>.f.</var> et <var>.g.</var> ita diÅ¿tent, vt globus circa eoÅ¿dem, in Å¿itu longitudinis mundi vol-<lb/>ui poſſit. </s> - <s xml:id="s2719" xml:space="preserve">HuiuÅ¿modi autem circulus, æquatoris deferens appelletur.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0229-01" xlink:href="fig-0229-01a"> - <image file="0229-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0229-01"/> - </figure> - </div> - <p> - <s xml:id="s2720" xml:space="preserve">Secundus autem circulus Å¿it <var>.h.e.K.</var> cum primo ad angulos rectos in puncto <var>.e.</var> & <lb/>in Å¿uo oppoÅ¿ito connexus, & is appellabitur æquator, & poli <var>.f.g.</var> primi poli dicentur.</s> - </p> - <p> - <s xml:id="s2721" xml:space="preserve">Circa huiuÅ¿modi duos circulos, alios <reg norm="etiam" type="context">etiã</reg> duos exiÅ¿tere <reg norm="vellem" type="context">vellẽ</reg> Å¿imul <reg norm="coniunctos" type="context context">cõiÅ©ctos</reg> medio <lb/>ad angulos rectos. </s> - <s xml:id="s2722" xml:space="preserve">In quibus <reg norm="quidem" type="context">quidẽ</reg> interÅ¿ectionis punctis Å¿int duo poli, qui hos duos <lb/>circulos cum Å¿ecundo priorum ideÅ¿t cum æquatore in duobus punctis inuicem op-<lb/>poÅ¿itis connectant; </s> - <s xml:id="s2723" xml:space="preserve">quæ æquatoris puncta à punctis interÅ¿ectionis eiuÅ¿dem cum Å¿uo <lb/>deferente, ratione vna quarta diÅ¿tent, quorum duorum circulorum primus Å¿it <var>.n.i.<lb/>m.</var> quem deferentem azimut appellabimus; </s> - <s xml:id="s2724" xml:space="preserve">Å¿ecundus <var>.r.n.s.m.</var> azimut locorum noÂ<lb/>minabimus. </s> - <s xml:id="s2725" xml:space="preserve">eorundem interÅ¿ectionis rectæ, puncta Å¿int <var>.n.</var> et <var>.m.</var> à quibus duo poli ex <lb/>aurichalco confecti Å¿imiles primis <var>.n.h.</var> et <var>.m.K.</var> vÅ¿que ad puncta <var>.h.</var> et <var>.K.</var> æquatoris <lb/>perueniant, qui Å¿pisÅ¿itudinem æquatoris diÅ¿tantem à puncto <var>.e.</var> vna quarta <reg norm="penetrent" type="context">penetrẽt</reg>, <lb/>ita vt æquator circum circa <var>.n.h.</var> et <var>.m.K.</var> in Å¿itu latitudinis mundi verti queat. </s> - <s xml:id="s2726" xml:space="preserve">Et <lb/>hos, Å¿ecundos polos nominabimus.</s> - </p> - <p> - <s xml:id="s2727" xml:space="preserve">Alius deinde circulus <var>.q.i.p.</var> duos poÅ¿teriores circulos ambiat, cum deferente ta-<lb/>men azimut mediantibus duobus polis in puncto <var>.i.</var> & in Å¿uo oppoÅ¿ito exÄ™quo diÅ¿tan <lb/>tibus à ſecundis polis vnius quartæ Å¿patio iungatur. </s> - <s xml:id="s2728" xml:space="preserve">Ita vt dictum deferens azimut <lb/>circa hos tertios polos volui poſſit, atque hunc circulum <var>.q.i.p.</var> orizontem vniuerÅ¿a <lb/>lem vocabimus. </s> - <s xml:id="s2729" xml:space="preserve">Hic vero orizon Å¿uper quatuor quartas circuli, aut Å¿uper quatuor <lb/>paruis columnis, ut fieri Å¿olet innixis Å¿uæ baÅ¿i, ita ponatur, vt moueri non poſſit.</s> - </p> - <p> - <s xml:id="s2730" xml:space="preserve">Primus autem circulus <var>.f.e.g.</var> deferens æquatoris in .4. partes æquales diuidatur, <lb/>quarum quælibet .90. gradibus conÅ¿tet, incipiendo ab interÅ¿ectionibus <var>.e.</var> & eius op <lb/>poÅ¿ito æquatoris, & numeri in polis <var>.f.</var> et <var>.g.</var> globi finem Å¿ortiantur. </s> - <s xml:id="s2731" xml:space="preserve">Diuidatur etiam <lb/>æquator <var>.h.e.K.</var> in .360. partes incipientes à puncto <var>.e.</var> verÅ¿us <var>.K.</var> deferens autem azi <lb/>mut <var>.n.i.m.</var> ab omni diuiÅ¿ione liber maneat, Å¿ed azimut <var>.n.s.m.r.</var> in .360. gradus inci <lb/>piendo à puncto <var>.n.</var> verÅ¿us <var>.r.</var> diuidatur.</s> - </p> - <p> - <s xml:id="s2732" xml:space="preserve">Orizon autem <var>.q.i.p.</var> diuidatur in quartas, quarum quælibet Å¿it nonaginta <reg norm="graduum" type="context">graduÅ©</reg> <lb/>incipiendo à puncto <var>.i.</var> & eius oppoÅ¿ito ideÅ¿t à polis poÅ¿tremis & terminando in pun-<lb/>ctis <var>.q.</var> et <var>.p.</var> in medio ipÅ¿orum polorum, & quarta <var>.i.p.</var> orientalis Å¿eptentrionalis, et <var>.i.<lb/>q.</var> orientalis meridiana appellentur. </s> - <s xml:id="s2733" xml:space="preserve">& Å¿ic ordine Å¿eruato occidentales.</s> - </p> - <p> - <s xml:id="s2734" xml:space="preserve">Præterea prÄ™parata Å¿it quædam quarta, ex aurichalco, circuli æqualis ipÅ¿i orizon-<lb/>ti, & in .90. gradus diÅ¿tincta quæ cum quauis Å¿uarum extremitatum ipÅ¿i zenit, in azi-<lb/>mut applicari poſſit, quemadmodum circa globos cÄ™leÅ¿tes fieri Å¿olet; </s> - <s xml:id="s2735" xml:space="preserve">quÄ™ quidem ad <lb/>cognoÅ¿cendam altitudinem poli ipÅ¿ius globi ab orizonte nobis inÅ¿eruiet.</s> - </p> - <p> - <s xml:id="s2736" xml:space="preserve">Atque hac ratione hanc noÅ¿tram machinam perfectè abÅ¿oluemus <reg norm="quam" type="context">quã</reg> appellan-<lb/>dam eſſe Armillam nauticam Å¿entio. </s> - <s xml:id="s2737" xml:space="preserve">Hic autem illud non omittam, concauum <lb/>duorum priorum circulorum à ſuperficie globi non nimis diÅ¿tare debere & con-<lb/>cauum aliorum à ſuperficie conuexa priorum longe poÅ¿itos eſſe <reg norm="non" type="context">nõ</reg> debere, & con <lb/>cauam orizontis à conuexa Å¿ecundorum procul abeſſe non debere.</s> - </p> - <p> - <s xml:id="s2738" xml:space="preserve">Neque illud etiam prætermittendum eÅ¿t, opere pretium fore Å¿i in interÅ¿ectione <lb/>e. priorum, erit foramen elicum, vt clauo elico ex aurichalco confecto, poſſimus <lb/>Å¿iÅ¿tere globum, quando oportuerit, ne amplius circa primos Å¿uos polos <var>.f.g.</var> circun-<lb/>uoluatur, cum <reg norm="Å¿itus" type="simple">Å¿itê°</reg> fuerit. </s> - <s xml:id="s2739" xml:space="preserve">Inde etiam laudo vt in azimut <var>.r.n.s.m.</var> è regione deferen <lb/>tis æquatoris, ideÅ¿t <var>.f.e.g.</var> aliud quoddam foramen huiuÅ¿modi Å¿it poÅ¿itum, in quo - <pb o="219" rhead="EPISTOL AE." n="231" file="0231" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0231"/> - clauus elicus vÅ¿que ad circulum <var>.f.e.g.</var> perueniens, æquatorem Å¿iÅ¿tere poſſit, ne circa <lb/>Å¿ecundos polos <var>.h.</var> et <var>.K.</var> amplius moueatur quum noluerimus eum mutare Å¿itum.</s> - </p> - <figure position="here"> - <image file="0231-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0231-01"/> - </figure> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head367" style="it" xml:space="preserve">DeuÅ¿u Armillæ nauticæ.</head> - <p> - <s xml:id="s2740" xml:space="preserve">V Tautem noÅ¿tra Armilla nautica vti poſſimus pyxidem nos prius oportebit <lb/>habere, diuerÅ¿am tamen ab ijs, quibus nautæ hactenus vÅ¿i fuere: </s> - <s xml:id="s2741" xml:space="preserve">nolo <lb/>enim vt <reg norm="tam" type="context">tã</reg> craſſa minerua beneficio <reg norm="ventorum" type="context">vẽtorum</reg> communium circa hanc rem nos gera <lb/>mus, Å¿ed ratione graduum orizontis in .360. partes diÅ¿tincti, atque ob hanc cauÅ¿am <lb/>Å¿entio, vt ima pars pyxidis penitus detecta videatur, & in .360. partes dinidatur, <reg norm="nilque" type="simple">nilqÌ</reg> <lb/>aliud quam quandam lanceo lam Å¿upra eius acum eſſe volo, quæ dum mouebitur na <lb/>uis, per gradus quamlibet orizontis partem oÅ¿tendet; </s> - <s xml:id="s2742" xml:space="preserve">hos autem .360. gradus, ita Å¿e <lb/>habere volo, vt quÄ™libet quarta .90. contineat, <reg norm="Å¿upputatioque" type="simple">Å¿upputatioqÌ;</reg> à linea meridiana inci-<lb/>piat, & in verticali deÅ¿inat, vt huiuÅ¿modi diuiÅ¿io cum ea, quæ eÅ¿t orizontis Armillæ <lb/>eadem Å¿it.</s> - </p> - <p> - <s xml:id="s2743" xml:space="preserve"><reg norm="Pręſupponantur" type="context">Pręſupponãtur</reg> nunc in globo duo loci extra æquatorem, & in diuerÅ¿is meridia-<lb/>nis quomodolibet a dinuicem diÅ¿tantes, à quorum vno ad alium Å¿it nauigandum iti-<lb/>nere quo ad fieri poterit breuiori, ideÅ¿t per gyrum circuli maioris, dixi autem extra <lb/>æquatorem, ideÅ¿t vt ambo, nec in æquatore, nec in uno <reg norm="eodemque" type="simple">eodemqÌ;</reg> meridiano <reg norm="exiÅ¿tant" type="context">exiÅ¿tãt</reg>, <lb/>quia vt alià s dixi in huiuÅ¿modi locis, vnico tantum vento comite, iter conficere <lb/>poſſumus.</s> - </p> - <pb o="220" rhead="IO. BAPT. BENED." n="232" file="0232" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0232"/> - <p> - <s xml:id="s2744" xml:space="preserve">Volo primum vt mediante <reg norm="circunuolutione" type="context">circÅ©uolutione</reg> globi circa primos polos <var>.f.g.</var> & æqua <lb/>toris circa Å¿ecundos <var>.h.K.</var> hoc eÅ¿t per longitudinem, & latitudinem, hi duo loci in <lb/>globo propoÅ¿iti Å¿ub azimut <var>.r.n.s.m.</var> Å¿ecundorum circulorum <reg norm="Å¿itum" type="context">Å¿itÅ©</reg> Å¿ortiantur, qui azi-<lb/>mut orizontem in punctis <var>.q.</var> et <var>.p.</var> Å¿emper ad angulos rectos diÅ¿peÅ¿cit <reg norm="ibique" type="simple">ibiq;</reg> globum <lb/>ita quieÅ¿cere vt circa polos <var>.f.g.</var> non voluatur, & æquatorem etiam Å¿ic Å¿irmare, vt cir-<lb/>ca Å¿ecundos polos <var>.h.K.</var> non vertatur faciamus.</s> - </p> - <p> - <s xml:id="s2745" xml:space="preserve">Quod cum factum fuerit, Å¿ecundorum circulorum primus, qui eÅ¿t <var>.n.i.m.</var> deferens <lb/>azimut, circa tertios polos <var>.i.</var> & eius oppoÅ¿itum, eo uÅ¿que voluatur quouÅ¿que prior <lb/>globi locus, ideÅ¿t is a quo iter eÅ¿t incohandum per .90. gradus azimut diÅ¿tet ab ori-<lb/>zonte, ideÅ¿t Å¿ub zenit orizontis <var>.q.i.p.</var> Å¿it poÅ¿itus, quemadmodum, exempli gratia, Å¿i <lb/>punctum <var>.a.</var> dicti primi loci globi rationem indueret, & borealius eſſet, mediante <lb/>circunuolutione circuli <var>.n.i.m.</var> circa dictos tertios polos æqualiter diÅ¿tans ab <var>.q.</var> et <var>.p.</var> <lb/>ideÅ¿t per .90. gradus poneretur Å¿ub <var>.r</var>.</s> - </p> - <p> - <s xml:id="s2746" xml:space="preserve">ConÅ¿ideretur deinde vbi æquator <var>.h.e.K.</var> Å¿ecundus circulus duorum primorum, <lb/>ab orizonte <var>.q.i.p.</var> Å¿ecabitur, exempli gratia, in puncto <var>.c.</var> quartÄ™ orientalis Å¿epten-<lb/>trionalis eiuÅ¿dem orizontis. </s> - <s xml:id="s2747" xml:space="preserve">Videatur deinde quot nam gradibus conÅ¿tabit ar-<lb/>cus <var>.i.c.</var> & per totidem gradus conÅ¿tituatur extremitas Å¿eptentrionalis lineæ meridia-<lb/>nÄ™ pyxidis nauticÄ™, diÅ¿tantis à cuÅ¿pide Å¿eptentrionali ipÅ¿ius lanceolæ orientem ver-<lb/>Å¿us, mediante nauis circunuolutione. </s> - <s xml:id="s2748" xml:space="preserve">vnde ipÅ¿amet nauis in huiuÅ¿modi Å¿itu azimut, <lb/>qui per duos hos loc<unsure/>os tranÅ¿it, dirigetur, eſſiciendo vt eius prora verÅ¿us <reg norm="locum" type="context">locÅ©</reg> ad <reg norm="quem" type="context">quẽ</reg> <lb/>voluerimus tendere dirigatur. </s> - <s xml:id="s2749" xml:space="preserve">Cum verò vela ventis dabimus, tot milliarium <lb/>Å¿eu leucarum iter conficiemus, quot quarta pars vnius gradus requirit. </s> - <s xml:id="s2750" xml:space="preserve">& dum <lb/>hociter abÅ¿oluitur, ille qui prÄ™eÅ¿t naui, defferentem azimut <var>.n.i.m.</var> circa Å¿uos polos <var>.i.</var> <lb/>& eius oppoÅ¿itum, Å¿ic circunuoluat, vt interÅ¿e ctio azimut <var>.r.n.s.m.</var> cum orizonte <var>.q.i.<lb/>p.</var> diÅ¿tet à prima ratione dictæ quartÄ™ partis vnius gradus, conÅ¿tituendo Å¿ecundum lo <lb/>cum, proximiorem zenit, ratione dictæ quartæ partis gradus azimut. </s> - <s xml:id="s2751" xml:space="preserve">HiÅ¿ce ita pera <lb/>ctis, obÅ¿eruetur deinde vbiæquator <var>.h.e.K.</var> hac Å¿ecunda vice interÅ¿ecabit orizontem <lb/><var>q.i.p.</var> quod quidem interÅ¿ectionis punctum Å¿emper appelletur <var>.c.</var> quod dico non am <lb/>plius in eadem diÅ¿tantia manÅ¿urum, ut prius à puncto <var>.i.</var> Å¿ed aut longius diÅ¿tabit, aut <lb/>propius accedet, vt in præſenti exemplo. </s> - <s xml:id="s2752" xml:space="preserve">quemadmodum ex Å¿e manifeÅ¿tum eÅ¿t, cú <lb/>poli globi, ideÅ¿t Ä™quatoris Å¿int extra azimut, vt præſupponitur, quia loci Å¿unt in diuer <lb/>Å¿is meridianis.</s> - </p> - <p> - <s xml:id="s2753" xml:space="preserve">Pro huiuÅ¿modi autem diÅ¿tantiæ ratione denuo dirigatur nauis prout æquator <var>.h.<lb/>e.K.</var> in orizonte <var>.q.i.p.</var> nobis oÅ¿tendet, atque hoc modo omnium iter quaÅ¿i breuiſſi-<lb/>mum fiet. </s> - <s xml:id="s2754" xml:space="preserve">dico autem, quaÅ¿i, quia omnibus modis neceſſariò conficitur iter contor-<lb/>tum & in formam Å¿erpigineæ lineæ. </s> - <s xml:id="s2755" xml:space="preserve">Applicantes deinde per vices extremitatem <lb/>quartæ appoÅ¿itæ (de qua Å¿uperius mentionem fecimus) ipÅ¿i zenit <var>.r.</var> efficientes ut per <lb/>Å¿itum poli globi pertranÅ¿eat, deueniemus in cognitionem altitudinis eiuÅ¿dem ab <lb/>orizonte, & per conÅ¿equens quantum itineris per latitudinem eiuÅ¿dem globi pere-<lb/>gerit. </s> - <s xml:id="s2756" xml:space="preserve">mediante deinde interÅ¿ectione orizontis <var>.q.i.p.</var> cum æquatore, cognoÅ¿cemus <lb/><reg norm="quantum" type="context">quãtum</reg> itineris per longitudinem eiuÅ¿dem globi, in ipÅ¿o Ä™quatore fuerit peractum,.</s> - </p> - <pb o="221" rhead="EPISTOLAE." n="233" file="0233" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0233"/> - </div> - <div type="letter" level="4" n="4"> - <head xml:id="head368" style="it" xml:space="preserve">Inctrumentum adortum, & occaÅ¿um LunacognoÅ¿cendum <lb/>qualibet anni die.</head> - <head xml:id="head369" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s2757" xml:space="preserve">ECce tibi vir IlluÅ¿triÅ¿s. <reg norm="modum" type="context">modÅ©</reg> <reg norm="conficiendi" type="context">conficiẽdi</reg> inÅ¿trumenti nuper à me <reg norm="inuenti" type="context">inuẽti</reg>, vt tibi Å¿i <lb/>gnificaui, quo Å¿cire poſſis fermè in dies, qua hora (de aÅ¿tronomicis loquor) ad <lb/>determinatum parallelum & abÅ¿que multa Å¿upputatione, etiam abÅ¿que AÅ¿trolabio <lb/>Luna oriatur <reg norm="occidatque" type="simple">occidatqÌ;</reg>. </s> - <s xml:id="s2758" xml:space="preserve">In quo inÅ¿trumento poteris etiam videre quo in Å¿igno Sol, <lb/>& ſæpius itidem Luna permeat, & huiuÅ¿ce aÅ¿pectus cum Sole, atque longitudinem <lb/>diei <reg norm="noctisque" type="simple">noctisqÌ;</reg> toto anni tempore exactè diÅ¿cernere.</s> - </p> - <p> - <s xml:id="s2759" xml:space="preserve">Circularis lamina ex argento, aut ære, aliaúe materia paranda eÅ¿t, in cu-<lb/>ius Å¿uperficie ambarum facierum Zodiacus delineabitur, modo inferius depicto, <lb/>deinde pro anno quinque circuli Å¿ibi inuicem <reg norm="concentrici" type="context">cõcentrici</reg>, at reÅ¿pectu Zodiaci excen <lb/>trici cÄ™labuntur in ea, adeo vt vtriuÅ¿que centri diÅ¿tantia Å¿it pro .32. parte Å¿emidia-<lb/>metri concauitatis Zodiaci è regione locis augis, temporis qui noÅ¿tra ætate circa Å¿i-<lb/>nem Å¿ecundi gradus cancri inuenitur, eandem viam, in hoc, Å¿equuti, quam Stofle-<lb/>rus in dorÅ¿o Aftrolabij docet. </s> - <s xml:id="s2760" xml:space="preserve">At nomina menÅ¿ium media ponantur inter duos <lb/>maiores circulos, poÅ¿tea inter Å¿ecundum, & tertium ab vna facierum laminæ, ar-<lb/>cus Å¿emidiurni, ab altera vero arcus Å¿eminocturni, per quinos quoÅ¿que dies collo-<lb/>centur, ita exactè, vt hic Å¿ubtus videbis. </s> - <s xml:id="s2761" xml:space="preserve">adeo vt numeri dierum & ipÅ¿orum dierum <lb/>Å¿igna Å¿int in interuallis vicinioribus centro communi dictorum quinque circulorum.</s> - </p> - <p> - <s xml:id="s2762" xml:space="preserve">PoÅ¿teaquam ab vna & altera <reg norm="facierum" type="context">facierÅ©</reg> laminæ hæc inÅ¿culpra fuerint, aliæ duæ circu <lb/>lares laminæ, magnitudinis Å¿emidiametri minimi quinque circulorum accipiantur: <lb/></s> - <s xml:id="s2763" xml:space="preserve">quarum vna pro ortu, & altera pro occaÅ¿u Lunæ deÅ¿eruiet. </s> - <s xml:id="s2764" xml:space="preserve">In qualibet ipÅ¿arum <lb/><reg norm="conÅ¿tituentur" type="context">conÅ¿tituẽtur</reg> circuli quatuor, eo modo qui paulo inferius cernitur, quos omnes diui-<lb/>demus in triginta Å¿pacia æqualia: </s> - <s xml:id="s2765" xml:space="preserve">& in interuallo <reg norm="quod" type="wordlist">qÄ‘</reg> inter duos primos circulos poÅ¿i <lb/>tum eÅ¿t, triginta dies <reg norm="annotabimus" type="simple">annotabimê°</reg> qui ipÅ¿os Lunætriginta dies <reg norm="pręſcribent" type="context">pręſcribẽt</reg>, vt in figara.</s> - </p> - <p> - <s xml:id="s2766" xml:space="preserve">PoÅ¿tmodum in lamina quæ ortus Lunæ indicabit, ac duorum maiorum <reg norm="circulo- rum" type="context">circulo-<lb/>rÅ©</reg> interuallo è regione numeri .1. videlicet primi diei, ponemus horas .12. & minuta <num value="48">.<lb/>48.</num> ex aduerÅ¿o diei Å¿ecundi ho .13. et min .36. ex oppoÅ¿ito tertij ho 14. min .24. & Å¿ic <lb/>Å¿ucceſſiuè augendo per min .48. & indicem è diuerÅ¿o diei .30. <reg norm="Å¿tatuendo" type="context">Å¿tatuẽdo</reg>, qui coitus <lb/>Lunæ cum Sole Å¿igniÅ¿icabit: </s> - <s xml:id="s2767" xml:space="preserve">atque lineas aÅ¿pectuum, vt inferius videre eÅ¿t facilè in <lb/>ueniemus.</s> - </p> - <p> - <s xml:id="s2768" xml:space="preserve">Altera in lamina quæ occaÅ¿um Lunæ indicabit, poÅ¿tquam diÅ¿tincta fuerit, vt alte-<lb/>ra .30. dies ac cÄ™teræ lineæ, eo modo quo in Å¿uperiori collocabuntur, at numeri in-<lb/>terualli maioris, aliter diÅ¿ponentur, vt potè ex aduerÅ¿o diei primi Å¿olum .48. minu-<lb/>ta deÅ¿cribi debent, è directo Å¿ecundi diei ponenda erit hora vna cum minutis .36. & <lb/>è regione tertij inÅ¿cribentur .2. horæ, & min .24. & Å¿ic ex ordine per .48. minuta au-<lb/>gendo.</s> - </p> - <p> - <s xml:id="s2769" xml:space="preserve">Nunc lamina ortus LunÄ™, cum anno arcuum Å¿eminocturnorum, & illa occaÅ¿us cum <lb/>anno arcuum Å¿emidiurnorum <reg norm="concentrari" type="context">concẽtrari</reg> debet, & ita noÅ¿trum inÅ¿trumentum perfe-<lb/>ctum erit & abÅ¿olutum.</s> - </p> - <p> - <s xml:id="s2770" xml:space="preserve">Quoties igitur voluerimus medio inÅ¿trumento dignoÅ¿cere fermè in tali orizonte <lb/>qua hora Luna oriatur, ita neceſſe erit volubilem rotam ortus flectere, ut index ve <lb/>niat è regione diei menÅ¿is in quo talis operatio fit & talirota Å¿irma manente perÅ¿pi- - <pb o="222" rhead="IO. BAPT. BENED." n="234" file="0234" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0234"/> - cere ex aduerÅ¿o diei Lunæ, numerum horarum & minutorum in maiori interuallo <lb/>ipÅ¿ius rotæ notatorum, qui cum arcu Å¿eminocturno anni, quo cum in ipÅ¿a rectitudi-<lb/>ne centri conueniet colligetur, & Å¿umma quæ ex tali Å¿upputatione proueniet aper-<lb/>tas faciet horas aÅ¿tronomicas, quibus ferè etÅ¿i non exactè in die propoÅ¿ito Luna <lb/>orietur. </s> - <s xml:id="s2771" xml:space="preserve">IdipÅ¿um fiet pro occaÅ¿u Lunæ.</s> - </p> - <p> - <s xml:id="s2772" xml:space="preserve"><emph style="sc">DIem</emph> ætatis Lunæ iam totus orbis Å¿cit inuenire, media Å¿upputatione nu-<lb/>meri EpactÄ™ currentis cum numero menÅ¿ium, Å¿umpto principio à Martio, <lb/>adiunctis diebus menÅ¿is currentis, & detracto numero .30. à ſumma prÄ™dicta, Å¿i ab ip <lb/>Å¿a dictus numerns .30. Å¿uperatur.</s> - </p> - <p> - <s xml:id="s2773" xml:space="preserve">Sed ne aliquis putet Å¿ufficere tantummodo additionem quatuor quintarum ho-<lb/>rÄ™ qualibet die. à nouilunio inchoando, Å¿ciendum eÅ¿t huiuÅ¿modi receſſum Lunæ <lb/>(quamuis non ita exactæ fiat) non <reg norm="computandum" type="context">computandÅ©</reg> eſſe ab orizonte aliquo, Å¿ed à recto, <lb/>Å¿eu à meridiano quod idem eÅ¿t, quemadmodum <reg norm="vnicuique" type="simple">vnicuiq;</reg> mediocriter erudito pa-<lb/>tere poteÅ¿t. </s> - <s xml:id="s2774" xml:space="preserve">At propoÅ¿itum nobis non eÅ¿t Å¿cire qua hora Luna in meridiano repe-<lb/>riatur, Å¿ed in noÅ¿tro obliquo orizonte, in parte orientali Å¿eu occidentali, </s> - <s xml:id="s2775" xml:space="preserve">propterea <lb/>igitur addendus eÅ¿t, ei Å¿ummæ temporis, qua Luna diÅ¿tat à meridiano, arcus Å¿emi-<lb/>diurnus, vel Å¿eminocturnus illius loci Zodiaci, in quo Luna reperitur illa die in pro <lb/>poÅ¿ito parallelo, vt Å¿ciatur proximÄ™, qua hora (ex aÅ¿tronomicis) Luna erit in ori-<lb/>zonte <reg norm="orientali" type="context">oriẽtali</reg>, vel occidentali dicti paralleli. </s> - <s xml:id="s2776" xml:space="preserve">Å¿upra dicta enim additio quatuor quin <lb/>tarum horæ tantummodo, Å¿ufficiens erit temporibus æquinoctij, Å¿ed aliis anni tem-<lb/>poribus falli ratione iam dicta.</s> - </p> - <pb o="223" n="235" file="0235" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0235"/> - <figure position="here"> - <image file="0235-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0235-01"/> - <caption style="it" xml:space="preserve">Pro Lunæ ortu. <lb/>Ad lati .45.</caption> - </figure> - <figure position="here"> - <image file="0235-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0235-02"/> - </figure> - <pb o="224" n="236" file="0236" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0236"/> - <figure position="here"> - <image file="0236-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0236-01"/> - <caption style="it" xml:space="preserve">Pro Lunæ occaÅ¿u. <lb/>Ad lati .45.</caption> - </figure> - <figure position="here"> - <image file="0236-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0236-02"/> - </figure> - <pb o="225" n="237" file="0237" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0237"/> - </div> - </div> - <div type="section" level="3" n="4"> - <div type="letter" level="4" n="1"> - <head xml:id="head370" xml:space="preserve">DE LVCERNA SPIRITALI QVAM SERENISS. <lb/>Sabaudiæ Duce <var>.D.</var> meo collendiÅ¿s. anno .1570. conÅ¿truxi.</head> - <head xml:id="head371" style="it" xml:space="preserve">CLARISS. FRANCISCO BARBARO VENETORVM <lb/>apud Sereniſſimum Sabaudiæ Ducem Oratori IlluÅ¿triſſimo.</head> - <p> - <s xml:id="s2777" xml:space="preserve"><emph style="sc">HEron</emph> varias ac diuerÅ¿as hydraulicas, & Å¿piritales machinas propoÅ¿uit, in <lb/>ter quastamen nullam Å¿imilem, ei <reg norm="quam" type="context">quã</reg> ego Sereniſſimo Sabaudiæ Duci <lb/>conÅ¿truxi, deÅ¿cribit, quæ quidem fuit Lucerna, & erat huiuÅ¿modi, vt à <lb/>magno aliquo vaÅ¿e oleo pleno Å¿upra alicuius triclinij tabulatum poÅ¿ito, <lb/>Å¿ubtilis quidam tubus perpendi-<lb/> - <anchor type="figure" xlink:label="fig-0237-01a" xlink:href="fig-0237-01"/> - culariter per tabulatum exiret, <lb/>& in dictum triclinium vÅ¿que ad <lb/>medium deÅ¿cenderet, ita tamcn <lb/>vt hic Å¿olus tubus, non item <lb/>vas oleo plenum cerneretur, cu-<lb/>ius <reg norm="quidem" type="context">quidẽ</reg> tubi inferiori extremi-<lb/>tati iunctum eſſet quoddam par-<lb/>uum receptaculum olei, Å¿imile co <lb/>operculo alicuius pyxidis, è cuius <lb/>ambitu prope baÅ¿im multi diuer-<lb/>Å¿i quæ tubi æquales & orizonta-<lb/>les, cuiuÅ¿uis longitudinis proÅ¿ili-<lb/>rent, quorum quilibet in extremi <lb/>tate Å¿ua, exiguam quandam pyra <lb/><reg norm="midem" type="context">midẽ</reg>, <reg norm="appensam" type="context context">appẽsã</reg> haberet, in qua elli <lb/><reg norm="chnum" type="context">chnÅ©</reg> eſſet <reg norm="cum" type="context">cÅ©</reg> mixo. </s> - <s xml:id="s2778" xml:space="preserve"><reg norm="oleum" type="context">oleÅ©</reg> deinde <lb/>medio <reg norm="perpendicularis" type="simple context">ê‘pẽdicularis</reg> tubi ad rece <lb/><reg norm="ptaculum" type="context">ptaculÅ©</reg> extrinÅ¿ece deÅ¿cendebat, <lb/>& per alios tubos ad nutriendas <lb/>flammas dum <reg norm="arderent" type="context">arderẽt</reg> ferebatur: <lb/></s> - <s xml:id="s2779" xml:space="preserve">at vero <reg norm="cum" type="context">cÅ©</reg> eædem erant extinctæ <lb/>ne minima quidem olei gutta de <lb/>Å¿cendebat: </s> - <s xml:id="s2780" xml:space="preserve">id quod eos qui aÅ¿ta-<lb/>bant in admirationem trahebat.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0237-01" xlink:href="fig-0237-01a"> - <image file="0237-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0237-01"/> - </figure> - </div> - <p> - <s xml:id="s2781" xml:space="preserve">Hæcautem lucerna Å¿ic erat <reg norm="con" type="context">cõ</reg> <lb/>Å¿tructa. </s> - <s xml:id="s2782" xml:space="preserve">Vas <reg norm="olearium" type="context">oleariÅ©</reg> cylindricum <lb/>vt in Å¿ubÅ¿cripta figura patet, cuiuÅ¿ <lb/>uis magnitudinis, omni ex parte <lb/>clauÅ¿um faciendum curaui, ita ta <lb/>men vt eius coopereulum <reg norm="aliquam" type="context">aliquã</reg> <lb/><reg norm="tulum" type="context">tulÅ©</reg> concauum eſſet, in cuius me <lb/>dio erat foramen <var>.e.</var> quod erat os <lb/>tubi <var>.e.g.</var> qui Å¿ub <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> vaÅ¿is fun-<lb/>do vÅ¿que ad <var>.g.</var> tranÅ¿ibat, Å¿ed po-<lb/>Å¿tea Å¿urÅ¿um, quaÅ¿i <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> ad cooper <lb/>culum in Å¿itu <var>.c.</var> ab inferiori parte <lb/>reflectebatur, & ibi <reg norm="terminabatur" type="simple">terminabat̃</reg>.</s> - <pb o="226" rhead="IO. BABPT. BENED." n="238" file="0238" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0238"/> - <s xml:id="s2783" xml:space="preserve">Vnde oleum quod in vas infundebatur per foramen <var>.e.</var> dictum vas poÅ¿tea ingredie-<lb/>batur per foramen <var>.c</var>. </s> - <s xml:id="s2784" xml:space="preserve">Habebat deinde tubum <var>.n.u.</var> rectum, qui à ſitu <var>.n.</var> propinquo co <lb/>operculo ad libellam extremi <var>.c.</var> incipiebat, & per fundum <reg norm="contignationemque" type="simple">contignationemqÌ;</reg> vÅ¿que <lb/>ad centrum Å¿upradictireceptaculi (circa quod <reg norm="tuborum" type="context">tuborÅ©</reg> ope appenfæ erant ellychnio-<lb/>rum pyramides) tranÅ¿ibat, atque huiuiuÅ¿modi tubi <var>.n.u.</var> extremitates tam Å¿uperius <lb/>quam inferius erant apertæ, & hic tubus aeris erat. </s> - <s xml:id="s2785" xml:space="preserve">Præterea aliud quoddam fora-<lb/> - <anchor type="figure" xlink:label="fig-0238-01a" xlink:href="fig-0238-01"/> - men in vaÅ¿is fundo feceram, cui <lb/>paruum <reg norm="tubum" type="context">tubÅ©</reg> <var>.a.o.t.</var> reflexum, ita <lb/>tamen, vt <var>.o.</var> altius eſſet quam <var>.g.</var> <lb/><reg norm="aptaueram" type="context">aptauerã</reg>, <reg norm="atque" type="simple">atq;</reg> <reg norm="per" type="simple">ê‘</reg> hunc reflexum tu <lb/>bum <var>.a.o.t.</var> oleum vaÅ¿is exibat, <reg norm="quod" type="simple">ê™</reg> <lb/>per oÅ¿culum <var>.t.</var> in quendam cana-<lb/>lem tubo <var>.n.u.</var> inÅ¿ertum, ab extra <lb/>oleum effundebat, & ab exteriori <lb/>parte arundinis <var>.n.u.</var> ingredieba-<lb/>tur receptaculum appenÅ¿um ex-<lb/>tremo dicti tubi <var>.n.u.</var> quod extre-<lb/>mum <reg norm="apertum" type="context">apertÅ©</reg> erat, ut dixi, & à fun-<lb/>do receptaculi tantum <reg norm="diÅ¿tans" type="context">diÅ¿tãs</reg>, <reg norm="quam" type="context">quã</reg> <lb/>tum volebamus oleum ab ipÅ¿ius <lb/>receptaculi fundo altum exiÅ¿tere <lb/>quod quidem oleum Å¿tatim vt ad <lb/>oÅ¿culum dicti tubi <var>.n.u.</var> accedebat <lb/>id claudebat. </s> - <s xml:id="s2786" xml:space="preserve">Vnde aeri <reg norm="ingrediem" type="context">ingrediẽ</reg> <lb/>di vas <var>.q.b.</var> non amplius patebat <lb/>aditus, & per conÅ¿equens, neque <lb/>amplius <reg norm="oleum" type="context">oleÅ©</reg> <reg norm="per" type="simple">ê‘</reg> <reg norm="tubum" type="context">tubÅ©</reg> <var>.a.o.t.</var> eflue-<lb/>bat, nec etiam per <reg norm="tubum" type="context">tubÅ©</reg> <var>.e.g.c.</var> aer <lb/>ingredi poterat, cum <var>.c.g.</var> Å¿emper <lb/>oleo exiÅ¿teret <reg norm="plenum" type="context">plenÅ©</reg> <reg norm="quum" type="context">quÅ©</reg> <var>.g.</var> magis <lb/><reg norm="quam" type="context">quã</reg> <var>.t.</var> <reg norm="adimum" type="context">adimÅ©</reg> deuergeret.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0238-01" xlink:href="fig-0238-01a"> - <image file="0238-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0238-01"/> - </figure> - </div> - <p> - <s xml:id="s2787" xml:space="preserve">Quoties deinde oleum in vas <lb/>infundere volebamus, oportebat <lb/><reg norm="cum" type="context">cÅ©</reg> fumitate digiti claudere oÅ¿cu-<lb/>lum <var>.t.</var> exiguitubi <var>.a.o.t.</var> vnde aer <lb/><reg norm="impulsus" type="simple">impulsê°</reg> ab oleo tubi <var>.e.g.c.</var> extra <lb/>per tubum <var>n.u.</var> quouÅ¿que oleum <lb/>vaÅ¿is ad Ä™quilibrium ipÅ¿ius <var>.n.</var> per-<lb/>ueniebat, <reg norm="pertubum" type="context">pertubÅ©</reg> <var>.n.u.</var> ingredie-<lb/>batu. </s> - <s xml:id="s2788" xml:space="preserve">& quando dictum oleum <reg norm="qui" type="simple">ê—</reg> <lb/>dictum tubum <var>.n.u.</var> extrinÅ¿ecè in-<lb/>trabat in receptaculum <var>.d.p.</var> nil <lb/>amplius olei in vas infundendum <lb/>erat, & oportebat alicuius digito <lb/>foramen <var>.u.</var> inferius arundinis <var>.n.<lb/>u.</var> claudi, & foramen <var>.t.</var> aperiri, vt <lb/><reg norm="per" type="simple">ê‘</reg> ipÅ¿um <var>.t.</var> aliqua portio olei exi-<lb/>ret, quia tunc quædam pars tubi <var>.e.g.</var> vacua reddebatur, & cum per <var>.t.</var> nil amplius - <pb o="227" rhead="EPISTOLAE." n="239" file="0239" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0239"/> - olei egrediebatur, aperiebatur <var>.u.</var> & peripÅ¿um <var>.t.</var> denuo tantum olei exire permitte-<lb/>bamus, quantum in receptaculo ad claudendum foramen <var>.u.</var> idoneum exiÅ¿teret. </s> - <s xml:id="s2789" xml:space="preserve">Ra-<lb/>tio vero, quÄ™ me mouit, ut punctum <var>.g.</var> inferius ipÅ¿ius <var>.t.</var> conÅ¿tituam, eÅ¿t, quia <reg norm="cum" type="context">cÅ©</reg> clau-<lb/>Å¿um erit <var>.u.</var> per dictum <var>.t.</var> oleum non amplius egredietur, quia pondus olei in tubo <var>.c.<lb/>g.</var> maius euadet oleo quod vÅ¿que ad <var>.t.</var> progrederetur, tubum autem <var>.e.g.c.</var> reflexum <lb/>facio, ne cogamur claudere foramen <var>.e.</var> quia hoc difficile præſtaretur, tubum etiam <lb/><var>a.o.t.</var> Å¿urÅ¿um verÅ¿usreflexum conÅ¿titui, vt aerem ab ingreſſu per foramen <var>.t.</var> <reg norm="arcerem" type="context">arcerẽ</reg>, <lb/>quia huiuÅ¿modi aer nunquam deÅ¿cendit Å¿i corpus magis denÅ¿um non deÅ¿cendat.</s> - </p> - <p> - <s xml:id="s2790" xml:space="preserve">Verum eÅ¿t, <reg norm="quod" type="simple">ê™</reg> melius erit, vt maiores difficultates euitemus, Å¿tatuere dictum <reg norm="tubum" type="context">tubÅ©</reg> <lb/><var>a.o.t.</var> ita curuum vt eÅ¿t .ω qui cum Å¿uo extremo inferiori ipÅ¿i <var>.n.u.</var> Å¿it contiguus ita ta-<lb/>men ut dictum extremum inferius <reg norm="non" type="context">nõ</reg> Å¿it inferius quam <var>.o.</var> quia totum oleum exiret.</s> - </p> - <p> - <s xml:id="s2791" xml:space="preserve">Volui etiam vt Å¿uperior extremitas <var>.n.</var> tubi <var>.n.u.</var> Å¿it in aere vaÅ¿is & non in oleo, ne <lb/>per eam oleum exeat, quia cum extremitas <var>.u.</var> inferior Å¿it <var>.g.</var> totum oleum quod Å¿u-<lb/>peraret oÅ¿culum <var>.n.</var> per dictum tubum <var>.n.u.</var> ratione maioris ponderis egrederetur, <lb/><reg norm="quemadmodum" type="wordlist">quẽadmodum</reg> cuilibet, vel mediocriter in philoÅ¿ophicis rebus verÅ¿ato innoteÅ¿cet.</s> - </p> - <pb o="228" n="240" file="0240" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0240"/> - </div> - </div> - <div type="section" level="3" n="5"> - <div type="letter" level="4" n="1"> - <head xml:id="head372" xml:space="preserve">DEFENSIO EPHEMERIDVM.</head> - <head xml:id="head373" style="it" xml:space="preserve">AD JLLVST.D. BERNARDVM <lb/>Trottum.</head> - <p> - <s xml:id="s2792" xml:space="preserve"><emph style="sc">EDita</emph> erant Å¿cripta quædam, quorum titulus <emph style="sc">animadver-<lb/>siones in ephemeridas</emph>. </s> - <s xml:id="s2793" xml:space="preserve">& breuis alia diÅ¿putatio de er <lb/>roribus calculorum AÅ¿tronomicorum. ac <reg norm="demum" type="context">demÅ©</reg> TheÅ¿es quæ-<lb/>dam typis datæ .11. AuguÅ¿ti .1581. quÄ™ omnia cum ad manus <lb/>meas perueniſſent, non potui, non eis animum admouere, <reg norm="cum" type="context">cÅ©</reg> <lb/>ibi de his Å¿tudijs <reg norm="ageretur" type="simple">ageret̃</reg>, in quibus partem non exiguam an-<lb/>norum meorum conÅ¿umpÅ¿i. </s> - <s xml:id="s2794" xml:space="preserve">nec tamen Å¿cribere aliquid Å¿ta-<lb/>tueram; </s> - <s xml:id="s2795" xml:space="preserve">tum quod exiÅ¿timarem viros AÅ¿tronomiæ peritos <lb/>facile, quanti facienda eſſent ea quæ edita erant iudicaturos, alijs verò haud gratam <lb/>futuram harum rerum tractationem. </s> - <s xml:id="s2796" xml:space="preserve">Tum quod Å¿i ingenue meam Å¿ententiam profer <lb/>re voluiſſem<unsure/>, non poteram, Å¿ine maxima authoris moleÅ¿tia ferè omnia reprobare. <lb/></s> - <s xml:id="s2797" xml:space="preserve">Quandoquidem Å¿olet vnuÅ¿quiÅ¿que indignationem concipere ex his, quæ Å¿uæ opi-<lb/>nioni repugnant, id omne maleuolentiæ potius quam veritatis Å¿tudio tribuens. </s> - <s xml:id="s2798" xml:space="preserve">Qui <lb/>nimo cum nec deeſſent qui dicerent in meipÅ¿um directa ea tela fuiſſe, nullam fidem <lb/>eis adhibendam duxi, nec enim qui in ephemeri das inuehitur, me arguere poteÅ¿t, <lb/>qui nullas ephemeridas Å¿cripÅ¿i, nec tabulas compoÅ¿ui. </s> - <s xml:id="s2799" xml:space="preserve">Nec Å¿i author quidpiam ex <lb/>his quæ à nobis edita fuiſſent impugnare voluiſſet, ægrè ferre debuiſſem, modo à ve <lb/>ritate nuÅ¿quam deuiaſſet. </s> - <s xml:id="s2800" xml:space="preserve">Liberum enim eÅ¿t cuique Å¿cribere quodlibet. </s> - <s xml:id="s2801" xml:space="preserve">nec Ari-<lb/>Å¿totelem afficit iniuria, quicunque illi fidem Å¿uam non accommodat, & Å¿i valdè ini-<lb/>quus Å¿it, quiÅ¿quis maiorum opiniones veras, & ab omnibus merito comprobatas <reg norm="non" type="context">nõ</reg> <lb/>admittit. </s> - <s xml:id="s2802" xml:space="preserve">Hinc mihi Å¿atis <reg norm="oimbus" type="context">oÄ©bus</reg> feciſſe videbar, cum his qui de Å¿criptis illis me <reg norm="inter- rogauerant" type="context">inter-<lb/>rogauerãt</reg> reÅ¿pondiſſem, ea non Å¿atis firmis eſſe innixa <reg norm="fundamentis" type="context">fundamẽtis</reg>, & quod ad talia <lb/>tractanda opus fuiſſet exercitatiore iudicio. </s> - <s xml:id="s2803" xml:space="preserve">Verumtamen cum tu vnus maxime om <lb/>nium deÅ¿ideres tibi clarius, quæ nam de his mea Å¿it Å¿ententia explicari, non tam tuis <lb/>precibus deuictus quam mea ipÅ¿ius cupiditate de te benemerendi impulÅ¿us, non Å¿u-<lb/>Å¿tineo diutius animum tuum hęſitantem relinquere. </s> - <s xml:id="s2804" xml:space="preserve">Atque vt tibi adeo honeÅ¿ta cu <lb/>pienti <reg norm="morem" type="context">morẽ</reg> geram paucis hiÅ¿ce Å¿criptis incultis quidem, vt ab homine omni pror-<lb/>Å¿us facundia deÅ¿tituto exaratis, Å¿ed ex quibus nihilominus facile, atque perÅ¿picue, vt <lb/>Å¿pero, conceptum animi noÅ¿tri percipere poſſis, Å¿i tamen eam præſtantis ingenij tui <lb/>aciem adhibueris, qua Å¿oles intima quæque Å¿cientiarum penetrare, noÅ¿træ opinio-<lb/>nis Å¿ummam perÅ¿trinxi, quÄ™ ad te mittere decreui. </s> - <s xml:id="s2805" xml:space="preserve">& quamuis ipÅ¿a res de qua agitur, <lb/>quæ exactiorem deÅ¿iderat expoÅ¿itionem, prolixiorem me eſſe cogerit quam voluiÅ¿-<lb/>Å¿em. </s> - <s xml:id="s2806" xml:space="preserve">multa tamen me obmiÅ¿iſſe intelliges, non admodum neceſſaria his quibus <lb/>AÅ¿trologiæ noti Å¿unt termini, vt tuarum occupationum rationem me etiam habere <lb/>intelligeres, at que vt Å¿ummam oblectationem <reg norm="concipiam" type="context">cõcipiam</reg> animo Å¿i me tibi aliqua ex <lb/>parte Å¿atisfeciſſe intellexero ita humanitati tuæ gratiam habebo, quÄ™ mihi occaÅ¿io <lb/><reg norm="nem" type="context">nẽ</reg> <reg norm="praebuit" type="simple">p̃buit</reg>, imò verò me impulit, ad ea <reg norm="proferenda" type="context">proferẽda</reg>: </s> - <s xml:id="s2807" xml:space="preserve">quæ grata eſſe poſſint tui Å¿imilibus, <lb/>ideÅ¿t <reg norm="praeclaro" type="simple">p̃claro</reg> & <reg norm="candido" type="context">cãdido</reg> ingenio præditis, <reg norm="atque" type="simple">atq;</reg> ad <reg norm="euellendam" type="context context">euellẽdã</reg> ex <reg norm="eorum" type="context">eorÅ©</reg> animis falÅ¿am opi- - <pb o="229" rhead="EPISTOL AE." n="241" file="0241" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0241"/> - nionem, Å¿i quam fortaſſe ex <reg norm="illorum" type="context">illorÅ©</reg> Å¿criptorum lectione <reg norm="conceperunt" type="context">cõceperunt</reg> circa ea, de qui-<lb/>bus nunc Å¿um acturus.</s> - </p> - <p> - <s xml:id="s2808" xml:space="preserve"><reg norm="Quemadmodum" type="context">QuemadmodÅ©</reg> igitur ab hoc authore ter Å¿criptum fuit de <reg norm="contradictionibus" type="context">cõtradictionibus</reg>, Å¿iue <lb/>erroribus Ephemeridum, & earum calculos Å¿equentium, & de ratione qua cognoÅ¿ci <lb/>poteÅ¿t Å¿itus & locus alicuius Å¿uperioris planetæ, diuerÅ¿us ab eo, qui ab ipÅ¿is Epheme <lb/>ridibus aſſignatus eÅ¿t, ita diÅ¿putationem hanc meam diuidam in tres partes, quo Å¿ci <lb/>licet minus confusè, & magis diÅ¿tinctè à me Å¿cribatur, <reg norm="praeÅ¿upponendo" type="simple">p̃ſupponendo</reg>, vt animaduer-<lb/>tere potes, huius Å¿criptoris intentionem, aliam non fuiſſe, <reg norm="quam" type="context">quã</reg> <reg norm="oÅ¿tendere" type="context">oÅ¿tẽdere</reg>, quod Å¿cripto <lb/>res Ephemeridum diuerÅ¿imode eiuÅ¿dem temporis locum planetæ aſſignauere, & <lb/>quod eum faciant modo nimium velociter currere, modo nimium in vno Å¿igno mo-<lb/>rari, vt (exempli gratia) Martem interdum faciunt morari Å¿ex, aut Å¿eptem menÅ¿ibus <lb/>in vno Å¿igno. </s> - <s xml:id="s2809" xml:space="preserve"><reg norm="Idque" type="simple">Idq;</reg> poÅ¿tea in cauÅ¿a eſſe ait, vt AÅ¿trologi indiciarij <reg norm="fallantur" type="context">fallãtur</reg>, & Å¿imul <lb/>careant certis fundamentis rationum quibus futura indicent, & prædicant. </s> - <s xml:id="s2810" xml:space="preserve">Primum <lb/>ergo videndum eÅ¿t, quam rectè hic vÅ¿us Å¿it arte, & Å¿cientia, vt aliorum opiniones, & <lb/>Å¿cripta redarguere poſſet. </s> - <s xml:id="s2811" xml:space="preserve">Deinde videbimus quomodo verum Å¿it, & poſſibile id <lb/>quod ab AÅ¿trologis hactenus creditum, atque traditum eÅ¿t, & qua ratione poſſint Å¿ie <lb/>ri veri calculi à peritis regularum Å¿cientiæ.</s> - </p> - <p> - <s xml:id="s2812" xml:space="preserve">In primo igitur tractatu inÅ¿cripto AnimadnerÅ¿iones, præſupponit Author pro-<lb/>feſſores huius Å¿cientiæ neÅ¿cire inuenire vera loca planetarum, quia vtuntur Ephe-<lb/>meridibus, in quibus eorum loca non rectè Å¿unt notata. </s> - <s xml:id="s2813" xml:space="preserve">Quod Å¿ecundum ipÅ¿um ori <lb/>tur, ex errore calculatorum, Å¿eu computiÅ¿tarum, potius quam ex varietate tabula-<lb/>rum, à quibus Ephemerides Å¿umptæ Å¿unt, hoc tamen verum non eÅ¿t, Ephemeridas, <lb/>Å¿cilicet, ita inter Å¿e differre, ratione errorum computiÅ¿tarum tantummodo, Å¿ed po-<lb/>tius ratione ipÅ¿arum tabularum, & Å¿i interdum contingere poſſit error aliquorum mi <lb/>nutorum, nec non graduum, non propterea Ephemerides ita Å¿pernendæ Å¿unt. </s> - <s xml:id="s2814" xml:space="preserve">In <lb/>multis enim calculis, tales errores excuÅ¿abiles Å¿unt, cum ab innumerabilibus propè <lb/>accidentibus oriri poſſint, præſertim in calculis prutenicis.</s> - </p> - <p> - <s xml:id="s2815" xml:space="preserve">Videatur deinde vbi is profert quinquageÅ¿imum enuntiatum centiloquij Ptole-<lb/>mÄ™i, Å¿atis mendoÅ¿e. </s> - <s xml:id="s2816" xml:space="preserve">PtolemÄ™us enim ibi Å¿ic ait.</s> - </p> - <quote> - <s xml:id="s2817" xml:space="preserve">Non obliuiÅ¿caris eſſe centum viginti coniunctiones, quæ Å¿unt in Å¿tellis erraticis, <lb/>in illis enim eÅ¿t maior Å¿cientia eorum quæfiunt in hunc <reg norm="mundum" type="context">mÅ©dum</reg> Å¿uÅ¿cipiendi incre-<lb/>mentum, & decrementum.</s> - </quote> - <p> - <s xml:id="s2818" xml:space="preserve">Nam, neque eo in loco, neque alibi, PtolemÄ™us quidquam eius dicere voluit <lb/>quod ab hoc profertur.</s> - </p> - <p> - <s xml:id="s2819" xml:space="preserve">Pergatur poÅ¿tea in pag .2. & videbitur hunc exiÅ¿timare abÅ¿urdum quod Saturni, <lb/>& Iouis coniunctio vera anni .1563. potuerit eſſe in Leone Å¿igno igneæ triplicitatis <lb/>cum eorum coniunctio vera anni .1544. fuerit in Scorpione, Å¿igno triplicitatis <lb/>aqueæ, & cum coitus eorum anni .1583. futurus Å¿it in PiÅ¿cibus, Å¿igno pariter tripli-<lb/>citatis aqueæ. </s> - <s xml:id="s2820" xml:space="preserve">Ita enim ait.</s> - </p> - <quote> - <s xml:id="s2821" xml:space="preserve">Nam poÅ¿tquam duæ Å¿tellæ coiuerint, non prius Å¿ub alio alterius triplicitatis Å¿igno <lb/>inter Å¿e Å¿unt conuenturæ, quam per omnia Å¿igna quæ eiuÅ¿dem ternarij cum primo ex <lb/>titerint prius <reg norm="coniungantur" type="context">coniungãtur</reg>. </s> - <s xml:id="s2822" xml:space="preserve">Ita Å¿entit PtolemÄ™us, <reg norm="cæterique" type="simple">cæteriqÌ;</reg> non aÅ¿pernendi nominis <lb/>Aftronomi.</s> - </quote> - <p> - <s xml:id="s2823" xml:space="preserve">Et <reg norm="tamen" type="wordlist">tamẽ</reg> PtolemÄ™us nunquam <reg norm="quidquam" type="context">quidquã</reg> huius rei attigit, & quamuis AlbumaÅ¿ar <lb/>& Alchibitius de eo <reg norm="loquantur" type="context">loquãtur</reg>, is tamen eosnon intellexit, cum illi ibi <reg norm="non" type="context">nõ</reg> agant de - <pb o="230" rhead="IO. BAPT. BENED." n="242" file="0242" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0242"/> - periodis apparentibus, aut veris, Å¿ed de mediocribus aut æqualibus, & quidem re-<lb/>ctè dicunt, quia lineæ eorum mediorum motuum non coeunt in aliquo Å¿igno alte-<lb/>rius triplicitatis, prius quam pertranÅ¿iuerint omnia Å¿igna illius, in qua incÄ™perunt. <lb/></s> - <s xml:id="s2824" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> nullum inconueniens Å¿equitur, Å¿i in veris <reg norm="coniunctionibus" type="context">cõiunctionibus</reg> non reperitur hæc re-<lb/>gula. </s> - <s xml:id="s2825" xml:space="preserve">Fieri enim poteÅ¿t, vt lineæ mediorum motuum coniungantur in vno Å¿igno, cor <lb/>pora verò eorum planetarum coeant in alio, cum rarò eueniat, vt linea medij mo-<lb/>tus, eadem Å¿it cum linea veri.</s> - </p> - <p> - <s xml:id="s2826" xml:space="preserve">Nunc quidem <reg norm="tamen" type="wordlist">tamẽ</reg> non affirmauerim, nec ne gauerim eorum coniunctionem an <lb/>ni .1563. fuiſſe potius in Cancro, quam in Leone. </s> - <s xml:id="s2827" xml:space="preserve">Sed tantum dicam <reg norm="vanum" type="context">vanÅ©</reg> eſſe cre <lb/>dere id eueniſſe propter Å¿imilem naturam, aut qualitatem Å¿ignorum. </s> - <s xml:id="s2828" xml:space="preserve">Hunc enim <lb/>reÅ¿pectum non habent illi planetæ in verisÅ¿uis coniunctionibus. </s> - <s xml:id="s2829" xml:space="preserve">Exempli autem <lb/>cauÅ¿a ponamus, quod rectè Å¿upputatæ fuerint coniunctiones annorum .1484. 1504. <lb/>& 1524. quod attinet ad differentiam duodecatemorij, Å¿cilicet prima in .24. gradu <lb/>Scorpij, Å¿ecunda in .20. Cancri, tertia in .10. PiÅ¿cium. </s> - <s xml:id="s2830" xml:space="preserve">Cum Å¿ecunda anticipauerit <lb/>trigonum perfectum cum prima, gradibus .4. & tertia anticipauerit trigonum perfe-<lb/>ctum <reg norm="cum" type="context">cÅ©</reg> Å¿ecunda gradibus .10. Å¿i forte prima vt facta fuit in .24. gradu Scorpij facta <lb/>fuiſſet in .2. gradu eiuÅ¿dem, <reg norm="planum" type="context">planÅ©</reg> eÅ¿t <reg norm="quod" type="simple">ê™</reg> <reg norm="Å¿ecunda" type="context">Å¿ecÅ©da</reg> facta fuiſſet in .28. gradu geminorum <lb/>& tertia in .18. Aquarij, quæ Å¿igna Å¿unt diuerſæ triplicitatis ab illa Cancri. </s> - <s xml:id="s2831" xml:space="preserve">InÅ¿uper <lb/>Å¿i coniunctio anni .1544. quæ fuit in .28. gradu Scorpij fuerit recta <reg norm="correÅ¿pondens" type="context">correÅ¿pondẽs</reg> <reg norm="prae cedenti" type="simple">prÄ™Â<lb/>cedenti</reg>, anni .1524. per gra .18. Å¿ine dubio Å¿i coniunctio anni .1524. facta fuiſſet in <lb/>18. gradu Aquarij, illa anni .1544. fuiſſet in .6. Scorpij Å¿igni alterius triplicitatis <reg norm="quam" type="context">quã</reg> <lb/>Å¿int Gemini. </s> - <s xml:id="s2832" xml:space="preserve">Præterea, vt anno .1544. <reg norm="coniunctio" type="context">cõiunctio</reg> facta eÅ¿t in .28. gra. Scorpij, & 1563 <lb/>in .29. Cancri, ponendo eas eſſe rectas, quod attinet ad Å¿uperandum trigonum vno <lb/>gradu, Å¿i anno .1544. facta fuiſſet in .30. Scorpij, anno .1563. proculdubio facta fuiÅ¿-<lb/>Å¿et in primo gradu Leonis. </s> - <s xml:id="s2833" xml:space="preserve">Et Å¿uppoÅ¿itis ijs interuallis, quæ Å¿uperÅ¿unt, aut deÅ¿unt per <lb/>fectis trigonis, Å¿i coniunctio anni .1524. fuiſſet in .20. gradu PiÅ¿cium, anno .1544. <lb/>fuiſſet in .8. Sagittarij. </s> - <s xml:id="s2834" xml:space="preserve">Quæ quidem omnia aduerÅ¿antur opinioni huius Å¿criptoris.</s> - </p> - <p> - <s xml:id="s2835" xml:space="preserve">Quodautem opinatur coniunctionem anni .1583. fore in Ariete, Å¿ic dicens pagi <lb/>na Å¿ecunda.</s> - </p> - <quote> - <s xml:id="s2836" xml:space="preserve">Non erit ab re Å¿i & eandem Saturni, & Iouis coniunctionem in primo igneæ tri-<lb/>plicitatis Å¿igno, quod eÅ¿t Aries futuram afferamus anno .1583. Å¿i ab accidentibus no <lb/>bis licet, vt ab omnibus paſſim conceditur, planetarum loca diÅ¿cernere.</s> - </quote> - <p> - <s xml:id="s2837" xml:space="preserve">In eo fallitur, <reg norm="nam" type="context">nã</reg> <reg norm="neque" type="simple">neq;</reg> Saturnus, <reg norm="neque" type="simple">neq;</reg> Iupiter, <reg norm="errant" type="context">errãt</reg> à vero per .9: nec .8. gra. ac ne <reg norm="per" type="simple">ê‘</reg> <lb/>4. <reg norm="quidem" type="context">quidẽ</reg> in <reg norm="qui" type="simple">ê—</reg> buÅ¿uis Ephemeridibus aut tabulis. </s> - <s xml:id="s2838" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> videbit eiuÅ¿modi <reg norm="coniunctionem" type="context context">cõiunctionẽ</reg> <lb/>contra <reg norm="Å¿ententiam" type="context">Å¿ententiã</reg> <reg norm="Å¿uam" type="context">Å¿uã</reg> fieri in PiÅ¿cibus, <reg norm="non" type="context">nõ</reg> <reg norm="autem" type="wordlist">aÅ©t</reg> in Ariete. (vt poÅ¿tea res ipÅ¿a nos docuit <lb/>Å¿ub <reg norm="menÅ¿e" type="context">mẽſe</reg> Aprili. poÅ¿t <reg norm="quidem" type="simple context">ê—dẽ</reg> <reg norm="Å¿criptam" type="context">Å¿criptã</reg> <reg norm="hanc" type="context">hãc</reg> <reg norm="epiÅ¿tolam" type="context">epiÅ¿tolã</reg>, <reg norm="vulgarique" type="simple">vulgariqÌ;</reg> Å¿ermone <reg norm="tranÅ¿miÅ¿sam" type="context context">trãſmiÅ¿sã</reg>, Å¿ed <reg norm="an- tequam" type="context">an-<lb/>tequã</reg> in <reg norm="latinum" type="context">latinÅ©</reg> <reg norm="tranÅ¿lata" type="context">trãſlata</reg>, & huic volumini inÅ¿erta <reg norm="cum" type="context">cÅ©</reg> alijs Typographo <reg norm="committeretur" type="context">cõmitteretur</reg>.)</s> - </p> - <p> - <s xml:id="s2839" xml:space="preserve">Vbi poÅ¿tea meminit magnæ periodi annorum .960. non tantum ei cogitandum <lb/>erat <reg norm="hanc" type="context">hãc</reg> fuiſſe opinionem <reg norm="antiquorum" type="context">antiquorÅ©</reg>, vt videri <reg norm="pont" type="context">põt</reg> apud <reg norm="AlbumaÅ¿arem" type="context">AlbumaÅ¿arẽ</reg> & <reg norm="Alchibicium" type="context">AlchibiciÅ©</reg>, <lb/>Å¿ed etiam perpendendum an eſſet vera, <reg norm="priuÅ¿quam" type="context">priuÅ¿quã</reg> ei adhæreret. </s> - <s xml:id="s2840" xml:space="preserve">Hic enim fuit vnus <lb/>ex erroribus illius ætatis, quæ nondum penetrauerat intima huius Å¿cientiæ. </s> - <s xml:id="s2841" xml:space="preserve">Sunt <lb/>tamen illi antiqui excuÅ¿atione aliqua digni. </s> - <s xml:id="s2842" xml:space="preserve">Ponebant enim vigeÅ¿imo quoque an-<lb/>no præcisè fieri mediam coniunctionem Saturni <reg norm="cum" type="context">cÅ©</reg> Ioue, & in quolibet Å¿igno eiuÅ¿-<lb/>dem triplicitatis <reg norm="coniungi" type="context">cõiungi</reg> quater. </s> - <s xml:id="s2843" xml:space="preserve">Itaque in qualibet triplicitate dicebant eos coire <lb/>duodecies.</s> - </p> - <pb o="231" rhead="EPISTOL AE." n="243" file="0243" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0243"/> - <p> - <s xml:id="s2844" xml:space="preserve">Quod Å¿ecundum primum Å¿uppoÅ¿itum finiebatur Å¿pacio annorum .240. qui nume <lb/>rus fit. ex .20. duodecies multiplicatis. </s> - <s xml:id="s2845" xml:space="preserve">Et quia triplicitates Å¿unt .4. ideo credebant <lb/>in Å¿pacio annorum .960. qui numerus fit ex .240. quater multiplicato, perfici .48. <reg norm="con- iunctiones" type="context">cõ-<lb/>iunctiones</reg>, priuÅ¿quam redirent ad Å¿e coniungendos in eodem loco, ubi prius iun-<lb/>cti fuiſſent. </s> - <s xml:id="s2846" xml:space="preserve">Primum autem Å¿uppoÅ¿itum, quod vigeÅ¿imo quoque anno <reg norm="iungerentur" type="context">iÅ©gerentur</reg>, <lb/>colligebant Å¿ic ratiocinantes. </s> - <s xml:id="s2847" xml:space="preserve">Si Saturnus annis .30. peragit Å¿uum curÅ¿um per om-<lb/>nia Å¿igna Zodiaci, Iupiter autem peragit eum annis .12. Saturnus ambulauerit .4. Å¿i <lb/>gna, et .4. quintas partes Å¿igni, Å¿iue gra .24. dum Iupiter peragit integrum ambitum <lb/>ideÅ¿t annis .12. </s> - <s xml:id="s2848" xml:space="preserve">Itaque deÅ¿unt ei anni .8. ad perueniendum ad .20. quibus .8. annis Sa-<lb/>turnus <reg norm="perambulat" type="simple">ê‘ambulat</reg> Å¿igna tria & <reg norm="quintam" type="context">quintã</reg> <reg norm="partem" type="context">partẽ</reg> unius Å¿igni <var>.i.</var> gradus .6. qui iuncti dictis Å¿i <lb/>gnis .4. & gra .24. faciunt Å¿igna .8. quæ Iupiter item percurrit in annis .8. atque ita in <lb/>annis .20. </s> - <s xml:id="s2849" xml:space="preserve">Iupiter percurrit .20. Å¿igna <reg norm="antequam" type="context">antequã</reg> perueniat ad <reg norm="Saturnum" type="context">SaturnÅ©</reg>, cum Saturnus eo <lb/>dem tempore perfecerit curÅ¿um Å¿ignorum .8. </s> - <s xml:id="s2850" xml:space="preserve">Eandem concluÅ¿ionem etiam fortaÅ¿ <lb/>Å¿e collegerant ex dictis Å¿uppoÅ¿itis, dicentes, Å¿i Saturnus annis .30. ambulat .12. Å¿igna <lb/>proculdubio annis .20. ambulat .8. Å¿igna, quo tempore Iupiter perambulat .20. ad ra <lb/>tionem .12. Å¿ignorum in annis .12.</s> - </p> - <p> - <s xml:id="s2851" xml:space="preserve">Verum hoc Å¿uppoÅ¿itum non eÅ¿t bonum, quoniam, Å¿i ita eſſet, coniunctiones <reg norm="horum" type="context">horÅ©</reg> <lb/>duorum planetarum nunquam exirent ex vna triplicitate, & non modo .960. <reg norm="quoque" type="simple">quoq;</reg> <lb/>anno, Å¿ed etiam Å¿exageÅ¿imo rurÅ¿us coniungerentur in eodem puncto. </s> - <s xml:id="s2852" xml:space="preserve">nec coniunctio <lb/>nes eorum (Å¿emper autem intelligo de medijs) unquam egrederentur ex illis tribus <lb/>Å¿ignis Zodiaci.</s> - </p> - <p> - <s xml:id="s2853" xml:space="preserve">Sed periodus æqualis Saturni, eÅ¿t dierum circiter .10740. atque ita minor an .30. <lb/>atque etiam .29. cum dimidio periodus autem æqualis Iouis, eÅ¿t circiter .4328. vt <lb/>ego eam comperio, quidquid alij <reg norm="dicant" type="context">dicãt</reg>, <reg norm="vtque" type="simple">vtqÌ;</reg> planius alias oÅ¿tendam. </s> - <s xml:id="s2854" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> hæc per <lb/>iodus Iouis, etiam minor eÅ¿t ann .12. prætermittendo in Å¿upputatione <reg norm="tam" type="context">tã</reg> Saturni <reg norm="quam" type="context">quã</reg> <lb/>Iouis quaÅ¿dam minutias horarum & earum partium, quæ hac in re pto nihilo habe-<lb/>ri poſſunt. </s> - <s xml:id="s2855" xml:space="preserve">Atque his duabus periodis eccentricorum duorum planetarum poſſu-<lb/>mus cognoÅ¿cere interuallum quod erit inter vtramque mediam coniunctionem, hoc <lb/>modo agendo, & ratiocinando.</s> - </p> - <p> - <s xml:id="s2856" xml:space="preserve">Si Saturnus diebus .10740. circuit gradus .360. diebus .4328. qui Å¿unt periodus <lb/>Iouis, conficiet gradus .145. & min .4. ideÅ¿t min .8704. & eadem regula inueniemus <lb/><reg norm="quod" type="simple">ê™</reg> Saturnus .30. <reg norm="quibuÅ¿que" type="simple">quibuÅ¿qÌ;</reg> diebus, conÅ¿iciet min .60. & Å¿ecunda .20. </s> - <s xml:id="s2857" xml:space="preserve">Iupiter autem Å¿in-<lb/>gulis .30. diebus, conficiet min .149. & Å¿ecunda .43. vnde Å¿ubtrahendo minuta Satur-<lb/>ni à minutis Iouis, Å¿upererunt min .89. cum Å¿ecun .23. </s> - <s xml:id="s2858" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> Iupiter .30. <reg norm="quibuÅ¿que" type="simple">quibuÅ¿qÌ;</reg> die-<lb/>bus velocitate curÅ¿us, Å¿uperabit Saturnum minutis .89. cum Å¿ecundis .23. </s> - <s xml:id="s2859" xml:space="preserve"><reg norm="Atque" type="simple">Atq;</reg> dicen-<lb/>do, Å¿i minuta .89. cum Å¿ecundis .23 dant nobis dies .30. Å¿upradicta, minuta .8704. da-<lb/>bunt nobis dies .2921. quibus iunctis cum diebus .4328. periodi Iouis, efficientur dies <lb/>7249. ideÅ¿t anni Aegiptij .19. cum diebus .314. & hæc erit æqua periodus temporis <lb/>inter vtranque coniunctionem horum duorum Altiorum planetarum. </s> - <s xml:id="s2860" xml:space="preserve">Vt autem pla <lb/>nius oÅ¿tendatur hanc operationem rectam eſſe (nam demonÅ¿trationem <reg norm="Å¿peculatiuam" type="context">Å¿peculatiuã</reg> <lb/>huius operationis in .113. Theoremate noÅ¿træ ArithmeticÄ™ <reg norm="cuique" type="simple">cuiq;</reg> videre licet) fieri <lb/>poteÅ¿t his alijs calculis.</s> - </p> - <p> - <s xml:id="s2861" xml:space="preserve">Si Saturnus diebus .10740. tranÅ¿it per gra .360. in Å¿pacio dierum .2921. tranÅ¿ibit <lb/>per gradus .97. min .54. quibus iunctis cum gra .145. min .4. Å¿upra notatis, efticientur <lb/>gra .242. min .58. </s> - <s xml:id="s2862" xml:space="preserve">Deinde, Å¿i Iupiter Å¿patio dierum .4328. tranÅ¿it per gra .360. igitur <lb/>Å¿patio .2921. per eandem regulam inueniemus eum tranÅ¿ire gradus .242. mi .58. qui <lb/>numerus par eÅ¿t illi Saturni. </s> - <s xml:id="s2863" xml:space="preserve">Cum ergo Iupiter confecerit vnum ambitum poÅ¿t con - <pb o="232" rhead="IO. BAPT. BENED." n="244" file="0244" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0244"/> - iunctionem cum Saturno, vt rurÅ¿us aſſequatur Saturnum, tranÅ¿eundum ei erit <reg norm="per" type="simple">ê‘</reg> gra <num value="242">.<lb/>242.</num> min .58. iter confectum à Saturno toto tempore annorum .19. & dierum .314. <lb/>ad rationem graduum .360. diebus .10740. (poſſumus <reg norm="etiam" type="context">etiã</reg> dicere gra .243. quia præ <lb/>termiÅ¿imus quaÅ¿dam exiguas particulas periodorum perfectorum cuiuÅ¿que planetÄ™ <lb/>in Å¿uperioribus Å¿upputationibus.) </s> - <s xml:id="s2864" xml:space="preserve">Illos verò gradus .43. Iupiter conÅ¿iciet diebus <num value="2921">.<lb/>2921.</num> ad rationem graduum .360. diebus .4328. </s> - <s xml:id="s2865" xml:space="preserve"><reg norm="Atque" type="simple">Atq;</reg> ita vt diximus, ab vna coniun-<lb/>ctione ad aliam intererunt anni .19. </s> - <s xml:id="s2866" xml:space="preserve">Aegiptij cum diebus .314. vel circa.</s> - </p> - <p> - <s xml:id="s2867" xml:space="preserve">Nunc autem vt videatur an tabulæ AlfonÅ¿i conueniant cum hoc <reg norm="numero" type="wordlist">nr̃o</reg> calculo, <reg norm="con- Å¿iderabimus" type="context">cõ-<lb/>Å¿iderabimus</reg>, <reg norm="quod" type="simple">ê™</reg> Era (vt vocant) dicti temporis annorum .19. cum diebus .314. eÅ¿t dua-<lb/>rum tertiarum Å¿exagenarum, Å¿ecundæ nullius, & 53. <reg norm="primarum" type="context">primarÅ©</reg> Å¿iue dierum. </s> - <s xml:id="s2868" xml:space="preserve">Et per <reg norm="hanc" type="context">hãc</reg> <lb/>Eram colligendo motum mediocrem, tum Saturni, tum Iouis, omiſſis radicibus, & in <lb/>cipiendo ab Ariete, comperiemus <reg norm="quod" type="simple">ê™</reg> <reg norm="vtriuÅ¿que" type="simple">vtriuÅ¿q;</reg> planetæ lineæ eiuÅ¿modi motus tranÅ¿i <lb/>bunt per min .56. tertij gradus Sagittarij, ideÅ¿t coniunctæ erunt.</s> - </p> - <p> - <s xml:id="s2869" xml:space="preserve">In fine poÅ¿tea Å¿ecundæ periodi, cuius era erit .4. tertiarum, Å¿ecundæ .1. et .47. pri-<lb/>marum Å¿exagenarum, locus mediocris <reg norm="vtriuÅ¿que" type="simple">vtriuÅ¿q;</reg> erit in min .56. gra. Å¿exti Leonis. </s> - <s xml:id="s2870" xml:space="preserve">In <lb/>fine verò tertiæ periodi, cuius era erit .6. tertiarum .2. Å¿ecundarum, et .41. primæ, lo-<lb/>cus eorum mediocris inuenietur in .56. minuto gradus .9. Arietis. </s> - <s xml:id="s2871" xml:space="preserve"><reg norm="Atque" type="simple">Atq;</reg> ita deinceps <lb/>in fine <reg norm="cuiusque" type="simple">cuiusqÌ;</reg> periodi, locus <reg norm="eorum" type="context">eorÅ©</reg> mediocris coniunctim Å¿emper diÅ¿tabit à loco me <lb/>diocri præcedentis coniunctionis gradibus .117. ideÅ¿t in trigono antecedenti, minus <lb/>gra .3. </s> - <s xml:id="s2872" xml:space="preserve">Vnde apparet has coniunctiones procedere in contrariam partem reÅ¿pectu or <lb/>dinis <reg norm="Å¿ignorum" type="context">Å¿ignorÅ©</reg> Zodiaci, Å¿ed reÅ¿pectu ordinis graduum <reg norm="Å¿ignorum" type="context">Å¿ignorÅ©</reg>, Å¿emper <reg norm="progrediuntur" type="simple">progrediunt̃</reg> or <lb/>dine per ternos gradus nunquam retrogradientes. </s> - <s xml:id="s2873" xml:space="preserve">Hinc Å¿e quitur, vt non duodecies <lb/>in omni triplicitate coniungantur hi duo planetÄ™, vt antiqui putauerunt, Å¿ed decies <lb/>tantum. </s> - <s xml:id="s2874" xml:space="preserve">& ad Å¿ummum ter in Å¿ingulo Å¿igno, Å¿patio annorum .198. & dierum .220. aut <lb/>circiter, non autem .240. nec .242. </s> - <s xml:id="s2875" xml:space="preserve">Atque decem vices comprehendunt gra .27. & <lb/>vltima vice inueniuntur in Å¿igno Å¿equenti alterius triplicitatis. </s> - <s xml:id="s2876" xml:space="preserve">Exempli gratia, po-<lb/>namus <reg norm="quod" type="simple">ê™</reg> prima vice <reg norm="coniungantur" type="context simple">cõiungant̃</reg> in gra .2. Arietis, Å¿ecunda coniunctio erit in .5. </s> - <s xml:id="s2877" xml:space="preserve">Sagit-<lb/>tarij, tertia. in .8. Leonis. quarta in .11. Arietis, quinta in .14. Sagit .6. in .17. Leonis. <lb/>Å¿eptima in .20. Arietis, octaua in .23. Sagittarij, nona in .26. Leonis, decima in .29. <lb/>Arietis, et vndecima erit in gra .2. Capricorni Å¿igni <reg norm="Å¿equentis" type="context">Å¿equẽtis</reg> triplicitatis. </s> - <s xml:id="s2878" xml:space="preserve">Decem igiÂ<lb/>tur interualla Å¿ingula annorum .19. & dierum .314. faciunt annos .198. & dies .220. <lb/>Immo pertabulas AlfonÅ¿i, eiuÅ¿modi periodus non modo non reperitur <reg norm="annorum" type="context">annorÅ©</reg> .242 <lb/>nec .240. vt antiqui credidere, Å¿ed tribus diebus minor annis .198. & diebus .220. id-<lb/>eÅ¿t per dictas tabulas inuenitur eſſe annorum .198. & dierum .217. tantum, qui nume <lb/>rus multiplicatus per .4. triplicitates, efficiet periodum maiorem, quæ erit annorum <lb/>794. & dierum .138. quo tempore dicti planetæ redeunt ad eundem locum vbi pri-<lb/>mum Å¿e coniunxere.</s> - </p> - <p> - <s xml:id="s2879" xml:space="preserve">Vt exempli gratia, locus mediocris Saturni & Iouis in fine annorum .198. dierum <lb/>217. reperitur in gradu .30. Sagittarij. </s> - <s xml:id="s2880" xml:space="preserve">Si quæſiuerimus hunc locum per <reg norm="aggregatum" type="context">aggregatÅ©</reg> <lb/>annorum .794. & dierum .138. cum annis .198. & diebus .217. quorum Å¿umma eÅ¿t <num value="992">.<lb/>992.</num> & dies .355. inuenietur locus mediocris ipÅ¿orum <reg norm="planetarum" type="context">planetarÅ©</reg> in dicto vltimo gra <lb/>du Sagittarij. </s> - <s xml:id="s2881" xml:space="preserve">Sed Å¿i quęſiuerimus eorum locum mediocrem per aggregatum anno <lb/>rum .198. & dierum .217. cum annis .960. quod erit Å¿umma annorum .1158. & <reg norm="dierum" type="context">dierÅ©</reg> <lb/>217. reperiemus Iouem in gradu .18. </s> - <s xml:id="s2882" xml:space="preserve">Sagittarij & Saturnum in .16. Leonis diÅ¿tanti-<lb/>bus inter Å¿e duabus eorum lineis motuum mediocrium gra. circiter .122. </s> - <s xml:id="s2883" xml:space="preserve"><reg norm="Atque" type="simple">Atq;</reg> Iupi-<lb/>ter præcedet, & oportebit <reg norm="quod" type="simple">ê™</reg> coniunctio eorum mediocris fuerit multis annis ante <lb/>omittendo (vt dixi) radices, quia Å¿atis eÅ¿t inuenire interuallum inter lineas eorum me <lb/>diorum motuum.</s> - </p> - <pb o="233" rhead="EPISTOL AE." n="245" file="0245" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0245"/> - <p> - <s xml:id="s2884" xml:space="preserve">Debebat igitur author animaduerÅ¿ionum non quaÅ¿i cæcus eæcos Å¿equi, Å¿ed prius <lb/>laborare, vt certior fieret, an interuallum annorum .960. </s> - <s xml:id="s2885" xml:space="preserve">Verum eſſet.</s> - </p> - <p> - <s xml:id="s2886" xml:space="preserve">Sed peius eÅ¿t, <reg norm="quod" type="simple">ê™</reg> idem author paulo inferius citat coniunctiones horum duorum <lb/>planetarum anni .1493. et .1512. quas neÅ¿cio vnde Å¿umpÅ¿erit.</s> - </p> - <p> - <s xml:id="s2887" xml:space="preserve">Nam, etÅ¿i inter hos annos eÅ¿t interuallum annorum .19. tamen tantum abeÅ¿t, vt <lb/>coiuerint dictis annis, vt Saturnus anno .1493. ante finem AuguÅ¿ti fuerit in .28. gra-<lb/>du Aquarij, Iupiter verò in .28. Leonis ex diametro oppoÅ¿iti. </s> - <s xml:id="s2888" xml:space="preserve">Et anno .1512. per to <lb/>tum menÅ¿em Iunium & AuguÅ¿tum, Saturnus fuerit in Libra, Iupiter verò in Ariete, <lb/>itaque inter Å¿e Å¿imiliter oppoÅ¿iti, & Å¿i perfecta oppoÅ¿itio non fuit poÅ¿tea niÅ¿i ad <reg norm="finem" type="context">finẽ</reg> <lb/>Iunij ann .1513. & locus Monteregij ab eo citatus, vbi ait eum ponere coniunctio-<lb/>nem anni .1484. in gra .23. min .4. </s> - <s xml:id="s2889" xml:space="preserve">Scorpij, eÅ¿t mendoÅ¿us. </s> - <s xml:id="s2890" xml:space="preserve">Nam ipÅ¿e Montere gius po <lb/>nit dictam coniunctionem in mi .42. gra .24. non autem in min .4. ipÅ¿ius gradus. </s> - <s xml:id="s2891" xml:space="preserve">Sed <lb/>hic error nullius eÅ¿t momenti, fortaſſe qui impræſſorum incuria irrepÅ¿it.</s> - </p> - <p> - <s xml:id="s2892" xml:space="preserve">Pergatur poÅ¿tea obÅ¿ecro ad paginam .3. ipÅ¿arum AnimaduerÅ¿ionum, vbi hic co-<lb/>natur oÅ¿tendere calculatores non obÅ¿eruaſſe verum modum, Å¿ic dicens.</s> - </p> - <quote> - <s xml:id="s2893" xml:space="preserve">Anno .1484. Nouembris .25. Saturno locum conÅ¿tituit Monteregius in grad .23. <lb/>min .4. Scorpij. </s> - <s xml:id="s2894" xml:space="preserve">Anno poÅ¿tmodum Å¿ubÅ¿equenti qui eÅ¿t .1485. eundem in min .7. Sa-<lb/>gitarij collocat .21. Februarij die. </s> - <s xml:id="s2895" xml:space="preserve"><reg norm="Interque" type="simple">InterqÌ;</reg> tempora duo interÅ¿unt menÅ¿es dies .26. <lb/></s> - <s xml:id="s2896" xml:space="preserve">At cum ex motus Å¿ui natura Saturnus hoctemporis Å¿pacio gradus .4. non debeat <reg norm="tran- Å¿cendere" type="context">trã-<lb/>Å¿cendere</reg>, Å¿it tamen inter <reg norm="vtrunque" type="simple">vtrunq;</reg> tempus differentia graduum .7. minutorum .3. quæ <lb/>ratione Å¿ui motus requirunt menÅ¿es .6. vt eos perficiat, conÅ¿tat pluÅ¿quam tribus men <lb/>Å¿ibus fallere nos Saturnum.</s> - </quote> - <p> - <s xml:id="s2897" xml:space="preserve">Hic videre licet quam veram viam hic Å¿ecutus Å¿it ad aperiendos errores Epheme <lb/>ridum, & miÅ¿eri Monteregij, qui Saturnum claudum facit tantum itineris conficere <lb/>tribus <reg norm="menÅ¿ibus" type="context">mẽſibus</reg>, <reg norm="quantum" type="context context">quãtÅ©</reg> vix confeciſſet <reg norm="menÅ¿ib" type="context">mẽſib</reg>. Å¿ex. </s> - <s xml:id="s2898" xml:space="preserve">Sed fortaſſe <reg norm="ratiocinatur" type="simple">ratiocinat̃</reg> hoc modo.</s> - </p> - <p> - <s xml:id="s2899" xml:space="preserve">Si motus naturalis Saturni facit vt circumeat totum cÄ™lum annis .30. igitur menÅ¿i-<lb/>bus .30. conficiet duodecimam partem circuitus, cum menÅ¿es .30. Å¿int duodecima <lb/>pars annorum .30. & quia duodecima pars circuitus cÄ™li intelligitur conÅ¿tare ex .30. <lb/>gradibus, igitur quilibet menÅ¿is poÅ¿tulabit gradum vnum. </s> - <s xml:id="s2900" xml:space="preserve">Ideo illi .6. aut .7. gradus <lb/>poÅ¿tulant tempus, amplius menÅ¿ium Å¿ex.</s> - </p> - <p> - <s xml:id="s2901" xml:space="preserve">Atque eiuÅ¿modi mira ratiocinatio poteÅ¿t in .2. exemplo eius, inÅ¿cripto.</s> - </p> - <quote> - <s xml:id="s2902" xml:space="preserve">Deeodem ex eodem</s> - </quote> - <p> - <s xml:id="s2903" xml:space="preserve">Vbi miratur, <reg norm="quod" type="simple">ê™</reg> Monteregius faciat Saturnum ambulare gra .9. min .10. in menÅ¿i-<lb/>bus .7. & diebus .6. </s> - <s xml:id="s2904" xml:space="preserve">Ad quod iter Saturnc Å¿eni opus eſſet Å¿altem menÅ¿ibus .9. eius <lb/>iudicio.</s> - </p> - <p> - <s xml:id="s2905" xml:space="preserve">Sed Å¿i hoc miratur, quid dicturus fuiſſet, Å¿i animaduertiſſet, quod idem calculator <lb/>Monteregius facit Saturnum ambulare immo volare gra .9. min .48. non in 7. Å¿ed in <lb/>2. menÅ¿ibus cum dimidio, videlicet à .10. die Iunij vÅ¿que ad .26. AuguÅ¿ti <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> an-<lb/>ni .1504.</s> - </p> - <p> - <s xml:id="s2906" xml:space="preserve">Quid Å¿i etiam animaduertiſſet <reg norm="quod" type="simple">ê™</reg> à .10. die Iunij Å¿upradicti vÅ¿que ad .16. Ianuarij <lb/>anni Å¿equentis, faciunt Saturnum, Å¿urÅ¿um, deorÅ¿um curÅ¿itare amplius gra .17. mi .54. <lb/>Imino Å¿i animaduertiſſet, quod anno .1524. </s> - <s xml:id="s2907" xml:space="preserve">Stoflerinus ab initio anni, <reg norm="vÅ¿quem" type="context">vÅ¿quẽ</reg> ad <lb/>medium Maium, ideÅ¿t menÅ¿ib .4. cum dimidio, facit Saturnum ambulare gra .15. </s> - <s xml:id="s2908" xml:space="preserve">Pro <lb/>fectò ob has velocitates, eius iudicio, tam abÅ¿urdas, obÅ¿tupuiſſet.</s> - </p> - <p> - <s xml:id="s2909" xml:space="preserve">Vbi autem in tergo eiuÅ¿dem paginÄ™ ait, quod gradibus .13. min .42. reÅ¿pondent <lb/>menÅ¿es .19. errauit in calculo, nam ex eiuÅ¿modi tempore Å¿ecundum eius regulam ef- - <pb o="234" rhead="IO. BAPT. BENED." n="246" file="0246" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0246"/> - ficerentnr Å¿inguli ambitus Saturni ad rationem annorum amplius .40.</s> - </p> - <p> - <s xml:id="s2910" xml:space="preserve">Videamus nunc vbi agit de Ioue, & reperiemus <reg norm="quod" type="simple">ê™</reg> in primo exemplo circa <reg norm="annum" type="context">annÅ©</reg>. <lb/>1484. reprÄ™hendit Monteregium, quia facit Iouem ambulare gradus .14. cum min <num value="6">.<lb/>6.</num> in menÅ¿ibus .2. diebus .4. ad quod iter, vt ipÅ¿e ait, opus eſſet Å¿altem <reg norm="menÅ¿ibus" type="context">mẽſibus</reg> .11. <reg norm="atque" type="simple">atq;</reg> <lb/>ita Å¿ecundum ipÅ¿um, Ioui opus eſſet anno vno pro Å¿ingulo medio Å¿igno. </s> - <s xml:id="s2911" xml:space="preserve">Vbi bonus <lb/>hic vir pariter cæcutit.</s> - </p> - <p> - <s xml:id="s2912" xml:space="preserve">Idem in Å¿ecundo exemplo Å¿umpto à Stoflerino ait, <reg norm="quod" type="simple">ê™</reg> Ioui ad curÅ¿um vnius gra-<lb/>dus, & min .5. opus eÅ¿t diebus .30. non autem menÅ¿ibus .7. & diebus .28. vbi oÅ¿tendit, <lb/>Å¿e paruum diÅ¿crimen facere inter Iouem, & Saturnum.</s> - </p> - <p> - <s xml:id="s2913" xml:space="preserve">Miratur poÅ¿tea <reg norm="quod" type="simple">ê™</reg> Stoflerinus faciat laborare generoÅ¿um Iouem ferè menÅ¿ibus <lb/>Å¿ex in vno gradu. </s> - <s xml:id="s2914" xml:space="preserve">Sed multo magis, vt puto, miratus eſſet, Å¿i vidiſſet, quod idem <lb/>Stoflerus in eodem anno facit, quod Iupiter die .4. </s> - <s xml:id="s2915" xml:space="preserve">Ianuarij Å¿it in eodem puncto, in <lb/>quo poÅ¿tea reperitur die vltima AuguÅ¿ti. </s> - <s xml:id="s2916" xml:space="preserve">At fortaſſe dici poſſet, quod Iupiter pro-<lb/>pter prudentiam, & bonitatem Å¿uam factus eÅ¿t R ex omnium Deorum, vt ait Home-<lb/>rus, & ideo expulit è Å¿ede Saturnum, & aÅ¿cendit in altiori cÅ“lo. </s> - <s xml:id="s2917" xml:space="preserve">Vnde euenit vt fa-<lb/>ctus fuerit lentior in curÅ¿u, Saturnus autem velocior. </s> - <s xml:id="s2918" xml:space="preserve">Aut iam tot annos eſſe na-<lb/>tum Iouem, vt iure credi poſſit eum iam factum eſſe Å¿enem, & pariter tardio-<lb/>rem in Å¿e <reg norm="mouendo" type="context">mouẽdo</reg>. </s> - <s xml:id="s2919" xml:space="preserve">aut <reg norm="tunc" type="context">tÅ©c</reg> temporis illum detentum fuiſſe in Å¿ibi dilecta Arcadia <reg norm="cum" type="context">cÅ©</reg> <lb/>CaliÅ¿to. </s> - <s xml:id="s2920" xml:space="preserve">Aut fortaſſe erat in alta Å¿pecula intentus audiendo ingenti certami-<lb/>ni Timoclis & Damidis, vnde pendebat exitium aut gloria familiæ Å¿uæ, nam alio-<lb/>quin Stoflerus non depræhendiſſet eum tam otioÅ¿um & morantem. </s> - <s xml:id="s2921" xml:space="preserve">Sediam relin-<lb/>quamus Saturnum & Iouem, & ad Martem veniamus.</s> - </p> - <p> - <s xml:id="s2922" xml:space="preserve">Ferox & inquietus Mars, qui Å¿emper bella & ignes Å¿pirare Å¿olet, etiam, & ipÅ¿e ab <lb/>AÅ¿trologis factus eÅ¿t piger, & languidus, vt velint eum nonnunquam commorari <lb/>in vno Å¿igno Å¿ex aut Å¿eptem menÅ¿ibus; </s> - <s xml:id="s2923" xml:space="preserve">quod nullo pacto placet authori Animaduer-<lb/>Å¿ionum, cum pag .4. ita Å¿cribat.</s> - </p> - <quote> - <s xml:id="s2924" xml:space="preserve">Quod citra notam, ab omnibus creditur poſſe obÅ¿eruari, quamuis à nobis non ac <lb/>cipiatur.</s> - </quote> - <p> - <s xml:id="s2925" xml:space="preserve">Itaque ei videtur impoſſibile. </s> - <s xml:id="s2926" xml:space="preserve">Quia Mars peragit Å¿uum eircuitum minus .2. an-<lb/>nis. </s> - <s xml:id="s2927" xml:space="preserve">Sed audacior fuiſſe videtur, qui voluerit arguere tot egregios viros antiquos, <lb/>& recentiores, qui vti diligentes rerum cÅ“leÅ¿tium obÅ¿eruatores, ipÅ¿is oculis certi fa <lb/>cti Å¿unt tam de his effectibus Martis, quam aliorum, vnde coacti Å¿unt fingere tantam <lb/>magnitudinem eius epicycli, cum ipÅ¿e nunquam obÅ¿eruauerit motum, nec huius nec <lb/>alterius planetæ, Å¿ed tantum viderit eius moram in Ephemeride Å¿criptam. </s> - <s xml:id="s2928" xml:space="preserve">Si enim <lb/>Å¿altem diceret, Å¿e aliquo tempore obÅ¿eruaſſe iter Martis, & comperuiſſe aliorum opi <lb/>nionem falÅ¿am, attuliſſet aliquem colorem Å¿ententiæ Å¿uæ. </s> - <s xml:id="s2929" xml:space="preserve">Sed Å¿i obÅ¿eruaſſet, non <lb/>Å¿cripÅ¿iſſet poÅ¿tea contra, vt puto. </s> - <s xml:id="s2930" xml:space="preserve">Res enim ita Å¿e habet, quod Mars in omni circui <lb/>tu Å¿ui epicycli tranÅ¿iens per inferiorem partem ipÅ¿ius epicycli, Å¿emper commoratur <lb/>multis menÅ¿ibus in vno duodecatemorio Zodiaci, Å¿cilicet .6. et .7. menÅ¿ibus, atque <lb/>etiam amplius, quod quidem ego ſæpe obÅ¿eruaui, præſertim anno .1565. et .1566. <lb/>hoc ordine. </s> - <s xml:id="s2931" xml:space="preserve">Primum inÅ¿piciens Ephemeridas Å¿tadij, reperi <reg norm="quod" type="simple">ê™</reg> Mars Å¿ecundum eum <lb/>egrediebatur retrogradationem circa diem .12. Ianuarij anni 1566. in .16. grad. <lb/>Geminorum. </s> - <s xml:id="s2932" xml:space="preserve">Et Å¿imiliter quod anno .1565. die vltima AuguÅ¿ti Mars futurus erat in <lb/>eodem Å¿upradicto loco, priuÅ¿quam retrogradi inciperet. </s> - <s xml:id="s2933" xml:space="preserve">PoÅ¿tea inueni, quod poÅ¿t <lb/>retrogradationem die .11. Aprilis .1566. </s> - <s xml:id="s2934" xml:space="preserve">Idem Mars futurus erat in gra .16. Cancri, <lb/><reg norm="itaque" type="simple">itaq;</reg> in his .30. gradibus à .16. Geminorum ad .16. Cancri conÅ¿umebatur Å¿patium <lb/>menÅ¿ium .7. & dierum .11.</s> - </p> - <pb o="235" rhead="EPISTOLAE." n="247" file="0247" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0247"/> - <p> - <s xml:id="s2935" xml:space="preserve">Quo Å¿upputato, Å¿umpÅ¿i inÅ¿trumenta, & ad experimentum me paraui, & vltima <lb/>nocte menÅ¿is AuguÅ¿ti anni .1565. reperi Martem eſſe in dicto gradu geminorum vt <lb/>Å¿cribebat Stadius. </s> - <s xml:id="s2936" xml:space="preserve">Deinde Å¿ingulis ebdomadibus obÅ¿eruans retrogradationem; </s> - <s xml:id="s2937" xml:space="preserve">vidi <lb/>circa finem Octobris quod retrogradi incipiebat, & ea retrogradatio perÅ¿euerauit <lb/>vÅ¿que ad medium menÅ¿em Ianuarium, aut circiter, anni .1566. obÅ¿eruaui poÅ¿tea <lb/>etiam Å¿itum eiuÅ¿dem planetæ die .11. </s> - <s xml:id="s2938" xml:space="preserve">Aprilis Å¿equentis <reg norm="eumque" type="simple">eumqÌ;</reg><unsure/> inueni in gradu .16. <lb/>Cancri, vti eum poÅ¿uerat Stadius. </s> - <s xml:id="s2939" xml:space="preserve">Atque ita experimentum meum conuenit cum <lb/>calculo Stadij, <reg norm="comperique" type="simple">comperiqÌ;</reg> eum non erraſſe: </s> - <s xml:id="s2940" xml:space="preserve">Et Å¿ic quiÅ¿que binis <reg norm="quibusque" type="simple">quibusqÌ;</reg> annis pote-<lb/>rit certior fieri de veritate. </s> - <s xml:id="s2941" xml:space="preserve">Si autem delectationis cauÅ¿a id experiri volueris, expe-<lb/>ctato primam retrogradationem Martis, cuius initium Å¿ecundum Stadium futurum <lb/>eÅ¿t circa diem .20. </s> - <s xml:id="s2942" xml:space="preserve">Nouembris anni .1582. & finis circa diem .10. Februar .1583. <lb/>circa grad .9. Cancri, & animaduerte quando Mars erit circa dictum gra .9. Can-<lb/>cri prius quam retrogradi incipiat, quod erit circa diem .19. Septem .1582. </s> - <s xml:id="s2943" xml:space="preserve">Dein-<lb/>de aÅ¿pice quum erit in grad .9. Leonis, quod erit circa diem .7. Mai .1583. & vide-<lb/>bis <reg norm="quod" type="simple">ê™</reg> ipÅ¿e Mars in his gra .30. <reg norm="morabitur" type="simple">morabit̃</reg> <reg norm="per" type="simple">ê‘</reg> menÅ¿es .7. & dies .18. <reg norm="atque" type="simple">atq;</reg> vt eius rei pericu-<lb/>lum facias, obÅ¿erua noctem præcedentem diei .19. </s> - <s xml:id="s2944" xml:space="preserve">Septem .1582. locum <reg norm="longitudinis" type="context">lõgitudinis</reg> <lb/>eius Å¿tellæ, & idem poÅ¿tea obÅ¿erua nocte præcedente diei .7. Mai, aut nocte Å¿equen-<lb/>ti .1583. & inter duos hoÅ¿ce terminos obÅ¿erua aliqua alia nocte Å¿tatum eius. </s> - <s xml:id="s2945" xml:space="preserve">Mani-<lb/><reg norm="feÅ¿toque" type="simple">feÅ¿toqÌ;</reg> videbis Martem conÅ¿umere totum dictum tempus in hoc duodecatemorio. <lb/></s> - <s xml:id="s2946" xml:space="preserve">Et quicunque aliquid intelligit in hac facultate quamuis non viderit PtolomÄ™i <lb/>AlmageÅ¿tum, minori labore poſſet per calculos Å¿cientificos colligere verita-<lb/>tem, Å¿uppoÅ¿itis tamen terminis Å¿criptis in theoricis planetarum. </s> - <s xml:id="s2947" xml:space="preserve">Qui enim vidit <lb/>AlmageÅ¿tum vel reuolutiones orbium cÅ“leÅ¿tium Nicolai Copernici, non poteÅ¿t de <lb/>hoc vllo pacto dubitare. </s> - <s xml:id="s2948" xml:space="preserve">Sed qui nondum tantopere progreſſus eÅ¿t, <reg norm="Å¿altem" type="context">Å¿altẽ</reg> capiat <reg norm="huius" type="simple">huiê°</reg> <lb/>rei notitiam vniuerÅ¿alem, hoc modo. </s> - <s xml:id="s2949" xml:space="preserve">Supponat primum eccentricitatem deferen-<lb/>tis epicycli Martis, eſſe .6. partium <reg norm="talium" type="context">taliÅ©</reg>, quales Å¿unt Å¿exageÅ¿imæ Å¿emidiametri ipÅ¿ius <lb/>deferentis, & Å¿emidiametrum epicycli eſſe, partium <reg norm="Å¿upradictarum" type="context">Å¿upradictarÅ©</reg> .39. cum dimidia, <lb/>& quod argumenta vera, in temporibus primarum Å¿tationum ( cum epicyclus eÅ¿t in <lb/>auge, aut in eius oppoÅ¿ito, aut in <reg norm="lungitudinib" type="context">lÅ©gitudinib</reg>. me dio cribus ) <reg norm="iam" type="context">iã</reg> ab antiquis rectè Å¿uppu <lb/>tata Å¿int, Å¿icuti Å¿unt. </s> - <s xml:id="s2950" xml:space="preserve">Et præſupponat motum diurnum centri epicycli. min .31. cum di <lb/>midio, quamuis reuera Å¿it min .31. & Å¿ecundorum .27. aut circiter, nunc <reg norm="quidem" type="context">quidẽ</reg> præter <lb/>mittens, quod vnus habeat reÅ¿pectum ad augem mediam epicycli, & alter ad cen-<lb/>trum æquantis. </s> - <s xml:id="s2951" xml:space="preserve">Atque his præſuppoÅ¿itis fingat ( exempli gratia ) quod centrum <lb/>epicycli Å¿it in quauis longitudinum mediarum, & Mars in prima maxima æqua-<lb/>tione argumenti, Å¿cilicet in prima linea, quæ attingens epicyclum, à centro mundi <lb/>pergat ad circunferentiam Zodiaci, quæ erit illa linea <reg norm="contingentiæ" type="context">cõtingentiæ</reg> a qua <reg norm="proficiÅ¿cens" type="context">proficiÅ¿cẽs</reg> <lb/>Mars perget ad lineam primæ Å¿tationis, vt poÅ¿tea retrogradiatur, veluti Å¿i in infrapo <lb/>Å¿ita figura maiori, <reg norm="centrum" type="context context">cẽtrÅ©</reg> <reg norm="mundi" type="context">mÅ©di</reg> eſſet <var>.o.</var> & vnus arcus <reg norm="eccentrici" type="context">eccẽtrici</reg> eſſet <var>.a.b.c.d.</var> & vna ex li <lb/>neis medio cribus longitudinum eſſet <var>.o.c.f.</var> & centrum epicycli <var>.c.</var> qui notabitur per <lb/><var>a.f.e.g.</var> & lineæ contingentes epicyclum in punctis <var>.i.</var> et <var>.t.</var> Å¿int notatæ <var>.o.i.</var> et <var>.o.t.</var> & li-<lb/>nea primæ Å¿tationis <var>.o.n.b.</var> & linea Å¿ecundæ <var>.o.u.d.</var> Å¿i igitur Mars eſſet in puncto <var>.i.</var> an-<lb/>gulus <var>.i.o.e.</var> maximæ æquationis argumenti eſſet gra .40. minut .55. <reg norm="quanuis" type="context">quãuis</reg> talis maxi <lb/>ma æquatio argumenti in longitudinibus mediocribus AlfonÅ¿i ponatur eſſe gra .41. <lb/>minut .10. quod euenit quia calculatores ipÅ¿arum tabularum interuallum <var>.o.c.</var> quod <lb/>in eo Å¿itu epicycli interponitur inter centrum mundi, & centrum dicti epicycli, ac-<lb/>ceperunt partium Å¿exaginta præcisè, nihili facientes minuta illa .18. aut circiter, quÄ™ <lb/>verè Å¿unt præter dictas partes .60. quandoquid em euenit vt dictum interuallum in - <pb o="236" rhead="IO. BAPT. BENED." n="248" file="0248" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0248"/> - tali Å¿itu epicycli Å¿it baÅ¿is vnius trianguli orthogonij, cuius vnum ex illis duobus late-<lb/>ribus eÅ¿t Å¿emidiameter eccentrici partium .60. prÄ™cisè, aliud eÅ¿t interuallum eccen-<lb/>tricitatis partium .6. eiuÅ¿modi. </s> - <s xml:id="s2952" xml:space="preserve">Angulus ergo <var>.i.o.c.</var> vt dixi, erit partium .40. minu <num value="55">.<lb/>55.</num> qui angulus continuò variatur Å¿ecundum Å¿itum epicycli. </s> - <s xml:id="s2953" xml:space="preserve">& cum centrum <lb/>eius eÅ¿t in auge eccentrici. eÅ¿t minimus <reg norm="quam" type="context">quã</reg> eſſe poſſit. </s> - <s xml:id="s2954" xml:space="preserve"><reg norm="eÅ¿tque" type="simple">eÅ¿tqÌ;</reg> tantum grad .36. min <num value="46">.<lb/>46.</num> & in oppoÅ¿ito ipÅ¿ius augis eÅ¿t grad .47. min .1. maximus quam alibi vnquam Å¿it, <lb/>& Å¿ic continuò variatur, Å¿ecundum Å¿itum, quem habet epicyclus in eccentrico. </s> - <s xml:id="s2955" xml:space="preserve">Qui <lb/>quidem angulus inuenitur per doctrinam .27. et .28. libri primi Monteregij de trian <lb/>gulis. </s> - <s xml:id="s2956" xml:space="preserve">Nam triangulus <var>.c.i.o.</var> eÅ¿t Å¿emper rectangulus in puncto <var>.i.</var> & latus <var>.c.i.</var> reÅ¿pectu <lb/>Å¿emidiametri eÅ¿t datum. </s> - <s xml:id="s2957" xml:space="preserve">Quod <var>.c.i.</var> erit veluti partium .39. cum dimidia, et dictum <lb/>interuallum <var>.o.c.</var> veluti pat<unsure/> cium .60. min .18. & quia datur nobis etiam eccentricitas <lb/>veluti partium .60. talium, & cum <var>.c.o.</var> Å¿it linea veri motus epicycli, & latus Å¿imiliter <lb/>vnius trian guli, cuius duo latera Å¿unt Å¿upradicta, Å¿cilicet Å¿emidiameter eccentrici, & <lb/>eccentricitas, inter Å¿e compræhendentes angulum datum. </s> - <s xml:id="s2958" xml:space="preserve">Nam Å¿emper præſuppo <lb/>nitur datus locus centri ipÅ¿ius epicycli, cum ipÅ¿e eÅ¿t extra augem aut oppoÅ¿itum eius <lb/>quia in auge linea <var>.o.c.</var> conÅ¿tat ex Å¿emidiametro eccentrici & interualli eccentricita-<lb/>tis. </s> - <s xml:id="s2959" xml:space="preserve">& in eius oppoÅ¿ito, ipÅ¿a linea <var>.o.c.</var> eÅ¿t minor dicto Å¿emidiametro eccentrici per in <lb/>teruallum dictæ eccentricitatis. </s> - <s xml:id="s2960" xml:space="preserve">Vnde etiam poſſumus extra augem, vel oppoÅ¿itum <lb/>eius cognoÅ¿cere <var>.o.c.</var> tanquam latus dicti trianguli duorum laterum <reg norm="cum" type="context">cÅ©</reg> angulo cogni <lb/>torum. </s> - <s xml:id="s2961" xml:space="preserve"><reg norm="Idque" type="simple">IdqÌ;</reg> per .49. propoÅ¿itionem libri primi <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> Monteregij cum Å¿cilicet dictus <lb/>angulus <reg norm="non" type="context">nõ</reg> fuerit rectus. </s> - <s xml:id="s2962" xml:space="preserve">Nam Å¿i fuerit rectus videbitur per .27. et .28. Å¿upra citatas.</s> - </p> - <p> - <s xml:id="s2963" xml:space="preserve">Cum igitur hab eamus angulum <var>.c.o.i.</var> gra .40. mi .55. angulus <var>.o.c.i.</var> tanquam reli-<lb/>quus exrecto, erit grad .49. mi .5. cui reÅ¿pondet arcus <var>.i.g.</var> epicycli confectus à Marte <lb/>in diebus circiter .105. ad rationem min .28. aut circiter in Å¿ingulos dies, prætermiÅ¿-<lb/>Å¿is nunc quidem minutijs cum exigui momenti Å¿it error .15. aut .20. dierum ad verifi <lb/>cationem longæ morÄ™ Martis in vno duodecatemorio, atque per hoc tempus cen-<lb/>trum epicycli conficit gradus .55. min .7. aut circiter, ad rationem minutorum .31. <reg norm="cum" type="context">cÅ©</reg> <lb/>dimidio in Å¿ingulos dies. qui numerus graduum .55. min .7: differt à numero <reg norm="graduum" type="context">graduÅ©</reg>. <lb/>40. min .55. maximæ æquationis argumenti gradibus .14. mi .12. nec refert quod gra <num value="55">.<lb/>55.</num> min .7. habeant reÅ¿pectum ad centrum æquantis, magis quam ad centrum <reg norm="mundi" type="context">mÅ©di</reg>, <lb/>quia differentia non eÅ¿t tanta, vt poſſit inducere errorem menÅ¿ium. </s> - <s xml:id="s2964" xml:space="preserve">Hinc Å¿equitur <lb/>quod in fine dictorum dierum .105. </s> - <s xml:id="s2965" xml:space="preserve">Mars erit in linea <var>.o.c.</var> veri motus epicycli, Å¿ed <lb/>gradibus .14. min .12. vlterius quam in primo loco, in quo erat in Zodiaco, & erit in <lb/>medio Å¿uæ retrogradationis. </s> - <s xml:id="s2966" xml:space="preserve">Sed quoniam Mars manifeÅ¿tè retrogradi non incipit <lb/>in puncto <var>.i.</var> conting entiæ, imo ab illo puncto vÅ¿que ad terminum primæ Å¿tationis li <lb/>neæ <var>.o.n.</var> interponitur arcus <var>.i.n.</var> epicycli, qui eÅ¿t graduum .32. minu .14. </s> - <s xml:id="s2967" xml:space="preserve"><reg norm="Idque" type="simple">IdqÌ;</reg> cogno-<lb/>Å¿citur Å¿ubtrahendo arcum <var>.f.i.n.</var> graduum .163. mi .9. qui eÅ¿t inter augem, & primam <lb/>Å¿tationem, à gradibus .180. ( qui arcus <var>.f.i.n.</var> erit verum argumentum, quod Å¿i-<lb/>militer variatur Å¿ecundum Å¿itum epicycli, etÅ¿i eiuÅ¿modi varietas, nobis <reg norm="non" type="context">nõ</reg> eÅ¿t magni <lb/>momenti, vnde poſſumus præſupponere, quod <var>.c.</var> centrum epicycli non alteret <reg norm="in- teruallum" type="context">in-<lb/>teruallÅ©</reg> <var>.c.o.</var> à centro <reg norm="mundi" type="context">mÅ©di</reg>, <reg norm="cum" type="context">cÅ©</reg> non posÅ¿it intercedere, error <reg norm="menÅ¿ium" type="context context">mẽſiÅ©</reg> reliquum verò <var>.g.<lb/>n.</var> graduum .16. min .51. Å¿ubtrahendo ex arcu <var>.g.i.</var> graduum .49. minuti .5. vnde reli-<lb/>quus nobis erit arcus <var>.n.i.</var> graduum .32. min .14. in eiuÅ¿modi tamen Å¿itu mediocrium <lb/>longitudinum. </s> - <s xml:id="s2968" xml:space="preserve">Nunc hic arcus epicycli graduum .32. mi .14. fit à ſtella Martis die-<lb/>bus .69. ad rationem Å¿upradictam, omittendo quod ipÅ¿a Å¿tella habeat reÅ¿pectum ad <lb/>augem mediocrem epicycli, & quod dicta aux mediocris mutet diÅ¿tantiam à vera <lb/>propter motum epicycli, quod nunc quidem parui refert, in quibus diebus .69. cen- - <pb o="237" rhead="EPISTOLAE." n="249" file="0249" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0249"/> - trum epicycli conficit gra .36. min .13. ad rationem Å¿upradictam. </s> - <s xml:id="s2969" xml:space="preserve">ReÅ¿tat nunc no-<lb/>bis inuenire angulum <var>.b.o.c.</var> in centro <reg norm="mundi" type="context">mÅ©di</reg> inter duas lineas, <var>b.o.</var> et <var>.c.o.</var> <reg norm="quarum" type="context">quarÅ©</reg> prior <lb/>eÅ¿t primæ Å¿tationis, altera eÅ¿t veri motus epicycli, quod facilè intelligemus per di-<lb/>ctam .49. lib. 1. Monteregij, cum duo latera <var>.n.c.</var> et <var>.c.o.</var> & angulus <var>.n.c.o.</var> Å¿int nobis no <lb/>ta. </s> - <s xml:id="s2970" xml:space="preserve">Hoc autem fiet fingendo lineam <var>.n.h.</var> perpendiculerem ad <var>.o.c.</var> quæ tanquam Å¿i-<lb/>nus anguli <var>.n.c.h.</var> erit partium .28986. talium qualium <var>.n.c.</var> eſſet partium .100000. & <lb/><var>c.h.</var> tanquam Å¿inus anguli <var>.c.n.h.</var> reÅ¿tantis ex vno recto, erit partium .95706. dicendo <lb/>poÅ¿tea Å¿i <var>.n.e.</var> tanquam Å¿inus totalis partium .100000. dat nobis <var>.n.h.</var> partium .28986 <lb/>quid dabit nobis diameter <var>.n.c.</var> tanquam partium .39. mi .30. inueniemus <var>.n.h.</var> venire <lb/>nobis ex partibus 11. mi .27. & idem faciendo de <var>.c.h.</var> inueniemus quod veniet no-<lb/>bis partium .37. mi .48. quibus Å¿ubtractis extota <var>.c.o.</var> quÄ™ eÅ¿t <reg norm="partium" type="context">partiÅ©</reg> .60. mi .18. reliqua <lb/>erit nobis <var>.h.o.</var> partium .22. min .30. capiendo poÅ¿tea radicem <reg norm="quadratam" type="context">quadratã</reg> Å¿ummæ qua <lb/>drati <var>.n.h.</var> cum quadrato <var>.h.o.</var> veniet nobis <var>.n.o.</var> partium .25. min .12. talium qualis <var>.n.<lb/>h.</var> eÅ¿t partium .11. min .27. Å¿i igitur ad <var>.o.n.</var> tanquam partium .25. min .12. reÅ¿pondet <var>.<lb/>n.h.</var> partium .11. minuti .27. linea <var>.n.h.</var> ad <var>.o.n.</var> tanquam partium 100000. reÅ¿pon <lb/>debit part .45436. tanquam Å¿inus anguli <var>.n.o.h.</var> qui angulus erit gra .27. minut .1. Å¿ub <lb/>tracto poÅ¿tea hoc angulo ab angulo <var>.c.o.i.</var> graduum .40. minut .55. remanebit an <lb/>gulus <var>.n.o.i.</var> graduum .13. minut .54. inter lineam contingentiæ, & lineam primæ <lb/>Å¿tationis in eiuÅ¿modi Å¿itu. </s> - <s xml:id="s2971" xml:space="preserve">Et ideo Mars acceſſerit ad lineam <var>.o.c.</var> veri motus epi-<lb/>cycli. </s> - <s xml:id="s2972" xml:space="preserve">Sed quia linea <var>.o.i.</var> contingentiæ, propter motum centri epicycli, in dictis die-<lb/>bus .69. confecerit gradus .36. minut .13. ( præſuppoÅ¿ita Å¿emper eadem di-<lb/>Å¿tantia <var>.o.c.</var> quamuis nonnulla Å¿it differentia, quam nunc prætermittemus ) & Mars in <lb/>dicto tempore retrogreſſus fuerit per dictum angulum gra .13. mi .54. quibus dedu-<lb/>ctis, ex .36. & min .13. reÅ¿tabunt gra .22. min .19. itaque in diebus .69. </s> - <s xml:id="s2973" xml:space="preserve">Mars promo-<lb/>tus fuerit a primo Å¿itu gra .22. min .19. aut circiter, prius quam retrogradatio eius in-<lb/>cipiat eſſe appa-<lb/>rens.</s> - </p> - <figure position="here"> - <image file="0249-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0249-01"/> - </figure> - <p> - <s xml:id="s2974" xml:space="preserve">Nunc à prima <lb/>Å¿tatione vÅ¿que ad <lb/><reg norm="lineam" type="context">lineã</reg> veri motus <lb/>epicycli Å¿unt gra <num value="16">.<lb/>16.</num> min .51. ipÅ¿ius <lb/>epicycli, vt Å¿upra <lb/><reg norm="vidimus" type="simple">vidimê°</reg> quos Mars <lb/>tranÅ¿it in diebus <lb/>36. aut circiter ad <lb/>rationem min .28. <lb/>in Å¿ingulos dies, <lb/>quo tempore cen <lb/>trum epicycli, in <lb/>tali diÅ¿tantia à <reg norm="cen- tro" type="context">cẽ-<lb/>tro</reg> mundi confice <lb/>ret gra .18. mi .54. <lb/>ad rationem min <num value="31">.<lb/>31.</num> cum dimidio <lb/>in Å¿ingulos dies, <lb/><reg norm="quibus" type="simple">quibê°</reg> deductis ex <lb/>gra .27. min .1. an- - <pb o="238" rhead="IO. BAPT. BENED." n="250" file="0250" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0250"/> - guli <var>.c.o.n.</var> remanebunt gra .8. min .7. pro numero dimidiæ retrogradationis quum <lb/>Mars erit in linea <var>.o.c.</var> veri motus epicycli. </s> - <s xml:id="s2975" xml:space="preserve">Quibus gradibus .8 min .7. Å¿ubtractis à <lb/>grad .22. minu .19. per quos Mars progreſſus erat, Å¿upererunt grad .14. minut .12. <lb/>quibus ipÅ¿e in media retrogradatione exiÅ¿tens in linea <var>.o.c.</var> veri motus epicycli pro-<lb/>motus erit à principio primi Å¿itus. </s> - <s xml:id="s2976" xml:space="preserve">Quod cum eo concordat quod Å¿upra diximus cir <lb/>ca hos gradus .14. min .12. vltra primum Å¿itum in Å¿patio dierum .105. vt Å¿upra, ad <reg norm="tam" type="context">tã</reg> <lb/>tum enim aÅ¿cendunt .69. et .36. nunc fingendo <reg norm="quod" type="simple">ê™</reg> Mars pergat in Å¿uo motu compo-<lb/>Å¿ito qui conÅ¿tat ex his duobus circulis, vi eccentrici, & epicycli ( quanquam <lb/>vt dixi omittimus illam Å¿ummam Å¿ubtilitatem Å¿eu Å¿crupuloÅ¿i<unsure/> atem <reg norm="continuæ" type="context">cõtinuæ</reg> inæqua-<lb/>litatis diÅ¿tantiæ centri epicycli à centro mundi, & præterimus etiam <reg norm="irregularitatem" type="context">irregularitatẽ</reg> <lb/>eius circa centrum mundi, propter regularitatem eius circa centrum æquantis, <lb/>atque etiam miſſum facimus motum epicycli recti à ſua media auge ) fingendo <lb/>inquam, quod Mars dicto motu Å¿uo pergat, vÅ¿que ad punctum Å¿ecundæ Å¿ta-<lb/>tionis, præteribunt alij dies .36. vt prius, quibus iunctis cum .105. fiet Å¿um-<lb/>ma dierum .141. & Mars retrogreſſus erit per alios grad .8. minut .7. quibus Å¿ub-<lb/>tractis a gradibus .14. m in .12. per quos progreſſus erat Å¿upererunt. gra .6. min .5. qui <lb/>bus ipÅ¿e Mars in fine Å¿uæ retrogradationis promotus erit à primo loco vnde moueri <lb/>cÅ“pit. </s> - <s xml:id="s2977" xml:space="preserve">Inter hanc igitur Å¿ecundam Å¿tationem lineæ <var>.o.u.d.</var> & lineam <var>.o.t.</var> Å¿ecundæ con <lb/>tingentiæ, Mars diebus .69. vt prius, confecerit gradus .32. min .14. Å¿ui epicycli, & eo-<lb/>dem tempore linea contingens <var>.o.t.</var> ambulauerit gra .36. min .13. vt prius, à quo itine <lb/>re Å¿ubtracto angulo Å¿ecundo <var>.d.o.t.</var> graduum .13. min .54. Å¿upererunt gra .22. min .19 <lb/>vti prius, quos Mars ambulauerit directè & apparenter, quibus additis ad gra .6. mi <lb/>nut .5. quibus Mars progreſſus erat à principio motus, fient gra .28. min. circiter .24. <lb/>quibus proceſſerit à priori loco in diebus .177. ideÅ¿t in .141. et .36. qui Å¿unt ferè m. n <lb/>Å¿es .6. </s> - <s xml:id="s2978" xml:space="preserve">Itaque Mars partim Å¿urÅ¿um partim, deorÅ¿um ambulans detentus erit <reg norm="menÅ¿ibus" type="context">mẽſibus</reg> <lb/>6. in grad .28. </s> - <s xml:id="s2979" xml:space="preserve">Zo-<lb/>diaci, <reg norm="atque" type="simple">atq;</reg> Å¿i finxe <lb/> - <anchor type="figure" xlink:label="fig-0250-01a" xlink:href="fig-0250-01"/> - rimas quod epicy <lb/>clus <reg norm="moueatur" type="simple">moueat̃</reg> ver-<lb/>Å¿us oppoÅ¿itum au-<lb/>gis, <reg norm="longior" type="context">lõgior</reg> erit mo <lb/>ra planetæ in eiuÅ¿ <lb/>modi duodecate-<lb/>morio, propter au <lb/>gumentum æqua-<lb/>tionis argumenti. <lb/></s> - <s xml:id="s2980" xml:space="preserve">Itaque probata à <lb/>nobis eÅ¿t poſſibi-<lb/>litas huius moræ <lb/>Mattis. </s> - <s xml:id="s2981" xml:space="preserve">Quod qui <lb/>dem mihi Å¿uffice-<lb/>re videtur <reg norm="non" type="context">nõ</reg> mo-<lb/>do cibi, Å¿ed etiam <lb/>cuiuis, qui harum <lb/>Å¿cientiarum prin-<lb/>cipia teneat. </s> - <s xml:id="s2982" xml:space="preserve">Ne-<lb/>que enim <reg norm="nunc" type="context">nÅ©c</reg> do-<lb/>cere volo eos qui - <pb o="239" rhead="EPISTOLAE." n="251" file="0251" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0251"/> - in ijs Å¿unt conÅ¿umati, nec curam mihi Å¿uÅ¿cipere erudiendi imperitos. </s> - <s xml:id="s2983" xml:space="preserve">Satis igitur Å¿it <lb/>oÅ¿tendiſſe, quod qui Å¿cripÅ¿it Martem commorari poſſe tam multos menÅ¿es in vno Å¿i <lb/>gno, non impoſſibilem rem tradidit. </s> - <s xml:id="s2984" xml:space="preserve">Immo per obÅ¿eruationes huius veritatis mil-<lb/>lies factas, AÅ¿trologi fecere Å¿upradictas Å¿uppoÅ¿itiones neceſſarias ad <reg norm="reducendum" type="context">reducẽdum</reg> in <lb/>Å¿uas cauÅ¿as, & ad regulam, eiuÅ¿modi veriſſimos effectus.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0250-01" xlink:href="fig-0250-01a"> - <image file="0250-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0250-01"/> - </figure> - </div> - <p> - <s xml:id="s2985" xml:space="preserve">Non oportebat autem Å¿criptorem harum animaduerÅ¿ionum tantopere eiuÅ¿modi <lb/>mora commoueri, Å¿ed cogitare <reg norm="quod" type="simple">ê™</reg> fortaſſe calculi facti fuerunt eo tempore quo mi-<lb/>Å¿er Mars à Vulcano rete vinctus erat. </s> - <s xml:id="s2986" xml:space="preserve">Vnde cum nonita celeriter Å¿e expedire poſſet <lb/>iter eius Å¿egnius peractum fuit. </s> - <s xml:id="s2987" xml:space="preserve">Aut <reg norm="quod" type="simple">ê™</reg> <reg norm="quum" type="context">quÅ©</reg> vulneratus fuit in bello Troiano, vis eius <lb/>& agilitas per aliquantulum temporis imminuta fuit. </s> - <s xml:id="s2988" xml:space="preserve">Atque Å¿i hic etiam intellexiÅ¿ <lb/>Å¿et eum <reg norm="aliquando" type="context">aliquãdo</reg> fuiſſe in poteÅ¿tate Othi, & Ephialtis <reg norm="vinctum" type="context">vinctÅ©</reg> & carceri incluÅ¿um <reg norm="men- Å¿es" type="context">mẽ-<lb/>Å¿es</reg> tredecim, dum ab Eribea Å¿olutus fuit, vt tu, antiquos Å¿equens, eleganter Å¿cribis <lb/>in illis tuis pulcherimis dialogis. non exiÅ¿timaſſet, credo, tam abÅ¿urdum quod alius <lb/><reg norm="eum" type="context">eÅ©</reg> detinuiſſet Å¿ex aut Å¿eptem menÅ¿ibus, Å¿ed operam dediſſet vt a te intelligeret quid <lb/>Å¿ibi vellet tam longa captiuitas.</s> - </p> - <p> - <s xml:id="s2989" xml:space="preserve">Sed vt ad rem redeamus. </s> - <s xml:id="s2990" xml:space="preserve">Idem pag .4. ait, quod verus motus Martis diÅ¿tat à me-<lb/>dio circiter dies .8. Å¿upponens medium motum eſſe dierum .683. <reg norm="quod" type="simple">ê™</reg> etiam falÅ¿um eÅ¿t. <lb/></s> - <s xml:id="s2991" xml:space="preserve">Sed vtcunque Å¿it, fallitur. </s> - <s xml:id="s2992" xml:space="preserve">Solet enim periodus veri motus Martis eſſe die-<lb/>rum circiter .708. modo paulo plus, modo paulo minus, & interdum poteÅ¿t etiam <lb/>eſſe multo breuior, Å¿icuti erit à die 3. </s> - <s xml:id="s2993" xml:space="preserve">Decembris anni .1593. <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> ad initium Iunij <num value="1595">.<lb/>1595.</num> </s> - <s xml:id="s2994" xml:space="preserve">Tunc enim erit tantum dierum .545. & non quidem Å¿ine ratione, nam dicto <lb/>initio Decembris Mars paulo ante cæperiteſſe directus, cum centrum epicycli erit <lb/>circa medium Tauri, & eius Å¿tella in principio Arietis & initio Iunij .1595. </s> - <s xml:id="s2995" xml:space="preserve">Mars pa <lb/>rum diÅ¿tabit ab initio retrogradationis, regreſſus tamen ad initium ipÅ¿ius Arietis, & <lb/>centrum epicycli erit circa medium Aquarij, in cuius Å¿igni medio, hac ætate repe-<lb/>ritur oppoÅ¿itum augis, & in quo Å¿itu, æquationes <reg norm="argumenti" type="context">argumẽti</reg> Å¿unt, quam maximÄ™ eſſe <lb/>poſſint, quum centrum epicycli circuiuerit Å¿olum circiter tres quartas totius ambi-<lb/>tus, & Mars circuiuerit per partem Å¿uperiorem epicycli circiter gradus .252. </s> - <s xml:id="s2996" xml:space="preserve">Hoc au <lb/>tem dico, vt oÅ¿tendam poſſibilitatem huius eius extraordinariæ velocitatis. </s> - <s xml:id="s2997" xml:space="preserve">Nam <lb/><reg norm="quicunque" type="simple">quicunq;</reg> voluerit poterit certior fieri, per calculum partium motus Martis.</s> - </p> - <p> - <s xml:id="s2998" xml:space="preserve">Vbiautem poÅ¿tea idem author miratur interualla, quæ <reg norm="ponuntur" type="context">ponÅ©tur</reg> inter coniunctio <lb/>nes Iouis, & Martis in eodem Å¿igno, <reg norm="eaque" type="simple">eaqÌ;</reg> vocat errores maximos, oÅ¿tendit Å¿e non re <lb/>ctè conÅ¿ideraſſe motus eorum. </s> - <s xml:id="s2999" xml:space="preserve">Et præcipuè primum miratur <reg norm="quod" type="simple">ê™</reg> inter annum .1528. <lb/>et .1553. </s> - <s xml:id="s3000" xml:space="preserve">Iupiter & Mars nunquam coeant in Leone, cum hæ duæ coniunctiones in <lb/>ter Å¿e diÅ¿tent ann .25. afferens pro ratione, quod hæc duo Å¿ydera, altero quoque anno <lb/>coniunguntur, Å¿ic dicens.</s> - </p> - <quote> - <s xml:id="s3001" xml:space="preserve">Qui Å¿ciet has duas Å¿tellas Å¿ecundo quoque anno inter Å¿e coniungendas, mirabitur <lb/>quomodo non poterunt numeratores, huiuÅ¿modi animaduertere errores.</s> - </quote> - <p> - <s xml:id="s3002" xml:space="preserve">Et præter hanc <reg norm="rationem" type="context">rationẽ</reg> fortaſſe <reg norm="ent" type="context">ẽt</reg> conÅ¿iderauit, <reg norm="quod" type="simple">ê™</reg> in dicto temporis interuallo Iu <lb/>piter <reg norm="semper" type="context">sẽper</reg> fuit in Leone, vt ann .1540. et .1541. </s> - <s xml:id="s3003" xml:space="preserve">Mars <reg norm="autem" type="wordlist">aÅ©t</reg> in eo ſæpe fuit. </s> - <s xml:id="s3004" xml:space="preserve">Vnde im <lb/>poſſibile eilvidetur eos non conueniſſe in dicto Å¿igno. </s> - <s xml:id="s3005" xml:space="preserve"><reg norm="Idemque" type="simple">IdemqÌ;</reg> dici poteÅ¿t de alijs <lb/>coniunctionibus eorundem planetarum, atque has differentias temporum inter di-<lb/>ctas coniunctiones ipÅ¿e tribuit erroribus calculorum Ephemeridum, non autem ta-<lb/>bularum, vt Å¿upra dixit. </s> - <s xml:id="s3006" xml:space="preserve">Å¿ed neÅ¿cio quare vellet dictos planetas coire in Leone, Å¿i <lb/>quum Iupiter in eo erat anno .1540. et .1541. & in eo <reg norm="deambulabatur" type="simple">deambulabat̃</reg>, Mars interea <lb/>erat <reg norm="mon" type="context">mõ</reg> in Libra, modo in Scorpione, Sagittario, Capricorno, & alijs Å¿ignis <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> ad <lb/>Cancrum, in quo cum repertus fuit anno .1541. cogitans congredi cum loue in Leo - <pb o="240" rhead="IO. BAPT. BENED." n="252" file="0252" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0252"/> - ne, comperit eum inde aufugiſſe. </s> - <s xml:id="s3007" xml:space="preserve"><reg norm="Idque" type="simple">IdqÌ;</reg> fortaſſe, Iupiter data opera fecit, vt huiuÅ¿-<lb/>modi AÅ¿trologos in admirationem induceret.</s> - </p> - <p> - <s xml:id="s3008" xml:space="preserve">Idem dico de alijs coniunctionibus horum duorum.</s> - </p> - <p> - <s xml:id="s3009" xml:space="preserve">Quod poÅ¿tea ait, eos Å¿ecundo quoque anno coniungi, animaduertendum eÅ¿t, <reg norm="quia" type="simple">ê—a</reg> <lb/>(vt <reg norm="iam" type="context">iã</reg> dixi) duæ Å¿unt Å¿pecies coniunctionum, quarum vna eÅ¿t linearum eorum me-<lb/>diorum motuum, altera corporum eorum, Å¿altem in longitudine, cum ambo inue-<lb/>niuntur in eodem circulo, qui tranÅ¿it per polos ecclipticæ, nam eos inueniri in <reg norm="eadem" type="context">eadẽ</reg> <lb/>linea recta <reg norm="tranÅ¿eunte" type="context context">trãſeÅ©te</reg> per centrum mundi, rarisÅ¿imum eÅ¿t. </s> - <s xml:id="s3010" xml:space="preserve">Atque coniunctio Å¿upra di-<lb/>ctarum linearum vocatur media, & inter Iouem & Martem fieri Å¿olet Å¿patio <reg norm="dierum" type="context">dierÅ©</reg>. <lb/>816. cum dimidio, aut circiter. </s> - <s xml:id="s3011" xml:space="preserve">Altera <reg norm="dicitur" type="simple">dicit̃</reg> vera, Å¿iue apparens, & irregulatiſſima, <lb/>quæ quidem non Å¿eruat tempus determinatum. </s> - <s xml:id="s3012" xml:space="preserve">Quare quamuis altero <reg norm="quoque" type="simple">quoq;</reg> an-<lb/>no coniungantur; </s> - <s xml:id="s3013" xml:space="preserve">& Iupiter duodenis annis tranÅ¿eat per totum Zodiacum, non ideo <lb/>neceſſe eÅ¿t, vt in Å¿patio .24. annorum coniungantur in Å¿ingulis Å¿ignis, nunquam in eo <lb/>deÅ¿icientes, vtipÅ¿e credit loquens de veris coniunctionibus apparentibus, eo quod <lb/>Å¿int irregulatiſſimæ, vt dixi.</s> - </p> - <p> - <s xml:id="s3014" xml:space="preserve">Atque Å¿i quis velit inuenire periodum coniunctionum mediocrium horum duo-<lb/>rum planetarum, ita faciendum erit. </s> - <s xml:id="s3015" xml:space="preserve">Sumat periodum motus mediocris Iouis, quÄ™ <lb/>eÅ¿t dierum .4328. & Martis, quæ eÅ¿t dierum .687. in quo tempore Martis, Iupiter am <lb/>bulat gra .57. min .8. & diebus .30. conficit. grad .2. minut .29. & Å¿ecun .23. ad ratio-<lb/>nem gra .360. in diebus .4328. </s> - <s xml:id="s3016" xml:space="preserve">Mars verò ad rationem <reg norm="graduum" type="context">graduÅ©</reg> .360. in diebus .687. <lb/>Å¿ingulis .30. diebus conficit. gra .15. mi .43. <reg norm="Å¿ecum" type="context">Å¿ecÅ©</reg> .14. vnde differentia inter eos eÅ¿t gra <lb/>duum .13. mi .15. <reg norm="Å¿ecum" type="context">Å¿ecÅ©</reg> .51. per quam diuidendo productum graduum .57. min .8. in <lb/>dies .30. <reg norm="obuenient" type="context">obueniẽt</reg> dies .129. & duæ tertiæ. </s> - <s xml:id="s3017" xml:space="preserve">quibus addendo periodum Martis fient <num value="816">.<lb/>816.</num> cum dimidio, aut circiter. </s> - <s xml:id="s3018" xml:space="preserve">Atque hÄ™c eÅ¿t periodus infallibilis mediarum con <lb/>iunctionum Iouis cum Marte.</s> - </p> - <p> - <s xml:id="s3019" xml:space="preserve">Nunc venientes ad tabulas AnimaduerÅ¿ionum, videbimus hæc mirabilia eius, in <lb/>quo conÅ¿iÅ¿tant & vbi Å¿int tam multi inÅ¿ignes errores.</s> - </p> - <p> - <s xml:id="s3020" xml:space="preserve">Primum igitur neminem later quod calculus Saturni, à Leouitio editus, difert à <lb/>calculo Stadij circiter gra .2. aut .3. cum Leouitius faciat eum progredi per <reg norm="tantum" type="context">tãtum</reg> in <lb/>teruallum, modo plus, modo minus, & Å¿imiliter Iouem. </s> - <s xml:id="s3021" xml:space="preserve">Å¿ed longe minori diffe-<lb/>rentia, & ſępe gra .1. minus, atque in alijs planetis differunt, modo plus, modo minus. <lb/></s> - <s xml:id="s3022" xml:space="preserve">Huic igitur <reg norm="mirum" type="context">mirÅ©</reg> videtur, quod vnus ex his calculatoribus detineat Saturnum plu-<lb/>ribus menÅ¿ibus in vno Å¿igno, & alter in alio, non animaduertens dictam <reg norm="differentiam" type="context">differentiã</reg> <lb/>eſſe eius rei cauÅ¿am. </s> - <s xml:id="s3023" xml:space="preserve">Miratur item, quod vnus ex is faciat Saturnum morari paucis <lb/>menÅ¿ibus in vno Å¿igno, alter vero eum ibi detineat integris annis. </s> - <s xml:id="s3024" xml:space="preserve">Vt exempli gra-<lb/>tia, verÅ¿us finem Å¿uæ tabulæ Saturni, dicit quod Leouitius eum carceri includit in <lb/>geminis annis .2. menÅ¿e vno, & diebus .9. </s> - <s xml:id="s3025" xml:space="preserve">Stadius vero clementior eum liberat intra <lb/>menÅ¿es .3. & dies .14. </s> - <s xml:id="s3026" xml:space="preserve">Sed hic non cogitat, quod Stadius facit eum ingredi in gemi-<lb/>nos anno .1559. die .10. Iunii, & ambulare directum vÅ¿que ad diem .6. Septembris, <lb/>eiuÅ¿dem anni gra .6. min .34. <reg norm="eumque" type="simple">eumqÌ;</reg> poÅ¿tea retrogradum inde exire die .22. Decem. <lb/>eiuÅ¿dem anni, cum ingreditur in Taurum, vbi partim retrogradus, & partim dire-<lb/>ctus manet vÅ¿que ad diem .20. Februa .1560. rediens poÅ¿tea in geminos, in quibus <lb/>manet vÅ¿que ad diem Iunii .1561. & inde ingreditur in Cancrum, <reg norm="ambulatque" type="simple">ambulatqÌ;</reg> dire-<lb/>ctus. gra .4. min .59. vÅ¿que ad diem .4. Octob. </s> - <s xml:id="s3027" xml:space="preserve">Vnde retrogradiens rurÅ¿um intratin Ge <lb/>minos die .28. Decemb. eiuÅ¿dem anni, at que ibi partim retrogradus partim directus <lb/>manet vÅ¿que ad diem .12. Apr .1562. itaque in pluribus vicibus facit eum morari in <lb/>Geminis dies circiter .816. ideÅ¿t circiter menÅ¿es .27. Å¿umpÅ¿it autem hic Å¿criptor bre - <pb o="241" rhead="EPISTOL AE." n="253" file="0253" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0253"/> - uiſſimam moram cauÅ¿a comparationis cum calculo Leouitij, vt faceret differentiam <lb/>apparere maiorem. </s> - <s xml:id="s3028" xml:space="preserve">Tamen in quouis dictorum temporum nunquam inuenietur <lb/>Leouitius differre à Stadio plus gradibus tribus integris. </s> - <s xml:id="s3029" xml:space="preserve">Idem fecit in multis alijs lo <lb/>cis dictorum virorum eos conferens tum in Saturno tum in Ioue, & Marte, <reg norm="putans" type="context">putãs</reg> ma <lb/>gnum eſſe errorem, <reg norm="quod" type="simple">ê™</reg> planeta non perambulet totum Å¿ignum, in quod eÅ¿t ingreſſus <lb/>vel directus vel totum retrogradus. </s> - <s xml:id="s3030" xml:space="preserve">Atque hæc opinio Å¿imilis eÅ¿t Å¿uperiori de con <lb/>iunctionibus veris Saturni, & Iouis, vbi dicit quod nunquam coniunguntur in vno Å¿i <lb/>gno alterius triplicitatis, niÅ¿i perfecerit coniunctionem in omnibus Å¿ignis primæ tri <lb/>plicitatis. </s> - <s xml:id="s3031" xml:space="preserve">Verum vt Å¿uperÅ¿edeam vlterius diÅ¿putare, mihi videtur, quod hactenus <lb/>dixi, poſſe tibi Å¿atisfacere, quod attinet ad Å¿ciendam Å¿ententiam mean Å¿uper dictis <lb/>AnimaduerÅ¿ionibus latinè Å¿criptis. </s> - <s xml:id="s3032" xml:space="preserve">Hoc tamen non prætermittam, <reg norm="quod" type="simple">ê™</reg> hic non ani-<lb/>maduertit, <reg norm="nempe" type="context">nẽpe</reg> <reg norm="quod" type="simple">ê™</reg> <reg norm="differentiæ" type="context">differẽtiæ</reg> locorum <reg norm="planetarum" type="context">planetarÅ©</reg> quæ <reg norm="suntinter" type="context">sÅ©tinter</reg> ephemeridas Leouitij <lb/>& Stadij, euenere, quia vnus Å¿upputat <reg norm="cum" type="context">cÅ©</reg> radicibus, & <reg norm="fundamentis" type="context">fundamẽtis</reg> AlfonÅ¿i alter verò <lb/>Reinoldi ex Copernico recentius obÅ¿eruatis, ita idem euenire poterit futuris tem-<lb/>poribus, Å¿i Å¿upputati fuerint dicti motus, & loci cum recentioribus obÅ¿eruationibus <lb/>cum impoſſibile Å¿it tam Å¿ubtiliter, tanqÌue perfectè Å¿upputare loca & <reg norm="motus" type="simple">motê°</reg> eorum, <lb/>vtlungo interuallo temporis non comperiantur in eis aliquæ differentiÄ™, cuius rei re <lb/>medium eÅ¿t Å¿emper Å¿equi recentiores obÅ¿eruationes & tabulas.</s> - </p> - <p> - <s xml:id="s3033" xml:space="preserve">Atque vt tibi Å¿atisfaciam etiam circa alia Å¿cripta vulgari lingua edita menÅ¿ibus .4 <lb/>poÅ¿t latina, etÅ¿i intelligere potes, qualia poſſint eſſe alia eius Å¿cripta, ex ijs quæ Å¿upra <lb/>dicta Å¿unt, atque etiam ex eo, quod dicit Å¿emiÅ¿iſſe multa exempla Å¿uarum Animad <lb/>uerÅ¿ionum in varias terras, illis qui profitentur has Å¿cientias, aut earum Å¿tudioÅ¿i Å¿unt, <lb/>nec <reg norm="quenquam" type="context">quenquã</reg> inueniſſe qui ad <reg norm="tam" type="context">tã</reg> laudabilem prouinciam motus Å¿it, nec vidiſſe, <reg norm="quod" type="simple">ê™</reg> ali <lb/>quis reÅ¿ponderit eius rationibus; </s> - <s xml:id="s3034" xml:space="preserve">laudabilem prouinciam, autem puto, <reg norm="quod" type="simple">ê™</reg> intelligat <lb/>correctionem ephemeridum, verens, ne culpa calculatorum, qui eas Å¿umpÅ¿ere e ta-<lb/>bulis, tam differentes Å¿int, vt <reg norm="quibuÅ¿dam" type="context">quibuÅ¿dã</reg> locis cap .1. </s> - <s xml:id="s3035" xml:space="preserve">Videtur, & præcipuè vbi Å¿ic ait.</s> - </p> - <quote xml:lang="it"> - <s xml:id="s3036" xml:space="preserve">Perche eſſendo impoſſibile alli Å¿tudioÅ¿i di dette Å¿cientiæ di non Å¿eruirÅ¿i delle <lb/>ephemeridi, maggiormente a quelli che non Å¿anno Å¿eruirÅ¿i delle tauole, e cono-<lb/>Å¿cendo d'incorrere in errori Å¿enza hauerui altro rimedio, Å¿arebbono forzati di ab <lb/>bandonare i Å¿tudij loro.</s> - </quote> - <p> - <s xml:id="s3037" xml:space="preserve">Quanquam circa finem dicti capitis redeat in meliorem viam & aduerÅ¿etur Å¿i-<lb/>bijpÅ¿i vbi Å¿ic ait.</s> - </p> - <quote xml:lang="it"> - <s xml:id="s3038" xml:space="preserve">Che poi eſſi poſſeſſori della Å¿cienza, & c.</s> - </quote> - <p> - <s xml:id="s3039" xml:space="preserve">Etiam aperiam tibi, quæ mea Å¿it de ijs Å¿ententia.</s> - </p> - <p> - <s xml:id="s3040" xml:space="preserve">Hicigitur in Å¿criptis Italicis, vt morderet aliquem ex ijs, qui eius Å¿uperiora Å¿cripta <lb/>non laudauerant, occaÅ¿ionem capit aperiendi aliquos illius errores, per editionem <lb/>collationis quorundam calculorum a Å¿e collectorum illius, atque etiam aliorum, cu <lb/>ius calculi Å¿unt in Å¿ecunda, & Å¿eptima figura. </s> - <s xml:id="s3041" xml:space="preserve">Sed prius quam veniamus ad defenÅ¿io <lb/>nem harum duarum figurarum vide obÅ¿ecro quam alienum ei videatur, quod alij <lb/>dixerint differentiam ephemeridum non eſſe magni momenti, non afferens reÅ¿pe-<lb/>ctum vllum, qui enim dixerunt eiuÅ¿modi differentiam non eſſe magni momen-<lb/>ti id dixerunt habito reÅ¿pectu ad Å¿ignum in quo eÅ¿t planeta, vt (exempli gra-<lb/>tia) quamuis in ponendo loco Saturni Leouitius interdum differat à Stadio gra <lb/>dibus .3. quum vterque eum ponat in eodem Å¿igno, tuncid nullius momenti eÅ¿t, <lb/>& Å¿ic in coniunctionibus aut alijs aÅ¿pectibus duo, aut .3. gradus non faciunt alteratio <lb/>nem Å¿enÅ¿ibilem, cum virtus coniunctionum, & aÅ¿pectuum inÅ¿it, & duret per mul-<lb/>tos gradus ante aut poÅ¿t ipÅ¿um punctum. </s> - <s xml:id="s3042" xml:space="preserve">Nec quicquam <reg norm="tamen" type="wordlist">tamẽ</reg> eÅ¿t qui dubitet, quin <lb/>præſtaret Å¿cire Å¿ubtiliter ipÅ¿um punctum. </s> - <s xml:id="s3043" xml:space="preserve">Nec vnquam fuit aliquis qui negauerit <reg norm="re ferre" type="conjecture" resp="paul">re - <pb o="242" rhead="IO. BAPT. BENED." n="254" file="0254" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0254"/> - ferre</reg> vt anni directionum correÅ¿pondeant gradibus æquatoris. </s> - <s xml:id="s3044" xml:space="preserve">Et præterea in ephe <lb/>meridibus videntur certè motus & aÅ¿pectus luminarium, quamuis inÅ¿it <reg norm="differentia" type="context">differẽtia</reg> mi <lb/>nutorum. </s> - <s xml:id="s3045" xml:space="preserve">Nam non differunt gradibus, præter Å¿itum parum diÅ¿tantem à vero om-<lb/>nium planetarum, quorum cognitio in cÅ“lo, quamuis circa eorum locum error eÅ¿-<lb/>Å¿et gra .10. tamen in hoc prodeſſet, & tempus aÅ¿pectus eorum, etÅ¿inon diei præcisè, <lb/>quia influentia eiuÅ¿modi a Å¿pectuum, præterquam Lunæ durat multis diebus, & non <lb/>vno tantum. </s> - <s xml:id="s3046" xml:space="preserve">præterquam quod ipſæ ephemerides <reg norm="oÅ¿tendunt" type="context">oÅ¿tendÅ©t</reg> nobis tempus <reg norm="ecclipÅ¿ium" type="context">ecclipÅ¿iÅ©</reg>, <lb/>in quo certènon differunt nec diebus nec multis horis, & itidem multa alia.</s> - </p> - <p> - <s xml:id="s3047" xml:space="preserve">Non Å¿unt igitur contemnendæ ephemerides, nec habendæ pro re nullius pretij, <lb/>vt hic ait.</s> - </p> - <p> - <s xml:id="s3048" xml:space="preserve">Quod attinet ad illa alia, quæ hic vocat errores ephemeridum, tam de apparenti <lb/>coniunctione Saturni cum Ioue in Å¿ignis alterius triplicitatis prius quam perÄ™gerit <lb/>præcedentem, quam de faciendo currere <reg norm="Saturnum" type="context">SaturnÅ©</reg>, & de retinendo Ioue, de <reg norm="detinem" type="context">detinẽ</reg> <lb/>do Marte .6. aut .7. menÅ¿ibus in vno Å¿igno, de Marte, & Ioue non coeuntibus Å¿ingu-<lb/>lis .24. annis in quolibet Å¿igno, & cius generis alia, minime verum eÅ¿t quod Å¿int er-<lb/>rores, quamuis huic præbuerint occaÅ¿ionem toties errandi.</s> - </p> - <p> - <s xml:id="s3049" xml:space="preserve">Comparatio poÅ¿tea inter eius calculos Å¿umptos partim ex tabulis Iunctini, & par <lb/>tim ex ephemeridibus Stadij tan quam calculis Copernici, & calculos figurarum Å¿u-<lb/>per eis poÅ¿itarum Å¿upputatarum à diuerÅ¿is per ephemeridas AlfonÅ¿inas, etiam pro-<lb/>poÅ¿ita ab eo eÅ¿t ad oÅ¿tendendum magnam & monÅ¿truoÅ¿am differentiam, vt ait cap <num value="2">.<lb/>2.</num> vbi miratur, quod cum ex communi Å¿ententia calculi Copernici meliores Å¿int, cal <lb/>culatores dictarum figurarum potius eos Å¿umpÅ¿erint à tabulis AlfonÅ¿i, quam Coper <lb/>nici. </s> - <s xml:id="s3050" xml:space="preserve">Quæ admiratio quam aliena Å¿it, conÅ¿iderandum permittam cuiuis intelligen-<lb/>ti harum Å¿acultatum, cum ſæpe accidere poſſit. </s> - <s xml:id="s3051" xml:space="preserve">vt cum aliquis velit Å¿cire Å¿olum vni <lb/>uerÅ¿alia alicuius geneÅ¿is, Å¿iue natiuitatis, cum non inueniantur ephemerides Coper-<lb/>nici, Å¿ed tantum AlfonÅ¿i, calculator vtatur tantum ephemeridibus, quas inuenit, <reg norm="tum" type="context">tÅ©</reg> <lb/>cauÅ¿a vitandi tædij calculi tabularum, qui magni laboris eÅ¿t, prÄ™cipuè in tabulis Pru <lb/>tenicis Reinoldi. </s> - <s xml:id="s3052" xml:space="preserve">tum quia Å¿uperflua ei eÅ¿t Å¿umma Å¿ubtilitas, cum non curet laborare <lb/>circa directiones vt <reg norm="factum" type="context">factÅ©</reg> eÅ¿t pro <reg norm="Å¿ecunda" type="context">Å¿ecÅ©da</reg> figura ab hoc propoÅ¿ita, quæ erat anni .1551 <lb/>quo non <reg norm="inueniebantur" type="context">inueniebãtur</reg> ephemerides Copernicæ, quæ non editæ Å¿unt ante annum <lb/>1554. præter quam quod ille nobilis vir pro quo Å¿upputata fuit dicta Å¿ecunda natiui <lb/>tas dubitabat de anno, vt hic Å¿imiliter Å¿cit. </s> - <s xml:id="s3053" xml:space="preserve">quare potuiſſet perdi tempus, & labor, Å¿i <lb/>Å¿upputata fuiſſet per tabulas Reinoldi, nam Iunctini tabulæ nondum editæ fuerant. <lb/></s> - <s xml:id="s3054" xml:space="preserve">Calculus poÅ¿tea Å¿eptimæ figuræ, qui erat reuolutio dictæ Å¿ecundæ natiuitatis, <reg norm="duabus" type="simple">duabê°</reg> <lb/>de cauÅ¿is non factus eÅ¿t per tabulas prutenicas, primum, quia eius anni .1580. non <lb/>inueniebantur amplius ephemerides Copernicæ. </s> - <s xml:id="s3055" xml:space="preserve">Ephemerides enim Stadij <reg norm="in- cipientes" type="context">in-<lb/>cipiẽtes</reg> ab ann .1554. <reg norm="deÅ¿inunt" type="context">deÅ¿inÅ©t</reg> ann .1576. & <reg norm="continuatæ" type="context">cõtinuatæ</reg> poÅ¿tea quæ perueniunt vÅ¿que <lb/>ad annum .1600. non peruenere ad manus calculatoris ante hunc annum .1581. </s> - <s xml:id="s3056" xml:space="preserve">Al-<lb/>tera ratio eÅ¿t, quia in reuolutionibus, quoniam in eis non fiunt directiones, non po-<lb/>nuntur à doctis, ne minuta quidem. </s> - <s xml:id="s3057" xml:space="preserve">quare non Å¿olum non curant eas Å¿upputare per <lb/>tabulas, Å¿ed nec exquiÅ¿itè quidem per ephemeridas. </s> - <s xml:id="s3058" xml:space="preserve">Calculi poÅ¿tea ab hoc Å¿umpti <lb/>ex tabulis Iunctini, & poÅ¿iti Å¿ub dicta Å¿ecunda figura, adeò rectè facti Å¿unt, vt cum Å¿e <lb/>cundum ip Å¿as tabulas oporteat Saturnum eſſe circa .32. minutum gradus .23. Aqua-<lb/>rij, ipÅ¿e eum Å¿cribat in gra .11. mi .3. dicti Å¿igni. Iupiter Å¿imiliter qui Å¿ecundum dictas <lb/>tabulas inuenitur circa finem gradus .5. Cancri, ab eo ponitur in min .28. gra .19. eiuÅ¿ <lb/>dem. ex quibus planetis Saturnus in figura poÅ¿itus eÅ¿t in min .27. grad .23. Arietis, Iu <lb/>piter autem in min .3. gra .6. Cancri, </s> - <s xml:id="s3059" xml:space="preserve">Vnde Å¿ecun dum verum, inter calculum Alfon - <pb o="243" rhead="EPISTOLAE." n="255" file="0255" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0255"/> - Å¿i & Iunctini in Saturno non erat differentia plus quam minu .5. & in Ioue min .4. Å¿ed <lb/>Å¿ecundum calculum huius in Saturno fuiſſet differentia gra .11. minu .54. & in Ioue <lb/>grad .13. min .35. </s> - <s xml:id="s3060" xml:space="preserve">Atque hæ Å¿unt quidem differentiæ magnæ, & monÅ¿truoſæ vtipÅ¿e <lb/>eas vocat, vt etiam eÅ¿tilla Veneris, & Mercurij inter tertiam figuram, & eius calcu-<lb/>lum Å¿umptum, non quidem à tabulis laborioÅ¿is, Å¿ed à ſimplicibus ephemeridibus Sta <lb/>dij, quæ differentia <reg norm="non" type="context">nõ</reg> eÅ¿t quidem paucorum graduum, cum Å¿it tertiæ partis cÅ“li in <lb/>quolibet dictotum planetarum. </s> - <s xml:id="s3061" xml:space="preserve"><reg norm="HuiuÅ¿modique" type="simple">HuiuÅ¿modiqÌ;</reg> monÅ¿tra certè non Å¿unt orta à ta-<lb/>bulis Å¿iue ephemeridibus diuerÅ¿is, Å¿ed Å¿unt partus huius authoris.</s> - </p> - <p> - <s xml:id="s3062" xml:space="preserve">Pergens poÅ¿tea aſſiduè bonus hic vir hominibus dare Å¿pecimen doctrinæ Å¿uæ ape <lb/>riendo (vt conatur) aliorum errores, proponit duas differentias inter primam fi-<lb/>guram, & Å¿uum calculum Å¿uppoÅ¿itum Saturni, & Iouis. </s> - <s xml:id="s3063" xml:space="preserve">Primum de Saturno ait, <reg norm="quod" type="simple">ê™</reg> <lb/>cum differentia Å¿it gra .1. min .30. oÅ¿tendit in directione, <reg norm="quod" type="simple">ê™</reg> accidens Å¿it euenturum <lb/>anno vno, & menÅ¿ibus Å¿ex ante, aut poÅ¿t, quaÅ¿i eiuÅ¿modi differentia eſſet partium æ-<lb/>quatoris, Å¿icuti eÅ¿t partium Zodiaci. </s> - <s xml:id="s3064" xml:space="preserve">Idem dico de differentia Iouis. </s> - <s xml:id="s3065" xml:space="preserve">Quod quidem, <lb/>manifeÅ¿tum eÅ¿t inditium Å¿cientiæ Å¿uæ, & quantum ea intelligat de quibus loquitur.</s> - </p> - <p> - <s xml:id="s3066" xml:space="preserve">Quod poÅ¿tea attinet ad differentiam inter Copernicum & AlfonÅ¿um, circa <reg norm="Solem" type="context">Solẽ</reg>, <lb/>nullus eÅ¿t harum Å¿cientiarum peritus, qui id neÅ¿ciat, & Å¿imiliter de differentia Å¿itus <reg norm="cae li" type="simple">cÄ™Â<lb/>li</reg> in reuolutionibus annuis.</s> - </p> - <p> - <s xml:id="s3067" xml:space="preserve">Quod vero ait Å¿eptimam Å¿iguram malè <reg norm="Å¿upputatam" type="context">Å¿upputatã</reg> fuiſſe, Å¿i non eÅ¿t maximus cer-<lb/>tè non eÅ¿t minimus monÅ¿truoÅ¿orum eius errorum. </s> - <s xml:id="s3068" xml:space="preserve">Vbi itidem videri poteÅ¿t, quam <lb/>alienus hic Å¿it ab hac Å¿cientia. </s> - <s xml:id="s3069" xml:space="preserve">Nam Å¿i Å¿altem curaſſet Å¿ibi ab aliquo Å¿upputandum <lb/>locum Solis per tabulas AlfonÅ¿i in inÅ¿tanti minutorum .36. pomeridianorum, certior <lb/>factus eſſet quod in illo puncto Sol inueniebatur in minu 54. grad .11. </s> - <s xml:id="s3070" xml:space="preserve">Geminorum, <lb/>ideÅ¿t præterierat gra .10. cum min .54. vel Å¿i curaſſet Å¿ibi inueniendum tempus, per <lb/>dictas tabulas cum grad .10. min .54. </s> - <s xml:id="s3071" xml:space="preserve">Geminorum vt faciendum eÅ¿t, Å¿equendo <reg norm="tamen" type="wordlist">tamẽ</reg> <lb/>AlfonÅ¿um, & non per calculum Solis poÅ¿itum in ephemeridibus, vt parum periti fa <lb/>cere Å¿olent, vidiſſet <reg norm="quod" type="simple">ê™</reg> inuenta eſſent min .36. pomeridiana. </s> - <s xml:id="s3072" xml:space="preserve">Leuis tamen occaÅ¿io hu <lb/>ic fuit Å¿uÅ¿picandi eiuÅ¿modi tempus eſſe falÅ¿um, quod viderit in illa figura <reg norm="Solem" type="context">Solẽ</reg> po <lb/>Å¿itum eſſe cum gra .11. & non cum gra .10. min .54. non animaduertens ita notatum <lb/>fuiſſe Solem vt omnes alios planetas, Å¿cilicet Å¿ine minutis, quum, vt dixi, in reuolu <lb/>tionibus non adhibeatur tanta Å¿cruploÅ¿itas.</s> - </p> - <p> - <s xml:id="s3073" xml:space="preserve">Quod deinde ait, in illa figura Solem poÅ¿itum eſſe in decima domo, & non in .9. <lb/>id relinquam iudicio eorum qui Å¿ciunt numerare domos, Å¿altem poÅ¿uiſſet authori-<lb/>tate Å¿ua Solem in dicta decima diuersè ab exemplo ei dato ab amico, vt oÅ¿tenderet <lb/>Å¿e dicere verum, vt in Å¿ecunda figura diÅ¿crepat ab ipÅ¿o exemplo in collocando Leo-<lb/>ne, Virgine, & Libra, & Scorpio, quos malè locauit, & Å¿i alii bene Å¿e habent.</s> - </p> - <p> - <s xml:id="s3074" xml:space="preserve">Atque quod hactenus à me dictum eÅ¿t, Å¿atis Å¿it ad <reg norm="intelligendum" type="context">intelligẽdum</reg> quale Å¿it <reg norm="reliquum" type="context">reliquÅ©</reg> <lb/>dictæ eius diÅ¿putationis. </s> - <s xml:id="s3075" xml:space="preserve">Sienim velim pergere notare omnia eius errorum loca, eÅ¿ <lb/>Å¿et mihi inanis labor, & tibi nimia moleÅ¿tia. </s> - <s xml:id="s3076" xml:space="preserve">Et quamuis non defuerint præſtantiſſi-<lb/>mi viri, qui viÅ¿is eius Å¿criptis familiariter eum monuere, & tu ipÅ¿e, vt audiui, cum <reg norm="in- Å¿trumento" type="context">in-<lb/>Å¿trumẽto</reg> theoricÄ™ in <reg norm="manibus" type="simple">manibê°</reg> ei <reg norm="oÅ¿tenderis" type="context">oÅ¿tẽderis</reg> quo <reg norm="mon" type="context">mõ</reg> Mars poſſit morari amplius Å¿ex <reg norm="mem" type="context">mẽ</reg> <lb/>Å¿ibus in vno Å¿igno. </s> - <s xml:id="s3077" xml:space="preserve">& præterea cum iam ab initio Taurinum aduenit, mecum com-<lb/>municauerit illa Å¿ua prima Å¿cripta, <reg norm="egoque" type="simple">egoqÌ;</reg> eum monuerim, quod in varijs <reg norm="rebus" type="simple">rebê°</reg> falleba <lb/>tur, <reg norm="diſſuaÅ¿erimque" type="simple">diſſuaÅ¿erimqÌ;</reg> ne ea imprimenda curaret, quia nullum honorem inde referret, <lb/>eum hortans, vt potius alijs rebus operam daret, atque ei dixerim quod ad animad-<lb/>uerÅ¿iones differentiarum ephemeridum attinet, quod id iam <reg norm="oens" type="context">oẽs</reg> animaduerterant. <lb/></s> - <s xml:id="s3078" xml:space="preserve">Mihi reÅ¿pondit Å¿e decreuiſſe illa edere, vt poÅ¿tea fecit, & tot admonitionibus non - <pb o="244" rhead="IO. BABPT. BENED." n="256" file="0256" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0256"/> - acqu<unsure/>ieÅ¿cens, die .11. </s> - <s xml:id="s3079" xml:space="preserve">AuguÅ¿ti edidit chartam illam impreſſam inuitans ad diÅ¿puta-<lb/>tionem quotquot adhÄ™rerent contrariæ Å¿ententiæ, volens Å¿uÅ¿tinere <reg norm="Martem" type="context">Martẽ</reg> non poÅ¿ <lb/>Å¿e commorari in vno Å¿igno amplius duobus menÅ¿ibus, Å¿upponens partem princi-<lb/>piorum ab omnibus admiſſorum, & in fine paginæ exponens modum, quo vtitur ad <lb/>probationem Å¿uæ intentionis. </s> - <s xml:id="s3080" xml:space="preserve">Puto autem quodÅ¿ecum ratiocinabatur de Marte, <lb/>vt fecit de Saturno in Å¿cripto latino, hoc modo. </s> - <s xml:id="s3081" xml:space="preserve">Si Mars in duobus annis ambulat <lb/>per omnia 12. Å¿igna, neceſſe eÅ¿t igitur, vtin menÅ¿ibus duobus ambulet per vnum Å¿i-<lb/>gum, cum menÅ¿es .2. Å¿int duo decima pars annorum duorum. </s> - <s xml:id="s3082" xml:space="preserve">Sedibi Å¿tatim in ipÅ¿o <lb/>initio commitcit errorem graduum ferè .7. dicens, quod medius motus Martis inue-<lb/>niebatur Å¿ignorum .4. & gra .17. cum eo tempore dictus medius motus non eſſet reue <lb/>ra plus quam Å¿ign .4. grad .10. mi .36. verum hoc ad ea, quæ Å¿equuntur exigui eÅ¿t mo <lb/>menti. </s> - <s xml:id="s3083" xml:space="preserve">Is poÅ¿tea particulatim colligit medium motum Martis ad diem .29. </s> - <s xml:id="s3084" xml:space="preserve">Mai an-<lb/>ni .15 14. quem ait eſſe Å¿ignorum .9. gra .27. min .53. & tamen reuera erat <reg norm="tantum" type="context">tantÅ©</reg> Å¿igno <lb/>rum .9. gra .21. mi .29. Å¿ed miſſum faciamus etiam hunc errorem <reg norm="tanquam" type="context">tanquã</reg> à primo pen <lb/>dentem. </s> - <s xml:id="s3085" xml:space="preserve">Cum deinde ibidem ponit centrum epicycli, Å¿imiliter errat, nam <lb/>centrum epicycli nunquam poni debet vbi eÅ¿t linea medij motus, niÅ¿i Å¿it in auge, aut <lb/>in oppoÅ¿ito aug is eccentrici, quia debebat collocare ipÅ¿um centrum <reg norm="tanto" type="context">tãto</reg> poÅ¿t linec<unsure/><reg norm="am" type="context">ã</reg> <lb/>medij motus, quanta erat æquatio centri, quia medium centrum Martistunc erat mi <lb/>nus Å¿ignis Å¿ex, & aux eccentrici eius erat in Å¿exto minuto grad .16. Leonis. </s> - <s xml:id="s3086" xml:space="preserve">Tamen <lb/>hoc etiam leue eÅ¿t. </s> - <s xml:id="s3087" xml:space="preserve">Præſupponamus igitur quod centrum epicycli cſſet in grad .28 <lb/>Capricorni, vt ipÅ¿e credidit, ideÅ¿t gradibus .7. vlterius quam erat reuera. </s> - <s xml:id="s3088" xml:space="preserve">Ait poÅ¿tea <lb/>Å¿e comperiſſe Martem ambulaſſe Å¿igna .4. & grad .22. eius epicycli, Å¿ed non explicat <lb/>an intelligat de argumento medio, an de vero, quod vocatur æquatum, nam Å¿i intel <lb/>ligatur de medio, hoc eſſe non poteÅ¿t, cum <reg norm="medium" type="context">mediÅ©</reg> eſſet Å¿ignorum .4. gra .24. mi .35. <lb/>Å¿ed Å¿i intelligatur de vero, vt iure credendum eÅ¿t (alioquin etiam erraſſet) cer-<lb/>tè falÅ¿um eÅ¿t. </s> - <s xml:id="s3089" xml:space="preserve"><reg norm="Nam" type="context">Nã</reg>, verum, erat Å¿ignorum .4. grad .29. minu .39. </s> - <s xml:id="s3090" xml:space="preserve">Itaque Mars non di-<lb/>Å¿tabat à linea veri motus epicycli amplius gradibus .30. & minu .21. ipÅ¿ius epicycli, <lb/>& æquatio argumentiÅ¿ecundo correcta erat gra .44. minu .2. à quo Å¿ubtracta æquatio <lb/>ne centri, quæ erat gr .5. minu .4. (cum centrum epicycli deberet tanto Å¿pacio eſſe <lb/>poÅ¿t lineam medij motus quantum Å¿upra dixi) Å¿upererant gra .38. minu .58. adden-<lb/>di gradibus, & minu. medij motus, qui cum reuera eſſent grad .21. & minu .29. Ca-<lb/>pricorni, perueniebant ad minu .27. grad .1. PiÅ¿cium. </s> - <s xml:id="s3091" xml:space="preserve">Sed præſuppoÅ¿ito <reg norm="Å¿ecundum" type="context">Å¿ecundÅ©</reg> <lb/>ipÅ¿um, quod medius motus eſſet grad .28. </s> - <s xml:id="s3092" xml:space="preserve">Capricorni, & quod Mars eſſet non <reg norm="Å¿olum" type="context">Å¿olÅ©</reg> <lb/>vbi hic ait, Å¿ed etiam in prima linea contingentiæ epicycli, ideÅ¿t in prima linea ma <lb/>ximæ æquationis argumenti, & præſuppoÅ¿ito etiam quod dicta æquatio eſſet æqua-<lb/>lis illi, quam haberet ad medium Aquarij Å¿cilicet grad .47. quum centrum epicycli <lb/>eÅ¿t in oppoÅ¿ito augis, manifeÅ¿tum eÅ¿t, quod eiuÅ¿modi linea contingentiæ non tranÅ¿i <lb/>ret vltra grad .15. PiÅ¿cium, & tamen hic ait, quodlinea veri motus Martis vadit ad <lb/>grad .16. Arietis. </s> - <s xml:id="s3093" xml:space="preserve">vnde oporteret, quod æquatio argumenti eſſet plus quam grad <num value="78">.<lb/>78.</num> </s> - <s xml:id="s3094" xml:space="preserve">Quod Å¿i verum eſſet, & <var>.o.c.</var> etiam eſſet partium .54. Å¿ecundum diÅ¿tantiam pro-<lb/>ximiorem centro mundi, Å¿emidiameter epicycli eſſet eiuÅ¿modi partium .52. minut <num value="49">.<lb/>49.</num> & quum Mars eſſet in <var>.g.</var> ideÅ¿t in oppoÅ¿ito veræ augis epicycli, dum centrum epi <lb/>cycli eſſet in eiuÅ¿modi diÅ¿tantia à terra, diÅ¿tantia <var>.o.g.</var> ideÅ¿t à terra ad Martem non <lb/>eſſet plus, quam vna Å¿ola pars ex dictis, cum minut .11. cum partes .52. minu .49. <lb/>ad .54. Å¿int vt Å¿inus anguli gra .78. qui eÅ¿t partium .97814. ad Å¿inum totalem partium <lb/>100000. </s> - <s xml:id="s3095" xml:space="preserve">Nam iam Å¿upra dixi, quod triangulus <var>.o.c.i.</var> eÅ¿t rectangulus. </s> - <s xml:id="s3096" xml:space="preserve">Hinc Å¿eque- - <pb o="245" rhead="EPISTOLAE." n="257" file="0257" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0257"/> - retur, quod in in-<lb/>teruallo <var>.o.g.</var> <reg norm="vnius" type="simple">vniê°</reg> <lb/> - <anchor type="figure" xlink:label="fig-0257-01a" xlink:href="fig-0257-01"/> - partis, & min .11. <lb/>reÅ¿pectu <var>.o.c.</var> par-<lb/>tium .54. </s> - <s xml:id="s3097" xml:space="preserve">Colloca <lb/>retur Å¿emidiame-<lb/>ter terrÄ™ cum Å¿piÅ¿-<lb/>Å¿itudine aeris, i-<lb/>gnis, cÄ™lorum Lu-<lb/>næ, Mercurij, Ve-<lb/>neris, & Solis, <reg norm="prae terquam" type="simple">prÄ™Â<lb/>terquam</reg> quod vt <lb/>inter Solem, & <lb/>terram Å¿unt circa <lb/>605. diametri ip <lb/>Å¿ius terræ, inter <lb/>terram, & Mar-<lb/>tem cum eſſet in <lb/>auge Å¿ui epicycli, <lb/>& epicyclus in au <lb/>ge eccentrici, in-<lb/>uenirentur cir--<lb/>ca .60000. dia--<lb/>metri eiuÅ¿dem ter <lb/>ræ, & tamen ea diÅ¿tantia Å¿iue interuallum non poteÅ¿t continere .5000. diametri ter-<lb/>ræ. </s> - <s xml:id="s3098" xml:space="preserve">Et quod plus eÅ¿t, hic tam vaÅ¿tum facit hunc Å¿uum epicyclum, vt ambiente Mar-<lb/>te per inferiorem eius partem, neceſſe ei eſſet manere in vno duodecatemorio mul-<lb/>to plus quam .7. aut .8. menÅ¿. </s> - <s xml:id="s3099" xml:space="preserve">vnde hic multo magis miraretur quam prius. </s> - <s xml:id="s3100" xml:space="preserve">Hinc cer-<lb/>nere licet quam rectè facti Å¿int hi eius calculi.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0257-01" xlink:href="fig-0257-01a"> - <image file="0257-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0257-01"/> - </figure> - </div> - <p> - <s xml:id="s3101" xml:space="preserve">Vt autem etiam hinc aliqua vtilitas capiatur (prætermiſſis inconuenientibus vna <lb/>cum falÅ¿is Å¿uppoÅ¿itis huius) Videamus ordine Å¿cientifico vbi poterat eſſe verus lo-<lb/>cus Martis, aut vero proximus, die .29. Mai anni .1514. quem hic exempli cauÅ¿a Å¿u-<lb/>mit. </s> - <s xml:id="s3102" xml:space="preserve"><reg norm="Idque" type="simple">IdqÌ</reg> tam ad defenÅ¿ionem tabularum AlfonÅ¿i, quam ephemeridum ex eis col <lb/>lectarum. </s> - <s xml:id="s3103" xml:space="preserve">quæ quidem exactæ Å¿unt, vt quiÅ¿que peritus Å¿acile videre poterit, non au-<lb/>tem calculatæ à tam Å¿tupidis hominibus, vt à vero aberrent etiam gradibus .46. vt <lb/>hic ait Å¿e depræhendiſſe.</s> - </p> - <p> - <s xml:id="s3104" xml:space="preserve">Primum igitur Å¿upponemus eoÅ¿dem illos terminos, quos ipÅ¿e nec d@bet, nec po <lb/>teÅ¿t negare, præter ea quæ Å¿upra Å¿uppoÅ¿ita Å¿unt, nempe quod Å¿emidiameter epicycli <lb/>Å¿it partium .39. minu .30. & eccentricitas partium .6. talium qualium eÅ¿t Å¿emidiame-<lb/>ter deferentis diuiÅ¿us in .60. & quod dicto tempore aux eccentrici Martis eſſet cir-<lb/>ca minutum .5. grad .16. Leonis, Å¿cilicet graduum .135. min .5. & quod linea motus <lb/>me<unsure/>diocris eſſer circa minu .30. gradus .22. </s> - <s xml:id="s3105" xml:space="preserve">Capricorni, & quod verum <reg norm="centrum" type="context">centrÅ©</reg> Mar <lb/>tis eſſet grad .151 minut .20. & quod argumentum verum eſſet grad .149. minu .39. <lb/><reg norm="atque" type="simple">atq;</reg> ita oÅ¿tendam, neque tabulas, neque ephemerides errare, ne quidem vno gra-<lb/>du, ac ne quidem multis minutis, non modò tam monÅ¿truoÅ¿a differentia, vt ipÅ¿e <lb/>ait.</s> - </p> - <p> - <s xml:id="s3106" xml:space="preserve">Quare primum nobis Å¿cientificè inueniendum eÅ¿t, quanta eſſet diÅ¿tantia <var>.o.c.</var> - <pb o="246" rhead="IO. BAPT. BENED." n="258" file="0258" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0258"/> - præciÅ¿e ideÅ¿t interuallum inter centrum mundi, & centrum epicycli Martis in huiuÅ¿-<lb/>modi Å¿itu.</s> - </p> - <p> - <s xml:id="s3107" xml:space="preserve">Fingemus igitur eccenticum Martis Å¿ignificatum per <var>.p.c.m.</var> cuius centrum Å¿it <var>.r.</var> <lb/>& lineam augis <var>.p.r.o.m.</var> in qua <reg norm="centrum" type="context">centrÅ©</reg> mundi Å¿it <var>.o.</var> centrum autem verum epicycli, <lb/>comprehendatur ab angulo <var>.p.o.c.</var> qui Å¿it graduum .151. min .30. Å¿ecundum Å¿uppoÅ¿i-<lb/>tum. </s> - <s xml:id="s3108" xml:space="preserve">Quare in puncto <var>.c.</var> erit centrum epicycli. </s> - <s xml:id="s3109" xml:space="preserve">Imaginemur ergo <var>.c.o.</var> productam à <lb/>parte <var>.o.</var> quouÅ¿que ab <var>.r.</var> centro deferentis veniat linea <var>.r.k.</var> perpendiculariter, faciens <lb/>angulum rectum in puncto. k & quoniam angulus <var>.r.o.c.</var> datur nobis graduum .151. <lb/>min .30. ideo cognoÅ¿cemus angulum <var>.r.o.k.</var> tanquam reliquum ex duobus rectis, qui <lb/>erit gra .28. min .30. & Å¿imiliter angu-<lb/> - <anchor type="figure" xlink:label="fig-0258-01a" xlink:href="fig-0258-01"/> - lum <var>.o.r.k.</var> tanquam reÅ¿iduum vnius <lb/>recti, qui erit gra .61. min .30. cuius Å¿i-<lb/>nus ideÅ¿t <var>.o.k.</var> erit partium .8788 1. et <var>.k.<lb/>r.</var> vt Å¿inus anguli <var>.r.o.k.</var> partium .47715 <lb/>talium qualium <var>.o.r.</var> eſſet 100000. Å¿ed <lb/>vt <var>.o.r.</var> eÅ¿t .6. latus <var>.o.k.</var> erit .5. & min .16 <lb/>et <var>.r.k.</var> partium .2. min .52. & quia <var>.r.c.</var> <lb/>cÅ¿t <reg norm="partium" type="context">partiÅ©</reg> 60. eiuÅ¿modi, Å¿i ab eius qua-<lb/>drato Å¿ubtractum fuerit quadratum ip <lb/>Å¿ius <var>.r.k.</var> reliquum erit nobis <reg norm="quadratum" type="context">quadratÅ©</reg> <lb/>ipÅ¿ius <var>.k.c.</var> cuius radix, ideÅ¿t <var>.k.</var> erit par-<lb/>tium .59. min .56. à qua <var>.c.k.</var> Å¿ubtrahen-<lb/>do poÅ¿tea <var>.k.o.</var> partium .5. minu .16. re-<lb/>manebit <var>.o.c.</var> partium .54. min .40. pro <lb/>diÅ¿tantia quæſita.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0258-01" xlink:href="fig-0258-01a"> - <image file="0258-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0258-01"/> - </figure> - </div> - <p> - <s xml:id="s3110" xml:space="preserve">Fingamus poÅ¿tea epicyclum <var>.f.n.g.</var> <lb/>in quo argumentum verum graduum <num value="149">.<lb/>149.</num> minu .39. Å¿it arcus <var>.f.n.</var> vbi Mars inueniatur in <var>.n.</var> per quem punctum tranÅ¿eat li-<lb/>nea <var>.o.n.</var> veri motus Martis. </s> - <s xml:id="s3111" xml:space="preserve">Deinde inueniamus angulum <var>.c.o.n.</var> æquationis <reg norm="argumem" type="context">argumẽ</reg> <lb/>ti, modo iam dicto, ideÅ¿t ducendo Å¿inum <var>.n.h.</var> arcus <var>.n.g.</var> qui arcus tanquam reliquus <lb/>argumenti veri, iam præſuppoÅ¿iti, ex dimidio circulo, erit graduum 30. minu .21. & <lb/><var>n.h.</var> eius Å¿inus partium .50528. Å¿inus Å¿imiliter anguli <var>.n.c.h.</var> et <var>.c.h.</var> tanquam Å¿inus an-<lb/>guli <var>.c.n.h.</var> reÅ¿tantis ex uno recto grad .59. minu .39. erit partium .86295. <reg norm="talium" type="context">taliÅ©</reg> qua-<lb/>lium <var>.c.n.</var> Å¿inus totus eſſet partium .100000. Å¿ed vt partium .39. & min .30. Å¿inus <var>.c.h.</var> <lb/>erit partium .34. min .5. et <var>.n.h.</var> partium .19. mi .57. reliquum poÅ¿tea <var>.h.o.</var> ex <var>.o.c.</var> par-<lb/>tium .20. min .35. quia iam Å¿upra inuenimus <var>.o.c.</var> eſſe partium eiuÅ¿modi .54. minu .40. <lb/>vnde <var>.o.n.</var> vt radix quadrata Å¿ummæ duorum <var>.n.h.</var> et <var>.h.o.</var> erit partium .28. minu .41. <lb/>talium qualium <var>.n.h.</var> inuenta fuit partium .19. min .57. quæ <var>.n.h.</var> erit poÅ¿tea partium, <lb/>69552. talium qualium <var>.n.o.</var> partium .100000. & Å¿umpta dicta <var>.n.h.</var> vt Å¿inus dictarum <lb/>partium, dabit nobis angulum <var>.n.o.h.</var> quæſitum gra .44. min .4. qui per tabulas Alfon <lb/>Å¿i inuentus eÅ¿t gra .44. min .2. par huic, vt dici poteÅ¿t. </s> - <s xml:id="s3112" xml:space="preserve">Quiangulus gra .44. minu .4. <lb/>collectus cum angulo veri centri iam Å¿uppoÅ¿ito graduum .151. minu .20. & cum an-<lb/>gulo augis eccentrici Martis, Å¿imiliter Å¿uppoÅ¿itæ grad .135. min .5. dabit nobis Å¿um-<lb/>mam veræ diÅ¿tantiæ Martis à principio Arietis grad .330. min .29. quod aliud non <lb/>Å¿ignificat, niÅ¿i quod Mars inuenietur in minu .29. primi gradus PiÅ¿cium. </s> - <s xml:id="s3113" xml:space="preserve">Et Stofle-<lb/>rus in Å¿uis ephemeridibus ponit eum in .22. minuto dicti primi gradus, cuius diffe- - <pb o="247" rhead="EPISTOL AE." n="259" file="0259" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0259"/> - rentia à tabulis <lb/>eÅ¿t minut .5. tan <lb/> - <anchor type="figure" xlink:label="fig-0259-01a" xlink:href="fig-0259-01"/> - <reg norm="tum" type="context">tÅ©</reg>, & à me<unsure/>o cal-<lb/>culo min .7. vide <lb/>licet minima.</s> - </p> - <div type="float" level="5" n="4"> - <figure xlink:label="fig-0259-01" xlink:href="fig-0259-01a"> - <image file="0259-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0259-01"/> - </figure> - </div> - <p> - <s xml:id="s3114" xml:space="preserve">Nunc autem <lb/>nolui Å¿umere ip-<lb/>Å¿um angulum æ-<lb/>quationis à tabu <lb/>lis propter duas <lb/>rationes, <reg norm="primum" type="context">primÅ©</reg> <lb/>quia ne hic qui-<lb/>dem repræhen-<lb/>for in hoc voluit <lb/>credere dictis ta <lb/>bulis. </s> - <s xml:id="s3115" xml:space="preserve">Sed id vo-<lb/>luit videre pro-<lb/>prijs oculis <reg norm="in" type="wordlist">Ä©</reg> Å¿ua <lb/>theorica Martis. <lb/></s> - <s xml:id="s3116" xml:space="preserve">Vbi <reg norm="inuenit" type="wordlist">Ä©uenit</reg> quod <lb/>linea <var>.o.n.</var> tranÅ¿it <lb/>per gra .16. </s> - <s xml:id="s3117" xml:space="preserve">Arie <lb/>tis. </s> - <s xml:id="s3118" xml:space="preserve">Secunda ra-<lb/>tio eÅ¿t, vt videa-<lb/>tur quod dictæ tabulæ rectè Å¿upputatæ Å¿unt, Å¿uper dictis Å¿uppoÅ¿itis.</s> - </p> - <p> - <s xml:id="s3119" xml:space="preserve">Sed vt videat quantus Å¿it medius motus Martis die .29. </s> - <s xml:id="s3120" xml:space="preserve">Mai colligit fruÅ¿tatim, <reg norm="quod" type="simple">ê™</reg> <lb/>eleganter colligere poterat, vna opera in columellis ipÅ¿ius medij motus eiuÅ¿modi <lb/>Å¿tellæ per eram eiuÅ¿dem temporis, quæ erat .2. primarum Å¿exagenarum .33. Å¿ecunda <lb/>rum .32. tertiarum, et .52. quartarum.</s> - </p> - <p> - <s xml:id="s3121" xml:space="preserve">Primum deinde Å¿uppoÅ¿itum quod Å¿cribit, Å¿cilicet, quod diameter epicycli Å¿um-<lb/>ptus in longitudine media Å¿it Å¿ignorum .2. & grad .19. vti Å¿uperfluum eÅ¿t, ita etiam fal <lb/>Å¿um, nam eiuÅ¿modi diameter in dicto loco, non occupat ad centrum mundi plus <reg norm="quam" type="context">quã</reg> <lb/>gra .66. min .28. ideÅ¿t Å¿igna .2. gra .6. min .28. quia proportio <var>.o.c.</var> ad Å¿emidiametrum <lb/>epicycli in eiuÅ¿modi loco eÅ¿t ut partium .60. minu .18. ad partes .39. min .30. quÄ™ duÄ™ <lb/>lineæ intellectÄ™, vt latera vnius trianguli rectanguli, habebunt pro baÅ¿i aliam lineam <lb/>partium Å¿imilium .72. mi .5. </s> - <s xml:id="s3122" xml:space="preserve">Quæ intellecta vt Å¿inus totus dabit Å¿emidiametrum epi-<lb/>cycli partium .54798. tanquam Å¿inum Å¿ubiectum angulo gra .33. min .14. pro medie <lb/>tate illius, quod quæritur.</s> - </p> - <p> - <s xml:id="s3123" xml:space="preserve">Nec prætermittenda mihi videtur ratio, qua credere poſſumus, hunc cogi-<lb/>taſſe, quod diameter epicycli compleat Å¿patium duorum Å¿ignorum cum gradibus. <lb/>19 quæ quidem ratio alia eſſe non poteÅ¿t, niÅ¿i quod cum iÅ¿te inuenerit, in <reg norm="comentarijs" type="context">comẽtarijs</reg> <lb/>Theoricarum, Å¿emidiametr um huiuÅ¿modi epicycli eſſe partium .39 min .30. talium <lb/>qualium Å¿unt .60. illæ quæ Å¿unt Å¿emidiametri huius eccentrici, dictas igitur partes .39 <lb/>mi .30. hic putauit eſſe gradus Zodiaci, & propterea dixit <reg norm="diametrum" type="context">diametrÅ©</reg> huiuÅ¿modi epicy <lb/>cli eſſe <reg norm="Å¿ignorum" type="context">Å¿ignorÅ©</reg> <reg norm="duorum" type="context">duorÅ©</reg>, & gra .19. qui numerus .79. duplus eÅ¿t numero .39. <reg norm="cum" type="context">cÅ©</reg> dimidio <lb/>hoc autem dixit accidere in longitudinibus medijs, quia Å¿i hic intellexiſſet de pro-<lb/>portione horum duorum diametrorum, quæ eÅ¿t ut .120. ad .79. non Å¿pecificaſſet lo- - <pb o="248" rhead="IO. BAPT. BENED." n="260" file="0260" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0260"/> - cum epicycli, cum ipÅ¿a proportio nullo modo alteratur exiÅ¿tente epicyclo vbi volue <lb/>ris ipÅ¿ius circunferentiæ eccentrici, Å¿ed angulus in centro mundi, cui Å¿ubiacet dictus <lb/>diameter epicycli, bene alteratur, propter inæqualem diÅ¿tantiam centri epicycli <lb/>ab ipÅ¿o centro mundi. </s> - <s xml:id="s3124" xml:space="preserve">At Å¿i de tali angulo inferre voluiſſet, iam probaui ipÅ¿um <reg norm="con- tinere" type="context">cõ-<lb/>tinere</reg> Å¿olum gra .66. minu .28. exiÅ¿tente centro epicycli in longitudinibus medijs & <lb/>non gra .79. vt ipÅ¿e dicit.</s> - </p> - <p> - <s xml:id="s3125" xml:space="preserve">Omitto poÅ¿tea, quod vbi mentionem facit coniunctionum Solis cum Marte au-<lb/>gium & earum oppoÅ¿itorum, non explicat an intelligat de veris an de medijs. </s> - <s xml:id="s3126" xml:space="preserve">Nam <lb/>Å¿i ex eius modo loquendi accipiatur eum loqui de veris multum erraret.</s> - </p> - <p> - <s xml:id="s3127" xml:space="preserve">Sed quia iam tibi moleÅ¿tum eſſe inciperet Å¿i diutius te detinerem in his conten-<lb/>tionibus aÅ¿tronomicis, vlterius non diÅ¿putabo. </s> - <s xml:id="s3128" xml:space="preserve">Satis enim hactenus explicaui <reg norm="Å¿enten- tiam" type="context">Å¿entẽ-<lb/>tiam</reg> meam, vt oÅ¿tendiſſe videor quam mihi iucundum Å¿it tibi <reg norm="morem" type="context">morẽ</reg> gerere. </s> - <s xml:id="s3129" xml:space="preserve">In quo <lb/>etiam hnmanitati tuæ gratiam habebo, <reg norm="quum" type="context">quÅ©</reg> petitione tua occaÅ¿ionem mihi de deris <lb/>efficiendi, vt tum amici tui (amant enim te omnia Å¿ublimia ingenia) tum alij, Å¿i <reg norm="quam" type="context">quã</reg> <lb/>falÅ¿am opinionem ex huius Benedicti Altæuillæ Å¿criptis Å¿umpÅ¿iſſent, <reg norm="eam" type="context">eã</reg> relinquant, <lb/>& per te hoc beneficium à me conÅ¿equantur, & huiuÅ¿modi occaÅ¿ionem, & iuuandi <lb/>hominum Å¿tudia & tibi gratum faciendi, honorificum, & per gratum mihi fuiſſe in-<lb/>telligant. </s> - <s xml:id="s3130" xml:space="preserve">Vale & me vt Å¿oles ama.</s> - </p> - <p> - <s xml:id="s3131" xml:space="preserve">Taurini pridie Kal. </s> - <s xml:id="s3132" xml:space="preserve">Octobris .1581.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head374" style="it" xml:space="preserve">De probatione diuiÅ¿ionis numerorum.</head> - <head xml:id="head375" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3133" xml:space="preserve">INter alia quæ à me Å¿cire cupis, vir doctiſſime, hoc vnum eÅ¿t, vt ex literis tuis ac-<lb/>cepi, vnde Å¿it vt priÅ¿ci noÅ¿tri probatione numeri nouenarij potius quam Å¿epte-<lb/>narij vÅ¿i fuerint, & qua ratione non idem proueniat ex probatione numerorum <lb/>octonarij, Å¿enarij, vel quinarij, aut cuiuslibet alterius: </s> - <s xml:id="s3134" xml:space="preserve">Vnde pariter oriatur quod in <lb/>partitionis probatione neceſſum Å¿it probationum euentus multiplicare cum proba-<lb/>tione diuiÅ¿oris, ac eam quæ eÅ¿t producti poÅ¿tea cum probatione fractionis in Å¿um-<lb/>mam colligere, & c. </s> - <s xml:id="s3135" xml:space="preserve">Ad hæc in primis reÅ¿pondeo, cum aliquoties accidere poſſit ta <lb/>les probationes nos fallere poſſe, <reg norm="idque" type="simple">idqÌ;</reg> fi in tali Å¿umma Å¿imilis numerus, ut puta Å¿e-<lb/>ptem, aut nouem, plus vel minus æquo iuÅ¿touè poÅ¿itus fuerit, attamen per raro eueni <lb/>re poteÅ¿t, vt quis per nouenarium potius quam per Å¿eptenarium decipiatur. </s> - <s xml:id="s3136" xml:space="preserve">Exem-<lb/>pli gratia, ponamus Å¿ummam eſſe .100. quam numerus nouenarius vndecies Å¿olum <lb/>ingreditur, at Å¿eptenarius quatuordecies, vnde quis <reg norm="ſępius" type="simple">ſępiê°</reg> ex Å¿eptenario, hacratione, <lb/>quam ex nouenario numero Å¿e poſſe errare facile depræhendet, efÅ¿i ex probatione <lb/>nouenarij magis quam Å¿eptenarij, vt practici Å¿cribunt, duabus de cauÅ¿is errare poſſi-<lb/>mus. </s> - <s xml:id="s3137" xml:space="preserve">Alia tamen ratio mihi Å¿uppetit, ob quam credibile eÅ¿t ipÅ¿os potius nouena-<lb/>rio adiutos fuiſſe, quam Å¿eptenario, quæ eÅ¿t ob Å¿ui cum velocitatem tum <reg norm="facilitatem" type="context">facilitatẽ</reg>, <lb/><reg norm="neque" type="simple">neq;</reg> enim in Å¿eptenario eÅ¿t adeo facilis. </s> - <s xml:id="s3138" xml:space="preserve"><reg norm="Nam" type="context">Nã</reg> quamuis, tam vna quam altera aliud <lb/>non Å¿it, quam numerorum ordines diuidere (Å¿i de Å¿ummis primo loquamur) aut è <lb/>Å¿umma Å¿uperÅ¿luum ordinum colligere, & videre an idemmet Å¿uperfluum ex eadem <lb/>Å¿umma emanet, attamen cum modus, qui in hoc adhiberi poteÅ¿t in nouenario <reg norm="quam" type="context">quã</reg> <lb/>in Å¿eptenario velocior Å¿it, & ob id probationem nouenarij Å¿eligunt potius quam <lb/>Å¿eptenarij.</s> - </p> - <pb o="249" rhead="EPISTOLAE." n="261" file="0261" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0261"/> - <p> - <s xml:id="s3139" xml:space="preserve">Verum nolo te in ea, quæfalÅ¿a eÅ¿t, opinione conÅ¿iÅ¿tere, nonidem, & cum octona-<lb/>rio, Å¿enario, vel quinario, aut quouis alio numero poſſe efficere, cum eademmet ra <lb/>tio, quæ in Å¿eptenario, aut nouenario, <reg norm="ent" type="context">ẽt</reg> in cæteris perhibeatur. </s> - <s xml:id="s3140" xml:space="preserve">Ponamus <reg norm="exemplum" type="context">exemplÅ©</reg> <lb/>hos tres or dinum numeros velle Å¿upputare, quorum primus Å¿it .679. Å¿ecundus .846. <lb/>& tertius .935. & illorum <reg norm="Å¿ummam" type="context">Å¿ummã</reg> .2460. nunc maiorem numerum primi ordinis ab <lb/>octonario menÅ¿i, proijciendo, remanebit .7. deinde maiorem numerum demendo à <lb/>Å¿ecundo or dine, reÅ¿iduum erit .6. ac Å¿i idem in tertio ordine fecerimus, erit nobis re-<lb/>liquum .7. </s> - <s xml:id="s3141" xml:space="preserve">Demum tria hæc reÅ¿idua in vnum collecta .20. efficient, à quibus Å¿i nume <lb/>rum maiorem ab octonario menÅ¿um dempÅ¿eris, Å¿upererunt .4. & totidem à ſumma <num value="2460">.<lb/>2460.</num> remanebunt, reiecto maiori numero ab octonario menÅ¿o. </s> - <s xml:id="s3142" xml:space="preserve">Atque idem me-<lb/>dio quouis alio numero, euenire poteÅ¿t.</s> - </p> - <p> - <s xml:id="s3143" xml:space="preserve">Cuius ratio tam perÅ¿e clara atque euidens eÅ¿t, quod Å¿i Å¿ummam trium <reg norm="reliquorum" type="context">reliquorÅ©</reg>, <lb/>quæ eÅ¿t .20. à ſumma .2460. Å¿ubduxeris, remanebunt .2440. pro Å¿umma trium nume <lb/>rorum dictorum trium ordinum ab octonario menÅ¿orum, cui numero addito .16. pro <lb/>maiori numero Å¿ummÄ™ <reg norm="reliquorum" type="context">reliquorÅ©</reg>, qui ab octonario menÅ¿us Å¿it, Å¿upererunt .4. </s> - <s xml:id="s3144" xml:space="preserve">At Å¿i per <lb/><reg norm="Å¿enarium" type="context">Å¿enariÅ©</reg> <reg norm="experimentum" type="context context">experimẽtÅ©</reg> feceris, remanebit <var>.o.</var> & Å¿ic de reliquis per ordinem <reg norm="procedendo" type="context">procedẽdo</reg>.</s> - </p> - <p> - <s xml:id="s3145" xml:space="preserve">Verum poſſes Å¿ciÅ¿citari, quare velocius, exceſſus ordinum, potius per <reg norm="noue- narium" type="context">noue-<lb/>nariÅ©</reg>, quam per cæteros numeros, prout <reg norm="docent" type="context">docẽt</reg> practici, inueniri queat, videlicet ag <lb/>gregando prius duas figuras numerorum primæ Å¿ummæ, deinde alias duas. </s> - <s xml:id="s3146" xml:space="preserve">Exem-<lb/>plum Å¿it primus ordo .679. colligendo .6. et .7. faciunt 13. & cum hæc Å¿umma Å¿it dua <lb/>rum figurarum, Å¿upputantur & ipſæ, è quibus prodeunt .4. & conÅ¿imilis erit proba-<lb/>tio numeri .67. facta per .9. quod idem eÅ¿t, ac Å¿i quis diuidat .67. per .9. ex quo reli-<lb/>qui erunt Å¿emper .4.</s> - </p> - <p> - <s xml:id="s3147" xml:space="preserve">At quo ratio huiuÅ¿ce perÅ¿picuè dignoÅ¿ci poſſit, in primis Å¿ciendum eÅ¿t, cuique <lb/>ex Å¿e cognitum, atque exploratum eſſe, denarium numerum vnitate nouenarium Å¿u <lb/>perare, & ex hoc Å¿equitur, Å¿ex denarios continere in Å¿e Å¿ex nouenarios, & Å¿ex vni-<lb/>tates.</s> - </p> - <p> - <s xml:id="s3148" xml:space="preserve">At Å¿ex vnitates, vna cum .7. faciunt .13. & quia in .13. eÅ¿t denarius, igitur in illo erit <lb/>vnitas Å¿upra .9. </s> - <s xml:id="s3149" xml:space="preserve">Quæ vnitas addita ternario, præbet nobis Å¿uperfluum, per quod .67. <lb/>Å¿uperat .54. iunctum cum .9. Å¿cilicet Å¿ummam .63.</s> - </p> - <p> - <s xml:id="s3150" xml:space="preserve">Idem dicinon poteÅ¿t de octonario, Å¿eptenario, vel Å¿enario, & de reliquis, quo-<lb/>niam numerus denariorum, in cæteris minoribus nouenario non præbet illico nu-<lb/>merum exceſſus maioris numeri, qui à numero probationis menÅ¿us eÅ¿t. </s> - <s xml:id="s3151" xml:space="preserve">Et quod di <lb/>co de probatione aggregationis, idem intelligo de alijs operationibus, vt puta Å¿ub-<lb/>tractionis, multiplicationis, & partitionis Å¿eu diuiÅ¿ionis.</s> - </p> - <p> - <s xml:id="s3152" xml:space="preserve">Vnde autem oriatur, vt in partitionis probatione opus Å¿it probationem euentus <lb/>cum diuiÅ¿ionis probatione multiplicare, & productum cum fractionis probatione <lb/>Å¿upputare, Å¿eu aggregare, tibi non erit ignotum, quoties animaduerteris, quod <lb/>productum ipÅ¿ius euentus cum diuiÅ¿ore, adiunctum fractioni, perpetuo Å¿e æquat nu <lb/>mero diuiÅ¿ibili. </s> - <s xml:id="s3153" xml:space="preserve">Et quoniam numeri probationum Å¿unt partes, quæ remanent ex <lb/>ipÅ¿is totis, detractis maioribus numeris ab eo dimenÅ¿is, quo pro communi men-<lb/>Å¿ura vtimur (prout .7. vel .9. aut alium numerum, quem voluerimus) par eÅ¿t vt ex ip-<lb/>Å¿arum remanentibus partibus, velut ex ipÅ¿is totis idem fiat.</s> - </p> - <pb o="250" rhead="IO. BAPT. BENED." n="262" file="0262" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0262"/> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head376" style="it" xml:space="preserve">De falacia operationis triangulorum Å¿phericorum.</head> - <head xml:id="head377" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3154" xml:space="preserve">QVod diebus præteris tibi Å¿ignificaui, idem nunc confirmo, Å¿cilicet Å¿phærico-<lb/>rum triangulorum operationem ſæpe nos fallere, vt exempli gratia, Å¿i pro <lb/>poÅ¿itus nobis fuiſſet triangulus <var>.A.B.C.</var> cuius angulus <var>.A.</var> nobis datus eſſet graduum <num value="114">.<lb/>114.</num> mi <var>.o.</var> & eius latus <var>.A.</var> B, graduum .67. min .5. & latus <var>.A.C.</var> graduum .45. mi .10. <lb/>Å¿i reliquos angulos cum tertio latere etiam cognoÅ¿cere voluerimus, ex methodo .11 <lb/>primi Copernici propoÅ¿itum obtinebimus. </s> - <s xml:id="s3155" xml:space="preserve">vnde latus <var>.B.C.</var> eſſet graduum .89. min <num value="30">.<lb/>30.</num> angulus vero <var>.C.</var> graduum .57. min .14. angulus autem <var>.B.</var> grad .48. min .38. </s> - <s xml:id="s3156" xml:space="preserve">Qua-<lb/>re vltimus hic angulus <var>.B.</var> falÅ¿us eſſet, eo quod operatio paruorum triangulorum in <lb/>cauÅ¿a eÅ¿t, quotieÅ¿cunque eorum latera tam breuia Å¿int, ut non eccedant vnum gra-<lb/>dum, quare ipÅ¿orum angulorum veram quantitatem non tribuunt. </s> - <s xml:id="s3157" xml:space="preserve">propterea igitur <lb/>cum voluerimus veram <reg norm="quantitatem" type="context">quãtitatem</reg> ipÅ¿ius anguli <var>.B.</var> oportet poÅ¿t quam inuenerimus <lb/>angulum <var>.C.</var> mediante arcu <var>.D.E.</var> Å¿upponere alium polum in <var>.B.</var> deinde producere. </s> - <s xml:id="s3158" xml:space="preserve">B <lb/>A. vÅ¿que ad <var>.d.</var> et <var>.B.C.</var> vÅ¿que ad <var>.e.</var> imaginando <var>.B.d.</var> et <var>.B.e.</var> duas quartas eſſe magno-<lb/>rum circulorum, extendendo poÅ¿tea <var>.d.e.</var> vÅ¿que ad interÅ¿ectionem cum <var>.A.C.</var> & <reg norm="eum" type="context">eÅ©</reg> <lb/>dem ordinem proÅ¿equendo, </s> - <s xml:id="s3159" xml:space="preserve">tunc <var>.e.d.</var> nobis oÅ¿tendet angulum <var>.B.</var> eſſe gra .40. mi .22 <lb/>quæ erit eius vera quantitas. </s> - <s xml:id="s3160" xml:space="preserve">Cuius quidem rei experientiam poſſumus etiam fa-<lb/>cere, hoc modo, eÅ¿to, exempli gratia, quod nobis datus Å¿it angulus <var>.C.</var> graduum .57. <lb/>min .14. cum latere <var>.A.C.</var> gra .45. min .10. & latus <var>.B.C.</var> gra .89. min .30. </s> - <s xml:id="s3161" xml:space="preserve">Tunc Å¿i ordi-<lb/>nem .11. dicti lib. Å¿e quemur, obtinebimus intentum, hoc modo Å¿cilicet Å¿upponendo <lb/>in <var>.A.</var> polum, & non in <var>.B.</var> ducendo etiam <var>.A.B.</var> et <var>.A.C.</var> Å¿ed <var>.A.B.</var> <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> ad gra .90. du-<lb/>cendo poÅ¿tea <var>.D.E.</var> ita quod ab omni parte concurrat cum latere <var>.B.C.</var> producto, vn <lb/>de tam <var>.f.C.B.F.</var> quam <var>.f.D.E.F.</var> erunt Å¿emicirculi magnorum circulorum. </s> - <s xml:id="s3162" xml:space="preserve">quare <var>.C.<lb/>D.</var> nobis cognitus erit gra .44. min .50. & Å¿ic etiam angulus <var>.D.C.f.</var> gra .57. min .14. ex <lb/>4. dicti lib. poÅ¿tea habebimus <var>.F.l.</var> gra .60. min .54. & angulum <var>.f.</var> gra .53. mi .24. aggre <lb/>gatum poÅ¿tea <var>.f.C.</var> cum <var>.C.B.</var> habebimus <var>.f.B.</var> gra .150. min .24. qui Å¿i a Å¿emicirculo <reg norm="dem" type="context">dẽ</reg> <lb/>ptus fuerit, nobis remanebit <var>.B.F.</var> gra .29. mi .36. cum angulo <var>.F.</var> cognito <reg norm="cum" type="context">cÅ©</reg> Å¿it æqua-<lb/> - <anchor type="figure" xlink:label="fig-0262-01a" xlink:href="fig-0262-01"/> - <anchor type="figure" xlink:label="fig-0262-02a" xlink:href="fig-0262-02"/> - lis <var>.f.</var> eius oppoÅ¿ito. </s> - <s xml:id="s3163" xml:space="preserve">Vnde ex dicta .4. co-<lb/>gnoÅ¿cemus angulum <var>.B.</var> gra .40. min .31. <lb/>qui ferè æ qualis eÅ¿t Å¿uperiori iam inuen-<lb/>to, nec ab ipÅ¿o differt niÅ¿i per min .9. quæ <lb/>quidem differentia parua eÅ¿t reÅ¿pectu al - <pb o="251" rhead="EPISTOLAE." n="263" file="0263" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0263"/> - terius differentiæ quam Å¿upra inuenerimus.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0262-01" xlink:href="fig-0262-01a"> - <image file="0262-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0262-01"/> - </figure> - <figure xlink:label="fig-0262-02" xlink:href="fig-0262-02a"> - <image file="0262-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0262-02"/> - </figure> - </div> - <p> - <s xml:id="s3164" xml:space="preserve">Superius enim dixinon eſſe ponendum polum in <var>.B.</var> eo quod <var>.B.C.</var> Å¿it gra .89. mi <num value="30">.<lb/>30.</num> vnde nobis prodijſſet triangulus <var>.f.C.D.</var> trium valde paruorum laterum, quorum <lb/>latus <var>.C.D.</var> eſſet gra <var>.o.</var> mi .30. & latus <var>.f.l.</var> gra <var>.o.</var> mi .55. & latus <var>.F.D.</var> gra <var>.o.</var> mi .47. vn-<lb/>de angulus <var>.f.</var> gra .32. min .40. falÅ¿us eſſet, qui <reg norm="quidem" type="context">quidẽ</reg> poÅ¿tea nobis daret <var>.D.E.</var> gra .45 <lb/>minu .16. falÅ¿um Å¿imiliter.</s> - </p> - </div> - <div type="letter" level="4" n="4"> - <head xml:id="head378" style="it" xml:space="preserve">De paßione circuli bactenus incognita.</head> - <head xml:id="head379" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3165" xml:space="preserve">DVbitandum quidem <reg norm="non" type="context">nõ</reg> eÅ¿t quin paſſiones circuli innumerabiles penè Å¿int, quÄ™ <lb/>quidem omnes ferè caÅ¿u inueniuntur, vt mihi nunc accidit, quam tibi mitto, <lb/>hæc autem eÅ¿t, quòd quadratum lineæ <var>.a.g.</var> in figura hic Å¿ubÅ¿cripta Å¿emper æquale <lb/>eÅ¿t ei producto, quod fit ex <var>.a.e.</var> in diametro circuli <var>.g.c.b.</var> Å¿imul Å¿umpto cum quadra <lb/>to inÅ¿criptibili in dicto circulo, & Å¿imul cum quadrato lineæ <var>.a.b.</var> <reg norm="contingentis" type="context">contingẽtis</reg> ipÅ¿um <lb/>circulum, Å¿upponendo <var>.a.g.</var> per centrum ipÅ¿ius circuli tranÅ¿ire.</s> - </p> - <p> - <s xml:id="s3166" xml:space="preserve">Pro cuius demonÅ¿tratione à centro <var>.e.</var> duco Å¿emidiametrum <var>.e.c.</var> <reg norm="perpendicularem" type="context">perpendicularẽ</reg> <lb/>ipÅ¿i <var>.g.a.</var> & à puncto <var>.c.</var> ad <var>.a.</var> duco <var>.c.a.</var> quæ Å¿ecabit circunferentiam ipÅ¿ius circuli in <reg norm="pum" type="context">pÅ©</reg> <lb/>cto <var>.d.</var> eo, quod angulus <var>.c.</var> acutus eÅ¿t. </s> - <s xml:id="s3167" xml:space="preserve">Nunc ex .35. tertij, productum <var>.c.a.</var> in <var>.a.d.</var> æqua <lb/>le eÅ¿t quadrato <var>.a.b.</var> productum autem <var>.a.c.</var> in <var>.d.c.</var> æquale eÅ¿t quadrato inÅ¿criptibili in <lb/>circulo <var>.g.c.b.</var> ex .130. primi Vitellionis, <reg norm="in" type="wordlist">Ä©</reg> qua propoÅ¿itione ipÅ¿e Vitellio Å¿upplet pro <lb/>eo, quod in quinta propoÅ¿itione libri de lineis Å¿pirabilibus Archimedis deÅ¿ideratur, <lb/>Å¿ed quadratum <var>.a.c.</var> æquale eÅ¿t ijs duobus productis. per .2. Å¿ecundi Eucli. ergo qua-<lb/>dratum <var>.a.c.</var> æquale erit quadrato inÅ¿criptibili in circulo <var>.d.c.g.</var> & quadrato <var>.a.b.</var> Å¿ed <lb/>quadratum lineæ <var>.a.c.</var> æquale eÅ¿t duobus quadratis, hoc eÅ¿t lineæ <var>.a.e.</var> & lineæ <var>.e.c.</var> ex <lb/>pitagorica, </s> - <s xml:id="s3168" xml:space="preserve">quare ex communi conceptu duo quadrata lineæ <var>.a.e.</var> & lineÄ™ <var>.e.c.</var> hoc eÅ¿t <lb/>lineæ <var>.e.g.</var> quod idem eÅ¿t, æqualia erunt duobus iam dictis, hoc eÅ¿t inÅ¿criptibili, <lb/>& ei, quod fit ex <var>.a.b.</var> Å¿ed quadratum lineæ <var>.a.g.</var> æquale eÅ¿t quadrato lineæ <var>.a.e.</var> & qua <lb/>drato quod fit ex <var>.e.g.</var> & duplo illius quod fit ex <var>.a.e.</var> in <var>.e.g.</var> hoc eÅ¿t producto <var>.a.e.</var> in <lb/>diametrum. </s> - <s xml:id="s3169" xml:space="preserve">Quare quadratum lineæ <var>.a.g.</var> æquale eÅ¿t quadrato circunÅ¿criptibili, & <lb/>quadrato lineæ <var>.a.b.</var> & producto lineæ <var>.a.e.</var> in diametrum circuli <var>.d.c.g</var>.</s> - </p> - <p> - <s xml:id="s3170" xml:space="preserve">Breuiori etiam methodo demonÅ¿trare poſſu <lb/> - <anchor type="figure" xlink:label="fig-0263-01a" xlink:href="fig-0263-01"/> - mus quadrata lineæ <var>.a.e.</var> et <var>.e.g.</var> æqualia eÅ¿-<lb/>Å¿e quadrato circunÅ¿criptibili, & quadrato lineæ <var>.<lb/>a.b.</var> ducendo lineam <var>.e.b.</var> quæ æqualis eÅ¿t lineæ <var>.<lb/>e.g.</var> tali methodo, hoc eÅ¿t, conÅ¿iderando, quod <lb/>quadratum inÅ¿criptibile Å¿emper duplum eÅ¿t qua <lb/>drato Å¿emidiametri, vel medietati circumÅ¿cri-<lb/>ptibili, quod quidem nihil aliud eÅ¿t, niÅ¿i æquale <lb/>eſſe ijs duobus quadratis, hoc eÅ¿t lineæ <var>.e.b.</var> & li-<lb/>neæ <var>.e.g.</var> Å¿ed quadratum lineæ <var>.a.e.</var> æquale eÅ¿t iis <lb/>duobus quadratis, hoc eÅ¿t lineæ <var>.a.b.</var> & lineæ <var>.b.e.</var> vnde quadrat um lineæ <var>.a.e.</var> cum <lb/>quadrato lineæ <var>.e.g.</var> æquale eÅ¿t quadrato circunÅ¿criptibili, Å¿imul collecto cum qua-<lb/>drato lineæ <var>.a.b</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0263-01" xlink:href="fig-0263-01a"> - <image file="0263-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0263-01"/> - </figure> - </div> - <pb o="252" rhead="IO. BAPT. BENED." n="264" file="0264" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0264"/> - </div> - <div type="letter" level="4" n="5"> - <head xml:id="head380" style="it" xml:space="preserve">Demonstrationes quarundam propoÅ¿itionum de quibus agit <lb/>Cardanus capite primo libro .16. de <lb/>Å¿ubtilitate.</head> - <head xml:id="head381" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3171" xml:space="preserve">EA quæ Cardanus in primo cap. lib. 16. de Å¿ubtilitate ita Å¿cribit, quod Å¿i diame-<lb/>tros producatur extra quantumlibet, alia verò diametro in centro Å¿ecetur ad <lb/>rectos, ex huius fine <reg norm="&c." type="unresolved">&c.</reg> quæ quidem Å¿ecundum illum eÅ¿t vndecima proprietas cir <lb/>culi, quoniam te id non intelligere Å¿cribis, <reg norm="idemque" type="simple">idemqÌ;</reg> dicis etiam de duodecima, & Å¿i-<lb/>militer de tribus illis paſſionibus, quas ipſæ communes facit circulo, defectioni, Å¿eu <lb/>ellipÅ¿i, & hyperboli, tibi breuiter reÅ¿pondebo.</s> - </p> - <p> - <s xml:id="s3172" xml:space="preserve">Circa vndecimam proprietatem circuli verum dicit. </s> - <s xml:id="s3173" xml:space="preserve">Imaginemur circulum <var>.p.<lb/>d.q.</var> à duabus diametris, inuicem ad angulos rectos coniunctis, diuiÅ¿um <var>.p.d.</var> et <var>.d.g.</var> di <lb/>uidatur enim quarta <var>.q.d.</var> per quot partes æquales volueris, mediantibus punctis <var>.b.a.<lb/>o.</var> <reg norm="ducanturque" type="simple">ducanturqÌ;</reg> ab ijÅ¿dem punctis tot perpendiculares diametro <var>.d.g.</var> quæ Å¿int <var>.b.m.a.n.</var> <lb/>et <var>.o.s.</var> quæ quidem erunt parallelæ diametro <var>.q.p.</var> coniungatur deinde extremitas <var>.d.</var> <lb/>diametri <var>.d.g.</var> cum primo puncto <var>.b.</var> & protrahatur <var>.d.b.</var> vÅ¿que ad concurÅ¿um cum diz <lb/>metro <var>.p.q.</var> protracto in puncto, h. </s> - <s xml:id="s3174" xml:space="preserve">Nunc dico <var>.q.h.</var> quæ adiacet diametro <var>.q.p.</var> æqua-<lb/>lem eſſe omnibus dictis perpendicularibus, quapropter coniungantur puncta <var>.m.a</var>: <lb/><var>n.o.</var> et <var>.s.q.</var> & producantur vÅ¿que ad adiacentem diametro <var>.q.p.</var> in punctis <var>.c.</var> et <var>.e.</var> vn <lb/>de habebimus angulos <var>.b.a.o.q.</var> inuicem æquales ex .26. tertij, cum verò <var>.o.s.a.n.</var> et <lb/><var>b.m.</var> parallelæ Å¿int ipÅ¿i <var>.p.h</var>. </s> - <s xml:id="s3175" xml:space="preserve">tunc anguli <var>.b.h.c</var>: <var>a.c.e</var>: et <var>.o.e.q.</var> æquales erunt angulis <var>.d.<lb/>b.m</var>: <var>m.a.n.</var> et <var>.n.o.s.</var> ex .29. primi: </s> - <s xml:id="s3176" xml:space="preserve">quare anguli <var>.h.c.e.q.</var> erunt inuicem æquales, vnde <lb/>ex .28. eiuÅ¿dem <var>.b.h</var>: <var>m.c</var>: <var>n.e.</var> et <var>.s.q.</var> erunt <reg norm="inuicem" type="context">inuicẽ</reg> parallelÄ™, & ex .34. <var>e.q.</var> æqualis erit <var>.<lb/>o.s.</var> et <var>.e.c.</var> æqualis <var>.n.a.</var> et <var>.m.b.</var> æqualis <var>.c.h.</var> verum eÅ¿t igitur propoÅ¿itum.</s> - </p> - <p> - <s xml:id="s3177" xml:space="preserve">Duodecima vero <reg norm="proprietas" type="simple">ê“prietas</reg> eÅ¿t, ut Å¿i fuerit circulus <var>.a.b.e.q.</var> cuius duo diametriad <lb/>rectos coniuncti Å¿int <var>.a.e.</var> et <var>.q.b.</var> & diameter <var>.a.e.</var> protractus indeterminatè ad partem <lb/>e. </s> - <s xml:id="s3178" xml:space="preserve">tunc Å¿i ab extremo <var>.b.</var> diametri <var>.q.b.</var> ducta fuerit <var>.b.n.u.</var> extra circulum, Å¿eu <var>.b.u.n.</var> in <lb/>tra circulum, vt in Å¿ubiecta figura patet, ita vt Å¿ecta Å¿it à circunferentia circuli in <reg norm="pum" type="context">pÅ©</reg> <lb/>cto <var>.n.</var> vel à diametro in puncto <var>.u.</var> Å¿emper id quod fit ex <var>.u.b.</var> in <var>.b.n.</var> æquale erit qua-<lb/>drato inÅ¿criptibili in dicto circulo, hoc autem diuerÅ¿imodè cognoÅ¿ci poteÅ¿t, tribus <lb/>enim modis ego inueni, quorum primus ita Å¿e habet. </s> - <s xml:id="s3179" xml:space="preserve">Nam Å¿i punctus <var>.u.</var> fuerit ex-<lb/>tra circulum, ducantur <var>.b.e.</var> et <var>.e.n.</var> & habebimus duos triangulos <var>.b.n.e.</var> et <var>.b.e.u.</var> Å¿imi <lb/>les inuicem, eo, quod angulus <var>.b.</var> communis ambobus exiÅ¿tit, & angulus <var>.b.n.e.</var> æqua <lb/>lis eÅ¿t angulo <var>.b.e.u.</var> quod ita probatur, nam angulus <var>.b.n.e.</var> cum angulo <var>.b.a.e.</var> (ducta <lb/>cum fuerit <var>.b.a.</var>) æquatur duobus rectis ex .21. tertij, Å¿ed ex quinta primi angulus <var>.b.<lb/>e.a.</var> Ä™qualis eÅ¿t angulo <var>.b.a.e</var>: </s> - <s xml:id="s3180" xml:space="preserve">quare angulus <var>.b.n.e.</var> cum angulo <var>.b.e.a.</var> Ä™quatur duobus <lb/>rectis, Å¿ed ex .13. eiuÅ¿dem angulus <var>.b.n.e.</var> cum angulo etiam <var>.e.n.u.</var> æquatur duobus re <lb/>ctis, ergo angulus <var>.e.n.u.</var> æquatur angulo <var>.b.e.a</var>. </s> - <s xml:id="s3181" xml:space="preserve">quare angulus <var>.b.n.e.</var> æquatur <reg norm="etiam" type="context">etiã</reg> an-<lb/>gulo <var>.b.e.u.</var> vnde ex .32. eiuÅ¿dem reliquus angulus <var>.b.u.e.</var> æqualis erit reliquo angulo <lb/><var>b.e.n.</var> latera igitur erunt proportionalia ex .4. Å¿exti, vnde ita Å¿e habebit <var>.u.b.</var> ad <var>.b.<lb/>e.</var> vt <var>.b.e.</var> ad <var>.b.n.</var> ex .16. Å¿exti igitur <reg norm="verum" type="context">verÅ©</reg> erit propoÅ¿itum.</s> - </p> - <p> - <s xml:id="s3182" xml:space="preserve">Sed Å¿i punctus <var>.u.</var> intra circulum fuerit, triangulus <var>.b.e.n.</var> Å¿imilis erit triangulo <var>.b.u.<lb/>e.</var> nam angulus <var>.b.</var> ambobus communis erit. </s> - <s xml:id="s3183" xml:space="preserve">Angulus vero <var>.b.n.e.</var> Ä™qualis eÅ¿t angulo <var>.<lb/>b.e.u.</var> ex .26. tertij, </s> - <s xml:id="s3184" xml:space="preserve">quare ex .32. primi reliquus angulus <var>.b.e.n.</var> æqualis erit reliquo - <pb o="253" rhead="EPISTOLAE." n="265" file="0265" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0265"/> - angulo <var>.b.u.e.</var> vnde ex .4. Å¿exti eadem proportio erit ipÅ¿ius <var>.b.n.</var> ad <var>.b.e.</var> quæ <var>.b.e.</var> ad <lb/><var>b.u</var>. </s> - <s xml:id="s3185" xml:space="preserve">quare ex .16. eiuÅ¿dem patebit propoÅ¿itum.</s> - </p> - <p> - <s xml:id="s3186" xml:space="preserve">Secundus autem modus ita Å¿e habet, ducta <var>.q.n.</var> habebimus duo triangula ortho-<lb/>gonia Å¿imilia inuicem <var>.b.q.n.</var> et <var>.b.u.o.</var> eo quod angulus <var>.b.</var> communis ambobus exi-<lb/>Å¿tit, </s> - <s xml:id="s3187" xml:space="preserve">quare ex .4. Å¿exti ita Å¿e habebit <var>.u.b.</var> ad <var>.b.o.</var> vt <var>.q.b.</var> ad <var>.b.n.</var> vnde ex .15. eiuÅ¿dem <lb/>quod fit ex <var>.u.b.</var> in <var>.b.n.</var> æquale erit ei, quod fit ex <var>.q.b.</var> in <var>.b.o</var>. </s> - <s xml:id="s3188" xml:space="preserve">Sed ex .16. eiuÅ¿dem, <reg norm="quod" type="simple">ê™</reg> <lb/>fit ex <var>.q.b.</var> in <var>.b.o.</var> Ä™quatur quadrato <var>.b.e.</var> quia <var>.b.e.</var> media proportionalis eÅ¿t inter dia <lb/>metrum & Å¿emidiametrum eiuÅ¿dem circuli. ex .4. eiuÅ¿dem, </s> - <s xml:id="s3189" xml:space="preserve">quare quod fit ex <var>.u.b.</var> in <lb/><var>b.n.</var> æquale erit quadrato ipÅ¿ius <var>.b.e</var>.</s> - </p> - <p> - <s xml:id="s3190" xml:space="preserve">Tertius modus adiungitur, & eÅ¿t quod cum quadratum <var>.u.b.</var> exiÅ¿tente <var>.u.</var> extra cir-<lb/>culum æquale Å¿it ei, quod Å¿it ex <var>.u.b.</var> in <var>.b.n.</var> Å¿imul Å¿umpto cum eo, <reg norm="quod" type="simple">ê™</reg> fit ex <var>.u.b.</var> in <var>.u.n.</var> <lb/>ex Å¿ecunda Å¿ecundi, & idem quadratum <var>.u.b.</var> æquale duobus quadratis <var>.u.o.</var> et <var>.o.b.</var> ex <lb/>penultima primi, ideo duo dicta producta æqualia erunt dictis duobus quadratis <var>.o.</var> <lb/> - <anchor type="figure" xlink:label="fig-0265-01a" xlink:href="fig-0265-01"/> - u. Å¿cilicet et <var>.o.b.</var> Å¿ed quadratum <lb/>o u. æquatur ei, quod fit ex <var>.a.u.</var> <lb/>in <var>.e.u.</var> & ei quod fit. ex <var>.o.e.</var> in Å¿e <lb/>ipÅ¿am ex .6. Å¿ecundi, </s> - <s xml:id="s3191" xml:space="preserve">quare duo <lb/><reg norm="iam" type="context">iã</reg> dicta producta æqualia erunt <lb/>duobus dictis quadratis, <var>o.b.</var> Å¿ci <lb/>licet. et <var>.o.e.</var> & ei quod fit ex <var>.a.<lb/>u.</var> in <var>.u.e.</var> Å¿ed quod fit ex <var>b.u.</var> in <var>.u <lb/>n.</var> æquale eÅ¿t ei quod fit ex <var>.a.u.</var> <lb/>in <var>.u.e.</var> ex .35. 3. <reg norm="relinquitur" type="simple">relinquit̃</reg> ergo <lb/>vt id <reg norm="quod" type="wordlist">qÄ‘</reg> fit ex <var>.u.b.</var> in <var>.b.n.</var> æqua-<lb/>le Å¿it <reg norm="duobus" type="simple">duobê°</reg> quadratis <var>.o.b.</var> et <var>.o.<lb/>e</var>. </s> - <s xml:id="s3192" xml:space="preserve">quare & quadrato ipÅ¿ius <var>.b.e.</var> <lb/>ex Pitagorica.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0265-01" xlink:href="fig-0265-01a"> - <image file="0265-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0265-01"/> - </figure> - </div> - <p> - <s xml:id="s3193" xml:space="preserve">Siautem <reg norm="punctum" type="context context">pÅ©ctÅ©</reg> <var>.u.</var> fuiſſet intra <lb/>circulum idem eueniret. </s> - <s xml:id="s3194" xml:space="preserve">Nam <lb/>quadrato <var>.b.e.</var> <reg norm="æquantur" type="context">æquãtur</reg> duo qua <lb/>drata <var>.o.b.</var> et <var>.o.e.</var> Å¿ed vice qua-<lb/>drati <var>.o.e.</var> dicemus <reg norm="quadratum" type="context">quadratÅ©</reg> <var>.o.<lb/>u.</var> cum eo quod fit ex <var>.a.u.</var> in <var>.u.e.</var> <lb/>ex .5. Å¿ecundi, id eÅ¿t quadratum <var>.<lb/>o.u.</var> <reg norm="cum" type="context">cÅ©</reg> eo quod fit ex <var>.b.u.</var> in <var>.u.<lb/>n.</var> ex .34. tertij, vnde quadratum <lb/><var>b.e.</var> æquale erit quadrato <var>.o.b.</var> <lb/>& quadrato <var>.o.u.</var> ideÅ¿t quadrato <lb/><var>b.u.</var> ex Pitagorica Å¿imul <reg norm="cum" type="context">cÅ©</reg> pro-<lb/>ducto <var>.b.u.</var> in <var>.u.n.</var> ideÅ¿t producto <lb/><var>n.b.</var> in <var>.b.u.</var> quod æquale eÅ¿t qua <lb/>drat <var>o.b.u.</var> cum producto <var>.b.u.</var> in <lb/><var>u.n.</var> ex .3. Å¿ecundi.</s> - </p> - <p> - <s xml:id="s3195" xml:space="preserve">Circa tres paſſiones commu-<lb/>nes poÅ¿tea circulo hyperboli, & <lb/>defectioni notandum eÅ¿t <reg norm="primam" type="context">primã</reg> <lb/>patere ex .36: primi Pergei, Å¿e- - <pb o="254" rhead="IO. BABPT. BENED." n="266" file="0266" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0266"/> - cundam verò ex .37. et .38. eiuÅ¿dem, </s> - <s xml:id="s3196" xml:space="preserve">propterea quod in .37. probat mediante maiori <lb/>diametro ipÅ¿ius hyperbolis & defectionis, In .38. autem mediante minori diametro <lb/>ordinatè ad maiorem.</s> - </p> - <p> - <s xml:id="s3197" xml:space="preserve">Tertia autem paſſio, non niÅ¿i circulo conuenit; </s> - <s xml:id="s3198" xml:space="preserve">pace ipÅ¿ius Cardani dictum Å¿it.</s> - </p> - <p> - <s xml:id="s3199" xml:space="preserve">Quapropter Å¿it circulus <var>.q.o.b.</var> cuius diameter Å¿it <var>.q.b.</var> contingentes vero ab extre <lb/>mitate diametri Å¿int <var>.d.b.</var> et <var>.q.g.</var> per punctum autem <var>.o.</var> quoduis, ipÅ¿ius <reg norm="circunferentiæ" type="context">circÅ©ferentiæ</reg>, <lb/>tranÅ¿eant <var>.b.o.g.</var> et <var>.q.o.d</var>. </s> - <s xml:id="s3200" xml:space="preserve">tunc dico productum <var>.q.o.</var> in <var>.q.d.</var> vel <var>.b.o.</var> in <var>.b.g.</var> Ä™quale eÅ¿-<lb/>Å¿e quadrato <var>.q.b.</var> quod ita probo.</s> - </p> - <p> - <s xml:id="s3201" xml:space="preserve">Nam angulus <var>.q.b.d.</var> Å¿eu <var>.b.q.g.</var> rectus eÅ¿t ex .17. tertij Eucli. et <var>.b.o.q.</var> Å¿imiliter re-<lb/>ctus ex .30. ipÅ¿ius lib. angulus verò <var>.b.q.d.</var> Å¿eu <var>.q.b.g.</var> communis eÅ¿t. </s> - <s xml:id="s3202" xml:space="preserve">quare <var>.b.q.</var> media <lb/>proportionalis erit inter dictas lineas <var>.q.d.</var> et <var>.q.o.</var> & inter <var>.b.g.</var> et <var>.b.o</var>. </s> - <s xml:id="s3203" xml:space="preserve">Vnde Å¿equetur <lb/>propoÅ¿itum ex .16.6. Eucli.</s> - </p> - <p> - <s xml:id="s3204" xml:space="preserve">Sed Å¿i circa diametrum <var>.q.b.</var> mente fingamus aliquam elipÅ¿im, quætangat ipÅ¿um <lb/> - <anchor type="figure" xlink:label="fig-0266-01a" xlink:href="fig-0266-01"/> - circulum duobus punctis me-<lb/>diantibus <var>.q.</var> et <var>.b.</var> (nam pluribus <lb/>eſſet impoſſibile, ex .27. quarti <lb/>Pergei) clarè patebit, quod <reg norm="pum" type="context">pÅ©</reg> <lb/>ctus <var>.o.</var> erit extra <reg norm="circunferentiam" type="context">circunferentiã</reg> <lb/>ipÅ¿ius defectionis, </s> - <s xml:id="s3205" xml:space="preserve">quare ipÅ¿a cir <lb/>cunferentia Å¿ecabit <var>.b.g.</var> vel <var>.q.<lb/>d.</var> in alio puncto, vnde ipÅ¿i non <lb/>occurret id quod probauimus <lb/>de circulo.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0266-01" xlink:href="fig-0266-01a"> - <image file="0266-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0266-01"/> - </figure> - </div> - <p> - <s xml:id="s3206" xml:space="preserve">Admiratus etiam Å¿um, ipÅ¿um <lb/>Cardanum dicere hyperbolem <lb/>ita vocari, eo quod angulus con <lb/>tentus ab axe ipÅ¿ius figuræ, & à <lb/>latere trigoni in hyperbole ma-<lb/>ior Å¿it quam in parabole, quod <lb/>eriam confirmat paulo inferius, <lb/>nam hoc verum non eÅ¿t, imo fal <lb/>Å¿iſſimum. </s> - <s xml:id="s3207" xml:space="preserve">Talis enim Å¿ectio ita <lb/>nominata fuit, hoc eÅ¿t hyperbo <lb/>les, Å¿imili ratione, qua elipÅ¿is Å¿eu <lb/>defectio etiam vocata fuit, nam <lb/>Å¿icut in ipÅ¿a defectione quadra-<lb/>tum ordinatÄ™ <var>.l.m.</var> minor eÅ¿t pro <lb/>ducto lineæ <var>.e.m.</var> in <var>.e.t.</var> per figu <lb/>ram Å¿imilcm producto <var>.d.e.</var> in <var>.e.<lb/>t.</var> quæ eandem obtineat <reg norm="altitu- dinem" type="context">altitu-<lb/>dinẽ</reg> ipÅ¿ius <var>.e.m.</var> vt ipÅ¿e Pergeus <lb/>monÅ¿trat in .13. primi lib. ita in <lb/>hyperbole <reg norm="dictum" type="context">dictÅ©</reg> quadratum ex <lb/>cedit quantitatem illius figuræ, <lb/>per Å¿imilem dictæ vt in .12. <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <lb/>Pergei facilè videre eÅ¿t. </s> - <s xml:id="s3208" xml:space="preserve">Å¿ed <reg norm="prae ter" type="simple">prÄ™Â<lb/>ter</reg> illas paſſiones, quas notat - <pb o="255" rhead="EPISTOL AE." n="267" file="0267" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0267"/> - Cardanus in Å¿upradicto capite, multæ aliæ Å¿unt, cum corollarium primæ tertij Eu-<lb/>cli. Å¿it paſſio propria ipÅ¿ius circuli, & idem dico de propoÅ¿itione .3. 4. 7. 8. 9. 11. 12. <lb/>13. 14. 15. 17. 18. 19. 20. 30. 31. ipÅ¿ius tertij lib. nec non de .8. 9. ct .10. tertijdecimi, & <lb/>de prima .3. 4. 5. 6. et .7. quartidecimi eiuÅ¿dem. </s> - <s xml:id="s3209" xml:space="preserve">Idem infero de ea quod Å¿crip Å¿i Ma <lb/>rio Nizzolio, FranciÅ¿co Vimercato, FranciÅ¿co Contareno, Angelo AgrimenÅ¿ori, & <lb/>de alijs nonnullis à me excogitatis.</s> - </p> - </div> - </div> - <div type="section" level="3" n="6"> - <div type="letter" level="4" n="1"> - <head xml:id="head382" xml:space="preserve">DE FINE CORPORVM COELESTIVM, <lb/>& eorum motu.</head> - <head xml:id="head383" style="it" xml:space="preserve">IlluÅ¿tri vire, Philiberto Pingonio Sabaudo CuÅ¿iacenÅ¿ium <lb/>Baroni.</head> - <p> - <s xml:id="s3210" xml:space="preserve">CVm antea meo nomine SebaÅ¿tianus noÅ¿ter omnia ferè tibi retuliſſet, inter alia, <lb/>quæ relin quebantur tibi <reg norm="dicenda" type="context">dicẽda</reg>, hoc vnum erat, quod Å¿i abÅ¿que lumine Å¿upe-<lb/>riori, in quem finem facta fuerint corpora cÅ“leÅ¿tia Å¿cire deÅ¿ideras, & humanam ra-<lb/>tionem Å¿equi volueris, putandum tibi non erit ea Å¿olum effecta eſſe, vt tam vile cor <lb/>pus, vt eÅ¿t terra aquis irrigata, animalia, & plantas regant, cum ea corpora Å¿int diuina, <lb/>in numero incomprÄ™henÅ¿ibilia, maximis ma gnitudinibus, & motibus velocisfimis, <lb/>prædita, id etiam minus putabunt hij, qui opinionem AriÅ¿tarchi Samij, & Nicolai <lb/>Copernici Å¿equuntur, quorum ratione fieri non poteÅ¿t, vt credant, eius, quod ex vni <lb/>uerÅ¿o reliquum eÅ¿t, alium finem non habere, quam regimen huius centri epicycli Lu <lb/>naris, vt illorum more loquar. </s> - <s xml:id="s3211" xml:space="preserve"><reg norm="Quam" type="context">Quã</reg> enim turpe eſſet Å¿i centra aliorum epicyclorum <lb/>planetarum tali regimine priuarentur, id quod nullo modo cum ratione conÅ¿entit, <lb/>Å¿i tam vera eÅ¿t ea opinio, quemadmodum <reg norm="rationabiliorem" type="context">rationabiliorẽ</reg> eam <reg norm="exiÅ¿timant" type="context">exiÅ¿timãt</reg>. </s> - <s xml:id="s3212" xml:space="preserve"><reg norm="Neque" type="simple">Neq;</reg> quid <lb/>quam valet opinio AriÅ¿totelis, qui corpora cÅ“leÅ¿tia, ab ortu, & interitu libera eſſe <lb/>fentit. </s> - <s xml:id="s3213" xml:space="preserve">dicens Å¿uperioribus fÄ™culis, à noÅ¿tris antiquis nullam vnquam animaduerÅ¿am <lb/>fuiſſe alterationem in cÅ“lo, cum non videat Å¿i quis eſſet in cÅ“lo, <reg norm="neque" type="simple">neq;</reg> etiam obÅ¿erua <lb/>re poſſet alterationes quæ in terra, & circa terram fiunt, quæ in partibus, & non in to <lb/>to Å¿pectantur: </s> - <s xml:id="s3214" xml:space="preserve">vnde etiam fieri poteÅ¿t, vt in cÅ“lo Å¿int particulares alterationes, <lb/>quæ à nobis tamen, qui ab illis longè diÅ¿tamus, non comprÄ™hendantur, terra, <reg norm="mareque" type="simple">mareqÌ;</reg> <lb/>(quamuis minimum reÅ¿pectu ipÅ¿ius terræ) ratione totius ita Å¿e Å¿emper <reg norm="habuerunt" type="context">habuerÅ©t</reg> <reg norm="quem" type="context">quẽ</reg> <lb/>admodum Å¿eÅ¿e habere corpora cÅ“leÅ¿tia videmus, Å¿ed alteratio, ratione tantum ali-<lb/>quarum minimarum partium quaÅ¿i inÅ¿enÅ¿ibilium, Å¿i cum toto comparentur fit. </s> - <s xml:id="s3215" xml:space="preserve">Quis <lb/>enim Å¿cit, vt iam tibi dixi, quin, quemadmodum Luna circa terram voluitur, <reg norm="ipÅ¿aque" type="simple">ipÅ¿aqÌ;</reg> <lb/>terra Å¿it veluti centrum epicycli maioris eiuÅ¿dem, vt AriÅ¿tarchus Samius, & Nico-<lb/>laus Copernicus cenÅ¿uerunt; </s> - <s xml:id="s3216" xml:space="preserve">Å¿ic etiam Saturnus, Iupiter, Mars, Venus, <reg norm="atque" type="simple">atq;</reg> Mercu <lb/>rius circa alia huiuÅ¿modi corpora, huic terræ Å¿imilia, in orbem agantur, quaÅ¿i Å¿pecu-<lb/>la, lumen Solis Å¿uo centro ex reflexione, deferentia (fuppoÅ¿ita dico vera illorum opi <lb/>nione) <reg norm="Nollem" type="context">Nollẽ</reg> tamen tibi è mente excidere, vt alià s te monui, <reg norm="quod" type="simple">ê™</reg> Å¿i communis opinio <lb/>vera eÅ¿t, neceſſario <reg norm="fatendum" type="context">fatendÅ©</reg> Å¿it corpus Å¿olare, <reg norm="dum" type="context">dÅ©</reg> in æquatore reperitur moru diurno <lb/>quolibet horÄ™ minuto, magis <reg norm="quam" type="context">quã</reg> <reg norm="decem" type="context">decẽ</reg> & <reg norm="Å¿eptem" type="context">Å¿eptẽ</reg> mille milliaria <reg norm="peragere" type="simple">ê‘agere</reg>, ideÅ¿t paulo mi <lb/>nus quam .18000. milliaria, Saturnum verò cum Å¿imiliter eÅ¿t in æquatore, eodem <reg norm="tem- poris" type="context">tẽ-<lb/>poris</reg> Å¿patio, quaÅ¿i tercentamille milliaria Italica conficere, & Å¿ic per gradus alia cor <lb/>pora velociora alijs moueri; </s> - <s xml:id="s3217" xml:space="preserve">quæ quidem omnia, <reg norm="cum" type="context">cÅ©</reg> Å¿implici gyro terræ circa Å¿uum - <pb o="256" rhead="IO. BABPT. BENED." n="268" file="0268" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0268"/> - axem (vt dicunt) tolluntur, quod Å¿ufficit ad recipiendum lumen, & influentias illo-<lb/>rum corporum. </s> - <s xml:id="s3218" xml:space="preserve">Et ita, veluti princeps corporum vniuerÅ¿i, intra vnum an-<lb/>num circa eam vertitur. </s> - <s xml:id="s3219" xml:space="preserve">Ita etiam Å¿uſſiceret, vt ipÅ¿a terra circa dictum diuinum <lb/>corpus Å¿olare, interÅ¿ecando axem diurnum cum axe annuali (cum ab eo lumen, ca-<lb/>lorem, & influentiam Å¿uÅ¿cipere debeat) <reg norm="circunuolueretur" type="simple">circunuolueret̃</reg>. </s> - <s xml:id="s3220" xml:space="preserve">Rationes autem a Ptolo-<lb/>meo in contrarium adductæ apud ipÅ¿os, nullÄ™ Å¿unt, quia quÄ™libet pars (vt inquiunt) <lb/>retinet naturam totius, præterquam <reg norm="quod" type="simple">ê™</reg> aer, & aqua, quæ ipÅ¿am terram <reg norm="circundant" type="context">circundãt</reg>, pla <lb/>nè eundem naturalem impetum motus obtineant, quitanto lentior eÅ¿t, quanto lon <lb/>gius diÅ¿tat aer ab ipÅ¿a terra, Å¿ecundum etiam talem opinionem, nulla neceſſitas fo-<lb/>ret, vt locus fixarum terminaretur aliquibus Å¿uperficiebus, conuexa Å¿cilicet, & de-<lb/>uexa.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head384" style="it" xml:space="preserve">De Luce, Lumine, & Colore, De obiectuoculi, De lumine <lb/>Luna, & Rubedine nubium.</head> - <head xml:id="head385" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3221" xml:space="preserve">QVod proximè quærebas, an Å¿it lux aliqua, quæ à corpore lucido non proue-<lb/>niat, mihi facilè ad conÅ¿iderandum videtur. </s> - <s xml:id="s3222" xml:space="preserve">hic enim oportet, vt nos ad id <lb/>quod perpetuò videmus referamus, exiÅ¿timo autem te velle dicere lumen, non lu-<lb/>cem, quia propriè lux, qualitas ea viÅ¿ibilis appellatur, quæ eÅ¿t in corpore lucido, à <lb/>quo quidem corpore lumen effunditur; </s> - <s xml:id="s3223" xml:space="preserve">lumen verò, ea qualitas eſſe dicitur, quæ ex <lb/>tra ipÅ¿um corpus reperitur, à luce, quæ in dicto corpore manet emanans. </s> - <s xml:id="s3224" xml:space="preserve">vnde pa-<lb/>tet, nullam lucem abÅ¿que corpore Å¿ubiecto eſſe poſſe, id quod cum fieri quîret, <reg norm="idem" type="context">idẽ</reg> <lb/>de quolibet alio accidente dici poſſet, id eÅ¿t quod ex Å¿e, & abÅ¿que aliquo Å¿ubiecto <lb/>Å¿ubÅ¿iÅ¿teret.</s> - </p> - <p> - <s xml:id="s3225" xml:space="preserve">Lumen deinde à luce proficiÅ¿ci patet, <reg norm="quod" type="simple">ê™</reg> penetrat diaphanum, neque aliquo mo-<lb/>do Å¿uum actum oÅ¿tendit, niÅ¿i, aut per incidentiam, aut ratione opaci, ex reflexione, <lb/>cuius Å¿uperficiei colorem induit. </s> - <s xml:id="s3226" xml:space="preserve">Atque hæc eÅ¿t cauÅ¿a, vt inter crepuÅ¿culum matu-<lb/>tinum, aut veÅ¿pertinum, nox etiam Å¿i Å¿it Å¿erena, adeo obÅ¿cura nobis appareat, quam-<lb/>uis totum vniuerÅ¿um diaphanum, extra conum vmbræ, quæ ex terra prouenit Å¿it vn <lb/><reg norm="dique" type="simple">diq;</reg> radijs luminoÅ¿is Solis colluÅ¿tratum; </s> - <s xml:id="s3227" xml:space="preserve">qui quidem radij, non niÅ¿i à ſuamet reflexio <lb/>ne à Luna, & ab alijs Å¿tellis (vt corporibus opacis, quæ reÅ¿iÅ¿tunt lumini, ne vlterius <lb/>penetrare poſſit, vnde retrò redit) comprehenduntur.</s> - </p> - <p> - <s xml:id="s3228" xml:space="preserve">Ais etiam propria viÅ¿us obiecta plura eſſe, nominans pro vno, colorem, & lucem <lb/>pro alio. </s> - <s xml:id="s3229" xml:space="preserve">Ego autem reÅ¿pondeo, obiectum oculi eſſe vnicum tantum, ideÅ¿t lumen. <lb/></s> - <s xml:id="s3230" xml:space="preserve">Quod ad lucem Å¿pectat, iam tibi dixi, eam eſſe quandam qualitatem in corpore luci <lb/>do, & non extra ipÅ¿um poÅ¿itam, à quo quidem corpore, cum non exeat, oculi obie-<lb/>ctum eſſe nequit, Å¿ed lumen quidem ab ipÅ¿a luce productum. </s> - <s xml:id="s3231" xml:space="preserve">Color etiam, qui eÅ¿t in <lb/>corpore colorato, obiectum oculi eſſe non poteÅ¿t, cum dictum corpus non deÅ¿erat, <lb/>Å¿ed lumen quidem ab eodem corpore reflexum, & huiuÅ¿modi corporis colore tin-<lb/>ctum: </s> - <s xml:id="s3232" xml:space="preserve">vnde tam lumen incidens, quam reflexum colore eÅ¿t Å¿emper imbutum.</s> - </p> - <p> - <s xml:id="s3233" xml:space="preserve">Illud quidem coloratum eÅ¿t qualitate lucis corporis lucidi, a ut medij, per quod <lb/>tranÅ¿it, Å¿ed hoc colore corporis, à quo reflectitur.</s> - </p> - <p> - <s xml:id="s3234" xml:space="preserve">Neque etiam te ignorare volo, lumina reflexa colorata, non reflecti à puris pro-<lb/><reg norm="priisque" type="simple">priisqÌ;</reg> Å¿uperficiebus corporum coloratorum, eo <reg norm="quod" type="wordlist">qÄ‘</reg> pauca corpora tam opaca repe-<lb/>riuntur, ut immediatè lumen à ſuperficie propriè <reg norm="reflectant" type="context">reflectãt</reg>, Å¿ed lumen penetrat alim - <pb o="259" rhead="EPISTOL AE." n="271" file="0271" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0271"/> - quantulum dicta corpora, & ita illorum colore afficitur, vbi verò non penetrat, non <lb/>coloratur colore corporis illius.</s> - </p> - <p> - <s xml:id="s3235" xml:space="preserve">Sed vt ad propoÅ¿itum redeamus, dico lum en tantum eſſe viÅ¿us obiectum, quod Å¿i <lb/>colore eÅ¿t imbutum, aut tale eÅ¿t ratione color ris lucis, quæ eum mittit, aut ratione me <lb/>dij per quem tranÅ¿it, aut ratione corporis, vnde reflectitur, etÅ¿i Å¿uperficies corporis <lb/>vnde lumen reflectitur eſſet omnino priuata colore, Å¿ub aÅ¿pectum non caderet, vt <lb/>etiam cum huiuÅ¿modi Å¿uperficies læuigata, & polita eÅ¿t Å¿ecundum <reg norm="continuitatem" type="context">continuitatẽ</reg> Å¿ua-<lb/>rum partium, videlicet, vt Å¿peculi radio tamen non profundante, & ideo perfectiffi <lb/>morum quorundam Å¿peculorum Å¿uperficies non cernuntur, Å¿ed lumen tantum re-<lb/>flexum, colore alicuius alterius Å¿uperficiei, aut à luce, corporis lucidi, aut à me-<lb/>dio per quem tranÅ¿it, conÅ¿picitur. </s> - <s xml:id="s3236" xml:space="preserve">Ego verò non aſſero colorem non eſſe quid di-<lb/>uerÅ¿um à lumine, Å¿ed imagineris lumen eſſe veluti animam, aut Å¿ubÅ¿tantiam & colo <lb/>rem corporis formam accidentalem, cum nullum lumen à ſenÅ¿u viÅ¿us percipi poÅ¿-<lb/>Å¿it, quod aliquo modo colore non Å¿it imbutum: </s> - <s xml:id="s3237" xml:space="preserve">& eundem reÅ¿pectum quem Å¿onus <lb/>ad auditum, lumen ad oculum habet, quia vt Å¿onus Å¿ecundum eam velocitatem, quæ <lb/>à motione aeris, aut aquæ, ex colliÅ¿ione <reg norm="aliorum" type="context">aliorÅ©</reg> corporum producitur ad euitan dum <lb/>vacuum, a cutus, vel grauis Å¿entitur, ita lumen originem ducens à corpore lucido per <lb/>medium diaphanum aeris, aut aquæ, aut alterius huiuÅ¿modi corporis ad oculum tran <lb/>Å¿it colorem lucis, aut medij per quod tranÅ¿it, aut vnde reflectitur induit.</s> - </p> - <p> - <s xml:id="s3238" xml:space="preserve">Quod verò Luna nullum ex Å¿e habeat lumen, Å¿ufficiens inditium eÅ¿t nos ipÅ¿am <lb/>tantò magis obÅ¿curam videre, quantò magis in cono vmbræ terræ immergitur, & <lb/>Å¿i eo tempore ipÅ¿am videmus rubeo colore affectam, hoc enim accidit, quia radij Å¿o <lb/>lares vndequaque refranguntur à vaporibus ipÅ¿am terram circundantibus, quæ qui-<lb/>dem refractio fit verÅ¿us axem coni vmbræ terræ, </s> - <s xml:id="s3239" xml:space="preserve">& propterea vmbra dicti coni non <lb/>eÅ¿t æqualiter obÅ¿cura, Å¿eu tenebroÅ¿a, circa vero <reg norm="axem" type="context">axẽ</reg> ipÅ¿ius coni, magis quam circa eius <lb/><reg norm="circunferentiam" type="context context">circÅ©ferentiã</reg>, obÅ¿cura <reg norm="verum" type="wordlist">vr̃</reg>, & quia corpus lunare tale eÅ¿t, vt facillimè recipiat <reg norm="qualecum" type="context">qualecÅ©</reg> <lb/>que lumen, quod etiam manifeÅ¿tè videtur dum ipÅ¿a Luna reperitur Å¿ecundum lon-<lb/>gitudinem inter Solem, & Venerem, quod pars Lunæ lumine Solis deÅ¿tituta, à lumi <lb/>ne Veneris aliquantulum illuÅ¿tratur, quod ego ſæpè vidi, & multis oÅ¿tendi. </s> - <s xml:id="s3240" xml:space="preserve">Propte-<lb/>rea dum ipÅ¿a Luna in cono vmbræ terræ reperitur adhuc videtur. </s> - <s xml:id="s3241" xml:space="preserve">Rubedo etiam il-<lb/>la nubium poÅ¿t Solis occaÅ¿um, vel ante ortum, aliunde non prouenit, niÅ¿i à qualitate <lb/><reg norm="vaporum" type="context">vaporÅ©</reg>, per quos Å¿olares radij tranÅ¿eunt, à quibus vaporibus, tali colore ipÅ¿i radij <lb/>afficiuntur, eomet modo quo radius, cuiuÅ¿uis corporis lucidit, <reg norm="tranÅ¿iens" type="context">trãſiens</reg> per vitrum, Å¿eu <lb/>aliud diaphanum coloratum.</s> - </p> - <pb o="258" rhead="IO. BABPT. BENED." n="270" file="0270" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0270"/> - </div> - </div> - <div type="section" level="3" n="7"> - <div type="letter" level="4" n="1"> - <head xml:id="head386" xml:space="preserve">DE ICTV BOMBARDAE SECVNDVM <lb/>diuerÅ¿as eleuationes. Et de quibuÅ¿dam erroribus Nico-<lb/>lai Tartaleæ, circa idem.</head> - <head xml:id="head387" style="it" xml:space="preserve">Fllustri D. IoÅ¿epho Cambiano ex Ruffia Dominis, aquiti <lb/>Å¿trenuo, & tormentis bellicis Serenißimi Ducis <lb/>Sabaudia Prafecto.</head> - <p> - <s xml:id="s3242" xml:space="preserve">EXcogitaui quÄ™dam dum ocio frui licuit per abÅ¿entiam Ducis Sereniſſimi, <lb/>quæ ad te Å¿cribere placuit, vt Å¿i probaueris in lucem quandoque profer-<lb/>re non dubitem, Å¿i deÅ¿pexeris, ocius Å¿upprimam, Å¿unt autem huiuÅ¿modi.</s> - </p> - <p> - <s xml:id="s3243" xml:space="preserve">Vnde fiat vt tormentum bellicum vehementi<unsure/> feriat ictu Å¿uperius delato <lb/>quam orizontali, vt Tartalea Å¿cribit, quæſito Å¿ecundo libr. primi quæſitorum, à ne-<lb/>mine adhuc (quod Å¿ciam) traditum eÅ¿t.</s> - </p> - <p> - <s xml:id="s3244" xml:space="preserve">Rationes verò Tartaleæ nullius Å¿unt momenti, quia Å¿i validæ eſſent, Å¿equeretur <lb/>vt inclinata bombarda, adeo vt angulus Å¿ub orizonte factus æqualis eſſet ei, qui Å¿u <lb/>pra orizontem eÅ¿t, ictum bombardÄ™ in vtroque huiuÅ¿modi Å¿itu eundem eſſe <reg norm="futurum" type="context">futurÅ©</reg>. <lb/></s> - <s xml:id="s3245" xml:space="preserve">& Å¿i aliqua differentia oriretur ratione granitatis pilæ ab ipÅ¿a bombarda emiſſæ, hoc <lb/>fieret, vt Å¿cilicet velocior eſſet in motu inclinato quam in eleuato cum pondus, mo-<lb/>tui adeo non opponatur. </s> - <s xml:id="s3246" xml:space="preserve">Id quod non ita se habet, vera enim cauÅ¿a vnde fiat, vt bom <lb/>barda eleuata vehementius feriat, quà m ea quæ eÅ¿t minus alta, eadem eÅ¿t ferè, in ge-<lb/>nere, cum ea, qua aliquod corpus materia magis denÅ¿a, Å¿ed Å¿imile & Ä™quale alteri cor <lb/>pori materiæ minus denſæ velocius mouetur ab vna eademqÌue, aut æquali vi <lb/>compulÅ¿um. </s> - <s xml:id="s3247" xml:space="preserve">EÅ¿t eadem etiam in Å¿pecie ei, qua maiorem effectum producit <lb/>puluis, qui in locis Å¿ubterraneis ponitur quum vaÅ¿is optimè colligatis ferro in-<lb/>cluditur. </s> - <s xml:id="s3248" xml:space="preserve">EÅ¿t etiam Å¿imilis ei, qua longius impellitur pila, qua ludimus, ab ali-<lb/>quo inÅ¿trumento ligneo, quando percutitur contra, quam cum Å¿ecundum Å¿uum mo-<lb/>tum proijcitur. </s> - <s xml:id="s3249" xml:space="preserve">Id quod inde fit, quia virtus mouens maiori vi, & intenÅ¿iori huiuÅ¿-<lb/>modi corpus percutit, quia corpus quod moueri debet, quanto magis reÅ¿iÅ¿tit virtu-<lb/>ti mouenti (certum tamen terminum præſcribendo) in exiguo eo temporis Å¿patio, <lb/>tanto maiorem virtutem colligit, quæ ipÅ¿um deinde tanto cum impetu mo-<lb/>uet, & tanto magis impellens concomitatur, vt maiorem effectum efficiat, quam Å¿i <lb/>ad mouendum Å¿eÅ¿e facilè reddidiſſet. </s> - <s xml:id="s3250" xml:space="preserve">Atque hoc Å¿upradictis ictibus eleuatis acci-<lb/>dit, quia grauitas pilæ, ea eÅ¿t quæ reÅ¿iÅ¿tens virtuti mouenti, dat ei commoditatem <lb/>colligendi dictam virtutem, multo magis quam eſſet ea, quæ ad depreſſiorem eleua <lb/>uationem eam <reg norm="immpelleret" type="context">Ä©mpelleret</reg>. </s> - <s xml:id="s3251" xml:space="preserve">Et quia huiuÅ¿modi multiplicatio virtutis, nullam propor <lb/>tionem cum pondere pilæ gerit, volo inferre quod dum colligitur tanta virtus, col-<lb/>ligitur multo plus eo, quod ad impellendam dictam pilam Å¿ufficeret, ratione magnæ <lb/>velocitatis augumenti, quia quanto plus temporis ei conceditur ad commutandam <lb/>puluerem in ignem, tanto maior quantitasignis progignitur, vnde fit, vt tanto ma-<lb/>iori loco indigeat, quamobrem tanto magis impellit, Å¿ed vt dixi, tanta cum veloci-<lb/>tate <reg norm="adaugetur" type="simple">adauget̃</reg>, vt huiuÅ¿modi virtus longè Å¿uperet <reg norm="reÅ¿iÅ¿tentiam" type="context context">reÅ¿iÅ¿tẽtiã</reg> <reg norm="ponderis" type="context">põderis</reg> pilæ, & Å¿ic eÅ¿t cau <lb/>fa, ut effectus, quod <reg norm="experientia" type="context">experiẽtia</reg> innot eÅ¿cit <reg norm="producatur" type="simple">producat̃</reg>. </s> - <s xml:id="s3252" xml:space="preserve">Sed ea ratio, qua Å¿eÅ¿e <reg norm="idem" type="context">idẽ</reg> author <lb/>in tertio quæſito ad aliquod impoſſibile, circa iter ipÅ¿ius pilæ Legatum HiÅ¿panum - <pb o="259" rhead="EPISTOL AE." n="271" file="0271" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0271"/> - reducere putat, nullo fundamento nititur, quia non eÅ¿t Å¿emper dicendum, quod <reg norm="quam" type="context">quã</reg> <lb/>to velocior Å¿it quædam pila, tanto rectius moueatur, quia ei dici poſſet, vÅ¿que ad cer <lb/>tum quendam terminum velocitatis, per tantum Å¿patij eam aptam eſſe, vt recta per <lb/>fectè moueatur, Å¿ed Å¿i velocius iret, non tamen futurum, vt per idem Å¿patium re-<lb/>ctius moueretur, Å¿ed quod per longius Å¿patium recta motum perageret, & Å¿ic nihil <lb/>haberet quod replicaret, præter quam quod ipÅ¿e Å¿upponit id quod in 18. quæſito <lb/>negat, in quo ait pilam uicinam orificio, non adeo uelocem eſſe, quam cum aliquan-<lb/>tulum ab eodem eÅ¿t rem ota, ratione reÅ¿iÅ¿tentiæ Å¿ui cyllindriaerei. </s> - <s xml:id="s3253" xml:space="preserve">Sed quod pila, <lb/>recta eat quanto altior, aut depreſſior <reg norm="bombarda" type="context">bõbarda</reg> erit, <reg norm="ion" type="context">iõ</reg> fit, quia linea inclinationis na <lb/>turalis cum linea inclinationis uiolentæ angulum rectum non facit, </s> - <s xml:id="s3254" xml:space="preserve">unde quanto lon <lb/>gius diÅ¿tat à recto huiuÅ¿modi <reg norm="angulus" type="simple">angulê°</reg>, Å¿iue Å¿it acutus Å¿iue obtuÅ¿us, tanto minorem uim <lb/>habet, eodem planè ferè modo quem tertio capite mei tractatus de rebus mechani <lb/>cis deÅ¿cripÅ¿i. </s> - <s xml:id="s3255" xml:space="preserve">Quia in ictibus eleuatis, iter inclinationis violentæ ipÅ¿ius pilæ verÅ¿us <lb/>terminum ad quem, incipiendo à loco ipÅ¿ius pilæ cum itinere inclinationis natura-<lb/>lis, angulum obtuÅ¿um, & in ictibus inclinatis acutum conÅ¿tituit. </s> - <s xml:id="s3256" xml:space="preserve">Neque etiam hic <lb/>prætermittam notatu dignum errorem, quem Tartalea eodem loco committit, <reg norm="cum" type="context">cÅ©</reg> <lb/>putet indifferenter aliquod corpus impellere, aut percutere maiori <reg norm="cum" type="context">cÅ©</reg> impetu cum <lb/>eÅ¿t in itinere recto. </s> - <s xml:id="s3257" xml:space="preserve">Quia Å¿equeretur quod aliquod corpus graue perpendiculari-<lb/>ter Å¿urÅ¿um verÅ¿us proiectum, in qualibet parte Å¿ui itineris, Å¿emper fortius percute-<lb/>ret, quam in qualibet parte itineris alterius cuiuÅ¿uis eleuationis obliquæ, quod <reg norm="quam" type="context">quã</reg> <lb/>Å¿it falÅ¿um, tibi conÅ¿iderandum relinquo.</s> - </p> - <p> - <s xml:id="s3258" xml:space="preserve">EÅ¿t etiam falÅ¿a ea ratio, quam in quarto quæſito idem adducit, quia aer in motu <lb/>non tantum durat, quantum ipÅ¿e putat, imò huiuÅ¿modi violenta agitatio, citò ceſſat <lb/>& citius etiam, quam Å¿i extra aliquam bombardam cum tanta violentia impulliſſet <lb/>Å¿aceum plumis plenum.</s> - </p> - <p> - <s xml:id="s3259" xml:space="preserve">Ratio etiam quam in .18. quæſito de eo, quod pila pertran Å¿eat illud corpus cyllin <lb/>dricum aereum adducit, eÅ¿t planè vana, quia Å¿tatim aer, qui prius in <reg norm="bombarda" type="context">bõbarda</reg> erat <lb/>incluÅ¿us, extra ipÅ¿am <reg norm="erumpit" type="context">erÅ©pit</reg>, cedit, à <reg norm="pilaque" type="simple">pilaqÌ;</reg> diuiditur, vt Å¿i nunquam eam figuram in-<lb/>duiſſet, neque aer ambiens ei reÅ¿iÅ¿tit. </s> - <s xml:id="s3260" xml:space="preserve">Sed quod velocior Å¿it in certa quadam diÅ¿tan <lb/>tia, quam in principio erat, Å¿i hoc <reg norm="verum" type="context">verÅ©</reg> eſſet, ab alia cauÅ¿a dependeret, quæ partim Å¿i <lb/>milis eſſet ei, quæ efficit, vt corpora in motibus naturalibus, cum longius diÅ¿tant à ter <lb/>mino vnde naturaliter Å¿eÅ¿e mouerunt, Å¿int velociora, quia per aliquod Å¿patium hu-<lb/>iuÅ¿modi corpus moueretur quemadmodum motu naturali cietur.</s> - </p> - <p> - <s xml:id="s3261" xml:space="preserve">Ratio autem eius quare pila, aut globus bombardæ Å¿ibiletab eodem in Å¿eptimo <lb/>quæſito nil valet, quia hoc fit cum pila aliquam paruam concauitatem habet.</s> - </p> - <p> - <s xml:id="s3262" xml:space="preserve">In .27. autem quæſito ait, quod retrotrahendo Å¿ignum, ictus altius tenderet, quod <lb/>poteÅ¿t etiam eſſe falÅ¿um, cum hocnon Å¿it neceſſarium, quia pila dum deÅ¿cendit, for-<lb/>taſſe tangeret Å¿copum.</s> - </p> - <pb o="260" rhead="IO. BAPT. BENED." n="272" file="0272" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0272"/> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head388" style="it" xml:space="preserve">Deerroribus Ioannis Stadij.</head> - <head xml:id="head389" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3263" xml:space="preserve">FIguram quam ponit Ioannes Stadius pag .147. in lib. Å¿uarum tabularum Prute-<lb/>nicarum, à Nicolao Copernico Å¿umpÅ¿it pag .64. à tergo in libr. reuolutionum <lb/>cÅ“leÅ¿tium, Å¿ed ipÅ¿e Stadius eam non intellexit, omitto, quod mutauerit characte-<lb/>res ipÅ¿ius figurÄ™, vt illa Å¿ua videatur, quod nihil refert, alterat etiam <reg norm="demonÅ¿trationem" type="context">demonÅ¿trationẽ</reg>, <lb/>Å¿ed ipÅ¿um putare <var>.i.K.</var> perpendicularem à centro circuli Å¿emper dependere, eÅ¿t intol <lb/>lerabilis error; </s> - <s xml:id="s3264" xml:space="preserve">nec vnquam verificatur hoc, niÅ¿i quando punctum <var>.K.</var> interÅ¿ectionis <lb/>diametrorum parallelorum, forte reperitur in axe mundi. </s> - <s xml:id="s3265" xml:space="preserve">Reliqua verò Å¿uæ demon <lb/>Å¿trationis, Å¿i non intelligis, minimè miror, eo quod ipÅ¿emet Stadius Å¿eipÅ¿um confun <lb/>dit. </s> - <s xml:id="s3266" xml:space="preserve">Veram autem demonÅ¿tratio nem huiuÅ¿modi figuræ in dicto libr. Copernici cla-<lb/>rè videbis. </s> - <s xml:id="s3267" xml:space="preserve">Quod verò diuersè cogitaui nunc acciptito.</s> - </p> - <p> - <s xml:id="s3268" xml:space="preserve">Cum nobis cognita Å¿it maxima ecclipticæ declinatio, vt puta <var>.a.c.</var> Å¿i latitudo <reg norm="etiam" type="context">etiã</reg> <lb/>Å¿tellæ nobis data fuerit, vt puta <var>.c.e.</var> cognitus nobis erit totalis arcus <var>.a.e.</var> & eius Å¿inus <var>.<lb/>e.m.</var> & quia notus etiam nobis eÅ¿t Å¿inus arcus <var>.a.c.</var> hoc eÅ¿t <var>.c.n.</var> & corda <var>.e.f.</var> medio eius <lb/>arcus <var>.e.p.f.</var> minoris media circunferentia, per duplum latitudinis datæ, vnde <var>.e.l.</var> eius <lb/>dimidium nobis cognitum erit, vel vt Å¿inus arcus <var>.e.p.</var> cognitus etiam nobis eÅ¿t Å¿inus <var>.<lb/>q.g.</var> declinationis <var>.a.g.</var> datæ, cui æqualis eÅ¿t <var>.m.t.</var> ex .34. primi Euclid. </s> - <s xml:id="s3269" xml:space="preserve">vnde <var>.e.t.</var> nobis <lb/>cognita remanet, cum verò duo trianguli <var>.i.c.n.</var> et <var>.t.e.K.</var> <reg norm="æquianguli" type="context">æquiãguli</reg> Å¿int, propter duas <lb/>parallelas <var>.e.m.</var> et <var>.n.c.</var> ex .28. primi, & propter duas <var>.a.b.</var> et <var>.g.h.</var> & propter duas <var>.c.d.</var> <lb/>et <var>.e.f.</var> eo quod ex communi Å¿cientia anguli <var>.c.</var> et <var>.e.</var> Å¿unt æquales, cum ex .29. dicti lib. <lb/><reg norm="vnuÅ¿quiÅ¿que" type="simple">vnuÅ¿quiÅ¿q;</reg> æqualis Å¿it angulo <var>.m.ω.i.</var> ita etiam infero de angulis <var>.e.K.t.</var> et <var>.c.i.n.</var> <reg norm="quorum" type="context">quorÅ©</reg> <lb/>vnuÅ¿quiÅ¿que æqualis eÅ¿t angulo <var>.ω.x.t.</var> & Å¿ic de alijs dico, co quod vnuÅ¿quiÅ¿que <reg norm="eorum" type="context">eorÅ©</reg> <lb/>æqualis eÅ¿t angulo <var>.m.</var> vnde cum cognitum nobis Å¿it latus <var>.n.c.</var> et <var>.c.i.</var> et <var>.t.e.</var> <reg norm="notum" type="context">notÅ©</reg> etiam <lb/>nobis erit <var>.e.K.</var> ex .19. Å¿eptimi, eo <reg norm="quod" type="simple">ê™</reg> <lb/>ex .4. Å¿exti Å¿unt inuicem proportio-<lb/> - <anchor type="figure" xlink:label="fig-0272-01a" xlink:href="fig-0272-01"/> - nalia, detrahendo poÅ¿tea <var>.e.K.</var> ab <var>.<lb/>e.l.</var> cognito, vel ècontra, hoc ab il-<lb/>lo, nobis innoteÅ¿cet <var>.K.l.</var> Å¿inus longi <lb/>tudinis Å¿tellæ.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0272-01" xlink:href="fig-0272-01a"> - <image file="0272-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0272-01"/> - </figure> - </div> - <p> - <s xml:id="s3270" xml:space="preserve">Valde etiam miror id, quod di-<lb/>ctus Stadius pag .9. illius libr. Å¿cri-<lb/>bit, hoc eÅ¿t, Solem maiorem <lb/>eſſe Luna, Å¿olum .1644. vici-<lb/>bus, </s> - <s xml:id="s3271" xml:space="preserve">propterea <reg norm="quod" type="simple">ê™</reg> cum affirmet So-<lb/>lem maiorem eſſe terra (vt etiam <lb/>in AlmageÅ¿to videre eÅ¿t) 166. vici <lb/>bus cum tribus quartis, terram ve-<lb/>ro maiorem Luna .39. vicibus cum <lb/>quarta parte, </s> - <s xml:id="s3272" xml:space="preserve">tunc Solem oporte-<lb/>ret maiorem eſſe Luna .6545. vici-<lb/>bus, & non .1644.</s> - </p> - <pb o="261" rhead="EPISTOL AE." n="273" file="0273" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0273"/> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head390" style="it" xml:space="preserve">Decognitione latitudinum stellarum.</head> - <head xml:id="head391" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3273" xml:space="preserve">AD cognoÅ¿cendam latitudinem Å¿tellæ, <reg norm="eiusque" type="simple">eiusqÌ</reg> declinationem, Monteregius in <lb/>10. propoÅ¿itione .8. li. </s> - <s xml:id="s3274" xml:space="preserve">AlmageÅ¿ti <reg norm="methodum" type="context">methodÅ©</reg> Å¿atis docuit, Å¿ed Å¿i alia aliqua metho <lb/>do hoc idem cognoÅ¿cere voluerimus, oportebit nos prius altitudinem poli cogno-<lb/>Å¿cere, deinde altitudinem meridianam ipÅ¿ius Å¿tellæ, nec non horam, <reg norm="quando" type="context">quãdo</reg> ipÅ¿a Å¿tel <lb/>la in meridiano Å¿upra terram reperitur, qua hora mediante, illicò cognoÅ¿cemus pun <lb/>ctum ecclipticæà meridiano interÅ¿ecto, eo tempore, quo Å¿tella cÅ“lum mediat Å¿u-<lb/>pra terram. </s> - <s xml:id="s3275" xml:space="preserve">Et quia ex cognita altitudine poli, illico cognoÅ¿citur altitudo æqua-<lb/>toris, cuius altitudinis differentia ab altitudine Å¿tellæ eÅ¿t declinatio ipÅ¿ius Å¿tellæ, ha-<lb/>bebimus ideo eius declinationem cognitam; </s> - <s xml:id="s3276" xml:space="preserve">qua mediante ad <reg norm="cognoÅ¿cendum" type="context">cognoÅ¿cendÅ©</reg> etiam <lb/>latitudinem ita faciemus.</s> - </p> - <p> - <s xml:id="s3277" xml:space="preserve">Sit exempli gratia <var>.p.o.u.</var> meridianus <var>.u.a.</var> verò æquator <var>.e.a.</var> autem eccliptica, & <lb/>o. centrum aÅ¿tri <var>.u.o.</var> verò eius declinatio ab æquatore, et <var>.e.a.</var> arcus Ä™cclipticæ inter <lb/>æquatorem, & meridianum, hoc eÅ¿t minor quarta, et <var>.a.u.</var> aÅ¿cenÅ¿io recta ipÅ¿ius arcus, <lb/>et <var>.u.e.</var> Å¿it declinatio puncti <var>.e.</var> Ä™cclipticÄ™ ab æquatore, <reg norm="reÅ¿iduum" type="context">reÅ¿iduÅ©</reg> vero declinationis Å¿tel-<lb/>lÄ™ Å¿it <var>.o.e.</var> quæ <reg norm="omnia" type="wordlist">oÄ©a</reg> nobis cognita erunt, <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.t.</var> polus Ä™cclipticus, à quo per <var>.o.</var> vÅ¿que ad <lb/>Ä™cclipticam tranÅ¿eat quarta <var>.t.i.</var> in qua quÄ™rendus erit arcus <var>.o.i.</var> hoc modo.</s> - </p> - <p> - <s xml:id="s3278" xml:space="preserve">Primum arcus <var>.o.u</var>: <var>e.u</var>: <var>e.o</var>: <var>a.e</var>: et <var>.a.u.</var> nobis cogniti Å¿unt, cum angulo <var>.a.</var> declinatio <lb/>nis Ä™clipticÄ™, & cum angulo <var>.u.</var> recto, vnde ex .4. primi Copernici, cognoÅ¿cemus angu <lb/>lum <var>.a.e.u.</var> collateralem, & eius <var>.o.e.i</var>. </s> - <s xml:id="s3279" xml:space="preserve">quare in triangulo <var>.o.e.i.</var> cognoÅ¿cemus <reg norm="angulum" type="context">angulÅ©</reg> <lb/>e. et deinde <var>.i.</var> vt <reg norm="rectum" type="context">rectÅ©</reg>, & latus <var>.o.e.</var> ergò ex <reg norm="eadem" type="context">eadẽ</reg> .4. cognoÅ¿cemus <reg norm="arcum" type="context">arcÅ©</reg> <var>.o.i.</var> quæſitum, <lb/>& Å¿imiliter arcum <var>.e.i.</var> qui coniunctus vel <reg norm="demptus" type="context">dẽptus</reg> ab <var>.a.e.</var> tribuet nobis longitudinem <lb/>Å¿tellÄ™, Å¿ed quia huiuÅ¿modi operatio in paruis triangulis valde fallit. </s> - <s xml:id="s3280" xml:space="preserve">Ideo tibi Å¿ua-<lb/>deo alia methodo, hoc facere, hoc eÅ¿t inuenire angulum <var>.o.</var> trianguli <var>.t.p.o.</var> cuius duo <lb/>latera <var>.t.p.</var> et <var>.p.o.</var> cognita nobis Å¿unt, cum angulo <var>.p</var>. </s> - <s xml:id="s3281" xml:space="preserve">Nam <var>.o.p.</var> eÅ¿t complementum de <lb/>clinationis Å¿tellæ, et <var>.p.t.</var> eÅ¿t arcus coluri Å¿olÅ¿titiorum inter duos polos, & angulus <var>.p.</var> <lb/>reÅ¿iduum ex recto <var>.t.p.a.</var> duorum colurum dempto angulo. a, <var>p.u.</var> cognito aÅ¿cenÅ¿ionis <lb/>recte, vnde angulus <var>.u.o.s.</var> vt contrapoÅ¿itus cognitus remanet. </s> - <s xml:id="s3282" xml:space="preserve">angulus verò <var>.u.</var> rectus <lb/>eÅ¿t, & arcus <var>.o.u.</var> cognitus, </s> - <s xml:id="s3283" xml:space="preserve">quare cognitus <lb/>nobis erit arcus <var>.u.s.</var> & angulus <var>.u.s.o.</var> vnde <lb/> - <anchor type="figure" xlink:label="fig-0273-01a" xlink:href="fig-0273-01"/> - arcus <var>.a.s.</var> nobis cognitus remanebit <reg norm="cum" type="context">cÅ©</reg> an-<lb/>gulo <var>.a.s.i.</var> reÅ¿iduo ex duobus rectis. </s> - <s xml:id="s3284" xml:space="preserve">Et quia <lb/>etiam angulus <var>.s.a.i.</var> cognitus eÅ¿t, cum Å¿it an <lb/>gulus maximÄ™ declinationis Zodiaci ab <lb/>æquatore. </s> - <s xml:id="s3285" xml:space="preserve">Ideo in triangulo <var>.a.s.i.</var> cuius <lb/>duo anguli <var>.a.</var> et <var>.s.</var> cum latere <var>.a.s.</var> dantur, fa <lb/>cilè inueniemus arcum <var>.s.i.</var> <reg norm="cum" type="context">cÅ©</reg> arcu <var>.a.i.</var> Å¿ed <lb/><var>a.i.</var> erit longitudinis Å¿tellæ dempto poÅ¿tea <var>.<lb/>s.i.</var> ex <var>.s.o.</var> iam inuento habebimus arcum <var>.i.<lb/>o.</var> latitudinis ipÅ¿ius Å¿tellæ.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0273-01" xlink:href="fig-0273-01a"> - <image file="0273-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0273-01"/> - </figure> - </div> - <p> - <s xml:id="s3286" xml:space="preserve">Hæc autem tibi Å¿cribo non vt ipÅ¿is vta-<lb/>ris, Å¿ed potius vt tibi morem <reg norm="geram" type="context">gerã</reg>, cum bre <lb/>uiſſima methodus Å¿it illa, <reg norm="quam" type="context">quã</reg> Monteregius <lb/>Å¿cripÅ¿it <reg norm="in" type="wordlist">Ä©</reg> .10. <reg norm="propoÅ¿itione" type="simple">ê“poÅ¿itione</reg> .8. li. in AlmageÅ¿t.</s> - </p> - <pb o="222" rhead="IO. BAPT. BENED." n="274" file="0274" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0274"/> - </div> - </div> - <div type="section" level="3" n="8"> - <div type="letter" level="4" n="1"> - <head xml:id="head392" style="it" xml:space="preserve">Qualiter circulus deÅ¿ignari poßit alios duos circulos <lb/>propoÅ¿itos includens.</head> - <head xml:id="head393" xml:space="preserve">CLARISS. PETRO PIZZAMANO.</head> - <p> - <s xml:id="s3287" xml:space="preserve">SVperioribus diebus per tuas literas à me quæſiuiÅ¿ti, vt modum tibi Å¿cribere vel-<lb/>lem, quo circulus deÅ¿ignari poſſit circunÅ¿cribens alios duos propoÅ¿itos circulos. <lb/></s> - <s xml:id="s3288" xml:space="preserve">Qua in re vt tibi Å¿atisfaciam quod maximè cupio ita rem accipe.</s> - </p> - <p> - <s xml:id="s3289" xml:space="preserve">PropoÅ¿iti circuli Å¿int, aut inter Å¿e contigui, aut interÅ¿ecantes vel Å¿eparati. </s> - <s xml:id="s3290" xml:space="preserve">EÅ¿to <reg norm="pri- mum" type="context">pri-<lb/>mÅ©</reg> contiguos eſſe, qui Å¿int <var>.d.b.</var> et <var>.f.q.</var> <reg norm="quorum" type="context">quorÅ©</reg> <var>.d.b.</var> maior Å¿it et <var>.f.q.</var> minor, <reg norm="eorum" type="context">eorÅ©</reg> vero <lb/>centra Å¿int .a et <var>.o.</var> <reg norm="punctum" type="context">punctÅ©</reg> autem <reg norm="contingentiÄ™" type="context">cõtingentiÄ™</reg> Å¿it <var>.i</var>. </s> - <s xml:id="s3291" xml:space="preserve"><reg norm="Nunc" type="context">NÅ©c</reg> <reg norm="protrahatur" type="simple simple">ê“trahat̃</reg>. <var>b.a.o.q.</var> per <reg norm="centra" type="context">cẽtra</reg> eo <lb/>rum ab vna circunferentia ad aliam, quÄ™ quidem linea tranÅ¿ibit per punctum <var>.i.</var> ex <lb/>11, tertij Eucli. </s> - <s xml:id="s3292" xml:space="preserve">deinde à diametro maiori abÅ¿cindatur <var>.i.e.</var> ad æqualitatem minoris <lb/>Å¿emidiametri, quo facto Å¿umatur diÅ¿tantia inter <var>.e.</var> et <var>.b.</var> circino mediante <reg norm="factoque" type="simple">factoqÌ</reg> cen <lb/>tro <var>.o.</var> Å¿cindatur, alio circini pede, circunferentia maioris circuli in puncto <var>.u.</var> à quo Å¿i <lb/>mente concipiemus duas lineas <var>.u.a.d.</var> et <var>.u.o.f.</var> tranÅ¿euntes per eorum centra <var>.a.</var> et <var>.o.</var> <lb/>vÅ¿que ad circunferentias in punctis <var>.d.</var> et <var>.f.</var> ipſę <reg norm="erunt" type="context">erÅ©t</reg> inuicem Ä™quales, eo quod <var>.e.i.</var> <reg norm="sum- pta" type="context">sÅ©-<lb/>pta</reg> fuit æqualis <var>.o.f.</var> et <var>.o.u.</var> æqualis <var>.e.b.</var> </s> - <s xml:id="s3293" xml:space="preserve">quare <var>.u.f.</var> æqualis erit <var>.b.i.</var> Å¿ed <var>u.d.</var> <reg norm="etiam" type="context">etiã</reg> æqua <lb/>lis <var>.b.i.</var> ergo <var>.u.d.</var> æqualis erit <var>u.f.</var> & circulus, cuius <var>u.d.</var> vel <var>.u.f.</var> erit Å¿emidiameter, con-<lb/>tiguus erit ipÅ¿is propoÅ¿itis circulis ex conuerÅ¿o .11. iam dictæ. </s> - <s xml:id="s3294" xml:space="preserve">Idem dico pro circu-<lb/>lis Å¿e inuicem Å¿ecantibus.</s> - </p> - <figure position="here"> - <image file="0274-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0274-01"/> - </figure> - <figure position="here"> - <image file="0274-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0274-02"/> - </figure> - <pb o="263" rhead="EPISTOL AE." n="275" file="0275" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0275"/> - <p> - <s xml:id="s3295" xml:space="preserve">Sed Å¿i circuli propoÅ¿iti Å¿eiuncti fuerint, Å¿umatur <var>.b.i.</var> diameter maioris, qui fiat Å¿e-<lb/>midiameter vnius circuli circa centrum <var>.o.</var> & hic circulus vocetur <var>.h.x.</var> coniunga-<lb/>tur deinde Å¿emidiameter <var>.o.i.</var> minoris circuli cum Å¿emidiametro <var>.a.i.</var> circuli maio-<lb/>ris, & ex huiuÅ¿modi compoÅ¿ita linea, fiat vnus Å¿emidiameter <var>.a.x.</var> circuli <var>.x.n.</var> concen <lb/>trici cum maiori, & à puncto <var>.x.</var> interÅ¿ectionis horum circulorum (poÅ¿ito quod Å¿e in-<lb/>uicem interÅ¿ecent) ducantur per eorum centra <var>.x.a.</var> et <var>.x.o.</var> vÅ¿que ad ipÅ¿orum circun-<lb/>ferentias in punctis <var>.d.</var> et <var>.f.</var> duÄ™ <lb/>lineæ, vnde habebimus <var>.x.d.</var> <lb/>æqualem <var>.x.f.</var> eo quod tam in <lb/> - <anchor type="figure" xlink:label="fig-0275-01a" xlink:href="fig-0275-01"/> - <var>x.d.</var> quam in <var>.x.f.</var> reperiuntur <lb/>diametri, & Å¿emidiametri am-<lb/>borum circulorum, facto deni <lb/>que centro <var>.x.</var> vnius circuli, cu <lb/>ius Å¿emidiameter Ä™qualis Å¿it <lb/>vni earum <var>.x.d.</var> vel <var>.x.f.</var> folu-<lb/>tum erit problema, dicta ra-<lb/>tione.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0275-01" xlink:href="fig-0275-01a"> - <image file="0275-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0275-01"/> - </figure> - </div> - <p> - <s xml:id="s3296" xml:space="preserve">Si verò diÅ¿tantia duorum <lb/>propoÅ¿itorum circulorum tanta fuerit, quod Å¿ecundi circuli nequeant Å¿e inuicem <lb/>tangere, vel Å¿ecare, tunc alia via incedendum erit, quÄ™ talis eÅ¿t & generalis. </s> - <s xml:id="s3297" xml:space="preserve">Diuida-<lb/>tur tota <var>.q.b.</var> per æqualia in puncto <var>.z.</var> circa quod <reg norm="Å¿ignentur" type="context">Å¿ignẽtur</reg> duo puncta ab ipÅ¿o Ä™quidi <lb/>Å¿tantia <var>.K.</var> et <var>.p.</var> diÅ¿tantia vero <var>.a.K.</var> facta Å¿it Å¿emidiameter eſſe vnius circuli <var>.K.x.</var> circa <lb/>centrum <var>.a.</var> diÅ¿tantia autem <var>.o.p.</var> Å¿emidiameter alterius circuli <var>.p.x.</var> circa cen-<lb/>trum <var>.o.</var> qui quidem circuli Å¿e inuicem Å¿ecent in puncto <var>.x.</var> à quo cum ductÄ™ fue-<lb/>rinc <var>.x.a.d.</var> et <var>.x.o.f.</var> per centra dictorum circulorum, ipÅ¿e erunt <reg norm="inuicem" type="context">inuicẽ</reg> Ä™quales, eo <reg norm="quod" type="wordlist">qÄ‘</reg> <lb/>cum <var>.b.K.</var> æqualis Å¿it <var>.q.p.</var> igitur <var>.x.d.</var> et <var>.q.p.</var> erunt inuicem Ä™quales, Å¿ed <var>.f.x.</var> æqualis eÅ¿t <lb/><var>q.p</var>. </s> - <s xml:id="s3298" xml:space="preserve">quare <var>.x.f.</var> æqualis erit <var>.x.d.</var> tunc Å¿i <var>.x.</var> centrum fuerit vnius circuli, cuius Å¿emidia-<lb/>mer Å¿it vna dictarum, problema Å¿olutum erit.</s> - </p> - <p> - <s xml:id="s3299" xml:space="preserve">Talis etiam Å¿oiutio commo-<lb/>da erit ad inueniendum dictum <lb/> - <anchor type="figure" xlink:label="fig-0275-02a" xlink:href="fig-0275-02"/> - circulum cuiuÅ¿uis magnitudinis, <lb/>dato tamen <reg norm="quod" type="simple">ê™</reg> eius diameter, ma <lb/>ior Å¿it <var>.b.z.</var> cum in noÅ¿tra poteÅ¿ta <lb/>te Å¿it accipere puncta <var>.K.</var> et <var>.p.</var> pro <lb/>xima vel remota ab ipÅ¿o <var>.z.</var> ad li-<lb/>bitum. </s> - <s xml:id="s3300" xml:space="preserve">Vnde abÅ¿que vlla diuiÅ¿io <lb/>neipÅ¿ius <var>.q.b.</var> per medium, Å¿atis <lb/>erit Å¿ignare puncta <var>.K.</var> et <var>.p.</var> dua-<lb/>bus diÅ¿tantijs mediantibus <var>.b.K.</var> <lb/>et <var>.q.p.</var> inuicem æqualibus, & <lb/>etiam propoÅ¿itis.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0275-02" xlink:href="fig-0275-02a"> - <image file="0275-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0275-02"/> - </figure> - </div> - <handwritten/> - <pb o="264" rhead="IO. BAPT. BENED." n="276" file="0276" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0276"/> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head394" style="it" xml:space="preserve">Figuram Å¿uperficialem ellipÅ¿i Å¿imilem, ex datis axibus cir-<lb/>cino mediante delineari poſſe.</head> - <head xml:id="head395" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3301" xml:space="preserve">FIguram Å¿uperficialem ellipÅ¿i Å¿imilem, ex datis axibus, circino mediante delinea <lb/>re cum volueris, ita facito.</s> - </p> - <p> - <s xml:id="s3302" xml:space="preserve">Sit <var>.e.c.</var> Å¿emiaxis maior <var>.a.e.</var> verò minor, ad angulum rectum inuicem coniuncti, <lb/>tunc <var>.a.e.</var> producatur vÅ¿que ad <var>.o</var>. </s> - <s xml:id="s3303" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> <var>.a.o.</var> maior Å¿it quam diÅ¿tantia inter <var>.o.</var> et <var>.c.</var> quÄ™ <lb/>quidem <var>.a.o.</var> poſſet etiam dari, deÅ¿cribatur poÅ¿tea circulus <var>.a.d.b.</var> circa centrum <var>.o.</var> à <lb/>quo puncto protrahatur Å¿emidiameter <var>.o.b.</var> quæ cum <var>.a.o.</var> angulum rectum conÅ¿ti-<lb/>tuat, quÄ™ <var>.o.b.</var> erit æquidiÅ¿tans <var>.e.c.</var> ex .28. primi, ducatur poÅ¿tea <var>.b.c.d.</var> et <var>.o.t.d.</var> vnde <lb/>angulus <var>.t.c.d.</var> Ä™qualis erit angulo <var>.o.b.d.</var> ex .29. eiuÅ¿dem. </s> - <s xml:id="s3304" xml:space="preserve">ex quinta autem anguli <var>.b.</var> <lb/>et <var>.d.</var> Å¿unt inuicem æquales, </s> - <s xml:id="s3305" xml:space="preserve">quare etiam <lb/>& anguli <var>.d.</var> et <var>.c.</var> inuicem Ä™quales erunt, <lb/> - <anchor type="figure" xlink:label="fig-0276-01a" xlink:href="fig-0276-01"/> - & ex .6. eiuÅ¿dem <var>.t.c.</var> Ä™qualis erit <var>.t.d.</var> duca <lb/>tur poÅ¿tea <var>.d.x.h.</var> perpendicularis lineæ <var>.c.<lb/>e.</var> ita diÅ¿tans Å¿ub ipÅ¿a <var>.c.e.</var> vt arcus circula-<lb/>ris circa <var>.t.</var> delineatus ex Å¿emidiametro <var>.t.<lb/>d.</var> aptus Å¿it eam Å¿ecare, Å¿umpto poÅ¿tea <var>.r.</var> <lb/>tam diÅ¿tante ab <var>.e.</var> vt <var>.t.</var> reperitur ab ipÅ¿o <lb/>e. et <var>.z.</var> ab <var>.e.</var> vt <var>.o.</var> ab eodem, ducendo po-<lb/>Å¿tea duos alios arcus magnitudinis <reg norm="priorum" type="context">priorÅ©</reg> <lb/>circa centra <var>.r.</var> et <var>.z.</var> habebimus propoÅ¿i-<lb/>tum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0276-01" xlink:href="fig-0276-01a"> - <image file="0276-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0276-01"/> - </figure> - </div> - <p> - <s xml:id="s3306" xml:space="preserve">Sed cum quis voluerit prius arcus mi-<lb/>norum circulorum delineare circa maio-<lb/>rem axem, fiant cuiuÅ¿uis magnitudinis, vt <lb/>in Å¿ecunda figura videre eÅ¿t, poÅ¿ito tamen quod eorum diameter, minor Å¿it minore <lb/>axe ipÅ¿ius figurÄ™, quorum circulorum vnus Å¿it <var>.c.d.</var> circa <var>.t.</var> eius centrum, deinde in axe <lb/>minori Å¿umatur <var>.a.x.</var> æqualis <var>.c.t.</var> & protrahatur <var>.t.x.</var> quÄ™ per Ä™qualia diuidatur in pun-<lb/>cto <var>.n.</var> à quo poÅ¿tea ducatur <var>.n.o.</var> ad angulos rectos <lb/> - <anchor type="figure" xlink:label="fig-0276-02a" xlink:href="fig-0276-02"/> - cum <var>.t.x.</var> vÅ¿que ad interÅ¿ectionem cum <var>.a.e.</var> in pun-<lb/>cto <var>.o.</var> minori axi producta cum oportuerit, quod <lb/>quidem punctum <var>.o.</var> centrum erit arcus <var>.d.a.</var> maio-<lb/>ris, eo quod <var>.o.t.</var> æqualis eſſet <var>.o.x.</var> ex .4. primi Eu-<lb/>cli. </s> - <s xml:id="s3307" xml:space="preserve">vnde <var>.o.d.</var> æqualis eſſet <var>.o.a.</var> & circuli etiam in-<lb/>uicem contingentes in puncto <var>.d.</var> ex .11. tertij tam <lb/>in prima, quam in Å¿ecunda figura, Å¿umpto <reg norm="denique" type="simple">deniq;</reg> <lb/>puncto <var>.s.</var> tam remoto ab <var>.e.</var> quam <var>.o.</var> reperitur ab <lb/>eodem, ipÅ¿um, centrum erit alterius arcus oppoÅ¿i-<lb/>ti, poſſemus etiam <reg norm="abÅ¿que" type="simple">abÅ¿q;</reg> diuiÅ¿ione ipÅ¿ius, <var>t.x.</var> conÅ¿ti <lb/>tuere angulum <var>.x.t.o.</var> <reg norm="æqualem" type="context">æqualẽ</reg> angulo <var>.t.x.o.</var> vnde ex <lb/>6. primi haberemus <var>.o.t.</var> æqualem <var>.o.x</var>.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0276-02" xlink:href="fig-0276-02a"> - <image file="0276-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0276-02"/> - </figure> - </div> - <pb o="265" rhead="EPISTOLAE." n="277" file="0277" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0277"/> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head396" style="it" xml:space="preserve">De inuentione axis propoÅ¿ite portionis datæ Å¿phæræ.</head> - <head xml:id="head397" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3308" xml:space="preserve">VTaxem propoÅ¿itæ alicuius datæ Å¿phæræ inuenire poſſis ita tibi operandum eÅ¿t <lb/>vt gratia exempli. </s> - <s xml:id="s3309" xml:space="preserve">PropoÅ¿ita nobis eÅ¿t Å¿phæra <var>.c.i.e.t.</var> diametri cognitæ. </s> - <s xml:id="s3310" xml:space="preserve">pro <lb/>poÅ¿ita etiam eÅ¿t nobis eius portio <var>.n.e.u.</var> axis <var>.e.a.</var> cognitæ minoris Å¿emidiametro, da-<lb/>ta etiam nobis eÅ¿t proportio alterius portionis minoris hemiÅ¿phærio <var>.i.e.t.</var> ad por-<lb/>tionem <var>.n.e.u.</var> quæritur nunc quantus Å¿it axis <var>.e.x.</var> Å¿ecundæ portionis hoc eÅ¿t deÅ¿idera-<lb/>mus cognoÅ¿cere proportionem <var>.e.x.</var> ad <var>.e.a.</var> vel ad diametrum ipÅ¿ius Å¿pheræ.</s> - </p> - <p> - <s xml:id="s3311" xml:space="preserve">Cuius gratia reperiatur primò proportio <reg norm="circunferentiæ" type="context">circÅ©ferentiæ</reg> maioris circuli ipÅ¿ius <reg norm="Å¿phae ræ" type="simple">Å¿phÄ™Â<lb/>ræ</reg> adeius diametrum, quæ ferè eÅ¿t vt .22. ad .7. ex Archimede.</s> - </p> - <p> - <s xml:id="s3312" xml:space="preserve">Quo facto, inueniatur quantitas Å¿uperficialis huiuÅ¿modi maioris circuli, quæ Å¿em-<lb/>per æqualis eÅ¿t producto quod fit ex Å¿emidiametro in dimidium circunferentiæ ip-<lb/>fius circuli, ex eodem Archimede. </s> - <s xml:id="s3313" xml:space="preserve">Et Å¿ic cognoÅ¿cemus quartam partem Å¿uperficiei <lb/>Å¿phæricæ Å¿phærÄ™ propoÅ¿ite ex .31. primi lib. de Å¿phæra, & cyllindro Archimedis.</s> - </p> - <p> - <s xml:id="s3314" xml:space="preserve">Deinde Å¿umatur tertia pars producti, quod fit ex Å¿emidiametro in Å¿uperficiem <lb/>maioris circuli, & habebimus conum, cuius baÅ¿is erit circulus maior, altitudo verò <lb/>Å¿emidiameter propoÅ¿itæ Å¿phæræ ex .9. duodecimi Eucli.</s> - </p> - <p> - <s xml:id="s3315" xml:space="preserve">Quadruplum poÅ¿tea huiuÅ¿modi coni, erit quantitas Å¿oliditatis, Å¿eu corporeitas to <lb/>tius Å¿phærÄ™ ex .32. dicti lib. Archimedis.</s> - </p> - <p> - <s xml:id="s3316" xml:space="preserve">Imaginemur poÅ¿tea <reg norm="in" type="wordlist">Ä©</reg> Å¿phærica portione <var>.n.e.u.</var> <reg norm="lineam" type="context">lineã</reg> <var>.e.u.</var> à <reg norm="summitate" type="context">sÅ©mitate</reg> ad <reg norm="extremitatem" type="context">extremitatẽ</reg> <lb/>baÅ¿is, cuius <var>.e.u.</var> quantitatem cognoÅ¿cemus, hoc modo Å¿cilicet, fumendo <reg norm="radicem" type="context">radicẽ</reg> qua-<lb/>dratam producti <var>.c.e.</var> in <var>.e.a.</var> eo quod <lb/>quadratum <var>.e.u.</var> æquale eÅ¿t quadrato <lb/> - <anchor type="figure" xlink:label="fig-0277-01a" xlink:href="fig-0277-01"/> - <var>a.u.</var> & quadrato <var>.a.e.</var> ex penultima <lb/>primi Eucli. </s> - <s xml:id="s3317" xml:space="preserve">hoc eÅ¿t producto quod <lb/>fit ex <var>.c.a.</var> in <var>.a.e.</var> ex .34. tertij <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg>, <lb/>& quadrato <var>.a.e.</var> hoc eÅ¿t producto, <lb/>quod fit ex <var>.c.e.</var> in <var>.e.a.</var> ex .3. Å¿ecundi <lb/>eiuÅ¿dem.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0277-01" xlink:href="fig-0277-01a"> - <image file="0277-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0277-01"/> - </figure> - </div> - <p> - <s xml:id="s3318" xml:space="preserve">Inuenta poÅ¿tea <var>.e.u.</var> ponamus eam <lb/>vnius circuli Å¿emidiametrum eſſe, cu <lb/>ius Å¿uperficialis quantitas etiam inue <lb/>niatur, vt Å¿upra dictum eÅ¿t, quæ qui <lb/><reg norm="dem" type="context">dẽ</reg> æqualis erit Å¿uperficiei portionis <lb/><var>n.e.u.</var> ex .40. primi li. </s> - <s xml:id="s3319" xml:space="preserve">Archimedis de <lb/>Å¿phæra, & cyllindro.</s> - </p> - <p> - <s xml:id="s3320" xml:space="preserve">Hæc autem quantitas vltimo <reg norm="inuem" type="context">inuẽ</reg> <lb/>ta multiplicetur cum tertia parte Å¿e-<lb/>midiametri datæ Å¿phæræ, & habebi-<lb/>mus Å¿oliditatem vnius coni æqualis <lb/>aggregato Å¿oliditatis portionis <var>.n.e.<lb/>u.</var> Å¿imul Å¿umptÄ™<unsure/>, <reg norm="cum" type="context">cÅ©</reg> Å¿oliditate vnius co <lb/>ni, cuius axis Å¿it <var>.a.o.</var> <reg norm="reÅ¿iduum" type="context">reÅ¿iduÅ©</reg> Å¿emidia-<lb/>metri noÅ¿træ Å¿phæræ dempta <var>.a.e.</var> ba - <pb o="266" rhead="IO. BABPT. BENED." n="278" file="0278" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0278"/> - Å¿is verò eadem quæ eÅ¿t portionis, cuius diameter eÅ¿t <var>.n.u.</var> ex .9. 12. Eucli. & ex .42. id-<lb/>eÅ¿t vltima primi Archimedis de Å¿phæra, & cyllindro.</s> - </p> - <p> - <s xml:id="s3321" xml:space="preserve">Nunc autem ex hoc aggregato iam vltimo dicto detrahatur conus, cuius <var>.o.a.</var> eÅ¿t <lb/>axis et <var>.n.u.</var> diameter baÅ¿is, qui quidem conus nobis cognitus eÅ¿t, cum <var>.a.n.</var> Å¿emidia-<lb/>meter eius baÅ¿is, nobis cognita Å¿it ex .34. 3. Eucli. </s> - <s xml:id="s3322" xml:space="preserve">& Å¿ic quantitas eius baÅ¿is, & ita ter-<lb/>tia pars <var>.a.o.</var> eius axis, quÄ™ multiplicata cum dicta baÅ¿i, cuius <var>.n.u.</var> eÅ¿t diameter, produ <lb/>cit dictum conum, qui quidem conus, vt diximus, demptus cum fuerit ex dicto ag-<lb/>gre gato, relinquet nobis Å¿oliditatem portionis <var>.n.e.u.</var> vnde cognoÅ¿cemus proportio <lb/>nem iÅ¿tius portionis ad totam Å¿phæram propoÅ¿itam.</s> - </p> - <p> - <s xml:id="s3323" xml:space="preserve">Sed cum nobis propoÅ¿ita Å¿it proportio portionis <var>.n.e.u.</var> ad portionem <var>.i.e.t.</var> cogno <lb/>Å¿cemus etiam Å¿oliditatem huius Å¿ecundÄ™ portionis <var>.i.e.t.</var> & Å¿imiliter <reg norm="proportionem" type="context">proportionẽ</reg> hu-<lb/>ius ad totam Å¿phÄ™ram, & ad <reg norm="reÅ¿iduum" type="context">reÅ¿iduÅ©</reg> <reg norm="etiam" type="context">etiã</reg> ipÅ¿ius Å¿phÄ™rÄ™ hoc eÅ¿t portioni <var>.i.c.t</var>.</s> - </p> - <p> - <s xml:id="s3324" xml:space="preserve">Protrahatur nunc diameter <var>.c.e.</var> à parte <var>.e.</var> <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> quo <var>.e.f.</var> æqualis Å¿it <var>.e.o.</var> Å¿emidiame <lb/>tro Å¿phÄ™rÄ™, quæ quidem <var>.f.e.</var> diuidatur in puncto <var>.h.</var> ita vt proportio <var>.f.h.</var> ad <var>.h.e.</var> æqua-<lb/>lis Å¿it proportioni portionis <var>.i.c.t.</var> ad portionem <var>.i.e.t.</var> quod quidem hoc modo efficieÂ<lb/>tur. </s> - <s xml:id="s3325" xml:space="preserve">applicabimus lineam <var>.f.q.</var> (indeterminatam) cum <var>.f.e.</var> ad quemuis angulum in <reg norm="pun- cto" type="context">pÅ©-<lb/>cto</reg> <var>.f.</var> in qua accipiemus duas lineas <var>.f.p.</var> et <var>p.q.</var> inuicem ita relatas, vt Å¿e habent in pro <lb/>portione duæ iam dictæ portiones, hoc eÅ¿t, vt <var>.i.c.t.</var> portio ad portionem <var>.i.e.t.</var> ducen <lb/>do poÅ¿tea <var>.q.e.</var> et <var>.p.h.</var> parallelam ad ipÅ¿am <var>.q.e.</var> diuiÅ¿am habebimus <var>.f.e.</var> in eadem pro <lb/>portione vt dictum eÅ¿t ex .2. Å¿exti, & .11 quinti Euclidis, vnde <var>.c.e</var>: <var>e.f.</var> et <var>.f.h.</var> nobis co <lb/>gnitÄ™ erunt.</s> - </p> - <p> - <s xml:id="s3326" xml:space="preserve">Oportebit nos nunc cognoÅ¿cere quantitatem <var>.c.x.</var> hoc modo, videlicet, quÄ™ramus <lb/>quadratum, cuius <var>.c.x.</var> eius Å¿it radix, cui quadratum lineÄ™ <var>.c.e.</var> cognitum, ita Å¿it propor-<lb/>tionatum, vt eÅ¿t linea <var>.x.f.</var> ad lineam <var>.f.h.</var> quÄ™ nobis cognita eÅ¿t, quod rectè factum erit <lb/>ex eo, quod Å¿cripÅ¿it Archimedes in .4. Å¿ecundi de Å¿phÄ™ra, & cyllindro.</s> - </p> - <p> - <s xml:id="s3327" xml:space="preserve">Sed quia Archimedes eo in loco Å¿upponit id, quod necipÅ¿e, nec alius adhuc inue <lb/>nit, niÅ¿i via naturali, hoc eÅ¿t tres partes Ä™quales ex proportione data effici, non erit in <lb/>conueniens etiam nobis hac via, circa hoc aliquid dicere.</s> - </p> - <p> - <s xml:id="s3328" xml:space="preserve">Accipiemus igitur diametrum <var>.c.e.</var> cum addita <var>.e.f.</var> eius Å¿emidiametro, diuidemusÂ<lb/>qÌue <var>.f.e.</var> in puncto <var>.h.</var> vt Å¿upra factum fuit, applicabimus poÅ¿tea <var>.c.m.</var> indeterminatam <lb/>angulariter ad <var>.c.e.</var> à qua <var>.c.m.</var> accipiemus <var>.c.g.</var> æqualem <var>.f.h.</var> quÄ™remus deinde natu-<lb/>rali via punctum <var>.b.</var> ita ut protrahendo à puncto <var>.e.</var> (altero extremo diametri) <var>e.m.</var> pa <lb/>rallelam ad <var>.b.g.</var> ductam, erigendo <var>.b.d.</var> perpendicularem ad <var>.c.e.</var> in puncto <var>.b.</var> protra <lb/><reg norm="ctaque" type="simple">ctaqÌ;</reg> <var>.d.c.</var> quæ à diametro <var>.e.c.</var> deducta ab <var>.c.</var> incohando vÅ¿que ad <var>.x.</var> relinquat nobis <var>.<lb/>x.f.</var> Ä™qualem <var>.c.m</var>.</s> - </p> - <p> - <s xml:id="s3329" xml:space="preserve">Cuius rei ratio eÅ¿t, quia quadratum <var>.c.e.</var> Å¿e habet ad quadratum <var>.c.d.</var> vt <var>.c.e.</var> ad <var>.c.<lb/>b.</var> ex .4. et .18. Å¿exti Eucl. </s> - <s xml:id="s3330" xml:space="preserve">Å¿ed ex .4. ita Å¿e habet <var>.m.c.</var> ad <var>.c.g.</var> vt <var>.e.c.</var> ad <var>.b.c.</var> & cum Å¿it <var>.c.<lb/>g.</var> Ä™q alis <var>.f.h.</var> Å¿i <var>.c.m.</var> Ä™qualis fuerit <var>.f.x.</var> habebimus propoÅ¿itum. </s> - <s xml:id="s3331" xml:space="preserve">Quod Å¿i quis per di-<lb/>Å¿cretum vel et hoc facere, ita ei agendum erit.</s> - </p> - <p> - <s xml:id="s3332" xml:space="preserve">Ponamus exempli gratia totum diametrum <var>.c.e.</var> propoÅ¿itæ Å¿phæræ eſſe ut decem, <lb/><reg norm="proportionemque" type="simple">proportionemqÌ;</reg> reÅ¿iduÄ™ portionis <var>.i.c.t.</var> ad Å¿ecundam <var>.i.e.t.</var> hoc eÅ¿t <var>.f.h.</var> ad <var>.h.e.</var> Å¿eÅ¿qui-<lb/>alteram eſſe, vnde <var>.e.h.</var> bis tertia erit ìpÅ¿ius <var>.f.h.</var> <reg norm="totaque" type="simple">totaqÌ;</reg> linea <var>.c.f.</var> erit .15. et <var>.f.h.</var> erit .3. <lb/>& quadratum lineæ <var>.c.e.</var> erit .100.</s> - </p> - <p> - <s xml:id="s3333" xml:space="preserve">Quærendo poÅ¿tea quadratum lineæ <var>.c.x.</var> cui quadratum <var>.c.e.</var> hoc eÅ¿t .100. ita pro-<lb/>portionatum Å¿it vt <var>.f.x.</var> ad <var>.f.h.</var> hoc eÅ¿t ad .3. Å¿i autem cogitauerimus <var>.c.x.</var> eſſe nouem <lb/>partium talium qualium <var>.c.e.</var> eÅ¿t decem, eius quadratum erit .81. et <var>.x.f.</var> erit .6. par-<lb/>tium talium qualium <var>.c.f.</var> eÅ¿t .15. dicendo poÅ¿tea Å¿i .100. dat .81. (ex regula de tribus) - <pb o="267" rhead="EPISTOL AE." n="279" file="0279" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0279"/> - <var>x.f.</var> hoc eÅ¿t .6. dabit .4. integra cum <num value="86">.<lb/> - <anchor type="figure" xlink:label="fig-0279-01a" xlink:href="fig-0279-01"/> - 86.</num> centeÅ¿imis, Å¿ed nos vellemus no-<lb/>bis prouenire tria, eo <reg norm="quod" type="simple">ê™</reg> ita eÅ¿t <var>.f.h.</var> <lb/>qua propter deÅ¿cendere nos oporte-<lb/>bit à nouem ad .8. & ab .8. ad .7. & à . <lb/>7. ad .6. </s> - <s xml:id="s3334" xml:space="preserve">tunc inueniemus <var>.c.x.</var> oporte-<lb/>re eſſe circiter quinque cum duabus <lb/>tertijs, <reg norm="operando" type="context">operãdo</reg> poÅ¿tea ex regula de <lb/>tribus, Å¿i dixerimus quando .100. no-<lb/>bis dat .32. cum nona parte integri, <lb/></s> - <s xml:id="s3335" xml:space="preserve">tunc nouem cum tertia parte integri <lb/>dabit .2. <reg norm="cum" type="context">cÅ©</reg> 296. de .300. hoc eÅ¿t .2. <lb/>cum circa .49. quinquageÅ¿imis, quæ <lb/>quidem quantitas, cum propinquiſſi <lb/>ma Å¿it lineæ <var>.f.h.</var> trium integrorum di <lb/>cemus <var>.c.x.</var> eſſe quinque integrorum <lb/>cum duabus tertijs partibus vnius in <lb/>tegri, et <var>.e.x.</var> reÅ¿iduum, hoc eÅ¿t axem <lb/>quæſitum portionis <var>.i.e.t.</var> eſſe circa .4 <lb/>integra cum tertia parte vnius inte-<lb/>gri.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0279-01" xlink:href="fig-0279-01a"> - <image file="0279-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0279-01"/> - </figure> - </div> - </div> - </div> - <div type="section" level="3" n="9"> - <div type="letter" level="4" n="1"> - <head xml:id="head398" xml:space="preserve">DE ERRORIBVS THOMAE PORCACHII <lb/>& Benedicti Bordonij in eorum inÅ¿ularijs.</head> - <head xml:id="head399" style="it" xml:space="preserve">Excellentißimo D. lo. BaptiÅ¿tæ Fæmello Ciui <reg norm="Decurionique" type="simple">DecurioniqÌê«</reg> Tau-<lb/>rinenÅ¿i PhiloÅ¿opho, Medico, & in Accademia eius <lb/>Ciuitatis Medicinæ Practicæ Ordinario, <reg norm="Pri- marioque" type="simple">Pri-<lb/>marioqÌê«</reg> profeßori celeberrimo.</head> - <p> - <s xml:id="s3336" xml:space="preserve">DIj perdant tuas adeo moleÅ¿tas, & aſſiduas curas, quæ te nimis à ſuauiori-<lb/>bus Å¿tudijs diÅ¿trahunt, & à nobis longius abducunt. </s> - <s xml:id="s3337" xml:space="preserve">Nam, ut tibi <reg norm="quietem" type="context">quietẽ</reg>, <lb/>ita mihi ingentem adimunt voluptatem. </s> - <s xml:id="s3338" xml:space="preserve">Sed ne in aliquo erga te defi-<lb/>cere videar, quæ tibi olim promiÅ¿i, nunc mitto.</s> - </p> - <p> - <s xml:id="s3339" xml:space="preserve">Negari quidem non poteÅ¿t, quin fuerit laborioÅ¿um opus Porcachij, & Benedicti <lb/>Bordonij, hoc eÅ¿t inÅ¿ularium, qui rectè etiam feciſſent, cum loqui eos oportebat de <lb/>terminis Å¿phæræ ratione Å¿itus locorum, Å¿i Å¿eipÅ¿os alicuius excellentis CoÅ¿mographi <lb/>conÅ¿ilio Å¿ubmiÅ¿iſſent. </s> - <s xml:id="s3340" xml:space="preserve">ConÅ¿idera quæſo, quomodo admitti poſſit, id quod ait Por-<lb/>cachius initio Å¿ui operis, ideÅ¿t IÅ¿landiam Å¿ub Polo arctico iacere, inter auÅ¿trum, & <lb/>boream: </s> - <s xml:id="s3341" xml:space="preserve">omittamus etiam quod idem in ProÄ™mio lib. Å¿ecundi, vbi ait Biarmiam, <lb/>(& non IÅ¿landiam) eſſe Å¿ub dicto polo arctico: </s> - <s xml:id="s3342" xml:space="preserve">in <reg norm="eodemque" type="simple">eodemqÌ;</reg> principio repetit ipÅ¿am <lb/>IÅ¿landiam inter auÅ¿trum, & boream per centum leucas Germanicas extendi, dein-<lb/>de verÅ¿us occidentem, ea duo Å¿tupenda miracula conÅ¿pici. </s> - <s xml:id="s3343" xml:space="preserve">Vide quæſo, quomodo <lb/>incolÄ™ Å¿ub aliquo ex polis, habere poſſint occidentem, orientem, magiÅ¿trum, <reg norm="auÅ¿trum" type="context">auÅ¿trÅ©</reg>, - <pb o="268" rhead="IO. BAPT. BENDE." n="280" file="0280" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0280"/> - & boream, & vt melius dicam aliquem rhombum. </s> - <s xml:id="s3344" xml:space="preserve">Sed quomodo fieri poteÅ¿t, vt in-<lb/>Å¿ula IÅ¿landiæ Å¿it Å¿ub polo, eius tamen dies, & nox maior non Å¿it longior Å¿patio <reg norm="trium" type="context">triÅ©</reg> <lb/>menÅ¿ium? </s> - <s xml:id="s3345" xml:space="preserve">vt ipÅ¿e pagina .62. in proÄ™mio Å¿ecundi lib. affirmat, quamuis hoc à Bordo <lb/>ne deÅ¿umat. </s> - <s xml:id="s3346" xml:space="preserve">In quo <reg norm="vterque" type="simple">vterq;</reg> fallitur, <reg norm="Å¿entientes" type="context">Å¿entiẽtes</reg> huiuÅ¿modi diem ab ingreſſu Solis, in <lb/>principium geminorum incipere, & in egreſſu à Leone terminari, ideÅ¿t à .12. Maij ad <lb/>14. AuguÅ¿ti, quaÅ¿i Å¿i ab æquatore finis Leonis ita declinaret, vt principium gemino-<lb/>rum, & finis Aquarij, vt initium Sagittarij, nam ratio poÅ¿tulat, tantum de-<lb/>clinari ab æquatore finem quantum initium diei, vbi maximus dies .24. horas ex <lb/>cedit, & Å¿ic dico de noctibus: </s> - <s xml:id="s3347" xml:space="preserve">vnde in huiuÅ¿modi regione, vbi per tres menÅ¿es conti <lb/>nuos Sol radios emittit, huiuÅ¿modi dies à medietate Tauri incipit, & in medietate <lb/>Leonis terminatur, quæ quidem loca æqualem declinationem habent, & Å¿ic nox <lb/>trium menÅ¿ium incipit à medietate Scorpionis, & in medietate Aquarij, eadem ra-<lb/>tione finitur.</s> - </p> - <p> - <s xml:id="s3348" xml:space="preserve">Septima verò pag. <reg norm="idem" type="context">idẽ</reg> ait, dies Å¿olÅ¿titiales eſſe circa .24. Iunij. </s> - <s xml:id="s3349" xml:space="preserve"><reg norm="quod" type="wordlist">QÄ‘</reg>, an <reg norm="tunc" type="context">tÅ©c</reg> eſſet <reg norm="verum" type="context">verÅ©</reg>, <lb/>tu ipÅ¿e videto. </s> - <s xml:id="s3350" xml:space="preserve">Is præterca modus <reg norm="quem" type="context">quẽ</reg> ad <reg norm="inueniendum" type="context context">inueniẽdÅ©</reg> <reg norm="orientem" type="context">orientẽ</reg>, & <reg norm="occidentem" type="context">occidẽtem</reg> præſcribit <lb/>in eodem proÄ™mio pag .63. eÅ¿t tædioÅ¿us, cum Å¿emper expectare nos cogat æquino-<lb/>ctij tempus, cum alij modi reperiantur breuiores, qui in qualibet reuolutione primi <lb/>mobilis obÅ¿eruari poſſunt, quorum vnus erit mediante inuentione lineæ meridiane <lb/>orizontalis, eo modo, quo Å¿criptum eÅ¿t ab antiquis mediante Sole, aut Luna, quæ <lb/>luminaria in quolibet alio loco, <reg norm="præterquam" type="context">præterquã</reg> Å¿ub polo efficiunt, vt extremitas vmbræ <lb/>rectæ <reg norm="gnomonum" type="context">gnomonÅ©</reg> <reg norm="gyrum" type="context">gyrÅ©</reg> <reg norm="oxigonium" type="context">oxigoniÅ©</reg>, Å¿eu <reg norm="eclipticum" type="context">eclipticÅ©</reg> ducat, ideÅ¿t in ijs locis, <reg norm="quorum" type="context">quorÅ©</reg> zenit eÅ¿t <lb/>inter polum, & circulum arcticum, quemadmodum facit, vt alijs, exiÅ¿tentibus ipÅ¿is <lb/>luminaribus extra æquatorem, & circulos arcticos gyrum hyperbolicum reddant. <lb/></s> - <s xml:id="s3351" xml:space="preserve">Sed id quod eidem Porcachio impoſſibile eſſe apud eos, qui habitant Å¿ub polo vi-<lb/>detur, ideÅ¿t vt multis rationibus, vt ipÅ¿e dicit, fieri non poſſit, ut fiat immediata quÄ™ <lb/>dam, & Å¿ubita mutatio à continuo die ad continuam noctem abÅ¿que eo quod ijs, <lb/>Å¿altem Å¿emel conceſſa Å¿int dies, & nox terminata duodecim horarum, eÅ¿t magis ad <lb/>mirandum impoſſibile, quod imaginari poſſimus, nam neceſſarium eſſet, ut orizon-<lb/>habitatorum Å¿ub polo Å¿ecaret æquatorem contra id, quod Å¿uperius admiÅ¿erat, id-<lb/>eÅ¿t <reg norm="orizontem" type="context context">orizõtẽ</reg> Biarmiæ, eſſe <reg norm="eundem" type="context context">eÅ©dẽ</reg> <reg norm="cum" type="context">cÅ©</reg> circulo æquinoctiali. </s> - <s xml:id="s3352" xml:space="preserve">Vide etiam quid is ab anti-<lb/>quis colligat, loquens de iis, quæ in inÅ¿ula Taprobana ad finem pag .186. admirabi <lb/>lia Å¿unt, Å¿cribens eiuÅ¿dem inÅ¿ulÄ™ habitatoribus, Lunam Å¿uper terram non apparere <lb/>ab octauo uÅ¿que ad decimumÅ¿extum diem: </s> - <s xml:id="s3353" xml:space="preserve">prÄ™ter quam, quod etiam Å¿cribit, in <lb/>eadem inÅ¿ula, tramuntanam non uideri, quod falÅ¿um eÅ¿t, quia hæc à polo arctico <lb/>circiter quatuor gradibus diÅ¿tat noÅ¿tris temporibus. </s> - <s xml:id="s3354" xml:space="preserve">unde ab ijs qui Å¿unt Å¿ub æqua-<lb/>tore, cum ea Å¿upra orizontem eÅ¿t, conÅ¿pici poteÅ¿t, cum ijÅ¿dem Å¿ingulis diebus oria-<lb/>tur, & occidat. </s> - <s xml:id="s3355" xml:space="preserve">Idem etiam pro re admirabili Å¿cribit, uideri Canopum, qui à po-<lb/>lo antarctico plus quam quadraginta gradibus diÅ¿tat.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head400" style="it" xml:space="preserve">De erroribus Lucilli Philalthæi.</head> - <head xml:id="head401" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3356" xml:space="preserve">QVod Lucillus PhilalthÄ™us tam eximius Mathematicus Å¿it, ut ipÅ¿um Anto-<lb/>nius Berga facit, ego quidem non uideo. </s> - <s xml:id="s3357" xml:space="preserve">In Å¿uis enim commentariis de <lb/>CÅ“lo, dicit primum, Pyramidem, quÄ™ inter corpora regularia primum locum tenet - <pb o="269" rhead="EPISTOL AE." n="281" file="0281" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0281"/> - Å¿ex baÅ¿ibus conÅ¿tare, pag.15. 583. 632. et .647. </s> - <s xml:id="s3358" xml:space="preserve">Omitto errorem ab eodem com-<lb/>miſſum in fine pag .39. ubi oleum grauius eſſe quam aquam fatetur, cum id ad res <lb/>mathematicas non Å¿pectet: </s> - <s xml:id="s3359" xml:space="preserve">Omitto etiam quod idem neget aÅ¿trologiam pag .74. <lb/>79. & quod etiam dicat pag .89. </s> - <s xml:id="s3360" xml:space="preserve">Deum eſſe ad orientem, non conÅ¿iderans aliqui-<lb/>bus populis noÅ¿trum orientem eſſe occidentem.</s> - </p> - <p> - <s xml:id="s3361" xml:space="preserve">Quod idem ait pag .241. </s> - <s xml:id="s3362" xml:space="preserve">AÅ¿trologiam eſſe antiquiorem AÅ¿tronomia eÅ¿t falÅ¿iſſi-<lb/>mum, quia iudiciaria Å¿emper præſupponit cognitionem Å¿itus Å¿tellarum, quæ ab A-<lb/>Å¿tronomia petitur. </s> - <s xml:id="s3363" xml:space="preserve">Mouebit tibi riÅ¿um quod ait pag .307. his verbis.</s> - </p> - <quote> - <s xml:id="s3364" xml:space="preserve">Verum propriè media dicitur illa, quæ rectam Å¿phæram omninò habet, quæ eun <lb/>dem polum orizontis & mundi obtinet, quæ orizontem habet diuidentem <reg norm="Å¿phæram" type="context">Å¿phærã</reg> <lb/>æquè Å¿ecundum angulos rectè.</s> - </quote> - <p> - <s xml:id="s3365" xml:space="preserve">Paulo inferius continuans Å¿ermonem de Å¿phæra recta, ait.</s> - </p> - <quote> - <s xml:id="s3366" xml:space="preserve">Et niÅ¿i tumor terræ, & gibum eſſet, ijs perpetuus eſſet dies Å¿ine nocte.</s> - </quote> - <p> - <s xml:id="s3367" xml:space="preserve">Linea verò .56. ait habitatores Å¿phÄ™rÄ™ rectè habere .4. Å¿olÅ¿titia, Å¿eÅ¿e ipÅ¿um huius <lb/>rei planè ignarum prodens .310. autem pag. Å¿ic Å¿cribit.</s> - </p> - <quote> - <s xml:id="s3368" xml:space="preserve">Quoniam repercutiuntur radij, & peridem centrum tranÅ¿eunt, ob id Å¿tupam ap <lb/>poÅ¿itam centro radius accendit.</s> - </quote> - <p> - <s xml:id="s3369" xml:space="preserve">Quem quidem errorem ab Euclide deÅ¿umit, et .15. linea pag .636. repetit.</s> - </p> - <p> - <s xml:id="s3370" xml:space="preserve">Si vis ridere, legito .16. primas lineas .357. pag. </s> - <s xml:id="s3371" xml:space="preserve">Quod idem deinde dicat circa fi-<lb/>nem 396. pag. lucem eſſe Å¿ubÅ¿tantiam corporis lucidi & corpoream, Å¿ubijciam tuo <lb/>iudicio, vt etiam quod ait .397. pag. his vetbis vtens.</s> - </p> - <quote> - <s xml:id="s3372" xml:space="preserve">Idcirco animalia illa, quæ nocte vagantur perpolita, dum volant, aerem terunt <lb/>nocturnum, & fulgent.</s> - </quote> - <p> - <s xml:id="s3373" xml:space="preserve">Et pag .398.</s> - </p> - <quote> - <s xml:id="s3374" xml:space="preserve">Multitudo radiorum non admodum facit ad excitandum calorem Å¿i Å¿olum inci-<lb/>dat Å¿ine repercuſſu, neorecta incidere iuuerit.</s> - </quote> - <p> - <s xml:id="s3375" xml:space="preserve">Quod falÅ¿um eÅ¿t cum radius incidens longè magis quam reflexus calefaciat. </s> - <s xml:id="s3376" xml:space="preserve">In fi <lb/>ne autem .405. Å¿ic Å¿cribit.</s> - </p> - <quote> - <s xml:id="s3377" xml:space="preserve">Sol in ortu & in occaÅ¿u longius apparet, iccircò reuolui creditur. </s> - <s xml:id="s3378" xml:space="preserve">Hinc etiam in <lb/>abÅ¿ide Å¿tare putatur, & in oppoÅ¿ito abÅ¿idis, vnde Å¿olÅ¿titia vocant, Å¿ed nobis in Can-<lb/>cro, antipodibus verò in Capricorno tum Sol abeſſe longius apparet vtriÅ¿que.</s> - </quote> - <p> - <s xml:id="s3379" xml:space="preserve">An hoc quid peius dici poteÅ¿t? </s> - <s xml:id="s3380" xml:space="preserve">Circa vero .40. lineam pag .459. Å¿ic Å¿cribit.</s> - </p> - <quote> - <s xml:id="s3381" xml:space="preserve">Si enim alij planetæ, & Å¿tellæ fixÄ™ reciperent à Sole lumen, dum accederent ad So <lb/>lem, vel recederent, aut contra, Sol ad eas appropinquaret, & abÅ¿cederet, eaÅ¿dem-<lb/>lucis viciſſitudinis Å¿ubiret, quas Luna.</s> - </quote> - <quote> - <s xml:id="s3382" xml:space="preserve">Hoc autem nondum deprÄ™henÅ¿um eÅ¿t, quin etiam Mercurius, Venus, Å¿uo interpo <lb/>Å¿itu, Solem occultarent nobis, vt Luna.</s> - </quote> - <quote> - <s xml:id="s3383" xml:space="preserve">Paulo inferius Å¿ic ait. </s> - <s xml:id="s3384" xml:space="preserve">RurÅ¿us æquè Saturnus, Jupiter, Mars, Å¿ubire deliquium, <lb/>more Lunæ, aut Å¿altem obiectu terræ inter Solem & ipÅ¿os, quia tum ob interpoÅ¿itam <lb/>terram non poſſent haurire lumen à Sole.</s> - </quote> - <p> - <s xml:id="s3385" xml:space="preserve">Hæc verò omnia, talia Å¿unt, qualia ab ijs qui incipiunt intelligere Å¿phæram non <lb/>proferrentur. </s> - <s xml:id="s3386" xml:space="preserve">Omittamus, quod ait deinde.</s> - </p> - <quote> - <s xml:id="s3387" xml:space="preserve">Accedit quod Å¿i aÅ¿tra lumen à Sole acciperent eiuÅ¿dem caloris eſſent. </s> - <s xml:id="s3388" xml:space="preserve">Itaque om <lb/>nia Å¿iccarent, & nulla eſſent frigidæ conÅ¿titutionis contra AÅ¿trologos.</s> - </quote> - <p> - <s xml:id="s3389" xml:space="preserve">Quia hac ratione, Luna, quæ negari non poteÅ¿t, quin ab ipÅ¿o Sole <reg norm="lumem" type="context">lumẽ</reg> accipiat, <lb/>eiuÅ¿dem caloris eſſet cum eodem Sole. </s> - <s xml:id="s3390" xml:space="preserve">Sunt ea etiam ridenda, quæ idem ait pag.<lb/>460. lineis .18. 19. 23. 26. 27. 29. quaÅ¿i ea lux infinita (vt ita dicam) magni Solis, non - <pb o="270" rhead="IO. BAPT. BENED." n="282" file="0282" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0282"/> - in alium finem Å¿it effecta quam ad illuminandam Å¿uperficiem huius excrementi ip-<lb/>Å¿ius vniuerÅ¿i ad vtilitatem hominum, imò, vt rectius dicam, <reg norm="animalium" type="context">animaliÅ©</reg>. </s> - <s xml:id="s3391" xml:space="preserve">vide etiam <lb/>pag .632. et .633. vbi AriÅ¿totelem de implendo loco non intellexit, cum citet Å¿phæ-<lb/>ram, loco pyramidis, & inter .46. et .47. lineas dicat <reg norm="quadratum" type="context">quadratÅ©</reg> eſſe quid multiplex, <lb/>cum Å¿it vnicum tantum in Å¿pecie, quia Å¿pecies eÅ¿t quadrilateri, & quadranguli, Å¿ed <lb/>vbi in .6. linea pag .633. ait.</s> - </p> - <p> - <s xml:id="s3392" xml:space="preserve">Item hexagonus.</s> - </p> - <p> - <s xml:id="s3393" xml:space="preserve">Magnum errorem committit, vt etiam cum .12. linea .636. pag. Å¿cribens.</s> - </p> - <p> - <s xml:id="s3394" xml:space="preserve">Pyramis, Å¿iue planum, Å¿iue Å¿olidum, habet acutiſſimum, & in .2. libr. de anima <lb/>pag .215. dicat de die poſſe videri Å¿tellas in Å¿peculo poÅ¿ito in vaÅ¿e aqua pleno, quod <lb/>reuera eÅ¿t valde abÅ¿urdum. </s> - <s xml:id="s3395" xml:space="preserve">Alios eiuÅ¿dem errores tibi non patefacio, quia iam ni-<lb/>hil amplius otij mihi eÅ¿t, Å¿ed eos tu ipÅ¿e perÅ¿picere, & cognoÅ¿cere facilè poteris, & <lb/>multò plures quidem, quam putas.</s> - </p> - </div> - </div> - <div type="section" level="3" n="10"> - <div type="letter" level="4" n="1"> - <head xml:id="head402" style="it" xml:space="preserve">Cur maius lumen extenuet minus.</head> - <head xml:id="head403" xml:space="preserve">PIRRO DE ARZONIS.</head> - <p> - <s xml:id="s3396" xml:space="preserve">EX tuis literis intellexi id, quod etiam Å¿ine ijs exploratum mihi erat. </s> - <s xml:id="s3397" xml:space="preserve">Sed conce <lb/>do tantum eſſe dicere vbi eÅ¿t maius lumen, minus non diÅ¿cerni, quantum inter <lb/>diu Å¿tellas non videri: </s> - <s xml:id="s3398" xml:space="preserve">immo eÅ¿t etiam magis vniuerÅ¿ale, quia idem multis aliis lumi-<lb/>nibus, præter ea quæ Å¿unt Å¿tellarum, ea ratione contingit, quia ingrediente per pupil-<lb/>lam, tam lumine maiori, quam minori, reflexum ipÅ¿ius maioris in oculo, in Å¿itu mino <lb/>ris, efficit, vt ipÅ¿um minus confundatur, & diÅ¿tingui nequeat, quemadmodum aper-<lb/>te cognoÅ¿ci poteÅ¿t in aliquo cubiculo, cuius parietes dealbati Å¿int, in quo, vnicum <lb/>tantum Å¿it exiguum foramen, per quod aliqua lumina reflexa ab obiectis extrinÅ¿ecis <lb/>intra ipÅ¿um cubiculum ingredi poſſint, vnde imagines <reg norm="obiectorum" type="context">obiectorÅ©</reg> in parietibus con-<lb/>Å¿piciuntur, Å¿ed Å¿i per idem foramen ingrederetur etiam primarius radius Solis, re-<lb/>flexus huiuÅ¿modi radij efficeret, vt dictæ imagines, magis aut minus euaneÅ¿cerent, <lb/>prout dictus reflexus radij Å¿olaris, maiori, minorÃue vi polleret.</s> - </p> - <p> - <s xml:id="s3399" xml:space="preserve">Ad hoc tamen propoÅ¿itum, nolo tibi Å¿ilentio inuolui mirabilem quendam effe-<lb/>ctum eiuÅ¿modi rei. </s> - <s xml:id="s3400" xml:space="preserve">Hoc eÅ¿t vt fiat foramen illud rotundum, magnitudinis tamen <lb/>vnius Å¿pecilli, quod foramen obturetur mediante vno illorum Å¿pecillorum, quæ pro <lb/>Å¿enibus (non breuis viÅ¿ionis) conficiuntur, hoc eÅ¿t quorum ambæ Å¿uperficies con <lb/>uexæ Å¿unt, non autem concauæ. </s> - <s xml:id="s3401" xml:space="preserve">Deinde opponatur folium album papiri, adeo di <lb/>Å¿tans à foramine, vt extrinÅ¿eca obiecta in eo appareant. </s> - <s xml:id="s3402" xml:space="preserve">Quæ quidem obiecta Å¿i à <lb/>Sole illuÅ¿trata fuerint, tam clara, & diÅ¿tincta videbuntur, vt nihil pulchrius dele-<lb/><reg norm="ctabiliusque" type="simple">ctabiliusqÌ;</reg> videri poterit, inuerÅ¿a tamen. </s> - <s xml:id="s3403" xml:space="preserve">Sed Å¿i ea directa videre voluerimus. </s> - <s xml:id="s3404" xml:space="preserve">hoc <lb/>optimè faciemus, mediante reflexione alicuius Å¿peculi plani.</s> - </p> - <pb o="271" rhead="EPISTOL AE." n="283" file="0283" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0283"/> - </div> - </div> - <div type="section" level="3" n="11"> - <div type="letter" level="4" n="1"> - <head xml:id="head404" style="it" xml:space="preserve">Cur byems valde frigida Å¿equatur ac<unsure/>tatem in qua <lb/>calor viguerit.</head> - <head xml:id="head405" xml:space="preserve">NOBILISSIMO, NECNON INGENIOSISSIMO <lb/>Gabrieli BuÅ¿chæ, MediolanenÅ¿i.</head> - <p> - <s xml:id="s3405" xml:space="preserve">QVod dixi hyemem valde frigidam Å¿equi <reg norm="æſtatem" type="context">æſtatẽ</reg>, in qua calor viguerit, inde na <lb/>Å¿citur, quia calor terrÄ™, aquæ, & aeris, non eÅ¿t naturalis horum corporum, vt <lb/>eÅ¿t frigus, cum calor à Sole procedat, qui ea calefacit Å¿uo lumine, vnde quod æſtate <lb/>Sol præter modum calefaciat <reg norm="terram" type="context">terrã</reg>, ideo <reg norm="contingit" type="context">cõtingit</reg>, quod minora <reg norm="impedimenta" type="context">impedimẽta</reg> contra <lb/>ria Å¿ortiatur, & cum eandem poÅ¿tea deÅ¿erit, ad aliam partem æquatoris <reg norm="tranÅ¿migrans" type="context">tranÅ¿migrãs</reg> <lb/>terra ad Å¿uam qualitatem reddit, maiori cum impetu, eo modo, quo res in mo-<lb/>tibus localibus naturalibus, qui etiam terminos Å¿ibi prÄ™fixos, & conÅ¿titutos exce-<lb/>dunt, hinc etiam hyeme fit glacies, ex calefacta prius aqua, quæ durior poÅ¿tea eÅ¿t <lb/>atque frigidior alia. </s> - <s xml:id="s3406" xml:space="preserve">AeÅ¿tas etià m quæ Å¿equitur hyemem valde frigidam, non <lb/>erit admodum calida, quia Sol inueniens contrarium naturale valde potens, non <lb/>tam facile illud pellere poteÅ¿t, vnde etiam Å¿i in Geminis, Cancro, & Leone, moram <lb/>trahat, non Å¿ufficit tamen ut magnum calorem imprimere poſſit. </s> - <s xml:id="s3407" xml:space="preserve">Vnde Å¿equitur duas <lb/>æſtates quarum una Å¿equatur aliam, in eodem loco, uehementi calore præditas eÅ¿-<lb/>Å¿e non poſſe, quemadmodum nec duas hyemes exceſſiuo frigore, remotis tamen <lb/>accidentibus uentorum, pluuiarum, & niuium.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head406" xml:space="preserve">QVOD MALE SENSERIT NICOLAVS TARTA-<lb/>lea circa attractionem machinæ tormentalis.</head> - <head xml:id="head407" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3408" xml:space="preserve">EFfectus, quem Å¿cribit Tattalea quęſito quinto primi lib. necnon quæſito <lb/>21. et .24. maxima cum ratione eſſe uidetur, non tamen ea quam ipÅ¿e in <lb/>quinto profert, quia uerum non eÅ¿t, vt quanto aliquid fit calidius, <reg norm="tanto" type="context">tãto</reg> ue-<lb/>hementius attrahat, eo quod Å¿i etiam huiuÅ¿modi res, in eodem calore, in <lb/>quo Å¿emel reperitur, firma maneret; </s> - <s xml:id="s3409" xml:space="preserve">neque attraheret, neque aliquid impelleret. <lb/></s> - <s xml:id="s3410" xml:space="preserve">Nam dum aliquod corpus calefit, dilatatur, & per conÅ¿equens circumcirca <reg norm="undique" type="simple">undiq;</reg> <lb/>trudit, & partes uaÅ¿is debiliores cedunt. </s> - <s xml:id="s3411" xml:space="preserve">dum uerò dictum corpus re frigeratur, re-<lb/>Å¿tringitur, & dum in unum cogitur, Å¿i reperiatur in uaÅ¿e, quod aer, aqua, aut aliud <lb/>aliquod corpus ingredi nequeat, dictum uas à quo circundatur frangit, ne aliqua <lb/>pars loci uacua remaneat, Å¿ed Å¿i aliquod corpus ingredi poteÅ¿t, illud ipÅ¿um ad Å¿e at-<lb/>trahit, quemadmodum uidere licet in cucurbitulis. </s> - <s xml:id="s3412" xml:space="preserve">Vnde Å¿equitur eam propoÅ¿i-<lb/>tionem, qua dicitur, calidi eÅ¿t attrahere, ueram non eſſe, quia Å¿i hoc fieret, quanto <lb/>aliquid calidus efficeretur, tanto magis attraheret, & ècontra, cum tamen planè <lb/>contrarium appareat, cum quanto magis aliquid calefit, tanto uehementius impel-<lb/>lat, & quanto magis frigefit, tanto plus attrahat. </s> - <s xml:id="s3413" xml:space="preserve">Quapropter uerius dicemus, fri-<lb/>gidi eſſe attrahere, calidi uerò expellere, quamuis per accidens. </s> - <s xml:id="s3414" xml:space="preserve">Ex quo Å¿equitur, ut <lb/>quanto calidior facta fuerit materia aliqua, aliquo loco determinata, redeundo po-<lb/>Å¿tea ad Å¿uam priorem frigiditatem, tanto minori loco indigeat, Å¿imiliter etiam <lb/>è conuerÅ¿o accidit, ut quanto frigidior <reg norm="reperitur" type="simple">reê‘itur</reg> talis materia, tanto maioriloco, po- - <pb o="272" rhead="IO. BAPT. BENED." n="284" file="0284" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0284"/> - Å¿tea egeat ipÅ¿a ualde calefacta. </s> - <s xml:id="s3415" xml:space="preserve">Quod Tartalea in quinto quęſito non animaduer-<lb/>terat.</s> - </p> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head408" style="it" xml:space="preserve">Solutiones aliqua, circa altimetriam.</head> - <head xml:id="head409" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3416" xml:space="preserve">TVas literas accepi, <reg norm="tuasque" type="simple">tuasqÌ;</reg> dubitationes conÅ¿ideraui, quas quidem non inutiles <lb/>inueni, quo uerò ad primam, dico te oportere illud Theorema Å¿peculari or <lb/>dine huiuÅ¿modi methodi, uidelicet quod <reg norm="quotieÅ¿cunque" type="simple">quotieÅ¿cunq;</reg> habuerimus <reg norm="angulum" type="context">angulÅ©</reg> <reg norm="aliquem" type="context">aliquẽ</reg> <lb/>cuiufuis amplitudinis, puta <var>.A.R.V.</var> cuius duo latera <var>.R.A.</var> et <var>.R.V.</var> indeterminata <lb/>intelligantur, Å¿i ab aliquo puncto inter ipÅ¿as poÅ¿ito, puta <var>.u.</var> quod etiam uocetur <var>.i.</var> du <lb/>ctÄ™ fuerint .4. lineÄ™ ipÅ¿is dictis lateribus, hac Å¿cilicet <reg norm="conditione" type="context">cõditione</reg>, <reg norm="quod" type="wordlist">qÄ‘</reg> duÄ™ ex dictis .4. Å¿int <lb/>parallelÄ™ ipfis <reg norm="lateribus" type="simple">lateribê°</reg>, puta <lb/> - <anchor type="figure" xlink:label="fig-0284-01a" xlink:href="fig-0284-01"/> - <var>u.e.</var> et <var>.u.E.</var> reliquÄ™ uero duÄ™ <lb/>Å¿eccent ipÅ¿a latera, ut <var>V.u.<lb/>a.</var> et <var>.I.u.A</var>. </s> - <s xml:id="s3417" xml:space="preserve">Dico nunc pro-<lb/>portionem <var>.e.A.</var> ad <var>.e.a.</var> ean <lb/>dem eſſe, quÄ™ <var>.E.V.</var> ad <var>.E.I.</var> <lb/>Nam Å¿cimus proportionem <lb/><var>E.i.</var> ad <var>.E.i.</var> eandem eſſe quÄ™ <lb/><var>e.i.</var> ad <var>.e.A.</var> ex fimilitudine <lb/><reg norm="triangulorum" type="context">triangulorÅ©</reg>, Å¿imiliter <reg norm="propor" type="simple">ê“por</reg> <lb/><reg norm="tionem" type="context">tionẽ</reg> <var>.E.u.</var> ad <var>.E.V.</var> <reg norm="eandem" type="context context">eãdẽ</reg> quÄ™ <lb/><var>e.a.</var> ad <var>.e.u.</var> </s> - <s xml:id="s3418" xml:space="preserve">quare aggregata <lb/>ex iÅ¿tis erunt inuicem <reg norm="aequa- lia" type="simple">Ä™qua-<lb/>lia</reg>, uel Å¿i mauis ex Ä™qua pro <lb/>portionalitate, quod idem <lb/>eÅ¿t, ita Å¿e habebit <var>.E.I.</var> ad <var>.<lb/>E.V.</var> ut <var>.e.a.</var> ad <var>.e.A</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0284-01" xlink:href="fig-0284-01a"> - <image file="0284-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0284-01"/> - </figure> - </div> - <p> - <s xml:id="s3419" xml:space="preserve">SuppoÅ¿ito nunc plano orizontali <var>.V.E</var>. </s> - <s xml:id="s3420" xml:space="preserve"><reg norm="Altitudineque" type="simple">AltitudineqÌ;</reg> inacceſſibili <var>.A.E</var>. </s> - <s xml:id="s3421" xml:space="preserve">DuÄ™ ue-<lb/>rò Å¿tationes oculorum Å¿int <var>.V.</var> et <var>.I.</var> lineÄ™ autem uiÅ¿uales Å¿int <var>.V.A.</var> et <var>.I.A</var>. </s> - <s xml:id="s3422" xml:space="preserve">Et quadra-<lb/>tum geometricum Å¿it <var>.b.e</var>. </s> - <s xml:id="s3423" xml:space="preserve">Supponatur nunc pro prima dubitatione, quod in am-<lb/>babus Å¿tationibus filum perpendiculare Å¿eccet latus <var>.e.c.</var> non autem <var>.b.c.</var> (nam quan-<lb/>do in ambabus Å¿tationibus filum Å¿ecat latus <var>.b.c.</var> nullum tibi dubium oritur, imo ma <lb/>nifeÅ¿tè patent partes lateris <var>.b.c.</var> terminatas à <var>.b.</var> & à filo proportionales eſſe <var>.V.E.</var> & <lb/><var>I.E.</var> Å¿umpto <var>.E.</var> pro <var>.b.</var> et <var>.I.V.</var> pro punctis Å¿ecatis à filo, ex <reg norm="euidenti" type="context">euidẽti</reg> Å¿imilitudine trian-<lb/>gulorum quadrati cum triangulis <var>.A.E.V.</var> et <var>.A.E.I.</var>) Sed cum in pręſenti caÅ¿u repe-<lb/>riatur triangulum <var>.u.e.a.</var> minus, in Å¿tatione remotiori, Å¿imile triangulo maiori <var>.V.E.<lb/>A.</var> & triangulum maius <var>.i.e.a.</var> proximioris Å¿tationis, Å¿imile triangulo minori <var>.I.E.A.</var> <lb/>(quod in alio iam dicto, caÅ¿u non accidit, ut unum triangulorum, minus Å¿cilicet, Å¿i-<lb/>mile Å¿it uno triangulorum, maiori Å¿cilicet & è conuerÅ¿o) Non omnino abÅ¿que ratio <lb/>ne dubitas quo pacto fieri poſſit ut <var>.a.e.</var> remotioris Å¿tationis ad <var>.a.e.</var> propinquioris ita <lb/>Å¿e habeat quema dmodum <var>.I.E.</var> ad <var>.E.V</var>. </s> - <s xml:id="s3424" xml:space="preserve">Quapropter Å¿i prÄ™cedentem figuram dili- - <pb o="273" rhead="EPISTOL AE." n="285" file="0285" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0285"/> - genter inÅ¿pexeris, omnis tua dubitatio euaneÅ¿cet. in qua figura apertè vi debis cor-<lb/>reÅ¿pondentiam talium triangulorum inter Å¿e, nec magis, nec minus quam in infra-<lb/>Å¿cripta hic figura cernere licet, quamuis in hac, triangula quadrati, Å¿eparata Å¿int ab <lb/>imaginarijs <var>.A.E.V.</var> et <var>.A.E.I.</var> in Å¿upradicta vero coniuncta, & inuicem communican <lb/>tia in puncto <var>.u.i.</var> quod quidem nihil refert. </s> - <s xml:id="s3425" xml:space="preserve">Dempta igitur <var>.a.e.</var> minori ex <var>.a.e.</var> ma-<lb/>iori, reliquum ita Å¿e habebit ad <var>.a.e.</var> minorem, vt, <var>V.I.</var> ad <var>.I.E.</var> quod nunc tibi <lb/>clarè patebit. </s> - <s xml:id="s3426" xml:space="preserve">Vnde ex te poteris ordinem operationis proÅ¿equi, vt in cognitionem <lb/>peruenias ipÅ¿ius <var>.I.E.</var> ipÅ¿ius <var>.A.E.</var> & ipÅ¿ius <var>.I.A.</var> vel <var>.V.A</var>.</s> - </p> - <figure position="here"> - <image file="0285-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0285-01"/> - </figure> - <p> - <s xml:id="s3427" xml:space="preserve">Sed <reg norm="quando" type="context">quãdo</reg> in proximiori Å¿tatione latus <var>.b.c.</var> in remotiori vero latus <var>.c.e.</var> Å¿ecatur à fi <lb/>lo (pro Å¿ecunda dubitatione) </s> - <s xml:id="s3428" xml:space="preserve">Tunc oportet imaginatione conÅ¿iderare latus <var>.b.c.</var> in <lb/>re motiori Å¿tatione diÅ¿tentum eſſe vÅ¿que ad filum in puncto <var>.n.</var> vbi videbis <reg norm="triangulum" type="context">triangulÅ©</reg> <var>.<lb/>u.b.n.</var> Å¿imile triangulo <var>.A.E.V.</var> ita vt <var>.i.b.a.</var> Å¿imile Å¿uo <var>.A.E.I.</var> reperitur, vbi tam in vno <lb/> - <anchor type="figure" xlink:label="fig-0285-02a" xlink:href="fig-0285-02"/> - <pb o="274" rhead="IO. BAPT. BENED." n="286" file="0286" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0286"/> - quam in altero <var>.i.b.</var> et <var>u.b.</var> correÅ¿pondebit ipÅ¿i <var>.A.E.</var> et <var>.b.n.</var> ipÅ¿i <var>.E.V.</var> et <var>.b.a.</var> ipÅ¿i <var>.E.I.</var> <lb/>quapropter iubeo, vt quæras quantum Å¿it latus <var>.b.n.</var> ex regula de tribus, dicens Å¿i <var>.a.e.</var> <lb/>tribuit mihi <var>.e.u.</var> quid mihi dabit <var>.u.b</var>? </s> - <s xml:id="s3429" xml:space="preserve">eo quod <var>.a.e.u.</var> Å¿imile eÅ¿t <var>.u.b.n.</var> reperto autem <lb/>latere <var>.b.n.</var> ex quo dempto <var>.b.a.</var> breuioris diÅ¿tantiÄ™, reÅ¿iduum reÅ¿pondebit ipÅ¿i <var>.I.V.</var> vt <lb/>Å¿cis, vnde proÅ¿equendo operationem tibi cognitam, obtinebis intentum, hoc eÅ¿t co <lb/>gnoÅ¿ces reliqua interualla. </s> - <s xml:id="s3430" xml:space="preserve">Nihil enim miror demonÅ¿trationem Tartaleæ circa hu <lb/>iuÅ¿modi operationem te minime Å¿atisfeciſſe.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0285-02" xlink:href="fig-0285-02a"> - <image file="0285-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0285-02"/> - </figure> - </div> - <p> - <s xml:id="s3431" xml:space="preserve">Quod autem quarta propoÅ¿itio illius Å¿criptoris, de quo nuper <reg norm="mecum" type="context">mecÅ©</reg> locutus es, vt <lb/>mihi dixiÅ¿ti, tua Å¿it, hoc enimego, nec affirmare, nec negare audeo, quamuis in mul <lb/>tis cum tua manuÅ¿cripta concordet. </s> - <s xml:id="s3432" xml:space="preserve">Nam ſępæ cogitationes hominum in idem co-<lb/>incidunt, vt pluries cenÅ¿uit AriÅ¿to.</s> - </p> - <figure position="here"> - <image file="0286-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0286-01"/> - </figure> - </div> - <div type="letter" level="4" n="4"> - <head xml:id="head410" style="it" xml:space="preserve">Demonstrationes quorundam problematum Nicolai Tartalea <lb/>cum <reg norm="alijs" type="lig">alijs</reg> operationibus circa eadem Å¿ubiecta.</head> - <head xml:id="head411" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3433" xml:space="preserve">AMor erga te meus Å¿anè Å¿ingularis, nullo modo <reg norm="permittit" type="simple">ê‘mittit</reg>, vt ea quæ TartaleÄ™ Å¿cri <lb/>pta <reg norm="examinando" type="context">examinãdo</reg> inuenerim, non tibi <reg norm="communicem" type="context">cõmunicem</reg>. </s> - <s xml:id="s3434" xml:space="preserve">Hæc autem Å¿unt circa quæ <lb/>dam illius Authoris problemata, quorum primum ab ipÅ¿o Tartalea <lb/>Å¿criptum in .3. quæſito libr .4. tale eÅ¿t, is vult locare .3500. homines, eodem modo, <lb/>quo præſupponit locatos eſſe .1000. ita vt quilibet hominum ordo Å¿iue vt vulgo di-<lb/>citur filtia Å¿it .49. quapropter multiplicat quadratum ipÅ¿ius .49. quod eÅ¿t .2401. <reg norm="cum" type="context">cÅ©</reg> <lb/>numero .3500. propoÅ¿ito, productum verò .8403500. diuidit per .1000. vt proue-<lb/>niat .8403. cuius radix quadrata eÅ¿t .91. pro numero hominum vniuÅ¿cuiuÅ¿que ordinis <lb/>propoÅ¿iti numeri .3500.</s> - </p> - <p> - <s xml:id="s3435" xml:space="preserve">Pro cuius operationis ratione, cogitemus rectangulum <var>.a.b.</var> 1000. hominum, et <var>.d.<lb/>b.</var> Å¿it vna filtia Å¿iue ordo .49. <reg norm="hominum" type="wordlist">hoÄ©um</reg>, cuius <reg norm="quadratum" type="context">quadratÅ©</reg> Å¿it <var>.b.c.</var> 2401. imaginemur <reg norm="etiam" type="context">etiã</reg> re <lb/>ctangulum <var>.A.B.</var> 3500. hominum, quod Å¿upponemus Å¿imile rectangulo <var>.a.b.</var> et <var>.B.C.</var> - <pb o="275" rhead="EPISTOL AE." n="287" file="0287" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0287"/> - Å¿it quadratum ipÅ¿ius <var>.D.B</var>. </s> - <s xml:id="s3436" xml:space="preserve">Nunc Å¿upponendo <var>.A.B.</var> Å¿imile <var>.a.b.</var> clarum erit ex diffini-<lb/>tione Å¿imilium figurarum, quod eadem proportio erit <var>.A.D.</var> ad <var>.D.B.</var> quÄ™ <var>.a.d.</var> ad <var>.d.<lb/>b.</var> hoc eÅ¿t <var>.A.D.</var> ad <var>.D.C.</var> vt <var>.a.d.</var> ad <var>.d.c.</var> hoc eÅ¿t <var>.A.B.</var> ad <var>.B.c.</var> vt <var>.a.b.</var> ad <var>.b.c.</var> ex prima <lb/>Å¿exti, vel .18. Å¿eu .19. Å¿eptimi, </s> - <s xml:id="s3437" xml:space="preserve">tunc cum dixerimus Å¿i <var>.a.b.</var> ita reÅ¿pondet ad <var>.b.c.</var> ergo <var>.A.<lb/>B.</var> correÅ¿pondet etiam ita ad <var>.B.C.</var> </s> - <s xml:id="s3438" xml:space="preserve">quare ex regula de tribus rectè fit multiplicando <var>.<lb/>A.B.</var> per <var>.b.c.</var> productum verò diuidendo per <var>.a.b.</var> ex .15. Å¿exti vel .20. Å¿eptimi, cuius <lb/>prouentus radix quadrata erit quod quærebatur.</s> - </p> - <p> - <s xml:id="s3439" xml:space="preserve">Sed aliter idem poſſe fieri Å¿peculatus Å¿um, hoc eÅ¿t multiplicando numerum .49. <lb/>ordinis .1000. hominum <reg norm="cum" type="context">cÅ©</reg> radice quadrata numeri .3500. propoÅ¿iti, productum ve-<lb/>rò diuidere per radicem quadratam ipÅ¿ius .1000. vnde prouentus .91. erit numerus <lb/>vnius ordinis .3500. numeri <reg norm="propoÅ¿iti" type="simple">ê“poÅ¿iti</reg>.</s> - </p> - <p> - <s xml:id="s3440" xml:space="preserve">Cuius <reg norm="operationis" type="simple">oê‘ationis</reg> Å¿peculatio eÅ¿t iÅ¿ta. <lb/> - <anchor type="figure" xlink:label="fig-0287-01a" xlink:href="fig-0287-01"/> - </s> - <s xml:id="s3441" xml:space="preserve">Sit <var>.a.b.</var> quadratum .1000. et <var>.a.c.</var> Å¿ua <lb/>radix et <var>.a.d.</var> rectangulum propoÅ¿i-<lb/>tum ipÅ¿ius .1000. et <var>.a.e.</var> vnus ordo. <lb/></s> - <s xml:id="s3442" xml:space="preserve">Sit etiam <var>.A.B.</var> quadratum .3500. & <lb/><var>A.C.</var> eius radix et <var>.A.D.</var> <reg norm="rectangulum" type="context">rectangulÅ©</reg> <lb/>ipÅ¿ius numeri .3500. propoÅ¿iti, Å¿imile <lb/>tamen rectangulo <var>.a.d.</var> et <var>.A.E.</var> eius <lb/>vnus ordo. </s> - <s xml:id="s3443" xml:space="preserve"><reg norm="Cum" type="context">CÅ©</reg> enim <var>.a.b.</var> æquale Å¿it <lb/><var>a.d.</var> et <var>.A.B</var>: <var>A.D.</var> <reg norm="tunc" type="context">tÅ©c</reg> <var>.a.c.</var> erit media <lb/>proportionalis inter <var>.a.e.</var> et <var>.e.d.</var> & Å¿ic <lb/><var>A.C.</var> erit etiam media proportiona <lb/>lis inter <var>.A.E.</var> et <var>.E.D.</var> per .16. Å¿exti, <lb/>Å¿eu .20. Å¿eptimi, & quia proporrio. A <lb/>E. ad <var>.E.D.</var> æqualis eÅ¿t proportioni <var>.<lb/>a.e.</var> ad <var>.e.d.</var> cum <var>.A.D.</var> Å¿upponatur Å¿i-<lb/>mile <var>.a.d.</var> ergo proportio <var>.A.E.</var> ad <var>.A<lb/>C.</var> Ä™qualis erit proportioni <var>.a.e.</var> ad <var>.a.<lb/>c.</var> quÄ™ medietates Å¿unt <reg norm="totorum" type="context">totorÅ©</reg> æqua-<lb/>lium, rectè igitur fiet Å¿i procedamus <lb/>ex regula de tribus, dicendo Å¿i <var>.a.c.</var> <lb/><reg norm="correÅ¿pondet" type="context">correÅ¿põdet</reg> <var>.a.e.</var> tùc <var>.A.C.</var> <reg norm="correÅ¿pon" type="context">correÅ¿põ</reg> <lb/>det <var>.A.E.</var> ex Å¿upradictis .15. Å¿exti. vel <lb/>20. Å¿eptimi.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0287-01" xlink:href="fig-0287-01a"> - <image file="0287-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0287-01"/> - </figure> - </div> - <p> - <s xml:id="s3444" xml:space="preserve">Ratio verò quarti quæſiti per Å¿e <lb/>patet, quod eÅ¿t inuenire <reg norm="pauimentum" type="context">pauimentÅ©</reg> <lb/>Å¿eu aream quadratam, in qua poſſint <lb/>locari quot homines volueris, ita in <lb/>ter Å¿e Å¿iti, ut vnuÅ¿quiÅ¿que occupet <num value="7">.<lb/>7.</num> pedes ipÅ¿ius areÄ™ in longitudinem <lb/>et .3. per latitudinem à lateribus.</s> - </p> - <p> - <s xml:id="s3445" xml:space="preserve">Seu ex propoÅ¿ito hominum nume <lb/>ro inuenire numerum ipÅ¿orum loca-<lb/>bilem in aliqua area quadrata, ita, <lb/>vt vnuÅ¿quiÅ¿que occupet .21. pedes <lb/>quadratos ipÅ¿ius areæ.</s> - </p> - <pb o="276" rhead="IO. BAPT. BENED." n="288" file="0288" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0288"/> - <p> - <s xml:id="s3446" xml:space="preserve">Sed aliter idem fieri poſſe inueni, hoc eÅ¿t <reg norm="multiplicando" type="context">multiplicãdo</reg> radicem quadratam pro-<lb/>poÅ¿iti numeri hominum per .21. & productum item multiplicando per eandem radi <lb/>cem, & huiuÅ¿modi producti radicem diuiden do per .3. vnde prouentus eſſet nume-<lb/>rus hominum vnius ordinis. </s> - <s xml:id="s3447" xml:space="preserve"><reg norm="Exempli" type="context">Exẽpli</reg> gratia proponu<unsure/>ntur .3600. homines, multiplica <lb/>bimus huiuÅ¿modi numeri radicem <lb/>quadratam hoc eÅ¿t .60. per .21. hoc <lb/> - <anchor type="figure" xlink:label="fig-0288-01a" xlink:href="fig-0288-01"/> - eÅ¿t per productum quod fit ex .7. <reg norm="cum" type="context">cÅ©</reg> <lb/>3. & reÅ¿ultabit nobis .1260. quod Å¿i <lb/>multiplicabitur, per .60. hoc eÅ¿t per <lb/>eandem radicem, reÅ¿ultabit nobis <num value="75600">.<lb/>75600.</num> cuius producti radix qua-<lb/>drata eÅ¿t ferè .275. qua diuiÅ¿a per .3 <lb/>proueniet nobis .91. pro hominum <lb/>numero vnius ordinis.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0288-01" xlink:href="fig-0288-01a"> - <image file="0288-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0288-01"/> - </figure> - </div> - <p> - <s xml:id="s3448" xml:space="preserve">Cuiusratio eÅ¿t iÅ¿ta, cogitemus nu <lb/>merum .3600. propoÅ¿itum eſſe qua <lb/>dratum <var>.a.b.</var> (Å¿ed non areæ) cuius ra <lb/>dix .60. Å¿it <var>.a.c.</var> & quia hic numerus <num value="60">.<lb/>60.</num> intelligitur eſſe hominum, quo-<lb/>rum <reg norm="vnuÅ¿quiÅ¿que" type="simple">vnuÅ¿quiÅ¿q;</reg> occupat .21. pedes <lb/>quadratos Å¿uperficiales ex Å¿uppoÅ¿i-<lb/>to, </s> - <s xml:id="s3449" xml:space="preserve">& propterea multiplicatur, 60. <lb/>cum .21. vnde nobis veniat .1260. <lb/>quadrati Å¿uperficiales pro vnoquo-<lb/>que ordine, & <reg norm="quia" type="simple">ê—a</reg>.b.c. vt. latus qua-<lb/>drati <var>.a.b.</var> habet tot ordines homi-<lb/>num Å¿imiliter, hoc eÅ¿t .60. igi-<lb/>tur multiplicando .60. cum .1260. <lb/>habebimus totalem Å¿uperficiem <var>.a.<lb/>b.</var> ex .75600. quadratis Å¿uperficiali-<lb/>bus, quæ quadrata imaginemur lo-<lb/>cata eſſe in quodam totali quadra-<lb/>to, quod Å¿it <var>.e.f.</var> cuius radix Å¿it <var>.e.</var> g <num value="275">.<lb/>275.</num> pedum qui diuidantur per .3. <lb/>hoc eÅ¿t per numerum pedum latitu-<lb/>dinis & prouenient nobis .91. pro <lb/>numero <reg norm="hominum" type="context">hominÅ©</reg> <reg norm="vniuÅ¿cuiuÅ¿que" type="simple">vniuÅ¿cuiuÅ¿q;</reg> ordi-<lb/>nis, diuidendo poÅ¿tea latus <var>.f.g.</var> per <lb/>numerum Å¿patij inter vnum, & <reg norm="alium" type="context">aliÅ©</reg> <lb/>ordinem, quod eÅ¿t .7. proueniet <lb/>nobis .39. pro numero ordinum.</s> - </p> - <p> - <s xml:id="s3450" xml:space="preserve">Aliter, & breuius etiam poſſumus idem inuenire, hoc eÅ¿t multiplicando <reg norm="nume- rum" type="context">nume-<lb/>rÅ©</reg> <reg norm="propoÅ¿itum" type="context">propoÅ¿itÅ©</reg> <reg norm="hominum" type="context">hominÅ©</reg> <reg norm="cum" type="context">cÅ©</reg> rectangulo .21. vnde venietnobis <reg norm="productum" type="simple context">ê“ductÅ©</reg> .75600 quod pro <lb/><reg norm="ductum" type="context">ductÅ©</reg> Å¿i accipiemus vt <reg norm="quadratum" type="context">quadratÅ©</reg>, cuius radix erit .275. quæ diuidatur <reg norm="per" type="simple">ê‘</reg> .3. habebi-<lb/>mus <reg norm="propofitum" type="simple context">ê“pofitÅ©</reg>. </s> - <s xml:id="s3451" xml:space="preserve">Cuius ratio <reg norm="pendet" type="context">pẽdet</reg> à ſupradicta, eo <reg norm="quod" type="wordlist">qÄ‘</reg> loco <reg norm="multiplicandi" type="context">multiplicãdi</reg> <var>.a.c.</var> (hoc eÅ¿t <num value="60">. - <pb o="277" rhead="EPISTOL AE." n="289" file="0289" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0289"/> - 60.</num>) per .21. deinde <reg norm="productum" type="context">productÅ©</reg> etiam multiplicare per <var>.b.c.</var> (hoc eÅ¿t .60.) breuius erit <lb/>multiplicare totum numerum .3600. per .21. cÄ™tera verò facere, vt diximus.</s> - </p> - <p> - <s xml:id="s3452" xml:space="preserve">Sed <reg norm="vnaquæque" type="simple">vnaquæq;</reg> iÅ¿tarum operationum, aliquid imperfectionis patitur, eo quod <reg norm="cum" type="context">cÅ©</reg> <lb/>aliquis cuperet quadratum perfectum Å¿uperficiale habere, <reg norm="abÅ¿que" type="simple">abÅ¿q;</reg> aliquo defectu, vel <lb/>exceÅ¿fu, aliquid aliud adhuc facere oporteret, hoc eÅ¿t, inuentum cum fuerit quadra <lb/>tum <var>.e.f.</var> cum Å¿uis radicibus <var>.e.g.</var> et <var>.g.f.</var> pedum .275. vnaquaque, vt in dicto exemplo <lb/>factum eÅ¿t, oportebit <reg norm="numerum" type="context">numerÅ©</reg> quærere minorem ipÅ¿o .275. Å¿ed proximiorem men-<lb/>furabilem ab .3. & ab .7. quod facilè fiet Å¿i diuiÅ¿erimus .275. per .21. detrahendo fra-<lb/>cta diuiÅ¿ionis ab ipÅ¿o .275. quæ quidem fracta in hoc exemplo Å¿unt .2. vnde remane-<lb/>bit .273. pro numero laterum quadrati Å¿uperficialis, in quo poſſent locari .3549. ho-<lb/>mines, eo ordine quo Å¿upra dictum eÅ¿t, quorum Å¿cilicet vnuÅ¿quiÅ¿que obtineat .21. <lb/>pedes Å¿uperÅ¿iciales.</s> - </p> - </div> - </div> - <div type="section" level="3" n="12"> - <div type="letter" level="4" n="1"> - <head xml:id="head412" xml:space="preserve">DE INTERVALLIS MVSICIS.</head> - <head xml:id="head413" style="it" xml:space="preserve">Cypriano Rorè MuÅ¿ico celeberrimo.</head> - <p> - <s xml:id="s3453" xml:space="preserve"><emph style="sc">OPinio</emph> Hectoris EuÅ¿onij Cypriane mi dilectiſſime, vera non eÅ¿t, quod ali <lb/>quisrectè poſſit intelligere rationes conÅ¿onantiarum muÅ¿icæ, ablque co <lb/>gnitione illarum mediante ipÅ¿o Å¿enÅ¿u, imo nemo <reg norm="pont" type="context">põt</reg> calere <reg norm="theoriam" type="context">theoriã</reg> mu <lb/>Å¿ices, niÅ¿i aliquo <reg norm="mon" type="context">mõ</reg> verÅ¿atus Å¿it in praxi. </s> - <s xml:id="s3454" xml:space="preserve"><reg norm="Quando" type="wordlist">QÅ©o</reg> enim cognoÅ¿ci <reg norm="poterunt" type="context">poterÅ©t</reg> <lb/>quid nam Å¿int diapaÅ¿on, diapente, diateſſeron, ditonus, Å¿emiditonus, hexacordum <lb/>maius, aut minus, & conÅ¿onantiæ ex ijs cum diapaÅ¿on compoÅ¿itæ, abÅ¿que earum <lb/>praxi? </s> - <s xml:id="s3455" xml:space="preserve">vnde Å¿equetur <reg norm="neque" type="simple">neq;</reg> etiam cognoÅ¿ci poſſe interualla diſſonantia. </s> - <s xml:id="s3456" xml:space="preserve">Et purus <lb/>practicus non intelliget quid Å¿it octaua, quinta, quarta, tertia maior, tertia minor, <lb/>Å¿exta maior, Å¿exta minor, decima maior, decima minor, vndecima, duode-<lb/>cima, decimatertia maior, aut minor, aut decimaquinta, & aliæ, ita vt ad <lb/>comparandam perfectionem muÅ¿icæ neceſſarium Å¿it, & thÄ™oriam & praxim ad-<lb/>diÅ¿cere. </s> - <s xml:id="s3457" xml:space="preserve">Cum prÄ™terea Ludouicus Folianus apertè monÅ¿trarit (etiam Å¿i id à diato-<lb/>nico Å¿intono Ptolomei deÅ¿umpÅ¿erit) reperiri duos tonos, maiorem, & minorem, id-<lb/>eÅ¿t Å¿eÅ¿quioctauum, & Å¿eÅ¿quinonum, & tria Å¿emitonia, maius, minus, & mini-<lb/>mum, ideÅ¿t Å¿eÅ¿quiquintumdecimum, qui eÅ¿t maius, Å¿eÅ¿quiuigeÅ¿imum quartum id-<lb/>eÅ¿t minimum, & mediocre, vt .27. ad .25. quæ proportio Å¿uperbipartiens vigeÅ¿i-<lb/>maÅ¿quintas appellatur, & cum cognouerit Å¿emiditonum conÅ¿onantem eſſe Å¿eÅ¿qui-<lb/>quintum, ditonum Å¿eÅ¿quiquartum, & hexachordum minus, vt .8. ad .5. quæ propor-<lb/>tio dicitur Å¿upertripartiens quintas, & hexachordum maius, vt .5. ad .3. hÄ™c autem vo <lb/>catur Å¿uperbipartienstertias; </s> - <s xml:id="s3458" xml:space="preserve">omnium Å¿implicium conÅ¿onantiarum cognitioni, ex-<lb/>tremam impoÅ¿uit manum. </s> - <s xml:id="s3459" xml:space="preserve">Et quia tibi etiam oÅ¿tendere promiÅ¿i in modulationibus - <pb o="278" rhead="IO. BAPT. BENED." n="290" file="0290" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0290"/> - hæc omnia interualla Å¿eruari, ideo ad te mitto Å¿eptem hic Å¿ubÅ¿cripta exempla, in <lb/>quorum primo, & Å¿ecundo, inter dieÅ¿im, et <var>.b.</var> in Å¿uperiori, agnoÅ¿ces interuallum mi <lb/>nimi Å¿emitonij, & Å¿i ibi Å¿it dieÅ¿is, tanquam terminus ad quem, et <var>.b.</var> tanquam termi-<lb/>nus à quo: </s> - <s xml:id="s3460" xml:space="preserve">quod autem inter dieÅ¿im et <var>.b.</var> Å¿it Å¿emitonium minimum, facilè agnoÅ¿ces <lb/>Å¿i Å¿ubtraxeris <reg norm="decimam" type="context">decimã</reg> <reg norm="minorem" type="context">minorẽ</reg> à maiori, <reg norm="quam" type="context">quã</reg> facit Å¿uperius <reg norm="cum" type="context">cÅ©</reg> inferiori, ideÅ¿t <reg norm="cum" type="context">cÅ©</reg> baſſu.</s> - </p> - <p> - <s xml:id="s3461" xml:space="preserve">Qua quidem modulatione tu etiam vÅ¿us es in cantilena illa, quæ Galica lingua <lb/>incipit. </s> - <s xml:id="s3462" xml:space="preserve">Hellas comment. </s> - <s xml:id="s3463" xml:space="preserve">Eadem, ego quoque in meis cantilenis latino Å¿ermo-<lb/>ne compoÅ¿itis, quæ Moreta vocantur aliquando vÅ¿us Å¿um.</s> - </p> - <p> - <s xml:id="s3464" xml:space="preserve">Sed in tertio exemplo inuenies Å¿emitonium maius, neceſſariò genitum in Å¿upe-<lb/>riori, Å¿i Å¿extam maiorem cum baſſu eſſicere volueris, quia tenor, à ditono cum <lb/>Å¿uperiori ad diapentem, & ad vniÅ¿onum cum baſſu procedit, vbi quieÅ¿cit, progre-<lb/>diendo poÅ¿tea baſſus ad Å¿emiditonum cum tenore, </s> - <s xml:id="s3465" xml:space="preserve">tunc Å¿i à proportione huius Å¿ep-<lb/>timæ, quæ eÅ¿t vt .9. ad .5. hoc eÅ¿t Å¿uperquadripartiensquintas demptum fuerit hexa-<lb/>chordum maius, Å¿eu Å¿exta maior, quæ eÅ¿t vt .5. ad .3. remanebit proportio .27. ad .25. <lb/>quæ maior eÅ¿t quam .32. ad .30.</s> - </p> - <p> - <s xml:id="s3466" xml:space="preserve">In quarto <reg norm="exemplo" type="context">exẽplo</reg> habebis Å¿emitonium minus in Å¿uperiori, quod quidem remanet <lb/>ex Å¿ubtractione ditoni <reg norm="conÅ¿onantis" type="context context">cõſonãtis</reg> ab diateſſaron <reg norm="comprÄ™henÅ¿a" type="context">cõprÄ™henÅ¿a</reg> à ſuperiori cum tenore.</s> - </p> - <p> - <s xml:id="s3467" xml:space="preserve">In quinto exemplo videbis tonum minorem, & tonum maiorem Å¿ucceſſiuè vnum <lb/>poÅ¿t alium in tenore, detrahendo primo <reg norm="Å¿emiditonum" type="context">Å¿emiditonÅ©</reg> à diateſſaron, quod Å¿uperius fa-<lb/>cit cum tenore, vel detrahendo diapente ab hexachordo maiori, quod facit tenor <lb/>cum baſſu, vnde remanet tonus minor Å¿eÅ¿quinonus, detrahendo poÅ¿tea diateſſaron <lb/>à diapente, quod Å¿uperius facit cum tenore, remanebit tonus maior Å¿eÅ¿quioctauus.</s> - </p> - <p> - <s xml:id="s3468" xml:space="preserve">In Å¿exto exemplo deinde videbis tenorem aÅ¿cendere per duos tonos minores Å¿uc <lb/>ceſſiuè vnum poÅ¿t alium in tenore, Å¿i <reg norm="demp" type="context">dẽp</reg> Å¿eris <reg norm="Å¿emiditonum" type="context">Å¿emiditonÅ©</reg> à diateſſaron <reg norm="cum" type="context">cÅ©</reg> Å¿uperiori.</s> - </p> - <p> - <s xml:id="s3469" xml:space="preserve">In .7. <reg norm="exemplo" type="context">exẽplo</reg> demum videbis <reg norm="Å¿uperiorem" type="context">Å¿uperiorẽ</reg> aÅ¿cendere per duos tonos maiores Å¿ucceÅ¿-<lb/>Å¿iuè <reg norm="vnum" type="context">vnÅ©</reg> poÅ¿t <reg norm="alium" type="context">aliÅ©</reg>, Å¿i dempÅ¿eris diateſſaron à diapente, quod facittenor <reg norm="cum" type="context">cÅ©</reg> Å¿uperiori.</s> - </p> - <figure position="here"> - <image file="0290-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0290-01"/> - </figure> - <pb o="279" rhead="EPISTOL AE." n="291" file="0291" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0291"/> - </div> - </div> - <div type="section" level="3" n="13"> - <div type="letter" level="4" n="1"> - <head xml:id="head414" style="it" xml:space="preserve">De eodem Å¿ubiecto.</head> - <head xml:id="head415" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3470" xml:space="preserve">QVod alià s tibi dixi, verum eÅ¿t, quod neceſſarium nullo modo Å¿it, vt modulan-<lb/>do, deÅ¿inat cantilena in eodem tono (quod Græci phthongum appel-<lb/>lant) à quo incÄ™pit. </s> - <s xml:id="s3471" xml:space="preserve">immo neceſſariò Å¿emper ferè, altius, aut depræſ-<lb/>Å¿ius terminatur, per differentiam alicuius interualli æqualis, vel multiplicis ipÅ¿i com <lb/>mati Å¿eÅ¿quioctuageÅ¿imæ, quod quidem comma, quamuis cantabile non Å¿it, inÅ¿enÅ¿i-<lb/>biliter tamen generatur, & toties ab aliqua parte ipÅ¿ius cantilenæ poſſet <reg norm="dictum" type="context">dictÅ©</reg> com-<lb/>ma gcnerari, verÅ¿us acutum, vel graue, quod in fine ipÅ¿ius cantilenÄ™, vocis phtongus <lb/>reperiatur diÅ¿tans à primo per interuallum alicuius toni Å¿eÅ¿quinoni, Å¿eu Å¿eÅ¿quioctaui <lb/>plus, minúsue, vt in Å¿ubÅ¿cripto exemplo clarè videre potes in prima figura, vbi Å¿u-<lb/>perius à <var>.g.</var> primÄ™ cellulæ ad <var>.g.</var> Å¿ecundæ, intereÅ¿t vnum <reg norm="comma" type="context">cõma</reg>, eo quod progrediens <lb/>Å¿uperius in prima cellula ipÅ¿ius cantilenæ à quarta ad quintam cum tenore, aÅ¿cendit <lb/>per tonum Å¿eÅ¿quioctauum, à prima cellula deinde ad Å¿ecundam, tenor aÅ¿cendit Å¿imi-<lb/>liter per tonum Å¿eÅ¿quioctauum cum tranÅ¿eat à quinta ad quartam, quod facit cum <lb/>Å¿uperiori, in Å¿ecunda cellula poÅ¿tea, cum Å¿uperius deÅ¿cendat à maiori Å¿exta ad quin <lb/>tam, quod facit cum baſſu, Å¿eu à quarta ad tertiam minorem, quod facit cum teno-<lb/>re, </s> - <s xml:id="s3472" xml:space="preserve">tunc deÅ¿cendit per tonum Å¿eÅ¿quinonum, ita quod non reuertitur ad <reg norm="eundem" type="context">eundẽ</reg> <reg norm="phthon" type="context">phthõ</reg> <lb/>gum, vbi prius erat in prima cellula, Å¿ed reperitur per <reg norm="vnum" type="context">vnÅ©</reg> <reg norm="coomma" type="context">coõma</reg> altius, <reg norm="quod" type="wordlist">qÄ‘</reg> <reg norm="quidem" type="context">quidẽ</reg> <lb/><reg norm="comma" type="context">cõma</reg> eÅ¿t differentia inter <reg norm="tonum" type="context">tonÅ©</reg> <reg norm="Å¿eÅ¿quioctauum" type="context">Å¿eÅ¿quioctauÅ©</reg> & <reg norm="Å¿eÅ¿quinonum" type="context">Å¿eÅ¿quinonÅ©</reg>, vt alias tibi <reg norm="demonÅ¿traui" type="context">demõſtraui</reg>.</s> - </p> - <p> - <s xml:id="s3473" xml:space="preserve">Progrediendo igitur hoc modo, videbis quod cum tenor à ſecunda cellula ad ter <lb/>tiam tranÅ¿eat à tertia minori ad quartam, quod facit cum Å¿uperiori, deÅ¿cendit per <lb/>tonum Å¿eÅ¿quinonum, vnde in tertia cellula altius remanet quam in prima per <reg norm="vnum" type="context">vnÅ©</reg> <lb/>comma, in qua tertia cellula, cum iterum tranÅ¿eat Å¿uperius à quarta ad quintam, <reg norm="quod" type="simple">ê™</reg> <lb/>facit cum tenore, eleuatur per tonum Å¿eÅ¿quioctauum, proÅ¿equendo deinde tali ordi <lb/>ne, vidc<unsure/>bis in quarta cellula cantilenam auctam per duo commata, in Å¿exta, <reg norm="autem" type="wordlist">aÅ©t</reg> cel-<lb/>lula per tria commata, in octaua verò per .4. commata, vnde hac merhodo, Å¿i can-<lb/>tilena prolixior debito eſſet, vel Å¿i talia interualla frequentiora reperirentur, poſſet <lb/>cantilena à principio ad finem differre per .9. commata, & plus etiam, quæ quidem <lb/> - <anchor type="figure" xlink:label="fig-0291-01a" xlink:href="fig-0291-01"/> - <pb o="280" rhead="IO. BABPT. BENED." n="292" file="0292" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0292"/> - interualla Å¿uperant tonum Å¿eÅ¿quinonum, & Å¿i eſſent .10. commata<unsure/> Å¿uperarent tonum <lb/>Å¿eÅ¿quioctauum, eo quod aggregatum ex .9. commatibus continetur Å¿ub iÅ¿tis duobus <lb/>terminis hoc eÅ¿t .150094635296999121. et .134217728000000000. quæ qui-<lb/>dem proportio maior eÅ¿t proportione Å¿eÅ¿quinona, Å¿umma verò .10. commatum con <lb/>tinetur Å¿ub .12157665459056928801. et .10737418240000000000. quæ pro <lb/>portio maior eÅ¿t tono Å¿eÅ¿quioctauo, quod autem dico de aÅ¿cenÅ¿u cantilenæ, idem aÅ¿-<lb/>Å¿ero de eiuÅ¿dem deÅ¿cenÅ¿u, & hoc non tantum per interuallum illius commatis, quod <lb/>eÅ¿t differentia toni maioris à minori, Å¿ed etiam per illud quod eÅ¿t differentia Å¿emito <lb/>nij maioris à minori, vt in Å¿ecundo exemplo hic Å¿ubÅ¿cripto videre eÅ¿t in deÅ¿cenÅ¿u <lb/>cantilenæ per comma & comma, vt differentia inter Å¿emitonia maiora<unsure/> & minora, <lb/>vbi in prima cellula diÅ¿cedens baſſus à quinta cum Å¿uperiori, & ab vniÅ¿ono cum te-<lb/>nore deÅ¿cendens ad tertiam minorem cum ipÅ¿o tenore, facit cum Å¿uperiori <reg norm="Å¿eptimam" type="context">Å¿eptimã</reg> <lb/>maiorem, quæ eÅ¿t vt .9. ad .5. Å¿uperquadripartiensquintas Å¿cilicet, à qua diÅ¿cedens <lb/>poÅ¿tea Å¿uperius, vt faciat cum baſſu Å¿extam maiorem, deÅ¿cendit per Å¿emitonium ma <lb/>ius, à qua Å¿exta maiori deÅ¿cendens baſſus, & aÅ¿cendens per quartam, efficit cum di-<lb/>cto Å¿uperiori <reg norm="tertiam" type="context">tertiã</reg> maiorem, à qua diÅ¿cedens Å¿uperius, vt efficiat quartam cum ipÅ¿o <lb/>baſſu (qui quidem baſſus tranÅ¿it in tenorem) aÅ¿cendit per Å¿emitonium minus, diffe-<lb/>rens à ſemitonio maiori per vnum comma, vnde cantilena remanet depreſſa per <lb/>vnum comma. </s> - <s xml:id="s3474" xml:space="preserve">cum deinde idem faciat inter tertiam, & quartam cellulam, per a-<lb/>liud comma deÅ¿cendit, & Å¿ic toties facere poſſet, vt poÅ¿tremo valde deprimatur <lb/>cantilena à primo phthongo.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0291-01" xlink:href="fig-0291-01a"> - <image file="0291-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0291-01"/> - </figure> - </div> - <figure position="here"> - <image file="0292-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0292-01"/> - </figure> - <p> - <s xml:id="s3475" xml:space="preserve">Quod autem hic Å¿upradictum eÅ¿t, clrca inÅ¿trumenta artificialia non accidit, qua <lb/>propter organa, & clauicimbula concordantur certo quodam ordine, ita vt omnes <lb/>conÅ¿onantiæ, excepta diapaÅ¿on, Å¿eu octaua, Å¿int imperfectæ, hoc eÅ¿t, aut diminutÄ™, <lb/>aut Å¿uperantes à inÅ¿to, vt exempli gratia, omnes quintæ Å¿unt diminutæ, quartæ verò <lb/>Å¿unt exceſſiuÄ™, quod quidem fit, vt tertiæ, & Å¿extæ, non multum auribus diſſonent, <lb/>eo quod Å¿i quintæ omnes, & quartæ, perfectæ eſſent, </s> - <s xml:id="s3476" xml:space="preserve">tunc omnes Å¿extÄ™, & tertiæ in-<lb/>tollerabiles eſſent, & à perfectis differrent per vnum comma, quod manifeÅ¿tum no-<lb/>bis erit hoc modo, accipiamus tres diapentes, Å¿eu quintas, conÅ¿equenter Å¿ucceſſiuas <lb/>vnam poÅ¿t aliam, hoc eÅ¿t tres proportiones Å¿eÅ¿quialteras, quarum aggregatum erit <lb/>vt .27. ad .8. quæ proportio, dicitur tripla Å¿upertripartiensoctauas, & quæ à practicis - <pb o="281" rhead="EPISTOL AE." n="293" file="0293" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0293"/> - appellaretur tertia decima maior, vt exempli gratia, eſſet Gamaut cum Å¿ecundo ela <lb/>mi, </s> - <s xml:id="s3477" xml:space="preserve">tunc talis tertiadecima valde odioÅ¿a eſſet Å¿enÅ¿ui auditus, à qua, Å¿i dempta Å¿uerit <lb/>diapaÅ¿on, Å¿eu octaua, remaneret quoddam hexachordum maius, Å¿eu Å¿exta maior, au-<lb/>ribus valde inimica, Å¿ub proportione .13. ad 8. Å¿ed hæc proportio differret à propor <lb/>tione Å¿uperbipartientetertias perfecti hexachordi maioris, hoc eÅ¿t Å¿extæ maioris <lb/>conÅ¿onantis, per proportionem Å¿eÅ¿quioctuageÅ¿imam, hoc eÅ¿t per vnum comma, <lb/>quod quidem eÅ¿t etiam differentia aggregati trium Å¿eÅ¿quialterarum, à tertiadeci-<lb/>ma maiori conÅ¿onanti, hoc eÅ¿t exceſſus proportionis triplæ Å¿upertripartientis octa-<lb/>uas, Å¿upra triplam Å¿eÅ¿quitertiam, quæ eÅ¿t Å¿umma ipÅ¿ius duplæ cum Å¿uperbi partien-<lb/>tetertias.</s> - </p> - <p> - <s xml:id="s3478" xml:space="preserve">A tali Å¿umma igitur trium Å¿eÅ¿quialterarum efficitur tertiadecima maior diſſonans <lb/>excedens conÅ¿onantem per vnum comma (cuius proportio eÅ¿t .81. ad .80.) quæ con-<lb/>fonans continetur in proportione .10. ad .3. vt Å¿upra dixi.</s> - </p> - <p> - <s xml:id="s3479" xml:space="preserve">Hæcigitur eÅ¿t vera ratio, propter quam debemus comma diÅ¿tribuere in organis <lb/>& clauicymbalis, cum ab aggregato trium quintarum producatur talis exceſſus Å¿u-<lb/>pra perfectam, Å¿eu conÅ¿onantem tertiamdecimam maiorem, quod quidem aggre-<lb/>gatum, cum demptum fuerit à quintadecima, relinquet nobis tertiam minorem <lb/>diſſonantem, & mancam, per eundem exceſſum à conÅ¿onanti. </s> - <s xml:id="s3480" xml:space="preserve">quæ quidem tertia <lb/>minor diſſonans Å¿ubtracta à diapente Å¿eu quinta perfecta, relinquet nobis tertiam <lb/>maiorem diſſonantem, qu@ conÅ¿onantem excedit per eundem exceſſum comma-<lb/>tis, & hæc demum tertia maior diſſonans, dempta ex diapaÅ¿on, Å¿eu octaua, relin-<lb/>quet nobis hexachordum minus, hoc eÅ¿t Å¿extam minorem diſſonantem, & muti-<lb/>lam à conÅ¿onanti per eundem exceſſum commatis. </s> - <s xml:id="s3481" xml:space="preserve">De huiuÅ¿modi verò commatis <lb/>diÅ¿tributione doctiſſimè Å¿cripÅ¿it Excellentiſſimus Zarlinus in Å¿ecunda parte InÅ¿titu-<lb/>tionum Harmonicarum.</s> - </p> - <p> - <s xml:id="s3482" xml:space="preserve">Sed quia Å¿enÅ¿us auditus non poteÅ¿t exactè cognoÅ¿cere debitam quantitatem ex-<lb/>ceſſus, vel defectus, intendendo vel remittendo chordas inÅ¿trumentorum, ideo hanc <lb/>viam Å¿equutus Å¿um.</s> - </p> - <p> - <s xml:id="s3483" xml:space="preserve">Sit exempli gratia, hic Å¿ubÅ¿criptus ordo lignorum tangentium Å¿eu pinarum inci-<lb/>piens ab <var>.G.</var> deÅ¿inens ad <var>.g.</var> ita quod inter ipÅ¿os terminos Å¿it ea conÅ¿onantia quæ vo-<lb/>catur vigeÅ¿imaÅ¿ecunda, quæro primum <var>.b.</var> inter <var>.D.E.</var> quod eÅ¿t nigrum ipÅ¿ius Ela-<lb/>mi grauiſſimum, quod groſſo modo facio conÅ¿onans cum <var>.G.</var> grauiſſimo per Å¿ex-<lb/>tam <reg norm="minorem" type="context">minorẽ</reg>, deinde <reg norm="cum" type="context">cÅ©</reg> ipÅ¿o pt<unsure/>imo <var>.b.</var> ipÅ¿ius elami concordo Å¿uum octauum & quin-<lb/>tumdecimum, quo perfectius poſſum, deinde accipio <var>.b.</var> molle Å¿ecundum ipÅ¿ius. b <lb/>fabmi quod concordo cum <var>.b.</var> primo ipÅ¿ius Elami per quintam imperfectam, dein-<lb/>de cum hoc <var>.b.</var> Å¿ecundo ipÅ¿ius bÅ¿abmi concordo Å¿ecundum <var>.f.</var> per quintam Å¿imiliter <lb/>imperfectam, cum quo <var>.f.</var> poÅ¿tea concordo tertium <var>.c.</var> per Å¿imilem quintam, quem <lb/>tertium <var>.c.</var> poÅ¿tea confero <reg norm="cum" type="context">cÅ©</reg> Å¿ecundo <var>.b.</var> ipÅ¿ius elami, ita quod inter Å¿e conÅ¿onent per <lb/>Å¿extam maiorem tolerabilem, & Å¿i Å¿ic inuenio, tunc nihil muto has treschordas hoc. <lb/> - <anchor type="table" xlink:label="table-0293-01a" xlink:href="table-0293-01"/> - <pb o="288" rhead="IO. BAPT. BENED." n="294" file="0294" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0294"/> - eÅ¿t <var>.b.</var> Å¿ecundum ipÅ¿ius bfabmi, f. Å¿ecundum, et <var>.c.</var> tertium, Å¿ed Å¿i dictum tertium <var>.c.</var> <lb/>valde diſſonans eſſet cum <var>.b.</var> Å¿ecundo ipÅ¿ius elami, </s> - <s xml:id="s3484" xml:space="preserve">tunc ipÅ¿um <var>.c.</var> intendo, aut re-<lb/>mitto, quouÅ¿que aliquo modo Å¿it conÅ¿onans per Å¿extam maiorem aliquantulum ex <lb/>ceſſiuam cum <var>.b.</var> Å¿ecundo ipÅ¿ius elami, cum quo poÅ¿tea <var>.c.</var> conÅ¿onare aliquantulum fa <lb/>cio <var>.f.</var> Å¿ecundum per quintam defectiuam, & cum hoc<unsure/> demum <var>.b.</var> Å¿ecundum ipÅ¿ius <lb/>bfabmi, quo facto concordo Å¿ecundum <var>.c.</var> cum tertio per octauam, cum quo Å¿ecun-<lb/>do <var>.c.</var> poÅ¿tea concordo tertium <var>.g.</var> per talem quintam, quod ipÅ¿um tertium <var>.g.</var> cum Å¿e-<lb/>cundo <var>.b.</var> ipÅ¿ius bfabmi conÅ¿onet tolerabiliter per Å¿extam maiorem <reg norm="aliquantulum" type="context">aliquãtulum</reg> ex-<lb/>ceſſiuam, </s> - <s xml:id="s3485" xml:space="preserve">deinde cum iÅ¿to tertio <var>.g.</var> concordo tertium <var>.d.</var> per talem quintam, ita quod <lb/>ipÅ¿um <var>.3.d.</var> concordet tolerabiliter cum <var>.2.f.</var> per Å¿extam maiorem exceſſiuam, poÅ¿tea <lb/>cum hoc <var>.3.d.</var> concordo <var>.2.d.</var> per octauam perfecte, cum quo <var>.2.d.</var> poÅ¿tea concordo <var>.<lb/>3.a.</var> per quintam, vt in alijs <reg norm="factum" type="context">factÅ©</reg> eÅ¿t, ita vt <reg norm="cum" type="context">cÅ©</reg> <var>.2.c.</var> conÅ¿onet talis Å¿exta maior, vt Å¿upra <lb/>dictum eÅ¿t, cum quo <var>.3.a.</var> poÅ¿tea concordo <var>.3.e.</var> per quintam, vt dictum eÅ¿t, ita quod <lb/>cum <var>.3.g.</var> faciat Å¿extam maiorem vt Å¿upra, poÅ¿tea cum hoc <var>.e.</var> concordo <var>.2.e.</var> per octa <lb/>uam, cum quo concordo <var>.b.</var> quadrum tertium per quintam, vt dictum eÅ¿t, ita quod <reg norm="cum" type="context">cÅ©</reg> <lb/><var>2.d.</var> faciat Å¿extam maiorem Å¿imilem alijs Å¿uperius dictis, cum quo <var>.b.</var> quadrato tertio <lb/>concordo tertium nigrum ipÅ¿ius <var>.f.</var> per quintam, ita quod cum <var>.3 a.</var> faciat Å¿extam ma-<lb/>iorem, vt Å¿upra, deinde cum hoc concordo <var>.2.f.</var> nigrum per octauam, cum quo, per <lb/>quintam concordo <var>3.c.</var> nigrum ita quod cum <var>.2.e.</var> faciat Å¿extam dictam, demum <reg norm="cum" type="context">cÅ©</reg> <lb/>hoc concordo <var>.4.g.</var> nigrum per quintam, ita quod faciat cum <var>.3.b.</var> quadrato Å¿extam <lb/>dictam, & Å¿ic ad vltimam quintam peruenio, Å¿upra quod <var>.g.</var> nigrum nulla quinta am-<lb/>plius reperitur, poÅ¿tea cum iÅ¿tis chordis concordo per octauas omnes alias ab acutis <lb/>ad graues.</s> - </p> - <div level="5" n="2" type="float"> - <xhtml:table xlink:label="table-0293-01" xlink:href="table-0293-01a"> - <xhtml:tr> - <xhtml:td xml:space="preserve">G A B b C * D b E F * g * a b b c * d b e f * g * a b b c * d b e f * g *</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 4.</xhtml:td> - </xhtml:tr> - </xhtml:table> - </div> - <xhtml:table> - <xhtml:tr> - <xhtml:td xml:space="preserve">G A B b C * D b E F * g * a b b c * d b e f * g * a b b c * d b e f * g *</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 4.</xhtml:td> - </xhtml:tr> - </xhtml:table> - <p> - <s xml:id="s3486" xml:space="preserve">Valde etiam admiratione dignum eÅ¿t, quod perÅ¿ectiores quæque conÅ¿onan <lb/>tiæ, ita in harmonica diuiÅ¿ione Å¿ibi inuicem conueniant, vt dia paÅ¿on cum diapente, <lb/>cum diapaÅ¿ondiapente, cum ditono, cum hexachordo maiori cum bisdiapaÅ¿on, <reg norm="cum" type="context">cÅ©</reg> <lb/>decimaleptima maiori. </s> - <s xml:id="s3487" xml:space="preserve">Nam in ipÅ¿a diapaÅ¿on, harmonicè <reg norm="locantur" type="context">locãtur</reg> diapente in par <lb/>te grauiori, & diateÅ¿laron in acutiori. </s> - <s xml:id="s3488" xml:space="preserve">In diapente verò harmonicè locantur ditonus <lb/>in parte grauiori, & Å¿emiditonus in acutiori. </s> - <s xml:id="s3489" xml:space="preserve">In ditono harmonicè locantur tonus <lb/>maior in parte grauiori, & tonus minor in acutiori. </s> - <s xml:id="s3490" xml:space="preserve">In hexachordo maiori, harmo-<lb/>nicè locantur diateſſaron in parte grauiori, & ditonus in acutiori. </s> - <s xml:id="s3491" xml:space="preserve">In diapaÅ¿ondia-<lb/>pente, harmonic<unsure/>è locantur diapaÅ¿on in parte grauiori, & diapente in acutiori. <lb/></s> - <s xml:id="s3492" xml:space="preserve">In bisdiapaÅ¿on, ha@monicè locantur decima maior in parte grauiori & hexachor-<lb/>dum minus in acutiori. </s> - <s xml:id="s3493" xml:space="preserve">In decimaÅ¿eptima maiori, harmonicè locantur diapaÅ¿ondia-<lb/>pente in parte grauiori, & hexachordum maius in parte acutiori. </s> - <s xml:id="s3494" xml:space="preserve">Ita quod ronus <lb/>Å¿eÅ¿quioctauus in ditono, proportionalis eÅ¿t ipÅ¿i ditono in diapente. </s> - <s xml:id="s3495" xml:space="preserve">Tonus verò Å¿eÅ¿-<lb/>quinonus in ipÅ¿o ditono, proportionalis eÅ¿t triemitonio, vel Å¿eÅ¿quitonio Å¿eu Å¿emidi-<lb/>tono (quod idem eÅ¿t) in diapente. </s> - <s xml:id="s3496" xml:space="preserve">Ditonus autem in diapente, proportionalis eÅ¿t <lb/>ipÅ¿i diapente in diapaÅ¿on. </s> - <s xml:id="s3497" xml:space="preserve">SeÅ¿quitonus verò in diapente, proportionalis eÅ¿t diateÅ¿-<lb/>Å¿aron in diapaÅ¿on. </s> - <s xml:id="s3498" xml:space="preserve">Et Å¿ic de Å¿ingulis. </s> - <s xml:id="s3499" xml:space="preserve">Ita quod tonus Å¿eÅ¿quioctauus in ditono, dito-<lb/>mus in diapente, diateſſaron in hexachordo maiori, diapente in diapaÅ¿on, diapaÅ¿on <lb/>in diapaÅ¿ondiapente, decimamaior in bisdiapaÅ¿on, diapaÅ¿ondiapente in decima- - <pb o="283" rhead="EPISTOL AE." n="295" file="0295" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0295"/> - Å¿eptima maiori, omnia Å¿ibi inuicem Å¿unt proportionalia, idem etiam dico de reli-<lb/>quis partibus, cum relatæ fuerint ad Å¿ua tota.</s> - </p> - <p> - <s xml:id="s3500" xml:space="preserve">Nec alienum mihi videtur à propoÅ¿ito inÅ¿tituto, Å¿peculari modum generationis <lb/>ipÅ¿arum Å¿implicium <reg norm="conÅ¿onantiarum" type="context">conÅ¿onantiarÅ©</reg>; </s> - <s xml:id="s3501" xml:space="preserve">qui quidem modus fit ex quadam æquatione per <lb/>@uſſionum, Å¿eu æquali concurÅ¿u vndarum aeris, vel conterminatione earum.</s> - </p> - <p> - <s xml:id="s3502" xml:space="preserve">Nam, nulli dubium eÅ¿t, quin vniÅ¿onus Å¿it prima principalis <reg norm="audituque" type="simple">audituqÌ</reg> amiciſſima, <lb/>nec non magis propria conÅ¿onantia; </s> - <s xml:id="s3503" xml:space="preserve">& Å¿i intelligatur, vt punctus in linea, vel vnitas <lb/>in numero, quam immediate Å¿equitur diapaÅ¿on, ei Å¿imillima, poÅ¿t hanc verò diapen <lb/>te, <reg norm="cæteræque" type="simple">cæteræqÌ;</reg>. </s> - <s xml:id="s3504" xml:space="preserve">Videamus igitur ordinem concurÅ¿us percuſſionum terminorum, Å¿eu <lb/>vndarum aeris, vnde Å¿onus generatur.</s> - </p> - <p> - <s xml:id="s3505" xml:space="preserve">Concipiatur igitur mente monochordus, hoc eÅ¿t chorda diÅ¿tenta, quæ cum diuiÅ¿a <lb/>fuerit in duas æquales partes à ponticulo, </s> - <s xml:id="s3506" xml:space="preserve">tunc <reg norm="vnaquæque" type="simple">vnaquæq;</reg> pars eundem Å¿onum pro-<lb/>feret, & ambæ formabunt vniÅ¿onum, quia eodem tempore, tot percuſſiones in aere <lb/>faciet vna partium illius chordæ, quot & altera: </s> - <s xml:id="s3507" xml:space="preserve">ita vt vndæ aeris Å¿imul eant, & æqua <lb/>liter concurrant, abÅ¿que ulla interÅ¿ectione, vel fractione illarum inuicem.</s> - </p> - <p> - <s xml:id="s3508" xml:space="preserve">Sed cum ponticulus ita diuiÅ¿erit chordam, vt relicta Å¿it eius tertia pars ab vno la-<lb/>tere, ab alio vero, duÄ™ tertiÄ™, </s> - <s xml:id="s3509" xml:space="preserve">tunc maior pars, dupla erit minori, & <reg norm="Å¿onabunt" type="context">Å¿onabÅ©t</reg> ipÅ¿am dia <lb/>paÅ¿on conÅ¿onantiam, percuſſiones vero terminorum ipÅ¿ius, tali proportione Å¿e inui-<lb/>cem habebunt, ut in qualibet Å¿ecunda percuſſione minoris portionis ipÅ¿ius chordæ, <lb/>maior percutiet, Å¿eu concurret cum minori, eodem temporis inÅ¿tanti, cum ne-<lb/>mo Å¿it qui neÅ¿ciat, quod quo longior eÅ¿t chorda, etiam tardius moueatur, </s> - <s xml:id="s3510" xml:space="preserve">quare <lb/>cum longior dupla Å¿it breuiori, & eiuÅ¿dem intenÅ¿ionis tam vna quam altera, tunc eo <lb/>tempore, quo longior vnum interuallum tremoris perfecerit, breuior duo interual-<lb/>la conficiet.</s> - </p> - <p> - <s xml:id="s3511" xml:space="preserve">Cum autem ponticulus ita diuiÅ¿erit chordam, ut ab uno latere relinquantur duæ <lb/>quintæ partes, ab alio verò tres quintæ, ex quibus partibus generatur conÅ¿onantia <lb/>diapente; </s> - <s xml:id="s3512" xml:space="preserve">tunc clarè patet, quod eadem proportione tardius erit vnum interuallu<unsure/>m <lb/>tremoris maioris portionis, vno interuallo tremoris minoris portionis, quam ma-<lb/>i<unsure/>or portio habet ad minorem; </s> - <s xml:id="s3513" xml:space="preserve">hoc eÅ¿t tempus maioris interualli ad tempus minoris <lb/>erit <reg norm="Å¿eÅ¿quialterum" type="context">Å¿eÅ¿quialterÅ©</reg> quare non <reg norm="conuenient" type="context">cõuenient</reg> Å¿imul, niÅ¿i perfectis tribus interuallis mino-<lb/>ris portionis, & duobus maioris; </s> - <s xml:id="s3514" xml:space="preserve">ita quod eadem proportio erit numeri interuallo-<lb/>rum minoris portionis ad interualla maioris, quæ longitudinis maioris portionis ad <lb/>longitudinem minoris; </s> - <s xml:id="s3515" xml:space="preserve">vnde productum numeri portionis minoris ipÅ¿ius chordæ <lb/>in numerum interuallorum motus ipÅ¿ius portionis, æquale erit producto numeri <lb/>portionis maioris in numerum interuallorum ipÅ¿ius maioris portionis; </s> - <s xml:id="s3516" xml:space="preserve">quæ quidem <lb/>producta ita Å¿e habebunt, vt in diapaÅ¿on, Å¿it binarius numerus; </s> - <s xml:id="s3517" xml:space="preserve">in diapente verò <lb/>Å¿enarius; </s> - <s xml:id="s3518" xml:space="preserve">in diateſſaron duodenarius, in hexachordo maiori quindenarius; </s> - <s xml:id="s3519" xml:space="preserve">in di-<lb/>tono vicenarius, in Å¿emiditono tricenarius, demum in hexachordo minori quadra <lb/>genarius: </s> - <s xml:id="s3520" xml:space="preserve">qui quidem numeri non abÅ¿que mirabili analogia conueniunt inuicem.</s> - </p> - <p> - <s xml:id="s3521" xml:space="preserve">Voluptas autem, quam auditui afferunt conÅ¿onantiæ fit, quia leniuntur Å¿enÅ¿us, <lb/>quemadmodum <reg norm="contra" type="context">cõtra</reg>, dolor qui à diſſonantijs oritur, ab aÅ¿peritate naÅ¿citur, id quod <lb/>facilè vide<unsure/>re poteris cum conchordantur organorum fiÅ¿tulæ.</s> - </p> - <pb o="284" rhead="IO. BAPT. BENED." n="296" file="0296" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0296"/> - </div> - </div> - <div type="section" level="3" n="14"> - <div type="letter" level="4" n="1"> - <head xml:id="head416" xml:space="preserve">DE IVSTITIA COMMVTATIVA.</head> - <head xml:id="head417" style="it" xml:space="preserve">FranciÅ¿co Ferrario AnciÅ¿a IuriÅ¿conÅ¿ulto <reg norm="Å¿enatorique" type="simple">Å¿enatoriqÌê«</reg> apud <lb/>Å¿ubalpinos grauißimo.</head> - <p> - <s xml:id="s3522" xml:space="preserve"><emph style="sc">SAepivs</emph> inter nos dum oportunitas vicinarum ædium, & amoris mutui <lb/>vis, ad familiaria trahunt colloquia ego de meis mathematicis, tu de tuis <lb/>legibus, in quibus tractandis magnum tibi nomen comparaÅ¿ti loquuti Å¿u <lb/>mus. </s> - <s xml:id="s3523" xml:space="preserve">Cum vero nonnunquam de mirabili iuÅ¿titiæ commutatÃuæ inÅ¿titu <lb/>to non ingratus incidiſſet Å¿ermo, dixi modum, quo formam Å¿uam à proportionali-<lb/>tate arithmetica diÅ¿iuncta, & non a coniuncta deÅ¿umat, à nemine literis proditum <lb/>eſſe, libet autem nunc per otium latius explicare. </s> - <s xml:id="s3524" xml:space="preserve">dixi enim à diÅ¿iuncta, & non con-<lb/>iuncta proportionalitate, quia in coniuncta, Å¿eu continua nullo pacto fieri poteÅ¿t <lb/>talis commutatio, cum Å¿emper quatuor terminos ad minus tranÅ¿eat, vt nunc vide-<lb/>bimus.</s> - </p> - <p> - <s xml:id="s3525" xml:space="preserve">Exempli gratia, Petrus ex Å¿uis bonis tribuat Ioanni aliquid valoris quinquagin <lb/>ta aureorum.</s> - </p> - <p> - <s xml:id="s3526" xml:space="preserve">Vnde priuÅ¿quam Ioannes aliquid ex Å¿uis bonis retribuat Petro, bona ipÅ¿ius Pe-<lb/>tri diminuta erunt per quinquaginta aureos, bona verò ipÅ¿ius Ioannis, aucta toti-<lb/>dem aureis.</s> - </p> - <p> - <s xml:id="s3527" xml:space="preserve">Ecce nunc quo pacto conftituti Å¿unt .4. termini in proportionalitate aritmetica, <lb/>per quos Å¿it talis permutatio, Å¿ed nondum æquata, niÅ¿i fiat æqualis retributio à Ioan-<lb/>ne ad Petrum, vt videbimus.</s> - </p> - <p> - <s xml:id="s3528" xml:space="preserve">Cogitentur itaque .4. termini aritmeticè proportionales <var>.C.A.B.D</var>. </s> - <s xml:id="s3529" xml:space="preserve">Ita quod <var>.A.</var> <lb/>mediante Å¿ignificentur bona Ioannis <var>.B.</var> vero Petri, prius quam Petrus aliquid ex bo <lb/>nis Å¿uis tribuat Ioanni. </s> - <s xml:id="s3530" xml:space="preserve">Tunc Petrus Å¿ecat partem vnam ex <var>.B.</var> <reg norm="eamque" type="simple">eamqÌ;</reg> dat ipÅ¿i Ioan-<lb/>ni, vnde ipÅ¿i Petro remanet <var>.D</var>. </s> - <s xml:id="s3531" xml:space="preserve">Ioanni autem <var>.C.</var> quatuor igitur termini conÅ¿tituti <lb/>Å¿unt <var>.B.D.C.A.</var> quorum <var>.B.</var> primus <var>.A.</var> quartus <var>.C.</var> uero tertius <var>.D.</var> <reg norm="autem" type="wordlist">aÅ©t</reg> Å¿ecundus, Å¿ed <lb/>B. et <var>.A.</var> Å¿unt in Å¿ua naturali mediocritate abÅ¿que defectu vel exceſſu Å¿ui ipÅ¿ius. </s> - <s xml:id="s3532" xml:space="preserve">Non <lb/>ita tamen Å¿e habet <var>.C.</var> et <var>.D.</var> quia <var>.D.</var> deficit <var>.C.</var> autem excedit à ſua priori quantitate. <lb/></s> - <s xml:id="s3533" xml:space="preserve">Nihilominus iÅ¿ti .4. termini conÅ¿tituti Å¿unt in ipÅ¿a aritmetica proportionalitate, nam <lb/>eadem quantitate qua <var>.D.</var> diminuta eÅ¿t à <var>.B.</var> eadem <var>.C.</var> aucta eÅ¿t Å¿upra <var>.A</var>.</s> - </p> - <p> - <s xml:id="s3534" xml:space="preserve">Sed quia <var>.B.</var> et <var>.A.</var> tantummodo iuÅ¿ti Å¿unt termini <var>.C.</var> uerò et <var>.D.</var> iniuÅ¿ti, vt ad Å¿uam <lb/>priorem æqualitatem reuertantur, oportebit ex <var>.C.</var> Å¿ecare aliquam partem æqualis <lb/>valoris ei, qua <var>.C.</var> Å¿uperat <var>.A.</var> vel qua <var>.D.</var> minor eÅ¿t <var>.B.</var> & ipÅ¿am partem addere ipÅ¿i <var>.D.</var> <lb/>vt bona Petri reuertantur ad priorem Å¿uam quantitatem ipÅ¿ius <var>.B.</var> & bona Ioannis <lb/>remaneant æqualia <var>.A.</var> vt prius.</s> - </p> - <p> - <s xml:id="s3535" xml:space="preserve">Quare neceſſarium non eÅ¿t, vt talis proportionalitas Å¿it coniuncta (vt inquit Eu <lb/>f<gap/>atius Å¿eu Michael EpheÅ¿ius, <lb/>Å¿uper quinto capite libr. quin-<lb/> - <anchor type="figure" xlink:label="fig-0296-01a" xlink:href="fig-0296-01"/> - ti Ethicorum) tribus terminis <lb/>contenta, imò oportet ut ipÅ¿a <lb/>diÅ¿iuncta Å¿it, ut diximus, vbi <lb/>non eÅ¿t neceſſe quod <var>.A.</var> æqualis Å¿it <var>.B.</var> aliquo modo.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0296-01" xlink:href="fig-0296-01a"> - <image file="0296-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0296-01"/> - </figure> - </div> - <pb o="285" rhead="EPISTOL AE." n="297" file="0297" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0297"/> - </div> - </div> - <div type="section" level="3" n="15"> - <div type="letter" level="4" n="1"> - <head xml:id="head418" xml:space="preserve">DE MOTV MOLAE, ET TROCHI, DE AMPVL-<lb/>lis aquæ, de claritate aeris, & Lunæ noctu fulgentis, de æter-<lb/>nitate temporis, & infinito Å¿pacio extra <lb/>CÅ“lum, CÅ“<reg norm="lique" type="simple">liqÌ;</reg> figura.</head> - <head xml:id="head419" style="it" xml:space="preserve">Illust. Ioanni Paulo Capra NouarienÅ¿i Sabaudia Ducis boÅ¿picij <lb/>Magistro, viro ingeny praÅ¿tantia, & morum cando-<lb/>re, non minus quam familia nobili-<lb/>tate conÅ¿picuo.</head> - <p> - <s xml:id="s3536" xml:space="preserve">SI vera eſſet animorum illa tranÅ¿migratio quam Å¿ibi Italicæ Å¿apientiæ Pa-<lb/>ter Pythagoras effinxerat, tuam, <reg norm="meamque" type="simple">meamqÌ;</reg> exiÅ¿timarem animam canis, <lb/>quandoque venatici fuiſſe.</s> - </p> - <p> - <s xml:id="s3537" xml:space="preserve">Quæris à me literis tuis, an motus circularis alicuius molæ molendina <lb/>rie, Å¿i Å¿uper aliquod punctum, quaÅ¿i <reg norm="mathematicum" type="context">mathematicÅ©</reg>, quieÅ¿ceret, poſſet eſſe perpetuus, <lb/>cum aliquando eſſet mota, Å¿upponendo etiam eandem eſſe perfectè rotundam, & <lb/>lÄ™uigatam. </s> - <s xml:id="s3538" xml:space="preserve">ReÅ¿pondeo huiuÅ¿modi motum nullo modo futurum perpetuum, nec <lb/>etiam multum duraturum, quia præterquam quod ab aere qui ei circumcirca <reg norm="aliquam" type="context">aliquã</reg> <lb/>reÅ¿iÅ¿tentiam facit Å¿tringitur, eÅ¿t etiam reÅ¿iÅ¿tentia partium illius corporis moti, quæ <lb/>cum motæ Å¿unt, natura, impetum habent efficiendi iter directum, vnde cum Å¿imul <lb/>iunctæ Å¿int, & earum vna continuata cum alia. </s> - <s xml:id="s3539" xml:space="preserve">dum circulariter mouentur patiuntur <lb/>violentiam, & in huiuÅ¿modi motu per vim vnitæ manent, quia quanto magis mo-<lb/>uentur, tanto magis in ijs creÅ¿cit naturalis inclinatio recta eundi, vnde tanto magis <lb/>contra Å¿uammet naturam voluuntur, ita vt Å¿ecundum naturam quieÅ¿cant, quia cum <lb/>eis proprium Å¿it, quando Å¿unt motæ, eundi recta, quanto violentius voluuntur, tan-<lb/>to magis vna reÅ¿iÅ¿tit alteri, & quaÅ¿i retrò reuocat eam, quam antea reperitur habere.</s> - </p> - <p> - <s xml:id="s3540" xml:space="preserve">Ab eiuÅ¿modi inclinatione rectitudinis motus partium alicuius corporis rotundi <lb/>fit, vt per aliquod temporis Å¿pacium, trochus cum magna violentia Å¿eipÅ¿um circun-<lb/>agens, omninò rectus quieÅ¿cat Å¿uper illam cuÅ¿pidem ferri quam habet, non incli-<lb/>nans Å¿e verÅ¿us mundi centrum, magis ad vnam <reg norm="partem" type="context">partẽ</reg>, quam ad aliam, cum quælibet <lb/>Å¿uarum partium in huiuÅ¿modi motu non inclinet omnino verÅ¿us <reg norm="mundi" type="context">mÅ©di</reg> centrum, Å¿ed <lb/>multo magis per tranÅ¿uerÅ¿um ad angulos rectos cum linea directionis, aut verticali, <lb/>aut orizontis axe, ita vt neceſſariò huiuÅ¿modi corpus rectum Å¿tare debeat. </s> - <s xml:id="s3541" xml:space="preserve">Et quod <lb/>dico ipÅ¿as partes non omninò inclinare verÅ¿us mundi centrum, id ea ratione dico, <lb/>quia non abÅ¿olutè Å¿unt unquam priuatæ huiuÅ¿modi inclinatione, quÄ™ efficit vt ipÅ¿um <lb/>corpus eo puncto nitatur. </s> - <s xml:id="s3542" xml:space="preserve">Verum tamen eÅ¿t, quod quanto magis eÅ¿t velox, tan-<lb/>to minus premit ipÅ¿um punctum, imò ipÅ¿um corpus <reg norm="tanto" type="context">tãto</reg> magis leue remanet. </s> - <s xml:id="s3543" xml:space="preserve">Id <reg norm="quod" type="wordlist">qÄ‘</reg> <lb/>apertè patet <reg norm="Å¿umendo" type="context">Å¿umẽdo</reg> <reg norm="exemplum" type="context context">exẽplÅ©</reg> pilÄ™ alicuius arcus, aut <reg norm="alicuius" type="simple">alicuiê°</reg> alterius <reg norm="inÅ¿trumenti" type="context">inÅ¿trumẽti</reg>, Å¿eu ma <lb/>chinæ miſſilis, quæ pila quanto eÅ¿t velocior, in motu violento, tanto maiorem pro-<lb/>penÅ¿ionem habet rectius eundi, vnde verÅ¿us mundi centrum tanto minus inclinat, <lb/>& hanc ob cauÅ¿am leuior redditur. </s> - <s xml:id="s3544" xml:space="preserve">Sed Å¿i clarius, hanc veritatem videre cupis, <lb/>cogita illud corpus, Trochum Å¿cilicet, dum velociſſime circunducitur Å¿ecari, Å¿eu <lb/>diuidi in multas partes, vnde uidebis illas omnes, non illico uerÅ¿us mundi centrum - <pb o="286" rhead="IO. BAPT. BENED." n="298" file="0298" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0298"/> - deÅ¿cendere, Å¿ed recta orizontaliter, vt ita dicam, moueri. </s> - <s xml:id="s3545" xml:space="preserve">Id quod à nemine ad-<lb/>huc (quod Å¿ciam) in trocho eÅ¿t obÅ¿eruatum. </s> - <s xml:id="s3546" xml:space="preserve">Ab huiuÅ¿modi motu trochi, aut hu-<lb/>ius generis corporis, clarè perÅ¿picitur, quà m errent peripatetici circa motum uio-<lb/>lentum alicuius corporis, qui exiÅ¿timant aerem qui Å¿ubintrat ad occupandum locum <lb/>à corpore relictum, ipÅ¿um corpus impellere, cum ab hoc, magis effectus contrarius <lb/>naÅ¿catur.</s> - </p> - <p> - <s xml:id="s3547" xml:space="preserve">Quod deinde ampullæ iungantur in aqua, non fit ratione Å¿impathiæ, de qua lo-<lb/>quitur FracaÅ¿torus, nam per accidens iunguntur, quia cum alia ad aliam accedit, quÄ™ <lb/>libet earum tentat aÅ¿cendere ab ea parte, à qua inuicem hærent, quemadmodum <lb/>efficiunt iuxta labrum vaÅ¿is, ea enim Å¿uperficies a quæ vicina circunferentiæ vaÅ¿is ali <lb/>quantulum aÅ¿cendit in vaÅ¿e, qui non eÅ¿t omnino plenus.</s> - </p> - <p> - <s xml:id="s3548" xml:space="preserve">Ad id deinde quod de claritate noctium Å¿cribis, miror cur non videas, quod <reg norm="quam" type="context">quã</reg> <lb/>to magis obÅ¿cura nox apparet, non dico ratione nubium, Å¿ed diÅ¿tantiæ Solis Å¿ub <lb/>orizonte ab eodem orizonte, tanto magis claram, & luminoÅ¿am Å¿eÅ¿e nobis oÅ¿ten-<lb/>dit Luna in quintadecima, quia cum Sol eÅ¿t in Sagittario, & Capricorno, Luna eÅ¿t <lb/>in Geminis, & in Cancro, vnde in media nocte, eius radius per valde exiguam <reg norm="quam" type="context">quã</reg> <lb/>titatem vaporum tranÅ¿it, quia tunc ipÅ¿a eÅ¿t valde propinqua axi orizontis, & præ-<lb/>terea in huiuÅ¿modi tempore anni & noctis, aer eÅ¿t magis purgatus, quà m in qualibet <lb/>alia temporis parte, quia hieme Sol non poteÅ¿t excitare multos vapores, & ij, qui <lb/>at tolluntur, nocte à frigore Å¿tatim congellati ratione grauitatis <reg norm="decidunt" type="context">decidÅ©t</reg>, </s> - <s xml:id="s3549" xml:space="preserve">unde rema-<lb/>net aer multo clarior, qua ratione apparent Å¿tellæ minutæ, & CÅ“lum ijÅ¿dem ma-<lb/>gis ornatum, quà m in quolibet alio anni tempore.</s> - </p> - <p> - <s xml:id="s3550" xml:space="preserve">Dicere deinde, quemadmodum hic mundus eÅ¿t ætatis Å¿eptem, aut octomillium <lb/>annorum, ita nunc potuiſſet eſſe (Å¿i Deus voluiſſet) ætatis quinquagintamillium; </s> - <s xml:id="s3551" xml:space="preserve">er <lb/>go erat tempus; </s> - <s xml:id="s3552" xml:space="preserve">ita Å¿e habet, ac Å¿i diceremus, quemadmodum hic mundus eÅ¿t tan-<lb/>tæ magnitudinis, ita etiam quinquagies maior eſſe potuiſſet, ergo eÅ¿t Å¿patium, aut <lb/>interuallum corporeum, quod eum capere potuiſſet.</s> - </p> - <p> - <s xml:id="s3553" xml:space="preserve">Illud, nihil, AriÅ¿totelis extra CÄ™lum, nullo modo nobis inÅ¿eruit pro eiuÅ¿dem CÅ“ <lb/>li Å¿phÄ™rica rotunditate, cum cuiuÅ¿que alterius ex infinitis figuris CÄ™lum ipÅ¿um eſſe <lb/>poſſit, Å¿ecundum Å¿uam Å¿uperficiem conuexam. </s> - <s xml:id="s3554" xml:space="preserve">Nam CÅ“lum ea ratione Å¿phÄ™ricum <lb/>non eÅ¿t, quod magis Å¿it capax, quia ei innumerahiles alias figuras adeo magnas po <lb/>terat concedere cauÅ¿a diuina: </s> - <s xml:id="s3555" xml:space="preserve">Å¿ed Å¿phæricum eÅ¿t effectum, ne partem aliquam habe <lb/>ret Å¿ui termini Å¿uperfluam, quia nullum corpus à breuiori termino quam à ſphærico <lb/>terminari poteÅ¿t.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head420" style="it" xml:space="preserve">Derèuolutione rota putealis & <reg norm="alijs" type="lig">alijs</reg> <lb/>problematibus.</head> - <head xml:id="head421" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3556" xml:space="preserve">FVnis cui appenÅ¿a eÅ¿t Å¿itula, longè facilius axi inuoluitur, Å¿i ipÅ¿i axi affixa Å¿it rota. <lb/></s> - <s xml:id="s3557" xml:space="preserve">atque item commodius eò fiet, quo amplior rota erit, & axis exilior. <lb/></s> - <s xml:id="s3558" xml:space="preserve">Commodiſſimè autem, Å¿i ipÅ¿a rotæ extrema circunferentia, ex materia minori, <lb/>& denÅ¿iori, ac proinde grauiori conÅ¿tabit. </s> - <s xml:id="s3559" xml:space="preserve">Cuius rei ratio multiplex eÅ¿t. </s> - <s xml:id="s3560" xml:space="preserve">Nem-<lb/>pe quia omne corpus graue, aut Å¿ui natura, aut vi motum, in Å¿e recipit impreſſio- - <pb o="287" rhead="EPISTOL AE." n="299" file="0299" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0299"/> - nem & impetum motus, ita vt Å¿eparatum à virtute mouente per aliquod temporis <lb/>Å¿patium ex Å¿eipÅ¿o moueatur. </s> - <s xml:id="s3561" xml:space="preserve">nam Å¿i Å¿ecundum naturam motu cieatur, Å¿uam veloci-<lb/>tatem Å¿emper augebit, cum in eo, impetus & impreſſio Å¿emper augeantur, quia <lb/>coniunctam habet per petuò virtutem mouentem. </s> - <s xml:id="s3562" xml:space="preserve">Vnde manu mouendo rotam, ab <lb/><reg norm="eaque" type="simple">eaqÌ;</reg> eam remouendo rota Å¿tatim non quieÅ¿cet, Å¿ed per aliquod temporis Å¿patium <lb/>circunuertetur.</s> - </p> - <p> - <s xml:id="s3563" xml:space="preserve">Secunda cauÅ¿a eÅ¿t, quia quoduis gr aue corpus, aut per naturam, aut per vim mo-<lb/>tum, rectitudinem itineris naturaliter appetat, quod clarè cognoÅ¿cere poſſumus, <lb/>proijciendo lapides funda, & circunducentes brachium, nam funes tanto maius <lb/>pondus acquirunt, & manum tanto magis onerant, quanto velocius voluitur funda, <lb/>& incitatur motus, quod ab appetitu naturali inÅ¿ito ei corpori per <reg norm="lineam" type="context">lineã</reg> rectam pro-<lb/>grediendi procedit. </s> - <s xml:id="s3564" xml:space="preserve">Vnde fit, vt pondus circunferentiæ ipÅ¿ius rotæ, tanto facilius cir-<lb/>cunuoluatur, & ex Å¿eipÅ¿o tanto longiori tempore moueatur, quanto longius diÅ¿tat à <lb/>centro, cum eius iter tanto minus Å¿it curuum. </s> - <s xml:id="s3565" xml:space="preserve">Hanc igitur ob cauÅ¿am, rota, quanto <lb/>maior erit, <reg norm="eiuÅ¿que" type="simple">eiuÅ¿qÌ;</reg> pondus tanto magis vicinum circunferentiæ, tanto magis durabit <lb/>impetus motus aſſumptus.</s> - </p> - <p> - <s xml:id="s3566" xml:space="preserve">Tertia cauÅ¿a eÅ¿t, quod funis dum circunuoluitur, vicinius axi mathematico reuo-<lb/>lutionis, quam corpus graue circunferentiæ rotæ, ratione vectis, cum rota eÅ¿t in mo <lb/>tu, eius impetus non obtinet reÅ¿iÅ¿tentiam æqualem à contrario pondere aquæ in Å¿itu <lb/>la poÅ¿itæ.</s> - </p> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head422" style="it" xml:space="preserve">De machina, qua aquam impellit & Å¿ubleuat.</head> - <head xml:id="head423" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3567" xml:space="preserve">VNde Å¿it vt in fonte mandauerim, <lb/> - <anchor type="figure" xlink:label="fig-0299-01a" xlink:href="fig-0299-01"/> - vas Å¿eu mortarium in quod in-<lb/>greditur inÅ¿trumentum, quod aquam <lb/>impellit, diametrum Å¿uæ concauitatis, <lb/>habere non oportere maiorem dia-<lb/>metro fiÅ¿tulæ, per quam debet aÅ¿cende <lb/>re aqua, ratio eÅ¿t, quia Å¿i maius eſſet, <lb/>neceſſarium eſſet aliquod inÅ¿trumen-<lb/>tum quo aqua impelleretur multo gra <lb/>uius toto corpore aqueo, quod aptum <lb/>eſſet implere aliquam fiÅ¿tulam adeo <lb/>altam, vt eÅ¿t fons, quæ tamen eſſet <lb/>adeo lata vt eÅ¿t mortarium.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0299-01" xlink:href="fig-0299-01a"> - <image file="0299-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0299-01"/> - </figure> - </div> - <p> - <s xml:id="s3568" xml:space="preserve">Sit exempli gratia, tota fiÅ¿tula, Å¿eu <lb/>hirundo, per quam aÅ¿cendit aqua <var>.f.</var> <lb/>mortarium verò Å¿it <var>.a.u.</var> quod tam <reg norm="altum" type="context">altÅ©</reg> <lb/>Å¿it vt <var>.f.</var> Å¿ed <var>.f.</var> anguÅ¿tior ipÅ¿o <var>.a.u</var>. </s> - <s xml:id="s3569" xml:space="preserve">Nunc <lb/> - <anchor type="figure" xlink:label="fig-0299-02a" xlink:href="fig-0299-02"/> - cum repleta fuerint hæc duo vaÅ¿a, ma-<lb/>nifeÅ¿tum erit, quod aqua ipÅ¿ius <var>.f.</var> Å¿uffi-<lb/>ciens erit ad <reg norm="reÅ¿iÅ¿tendum" type="context">reÅ¿iÅ¿tẽdum</reg> toti aquæ <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <lb/><var>a.u.</var> & aqua <var>.a.u.</var> reÅ¿iÅ¿tet aquæ <var>.f.</var> quam-<lb/>uis aqua <var>.a.u.</var> maioris quantitatis Å¿it, & <lb/>ponderis ipÅ¿a <var>.f.</var> hoc autem euenit ex <lb/>eo quod aqua <var>.a.u.</var> <reg norm="non" type="context">nõ</reg> impellit aquam - <pb o="288" rhead="IO. BAPT. BENED." n="300" file="0300" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0300"/> - f. toto Å¿uo pondere, </s> - <s xml:id="s3570" xml:space="preserve">propterea quod pondus diuiditur proportionaliter Å¿upra ba-<lb/>Å¿im vaÅ¿is.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0299-02" xlink:href="fig-0299-02a"> - <image file="0299-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0299-02"/> - </figure> - </div> - <p> - <s xml:id="s3571" xml:space="preserve">Sit exempli <reg norm="gratia" type="wordlist">gr̃a</reg> vas aliquod <var>.b.d.n.m.</var> conicæ figuræ, Å¿eu <reg norm="truncus" type="context">trÅ©cus</reg> coni concaui aqua <lb/>plenus, cuius orificij diameter Å¿it <var>.b.d.</var> & multiplex diametro <var>.m.n.</var> infimæ baÅ¿is. </s> - <s xml:id="s3572" xml:space="preserve">co-<lb/>gitemus etiam <var>.b.d.</var> diuiÅ¿um in tot partes, quarum <reg norm="vnaquæque" type="simple">vnaquæq;</reg> æqualis Å¿it <var>.m.n.</var> imagi-<lb/><reg norm="nemurque" type="simple">nemurqÌ;</reg> tot lineas perpendiculares deÅ¿cendere verÅ¿us mundi centrum ad puncta <var>r.<lb/>c.m.</var> et <var>.t.x.m.</var> vt in Å¿ubÅ¿cripta hic figura videre eÅ¿t, per quas cogitemus tot Å¿uperfi-<lb/>cies curuas <reg norm="conicasque" type="simple">conicasqÌ;</reg>, inter quas, mente concipienda eÅ¿t aqua, quÄ™ pondere Å¿uo quie <lb/>Å¿cet Å¿upra maiorem Å¿uperficiem illa, quæ æque diÅ¿tans eſſet mundi centro, Å¿eu quam <lb/>Å¿upra baÅ¿im <var>.m.n.</var> vt exempli gratia conÅ¿ideretur aqua inter <var>.g.m.</var> et <var>.s.r.</var> cuius pondus <lb/>diÅ¿tribuitur fecundum latitudinem <var>.m.r.</var> quæ maior eÅ¿t <var>.g.s.</var> cogitemus igitur <var>.m.c.</var> æ-<lb/>qualem eſſe <var>.g.s.</var> manifeÅ¿tum erit, quod <var>.m.c.</var> non Å¿uÅ¿tinebit totum pondus a quæ, quæ <lb/>inter <var>.g.m.</var> et <var>.s.r.</var> reperitur, eo quod omnis pars aquæ ad perpendiculum inclinat ver-<lb/>Å¿us mundi centrum, quapropter fundus Å¿eu baÅ¿is <var>.m.n.</var> non Å¿uÅ¿tinet aliud pondus <reg norm="quam" type="context">quã</reg> <lb/>aquæ <var>.f.m.</var> Å¿ed Å¿i quis hoc in dubium reuocaret dicens, quod aqua circunÅ¿cribens Å¿i-<lb/>tum corporis aquei <var>.f.m.</var> impellit lateraliter dictum corpus aqueum, reÅ¿pondendum <lb/>eÅ¿t, quod ex æquo huius corporis <var>.f.m.</var> aqua impellit etiam aquam circunÅ¿tantem, <lb/>eo, quod Å¿unt corpora homogenea, cum in corporibus homogeneis æquales partes <lb/>habeant æquales vires.</s> - </p> - <p> - <s xml:id="s3573" xml:space="preserve">Sed redeundo ad vaÅ¿a <var>.a.u.</var> et <var>.f.</var> dico quod Å¿icut aqua <var>.f.</var> Å¿ufficit ad <reg norm="reÅ¿iÅ¿tendum" type="context">reÅ¿iÅ¿tendÅ©</reg> aquæ <lb/><var>a.u.</var> ita quodlibet aliud pondus Ä™quale <var>.f.</var> cuiuÅ¿uis materiæ, in fiÅ¿tula <var>.f.</var> poÅ¿itum, Å¿uffi-<lb/>ciens erit, dummodo illud corpus ita Å¿it adæquatum concauitati fiÅ¿tulæ <var>.f.</var> quod non <lb/>permittat tranÅ¿itum aliquem aquæ vel <lb/> - <anchor type="figure" xlink:label="fig-0300-01a" xlink:href="fig-0300-01"/> - aeris inter conuexum ipÅ¿ius corporis, <lb/>& deuexum fiÅ¿tulæ <var>.f.</var> & hoc ex Å¿e Å¿atis <lb/>patet, Å¿ed in vaÅ¿e <var>.a.u.</var> cum ex hypothe <lb/>Å¿i latius Å¿it ipÅ¿o <var>.f.</var> nullum aliud corpus <lb/>Å¿ufficiens erit ad reÅ¿iÅ¿tendum aquæ ip-<lb/>Å¿ius <var>.f.</var> quin tam graue Å¿it, quam tota <lb/>aqua <var>.a.u.</var> exiÅ¿tente <var>.a.u.</var> tam alto quam <lb/>f. </s> - <s xml:id="s3574" xml:space="preserve">Vnde Å¿i aqua ipÅ¿ius <var>.f.</var> nil plus eſſet <lb/>quam vna tantummodo libra, & vas <var>.a.<lb/>u.</var> exiÅ¿teret latius ipÅ¿o <var>.f.</var> in decupla pro <lb/>portione, </s> - <s xml:id="s3575" xml:space="preserve">tunc in ipÅ¿o <var>.a.u.</var> oporteret <lb/>corpus adæquatum ipÅ¿i concauitati po <lb/>nere, cuius pondus eſſet decem libra-<lb/>rum, vt Å¿ufficeret ad Å¿uÅ¿tinendum <reg norm="aquam" type="context">aquã</reg> <lb/>ipÅ¿ius <var>.f.</var> & ad im<unsure/><reg norm="pellendum" type="context">pellendÅ©</reg> ipÅ¿am <reg norm="aquam" type="context">aquã</reg> <var>.<lb/>f.</var> deberet eſſe plus quam decem libra-<lb/>rum. </s> - <s xml:id="s3576" xml:space="preserve">Ponamus nunc illud corpus, ita <lb/> - <anchor type="figure" xlink:label="fig-0300-02a" xlink:href="fig-0300-02"/> - denÅ¿ius eſſe aqua, vt maius <reg norm="interuallum" type="context">interuallÅ©</reg> <lb/>non occupet, quam <var>.o.e.</var> corpus igitur <lb/><var>o.e.</var> Å¿ufficiens erit ad impellendum <lb/>aquam <var>.f.</var> & non eo minus.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0300-01" xlink:href="fig-0300-01a"> - <image file="0300-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0300-01"/> - </figure> - <figure xlink:label="fig-0300-02" xlink:href="fig-0300-02a"> - <image file="0300-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0300-02"/> - </figure> - </div> - <pb o="289" rhead="EPISTOL AE." n="301" file="0301" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0301"/> - </div> - </div> - <div type="section" level="3" n="16"> - <div type="letter" level="4" n="1"> - <head xml:id="head424" xml:space="preserve">NOVA SOLVTIO PROBLEMATIS DE VASE <lb/>pleno liquoris.</head> - <head xml:id="head425" style="it" xml:space="preserve">Nicolao Caluxio Serenißimi Ducis Sabaudia à ſecretis.</head> - <p> - <s xml:id="s3577" xml:space="preserve"><emph style="sc">QVod</emph> à me poÅ¿tulas eÅ¿t problema ab alijs iam Å¿criptum, Å¿ed illud tibialio <lb/>medio Å¿oluam.</s> - </p> - <p> - <s xml:id="s3578" xml:space="preserve">Proponitur vas <reg norm="plenum" type="context">plenÅ©</reg> liquore aliquo, puta aqua, <reg norm="quod" type="simple">ê™</reg> tres habeat fiÅ¿tulas <lb/>ad baÅ¿im, quarum vnaquæque poſſit euacuare ipÅ¿um vas, inæquales ta-<lb/>men, ita quod prima tam lata Å¿it, vt Å¿patio vnius horæ poſſit ipÅ¿um euacuare to-<lb/>tum; </s> - <s xml:id="s3579" xml:space="preserve">Å¿ecunda vero Å¿patio duarum horarum, tertia autem Å¿patio trium hora-<unsure/>-<lb/>rum. </s> - <s xml:id="s3580" xml:space="preserve">Tunc quæritur quanto tempore omnes tres fiÅ¿tulæ Å¿imul apertæ euacua-<lb/>bunt ipÅ¿um vas. </s> - <s xml:id="s3581" xml:space="preserve">Ad hoc volo vt quæratur primo quanta pars aquæ vnaquÄ™quÄ™ fi-<lb/>Å¿tula euacuabit in aliquo dato tempore, quod facilè eÅ¿t, vt puta, prima fiÅ¿tu-<lb/>la, Å¿patio dimidiæ horæ euacuabit dimidium vas, eo quod Å¿patio integræ horæ po-<lb/>teÅ¿t totum euacuare, Å¿ecunda fiÅ¿tula, eodem temporis Å¿patio, euacuabit quartam <lb/>partem ipÅ¿ius vaÅ¿is, tertia verò fiÅ¿tula, eodemmet Å¿patio temporis dimidiæ horæ, <lb/>euacuabit Å¿extam partem ipÅ¿ius vaÅ¿is, quæ omnia fracta Å¿imul collecta faciunt vnde-<lb/>cim duodecimas partes totius vaÅ¿is, vnde manifeÅ¿tum erit, quod omnes fiÅ¿tulæ pari-<lb/>ter apertæ, Å¿patio dimidiÄ™ horæ euacuabunt vndecim duodecimas partes totius a-<lb/>quæ, Å¿ed nos cupimus Å¿cire, quanto tempore, totum vas euacuabitur, apertis omni <lb/>bus fiÅ¿tulis, quapropter dicemus ita; </s> - <s xml:id="s3582" xml:space="preserve">Si vndecim duodecimæ partes conÅ¿umunt mi-<lb/>nuta .30. temporis, quantum conÅ¿ument omnes partes aquæ? </s> - <s xml:id="s3583" xml:space="preserve">quæ Å¿unt .12. quare ex <lb/>regula de tribus prouenient nobis minuta .32. cum .8. vndecimis vnius minuti, hoc <lb/>eÅ¿t cum .43. Å¿ecundis horæ ferè, vel Å¿iaccipiemus tres quartas vnius horæ, </s> - <s xml:id="s3584" xml:space="preserve">tunc pri-<lb/>ma fiÅ¿tula emittet tres quartas partes totius aquæ, Å¿ecunda, tres octauas <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> aquÄ™, <lb/>tertia verò, quarta pars, tunc omnia, hæc collecta, faciunt vnum integrum cum tri <lb/>bus octauis. </s> - <s xml:id="s3585" xml:space="preserve">Si dixerimus igitur quando vnum integrum cum tribus octauis abÅ¿u-<lb/>mit .45. minuta temporis, ergo illud Å¿olum integrum abÅ¿umet idem vt Å¿upra hoc eÅ¿t <lb/>min .32. cum .8. vndecimis vnius minuti vel .43. Å¿ecundis. </s> - <s xml:id="s3586" xml:space="preserve">Cuius rei Å¿peculatio <reg norm="tam" type="context">tã</reg> con <lb/>iuncta eÅ¿t operationi, quòd vna cognita, reliqua Å¿tatim cognoÅ¿citur.</s> - </p> - <p> - <s xml:id="s3587" xml:space="preserve">Idem eueniet de implendo vaÅ¿e tribus Å¿imilibus fiÅ¿tulis mediantibus.</s> - </p> - <p> - <s xml:id="s3588" xml:space="preserve">Secundum quæſitum ab alijs traditum, tuum etiam, aliter quoque poteÅ¿t Å¿olui, <lb/>propterea non prÄ™termittam tibi Å¿atisfacere.</s> - </p> - <p> - <s xml:id="s3589" xml:space="preserve">Problema itaque tale eÅ¿t, vt Å¿it vas aliquod in <reg norm="quod" type="wordlist">qÄ‘</reg> infunditur aqua per tres fiÅ¿tu-<lb/>las, Å¿ed dum infunditur aqua, eadem egreditur per duas alias fiÅ¿tulas in fundo <lb/>vaÅ¿is poÅ¿itas, Å¿ed tres Å¿uperiores Å¿int inuicem proportionatæ, vt Å¿upradictum <lb/>eÅ¿t, primaqÌue inferiorum talis Å¿it, vt Å¿patio .4. horarum poſſit totum vas euacua-<lb/>re, Å¿ecunda autem poſſit Å¿patio .6. horarum idem facere, vnde ex Å¿upradictis, vas im <lb/>plebitur à tribus fiÅ¿tulis Å¿uperioribus, clauÅ¿is exiÅ¿tentibus inferioribus, Å¿patio tempo <lb/>ris <reg norm="minutorum" type="context">minutorÅ©</reg> .32. <reg norm="cum" type="context">cÅ©</reg> .8. vndecimis hoc eÅ¿t min .32. cum .43. Å¿ecundis, deinde per duas <lb/>fiÅ¿tulas inferiores poſſet euacuari Å¿patio <reg norm="temporis" type="context">tẽporis</reg> horarum .2. et. mi .24. exÅ¿upradictis.</s> - </p> - <p> - <s xml:id="s3590" xml:space="preserve">Supponamus igitur omnes fiÅ¿tulas operari Å¿patio temporis minutorum .32. cum <lb/>Å¿ecundis .43. </s> - <s xml:id="s3591" xml:space="preserve">tunc manifeÅ¿tum eÅ¿t quod vas non implebitur, eo Å¿patio min .32. cum <lb/>Å¿ecundis .43. Å¿ed tanta aqua deficiet, quanta ab inferioribus fiÅ¿tulis eo Å¿patio tempo <lb/>ris min .32. Å¿ecun .43. poteÅ¿t euacuari, </s> - <s xml:id="s3592" xml:space="preserve">quare proportio partis vaÅ¿is vacuæ, ad totum <lb/>vas, erit vt min .33. ferè ad horas .2. min .24. quod per Å¿e patet, </s> - <s xml:id="s3593" xml:space="preserve">tunc Å¿i demptum fue- - <pb o="290" rhead="IO. BAPT. BENED." n="302" file="0302" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0302"/> - rit tempus .33. minutorum ex h oris .2. min .24. reliquum erit hora .1. min .51. vnde <lb/>proportio aquæ, quæ in vaÅ¿e reperitur, ad eam, quæ totum vas implet, erit vt .111. <lb/>ad .144. </s> - <s xml:id="s3594" xml:space="preserve">Quare nunc poſſumus rectè dicere ex regula de tribus Å¿i .111. indigent mi-<lb/>nuta .33. temporis, ergo .144. indigent min .43. horæ, in quo tempore implebitur to-<lb/>tum vas omnibus fiÅ¿tulis operantibus.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head426" style="it" xml:space="preserve">Aliæ circuli noua paßiones.</head> - <head xml:id="head427" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3595" xml:space="preserve">VTad aÅ¿cendendum ignis, & ad <reg norm="deÅ¿cendendum" type="context">deÅ¿cendendÅ©</reg> quicquid graue natum eÅ¿t, ita ad <lb/>Å¿peculandum humanus intellectus. </s> - <s xml:id="s3596" xml:space="preserve">nec quieÅ¿cit, dum poteÅ¿t, eÅ¿t enim ver-<lb/>Å¿atile, <reg norm="agitandoque" type="simple">agitandoqÌ;</reg> Å¿eÅ¿e cauÅ¿is rerum immiÅ¿cere, & abditum aliquid rimari, <lb/>conatur, & eÅ¿t in nobis, quaÅ¿i Diogenes quidam in Dolio.</s> - </p> - <p> - <s xml:id="s3597" xml:space="preserve">Tibi igitur mitto quod vltimò inueni, alias Å¿cilicet nouas circuli paſſiones, <lb/>quæ ita Å¿e <reg norm="habent" type="context">habẽt</reg>. </s> - <s xml:id="s3598" xml:space="preserve">Sit circulus <var>.a.b.c.</var> in quo Å¿it <var>.a.d.</var> latus quadrati inÅ¿criptibilis in ipÅ¿o <lb/>circulo, ct <var>.b.c.</var> Å¿it diameter ad rectos cum <var>.a.d.</var> in puncto <var>.e.</var> quod medium erit inter <lb/>a. et <var>.d.</var> ex .3. tertij Eucli. Å¿it Å¿imiliter <var>.a.f.</var> contingens ipÅ¿um circulum in puncto <var>.a.</var> quæ <lb/>protracta Å¿it vÅ¿que ad punctum <var>.f.</var> interÅ¿ectionis cum diametro protracto, quod ita <lb/>eueniet cum anguli <var>.a.e.f.</var> et <var>.f.a.e.</var> minores Å¿int duobus rectis, eo quod angulus <var>.f.a.e.</var> <lb/>acutus Å¿it, cum <var>.a.d.</var> tranÅ¿eat inter centrum et <var>.f</var>.</s> - </p> - <p> - <s xml:id="s3599" xml:space="preserve">Dico nunc quod productum diametri <var>.b.c.</var> in parte <var>.c.e.</var> ipÅ¿ius, æqualis erit produ-<lb/>cto ipÅ¿ius <var>.c.f.</var> in <var>.a.d</var>. </s> - <s xml:id="s3600" xml:space="preserve">Protrahatur imaginatione <var>.b.a.</var> et <var>.a.c.</var> </s> - <s xml:id="s3601" xml:space="preserve">vnde ex .26. tertij Euclid. <lb/>habebimus angulum <var>.d.a.c.</var> æqualem angulo <var>.a.b.c</var>. </s> - <s xml:id="s3602" xml:space="preserve">Å¿ed ex .31. eiuÅ¿dem angulus <var>.f.a.<lb/>c.</var> æqualis eÅ¿t angulo <var>.b</var>. </s> - <s xml:id="s3603" xml:space="preserve">quare æqualis erit angulo <var>.d.a.c.</var> & ita habebimus per .3. Å¿exti <lb/>eandem proportionem <var>.f.c.</var> ad <var>.c.e.</var> quæ <var>.f.a.</var> ad <var>.a.e.</var> Å¿ed <var>.a.f.</var> eÅ¿t æqualis Å¿emidiametro <lb/>circuli propoÅ¿iti, </s> - <s xml:id="s3604" xml:space="preserve">propterea quod Å¿i producta fuerit à puncto <var>.a.</var> ad centrum <var>.o.</var> Å¿emi <lb/>diameter <var>.a.o.</var> hæc cum <var>.o.e.</var> faciet dimidium angulirecti, cum ex Å¿uppoÅ¿ito <var>.a.d.</var> la-<lb/>tus Å¿it quadrati inÅ¿criptibilis in ipÅ¿o circulo. </s> - <s xml:id="s3605" xml:space="preserve">& cum <var>.a.f.</var> rectum ex .17. tertij, vnde an <lb/>gulus <var>.f.</var> erit Å¿imiliter medietas recti ex .32. primi, </s> - <s xml:id="s3606" xml:space="preserve">quare ex .6. eiuÅ¿dem <var>.a.f.</var> æqualis <lb/>erit <var>.a.o</var>. </s> - <s xml:id="s3607" xml:space="preserve">Ergo cum proportio <var>.f.c.</var> ad <var>.c.e.</var> Å¿it. vt <var>.f.a.</var> ad <var>.a.e.</var> erit Å¿imiliter vt <var>.b.c.</var> ad <var>.a.d.</var> <lb/>hoc eÅ¿t ut dupli ad duplum, vnde ex .15. Å¿exti <lb/>manifeÅ¿tum erit propoÅ¿itum, ex quo alia paÅ¿-<lb/> - <anchor type="figure" xlink:label="fig-0302-01a" xlink:href="fig-0302-01"/> - Å¿io oritur, hoc eÅ¿t, quod productum <var>.f.c.</var> in <var>.a.<lb/>d.</var> æ quale Å¿it qua drato ipÅ¿ius <var>.a.c.</var> ratio eÅ¿t, quia <lb/>quadratum <var>.a.c.</var> æ quale eÅ¿t producto <var>.b.c.</var> in <var>.c.<lb/>e.</var> eo quod <var>.a.c.</var> media proportionalis eÅ¿t inter <var>.<lb/>b.c.</var> et <var>.c.e.</var> ex Å¿imilitudine triangulorum <var>.a.b.c.</var> <lb/>et <var>.e.a.c.</var> nam anguli <var>.b.a.c.</var> et <var>.a.e.c.</var> recti Å¿unt <lb/>et <var>.c.</var> <reg norm="communis" type="context">cõmunis</reg>, vnde <var>.b.</var> erit æqualis <var>.e.a.c.</var> ex .32 <lb/>primi, Å¿equitur etiam, quod <var>.a.c.</var> Å¿it media pro <lb/>portionalis inter <var>.a.d.</var> et <var>.f.c.</var> & hæc etiam erit <lb/>alia circuli paſſio, & quia <var>.a.c.</var> eÅ¿t latus octago-<lb/>ni igitur tale latus <reg norm="medium" type="context">mediÅ©</reg> proportionale erit <lb/>inter latus quadrati. et <var>.f.c.</var> <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> circuli, quÄ™ <lb/>quidem <var>.f.c.</var> eÅ¿t una portio diametri quadrati circunÅ¿criptibilis ipÅ¿um circulum inter <lb/>circulum & angulum ipÅ¿ius quadrati.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0302-01" xlink:href="fig-0302-01a"> - <image file="0302-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0302-01"/> - </figure> - </div> - <pb o="291" rhead="EPISTOLAE." n="303" file="0303" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0303"/> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head428" style="it" xml:space="preserve">Quod incendium, ex reflexione radiorum Å¿olarium, non fiat in cen <lb/>tro Å¿peculi Å¿pharici, & aliquid contra Cardanum, & <lb/>de moturadiorum Å¿olarium.</head> - <head xml:id="head429" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3608" xml:space="preserve">ITerum tibi dico, quod radij illi Å¿olares, quià diuerÅ¿is punctis ipÅ¿ius Å¿olaris corpo-<lb/>ris veniunt, tranÅ¿euntes per centrum Å¿peculi Å¿phærici concaui, quamuis à ſuper-<lb/>ficie Å¿peculi ad centrum ipÅ¿um reflectantur, vt alÃâs tibi dixi, nihilominus nullo mo <lb/>do poſſunt aliquod obiectum incendere duabus ex cauÅ¿is, quarum vna eÅ¿t, quia cum <lb/>Sol valde remotus Å¿it à nobis, valde etiam acutus generatur angulus coni radiorum <lb/>in centro Å¿peculi, vnde à parua Å¿uperficie ipÅ¿ius Å¿peculi reflectuntur, </s> - <s xml:id="s3609" xml:space="preserve">quare pauciſſi-<lb/>mi radij Å¿unt qui reflectantur in ipÅ¿o centro, & propterea non Å¿ufficiunt ad combu <lb/>Å¿tionem alicuius obiecti. </s> - <s xml:id="s3610" xml:space="preserve">Alia verò cauÅ¿a eÅ¿t, quod quamuis multi, & Å¿ufficientes <lb/>radij fuiſſentad <reg norm="comburendum" type="context context">cõburendÅ©</reg> velociter quoduis obiectum. </s> - <s xml:id="s3611" xml:space="preserve">impoſſibile tamen omnino <lb/>eÅ¿set, vt aliquod obiectum comburerent, propterea quod cum radij incidentes de-<lb/>beant per centrum tranÅ¿ire, obiectum combuÅ¿tibile, vt opacum, obÅ¿taret ipÅ¿is radijs, <lb/>ne vlterius tranÅ¿irent, vnde nulla fieret reflexio, Å¿ed etiam Å¿i dicti radij in centro re <lb/>flexi, Å¿ufficerent ad combuÅ¿tionem, incidentes hoc magis eſſicerent. </s> - <s xml:id="s3612" xml:space="preserve">& ita abÅ¿que <lb/>vllo Å¿peculo, omnia & in quolibet loco comburerentur, quod manifeÅ¿tè falÅ¿um eÅ¿t. <lb/></s> - <s xml:id="s3613" xml:space="preserve">DeÅ¿ine igitur mihi citate Lucillum Philalteum, qui in philoÅ¿ophia mathematica <lb/>fuit omnium imperitiſſimus. </s> - <s xml:id="s3614" xml:space="preserve">Verum Å¿peculum vÅ¿torium illud eÅ¿t quod ab Alhazem <lb/></s> - <s xml:id="s3615" xml:space="preserve">Deinde à Vitellìone deÅ¿cribitur.</s> - </p> - <p> - <s xml:id="s3616" xml:space="preserve">Quod deinde verum Å¿it, <reg norm="vmbram" type="context">vmbrã</reg> vniuÅ¿cuiuÅ¿que corporis opaci à Sole productam <lb/>Å¿emper eſſe centum <reg norm="nouemque" type="simple">nouemqÌ;</reg> vicibus maiorem diametro eiuÅ¿dem corporis, nego.</s> - </p> - <p> - <s xml:id="s3617" xml:space="preserve">Imaginemur <var>.s.</var> 1. diametrum eſſe illius circuli, quo vltimi radij Å¿olares veniunt tan <lb/>gentes corpus cuius diameter Å¿it <var>.c.e.</var> et <var>.a.i.</var> Å¿it diameter alterius circuli eiuÅ¿dem cor-<lb/>poris Å¿olaris à quo vltimi radij veniunt tangentes corpus, cuius diameter Å¿it <var>.f.g.</var> in <lb/>eadem diÅ¿tantia, & eodem Å¿itu prioris corporis. </s> - <s xml:id="s3618" xml:space="preserve">Tunc conus vmbræ ipÅ¿ius <var>.f.g.</var> Å¿it <var>.f.<lb/>g.q.</var> & ipÅ¿ius <var>.c.e.</var> Å¿it <var>.c.n.e.</var> centrum autem Å¿olare Å¿it <var>.o.</var> conorum verò axes Å¿int <var>.t.n.q</var>. <lb/></s> - <s xml:id="s3619" xml:space="preserve">tunc ex Å¿uppoÅ¿ito <var>.q.f.a</var>: <var>n.c.s</var>: <var>n.e.l</var>: et <var>.q.g.i.</var> erunt omnes contigui corpori Å¿olari, vn-<lb/>de ex .17. tertij Eucli. anguli <var>.o.a.q.</var> et <var>.o.s.n.</var> erunt recti. </s> - <s xml:id="s3620" xml:space="preserve">protracta deinde cum fu erit <lb/><var>a.s.</var> habebimus angulos <var>.u.a.s.</var> et <var>.u.s.a.</var> minores duobus rectis. </s> - <s xml:id="s3621" xml:space="preserve">Quare <var>.n.s.</var> concurret <lb/> - <anchor type="figure" xlink:label="fig-0303-01a" xlink:href="fig-0303-01"/> - <pb o="292" rhead="IO. BAPT. BENED." n="304" file="0304" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0304"/> - cum <var>.a.q.</var> in puncto <var>.u</var>. </s> - <s xml:id="s3622" xml:space="preserve">Nunc verò Å¿i vmbra <var>.t.q.</var> tanto maior eÅ¿t <var>.f.g.</var> quanto .109. eÅ¿t <lb/>vno et <var>.t.n.</var> etiam <reg norm="tanto" type="context">tãto</reg> maior <var>.c.e.</var> ergò eadem proportio erit <var>.q.t.</var> ad <var>.t.f.</var> quÄ™ <var>.n.t.</var> ad <var>.t.<lb/>c.</var> Å¿ed cum angulus <var>.t.</var> communis Å¿it ambobus triangulis <var>.q.t.f.</var> et <var>.n.t.c.</var> Å¿equitur ex .6. <lb/>Å¿exti dictos triangulos æ quiangulos eſſe. </s> - <s xml:id="s3623" xml:space="preserve">Vnde Å¿i anguli <var>.t.n.c.</var> et <var>.t.q.f.</var> æ quales inui <lb/>cem Å¿unt, ergo <var>.q.f.</var> æquidiÅ¿tans erit <var>.n.c.</var> quod eÅ¿t impoſſibile, quia nunc demonÅ¿tra-<lb/>uimus ipÅ¿as concurrere in puncto <var>.u</var>. </s> - <s xml:id="s3624" xml:space="preserve">Quare non eÅ¿t eadem proportio <var>.q.t.</var> ad <var>.t.f.</var> quæ <lb/><var>n.t.</var> ad <var>.t.c.</var> decipitur ergo Cardanus in .4. lib. de Å¿ubtilitate.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0303-01" xlink:href="fig-0303-01a"> - <image file="0303-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0303-01"/> - </figure> - </div> - <p> - <s xml:id="s3625" xml:space="preserve">Circa illud deinde quod à me quæris, hoc eÅ¿t, quæ Å¿it cauÅ¿a, quod nos videmus <lb/>radium Å¿olarem tardiſſimè moueri, cum alias tibi dixerim ipÅ¿um qualibet hora cir-<lb/>ca terram quindecim gradus perficere, reÅ¿pondeo, quod radius ille quem videmus, <lb/>exempli gratia, in aliquo cubiculo, nunquam eÅ¿t idem numero, Å¿ed quia ipÅ¿i radij <lb/>nullo modo differunt inter Å¿e, niÅ¿i in numero, proptera putamus eundem Å¿emper eſſe, <lb/>cum Å¿emper alius, atque alius Å¿it, quorum vnuÅ¿quiÅ¿que (de illis loquor, qui ad hunc <lb/>terræ globum perueniunt) circa terram reuoluitur Å¿patio .24. horarum, & cum quili <lb/>bet circulus diuidatur in .360. gradus, quorum vigeÅ¿imaquarta pars eÅ¿t .15. verum <lb/>eÅ¿t igitur, quod tibi iam dixeram.</s> - </p> - <figure position="here"> - <image file="0304-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0304-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="17"> - <div type="letter" level="4" n="1"> - <head xml:id="head430" xml:space="preserve">OPERATIONES DIVERSAE AB ALIIS <lb/>Michaelis Stifelij.</head> - <head xml:id="head431" style="it" xml:space="preserve">Conrado Terl.</head> - <p> - <s xml:id="s3626" xml:space="preserve"><emph style="sc">QVod</emph> in .2. exemplo. II. cap. Stifelius Å¿cribit in .3. lib. pag .282. non nego <lb/>quin pulchrum Å¿it, Å¿ed alijs pulchrioribus modis poſſumus illud idem de-<lb/>monÅ¿trare; </s> - <s xml:id="s3627" xml:space="preserve">cogita igitur Å¿uperficiem rectangulam, cuius medietas Å¿it <reg norm="triam" type="context">triã</reg> <lb/>gulus rectangulus <var>.a.b.g.</var> vnde ex Å¿uppoÅ¿ito nobis cognita erit Å¿uperficies <lb/>ipÅ¿ius trianguli, tanquam dimidium totius parallelogrammi rectanguli cogniti. <lb/></s> - <s xml:id="s3628" xml:space="preserve">Quare ex .25. Å¿ecundi triangulorum <reg norm="Monteregij" type="context">Mõteregij</reg>, cognita nobis <reg norm="erunt" type="context">erÅ©t</reg> latera <var>.a.b.</var> et <var>.b.g</var>.</s> - </p> - <p> - <s xml:id="s3629" xml:space="preserve">Alia etiam breuiori methodo idem poſſumus eſſicere, mediante angulo <var>.b.</var> recto, <lb/>eo quod cum nobis cognita Å¿it Å¿uperficies trianguli Å¿imul <reg norm="cum" type="context">cÅ©</reg> baÅ¿i <var>.a.g.</var> cognita etiam <lb/>nobis fit perpendicularis <var>.b.d.</var> à puncto <var>.b.</var> ad baÅ¿im, & conÅ¿equenter cognitum no-<lb/>bis erit productum ipÅ¿ius <var>.a.d.</var> in <var>.d.g.</var> & quia nobis cognita eÅ¿t <var>.a.g.</var> & eius medietas, - <pb o="293" rhead="EPISTOL AE." n="305" file="0305" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0305"/> - ideo vnaquæque eius pars <var>.a.d.</var> et <var>.d.g.</var> Å¿imiliter nobis cognita erit ex quinta Å¿ecundi <lb/>Eucl. </s> - <s xml:id="s3630" xml:space="preserve">vnde ex penultima primi habebimus propoÅ¿itum.</s> - </p> - <p> - <s xml:id="s3631" xml:space="preserve">Poſſumus item circulum mente concipere cuius <var>.a.g.</var> Å¿it diameter, & ab eius cen-<lb/>tro <var>.e.</var> protracta cum fuerit <var>.e.b.</var> quæ nobis cognita erit, vt medietas ipÅ¿ius <var>.a.g.</var> de cu <lb/>ius potentia, dempta <reg norm="cum" type="context">cÅ©</reg> fuerit potentia <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <var>b.o.</var> remanebit nobis potentia ipÅ¿ius <var>.d.<lb/>e.</var> & ita eius longitudo, quæ addita medietati <var>.e.g.</var> & detracta à dimidio <var>.e.d.</var> erunt <lb/>nobis cognitæ <var>.a.d.</var> et <var>.d.g.</var> vnde <var>.b.g.</var> et <var>.b.d.</var> remanebunt nobis cognitæ ex dicta pe-<lb/>nultima primi Eucli. </s> - <s xml:id="s3632" xml:space="preserve">huiuÅ¿modi figuram videbis in dicto .25. problemate .2. li. Mon-<lb/>tisregij.</s> - </p> - <p> - <s xml:id="s3633" xml:space="preserve">Aliter etiam poſſumus hoc idem efficere.</s> - </p> - <p> - <s xml:id="s3634" xml:space="preserve">Sit rectangulus hic Å¿ubÅ¿criptus <var>.a.b.c.u.</var> Å¿uperficiei cognitÄ™ Å¿imul cum diametro <var>.a.<lb/>c.</var> extendatur imaginatione <var>.b.c.</var> vÅ¿que ad, f. ita quod <var>.c.f.</var> æqualis Å¿it <var>.c.u.</var> intelligan-<lb/><reg norm="turque" type="simple">turqÌ;</reg> quadrata <var>.g.f</var>: <var>g.u.</var> ct <var>.u.f.</var> vnde <reg norm="summa" type="context">sÅ©ma</reg> <reg norm="quadratorum" type="context">quadratorÅ©</reg> <var>.g.u</var>:u.f. cognita nobis erit ex <lb/>penultima primi. </s> - <s xml:id="s3635" xml:space="preserve">nam <var>.a.c.</var> data nobis fuit, quare <reg norm="Å¿ummam" type="context">Å¿ummã</reg> <var>.g.u</var>:u.b: et <var>.u.f.</var> cognoÅ¿ce-<lb/>mus, cui <reg norm="summæ" type="context">sÅ©mæ</reg> addito Å¿uplemento <var>.d.e.</var> æ quali <var>.u.<lb/>b.</var> dabit nobis <reg norm="cognitum" type="context">cognitÅ©</reg> quadrarum <var>.g.f.</var> totale, qua <lb/> - <anchor type="figure" xlink:label="fig-0305-01a" xlink:href="fig-0305-01"/> - re cognoÅ¿cetur eius radix <var>.b.f.</var> cognita igitur <var>.b.f.</var> <lb/>cum pro ducto <var>.b.u.</var> illico ex .5. Å¿ecundi cognoÅ¿ce-<lb/>tur <var>.b.c.</var> et <var>.c.f.</var> forte cognita <var>.b.f.</var> diuiÅ¿a <reg norm="per" type="simple">ê‘</reg> æqualia <lb/>in puncto <var>.t.</var> & per inæqualiz in <reg norm="puncto" type="context">pÅ©cto</reg> <var>.c</var>. </s> - <s xml:id="s3636" xml:space="preserve">Nam qua <lb/><reg norm="dratum" type="context">dratÅ©</reg> ipÅ¿ius <var>.t.f.</var> cognitum, Ä™quatur <reg norm="rectangulo" type="context">rectãgulo</reg> <var>.b.u.</var> <lb/><reg norm="cum" type="context">cÅ©</reg> quadrato ipÅ¿ius <var>.t.c.</var> <reg norm="dempto" type="context">dẽpto</reg> igitur rectangulo, <var>b.<lb/>u.</var> ex quadrato ipÅ¿ius <var>.t.f.</var> relinquetur quadratum <lb/><reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <var>.t.c.</var> cognitum & eius radix <var>.t.c.</var> qua addita ipÅ¿i <lb/>medietati <var>.b.t.</var> & <reg norm="dempta" type="context">dẽpta</reg> ex medietate <var>.f.t.</var> relinque-<lb/>tur propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0305-01" xlink:href="fig-0305-01a"> - <image file="0305-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0305-01"/> - </figure> - </div> - <p> - <s xml:id="s3637" xml:space="preserve">Similiter de tertio exemplo eiuÅ¿dem Stifelij <lb/>infero.</s> - </p> - <p> - <s xml:id="s3638" xml:space="preserve">Sit rectangulus <var>.a.b.c.u.</var> cuius diametri <var>.a.c.</var> quantitas, Å¿imul cum proportione late <lb/>rum <var>.b.c.</var> et <var>.b.a.</var> nobis data Å¿it. </s> - <s xml:id="s3639" xml:space="preserve">cum autem Å¿cire voluerimus eius Å¿uperficiem <var>.b.u.</var> cla-<lb/>rum eÅ¿t, quod cum nobis data Å¿it proportio <var>.b.c.</var> ad <var>.b.a.</var> illico cognoÅ¿cemus <reg norm="etiam" type="context">etiã</reg> pro-<lb/>portionem quadrati ipÅ¿ius <var>.b.c.</var> ad quadratum ip-<lb/> - <anchor type="figure" xlink:label="fig-0305-02a" xlink:href="fig-0305-02"/> - Å¿ius <var>.b.a.</var> cum dupla Å¿it ei quæ <var>.b.c.</var> ad <var>.b.a.</var> ita etiam <lb/>& aggregati dictorum quadratorum ad quadra-<lb/>tum ipÅ¿ius <var>.b.a.</var> hoc eÅ¿t nota erit nobis proportio <lb/>quadrati ipÅ¿ius <var>.a.c.</var> diagonalis ad quadratum ip-<lb/>Å¿ius <var>.a.b.</var> idem dico de quadrato <var>.b.c.</var> ideÅ¿t quod <lb/>proportio quadrati ipÅ¿ius <var>.a.c.</var> ad quadratum <var>.b.c.</var> <lb/>cognita nobis erit, Å¿ed <var>.a.c.</var> data nobis fuit, qua-<lb/>re cognoÅ¿cemus etiam omnia dicta quadrata eo-<lb/><reg norm="rumque" type="simple">rumqÌ;</reg> radices <var>.a.b.</var> et <var>.b.c.</var> </s> - <s xml:id="s3640" xml:space="preserve">quare & Å¿uperficiem re-<lb/>ctanguli quæſitam.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0305-02" xlink:href="fig-0305-02a"> - <image file="0305-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0305-02"/> - </figure> - </div> - <p> - <s xml:id="s3641" xml:space="preserve">Quartum exemplum etiam faciliori via poteÅ¿t <lb/>Å¿olui, propterea, quod cum nobis cognita Å¿it ba-<lb/>Å¿is trianguli cum Å¿umma reliquorum laterum, & <lb/><reg norm="cum" type="context">cÅ©</reg> angulo oppoÅ¿ito baÅ¿i ipÅ¿ius reliqua cognita no <lb/>bis emergunt ex .15. problemate Å¿ecundi lib. de Triangulis ipÅ¿ius Monteregii.</s> - </p> - <pb o="294" rhead="IO. BAPT. BENED." n="306" file="0306" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0306"/> - <p> - <s xml:id="s3642" xml:space="preserve">Vel Å¿i tibi placet, accipe hanc aliam methodum à me excogitatum.</s> - </p> - <p> - <s xml:id="s3643" xml:space="preserve">Duplicetur <reg norm="triangulum" type="context">triangulÅ©</reg> <var>.a.b.c.</var> <reg norm="orthogonium" type="context">orthogoniÅ©</reg>, & fiat <reg norm="rectangulum" type="context">rectangulÅ©</reg> <var>.b.u.</var> vt in mea figura <lb/>Å¿ecundi exempli hic vides. </s> - <s xml:id="s3644" xml:space="preserve"><reg norm="producaturque" type="simple">producaturqÌ;</reg> <var>.b.c.</var> quouÅ¿que <var>.c.f.</var> æqualis Å¿it <var>.c.u.</var> vnde <var>.b.f.</var> <lb/>cognita nobis erit ex hypotheÅ¿i, </s> - <s xml:id="s3645" xml:space="preserve">quare cognoÅ¿cemus etiam quadratum <var>.g.f.</var> à quo <lb/><reg norm="demptum" type="context">demptÅ©</reg> cum fuerit <reg norm="aggregatum" type="context">aggregatÅ©</reg> quadratorum <var>.g.u.</var> et <var>.u.f.</var> nobis <reg norm="cognitum" type="context">cognitÅ©</reg> (nam quadra <lb/>ta <var>.g.u.</var> et <var>.u.f.</var> æqualia Å¿unt quadrato ipÅ¿ius <var>.a.c.</var> diagonalis datÄ™) remanebit aggrega-<lb/>tum <reg norm="Å¿upplementorum" type="context context">Å¿upplemẽtorÅ©</reg> cognitum, </s> - <s xml:id="s3646" xml:space="preserve">quare eius medietas cognoÅ¿cetur ideÅ¿t <var>.b.u.</var> vndæ ex <num value="5">.<lb/>5.</num> Å¿ecundi Eucli. vt Å¿uperius diximus cognoÅ¿cetur etiam <var>.b.c.</var> et <var>.c.f.</var> diÅ¿tinctæ.</s> - </p> - <p> - <s xml:id="s3647" xml:space="preserve">Idem aſſero de <reg norm="exemplo" type="context">exẽplo</reg> Gemmæ FriÅ¿ij à Stifelio citato in Appendice regulæ falÅ¿i.</s> - </p> - <p> - <s xml:id="s3648" xml:space="preserve">Sit gratia exempli rectangulum hicÅ¿ubÅ¿criptum <var>.a.b.</var> datæ Å¿uperficiei data etiam <lb/>nobis Å¿it proportio <var>.a.e.</var> ad <var>.e.b.</var> laterum producentium, <reg norm="cogitemusque" type="simple">cogitemusqÌ;</reg> <var>.a.e.</var> producta <lb/>vÅ¿que ad <var>.o.</var> ita vt <var>.e.o.</var> æqualis Å¿it ipÅ¿i <var>.e.b.</var> imagine <lb/> - <anchor type="figure" xlink:label="fig-0306-01a" xlink:href="fig-0306-01"/> - mus <reg norm="etiam" type="context">etiã</reg> perfectum eſſe quadratum <var>.b.o.</var> vnde ex <lb/>prima Å¿exti Å¿eu .18. vel .19. Å¿eptimi vel .15. quinti <lb/>eadem proportio erit ipÅ¿ius <var>.a.b.</var> ad <var>.b.o.</var> vt <var>.a.e.</var> ad <lb/><var>e.o.</var> vel ad <var>.e.b.</var> </s> - <s xml:id="s3649" xml:space="preserve">quare ex regula de tribus, cogno-<lb/>Å¿cemus quadratum <var>.b.o.</var> & eius <reg norm="radicem" type="context">radicẽ</reg> <var>.e.o.</var> & ex ea <lb/>demregula cognoÅ¿cemus <var>.a.e.</var> cum cognita nobis Å¿it <var>.e.o.</var> Å¿imul cum proportione <var>.e.o.</var> <lb/>ad <var>.e.a</var>.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0306-01" xlink:href="fig-0306-01a"> - <image file="0306-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0306-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head432" style="it" xml:space="preserve">Quod circulus Å¿it figura infinitorum angulorum hoc eÅ¿t <lb/>ultima poligoniarum.</head> - <head xml:id="head433" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3650" xml:space="preserve">SEd quod idem Stifelius in Appendice Å¿ecundi libri dicat circulum eſſe figuram <lb/>poligoniam, non eÅ¿t ita mirandum, nam & alij multi doctiſſimi viri hanc <lb/>veritatem cognouerunt, de Leone BaptiÅ¿ta Alberto nihil dicam, cum ipÅ¿e fateatur <lb/>hoc accepiſſe à philoÅ¿ophis, vt etiam refert AriÅ¿t. de Å¿phæratertio de cÅ“lo. </s> - <s xml:id="s3651" xml:space="preserve">conÅ¿i-<lb/>dera quæſo in circulo, quod cum angulus contingentiæ Å¿it angulus, quamuis <reg norm="omnium" type="context">omniÅ©</reg> <lb/>acutorum rectilineorum anguÅ¿tiſſimus, vnde ex communi ratione Å¿equitur reliquum <lb/>ex duobus rectis rectilineis eſſe angulum, & Å¿i omnium obtuÅ¿orum rectilineorum Å¿it <lb/>ampliſſimum, tanto magis igitur erit angulus, id quod remanet ex duobus rectis re <lb/>ctilineis, detractis <reg norm="cum" type="context">cÅ©</reg> fuerint duobus angulis contingentiæ, qui quidem angulus erit <lb/>in quouis puncto circunferentiæ ipÅ¿ius circuli, idem intelligendum eÅ¿t de Å¿phæra, <lb/>cuius angulus eÅ¿t reÅ¿iduum ex quatuor rectis Å¿olidis, detractis cum fuerint quatuor <lb/>angulis contingentiæ <reg norm="Å¿olidisque" type="simple">Å¿olidisqÌ;</reg>.</s> - </p> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head434" style="it" xml:space="preserve">Explanatio .25. Problematis lib. 2. Monteregij.</head> - <head xml:id="head435" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3652" xml:space="preserve">QVod in .25. problemate .2. lib. de triangulis Monteregium non intelligas, mi-<lb/>rum non eÅ¿t, eo quod quandoque bonus dormitat Homerus. </s> - <s xml:id="s3653" xml:space="preserve">Puto enim il-<lb/>lud problema ab ipÅ¿o Monteregio non fuiſſe viÅ¿itatum. </s> - <s xml:id="s3654" xml:space="preserve">Sed ne me aliquo modo <lb/>culpes, accipe hanc <reg norm="aliam" type="context">aliã</reg> <reg norm="methodum" type="context">methodÅ©</reg> a me aliter <reg norm="etiam" type="context">etiã</reg> <reg norm="excogitatam" type="context">excogitatã</reg> in eadem ipÅ¿ius figura.</s> - </p> - <pb o="295" rhead="EPISTOL AE." n="307" file="0307" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0307"/> - <p> - <s xml:id="s3655" xml:space="preserve"><reg norm="PropoÅ¿itum" type="context">PropoÅ¿itÅ©</reg> Å¿it nobis triangulum <var>.a.b.g.</var> cuius baÅ¿is data Å¿it cum area, Å¿eu perpendi-<lb/>culari <var>.a.d.</var> cum angulo etiam <var>.a.</var> ad cognoÅ¿cendum autem <var>.a.b.</var> et <var>.b.g.</var> cogitemus circu <lb/>lum <var>.a.b.q.g.</var> circunÅ¿cribere ipÅ¿um triangulum cuius diameter <var>.p.q.</var> ad rectos Å¿e-<lb/>cet baÅ¿im <var>.b.g.</var> in puncto <var>.m.</var> cogitemus etiam <var>.b.p.</var> et <var>.p.g.</var> vnde ex .20. ter-<lb/>tij Euclid. angulus <var>.b.p.g.</var> æqualis erit <lb/> - <anchor type="figure" xlink:label="fig-0307-01a" xlink:href="fig-0307-01"/> - angulo <var>.a.</var> & angulus <var>.m.p.b.</var> erit eius di <lb/>midium, quod ex te ipÅ¿o cognoÅ¿ces, & <lb/><reg norm="angulus" type="simple">angulê°</reg> <var>.p.b.m.</var> Å¿imiliter cognoÅ¿cetur, <lb/></s> - <s xml:id="s3656" xml:space="preserve">quare ex .29. primi eiuÅ¿dem Montere <lb/>gij cognoÅ¿cemus <var>.p.m.</var> et <var>.p.b.</var> (nam <var>.b.<lb/>m.</var> datum fuit, vt dimidium totius ba-<lb/>Å¿is <var>.b.g.</var>) ducta poÅ¿tea <var>.b.q.</var> ex <reg norm="eadem" type="context">eadẽ</reg> .29. <lb/>cognoÅ¿cemus <var>.p.q.</var> cum <var>.p.b.</var> iam cogni <lb/>ta fuerit, à qua <var>.p.q.</var> (diametro) <reg norm="dempta" type="context">dẽpta</reg> <lb/><var>p.m.</var> remanebit <var>.q.m.</var> cognita, <reg norm="cum" type="context">cÅ©</reg> qua <lb/>iuncta cum fuerit <var>.m.t.</var> æquali <var>.a.d.</var> per <lb/>pendiculari, dabitur <var>.q.t.</var> et <var>.t.p.</var> inter <lb/>quas <var>.a.t.</var> media proportionalis loca-<lb/>tur, </s> - <s xml:id="s3657" xml:space="preserve">quare cognoÅ¿cemus <var>.a.t.</var> quæ Å¿inus <lb/>eÅ¿t arcus <var>.a.p.</var> vnde cognitus erit arcus <lb/><var>a.p.</var> Å¿ed arcus <var>.p.g.</var> cognitus eÅ¿t median <lb/>te angulo <var>.p.b.g.</var> cognito, qui quidem <lb/>arcus <var>.p.g.</var> Å¿i coniunctus fuerit cum arcu <var>.p.a.</var> cognoÅ¿cemus compoÅ¿itum <var>.a.g.</var> & eius <lb/>chorda Å¿imiliter (hoc eÅ¿t <reg norm="Å¿ecundum" type="context">Å¿ecundÅ©</reg> latus) qua cognita, illico cognoÅ¿cemus chordam <lb/><var>a.b.</var> hoc eÅ¿t tertium latus trianguli propoÅ¿iti.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0307-01" xlink:href="fig-0307-01a"> - <image file="0307-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0307-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="4"> - <head xml:id="head436" style="it" xml:space="preserve">Quæ<unsure/>dam not and a in Federicum Comandinum.</head> - <head xml:id="head437" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3658" xml:space="preserve">PVtabas enim me ioco dixiſſe Federicum Comandinum non omnino irrepræ-<lb/>henÅ¿ibilem eſſe, vide igitur, quod Å¿cribit in quinto lemmate in decimam <lb/>propoÅ¿itionem libr .2. de inÅ¿identibus aquæ Archimedis, volens demonÅ¿tra-<lb/>re eandem eſſe proportionem <var>.l.b.</var> ad <var>.b.m.</var> quæ <var>.c.e.</var> ad <var>.e.a.</var> vbi eÅ¿t aliquo modo pro-<lb/>lixum, mediante linea <var>.c.p.</var> cum Å¿uis partibus, citans etiam antecedens lemma extra <lb/>propoÅ¿itum, eo quod nec in antecedente lemmate, nec in alio, ipÅ¿e vnquam proba <lb/>uerit proportionem <var>.c.d.</var> ad <var>.d.q.</var> eſſe, vt <var>.l.b.</var> @d.b.m. Å¿ed ne putes me falli, tibi demon <lb/>Å¿trabo non eſſe neceſſarium ducere lineam <var>.c.m.p.</var> vel <var>.q.p.</var> eo quod <reg norm="cum" type="context">cÅ©</reg> per quintam <lb/>lib. de quadratura parabolÄ™ Archimedis, ita Å¿it <var>.c.d.</var> ad <var>.d.e.</var> vt <var>.l.b.</var> ad <var>.b.m.</var> exiÅ¿tente <lb/><var>a.c.</var> dupla ipÅ¿i <var>.d.c.</var> et <var>.e.c.</var> dupla ipÅ¿i <var>.g.c.</var> et <var>.l.d.</var> dupla ipÅ¿i <var>.l.b</var>: erit, primo componen-<lb/>do <var>.c.e.</var> ad <var>.e.d.</var> vt <var>.l.d.</var> ad <var>.d.m.</var> & per æqualitatem proportionum, ita erit <var>.e.g.</var> ad <var>.e.d.</var> <lb/>vt <var>.b.d.</var> 2d.d.m. & per .19. quinti Eucli. ita erit <var>.e.g.</var> ideÅ¿t <var>.g.c.</var> ad <var>.g.d.</var> vt <var>.b.d.</var> ideÅ¿t <var>.l.b.</var> <lb/>ad <var>.b.m.</var> Å¿ed <var>.c.g.</var> ad <var>.g.d.</var> eft vt <var>.c.e.</var> ad <var>.e.a.</var> ratio eÅ¿t, quia componendo ita eÅ¿t <var>.c.d.</var> ad <var>.d.<lb/>g.</var> vt <var>.c.a.</var> ad <var>.a.e.</var> & hoc eÅ¿t, quia permutando, ita eÅ¿t <var>.a.c.</var> ad <var>.d.c.</var> vt <var>.a.e.</var> ad <var>.d.g.</var> & hoc <lb/>verum eÅ¿t ex .19. quinti eo quod totius <var>.a.c.</var> ad totum <var>.d.c.</var> eft vt abÅ¿ciſſi <var>.e.c.</var> ad abÅ¿ciÅ¿ <lb/>Å¿um <var>.g.c.</var> vt Å¿upradixi.</s> - </p> - <pb o="296" rhead="IO. BAPT. BENED." n="308" file="0308" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0308"/> - <p> - <s xml:id="s3659" xml:space="preserve">Sed etiam alio vniuerÅ¿aliori modo potes probare, quod ita Å¿it <var>.u.x.</var> ad <var>.x.y.</var> vt <var>.c.e.</var> <lb/>ad <var>.e.a.</var> cogitando in linea <var>.c.a.</var> punctum quoddam quod vocabimus Å¿imiliter <var>.y.</var> in <lb/>tali Å¿itu locatum, quod diuidat <var>.c.a.</var> eadem proportione qua <var>.y.</var> diuidit <var>.u.s.</var> vnde cum <lb/><var>e.s.</var> diuiÅ¿a eodem modo etiam Å¿it à puncto <var>.s.</var> ex Å¿upradicta quinta lib. de quadratura <lb/>parabolæ, erit igitur proportio <var>.a.y.</var> ad <var>.y.c.</var> vt <var>.e.s.</var> ad <var>.s.c.</var> per .11. quinti Eucli. </s> - <s xml:id="s3660" xml:space="preserve">& com <lb/>ponendo ita erit <reg norm="totius" type="simple">totiê°</reg> <var>.a.c.</var> ad totum <var>.y.c.</var> vt abÅ¿cisÅ¿i <var>.s.c.</var> ad abÅ¿ciÅ¿sum <var>.s.c.</var> </s> - <s xml:id="s3661" xml:space="preserve">quare reÅ¿idui <lb/><var>a.e.</var> ad reÅ¿iduum <var>.y.s.</var> erit vt totius <var>.a.c.</var> ad totum <var>.y.c.</var> & permutando, ita erit <var>.a.c.</var> ad <var>.a.<lb/>e.</var> vt <var>.y.c.</var> ad <var>.y.s.</var> & diuidendo, ita erit <var>.<lb/>c.e.</var> ad <var>.e.a.</var> ut <var>.c.s.</var> ad <var>.s.y.</var> & quia pun-<lb/> - <anchor type="figure" xlink:label="fig-0308-01a" xlink:href="fig-0308-01"/> - ctum <var>.s.</var> diuidit <var>.c.a.</var> eodem modo, quo <lb/>x. diuidit <var>.u.s.</var> per Å¿upradictam <reg norm="quintam" type="context">quintã</reg>, <lb/>ergo ita erit <var>.c.s.</var> ad <var>.s.y.</var> in linea <var>.c.a.</var> vt <lb/><var>u.x.</var> ad <var>.x.y.</var> </s> - <s xml:id="s3662" xml:space="preserve">vnde ex .11. quinti <var>.c.e.</var> ad <lb/><var>e.a.</var> erit, vt <var>.u.x.</var> ad <var>.x,y</var>. </s> - <s xml:id="s3663" xml:space="preserve">quare Å¿equitur, <lb/>primum, Å¿ecundum, tertium, & quartum lemma Å¿uperflua eſſe.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0308-01" xlink:href="fig-0308-01a"> - <image file="0308-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0308-01"/> - </figure> - </div> - <p> - <s xml:id="s3664" xml:space="preserve">Quod deinde ponit pro corellario in fine .6. lemmatis, aliter quam per .6. lemma <lb/>poteÅ¿t demonÅ¿trari, hoc mode. </s> - <s xml:id="s3665" xml:space="preserve">Nam Å¿uperius demonÅ¿trauimus eandem propor-<lb/>tionem eſſe <var>.l.b.</var> ad <var>.b.m.</var> quæ <var>.c.e.</var> ad <var>.e.a.</var> <reg norm="idem" type="context">idẽ</reg> dico de proportione <var>.u.x.</var> ad <var>.x.y.</var> & om-<lb/>nium æquidiÅ¿tantium ad <var>.h.e.</var> quibus rationibus mediantibus codem modo Å¿cies, <reg norm="quod" type="simple">ê™</reg> <lb/><var>u.y.</var> ad <var>.y.r.</var> erit, vt <var>.c.d.</var> ad <var>.d.c.</var> & ita dico de omnibus <reg norm="æquidiÅ¿tantibus" type="context">æquidiÅ¿tãtibus</reg>. ad <var>.h.e.</var> </s> - <s xml:id="s3666" xml:space="preserve">vnde <var>.l.b.</var> <lb/>ad <var>.b.m.</var> erit vt <var>.u.x.</var> ad <var>.x.y.</var> et <var>.l.m.</var> ad <var>.m.d.</var> vt <var>.u.y.</var> ad <var>.y.r.</var> per .11. quinti, Å¿ed cum Å¿it <var>.l.<lb/>b.</var> ad <var>.b.m.</var> vt <var>.u.x.</var> ad <var>.x.y.</var> componendo erit <var>.l.m.</var> ad <var>.b.m.</var> vt <var>.u.x.</var> ad <var>.x.y.</var> & euerÅ¿im <var>.b.<lb/>m.</var> ad <var>.m.b.</var> erit, vt <var>.x.y.</var> ad <var>.y.u.</var> & per æquam proportionalitatem erit <var>.b.m.</var> ad <var>.m.d.</var> vt<unsure/> <lb/><var>x.y.</var> ad <var>.y.r.</var> quod eÅ¿t propoÅ¿itum.</s> - </p> - <p> - <s xml:id="s3667" xml:space="preserve">Non video etiam, quare ipÅ¿e ducat lineam <var>.s.r.</var> cum in ipÅ¿o contextu nihil Å¿aciac <lb/>de dicta <var>.s.r</var>.</s> - </p> - <p> - <s xml:id="s3668" xml:space="preserve">Comentum poÅ¿tea contextus <var>.P.</var> pulchrius eſſet, Å¿i diceret, quod cum ita Å¿it totius, <lb/><var>l.a.</var> ad totum <var>.a.d.</var> Å¿ic Å¿e habebit abÅ¿ciſſum <var>.a.i.</var> ad abÅ¿ciſſum <var>.a.z.</var> eo quod ita eÅ¿t, vt Å¿cis, <lb/>hoc eÅ¿t in proportione dupla, ergo reÅ¿idui <var>.i.l.</var> ad reÅ¿iduum <var>.d.z.</var> erit vt totius <var>.a.l.</var> ad <lb/>totum <var>.a.d.</var> hoc eÅ¿t in proportione dupla.</s> - </p> - </div> - <div type="letter" level="4" n="5"> - <head xml:id="head438" style="it" xml:space="preserve">De ViÅ¿u.</head> - <head xml:id="head439" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3669" xml:space="preserve">RAtio vnde Å¿iat, vt videamus diÅ¿tinctè omnes eolores, cum in qualibet aeris par <lb/>te, quo lumina reÅ¿lexa poſſunt peruenire mixta Å¿int, & non diÅ¿tincta, oritur à <lb/>paruitate ipÅ¿ius pupillæ oculorum, & à magna expanÅ¿ione virtutis viÅ¿iuæ in Å¿uperÅ¿i-<lb/>cie concaua orbis continentis humores diaphanos oculorum per ramuÅ¿culos nerui <lb/>optici remotè ab ipÅ¿a pupilla. </s> - <s xml:id="s3670" xml:space="preserve">& quamuis radii luminoÅ¿i frangantur ab vnoquoque <lb/>humore diuerÅ¿imodè, hoc nihilominus maximè iuuat ad diÅ¿tinctionem radiorum, <lb/>Å¿ed & Å¿i directè procederent, idem ferè eueniret, non tamen Å¿uis locis, cogita exem-<lb/>pli gratia lineam <var>.a.u.e.</var> vt communis Å¿ectio cuiuÅ¿dam plani Å¿ecantis Å¿phæram oculi, <lb/>per centrum ipÅ¿ius, & pupillæ, et <var>.o.</var> punctum Å¿it proximum centro ipÅ¿ius pupillæ, <lb/>Å¿ed interius aliquantulum, extra <reg norm="autem" type="context">autẽ</reg> <reg norm="oculum" type="context">oculÅ©</reg>, Å¿int varij colores, vt <var>.c.n.t.</var> in dicto plano.</s> - </p> - <p> - <s xml:id="s3671" xml:space="preserve">Iam nulli dubium eÅ¿t quod lumina quæ producuntur ab <var>.c.n.t.</var> ad <var>.o.</var> in ipÅ¿o <var>.o.</var> mi- - <pb o="297" rhead="EPISTOL AE." n="309" file="0309" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0309"/> - xta, & non diÅ¿tincta, procedendo igitur vlteriusipÅ¿i radij citra <var>.o</var>. </s> - <s xml:id="s3672" xml:space="preserve">tunc <reg norm="diÅ¿gregantur" type="context">diÅ¿gregãtur</reg>, <lb/>& Å¿eparantur abinuicem, & <reg norm="cum" type="context">cÅ©</reg> perueniunt ad lineam <var>.a.u.e.</var> Å¿entiuntur diÅ¿tincti alij <lb/>ab alijs. </s> - <s xml:id="s3673" xml:space="preserve">Cuius quidem rei, exemplum manifeÅ¿tum accipere poſſumus à quouis <lb/>cubiculo ex omni parte clauÅ¿o, quod tranÅ¿itum <reg norm="nullum" type="context">nullÅ©</reg> permittat radijs luminoÅ¿is, ni <lb/>Å¿i per aliquod paruum foramen, in quo foramine, & extra ipÅ¿um cubiculum, omnes <lb/>radij mixti erunt, Å¿ed in obiecto pariete ipÅ¿ius cu-<lb/> - <anchor type="figure" xlink:label="fig-0309-01a" xlink:href="fig-0309-01"/> - biculi videb untur diÅ¿tincti, vnde Å¿equitur, quòd <lb/>quo remotius erit obiectum <var>.c.n.t.</var> ab <var>.o.</var> tanto acu-<lb/>tior erit angulus <var>.c.o.t.</var> & Å¿uus contrapoÅ¿itus Å¿imili-<lb/>ter, & per conÅ¿equens linea <var>.e.u.a.</var> breuior erit, & <lb/><reg norm="punctum" type="context">punctÅ©</reg> <var>.o.</var> propinquius etiam erit ipÅ¿i lineæ <var>.a.u.e.</var> <lb/>quæ omnia efficiunt, vt nobis obiectum <var>.c.t.</var> <reg norm="paruum" type="context">paruÅ©</reg>, <lb/>& minus diÅ¿tinctum, Å¿eu magis confuÅ¿um appareat.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0309-01" xlink:href="fig-0309-01a"> - <image file="0309-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0309-01"/> - </figure> - </div> - </div> - </div> - <div type="section" level="3" n="18"> - <div type="letter" level="4" n="1"> - <head xml:id="head440" xml:space="preserve">DE APPARENTI DISTANTIA PARTIV M <lb/>hæmiÅ¿phærij.</head> - <head xml:id="head441" style="it" xml:space="preserve">AnÅ¿elmo Fucaro.</head> - <p> - <s xml:id="s3674" xml:space="preserve">GRatæ mihi tuæ literæ <reg norm="fuerunt" type="context">fuerÅ©t</reg>, quibus <reg norm="oÅ¿tendis" type="context">oÅ¿tẽdis</reg> non <reg norm="paruum" type="context">paruÅ©</reg> <reg norm="deÅ¿iderium" type="context">deÅ¿ideriÅ©</reg> <reg norm="Å¿ciendi" type="context">Å¿ciẽdi</reg> vnde <lb/>fiat, quod cum dies illuceÅ¿cit, & eÅ¿t Å¿erena pars CÅ“li, circa axem orizontis <lb/>demiſſior appareat, quam aliæ partes, <reg norm="cum" type="context">cÅ©</reg> ab alijs (quod <reg norm="Å¿ciam" type="context">Å¿ciã</reg>) Å¿atis expreſſum <reg norm="non" type="context">nõ</reg> fue <lb/>rit, Å¿ed quia de eo à me aliquid Å¿cire deÅ¿ideras dicam quod mihi <reg norm="verum" type="wordlist">vr̃</reg>. </s> - <s xml:id="s3675" xml:space="preserve">Scias non <reg norm="Å¿olum" type="context">Å¿olÅ©</reg> <lb/><reg norm="multitudinem" type="context">multitudinẽ</reg> <reg norm="obiectorum" type="context">obiectorÅ©</reg> <reg norm="oppoÅ¿itorum" type="context">oppoÅ¿itorÅ©</reg> efficere, vt aliqua res alia longius diÅ¿tare videat, <lb/>vt alij <reg norm="putarunt" type="context">putarÅ©t</reg>, Å¿ed etiam diuerÅ¿itates colorum, quamobrem cum decipiamur, cre-<lb/>dentes CÅ“lum eſſe præditum colore cÄ™ruleo, cum is color, aeri, non CÅ“lo <lb/>conueniat, & videntes huiuÅ¿modi colorem circa axem <reg norm="orizontis" type="context">orizõtis</reg> magis denÅ¿um, <reg norm="quam" type="context">quã</reg> <lb/>verÅ¿us ipÅ¿um orizontem, ratione exiguæ reÅ¿lexionis, à pauca quantitate vaporum <lb/>inter noÅ¿trum Å¿itum, & reÅ¿lexionis locum, iudicamus CÅ“lum proximiorem eſſe cir-<lb/>ca dictum axem, quam Å¿int aliæ partes; </s> - <s xml:id="s3676" xml:space="preserve">præterquam, quod is color, qui videtur <lb/>terminare, aut impedire radium viÅ¿ualem (aduertas tamen me hac in re platonicum <lb/>non eſſe) eo Å¿emper propinquior eſſe videtur, qui ei locum dat, & hanc ob cauÅ¿am <lb/>videntes nos <reg norm="denÅ¿itatem" type="context context">dẽſitatẽ</reg> cÄ™rulei circa axem orizontis, & cernentes amplitudinem gy <lb/>ri aliarum partium, adducimur, vt putemus <reg norm="eam" type="context">eã</reg> <reg norm="partem" type="context">partẽ</reg> <lb/> - <anchor type="figure" xlink:label="fig-0309-02a" xlink:href="fig-0309-02"/> - viciniorem eſſe. </s> - <s xml:id="s3677" xml:space="preserve"><reg norm="Neque" type="simple">Neq;</reg> illud <reg norm="etiam" type="context">etiã</reg> <reg norm="omittam" type="context">omittã</reg> hoc <reg norm="etiam" type="context">etiã</reg> fie <lb/>ri ratione imaginationis, vnde <reg norm="etiam" type="context">etiã</reg> multis <reg norm="contrarium" type="context">contrariÅ©</reg> <lb/>euenire poteÅ¿t, ideÅ¿t vt eis magis profundum videa <lb/>tur <reg norm="CÅ“lum" type="context">CÅ“lÅ©</reg>, circa axem orizontis, quam vicinum gy <lb/>ro <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> <reg norm="orizontis" type="context">orizõtis</reg>, iudicantibus <reg norm="eam" type="context">eã</reg> <reg norm="partem" type="context">partẽ</reg> <reg norm="longinquio" type="context">lõginquio</reg> <lb/><reg norm="rem" type="context">rẽ</reg> eſſe, quæ Å¿eÅ¿e magis <reg norm="obÅ¿curam" type="context">obÅ¿curã</reg> oculo <reg norm="demonÅ¿trat" type="context">demõſtrat</reg>, & <lb/><reg norm="eam" type="context">eã</reg> <reg norm="propinquiorem" type="context">propinquiorẽ</reg> quÄ™ Å¿eÅ¿e <reg norm="clariorem" type="context">clariorẽ</reg> oÅ¿tendit, vt ei <reg norm="ent" type="context">ẽt</reg> <lb/>cont<unsure/>ingere poteÅ¿t, qui <reg norm="Å¿ubÅ¿criptam" type="context">Å¿ubÅ¿criptã</reg> <reg norm="Å¿iguram" type="context">Å¿igurã</reg> <reg norm="cubicam" type="context">cubicã</reg> non <lb/><reg norm="quidem" type="context">quidẽ</reg> <reg norm="ductam" type="context">ductã</reg> Å¿ecundú <reg norm="ordinem" type="context">ordinẽ</reg> opticè, Å¿ed ita, vt om-<lb/>nia latera oppoÅ¿ita <reg norm="inuicem" type="context">inuicẽ</reg> Å¿int parallela, proÅ¿piciet, <lb/>ideÅ¿t <var>.a.i.</var> ad <var>.e.t.</var> et <var>.c.u.</var> ad <var>.o.n.</var> et <var>.a.i.</var> ad <var>.c.u.</var> et <var>.e.t.</var> ad <lb/><var>o.n.</var> vnde Å¿equitur, vt aliquando quadratum <var>.a.o.</var> <lb/>videbitur citra, et <var>.i.n.</var> vltra <reg norm="dictum" type="context">dictÅ©</reg> cubum aliquando <lb/>verò èconuerÅ¿o.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0309-02" xlink:href="fig-0309-02a"> - <image file="0309-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0309-02"/> - </figure> - </div> - <pb o="298" rhead="IO. BAPT. BENED." n="310" file="0310" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0310"/> - </div> - </div> - <div type="section" level="3" n="19"> - <div type="letter" level="4" n="1"> - <head xml:id="head442" xml:space="preserve">DE PHILOSOPHIA MATHEMA TICA.</head> - <head xml:id="head443" style="it" xml:space="preserve">Dominico PiÅ¿ano.</head> - <p> - <s xml:id="s3678" xml:space="preserve">SI omnia vno colore conÅ¿tarent, & corporum vmbræ à luminibus non di-<lb/>Å¿tinguerentur, neque diuerÅ¿itas Å¿itus, lumina, quæ veniunt ad oculum non al-<lb/>teraret; </s> - <s xml:id="s3679" xml:space="preserve">perinde eſſet, ac Å¿i eſſemus cÅ“ci. </s> - <s xml:id="s3680" xml:space="preserve">Miror quod cum in AriÅ¿totele Å¿is <lb/>verÅ¿atus, in tuis tamen Å¿criptis philoÅ¿ophum à Mathematico Å¿epares, quaÅ¿i mathe-<lb/>maticus non Å¿it adeò philoÅ¿ophus, vt eÅ¿t naturalis, & metaphyÅ¿icus, cum multo ma <lb/>gis quam ij philoÅ¿ophus Å¿it appellandus, Å¿i ad veritatem Å¿uarum concluÅ¿ionum reÅ¿pi <lb/>ciamus. </s> - <s xml:id="s3681" xml:space="preserve">Verum <reg norm="quidem" type="context">quidẽ</reg> eÅ¿t, te in huiuÅ¿modi errore <reg norm="Å¿olum" type="context">Å¿olÅ©</reg> non verÅ¿ari; </s> - <s xml:id="s3682" xml:space="preserve">Å¿ed grauius eÅ¿t, <lb/>quod cum vos videatis etiam res morales Å¿ub philoÅ¿ophiÄ™ <reg norm="appellationem" type="context">appellationẽ</reg> cadere, non <lb/>animaduertatis diuinas Å¿cientias mathematicas etiam philoÅ¿ophiæ nomine ornan-<lb/>das eſſe. </s> - <s xml:id="s3683" xml:space="preserve">Quod Å¿i eiuÅ¿dem nomen penitius conÅ¿iderare velimus, inueniemus aper-<lb/>tè, mathematico magisillud ipÅ¿um quà m cuilibet alio conuenire, cum nullus ex alijs <lb/>tam certo Å¿ciat id quod affirmat quam mathematicus, neque aliquis Å¿it, qui in co-<lb/>gnitionis, & Å¿cientiæ cupiditatem magis ducatur, vt apertè patet, cum nec etiam ipÅ¿i <lb/>Å¿enÅ¿ui det locum, neque aliquid præſupponat, quod non Å¿it ita verum & intellectui <lb/>notum, vt nulla quæuis porentia, illud eſſe falÅ¿um oÅ¿tendere queat. </s> - <s xml:id="s3684" xml:space="preserve">Sed quia <lb/>Græci, qui ad placitum nomina rebus impoÅ¿uerunt, voluerunt etiam, non Å¿olum <lb/>mathematica, Å¿ed etiam naturalia, metaphyÅ¿ica, & moralia, Å¿ub communi philoÅ¿o-<lb/>phiæ nomine contineri. </s> - <s xml:id="s3685" xml:space="preserve"><reg norm="Vtaunt" type="context">VtaÅ©t</reg> tibi Å¿atisfaciam authoritate AriÅ¿totelis, quem tanto-<lb/>pere colis, primum conÅ¿idera, nunquam eum de philoÅ¿opho <reg norm="mentionem" type="context">mẽtionem</reg> facere quin <lb/>prius aperiat de quo philoÅ¿opho loquatur, atque hoc Å¿emper præſtat, exceptis qui-<lb/>buÅ¿dam locis, vt cap .2. lib. 4. <reg norm="MetaphyÅ¿icorum" type="context">MetaphyÅ¿icorÅ©</reg>, vbi de philoÅ¿opho in genere <reg norm="loquens" type="context">loquẽs</reg>, <lb/>ait, proprium eſſe philoÅ¿ophi. vt res omnes Å¿peculetur atque hoc in principio quin <lb/>ti textus aſſerit, cum in quarto iam oÅ¿tenderit mathematicum eſſe philoÅ¿ophum: <lb/></s> - <s xml:id="s3686" xml:space="preserve">omitto quod in .2. textu Å¿ecundi phyÅ¿icorum idem affirmet, æquum eſſe appellare <lb/>philoÅ¿ophiam Å¿cientiam veritatis, & finem Å¿peculatiuæ exiÅ¿tere veritatem. </s> - <s xml:id="s3687" xml:space="preserve">An non <lb/>idem in primo cap .6. metaphiÅ¿icæ philoÅ¿ophiam Å¿peculatiuam, mathematicis phy <lb/>Å¿icis & Å¿upernaturalibus rebus contineri? </s> - <s xml:id="s3688" xml:space="preserve">An non idem paulo inferius Å¿cribit phyÅ¿i-<lb/>cam primam futuram, Å¿i aliæ Å¿ubÅ¿tantiæ quam naturales non reperirentur? </s> - <s xml:id="s3689" xml:space="preserve">conÅ¿idera <lb/>deinde quid dicat in fine tertij cap. lib. 11. quo loco nil clarius eſſe poteÅ¿t, lege etiam <lb/>quæ .6. cap. eiuÅ¿dem libri ab eodem adducuntur, & quæ in .8. cap .12. libri textu .44. <lb/>apertè ponuntur. </s> - <s xml:id="s3690" xml:space="preserve">Quod Å¿i hæc tibi non Å¿ufficiunt, vereor ne tuus morbus deÅ¿pe-<lb/>ratus euadat.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head444" style="it" xml:space="preserve">De imaginatione Å¿pecierum.</head> - <head xml:id="head445" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3691" xml:space="preserve">QVod dixi domino Tadeo eÅ¿t, quod aliquas particularium Å¿pecies, perfectè & <lb/>integrè imaginari poſſumus, alias non item, id tibi melius exemplo innote-<lb/>Å¿c<unsure/>et. </s> - <s xml:id="s3692" xml:space="preserve">Proponatur tibi triangulus æquilaterus datæ magnitudinis, <reg norm="datique" type="simple">datiqÌ;</reg> coloris, hu <lb/>iuÅ¿modi enim particularis, potes imaginatione tibi fingere integram Å¿peciem, tota <lb/><reg norm="lemque" type="simple">lemqÌ;</reg> ei adæquatam, Å¿ed Å¿i aliquam Å¿peciem aliquando vniuerÅ¿aliorem imaginatio - <pb o="299" rhead="EPISTOL AE." n="311" file="0311" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0311"/> - ne concipere velles, quemadmodum vnius trianguli Ä™quilateri, tali magnitudine, <lb/>Å¿ed non præfinito colore conÅ¿tantis, hoc minime præſtare poſſes. </s> - <s xml:id="s3693" xml:space="preserve">quia nullam rem <lb/>viÅ¿ibilem priuatam colore imaginari poſſumus. </s> - <s xml:id="s3694" xml:space="preserve">nec etiam potes imaginari <reg norm="Å¿peciem" type="context">Å¿peciẽ</reg> ali <lb/>cuius trianguli æquilateri, indeterminatæ magnitudinis, & indefiniti coloris, quæ <lb/>cuilibet particulari cuiuÅ¿uis magnitudinis, & coloris poÅ¿tea applicari queat. </s> - <s xml:id="s3695" xml:space="preserve">Species <lb/>deinde alicuius trianguli Ä™quicruri, aut vnius trianguli laterum inÄ™qualium, aut <reg norm="triam" type="context">triã</reg> <lb/>guli in genere, aut tandem figuræ, conÅ¿iderato tu ipÅ¿e, an poſſit Å¿ub imaginationem <lb/>cadere. </s> - <s xml:id="s3696" xml:space="preserve">Poſſumus quidem huiuÅ¿modi Å¿peciem (ratione mediante) intelligere, vn <lb/>de quamlibet Å¿peciem rei particularis viÅ¿ibilis, compoÅ¿itæ, ex figura, magnitudine, <lb/>& colore, perfectè imaginari poſſimus, & huiuÅ¿modi conceptus erit Å¿pecialiſſima <lb/>Å¿pecies, quia in infinito Å¿uorum indiuiduorum, nunquam fiet, vt aliquod eorum, ali <lb/>quo modo ab alijs differre poſſit; </s> - <s xml:id="s3697" xml:space="preserve">admonens te, nil reperiri, quod differat, aut in Å¿e <lb/>partem aliquam habeat, quod aliquid aliud non obtineat, quin dicta differentia Å¿it <lb/>Å¿pecifica, eius tamen Å¿olum partis quæ differt ab alia duorum indiuiduorum, vnius, <lb/><reg norm="eiuÅ¿demque" type="simple">eiuÅ¿demqÌ;</reg> Å¿peciei. </s> - <s xml:id="s3698" xml:space="preserve">quia Å¿i eÅ¿t in magnitudine nulla planè magnitudo reperitur, quÄ™ <lb/>Å¿ua Å¿pecie non Å¿it dotata, quod Å¿i non eſſet, inter res omnes nulla æqualitas <lb/>eluceret: </s> - <s xml:id="s3699" xml:space="preserve">& Å¿i in figura, & colore, idem affirmo, aliter nulla res fimilis eſſet <lb/>alteri, neque aliqua Å¿imilitu lo reperiretur. </s> - <s xml:id="s3700" xml:space="preserve">Idem de quolibet alio obiecto <lb/>Å¿enÅ¿ibili dico. </s> - <s xml:id="s3701" xml:space="preserve">Ratio autem eorum omnium quæ dixi eÅ¿t, quia imaginatiua nihil <lb/>aliud intellectui oÅ¿tendere poteÅ¿t, quam id quod recipit à ſenÅ¿u, & cum Å¿enÅ¿us, <lb/>alio modo moueri non poſſit quam Å¿upradicto, hanc ob cau Å¿am verum eÅ¿t, <reg norm="quicquid" type="simple">ê—cquid</reg> <lb/>Å¿cripÅ¿i. </s> - <s xml:id="s3702" xml:space="preserve">Vnde triangulum Ä™quilaterum datÄ™ magnitudinis, erit genus triangulorum <lb/><reg norm="Ä™quilaterum" type="context">Ä™quilaterÅ©</reg> <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> datæ magnitudinis, Å¿ed <reg norm="diuerÅ¿orum" type="context">diuerÅ¿orÅ©</reg> <reg norm="colorum" type="context">colorÅ©</reg>, erit <reg norm="etiam" type="context">etiã</reg> Å¿pecies trianguli <lb/>æquilateri indeterminatæ magnitudinis, & hic deinde erit Å¿pecies trianguli, & hic <lb/>poÅ¿tea Å¿pecies figuræ. </s> - <s xml:id="s3703" xml:space="preserve">Idem de alijs omnibus rebus per gradus dico, quÄ™ Å¿icut à ſen <lb/>Å¿u, ita etiam ab imaginatione longè recedunt, adeo vt has Å¿pecies Å¿pecialiſſimas tan-<lb/>tum, ideÅ¿t eas Å¿olùm, quas hic Å¿uperius deÅ¿cripÅ¿i, integrè capere poſſit: </s> - <s xml:id="s3704" xml:space="preserve">at verò gene <lb/>ra, quanto vniuerÅ¿aliora Å¿unt, ab eadem imaginatione, tanto longius diÅ¿tant.</s> - </p> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head446" style="it" xml:space="preserve">De maculis Lunæ, & eius lumine.</head> - <head xml:id="head447" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3705" xml:space="preserve">MAculæ Lunæ, nihil aliud Å¿unt, quà m partes ipÅ¿ius LunÄ™ magis perÅ¿picuæ, à qui <lb/>bus, lumen non refleſſum, Å¿ed penetrans, nobis occultatur; </s> - <s xml:id="s3706" xml:space="preserve"><reg norm="quemadmodum" type="context">quemadmodÅ©</reg> <lb/>via lactea, nihil aliud eÅ¿t, quam pars octaui orbis magis opaca, à qua lumen Solis re-<lb/>fleſſum, Å¿eÅ¿e nobis oÅ¿tendit. </s> - <s xml:id="s3707" xml:space="preserve">Quod autem Maurolicus Å¿cribit folio .64. cap. de aÅ¿tro <lb/>rum fulÅ¿ionibus, circa <reg norm="Lunam" type="context">Lunã</reg>, eÅ¿t falÅ¿um; </s> - <s xml:id="s3708" xml:space="preserve">primo, quia non conÅ¿iderat <reg norm="differentiam" type="context">differentiã</reg> inten-<lb/>Å¿ionis luminum inter Venerem, & Lunam, cum <reg norm="lumem" type="context">lumẽ</reg> illius Å¿it magis intenÅ¿um, quam <lb/>Lunæ, quia quilibet qui Å¿ano Å¿it <reg norm="onculo" type="context">õculo</reg>, facile poteÅ¿t compræhendere, Å¿i Lu-<lb/>na eſſet, vbi eÅ¿t Venus, aut Venus vbireperitur Luna (quibus in locis eiuÅ¿dem ma <lb/>gnitudinis nobis apparerent) ipÅ¿a Luna à Venere longè Å¿uperaretur, & excedere-<lb/>tur Å¿plendore, & lumine, ita vt Å¿i etiam verum eſſet, quod per tres gradus inter-<lb/>ualli Å¿eÅ¿e nobis proderet Å¿exageÅ¿ima pars luminis (quod in quadraturis nec in vllo <lb/>alio Å¿itu verum e<unsure/>uadit, reÅ¿pectu ad Solem, ideÅ¿t vt tres gradus differentiæ Å¿itus, con <lb/>Å¿tituant Å¿exag eÅ¿imam partem differentiæ luminis reÅ¿pectu noÅ¿tri) non ideo tamen - <pb o="300" rhead="IO. BAPT. BENED." n="312" file="0312" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0312"/> - dictum lumen conÅ¿piceretur, quia non Å¿ufficit extenÅ¿io luminis, cum eiuÅ¿dem inten <lb/>Å¿io Å¿it etiam neceſſaria. </s> - <s xml:id="s3709" xml:space="preserve">Sed id quoque tibi dico, quod etiam Å¿i dicta Å¿exageÅ¿ima <lb/>pars totius luminis lunaris, eadem intenÅ¿ione Å¿plendoris, & luminis Veneris, in tali <lb/>diÅ¿tantia trium graduum à Sole prædita eſſet, non eam <reg norm="tamen" type="wordlist">tamẽ</reg> videremus, ratione ob <lb/>liquitatis curuÄ™, & Å¿phæricÄ™ Å¿uperficiei Lunæ, reÅ¿pectu noÅ¿tri, in huiuÅ¿modi Å¿itu: </s> - <s xml:id="s3710" xml:space="preserve">id <reg norm="qui" type="simple">ê—</reg> <lb/>tibi ita demonÅ¿tratum volo.</s> - </p> - <p> - <s xml:id="s3711" xml:space="preserve">Pars Å¿uperficialis lunaris globi, quæ nos reÅ¿picit Å¿it <var>.a.p.u.</var> quam accipere poſſu-<lb/>mus pro medietate ipÅ¿ius Å¿uperficiei totalis, eo quod reÅ¿pectu noÅ¿tri viÅ¿us, inÅ¿enÅ¿ibi <lb/>liter, ab ipÅ¿a medietate differat, pars autem à Sole viÅ¿a Å¿it <var>.u.q.a.</var> cogitemus etiam cir <lb/>culum <var>.a.p.u.q.</var> vnum eſſe ex maioribus ipÅ¿ius globi, cuius Å¿uperficies <reg norm="tranÅ¿eat" type="context">trãſeat</reg> per ocu <lb/>lum vidontis, vnde pars eius <var>.a.p.u.</var> diuidet vmbram per æqualia, reliqua verò pars <var>.<lb/>a.q.u.</var> diuidet per æqualia lumen ipÅ¿ius Lunæ à Sole receptum, ita quod pars illumi <lb/>nata, erit medietas <var>.u.q.a.</var> exceſſus verò, cum noÅ¿tro viÅ¿ui incompræhenÅ¿ibilis Å¿it, pro <lb/>nihilo reputetur, cuius cauÅ¿a eÅ¿t, maxima illa diÅ¿tantia, quæ inter Solem, & Lunam <lb/>reperitur, quamuis Sol maior Å¿it Luna multis millibus vicium, eo quod tunc inter So <lb/>lem, & Lunam reperiantur plus quam .570. diametri terræ.</s> - </p> - <p> - <s xml:id="s3712" xml:space="preserve">Supponamus nunc Lunam remotam eſſe à loco ipÅ¿ius <reg norm="coniunctionis" type="context">cõiunctionis</reg> cum Sole per <lb/>3. gradus. </s> - <s xml:id="s3713" xml:space="preserve">vnde <reg norm="quemadmodum" type="wordlist">quẽadmodum</reg> prius <lb/> - <anchor type="figure" xlink:label="fig-0312-01a" xlink:href="fig-0312-01"/> - lumen erat in gyro <var>.a.q.u.</var> nunc re-<lb/>periatur in gyro <var>.x.q.t.</var> ita quod <var>.t.u.</var> <lb/>erit Å¿exageÅ¿ima pars ipÅ¿ius <var>.a.p.u.</var> <reg norm="quod" type="wordlist">qÄ‘</reg> <lb/>à vero Å¿enÅ¿ibiliter non diÅ¿cedit. <lb/></s> - <s xml:id="s3714" xml:space="preserve">Imaginentur nunc duæ rectæ lineæ <lb/>ductæ ab oculo <var>.d.</var> ad puncta <var>.t.</var> et <var>.u.</var> <lb/>verum tamen eÅ¿t quod linea <var>.d.u.</var> Å¿e-<lb/>cabit <reg norm="arcum" type="context">arcÅ©</reg> <var>.t.u.</var> Å¿ed ita propinqua <reg norm="pum" type="context">pÅ©</reg> <lb/>cto <var>.u.</var> quod erit ei ferè contingens, <lb/>vnde abÅ¿que Å¿enÅ¿ibili errore poſſu-<lb/>mus arcum <var>.t.u.</var> intelligere inter duas <lb/>lineas <var>.d.t.</var> et <var>.d.u.</var> quapropter tale lu-<lb/>men compræhendetur, ferè, Å¿ub an-<lb/>gulo <var>.t.d.u.</var> quem quidem angulum <lb/>oportet nos videre, cuius magnitu-<lb/>dinis exiÅ¿tat, reÅ¿pectu totalis anguli <lb/><var>a.d.u.</var> protracta cum fuerit <var>.d.a</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0312-01" xlink:href="fig-0312-01a"> - <image file="0312-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0312-01"/> - </figure> - </div> - <p> - <s xml:id="s3715" xml:space="preserve">Producatur primo <var>.d.t.</var> vÅ¿que ad <lb/>diametrum in puncto <var>.i.</var> deinde per <lb/>puncta <var>.a.</var> et <var>.u.</var> ducatur arcus <var>.a.e.u.</var> cir <lb/>ca <var>.d.</var> <reg norm="centrum" type="context">cẽtrum</reg>, ad quem ducatur linea <var>.<lb/>d.t.i.</var> in puncto <var>.e.</var> Å¿ed quia, cum dia-<lb/>meter <var>.a.u.</var> tam breuis Å¿it reÅ¿pectu di <lb/>Å¿tantiæ à terra, tempore interlunij, <lb/>vnde minor <reg norm="centeÅ¿ima" type="context">cẽteÅ¿ima</reg> parte ipÅ¿ius di-<lb/>Å¿tantiæ exiÅ¿tit, <reg norm="Å¿equitur" type="simple">Å¿equit̃</reg> nos poſſe <reg norm="abÅ¿que" type="simple">abÅ¿q;</reg> <lb/>Å¿enÅ¿ibili errore cogitare, à puncto <var>.d.</var> <lb/>ad quoduis punctum ipÅ¿ius diametri <lb/>omnes lineas ad angulos rectos cum <lb/>ipÅ¿o diametro, & inÅ¿enÅ¿ibilis inæqua - <pb o="301" rhead="EPISTOL AE" n="313" file="0313" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0313"/> - litatis à linea <var>.d.o</var>. </s> - <s xml:id="s3716" xml:space="preserve">Accipiemus igitur <var>.t.i.</var> pro Å¿inu arcus <var>.t.u.</var> qui eÅ¿t graduum .3. hoc <lb/>eÅ¿t Å¿exageÅ¿ima pars Å¿emicirculi graduum .180. quapropter <var>.t.i.</var> erit partium .5233. ta-<lb/>lium qualium <var>.o.u.</var> eÅ¿t .100000. cuius <var>.t.i.</var> quadratum demptum cum fuerit à quadra-<lb/>to Å¿emidiametri <var>.o.t.</var> relinquet nobis quadratum ipÅ¿ius <var>.o.i.</var> quæ quidem <var>.o.i.</var> vt radix <lb/>quadrata, erit partium .99862. talium qualium Å¿emidiameter eÅ¿t .100000. vnde <var>.i.u.</var> <lb/>reÅ¿iduum diametri, remanebit partium .138. </s> - <s xml:id="s3717" xml:space="preserve">Vel Å¿ic, cum cognitus Å¿it nobis arcus <var>.<lb/>t.u.</var> illicò cognoÅ¿cemus Å¿inum arcus <var>.p.t.</var> <reg norm="complementum" type="context">complementÅ©</reg> vnius quartæ, qui Å¿inus æ qua-<lb/>lis erit ferè arcui <var>.o.i.</var> partium .99862. vnde <var>.i.u.</var> erit, vt dictum eÅ¿t, partium .138. quÄ™ <lb/>quidem <var>.i.u.</var> æqualis eÅ¿t ferè Å¿inui arcus <var>.u.e.</var> & ita etiam <var>.u.e.</var> </s> - <s xml:id="s3718" xml:space="preserve">quare Å¿i diuiÅ¿a fuerit to-<lb/>ta <var>.a.u.</var> <reg norm="partium" type="context">partiÅ©</reg> .200000. per .138. proueniet nobis .1449. & Å¿ic angulus <var>.t.d.u.</var> erit vna <lb/>partium .1449. anguli <var>.a.d.u</var>. </s> - <s xml:id="s3719" xml:space="preserve">Confideremus igitur quomodo fieri poteÅ¿t, vt oculo <lb/>compræhendatur hæc tam parua particula luminis lunaris.</s> - </p> - </div> - </div> - <div type="section" level="3" n="20"> - <div type="letter" level="4" n="1"> - <head xml:id="head448" xml:space="preserve">SOLVTIONES ALIQVAE.</head> - <head xml:id="head449" style="it" xml:space="preserve">Paulo Aemilio Raifestaim.</head> - <p> - <s xml:id="s3720" xml:space="preserve">PoÅ¿t eas literas quas proximè ad te dedi, FranciÅ¿cus Monardus mihi retulit tuas <lb/>nonnullas dubitationes circa noÅ¿trum Theorema Arithmeticum .116. quarum <lb/>prima eÅ¿t, quod Å¿i numerus <var>.a.</var> <reg norm="cogitatus" type="simple">cogitatê°</reg>, eſſet æqualis .4. </s> - <s xml:id="s3721" xml:space="preserve">tunc ipÅ¿e non eſſet multiplex <lb/>ipÅ¿i .4. de quo tamen nullam mentionem feci. </s> - <s xml:id="s3722" xml:space="preserve">Idem etiam inquis, Å¿i <var>.a.</var> fuiſſet .5. 6. 7. <lb/>nec non .1. 2. et .3. </s> - <s xml:id="s3723" xml:space="preserve">Cui reÅ¿pondi, quod <reg norm="quanuis" type="context">quãuis</reg> nullam fecerim mentionem de æqua <lb/>litate ipÅ¿ius <var>.a.</var> cum .4. nihiltamen refert, </s> - <s xml:id="s3724" xml:space="preserve">propterea quod quando ita fuiſſet, nihi-<lb/>lominus eaÅ¿dem conditiones Å¿ubiret, <reg norm="quemadodum" type="context">quemadodÅ©</reg> Å¿i fuiſſet duplus, triplus, aut qua <lb/>druplus. eo quod à genere multiplici, æqualitas, formam diuerÅ¿am non induat. </s> - <s xml:id="s3725" xml:space="preserve">Qua <lb/>re idem eueniet Å¿i <var>.a.</var> fuerit .4. 5. 6. 7. vt Å¿i eſſet .8. 9. 10. et .11. & Å¿ic de cæteris, excepto <lb/>quod in proprijs multiplicibus, vel in Å¿uperantibus ipÅ¿is multiplicibus <var>.a.</var> menÅ¿urare <lb/>tur ab ipÅ¿o .4. plus quam Å¿emel. </s> - <s xml:id="s3726" xml:space="preserve">Quod autem dicis. de .1. 2. et .3. nihil eÅ¿t, quia, <lb/>vt in Å¿ecunda Å¿umma, hoc eÅ¿t in tertio termino maximo, reliquus tertius terminus, <lb/>ideÅ¿t .9. non compræhendetur, ita nobis indicabit primum numerum Å¿umptum mi <lb/>norem eſſe quaternario. </s> - <s xml:id="s3727" xml:space="preserve">Quæ omnia, exipÅ¿a noÅ¿tra thÄ™oria ibidem expreſſa ma-<lb/>nifeÅ¿tantur. </s> - <s xml:id="s3728" xml:space="preserve">Quid autem circa hoc Frater Lucas dicat, neÅ¿cio, quia ipÅ¿ius opus <lb/>ad manus meas nunquam peruenit, Å¿atis enim mihi fuit, in Tartalea hanc praxim <lb/>vidiſſe, ratio vero nullibi à me reperta fuit. </s> - <s xml:id="s3729" xml:space="preserve">Tartalea enim multos citat authores, <lb/>quorum Å¿cripta ego nunquam vidi, vt Leonardi PiÅ¿ani, ProÅ¿docimi, Petri Borghi, <lb/>Fratris Lucæ, Ioannis Sfortunati, <reg norm="cæterorumque" type="simple">cæterorumqÌ;</reg> Å¿imilium.</s> - </p> - <pb o="302" rhead="IO. BABPT. BENED." n="314" file="0314" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0314"/> - </div> - </div> - <div type="section" level="3" n="21"> - <div type="letter" level="4" n="1"> - <head xml:id="head450" xml:space="preserve">ELIPSIM PROPOSITAM QVALITER</head> - <head xml:id="head451" xml:space="preserve">quadrare valeamus.</head> - <head xml:id="head452" style="it" xml:space="preserve">IlluÅ¿tri Uiro FranciÅ¿co Mendo Zzæ</head> - <p> - <s xml:id="s3730" xml:space="preserve">QVod antea tuo nomine fecerat Marcus Antonius amicus noÅ¿ter Å¿ufficie-<lb/>bat. </s> - <s xml:id="s3731" xml:space="preserve">Sed quia, quæ nunc à me petis, talia Å¿unt, vt Å¿ine tripartita <reg norm="aequa- liter" type="simple">Ä™qua-<lb/>liter</reg> aliqua data proportione non poſſit aliquis exactè intentum perfice-<lb/>re, nihilominus, Å¿uppoÅ¿ita di <lb/> - <anchor type="figure" xlink:label="fig-0314-01a" xlink:href="fig-0314-01"/> - cta diuiÅ¿ione, reliqua facilia <reg norm="erunt" type="context">erÅ©t</reg>. </s> - <s xml:id="s3732" xml:space="preserve"><reg norm="Primum" type="context">PrimÅ©</reg> <lb/>enim eÅ¿t. </s> - <s xml:id="s3733" xml:space="preserve">PropoÅ¿itam EllipÅ¿im qua-<lb/>drare.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0314-01" xlink:href="fig-0314-01a"> - <image file="0314-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0314-01"/> - </figure> - </div> - <p> - <s xml:id="s3734" xml:space="preserve">Sit <reg norm="igitur" type="simple">igit̃</reg> EllipÅ¿is propoÅ¿ita <var>.a.b.d.c.</var> cu-<lb/>ius axes Å¿int <var>.a.b.</var> et <var>.d.c.</var> dati, Å¿eu <reg norm="reperti" type="simple">reê‘ti</reg> ex <lb/>47. <reg norm="Å¿ecundi" type="context">Å¿ecÅ©di</reg> Pergei, <reg norm="Å¿intque" type="simple">Å¿intqÌ;</reg> duo circuli <var>.a.e.<lb/>b.f.</var> et <var>.g.d.h.c.</var> circa eaÅ¿dem diametros, <lb/><reg norm="tunc" type="context">tÅ©c</reg> proportio <var>.a.b.</var> ad <var>.d.c.</var> <reg norm="dimidium" type="context">dimidiÅ©</reg> erit <lb/>proportionis circulorum ex .2. 12. Eu-<lb/>clid. </s> - <s xml:id="s3735" xml:space="preserve">Å¿ed proportio <var>.a.b.</var> ad <var>.d.c.</var> æqualis <lb/>eÅ¿t proportioni maioris circuli ad Elli <lb/>pÅ¿im .ex .5. Archimedis in lib. de conoÂ<lb/>idalibus, quapropter proportio Elli-<lb/>pÅ¿is ad minorem circulum altera me-<lb/>dietas erit totius proportionis circulo-<lb/>rum, hoc eÅ¿t maioris ad minorem, qua <lb/>re EllipÅ¿is media proportionalis erit <lb/>inter eos circulos. </s> - <s xml:id="s3736" xml:space="preserve">Nunc verò cum <lb/>ex Archimede repertÄ™ fuerint duæ fi-<lb/>guræ rectilineæ æquales duobus circu <lb/>lis iam dictis, & inter has, reperta fue <lb/>rit alia media proportionalis propoÅ¿i-<lb/>tum obtinebimus.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head453" style="it" xml:space="preserve">Spheroidem propoÅ¿itam cubare.</head> - <head xml:id="head454" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3737" xml:space="preserve">PRopoÅ¿ita Å¿phæroides erit, aut prolata, aut oblonga, Å¿it prius prolata, <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.a.b.</var> <lb/>diameter circuli, qui eam per æqualia Å¿ecat, circa quam <var>.a.b.</var> vt circa axem in-<lb/>telligatur Å¿phæroides oblonga, cuius Å¿piſſitudo Å¿it <var>.d.c.</var> axis prolatæ, cogitemus <reg norm="nunc" type="context">nÅ©c</reg> <lb/>duas Å¿phæras <var>.a.e.b.f.</var> et <var>.g.d.h.c.</var> circa dictos axes. </s> - <s xml:id="s3738" xml:space="preserve">Vnde quatuor corpora habebi-<lb/>mus, hoc eÅ¿t duas Å¿phæras, & duas Å¿phæroides, quas probabo continuas proportio-<lb/>nales inuicem eſſe.</s> - </p> - <p> - <s xml:id="s3739" xml:space="preserve">ConÅ¿ideremus igitur duos conos rectos, quorum <var>.a.b.</var> diameter Å¿it eorum baÅ¿ium, <lb/>altitudo autem maioris, æqualis Å¿it Å¿emidiametro majori, hoc eÅ¿t medietati <var>.a.b.</var> al- - <pb o="303" rhead="EPISTOLAE." n="315" file="0315" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0315"/> - titudo verò minoris, æqualis Å¿it Å¿emidiametro minori, hoc eÅ¿t medietati <var>.d.c.</var> vnde <lb/>habebimus proportionem coni maioris ad conum minorem, <reg norm="eandem" type="context">eãdem</reg> quæ eÅ¿t diame <lb/>tri maioris ad diametrum minorem, quod ex .2. parte .11. duodecimi Eucli. </s> - <s xml:id="s3740" xml:space="preserve">nec non <lb/>ex .9. eiuÅ¿dem manifeÅ¿tum eÅ¿t, Å¿ed conus minor, eÅ¿t quarta pars Å¿phæroidis prolatæ <lb/>ex .29. </s> - <s xml:id="s3741" xml:space="preserve">Archimedis in lib. de conoidalibus, & conus maior, eÅ¿t etiam quarta pars <lb/>Å¿phæræ, ex .32. primi lib. de Å¿phæra, & cyllindro, </s> - <s xml:id="s3742" xml:space="preserve">quare ex communi Å¿cientia, <reg norm="eadem" type="context">eadẽ</reg> <lb/>proportio erit Å¿phæræ maioris ad Å¿phæroidem prolatam, quæ <var>.a.b.</var> ad <var>.d.c.</var> Å¿ed pro-<lb/>portio <var>.a.b.</var> ad <var>.d.c.</var> eÅ¿t tertia pars proportionis maioris Å¿phæræ ad <reg norm="minorem" type="context">minorẽ</reg>. </s> - <s xml:id="s3743" xml:space="preserve">ConÅ¿idere <lb/>mus <reg norm="nunc" type="context">nÅ©c</reg> alios duos conos rectos, vnius & <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> baÅ¿is, <reg norm="cuius" type="simple">cuiê°</reg> diameter Å¿it <var>.d.c.</var> Å¿ed altitu <lb/>do maioris, æqualis Å¿it Å¿emidiametroÅ¿phÄ™rÄ™ maioris, altitudo verò minoris, Å¿it æqua <lb/>lis Å¿emidiametro minoris Å¿phæræ, vnde ex dictis rationibus habebimus <reg norm="proportio- nem" type="context">proportio-<lb/>nẽ</reg> maioris coni ad <reg norm="minorem" type="context">minorẽ</reg>, vt quæ eÅ¿t <var>.o.b.</var> ad <var>.o.d.</var> hoc eÅ¿t vt <var>.a.b.</var> ad <var>.d.c.</var> & ex dictis <reg norm="pro poÅ¿itionibus" type="simple">ê“Â<lb/>poÅ¿itionibus</reg> ita Å¿e habebit Å¿phæroides oblonga ad Å¿phÄ™ram minorem vt <var>.a.b.</var> ad <var>.d.<lb/>c.</var> hoc eÅ¿t tertia pars proportionis Å¿phæræ maioris ad minorem. </s> - <s xml:id="s3744" xml:space="preserve">Quare proportio <lb/>Å¿phæroidis prolatæ ad oblongam, erit reliqua tertia pars proportionis maioris <reg norm="Å¿phae ræ" type="simple">Å¿phÄ™Â<lb/>ræ</reg> ad minorem. </s> - <s xml:id="s3745" xml:space="preserve">Quapropter hæc quatuor corpora continua proportionalia inui-<lb/>cem erunt.</s> - </p> - <p> - <s xml:id="s3746" xml:space="preserve">Nunc verò quærenda eÅ¿t inter <var>.a.b.</var> & Å¿uas duas tertias partes vna media pro por-<lb/>tionalis, quæ Å¿it <var>.K.</var> & ex Archimede, inuentum Å¿it quadratum Ä™quale circulo, cuius <lb/>Å¿it <var>.K.</var> diameter. </s> - <s xml:id="s3747" xml:space="preserve">Vnde proportio circuli (cuius <var>.a.b.</var> eÅ¿t diameter) ad circulum cu-<lb/>ius <var>.K.</var> eÅ¿t diameter, Å¿eÅ¿quialtera erit ex .2. 12. Eucli.</s> - </p> - <p> - <s xml:id="s3748" xml:space="preserve">Ducatur deinde quadratum lineæ <var>.K.</var> in lineam <var>.a.b.</var> & proueniet nobis cor-<lb/>pus quoddam, quod æquale erit Å¿phærÄ™ maiori, ex corellario .32. primi de Å¿phÄ™ra & <lb/>cyllindro, cuius corporis, latus cubus Å¿it <var>.m</var>.</s> - </p> - <p> - <s xml:id="s3749" xml:space="preserve">Idem facere oportebit mediante <var>.d.c.</var> minoris Å¿phærÄ™, cuius corporis cubica ra-<lb/>dix Å¿it <var>.n</var>.</s> - </p> - <p> - <s xml:id="s3750" xml:space="preserve">Nunc verò inter <var>.m.</var> et <var>.n.</var> inueniantur duÄ™ mediÄ™ proportionales <var>.s.t.</var> & ex <var>.s.</var> pro-<lb/>ducatur cubus, qui Ä™qualis erit Å¿phÄ™roidi prolatæ propoÅ¿iti, cubus vero <var>.t.</var> æqualis <lb/>erit Å¿phÄ™roidi oblongÄ™, cuius axis eſſet <var>.a.b</var>.</s> - </p> - <p> - <s xml:id="s3751" xml:space="preserve">Si autem Å¿phÄ™roides oblonga nobis propoÅ¿ita fuiſſet, eodem methodo Å¿oluere-<lb/>tur problema.</s> - </p> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head455" style="it" xml:space="preserve">Quadratum circulis mediantibus deÅ¿ignare.</head> - <head xml:id="head456" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3752" xml:space="preserve">MOdus autem conficiendi quadratum ex circulis Å¿upra datam lineam, vt Do-<lb/>minum GaÅ¿parem docui, facillimus eÅ¿t.</s> - </p> - <p> - <s xml:id="s3753" xml:space="preserve">Sit enim linea <var>.b.a.</var> 46. propoÅ¿itionis primi Euclidis, <reg norm="poÅ¿itoque" type="simple">poÅ¿itoqÌ;</reg> pede immobli circi-<lb/>ni in puncto <var>.a.</var> Å¿ecundum quantitatem lineæ <var>.a.b.</var> propoÅ¿itÄ™ fiat circulus, Å¿imiliter cir-<lb/>ca punctum <var>.b.</var> alius circulus eiuÅ¿dem magnitudinis, </s> - <s xml:id="s3754" xml:space="preserve">erecta deinde Å¿ola <var>.a.c.</var> perpendi <lb/>culari ipÅ¿i <var>.a.b.</var> ex puncto <var>.a.</var> ipÅ¿a Å¿ecabitur à circunferentia circuli. cuius centrum eÅ¿t <var>.<lb/>a.</var> in puncto <var>.c.</var> vnde <var>.a.c.</var> æqualis erit <var>.a.b.</var> poÅ¿ito demum pede immobili ipÅ¿ius circi <lb/>ni in puncto <var>.c.</var> Å¿ecundum longitudinem ipÅ¿ius <var>.c.a.</var> fiat alius circulus, qui æqualis erit <lb/>reliquis duobus circulis cum eorum Å¿emidiametri æquales Å¿int, & hic vltimo factus <lb/>Å¿ecabit circulum, cuius <reg norm="centrum" type="context">centrÅ©</reg> eÅ¿t <var>.b.</var> in <reg norm="puncto" type="context">pÅ©cto</reg> <var>.d.</var> à quo cum ductæ fuerint <var>.d.c.</var> et <var>.d.b.</var> - <pb o="304" rhead="IO. BAPT. BENED." n="316" file="0316" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0316"/> - rectè habebimus quod volumus. </s> - <s xml:id="s3755" xml:space="preserve">nam omnia latera Å¿unt inuicem Ä™qualia ex condi-<lb/>ti onibus circuli, angulus autem <var>.a.</var> rectus effectus fuit, </s> - <s xml:id="s3756" xml:space="preserve">tunc Å¿i imaginatione cogita-<lb/>ta fuerit diameter <var>.b.c.</var> ex .8. primi, concludemus angulum <var>.d.</var> eſſe rectum </s> - <s xml:id="s3757" xml:space="preserve">deinde ex .5 <lb/>et .32. eiuÅ¿dem concludemus etiam reliquos angulos rectos eſſe.</s> - </p> - <p> - <s xml:id="s3758" xml:space="preserve">Circa verò id quod mihi Å¿cripÅ¿iÅ¿ti de igne perpetuo putans nugas eſſe, quod Ro-<lb/>mæ inuentæ fuerint lucernÄ™ ardentes in Å¿epulchris antiquorum. </s> - <s xml:id="s3759" xml:space="preserve">Ego quid em mi-<lb/>nimè puto eas nugas eſſe, propterea quod tales lucernas non vnus tantum aut duo <lb/>viderint, Å¿ed multi homines fide digniſſimi. </s> - <s xml:id="s3760" xml:space="preserve">PrÄ™tera cum aisid nulla ratione poſſe <lb/>fieri. </s> - <s xml:id="s3761" xml:space="preserve">ReÅ¿pondeo quod maxima ratione poſſibile eſſe puto, quam quidem ra-<lb/>tionem ita eſſe oportet, quod primum lucerna Å¿it perfectè circuncluÅ¿a, vt <lb/>materia in ea conÅ¿tituta nullo modo exire poſſit, </s> - <s xml:id="s3762" xml:space="preserve">deinde quod materia inflamabilis <lb/>talis Å¿it, vt excrementum fuliginoÅ¿um ex flamma tranÅ¿miſſum, tangendo Å¿uperfi-<lb/>ciem deuexam ipÅ¿ius lucernæ, aptum Å¿it in <reg norm="priÅ¿tinum" type="context">priÅ¿tinÅ©</reg> <reg norm="humorem" type="context">humorẽ</reg> conge<unsure/>lari, Å¿iue transfor-<lb/>mari, vnde materia prima per tres formas perpetuò tranÅ¿ibit, hoc eÅ¿t per humorem, <lb/>Å¿iue oleum tale, vt diximus, per ignem, Å¿eu flammam, & per vaporem, Å¿eu exhala-<lb/>tionem fuliginoÅ¿am aptam condenÅ¿ari, atque in priorem humorem illicò reuerti.</s> - </p> - </div> - </div> - <div type="section" level="3" n="22"> - <div type="letter" level="4" n="1"> - <head xml:id="head457" xml:space="preserve">DE DIVISIONE TRIANGVLI SECVNDVM <lb/>propoÅ¿itam proportionem.</head> - <head xml:id="head458" style="it" xml:space="preserve">Michaeli Angelo MuciaÅ¿co.</head> - <p> - <s xml:id="s3763" xml:space="preserve"><emph style="sc">QVod</emph> mihi proponis, tale eÅ¿t, vt Å¿cilicet tibi modum Å¿cribam diuidendi <lb/>triangulum propoÅ¿itum Å¿ecundum datam proportionem à linea tranÅ¿eun <lb/>te per punctum notatum extra triangulum.</s> - </p> - <p> - <s xml:id="s3764" xml:space="preserve">Tr<unsure/><reg norm="iangulum" type="context">iangulÅ©</reg> <reg norm="igitur" type="simple">igit̃</reg> à te mihi propoÅ¿itum Å¿it <var>.n.o.u.</var> conÅ¿idero <reg norm="primum" type="context">primÅ©</reg> quod Å¿i <lb/>quis ipÅ¿um diuiÅ¿erit in duas partes mediante <var>.e.s.</var> parallela ad <var>.n.u.</var> ea proportione, <lb/>quam mihi proponis. </s> - <s xml:id="s3765" xml:space="preserve">deinde inuenerit in dicta <var>.e.s.</var> punctum <var>.r.</var> per quod tranÅ¿iens <lb/>alia linea à puncto <var>.p.</var> propoÅ¿ito, ita quod efficiat duo triangula <var>.m.r.e.</var> et <var>.r.s.x.</var> inui-<lb/>cem æqualia, problema Å¿olutum erit. <lb/> - <anchor type="figure" xlink:label="fig-0316-01a" xlink:href="fig-0316-01"/> - eo quod triangulum <var>.m.o.x.</var> æquale <lb/>eſſet triangulo <var>.e.o.s.</var> & quadrilate-<lb/>rum reÅ¿iduum <var>.m.n.u.x.</var> etiam Ä™quale <lb/>eſſet quadrilatero <var>.e.n.u.s</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0316-01" xlink:href="fig-0316-01a"> - <image file="0316-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0316-01"/> - </figure> - </div> - <p> - <s xml:id="s3766" xml:space="preserve">Sed dum punctum <var>.r.</var> uenarer, alia <lb/> - <anchor type="figure" xlink:label="fig-0316-02a" xlink:href="fig-0316-02"/> - via mihi in mentem venit, cognoui <lb/>igitur quod quum propoÅ¿itum expe-<lb/>ditum fuiſſer, hoc eÅ¿t, <reg norm="quod" type="simple">ê™</reg> Å¿i à puncto p. <lb/>protracta eſſet linea <var>.p.m.</var> quÄ™ trian-<lb/>gulum <var>.n.o.u.</var> in duas partes inuicem <lb/>ita proportionatas diuiÅ¿iſſet, vt Å¿e ha <lb/>bet <var>.A.</var> et <var>.B.</var> ita Å¿e haberet <reg norm="productum" type="context">productÅ©</reg> <lb/> - <anchor type="figure" xlink:label="fig-0316-03a" xlink:href="fig-0316-03"/> - <var>n.o.</var> in <var>.o.u.</var> ad productum <var>.m.o.</var> in <var>.o.<lb/>x.</var> vt trianguli <var>.n.o.u.</var> ad triangulum <lb/><var>m.o.x.</var> quod quidem non eÅ¿t diffi-<lb/>cile Å¿peculari, ex methodo .24. Å¿exti, - <pb o="305" rhead="EPISTOLAE." n="317" file="0317" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0317"/> - eo quod tam proportio producti <var>.n.o.</var> in <var>.o.u.</var> ad productum <var>.m.o.</var> in <var>.o.x.</var> quam pro-<lb/>portio trianguli <var>.n.o.u.</var> ad triangulum <var>.m.o.x.</var> componitur ex proportione <var>.u.o.</var> ad <var>.o.<lb/>x.</var> & ex proportion <var>e.n.o.</var> ad <var>.m.o.</var> vnde proportio dictorum productorum nobis co-<lb/>gnita erit, eo quod cum nobis cognita Å¿it proportio <var>.A.</var> ad <var>.B.</var> vt data, cognita etiam <lb/>nobis erit coniuncta, hoceÅ¿t <var>.A.B.</var> ad <var>.B</var>. </s> - <s xml:id="s3767" xml:space="preserve">& propterea ea quæ trianguli <var>.n.o.u.</var> ad <reg norm="trian- gulum" type="context">triã-<lb/>gulum</reg> <var>.m.o.x.</var> & Å¿imiliter productorum. </s> - <s xml:id="s3768" xml:space="preserve">Quæſiui poÅ¿tea modum inueniendi duas <lb/>dictas lineas <var>.m.o.</var> et <var>.o.x.</var> & cognoui quod Å¿i producta fuerit <var>.p.i.</var> æquidiÅ¿tans li-<lb/>neæ <var>.o.x.</var> <reg norm="producendoque" type="simple">producendoqÌ</reg> <var>.o.n.</var> quouÅ¿que cum <var>.p.i.</var> Å¿e interÅ¿ecarent in puncto <var>.i.</var> inuenien <lb/>do poÅ¿tea lineam quandam, quæ ducta cum <var>.p.i.</var> efficeret rectangulum æquale rectan <lb/>gulo cognito quod ex <var>.m.o.</var> in <var>.o.x.</var> poteÅ¿t fieri, quod cognitum dico, eo quod nobis <lb/>cognita eÅ¿t proportio data, & rectangulum etiam <var>.n.o.</var> in <var>.o.u.</var> deinde Å¿ecando ab <var>.o.<lb/>n.</var> partem æqualem lineæ iam inuentæ, quæ Å¿it <var>.o.t</var>. </s> - <s xml:id="s3769" xml:space="preserve">Inueniendo poÅ¿tea, ex .28. Å¿exti <lb/>lineam <var>.o.m.</var> cuius productum in <var>.m.t.</var> æquale Å¿it producto <var>.t.o.</var> in <var>.o.i.</var> vnde ex .15. eiuÅ¿ <lb/>dem proportio <var>.o.i.</var> ad <var>.m.o.</var> eadem eſſet, quæ <var>.m.t.</var> ad <var>.o.t.</var> & componendo, ita Å¿e ha-<lb/>beret <var>.m.i.</var> ad <var>.m.o.</var> vt <var>.m.o.</var> ad <var>.o.t.</var> Å¿ed ex .4. Å¿exti, ita eſſet <var>.p.i.</var> ad <var>.o.x.</var> vt <var>.m.i.</var> ad <var>.m.o</var>. <lb/></s> - <s xml:id="s3770" xml:space="preserve">quare ex .11. quinti, ita eſſet <var>.p.i.</var> ad <var>.o.x.</var> vt <var>.m.o.</var> ad <var>.o.t.</var> vnde ex .15. Å¿exti productum <var>.<lb/>o.x.</var> in <var>.m.o.</var> æquale eſſet producto. p, i. in <var>.o.t.</var> & Å¿ic haberemus intentum.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0316-02" xlink:href="fig-0316-02a"> - <image file="0316-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0316-02"/> - </figure> - <figure xlink:label="fig-0316-03" xlink:href="fig-0316-03a"> - <image file="0316-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0316-03"/> - </figure> - </div> - <p> - <s xml:id="s3771" xml:space="preserve">Sed Å¿i punctum <var>.m.</var> caderet in punctum <var>.n.</var> idem eſſet, Å¿i vorò punctum <var>.m.</var> tranÅ¿iret <lb/>n. oporteret nos facere hoc in latere <var>.n.u.</var> ipÅ¿um quærendo in linea <var>.n.u.</var> ducendo pri <lb/>mum lineam <var>.p.i.</var> <reg norm="æquidiÅ¿tantem" type="context">æquidiÅ¿tantẽ</reg> <var>.u.x.</var> & producendo <var>.u.n.</var> ad partem <var>.u.</var> proÅ¿equendo, <reg norm="quod" type="simple">ê™</reg> <lb/>Å¿uperius iam dictum eÅ¿t.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head459" style="it" xml:space="preserve">Idem facere de parallelogr ammo.</head> - <head xml:id="head460" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3772" xml:space="preserve">DAtum parallelogrammum in duas partes diuidere, Å¿ecundum aliquam datam <lb/>proportionem à linea tranÅ¿eunte per punctum propoÅ¿itum.</s> - </p> - <p> - <s xml:id="s3773" xml:space="preserve">Sit exempli gratia, datum parallelogrammum <var>.b.u.</var> datum verò punctum <var>.o.</var> extra <lb/>figuram, proportio autem ea Å¿it, quæ <var>.A.</var> ad <var>.B.</var> vt Å¿upra. </s> - <s xml:id="s3774" xml:space="preserve">Nunc diuidatur primò re-<lb/>ctangulum datum per æqualia, mediante linea <var>.r.c.</var> parallela ambobus lateribus <var>.b.x.</var> <lb/>et <var>.s.u.</var> quæ quidem linea diuidatur in puncto <var>.i.</var> ita quod eadem proportio Å¿it <var>.r.i.</var> ad <var>.<lb/>i.c.</var> vt <var>.A.</var> ad <var>.B.</var> protrahatur deinde à puncto <var>.o.</var> linea <var>.o.i.q.</var> quæ Å¿ecabit ambo duo la-<lb/>tera <var>.b.x.</var> vel <var>.s.u.</var> intra terminos eorum, vel tantum <var>.b.x.</var> reliquum verò extra termi-<lb/>nos <var>.s.u</var>.</s> - </p> - <p> - <s xml:id="s3775" xml:space="preserve">Nunc autem Å¿i intra dictos terminos tranÅ¿ibit, vt in prima figura videre potes, <lb/>problema Å¿olutum erit, eo quod <lb/> - <anchor type="figure" xlink:label="fig-0317-01a" xlink:href="fig-0317-01"/> - Å¿i à puncto <var>.i.</var> protracta fuerit <var>.p.<lb/>d.</var> pa rallela ad <var>.u.x.</var> habebimus <lb/>ex prima Å¿exti eandem propor-<lb/>tionem <var>.s.d.</var> ad <var>.p.x.</var> ut <var>.r.i.</var> ad <var>.i.c.</var> <lb/>hoc eÅ¿t vt <var>.A.</var> ad <var>.B.</var> Å¿ed <reg norm="triangulus" type="context">triãgulus</reg> <lb/><var>i.e.d.</var> æqualis eÅ¿t triangulo <var>.i.q.p.</var> <lb/>vt tibi facilè patebit, vnde qua-<lb/>drilaterum <var>.e.q.u.x.</var> æquale erit <lb/>quadrilatero <var>.d.u.</var> ex communi - <pb o="306" rhead="IO. BAPT. BENED." n="318" file="0318" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0318"/> - Å¿cientia. </s> - <s xml:id="s3776" xml:space="preserve">Quare ex .9. quinti, ita erit <var>.s.d.</var> ad dictum <var>.d.u.</var> vt ad quadrilaterum <var>.e.q.u.<lb/>x.</var> hoc eÅ¿t vt <var>.A.</var> ad <var>.B.</var> ex .11. eiuÅ¿dem.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0317-01" xlink:href="fig-0317-01a"> - <image file="0317-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0317-01"/> - </figure> - </div> - <p> - <s xml:id="s3777" xml:space="preserve">Sed Å¿i punctum <var>.q.</var> fuerit extra ut in .2. figura videre eÅ¿t. </s> - <s xml:id="s3778" xml:space="preserve">tunc manifeÅ¿tum erit, <reg norm="quod" type="simple">ê™</reg> <lb/>triangulus <var>.e.x.t.</var> maior erit pa-<lb/>rallelogrammo <var>.d.u.</var> per triangu <lb/> - <anchor type="figure" xlink:label="fig-0318-01a" xlink:href="fig-0318-01"/> - lum <var>.q.t.u.</var> cum triangulus <var>.q.i.p.</var> <lb/>æqualis triangulo <var>.d.i.e.</var> excedat <lb/>quadrilaterum <var>.i.t.u.p.</var> per trian <lb/>gulum <reg norm="dictum" type="context">dictÅ©</reg> <var>.q.t.u.</var> quapropter <lb/>cum diuiÅ¿us fuerit triangulus <var>.e.<lb/>x.t.</var> mediante linea <var>.o.n.K.</var> ita <reg norm="quod" type="simple">ê™</reg> <lb/><reg norm="quadrilaterum" type="context">quadrilaterÅ©</reg> <var>.e.n.K.t.</var> Å¿it æquale <lb/>triangulo <var>.q.t.u.</var> ex doctrina præ <lb/>cedenti, habebimus propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0318-01" xlink:href="fig-0318-01a"> - <image file="0318-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0318-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head461" style="it" xml:space="preserve">Idem de frusto trianguli.</head> - <head xml:id="head462" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3779" xml:space="preserve">SEd Å¿i quadrilaterum dictum eſſet fruÅ¿tum alicuius <reg norm="trianguli" type="context">triãguli</reg> ut in figura <var>.A.</var> hic Å¿ub <lb/>Å¿cripta videre eÅ¿t, Å¿uppoÅ¿ita, <var>b.d.</var> parallela ad <var>.u.p.</var> ita faciendum eſſet, ducendo <lb/>Å¿cilicet parallelam <var>.u.x.</var> ad <var>.b.p.</var> quæ producatur vÅ¿que ad concurÅ¿um cum <var>.b.d.</var> <lb/>in puncto <var>.x.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> proportio data inter <var>.t.a.</var> et <var>.a.e.</var> quas duas lineas cogitemus inuicem <lb/>directè coniunctas, </s> - <s xml:id="s3780" xml:space="preserve">tunc diuidatur tota <var>.t.e.</var> <lb/> - <anchor type="figure" xlink:label="fig-0318-02a" xlink:href="fig-0318-02"/> - in puncto <var>.i.</var> ita vt <var>.t.i.</var> ad <var>.i.e.</var> Å¿it vt quadrilate <lb/>ri <var>.p.d.</var> ad trigonum <var>.u.d.x</var>. </s> - <s xml:id="s3781" xml:space="preserve">deinde diuidatur <lb/><var>t.i.</var> in puncto r. tali modo vt <var>.t.r.</var> ad <var>.r.i.</var> Å¿e ha-<lb/>beat vt <var>.t.a.</var> ad <var>.a.e.</var> quo facto ex doctrina <reg norm="prae cedenti" type="simple">prÄ™Â<lb/>cedenti</reg> diuidatur totum parallelogram--<lb/>mum <var>.p.x.</var> mediante linea <var>.o.q.</var> Å¿ecundum <lb/>quod Å¿e habet <var>.t.r.</var> ad <var>.r.e</var>. </s> - <s xml:id="s3782" xml:space="preserve">Atque ita Å¿olu-<lb/>tum erit problema, vt exte ipÅ¿o ratiotina-<lb/>ri facile potes.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0318-02" xlink:href="fig-0318-02a"> - <image file="0318-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0318-02"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="4"> - <head xml:id="head463" style="it" xml:space="preserve">Fdem de quadrilatero in genere.</head> - <head xml:id="head464" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3783" xml:space="preserve">SEd Å¿i nullum latus parallelum reliquo erit, ita faciendum erit. </s> - <s xml:id="s3784" xml:space="preserve">Å¿i Å¿it tale quadrila <lb/>terum <var>.b.d.u.p.</var> oportet vt ipÅ¿um conuertamus in triangulum, producendo duo <lb/>quÄ™uis eius latera oppoÅ¿ita uÅ¿que ad interÅ¿ectionem ut pote <var>.u.p.</var> et <var>.d.b.</var> in puncto <var>.x.</var> <lb/>quo facto, Å¿upponemus <var>.o.</var> eſſe punctum datum, proportio verò data Å¿it <var>.t.r.</var> ad <var>.r.i.</var> ad <lb/>iungatur deinde <var>.i.e.</var> ad <var>.t.i.</var> ad quam <var>.e.i.</var> ipÅ¿a <var>.t.i.</var> Å¿e habeat vt quadrilaterum <var>.b.d.u.p.</var> - <pb o="307" rhead="EPISTOLAE." n="319" file="0319" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0319"/> - Å¿e habet ad triangulum <var>b.p.x.</var> ducatur poÅ¿tea <var>.o.q.</var> quæ diuidat totale triangulum <var>.d.<lb/>u.x.</var> in duas partes inuicem ita proportionatas, ut Å¿e habent <var>t.r.</var> et <var>.r.e.</var> quæ quidem <lb/>partes Å¿int <var>.c.d.u.q.</var> et <var>.c.q.x.</var> ut in primo problemate tibi monÅ¿traui, & habebis pro-<lb/>poÅ¿itum, dato quod punctum <var>.c.</var> Å¿it inter <lb/>b. et <var>.d</var>.</s> - </p> - <p> - <s xml:id="s3785" xml:space="preserve">Sed Å¿i forte linea <var>.o.q.</var> Å¿ecabit <var>.b.x.</var> hoc <lb/> - <anchor type="figure" xlink:label="fig-0319-01a" xlink:href="fig-0319-01"/> - eÅ¿t Å¿i punctum <var>.c.</var> eſſet inter <var>.b.</var> et <var>.x.</var> mani-<lb/>feÅ¿tum eÅ¿t, quod <var>.c.q.</var> Å¿ecaret <var>.b.p.</var> in pun-<lb/>cto <var>.y.</var> vnde in tali caÅ¿u, alio modo ope-<lb/>randum eſſet, hoc eÅ¿t ducendo <var>.b.u.</var> quæ <lb/>diuideret quadrilaterum in duo triangu-<lb/>la, & ut Å¿e haberet triangulum <var>.b.d.u.</var> ad <lb/>triangulum <var>.b.p.u.</var> vellem vt ita Å¿ecaretur <lb/><var>t.i.</var> in puncto <var>.n.</var> vt ita Å¿e haberet <var>.t.n.</var> ad <var>.n.<lb/>i.</var> ut dictum eÅ¿t de iÅ¿tis duobus triangulis, <lb/></s> - <s xml:id="s3786" xml:space="preserve">deinde prout Å¿e habet <var>.n.r.</var> ad <var>.r.i.</var> ita Å¿eca-<lb/>res triangulum <var>.b.p.u.</var> mediante linea <var>.o.<lb/>K.</var> ex doctrina primi problematis, & ita haberes propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0319-01" xlink:href="fig-0319-01a"> - <image file="0319-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0319-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="5"> - <head xml:id="head465" style="it" xml:space="preserve">Idem de Pentagono, Exagono, & de reliquis.</head> - <head xml:id="head466" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3787" xml:space="preserve">PEntagonum, Å¿eu hexagonum, vel alias quaÅ¿uis multilateras figuras propoÅ¿itas its <lb/>diuidere, vt dictum eÅ¿t de trilateris, & quadrilateris.</s> - </p> - <p> - <s xml:id="s3788" xml:space="preserve">Sit exempli gratia pentagonus <var>.a.d.u.p.b.</var> quem Å¿ecare volumus <reg norm="mediante" type="context">mediãte</reg> linea <var>.o.<lb/>q.</var> in duas partes inuicem Å¿e habentes, vt Å¿e habent <var>.t.r.</var> et <var>.r.i.</var> oportet igitur ut ipÅ¿um <lb/>pentagonum reducas ad quadrilaterum <var>.x.a.d.u.</var> quod diuidatur Å¿ecundum præce-<lb/>dentem doctrinam, vt Å¿e habet <var>.t.r.</var> ad <var>.r.e.</var> <lb/>vnde Å¿i punctum <var>.q.</var> incidit inter <var>.p.</var> et <var>.u</var>. </s> - <s xml:id="s3789" xml:space="preserve">tunc <lb/>habebis propoÅ¿itum, Å¿i verò incidet inter <var>.<lb/> - <anchor type="figure" xlink:label="fig-0319-02a" xlink:href="fig-0319-02"/> - p.</var> et <var>.x.</var> clarum erit quod linea <var>.o.q.</var> Å¿ecabit <lb/>latus <var>.p.b.</var> trianguli <var>.b.x.p.</var> in puncto <var>.y.</var> qua-<lb/>propter duces lineam <var>.a.p.</var> vt claudat trian-<lb/>gulum <var>.a.b.p.</var> <reg norm="diuidaturque" type="simple">diuidaturqÌ;</reg> <var>.t.i.</var> in puncto <var>.n.</var> ita <lb/>vt <var>.t.n.</var> ad <var>.n.i.</var> Å¿e habeat, vt quadrilaterum. a<unsure/> <var>.<lb/>d.u.p.</var> ad <reg norm="triangulum" type="context">triãgulum</reg> <var>.a.b.p</var>. </s> - <s xml:id="s3790" xml:space="preserve">deinde <reg norm="hunc" type="context">hÅ©c</reg> trian <lb/>gulum <var>.a.b.p.</var> diuidas mediante linea <var>.o.K.</var> <lb/>vt <var>.n.r.</var> ad <var>.r.i.</var> ex doctrina primi problematis <lb/>& habebis propoÅ¿itum. </s> - <s xml:id="s3791" xml:space="preserve">Idem dico de hexa <lb/>gono, reducendo ipÅ¿um ad pentagonum, & <lb/>item de eptagono, ipÅ¿um reducendo ad exa <lb/>gonum, & idem infero de infinito ipÅ¿arum <lb/>Å¿uperficialium figurarum rectilinearum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0319-02" xlink:href="fig-0319-02a"> - <image file="0319-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0319-02"/> - </figure> - </div> - <pb o="308" rhead="IO. BAPT. BENED." n="320" file="0320" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0320"/> - </div> - <div type="letter" level="4" n="6"> - <head xml:id="head467" style="it" xml:space="preserve">De duobus triangulis equalibus inter lineas <lb/>inuicem inclinatas.</head> - <head xml:id="head468" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3792" xml:space="preserve">TV mihi vltimò proponis duas lineas rectas <var>.b.f.</var> et <var>.q.s.</var> in eadem Å¿uperficie pla-<lb/>na, non tamen inuicem æqu idiÅ¿tantes, proponis etiam <var>.n.t.</var> in eadem Å¿uperfi-<lb/>cie, quæ vnamquamque priorum Å¿ecat, proponis etiam lineam <var>.h.</var> tali conditione, <lb/>quod nulli dictarum Å¿it parallela, </s> - <s xml:id="s3793" xml:space="preserve">deinde Å¿cire cupis qua arte aliquis poſſet ducere <var>.<lb/>c.u.</var> parallelam ad <var>.h.</var> ita quod Å¿ecando <var>.n.t.</var> conÅ¿tituat duos triangulos <var>.n.o.u.</var> et <var>.t.o.e.</var> <lb/>inuicem æquales.</s> - </p> - <p> - <s xml:id="s3794" xml:space="preserve">Facita, producas primò duas primas lineas à parte, in qua inuicem inclinantur, <lb/>vÅ¿que ad concurÅ¿um in puncto <var>.i</var>. </s> - <s xml:id="s3795" xml:space="preserve">deinde à puncto <var>.n.</var> duces <var>.n.c.</var> <reg norm="parallelam" type="context">parallelã</reg> ad <var>.h.</var> poÅ¿tea <lb/>ex .25. Å¿exti Eucli. conÅ¿titues <reg norm="triangulum" type="context">triãgulum</reg> <var>.i.u.e.</var> Å¿imile triangulo <var>.i.c.n.</var> æquale tamen <lb/>triangulo <var>.i.t.n.</var> & Å¿olutum erit problema. <lb/></s> - <s xml:id="s3796" xml:space="preserve">VelÅ¿ic, inuenies <var>.i.e.</var> mediam <lb/> - <anchor type="figure" xlink:label="fig-0320-01a" xlink:href="fig-0320-01"/> - proportionalem inter <var>.i.c.</var> & <lb/><var>i.t.</var> duces poÅ¿tea <var>.e.u.</var> paralle-<lb/>lam lineÄ™ <var>.h.</var> vel <var>.c.n.</var> quod <reg norm="idem" type="context">idẽ</reg> <lb/>erit ex .30. primi Eucli. </s> - <s xml:id="s3797" xml:space="preserve">& Å¿o-<lb/>lutum erit problema.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0320-01" xlink:href="fig-0320-01a"> - <image file="0320-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0320-01"/> - </figure> - </div> - <p> - <s xml:id="s3798" xml:space="preserve">Nam ex .17. Å¿exti eadem <lb/>proportio erittrianguli <var>.i.c.<lb/>n.</var> ad triangulum <var>.i.e.u.</var> ut <var>.i.c.</var> <lb/>ad <var>.i.t</var>. </s> - <s xml:id="s3799" xml:space="preserve">Quare ut trianguli <var>.i.<lb/>c.n.</var> ad <reg norm="triangulum" type="context">triangulÅ©</reg> <var>.i.t.n.</var> ex pri-<lb/>ma Å¿exti, et .11. quinti. </s> - <s xml:id="s3800" xml:space="preserve">Vnde <lb/>ex .9. eiuÅ¿dem <var>.i.e.u.</var> æqualis <lb/>erit <var>.i.t.n</var>. </s> - <s xml:id="s3801" xml:space="preserve">Quapropter <var>.o.n.u.</var> <lb/>æqualis etiam erit <var>.o.e.t</var>.</s> - </p> - </div> - </div> - <div type="section" level="3" n="23"> - <div type="letter" level="4" n="1"> - <head xml:id="head469" xml:space="preserve">SOLVTIONES NONNVLLAE QVOR VNDAM <lb/>problematum.</head> - <head xml:id="head470" style="it" xml:space="preserve">ThaodoÅ¿io à Raifestaim.</head> - <p> - <s xml:id="s3802" xml:space="preserve"><emph style="sc">DVritandvm</emph> profecto non eÅ¿t, quin quotidie hominibus Å¿tudioÅ¿is ali-<lb/>quid noui deÅ¿it, quemadmodum, quod tibi nunc occurrit, mihi non-<lb/>nunquam accidit, hoc eÅ¿t inuenire orizontem, cui aliqua propoÅ¿ita Å¿tel <lb/>la oriatur cum gradu ipÅ¿ius longitudinis. </s> - <s xml:id="s3803" xml:space="preserve">pro <reg norm="cuius" type="simple">cuiê°</reg> rei operatione te prius <lb/>Å¿cire oportebit vtrum Å¿tella in Å¿ignis aÅ¿cendentibus, vel deÅ¿cendentibus reperiatur, <lb/>hoc eÅ¿t in Å¿ignis, quÄ™ à Capricorno ad Cancrum procedunt, vel in illis, quÄ™ à Can-<lb/>cro ad Capricornum numerantur, </s> - <s xml:id="s3804" xml:space="preserve">propterea quod Å¿i in Å¿ignis aÅ¿cendentibus inue-<lb/>nitur, Å¿ciendum eÅ¿t, quod Å¿upra talem orizontem polus mundi auÅ¿tralis attollitur, <lb/>Å¿ed Å¿i in Å¿ignis deÅ¿cendentibus reperitur, </s> - <s xml:id="s3805" xml:space="preserve">tunc polus borealis eleuatur Å¿upra dictum - <pb o="309" rhead="EPISTOL AE." n="321" file="0321" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0321"/> - orizontem, vt exempli gratia, canicula quæ à Græcis <reg norm="Prochyon" type="context">Prochyõ</reg> vocatur, reperitur in <lb/>24. minuto vigeÅ¿imi gradus Cancri, quapropter polus borealis eleuatur Å¿upra ori-<lb/>zontem, cui ipÅ¿a oritur cum eodem gradu, & minuto eclipticÄ™ illius Å¿igni. </s> - <s xml:id="s3806" xml:space="preserve">Å¿ed <lb/>quia volumus etiam Å¿cire veram quantitatem arcus eleuationis huiuÅ¿modi poli, pro <lb/>pterea accipiemus in tabula generali Monteregij numerum qui vocatur radix aÅ¿cen <lb/>Å¿ionum, èregione numeri longitudinis ipÅ¿ius Å¿tellæ, qui quidem numerus in præſen <lb/>ti exemplo erit gra .107. cum minutis .53. qui eÅ¿t <reg norm="cuiuÅ¿dam" type="context">cuiuÅ¿dã</reg> arcus æquatoris, qui inci-<lb/>pit in principio Arietis, & in circulo latitudinis deÅ¿init, hoc eÅ¿t ab orizonte quæſi-<lb/>to, ita quod talis numerus erit aÅ¿cenÅ¿io obliqua huiuÅ¿modi puncti eclipticæ illi ori-<lb/>zonti, qua aÅ¿cenÅ¿ione mediante, Å¿imul cum gradu, & minuto longitudinis in tabulis <lb/>aÅ¿cenÅ¿ionum obliquarum, inueniemus gradum, & minutum altitudinis pollaris, <reg norm="quod" type="wordlist">qÄ‘</reg> <lb/>quærebatur, eodem ordine ac methodo, quo vtimur ad inueniendum in tabulis po-<lb/>Å¿itionum, polum circuli poÅ¿itionis alicuius aÅ¿tri, mediante declinatione & diÅ¿tantia <lb/>à meridiano ciuÅ¿dem aÅ¿tri, vt Å¿cis. </s> - <s xml:id="s3807" xml:space="preserve">Vnde in præſenti exemplo eleuatio poli borea <lb/>lis Å¿upra talem orizontem erit gra .7. cum minutis .45.</s> - </p> - <p> - <s xml:id="s3808" xml:space="preserve">Sed Å¿i Å¿tella fuerit in medietate aÅ¿cendente, tunc certi erimus polum auÅ¿tralem Å¿u <lb/>per dictum orizontem attolli, nam idem eÅ¿t quærere altitudinem vnius <reg norm="polorum" type="context">polorÅ©</reg> mun <lb/>di à tali orizonte, quod diÅ¿tantiam dicti poli à circulo Å¿ecundum quem longitudo <lb/>terminatur, qui etiam latitudinis dicitur, eo quod tunc temporis talis circulus vnus <lb/>& idem eÅ¿t cum orizonte. </s> - <s xml:id="s3809" xml:space="preserve">Sumatur ergo exempli gratia Å¿tella, quæ in ore piÅ¿cis au <lb/>Å¿tralis eÅ¿t, quÄ™, pro nunc, Å¿it in gradu .20. cum minutis .14. </s> - <s xml:id="s3810" xml:space="preserve">Aquarij longitudinis, & <lb/>in gradu .23. cum nullo minuto meridianæ latitudinis. </s> - <s xml:id="s3811" xml:space="preserve">Tunc certi erimus orizon-<lb/>tem, cui dicta Å¿tella oritur cum eiuÅ¿modi puncto eclipticæ, depreſſum eſſe à parte <lb/>auÅ¿trali Å¿ub <reg norm="illoque" type="simple">illoqÌ</reg> polo, Å¿ed quia propoÅ¿itum eÅ¿t Å¿cire etiam quantitatem huiuÅ¿mo-<lb/>di depræſſionis, reperiemus in tabula generali gradum, & minutum æquatoris, cor-<lb/>reÅ¿pondentem tali puncto longitudinis à circulo latitudinis terminato, qui quidem <lb/>numerus in præſenti exemplo erit gra .317. cum minutis .46. & hic numerus, vt dixi <lb/>mus eÅ¿t aÅ¿cen. obli. ad dictum orizontem, vbi polus auÅ¿tralis attollitur, & deÅ¿cenÅ¿io <lb/>obliqua, vbi polus borealis eleuatur. </s> - <s xml:id="s3812" xml:space="preserve">Quapropter Å¿i à .317. gradibus cum minutis <lb/>46. demptus fuerit dimidius circulus gra .180. remanebunt gra .137. cum minutis .46 <lb/>& punctus oppoÅ¿itus gradibus .20. cum .14. minutis Aquarij eÅ¿t in eodem numero <lb/>Leonis, & mediantibus iÅ¿tis gradibus .137. min .46. aÅ¿cenÅ¿ionis, cum grad .20. min <num value="14">.<lb/>14.</num> Leonis inueniemus eleuationem poli borealis ab orizonte in tabulis aÅ¿cenÅ¿io-<lb/>num obliquarum Monteregij, hoc eÅ¿t gra .17. min .53. & eadem altitudo erit poli <lb/>auÅ¿tralis Å¿upra orizontem à quo Fomahant cum dicto puncto eclipticæ oritur, in qua <lb/>longitudine dicta Å¿tella reperitur.</s> - </p> - <p> - <s xml:id="s3813" xml:space="preserve">Sed Å¿i propoÅ¿itus nobis fuerit punctus eclipticæ, cum quo aliqua Å¿tella oritura Å¿it, <lb/>& oporteat inuenire vbi, hoc eÅ¿t orizontem huiuÅ¿modi ortus, eleuatione poli arti <lb/>ci, Å¿eu antarctici Å¿upra talem orizontem, ita operandum eſſet.</s> - </p> - <pb o="310" rhead="IO. BAPT. BENED." n="322" file="0322" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0322"/> - <p> - <s xml:id="s3814" xml:space="preserve">Sit exempli gratia Å¿tella <var>.o.</var> ecli <lb/> - <anchor type="figure" xlink:label="fig-0322-01a" xlink:href="fig-0322-01"/> - ptica verò <var>.d.q.</var> æquator autem <var>.<lb/>p.q.</var> punctus verò eclipticæ, cum <lb/>quo Å¿tella oritura Å¿it <var>.e.</var> orizon de <lb/><reg norm="mum" type="context">mÅ©</reg> <var>.o.e.</var> vbi Å¿tella oriri poſſit <reg norm="cum" type="context">cÅ©</reg> <lb/>puncto <var>.e</var>. </s> - <s xml:id="s3815" xml:space="preserve">Nam cum Å¿tella pro-<lb/>ponitur, datur etiam eius longi-<lb/>tudo, nec non latitudo, </s> - <s xml:id="s3816" xml:space="preserve">quare ar-<lb/>cus <var>.a.q.</var> & arcus <var>.a.o.</var> nobis cogni <lb/>tus erit, cum Å¿upponatur arcus <var>.a.<lb/>o.</var> eſſe circuli latitudinis, et <var>.a.o.</var> <lb/>Iatitudo ipÅ¿ius Å¿tellæ, & angulus <lb/>a. rectus erit, & quia punctum <var>.e.</var> <lb/>datur, ergo arcus <var>.a.o.</var> & arcus <var>.a.<lb/>e.</var> Å¿imul <reg norm="cum" type="context">cÅ©</reg> angulo <var>.a.</var> recto cogni-<lb/>ti Å¿unt, vnde ex .11. primi lib. coÂ<lb/>pernici, angulus <var>.a.e.o.</var> cognoÅ¿ce-<lb/>tur, & angulus <var>.q.e.o.</var> Å¿imiliter, vt <lb/>reÅ¿iduum ex duobus rectis quo. e <lb/>mediante cum angulo <var>.q.</var> declina <lb/>tionis ab æquatore, <reg norm="medianteque" type="simple">medianteqÌ;</reg> <lb/>latere <var>.q.e.</var> cognito, cognitus quo <lb/>que nobis erit angnlus <var>.e.t.q.</var> ex <lb/>12. eiuÅ¿dem. </s> - <s xml:id="s3817" xml:space="preserve">qui quidem angulus <lb/>erit altitudinis æquatoris ab ori-<lb/>zonte quæſito, qui demptus à <lb/>90. gradibus, dabit altitudinem <lb/>poli ab orizonte quæſito.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0322-01" xlink:href="fig-0322-01a"> - <image file="0322-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0322-01"/> - </figure> - </div> - <p> - <s xml:id="s3818" xml:space="preserve">Inuenire poÅ¿tea gradum eclipticÄ™, cum quo Å¿tella data oriatur ad orizontem pro <lb/>poÅ¿itum, nullius eÅ¿t difficul@atis.</s> - </p> - <p> - <s xml:id="s3819" xml:space="preserve">Ponamus exempli gratia, aliquem Å¿cire velle gradum eclipticæ, cum quo canicu-<lb/>la oritur ad orizontem, cui polus boreus eleuatur per gradus .44. quæ canicula Å¿up-<lb/>ponatur habere gradus .19. cum min .24. </s> - <s xml:id="s3820" xml:space="preserve">Cancri longitudinis, & gra .16. min .10. lati-<lb/>tudinis meridianæ, quærere primum oportet eius declinationem ex doctrina .2. pro <lb/>blematis tabularum directionum Monteregij, quæ erit graduum .6. cum minutis .5. <lb/>Å¿eptentrionalis, </s> - <s xml:id="s3821" xml:space="preserve">deinde inuenire eius aÅ¿cenÅ¿ionem rectam ex doctrina .4. problema <lb/>tis eiuÅ¿dem Monteregij, quæ erit gra .108. mi .42. </s> - <s xml:id="s3822" xml:space="preserve">deinde <reg norm="mediante" type="context">mediãte</reg> declinatione iam <lb/>inuenta in tabulis differentiarum aÅ¿cenÅ¿ionalium Å¿ub polo .44. accipiemus differen-<lb/>tiam aÅ¿cenÅ¿ionum, qua differt recta ab obliqua, quæ in præſenti exemplo erit gra <num value="5">.<lb/>5.</num> min .55. quæ dempta ab aÅ¿cenÅ¿ione recta Å¿tellæ, vt præſens exemplum exigit, relin <lb/>quet nobis aÅ¿cenÅ¿ionem obliquam Å¿tellæ propoÅ¿itæ ad polum. gra .44. quæ erit gra <num value="102">.<lb/>102.</num> minu .47. qua mediante, in tabulis aÅ¿cenÅ¿ionum obliquarum poli .44. habebi-<lb/>mus gradum & minutum eclipticÄ™ cum quo Å¿tella oritur. </s> - <s xml:id="s3823" xml:space="preserve">quod in caÅ¿u noÅ¿tro erit <lb/>gra .1. min .8. Leonis, Å¿ed Å¿i tecum non fuerint tabulæ dictæ, potes eleganter omnia <lb/>hæc perficere via triangulorum Å¿phæricorum.</s> - </p> - <pb o="311" rhead="EPISTOL AE." n="323" file="0323" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0323"/> - <p> - <s xml:id="s3824" xml:space="preserve">Via triangul@rum<unsure/> idem facere.</s> - </p> - <p> - <s xml:id="s3825" xml:space="preserve">Sit <reg norm="exempli" type="context">exẽpli</reg> gr@tia <var>.q.b.</var> æquator, ecliptica verò <var>.q.a.</var> propoÅ¿itus <reg norm="autem" type="wordlist">aÅ©t</reg> orizon Å¿it <var>.o.c.d.</var> <lb/>& Å¿tella data Å¿it <var>.o.</var> in orientali parte orizontis, circulus verò <var>.a.o.</var> ille Å¿it, qui <reg norm="tranÅ¿iens" type="context">tranÅ¿iẽs</reg> <lb/>per polos eclipti<unsure/>cæ & per centrum Å¿tellæ terminat longitudinem ipÅ¿ius Å¿tellæ, & in <lb/>ipÅ¿o Å¿it eius latitudo. </s> - <s xml:id="s3826" xml:space="preserve">Nunc propoÅ¿itum Å¿it inuenire arcum <var>.d.q.</var> eo quod illicò Å¿cie <lb/>mus punctum <var>.d.</var> qua propter oportet nos prius cognoÅ¿cere arcum <var>.d.a.</var> qui demptus, <lb/>vel additus arcui <var>.a.q.</var> prius cognito ex Å¿uppoÅ¿ito (nam data nobis eÅ¿t longitudo, & <lb/>latitudo Å¿tellæ) dabit nobis <var>.d.q</var>.</s> - </p> - <p> - <s xml:id="s3827" xml:space="preserve">Cum igitur voluerimus arcum <var>.d.a.</var> cognoÅ¿cere, ita faciemus. </s> - <s xml:id="s3828" xml:space="preserve">nam <var>.q.a.</var> cognitus <lb/>nobis eÅ¿t ex Å¿uppoÅ¿ito vt dictum eÅ¿t. </s> - <s xml:id="s3829" xml:space="preserve">angulus quoque <var>.a.q.b.</var> qui declin tionis eclipti <lb/>cæ ab æquatore eÅ¿t, angulus deinde <var>.a.</var> (trianguli <var>.a.b.q.</var>) rectus eÅ¿t, ergo ex .4. primi <lb/>copernici cogn@tus nobis erit arcus <var>.<lb/>a.b.</var> nec non angulus <var>.a.b.q.</var> vnde an-<lb/> - <anchor type="figure" xlink:label="fig-0323-01a" xlink:href="fig-0323-01"/> - gulus <var>.o.b.e.</var> reÅ¿iduus ex duobus re-<lb/>ctis in duobus primis hic Å¿ubÅ¿criptis <lb/>figuris nobis itidem cognitus erit, <lb/>etiam & arcus <var>.b.o.</var> reÅ¿iduus Å¿iue com <lb/>poÅ¿itus ex ar cu <var>.a.o.</var> cognito ex Å¿up-<lb/>poÅ¿ito <reg norm="cum" type="context">cÅ©</reg> Å¿it arcus latitud nis ab ecli-<lb/>ptica. </s> - <s xml:id="s3830" xml:space="preserve">Tunc<unsure/> in triangulo <var>.o.b.e.</var> co-<lb/>gnoÅ¿cimus latus <var>.o.b.</var> & angulum <var>.o.<lb/>b.e.</var> nec non angulum <var>.b.e.o.</var> qui eÅ¿t <lb/>altitudinis æquatoris ab orizonte , <lb/></s> - <s xml:id="s3831" xml:space="preserve">quare ex .12. dicti lib. cognitus nobis <lb/>erit angulus <var>.b.o.e</var>. </s> - <s xml:id="s3832" xml:space="preserve">ConÅ¿ideremus <lb/>deinde triangulum <var>.a.o.d.</var> cuius angu <lb/>lus <var>.a.</var> rectus eÅ¿t, & angulus <var>.a.o.d.</var> <reg norm="cum" type="context">cÅ©</reg> <lb/>latere <var>.a.o.</var> etiam cognitus, vnde ex <lb/>Å¿upradicta .4. nobis cognitus erit ar-<lb/>cus <var>.a.d.</var> & conÅ¿equenter cognoÅ¿ce-<lb/>mus at cum <var>.d.q.</var> eius reÅ¿iduum, Å¿eu <lb/>compoÅ¿itum, quem quærebamus.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0323-01" xlink:href="fig-0323-01a"> - <image file="0323-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0323-01"/> - </figure> - </div> - <p> - <s xml:id="s3833" xml:space="preserve">Sed Å¿i hac via inuenire deÅ¿ideras, <lb/>cui orizonti propoÅ¿ita Å¿tella oriatur <lb/>cum eodem eclipticÄ™ puncto <var>.a.</var> lon-<lb/>gitudinis, hoc aliud nihil eſſet, quam <lb/>cognoÅ¿cere amplitudinem anguli <var>.a.<lb/>b.q.</var> eo quod talis orizon, idem cir-<lb/>culus eſſet <var>.a.b.o.</var> vnde cum quis Å¿ci-<lb/>ret vnum illorum angulorum quem <lb/>æquator efficit cum orizonte, reli-<lb/>qua illicò ei innoteÅ¿cent, Å¿ed dictus <lb/>angulus <var>.b.</var> iam diximus quomodo <lb/>cognoÅ¿catur.</s> - </p> - <pb o="312" rhead="IO. BAPT. BENED." n="324" file="0324" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0324"/> - <p> - <s xml:id="s3834" xml:space="preserve">Ponamus nos Å¿cire velle <reg norm="punctum" type="context">pÅ©ctum</reg> <lb/> - <anchor type="figure" xlink:label="fig-0324-01a" xlink:href="fig-0324-01"/> - eclipticÄ™, cum quo Procyon oritur <lb/>polo .44. o. dato, quod Å¿tella in gra <num value="19">.<lb/>19.</num> cum min .24. </s> - <s xml:id="s3835" xml:space="preserve">Cancri, reperiatur <lb/>diÅ¿tans ab ecliptica per gra .16. min <num value="10">.<lb/>10.</num> meridiem verÅ¿us. </s> - <s xml:id="s3836" xml:space="preserve">vnde <reg norm="arcus" type="simple">arcê°</reg> <var>.a.q.</var> <lb/>erit gra .70. min .36. <reg norm="eiusque" type="simple">eiusqÌ;</reg> Å¿inus par-<lb/>tium 94321. talium qualium totalis <lb/>eÅ¿t .100000. arcus verò <var>.a.o.</var> gra .16. <lb/>minut .10. Å¿inus erit 27845. angu-<lb/>lus autem <var>.a.q.e.</var> declinationis zodia <lb/>ci ab Ä™quatore grad .23. min .30. cu-<lb/>ius Å¿inus eÅ¿t .39875. </s> - <s xml:id="s3837" xml:space="preserve">Quare ex Å¿upra-<lb/>dictis rationibus angulus <var>.a.b.q.</var> erit <lb/>gra .82. mi .24. cuius <reg norm="Å¿inus" type="simple">Å¿inê°</reg> erit .99122. <lb/>arcus vero <var>.a.b.</var> gra .22. minu .17. cu-<lb/>ius Å¿inus erit .37945. </s> - <s xml:id="s3838" xml:space="preserve">angulus deinde <lb/><var>o.e.b.</var> trianguli <var>.o.e.b.</var> eÅ¿t gra .46. mi <var>.<lb/>o.</var> altitudinis æquatoris ab orizonte, <lb/>cuius Å¿inus eÅ¿t .71934. angulus Å¿imili <lb/>ter <var>.o.b.e.</var> medio coniuncti, quibus <lb/>rectus perÅ¿icitur, arcus etiam <var>.o.b.</var> no <lb/>tus eÅ¿t grad .6. min .7. cuius Å¿inus eÅ¿t <num value="10655">.<lb/>10655.</num> cum Å¿it differentia inter ar-<lb/>cus <var>.a.b.</var> et <var>.a.o.</var> cognitos.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0324-01" xlink:href="fig-0324-01a"> - <image file="0324-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0324-01"/> - </figure> - </div> - <p> - <s xml:id="s3839" xml:space="preserve">Quare ex .12. iam Å¿upradicta an-<lb/>gulus <var>.e.o.b.</var> hoc eÅ¿t <var>.a.o.d.</var> erit. grad <num value="36">.<lb/>36.</num> min .39. cuius Å¿inus erit .59693. <lb/></s> - <s xml:id="s3840" xml:space="preserve">deinde per .4. cognitus erit nobis an <lb/>gulus <var>.a.d.o.</var> gra .55. min .5. cuius <reg norm="Å¿inus" type="simple">Å¿inê°</reg> <lb/>erit .81998. arcus verò <var>.d.o.</var> gra .19. <lb/>min .51. cuius Å¿inus erit .33957 ar-<lb/>cus autem gra .11. min .42. cuius <reg norm="Å¿inus" type="simple">Å¿inê°</reg> <lb/>erit .20270. vnde <reg norm="arcus" type="simple">arcê°</reg> <var>.d.q.</var> reÅ¿iduus <lb/>ex <var>.a.q.</var> erit gra .58. min .54. complementum <reg norm="autem" type="wordlist">aÅ©t</reg> quartæ erit gra .31. mi .6. hoc eÅ¿t gra <num value="1">.<lb/>1.</num> Å¿igni Leonis. cum min .6.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head471" style="it" xml:space="preserve">De Å¿phæroide duplæ Å¿pbær æ propoÅ¿it æ.</head> - <head xml:id="head472" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3841" xml:space="preserve">MOdus autem inueniendi Å¿phæroidem ex dato axe, quod duplum Å¿it Å¿phæ-<lb/>ra propoÅ¿ita, talis eÅ¿t.</s> - </p> - <p> - <s xml:id="s3842" xml:space="preserve">Sit exempli gratia <var>.a.b.c.</var> Å¿phæra propoÅ¿ita. cuius Å¿emidia meter Å¿it <var>.o.c.</var> Å¿emiaxis <lb/>vero Å¿phæroidis Å¿it <var>.d.x.</var> cuius dimidium Å¿it <var>.u.x.</var> </s> - <s xml:id="s3843" xml:space="preserve">tunc ex doctrina .9. Å¿exti Euclid. inue <lb/>niatur <var>.g.h.</var> media proportionalis inter <var>.u.x.</var> et <var>.c.o.</var> </s> - <s xml:id="s3844" xml:space="preserve">deinde Å¿icut Å¿e habet <var>.u.x.</var> ad <var>.g.h.</var> - <pb o="313" rhead="EPISTOL AE." n="325" file="0325" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0325"/> - faciemus, quod diameter <var>.a.b.</var> dictæ Å¿phæræ ita Å¿e habcat ad <var>.e.f.</var> ex .10. Å¿exti, quæ <lb/><var>e.f.</var> erit reliqua axis quæſita. </s> - <s xml:id="s3845" xml:space="preserve">Vnde conÅ¿tituta cum fuerit ellipÅ¿is <var>.d.f.t.e.</var> ex dictis axi-<lb/>bus, </s> - <s xml:id="s3846" xml:space="preserve">deinde circumuertendo ellipÅ¿im circa maiorem axem, conÅ¿tituemus Å¿phæroi-<lb/>dem oblongam, Å¿i autem circumuertemus ipÅ¿am circa minorem axim conÅ¿tituemus <lb/>Å¿phæroidem prolatam.</s> - </p> - <p> - <s xml:id="s3847" xml:space="preserve">Quod autem talis operatio rationalis Å¿it, nulli dubium erit, que<unsure/>tieÅ¿cunque co-<lb/>gnoÅ¿cet conum rectum <var>.e.u.f.</var> æqualem eſſe cono recto <var>.a.c.b.</var> ex .2. parte .12. duodeci <lb/>mi Euclid. </s> - <s xml:id="s3848" xml:space="preserve">& quod cum conus <var>.e.d.f.</var> duplus Å¿it cono <var>.e.u.f.</var> ex lemmate collecto ab <lb/>11. duodecimi, conus <var>.e.d.f.</var> duplus exiÅ¿tit etiam cono <var>.a.c.b.</var> ex .7. quinti. </s> - <s xml:id="s3849" xml:space="preserve">Cum de-<lb/>inde ex .32. primi lib. de Å¿phæra, & cyllindro Å¿phæra <var>.a.c.b.q.</var> quadrupla Å¿it cono <var>.a.<lb/>c.b.</var> ipÅ¿a conÅ¿equenter dupla erit cono <var>.e.d.f.</var> Å¿ed ex .29. primi de conoidalibus, dimi <lb/>dium Å¿phæroidis <var>.e.d.f.t.</var> hoc eÅ¿t <var>.e.d.f.</var> dupla eÅ¿t cono <var>.e.d.f</var>. </s> - <s xml:id="s3850" xml:space="preserve">Quare talis medietas <lb/>æqualis eÅ¿t Å¿phæræ propoÅ¿itæ, totaqÌue Å¿phæroides dupla erit Å¿phærÄ™ datæ. </s> - <s xml:id="s3851" xml:space="preserve">Quod <lb/>autem dico de proportione dupla, idem infero de qualibet alia, Å¿umendo <var>.u.x.</var> ita pro <lb/>portionatam ad <var>.d.x.</var> vt proponitur.</s> - </p> - <p> - <s xml:id="s3852" xml:space="preserve">SphÄ™ram autem inuenire quæ dimidia Å¿it Å¿phæroidis propoÅ¿itæ nullius erit nego-<lb/>tij, quotieÅ¿cunque inuentus fuerit modus diuidendi vnam datam proportionem in <lb/>tres æquales partes.</s> - </p> - <p> - <s xml:id="s3853" xml:space="preserve">Sit propoÅ¿ita Å¿phæroides <var>.e.f.d.t.</var> cuius axes ex conÅ¿equentia dantur <var>.e.f.</var> et <var>.d.t.</var> quÄ™ <lb/>quidem Å¿phæroides Å¿it primo oblonga, et <var>.u.x.</var> Å¿it dimidium axis maioris. </s> - <s xml:id="s3854" xml:space="preserve">imagine-<lb/>tur etiam conus <var>.e.u.f.</var> vt Å¿upra. </s> - <s xml:id="s3855" xml:space="preserve">Imaginetur etiam factum eſſe, quod proponitur, hoc <lb/>eÅ¿t, vt Å¿phæra <var>.a.b.c.q.</var> Å¿it dimidium ipÅ¿ius Å¿phæroidis, vnde conus <var>.a.c.b.</var> æqualis erit <lb/>cono <var>.e.u.x.</var> vt Å¿upra demonÅ¿tratum eÅ¿t, & Å¿it <var>.g.h.</var> media proportionalis inter <var>.u.x.</var> et <lb/><var>o.c</var>. </s> - <s xml:id="s3856" xml:space="preserve">Iam viÅ¿um Å¿uperius fuit, quod eadem proportio erat ipÅ¿ius <var>.u.x.</var> ad <var>.g.h.</var> quæ <var>.a.b.</var> <lb/>ad <var>.e.f.</var> </s> - <s xml:id="s3857" xml:space="preserve">quare eadem quæ <var>.o.b.</var> ad <var>.e.x.</var> Å¿ed <var>.u.x.</var> et <var>.e.x.</var> dantur. </s> - <s xml:id="s3858" xml:space="preserve">inter quas <var>.g.h.</var> et <var>.o.b.</var> vel <lb/><var>o.c.</var> (nam <var>.o.c.</var> æqualis eÅ¿t <var>.o.b.</var>) mediÄ™ proportionales Å¿unt, eo quod cum <var>.g.h.</var> media <lb/>proportionalis Å¿it inter <var>.u.x.</var> et <var>.o.c.</var> & proportio <var>.o.b.</var> ad <var>.e.x.</var> æqualis Å¿it ei, quæ <var>.u.x.</var> <lb/>ad <var>.g.h.</var> hoc eÅ¿t ei quæ <var>.g.h.</var> ad <var>.o.c.</var> vel. ad <var>.o.b.</var> </s> - <s xml:id="s3859" xml:space="preserve">quare quotieÅ¿cunque inuentæ fuerint <var>.<lb/>g.h.</var> et <var>.o.c.</var> vel <var>.o.b.</var> mediæ proportionales inter <var>.d.x.</var> et <var>.x.e.</var> ipÅ¿a <var>.o.c.</var> vel <var>.o.b.</var> erit Å¿emi <lb/>diameter Å¿phæræ quæſitÄ™. </s> - <s xml:id="s3860" xml:space="preserve">eodem modo faciendum erit Å¿i Å¿phÄ™roides fuerit prolata.</s> - </p> - <figure position="here"> - <image file="0325-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0325-01"/> - </figure> - <pb o="314" rhead="IO. BABPT. BENED." n="326" file="0326" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0326"/> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head473" style="it" xml:space="preserve">Modus inueniendi duo triangula <reg norm="varijs" type="lig">varijs</reg> conditionibus <lb/>affecta.</head> - <head xml:id="head474" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3861" xml:space="preserve">QVod etiam quæris ita Å¿e habet, duo Å¿cilicet triangula inuenire, æqualia dua-<lb/>bus Å¿uperficiebus rectilineis propoÅ¿itis, quÄ™ quidem triangula Å¿int eiuÅ¿dem <lb/>alritudinis, & quod <reg norm="vnunquodque" type="context">vnÅ©quodque</reg> habeat angulum æqualem angulo pro<unsure/>poÅ¿ito, & <reg norm="quod" type="simple">ê™</reg> <lb/>alius angulus vnius, cum alio alterius, æquetur duobus rectis.</s> - </p> - <p> - <s xml:id="s3862" xml:space="preserve">Sint exempli gratia duæ propoÅ¿itæ Å¿uperflcies <var>.c.y.</var> duo verò anguli dati Å¿int <var>.r.s.</var> <lb/>cum voluerimus inuenire duo triangula (quæ Å¿int <var>.a.i.u.</var> et <var>.n.t.x.</var> ) tali conditio-<lb/>ne prædita, quod angulus, a. æqualis Å¿it angulo <var>.s.</var> & angulus <var>.t.</var> angulo <var>.r.</var> & quod <lb/>angulus <var>.x.</var> Å¿imul cum angulo <var>.u.</var> <reg norm="æ- quentur" type="context">æ-<lb/>quẽtur</reg> duobus rectis, & quod <reg norm="triam" type="context">triã</reg> <lb/> - <anchor type="figure" xlink:label="fig-0326-01a" xlink:href="fig-0326-01"/> - <reg norm="gulum" type="context">gulÅ©</reg> <var>.a.i.u.</var> æquale Å¿it Å¿uperficiei <var>.<lb/>c.</var> reliquum verò Å¿uperficiei <var>.y.</var> <lb/>Ex duabus Å¿uperficiebus <var>.c.</var> et <var>.y.</var> <lb/>conÅ¿tituemus duo quadrata, per vl <lb/>timam Å¿ecundi Eucli. accipiemus, <lb/></s> - <s xml:id="s3863" xml:space="preserve">deinde duo latera tetragonica ip-<lb/>Å¿orum quadratorum, & inuenie-<lb/>mus tertiam lineam in continua <lb/>proportionalitate cum illis lateri-<lb/>bus ex .10. Å¿exti, Å¿eruabimus po-<lb/>Å¿tea extremas illarum, quæ Å¿int <var>.z.</var> <lb/>et <var>.l.</var> quarum proportio, <reg norm="eadem" type="context">eadẽ</reg> erit, <lb/>quæ inter duas propoÅ¿itas Å¿uperfi-<lb/>cies reperitur ex .18. Å¿exti, accipie <lb/>mus, deinde lineam aliquam cu-<lb/>inÅ¿uis longitudinis, quæ Å¿it <var>.q.g.</var> Å¿u-<lb/>pra quam conÅ¿tituemus in puncto <lb/>q. angulum <var>.m.q.g.</var> Ä™qualem angu-<lb/>lo <var>.s.</var> & angulum <var>.m.q.K.</var> æqualem <lb/>angulo <var>.r.</var> ex .23. primi, poÅ¿tea ve-<lb/>rò à quouis puncto ipÅ¿ius lineæ <var>.q.<lb/>m.</var> puta <var>.o.</var> ducetur <var>.o.f.</var> vÅ¿que ad <var>.q.<lb/>g.</var> quorÅ¿um volueris, producendo <lb/>ipÅ¿am <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg>. ad <var>.d.</var> ita quod propor-<lb/>tio <var>f.o.</var> ad <var>.o.d.</var> Å¿it vt <var>.z.</var> ad <var>.l.</var> ex .10. <lb/>Å¿exti, ducendo poÅ¿tea à puncto <var>.d.</var> <lb/>lineam <var>.d.h.E.</var> parallelam lineæ <var>.q.<lb/>g.</var> & quia ex .2. primi Vitellionis <var>.<lb/>h.E.</var> Å¿ecatur ab <var>.q.K.</var> in puncto <var>.b.</var> <lb/>protrahemus <var>.b.o.p.</var> vnde ex Å¿imi-<lb/>litudine triangulorum habebimus <lb/>proportionem <var>.p.o.</var> ad <var>.o.b.</var> vt <var>.f.o.</var> - <pb o="315" rhead="EPISTOL AE." n="327" file="0327" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0327"/> - ad <var>.o.d.</var> hoc eÅ¿t vt <var>.z.</var> ad <var>.l.</var> hoc eÅ¿t vt <var>.c.</var> ad <var>.y</var>. </s> - <s xml:id="s3864" xml:space="preserve">quare <reg norm="triangulum" type="context">triangulÅ©</reg> <var>.p.q.o.</var> ita erit proportio <lb/><reg norm="natum" type="context">natÅ©</reg> triangulo <var>.o.q.b.</var> vt <var>.c.</var> ad <var>.y.</var> conÅ¿tituo deinde ex .25. fexti duo triangula Å¿imi-<lb/>lia duobus <var>.p.q.o.</var> et <var>.o.q.b.</var> <reg norm="æqualiaque" type="simple">æqualiaqÌ</reg> <var>.c.</var> et <var>.y.</var> quÄ™ Å¿int <var>.a.i.u.</var> et <var>.n.t.x.</var> Å¿ecetur poÅ¿tea <var>.q.<lb/>g.</var> in puncto <var>.æ.</var> ita, quod <var>.q.æ.</var> æqualis Å¿it <var>.i.a.</var> duco poÅ¿tea <var>.æ.e.</var> æquidiÅ¿tantem. ad <var>.p.b.</var> <lb/>& Å¿ic habebimus duo triangula <var>.q.x.æ.</var> ct <var>.q.x.</var> e, vt quærebantur, quamuis duo trian <lb/>gula <var>.a.i.u.</var> et <var>.t.n.x.</var> eaÅ¿dem habeant conditiones.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0326-01" xlink:href="fig-0326-01a"> - <image file="0326-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0326-01"/> - </figure> - </div> - </div> - </div> - <div type="section" level="3" n="24"> - <div type="letter" level="4" n="1"> - <head xml:id="head475" xml:space="preserve">DE IMPERFECTA SOLVTIONE PROBLE-<lb/>matis Nicolai Tartaleæ ad Cardanum. De animad-<lb/>uerÅ¿ione in Ptolomeum. Deincendio carbo-<lb/>num à vento.</head> - <head xml:id="head476" style="it" xml:space="preserve">Clariβimo Dominico MoreÅ¿ino.</head> - <p> - <s xml:id="s3865" xml:space="preserve"><emph style="sc">SCio</emph> propoÅ¿itam tibi quæſtionem te diu agitauiſſe, nectamen Å¿olutio-<lb/>nem aſſequi potuiſſe, aduerte igitur ipÅ¿am falÅ¿am, ideÅ¿t impoſſibilem <lb/>eſſe, quemadmodum etiam decimumoctauum quæſitum propoÅ¿itum à <lb/>Cardano Tartaleæ, ab ipÅ¿o Tarralea Å¿olutum minimè fuit. </s> - <s xml:id="s3866" xml:space="preserve">Quiquidem <lb/>Tartalea vult circulum deÅ¿cribi circa triangulum per quintam libri quarti Euclidis, <lb/>vt in fine ferè quintæ partis Å¿uarum menÅ¿urarum affirmat, neque videt in quinta <lb/>quarti Euclidem vti vndecima primi, & in vndecima primi, quarta aut octaua eiuÅ¿-<lb/>dem, quas, ipÅ¿e Euclides oÅ¿tenÅ¿iuè non demonÅ¿trauit. </s> - <s xml:id="s3867" xml:space="preserve">Quapropter oportebat Tar-<lb/>taleam demonÅ¿traſſe omnes propoÅ¿itiones ad hoc neceſſarias oÅ¿tenÅ¿iuè <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> ad pri-<lb/>mas indemonÅ¿trabiles, quia ad demonÅ¿trandam Å¿cientificè <reg norm="aliquam" type="context">aliquã</reg> propoÅ¿itionem, <lb/>aut à propoÅ¿itione in propoÅ¿itionem vÅ¿que ad prima principia vniuerÅ¿alia ( vt ali-<lb/>quando ego feci) eÅ¿t retrogradandum, aut ab ipÅ¿is principijs incipiendum Å¿ucceſſi-<lb/>uè eouÅ¿que progrediendo donec ad propoÅ¿itionem quam demonÅ¿trare volumus <lb/>perueniamus.</s> - </p> - <p> - <s xml:id="s3868" xml:space="preserve">Quod ad Ptolomeum in geographia attinet, dico eum mihi non Å¿atisfacere, cum <lb/>Å¿umit portionem arcus circuli maioris inter vnam ciuitatem, & aliam, ea ratione <lb/>quam deÅ¿cribit. </s> - <s xml:id="s3869" xml:space="preserve">Quod Å¿i vÅ¿us fuiſſet modo Menelai, ab ipÅ¿omet deinde in <reg norm="Å¿uum" type="context">Å¿uÅ©</reg> Al-<lb/>mageÅ¿tum vÅ¿urpato, aut Monteregij triangulorum Å¿phÄ™ricorum, quem Copernicus <lb/>adhibuit (qui tamen modus, tempore Ptolomei, nondum fortaſſe in lucem vene-<lb/>rat) bene egiſſet.</s> - </p> - <p> - <s xml:id="s3870" xml:space="preserve">Quod deinde ad Å¿uum illud inÅ¿trumentum geometricum attinet, eÅ¿t <reg norm="imperfectum" type="context">imperfectÅ©</reg>, <lb/>vt oÅ¿tendi domino Pandulfo.</s> - </p> - <p> - <s xml:id="s3871" xml:space="preserve">Motum autem aeris, aut mauis ventum, accendere ignem, non Å¿olum ratione an <lb/>tiperiÅ¿taÅ¿is, quam affers euenit, Å¿ed etiam quia à carbonibus accenÅ¿is totam excre <lb/>mentitiam materiam, quæ eos circundat, auferat.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head477" style="it" xml:space="preserve">Alia dilucidatio propoÅ¿itionis .25. lib. 2. Monteregij.</head> - <head xml:id="head478" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3872" xml:space="preserve">SCribiste non intelligere .25. propoÅ¿itionem lib. 2. Monteregij. cum necÅ¿cias <lb/>reperire diametrum circuli circunÅ¿criptibilis circa propoÅ¿itum triangu- - <pb o="316" rhead="IO. BAPT. BENED." n="328" file="0328" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0328"/> - lum, cuius data Å¿it b aÅ¿is tantummodo Å¿imul cum angulo, qui ipÅ¿i baÅ¿i opponitur.</s> - </p> - <p> - <s xml:id="s3873" xml:space="preserve">Imagineris igitur triangulum datum eſſe obtuÅ¿iangulum <var>.a.b.g.</var> cuius baÅ¿i <var>.b.<lb/>g.</var> Å¿it nobis data Å¿imul cum angulo <var>.a.</var> ei oppoÅ¿ito, obtuÅ¿oqÌue; </s> - <s xml:id="s3874" xml:space="preserve">ConÅ¿idera etiam cir-<lb/>culum <var>.a.b.g.q.</var> ipÅ¿um trian gulum circunÅ¿cribentem, cuius diameter <var>.q.e.p.</var> tranÅ¿eat <lb/>per <var>.m.</var> punctum medium ipÅ¿ius <var>.b.g.</var> <reg norm="tunc" type="context">tÅ©c</reg> protractis imaginatione <var>.e.g.</var> et <var>.g.p.</var> certi eri-<lb/>mus angulos. circa <var>.m.</var> rectos eſſe ex .3 tertij Eucli. <reg norm="angulumque" type="simple">angulumqÌ</reg> <var>.q.e.g.</var> duplum eſſe an <lb/>gulo <var>.q.p.g.</var> ex .19. eiuÅ¿dem, vnde æqualem angulo <var>.a.</var> qui etiam duplus eÅ¿t angulo <var>.q.<lb/>p.g.</var> quapropter proportio arcus <var>.q.g.</var> ad arcum <lb/> - <anchor type="figure" xlink:label="fig-0328-01a" xlink:href="fig-0328-01"/> - <var>g.p.</var> tibi cognita erit, & proportio etiam chor-<lb/>de <var>.p.g.</var> ad Å¿inum <var>.m.g.</var> arcus <var>.g.p.</var> & quia <var>.m.g.</var> vt <lb/>dimidium ipÅ¿ius <var>.b.g.</var> tibi data eÅ¿t, cognoÅ¿ces <lb/>etiam <var>.p.g.</var> vt <var>.m.g.</var> & Å¿ic tertium latus <var>.m.p.</var> trian-<lb/>guli orthogonij <var>.p.m.g.</var> & q<unsure/>a ex .34. tertij quod <lb/>fit ex <var>.p.m.</var> in <var>.m.q.</var> eÅ¿t æquale ei quod fit ex <var>.b.m.</var> <lb/>in <var>.m.g.</var> ideo cum diuiÅ¿um fuerit productum <var>.b.</var> <lb/>m in <var>.m.g.</var> per <var>.p.m.</var> proueniet <var>.m.q.</var> quapropter <lb/>habebis totum <var>.q.p</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0328-01" xlink:href="fig-0328-01a"> - <image file="0328-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0328-01"/> - </figure> - </div> - <p> - <s xml:id="s3875" xml:space="preserve">Idem efficies, Å¿i <reg norm="cum" type="context">cÅ©</reg> angulus <var>.a.</var> acutus fuiſſet.</s> - </p> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head479" style="it" xml:space="preserve">Modus inueniendi puncta elliptica via Pergei.</head> - <head xml:id="head480" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3876" xml:space="preserve">MOdus inueniendi puncta elliptica, via .21. primi lib. Pergei ex datis axibus, <lb/>vt vbi alias Å¿ignificati, talis eÅ¿t. <lb/> - <anchor type="figure" xlink:label="fig-0328-02a" xlink:href="fig-0328-02"/> - </s> - <s xml:id="s3877" xml:space="preserve">Sit exempli gratia maior axis propo-<lb/>Å¿itus <var>.a.c.</var> minor autem <var>.b.d.</var> cum ergo <lb/>volueris inuenire punctum circunfe-<lb/>rentiæ correÅ¿pondentem puncto <var>.e.</var> <lb/>maioris axis, inueniemus primò la-<lb/>tus tetragonicum producti <var>.a.g.</var> in <var>.g.<lb/>c.</var> quod Å¿it <var>.h.</var> <reg norm="latusque" type="simple">latusqÌ</reg> <reg norm="tetragonicum" type="context">tetragonicÅ©</reg> pro-<lb/>ducti <var>.a.e.</var> in <var>.e.c.</var> quod Å¿it <var>.i</var>. </s> - <s xml:id="s3878" xml:space="preserve">deinde in-<lb/>ueniemus lineam <var>.K.</var> tertiam in con-<lb/>tinua proportionalitate cum <var>.h.</var> et <var>.i.</var> <lb/>vnde <var>.i.</var> erit media proportionalis in-<lb/>ter <var>.h.</var> et <var>.K.</var> & vt <var>.h.</var> proportionalis erit <lb/>ad <var>.K.</var> inueniemus <var>.e.f.</var> cui <var>.g.d.</var> medie-<lb/>tas Å¿ecundi axis ita Å¿e habeat, quæ po <lb/>Å¿tea iuncta axi maiori, ad angulosrectos in puncto <var>.e.</var> dabit Å¿itum puncti <var>.f.</var> quæſiti ex <lb/>dicta .21. primi lib. Pergei, Å¿ed talis modus prolixus eÅ¿t.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0328-02" xlink:href="fig-0328-02a"> - <image file="0328-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0328-02"/> - </figure> - </div> - <pb o="317" rhead="EPISTOL AE." n="329" file="0329" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0329"/> - <p> - <s xml:id="s3879" xml:space="preserve">Accipeigitur huncalium.</s> - </p> - <p> - <s xml:id="s3880" xml:space="preserve">Sit propoÅ¿itus maior axis <var>.q.p.</var> minor verò <var>.e.c.</var> ad angulos rectos Å¿e inuicem <lb/>Å¿ecantes in puncto <var>.o.</var> deÅ¿cribatur circulus <var>.q.n.p.a.</var> cuius diameter Å¿it axis maior, in <lb/>quo accipiatur punctum, quod volueris, vt puta <var>.u.</var> à quo protrahatur <var>.u.b.</var> paralle-<lb/>la ad <var>.o.c.n.</var> deÅ¿ignetur poÅ¿tea Å¿eparatim circulus <var>.u.b.n.</var> cuius diameter æqualis Å¿it Å¿e <lb/>midiametro prioris circuli, ita etiam fiat circulus <var>.u.i.c.</var> contingens circulum <var>.u.b.n.</var> <lb/>in puncto <var>.u.</var> cuius diameter Å¿it <var>.u.c.</var> æqualis dimidio axi minori. </s> - <s xml:id="s3881" xml:space="preserve">accipiatur deinde in <lb/>circulo maximo longitudo <var>.u.b.</var> quæ collocetur in circulo mediocri à puncto <var>.u.</var> quæ <lb/>Å¿ecabitur à minimo circulo in puncto <var>.i.</var> cum itaque longitudo <var>.u.i.</var> menÅ¿urata fue-<lb/>rit in <var>.u.b.</var> maximi circuli à puncto <var>.u.</var> habebimus propoÅ¿itum.</s> - </p> - <p> - <s xml:id="s3882" xml:space="preserve">Cuius reiratio eÅ¿t, quia <var>.u.b.</var> mediocris circuli diuiditur à gyro minimi in puncto <lb/>i. eadem proportione, qua diuiÅ¿a eÅ¿t <var>.u.n.</var> in puncto <var>.c.</var> quod manifeÅ¿tum eÅ¿t exÅ¿imi-<lb/>litudine triangulorum <var>.u.b.n.</var> et <var>.u.i.c.</var> imaginatæ cum fuerint duæ <var>.b.n.</var> et <var>.i.c.</var> Å¿ed ita <lb/>eſſe oportet parallelas maximi circuli, quotieÅ¿cunque circunferentia ipÅ¿ius ellipÅ¿is <lb/>tranÅ¿itura Å¿it per <var>.c.</var> vt in .51. cap. meæ gnomonicæ oÅ¿tenÅ¿um fuit.</s> - </p> - <figure position="here"> - <image file="0329-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0329-01"/> - </figure> - </div> - <div type="letter" level="4" n="4"> - <head xml:id="head481" style="it" xml:space="preserve">Modus deÅ¿ignandi angulum, certo modo conditionatum.</head> - <head xml:id="head482" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3883" xml:space="preserve">NVllius reuera difficultatis mihi videtur eſſe, quotieÅ¿cunque nobis propoÅ¿ita <lb/>fuerint duo puncta <var>.a.</var> et <var>.b.</var> Å¿imul cum <lb/>angulo <var>.d.</var> <reg norm="necnon" type="context">necnõ</reg> linea <var>.g.</var> ducere duas lineas <lb/> - <anchor type="figure" xlink:label="fig-0329-02a" xlink:href="fig-0329-02"/> - à dictis punctis terminatas, quæ <reg norm="conÅ¿tituant" type="context">conÅ¿tituãt</reg> <lb/>angulum æqualem dato, & ipſæ directè <reg norm="con" type="context">cõ</reg> <lb/>iunctæ conÅ¿tituant lineam æqualem da-<lb/>tæ. </s> - <s xml:id="s3884" xml:space="preserve">Nam ducatur linea indefinita per <lb/>puncta propoÅ¿ita, cuius lineæ, pars illa, quÄ™ <lb/>intercepta fuerit inter dicta puncta, diui-<lb/>datur per æqualia in puncto <var>.o.</var> etiam & li-<lb/>nea data, quarum medietates accipio in <lb/>linea indefinitè protracta à puncto <var>.o.</var> me- - <pb o="318" rhead="IO. BAPT. BENED." n="330" file="0330" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0330"/> - dio, vt vna earum Å¿it <var>.o.c.</var> reliqua verò Å¿it <var>.o.e.</var> </s> - <s xml:id="s3885" xml:space="preserve">deinde aperiatur circinus quantum <var>.<lb/>o.c.</var> <reg norm="poÅ¿itoque" type="simple">poÅ¿itoqÌ;</reg> vno pede in <var>.b.</var> <reg norm="deÅ¿ignentur" type="context">deÅ¿ignẽtur</reg> cum altero duo arcus <var>.n.K.</var> poÅ¿ito iterum vno <lb/>pede in <var>.a.</var> deÅ¿ignentur alij duo arcus inter-<lb/>Å¿ecantes primos in punctis <var>.n.K</var>. </s> - <s xml:id="s3886" xml:space="preserve">Deinde à <lb/> - <anchor type="figure" xlink:label="fig-0330-01a" xlink:href="fig-0330-01"/> - puncto <var>.n.</var> ad <var>.K.</var> ducetur linea <var>.n.K.</var> quæ per <lb/>punctum <var>.o.</var> tranÅ¿ibit, quam <var>.n.K.</var> mente <reg norm="con- cipio" type="context">cõ-<lb/>cipio</reg>, vt axis minor <reg norm="vnius" type="simple">vniê°</reg> ellipÅ¿is, cuius <var>.e.c.</var> <lb/>Å¿it axis maior, quibus axibus mediantibus <lb/>deÅ¿ignetur ellipÅ¿is <var>.n.c.K.e.</var> conÅ¿idero dein <lb/>de <var>.a.b.</var> vt <reg norm="chordanvnius" type="context">chordãvnius</reg> circuli, Å¿eu portio-<lb/>nis circularis, quæ capax Å¿it vnius anguli <lb/>æqualis angulo <var>.d.</var> propoÅ¿ito, ex .32. tertij <lb/>Euclid. cuius circunferentia, circunferen-<lb/>tiam ipÅ¿ius ellipÅ¿is Å¿ecabit in <reg norm="duobus" type="simple">duobê°</reg> punctis <lb/>quorum <reg norm="vnum" type="context">vnÅ©</reg> Å¿it <var>.i.</var> à quo protractæ cum fue <lb/>rint duæ lineæ <var>.a.i.</var> et <var>.i.b.</var> habebis propoÅ¿itum, cum <var>.a.i.</var> iuncta cum <var>.i.b.</var> æquetur <var>.e.</var> c, <lb/>ex .52. tertij Pergei.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0329-02" xlink:href="fig-0329-02a"> - <image file="0329-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0329-02"/> - </figure> - <figure xlink:label="fig-0330-01" xlink:href="fig-0330-01a"> - <image file="0330-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0330-01"/> - </figure> - </div> - </div> - </div> - <div type="section" level="3" n="25"> - <div type="letter" level="4" n="1"> - <head xml:id="head483" xml:space="preserve">EMENDATIO CVIVSDAM FALSI MODI <lb/>delineandi horologia Italica orizontalia.</head> - <head xml:id="head484" style="it" xml:space="preserve">Foanni Paulo Dardano.</head> - <p> - <s xml:id="s3887" xml:space="preserve"><emph style="sc">MOdvs</emph> delineandi horologia Italica orizontalia, quem tibi monÅ¿trauit <lb/>neÅ¿cio quis, ni fallor, talis eÅ¿t. </s> - <s xml:id="s3888" xml:space="preserve">DeÅ¿ignato meridiano <var>.l.b.m.q.</var> <reg norm="ductisque" type="simple">ductisqÌ;</reg> duo <lb/>bus diametris <var>.l.m.</var> et <var>.b.</var> q, inuicem ad rectos in centro <var>.g.</var> quorum <var>.l.m.</var> Å¿it <lb/>verticalis <var>.b.q.</var> vero <reg norm="orizontalis" type="context">orizõtalis</reg>, <reg norm="ductoque" type="simple">ductoqÌ;</reg> diametro <var>.f.h.</var> tropici Cancri Å¿e-<lb/>cundum altitudinem poli datam, <reg norm="deÅ¿criptoque" type="simple">deÅ¿criptoqÌ;</reg> dimidio circulo <var>.f.z.h.</var> ipÅ¿ius paralleli, <lb/><reg norm="inuentoque" type="simple">inuentoqÌ;</reg> puncto <var>.z.</var> horæ propoÅ¿itÄ™, & ab eo ducta per pendiculari <var>.z.r.</var> ad <var>.f.h.</var> & à <lb/>puncto <var>.r.</var> ducta <var>.r.o.y.</var> parallela ad diametrum <var>.q.b.</var> orizontalem, ducis poÅ¿tea <var>.f.ω.</var> et <var>.<lb/>r.t.</var> vÅ¿que ad orizontalem <var>.q.b.</var> parallelas ad diametrum <var>.l.m.</var> verticalem. </s> - <s xml:id="s3889" xml:space="preserve">Determi-<lb/>nato poÅ¿tea gnomone <var>.g.s.</var> in orizontis axe, <reg norm="ductaque" type="simple">ductaqÌ;</reg> vmbrarum linea <var>.s.K.</var> parallela <lb/>orizontali, <reg norm="ductaque" type="simple">ductaqÌ;</reg> <var>.y.g.K.</var> ad terminandam <var>.s.K.</var> delineas deinde Å¿eparatim circulum <lb/><var>q.x.b.n.</var> magnitudinis prioris, qui quidem circulus Å¿ignificet orizontem ipÅ¿um, in <lb/>quo ductis diametris <var>.q.g.b.</var> et <var>.l.g.m.</var> accipis in diametro <var>.q.g.b.</var> puncta <var>.a.</var> et <var>.ω.</var> ita à <reg norm="cen tro" type="context">cẽÂ<lb/>tro</reg> <var>.g.</var> diÅ¿tantia, vt Å¿unt in diametro orizontali prioris circuli, ducis poÅ¿tea per pun-<lb/>ctum <var>.a.</var> lineam <var>.x.a.n.</var> ad rectos cum dicto diametro, </s> - <s xml:id="s3890" xml:space="preserve">deinde per tria puncta <var>.n.ω.x.</var> <lb/>tranÅ¿ire facis circunferentiam circuli per <reg norm="quintam" type="context">quintã</reg> quarti Euclidis, poÅ¿tea in dicto dia-<lb/>metro accipis punctum <var>.t.</var> ita diÅ¿tans à centro, & ex eadem parte, vt in priori circulo, <lb/>à quo puncto ducis <var>.t.u.</var> parallelam <var>.x.n.</var> vÅ¿que ad circunferentiam <var>.x.ω.n.</var> in puncto <var>.u.</var> <lb/>quo facto, ducis à centro <var>.g.</var> per punctum <var>.u.</var> ipÅ¿ius circularis circunferentiæ <var>.g.u.</var> inde-<lb/>terminatam, quam poÅ¿tea terminas in puncto <var>.K.</var> ita quod <var>.g.K.</var> æqualis Å¿it <var>.s.K</var>. </s> - <s xml:id="s3891" xml:space="preserve">Dicis <lb/>poÅ¿tea punctum <var>.K.</var> in eodem Å¿itu reperiri, reÅ¿pectu duorum diametrorum <var>.q.b.</var> me-<lb/>ridiani. et <var>.l.m.</var> verticalis, vt decet, & oportet punctum horæ propoÅ¿itÄ™ exiÅ¿tere.</s> - </p> - <p> - <s xml:id="s3892" xml:space="preserve">Quod quidem dico eſſe falÅ¿um, propterea quod perpendiculares quas cogita-<lb/>mus cadere à punctis circunferentiæ cuiuÅ¿uis paralleli Å¿upra quemuis orizontem ob - <pb o="319" rhead="EPISTOL AE." n="331" file="0331" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0331"/> - liquum Å¿ecantem æquatorem, omnes caduntin gyro elliptico, oxygonio, Å¿eu de-<lb/>fectionali, & non circulari. </s> - <s xml:id="s3893" xml:space="preserve">Vnde per Å¿upradicta tria puncta <var>.n.ω.x.</var> oporteret tranÅ¿i <lb/>re talem circunferentiam, & non <reg norm="circularem" type="context">circularẽ</reg>, quæ circunferentia eſſet vnius ellipÅ¿is, <lb/>cuius minor axis in diametro <var>.b.q.</var> eſſet .ab <var>.ω.</var> vÅ¿que ad <var>.i.</var> terminum Å¿ini <var>h.i.</var> arcus <var>.h.b.</var> <lb/>in analemate, maior verò axis eſſet magnitudinis <var>.f.h.</var> diametri paralleli, quæ <reg norm="tranÅ¿iÅ¿- Å¿et" type="context">trãſiÅ¿-<lb/>Å¿et</reg> per punctum <var>.c.</var> medium inter <var>.ω.</var> et <var>.i.</var> quÄ™ quidem circunfere ntia tota eſſet intra cir <lb/>culum <var>.q.n.b.x.</var> <reg norm="con" type="context">cõ</reg><reg norm="tiguaque" type="simple">tiguaqÌ;</reg> gyro <var>.q.n.b.x.</var> in punctis <var>.n.x</var>.</s> - </p> - <p> - <s xml:id="s3894" xml:space="preserve">Si ergò circunferentia <var>.n.ω.x.</var> eſſet elliptica tunc punctum <var>.u.</var> in orizonte illud eſſet <lb/>vbi caderet Å¿inus altitudinis horÄ™, et <var>.t.u.</var> æqualis eſſet <var>.r.z.</var> communi Å¿ectioni paralle <lb/>li cum almicantarat ex .34. primi Euclid. et <var>.u.g.</var> æqualis eſſet <var>.o.y.</var> communi Å¿ectioni <lb/>almicantarat cum meridiano, vel cum azimut illius horæ ex .4. primi, cum <var>.g.t.</var> æqua <lb/>lis Å¿it ipÅ¿i <var>.o.r.</var> et <var>.t.u.</var> ipÅ¿i <var>.r.z.</var> & angu <lb/> - <anchor type="figure" xlink:label="fig-0331-01a" xlink:href="fig-0331-01"/> - lus. t trianguli <var>.g.t.u.</var> rectus, quem-<lb/>admodum <var>.r.</var> qui compræhenditur <lb/>ab <var>.z.r.</var> et <var>.r.o.</var> vnde anguli <var>.K.g.m.</var> <lb/>et <var>.K.g.b.</var> rectè Å¿e haberent, diÅ¿tan-<lb/>tia verò inter <var>.K.</var> et <var>.g.</var> <reg norm="iam" type="context">iã</reg> rectè Å¿um-<lb/>pta fuit.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0331-01" xlink:href="fig-0331-01a"> - <image file="0331-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0331-01"/> - </figure> - </div> - <p> - <s xml:id="s3895" xml:space="preserve">Sed quia punctum <var>.u.</var> vt <reg norm="plurimum" type="context">plurimÅ©</reg> <lb/>(in gyro circulari Å¿umptum) extra <lb/>puncta interÅ¿ectionum ipÅ¿ius circu <lb/>laris gyri cum elliptico reperìtur, <lb/></s> - <s xml:id="s3896" xml:space="preserve">propterea efficit angulos <var>.K.g.m.</var> <lb/>et <var>.K.g.b.</var> falÅ¿os, & non æquales il-<lb/>lis, qui fiunt ab azimut horæ cum <lb/>verticali, & cum meridiano, quæ <lb/>omnia ex cap .52. meæ gnomonicÄ™ <lb/>facilè videre potes.</s> - </p> - <p> - <s xml:id="s3897" xml:space="preserve">Nectacere volo quod punctum <lb/>u. verum, hoc eÅ¿t ellipticum, inue-<lb/>niri poſſet ea via quam Å¿cripÅ¿i in <lb/>eodem .52. cap. qua mediante do-<lb/>cui demum inuenire punctum <var>.Ï€.</var> <lb/>orizontis, quamuis in præſenti caÂ<lb/>Å¿u <var>.ω.λ.</var> perpendicularis eſſet Å¿upra <lb/>minorem axem ipÅ¿ius ellipÅ¿is, <reg norm="quan- uis" type="context">quã-<lb/>uis</reg> Å¿upra maiorem axem, quod ta-<lb/>men minimè mutat ordinem, imò <lb/>rationes eÄ™dem Å¿unt, tam in vna, <lb/>quam in alia operatione, Å¿ed vt il-<lb/>licò <reg norm="idipsum" type="context">idipsÅ©</reg> habeas, fac vt <var>.t.u.</var> æqua <lb/>lis. Å¿it <var>.r.z</var>. </s> - <s xml:id="s3898" xml:space="preserve">& tunc punctum <var>.K.</var> erit <lb/><reg norm="quæſitum" type="context">quæſitÅ©</reg>, quod ego in .52. cap. meÄ™ <lb/>gnomonicÄ™, ijs verbis Å¿ignificaui.</s> - </p> - <quote> - <s xml:id="s3899" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> mediis binis triangulis ijs, <lb/><reg norm="medioque" type="simple">medioqÌ;</reg> azimut Solis pariter ho-<lb/>rologia fabricari poterunt.</s> - </quote> - <pb o="320" rhead="IO. BAPT. BENED." n="332" file="0332" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0332"/> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head485" style="it" xml:space="preserve">De Horologio perpendiculari ad oriz ontem rectum.</head> - <head xml:id="head486" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3900" xml:space="preserve">MOdus quem tibi Å¿cribere promiÅ¿i delineandi lineas horarias communes in <lb/>pariete perpendiculariter ad orizontem rectum, declinantem à meridiano, <lb/>Å¿umendus eÅ¿t ex .46. cap. meæ gnomonicæ, hocÅ¿cilicet ordine.</s> - </p> - <p> - <s xml:id="s3901" xml:space="preserve">Sit exempli gratia, orizon hic Å¿ubÅ¿criptus. or. oc <var>.M.S.</var> diuiÅ¿us à meridiana <var>.M.<lb/>S.</var> et verticali Å¿eu æquinoctiali. or. oc. </s> - <s xml:id="s3902" xml:space="preserve">SitqÌue <var>.e.t.</var> communis Å¿ectio muri cum ori-<lb/>zonte, et <var>.g.n.</var> Å¿it gnomon perpendicularis ipÅ¿i muro, vnde ex dictis in mea gnomo-<lb/>nica, cognoÅ¿cemus in ipÅ¿a murali orizontali totam <var>.e.t.</var> inter meridianam orizonta-<lb/>lem, & æquinoctialem orizontalem, cognoÅ¿cemus etiam partem <var>.g.t.</var> ipÅ¿ius æ-<lb/>quinoctialis orizontalis, quam quidem accipiamus in rectitudine ipÅ¿ius muralis ori <lb/>zontalis, quæ quidem Å¿it <var>.t.G.</var> quo facto erigatur <var>.G.A.</var> ad rectos cum <var>.G.t.e.</var> & cir-<lb/>cum <var>.G.e.</var> deÅ¿ignetur vna medietas circuli verÅ¿us <var>.e.</var> cuiuÅ¿uis magnitudinis, quæ di-<lb/>uiÅ¿a in .12. partes Ä™quales, Å¿ignificabit <reg norm="medietatem" type="context">medietatẽ</reg> æquatoris, <reg norm="protrahanturque" type="simple">protrahanturqÌ;</reg> lineæ oc <lb/>cultæ à centro <var>.G.</var> per Å¿ectiones circunferentiÄ™ dimidij circuli, quæ fignificabunt <reg norm="con" type="context">cõ</reg> <lb/>munes Å¿ectiones æquatoris cum circulis horarijs communibus, quo facto oportet, vt <lb/>à puncto <var>.t.</var> protrahatur <var>.t.s.</var> ad rectos cum murali orizontali, quæ quidem <var>.t.s.</var> Å¿ignifi-<lb/>cabit communem Å¿ectionem æquatoris cum muro propoÅ¿ito, & erit Ä™quÄ™diÅ¿tans me <lb/>ridianæ murali ex .6. vndecimi Eucli. eo quod ex .19. eiuÅ¿dem <reg norm="vnaquæque" type="simple">vnaquæq;</reg> illarum, per <lb/>pendicularis eÅ¿t tali orizonti. </s> - <s xml:id="s3903" xml:space="preserve">Videantur nunc puncta communia iÅ¿ti <var>.t.s.</var> & occultis <lb/>protractis à centro <var>.G.</var> medietatis circularis, per quæ puncta protrahantur à puncto <var>.<lb/>e.</var> tot lineæ, punctum enim <var>.e.</var> Å¿ignificat punctum axis mundi, & meridianæ in mu-<lb/>ro propoÅ¿ito, eo quod in tali Å¿itu Å¿phæræ rectæ, dictum punctum reperitur in orizon <lb/>re, cum <var>.M.s.</var> non Å¿olum Å¿it meridiana orizontalis, Å¿ed etiam axis mundi, </s> - <s xml:id="s3904" xml:space="preserve">deinde nul <lb/>li dubium eÅ¿t quin meridiana muralis Å¿it perpendicularis orizontali murali <var>.e.t.</var> à <lb/>puncto <var>.e</var>. </s> - <s xml:id="s3905" xml:space="preserve">Sed quia dimidium <lb/> - <anchor type="figure" xlink:label="fig-0332-01a" xlink:href="fig-0332-01"/> - harum <reg norm="linearum" type="context">linearÅ©</reg> horariarum erit <lb/>Å¿ub orizontali <var>.e.t.G.</var> alterum <lb/>vero <reg norm="dimidium" type="context">dimidiÅ©</reg> Å¿upra ipÅ¿am, opor <lb/>tet quod quÄ™ Å¿upra Å¿unt produ-<lb/>cantur à parte .oe. Å¿ub orizonta-<lb/>li, ab alia parte meridianæ, & <lb/>talis erit effigies horologij mura <lb/>lis in hoc Å¿phÄ™rÄ™ Å¿itu, hoc eÅ¿t ver <lb/>Å¿us quartam orientalem auÅ¿tra-<lb/><reg norm="lemque" type="simple">lemqÌ;</reg>, vnde orizontalis <var>.e.t.</var> <lb/>erit Å¿emper horæ .6. matutinæ, <lb/>Å¿ecunda verò ab ipÅ¿a erit horæ. <lb/>7. tertia <reg norm="autem" type="context">autẽ</reg> horæ .8. & Å¿ic dein-<lb/>ceps.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0332-01" xlink:href="fig-0332-01a"> - <image file="0332-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0332-01"/> - </figure> - </div> - <p> - <s xml:id="s3906" xml:space="preserve">QuotieÅ¿cunque verò angu-<lb/>lus <var>.n.g.e.</var> minor erit maxima <lb/>Solis declinatione, & Sol fuerit <lb/>in parte auÅ¿trali ab æquatore <reg norm="cum" type="context">cÅ©</reg> - <pb o="321" rhead="EPISTOL AE." n="333" file="0333" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0333"/> - maiori numero declinationis quam fuerit angulus <var>.n.g.e</var>. </s> - <s xml:id="s3907" xml:space="preserve">tunc talis paries illumina-<lb/>bitur ab ipÅ¿o Sole à mane vÅ¿que ad veÅ¿peram.</s> - </p> - <p> - <s xml:id="s3908" xml:space="preserve">Huius quidem rei Å¿peculatio, vnicuique manifeÅ¿ta erit, qui rationes .46. cap. no-<lb/>Å¿træ gnomonicæ prius intellexerit, vbi manifeÅ¿tè apparet proportionem Å¿emidiame <lb/>tri horologij (Å¿i ita eam appellare licet) ad Å¿emidiametrum æquatoris horarij Å¿em-<lb/>per eſſe, vt <var>.e.t.</var> ad <var>.t.g.</var> hoc eÅ¿t proportio maioris inæqualitatis. </s> - <s xml:id="s3909" xml:space="preserve">nolo etiam prætermic <lb/>tere. </s> - <s xml:id="s3910" xml:space="preserve">quin te admoneam, vt nullo pacto confidas in longioribus vmbris, eo quod val <lb/>de nos decipiant, cum Å¿emper iuÅ¿to breuiores Å¿int.</s> - </p> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head487" style="it" xml:space="preserve">Declar atio quorundam verborum noÅ¿træ<unsure/> Gnomonicæ. <lb/><reg norm="DefenÅ¿ioque" type="simple">DefenÅ¿ioqÌê«</reg> nostra contra Christophorum Clauium.</head> - <head xml:id="head488" xml:space="preserve">AD EVNDEM DARDANVM.</head> - <p> - <s xml:id="s3911" xml:space="preserve">TVas demum accepi literas, qui <lb/> - <anchor type="figure" xlink:label="fig-0333-01a" xlink:href="fig-0333-01"/> - bus mihi Å¿ignificas te totum <num value="52">.<lb/>52.</num> caput meæ gnomonicæ intelle-<lb/>xiſſe, præter illa verba, quæ etiam <lb/>Å¿uperioribus diebus ad te Å¿cripÅ¿i, <lb/>hoc eÅ¿t.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0333-01" xlink:href="fig-0333-01a"> - <image file="0333-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0333-01"/> - </figure> - </div> - <quote> - <s xml:id="s3912" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> medijs binis triangulis ijs, <lb/><reg norm="medioque" type="simple">medioqÌ;</reg> azimut Solis, pariter ho-<lb/>rologia fabricari poterunt.</s> - </quote> - <p> - <s xml:id="s3913" xml:space="preserve">Quapropter nealiquid tibi de-<lb/>Å¿it, Å¿cire debes, menihil aliud, eo <lb/>in loco inferre voluiſſe, quà m <reg norm="quod" type="wordlist">qÄ‘</reg> <lb/>punctum horæ propoÅ¿itæ in plano <lb/>horologij orizontali reperiri po-<lb/>teÅ¿t, ope longitudinis vmbræ gno-<lb/>monis, & eius declinationis à ver-<lb/>ticali linea, Å¿eu à meridiana orizon <lb/>tali, iam in ipÅ¿o horologij plano <lb/>ductis.</s> - </p> - <p> - <s xml:id="s3914" xml:space="preserve">Exempli gratia, Å¿it analemma <var>.<lb/>l.q.m.b.</var> in quo <var>.l.m.</var> Å¿it verticalis <var>.q.<lb/>b.</var> verò orizontalis <var>.f.n.h.</var> autem Å¿it <lb/>Å¿emicirculus, cuiuÅ¿uis paralleli æqui <lb/>noctiali, cuius diameter fit <var>.f.h.</var> et <var>.<lb/>n.</var> Å¿it Solis locus in ipÅ¿o parallelo: </s> - <s xml:id="s3915" xml:space="preserve">n.<lb/>r. autem Å¿it rectus Å¿inus arcus <var>.f.n</var>: et <var>.<lb/>r.o.z.</var> Å¿ectio communis ipÅ¿ius almi-<lb/>cantarat cum meridiano, et <var>.s.a.</var> <reg norm="con" type="context">cõ</reg> <lb/>munis Å¿ectio azimut Solis cum pla-<lb/>no horologij, et <var>.s.g.</var> gnomon, et <var>.x.<lb/>g.a.</var> radius Solis <var>.z.u.</var> verò Å¿inus alti-<lb/>tudinis ipÅ¿ius Solis, vbi videre po-<lb/>tes duo triangula dicta eſſe <var>.z.u.g.</var> - <pb o="322" rhead="IO. BAPT. BENED." n="334" file="0334" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0334"/> - et <var>.g.s.a.</var> quibus mediantibus cognoÅ¿citur longitudo vmbræ gnomonis hoc eÅ¿t <var>.s.a</var>.</s> - </p> - <p> - <s xml:id="s3916" xml:space="preserve">Cum autem dico, <reg norm="medioque" type="simple">medioqÌ;</reg> azimut Solis, nihil aliud Å¿igniÅ¿icare volo, niÅ¿i angu-<lb/>lum, quem terminat linea azimutalis horologij, hoc eÅ¿t vmbra gnomonis cum li-<lb/>nea meridiana, Å¿eu cum verticali in ipÅ¿o plano horologij. </s> - <s xml:id="s3917" xml:space="preserve">qui quidem anguli, æqua <lb/>les Å¿unt ijs, qui in triangulo conÅ¿tituto ex <var>.n.r.</var> ex <var>.r.o.</var> & ex <var>.o.z.</var> reperiuntur, cuius qui <lb/>dem trianguli, angulus puncti <var>.r.</var> rectus eÅ¿t, angulus verò terminatus ab <var>.n.r.</var> et <var>.o.z.</var> il <lb/>le eÅ¿t quem conÅ¿tituit azimut cum verticali, vel ipÅ¿i æqualis, vt coalternus, reliquus <lb/>verò in <reg norm="puncto" type="context">pÅ©cto</reg> <var>.o.</var> ille eÅ¿t <reg norm="quem" type="context">quẽ</reg> azimut facit <reg norm="cum" type="context">cÅ©</reg> meridiano, vel ipÅ¿i Ä™qualis vt coalternus.</s> - </p> - <p> - <s xml:id="s3918" xml:space="preserve">Vnde quotieÅ¿cunque volueris in aliquo plano, orizonti parallelo, lineas hora-<lb/>rias ducere, iudico optimum fore Å¿i Å¿eparatim deÅ¿ignatæ fuerint hæ tres figuræ, hoc <lb/>eÅ¿t analemma meridianum, vel azimutale, vt ita dicam, </s> - <s xml:id="s3919" xml:space="preserve">deinde parallelus <reg norm="inÅ¿eruiens" type="context">inÅ¿eruiẽs</reg> <lb/>pro tropicis, vt ego feci cap .51. meæ gnomonicæ, quæ duæ figuræ, <reg norm="Å¿ufficientes" type="context">Å¿ufficiẽtes</reg> <reg norm="erunt" type="context">erÅ©t</reg> <lb/>pro omnibus horologijs, tam ori-<lb/> - <anchor type="figure" xlink:label="fig-0334-01a" xlink:href="fig-0334-01"/> - zontalibus quam muralibus, non <lb/>tamen omninò, ideo pro orizon-<lb/>talibus, tertiam figuram Å¿eparatam <lb/>deÅ¿ignaui, quÄ™ erit circulus <var>.H.I.<lb/>K.</var> eiuÅ¿dem magnitudinis cum ana-<lb/>lemmate, in quo, ductis duobus <lb/>dia metris inuicom ad rectos, quo-<lb/>rum vnus <var>.H.I.</var> Å¿ignificet orizon-<lb/>talem lineam, reliqua verò <reg norm="ver- ticalem" type="context">ver-<lb/>ticalẽ</reg>, ducatur poÅ¿tea <var>.s.a.</var> tam di-<lb/>Å¿tans ab <var>.H.I.</var> quanta eÅ¿t longitudo <lb/>gnomonis horologij orizontalis, <lb/>cogitemus, </s> - <s xml:id="s3920" xml:space="preserve">deinde hunc circulum <lb/>communem eſſe omnibus azimut <lb/>necnon plano horologij, in cuius <lb/>circunferentia à puncto <var>.k.</var> nadir ip-<lb/>Å¿ius zenit, accipiantur arcus æqua-<lb/>les ijs ipÅ¿orum azimut, quos termi-<lb/>nat zenit, & ipÅ¿i almicantarat, vt <lb/>exempli gratia, accipiemus arcum <lb/><var>k.L.</var> æqualem arcui <var>.L.z.</var> ipÅ¿ius ana-<lb/>lemmatis, ducta poÅ¿tea linea occul <lb/>ta <var>.o.L.</var> Å¿ignabimus azimutalem <var>.s.a.</var> <lb/>in puncto <var>.a.</var> vbi hæ duæ lineæ Å¿ein <lb/>uicem Å¿ecant, & Å¿ic habebimus iu-<lb/>Å¿tam <reg norm="quantitatem" type="context">quãtitatem</reg> ipÅ¿ius vmbræ gno <lb/>monis <var>.s.o.</var> tali hora, </s> - <s xml:id="s3921" xml:space="preserve">deinde in ori-<lb/>zontali <var>.H.I.</var> Å¿umatur <var>.o.r.</var> à centro <var>.<lb/>o.</var> Ä™qualis ei quÄ™ in analemate repe <lb/>ritur, quæ vna portio eÅ¿t commu-<lb/>nis Å¿ectionis meridiani cum almi-<lb/>cantarat, terminata ab axe orizon <lb/>tis, & à diametro paralleli. </s> - <s xml:id="s3922" xml:space="preserve">Deinde - <pb o="323" rhead="EPISTOL AE." n="335" file="0335" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0335"/> - <reg norm="ducatur" type="simple">ducat̃</reg>. <var>r.V.</var> ad rectos cum <var>.H.I.</var> vÅ¿que ad circunferentiam, in qua accipiatur <var>.r.n.</var> æqua <lb/>lis ei quæ eÅ¿t in parallelo, ducatur poÅ¿tea <var>.o.n.M.</var> & habe bimus triangulum <var>.o.r.n.</var> Å¿i-<lb/>milem <reg norm="æqualemque" type="simple">æqualemqÌ;</reg> triangulo iam Å¿upradicto. </s> - <s xml:id="s3923" xml:space="preserve">Vnde angulus <var>.H.o.M.</var> ei æqualis erit, <lb/>quem azimut facit cum meridiano, & angulus <var>.M.o.k.</var> ei Ä™qualis, quem azimut con-<lb/>Å¿tituit cum verticali, ita quod Å¿i talis circulus <var>.H.k.I.</var> eſſet planum horologij orizon-<lb/>talis, Å¿uppoÅ¿ito <var>.o.</var> pro pede gnomonis, Å¿ecando poÅ¿tea <var>.o.M.</var> in puncto <var>.i.</var> ita vt <var>.o.i.</var> <lb/>æqualis eſſet <var>.s.a.</var> dato quod <var>.o.M.</var> ducta Å¿it ad partem Å¿ibi conuenientem, reÅ¿pectu <var>.o.<lb/>k.</var> ipÅ¿a pro verticali Å¿uppoÅ¿ita, quod tibirelinquo, cum hoc facillimum Å¿it, tunc pun <lb/>ctum <var>.i.</var> eſſet quod quærebamus. </s> - <s xml:id="s3924" xml:space="preserve">Quod verò de vno puncto dico, idem de omni-<lb/>bus infero.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0334-01" xlink:href="fig-0334-01a"> - <image file="0334-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0334-01"/> - </figure> - </div> - <p> - <s xml:id="s3925" xml:space="preserve">Vbi verò mihi Å¿ignificas ChriÅ¿tophorum Clauium, me duobus in locis meæ gno <lb/>monicæ redarguere, iam vidi. </s> - <s xml:id="s3926" xml:space="preserve">Circa primum locum igitur, qui eÅ¿t in pagin .161. <lb/>ita inquit.</s> - </p> - <quote> - <s xml:id="s3927" xml:space="preserve">Non enim deÅ¿unt, qui vel omninò negent, inter quos eÅ¿t Ioannes BaptiÅ¿ta Be-<lb/>nedictus in Å¿ua gnomonica cap .70. et .71. vbialia, & multo longiore ratione cona-<lb/>tur arcus Å¿ignorum deÅ¿cribere, vel certe dubitent, hoc modo rectè poſſe deÅ¿cribi ar-<lb/>cus Å¿ignorum, cum rationem non videant, qua hæc noÅ¿tra deÅ¿criptio quam quidem <lb/>omnes Å¿criptores Å¿ine vlla demonÅ¿tratione tradunt nitatur.</s> - </quote> - <p> - <s xml:id="s3928" xml:space="preserve">AbÅ¿que dubio raptim tranÅ¿currit illa capita .70. 71. </s> - <s xml:id="s3929" xml:space="preserve">Reuerendus Clauius alio quin <lb/>non Å¿cripÅ¿iſſet, quòd ego alia & multo longiore ratione conatus Å¿im arcus Å¿ignorum <lb/>deÅ¿cribere & c. præſertim cum eadem prorÅ¿us ratio, quæ ibi à me tradita eÅ¿t, illa Å¿it, <lb/>quam ip Å¿e Å¿uis Å¿criptis inÅ¿eruit.</s> - </p> - <p> - <s xml:id="s3930" xml:space="preserve">Meus igitur modus in dictis capitibus traditus, minime diÅ¿crepat ab eo, Å¿ed ab il-<lb/>lorum modo, quorum opinio eÅ¿t interualla <var>.e.h</var>: <var>h.u</var>: u. n; </s> - <s xml:id="s3931" xml:space="preserve">n.m. et <var>.m.d.</var> meæ figuræ in <lb/>pagi .75. poÅ¿itæ, æqualia eſſe interuallis <var>.e.h</var>:h.u:u.n:n.m. et <var>.m.d.</var> præcedentis figuræ, <lb/>qui etiam Å¿upponunt <var>.t.e.</var> meæ figuræ .75. eſſe directè coniuncta cum linea <var>.e.h.u.n.<lb/>m.d.</var> & propterea verÅ¿us finem .73. pag. dixi.</s> - </p> - <quote> - <s xml:id="s3932" xml:space="preserve">Aduertat autem quam diligentiſſime quiÅ¿que ne Å¿e decipi patiatur à ſubÅ¿cripta fi <lb/>gura Å¿emicirculi <var>.Q.æ.m.</var> cum reliquis lineis ductis, ex antiquorum more, & c.</s> - </quote> - <p> - <s xml:id="s3933" xml:space="preserve">Eo quod non defuerunt aliqui, ex vetuÅ¿tioribus (quorum Å¿cripta ad meas manus <lb/>peruenerunt) qui Å¿umentes interualla <var>e.h</var>:h.u. & c. figuræ. pag .75. æqualia illis figu-<lb/>ræ pag .74. <reg norm="putauerunt" type="context">putauerÅ©t</reg> lineam <var>.t.e.</var> directè coniunctam eſſe cum <var>.e.h.</var> & c. quod quidem <lb/>maximi erroris cauÅ¿a erat, </s> - <s xml:id="s3934" xml:space="preserve">& propterea cap .71. verum modum oÅ¿tendi, Å¿eruando il <lb/>lam eandem Å¿uppoÅ¿itionem, hoc eÅ¿t quod interÅ¿titia <var>.e.h</var>: <var>h.u</var>: & c. figurÄ™ pag .75. æqua <lb/>lia Å¿int interÅ¿titijs <var>.e.h</var>: <var>h.u.</var> & c. præcedentis figuræ, & ideò in dicto cap .71. dixi.</s> - </p> - <quote> - <s xml:id="s3935" xml:space="preserve">SuppoÅ¿ito deinde <var>.f.e.b.</var> lineam meridianam eſſe in plano orizontali, cÄ™terÄ™ lineÄ™ <lb/>horariÄ™ erunt prÄ™dictÄ™.</s> - </quote> - <p> - <s xml:id="s3936" xml:space="preserve">Stantibus igitur his Å¿uppoÅ¿itis, vt habeantur omnia Å¿cientificè, volui, vt intellige-<lb/>retur pyra mis qua drilatera, eo modo quo dixi, cap .71. vbi clarè patet eandem pyra <lb/>midem eſſe, quam Pater Clauius (tacitè) poÅ¿uit in figura horologij, vt ipÅ¿e docuit <lb/>propoÅ¿itione Å¿ecunda, lib. Å¿ecundi, cuius baÅ¿is eÅ¿t triangulum <var>.H.I.F.</var> Å¿uæ figurÄ™ (exem <lb/>pli gratia pro quinta hora poÅ¿t meridiana) Alterum verò triangulum à me cogita-<lb/>tum, terminatum ab <var>.t.e</var>: <var>e.d</var>: et. ab <var>.t.d.</var> eleuata in mea figura, eÅ¿t in Å¿ua <reg norm="triangulum" type="context">triãgulum</reg> <var>.D.<lb/>I.F.</var> & propterea dixi.</s> - </p> - <quote> - <s xml:id="s3937" xml:space="preserve">Nam <var>.t.e.</var> et <var>.e.d.</var> <reg norm="vtræque" type="simple">vtræq;</reg> in plano horologii non Å¿unt, quamuis in plano æquatoris <lb/>tres Å¿int, & c.</s> - </quote> - <p> - <s xml:id="s3938" xml:space="preserve">Angulus verò <var>.e.</var> quem dico rectum eſſe, in Å¿ua figura eÅ¿t angulus <var>.D.I.F.</var> & mea - <pb o="324" rhead="IO. BAPT. BENED." n="336" file="0336" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0336"/> - <var>t.d.</var> imaginata, eÅ¿t Å¿ua <var>.D.F</var>. </s> - <s xml:id="s3939" xml:space="preserve">Tertium deinde triangulum, quod in mea figura ter-<lb/>minatur ab <var>.t.d.</var> ab <var>.f.d.</var> & ab <var>f.t.</var> in Å¿ua eÅ¿t triangulum <var>.D.F.H.</var> vnde mea <var>.f.t.</var> reÅ¿pon-<lb/>det Å¿uÄ™ <var>.H.D.</var> & mea <var>.f.d.</var> Å¿uæ <var>.H.F.</var> & mea <var>.t.d.</var> Å¿uÄ™ <var>.D.F</var>. </s> - <s xml:id="s3940" xml:space="preserve">Quartum autem triangulum <lb/><var>f.t.e.</var> in mea figura, reÅ¿pondet Å¿uo <var>.H.D.I.</var> & meum punctum <var>.t.</var> Å¿uo. </s> - <s xml:id="s3941" xml:space="preserve">D, Nunc triangu <lb/>lum rectangulum, quod dico Å¿eparatim conÅ¿tituere, eÅ¿t illud tertium dictum corre-<lb/>Å¿pondens Å¿uo <var>.D.F.H.</var> vt ipÅ¿e facit in <reg norm="Å¿equenti" type="context">Å¿equẽti</reg> figura, quod ipÅ¿e vocat <var>.D.C.H.</var> & <reg norm="meus" type="simple">meê°</reg><unsure/> <lb/>radius <var>.t.x.</var> in Å¿ua figura, ille eÅ¿t qui terminatur ab <var>.D.</var> & ab initio Tauri, & Virginis.</s> - </p> - <p> - <s xml:id="s3942" xml:space="preserve">Et quamuis ego non Å¿cripÅ¿erim talem Å¿iguram, vt ipÅ¿e fecit, nihilominus ipÅ¿am <lb/>verbis deÅ¿cripÅ¿i eomet modo, & propterea dixi.</s> - </p> - <quote> - <s xml:id="s3943" xml:space="preserve">Quam <reg norm="diuiÅ¿ionem" type="context">diuiÅ¿ionẽ</reg>, Å¿i in triangulo Å¿eorÅ¿um deÅ¿cripto inuenire voluerimus, res erit <lb/>inuentu facillima, cum rectum angulum <var>.f.t.d.</var> (reÅ¿pondentem Å¿uo <var>.H.D.C.</var>) prÄ™dicti <lb/>trianguli tertij ea ratione diuiÅ¿erimus, & c.</s> - </quote> - <p> - <s xml:id="s3944" xml:space="preserve">Quapropter Reuerendus Clauius non animaduertit meam rationem aliam non <lb/>eſſe, nec puncto longiorem Å¿ua, cum eademmet ipÅ¿a Å¿it.</s> - </p> - <p> - <s xml:id="s3945" xml:space="preserve">Citaui etiam MunÅ¿terum cap .30. eo quod in ea impreſſione, quam tunc prÄ™ mani <lb/>bus habui, vidi in ea figura, quam ipÅ¿e vocat fundamentum horologiorum, literam <lb/>c. poÅ¿itam eſſe loco <var>.f.</var> et <var>.f.</var> loco <var>.c.</var> quod cauſæ fuit, vt omnia mendoÅ¿a viderentur, re <lb/>centiores autem impreſſiones correctæ Å¿unt.</s> - </p> - <p> - <s xml:id="s3946" xml:space="preserve">RurÅ¿us alio in loco mihi accidit vt reprÄ™henderim Alexandrum Piccolomineum <lb/>in libris de Å¿phÄ™ra, qui quidem dicebat eas figuras Å¿uperficiales, quæ paucioribus an <lb/>gulis circunÅ¿criberentur, capaciores eſſe alijs, dummodo earum periphæriæ eſſent <lb/>æquales.</s> - </p> - <p> - <s xml:id="s3947" xml:space="preserve">Nunc autem correctÄ™ Å¿unt eo in loco impreſſiones, & qui non viderit primas, pu-<lb/>tabit me immeritò ipÅ¿um repræhendere.</s> - </p> - <p> - <s xml:id="s3948" xml:space="preserve">Idem etiam dico de eo capite ipÅ¿ius Piccolominei, in ijÅ¿dem libris, vbi tractat de <lb/>modo, quo vÅ¿i Å¿unt antiqui ad diuidendum zodiacum in .12. Å¿igna, quod erat circa <lb/>finem quarti libri.</s> - </p> - <p> - <s xml:id="s3949" xml:space="preserve">Nunc verò, in recentioribus impreſſionibus, illud caput poÅ¿itum non eÅ¿t. </s> - <s xml:id="s3950" xml:space="preserve">ImpreÅ¿ <lb/>Å¿iones autem illæ, vbi talia dixit, duæ fuerunt, quarum prima erat anni .1540. Å¿ecun <lb/>da verò .1552 Venetijs apud Andream Puteum.</s> - </p> - <p> - <s xml:id="s3951" xml:space="preserve">Alius verò locus ipÅ¿ius Reuerendi Clauij, contra meas reprÄ™henÅ¿iones, eÅ¿t circa <lb/>finem pag .298. & circa .299. vbi ita Å¿cribit.</s> - </p> - <quote> - <s xml:id="s3952" xml:space="preserve">Ex his liquido conÅ¿tat, non rectè à Ioan. BaptiÅ¿ta Benedicto in Å¿ua gnomonica ca <lb/>pit .49. reprÄ™hendi hancrationem deÅ¿cribendi horologij declinantis, qua omnes fe-<lb/>rè alij Å¿criptores vtuntur, quoniam, vt ex demonÅ¿tratione à nobis allata conÅ¿tat, re-<lb/>ctè per eam lineæ ho rariÄ™ in plano, quod à verticali declinat ducuntur. </s> - <s xml:id="s3953" xml:space="preserve">Modus au <lb/>tem quem eo loco pręſcribit differentem ab eo, quem nos tradidimus certus etiam <lb/>eÅ¿t, Å¿ed nulla ratione noÅ¿tro contrarius, quia nos conÅ¿tituimus <var>.D.E.F.</var> angulum de-<lb/>clinationis plani à verticali circulo propriè dicto, ipÅ¿e autem loco huius anguli aſſu-<lb/>mit angulum declinationis eiuÅ¿dem plani à Meridiano circulo, vnde mirum non eÅ¿t <lb/>modum ipÅ¿ius à noÅ¿tro diÅ¿crepare. </s> - <s xml:id="s3954" xml:space="preserve">Quod Å¿i <reg norm="conÅ¿titueremus" type="context">cõſtitueremus</reg> <var>.D.E.F.</var> angulum decli-<lb/>natio nis plani à Meridiano, ut ipÅ¿e (quemadmodum forÅ¿itan ab alijs putauit fieri) <lb/>& in reliqua deÅ¿criptione progrederemur, vt tradidimus, proculdubio <reg norm="horologium" type="context">horologiÅ©</reg> <lb/>declinans perperam deÅ¿criberetur, vt rectè docet.</s> - </quote> - <p> - <s xml:id="s3955" xml:space="preserve">Optimè Å¿cripÅ¿iſſet Reuerendus Clauius, Å¿i verum fuiſſet, quod antiqui Å¿umerent <lb/>declinationem Å¿uperius <reg norm="dictam" type="context">dictã</reg> à verticali propriè dicto, & non à meridiano. </s> - <s xml:id="s3956" xml:space="preserve">Sed ego <lb/>dico, authores à me citatos. capit .49. meÄ™ gnomonicÄ™ Å¿umere dictam declinatio- - <pb o="325" rhead="EPISTOLAE." n="337" file="0337" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0337"/> - nem planià meridiano, & non à dicto verticali.</s> - </p> - <p> - <s xml:id="s3957" xml:space="preserve">Con Å¿idera primum in MunÅ¿tero cap .16. Å¿uæ horologiographiæ, vbi clarè docet <lb/>accipere angulum compræhenÅ¿um inter meridianum, & planum propoÅ¿itum, vbi <lb/>etiam ponit quandam figuram ædificij cum pariete Å¿uper quo deÅ¿ignatum eÅ¿t quod <lb/>dam horologium, & vbi Å¿e manifeÅ¿tè declarar, ita dicens.</s> - </p> - <quote> - <s xml:id="s3958" xml:space="preserve">Nam ipÅ¿arum partium complementum. propoÅ¿itum indicabit angulum, quan-<lb/>tus videlicet fuerit arcus eiuÅ¿dem circuli <var>.d.e.f.g.</var> à puncto <var>.g.</var> vÅ¿que ad productam li-<lb/>neam meridianam interceptus, qui vnà cum ipÅ¿o <var>.f.g.</var> quadrantem integrare videtur, <lb/>vt in Å¿equenti figura: </s> - <s xml:id="s3959" xml:space="preserve">quoniam arcus <var>.f.g.</var> eÅ¿t Å¿exaginta partium, qualium <var>.e.</var> f<unsure/>. <reg norm="quadrans" type="context">quadrãs</reg> <lb/>nonaginta, vnde concluditur reliquam partem hoc eÅ¿t, datum inclinationis <reg norm="angulum" type="context">angulÅ©</reg>, <lb/>fore partium triginta Å¿imilium.</s> - </quote> - <p> - <s xml:id="s3960" xml:space="preserve">Orontius verò cap .13. <reg norm="ijÅ¿dem" type="context">ijÅ¿dẽ</reg> vtitur verbis, cum figura Å¿imili ad reliqua autem ipÅ¿ius <lb/>R. Clauij, videnda nondum mihi otium fuit. </s> - <s xml:id="s3961" xml:space="preserve">quod Å¿i dabitur, tibi libenter<unsure/> dicam <lb/>quid Å¿entiam.</s> - </p> - </div> - </div> - <div type="section" level="3" n="26"> - <div type="letter" level="4" n="1"> - <head xml:id="head489" xml:space="preserve">DE MODO DVCENDI LINEAS HORARIAS <lb/>Å¿uper cyllindro immobili.</head> - <head xml:id="head490" style="it" xml:space="preserve">Hieronymo Ferrerio artium & Medicina Doctori peritißimo.</head> - <p> - <s xml:id="s3962" xml:space="preserve"><emph style="sc">DEsignare</emph> horarias lineas Å¿uper cyllindro immobili, ad orizontemqÌue <lb/>perpendiculariter erecto difficile tibi non erit, (quod à me poÅ¿tulaÅ¿ti) Å¿i <lb/>modum .53. cap. meæ gnomonicæ obÅ¿eruaueris, accipiendo tamen pro <lb/>linea orizontali in tabula non aliquam rectam lineam, Å¿ed circularem, <lb/>Å¿imilemque circunferentiæ ipÅ¿ius cyllindri, dico autem Å¿imilem, eo quod Å¿i gn o-<lb/>mon <var>.o.x.</var> Å¿upra tabulam Å¿ignatus, & perpendicularis ipÅ¿i orizontali circulari <var>.b.i.x.</var> <lb/>eſſet dimidia, vel tertia vel quarta pars gnomonis cyllindro infixi, oporteret, vt <lb/>Å¿emidiameter circuli <var>.b.i.x.</var> etiam <lb/>eſſet medietas, vel tertia, aut quar <lb/> - <anchor type="figure" xlink:label="fig-0337-01a" xlink:href="fig-0337-01"/> - ta pars Å¿emidiametri cyllindri, vt <lb/>omnes arcus huiuÅ¿modi circuli in <lb/>ter ipÅ¿os azimut intercepti Å¿imiles <lb/>Å¿int arcubus cyllindri, quod à <lb/>te ipÅ¿o facilè videre Å¿cientificè po <lb/>teris. </s> - <s xml:id="s3963" xml:space="preserve">reliqua nihil mutanda erunt <lb/>ab eo, quod Å¿cripÅ¿i circa figuram <num value="53">.<lb/>53.</num> cap. vt dixi. </s> - <s xml:id="s3964" xml:space="preserve">Vnde inuenta <lb/>cum fuerit diÅ¿tantia orizontalis <lb/>puncti <var>.b.</var> à pede gnomonis <var>.x.</var> nec <lb/>non quantitas azi mutalis muralis <lb/><var>b.t.</var> quæ Å¿emper ab orizontali per <lb/>pendiculariter deÅ¿cendit, illicò <lb/>punctum <var>.t.</var> horæ propoÅ¿itæ in cy-<lb/>lindro inuenietur.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0337-01" xlink:href="fig-0337-01a"> - <image file="0337-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0337-01"/> - </figure> - </div> - <pb o="326" rhead="IO. BAPT. BENED." n="338" file="0338" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0338"/> - <p> - <s xml:id="s3965" xml:space="preserve">Nunc verò cum duo puncta alicuius horariÄ™ lineæ inuenta fuerint, quæ à Solis Å¿i-<lb/>tu in diuerÅ¿is parallelis efficiuntur, Å¿i voluerimus ipÅ¿am lineam <reg norm="horariam" type="context">horariã</reg> ducere, Å¿cien <lb/>dum primò eÅ¿t ipÅ¿am lineam horariam eſſe communem Å¿ectionem circuli horarij, <lb/>illius horæ cum Å¿uperficie cyllindrica, </s> - <s xml:id="s3966" xml:space="preserve">& propterea ellipticam, vt oÅ¿tendit Serenus <lb/>in .19. primi lib. quod etiam ellicere poſſumus ab eo, quod Archimedes in .10. pro-<lb/>poÅ¿itione libr. de conoidalibus, Å¿cribit. </s> - <s xml:id="s3967" xml:space="preserve">Quapropter oporter nos inÅ¿trumen-<lb/>tum prius componere, modo circini, Å¿ed trium crurum, quæ omnia in eadem <lb/>plana Å¿uperficie Å¿int, ea tamen arte factum, vt quodlibet illorum poſſimus pro-<lb/>longare, necnon contrahere, ut cum duo extrema firmata fuerint, media poÅ¿-<lb/>Å¿it circunduci circa centrum, Å¿eu punctum commune illarum interÅ¿ectionum <reg norm="Å¿imulque" type="simple">Å¿imulqÌ;</reg> <lb/>poſſit produci, necnon abbreuiari vel augeri, & diminui, vt mediante Å¿ua extremi-<lb/>tate inÅ¿eriori poſſimus delineare gyrum ellipticum horarium, dum <reg norm="centrum" type="context">cẽtrum</reg> ipÅ¿orum <lb/>crurum adhæreat extremitati gnomonis, reliquæ vero extremitates ipÅ¿orum <reg norm="crurum" type="context">crurÅ©</reg> <lb/>Å¿int Å¿upra puncta inuenta ipÅ¿ius horæ. </s> - <s xml:id="s3968" xml:space="preserve">oportet etiam vt hoc inÅ¿trumentum à tergo <lb/>ipÅ¿orum crurum habeat in Å¿uperiori parte Å¿uperficiem quandam <reg norm="Å¿emicircularem" type="context">Å¿emicircularẽ</reg>, quÄ™ <lb/>Å¿it vice vnius partis illius Å¿uperficiei, in qua Å¿upponuntur omnia crura inÅ¿trumenti, <lb/>& hoc quantum fieri poteÅ¿t, quod quidem fieri debet, ne crus medium, hoc eÅ¿t mo <lb/>bile, exeat à tali Å¿uperficie, Å¿eu declinet ab ea, quæ Å¿emper Å¿upponitur in Å¿itu circuli <lb/>horarij talis horæ. </s> - <s xml:id="s3969" xml:space="preserve">oportet etiam, vt iuxta circunferentiam dimidij circuli Å¿int duo <lb/>gyri eiuÅ¿dem materiæ inter Å¿e parum diÅ¿tantes, ita ut crura poſſint moueri, intra hos <lb/>gyros, & dimidium circulum, & quod inter hos gyros locatæ Å¿int duæ cochleæ, Å¿eu <lb/>duo helices, vt quando voluerimus, poſſimus fir-<lb/>mare ipÅ¿a crura extrema, dum eorum extremitates <lb/> - <anchor type="figure" xlink:label="fig-0338-01a" xlink:href="fig-0338-01"/> - fuerint Å¿upra puncta inuenta illius horæ, </s> - <s xml:id="s3970" xml:space="preserve">deinde in <lb/>dorÅ¿o iÅ¿tius inÅ¿trumenti, circa centrum coniunctio <lb/>nis, rectè factum erit Å¿i aliqua concauitas fuerit, in <lb/>qua, extremitas gnomonis poſſit locari, dum duce-<lb/>re voluerimus aliquam horariam lineam.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0338-01" xlink:href="fig-0338-01a"> - <image file="0338-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0338-01"/> - </figure> - </div> - <p> - <s xml:id="s3971" xml:space="preserve">Tale inÅ¿trumentum excogitaui ad fugiendum <lb/>tædium inueniendi dictam ellipticam ex punctis.</s> - </p> - <p> - <s xml:id="s3972" xml:space="preserve">Nunc autem Å¿ciendum eÅ¿t, quod vnus tantum-<lb/>modo gnomon Å¿ufficiens non erit pro tota die æſti-<lb/>ua, neque duo, niÅ¿i valde breues fuerint reÅ¿pectu <lb/>Å¿emidiametri cyllindri, & in Å¿itu medio quartarum <lb/>meridionalium noÅ¿tro orizonti, quorum autem <lb/>longitudo ita inuenienda eſſet.</s> - </p> - <p> - <s xml:id="s3973" xml:space="preserve">Sit exempli gratia circulus <var>.a.b.e.u.</var> cyllindri ori <lb/>zontis vice, <reg norm="diuiÅ¿usque" type="simple">diuiÅ¿usqÌ;</reg> à duobus diametris <var>.d.e.</var> et <var>.c.<lb/>f.</var> quarum <var>.c.f.</var> Å¿it pro meridiana: </s> - <s xml:id="s3974" xml:space="preserve">d.e. autem pro verticali, <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> e. punctus orientalis: </s> - <s xml:id="s3975" xml:space="preserve">d. <lb/>verò <reg norm="occidentalis" type="context">occidẽtalis</reg> <var>.f.</var> autem meridionalis. et <var>.c.</var> Å¿eptentrionalis, <reg norm="computeturque" type="simple">computeturqÌ;</reg> maxima. <lb/></s> - <s xml:id="s3976" xml:space="preserve">Solis amplitudo ab <var>.f.</var> verÅ¿us <var>.e.</var> quæ terminetur ab <var>.q.</var> ita <reg norm="quodarcus" type="simple">quodarcê°</reg> <var>.f.q.</var> minor Å¿it <reg norm="quam" type="context">quã</reg> <lb/>graduum .45. aliter impoſſibile eſſet duobus <reg norm="tantummodo" type="context">tantÅ©modo</reg> gnomonibus mediantibus <lb/>tota die æſtiua horas videre.</s> - </p> - <p> - <s xml:id="s3977" xml:space="preserve">Quo facto ducatur ab .q: <var>q.p.</var> contingens circulum & à centro circuli <var>.o.</var> per pun-<lb/>ctum <var>.u.</var> medium quartæ ducatur <var>.o.u.i.</var> vÅ¿que ad contingentem <var>.q.p.</var> vnde <var>.u.i.</var> longitu <lb/>do erit vniuÅ¿cuiuÅ¿que gnomonis, qui gnomones infixi erunt in medio dictarum <lb/>quartarum.</s> - </p> - <pb o="327" rhead="EPISTOL AE." n="339" file="0339" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0339"/> - <p> - <s xml:id="s3978" xml:space="preserve">HuiuÅ¿modi rei ratio per Å¿e nota erit quotieÅ¿cunque cogitauerimus verum arcum <lb/><var>e.b.</var> amplitudinis æſtiuæ, <reg norm="protractaque" type="simple">protractaqÌ;</reg> <var>.o.b.</var> quÄ™ parallela erit <var>.q.p.</var> vnde cum Sol tem-<lb/>pore æſtiuo orietur, tunc radios Å¿uos emittet via iÅ¿tarum æquidiÅ¿tantium linearum.</s> - </p> - <p> - <s xml:id="s3979" xml:space="preserve">Sed Å¿i longiores gnomones cuperes, oportebit eos tres eſſe, quorum vnus erit <lb/>orientalis in puncto <var>.e.</var> alter occi-<lb/>dentalis in puncto <var>.d.</var> reliquus ve-<lb/> - <anchor type="figure" xlink:label="fig-0339-01a" xlink:href="fig-0339-01"/> - rò meridionalis in puncto <var>.f.</var> quo-<lb/>rum <reg norm="vnuÅ¿quiÅ¿que" type="simple">vnuÅ¿quiÅ¿q;</reg> poteÅ¿t eſſe maior <lb/>tertia parte Å¿emidiametri cyllin-<lb/>dri, Å¿ed Å¿i voluerimus Å¿cire <reg norm="quan- tum" type="context">quan-<lb/>tÅ©</reg> ad plus poſſit eſſe longus vnuÅ¿-<lb/>quiÅ¿que illorum, ita faciendum <lb/>erit.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0339-01" xlink:href="fig-0339-01a"> - <image file="0339-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0339-01"/> - </figure> - </div> - <p> - <s xml:id="s3980" xml:space="preserve">Faciemus <reg norm="quadratum" type="context">quadratÅ©</reg> <var>.o.a.h.u.</var> ex <lb/>Å¿emidiametro dicti circuli, a dia-<lb/>metro poÅ¿tea <var>.o.h.</var> huiuÅ¿modi qua <lb/>drati Å¿ubtrahatur Å¿emidiameter <var>.<lb/>o.e.</var> circuli, reÅ¿iduum verò <var>.e.h.</var> ip-<lb/>Å¿ius diametri <var>.o.h.</var> quadrati, erit <reg norm="lon" type="context">lõ</reg> <lb/>gitudo gnomonis, vbi Å¿imul appa <lb/>ret huiuÅ¿modi rei ratio, eo quod <lb/>cum gnomon <var>.e.h.</var> orientalis deÅ¿i-<lb/>net operari, illico meridionalis <var>.<lb/>f.g.</var> Å¿ubintrabit, poÅ¿t hunc verò occidentalis <var>.d.K.</var> monÅ¿trabit reliquum diei.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head491" style="it" xml:space="preserve">Earundem line arum deÅ¿criptio Å¿aper conum rectum.</head> - <head xml:id="head492" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s3981" xml:space="preserve">CVm Å¿uper datum conum re-<lb/> - <anchor type="figure" xlink:label="fig-0339-02a" xlink:href="fig-0339-02"/> - ctum idem facere volueris <lb/>eÅ¿to conus <var>.A.</var> & R. qui diuiÅ¿us ima <lb/>ginatione Å¿it à quodam plano per <lb/>axem, & communis Å¿ectio Å¿it trian <lb/>gulus <var>.A.</var> & R. in quo plano cogite <lb/>mus gnomonem infixum ad rectos <lb/>vbi volueris, qui Å¿it <var>.p.t.o.</var> cogitem <lb/>etiam <var>.l.t.m.</var> aliud eſſe planum (in <lb/>quo Å¿it gnomon) quod conum Å¿e <lb/>cet, quæ quidem Å¿ectio, circularis <lb/>erit, ex .4. primi Pergei. </s> - <s xml:id="s3982" xml:space="preserve">imagine-<lb/>mur etiam Å¿uperficiem <var>.p.s.</var> eſſe azi <lb/>mut in quo gnomonreperirur, Å¿u-<lb/><reg norm="perficiemque" type="simple">perficiemqÌ;</reg> <var>.e.s.</var> azimut propoÅ¿itæ <lb/>horÄ™, <reg norm="angulumque" type="simple">angulumqÌ;</reg> <var>.e.o.a.</var> contrapoÅ¿i <lb/><reg norm="tum" type="context">tÅ©</reg> angulo altitudinis Solis ab ori- - <pb o="328" rhead="IO. BAPT. BENED." n="340" file="0340" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0340"/> - zonte; </s> - <s xml:id="s3983" xml:space="preserve">cogitemus etiam lineam <var>.A.t.i.x.</var> illud coni latus eſſe, qu od à ſummitate verÂ<lb/>Å¿us baÅ¿im tranÅ¿it per medium latitudinis ipÅ¿ius gnomonis, concipiamus etiam mente <lb/><var>e.a.</var> communem Å¿ectionem eſſe trianguli Å¿upra dicti cum azimut horæ, necnon pun-<lb/>ctum <var>.K.</var> eſſe commune radio Solis <var>.o.a.</var> & Å¿uperficiei conicæ, quod quidem eÅ¿t illud <lb/>quod quæritur, hoc Å¿cilicet modo. </s> - <s xml:id="s3984" xml:space="preserve">Primum cognoÅ¿cimus angulum <var>.p.A.t.</var> vt medie <lb/>tas anguli totius coni, & angulum <var>.p.</var> rectum, vnde <var>.t.</var> tam intrinÅ¿ecus, quam extrinÅ¿e-<lb/>custrianguli <var>.A.p.t.</var> nobis cognitus erit. </s> - <s xml:id="s3985" xml:space="preserve">Nunc cum angulus <var>.A.t.o.</var> cognoÅ¿catur, Å¿i <lb/>gnomon <var>t.o.</var> fixus fuerit in Å¿uperficie conica, ita qd cum latere <var>.A.t.</var> eſſiciat <reg norm="angulum" type="context">angulÅ©</reg> <lb/><var>A.t.o.</var> & lateraliter faciat angulosrectos cum Å¿uperficie conica, ad quod efficiendum <lb/>nulla eÅ¿t difficultas, cognoÅ¿cendo deinde <var>.A.t.</var> Å¿imul cum angulis <var>.A.</var> et <var>.t.</var> intrinÅ¿ecis <lb/>trianguli ortogonij <var>.A.p.t.</var> cognoÅ¿cemus <var>.p.t.</var> et <var>.A.p.</var> vnde etiam tota <var>.o.p.</var> Å¿ed cogno <lb/>Å¿cendo <var>.o.p.</var> cum angulo <var>.p.o.e.</var> (angulus enim <var>.p.o.e.</var> cognoÅ¿citur ex hypotheÅ¿i cum <lb/>Å¿it inter azimut Solis & azimut gnomonis) cum angulo <var>.o.p.e.</var> recto cognoÅ¿cemus <var>.p.<lb/>e.</var> et <var>.o.e.</var> </s> - <s xml:id="s3986" xml:space="preserve">deinde cum nobis nota Å¿it <var>.o.e.</var> cum angulo altitudinis Solis <var>.e.o.a.</var> & angu-<lb/>lo <var>.o.e.a.</var> recto cognoÅ¿c emus longitudinem azimutalis <var>.e.a.</var> necnon quantitatem <var>.a.o.</var> <lb/>Imaginata poÅ¿tea <var>.a.q.</var> æquidiÅ¿tante <var>.e.p.</var> habebimus <var>.p.q.</var> æqualem <var>.a.e.</var> ex .34. primi <lb/>Eucli. </s> - <s xml:id="s3987" xml:space="preserve">Vnde duabus <var>.o.p.</var> et <var>.p.q.</var> mediantibus, <reg norm="cognitiÅ¿que" type="simple">cognitiÅ¿qÌ;</reg> cum angulo recto <var>.p.</var> cogno <lb/>Å¿cemus <var>.o.q.</var> nec non angulum <var>.o.q.<lb/>p.</var> quo mediante, necnon median-<lb/>te angulo <var>.q.A.t.</var> et <var>.A.q.</var> cognita, co <lb/> - <anchor type="figure" xlink:label="fig-0340-01a" xlink:href="fig-0340-01"/> - gnoÅ¿cemus <var>.A.i.</var> et <var>.q.i.</var> quÄ™ <var>.q.i.</var> dem <lb/>pta à <var>.q.o.</var> relinquet nobis <reg norm="cognitam" type="context">cognitã</reg> <lb/><var>i.o</var>. </s> - <s xml:id="s3988" xml:space="preserve">Et quia <var>.o.i.q.</var> et <var>.o.K.a.</var> Å¿emper <lb/>Å¿unt in eadem Å¿uperficie Å¿ecante co <lb/>num, quæ etiam Å¿ecat Å¿uperficiem <lb/>trianguli <var>.A.q.x.</var> ad rectos ex .18. vn <lb/>decimi, cum linea <var>.u.n.</var> perpendicu <lb/>laris Å¿it Å¿uperficiei trianguli <var>.A.q.i.</var> <lb/>ex .8. dicti, quia parallela eÅ¿t <var>.l.p.</var> quÄ™ <lb/>perpendicularis eÅ¿t Å¿uperficiei <reg norm="trian- guli" type="context">triã-<lb/>guli</reg> <var>.o.p.q.</var> ex .4. eiuÅ¿dem, Å¿equitur, <lb/>quod talis Å¿ectio ( quæ intelligatur <lb/>per <var>.u.K.i.n.</var>) Å¿emper erit elliptica, <lb/>vel parabole, Å¿eu hyperbole, <reg norm="prout" type="simple">ê“ut</reg> <lb/>linea <var>.o.i.q.</var> Å¿ecabit latus coni, oppo <lb/>Å¿itum lateri <var>.A.i.</var> diÅ¿tento in ipÅ¿a Å¿uperficie conica, Å¿eu ad Å¿uperiorem partem produ <lb/>ctum, velipÅ¿i parallelum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0339-02" xlink:href="fig-0339-02a"> - <image file="0339-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0339-02"/> - </figure> - <figure xlink:label="fig-0340-01" xlink:href="fig-0340-01a"> - <image file="0340-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0340-01"/> - </figure> - </div> - <p> - <s xml:id="s3989" xml:space="preserve">Supponamus nunc dictam lineam <var>.o.q.</var> Å¿ecare dictum oppoÅ¿itum latus lateri <var>.A.i.</var> <lb/>verÅ¿us baÅ¿im, vnde Å¿ectio <var>.u.K.i.n.</var> erit elliptica. </s> - <s xml:id="s3990" xml:space="preserve">quod facile cognitu eÅ¿t <reg norm="mediante" type="context">mediãte</reg> com <lb/>paratione angulorum <var>.A.q.i.</var> et <var>.q.A.i.</var> interÅ¿e, eo quod Å¿i eſſent Ä™quales, dicta Å¿ect o <lb/>barabola eſſet ex .27. primi Eucli. et .11. primi Pergei, Å¿ed Å¿i angulus <var>.A.q.i.</var> maior eÅ¿-<lb/>Å¿et angulo <var>.q.A.i.</var> Å¿ectio eſſet ellipÅ¿is, ex ultimo poÅ¿tulato primi Euclid. </s> - <s xml:id="s3991" xml:space="preserve">& ex .13. pri-<lb/>mi Pergei, Å¿ed Å¿i dictus angulus <var>.A.q.i.</var> minor eſſet angulo <var>.A.</var> tunc Å¿ectio eſſet hyper-<lb/>bole ex dicto poÅ¿tulato & ex .12. primi Pergei. </s> - <s xml:id="s3992" xml:space="preserve">Sit ergo primum vt <reg norm="dictum" type="context">dictÅ©</reg> eÅ¿t, hoc <lb/>eÅ¿t, quod Å¿ectio eſſet oxygonia, ideÅ¿t elliptica, Å¿eu defectio (quod idem eÅ¿t,) Å¿epa-<lb/>ratim oportebit nos ellipÅ¿im deÅ¿ignare <reg norm="Å¿imilem" type="context">Å¿imilẽ</reg> <reg norm="Ä™qualemque" type="context simple">Ä™qualẽqÌ;</reg> ei, quæ eÅ¿t <var>.u.K.i.n.</var> <reg norm="quod" type="wordlist">qÄ‘</reg> <reg norm="quidem" type="context">quidẽ</reg> <lb/>difficile non erit, quotieÅ¿cunque Å¿uos axes inuenerimus, maiorem Å¿cilicet, & mino- - <pb o="329" rhead="EPISTOL AE." n="341" file="0341" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0341"/> - rem, quæ ita reperientur, efficiemus primo anguium coni, qui Å¿it <var>.i.A.b.</var> quem diui-<lb/>demus per æqualia mediante <var>.A.q.</var> conÅ¿tituendo <var>.A.i.</var> huius anguli æqualem <var>.A.i.</var> Å¿u-<lb/>perficiei conicæ et <var>.A.q.</var> diuidentem, æqualem parti <var>.A.q.</var> axis coni, ducendo poÅ¿tea <lb/>ab <var>.i.</var> per <var>.q.</var> lineam vnam quouÅ¿que concurrat <var>.A.b.</var> in puncto <var>.b.</var> habebimus <var>.i.b.</var> pro <lb/>maiori axi ipÅ¿i ellipÅ¿is, quod per Å¿e clarum eÅ¿t, cuius medietas Å¿it <var>.i.c.</var> Å¿ed <var>.i.q.</var> ipÅ¿ius <var>.i.<lb/>b.</var> æqualis eÅ¿t ipÅ¿i <var>.q.i.</var> ipÅ¿ius coni, ex quarta primi Eucli. et <var>.q.b.</var> ipÅ¿ius <var>.i.b.</var> æqualis alte <lb/>ri parti inuiÅ¿ibili. </s> - <s xml:id="s3993" xml:space="preserve">Reliquum eÅ¿t, vt reperiamus minorem axem, quem vocabimus <var>.<lb/>f.r.</var> ducatur ergo primum <var>.q.a.u.n.</var> ad rectos cum <var>.i.b.</var> <reg norm="æqualisque" type="simple">æqualisqÌ;</reg> ei quæ eÅ¿t coni, & diui <lb/>Å¿a Å¿imiliter in <var>.a.</var> quæ <var>.u.n.</var> ipÅ¿ius coni nobis cognita eÅ¿t ex lateribus <var>.A.u.</var> et <var>.A.n.</var> & ex <lb/>angulo coni, et <var>.a.q.</var> æqualis eÅ¿t <var>.e.p.</var> ex .34. primi. </s> - <s xml:id="s3994" xml:space="preserve">Nunc certi erimus ex .21. primi <lb/>Pergei, quod eadem proportio erit quadrati <var>.u.q.</var> ad quadratum ipÅ¿ius <var>.f.c.</var> quæ pro-<lb/>ducti ipÅ¿ius <var>.i.q.</var> in <var>.q.b.</var> ad productum ipÅ¿ius <var>.i.c.</var> in <var>.c.b.</var> & cum cognita nobis Å¿int <lb/>hæc tria producta hoc eÅ¿t <var>.i.q.</var> in <var>.q.b.</var> et <var>.i.c.</var> in <var>.c.b.</var> et <var>.u.q.</var> in Å¿eipÅ¿a, cognoÅ¿cemus <reg norm="etiam" type="context">etiã</reg> <lb/>quartum ipÅ¿ius <var>.f.c.</var> & fic <var>.f.c.</var> <reg norm="eiuÅ¿que" type="simple">eiuÅ¿qÌ;</reg> duplum <var>.f.r.</var> cogniti nobis itaque cum Å¿int hi duo <lb/>axes <var>.i.b.</var> et <var>.f.r.</var> formabimus ellipÅ¿im. </s> - <s xml:id="s3995" xml:space="preserve">Deinde producemus axim <var>.b.i.</var> à part <var>e.i.</var> quo-<lb/>uÅ¿que <var>.i.o.</var> æqualis Å¿it ei quæ extra conum eÅ¿t, dein-<lb/> - <anchor type="figure" xlink:label="fig-0341-01a" xlink:href="fig-0341-01"/> - de ducemus <var>.o.a.</var> quæ circunferentiam ellipticam <lb/>Å¿ecabit in puncto <var>.K.</var> vnde habebimus quantita-<lb/>tem ipÅ¿ius <var>.o.K.</var> et <var>.K.i.</var> rectam. </s> - <s xml:id="s3996" xml:space="preserve">inde mediante cir-<lb/>cino Å¿i acceperimus rectam diÅ¿tantiam ab <var>.i.</var> ad <var>.K.</var> <lb/>in ellipÅ¿i, </s> - <s xml:id="s3997" xml:space="preserve">deinde firmando pedem circini in pun-<lb/>cto <var>.i.</var> in Å¿uperficie conica, & cum alio Å¿ignando <lb/>lineam vnam curuam ad partem <var>.K.</var> in Å¿uperficie <lb/>conica, Å¿umendo poÅ¿tea interuallum <var>.o.K.</var> extra el <lb/>lipÅ¿im, </s> - <s xml:id="s3998" xml:space="preserve">deinde firmando vnum pedem circini in <lb/>extre mitate gnomonis, cum alio poÅ¿tea Å¿ignan-<lb/>do aliam lineam curuam in Å¿uperficie ipÅ¿ius coni, <lb/>quæ primam Å¿e cet in puncto <var>.K.</var> hoc erit punctum <lb/>quæſitum horÄ™ propoÅ¿itæ in Å¿uperficie conica <lb/>propoÅ¿ita.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0341-01" xlink:href="fig-0341-01a"> - <image file="0341-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0341-01"/> - </figure> - </div> - <p> - <s xml:id="s3999" xml:space="preserve">Sed Å¿i talis Å¿ectio fuerit parabole, vel hyperbo <lb/>le, tunc mediante Å¿uo diametro <var>.i.q.</var> cum baÅ¿i <var>.u.<lb/>q.n.</var> cognita, deÅ¿ignabimus ipÅ¿am Å¿ectionem <var>.u.i.</var> n <lb/>ope mei <reg norm="inÅ¿trumenti" type="context">inÅ¿trumẽti</reg> in calce meÄ™ gnomonicæ de <lb/>Å¿cripti, </s> - <s xml:id="s4000" xml:space="preserve">deinde diuiÅ¿a <var>.u.q.</var> in <var>.a.</var> <reg norm="pro" type="simple">ê“</reg><reg norm="ductaque" type="simple">ductaqÌ;</reg> <var>q.i.</var> <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> <lb/> - <anchor type="figure" xlink:label="fig-0341-02a" xlink:href="fig-0341-02"/> - ad <var>.o.</var> <reg norm="ductaque" type="simple">ductaqÌ;</reg> <var>.o.a.</var> habebimus punctum <var>.K</var>. </s> - <s xml:id="s4001" xml:space="preserve">Reli-<lb/>qua facienda Å¿unt, vt dictum eÅ¿t de ellipÅ¿i.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0341-02" xlink:href="fig-0341-02a"> - <image file="0341-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0341-02"/> - </figure> - </div> - <p> - <s xml:id="s4002" xml:space="preserve">Inuenta modo cum fuerint duo puncta eiuÅ¿-<lb/>dem horæ propoÅ¿itÄ™, ducemus ab vno ad a-<lb/>liud, lineam horariam mediante circino trium <lb/>crurum, quem tibi Å¿cripÅ¿i nudius tertius pro cyl <lb/>lindro, quæ <reg norm="quidem" type="context">quidẽ</reg> linea crit portio gyri ellipÅ¿is, <lb/>Å¿eu hyperbolÄ™, vel parabolÄ™, vt à te ipÅ¿o cogi-<lb/>tare potes.</s> - </p> - <pb o="330" rhead="IO. BAPT. BENED." n="342" file="0342" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0342"/> - </div> - </div> - <div type="section" level="3" n="27"> - <div type="letter" level="4" n="1"> - <head xml:id="head493" xml:space="preserve">QVAEDAM NOTATV DIGNA IN <lb/>Ptolomeum.</head> - <head xml:id="head494" style="it" xml:space="preserve">Bartolomeo Christino Serenißimi SabaudiÅ“ Ducis apparitore.</head> - <p> - <s xml:id="s4003" xml:space="preserve">EX tuis literis cognoui quo erga me animo eſſes, <reg norm="qualique" type="simple">qualiqÌ;</reg> voluntate, Å¿ed ne <lb/>tua pulcherrima Å¿tudia aliquo modo imperfecta <reg norm="relinquantur" type="simple">relinquant̃</reg>, vel ego tibi <lb/>deeſſe videar, dum Problemata geographica Magni Ptolomei conÅ¿i-<lb/>deras, aduerte, quod Å¿i putares in figura .6. cap. libr .7. geographiÄ™ eiuÅ¿-<lb/>dem (vt multi credunt) lineam <var>.V.*.</var> Å¿ecare circunferentiam <var>.A.D.</var> in puncto <var>.G.</var> ita <lb/>vt punctus <var>.G.</var> Å¿it tropici æſtiui, ideÅ¿t arcum <var>.D.G.</var> eſſe graduum .24. cum illis inci-<lb/>deres in maximum errorem. </s> - <s xml:id="s4004" xml:space="preserve">Quapropter conÅ¿idera quæ nunc tibi Å¿cribo.</s> - </p> - <p> - <s xml:id="s4005" xml:space="preserve">Sit circulus <var>.A.B.C.D.</var> huius centrum <var>.E.</var> <reg norm="Å¿upponaturque" type="simple">Å¿upponaturqÌ;</reg> Å¿emidiameter <var>.E.D.</var> eſſe <lb/>partium 120. quarum <var>.E.Z.</var> in alio Å¿emidiametro <var>.C.E.</var> ei orthogonaliter coniuncto, <lb/>talium Å¿it .17. in Å¿emidiametro vero <var>.E.A.</var> accipiatur <var>.E.S.</var> talium .24. et <var>.E.V.</var> 64. vn <lb/>de <var>.S.V.</var> erit partium 40. Å¿imilium.</s> - </p> - <p> - <s xml:id="s4006" xml:space="preserve">Erigatur deinde <var>.S.*.</var> ad rectos cum <var>.E.A.</var> in puncto <var>.S.</var> quæ terminetur ab inter-<lb/>Å¿ectione lineÄ™ ductæ per puncta <var>.Z.D.</var> in puncto <var>.*.</var> ducatur demum <var>.V.*.</var> quæ Å¿eca-<lb/>bit circunf rentiam <var>.A.D.</var> in puncto <var>.G</var>. </s> - <s xml:id="s4007" xml:space="preserve">Quæratur nunc quantitas ipÅ¿ius <var>.G.D.</var> <lb/>Ad quod efficiendum quærenda primum eÅ¿t quantitas ipÅ¿ius <var>.S.*.</var> quam illico co <lb/>gnoÅ¿cemus ex regula de tribus, cum dixerimus, Å¿i@ 17. dat nobis .120. quid dabit .41. <lb/>(nam duo triangula <var>.Z.E.D.</var> et <var>.Z.S.*.</var> Å¿unt inuicem Å¿imilia, cum <var>.S.*.</var> parallela Å¿it <lb/>ipÅ¿i <var>.E.D.</var>) vnde <var>.S.*.</var> proueniet nobis ex Å¿imilibus partibus .289. cum fracto, quod <lb/>r<unsure/>eijciamus ob minorem laborem.</s> - </p> - <p> - <s xml:id="s4008" xml:space="preserve">Producantur poÅ¿tea <var>.V.*.</var> et <var>.E.D.</var> vÅ¿que ad eorum concurÅ¿um in puncto <var>.ω.</var> quæ-<lb/><reg norm="remusque" type="simple">remusqÌ;</reg> quanta Å¿it <var>.E.ω.</var> ex eadem regula, cum dixerimus, Å¿i .40. dat nobis .289. quid <lb/> - <anchor type="figure" xlink:label="fig-0342-01a" xlink:href="fig-0342-01"/> - <pb o="331" rhead="EPISTOL AE." n="343" file="0343" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0343"/> - dabit .64. (nam duo triangula <var>.V.S.*.</var> et <var>.V.E.ω.</var> Å¿unt inuicem Å¿imilia eadem ratio-<lb/>ne) vnde <var>.E.ω.</var> veniet nobis extalibus partibus .462.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0342-01" xlink:href="fig-0342-01a"> - <image file="0342-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0342-01"/> - </figure> - </div> - <p> - <s xml:id="s4009" xml:space="preserve">Coniungatur nunc quadratum ipÅ¿ius <var>.E.V.</var> quod eÅ¿t .4096. cum quadrato ipÅ¿ius <var>.<lb/>E.ω.</var> quod eÅ¿t .213444. & habebimus quadratum ipfius <var>.V.ω.</var> talium <reg norm="partium" type="context">partiÅ©</reg> .217540.</s> - </p> - <p> - <s xml:id="s4010" xml:space="preserve">Dicemus poÅ¿tea Å¿i .217540. dat nobis .4096. quid dabit quadratum ipÅ¿ius <var>.V.ω.</var> vt <lb/>Å¿inus totus quod eÅ¿t .10000000000. vnde veniet pro quadrato ipÅ¿ius <var>.V.E.</var> talium <lb/>partium, Å¿uperficialium Å¿cilicet .18827211. cuius radix erit .13721. & erit Å¿inus an-<lb/>guli <var>.V.ω.E.</var> qui erit grad .7. min .53. vnde angulus <var>.ω.V.E.</var> erit grad .82. min .7. eius <lb/>vero Å¿inus erit partium .99054.</s> - </p> - <p> - <s xml:id="s4011" xml:space="preserve">Nunc autem quia angulus <var>.E.V.ω.</var> eÅ¿t acutus, imaginemur <var>.E.Å”.</var> ductam eſſe ad reÂ<lb/>ctos ipÅ¿i <var>.V.ω.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> etiam ducta ipÅ¿a <var>.E.G</var>. </s> - <s xml:id="s4012" xml:space="preserve">Vnde habebimus angulum <var>.Å”.E.V.</var> gra-<lb/>duum .7. min .53. eius vero Å¿inus <var>.Å”.V.</var> partium .13721. (propter Å¿imilitudinem trianÂ<lb/>gulorum <var>.E.Å”.V.</var> et <var>.ω.E.V.</var>) talium Å¿cilicet, qualium <var>.E.V.</var> fuerit .100000. </s> - <s xml:id="s4013" xml:space="preserve">Sed qua-<lb/>lium <var>.E.V.</var> eÅ¿t .64. talium erit .8. cum tribus quartis, cuius <var>.Å”.V.</var> quadratum erit parÂ<lb/>tium .76. cum dimidio Å¿imilium Å¿ed Å¿uperÅ¿icialium, quo quidem quadrato dempto <lb/>ex quadrato ipÅ¿ius .64. quod eÅ¿t .4096. remanebit quadratum ipÅ¿ius <var>.E.Å”.</var> partium <num value="2871">.<lb/>2871.</num> quo etiam quadrato <var>.E.Å”.</var> dempto ex quadrato <var>.E.G.</var> partium .14400. remaneÂ<lb/>bit quadratum ipÅ¿ius <var>.Å”.G.</var> partium .11529. cuius radix <var>.Å”.G.</var> erit partium .107. <reg norm="talium" type="context">taliÅ©</reg> <lb/>qualium <var>.E.G.</var> eÅ¿t .120. Å¿ed qualium <var>.E.G.</var> erit .100000. talium <var>.Å”.G.</var> erit partium <num value="89166">.<lb/>89166.</num> quæ vt Å¿inus anguli <var>.Å”.E.G.</var> habebit pro ipÅ¿o angulo, gra .63. min .5. qui colleÂ<lb/>cti cum gra .7. min .53. anguli <var>.V.E.Å”.</var> dabunt totum angulum <var>.A.E.G.</var> grad .70. <lb/>min .58. cuius complementum ex grad .90. erit <var>.G.D.</var> graduum .19. min .2. & non .24. <lb/>vt omnes ferè putant.</s> - </p> - </div> - </div> - <div type="section" level="3" n="28"> - <div type="letter" level="4" n="1"> - <head xml:id="head495" xml:space="preserve">DE REFLEXIONIBVS RADIORVM.</head> - <head xml:id="head496" style="it" xml:space="preserve">Excellentißimo PhiloÅ¿opbo FranciÅ¿co Vimercato.</head> - <p> - <s xml:id="s4014" xml:space="preserve"><emph style="sc">QVoniam</emph> non videbatur quieÅ¿cere animus tuus, cum paucis ab hinc die-<lb/>bus tibi Å¿iÅ¿citanti reÅ¿pondiſſem, nec tamen rationem omnium, quæ dixe-<lb/>ram exactè explicare per tem poris anguÅ¿tiam potuiſſem, cogitaui ad te <lb/>per hanc occaÅ¿ionem Å¿cribens, & iam dicta repetere, & omnium tibi ra-<lb/>tionem Å¿ubiungere, & vt mihi plenius Å¿atisfaciam, & tibi commodè perlegenti faci <lb/>lius Å¿it veritatem intueri. </s> - <s xml:id="s4015" xml:space="preserve">ScripÅ¿iÅ¿ti enim in tuis diÅ¿putationibus, vir doctiſſime, quod <lb/>omnis res viÅ¿a per <reg norm="Å¿peculum" type="context">Å¿peculÅ©</reg> <reg norm="quodcunque" type="context">quodcÅ©que</reg>, Å¿ub breuiſſimis lineis <reg norm="compræhendatur" type="context">cõpræhendatur</reg> à vifu.</s> - </p> - <p> - <s xml:id="s4016" xml:space="preserve">PropoÅ¿itio hæc non eÅ¿t vniuerÅ¿aliter vera (quamuis etiam ab alijs omnibus pro ta <lb/>li poÅ¿ita Å¿it) cum in Å¿peculis concauis non Å¿emper verificetur, vt nunc tibi demon-<lb/>Å¿trabo.</s> - </p> - <p> - <s xml:id="s4017" xml:space="preserve">EÅ¿to quod linea recta <var>.b.d.</var> tangat circulum <lb/> - <anchor type="figure" xlink:label="fig-0343-01a" xlink:href="fig-0343-01"/> - <var>b.o.q.n.</var> qui Å¿it communis Å¿ectionis Å¿up crficiei re <lb/>flexionis, & Å¿phæricÄ™ alicuius Å¿peculi Å¿phærici <lb/>concaui, & punctum contingentiæ Å¿it <var>.b.</var> à quo <lb/>exeant duæ lineæ <var>.b.q.</var> et <var>.b.n.</var> efficientes duos an <lb/>gulos inuicem æquales circa perpendicularem <var>.<lb/>b.c.</var> res autem viÅ¿a primò Å¿it in ipÅ¿a circunferen-<lb/>tia huiuÅ¿modi circuli in puncto <var>.n.</var> oculus vero in <lb/>puncto <var>.q.</var> ipÅ¿ius circunferentiÄ™. </s> - <s xml:id="s4018" xml:space="preserve">Dico nunc duas - <pb o="332" rhead="IO. BAPT. BENED." n="344" file="0344" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0344"/> - lineas <var>.b.q.</var> et <var>.b.n.</var> Å¿imul Å¿umptas longiores eſſe omnibus alijs lineis exeuntibus ab ip <lb/>Å¿is punctis <var>.q.n.</var> quæ in aliquo puncto dictæ circunferentiæ Å¿imul concurrant.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0343-01" xlink:href="fig-0343-01a"> - <image file="0343-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0343-01"/> - </figure> - </div> - <p> - <s xml:id="s4019" xml:space="preserve">Sint igitur aliæ duæ <var>.q.o.</var> et <var>.n.o.</var> quas probare volo Å¿imul Å¿umptas, eſſe minores dua <lb/>bus Å¿imul Å¿umptis <var>.q.b.</var> et <var>.n.b</var>. </s> - <s xml:id="s4020" xml:space="preserve">Nam ex .20. tertij Eucli. cognoÅ¿cimus angulos <var>.q.b.n.</var> <lb/>et <var>.q.o.n.</var> inuicem æquales eſſe, & Å¿imiliter angulos <var>.b.n.o.</var> et <var>.b.q.o</var>. </s> - <s xml:id="s4021" xml:space="preserve">deinde ex .15. pri <lb/>mi eiuÅ¿dem habemus angulos contra Å¿e poÅ¿itos, <lb/>circa <var>.a.</var> eſſe etiam inuicem Ä™quales. </s> - <s xml:id="s4022" xml:space="preserve">Vnde ex .4 <lb/> - <anchor type="figure" xlink:label="fig-0344-01a" xlink:href="fig-0344-01"/> - Å¿exti, habebimus proportionem <var>.a.b.</var> ad .a<lb/>o. eandem eſſe, quæ <var>.a.n.</var> ad <var>.a.q.</var> & Å¿ic .b<lb/>n. ad <var>.o.q</var>. </s> - <s xml:id="s4023" xml:space="preserve">Quare ita erit <var>.a.b.n.</var> ad <var>.a.o.q.</var> vt <var>.a.n</var> <lb/>ad <var>.a.q.</var> Å¿ed cum <var>.a.n.</var> maior Å¿it <var>.q.a.</var> ex .18. primi, <lb/>eo quod angulus <var>.b.q.n.</var> (qui æqualis eÅ¿t angulo <var>.<lb/>b.n.q.</var> ex .5. eiuÅ¿dem) maior eÅ¿t angulo <var>.a.n.q.</var> <lb/>qui pars eÅ¿t ipÅ¿ius <var>.b.n.q.</var> ergo latera Å¿imul Å¿um-<lb/>pta <var>.a.b.n.</var> maiora erunt lateribus <var>.a.o.q.</var> Å¿ed ex <num value="20">.<lb/>20.</num> primi <var>.a.b.n.</var> <reg norm="etiam" type="context">etiã</reg> maior erit <var>.a.n.</var> vnde ex .25. <lb/>quinti <var>.q.a.b.n.</var> maior erit <var>.n.a.o.q</var>. </s> - <s xml:id="s4024" xml:space="preserve">quare Å¿equi-<lb/>tur verum eſſe propofitum.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0344-01" xlink:href="fig-0344-01a"> - <image file="0344-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0344-01"/> - </figure> - </div> - <p> - <s xml:id="s4025" xml:space="preserve">Sed Å¿i oculus eſſet in <var>.u.</var> quemadmodum in Å¿ubÅ¿cripta hic <reg norm="Å¿ecunda" type="context">Å¿ecÅ©da</reg> figura videre eÅ¿t, <lb/>res autem viÅ¿ibilis in <var>.n.</var> ambo extra dictum circulum, eÅ¿to etiam primum <var>.b.u.</var> æqua-<lb/>lis <var>.b.n.</var> probabo Å¿imiliter <var>.u.b.n.</var> maiores eſſe <var>.u.o.n</var>. </s> - <s xml:id="s4026" xml:space="preserve">Nam angulus <var>.o.</var> maior eÅ¿t angu-<lb/>lo <var>.b.</var> eo quod Å¿i circulum <var>.u.b.n.</var> cogitemus circunÅ¿cribere triangulum <var>.u.b.n.</var> ducen-<lb/>do vÅ¿que ad Å¿uam circunferentiam <var>.o.n.</var> in puncto <var>.s.</var> deinde ducendo <var>.u.s.</var> habebimus <lb/>ex .20. tertij angulum <var>.u.s.n.</var> <reg norm="æqualem" type="context">æqualẽ</reg> angulo <var>.u.b.n.</var> Å¿ed <reg norm="cum" type="context">cÅ©</reg> angulus <var>.u.o.n.</var> exterior trian <lb/>guli <var>.u.o.s.</var> exiÅ¿tat, ipÅ¿e maior erit angulo <var>.s.</var> ex .16. primi. </s> - <s xml:id="s4027" xml:space="preserve">duco poÅ¿tea <var>.o.q.</var> parallelam <lb/>ad <var>.u.s.</var> quæ Å¿ecabit <var>.a.u.</var> in puncto <var>.q.</var> & habebimus angulum <var>.a.o.q.</var> Ä™qualem angulo <var>.<lb/>n.s.u.</var> ex .29. eiuÅ¿dem, hoc eÅ¿t angulo <var>.n.b.u.</var> fed ex Å¿u-<lb/> - <anchor type="figure" xlink:label="fig-0344-02a" xlink:href="fig-0344-02"/> - pradictis rationibus, lineæ <var>.q.b.n.</var> Å¿imul Å¿umptæ maio-<lb/>rem efficient longitudinem, quam <var>.q.o.n</var>. </s> - <s xml:id="s4028" xml:space="preserve">Nunc cum <lb/>ipÅ¿i <var>.q.b.</var> addita fuerit <var>.u.q.</var> & vice <var>.q.o.</var> Å¿umpta fuerit ali-<lb/>qua linea minor ipÅ¿a <var>.u.q.o.</var> eo amplius <var>.u.q.b.n.</var> maior <lb/>erit, quod quidem hoc modo faciendum. </s> - <s xml:id="s4029" xml:space="preserve">Acci-<lb/>piatur <var>.o.u.</var> vt comes <var>.o.n.</var> quæ minor eÅ¿t ambabus <var>.o.<lb/>q.</var> et <var>.q.u.</var> ex .20. primi, ita enim habebimus <reg norm="propoÅ¿itum" type="context">propoÅ¿itÅ©</reg>. <lb/></s> - <s xml:id="s4030" xml:space="preserve">Å¿ed breuiori modo hoc ipÅ¿um videbis ex prÄ™cedenti, <lb/>& ex .21. primi Euclid. </s> - <s xml:id="s4031" xml:space="preserve">Nam ex præcedenti <var>.u.b.n.</var> lon-<lb/>gior eÅ¿t ipÅ¿a <var>.u.s.n.</var> ex .21. autem primi <var>.u.s.n.</var> longior eÅ¿t <lb/>ipÅ¿a <var>.u.o.n.</var> ergo verum eÅ¿t propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0344-02" xlink:href="fig-0344-02a"> - <image file="0344-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0344-02"/> - </figure> - </div> - <figure position="here"> - <image file="0344-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0344-03"/> - </figure> - <p> - <s xml:id="s4032" xml:space="preserve">Si verò radius incidentiæ <reg norm="non" type="context">nõ</reg> fuerit æqualis radio <lb/>reflexionis, Å¿it vt in hac Å¿ubÅ¿cripta tertia figura vide <lb/>re eÅ¿t <var>.u.b.p</var>.</s> - </p> - <p> - <s xml:id="s4033" xml:space="preserve">Cum autem probauerim longitudinem <var>.u.b.n.</var> ma <lb/>iorem eſſe longitudine <var>.u.o.n.</var> coniungatur <var>.n.p.</var> cum <lb/><var>u.b.n</var>. </s> - <s xml:id="s4034" xml:space="preserve">deinde. ab <var>.o.</var> ad <var>.p.</var> ducatur <var>.o.p.</var> quæ minor <lb/>erit longitudine <var>.o.n.p.</var> ex .20. primi, & illicò <lb/>manifeÅ¿tabitur verum eſſe propoÅ¿itum, etiam hoc <lb/>tertio modo.</s> - </p> - <pb o="333" rhead="EPISTOL AE." n="345" file="0345" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0345"/> - <p> - <s xml:id="s4035" xml:space="preserve">Si <reg norm="autem" type="context">autẽ</reg> res viÅ¿ibilis <reg norm="oculusque" type="simple">oculusqÌ;</reg> ambo fuerint intra circulum, <reg norm="tunc" type="context">tÅ©c</reg> poſſibile eſſet quod <lb/><reg norm="longitudo" type="context">lõgitudo</reg> <var>.u.b.n.</var> modo maior, modo minor, modo verò æqualis eſſet ipÅ¿a <var>.u.o.n.</var> <reg norm="nunc" type="context">nÅ©c</reg>. <lb/></s> - <s xml:id="s4036" xml:space="preserve">Quod etiam affirmo de <var>.u.b.p.</var> Å¿imiliter etiam eueniet Å¿i vnus terminorum <var>.u.</var> vel <var>.n.</var> <lb/>fuerit intra circunferentiam, reliquus verò extra ipÅ¿am.</s> - </p> - <p> - <s xml:id="s4037" xml:space="preserve">ConÅ¿ideremus nunc hic inÅ¿raÅ¿criptam .4. figuram vbi <var>.d.b.p.</var> Å¿it circunferentia oxy <lb/>gonia Å¿eu elliptica (quod idem eÅ¿t) cuius maior axis Å¿it <var>.d.p.</var> in quo, duo termini <var>.u.n.</var> <lb/>Å¿int centra eius generationis: </s> - <s xml:id="s4038" xml:space="preserve">b.x. verò Å¿it minor axis. </s> - <s xml:id="s4039" xml:space="preserve">Imaginemur etiam circulum <var>.<lb/>b.o.x.</var> cuius Å¿emidiameter Å¿it <var>.c.b.</var> non maior medietate minoris axis, ne circunferen-<lb/>tia huiuÅ¿modi circuli Å¿ecet circunferentiam oxygoniam. </s> - <s xml:id="s4040" xml:space="preserve">Cogitemus etiam circu-<lb/>lum <var>.b.e.</var> cuius Å¿emidiameter, minor non Å¿it minori axe <var>.b.x.</var> ipÅ¿ius oxygoniæ, ne Å¿e <lb/>inuicem Å¿ecent huiuÅ¿modi circunferentiæ, Å¿int etiam ambo eorum centra in linea <var>.b.<lb/>x.</var> minoris axis, & punctum <var>.b.</var> Å¿it commune vnicuique earum periphæriarum, vnde <lb/>minor circulus, totus intra, maior autem, totus extra ipÅ¿am <reg norm="figuram" type="context">figurã</reg> oxygoniam erit. <lb/></s> - <s xml:id="s4041" xml:space="preserve">Nunc ad partem <var>.o.r.e.</var> vbi non communicant inuicem ipſæ circunferentiæ ducan-<lb/>tur <var>.n.o.r.e</var>: <var>u.o</var>: <var>u.r</var>: et <var>.u.e.</var> & per <var>.b.</var> et <var>.r.</var> cogitetur tranÅ¿ire alium circulum, cuius cen-<lb/>trum in axe <var>.b.x.</var> Å¿it <var>.t.</var> <reg norm="omnesque" type="simple">omnesqÌ;</reg> iÅ¿ti circuli imaginentur trium diuerÅ¿orum Å¿phærico-<lb/>rum Å¿peculorum, vnde pro genera <lb/>tione <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> oxygoniÄ™, Å¿eu ex .52. ter <lb/>tij Pergei, habebis longitudinem <var>.<lb/> - <anchor type="figure" xlink:label="fig-0345-01a" xlink:href="fig-0345-01"/> - u.r.n.</var> Ä™qualem eſſe longitudini <var>.u.b.<lb/>n.</var> & ei, quæ eÅ¿t <var>.u.o.n.</var> (vt minor ip <lb/>Å¿a <var>.u.r.n.</var> ex .21. primi Euclidis) mi-<lb/>nor ipÅ¿a <var>.u.b.n.</var> & longitudinem <var>.u.<lb/>e.n.</var> (vt maior ipÅ¿a <var>.u.r.n.</var> ex eadem <num value="21">.<lb/>21.</num> primi Eucli.) maior ipÅ¿a <var>.u.b.n</var>. <lb/></s> - <s xml:id="s4042" xml:space="preserve">Sed Å¿i quis vellet hoc demonÅ¿trare <lb/>ope circuli, <reg norm="vnius" type="simple">vniê°</reg> <reg norm="tantummodo" type="context context">tãtÅ©modo</reg> Å¿peculi, <lb/><reg norm="multiplicando" type="context">multiplicãdo</reg> ipÅ¿as oxygonias <reg norm="quem- admodum" type="wordlist">quẽ-<lb/>admodum</reg> de ipÅ¿is circulis fecimus, obtineret Å¿imiliter propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="4"> - <figure xlink:label="fig-0345-01" xlink:href="fig-0345-01a"> - <image file="0345-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0345-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head497" style="it" xml:space="preserve">Solutio dubitationis.</head> - <head xml:id="head498" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4043" xml:space="preserve">RAtionalis eÅ¿t dubitatio tua, <lb/> - <anchor type="figure" xlink:label="fig-0345-02a" xlink:href="fig-0345-02"/> - vtrum (<reg norm="cum" type="context">cÅ©</reg> circulus minor hoc <lb/>eÅ¿t <var>.b.o.</var> habeat Å¿uum centrum in mi <lb/>nori axe inter centrum oxygoniæ, <lb/>et .b: exiÅ¿tente <var>.b.</var> extremo axis mi-<lb/>noris, <reg norm="communeque" type="simple">communeqÌ;</reg> ambobus circun-<lb/>ferentijs circuli Å¿cilicet & oxigoniÄ™) <lb/>dictus circulus minor, plura puncta <lb/>communia habeat cum ipÅ¿is circun-<lb/>ferentijs.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0345-02" xlink:href="fig-0345-02a"> - <image file="0345-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0345-02"/> - </figure> - </div> - <p> - <s xml:id="s4044" xml:space="preserve">Cui dubitationi <reg norm="reÅ¿pondeo" type="context">reÅ¿põdeo</reg> quod <lb/>quotieÅ¿cunque centrum alicuius cir <lb/>culi fuerit idem cum <var>.c.</var> centro oxy-<lb/>goniæ, vel inter <var>.c.</var> et <var>.b.</var> in interual-<lb/>lo Å¿cilicet minoris axis, exiÅ¿tente <var>.b.</var> <lb/>Å¿ua extremitate communi ambabus - <pb o="334" rhead="IO. BABPT. BENED." n="346" file="0346" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0346"/> - circunferentijs, ipÅ¿as circunferentias inuicem contiguas eſſe oportebit in puncto <var>.b.</var> <lb/>tantummodo.</s> - </p> - <p> - <s xml:id="s4045" xml:space="preserve">EÅ¿to primum quod centrum <var>.c.</var> commune exiÅ¿tat, vt dictum eÅ¿t. </s> - <s xml:id="s4046" xml:space="preserve">Å¿it etiam centrum <lb/>vnius circuli, cuius diameter Å¿it <reg norm="idem" type="context">idẽ</reg> <reg norm="cum" type="context">cÅ©</reg> maiori axe <var>.d.p.</var> & in gyro oxygoniæ accipia-<lb/>tur punctum <var>.f.</var> proximum <var>.b.</var> quantum fieri poterit, </s> - <s xml:id="s4047" xml:space="preserve">tunc protrahatur <var>.f.a.e.</var> parallela <lb/>ipÅ¿i <var>.g.c.</var> vÅ¿que ad gyrum maioris circuli in puncto <var>.e.</var> quæ cum <var>.d.p.</var> rectos efficiec <lb/>angulos. ex .29. primi Eucli. </s> - <s xml:id="s4048" xml:space="preserve"><reg norm="Å¿ecabitque" type="simple">Å¿ecabitqÌ;</reg> gyrum circuli <var>.b.o.</var> minoris in puncto <var>.t.</var> quod di <lb/>co eſſe intra oxygoniam, <reg norm="Å¿eparatumque" type="simple">Å¿eparatumqÌ;</reg> ab <var>.f</var>. </s> - <s xml:id="s4049" xml:space="preserve">Quapropter duco <var>.c.e.</var> quæ Å¿ecabit cir-<lb/>cunferentiam circuli minoris in <reg norm="puncto" type="context">pÅ©cto</reg> <var>.o.</var> à quo puncto duco etiam <var>.o.i.</var> parallelam ad <lb/><var>e.a</var>. </s> - <s xml:id="s4050" xml:space="preserve">Deinde conÅ¿idero, quod ex ra-<lb/>tionibus ab Archimede adductis in <lb/> - <anchor type="figure" xlink:label="fig-0346-01a" xlink:href="fig-0346-01"/> - quinta propoÅ¿itione libri de conoi-<lb/>dalibus, & Å¿phæroidibus, eadem <lb/>proportio erit <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <var>.g.c.</var> ad <var>.b.c.</var> quÄ™ <lb/>ipÅ¿ius <var>.e.a.</var> ad <var>.f.a.</var> vnde permutando <lb/>ita erit ipÅ¿ius <var>.g.c.</var> ad <var>.e.a.</var> vel <var>.b.c.</var> ad <lb/><var>f.a.</var> hoc eÅ¿t ipÅ¿ius <var>.e.c.</var> ad <var>.e.a.</var> vt <var>.o.c.</var> <lb/>ad <var>.f.a.</var> Å¿ed ex Å¿imilitudine triangu-<lb/>lorum, & ex .11. quinti, ita <reg norm="etiam" type="context">etiã</reg> erit <lb/>ipÅ¿ius <var>.o.c.</var> ad <var>.o.i.</var> vt <var>.o.c.</var> ad <var>.f.a</var>. </s> - <s xml:id="s4051" xml:space="preserve">Vn-<lb/>de Å¿equitur <var>.o.i.</var> æqualem eſſe <var>.f.a.</var> <lb/>Å¿ed ex .14. tertij Eucli <var>.t.a.</var> minor eÅ¿t <var>.<lb/>o.i</var>. </s> - <s xml:id="s4052" xml:space="preserve">Quare minor etiam erit ipÅ¿a <var>.f.<lb/>a</var>. </s> - <s xml:id="s4053" xml:space="preserve">Vnde punctum <var>.t.</var> intra oxygo-<lb/>niam erit, & conÅ¿equenter Å¿epara-<lb/>tum .ab <var>.f</var>.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0346-01" xlink:href="fig-0346-01a"> - <image file="0346-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0346-01"/> - </figure> - </div> - <p> - <s xml:id="s4054" xml:space="preserve">Sed Å¿i centrum circuli minoris <lb/>fuerit inter <var>.c.</var> et <var>.b.</var> hoc eÅ¿t eccentri-<lb/>cum ipÅ¿ius oxygoniæ, ipÅ¿e tanget concentricum in puncto <var>.b.</var> tantummodò, vt in .3. <lb/>Euclidis libro probatur. </s> - <s xml:id="s4055" xml:space="preserve">Vnde tanto magis diÅ¿tans erit punctum <var>.t.</var> à puncto <var>.f.</var> quod <lb/>erit propoÅ¿itum.</s> - </p> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head499" style="it" xml:space="preserve">Alterius dubitationis Å¿olutio.</head> - <head xml:id="head500" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4056" xml:space="preserve">VNde autem fiat, quod à ſpeculis planis, obiectorum imagines, ita diÅ¿tantes <lb/>vltra Å¿uperficiem ipÅ¿ius Å¿peculi videantur, vt obiecta citra ipÅ¿am Å¿uperficiem <lb/>reperiuntur.</s> - </p> - <p> - <s xml:id="s4057" xml:space="preserve">Pro cuius rei Å¿cientia, tres cognitiones nos primum habere oportet, quarum pri-<lb/>ma eÅ¿t. </s> - <s xml:id="s4058" xml:space="preserve">Vnde fiat, quod obiecti imago in catheto incidentiæ videatur. </s> - <s xml:id="s4059" xml:space="preserve"><reg norm="Secunda" type="context">SecÅ©da</reg>. </s> - <s xml:id="s4060" xml:space="preserve">vn-<lb/>de efficiatur, quod angulus reflexionis, Å¿emper æqualis Å¿it angulo incidentiæ.</s> - </p> - <p> - <s xml:id="s4061" xml:space="preserve">Terria demum. </s> - <s xml:id="s4062" xml:space="preserve">Vnde naÅ¿catur quod radius incidentiæ Å¿imul cum radio reflexio-<lb/>nis Å¿it in quodam plano Å¿ecante Å¿uperficiem Å¿peculi Å¿emper ad rectos, quod qui-<lb/>dem planum vocatur Å¿uperficies reflexionis. </s> - <s xml:id="s4063" xml:space="preserve">HuiuÅ¿modi tres paſſiones, ab omnibus <lb/>Å¿pecularijs conÅ¿ideratæ Å¿unt, Å¿ed rationes ab illis traditæ, mihi non Å¿atisfaciunt.</s> - </p> - <pb o="335" rhead="EPISTOLAE." n="347" file="0347" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0347"/> - <p> - <s xml:id="s4064" xml:space="preserve">Nam circa æqualitatem angulorum reflexionis & incidentiæ, iam tibi probaui <lb/>illud non vniuerÅ¿aliter euenire à breuitate aggregati radiorum incidentiæ reflexio-<lb/><reg norm="nisque" type="simple">nisqÌ;</reg>. </s> - <s xml:id="s4065" xml:space="preserve">Sed hoc naÅ¿citur potius ab eo, quod cum radius incidentiæ non poſſit Å¿uper <lb/>ficiem corporis opaci penetrare, reflectit, vt citra ipÅ¿am <reg norm="cum" type="context">cÅ©</reg> angulo æquali ei, quem <lb/>faceret cum eadem Å¿uperficie vltra ipÅ¿am Å¿i tranÅ¿iuiſſet.</s> - </p> - <p> - <s xml:id="s4066" xml:space="preserve">Exempli gratia Å¿it <var>.a.</var> obiectum <var>.b.</var> <reg norm="autem" type="context">autẽ</reg> oculus in figura <var>.A.</var> et <var>.c.e.</var> Å¿uperficies ipÅ¿ius <lb/>Å¿peculi <var>.d.</var> verò Å¿it punctum ipÅ¿ius Å¿uperficiei, à quo ad oculum reflectitur imago ip-<lb/> - <anchor type="figure" xlink:label="fig-0347-01a" xlink:href="fig-0347-01"/> - Å¿ius <var>.a</var>. </s> - <s xml:id="s4067" xml:space="preserve">Nunc Å¿i radius <var>.a.d.</var> incidentiæ, recta <lb/>incederet Å¿ub <var>.c.e.</var> efficeret angulum <var>.e.d.h.</var> <lb/>æqualem angulo <var>.c.d.a.</var> eius contrapoÅ¿ito, <lb/>Å¿ed quia impeditur ipſæ radius ab opacitate <lb/>ipÅ¿ius Å¿peculi <var>.c.e.</var> ne vlterius incedat, propte <lb/>rea reflectitur ab ipÅ¿a Å¿uperficie Å¿peculi, con-<lb/>Å¿tituens cum ipÅ¿a angulum <var>.e.d.b.</var> æqualem <lb/>angulo <var>.e.d.h.</var> Å¿ed quia angulus <var>.c.d.a.</var> eÅ¿t <reg norm="etiam" type="context">etiã</reg> <lb/>Ä™qualis ipÅ¿i angulo <var>.e.d.h.</var> propterea angulus <var>.e.d.b.</var> Ä™qualis exiÅ¿tit angulo <var>.c.d.</var> a; </s> - <s xml:id="s4068" xml:space="preserve">per <lb/>accidens igitur Å¿equitur <var>.a.d.</var> et <var>.d.b.</var> Å¿imul Å¿umptas, breuiorem facere longiludinem <lb/>omni alia, quæ ab ipÅ¿a Å¿uperficie <var>.c.e.</var> ad eadem puncta <var>.a.b.</var> ducta eſſet, </s> - <s xml:id="s4069" xml:space="preserve">quare natu-<lb/>ræintentio eÅ¿t efficere angulum <var>.e.d.b.</var> æqualem angulo <var>.e.d.h.</var> vnde ex accidenti po <lb/>Å¿tea Å¿equitur, ipÅ¿um æqualem eſſe angulo <var>.c.d.a.</var> & deinde <reg norm="quod" type="wordlist">qÄ‘</reg> lineæ <var>.a.d.</var> et <var>.d.b.</var> con-<lb/>Å¿tituant longitudinem breuiorem. </s> - <s xml:id="s4070" xml:space="preserve">Quare illud quod omnes putabant eſſe primum <lb/>& perÅ¿e, vltimum eÅ¿t, & exaccidenti.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0347-01" xlink:href="fig-0347-01a"> - <image file="0347-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0347-01"/> - </figure> - </div> - <p> - <s xml:id="s4071" xml:space="preserve">Quare vero Å¿uperficies, quæ vocatur reflexionis, in qua Å¿unt duæ lineÄ™, hoc eÅ¿t <lb/>incidentiÄ™, <reg norm="reflexionisque" type="simple">reflexionisqÌ;</reg>, Å¿emper Å¿it perpendicularis Å¿uperficiei ipÅ¿ius Å¿peculi: </s> - <s xml:id="s4072" xml:space="preserve">Hæc <lb/>eÅ¿t ratio, quia cum quilibet radius incidentiæ, perpendicularis ipÅ¿i Å¿uperficiei Å¿pe-<lb/>culi, in Å¿eipÅ¿o reflectit, ex ijÅ¿dem dictis rationibus, hoc eÅ¿t, quia cum tali angulo vult <lb/>reflecti, cum quali tranÅ¿iret, ita etiam purandum eÅ¿t, quodradius incidens obliquus, <lb/>cum in Å¿eipÅ¿um non poſſit redire, quia non eÅ¿t perpendicularis Å¿uperficiei Å¿peculi, <lb/>reflectitur tamen per planum erectum ipÅ¿i Å¿uperficiei Å¿peculi, vt in eo, cui magis re-<lb/>Å¿iÅ¿tit Å¿uperficies corporis opaci, quà m alicui alij plano ipÅ¿ius infiniti inclinatorum <lb/>planorum, ab vtraque parte ipÅ¿ius plani perpendicularis, quod vnum etiam tan-<lb/>tummodo eÅ¿t, & in quo, radius maiorem vim obtinet reflectendi, Å¿eu in eo, in quo <lb/>radius ipÅ¿e cum maiori reÅ¿iÅ¿tentia repercutitur à ſuperficie corporis opaci.</s> - </p> - <p> - <s xml:id="s4073" xml:space="preserve">PoÅ¿tremo <reg norm="Å¿ciendum" type="context context">Å¿ciẽdÅ©</reg> vnde oriatur, <reg norm="quod" type="simple">ê™</reg> rei viÅ¿ibilis imago, à ſpeculo plano reflexa, Å¿em <lb/>per in catheto incidentiæ videatur.</s> - </p> - <p> - <s xml:id="s4074" xml:space="preserve">Pro cuius rei ratione cognoÅ¿cendum primò eÅ¿t, quo modo fit perfecta <reg norm="Å¿implexque" type="simple">Å¿implexqÌ;</reg> <lb/>viÅ¿io, & non reflexa, deinde proÅ¿equemur ad reliqua huius tertiæ propoÅ¿itionis.</s> - </p> - <p> - <s xml:id="s4075" xml:space="preserve">Animaduertendum igitur eÅ¿t, quod <reg norm="quotieÅ¿cunque" type="simple">quotieÅ¿cunq;</reg> obiectum aliquod viÅ¿ibile aÅ¿pi <lb/>cimus, nos nunquam perfectè illud comprehendere poſſumus, niÅ¿i in puncto con-<lb/>curÅ¿us, Å¿eu interÅ¿ectionis axium viÅ¿ualium, Å¿eu radialium ( vt ita loquar ) <reg norm="quam" type="context">quã</reg> <reg norm="inter- Å¿ectionem" type="context">inter-<lb/>Å¿ectionẽ</reg>, nos efficimus ope reuolutionis oculorum <reg norm="adinuicem" type="context">adinuicẽ</reg>, hoc eÅ¿t voluendo vnum <lb/>verÅ¿us alium, ita vt in Å¿itu ipÅ¿ius obiecti, Å¿einuicem Å¿ecent axes iam dicti, </s> - <s xml:id="s4076" xml:space="preserve">tunc enim <lb/>vtroque oculo mediante, exacte rem perÅ¿picimus, cÄ™teris .8. circunÅ¿tantijs non ob-<lb/>Å¿tantibus.</s> - </p> - <p> - <s xml:id="s4077" xml:space="preserve">Vnde Å¿tantibus oculis in tali Å¿itu, altero reÅ¿pectu alterius, Å¿i eorum alter tectus; <lb/></s> - <s xml:id="s4078" xml:space="preserve">Å¿eu velatus fuerit, tune alio tantummodo oculo mediante, videbimus obiectum, <lb/>in ea diÅ¿tantia, exactius, quam in quauis alia propinquiori, & remotiori.</s> - </p> - <pb o="336" rhead="IO. BAPT. BENED." n="348" file="0348" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0348"/> - <p> - <s xml:id="s4079" xml:space="preserve">Animal igitur, Å¿ecundum diÅ¿tantiam obiecti, oculum accommodat ad recipien-<lb/>dum quam exactiſſimè Å¿peciem ipÅ¿ius obiecti, & hoc voluendo ambos oculos, vnum <lb/>verÅ¿us alium, ita quod interÅ¿ectio axium Å¿it in Å¿itu Å¿eu loco dicti obiecti, nam tunc vi <lb/>dent ambo vel aliquis eorum Å¿olus, in tali diÅ¿tantia exactè obiectum videbit.</s> - </p> - <p> - <s xml:id="s4080" xml:space="preserve">Vnde Å¿equitur obiectum viÅ¿ibile, compræhenÅ¿ibile non eſſe ab vno tantummodo <lb/>oculo in quolibet Å¿itu axis ipÅ¿ius oculi, Å¿ed in eo, vbi alius axis interÅ¿ecatur à dicto. <lb/></s> - <s xml:id="s4081" xml:space="preserve">Quæ quidem interÅ¿ectio poteÅ¿t fieri propinqua, vel remota à viÅ¿u, ad certos tamen <lb/>terminos vÅ¿que.</s> - </p> - <p> - <s xml:id="s4082" xml:space="preserve">De huiuÅ¿modi axium viÅ¿ualium interÅ¿ectione Å¿cribit Alhazem in .2. et .15. propo <lb/>Å¿itione tertij lib. Vitellio verò in .32. et .45. eiuÅ¿dem.</s> - </p> - <p> - <s xml:id="s4083" xml:space="preserve">Quod igitur dico, verum eÅ¿t, ideÅ¿t, quod Å¿i vno tantummodo oculo aÅ¿piciemus <lb/>obiectum aliquod, ipÅ¿um nunquam perfectè proÅ¿picietur, niÅ¿i cum oculus ita Å¿itus <lb/>fuerit, vt eius axis cum axe alterius in loco obiecti Å¿e inuicem Å¿ecent, quamuis alter <lb/>oculus nihil videat, <reg norm="cum" type="context">cÅ©</reg> <reg norm="autem" type="wordlist">aÅ©t</reg> duobus oculis in tali Å¿itu <reg norm="conÅ¿titutis" type="context">cõſtitutis</reg> <reg norm="obiectum" type="context">obiectÅ©</reg> videmus, vnum <lb/>tantummodo nobis cernere videbimur, & Å¿i extra talem punctum interÅ¿ectionis ip-<lb/>Å¿um obiectum poÅ¿itum fuerit, tunc duo talia, obiecta nobis apparebunt, Å¿ed huiuÅ¿ <lb/>modi rei cauÅ¿am alias tibi manifeÅ¿tabo.</s> - </p> - <p> - <s xml:id="s4084" xml:space="preserve">His igitur cognitis, ponamus aliquam <lb/> - <anchor type="figure" xlink:label="fig-0348-01a" xlink:href="fig-0348-01"/> - Å¿peculi Å¿uperficiem eſſe <var>.g.h.</var> in figura <var>.B.</var> <lb/>obiectum autem viÅ¿ibile <var>.b.</var> oculos vero <var>.a.</var> <lb/>et <var>.u.</var> punctum autem <var>.n.</var> in Å¿uperficie Å¿pecu <lb/>li, à quo imago ipÅ¿ius <var>.b.</var> reflectit ad <var>.a.</var> & <lb/>punctum <var>.t.</var> à quo reflectitur ad <var>.u.</var> et <var>.c.e.</var> <lb/>Å¿it <reg norm="communis" type="context">cõmunis</reg> Å¿ectio Å¿uperficiei reflexionis <lb/>radiorum <var>.b.n.a.</var> et <var>.c.f.</var> Å¿it communis Å¿ectio <lb/>Å¿uperficiei reflexionis radiorum <var>.b.t.u.</var> qua <lb/>rum <reg norm="vnaquæque" type="simple">vnaquæq;</reg> Å¿uperficies reflexionis, ere-<lb/>cta eÅ¿t ad Å¿uperficiem Å¿peculi <var>.g.h.</var> vt Å¿upra <lb/>diximus. </s> - <s xml:id="s4085" xml:space="preserve">Nunc ex .19. vndecimi Eucl. Å¿equitur communem Å¿ectionem harum dua-<lb/>rum Å¿uperficierum. (b.c.d. Å¿cilicet) ad rectos etiam eſſe Å¿upra Å¿uperficiem Å¿peculi <var>.g.<lb/>h.</var> cum qua <var>.b.c.</var> quælibet linearum <var>.a.n.</var> vel <var>.u.t.</var> reflexarum ( productÄ™ cum fuerint ) <lb/>Å¿einuicem interÅ¿ecabunt eo quod duo anguli <var>.d.c.n.</var> et <var>.d.n.c.</var> Å¿imul collecti minores <lb/>Å¿unt duobus rectis, & ita <var>.d.c.t.</var> cum <var>.d.t.c.</var> cum anguli <var>.a.n.e.</var> et <var>.u.t.f.</var> reflexi, ipÅ¿is con-<lb/>trapoÅ¿iti, æquales Å¿int angulis <var>.b.n.c.</var> et <var>.b.t.c.</var> incidentiæ, quorum <reg norm="vnuÅ¿quiÅ¿que" type="simple">vnuÅ¿quiÅ¿q;</reg> ex .32. <lb/>primi, minor eÅ¿t recto.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0348-01" xlink:href="fig-0348-01a"> - <image file="0348-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0348-01"/> - </figure> - </div> - <p> - <s xml:id="s4086" xml:space="preserve">Dico etiam quod in eodem puncto huiuÅ¿modi catheti <var>.b.c.d.</var> in quo interÅ¿ecabi-<lb/>tur à linea <var>.a.n.</var> in eodem Å¿ecabitur à linea <var>.u.t.</var> & quod punctum dicti concurÅ¿us, tan-<lb/>tum depreſſum erit Å¿ub Å¿uperficie Å¿peculi <var>.g.h.</var> quantum <var>.b.</var> Å¿upra ipÅ¿am reperietur. <lb/></s> - <s xml:id="s4087" xml:space="preserve">Nam anguli <var>.b.n.c.</var> et <var>.d.n.c.</var> Å¿unt inuicem æquales, <reg norm="angulique" type="simple">anguliqÌ;</reg> <var>.b.c.n.</var> et <var>.d.c.n.</var> recti <var>.c.n.</var> <lb/>verò communis ambobus triangulis <var>.b.c.n.</var> et <var>.d.c.n.</var> vnde ex .26. primi Eucli. latus <var>.d.<lb/>c.</var> commune, vt trianguli <var>.d.c.n.</var>æquale erit lateri communi <var>.b.c.</var> vt trianguli <var>.b.c.n.</var> <lb/>Idem etiam dico de latere <var>.d.c.</var> vt ipÅ¿ius trianguli <var>.d.c.t.</var> quod æquatur lateri <var>.b.c.</var> vt <lb/>trianguli <var>.b.c.t</var>. </s> - <s xml:id="s4088" xml:space="preserve">Vnde cum <var>.b.c.</var> vnum, & idem Å¿it: </s> - <s xml:id="s4089" xml:space="preserve">d.c. igitur etiam erit, & ipÅ¿um <reg norm="vnum" type="context">vnÅ©</reg> <lb/>& idem, quod erit propoÅ¿itum.</s> - </p> - <p> - <s xml:id="s4090" xml:space="preserve">Nunc autem cum hi duo radij Å¿einuicem Å¿ecent in puncto <var>.d.</var> ergo in ipÅ¿o puncto <var>.<lb/>d.</var> videbimur nobis videre <reg norm="imaginem" type="context">imaginẽ</reg> obiecti .b: <reg norm="cum" type="context">cÅ©</reg> ope <reg norm="duorum" type="context">duorÅ©</reg> <reg norm="iÅ¿torum" type="context">iÅ¿torÅ©</reg> <reg norm="radiorum" type="context">radiorÅ©</reg> <var>.n.a.</var> et <var>.t.<lb/>u.</var> ita inuicem <reg norm="Å¿itorum" type="context">Å¿itorÅ©</reg>, videamur nobis <reg norm="imaginem" type="context">imaginẽ</reg> proÅ¿picere. </s> - <s xml:id="s4091" xml:space="preserve">Vnde Å¿i in tali caÅ¿u, vnus - <pb o="337" rhead="EPISTOLAE." n="349" file="0349" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0349"/> - oculorum clauderetur, nihilominus cum reliquo obiectum vidiſſemus in <reg norm="eodem" type="context">eodẽ</reg> ipÅ¿o <lb/>loco <var>.d.</var> & non in alio ex Å¿uperius dictis rationibus.</s> - </p> - <p> - <s xml:id="s4092" xml:space="preserve">Et Å¿i Å¿tantibus ijs terminis volueremus pupillam oculi <var>.u.</var> verÅ¿us aliam <var>.a.</var> ad aÅ¿pi-<lb/>ciendum punctum <var>.n.</var> in Å¿uperficie <var>.g.h.</var> ipÅ¿ius Å¿peculi, hoc eÅ¿t Å¿i fecerimus quod axes <lb/>viÅ¿uales Å¿einuicem Å¿ecarent in ipÅ¿o puncto <var>.n</var>. </s> - <s xml:id="s4093" xml:space="preserve">tunc videremur nobis videre duas <lb/>imagines ipÅ¿ius obiecti <var>.b.</var> intra Å¿peculum, eo quod obiectum, propter hoc non <lb/>ceſſaret reflectere ad oculos ab ipÅ¿is punctis <var>.n.</var> et <var>.t.</var> quapropter recipiendo ra-<lb/>dium <var>.t.u.</var> in Å¿itu axis oculi <var>.u.</var> & radium <var>.n.a.</var> in Å¿itu axis oculi <var>.a.</var> hi axes ex neceſſitate <lb/>(vt probauimus ) Å¿einuicem Å¿ecant in puncto <var>.d.</var> vnde vnam tantummodo imaginem <lb/>ipÅ¿ius obiecti nobis apparebit.</s> - </p> - <p> - <s xml:id="s4094" xml:space="preserve">Ex his igitur omnibus potes facilè videre omnem imaginem, cuiuÅ¿uis obiecti, re-<lb/>flexam à ſpeculo, reperiri in ipÅ¿o catheto incidentiæ, cum ipÅ¿e Å¿emper Å¿it communis <lb/>Å¿ectio duarum Å¿uperficierum reflexionis, in quo catheto concurrunt ipſæ axes vi-<lb/>Å¿uales.</s> - </p> - <p> - <s xml:id="s4095" xml:space="preserve">ExijÅ¿dem etiam dictis rationibus facile compræhendere poteris, vnde fiat, vt vi-<lb/>deamus imaginem reflexam à ſpeculis Å¿phÄ™ricis concauis citra <reg norm="ipÅ¿orum" type="context">ipÅ¿orÅ©</reg> Å¿uperficiem, & <lb/>non vltra. </s> - <s xml:id="s4096" xml:space="preserve">Quod <reg norm="nunquam" type="context">nunquã</reg> euenit, niÅ¿i quando <reg norm="punctum" type="context">punctÅ©</reg> <var>.d.</var> interÅ¿ectionis <reg norm="ipÅ¿orum" type="context">ipÅ¿orÅ©</reg> <reg norm="radiorum" type="context">radiorÅ©</reg> <lb/>viÅ¿ualium (quod alio in loco non fit, niÅ¿i in catheto incidentiæ hoc eÅ¿t in communi <lb/>Å¿ectione duarum Å¿uperficierum reflexionis. </s> - <s xml:id="s4097" xml:space="preserve">Dato quod obiectum non Å¿it in vna ea-<lb/>demqÌue Å¿uperficie, in qua reperti fuerint axes viÅ¿uales, hoc eÅ¿t dato, <reg norm="quod" type="simple">ê™</reg> ambo axes <lb/>viÅ¿uales non Å¿int in vna <reg norm="eademque" type="simple">eademqÌ;</reg> Å¿uperficie reflexionis) reperitur citra & non vltra Å¿u <lb/>perficiem ipÅ¿ius Å¿peculi.</s> - </p> - <p> - <s xml:id="s4098" xml:space="preserve">Ad cuius rei euidentiam non <reg norm="prætermittam" type="context">prætermittã</reg> dicere, quod cum debeant Å¿emper Å¿u-<lb/>perficies reflexionum perpendiculares eſſe, velad rectos Å¿ecare Å¿uperficiem ipÅ¿ius <lb/>Å¿peculi, ipÅ¿arum communes Å¿ectiones cum Å¿uperficie Å¿peculi Å¿phÄ™rici, Å¿emper erunt <lb/>circunferentiæ magnorum circulorum illius Å¿phæræ, cuius portio eÅ¿t Å¿peculum <lb/>propoÅ¿itum, vt etiam Vitellio affirmat in prima Å¿exti libri. </s> - <s xml:id="s4099" xml:space="preserve">Vnde vnuÅ¿quiÅ¿que ca-<lb/>thetus incidentiæ tranÅ¿ibit per centrum Å¿peculi, cum ipÅ¿e Å¿it communis Å¿ectio dua-<lb/>rum Å¿uperficierum reflexionis, </s> - <s xml:id="s4100" xml:space="preserve">quare in ipÅ¿o catheto erit punctum interÅ¿ectionis ip <lb/>Å¿orum axium viÅ¿ualium ex neceſſitate, vt videbimus, Å¿i vnam tantummodo <reg norm="imaginem" type="context">imaginẽ</reg> <lb/>obiecti nobis videremur videre.</s> - </p> - <p> - <s xml:id="s4101" xml:space="preserve">Exempli gratia, Å¿int duæ Å¿uperficies reflexionis Å¿peculi Å¿phærici concaui <var>.b.n.c.a.</var> <lb/>et <var>.b.t.c.u.</var> <reg norm="obiectumque" type="simple">obiectumqÌ;</reg> Å¿it <var>.b.</var> oculi autem Å¿int <var>.a.u.</var> punctum verò Å¿uperficiei Å¿peculi, à <lb/>quo obiectum emittit reflexionem Å¿uÄ™ <lb/>imaginis ad oculum <var>.a.</var> Å¿it <var>.n.</var> <reg norm="punctum" type="context">pÅ©ctum</reg> au-<lb/> - <anchor type="figure" xlink:label="fig-0349-01a" xlink:href="fig-0349-01"/> - <anchor type="figure" xlink:label="fig-0349-02a" xlink:href="fig-0349-02"/> - tem à quo eandem reflectit oculo <var>.u.</var> Å¿it <lb/>t. communis autem Å¿ectio harum dua-<lb/>rum Å¿uperficierum Å¿it <var>.b.c.</var> Å¿ed <var>.x.</var> <reg norm="centrum" type="context">centrÅ©</reg> <lb/>Å¿it Å¿peculi, radius verò incidentiÄ™ Å¿uper <lb/>ficiei <var>.b.n.c.</var> erit <var>.b.n.</var> cuius reflexus Å¿it <var>.n.<lb/>a.</var> radij autem alterius Å¿uperficiei erunt <lb/><var>b.t.</var> et <var>.t.u</var>. </s> - <s xml:id="s4102" xml:space="preserve">Imaginemur nunc duos Å¿emi <lb/>diametros <var>.x.n.</var> et <var>.x.t.</var> quæ angulos <var>.b.n.<lb/>a.</var> et <var>.b.t.u.</var> per æqualia diuidant ex Å¿up-<lb/>poÅ¿ito.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0349-01" xlink:href="fig-0349-01a"> - <image file="0349-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0349-01"/> - </figure> - <figure xlink:label="fig-0349-02" xlink:href="fig-0349-02a"> - <image file="0349-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0349-02"/> - </figure> - </div> - <p> - <s xml:id="s4103" xml:space="preserve">Nunc ijs Å¿uppoÅ¿itis, Å¿i vnam tantum-<lb/>modo obiecti imaginem videbimus, - <pb o="338" rhead="IO. BAPT. BENED." n="350" file="0350" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0350"/> - clarum erit ex rationibus Å¿upradictis nos ipÅ¿am videre in <reg norm="communi" type="context">cõmuni</reg> concurÅ¿u ipÅ¿orum <lb/>axium viÅ¿ualium, qui axes cum reperiantur vnà cum ipÅ¿is radijs reflexis <var>.n.a.</var> et <var>.t.u.</var> <lb/>ex neceſſitate Å¿einuicem <reg norm="Å¿ecabunt" type="context">Å¿ecabÅ©t</reg> in catheto <var>.b.c.</var> cum extendantur in ipÅ¿is Å¿uperficie-<lb/>bus reflexionum, quæ Å¿uperficies nihil aliud commune inuicem habent, quam cathe <lb/>tum dictum <var>.b.c.</var> Å¿it igitur in puncto <var>.d</var>.</s> - </p> - <p> - <s xml:id="s4104" xml:space="preserve">Ex his dictis alia oritur neceſſitas, hoc eÅ¿t, quod quotieÅ¿cunque vnam tantummo <lb/>do imaginem obiecti <var>.b.</var> videmus, dato quod duæ Å¿uperficies reflexionis Å¿int, & non <lb/>vna tantum, tunc angulos <var>.n.</var> et <var>.t.</var> Å¿emper inuicem æquales eſſe oportebit. </s> - <s xml:id="s4105" xml:space="preserve">Vnde ar-<lb/>cus <var>.n.c.</var> et <var>.t.c.</var> ex neceſſitate inuicem æquales erunt.</s> - </p> - <p> - <s xml:id="s4106" xml:space="preserve">Scimas enim ex .3. Å¿exti Euclid. quod eadem proportio erit ipÅ¿ius <var>.b.n.</var> ad <var>.n.<lb/>d.</var> quÄ™ ipÅ¿ius <var>.b.x.</var> ad <var>.x.d.</var> & ipÅ¿ius <var>.b.t.</var> ad <var>.t.d.</var> Å¿imiliter, </s> - <s xml:id="s4107" xml:space="preserve">quare ipÅ¿iusb <var>.n.</var> ad <var>.n.<lb/>d.</var> erit vt ipÅ¿ius <var>.b.t.</var> ad <var>.t.d</var>. </s> - <s xml:id="s4108" xml:space="preserve">Vnde Å¿equitur <var>.b.n.</var> æqualem eſſe ipÅ¿i <var>.b.t.</var> et <var>.n.d.</var> <lb/>ipÅ¿i <var>.t.d.</var> vt à medio circulo <var>.E.</var> potes videre, quamuis etiam <var>.b.</var> non eſſet extremum <lb/>diametri, Å¿ed vbicunque volueris in ipÅ¿o diametro, vel <reg norm="etiam" type="context">etiã</reg> protracta, eo quod pun-<lb/>ctum <var>.n.</var> & punctum <var>.t.</var> in eodem Å¿emicirculo, vel in æqualibus Å¿emicirculis, non <reg norm="poÅ¿sent" type="context">poÅ¿sẽt</reg> <lb/>aliter in ipÅ¿a circunferentia locari, <reg norm="eandem" type="context">eãdem</reg> Å¿eruando proportionem <var>.b.n.</var> ad <var>.n.d.</var> vt <var>.b.<lb/>t.</var> ad <var>.t.d.</var> </s> - <s xml:id="s4109" xml:space="preserve">propterea quod in omni alio Å¿itu exiÅ¿tente puncto <var>.t.</var> ipÅ¿a <var>.b.t.</var> eſſet aut maior <lb/>aut minor ipÅ¿a <var>.b.n.</var> et <var>.t.d.</var> aut minor, aut maior ipÅ¿a <var>.t.d.</var> ex .7. & 14. tertij Eucli. </s> - <s xml:id="s4110" xml:space="preserve">vnde <lb/>aut maior, aut minor proportio eſſet ipÅ¿ius <var>.b.t.</var> ad <var>.t.d.</var> quam ipÅ¿ius <var>.b.n.</var> ad <var>.n.d.</var> & non <lb/>eadem.</s> - </p> - <p> - <s xml:id="s4111" xml:space="preserve">Nunc è conuerÅ¿o Å¿i <var>.b.n.</var> et <var>.b.t.</var> Å¿unt Å¿ibi inuicem æquales, & Å¿ic <var>.n.d.</var> cum <var>.t.d.</var> Å¿equi-<lb/>tur ex .8. primi Eucli. angulos <var>.n.</var> et <var>.t.</var> inuicem æquales eſſe.</s> - </p> - <p> - <s xml:id="s4112" xml:space="preserve">Ab ijÅ¿dem Å¿peculationibus potes etiam videre vnde accidat quod partes Å¿uperio <lb/>res alicuius obiecti reflexæ à tali Å¿peculo concauo videntur nobis inferiores eſſe, & <lb/>inferiores appareant Å¿uperiores, & dextræ Å¿iniÅ¿træ, & Å¿iniÅ¿træ dextræ. </s> - <s xml:id="s4113" xml:space="preserve">quod autem <lb/>hucuÅ¿que demonÅ¿traui de Å¿peculis planis, & Å¿phæricis concauis, ratiocinare tu ijÅ¿dem <lb/>medijs circa Å¿phærica conuexa, vbi clarè videbis puncta huiuÅ¿modi Å¿peculi conuexi, <lb/>à quibus reflectitur imago obiecti ad ambos oculos, Å¿emper oportere æquidiÅ¿tantia <lb/>eſſe à <reg norm="puncto" type="context">pÅ©cto</reg> communi ipÅ¿ius Å¿uperficiei Å¿peculi, & catheto incidentiæ, dum unam tan <lb/>tummodo imaginem ipÅ¿ius obiecti videmus, & à diuerÅ¿is Å¿uperficiebus reflexionum.</s> - </p> - <p> - <s xml:id="s4114" xml:space="preserve">Nolo etiam prætermittere, quod nunc mihi Å¿uccurrit, hoc eÅ¿t quod poſſet ali-<lb/>quis duos Å¿itus inuenire, vnum pro oculo, alterum verò pro obiecto, reÅ¿pectu alicu-<lb/>ius Å¿peculi concaui, Å¿phÄ™roidis prolatæ, vt reflexio ipÅ¿ius obiecti videretur, vt linea <lb/>diuidens per æqualia ipÅ¿um Å¿peculum. </s> - <s xml:id="s4115" xml:space="preserve">ReÅ¿pectu verò alicuius Å¿peculi concaui Å¿phæ-<lb/>roidis oblongæ, vt reflexio obiecti ad oculum veniret à tota Å¿uperficie ipÅ¿ius Å¿pecu-<lb/>li, vnde tota Å¿uperficies ipÅ¿ius Å¿peculi videretur colorata illo colore cuius eſſet <lb/>obiectum, quæ quidem paſſiones <reg norm="pendent" type="context">pendẽt</reg> à .48. tertij lib. ipÅ¿ius Pergei, vt ex te ipÅ¿o fa <lb/>cile videre potes, propter æqualitatem angulorum reflexionis, & incidentiæ.</s> - </p> - <p> - <s xml:id="s4116" xml:space="preserve">Opinio autem mea, quam Å¿cire cupis de imagine obiecti reflexa, quam putas eÅ¿-<lb/>Å¿e in Å¿uperficie Å¿peculi, hæc eÅ¿t, quod nec in Å¿uperficie, nec ultra, nec citra eam eÅ¿t ip <lb/>Å¿a imago, quod autem vltra non Å¿it, hoc puto nulli dubium eſſe. </s> - <s xml:id="s4117" xml:space="preserve">eadem etiam ra-<lb/>tione non erit citra Å¿uperficiem Å¿peculi concaui, quamuis ipÅ¿am nos compræhenda-<lb/>mus in concurÅ¿u radiorum viÅ¿ualium, tam ab vno Å¿peculo quam ab alio reflexione <lb/>facta. </s> - <s xml:id="s4118" xml:space="preserve">Sed quòd ipÅ¿a neque Å¿it in ipÅ¿a Å¿peculi Å¿uperficie, manifeÅ¿tum erit ex hoc, <reg norm="quod" type="simple">ê™</reg> <lb/>duo Å¿pectantes in eodem Å¿peculo, duas diuerÅ¿as imagines vident, tres, <reg norm="autem" type="wordlist">aÅ©t</reg> tres, qua-<lb/>tuor, quatuor, & Å¿ic deinceps, vnde tot eſſent imagines Å¿upra Å¿uperficiem Å¿peculi, <lb/>quot obiecta, <reg norm="quod" type="simple">ê™</reg> tamen ita non eÅ¿t, nec plus eÅ¿t in vno loco ipÅ¿a imago, quam in alio, - <pb o="339" rhead="EPISTOLAE." n="351" file="0351" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0351"/> - niÅ¿i in obiecto ipÅ¿o, lumen enim abipÅ¿o obiecto reflexum, Å¿eipÅ¿um diffundit vndi-<lb/>que, & radijipÅ¿ius lu<unsure/>minis reflexi, vt plurimum Å¿einuicem Å¿ecant. </s> - <s xml:id="s4119" xml:space="preserve">Vnde in ipÅ¿o ae-<lb/>re funt omnes miÅ¿ti. </s> - <s xml:id="s4120" xml:space="preserve">Quapropter natura Å¿agacifÅ¿ima pupillam oculi animalibus tam <lb/>paruam conÅ¿truxit ad Å¿uperficiem tam amplæ Å¿phæræ ipÅ¿ius oculi, vt diÅ¿tinctæ vide-<lb/>rentur omnia obiecta.</s> - </p> - <p> - <s xml:id="s4121" xml:space="preserve">Nolo etiam tibi tacere, quod <reg norm="quotieÅ¿cunque" type="simple">quotieÅ¿cunq;</reg> oculorum pupillæ poÅ¿itæ fuerint inter <lb/>cathetum incidentiæ, & Å¿uperficiem <lb/> - <anchor type="figure" xlink:label="fig-0351-01a" xlink:href="fig-0351-01"/> - <anchor type="figure" xlink:label="fig-0351-02a" xlink:href="fig-0351-02"/> - Å¿peculi Å¿phærici concaui, vt puta in li-<lb/>neis <var>.d.t.</var> et <var>.t.n.</var> in figura <var>.D</var>. </s> - <s xml:id="s4122" xml:space="preserve">tunc nullo <lb/>pacto poſſemus videre vnam imagi-<lb/>nem obiecti, Å¿ed duas nec non confu-<lb/>sè, </s> - <s xml:id="s4123" xml:space="preserve">propterea <reg norm="quod" type="simple">ê™</reg> nullo pacto radij <var>.t.d.</var> <lb/>et <var>.n.t.</var> reflexi poterint. </s> - <s xml:id="s4124" xml:space="preserve">ambo vniri <reg norm="cum" type="context">cÅ©</reg> <lb/>ambobus axibus viÅ¿ualibus, eo quod <lb/>axes vifuales nunquam poſſunt inui-<lb/>cem interÅ¿ecari poÅ¿t viÅ¿um, Å¿ed Å¿em-<lb/>per ante ipÅ¿um, vnde nec inuicem pa-<lb/>ralleli poſſunt eſſe.</s> - </p> - <div type="float" level="5" n="4"> - <figure xlink:label="fig-0351-01" xlink:href="fig-0351-01a"> - <image file="0351-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0351-01"/> - </figure> - <figure xlink:label="fig-0351-02" xlink:href="fig-0351-02a"> - <image file="0351-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0351-02"/> - </figure> - </div> - <p> - <s xml:id="s4125" xml:space="preserve">Dico etiam, quod Å¿i obiectum inci <lb/>derit in eadem Å¿uperficie, in qua duo <lb/>axes viÅ¿uales, vel radij reflexi <reg norm="reperiuntur" type="context">reperiÅ©tur</reg>, hoc eÅ¿t in vna <reg norm="eademque" type="simple">eademqÌ;</reg> Å¿uperficie reflexio-<lb/>nis, </s> - <s xml:id="s4126" xml:space="preserve">tunc locus imaginis non erit in catheto incidentiæ, eo quod interfectio axium <lb/>uifualium non erit in ipÅ¿o catheto Å¿ed extra, in qua interÅ¿ectione fit viÅ¿io vnius tan-<lb/>tummodo imaginis, quod antiqui non animaduerterunt. </s> - <s xml:id="s4127" xml:space="preserve">Hoc autem dico deÅ¿pe-<lb/>culo Å¿phærico concauo.</s> - </p> - </div> - <div type="letter" level="4" n="4"> - <head xml:id="head501" style="it" xml:space="preserve">Speculatio cuiuÅ¿dam propoÅ¿itionis aritbmetica.</head> - <head xml:id="head502" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4128" xml:space="preserve">SPeculatio vltimæ propoÅ¿itionis quam numerorum via inueni, hæc eÅ¿t. </s> - <s xml:id="s4129" xml:space="preserve">Imagi-<lb/>nemur triangulum <var>.r.e.o.</var> abÅ¿ciÅ¿um à circulo, in cuius circunferentia Å¿it punctum <lb/>r. Å¿uperioris anguli ipÅ¿ius trianguli, vel etiam non Å¿it abÅ¿ciÅ¿um dummodo protrahan <lb/>tur lineæ <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> ad circunferentiam, à quo ad oppoÅ¿itum latus <reg norm="defcendant" type="context">defcẽdant</reg> duæ <var>.r.K.</var> et <var>.r.<lb/>f.</var> ita <reg norm="quod" type="simple">ê™</reg>. <var>K.o.</var> æqualis Å¿it <var>.f.e.</var> vnde hæc .4. lineæ Å¿ecabuntur à circulo dicto in punctis <var>.n.<lb/>c.b.u</var>. </s> - <s xml:id="s4130" xml:space="preserve">Dico nunc <reg norm="quod" type="simple">ê™</reg> producta <var>.o.r.n.</var> et <var>.e.r.u.</var> æqualia erunt productis <var>.K.r.c.</var> et <var>.f.r.b.</var> <lb/> - <anchor type="figure" xlink:label="fig-0351-03a" xlink:href="fig-0351-03"/> - <anchor type="figure" xlink:label="fig-0351-04a" xlink:href="fig-0351-04"/> - <pb o="340" rhead="IO. BAPT. BENED." n="352" file="0352" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0352"/> - quapropter cogitemus <var>.r.a.</var> indeterminatam tranÅ¿ire per centrum <var>.s.</var> ipÅ¿ius circuli, Å¿i-<lb/>militer etiam <var>.r.i.</var> ad punctum medium lateris <var>.e.o.</var> deinde à tribus punctis, <var>e.i.o.</var> ima-<lb/>ginemur tres perpendiculares ad <var>.r.a.</var> hoc eÅ¿t <var>.e.a</var>: <var>i.d.</var> et <var>.o.q.</var> & vbi circulus Å¿ecat <var>.r.a.</var> <lb/>fit punctum <var>.g.</var> protractis deinde <var>.g.n</var>: <var>g.x</var>: et <var>.g.u.</var> habebimus triangulum <var>.a.e.r.</var> Å¿imi-<lb/>lem triangulo <var>.g.u.r.</var> vnde clarum erit productum <var>.g.r.a.</var> æquale eſſe producto <var>.e.r.u.</var> <lb/><reg norm="productumque" type="simple">productumqÌ</reg> <var>.g.r.q.</var> æquale eſſe producto <var>.o.r.n.</var> nam trianguli <var>.g.r.n.</var> et <var>.o.r.q.</var> Å¿unt in-<lb/>u<unsure/>icem Å¿imiles, Å¿ed productum <var>.g.r.a.</var> Å¿imul cum producto <var>.g.r.q.</var> duplum eÅ¿t producto <var>.<lb/>g.r.d.</var> ex prima Å¿exti, eo quod <var>.a.r.q.</var> dupla eÅ¿t <var>.d.r.</var> & ideo productum <var>.e.r.u.</var> Å¿imul <reg norm="cum" type="context">cÅ©</reg> <lb/>producto <var>.o.r.n.</var> duplum erit producto <var>.i.r.x.</var> quod quidem æquale eÅ¿t producto <var>.g.r.<lb/>d.</var> ex Å¿imilibus rationibus iam Å¿upradictis. </s> - <s xml:id="s4131" xml:space="preserve">Nunc ex Å¿imilibus rationibus producta <var>.f.<lb/>r.b.</var> et <var>.K.r.c.</var> dupla erunt producto <var>.i.r.x</var>. </s> - <s xml:id="s4132" xml:space="preserve">quare prima producta æqualia erunt Å¿ecun-<lb/>dis. </s> - <s xml:id="s4133" xml:space="preserve">Quod eÅ¿t propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0351-03" xlink:href="fig-0351-03a"> - <image file="0351-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0351-03"/> - </figure> - <figure xlink:label="fig-0351-04" xlink:href="fig-0351-04a"> - <image file="0351-04" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0351-04"/> - </figure> - </div> - <p> - <s xml:id="s4134" xml:space="preserve">Ab huiuÅ¿modi demonÅ¿trat ione facilè videre poteris non eſſe generaliter verum, <lb/>id quod Nicolaus Tartalea inquit .43. quæſito vltimæ partis Å¿uorum tractatuum, hoc <lb/>eÅ¿t centrum circuli <var>.r.n.g.</var> Å¿emper eſſe in perpendiculari, quæ à puncto <var>.r.</var> ad lineam <var>.e.<lb/>o.</var> tranÅ¿it, protracta ipÅ¿a <var>.e.o.</var> quantum volueris, imò in quacunque alia linea ipÅ¿um eÅ¿ <lb/>Å¿e poteÅ¿t, nec non in aliqua parallela ipÅ¿i <var>.e.o.</var> quemadmodum ex te ipÅ¿o, medianti-<lb/>bus, hic Å¿upradictis rationibus videre poteris, vnde ex neceſſitate Å¿equitur illud pro <lb/>blema Å¿emper ferè falÅ¿um eſſe.</s> - </p> - <figure position="here"> - <image file="0352-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0352-01"/> - </figure> - <figure position="here"> - <image file="0352-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0352-02"/> - </figure> - </div> - <div type="letter" level="4" n="5"> - <head xml:id="head503" style="it" xml:space="preserve">Alia Å¿peculatio circa breuitatem radiorum incidentium <lb/>& reflexorum.</head> - <head xml:id="head504" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4135" xml:space="preserve">ALius modus quem exercitationis gratia vltimò cogitaui, ad demonÅ¿trandum <lb/>breuitatem radiorum incidentium, & reflexorum in Å¿peculo plano, nunc ad <lb/>te Å¿cribo, quamuis prolixior ali quantulum Å¿it eo, quod ab antiquis traditus eÅ¿t.</s> - </p> - <p> - <s xml:id="s4136" xml:space="preserve">Imaginemur itaque <reg norm="lineam" type="context">lineã</reg> <var>.p.h.</var> pro <reg norm="communi" type="context">cõmuni</reg> Å¿ectione Å¿uperficiei reflexionis <reg norm="cum" type="context">cÅ©</reg> Å¿pe-<lb/>culo <var>.r.a.</var> verò et <var>.a.b.</var> pro radijs dictis, qui Å¿emper <reg norm="faciunt" type="context">faciÅ©t</reg> angulos <var>.b.a.h.</var> et <var>.r.a.p.</var> <reg norm="inuicem" type="context">inuicẽ</reg> - <pb o="341" rhead="EPISTOL AE." n="353" file="0353" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0353"/> - æquales. </s> - <s xml:id="s4137" xml:space="preserve">Nunc protrahantur duæ <var>.r.o.</var> et <var>.b.o.</var> ab iiÅ¿dem punctis <var>.b.r.</var> ad aliud punctum, <lb/>quod volueris ipÅ¿ius lineæ <var>.p.h.</var> quas probabo <reg norm="longiores" type="conjecture">longiores'</reg> (Å¿imul Å¿umptas) eſſe priori-<lb/>bus. </s> - <s xml:id="s4138" xml:space="preserve">Imaginemur igitur duas perpendiculares, Å¿eu cathetos <var>.b.i.</var> et <var>.q.r.a.</var> punctis <var>.b.<lb/>r.</var> ad <var>.p.h.</var> <reg norm="abÅ¿ciſſaque" type="simple">abÅ¿ciſſaqÌ</reg> Å¿it linea <var>.o.b.</var> in puncto <var>.x.</var> ita quod <var>.b.x.</var> æqualis Å¿it ipÅ¿i <var>.b.a.</var> quod <lb/>nulli dubium erit poſſe effici, cum <var>.o.b.</var> <reg norm="longiot" type="context">lõgiot</reg> Å¿it <var>.b.a.</var> co quod opponatur angulo ob-<lb/>tuÅ¿o ipÅ¿ius trianguli <var>.b.a.o.</var> quÄ™ <var>.o.b.</var> Å¿imiliter protrahatur vÅ¿que ad <var>.d.</var> ita quod <var>.b.d.</var> <lb/>æqualis Å¿it <var>.x.b.</var> </s> - <s xml:id="s4139" xml:space="preserve">deinde protrahatur <var>.o.i.</var> quouÅ¿que <var>.i.h.</var> æqualis Å¿it <var>.a.i</var>. </s> - <s xml:id="s4140" xml:space="preserve">In alia parte po-<lb/>Å¿tea idem faciendum eÅ¿t Å¿ecando <var>.a.r.</var> in puncto <var>.u.</var> ita quod <var>.u.r.</var> æqualis Å¿it <var>.r.o.</var> efficien <lb/>do <var>.r.s.</var> æqualem <var>.r.u.</var> et <var>.q.p.</var> æquale <var>.q.o.</var> vnde habebimus <reg norm="productum" type="context">productÅ©</reg> <var>.o.d.</var> in <var>.o.x.</var> æqua <lb/>le producto <var>.o.h.</var> in <var>.o.a.</var> & productum <var>.a.s.</var> in <var>.a.u.</var> æquale producto <var>.a.p.</var> in <var>.a.o.</var> exiÅ¿tis <lb/>rationibus. </s> - <s xml:id="s4141" xml:space="preserve">Nam cum quadratum ipÅ¿ius <var>.o.b.</var> æquale Å¿it duobus quadratis <var>.o.i.</var> et <var>.i.<lb/>b.</var> ex penultima primi Eucli. ipÅ¿a quadrata <var>.o.i.</var> et <var>.i.b.</var> æqualia erunt producto <var>.o.d.</var> in <lb/><var>o.x.</var> Å¿imul Å¿umpto cum quadrato <var>.b.x.</var> ex .6. Å¿ecundi, hoc eÅ¿t ipÅ¿i producto Å¿imul Å¿um-<lb/>pto cum quadrato <var>.b.a.</var> hoc eÅ¿t ipÅ¿i producto Å¿imul Å¿umpto cum duobus quadratis <var>.a.<lb/>i.</var> et <var>.i.b.</var> Å¿ed quia productum <var>.o.h.</var> in <var>.o.a.</var> Å¿imul Å¿umpto cum quadrato <var>.a.i.</var> Ä™quatur qua <lb/>drato <var>.o.i.</var> ideo productum <var>.o.h.</var> in <var>.o.a.</var> Å¿imul Å¿umptum cum quadrato <var>.a.i.</var> & cum qua-<lb/>drato <var>.i.b.</var> æquale erit producto <var>.o.d.</var> in <var>.o.x.</var> Å¿imul Å¿umpto <reg norm="cum" type="context">cÅ©</reg> duobus quadratis dictis <lb/>hoc eÅ¿t ipÅ¿ius <var>.a.i.</var> et <var>.i.b.</var> quÄ™ quadrata dempta cum fuerint ab vtraque parte, tunc cer <lb/>ti erimus producta eſſe inuicem æqualia. </s> - <s xml:id="s4142" xml:space="preserve">Idem dico de alijs ex altera parte. </s> - <s xml:id="s4143" xml:space="preserve">Nunc <lb/>imaginemur protractam eſſc <var>.a.e.</var> parallelam ipÅ¿i <var>.o.b.</var> & habebimus proportionem <lb/>ipÅ¿ius <var>.a.b.</var> ad <var>.a.i.</var> maiorem eſſe ea quæ eÅ¿t ipÅ¿ius <var>.a.e.</var> ad eandem <var>.a.i.</var> cum <var>.a.b.</var> maior <lb/>Å¿it ipÅ¿a <var>.a.e.</var> vt oppoÅ¿ita angulo obtuÅ¿o, quapropter proportio <var>.x.b.</var> ad <var>.a.i.</var> maior erit <lb/>ea quæ eÅ¿t <var>.o.b.</var> ad <var>.o.i</var>. </s> - <s xml:id="s4144" xml:space="preserve">Iam enim Å¿cis proportionem <var>.o.b.</var> ad <var>.o.i.</var> eſſe, vt <var>.a.e.</var> ad <var>.a.i.</var> ex <lb/>Å¿imilitudine triangulorum. </s> - <s xml:id="s4145" xml:space="preserve">quare proportio <var>.b.d.</var> ad <var>.i.h.</var> maior erit proportione <var>.o.b.</var> <lb/>ad <var>.o.i.</var> <reg norm="tunc" type="context">tÅ©c</reg> ex .27. quinti <reg norm="permutando" type="simple context">ê‘mutãdo</reg> <reg norm="proportio" type="simple">ê“portio</reg> <var>.b.d.</var> ad <var>.b.o.</var> maior erit proportione <var>.i.h.</var> <lb/>ad <var>.i.o.</var> & ex .26. <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> <reg norm="componendo" type="context context">cõponẽdo</reg> maior <reg norm="proportio" type="simple">ê“portio</reg> erit <var>.o.d.</var> ad <var>.o.b.</var> ea quÄ™ eÅ¿t <var>.o.h.</var> ad. o <lb/>i. & <reg norm="permutando" type="context">permutãdo</reg> maior ipÅ¿ius <var>.o.d.</var> ad <var>.o.h.</var> ea quæ <var>.o.b.</var> ad <var>.o.i.</var> & ex .33. maior ipÅ¿ius <var>.b.<lb/>d.</var> ad <var>.i.h.</var> ea quæ <var>.o.d.</var> ad <var>.o.h</var>. </s> - <s xml:id="s4146" xml:space="preserve">Sed vt <var>.b.a.</var> ad <var>.a.i.</var> ita eÅ¿t <var>.a.r.</var> ad <var>.a.q.</var> ex Å¿imilitudine <reg norm="triam" type="context">triã</reg> <lb/>gulorum. </s> - <s xml:id="s4147" xml:space="preserve">Erit igitur <var>.a.r.</var> ad <var>.a.q.</var> maior proportio, ea quæ eÅ¿t <var>.o.b.</var> ad <var>.o.i.</var> & exijÅ¿dem <lb/>Å¿upradictis rationibus maior erit proportio ipÅ¿ius <var>.s.a.</var> ad <var>.p.a.</var> ea quæ eÅ¿t <var>.a.r.</var> ad <var>.a.q.</var> <lb/>Å¿ed cum iam probatum fuit proportio <lb/> - <anchor type="figure" xlink:label="fig-0353-01a" xlink:href="fig-0353-01"/> - nem <var>.b.d.</var> ad <var>.i.h.</var> hoc eÅ¿t <var>.a.b.</var> ad <var>.a.i.</var> ma <lb/>iorem eſſe <var>.o.d.</var> ad <var>.o.h.</var> ergo eo ma-<lb/>gis maior erit proportio ipÅ¿ius <var>.a.s.</var> ad <lb/><var>a.p.</var> ca quæ <var>.o.d.</var> ad <var>.o.h.</var> Å¿ed cum ex .15 <lb/>Å¿exti, eadem Å¿it proportio <var>.o.d.</var> ad <var>.o.a.</var> <lb/>quæ <var>.o.h.</var> ad <var>.o.x.</var> et <var>.s.a.</var> ad <var>.o.a.</var> quÄ™ <var>a.p.</var> <lb/>ad <var>.a.u.</var> </s> - <s xml:id="s4148" xml:space="preserve">tunc erit <reg norm="permutando" type="context">permutãdo</reg> eadem <lb/>proportio ipÅ¿ius <var>.o.d.</var> ad <var>.o.h.</var> quæ <var>.o.a.</var> <lb/>ad <var>.o.x.</var> & ipÅ¿ius <var>.a.o.</var> ad <var>.a.u.</var> quemad-<lb/>modum ipÅ¿ius <var>.a.s.</var> ad <var>.a.p</var>. </s> - <s xml:id="s4149" xml:space="preserve">Quare maior proportio erit ipÅ¿ius <var>.a.o.</var> ad <var>.a.u.</var> quam <var>.a.</var> o<unsure/>. <lb/>ad <var>.o.x</var>. </s> - <s xml:id="s4150" xml:space="preserve">Vnde Å¿equitur <var>.o.x.</var> maiorem eſſe <var>.a.u.</var> ex .8. quinti, ergo <var>.b.x.o.r.</var> longior erit <lb/>ipÅ¿a <var>.b.a.u.r</var>. </s> - <s xml:id="s4151" xml:space="preserve">Quod eÅ¿t propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0353-01" xlink:href="fig-0353-01a"> - <image file="0353-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0353-01"/> - </figure> - </div> - <pb o="342" rhead="IO. BAPT. BENED." n="354" file="0354" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0354"/> - <p> - <s xml:id="s4152" xml:space="preserve">Alia etiam via poſſumus idem concludere. </s> - <s xml:id="s4153" xml:space="preserve">Imaginemur maiorem axem alicu-<lb/>ius ellipÅ¿is tranÅ¿ire per duo puncta <var>.r.</var> et <var>.b.</var> Å¿upponendo ipÅ¿a puncta, ea eÅ¿le, quæ ita <lb/>axem diuidunt, vt Å¿ingula produ-<lb/> - <anchor type="figure" xlink:label="fig-0354-01a" xlink:href="fig-0354-01"/> - cta fectionum Å¿int, vt inquit Per-<lb/>geus. </s> - <s xml:id="s4154" xml:space="preserve">imaginemur, etiam <var>.p.h.</var> con <lb/>tiguam eſſe ipÅ¿i ellipÅ¿i in <reg norm="puncto" type="context">pÅ©cto</reg> <var>.a.</var> <lb/>vnde Å¿i protractæ fuerint duæ <var>.r.a.</var> <lb/>et <var>.b.a.</var> habebimus ex .48. tertijip-<lb/>Å¿ius Pergei angulos <var>.b.a.h.</var> et <var>.r.a.<lb/>p.</var> inuicem æquales. </s> - <s xml:id="s4155" xml:space="preserve">Ducendo <lb/>poÅ¿tea ad quoduis punctum ipÅ¿ius <lb/><var>p.h.</var> duas <var>.b.o.</var> et <var>.r.o.</var> certi erimus, <lb/>quod Å¿ecabuntur à gyro oxygo-<lb/>nio, quarum vna Å¿ecta Å¿it in pun-<lb/>cto <var>.i.</var> ducta poÅ¿tea <var>.i.r.</var> clarum erit ex .52. dicti, quod longitudo <var>.b.i.r.</var> æqualis erit lon <lb/>gitudini <var>.b.a.r.</var> & minor ipÅ¿a <var>.b.o.r.</var> ex .21. primi Euclid.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0354-01" xlink:href="fig-0354-01a"> - <image file="0354-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0354-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="6"> - <head xml:id="head505" style="it" xml:space="preserve">Deerrore Euclidis circa Å¿peculum vstorium.</head> - <head xml:id="head506" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4156" xml:space="preserve">VErum Å¿peculum vÅ¿torium, illud non eÅ¿t, quod ab Euclide traditum fuit, & <reg norm="quod" type="simple">ê™</reg> <lb/>tu etiam putas, Nam Euclides errat, cum credat radios reflexos à ſuperficie <lb/>Å¿phærica concaua Å¿einuicem in centro Å¿peculi interÅ¿ecare. </s> - <s xml:id="s4157" xml:space="preserve">Nam cum omnes lineÄ™ <lb/>recte à centro, & cir cunferentia alicuius Å¿phæræ terminatæ, Å¿int eidem circunferen-<lb/>tiæ perpendiculares, Å¿equeretur ex neceſſitate radios incidentiæ etiam perpendicu <lb/>lares eidem Å¿uperficiei eſſe, cum anguli incidentiæ Å¿emper æquales Å¿int angulis re-<lb/>flexionis, vnde etiam ex neceſſitate Å¿equeretur punctum corporis lucidi, à quo radij <lb/>luminoÅ¿i excunt, in centro Å¿peculi reperiri. </s> - <s xml:id="s4158" xml:space="preserve">quod quidem falÅ¿iſſimum eÅ¿t.</s> - </p> - <p> - <s xml:id="s4159" xml:space="preserve">Alia etiam via poſſum hanc oÅ¿tendere impoſſibilitatem, & tibi probabo, quod <lb/>in nullo aliquo puncto poſſunt inuicem conuenire ipÅ¿i radijrefle xi omnes.</s> - </p> - <p> - <s xml:id="s4160" xml:space="preserve">Sit igitur <var>.l.a.c.</var> <reg norm="conis" type="context">cõis</reg> Å¿ectio Å¿uperficiei reflexionis cum Å¿peculo, cuius centrum Å¿it <var>.o.</var> <lb/>punctum verò lucidum Å¿it <var>.g.</var> <reg norm="protrahaturque" type="simple">protrahaturqÌ</reg> <var>.g.o.a</var>. </s> - <s xml:id="s4161" xml:space="preserve">Nunc autem primum dico, quod <lb/>radij reflexi à punctis diuerÅ¿arum <reg norm="diÅ¿tantiarum" type="context">diÅ¿tantiarÅ©</reg> ab <var>.a.</var> non <reg norm="coincident" type="context">coincidẽt</reg> inuicem in aliquo <lb/>puncto lineÄ™ <var>.g.o.a</var>: Å¿int ergo duo puncta <var>.u.</var> et <var>.r.</var> diuerÅ¿arum <reg norm="diÅ¿tantiarum" type="context">diÅ¿tantiarÅ©</reg> ab <var>.a.</var> à quibus <lb/>veniant duo radij incidentiæ <var>.g.r.</var> et <var>.g.u.</var> radius verò reflexus ab <var>.r.</var> Å¿it <var>.r.e.</var> protrahatur <lb/><var>u.e.</var> quam dico effe non poſſe radium reflexum ab <var>.u.</var> quotieÅ¿cunque eius incidens <lb/>deÅ¿cendat ab <var>.g</var>. </s> - <s xml:id="s4162" xml:space="preserve">Protrahantur ergo duæ lineæ <var>.o.r.</var> et <var>.o.u.</var> vnde cum dixerit aliquis <lb/><var>u.e.</var> <reg norm="reflexum" type="context">reflexÅ©</reg> eſſe ipÅ¿ius <var>.g.u.</var> igitur anguli <var>.g.u.o.</var> et <var>.o.u.e.</var> erunt inuicem æquales, & Å¿ic <lb/>etiam erunt duo <var>.g.r.o.</var> et <var>.o.r.e.</var> vnde ex tertia Å¿exti & .11. quinti Eucli. proportio <var>.g.<lb/>u.</var> ad <var>.u.e.</var> æqualis eſſet ei, quæ <var>.g.r.</var> ad <var>.r.e.</var> quod quidem impoſſibile eſſe demonÅ¿tra-<lb/>bo, eo quod cum <var>.g.u.</var> maior Å¿it <var>.g.r.</var> ex .8. tertij, erit ex .8. quinti proportio ipÅ¿ius <var>.g.u.</var> <lb/>ad <var>.r.e.</var> maior proportione ipÅ¿ius <var>.g.r.</var> ad <var>.r.e.</var> Å¿ed ex .7. tertij <var>.u.e.</var> minor eÅ¿t <var>.r.e.</var> erit igi-<lb/>tur ex dicta .8. quinti maior proportio <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <var>.g.u.</var> ad <var>.u.e.</var> quam <var>.g.u.</var> ad <var>.r.e.</var> vnde eo ma - <pb o="343" rhead="EPISTOL AE." n="355" file="0355" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0355"/> - gis erit maior proportio ipÅ¿ius <var>.g.u.</var> ad <var>.u.e.</var> <lb/> - <anchor type="figure" xlink:label="fig-0355-01a" xlink:href="fig-0355-01"/> - quam ipÅ¿ius <var>.g.r.</var> ad <var>.r.e.</var> ergo non æqualis, <lb/>quapropter impoſſibile eÅ¿t <var>.u.e.</var> eſſe radium <lb/>reflexum incidentis radij <var>.g.u</var>. </s> - <s xml:id="s4163" xml:space="preserve">Vnde Å¿equi <lb/>tur concurÅ¿um radiorum reflexorum à ſpe-<lb/>culo Å¿phærico concauo non eſſe in vno, & <lb/>e<unsure/>odem puncto ipÅ¿ius catheti incidentiæ, <lb/>quando à ſitu non æquidiÅ¿tanti ab ipÅ¿o ca-<lb/>theto <reg norm="reflectuntur" type="context">reflectÅ©tur</reg>, ex hac Å¿peculatione <reg norm="etiam" type="context">etiã</reg> <lb/>videre licet, verum eſſe id quod in .3. EpiÅ¿to <lb/>la tibi Å¿cripÅ¿i nempe, quod quotieÅ¿cunque <lb/>axes viÅ¿uales, vel radij reflexi, in vna ea-<lb/><reg norm="demque" type="simple">demqÌ</reg> Å¿uperficie reflexionis fuerint, </s> - <s xml:id="s4164" xml:space="preserve">tunc <lb/>imago obiecti nullo modo videbitur in ca-<lb/>theto incidentiæ, in Å¿peculo Å¿phærico con-<lb/>cauo.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0355-01" xlink:href="fig-0355-01a"> - <image file="0355-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0355-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="7"> - <head xml:id="head507" style="it" xml:space="preserve">Alterius dubit ationis Å¿olutio.</head> - <head xml:id="head508" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4165" xml:space="preserve">NOn abÅ¿que ratione dubitas, vtrum etiam in Å¿phæricis Å¿peculis conuexis idem <lb/>accidat, hoc eÅ¿t, an radij reflexi à punctis inÄ™qualis diÅ¿tantiæ à catheto inciden <lb/>tiæ conueniant inuicem in eodem catheto.</s> - </p> - <p> - <s xml:id="s4166" xml:space="preserve">Ad quod reÅ¿pondeo, non concurrere in dicto catheto, Å¿ed extra ipÅ¿um, & Å¿imi-<lb/>liter extra ipÅ¿um vide bitur imago.</s> - </p> - <p> - <s xml:id="s4167" xml:space="preserve">Pro cuius rei ratione, imaginemur Å¿uperficiem reflexionis alicuius Å¿peculi Å¿phæ-<lb/>rici conuexi <var>.b.d.h.g.</var> cuius communis Å¿ectio cum Å¿uperficie Å¿phærica Å¿it linea <lb/>circularis <var>.d.e.h.</var> et <var>.o.</var> eius <reg norm="centrum" type="context">cẽtrum</reg>, à quo protrahatur <var>.g.b.</var> indeterminata, et <var>.o.g.</var> Å¿it Å¿e <lb/>midiameter circuli <var>.d.g.h.</var> et <var>.o.c.</var> Å¿it plus medietate ipÅ¿ius <var>.o.g.</var> <reg norm="accipiaturque" type="simple">accipiaturqÌ;</reg> linea <var>.e.c.</var> <lb/>minor ipÅ¿a <var>.o.c.</var> Å¿ed maior ipÅ¿a <var>.c.g.</var> <lb/> - <anchor type="figure" xlink:label="fig-0355-02a" xlink:href="fig-0355-02"/> - quod difficile non erit, locando im <lb/>mobilem pedem circini in puncto <var>.<lb/>c.</var> aperiendo ipÅ¿um aliquantulum <lb/>plus quam <var>.c.g.</var> Å¿ed minus quam <var>.c.<lb/>o.</var> Å¿ignando circunferentiam <var>.d.e.h.</var> <lb/>in puncto <var>.e.</var> quod ex .7. tertij poſſi-<lb/>bile eÅ¿t, protrahatur poÅ¿tea <var>.o.e.f.</var> <lb/>indeterminatè. </s> - <s xml:id="s4168" xml:space="preserve">Facicmus deinde <lb/>angulum <var>.f.e.b.</var> æqualem angulo <var>.o.<lb/>e.c.</var> protracta poÅ¿tea cum fuerit <var>.c.<lb/>e.K.</var> indeterminatè, <reg norm="habebimus" type="simple">habebimê°</reg> duos <lb/>angulos <var>.b.e.f.</var> et <var>.f.e.K.</var> æquales in-<lb/>uicem mediante .15. primi, ita <reg norm="quod" type="wordlist">qÄ‘</reg> Å¿i <lb/>radius incidens veniet à puncto <var>.b.</var> <lb/>ad <var>.e.</var> reflexus erit <var>.e.K.</var> qui quidem - <pb o="344" rhead="IO. BAPT. BENED." n="356" file="0356" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0356"/> - refleyus Å¿ecabit cathetum <var>.b.o.</var> in puncto <var>.c.</var> intra Å¿peculum, nec dubitandum eÅ¿t quin <lb/>linea <var>.e.b.</var> Å¿ectura Å¿it <var>.b.o.</var> eo quod cum angulus <var>.o.e.c.</var> Å¿it maior angulo <var>.e.o.c.</var> ex .19. <lb/>primi, & Å¿imiliter angulus <var>.b.e.f.</var> Å¿equitur ex .13. dicti, angulos <var>.b.e.o.</var> et <var>.e.o.b.</var> eſſe mi <lb/>nores duobus rectis, vnde ex penultima petitione primi, duæ lineæ <var>.b.e.</var> et <var>.o.b.</var> <reg norm="inuicem" type="context">inuicẽ</reg> <lb/>concurrent. </s> - <s xml:id="s4169" xml:space="preserve">Quare poſſumus ex hoc, quoddam corollarium extrahere, hoc eÅ¿t <lb/><reg norm="neceſſarium" type="context">neceſſariÅ©</reg> <reg norm="semper" type="context">sẽper</reg> exiÅ¿tat, vt linea <var>.c.e.</var> minor eſſe linea <var>.c.o</var>. </s> - <s xml:id="s4170" xml:space="preserve">Sed vnde eueniat quod ip <lb/>Å¿a neceſſariò debeat Å¿emper maior eſſe ipÅ¿a <var>.c.g.</var> clarum eÅ¿t ex .7. tertij Eucli. </s> - <s xml:id="s4171" xml:space="preserve">Nunc <lb/>imaginemur ductas eſſe duas <reg norm="tangentes" type="context">tãgentes</reg> <var>.b.d.</var> et <var>.b.h.</var> & ab <var>.e.</var> <reg norm="ipsam" type="context">ipsã</reg> <var>.e.i.</var> vnde certi erimus, <lb/>quod ab interuallo inter <var>.h.</var> et <var>.d.</var> punctum <var>.b.</var> <reg norm="ponſſibile" type="context">põſſibile</reg> Å¿it vt reflectatur. </s> - <s xml:id="s4172" xml:space="preserve">Accipiamus <lb/>nunc <var>.p.c.</var> minorem medietate ipÅ¿ius <var>.b.c.</var> & à puncto <var>.p.</var> imaginemur tangentem <var>.p.q.</var> <lb/>in puncto <var>.q.</var> prorractaqÌue Å¿it <var>.b.q.</var> vt radius incidentiæ, </s> - <s xml:id="s4173" xml:space="preserve">tunc dico, radium reflexum <lb/>ipÅ¿ius <var>.b.q.</var> <reg norm="non" type="context">nõ</reg> concurrere in eodem puncto <var>.c.</var> ipÅ¿ius catheti, Å¿i vero dixeris <reg norm="quod" type="simple">ê™</reg> Å¿ic. </s> - <s xml:id="s4174" xml:space="preserve">EÅ¿to <lb/><reg norm="igitur" type="simple">igit̃</reg> radius dictus <var>.c.q.s</var>. </s> - <s xml:id="s4175" xml:space="preserve">Imaginemur <reg norm="tangentem" type="context context">tãgentẽ</reg> <var>.e.i.</var> in puncto <var>.e.</var> vnde ex .18. quinti Alha <lb/>zem, vel .12. Å¿exti Vitellionis proportio <var>.b.i.</var> ad <var>.i.c.</var> erit, vt <var>.b.o.</var> ad <var>.o.c.</var> & Å¿imiliter erit <lb/>ipÅ¿ius <var>.b.p.</var> ad <var>.p.c.</var> vt <var>.b.o.</var> ad <var>.o.c.</var> ex eadem. </s> - <s xml:id="s4176" xml:space="preserve">Quare ex .11. quinti Eucli. proportio ip <lb/>Å¿ius <var>.b.p.</var> ad <var>.p.c.</var> erit vt ipÅ¿ius <var>.b.i.</var> ad <var>.i.c.</var> Å¿ed quia <var>.p.b.</var> vt pars ipÅ¿ius <var>.b.i.</var> minor eÅ¿t ip-<lb/>Å¿a, ergo ex .14. dicti <var>.p.c.</var> minor erit ipÅ¿a <var>.c.i.</var> hoc eÅ¿t totum minus Å¿ua parte, quod eÅ¿t <lb/>impoſſibile, </s> - <s xml:id="s4177" xml:space="preserve">quare non in ipÅ¿o catheto videbitur imago ipÅ¿ius obiecti.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0355-02" xlink:href="fig-0355-02a"> - <image file="0355-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0355-02"/> - </figure> - </div> - <p> - <s xml:id="s4178" xml:space="preserve">Aliud notandum etiam cernere potes ex ipÅ¿is Å¿peculis Å¿phæricis conuexis, hoc eÅ¿t <lb/>quod poſſibile Å¿it aliquoties, radium reflexum concurrere cum catheto incidentiæ <lb/>extra Å¿peculum inter puncta <var>.g.</var> et <var>.p.</var> vt exempli gratia .Å¿i punctus <var>.p.</var> eſſet exactè <lb/>in medio inter <var>.b.</var> et g. </s> - <s xml:id="s4179" xml:space="preserve">tunc punctum <var>.c.</var> ipÅ¿ius concurÅ¿us cum catheto incidentiæ eſſet <lb/>inter <var>.g.</var> et <var>.p.</var> eo quod <reg norm="cum" type="context">cÅ©</reg> linea <var>.p.q.</var> debeat @iui lere <reg norm="angulum" type="context">angulÅ©</reg> <var>.b.</var> q, c. <reg norm="per" type="simple">ê‘</reg> Ä™qualia, oportebit <lb/>c. poÅ¿itum eſſe inter <var>.g.</var> et <var>.p.</var> quia angulus <var>.g.q.p.</var> maior eÅ¿t angulo <var>.p.q.b.</var> vt per te faci <lb/>le potes ratiotinari, imaginando cir <lb/> - <anchor type="figure" xlink:label="fig-0356-01a" xlink:href="fig-0356-01"/> - culum circa <reg norm="triangulum" type="context">triãgulum</reg> <var>.g.q.b.</var> & dia <lb/>merrum perpendicularem .ad <var>.g.b.</var> <lb/>in puncto <var>.p.</var> producendo poÅ¿tea <var>.q.<lb/>p.</var> <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> ad <reg norm="alteram" type="context">alterã</reg> <reg norm="partem" type="context">partẽ</reg> circunferen-<lb/>tiæ ipÅ¿ius circuli. </s> - <s xml:id="s4180" xml:space="preserve"><reg norm="argumentando" type="context context">argumẽtãdo</reg> dein-<lb/>de mediante vltima Å¿exti, illud <reg norm="idem" type="context">idẽ</reg> <lb/>po@es etiam Å¿cire ex .22. quinti Alha <lb/>zeni. & ex .26. Å¿exti Vitellionis. </s> - <s xml:id="s4181" xml:space="preserve">vn-<lb/>de Å¿i ad ambas pupillas venerint ra <lb/>dij reflexi ipÅ¿ius obiecti <var>.b.a.</var> duobus <lb/>punctis huiuÅ¿modi Å¿peculi, ita di-<lb/>Å¿tantibus à puncto <var>.g.</var> vt <var>.q</var>. </s> - <s xml:id="s4182" xml:space="preserve">tunc com <lb/>mune punctum concurÅ¿us axium vi <lb/>Å¿ualium erit in catheto inter <var>.g.p.</var> <lb/>vbi apparebit imago ex Å¿uperius di <lb/>ctis rationibus, ita vt <reg norm="non" type="context">nõ</reg> Å¿olum con <lb/>cauis, Å¿ed etiam conuexis hoc accidere poſſit.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0356-01" xlink:href="fig-0356-01a"> - <image file="0356-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0356-01"/> - </figure> - </div> - <p> - <s xml:id="s4183" xml:space="preserve">In planis autem <reg norm="nunquam" type="context">nunquã</reg> hoc poteÅ¿t euenire, vt tibi alias dixi, eo quod Å¿i <reg norm="acceperi- mus" type="conjecture">accéperi-<lb/>mus</reg> <reg norm="rectam" type="context">rectã</reg> <var>.m.r.</var> pro <reg norm="coni" type="context">cõi</reg> Å¿ectione <reg norm="Å¿uperficiei" type="simple">Å¿uê‘ficiei</reg> <var>.l.t.x.</var> reflexionis & <reg norm="Å¿uperficiei" type="simple">Å¿uê‘ficiei</reg> Å¿peculi, <reg norm="punctumque" type="context context simple">pÅ©ctÅ©qÌ;</reg> <lb/>lucidum <var>.l.</var> <reg norm="protractoque" type="simple">protractoqÌ;</reg> catheto <var>.l.r.t.</var> <reg norm="lineisque" type="simple">lineisqÌ;</reg> incidentiæ <var>.l.x.</var> et <var>.l.m.</var> reflexionis etiam <lb/><var>x.y.</var> et <var>.m.z.</var> cum anguli <var>.l.x.r.</var> et <var>.y.x.h.</var> et <var>.r.x.t.</var> æquales inuicem Å¿int, & Å¿ic anguli <var>.l.m.<lb/>r.</var> et <var>.z.m.h.</var> et <var>.r.m.t.</var> erit <var>.r.t.</var> tam pro triangulo <var>.r.x.t.</var> quam pro triangulo <var>.r.m.t.</var> æqua <lb/>lis <var>.r.l.</var> ex .26. primi, ita quod Å¿emper in puncto <var>.t.</var> <reg norm="conuenient" type="context">conueniẽt</reg> omnes radij reflexi ipÅ¿ius - <pb o="345" rhead="EPISTOL AE." n="357" file="0357" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0357"/> - puncti <var>.l.</var> clarum igitur nunc habes, quod in Å¿phærico concauo, Å¿eu conuexo, non <lb/>omnes radij reflexi conueniunt in vno, <reg norm="eodemque" type="simple">eodemqÌ;</reg> puncto catheti incidentiæ, quemad <lb/>modum in planis accidit, in quibus Å¿emper vnum, & idem punctum eÅ¿t ipÅ¿is commu <lb/>ne in ipÅ¿o incidentiæ catheto.</s> - </p> - <p> - <s xml:id="s4184" xml:space="preserve">Non prætermittam etiam hunc alium breuiorem modum Å¿peculandi <reg norm="æqualita- tem" type="context">æqualita-<lb/>tẽ</reg> depreſſionis imaginis Å¿ub Å¿peculo plano, ei quæ Å¿upra reperitur ipÅ¿ius obiecti, in ca <lb/>theto incidentiæ, quemadmodum nu nc <lb/>vltimò diximus, hoc eÅ¿t quod cum <lb/> - <anchor type="figure" xlink:label="fig-0357-01a" xlink:href="fig-0357-01"/> - imago obiecti <var>.l.</var> reflexa à puncto <var>.<lb/>x.</var> reperiatur in linea <var>.y.x.t.</var> & ima-<lb/>go eiuÅ¿dem obiecti reflexa à pun-<lb/>cto <var>.m.</var> reperiatur in linea <var>.z.m.t.</var> & <lb/>iÅ¿tæ duæ lineÄ™ Å¿einuicem Å¿ecent in <lb/>puncto <var>.t.</var> ipÅ¿ius catheti, exiÅ¿tente <var>.<lb/>r.t.</var> æquali <var>.r.l.</var> vt nunc vidimus, er-<lb/>go Å¿emper imago reflexa à ſpecu-<lb/>lo plano, nobis apparebit <reg norm="in" type="wordlist">Ä©</reg> ipÅ¿o ca <lb/>theto, tam vltra Å¿peculum, quam ci <lb/>tra ipÅ¿um, <reg norm="repertum" type="simple context">reê‘tÅ©</reg> fuerit <reg norm="ipsum" type="context">ipsÅ©</reg> <reg norm="obiectum" type="context">obiectÅ©</reg> <lb/>quod nec Alhazem, nec Vitellio, <lb/>nec alius aliquis (quod Å¿ciam) ad huc Å¿cientificè demonÅ¿trauit. </s> - <s xml:id="s4185" xml:space="preserve">exempla enim vel ex <lb/>perientia non faciunt Å¿cire. </s> - <s xml:id="s4186" xml:space="preserve">Credo etiam te non dubitare quin duæ lineæ <var>.y.x.</var> et <var>.z.<lb/>m.</var> inuicem concurrant, cum anguli <var>.t.x.m.</var> et <var>.t.m.x.</var> minores Å¿int duobus rectis cum <lb/>æquales Å¿int angulis <var>.l.x.m.</var> et <var>.l.m.x</var>.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0357-01" xlink:href="fig-0357-01a"> - <image file="0357-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0357-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="8"> - <head xml:id="head509" style="it" xml:space="preserve">De rotunditate vmbræterræ in ecclipÅ¿ibus Lunaribus.</head> - <head xml:id="head510" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4187" xml:space="preserve">ROtunditas vmbræ in ecclipÅ¿i-<lb/> - <anchor type="figure" xlink:label="fig-0357-02a" xlink:href="fig-0357-02"/> - bus lunaribus oritur <reg norm="tam" type="context">tã</reg> à rotun <lb/>ditate maris, <reg norm="quam" type="context">quã</reg> terræ, & Å¿i terra eÅ¿-<lb/>Å¿et <reg norm="etiam" type="context">etiã</reg> cuiuÅ¿uis alterius figurÄ™, <reg norm="quam" type="context">quã</reg> <lb/>Å¿phæricÄ™, dummodo aqua impleret <lb/><reg norm="locum" type="context">locÅ©</reg> Å¿phÄ™riceitatis à terra <reg norm="derelictum" type="context">derelictÅ©</reg>, <lb/>nihilominus vmbra eſſet rotunda, <lb/>quÄ™ quidem ab aqua produceretur, <lb/><reg norm="quanuis" type="context">quãuis</reg> Alexander Piccolhomineus <lb/> - <anchor type="figure" xlink:label="fig-0357-03a" xlink:href="fig-0357-03"/> - aliter Å¿entiat in libro de magnitudi-<lb/>ne terrÄ™, & aquæ.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0357-02" xlink:href="fig-0357-02a"> - <image file="0357-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0357-02"/> - </figure> - <figure xlink:label="fig-0357-03" xlink:href="fig-0357-03a"> - <image file="0357-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0357-03"/> - </figure> - </div> - <p> - <s xml:id="s4188" xml:space="preserve"><reg norm="Sciendum" type="context context">SciẽdÅ©</reg> enim eÅ¿t, quod omne cor <lb/>pus in Å¿e habens <reg norm="aliquantulum" type="context">aliquantulÅ©</reg> opaci-<lb/>tatis, Å¿emper debilitat <reg norm="radium" type="context">radiÅ©</reg> lumino <lb/>Å¿um, & <reg norm="tanto" type="context">tãto</reg> magis, <reg norm="quanto" type="context">quãto</reg> magis in ip <lb/>Å¿o corpore radius penetrat, <reg norm="etiam" type="context">etiã</reg> & Å¿i <lb/>ad rectos incideret ipÅ¿e radius Å¿upra <lb/><reg norm="Å¿uperficiem" type="context">Å¿uperficiẽ</reg> <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> corporis. </s> - <s xml:id="s4189" xml:space="preserve"><reg norm="Exempli" type="context">Exẽpli</reg> gra <lb/>tia, eÅ¿to <var>.q.p.</var> corpus a <reg norm="queum" type="context">queÅ©</reg>, cuius pro <lb/>funditas diuidatur in partibus <var>.d.K</var>: <lb/><var>K.s</var>: et <var>.s.f.</var> à puncto verò lucido <var>.b.</var> - <pb o="346" rhead="IO. BABPT. BENED." n="358" file="0358" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0358"/> - deÅ¿cendat radius <var>.b.d.K.s.f.</var> ad <reg norm="libitum" type="context">libitÅ©</reg> hoc eÅ¿t rectè vel obliquè, cuius pars <var>.b.d.</var> in ipÅ¿o <lb/>aere exiÅ¿tat. </s> - <s xml:id="s4190" xml:space="preserve">Nunc manifeÅ¿tum erit partem <var>.b.d.</var> ipÅ¿ius radij clariorem Å¿eu minus im <lb/><reg norm="peditam" type="context">peditã</reg> eſſe quam <var>.d.K.</var> quod ex eo etiam cognoÅ¿cere poſſumus quia <var>.b.d.</var> reflectitur à <lb/>puncto <var>.d.</var> Å¿uperficiei corporis a quei, quapropter minus luminoÅ¿a remanebit pars <var>.d.<lb/>K.</var> cum non tota claritas <var>.b.d.</var> deÅ¿cendat in corpore aqueo, Å¿ed vna eius pars reflecta-<lb/>tur, reliqua verò tantummodò deÅ¿cendat, </s> - <s xml:id="s4191" xml:space="preserve">deinde pars <var>.K.s.</var> ex neceſſitate debilior <lb/>erit ipÅ¿a <var>.d.K.</var> eo quod Å¿uccedit poÅ¿t ipÅ¿am <var>.d.K.</var> propter hoc etiam, quia cum corpus <lb/>aqueum habeat aliquantulum opacitatis, radius <var>.d.K.</var> ab omni puncto ipÅ¿ius Å¿piſſitu-<lb/>dinis a quæ continuo reflectitur, quÄ™ quidem reflexio eÅ¿t illud lumen cÄ™ruleum, quod <lb/>in profunditate ipÅ¿ius aquÄ™ nobis apparet. </s> - <s xml:id="s4192" xml:space="preserve">Cum igitur reflexio ipÅ¿a Å¿emper detra-<lb/>hat ab ipÅ¿o radio luminoÅ¿o, reÅ¿iduum verò Å¿it id quod penetrat, ideo <var>.K.s.</var> erit vna <lb/>pars tantummodò luminis ipÅ¿ius <var>.d.K</var>: in <var>.s.f.</var> verò aliqua pars luminis ipÅ¿ius <var>.K.s.</var> & Å¿ic <lb/>continuò debilitatur radius, ita quod ad nihilum vÅ¿que deuenit, & vltra tale cor-<lb/>pus remanebit vmbra, quaÅ¿i Å¿i ipÅ¿um corpus eſſet perfectè opacum, cuius rei cauÅ¿a, <lb/>eÅ¿t illa continua reflexio, vt diximus, quæ continuò adimit aliquid ex ipÅ¿o radio, <lb/>nec permittit eum totum tranÅ¿ire.</s> - </p> - <p> - <s xml:id="s4193" xml:space="preserve">Quapropter mirandum non eÅ¿t eos, qui margaritas quærunt in fundo maris nul-<lb/>lum ibi videre lumen. </s> - <s xml:id="s4194" xml:space="preserve">Nihilominus vmbra maris, quam dico nos poſſe videre in <lb/>Å¿uperficie corporis lunaris, ab alia etiam ratione prouenire poſſet. </s> - <s xml:id="s4195" xml:space="preserve">Imaginemur enim <lb/>aggregatum terrÄ™, <reg norm="marisque" type="simple">marisqÌ;</reg> eſſe tantummodò aqueum, quod quidem eſſet perfectè <lb/>Å¿phæricum ratione centri grauitatis, <reg norm="Å¿upponamusque" type="simple">Å¿upponamusqÌ;</reg> <reg norm="ipsum" type="context">ipsÅ©</reg> eſſe valde diaphanum, ita <lb/>quod radij Å¿olares ipÅ¿um penetraſſent. </s> - <s xml:id="s4196" xml:space="preserve">Tunc dico quod in Å¿uperficie corporis luna-<lb/>ris produceret vmbram. </s> - <s xml:id="s4197" xml:space="preserve">Pro cuius intelligentia cogitemus Å¿ubÅ¿criptam hic figuram <lb/><var>b.h.q.a.e.</var> eſſe Å¿phÄ™ram aliquam cryÅ¿tallinam, & ad partem <var>.b.h.q.</var> Å¿it radius lumino-<lb/>Å¿us Å¿olaris qui ipÅ¿am illuminet, cuius radij extremitates Å¿int <var>.d.b.l.</var> et <var>.p.q.r.</var> Å¿upponen-<lb/>do <var>.d.l.</var> et <var>.p.r.</var> terminos eſſe vnius plani Å¿ecantis ipÅ¿um radium per axem, </s> - <s xml:id="s4198" xml:space="preserve">tunc vide-<lb/>bis ipÅ¿um radium <var>.b.p.q.d.</var> <reg norm="tranÅ¿eun- tem" type="context">tranÅ¿eÅ©-<lb/> - <anchor type="figure" xlink:label="fig-0358-01a" xlink:href="fig-0358-01"/> - tem</reg> ipÅ¿am Å¿phæram, congregari Å¿eu <lb/><reg norm="condenÅ¿ari" type="context">condẽſari</reg>, ob vniformem refractio-<lb/>nem, vÅ¿que ad punctum <var>.o.</var> deinde; <lb/>propter rectitudinem ipÅ¿ius diffu-<lb/>Å¿ionis, vltra punctum <var>.o.</var> ipÅ¿um dila-<lb/>tari, diÅ¿gregari, Å¿eu rarefieri, <reg norm="quouÅ¿que" type="simple">quouÅ¿q;</reg> <lb/>nullius illuminationis actum habeat .<lb/>vt: exempli gratia <var>.o.t.</var> et <var>.o.s.</var> eius par <lb/>tes, ita quod interualla <var>.c.o.b.</var> et <var>.u.</var> <lb/> - <anchor type="figure" xlink:label="fig-0358-02a" xlink:href="fig-0358-02"/> - <var>o.q.</var> relinquerentur priuata lumini-<lb/>bus, vnde vmbroÅ¿a remanerent. di-<lb/><reg norm="Å¿tantiaque" type="simple">Å¿tantiaqÌ;</reg> ab <var>.o.</var> ad Å¿uperficiem Å¿phÄ™ri <lb/>cam corporis <var>.b.e.d.q.</var> non Å¿olum <reg norm="non" type="context">nõ</reg> <lb/>maior eÅ¿t diametro ipÅ¿ius Å¿phæræ; <lb/></s> - <s xml:id="s4199" xml:space="preserve">imo minor, vt à te ipÅ¿o experiri po-<lb/>tes. </s> - <s xml:id="s4200" xml:space="preserve">PoÅ¿ito igitur aliquo obiecto <lb/>opaco in loco <var>.K.o.g.</var> eius Å¿uperficies <lb/>intercepta inter <var>.K.</var> et <var>.g.</var> adumbrata <lb/>erit, excepto puncto <var>.o</var>. </s> - <s xml:id="s4201" xml:space="preserve">PoÅ¿ito dein <lb/>de ipÅ¿o obiecto in loco <var>.n.y.x.m.</var> <reg norm="eius" type="simple">eiê°</reg> <lb/>partes <var>.y.n.</var> et <var>.x.m.</var> remanebunt lu- - <pb o="347" rhead="EPISTOL AE." n="359" file="0359" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0359"/> - mine deÅ¿titutæ <reg norm="interuallumque" type="simple">interuallumqÌ;</reg> tantummodò inter <var>.y.x.</var> illuminatum erit, Å¿ed Å¿i in <lb/>loco <var>.c.u.</var> poÅ¿itum fuerit, </s> - <s xml:id="s4202" xml:space="preserve">tunc totum <var>.c.u.</var> illuminatum erit, Å¿ed debili modo propter <lb/>detractionem factam à reflexione in Å¿uperficie corporis Å¿phærici, vt Å¿upra diximus.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0358-01" xlink:href="fig-0358-01a"> - <image file="0358-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0358-01"/> - </figure> - <figure xlink:label="fig-0358-02" xlink:href="fig-0358-02a"> - <image file="0358-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0358-02"/> - </figure> - </div> - <p> - <s xml:id="s4203" xml:space="preserve">PoÅ¿ito deinde obiecto in loco <var>.i.z.H.f.</var> tunc partes <var>.z.i.</var> et <var>.H.f.</var> rectos Solis radios <lb/>habebunt cum aliquibus refractis, Å¿ed <var>.z.H.</var> pauciſſimum habebit lumen, pro-<lb/>pter diÅ¿gregationem radiorum. </s> - <s xml:id="s4204" xml:space="preserve">PoÅ¿ito poÅ¿tea ipÅ¿o obiecto in loco <var>.t.l.r.s.</var> tanto <lb/>minus lumen habebit pars <var>.l.r.</var> propter dictam <reg norm="diÅ¿gregationem" type="context">diÅ¿gregationẽ</reg>, Å¿eu <reg norm="diſſipationem" type="context">diſſipationẽ</reg> radio <lb/>rum, & Å¿ic Å¿ucceſſiuè quanto remotius poÅ¿itum fuerit ipÅ¿um obiectum, tanto minus <lb/>illuminabitur. </s> - <s xml:id="s4205" xml:space="preserve">vnde ita remotum poterit locari, ut nullus actus luminis in eo <lb/>videatur, de radijs Å¿cilicet, qui per Å¿phæram chryÅ¿tallinam tranÅ¿ibunt, Å¿ed videbi-<lb/>tur vmbra ipÅ¿ius Å¿phÄ™rÄ™ in obiecto propoÅ¿ito, cum nullum actum illuminationis in <lb/>eo loco obiecti habeant radij tranÅ¿euntes per dictam Å¿phÄ™ram. </s> - <s xml:id="s4206" xml:space="preserve">quapropter partes <var>.<lb/>t.l.</var> et <var>.r.s.</var> illuminatæ erunt à Sole, et <var>.l.r.</var> omnino lumine deÅ¿tituta.</s> - </p> - <p> - <s xml:id="s4207" xml:space="preserve">Quòd vero tolerabilior Å¿it oculis radius reflexus Solis à ſuperÅ¿icie aquæ, quà m <lb/>à ſuperficie alicuius Å¿peculi, oritur ab eo, quod Å¿upra diximus, hoc eÅ¿t, quod ma-<lb/>gna parsipÅ¿ius luminis penetrat in aquam, & non totum reflectit, quod quidem non <lb/>accidit Å¿peculis opacis.</s> - </p> - </div> - </div> - <div type="section" level="3" n="29"> - <div type="letter" level="4" n="1"> - <head xml:id="head511" xml:space="preserve">DE LONGITVDINE DVORVM LATERVM <lb/>cuiuÅ¿uis trianguli Å¿upra tertium.</head> - <head xml:id="head512" style="it" xml:space="preserve">Hieronymo Fenarolo.</head> - <p> - <s xml:id="s4208" xml:space="preserve"><emph style="sc"><reg norm="QVod" type="conjecture">QVo'd</reg></emph> quælibet duo latera continentia rectum angulum cuiuÅ¿uis triangu-<lb/>li orthogonij, longiora Å¿int tertio latere, per diametrum circuli in eo in-<lb/>Å¿cripti, ab alijs iam demonÅ¿tratum fuit. </s> - <s xml:id="s4209" xml:space="preserve">Sed quòd quælibet duo latera <lb/>cuiuÅ¿uis trianguli longiora Å¿int tertio per latus tetragonicum, quadrupli <lb/>producti cuiuÅ¿uis lineæ deÅ¿cendentis ab angulo contento à dictis duobus lateribus <lb/>ad oppoÅ¿itam partem circuli inÅ¿cripti, in partem extrinÅ¿ecam ipÅ¿ius lineæ, nullus <lb/>(quod Å¿ciam) vnquam Å¿cripÅ¿it, vel animaduertit.</s> - </p> - <p> - <s xml:id="s4210" xml:space="preserve">Sit exempli gratia triangulus <var>.a.b.c.</var> quem volueris, in quo deÅ¿cribatur circulus <var>.<lb/>u.s.n.</var> & puncta contingentiæ Å¿int eadem <var>.u.s.n.</var> à puncto vero <var>.a.</var> deÅ¿cendat linea <var>.a.<lb/>i.e.</var> quæ terminetur à circunferentia in puncto <var>.e.</var> ipÅ¿ius circunferentiæ, vbi volue-<lb/>ris. </s> - <s xml:id="s4211" xml:space="preserve">Dico nunc latera <var>.a.b.</var> et <var>.a.c.</var> longiora eſſe latere <var>.b.c.</var> per latus <reg norm="tetragonicum" type="context">tetragonicÅ©</reg> qua-<lb/>drupli producti ipÅ¿ius <var>.a.e.</var> in <var>.a.i</var>. </s> - <s xml:id="s4212" xml:space="preserve">Nam certi Å¿amus ex vltima parte penultimæ ter-<lb/>tij Eucli <var>.n.c.</var> et <var>.s.c.</var> æquales inuicem eſſe, & Å¿imiliter <var>.b.s.</var> et <var>.b.u.</var> vnde ex communi <lb/>conceptu dicta latera maiora erunt <lb/> - <anchor type="figure" xlink:label="fig-0359-01a" xlink:href="fig-0359-01"/> - ipÅ¿o <var>.b.c.</var> per <var>.a.u.</var> et <var>.a.n.</var> quæ duæ <lb/>partes Å¿unt inuicem æquales di-<lb/>cta ratione, & quadratum lineæ <lb/>æqualis aggregato earum, eſſet qua <lb/>druplum quadrato cuiuÅ¿uis earum <lb/>ex .4. Å¿ecundi, Å¿ed ex penultima ter <lb/>tij, productum <var>.a.e.</var> in <var>.a.i.</var> æquale eÅ¿t <lb/>quadrato ipÅ¿ius <var>.a.u.</var> vel ipÅ¿ius <var>.a.n</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0359-01" xlink:href="fig-0359-01a"> - <image file="0359-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0359-01"/> - </figure> - </div> - <pb o="348" rhead="IO. BAPT. BENED." n="360" file="0360" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0360"/> - <p> - <s xml:id="s4213" xml:space="preserve">Verum eÅ¿t igitur quod <var>.a.b.</var> cum <var>.a.c.</var> longiores Å¿int ipÅ¿a <var>.b.c.</var> per latus terrago. <lb/>nicum quadrupli eius quod fit. ex <var>.a.e.</var> in <var>.a.i.</var> quod fuit propoÅ¿itum.</s> - </p> - <p> - <s xml:id="s4214" xml:space="preserve">Illud etiam non eÅ¿t Å¿pernendum, quod quotieÅ¿cunque data fuerint omnia latera <lb/>alicuius trianguli, illicò poſſumus cognoÅ¿cere puncta <var>.u.n.s.</var> contingentiæ circuli in <lb/>Å¿cripti, ope vltimæ partis penultimæ tertij, eo quod ex illa iam Å¿cimus, quod de-<lb/>trahendo <var>.b.c.</var> ex aggregato aliorum duorum laterum, remanebit <var>.u.a.</var> et <var>.a.n.</var> qua-<lb/>rum vnaquÄ™que nota erit, cum illarum quælibet, medietas Å¿it reÅ¿idui cogniti, detra <lb/>hendo poÅ¿tea vnam <reg norm="illarum" type="context">illarÅ©</reg> ab altero <lb/> - <anchor type="figure" xlink:label="fig-0360-01a" xlink:href="fig-0360-01"/> - duorum laterum <var>.a.b.</var> vel <var>.a.c.</var> rema <lb/>nebit <var>.u.b.</var> vel <var>.c.n.</var> Ä™qualis <var>.b.s.</var> vel <var>.c.<lb/>s.</var> vnde Å¿imiliter nobis innoteÅ¿cet <lb/>punctum <var>.s.</var> cum duobus punctis <var>.u.</var> <lb/>ct <var>.n.</var> à quibus duobus punctis, Å¿i <lb/>duÄ™ perpendiculares ad talia latera <lb/>ductæ fuerint, vbi hæe perpendicu <lb/>lares Å¿einuicem Å¿ecabunt, ibi <reg norm="cen- trum" type="context">cen-<lb/>trÅ©</reg> circuli inÅ¿criptibilis erit in trian <lb/>gulo propoÅ¿ito.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0360-01" xlink:href="fig-0360-01a"> - <image file="0360-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0360-01"/> - </figure> - </div> - <p> - <s xml:id="s4215" xml:space="preserve">Inter alia, quæ tibi dixi de Iride, quod memoria non tenes, nihil aliud eÅ¿t niÅ¿i <lb/>quod cum Iris videtur, non eodem loco ab omnibus videtur, quia reflexio eÅ¿t, & <lb/>vt reflexio luminis à ſpeculo non omnibus ab eodem puncto fit, ita etiam tibi dixi <lb/>de Iride.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head513" style="it" xml:space="preserve">De InÅ¿trumento oxygonio, Å¿eu elliptico.</head> - <head xml:id="head514" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4216" xml:space="preserve">QVod aliquando à me audiuiÅ¿ti falÅ¿um non eÅ¿t, Å¿cilicet poſſibile eſſe (vt <lb/>Å¿peculatus Å¿um) particulare inÅ¿trumentum fabricari ad deÅ¿ignandum oxy-<lb/>goniam, Å¿eu ellipticam Å¿ectionem, quæ à Pergeo defectio appellatur, quod quidem <lb/>inÅ¿trumentum valde diuerÅ¿um eÅ¿t ab alijs, quÄ™ alià s inueni, pro ipÅ¿is conicis Å¿ectio <lb/>nibus delineandis. </s> - <s xml:id="s4217" xml:space="preserve">OccaÅ¿ionem <reg norm="autem" type="wordlist">aÅ©t</reg> huiuÅ¿imodi inÅ¿trumenti inueniendi mihi præ <lb/>buit <reg norm="Å¿ecunda" type="context">Å¿ecÅ©da</reg> dubij Å¿olutio <reg norm="quam" type="context">quã</reg> feci ann .1568. grauiſſ. philoÅ¿opho FranciÅ¿co Vimer <lb/>cato, <reg norm="nam" type="context">nã</reg> <reg norm="cum" type="context">cÅ©</reg> viderim in ea figura <var>.f.a.</var> Å¿emper <reg norm="æqualem" type="context">æqualẽ</reg> eſſe <var>.o.i.</var> Å¿uæ parallelæ Å¿cilicet, <lb/>vnde cum recta linea fuerit protracta per <var>.o.</var> et <var>.f.</var> ipÅ¿a foret Å¿emper <reg norm="Ä™quidiÅ¿tans" type="context">Ä™quidiÅ¿tãs</reg> <var>.d.p.</var> ex <lb/>33. primi Eucli. </s> - <s xml:id="s4218" xml:space="preserve">Venit mihi in mentem modus conÅ¿truendi hoc Å¿ubÅ¿criptum inÅ¿tru-<lb/>mentum, tali ordine, videlicet, <reg norm="coniungendo" type="context">coniungẽdo</reg> Å¿eptem hic Å¿ubnotatas lineas materia-<lb/>les <var>.z.r</var>: <var>u.n</var>: <var>e.h</var>: <var>e.c</var>: <var>c.l</var>: <var>l.s.</var> et <var>.s.e.</var> Å¿imul, hoc modo, Å¿cilicet <reg norm="Å¿abricando" type="context">Å¿abricãdo</reg> quadrila-<lb/>terum æquilaterum <var>.c.e.s.l.</var> hac conditione, quod immobili exiÅ¿tente puncto <var>.c.</var> in li <lb/>nea <var>.z.r.</var> reliqua omnia mobilia exiÅ¿tant, hoc eÅ¿t quod <reg norm="punctum" type="context">punctÅ©</reg> <var>.s.</var> moueatur per di-<lb/>ctam lineam <var>.z.r.</var> & immobili exiÅ¿tente puncto <var>.e.</var> vt extremum lineæ <var>.e.h.</var> hoc eÅ¿t <lb/>coniuncto extremo <var>.e.</var> lineæ <var>.e.h.</var> cum angulo <var>.c.e.s.</var> reliqua puncta lineæ ipÅ¿ius <var>.e.h.</var> <lb/>moueantur per <var>.l.</var> & per duas parallelas <var>.u.n.</var> et <var>.z.r.</var> longitudo vero <var>.e.h.</var> Å¿it compo-<lb/>Å¿ita ex duplo vnius lateris ipÅ¿ius quadrilateris. </s> - <s xml:id="s4219" xml:space="preserve">Oportet deinde quod punctum <var>.f.</var> <lb/>Å¿emper vnum, & idem Å¿it ipÅ¿ius parallelæ <var>.u.n.</var> moueatur tamen per <var>.e.h.</var> quod qui-<lb/>dem punctum illud erit, quod vnam <reg norm="portionem" type="context">portionẽ</reg> circunferentiæ oxygoniæ Å¿ectonis - <pb o="349" rhead="EPISTOL AE." n="361" file="0361" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0361"/> - deÅ¿ignabit, puncta verò <var>.o.</var> et <var>.K.</var> vt puncta laterum <var>.c.e.</var> et <var>.s.e.</var> æquædiÅ¿tantia à <lb/>punctis <var>.c.</var> et <var>.s.</var> <reg norm="eadem" type="context">eadẽ</reg> Å¿emper Å¿int, ita tamen vt puncta lineæ <var>.u.n.</var> Å¿emper diuerÅ¿a exi <lb/><reg norm="Å¿tant" type="context">Å¿tãt</reg>, & quodlibet ipÅ¿ius quadrilateri latus, æquale Å¿it medietati maioris axis ipÅ¿ius <lb/>oxygoniæ Å¿ectionis delineandæ, et <var>.c.o.</var> Å¿eu <var>.s.K.</var> (quod idem eÅ¿t) Å¿it æqualis medie <lb/>tati axis minoris dictæ Å¿ectionis, et <var>.z.r.</var> æqualis duplo <var>.e.h.</var> vnde, quando puncta <var>.e.</var> et <var>.<lb/>l.</var> coniuncta Å¿imul erunt, Å¿imiliter coniunctæ Å¿imul erunt <var>.c.e.</var> et <var>.e.s.</var> cum <var>.c.l.</var> et <var>.l.s.</var> - <anchor type="figure" xlink:label="fig-0361-01a" xlink:href="fig-0361-01"/> - <anchor type="figure" xlink:label="fig-0361-02a" xlink:href="fig-0361-02"/> - </s> - <pb o="350" rhead="IO. BAPT. BENED." n="362" file="0362" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0362"/> - <s xml:id="s4220" xml:space="preserve">Quapropter puncta <var>.e.l.f.</var> et <var>.p.</var> extremum axis maioris, in eodem met loco erunt, <lb/>hoc eÅ¿t in aliquo extremorum maioris axis, & cum punct is <var>.s.</var> coniunct is fuerit cum <lb/>centro <var>.c.</var> punctus <var>.f.</var> parallelÄ™ <var>.u.p.</var> in extremo axis minoris erit, & in eodem loco erit <lb/>cum <var>.o.</var> & cum <var>.K</var>. </s> - <s xml:id="s4221" xml:space="preserve">In extremitatibus verò lineæ <var>.z.r.</var> neceſſe eÅ¿t, vt Å¿int duo <reg norm="puncta" type="context">pÅ©cta</reg> fer <lb/>rea, ad firmandum ipÅ¿am <var>.z.r.</var> Å¿uper Å¿ubiectam lineam Å¿ignificantem maiorem axem <lb/>propoÅ¿itÄ™ Å¿ectionis.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0361-01" xlink:href="fig-0361-01a"> - <image file="0361-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0361-01"/> - </figure> - <figure xlink:label="fig-0361-02" xlink:href="fig-0361-02a"> - <image file="0361-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0361-02"/> - <caption style="it" xml:space="preserve">Instrumentum <lb/>oxigonium</caption> - </figure> - </div> - <figure position="here"> - <image file="0362-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0362-01"/> - </figure> - <figure position="here"> - <image file="0362-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0362-02"/> - </figure> - <pb o="351" rhead="EPISTOL AE." n="363" file="0363" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0363"/> - <p> - <s xml:id="s4222" xml:space="preserve">Volo etiam quod ad partem <var>.c.l.s.</var> quadrilateri conÅ¿tituta Å¿it alia parallela ad <var>.z.<lb/>r.</var> & in æquali diÅ¿tantia ab ipÅ¿a quemadmodum <var>.u.n.</var> diÅ¿tat ad eademmet <var>.z.r.</var> ad ean <lb/>dem operationem faciendam. </s> - <s xml:id="s4223" xml:space="preserve">Vnde in vno tantummodo itinere puncti <var>.s.</var> ab <var>.r.</var> <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> <lb/>ad <var>.c.</var> deÅ¿ignabimus quartam partem Å¿ectionis, conuerÅ¿o poÅ¿tea inÅ¿trumento, hoc eÅ¿t <lb/>poÅ¿ito puncto <var>.r.</var> vbi prius erat <var>.z.</var> et <var>.z.</var> vbi erat <var>.r.</var> aliam delineabimus quartam, & <lb/>Å¿ic ad oppoÅ¿itam partem ipÅ¿ius <var>.z.r.</var> faciendum erit. </s> - <s xml:id="s4224" xml:space="preserve">Hoc inÅ¿trumentum poſſumus <lb/>etiam ita conÅ¿truere, vt puncta <var>.o.</var> et <var>.K.</var> poſſint collocari in laterihus <var>.c.e.</var> et <var>.e.s.</var> vbi no <lb/>bis magis libuerit, ita vt licebit in qualibet proportione <reg norm="axium" type="context">axiÅ©</reg> propoÅ¿ita, oxygoniam <lb/>deÅ¿ignare. </s> - <s xml:id="s4225" xml:space="preserve">Nam <var>.c.o.</var> erit longitudo dimidij axis minoris, et <var>.c.e.</var> dimidij maioris.</s> - </p> - </div> - </div> - <div type="section" level="3" n="30"> - <div type="letter" level="4" n="1"> - <head xml:id="head515" xml:space="preserve">DE CONSTITVTIONE TRIANGVLI <lb/>orthogonij conditionati.</head> - <head xml:id="head516" style="it" xml:space="preserve">Domino Ludouico de Rocchaforte.</head> - <p> - <s xml:id="s4226" xml:space="preserve"><emph style="sc">QVod</emph> à me poÅ¿tulas, non eÅ¿t admodum difficile, cupis enim triangulum <lb/>orthogonium, exempli gratia <var>.o.i.e.</var> in figura <var>.A.</var> ita conÅ¿tituere, vt di-<lb/>uiÅ¿um Å¿it à perpendiculari <var>.a.i.</var> & quod proportio <var>.o.e.</var> ad <var>.o.i.</var> Å¿it vt <var>.o.i.</var> ad <lb/><var>i.e.</var> & quod quadrati <var>.o.i.</var> ad quadratum <var>.o.a.</var> Å¿it vt <var>.e.i.</var> ad <var>.e.a.</var> & quadra <lb/>tum <var>.o.i.</var> ad quadratum <var>.e.i.</var> Å¿it .ut <var>.o.a.</var> ad <var>.e.a</var>. </s> - <s xml:id="s4227" xml:space="preserve">Quæ omnia in promptu veniunt, quo <lb/>tieÅ¿cunque <var>.o.e.</var> fuerit diameter alicuius circuli, <reg norm="diuiÅ¿aque" type="simple">diuiÅ¿aqÌ;</reg> in puncto <var>.a.</var> Å¿ecundum pro <lb/>portionem habentem medium <reg norm="duoque" type="simple">duoqÌ;</reg> extrema, protracta deinde perpendiculari <var>.a.<lb/>i.</var> ad <var>o.e.</var> uÅ¿que ad circunferentiam, <reg norm="coniunctæque" type="simple">coniunctæqÌ;</reg> <var>.o.i.</var> et <var>.i.e</var>: tale triangulum, omnia <lb/>Å¿upradicta in Å¿e continebit.</s> - </p> - <p> - <s xml:id="s4228" xml:space="preserve">Nam ex .30. tertij angulus <var>.i.</var> rectus erit, & ex .8. Å¿exti <var>.o.i.</var> erit media proportio-<lb/>nalis inter <var>.o.e.</var> et <var>.o.a.</var> et <var>.e.i.</var> inter <var>.o.e.</var> <lb/> - <anchor type="figure" xlink:label="fig-0363-01a" xlink:href="fig-0363-01"/> - et <var>.a.e.</var> Å¿ed quia ex diuiÅ¿ione facta in <reg norm="pum" type="context">pÅ©</reg> <lb/>cto <var>.a.</var> etiam <var>.o.a.</var> erit media proportio-<lb/>nalis inter totum & reÅ¿iduum, ideo ex <num value="11">.<lb/>11.</num> quinti ita erit <var>.o.e.</var> ad <var>.e.i.</var> vt <var>.o.e.</var> ad <var>.<lb/>o.a.</var> vnde ex .9. eiuÅ¿dem <var>.a.o.</var> erit æqua-<lb/>lis <var>.e.i.</var> & ideo <var>.o.i.</var> erit media proportio <lb/>nalis inter <var>.o.e.</var> et <var>.e.i</var>. </s> - <s xml:id="s4229" xml:space="preserve">Sed quia propor-<lb/>tio <var>.e.i.</var> ad <var>.a.e.</var> <reg norm="eadem" type="context">eadẽ</reg> eÅ¿t, quÄ™ ipÅ¿ius <var>.o.e.</var> ad <lb/><var>o.a</var>. </s> - <s xml:id="s4230" xml:space="preserve">tunc videbis ex .18. Å¿exti, quod pro <lb/>portio quadrati <var>.o.i.</var> ad quadratum <var>.o.a.</var> <lb/>erit vt <var>.e.i.</var> ad <var>.e.a.</var> cum vero duo trian-<lb/>guli <var>.o.i.a.</var> et <var>.a.i.e.</var> Å¿int inuicem Å¿imiles <lb/>ex Å¿upradicta .8. Å¿exti, </s> - <s xml:id="s4231" xml:space="preserve">tunc videbis ex <lb/>18. et .17. eiuÅ¿dem dictos <reg norm="triangulos" type="context">triãgulos</reg> ean <lb/>dem habere inter Å¿e proportionem, quÄ™ <lb/>eÅ¿t inrer quadrata ipÅ¿ius <var>.o.i.</var> et <var>.i.e.</var> vnde <lb/>ex prima Å¿exti ita Å¿e inuicem habebunt <var>.<lb/>a.o.</var> et <var>.a.e</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0363-01" xlink:href="fig-0363-01a"> - <image file="0363-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0363-01"/> - </figure> - </div> - <p> - <s xml:id="s4232" xml:space="preserve">Circa eam verò difficultatem quam - <pb o="352" rhead="I O. BAPT. BENED." n="364" file="0364" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0364"/> - habes in circulo .ω vbi fateris te non videre qua ratione eadem proportio Å¿it qua-<lb/>drati <var>.u.o.</var> ad quadratum <var>.o.n.</var> vt lineæ <var>.o.a.</var> ad lineam <var>.o.e.</var> partes diametri <var>.o.i.</var> ipÅ¿ius <lb/>circuli, terminatæ à perpendicularibus <var>.u.a.</var> et <var>.n.e</var>.</s> - </p> - <p> - <s xml:id="s4233" xml:space="preserve">Hoc neceſſario contingit, propterea quod cum fuerint protractæ <var>.u.i.</var> et <var>.n.i.</var> <reg norm="tunc" type="context">tÅ©c</reg> <lb/>habebimus ad partem <var>.o.u.i.</var> triangulum <var>.o.u.i.</var> diuiÅ¿um in duo triangula Å¿imilia ipÅ¿i <lb/>totali triangulo. </s> - <s xml:id="s4234" xml:space="preserve">Idem etiam dico ad partem <var>.o.n.i.</var> vnde ex tali Å¿imilitudine habe-<lb/>bimus <var>.o.u.</var> mediam proportionalem inter <var>.o.i.</var> et <var>.o.a.</var> et Å¿ic <var>.o.n.</var> erit media proportio <lb/>nalis inter <var>.o.i.</var> et <var>.o.e.</var> </s> - <s xml:id="s4235" xml:space="preserve">quare ex .16. Å¿exti, quadratum <var>.o.u.</var> æquale erit producto ipÅ¿ius <lb/><var>o.i.</var> in <var>.o.a.</var> & quadratum <var>.o.n.</var> æquale producto <var>.o.i.</var> in <var>.o.e.</var> Å¿ed ex prima eiuÅ¿dem, ea <lb/>dem proportio eÅ¿t ipÅ¿ius <var>.o.a.</var> ad <var>.o.e.</var> quæ producti ipÅ¿ius <var>.o.i.</var> in <var>o.a.</var> ad productum <var>.o.<lb/>i.</var> in <var>.o.e.</var> </s> - <s xml:id="s4236" xml:space="preserve">quare, ex <reg norm="communi" type="context">cõmuni</reg> conceptu, ita erit quadrati <var>.o.u.</var> ad quadratúm.o.n. </s> - <s xml:id="s4237" xml:space="preserve">Et hÄ™c <lb/>eÅ¿t alia circuli paſſio.</s> - </p> - <p> - <s xml:id="s4238" xml:space="preserve">Reliqua verò difficultas quam te habere Å¿cribis, eÅ¿t, quare cum duæ lineÄ™ <lb/><var>a.u.</var> et <var>.b.s.i.</var> Å¿int inuicem Ä™quales, diuiſæ verò non æquali modo, Å¿ed tali, quod <var>.a.</var> <lb/>maior Å¿it quam <var>.u.</var> et <var>.b.s.</var> maior quam <var>.i.</var> quomodo poteÅ¿t fieri, quod Å¿i <var>.u.</var> maior fue-<lb/>rit <var>.i.</var> proportio <var>.a.</var> ad <var>.i.</var> maior Å¿it quam ipÅ¿ius <var>.b.s.</var> ad <var>.u</var>.</s> - </p> - <p> - <s xml:id="s4239" xml:space="preserve">Hoc etiam ex neceſſitate cuenit, eo <lb/>quod Å¿i accepta fuerit <var>.t.n.</var> æqualis <var>.u.</var> ab <lb/> - <anchor type="figure" xlink:label="fig-0364-01a" xlink:href="fig-0364-01"/> - <reg norm="ipÅ¿aque" type="simple">ipÅ¿aqÌ;</reg> abÅ¿ciÅ¿a fuerit <var>.t.</var> æqualis <var>.i.</var> & ab <var>.<lb/>b.s.</var> abÅ¿ciÅ¿a <var>.s.</var> æqualis <var>.n.</var> habebimus <var>.a.</var> et <lb/>b. inuicem æ quales, vnde habebis ma-<lb/>iorem propor tionem ipÅ¿ius <var>.b.</var> ad <var>.t.</var> <reg norm="quam" type="context">quã</reg> <lb/>s. ad <var>.n.</var> quod cum clarum per Å¿e Å¿it, tibi <lb/>relinquo. </s> - <s xml:id="s4240" xml:space="preserve">Å¿ed ex .27. quinti, proportio <lb/>b. ad. s, maior erit quam <var>.t.</var> ad <var>.n.</var> & ex <lb/>28. <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> <reg norm="proportio" type="simple">ê“portio</reg> <var>.b.s.</var> ad <var>.s.</var> maior erit, <lb/>quam <var>.t.n.</var> ad <var>.n.</var> & ex .27. maior propor <lb/>tio erit ipÅ¿ius <var>.b.s.</var> ad <var>.n.t.</var> quam <var>.s.</var> ad <var>.n.</var> <lb/>ergo ex .33. maior erit ipÅ¿ius <var>.b.</var> ad <var>.t.</var> <reg norm="quam" type="context">quã</reg> <lb/><var>b.s.</var> ad <var>.n.t.</var> hoc eÅ¿t maior ipÅ¿ius <var>.a.</var> ad <lb/>i. quam <var>.b.s.</var> ad <var>.u.</var> quod eÅ¿t propo-<lb/>Å¿itum.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0364-01" xlink:href="fig-0364-01a"> - <image file="0364-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0364-01"/> - </figure> - </div> - <p> - <s xml:id="s4241" xml:space="preserve">Id verò de quo me interrogas <reg norm="nempe" type="context">nẽpe</reg> de <lb/>diÅ¿tinctione orbium cÄ™leÅ¿tium, ortum <lb/>habet à communi opinione motuum <lb/>fixarum. </s> - <s xml:id="s4242" xml:space="preserve">Nam cum putauerint philo-<lb/>Å¿ophi ipÅ¿as moueri, Å¿emper eandem <reg norm="Å¿eruando" type="context">Å¿eruãdo</reg> inuicem diÅ¿tantiam, non Å¿ine ratione <lb/>crediderunt eas fixas eſſe eodem in orbe, idem etiam poÅ¿tea de planetis opinaue-<lb/>runt. </s> - <s xml:id="s4243" xml:space="preserve">Hoc eÅ¿t, vnumquemque, aliquo in orbe, fixo exiÅ¿tere.</s> - </p> - <pb o="353" rhead="EPISTOL AE." n="365" file="0365" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0365"/> - </div> - </div> - <div type="section" level="3" n="31"> - <div type="letter" level="4" n="1"> - <head xml:id="head517" xml:space="preserve">DE MODO DIVIDENDI PARABOLAM <lb/>propoÅ¿itam Å¿ecundum datam proportionem.</head> - <head xml:id="head518" style="it" xml:space="preserve">Pamphilo Gothfrid.</head> - <p> - <s xml:id="s4244" xml:space="preserve"><emph style="sc">QVod</emph> à me quæris, eÅ¿t quidem poſſibile, non tamen adhuc inuentum, quo <lb/>niam nemo ad <reg norm="hunc" type="context">hÅ©c</reg> vÅ¿que diem diuiÅ¿it vnam datam proportionem in tres <lb/>æquales partes, Å¿ed Å¿i hoc pro facto conceſſeris, nunc tibi morem geram. <lb/></s> - <s xml:id="s4245" xml:space="preserve">Nam proponis n. ihi parabolem <var>.x.b.e.</var> cum proportione <var>.p.</var> ad <var>.q.</var> <reg norm="cupiÅ¿que" type="simple">cupiÅ¿qÌ;</reg> <lb/>Å¿cire modum diuidendi ipÅ¿am parabolem vna mediante linea parallela ipÅ¿i baÅ¿i, ita <lb/>vt eandem habeat proportionem tota parabola ad partem abÅ¿ciſſam, quæ eÅ¿t inter <var>.<lb/>p.</var> et <var>.q</var>. </s> - <s xml:id="s4246" xml:space="preserve">Ad quod faciendum, Å¿upponendum primò datam proportionem inter <var>.<lb/>p.</var> et <var>.q.</var> diuiÅ¿am eſſe in tres partes æquales, duabus lineis mediantibus <var>.n.</var> et <var>.u.</var> quæ me <lb/>diæ proportionales vocabuntur inter <var>.p.</var> et <var>.q</var>. </s> - <s xml:id="s4247" xml:space="preserve">deinde à quouis puncto circunferentiÄ™ <lb/>ipÅ¿ius figuræ ducatur parallela baÅ¿i <var>.x.e.</var> poÅ¿tea verò per puncta media harum dua-<lb/>rum <reg norm="æquidiÅ¿tantium" type="context">æquidiÅ¿tantiÅ©</reg> protrahatur <var>.g.b.</var> quæ diameter erit Å¿ectionis, ex 28. Å¿ecundi Per-<lb/>gei, </s> - <s xml:id="s4248" xml:space="preserve">diuidatur deinde hæc diameter in puncto <var>.a.</var> ita quod eadem proportio Å¿it ipÅ¿ius <lb/><var>b.g.</var> ad <var>.b.a.</var> quæ ipÅ¿ius <var>.p.</var> ad <var>.u.</var> quod tibi facile erit, Å¿ecando à linea <var>.p.</var> partem <var>.i.</var> æqua <lb/>lem ipÅ¿i <var>.u.</var> tali modo poÅ¿tea diuidendo <var>.b.g.</var> ex .12. Å¿exti, ducatur a puncto <var>.a.</var> ipÅ¿a <var>.d.<lb/>h.</var> parallclam ipÅ¿i <var>.x.e.</var> & habebitur propoÅ¿itum.</s> - </p> - <p> - <s xml:id="s4249" xml:space="preserve">Pro cuius reiratione, Å¿cies primum quod <var>.h.d.</var> diuiÅ¿a erit à diametro <var>.b.g.</var> per æqua <lb/>lia ex .7. primi Pergei, vel Å¿i cogitabimus aliquam lineam tangentem ipÅ¿am parabo <lb/>lam in puncto <var>.b</var>. </s> - <s xml:id="s4250" xml:space="preserve">tunc ex quinta Å¿ecundi ipÅ¿ius Pergei habebimus ipÅ¿am eſſe paralle-<lb/>lam <var>.e.x.</var> & ex .30. primi Eucli. erit Å¿imiliter æquidiÅ¿tans <var>.d.h.</var> vnde ex .46. primi eiuÅ¿-<lb/>dem Pergei <var>.h.a.</var> æqualis erit <var>.d.a</var>. </s> - <s xml:id="s4251" xml:space="preserve">Protrahatur deinde <var>.e.b</var>: d b: <var>x.b.</var> et <var>.h.b.</var> vnde ex .17 <lb/>lib. de quadratura parabolæ Archimedis, habebimus eandem proportionem Å¿uper <lb/>ficiei totalis parabolæ <var>.x.b.e.</var> ad trigonum <var>.x.b.e.</var> quæ portionis <var>.h.b.d.</var> ad Å¿uum <reg norm="tri- gonum" type="context">tri-<lb/>gonÅ©</reg>, eo quod <reg norm="tam" type="context">tã</reg> vna quà m alia erit Å¿eÅ¿quitertia, <reg norm="eius" type="simple">eiê°</reg> <reg norm="etiam" type="context">etiã</reg> medietates Å¿ic Å¿e <reg norm="habebunt" type="context">habebÅ©t</reg>.</s> - </p> - <p> - <s xml:id="s4252" xml:space="preserve">Vnde permutando, proportio medietatis totalis parabolÄ™ ad medietatem partia <lb/>lem ipÅ¿ius, æqualis erit proportioni trianguli <lb/><var>g.b.e.</var> ad triangulum <var>.a.b.d.</var> Å¿ed ex .20. primi <lb/>Pergei, eadem eÅ¿t proportio quadrati ipÅ¿ius <var>.<lb/> - <anchor type="figure" xlink:label="fig-0365-01a" xlink:href="fig-0365-01"/> - g.e.</var> ad quadratum ipÅ¿ius <var>.a.d.</var> quæ <var>.b.g.</var> ad <var>.b.a.</var> <lb/>hoc eÅ¿t, vt <var>.g.e.</var> ad <var>.a.o.</var> ex Å¿imilitudine triangu-<lb/>lorum, & quia <var>.b.g.</var> ad <var>.b.a.</var> eÅ¿t Å¿icut <var>.p.</var> ad <var>.u.</var> ita <lb/>igitur erit quadrati ipÅ¿ius <var>.g.e.</var> ad quadratum <lb/>ipÅ¿eus <var>.a.d.</var> </s> - <s xml:id="s4253" xml:space="preserve">quare <var>.g.e.</var> ad <var>.a.d.</var> erit ut p. ad <var>.n.</var> <lb/>ex .18. Å¿exti Euclid. </s> - <s xml:id="s4254" xml:space="preserve">Å¿ed cum ex .24. eiuÅ¿dem <lb/>proportio trianguli <var>.b.g.e.</var> ad triangulum <var>.b.<lb/>a.d.</var> compoÅ¿ita Å¿it ex proportione <var>.g.e.</var> ad <var>.a.<lb/>d.</var> er. ex <var>.g.b.</var> ad <var>.b.a.</var> hoc eÅ¿t <var>.g.e.</var> ad <var>.a.o.</var> & <lb/>quia <reg norm="proportio" type="simple">ê“portio</reg> <var>.g.e.</var> ad <var>.a.o.</var> æqualis eÅ¿t ei quæ <var>.p.</var> <lb/>ad. u ex .11. quinti Euclid. </s> - <s xml:id="s4255" xml:space="preserve">& proportio <var>.g.e.</var> <lb/>ad <var>.a.d.</var> æqualis eÅ¿t ei quæ <var>.p.</var> ad <var>.n.</var> hoc eÅ¿t vt <var>.u.</var> <lb/>ad <var>.q.</var> ergo proportio trianguli <var>.b.g.e.</var> ad trian-<lb/>gulum <var>.b.a.d.</var> compoÅ¿ita erit ex ca quÄ™ <var>.p.</var> ad <var>.u.</var> <lb/>& ex ea quæ <var>.u.</var> ad <var>.q.</var> æqualis ergo erit ei, quæ <lb/>p. ad <var>.q.</var> & ita medietates parabolarum, & eorum dupla.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0365-01" xlink:href="fig-0365-01a"> - <image file="0365-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0365-01"/> - </figure> - </div> - <pb o="354" rhead="IO. BABPT. BENED." n="366" file="0366" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0366"/> - <head xml:id="head519" style="sc" xml:space="preserve">COROLLARIVM.</head> - <p> - <s xml:id="s4256" xml:space="preserve">Proportio maioris portionis ad minorem Å¿emper erit Å¿eÅ¿quialtera proportioni <lb/>ipÅ¿ius <var>.b.g.</var> ad <var>.a.b.</var> eo quod cum Å¿it proportio totalis portionis ad partialem vt trian-<lb/>guli <var>.b.g.e.</var> ad <var>.b.a.d.</var> & hæc Å¿eÅ¿quialtera proportioni ipÅ¿ius <var>.g.e.</var> ad <var>.a.o.</var> hoc eÅ¿t vt ip-<lb/>Å¿ius <var>.b.g.</var> ad <var>.b.a.</var> ideo proportio ipÅ¿arum portionum erit Å¿imiliter Å¿eÅ¿quialtera pro-<lb/>portioni diametrorum.</s> - </p> - <p> - <s xml:id="s4257" xml:space="preserve">Deinde Å¿i protractæ fuerint <var>.b.d.</var> et <var>.g.e.</var> quouÅ¿que conueniant in puncto <var>.z.</var> habe <lb/>bis inter <var>.g.z.</var> et <var>.a.o.</var> duas <var>.g.e.</var> et <var>.a.d.</var> medias proportionales in proportionalitate con <lb/>tinua, eo quod cum (ex ijs quæ Å¿upra diximus.). <var>a.d.</var> media proportionalis Å¿it inter <var>.<lb/>g.e.</var> et <var>.a.o.</var> & proportio <var>.g.z.</var> ad <var>.g.e.</var> vt ipÅ¿ius <var>.a.d.</var> ad <var>.a.o.</var> eo quodipÅ¿ius <var>.g.z.</var> ad <var>.a.d.</var> <lb/>& ipÅ¿ius <var>.g.e.</var> ad <var>.a.o.</var> eÅ¿t vt ipÅ¿ius <var>.b.g.</var> ad <var>.b.a.</var> ex Å¿imilitudine triangulorum, ideo di-<lb/>ctæ <reg norm="proportiones" type="simple">ê“portiones</reg> erunt <reg norm="inuicem" type="context">inuicẽ</reg> æquales. </s> - <s xml:id="s4258" xml:space="preserve">Vnde permutatim ita erit ipÅ¿ius <var>.g.z.</var> ad <var>.g.e.</var> <lb/>vt ipÅ¿ius <var>.a.d.</var> ad <var>.a.o.</var> & ut ipÅ¿ius <var>.g.e.</var> ad <var>.a.d</var>.</s> - </p> - <p> - <s xml:id="s4259" xml:space="preserve">Amplius etiam dico, quod proportio pa <lb/> - <anchor type="figure" xlink:label="fig-0366-01a" xlink:href="fig-0366-01"/> - rabolæ totalis ad partialem, eadem eÅ¿t, quÄ™ <lb/>cubi ipÅ¿ius <var>.g.e.</var> ad cubum ipÅ¿ius <var>.a.d.</var> & ex <reg norm="con" type="context">cõ</reg> <lb/>Å¿equenti, vt cuborum earundem baÅ¿ium, eo <lb/>quod cum Å¿it, ex .36. vndecimi Euclid. pro-<lb/>portio cubi ipÅ¿ius <var>.g.e.</var> ad cubum ipÅ¿ius <var>.a.d.</var> <lb/>tripla ei quæ ipÅ¿ius <var>.g.e.</var> ad <var>.a.d.</var> ideo æqualis <lb/>erit ei quÄ™ trianguli <var>.b.g.e.</var> ad triangulum <var>.b.<lb/>a.d.</var> cum proportio horum duorum triangu <lb/>lorum compoÅ¿ita Å¿it (vt Å¿upra vidimus) ex <lb/>ea quæ <var>.g.e.</var> ad <var>.a.o.</var> & ex ea quæ <var>.g.e.</var> ad <var>.a.d.</var> <lb/>& hæc medietas illius, Å¿ed trianguli ita Å¿e in <lb/>uicem habenr, vt parabolÄ™, </s> - <s xml:id="s4260" xml:space="preserve">quare ipſæ para-<lb/>bolæ Å¿einuicem habebunt, vt cubi ipÅ¿arum <lb/>baÅ¿ium.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0366-01" xlink:href="fig-0366-01a"> - <image file="0366-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0366-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head520" style="it" xml:space="preserve">Cubum fabricare æqualem pyramidi propoÅ¿itæ.</head> - <head xml:id="head521" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4261" xml:space="preserve">CVbum fabricare æqualem propoÅ¿itæ pyramidi quadrilateræ, nullius erit diffi-<lb/>cultatis, Å¿uppoÅ¿ita tamen pro reperta diuiÅ¿ione cuiuÅ¿uis datæ proportionis in <lb/>tres partes æquales. </s> - <s xml:id="s4262" xml:space="preserve">Nam ex .6. duodecimi Eucli. patet omne corpus Å¿erratile d-ui <lb/>Å¿ibile eſſe in tres pyramides quadrilateras æquales, Å¿cimus etiam quod cuilibet py-<lb/>ramidi quadrilateræ poteÅ¿t reperiri Å¿uum Å¿erratile. </s> - <s xml:id="s4263" xml:space="preserve">Sit igitur propoÅ¿ita pyramis qua <lb/>drilatera <var>.m.g.f.h.</var> cuius Å¿erratile ita inueniemus, ducendo primum <var>.h.i.</var> parallelam <lb/>ipÅ¿i <var>.g.f.</var> et <var>.f.i.</var> ipÅ¿i <var>.g.h.</var> in Å¿uperficie trianguli <var>.f.g.h.</var> et <var>.m.K.</var> ipÅ¿i <var>.g.h.</var> in Å¿uperficie <lb/>trianguli <var>.m.g.h.</var> & æqualem dictæ <var>.g.h.</var> ducetur poÅ¿tea <var>.K.h.</var> et <var>.K.i.</var> & habebimus cor <lb/>pus <var>.f.K.g.</var> Å¿erratile, & triplum pyramidi propoÅ¿itæ. </s> - <s xml:id="s4264" xml:space="preserve">Nunc duplicemus ipÅ¿um, du-<lb/>cendo <var>.K.x.</var> in Å¿uperficie trianguli <var>.i.k.h.</var> parallelam, <reg norm="æqualemque" type="simple">æqualemqÌ;</reg> ipÅ¿i <var>.i.h.</var> et <var>.m.y.</var> <lb/>in Å¿uperficie trianguli <var>.f.m.g.</var> parallelam, Ä™<reg norm="qualemque" type="simple">qualemqÌ;</reg> ipÅ¿i <var>.f.g.</var> ducatur poÅ¿tea <var>.g.y.</var> et <var>.h.<lb/>x.</var> quarum <reg norm="vnaquæque" type="simple">vnaquæq;</reg> æqualis erit ipÅ¿i <var>.f.m.</var> vnde habebimus corpus <var>.f.x.</var> parallelepe-<lb/>pidum, & Å¿excuplum ipÅ¿i pyramidi propoÅ¿itæ.</s> - </p> - <pb o="355" rhead="EPISTOL AE." n="367" file="0367" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0367"/> - <p> - <s xml:id="s4265" xml:space="preserve">Inueniatur nunc quadratum <var>.u.n.</var> æquale Å¿extæ parti Å¿uperficiei <var>.f.i.g.h.</var> quod per <lb/>Å¿e facile erit, </s> - <s xml:id="s4266" xml:space="preserve">deinde accipiatur altitudo corporis <var>.f.x.</var> ducendo vnam perpendicula <lb/>rem à puncto <var>.m.</var> ad baÅ¿im <var>.f.g.h.</var> quÄ™ Å¿it <var>.n.e.</var> qua mediante, cum quadrato <var>.u.n.</var> fabri <lb/>cetur Å¿olidum parallelepepidum <var>.u.e.</var> quod erit æquale dictæ pyramidi ex .33. vnde-<lb/>cimi Euclid.</s> - </p> - <p> - <s xml:id="s4267" xml:space="preserve">Repertæ nunc Å¿int duæ mediæ proportionales <var>.r.s.</var> inter <var>.n.e.</var> et <var>.n.p.</var> quarum <var>.s.</var> Å¿it <lb/>proximior ipÅ¿i <var>.u.p.</var> ex qua <var>.s.</var> Å¿i conÅ¿titutus fuerit cubus, habebimus propoÅ¿itum.</s> - </p> - <p> - <s xml:id="s4268" xml:space="preserve">Pro cuius rei ratione, cogitemus corpus <var>.u.e.</var> productum eſſe vÅ¿que ad <var>.a.o.</var> per lon-<lb/>gitudem <var>.s.</var> latus dicti cubi, qui quidem cubus Å¿it <var>.d.b.</var> vnde proportio corporis <var>.u.e.</var> <lb/>ad corpus <var>.e.o.</var> erit, vt Å¿uperficiei <var>.p.e.</var> ad Å¿uperficiem <var>.t.e.</var> ex .33. undecimi, ipſæ verò <lb/>Å¿uperficies Å¿ibi inuicem erunt vt <var>.n.e.</var> ad <var>.e.a.</var> ex prima Å¿exti, </s> - <s xml:id="s4269" xml:space="preserve">quare proportio corpo <lb/>ris <var>.u.e.</var> ad corpus <var>.e.o.</var> dupla erit proportioni ipÅ¿ius <var>.s.</var> ad <var>.n.p.</var> Å¿ed cum ex .33 vndeci-<lb/>mi, proportio cubi <var>.d.b.</var> ad corpus <var>.e.o.</var> Å¿it vt <reg norm="quadratum" type="context">quadratÅ©</reg> <var>.q.b.</var> ad quadratum <var>.o.a.</var> & cum <lb/>proportio <var>.q.b.</var> ad <var>.o.a.</var> dupla Å¿it ei quæ <var>.q.o.</var> ad <var>.o.t.</var> ex .18. Å¿exti, erit igitur proportio <lb/>cubi <var>.d.b.</var> ad corpus <var>.e.o.</var> dupla ei quæ <var>.q.o.</var> ad <var>.o.t.</var> hoc eÅ¿t ei quæ <var>.s.</var> ad <var>.n.p.</var> Å¿ed ita erat <lb/>corporis <var>.u.e.</var> ad corpus <var>.e.o.</var> </s> - <s xml:id="s4270" xml:space="preserve">quare ex .9. quinti, cubus <var>.d.b.</var> æqualis erit corpor<unsure/>i.u.e. <lb/>hoc eÅ¿t pyramidi propoÅ¿itæ.</s> - </p> - <p> - <s xml:id="s4271" xml:space="preserve">Sed Å¿i oportebit cubum maiorem vel minorem ipÅ¿a pyramide reperire, in qua <lb/>proportione tibi placuerit, </s> - <s xml:id="s4272" xml:space="preserve">tunc opus erit aliud quadratum inuenire, quod in ea <lb/>proportione Å¿e habeat ad quadratum <var>.u.n.</var> quam volueris, quo mediante Å¿imul cum <lb/>altitudine pyramidis conÅ¿equemur propoÅ¿itum.</s> - </p> - <p> - <s xml:id="s4273" xml:space="preserve">Aduertendum tamen quod fabri-<lb/>care ipÅ¿um corpus Å¿erratile <var>.k.f.h.</var> & Å¿o <lb/> - <anchor type="figure" xlink:label="fig-0367-01a" xlink:href="fig-0367-01"/> - lidum <var>.f.x.</var> neceſſarium non eÅ¿t, niÅ¿i pro <lb/>demonÅ¿tratione. </s> - <s xml:id="s4274" xml:space="preserve"><reg norm="idemque" type="simple">idemqÌ;</reg> dico de alijs <lb/>Å¿olidis, nam pro Å¿implici operatione <lb/>huiuÅ¿modi problematis, abÅ¿que ali-<lb/>qua re neceſſaria ad Å¿peculandum, ita <lb/>faciendum erit.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0367-01" xlink:href="fig-0367-01a"> - <image file="0367-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0367-01"/> - </figure> - </div> - <p> - <s xml:id="s4275" xml:space="preserve">Data pyramide <var>.m.f.g.h.</var> accipe <reg norm="eius" type="simple">eiê°</reg> <lb/>alitudinem à <reg norm="puncto" type="context">pÅ©cto</reg> <var>.m.</var> vÅ¿que ad Å¿uper <lb/>ficiem baÅ¿is <var>.f.g.h.</var> quæ Å¿it <var>.n.e.</var> accipe <lb/>deinde latus letragonicum quadrati <var>.<lb/> - <anchor type="figure" xlink:label="fig-0367-02a" xlink:href="fig-0367-02"/> - u.n.</var> æqualis tertiæ partis ipÅ¿ius baÅ¿is <var>.f.<lb/>g.h.</var> quod latus Å¿it <var>.n.p.</var> inter quod, et <var>.<lb/>n.e.</var> inuentæ cum fuerint duæ lineæ <lb/>mediæ proportiona es <var>.s.</var> et <var>.r.</var> <reg norm="quarum" type="context">quarÅ©</reg> <var>.<lb/>s.</var> proximior Å¿it <var>.n.p.</var> quæ <reg norm="quidem" type="context">quidẽ</reg> <var>.s.</var> erit <lb/>latus cubi quæſiti.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0367-02" xlink:href="fig-0367-02a"> - <image file="0367-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0367-02"/> - </figure> - </div> - <pb o="356" rhead="IO. BAPT. BENED." n="368" file="0368" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0368"/> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head522" style="it" xml:space="preserve">Duplex modus par allelam orizontalem alicui muro propoÅ¿ito <lb/>una tantummodo statione ducendi.</head> - <head xml:id="head523" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4276" xml:space="preserve">DVcere parallelam orizontalem alicui muro recto propoÅ¿ito vna tantummodò <lb/>Å¿tatione, non Å¿olum poſſibile eÅ¿t Å¿ed etiam facile.</s> - </p> - <p> - <s xml:id="s4277" xml:space="preserve">Sit exempli gratia murus rectus <var>.a.d.</var> Å¿itus verò <var>.o.n</var>. </s> - <s xml:id="s4278" xml:space="preserve">Si cupimus ducere <var>.n.u.</var> <lb/>parallelam dicto muro, accipiatur quadratum geometricum, Å¿eu Å¿cala altimetra <lb/>vel aliquod Å¿imile inÅ¿trumentum, quo mediante à ſitu <var>.o.</var> videbimus punctum <var>.q.</var> <lb/>quod volueris ipÅ¿ius muri, <reg norm="dexteram" type="context">dexterã</reg> <lb/>verÅ¿us, inferius tamen. ipÅ¿o <var>.o.</var> vnde <lb/> - <anchor type="figure" xlink:label="fig-0368-01a" xlink:href="fig-0368-01"/> - formatum habebimus triangulum <var>.<lb/>n.o.q</var>. </s> - <s xml:id="s4279" xml:space="preserve">Quo facto ad partem <reg norm="Å¿iniÅ¿tram" type="context">Å¿iniÅ¿trã</reg> <lb/>cum eodem angulo <var>.n.o.q.</var> oporte-<lb/>bit nos inuenire punctum aliquod <var>.<lb/>p.</var> in dicta Å¿uperficie muri, </s> - <s xml:id="s4280" xml:space="preserve">& tunc <lb/>habebimus angulum <var>.n.o.p.</var> æqua-<lb/>lem angulo <var>.n.o.q.</var> vnde angulus <var>.q.<lb/>n.p.</var> nobis cognitus erit, <reg norm="duoque" type="simple">duoqÌ;</reg> late <lb/>ra <var>.n.q.</var> et <var>.n.p.</var> erunt inuicem æqua-<lb/>lia, ex .26. primi Euclid. cum angu-<lb/>li <var>.q.o.n.</var> et <var>.q.n.o.</var> Å¿int æquales angu <lb/>lis <var>.p.o.n.</var> et <var>.p.n.o.</var> & latus <var>.o.n.</var> com <lb/>mune, vnde angulus <var>.q.n.g.</var> extrinÅ¿e <lb/>cus trianguli <var>.p.q.n.</var> <reg norm="reÅ¿iduusque" type="simple">reÅ¿iduusqÌ;</reg> ex <lb/>duobus rectis nobis cognitus erit, <lb/>etiam & eius medictas <var>.q.n.u.</var> æqua <lb/>lis angulo <var>.p.q.n.</var> eo quod ex .5. pri-<lb/>mi, anguli <var>.q.p.</var> Å¿unt inuicem æquales, & ex .32. eiuÅ¿dem, æquales Å¿unt extrinÅ¿eco <var>.q.n.<lb/>g.</var> & ex 27. <var>n.u.</var> erit parallela ipÅ¿i <var>.q.p</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0368-01" xlink:href="fig-0368-01a"> - <image file="0368-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0368-01"/> - </figure> - </div> - <p> - <s xml:id="s4281" xml:space="preserve">Aliter etiam poſſumus idem efficere, Å¿umendo duo illa puncta in Å¿uprem a linea <lb/>orizontali ipÅ¿ius muri ad Å¿uperiorem partem aÅ¿piciendo, quemadmodum ad infe-<lb/>riorem, quod vnum & idem erit, dummodò non aÅ¿piciamus orizontaliter, eo quod <lb/>nos oportet Å¿uperficiem conicam producere, linea viÅ¿uali mediante. </s> - <s xml:id="s4282" xml:space="preserve">cognoÅ¿cere auÂ<lb/>tem angulum <var>.q.n.p.</var> facile erit, conÅ¿tituendo primò inÅ¿trumentum in Å¿itu trianguli <var>.<lb/>o.n.q.</var> <reg norm="aÅ¿piciendoque" type="simple">aÅ¿piciendoqÌ;</reg> punctum <var>.c.</var> in Å¿uperficie <var>.n.q.o.</var> & Å¿ic in alia parte, exiÅ¿tente in-<lb/>Å¿trumento in Å¿itu trianguli <var>.o.p.n.</var> aÅ¿picere oportet punctum <var>.e.</var> proximum puncto <var>.n.</var> <lb/>vbi poſſit metiri angulum <var>.c.n.e</var>.</s> - </p> - <p> - <s xml:id="s4283" xml:space="preserve">Sed Å¿i Å¿itus puncti <var>.n.</var> talis eſſet, vt ab eo non poſſet aliquis murum videre ad re-<lb/>ctos angulos, aÅ¿piceremus punctum <var>.q.</var> Å¿ub orizontali ab oculis noÅ¿tris, in orizontali <lb/>tamen puncti <var>.n.</var> ita quod angulus <var>.o.n.q.</var> rectus exiÅ¿tat, quo facto obÅ¿eruando angu-<lb/>lum <var>.n.o.q.</var> eo mediante, medianteqÌue <var>.n.o.</var> cum angulo <var>.o.n.q.</var> cognoÅ¿cemus <lb/>quantitatem diÅ¿tantiæ <var>.n.q.</var> idem etiam faciendum eÅ¿t cum alio puncto <var>.p.</var> quod <lb/>volueris, & mediantibus duobus punctis inuicem proximis <var>.c.e.</var> cognoÅ¿catur an- - <pb o="357" rhead="EPISTOL AE." n="369" file="0369" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0369"/> - gulus <var>.p.n.q.</var> vnde ex methodo .56. <lb/> - <anchor type="figure" xlink:label="fig-0369-01a" xlink:href="fig-0369-01"/> - primi triangulorum Monteregij, <lb/>cognoÅ¿cemus reliqua trianguli <var>.<lb/>q.p.n</var>. </s> - <s xml:id="s4284" xml:space="preserve">ConÅ¿tituendo poÅ¿tea angu-<lb/>lum <var>.q.n.u.</var> æqualem angulo <var>.n.q.p.</var> <lb/>propoÅ¿itum habebimus.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0369-01" xlink:href="fig-0369-01a"> - <image file="0369-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0369-01"/> - </figure> - </div> - <p> - <s xml:id="s4285" xml:space="preserve">Si etiam puncta <var>.q.p.</var> lineæ <var>.q.p.</var> <lb/>orizontali in eodem plano non exi <lb/>Å¿terent cum puncto <var>.n.</var> nihil refer-<lb/>ret, dummodo in pauimento <reg norm="notem" type="context">notẽ</reg> <lb/>tur <reg norm="puncta" type="context">pÅ©cta</reg> <var>.c.e.</var> proxima <var>.n.</var> in ijÅ¿dem <lb/>Å¿uperficiebus triangulorum <var>.n.o.p.</var> <lb/>et <var>.n.o.q.</var> vnde <var>.n.c.</var> et <var>.n.e.</var> erunt <reg norm="com- munes" type="context">cõ-<lb/>munes</reg> Å¿ectiones dictarum Å¿uperficierum cum Å¿uperficie pauimenti Å¿upra quam fit <lb/>Å¿tatio.</s> - </p> - </div> - </div> - <div type="section" level="3" n="32"> - <div type="letter" level="4" n="1"> - <head xml:id="head524" xml:space="preserve">CONI RECTI DIVISIO A PLANO <lb/>parallelo baÅ¿i Å¿ecundum datam proportionem.</head> - <head xml:id="head525" style="it" xml:space="preserve">Rapbaeli de Auria.</head> - <p> - <s xml:id="s4286" xml:space="preserve"><emph style="sc">QVotiescvnqve</emph> volueris conum rectum diuidere à plano parallelo ba-<lb/>Å¿i Å¿ecundum vnam datam proportionem, nullius tibi erit difficultatis, con <lb/>ceſſa <reg norm="tamen" type="wordlist">tamẽ</reg> pro inuenta diuiÅ¿ione cuiuÅ¿uis propoÅ¿itÄ™ proportionis per tres <lb/>æquales partes.</s> - </p> - <p> - <s xml:id="s4287" xml:space="preserve">Sit exempli gratia conus rectus <var>.a.b.c.</var> Å¿ecandus vt dictum eÅ¿t, accipiatur latus <lb/>ipÅ¿ius, quod Å¿it <var>.a.c.</var> <reg norm="ipÅ¿umque" type="simple">ipÅ¿umqÌ;</reg> diuidatur in puncto <var>.d.</var> Å¿ecundum illam proportionem <lb/>quam deÅ¿ideras, hoc eÅ¿t ipÅ¿ius <var>.a.c.</var> ad <var>.a.d.</var> quo facto, inter totum <var>.a.c.</var> et <var>.a.d.</var> inuenian <lb/>tur duæ lineæ proportionales, quarum maior Å¿it <var>.a.i.</var> </s> - <s xml:id="s4288" xml:space="preserve">tunc Å¿i conus <var>.a.b.c.</var> Å¿ectus fue-<lb/>rit à plano per punctum <var>.i.</var> parallelo baÅ¿i, habebimus quod quærebamus.</s> - </p> - <p> - <s xml:id="s4289" xml:space="preserve">Cuius rei ratio, primò eÅ¿t, quia quotieÅ¿cunque conus aliquis Å¿ectus fuerit ab ali-<lb/>quo plano parallelo baÅ¿i ipÅ¿ius, pars Å¿uperior Å¿imilis Å¿emper erit totali cono, quod <lb/>ita probo, cogitemus conum Å¿ectum eſſe <lb/>à plano per axem <var>.a.l.</var> vnde ex .3. primi <lb/> - <anchor type="figure" xlink:label="fig-0369-02a" xlink:href="fig-0369-02"/> - Pergei, talis Å¿ectio triangularis erit, quæ <lb/>Å¿it <var>.a.b.c.</var> et <var>.b.c.</var> diameter erit baÅ¿is.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0369-02" xlink:href="fig-0369-02a"> - <image file="0369-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0369-02"/> - </figure> - </div> - <p> - <s xml:id="s4290" xml:space="preserve">Imaginemur deinde <var>.K.i.</var> communem <lb/>eſſe Å¿ectionem huiuÅ¿modi trianguli cum <lb/>plano parallelo ipÅ¿i baÅ¿i, </s> - <s xml:id="s4291" xml:space="preserve">tunc tale <reg norm="planum" type="context">planÅ©</reg>, <lb/>circulare erit ex .4. primi ipÅ¿ius Pergei <var>.K.<lb/>i.</var> verò, eius diameter erit, et <var>.a.m.</var> <reg norm="Å¿uus" type="simple">Å¿uê°</reg> axis.</s> - </p> - <p> - <s xml:id="s4292" xml:space="preserve">Cum verò <var>.a.l.</var> Å¿it perpendicularis ipÅ¿i <lb/>baÅ¿i conitotalis, eo quod rectus Å¿upponi-<lb/>tur, ideo eadem <var>.a.m.l.</var> erit perpendicula <lb/>ris eriam ipÅ¿i Å¿ecundo plano circulari, ex <lb/>conuerÅ¿a .14. vndecimi Euclid. </s> - <s xml:id="s4293" xml:space="preserve">vnde ex - <pb o="358" rhead="IO. BAPT. BENED." n="370" file="0370" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0370"/> - Å¿ecunda definitione eiuÅ¿dem libr <var>.a.m.l.</var> efficiet angulos rectos cum duabus <var>.b.c.</var> et <var>.K.<lb/>i.</var> in punctis <var>.m.</var> et <var>.l.</var> et <var>.k.i.</var> parallela erit ipÅ¿i <var>.b.c.</var> ex .28. primi, quod etiam poteÅ¿t con <lb/>cludi mediante .16. vndecimi, cum <var>.k.i.</var> et <var>.b.c.</var> Å¿int communes Å¿ectiones duorum pla <lb/>norum cum triangulari. </s> - <s xml:id="s4294" xml:space="preserve">Deinde ex .29. primi anguli <var>.a.i.m.</var> et <var>.a.c.l.</var> erunt inuicem <lb/>æquales, idem etiam dico de angulis <var>.a.k.i.</var> et <var>.a.b.c.</var> anguli poÅ¿tea ad <var>.a.</var> communes <lb/>Å¿unt triangulis <var>.l.a.c.</var> et <var>.m.a.i.</var> vt triangulis <var>.l.a.b.</var> et <var>.m.a.k</var>. </s> - <s xml:id="s4295" xml:space="preserve">Vnde ex .4. Å¿exti, eadem <lb/>proportio erit ipÅ¿ius <var>.m.i.</var> ad <var>.l.c.</var> & ipÅ¿ius <var>.m.k.</var> ad <var>.l.b.</var> vt ipÅ¿ius <var>.a.m.</var> ad <var>.a.l</var>. </s> - <s xml:id="s4296" xml:space="preserve">Quare ex <lb/>vndecima quinti, ita erit ipÅ¿ius <var>.m.k.</var> ad <var>.l.b.</var> vt ipÅ¿ius <var>.m.i.</var> ad <var>.l.c.</var> & ex .13. eiuÅ¿dem, ita <lb/>erit ipÅ¿ius <var>.k.i.</var> ad <var>.b.c.</var> vt <var>.m.i.</var> ad <var>.l.c.</var> Å¿ed ipÅ¿ius <var>.m.i.</var> ad <var>.l.c.</var> eÅ¿t vt ipÅ¿ius <var>.a.m.</var> ad <var>.a.l.</var> quod <lb/>iam dictum eÅ¿t, vnde ex .11. dicta, ita erit ipÅ¿ius <var>.k.i.</var> ad <var>.b.c.</var> vt ipÅ¿ius <var>.a.m.</var> ad <var>.a.l.</var> & ex <lb/>16. dicti ita erit ipÅ¿ius <var>.a.m.</var> ad <var>.k.i.</var> vt ipÅ¿ius <var>.a.l.</var> ad <var>.b.c</var>. </s> - <s xml:id="s4297" xml:space="preserve">Quare ex definitione ab Eu-<lb/>cli. poÅ¿ita in .11, lib. pars coni Å¿uperior Å¿imilis erit cono totali.</s> - </p> - <p> - <s xml:id="s4298" xml:space="preserve">Deinde Å¿ciendum eÅ¿t illud quod Euclid. Å¿cribit in .10. duodecimi lib. hoc eÅ¿t, <reg norm="quod" type="simple">ê™</reg> <lb/>proportio duarum pyramidum inuicem <lb/>Å¿imilium, triplicata eÅ¿t ei diametrorum <lb/> - <anchor type="figure" xlink:label="fig-0370-01a" xlink:href="fig-0370-01"/> - Å¿uarum baÅ¿ium, hoc eÅ¿t, quod proportio <var>.<lb/>b.c.</var> ad <var>.k.i.</var> tertia pars erit proportionis to <lb/>tius pyramidis <var>.a.b.c.</var> partiali pyramidi <var>.a.<lb/>k.i.</var> Å¿ed ita eÅ¿t ipÅ¿ius <var>.a.c.</var> ad <var>.a.i.</var> vt ipÅ¿ius <var>.b.<lb/>c.</var> ad <var>.k.i.</var> ex .4. Å¿exti cum trianguli <var>.a.b.c.</var> <lb/>et <var>.a.k.i.</var> Å¿int æquianguli, quod ex ijs, quÄ™ <lb/>Å¿uperius diximus facile comprÄ™henditur. <lb/></s> - <s xml:id="s4299" xml:space="preserve">Quare <reg norm="proportio" type="simple">ê“portio</reg> <var>.a.c.</var> ad <var>.a.i.</var> tertia pars erit <lb/>proportionis totius coni <var>.a.b.c.</var> ad eius par <lb/>tem abÅ¿ciſſam <var>.a.k.i.</var> Å¿ed eadem proportio <lb/>ipÅ¿ius <var>.a.c.</var> ad <var>.a.i.</var> erat etiam tertia pars pro <lb/>portionis ipÅ¿ius <var>.a.c.</var> ad <var>.a.d</var>. </s> - <s xml:id="s4300" xml:space="preserve">Quare ex com <lb/>muni conceptu, proportio totius pyramidis, ad partem abÅ¿ciſſam, æqualis erit pro-<lb/>portioni ipÅ¿ius <var>.a.c.</var> ad <var>.a.d</var>.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0370-01" xlink:href="fig-0370-01a"> - <image file="0370-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0370-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head526" style="it" xml:space="preserve">De differentia caloris Solis propter vaporum<unsure/> <lb/>altitudinem.</head> - <head xml:id="head527" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4301" xml:space="preserve">NOlo, mihi credas, Å¿ed ex rationibus, quas tibi Å¿cribo conÅ¿idera, quod quo <lb/><reg norm="tieÅ¿cunque" type="simple">tieÅ¿cunq;</reg> craſſities vel <reg norm="denÅ¿itas" type="context">dẽſitas</reg> <reg norm="vaporum" type="context">vaporÅ©</reg>, Å¿eu altitudo, maior eſſet ea, quÄ™ nunc re-<lb/>peritur, </s> - <s xml:id="s4302" xml:space="preserve">tunc minor differentia eſſet inter maiorem <reg norm="minoremque" type="simple">minoremqÌ;</reg> calorem Solis, quam <lb/>nunc Å¿entiamus. </s> - <s xml:id="s4303" xml:space="preserve">Pro cuius rei euidentia, imaginemur in hac Å¿ubÅ¿cripta figura, li-<lb/>neam <var>.o.a.</var> pro Å¿emidiametro terræ, et <var>.a.c.</var> pro craſſitie vaporum, vt nunc Å¿e <lb/>habet, et <var>.a.d.</var> pro maiori craſſitie, imaginemurqÌue lineam <var>.a.b.</var> quaÅ¿i perpen-<lb/>dicularem ad <var>.o.a.</var> quæ abÅ¿ciſſa Å¿it in puncto u. à circunferentia <var>.c.u.</var> inferiori prio-<lb/>rum vaporum.</s> - </p> - <p> - <s xml:id="s4304" xml:space="preserve">Tunc dico minorem eſſe proportionem ipÅ¿ius <var>.a.b.</var> ad <var>.a.d.</var> quam ipÅ¿ius <var>.a.u.</var> ad <var>.a.<lb/>c.</var> cogitemus ergo protractas eſſe lineas <var>.o.b</var>: <var>d.b</var>: <var>c.u.</var> et <var>.c.n.</var> quæ <var>.c.n.</var> Å¿ecabit <var>.a.u.</var> in - <pb o="359" rhead="EPISTOL AE." n="371" file="0371" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0371"/> - puncto <var>.i.</var> ex communi conceptu, & <lb/> - <anchor type="figure" xlink:label="fig-0371-01a" xlink:href="fig-0371-01"/> - parallcla erit ipÅ¿i <var>.d.b.</var> ex. Å¿ecunda par-<lb/>te Å¿ecundæ Å¿exti, vnde ex prima parte <lb/>ciuÅ¿dem, ita eritipÅ¿ius <var>.b.i.</var> ad <var>.i.a.</var> vt <var>.d.<lb/>c.</var> ad <var>.c.a.</var> & coniunctim ita erit ipÅ¿ius <var>.b.<lb/>a.</var> ad <var>.a.i.</var> vt ipÅ¿ius <var>.d.a.</var> ad <var>.a.c.</var> & permu <lb/>tatim ipÅ¿ius <var>.a.b.</var> ad <var>.a.d.</var> erit, vt <var>.a.i.</var> <lb/>ad <var>.a.c.</var> Å¿ed cum <var>.a.u.</var> maior Å¿it ipÅ¿a <var>.a.i.</var> <lb/>vt omne totum maius eÅ¿t Å¿ua parte. <lb/></s> - <s xml:id="s4305" xml:space="preserve">maior proportio erit ipÅ¿ius <var>.a.u.</var> ad <var>.a.<lb/>c.</var> quam ipÅ¿ius <var>.a.i.</var> ad <var>.a.c.</var> hoc eÅ¿t quam <lb/>ipÅ¿ius <var>.a.b.</var> ad <var>.a.d</var>. </s> - <s xml:id="s4306" xml:space="preserve">Verum igitur eÅ¿t <lb/>propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0371-01" xlink:href="fig-0371-01a"> - <image file="0371-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0371-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head528" style="it" xml:space="preserve">De differentia caloris Solis reÅ¿pectu altitudinis ipÅ¿ius.</head> - <head xml:id="head529" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4307" xml:space="preserve">QVodà me poÅ¿tulas deinde, ita Å¿e habet. </s> - <s xml:id="s4308" xml:space="preserve">Inquis enim, quod cum differentia <lb/>inter maiorem, <reg norm="minoremque" type="simple">minoremqÌ;</reg> calorem, oriatur etiam ex differentia maioris <lb/>quantitatis vaporum ad minorem, per quam quantitatem vaporum rranÅ¿it lumen <lb/>Solis (vt alias etiam tibi dixi) velles nunc Å¿cire quantitatem ipÅ¿ius differentiÄ™, quæ <lb/>inter duas Solis datas altitudines Å¿upra orizontem reperitur.</s> - </p> - <p> - <s xml:id="s4309" xml:space="preserve">Quapropter imaginemur circulum <var>.a.e.</var> pro magno terræ, et <var>.z.b.d.</var> pro magno <lb/>vaporum, Å¿upponatur etiam quod angulus <var>.z.o.d.</var> vel <var>.z.a.b.</var> qui Å¿unt inuicem fe-<lb/>rè æquales, Å¿it angulus diÅ¿tantiæ Solis à zenit, <var>z.a.</var> verò Å¿it Å¿piſſitudo vaporum, et <var>.a.<lb/>b.</var> radius tranÅ¿iens per vapores dictos. </s> - <s xml:id="s4310" xml:space="preserve">nunc <lb/> - <anchor type="figure" xlink:label="fig-0371-02a" xlink:href="fig-0371-02"/> - quæratur proportio, quæ eÅ¿t inter <var>.a.b.</var> et <var>.a.<lb/>z.</var> qua inuenta, angulo <var>.z.a.b.</var> mediante, <lb/>quæremus eandem mediante angulo <var>.z.a.b.</var> <lb/>maiore priori, velipÅ¿o minore, vnde cogno <lb/>Å¿cemus differentiam duarum <var>.a.b.</var> quæ qui-<lb/>dem inæquales inuicem erunt, eo quod Å¿up <lb/>ponatur <var>.a.z.</var> immutabilis, & hoc ita facie-<lb/>mus. </s> - <s xml:id="s4311" xml:space="preserve">Imaginabimur <var>.o.b.</var> quæ claudat trian <lb/>gulum <var>.a.b.o.</var> & quia <var>.a.z.</var> cognita eÅ¿t quam <lb/>Alhazem docetinuenire, cognoÅ¿cimus <reg norm="etiam" type="context">etiã</reg> <lb/><var>o.a.</var> vt Å¿emidiametrum terræ, vnde <var>.o.b.</var> et <var>.<lb/>o.a.</var> duo latera trianguli <var>.a.o.b.</var> cognita <reg norm="erunt" type="context">erÅ©t</reg> <lb/>Å¿imul cum angulo <var>.o.a.b.</var> reÅ¿iduo duorum re <lb/>ctorum, eo quod reliquus <var>.z.a.b.</var> datus eÅ¿t. <lb/></s> - <s xml:id="s4312" xml:space="preserve">Quare <var>.a.b.</var> cognita erit reÅ¿pectu <var>.o.a.</var> et <var>.o.<lb/>b.</var> et <var>.a.z.</var> quæ eÅ¿t eorum differentia. </s> - <s xml:id="s4313" xml:space="preserve">Nunc <lb/>Å¿i idem faciemus cum alia <var>.a.b.</var> Å¿ub diuerÅ¿o <lb/>angulo, habebimus propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0371-02" xlink:href="fig-0371-02a"> - <image file="0371-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0371-02"/> - </figure> - </div> - <pb o="360" rhead="IO. BAPT. BENED." n="372" file="0372" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0372"/> - </div> - </div> - <div type="section" level="3" n="33"> - <div type="letter" level="4" n="1"> - <head xml:id="head530" xml:space="preserve">NOTABILES ERRORES ORONTII <lb/>& Tartaleæ.</head> - <head xml:id="head531" style="it" xml:space="preserve">Cornelio Bitonto.</head> - <p> - <s xml:id="s4314" xml:space="preserve"><emph style="sc">PArvvs</emph> error non fuit, vt putabat Orontius, quodanguli triangulorum <lb/>æquicrurium inuicem æqualium, baÅ¿ibus oppoÅ¿iti, ijÅ¿dem baÅ¿ibus propor <lb/>tionales eſſent, cuius opinionis cauÅ¿a fuit quod nunquam viderit vel me <lb/>minerit eius quod Ptolomeus Å¿cripÅ¿it lib. primo AlmageÅ¿ti, vbi de diÅ¿pro <lb/>portionalitate chordarum <reg norm="arcuumque" type="simple">arcuumqÌ;</reg> tractat, vel quod Å¿cribit Vitellio lib. primo pro <lb/>poÅ¿itione .35. Å¿eu lib. quarto, propoÅ¿itione .21. quod idem eÅ¿t. </s> - <s xml:id="s4315" xml:space="preserve">Sed nec ego tibi pro <lb/>ponam id quod Å¿cribit Nicolaus Tartalea diuiÅ¿ioni .28. quinti capitis quartæ partis <lb/>Å¿uorum tractatuum, eo quod non exactè Å¿cientificè Å¿cripÅ¿erit, nec vniuerÅ¿aliter, <reg norm="quan- uis" type="context">quã-<lb/>uis</reg> talis propoÅ¿itio poſſit Å¿cientificè Å¿cribi, accipiendo <var>.b.c.</var> in eius figura, pro latere <lb/>octagoni, vnde angulus <var>.a.e.b.</var> duplum foret angulo <var>.b.e.c.</var> collocato poÅ¿tea <var>.b.c.</var> in <lb/>arcu <var>.a.b.</var> punctum <var>.c.</var> medium fuiſſet dicti arcus, et <var>.e.c.</var> diuideret <var>.a.b.</var> per æqualia, <lb/>ex quinta primi, nec non ad rectos ex .3. tertij, vnde ex .18. primi, clare vidiſſemus <lb/>non eſſe proportionem <var>.a.b.</var> ad <var>.b.c.</var> vt anguli ad angulum. </s> - <s xml:id="s4316" xml:space="preserve">Sed vniuerÅ¿aliori modo <lb/>poſſumus hoc Å¿peculari. </s> - <s xml:id="s4317" xml:space="preserve">Nam manifeÅ¿tè Å¿cimus, eandem eſſe proportionem circun <lb/>ferentiæ ad diametrum in omnibus circulis tam maioribus, quam minoribus. <lb/></s> - <s xml:id="s4318" xml:space="preserve">Sint igitur duo anguli <var>.a.e.b.</var> et <var>.c.e.b.</var> cuiuÅ¿uis amplitudinis, quorum latera <var>.e.a</var>: <var>e.b</var>: <lb/>et <var>.e.c.</var> Å¿int inuicem æqualia, protrahatur <var>.b.a.</var> et <var>.b.c</var>. </s> - <s xml:id="s4319" xml:space="preserve">Tunc dico maiorem proportio <lb/>nem eſſe anguli <var>.a.e.b.</var> ad angulum <var>.b.e.c.</var> quam <var>.a.b.</var> ad <var>.c.b.</var> ducatur enim <var>.b.g.</var> ita <reg norm="quod" type="simple">ê™</reg> <lb/>faciat angulum <var>.g.b.c.</var> æqualem angulo <var>.e.b.a.</var> protracta poÅ¿tea <var>.c.g.</var> quæ idem faciat <lb/>in puncto <var>.c.</var> vnde <var>.g.b.</var> et <var>.g.c.</var> æquales inuicem erunt ex .6. primi, & quia angulus <var>.a.</var> <lb/>æqualis eÅ¿t angulo <var>e.b.a.</var> ex quinta eiuÅ¿dem, ideo ex .32. dicti, et .4. Å¿exti, horum <lb/>duorum triangulorum latera, erunt inuicem proportionalia. </s> - <s xml:id="s4320" xml:space="preserve">ConÅ¿tituto deinde <var>.g.</var> <lb/>centro, & Å¿ecundum Å¿emidiametrum <var>.g.b.</var> vel <var>.g.c.</var> quod idem eÅ¿t, deÅ¿cripto circu-<lb/>lo <var>.b.i.c.</var> necnon circulo <var>.b.c.a.</var> circa centrum <var>.e.</var> ope Å¿emidiametri <var>.e.b.</var> et <var>.e.a.</var> vn <lb/>de iÅ¿te circulus eritillo maior, cum <var>.e.b.</var> maior Å¿it <var>.g.b.</var> ex .14. quinti. cum ex .14. tertij <lb/><var>a.b.</var> longior Å¿it <var>.c.b.</var> Å¿ed ex vltima definitione tertij, arcus <var>.b.i.c.</var> et <var>.b.c.a.</var> erunt in-<lb/>uicem Å¿imiles, hoc eÅ¿t proportio totius cir-<lb/>cunferentiæ circuli <var>.b.i.c.</var> ad arcus <var>.b.i.c.</var> ea-<lb/> - <anchor type="figure" xlink:label="fig-0372-01a" xlink:href="fig-0372-01"/> - dem erit, quæ totius circunferentiæ circuli <lb/><var>b.c.a.</var> ad arcus <var>.b.c.a.</var> Å¿ed proportio diame-<lb/>tri ad circunferentiam eÅ¿t vt diametri ad cir <lb/>cunferentiam, vt Å¿upra diximus; </s> - <s xml:id="s4321" xml:space="preserve">Quare ex <lb/>proportionum æqualitate, vt Å¿emidiametri <lb/>ad circunferentiam erit, vt Å¿emidiametri <lb/>ad circunferentiam, & per eandem propor <lb/>tionum Ä™qualitatem, proportio <var>.e.b.</var> ad <reg norm="arcum" type="context">arcÅ©</reg> <lb/><var>b.c.a.</var> erit, vt <var>.g.b.</var> ad arcum <var>.b.i.c.</var> & per ean <lb/>dem æqualitatem, ita erit <var>.a.b.</var> chordæ ad ar <lb/>cum <var>.b.c.a.</var> vt <var>.c.b.</var> chordæ ad arcum <var>.b.i.c.</var> <lb/>& permutando, ita erit chordæ <var>.a.b.</var> ad chor <lb/>dam <var>.c.b.</var> vt arcus <var>.b.c.a.</var> ad arcum <var>.b.i.c.</var> Å¿ed <lb/>arcus <var>.b.i.c.</var> maior eÅ¿t arcu <var>.b.d.c.</var> ex commu - <pb o="361" rhead="EPISTOL AE." n="373" file="0373" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0373"/> - ni Å¿cicntia. </s> - <s xml:id="s4322" xml:space="preserve">Quare maior proportio erit acus <var>.b.c.a.</var> ad arcum <var>.b.d.c.</var> quam ad arcum <lb/><var>b.i.c.</var> ex .8. quinti. </s> - <s xml:id="s4323" xml:space="preserve">Vnde ex vltima Å¿exti et .12. quinti, proportio anguli <var>.a.e.b.</var> ad an-<lb/>gulum <var>.c.e.b.</var> maior erit quam chordæ, Å¿iue baÅ¿is <var>.a.b.</var> ad chordam Å¿iue baÅ¿im <var>.c.b</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0372-01" xlink:href="fig-0372-01a"> - <image file="0372-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0372-01"/> - </figure> - </div> - </div> - </div> - <div type="section" level="3" n="34"> - <div type="letter" level="4" n="1"> - <head xml:id="head532" xml:space="preserve">DE CAVSA SVSPENSIONIS NVBIVM <lb/>in aere contra Antonium Bergam.</head> - <head xml:id="head533" style="it" xml:space="preserve">Clarißimo FranciÅ¿co Venerio.</head> - <p> - <s xml:id="s4324" xml:space="preserve">EGo enim non tantum miror ea quæ mihi Å¿cripÅ¿iÅ¿ti de opinione OrtenÅ¿ij <lb/>quantum quod Antonius Berga putat nubes à Sole Å¿upenÅ¿as teneri, id pla <lb/>nè falÅ¿um eÅ¿t, vera cauÅ¿a huiuÅ¿modi effectus, alia nulla eÅ¿t, niÅ¿i earundem <lb/>raritas hoc eÅ¿t, cum rariores Å¿int ipÅ¿o aere Å¿ubiecto, </s> - <s xml:id="s4325" xml:space="preserve">propterea Å¿upra <reg norm="ipsum" type="context">ipsÅ©</reg> <lb/>natant & Å¿tant Å¿ub eo qui rarior ipÅ¿is eÅ¿t, eo quod corpora rariora poÅ¿ita in medio <lb/>non tam raro, aÅ¿cendunt, & denÅ¿iora in medio minus denÅ¿o deÅ¿cendunt. </s> - <s xml:id="s4326" xml:space="preserve">Nam Å¿i <lb/>Sol ipÅ¿as nubes Å¿uÅ¿penÅ¿as in aere teneret, hoc interdiu tantummodo fieret, Å¿ed no <lb/>ctu, cur non deÅ¿cendunt vÅ¿que ad terram, & in eodem loco Å¿emper manent? </s> - <s xml:id="s4327" xml:space="preserve">Scien-<lb/>dum igitur eÅ¿t nubes aÅ¿cendere in altum quouÅ¿que inueniant aerem eiuÅ¿dem ra-<lb/>ritatis cuius ipſæ Å¿unt. </s> - <s xml:id="s4328" xml:space="preserve">Raritas enim & denÅ¿itas non Å¿unt res viÅ¿ibiles niÅ¿i per acci-<lb/>dens, quemadmodum etiam leuitas, & grauitas, opacitas verò & diaphaneitas ma <lb/>gis <reg norm="compræhenduntur" type="context">compræhendÅ©tur</reg>, opacitas enim ex reflexione radiorum luminoÅ¿orum, diapha <lb/>neitas verò compræhenditur ex penetratione ipÅ¿orum radiorum, opacitas autem nu <lb/>bis non eÅ¿t denÅ¿itas, cum valde diuerÅ¿a Å¿it denÅ¿itas ab opacitate, Å¿icut raritas ab dia-<lb/>phaneitate, vt alià s dixi. </s> - <s xml:id="s4329" xml:space="preserve">Et quando dicit, quod Sol calefaciendo aerem ipÅ¿am nu <lb/>bem ambientem, rarefaciat eum magis quam ipÅ¿am nubem reÅ¿pondeo, hoc verum <lb/>non eſſe, proptere<unsure/>a quodradius Solis non multum calefacit ea corpora, quæ ip Å¿i per <lb/>mittunt liberum tranÅ¿itum. </s> - <s xml:id="s4330" xml:space="preserve">vnde corpora quanto magis diaphana Å¿unt tanto minus <lb/>ab ipÅ¿o radio luminoÅ¿o calefiunt, Å¿ed ea quæ magis opaca Å¿unt, magis etiam calefiunt <lb/>& per conÅ¿equens magis rarefiunt, cum calidi Å¿it per Å¿e rarefacere, & non attrahere, <lb/>vt ipÅ¿e & ferè omnes alij putant.</s> - </p> - </div> - </div> - <div type="section" level="3" n="35"> - <div type="letter" level="4" n="1"> - <head xml:id="head534" xml:space="preserve">DE RATIONE EXTENSIONIS FVNIS <lb/>cuiuÅ¿dam libramenti, & de quadam Å¿imboleita-<lb/>te circuli cum ellipÅ¿i.</head> - <head xml:id="head535" style="it" xml:space="preserve">Angelo Ferrario Serenißimi Ducis Sabaudia <lb/>AgrimenÅ¿ori expertißimo.</head> - <p> - <s xml:id="s4331" xml:space="preserve"><emph style="sc">TIbi</emph> in mentem veniet, quod cum Å¿uperioribus diebus in villa lucenti, in <lb/>qua degebat Sereniſſimus Dux noÅ¿ter, dum viridarium ad æquilibrium <lb/>reducebas, eſſemus, à te quæſiui an Å¿cires vnde fieret, vt Å¿tante libramen-<lb/>to ad angulos rectos Å¿upra Å¿uum pedem, funis quæ extrema eiuÅ¿dem li-<lb/>bramenti cum pede in formam trianguli æquicruris coniungit, magis diÅ¿tentus exi-<lb/>Å¿teret, quam cum dictum libramentum cum pede obliquum remanet, ita vt huiuÅ¿- - <pb o="362" rhead="IO. BAPT. BENED." n="374" file="0374" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0374"/> - modifunis cum libramento triangulum Å¿calenum conÅ¿titueret.</s> - </p> - <p> - <s xml:id="s4332" xml:space="preserve">Exempli gratia, ponamus lineam <var>.d.b.c.</var> eſſe libramentum .et <var>.b.e.u.</var> eius pedem, <lb/>funem autem, qui aliquando cum libramento facit triangulum iÅ¿ocellum, & aliquan <lb/>do Å¿calenum, eſſe <var>.d.e.c.</var> eÅ¿to etiam quod in figura <var>.A.</var> dictus triangulus <var>.d.e.c.</var> Å¿it iÅ¿o-<lb/>cellus, & in figura <var>.B.</var> Å¿calenus. </s> - <s xml:id="s4333" xml:space="preserve">Tunc quæſiui à te an Å¿cires rationem, quare <lb/>funis <var>.d.e.c.</var> in figura <var>.A.</var> eſſet diÅ¿tenÅ¿us, & in figura <var>.B.</var> laxus quemadmodum vide-<lb/>bamus. </s> - <s xml:id="s4334" xml:space="preserve">cum mihireÅ¿ponderis, neÅ¿cio quid, quod nunc memoria <reg norm="non" type="context">nõ</reg> teneo, Å¿ed quia <lb/>pollicitus Å¿um metibi eam afferre, propterea nunc ad te mitto. </s> - <s xml:id="s4335" xml:space="preserve">Scias ergo huiuÅ¿-<lb/>modirationem nihil aliud eſſe niÅ¿i quod in figura <var>.A.</var> duæ lineæ <var>.c.e.</var> et <var>.d.e.</var> Å¿imul è <lb/>directo iunctæ longiores Å¿int illis, quÄ™ reperiuntur in figura <var>.B.</var> Å¿ed quia funis tam in <lb/>figura <var>.B.</var> quam in figura <var>.A.</var> vnus, & idem eÅ¿t, ideo in figura <var>.B.</var> laxatus eÅ¿t, & non in <lb/>tenÅ¿us, ut in figura <var>.A</var>. </s> - <s xml:id="s4336" xml:space="preserve">Sed vt huiuÅ¿modi veritatis certam notitiam habeas, infraÅ¿cri <lb/>ptum circulum mente concipe <var>.f.e.i.</var> cuius Å¿emidiameter, æqualis Å¿it <var>.b.e.</var> & diame-<lb/>ter Å¿it <var>.f.i.</var> in quo imaginare eſſe tuum <lb/>libramentum <var>.d.b.c.</var> & figuras <var>.A.</var> et <var>.B.</var> <lb/> - <anchor type="figure" xlink:label="fig-0374-01a" xlink:href="fig-0374-01"/> - <anchor type="figure" xlink:label="fig-0374-02a" xlink:href="fig-0374-02"/> - & pr obabo lineas <var>.d.e.c.</var> figurÄ™ <var>.A.</var> lon <lb/>giores eſſe lineis <var>.d.e.c.</var> figuræ <var>.B</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0374-01" xlink:href="fig-0374-01a"> - <image file="0374-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0374-01"/> - </figure> - <figure xlink:label="fig-0374-02" xlink:href="fig-0374-02a"> - <image file="0374-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0374-02"/> - </figure> - </div> - <p> - <s xml:id="s4337" xml:space="preserve">Imaginemur igitur lineam <var>.b.e.</var> eſſe <lb/>dimidium minoris axis <reg norm="alicuius" type="simple">alicuiê°</reg> ellipÅ¿is <lb/>cuius quidem figuræ ponamus <var>.d.</var> et <var>.c.</var> <lb/>centra ipÅ¿ius circunÅ¿criptionis eſſe, cu <lb/>ius <reg norm="circunferentia" type="context">circunferẽtia</reg>, nullidubium eÅ¿t, quin <lb/>extra propoÅ¿itum circulum tranÅ¿itura, <lb/>& in vno tantummodo puncto ipÅ¿um <lb/>circulum tactura Å¿it, qui exiÅ¿tat <var>.e.</var> <lb/> figuræ <var>.A.</var> Å¿eparatum tamen à puncto <lb/>e. figuræ <var>.B</var>. </s> - <s xml:id="s4338" xml:space="preserve">Tunc Å¿i protracta fue-<lb/>rit linea <var>.d.e.</var> figuræ <var>.B.</var> vÅ¿que ad gi <lb/> - <anchor type="figure" xlink:label="fig-0374-03a" xlink:href="fig-0374-03"/> - rum ellipticum in puncto <var>.g.</var> à quo <lb/>ad punctum <var>.c.</var> ducta etiam Å¿it linea <lb/><var>g.c</var>. </s> - <s xml:id="s4339" xml:space="preserve">tunc <reg norm="manifeÅ¿tum" type="context">manifeÅ¿tÅ©</reg> erit duas lineas <lb/><var>d.e.</var> et <var>.e.c.</var> figuræ <var>.A.</var> Å¿imul iunctas, <lb/>æquales eſſe duabus <var>.d.g.</var> et <var>.g.c.</var> Å¿i-<lb/>mul poÅ¿itis, vt etiam ex .52. tertij <lb/>Pergei facilè videre eÅ¿t, Å¿ed ex .21. <lb/>primi Euclid. iam certò Å¿cimus <var>.d.g.c.</var> longiores eſſe <var>.d.e.c.</var> Å¿iguræ <var>.B.</var> ergo <var>.d.e.c.</var> figu-<lb/>ræ <var>.A.</var> longiores Å¿unt <var>.d.e.c.</var> figuræ <var>.B.</var> quod eÅ¿t propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0374-03" xlink:href="fig-0374-03a"> - <image file="0374-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0374-03"/> - </figure> - </div> - <p> - <s xml:id="s4340" xml:space="preserve">Quod etiam mihinunc circa hoc Å¿uccurrit, tibi libenter Å¿ignifico, hoc eÅ¿t, quod <lb/>Å¿icut in ellipÅ¿i duæ lineæ <var>.d.e.e.c.</var> figuræ <var>.A.</var> Å¿imul iunctæ, Å¿unt Å¿emper æquales duabus <lb/>lineis <var>.d.g.g.c.</var> in longitudine, ita in circulo duæ <var>.d.e.e.c.</var> figuræ <var>.A.</var> æquales Å¿unt in <lb/>potentia duabus <var>.d.e.e.c.</var> figurÄ™ <var>.B</var>.</s> - </p> - <p> - <s xml:id="s4341" xml:space="preserve">ManifeÅ¿tum enim primum eÅ¿t ex penultima primi in figura <var>.A.</var> quadratum <var>.e.c.</var> <lb/>æquale eſſe duobus quadratis Å¿cilicet <var>.e.b.</var> et <var>.b.c.</var> & quadratum <var>.e.d.</var> æquale duobus <var>.<lb/>e.b.</var> et <var>.b.d</var>. </s> - <s xml:id="s4342" xml:space="preserve">Quare quadrata <var>.e.c.</var> et <var>.e.d.</var> æqualia Å¿unt quadratis <var>.e.b.</var> figuræ <var>.A.</var> et <var>.e.<lb/>b.</var> figurÄ™. B et <var>.b.c.</var> et <var>.b.d.</var> hoc eÅ¿t duplo quadrati <var>.e.a.</var> (ducta cum fuerit <var>.e.a.</var> perpen-<lb/>dicularis ad <var>.c.b.d.a.</var>) duplo quadrati <var>.a.b.</var> ex penultima primi, & duplo quadrati <var>.b.<lb/>c</var>. </s> - <s xml:id="s4343" xml:space="preserve">Sed quadrata <var>.d.e.</var> et <var>.e.c.</var> figurÄ™ <var>.B.</var> æqualia Å¿unt duplo quadrati <var>.a.e.</var> & quadrato <var>a.d.</var> - <pb o="363" rhead="EPISTOL AE." n="375" file="0375" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0375"/> - & qua drato <var>.a.c.</var> ex <reg norm="eadem" type="context">eadẽ</reg>. </s> - <s xml:id="s4344" xml:space="preserve">Nunc videndum eÅ¿t <reg norm="vtrum" type="context">vtrÅ©</reg> <reg norm="duplum" type="context">duplÅ©</reg> quadrati <var>.a.e.</var> <reg norm="cum" type="context">cÅ©</reg> duplo qua <lb/>drati <var>.b.a.</var> <reg norm="cum" type="context">cÅ©</reg> duplo quadrati <var>.b.c.</var> Å¿it æquale duplo quadrati <var>.a.e.</var> <reg norm="cum" type="context">cÅ©</reg> quadrato <var>.a.d.</var> & <lb/>cum quadrato <var>.a.c</var>. </s> - <s xml:id="s4345" xml:space="preserve">Sed quia tam ex vna parte quà m ex alia habemus duplum qua-<lb/>drati <var>.a.e</var>. </s> - <s xml:id="s4346" xml:space="preserve">Videndum igitur erit vtrum duplum quadrati <var>.a.b.</var> Å¿imul cum duplo qua-<lb/>drati <var>.b.c.</var> Ä™quale Å¿it quadrato <var>.a.c.</var> cum quadrato <var>.a.d.</var> Å¿ed hoc manifeÅ¿tum eÅ¿t .ex .10. <lb/>Å¿ecundi Euclidis, dato quod <reg norm="punctum" type="context">punctÅ©</reg> <var>.a.</var> Å¿it inter <var>.f.</var> et <var>.d.</var> Å¿ed Å¿i fuerit inter <var>.d.</var> et <var>.b.</var> hoc <lb/>manifeÅ¿tum erit ex .9. Å¿ecundi dicti, nihilominus accipe hunc alium modum.</s> - </p> - <p> - <s xml:id="s4347" xml:space="preserve">Sit hic Å¿ubÅ¿criptum quadratum <var>.D.</var> ex <var>.a.c.</var> in Å¿eipÅ¿a producta, cuius diameter Å¿it <lb/><var>a.n.</var> <reg norm="protrahanturque" type="simple">protrahanturqÌ</reg> parallelÄ™ <var>.d.h</var>: <var>b.K</var>: <var>l.m.o.</var> et <var>.r.q.s.</var> <reg norm="eique" type="simple">eiqÌ;</reg> addatur <var>.c.p.</var> ad <var>.a.c.</var> æqua-<lb/>lis tamen <var>.d.a.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> protracta <var>.p.u.</var> vÅ¿que ad <var>.m.o.u.</var> vnde habebimus <var>.a.n.</var> pro totali <lb/>quadrato, et <var>.p.s.</var> pro partiali, & æquali quadrato lineæ <var>.a.d</var>. </s> - <s xml:id="s4348" xml:space="preserve">Videndum nunc eÅ¿t, <reg norm="vtrum" type="context">vtrÅ©</reg> <lb/>hÄ™c duo quadrata æqualia Å¿int duobus quadratis lineæ <var>.a.b.</var> & duobus lineæ <var>.b.c.</var> <reg norm="Nam" type="context">Nã</reg> <lb/>duo quadrata lineæ <var>.b.c.</var> Å¿int <var>.K.o.</var> et <var>.h.l.</var> videndum nunc eÅ¿t utrum reÅ¿iduum Ä™quale <lb/>Å¿it duobus quadratis lineÄ™ <var>.a.b.</var> quorum vnum Å¿it <var>.m.b.</var> alterum verò <var>.l.p.</var> quod Å¿upe-<lb/>rat <var>.l.c.</var> et <var>.s.p.</var> figuræ <var>.D.</var> per Å¿upplementum <var>.o.t.</var> cui æquale eÅ¿t parallelogrammum <var>.h.<lb/>m.</var> figuræ <var>.D.</var> Å¿ed Å¿i punctus <var>.a.</var> poÅ¿itus fuerit inter <var>.d.</var> et <var>.b.</var> conÅ¿tituto quadrato <var>.d.u.</var> <reg norm="cum" type="context">cÅ©</reg> <lb/>omnibus parallelis, vtin figura <var>.C.</var> viderelicet, in qua figura videbimus quadrata <var>.r.<lb/>n.</var> et <var>.d.r.</var> Ä™quari duplo quadratorum <var>.l.n.</var> et <var>.r.l.</var> nam in quadrato <var>.r.n.</var> ipÅ¿a duo quadra-<lb/>ta <var>.l.n.</var> et <var>.r.l.</var> capiuntur, reliquum eÅ¿t igitur vt videamus an duo Å¿upplementa <var>.l.t.</var> et <var>.l.<lb/>s.</var> cum quadrato <var>.d.r.</var> Å¿int æqualia dictis <reg norm="quadratis" type="typo">qÌuadratis</reg> <var>.l.n.</var> et <var>.r.l.</var> Å¿ed quadratum <var>.d.l.</var> æ qua-<lb/>tur quadrato <var>.l.n.</var> videndum igitur eÅ¿t, <lb/>an duo Å¿upplementa <var>.l.t.</var> et <var>.l.s.</var> cum qua <lb/> - <anchor type="figure" xlink:label="fig-0375-01a" xlink:href="fig-0375-01"/> - drato <var>.d.r.</var> Å¿int æqualia duobus quadra <lb/>tis <var>.d.l.</var> et <var>.r.l.</var> Å¿ed quadratum <var>.d.l.</var> æqua-<lb/>tur quadrato <var>.d.r.</var> & Å¿upplemento <var>.l.t.</var> <lb/>mediante <var>.q.l.</var> & Å¿upplemento <var>.r.b.</var> Å¿up-<lb/>plementum verò <var>.l.s.</var> Å¿uperat <reg norm="Å¿upplemem" type="context">Å¿upplemẽ</reg> <lb/>tum <var>.r.b.</var> per quantitatem <reg norm="æqualem" type="context">æqualẽ</reg> qua-<lb/>drato <var>.r.l.</var> </s> - <s xml:id="s4349" xml:space="preserve">quare duo Å¿upplementa <var>.l.t.</var> <lb/>et <var>.l.s.</var> cum quadrato <var>.d.r.</var> æquantur qua <lb/>drato <var>.d.l.</var> <reg norm="cum" type="context">cÅ©</reg> quadrato <var>.l.r.</var> verum igitur eÅ¿t duas <var>.d.e.e.c.</var> figuræ <var>.A.</var> æquales eſſe in <lb/>potentia duabus <var>d.e.e.c.</var> figurÄ™ <var>.D.</var> quæ quidem affectio circuli, à nemine fuit adhuc <lb/>(quod Å¿ciam) detecta.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0375-01" xlink:href="fig-0375-01a"> - <image file="0375-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0375-01"/> - </figure> - </div> - <pb o="364" rhead="IO. BAPT. BENED." n="376" file="0376" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0376"/> - </div> - </div> - <div type="section" level="3" n="36"> - <div type="letter" level="4" n="1"> - <head xml:id="head536" xml:space="preserve">DE AVGMENTO PONDERIS CORPORIS <lb/>ad Å¿tateram appenÅ¿i, & quadam alia demonÅ¿tratione, <lb/>& quibuÅ¿dam erroribus Tartaleæ.</head> - <head xml:id="head537" style="it" xml:space="preserve">Mutio Groto.</head> - <p> - <s xml:id="s4350" xml:space="preserve">SI ea quæ à me audiuiÅ¿ti non credis, conÅ¿idera quæſo libram Å¿eu Å¿tateram <lb/><var>o.a.</var> cuius centrum non longitudinis Å¿ed ponderum Å¿it <var>.i.</var> quÄ™ Å¿tatera, vt ori <lb/>zontaliter conÅ¿iÅ¿tat, oportebit pondus extremitatis <var>.o.</var> ita Å¿e habere <lb/>ad pondus extremitatis <var>.a.</var> ut <var>.a.i.</var> Å¿e habet ad <var>.o.i.</var> quod te Å¿cire puto, ima <lb/>ginemur nunc d uas lineas <var>.a.e.</var> et <var>.o.n.</var> paralle las <reg norm="infinitasque" type="simple">infinitasqÌ;</reg> & à puncto <var>.n.</var> immobili, <lb/>& fixo extra Å¿tateram, tranÅ¿eat per <var>.i.</var> linea <var>.n.i.e</var>. </s> - <s xml:id="s4351" xml:space="preserve">Cogitemus etiam punctum <var>.e.</var> inter <lb/>Å¿ectionis ipÅ¿ius <var>.n.i.e.</var> cum <var>.a.e.</var> progredi vniformiter <reg norm="continuòque" type="simple">continuòqÌ;</reg> ab <var>.a.</var> per lineam <var>.a.e.</var> <lb/>vnde punctum <var>.i.</var> interÅ¿ectionis ipÅ¿ius <var>.n.i.e.</var> cum <var>.a.i.o.</var> Å¿emper vicinius fiet puncto <var>.o.</var> <lb/>nec unquam cum illo vnum erit, quamuis moueatur tempore infinito. </s> - <s xml:id="s4352" xml:space="preserve">Nunc autem <lb/>dico, quod cum Å¿tateram <var>.o.i.a.</var> oporteat Å¿emper orizontalem eſſe virtute ponderis, <lb/>o. oportebit pundus <var>.o.</var> in infinitum etiam augeri, <reg norm="quotieÅ¿cunque" type="simple">quotieÅ¿cunq;</reg> pondus <var>.a.</var> nunquam <lb/>diminui voluerimus vel econtra hoc in infinitum diminui, Å¿i illud nunquam augeri <lb/>voluerimus.</s> - </p> - <p> - <s xml:id="s4353" xml:space="preserve">Sedre vera non putabam te indigere aliqua demonÅ¿tratione, quod linea <var>.b.h.</var> di-<lb/>uiÅ¿a Å¿it per æqualia à <unsure/> linea <var>.c.a.</var> cum hæc perpendicularis Å¿it ab <var>.a.</var> ad baÅ¿im <var>.g.d.</var> in <reg norm="triam" type="context">triã</reg> <lb/>gulo orthogonio <var>.g.a.d.</var> & cum Å¿it <var>.b.h.</var> perpendicularis ad <var>.a.o.</var> ex Å¿uppoÅ¿ito quæ <var>.a.<lb/>o.</var> in Å¿e habet punctum medium baÅ¿is <var>.g.d.</var> nec <reg norm="non" type="context">nõ</reg> illud anguli recti <var>.a.</var> quod per Å¿e cla <lb/>riſſimum eÅ¿t, cum iam Å¿cis <var>.o.</var> eſſe centrum circuli circundantis triangulum <var>.g.a.d.</var> or-<lb/>thogonium, et <var>.g.d.</var> eius diameter, vnde <var>.o.a.</var> æquabitur ipÅ¿i <var>.o.g.</var> quapropter angulus <lb/>o. <reg norm="am" type="context">ã</reg>. g. æquabitur angulo <var>.g.</var> ex quinta primi, </s> - <s xml:id="s4354" xml:space="preserve">deinde ex .32. eiuÅ¿dem, angulus <var>.h.</var> æqua <lb/>bitur angulo <var>.d.</var> eo quod an gulus <var>.e.</var> rectus eÅ¿t, quemadmodum et <var>.a.</var> Å¿ed angulus <var>.d.</var> <lb/>æqualis eÅ¿t angulo <var>.g.a.c</var>. </s> - <s xml:id="s4355" xml:space="preserve">& propterea angulus <var>.h.</var> erit etiam æqualis angulo <var>.h.a.u.</var> <lb/>vnde <var>.h.u.</var> æqualis erit ipÅ¿i <var>.u.<lb/>a.</var> ex .6. primi, cum poÅ¿tea angulus <var>.<lb/> - <anchor type="figure" xlink:label="fig-0376-01a" xlink:href="fig-0376-01"/> - o.a.d.</var> æqualis Å¿itangulo <var>.d.</var> ex quinÂ<lb/>ta primi erit angulus <var>.a.b.e.</var> æqua-<lb/>lis angulo <var>.g.</var> ex .32. dicta, eo quod <lb/>e. rectus eÅ¿t, & ex eadem æqualis <lb/>erit angulo <var>.d.a.c.</var> vnde <var>.u.b.</var> erit <lb/>æqualis ipÅ¿i <var>.u.a.</var> ex .6. dicti, & ideo <lb/>æqualis eric ipÅ¿i <var>.u.h</var>. </s> - <s xml:id="s4356" xml:space="preserve">Reliqua ve-<lb/>rò illius propoÅ¿itionis credo ex te <lb/>omnia poſſe <reg norm="intelligere" type="context">Ä©telligere</reg>, excepto, <reg norm="quod" type="simple">ê™</reg> <lb/>vt tibi Å¿ignificaui Å¿i à <reg norm="puncto" type="context">pÅ©cto</reg> <var>.i.</var> com-<lb/>muni ipÅ¿i <var>.a.c.u.</var> & circunferentiæ, <lb/>ducta fuerit <var>.i.x.</var> ad <reg norm="punctum" type="context">pÅ©ctum</reg> <var>.x.</var> com <lb/>mune vni parallelæ à <reg norm="puncto" type="context">pÅ©cto</reg> <var>.g.</var> ipÅ¿i <lb/><var>h.b.</var> & circunferentiæ, quod di-<lb/>cta <var>.i.x.</var> ad rectos erit ipÅ¿i <var>.a.b.d.</var> eo <lb/>quod cum angulus <var>.a.g.x.</var> æqualis - <pb o="365" rhead="EPISTOL AE." n="377" file="0377" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0377"/> - Å¿it angulo <var>.a.h.b.</var> propter æquidiÅ¿tantiam dictam, æqualis etiam erit angulo <var>.d.</var> & ar-<lb/>cus <var>.a.x.</var> æqualis arcui <var>.a.g.</var> vnde angulus <var>.a.i.x.</var> æqualis erit <var>.d.</var> Å¿ed angulus <var>.i.a.d.</var> com-<lb/>munis eÅ¿t triangulis <var>.c.a.d.</var> et <var>.i.a.t</var>. </s> - <s xml:id="s4357" xml:space="preserve">quare angulus <var>.a.t.i.</var> rectus erit, vt <var>.c.</var> hoc eÅ¿t <var>.i.x.</var> per <lb/>pendicularis erit ipÅ¿i <var>.a.d</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0376-01" xlink:href="fig-0376-01a"> - <image file="0376-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0376-01"/> - </figure> - </div> - <p> - <s xml:id="s4358" xml:space="preserve">Sed vbitibi Å¿cripÅ¿i circa finem illius epiÅ¿tolæ, Tartaleam erraſſe in quinta propo-<lb/>Å¿itione primi lib. Å¿uæ nouæ Å¿cientiæ, non Å¿ine ratione illud Å¿cripÅ¿i. </s> - <s xml:id="s4359" xml:space="preserve">Nam, inquit ipÅ¿e, <lb/>nullum corpus æquè graue poteÅ¿t in aliquo temporis Å¿patio moueri motu naturali, <lb/><reg norm="violentoque" type="simple">violentoqÌ;</reg> Å¿imul miÅ¿tis. </s> - <s xml:id="s4360" xml:space="preserve">Vbi decipitur, eo quod non animaduertit incrementum ve <lb/>locitatis vnius motus, Å¿imul eſſe cum decremento velocitatis alterius, <reg norm="eodemque" type="simple">eodemqÌ;</reg> tem <lb/>pore, vt manifeÅ¿tè patet in itinere corporis, ab ipÅ¿o pro exemplo aſſumpto, hoc eÅ¿t <lb/>quod velocitas motus in Å¿patio <var>.c.d.</var> creÅ¿cit vt naturalis, & decreÅ¿cit vt violenta. </s> - <s xml:id="s4361" xml:space="preserve"><reg norm="nam" type="context">nã</reg> <lb/>creÅ¿cit orizontem verÅ¿us & decreÅ¿cit in remotione à linea <var>.a.b.</var> Å¿ed Å¿i à puncto <var>.c.</var> ad <lb/>punctum <var>.d.</var> motus eſſet purè violentus, vt putat Tartalea, corpus illud minimè de-<lb/>Å¿cenderet, eo quod uirtus mouens, in <var>.a.</var> poÅ¿ita, nullo pacto poteÅ¿t talem effectum ef-<lb/>ficere, vnde ab ipÅ¿a natura prouenit deÅ¿cenÅ¿io illius corporis propter <reg norm="grauitatem" type="context">grauitatẽ</reg>, <reg norm="quam" type="context">quã</reg> <lb/>dictum corpus habet in tali medio, aeris Å¿cilicet, & non ex violentia aliqua. </s> - <s xml:id="s4362" xml:space="preserve">Sed Å¿i <lb/>dixiſſet ipÅ¿e, illum motum eſſe purum naturalem, hoc eſſet falÅ¿um, eo quod purus <lb/>naturalis motus alicuius corporis non impediti, extra locum Å¿uum, Å¿it per lineam re <lb/>ctam, & non per curuam, vt videre eÅ¿t inter <var>.c.</var> et <var>.d</var>.</s> - </p> - <p> - <s xml:id="s4363" xml:space="preserve">In vltima propoÅ¿itione deinde eiuÅ¿dem lib. quæ .6. eÅ¿t decipitur Å¿imiliter, & hæc <lb/>deceptio oritur ab ignoratione quintæ, & à putando motum naturalem non eſſe cau <lb/>Å¿am ipÅ¿ius deÅ¿cenÅ¿us per Å¿patium <var>.c.d</var>. </s> - <s xml:id="s4364" xml:space="preserve">Sed quia tibi Å¿ignificaui expeditiorem viam <lb/>repeririad <reg norm="cognoÅ¿cendam" type="context">cognoÅ¿cendã</reg> proportionem inter <var>.a.h.</var> et <var>.a.e.</var> in vltima propoÅ¿itione Å¿e-<lb/>cundi lib. ipÅ¿ius Tartaleæ, ipÅ¿am nunc tibi Å¿cribo. </s> - <s xml:id="s4365" xml:space="preserve"><reg norm="Nam" type="context">Nã</reg> iam Å¿cis angulum <var>.h.l.i.</var> diui-<lb/>Å¿um eſſe per æqualia ab <var>.P.l.</var> & quod <var>.a.h.</var> et <var>.h.p.</var> Ä™quales inuicem Å¿unt ex .6. primi Eu-<lb/>cli. </s> - <s xml:id="s4366" xml:space="preserve">vnde <var>.p.i.</var> et <var>.a.h.</var> æquales erunt inuicem Å¿imiliter, Å¿ed ex .3. Å¿exti ita eÅ¿t ipÅ¿ius <var>.a.l.</var> <lb/>ad <var>.l.i.</var> vt ipÅ¿ius <var>.a.p.</var> ad <var>.p.i.</var> & coniunctim ita erit <var>.a.l.i.</var> ad <var>.l.i.</var> vt <var>.a.i.</var> ad <var>.p.i.</var> Å¿ed <var>.a.l.</var> cogni <lb/>ta eÅ¿t ex eius quadrato, et <var>.l.i.</var> etiam, cum æqualis Å¿it ipÅ¿i <var>.a.i.</var> vnde ex regula de tribus <lb/>notam habebimus <var>.p.i.</var> reÅ¿pectu <var>.a.i.</var> & ita reÅ¿pectu <var>.a.e.</var> Å¿i hypotheÅ¿es ipÅ¿ius Tartaleæ <lb/>veræ Å¿unt.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head538" style="it" xml:space="preserve">Alia demonstratio impoßibilitatis diuidendi per æqualia <lb/>proportionem Å¿uperparticularem in <lb/>diÅ¿cretis.</head> - <head xml:id="head539" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4367" xml:space="preserve">QVod à me poÅ¿tulas, hoc eÅ¿t Å¿cientiam impoſſibilitatis diuidendi per æqualia <lb/>proportionem Å¿uperparticularem in numeris Å¿atis à Campano in .8. octaui <lb/>potes habere, Iacobus Faber StapulenÅ¿is etiam idem tractat<unsure/> in libello Å¿uÄ™ muÅ¿icæ <lb/>demonÅ¿tratæ. </s> - <s xml:id="s4368" xml:space="preserve">Sed Å¿i etiam alia via idem deÅ¿ideras, quamuis longiori, nih<unsure/>ilomi-<lb/>nus vniuerÅ¿aliori, conÅ¿idera duos numeros <var>.g.</var> et <var>.h.</var> inuicem relatos Å¿ecundum pro-<lb/>portionem Å¿uperparcicularem, quam volueris. </s> - <s xml:id="s4369" xml:space="preserve">Tunc dico impoſſibile eÅ¿le, vt per <lb/>æqualia diuidatur, quod Å¿i dixeris poſſibile eſſe, Å¿it per te <var>.K.</var> medius numerus - <pb o="366" rhead="IO. BAPT. BENED." n="378" file="0378" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0378"/> - proportionalis inter <var>.g.</var> et <var>.h</var>. </s> - <s xml:id="s4370" xml:space="preserve">quare <var>.g.</var> et <var>.h.</var> non erunt minimi in ea proportione, quia <lb/>vnitas diuiÅ¿ibilis eſſet Å¿i <var>.g.h.</var> minimi fuiſſent, quod non conceditur, Å¿int igitur mini <lb/>mi in dicta proportione <var>.a.</var> et <var>.b.</var> quorum differentia erit vnitas, vt Å¿cis, <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.c.</var> quadra <lb/>tum ipÅ¿ius <var>.g.</var> et <var>.d.</var> quadratum ipÅ¿ius <var>.K</var>. </s> - <s xml:id="s4371" xml:space="preserve">tunc clarum erit ex .11. octaui, quod propor-<lb/>tio ipÅ¿ius c. ad <var>.d.</var> eadem erit quæ <var>.g.</var> ad <var>.h.</var> hoc eÅ¿t vt ipÅ¿ius <var>.a.</var> ad <var>.b.</var> vnde Å¿i vnus termi. <lb/>norum <var>.a.</var> vel <var>.b.</var> eſſet quadratus, reliquus etiam quadratus eſſet ex .22. octaui, & ex <lb/>16. eiuÅ¿dem, inter <var>.a.</var> et <var>.b.</var> reperiretur aliquis medius numerus proportionalis, quod <lb/>fieri non poteÅ¿t ex hypotheÅ¿i, cum inter <var>.a.</var> et <var>.b.</var> nullus Å¿it numerus, quia differunt in <lb/>ter Å¿e per vnitatem tantummodo. </s> - <s xml:id="s4372" xml:space="preserve">Nunc autem cum nullus numerorum <var>.a.</var> vel <var>.b.</var> qua <lb/>dratus Å¿it, ponatur quod <var>.f.</var> quadratus Å¿it ipÅ¿ius <var>.b.</var> et <var>.e.</var> Å¿it productum ipÅ¿ius <var>.a.</var> in <var>.b.</var> vn <lb/>de ex .18. Å¿eptimi, proportio ipÅ¿ius <var>.e.</var> ad <var>.f.</var> erit vt. ipÅ¿ius <var>.a.</var> ad <var>.b.</var> hoc eÅ¿t vt ipÅ¿ius <var>.c.</var> ad <lb/>d. quapropter <var>.e.</var> erit quadratus ex .22. octaui, cuius latus tetragonicum eſſet <reg norm="medium" type="context">mediÅ©</reg> <lb/>proportionale inter <var>.a.</var> et <var>.b.</var> ex .20. Å¿eptimi, quod eÅ¿t impoſſibile, vt iam dixi, cum <var>.a.</var> <lb/>et <var>.b.</var> Å¿int inui cem conÅ¿equentes, vnus poÅ¿t alium immediatè.</s> - </p> - <p> - <s xml:id="s4373" xml:space="preserve">Superius enim dixi hunc modum eſſe vniuerÅ¿alem, <lb/>hoc eÅ¿t quod hac methodo poſſumus in cognitionem <lb/>vcnire, quod non Å¿olum in duas æquales partes diui-<lb/> - <anchor type="figure" xlink:label="fig-0378-01a" xlink:href="fig-0378-01"/> - di non poſſit, Å¿ed nec in tres, nec quatuor nec quot vo <lb/>lueris. </s> - <s xml:id="s4374" xml:space="preserve">Primum enim quod non in tres diuidatur à te <lb/>ipÅ¿o cognoÅ¿ces ope <reg norm="cuborum" type="context">cuborÅ©</reg> vice <reg norm="quadratorum" type="context">quadratorÅ©</reg>, opevero <lb/><reg norm="cenÅ¿uum" type="context">cenÅ¿uÅ©</reg> <reg norm="cenÅ¿uum" type="context context">cẽſuÅ©</reg>, ve<unsure/>l qui cognouerit eam <reg norm="proportionem" type="context">proportionẽ</reg> <lb/>eſſe indiuiÅ¿ibilem per æqualia, illicò etiam cognoÅ¿cet <lb/>indiuiÅ¿ibilem eſſe per quatuor partes, ope verò pri-<lb/>morum relatorum, cognoÅ¿cet non eſſe diuiÅ¿ibilem per <lb/><reg norm="quinque" type="simple">quinq;</reg> partes, & Å¿ic de cÄ™teris, Å¿ed mediantibus ijs <lb/>quas Å¿cripÅ¿i de iÅ¿tis dignitatibus in libro <reg norm="ThÄ™orematum" type="context">ThÄ™orematÅ©</reg> <lb/>arithmeticorum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0378-01" xlink:href="fig-0378-01a"> - <image file="0378-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0378-01"/> - </figure> - </div> - <p> - <s xml:id="s4375" xml:space="preserve">Id autem quod IlluÅ¿triſſimus Daniel Barbarus Å¿cri <lb/>bit in quinta parte Å¿uæ perÅ¿pectiuæ, Å¿i Å¿upra aliquo im <lb/>mobili, atque magno pariete facere volueris, te opor <lb/>tebit hoc ex reflexione radij Å¿olaris à ſpeculo plano <lb/>perficere.</s> - </p> - </div> - </div> - <div type="section" level="3" n="37"> - <div type="letter" level="4" n="1"> - <head xml:id="head540" xml:space="preserve">DE INVENTIONE DIAMETRI <lb/>circuli circunÅ¿cribentis triangulum.</head> - <head xml:id="head541" style="it" xml:space="preserve">Francbino Triuultio.</head> - <p> - <s xml:id="s4376" xml:space="preserve"><emph style="sc">QVod</emph> mihi nunc proponis eÅ¿t triangulum, cuius baÅ¿is cum angulo Å¿ibi op <lb/>poÅ¿ito dantur. </s> - <s xml:id="s4377" xml:space="preserve"><reg norm="Vellesque" type="simple">VellesqÌ;</reg> diametrum circuli apti eum triangulum circnn-<lb/>Å¿cribere inuenire in diÅ¿creto.</s> - </p> - <p> - <s xml:id="s4378" xml:space="preserve">Sit igitur triangulum <var>.a.b.g.</var> cuius baÅ¿is <var>.b.g.</var> Å¿imul cum angulo <var>.a.</var> ei op-<lb/>poÅ¿ito data Å¿it in numeris. </s> - <s xml:id="s4379" xml:space="preserve">Imaginetur ergo circulas circunÅ¿eribens ipÅ¿um triangu-<lb/>lum <var>.b.p.g.q.</var> cuius diameter Å¿it <var>.q.p.</var> perpendicularis eius baÅ¿i <var>.b.g.</var> vnde <var>.b.g.</var> diuiÅ¿a <lb/>erit per æqualia ab ipÅ¿o diametro in puncto <var>.m.</var> per tertiam tertij, protrahatur etiam - <pb o="367" rhead="EPISTOL AE." n="379" file="0379" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0379"/> - e<unsure/> <var>.g.</var> vnde angulus <var>.g.e.q.</var> æqualis erit angulo <var>.b.a.g.</var> portionis, cum duplus Å¿it angulo <lb/><var>q.p.g.</var> medietati anguli ipÅ¿ius portionis ex .19. tertij, ita quod angulus <var>.q.e.g.</var> nobis <lb/>cognitus erit, & Å¿imiliter arcus <var>.g.q.</var> & conÅ¿equenter ar-<lb/>cus <var>.p.g.</var> reÅ¿iduum medij circuli, & Å¿ic <var>.m.g.</var> eius Å¿inus re <lb/> - <anchor type="figure" xlink:label="fig-0379-01a" xlink:href="fig-0379-01"/> - ctus, & etiam chorda <var>.p.g.</var> vt dupla Å¿inus dimidij arcus <var>.<lb/>p.g.</var> & Å¿ic <var>.p.m.</var> eius Å¿inus verÅ¿us, vel vt tertium latus trian <lb/>guli orthogonij <var>.p.g.m.</var> vnde nobis cognita erit propor <lb/>tio ipÅ¿ius <var>.b.g.</var> (quæ dupla eÅ¿t ipÅ¿i <var>.m.g.</var>) ad <var>.m.p.</var> & quia <lb/>productum <var>.p.m.</var> in <var>.m.q.</var> æquale eÅ¿t ei, quod fit ex <var>.b.m.</var> <lb/>in <var>m.g.</var> ex .34. tertij, quapropter nobis cognita erit pars <lb/><var>q.m.</var> quæ cum <var>.p.m.</var> complet totum diametrum <var>.q.p.</var> vn <lb/>de nobis cognita erit proportio ipÅ¿ius <var>.b.g.</var> ad <var>.q.p.</var> qua <lb/>mediante cognoÅ¿cemus diametrum Å¿ecundum partes il <lb/>las quibus propoÅ¿ita Å¿uerit <var>.b.g</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0379-01" xlink:href="fig-0379-01a"> - <image file="0379-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0379-01"/> - </figure> - </div> - <p> - <s xml:id="s4380" xml:space="preserve">Hoc autem problema non in numeris Å¿ed in continuo ab Euclid. ponitur in .32<unsure/>. <lb/>tertij.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head542" style="it" xml:space="preserve">De inuentione alterius trianguli conditionati.</head> - <head xml:id="head543" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4381" xml:space="preserve">QVotieÅ¿cunque etiam inuenire voluerimus triangulum aliquem, puta <var>.n.q.o.</var> <lb/>æqualem triangulo <var>.t.</var> (exempli gratia) propoÅ¿ito, qui habeat angulum <var>.n.</var> æ-<lb/>qualem angalo <var>.a.</var> dato, latera vero continentia ipÅ¿um angulum <var>.n.</var> Å¿int inuicem pro-<lb/>portionata vt <var>.x.</var> et <var>.y.</var> ita faciemus, accipiemus lineam <var>.n.m.</var> cuius volueris magnitu-<lb/>dinis, Å¿upra quam conÅ¿tituemus triangulum <var>.m.n.p.</var> æqualem triangulo <var>.t.</var> hac metho-<lb/>do, hoc eÅ¿t prolungando latus <var>.r.z.</var> trianguli <var>.t.</var> quod Å¿it <var>.r.e.</var> ita vt duplum Å¿it ipÅ¿i <var>.r.z.</var> <lb/>ducendo poÅ¿tea <var>.c.e.</var> habebimus ex .38. primi triangulum <var>.t.</var> eſſe dimidium totius <lb/>trianguli <var>.r.c.e.</var> deÅ¿ignabimus deinde ex .44. dicti Å¿uperficiem <var>.p.n.m.b.</var> parallelo <lb/>grammam <reg norm="æqualemque" type="simple">æqualemqÌ;</reg> triangu <lb/>lo <var>.r.c.e.</var> habentem angulum <var>.<lb/> - <anchor type="figure" xlink:label="fig-0379-02a" xlink:href="fig-0379-02"/> - n.</var> æqualem angulo <var>.a.</var> ducatur <lb/>poÅ¿tea <var>.p.m.</var> & habebimus <reg norm="triam" type="context">triã</reg> <lb/>gulum <var>.m.n.p.</var> æqualem <var>.t.</var> cum <lb/>angulo <var>.n.</var> æquali angulo <var>.a.</var> pro <lb/>ducatur poÅ¿tea <var>.n.p.</var> ita vt <var>.n.K.</var> <lb/>Å¿e habeat .ad <var>.n.m.</var> quemadmo <lb/>dum <var>.x.</var> ad <var>.y.</var> quod erit facilli-<lb/>mum producendo <var>.n.m.</var> et <var>.n.<lb/>K.</var> indeterminatè Å¿i oportuerit, <lb/></s> - <s xml:id="s4382" xml:space="preserve">deinde eas ad æqualitatem Å¿e-<lb/>can<unsure/>do ipÅ¿is <var>.x.</var> et <var>.y.</var> efficiendo <lb/>exempli gratia quod <var>.n.i.</var> Å¿it <lb/>æqualis ipÅ¿i <var>.x.</var> et <var>.n.u.</var> ipÅ¿i <var>.y.</var> du <lb/>cendo poÅ¿tea <var>.u.i.</var> deinde à puncto <var>.m.</var> ducendo <var>.m.K.</var> æquidiÅ¿tanter <var>.u.i.</var> ex .31. <lb/>primi. </s> - <s xml:id="s4383" xml:space="preserve">& Å¿ic habebimus ex .4. Å¿exti proportionem <var>.x.</var> ad <var>.y.</var> eſſe inter <var>.n.K.</var> et <var>.n.</var> - <pb o="368" rhead="IO. BAPT. BENED." n="380" file="0380" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0380"/> - m. inuenies poÅ¿tea ex .9. eiuÅ¿-<lb/> - <anchor type="figure" xlink:label="fig-0380-01a" xlink:href="fig-0380-01"/> - dem lineam aliquam mediam <lb/>proportionalem inter <var>.n.K.</var> et <var>.<lb/>n.p.</var> quæ Å¿it <var>.n.o.</var> duces poÅ¿tea <lb/><var>o.q.</var> parallelam ipÅ¿i <var>.m.K.</var> & ha <lb/>bebis propoÅ¿itum, eo quod <reg norm="cum" type="context">cÅ©</reg> <lb/>Å¿it proportio trianguli <var>.n.m.K.</var> <lb/>ad triangulum <var>.n.m.p.</var> vt <var>.n.K.</var> <lb/>ad <var>.n.p.</var> ex prima Å¿exti, duo <reg norm="triam" type="context">triã</reg> <lb/>guli <var>.m.p.n.</var> et <var>.n.q.o.</var> æquales <lb/>erunt inuicem, ex .17. eiuÅ¿dem <lb/>& ex .9. quinti, & proportio <var>.<lb/>o.n.</var> ad <var>.n.q.</var> erit, vt <var>.x.</var><unsure/> ad <var>.y.</var> ex <num value="11">.<lb/>11.</num> dicti, cum ex .4. Å¿exti Å¿it vt <var>.<lb/>n.k.</var> ad <var>.n.m</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0379-02" xlink:href="fig-0379-02a"> - <image file="0379-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0379-02"/> - </figure> - <figure xlink:label="fig-0380-01" xlink:href="fig-0380-01a"> - <image file="0380-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0380-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head544" style="it" xml:space="preserve">De producto conditionato.</head> - <head xml:id="head545" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4384" xml:space="preserve">PRoponis deinde mihi duas rectas lineas, vni quarum, vis vt aliam quandam di-<lb/>rectè coniungam, ita quod productum huius aggregati in lineam adiunctam <lb/>æquale Å¿it quadrato alterius.</s> - </p> - <p> - <s xml:id="s4385" xml:space="preserve">Vt exempli gratia Å¿i fuerint duæ lineæ <var>.e.d.</var> et <var>.e.f.</var> opor-<lb/><reg norm="teretque" type="simple">teretqÌ;</reg> nos ad lineam <var>.e.f.</var> aliam lineam puta <var>.f.c.</var> vel <var>.e.b.</var> <reg norm="iun gere" type="context">iÅ©Â<lb/> - <anchor type="figure" xlink:label="fig-0380-02a" xlink:href="fig-0380-02"/> - gere</reg>, ita longam, vt productum totius compoÅ¿iti <var>.e.c.</var> vel <var>.<lb/>f.b.</var> in <var>.f.c.</var> vel <var>.e.b.</var> eſſet æquale quadrato ipÅ¿ius <var>.e.d</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0380-02" xlink:href="fig-0380-02a"> - <image file="0380-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0380-02"/> - </figure> - </div> - <p> - <s xml:id="s4386" xml:space="preserve">Hoc enim nu llius eſſet difficultatis, eo quod <reg norm="quotieÅ¿cun- que" type="context">quotieÅ¿cÅ©-<lb/>que</reg> <var>.e.d.</var> coniuncta erit cum <var>.e.f.</var> ad rectos, <reg norm="diuiÅ¿aque" type="simple">diuiÅ¿aqÌ;</reg> per me <lb/>dium à puncto <var>.a.</var> à quo ducta <var>.a.d.</var> deinde Å¿ecundum Å¿emi-<lb/>diametrum <var>.a.d.</var> deÅ¿ignato circulo <var>.b.d.c.</var> & protracta <var>.e.f.</var> <lb/>à qua volueris parte vÅ¿que ad circunferentiam in <reg norm="puncto" type="context">pÅ©cto</reg> <var>.c.</var> <lb/>Å¿eu in puncto <var>.b.</var> habebimus intentum, eò quod Å¿i produ-<lb/>cta fuerit <var>.e.f.</var> etiam ab alia parte, vÅ¿que ad circunferentiam, habebimus <var>.b.e.</var> æqua-<lb/>lem ipÅ¿i <var>.f.c.</var> ex communi conceptu, & productum <var>.e.c.</var> in <var>.e.b.</var> æqualem quadra-<lb/>to ipÅ¿ius <var>.e.d.</var> ex .34. tertij, cum ex .3. eiuÅ¿dem <var>.e.d.</var> medietas Å¿it chordæ arcus dupli <lb/><var>b.d</var>.</s> - </p> - <p> - <s xml:id="s4387" xml:space="preserve">De lapÅ¿u verò lapidis verÅ¿us mundi centrum, dum ipÅ¿um attingere, ac præterire <lb/>poſſet, de quo me interrogas. </s> - <s xml:id="s4388" xml:space="preserve">Dico Nicolaum Tartaleam, nec non FranciÅ¿cum <lb/>Maurolicum rectè Å¿enÅ¿iſſe, malè verò Alexandrum Piccolhomineum, & exemplum <lb/>Maurolici optimum eſſe, quod tamen Å¿i capere non potes, crede Å¿altem authoritati <lb/>bus talium virorum, qui tantum in ijs Å¿cientijs Å¿uperant ipÅ¿um Alexandrum Piccol-<lb/>homineum, quantum à Sole cætera Å¿uperantur aÅ¿tra.</s> - </p> - <p> - <s xml:id="s4389" xml:space="preserve">Lapis igitur ille tranÅ¿iret centrum, <reg norm="reddiretque" type="simple">reddiretqÌ;</reg>, cum diminutione tamen motus im <lb/>preſſi, eo fermè modo vt Å¿cribunt iudicioÅ¿iſſimi illi viri, donec poÅ¿t multas reddi-<lb/>tiones Å¿urÅ¿um, <reg norm="deorÅ¿umque" type="simple">deorÅ¿umqÌ;</reg> quieÅ¿ceret circa centrum mundi. </s> - <s xml:id="s4390" xml:space="preserve">Lucidioris tamen intelli - <pb o="369" rhead="EPISTOL AE." n="381" file="0381" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0381"/> - gentiæ gratia cogita Å¿ilum illum (exempli adducti ab illis doctiſſimis viris) cui pon <lb/>dus appenÅ¿um eÅ¿t, æqualem eſſe axi orizontis, hoc eÅ¿t eius extremitatem immobi-<lb/>lem eſſe in primo mobili, & in ipÅ¿o zenit tui orizontis, </s> - <s xml:id="s4391" xml:space="preserve">tunc arcus motionis ipÅ¿ius la <lb/>pidis per tantum interuallum, quantum eÅ¿t diameter terræ, inÅ¿enÅ¿ibiliter differret à <lb/>linea recta, & cum lapis diÅ¿tans à centro mundi per Å¿emidiametrum terræ, iret re-<lb/><reg norm="diretque" type="simple">diretqÌ;</reg>, vt Å¿cis, ergo idem faceret Å¿i Å¿ilum longius eſſet per dictum terræ Å¿emidia-<lb/>metrum, ita vt poſſetipÅ¿um centrum attingere, nam differentia illa Å¿emidiametri <lb/>terræ, ferè nulla eÅ¿t reÅ¿pectu Å¿emidiametri ipÅ¿ius primi mobilis.</s> - </p> - </div> - </div> - <div type="section" level="3" n="38"> - <div type="letter" level="4" n="1"> - <head xml:id="head546" xml:space="preserve">AN PENTAGONVS AB ALBERTO DVRERO <lb/>deÅ¿criptus æquiangulus Å¿it.</head> - <head xml:id="head547" style="it" xml:space="preserve">Conrado Neubart.</head> - <p> - <s xml:id="s4392" xml:space="preserve">SI non credis Pentagonum ab Alberto Durero Å¿uper datam lineam deÅ¿i-<lb/>gnatum, æquiangulum non eſſe. </s> - <s xml:id="s4393" xml:space="preserve">Fingamus hic Å¿ubiectam Å¿iguram Å¿imi-<lb/>lem ei quæ à Durero ponitur, in qua primò, ducta Å¿it linea <var>.o.a.</var> & habe <lb/>bimus angulum <var>.a.o.b.</var> graduum .60. talium qualium duo recti fuerint gra <num value="360">.<lb/>360.</num> vel .30. talium qualium duo recti fuerint .180. nam ex Å¿uppoÅ¿ito, arcus <var>.a.b.</var> eÅ¿t <lb/>Å¿exta pars totius circunferentiæ, angulus vero <var>.b.o.d.</var> rectus eÅ¿t, eo quod <var>.b.o.q.</var> rectus <lb/>etiam Å¿it, </s> - <s xml:id="s4394" xml:space="preserve">quare angulus <var>.d.o.a.</var> reÅ¿iduus ex recto erit graduum .60. talium, ut rectus <lb/>eÅ¿t .90. angulus verò <var>.o.a.c.</var> erit gra .15. eorundem.</s> - </p> - <p> - <s xml:id="s4395" xml:space="preserve">Ducatur deinde perpendicularis <var>.a.e.</var> ad <var>.o.d.</var> quæ vt Å¿inus anguli <var>.a.o.e.</var> erit par-<lb/>tium .86602. talium qualium <var>.a.o.</var> erit .100000. quæ quidem <var>.o.a.</var> vt chorda arcus <var>.a.<lb/>o.</var> eÅ¿t partium .51762. talium qualium <var>.a.d.</var> vel <var>.a.c.</var> Å¿emidiameter eÅ¿t .100000.</s> - </p> - <p> - <s xml:id="s4396" xml:space="preserve">Nam Å¿inus dimidij arcus <var>.a.o.</var> (exi <lb/>Å¿tente <var>.a.o.</var> graduum .30.) eÅ¿t <reg norm="partium" type="context">partiÅ©</reg> <num value="25881">.<lb/> - <anchor type="figure" xlink:label="fig-0381-01a" xlink:href="fig-0381-01"/> - 25881.</num> ex quo <var>.a.e.</var> erit partium <num value="44827">.<lb/>44827.</num> talium qualium <var>.a.d.</var> erit <lb/>100000. vnde angulus <var>.a.d.o.</var> cuius Å¿i <lb/>nus eÅ¿t <var>.a.e.</var> erit graduum .26. min .38 <lb/>qui quidem angulus, Å¿umptus cum an <lb/>gulo <var>.a.o.d.</var> erit gra .86. min .38. </s> - <s xml:id="s4397" xml:space="preserve">Dem <lb/>pta denique hac Å¿umma ex duobus <lb/>rectis gra .180. reliquum erit gra .93. <lb/>min .22. ideÅ¿t angulus <var>.o.a.d.</var> cui addi <lb/>tus cum fuerit angulus <var>.o.a.c.</var> gra .15. <lb/>talium, habebimus angulum <var>.c.a.d.</var> <lb/>graduum .108. min .22. exuperantem <lb/>verum angulum pentagoni per min <num value="22">.<lb/>22.</num> vel Å¿ic, cum inuentus fuerit angu-<lb/>lus <var>.a.d.o.</var> gra .26. min .38. Å¿i ex vno re <lb/>cto demptus fuerit, relinquetur an-<lb/>gulus <var>.d.a.e.</var> gra .63. min .22. qui qui-<lb/>dem collectus cum fuer it cum angu-<lb/>lo <var>.e.a.o.</var> reÅ¿iduo ex re cto dempto angulo <var>.a.o.e.</var> grad .60. qui <var>.e.a.o.</var> eÅ¿t grad .30. & - <pb o="370" rhead="IO. BAPT. BENED." n="382" file="0382" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0382"/> - e<unsure/>tiam collectus cum angulo <var>.o.a.c.</var> grad .15. hi tres anguli efficient angulum <var>.d.a.c.<lb/>d.</var> ctum grad .108. min .22.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0381-01" xlink:href="fig-0381-01a"> - <image file="0381-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0381-01"/> - </figure> - </div> - <head xml:id="head548" style="it" xml:space="preserve">Examinatio anguli <var>.u.</var></head> - <p> - <s xml:id="s4398" xml:space="preserve">Ducatur <var>.d.n.</var> quam quidem <var>.d.n.</var> cognoÅ¿cemus vt Å¿inus anguli <var>.d.o.n.</var> gra .45. nam <lb/>angulus ei contrapoÅ¿itus <var>.q.o.p.</var> eÅ¿t dimidium recti, </s> - <s xml:id="s4399" xml:space="preserve">quare <var>.d.n.</var> erit partium .70710. <lb/>talium qualium <var>.d.o.</var> fuerit .100000. Å¿ed <var>.d.o.</var> eÅ¿t partium .115270. qualium <var>.a.d.</var> eÅ¿t <num value="100000">.<lb/>100000.</num> nam <var>.e.d.</var> vt Å¿inus anguli <var>.e.a.d.</var> gra .63. min .22. eÅ¿t partium .89389. <var>o.e.</var> ve-<lb/>ro eÅ¿t partium .50000. talium qualium <var>.a.o.</var> eÅ¿t .100000. vt Å¿inus anguli <var>.e.a.o.</var> gra .30. <lb/>Å¿ed vt <var>.a.o.</var> eÅ¿t <reg norm="partium" type="context">partiÅ©</reg> .51762. </s> - <s xml:id="s4400" xml:space="preserve">hoc eÅ¿t vt <var>.a.d.</var> eÅ¿t .100000. ipÅ¿a <var>.o.e.</var> erit <reg norm="partium" type="context">partiÅ©</reg> .25881 <lb/>quæ iuncta cum fuerit cum <var>.e.d.</var> efficiet <var>.d.o.</var> partium .115270. vt dictum eÅ¿t, quapro-<lb/>pter cum <var>.d.n.</var> Å¿it partium .70710. talium qualium <var>.d.o.</var> fuerit .100000. ipÅ¿a <var>.d.n.</var> erit <lb/>partium .81507. talium qualium <var>.d.o.</var> erit .115270. ideÅ¿t qualium <var>.d.a.</var> vel <var>.d.u.</var> erit <lb/>100000. quæ quidem <var>.d.n.</var> eÅ¿t Å¿inus anguli <var>.d.u.n.</var> graduum Å¿cilicet .54. 36. cuius du-<lb/>plum erit gra .109. mi .12. debebattamen eſſe .108. <var>m.o</var>.</s> - </p> - <head xml:id="head549" style="it" xml:space="preserve">Examinatio anguli <var>.d.</var></head> - <p> - <s xml:id="s4401" xml:space="preserve">Accipe angulum <var>.a.d.o.</var> gra .26. <lb/> - <anchor type="figure" xlink:label="fig-0382-01a" xlink:href="fig-0382-01"/> - min .38. vt Å¿upra, cui applica angu-<lb/>lum <var>.o.d.n.</var> gra .45. min <var>.o.</var> Å¿imul cum <lb/>angulo <var>.u.d.n.</var> reÅ¿iduo ex recto gra <lb/>duum .35. minu .24. & conficies an-<lb/>gulum <var>.a.d.u.</var> grad .107. minu .2. & <lb/>habebis propoÅ¿itum, quem tamen <lb/>oportebat eſſe gra .108. min <var>.o</var>.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0382-01" xlink:href="fig-0382-01a"> - <image file="0382-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0382-01"/> - </figure> - </div> - <p> - <s xml:id="s4402" xml:space="preserve">Quod autem omnia rectè Å¿uppu-<lb/>tata Å¿int, ex Å¿umma omnium angulo <lb/>rum patere poteÅ¿t. </s> - <s xml:id="s4403" xml:space="preserve">nam collectis om <lb/>nibus quinque angulis <var>.a.c.d.f.u.</var> Å¿i-<lb/>mul, hoc eÅ¿t grad .108. minu .22. cum <lb/>gra .107. min .2. cum grad .12. effi-<lb/>cient grad .540. min <var>.o.</var> Å¿umma æqua <lb/>lis Å¿ex angulis rectis.</s> - </p> - <pb o="371" rhead="EPISTOL AE." n="383" file="0383" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0383"/> - </div> - </div> - <div type="section" level="3" n="39"> - <div type="letter" level="4" n="1"> - <head xml:id="head550" xml:space="preserve">ALIA DEMONSTRATIO NONÆ, ET DECIMÆ <lb/>Å¿ecundi Euclidis.</head> - <head xml:id="head551" style="it" xml:space="preserve">Petro Catenæ.</head> - <p> - <s xml:id="s4404" xml:space="preserve"><emph style="sc">QVamvis</emph> nona ac decima Å¿ecundi Euclid. aliter à Comandino & Mau-<lb/>rolico demonÅ¿tratæ fuerint, nihilominus mihi etiam viÅ¿um eÅ¿t non nihil <lb/>meo moræ in eas tibi Å¿cribere, vt Å¿enÅ¿ibiliter quoque cognoÅ¿cas il-<lb/>las veras eſſe.</s> - </p> - <p> - <s xml:id="s4405" xml:space="preserve">EÅ¿to linc<unsure/>a.a.b. pro nona propoÅ¿itione, diuiÅ¿a per æqualia in <var>.c.</var> per inæqualia verò <lb/>in <var>.d.</var> quadratum autem <var>.a.d.</var> Å¿it <var>.d.e.</var> quadratum verò <var>.d.b.</var> Å¿it <var>.d.i.</var> quadratum <var>.a.c.</var> <lb/>Å¿it <var>.c.f.</var> & quadratum <var>.c.d.</var> Å¿it <var>.c.K.</var> clarum enim erit <var>.K.h.</var> æqualem exiÅ¿tere ipÅ¿i <var>.a.c.</var> <lb/>Å¿eccetur igitur <var>.e.h.</var> in <var>.g.</var> ita vt <var>.h.g.</var> Ä™qualis exiÅ¿tat ipÅ¿i <var>.K.h.</var> vnde <var>.g.e.</var> æqualls erit <lb/><var>c.d.</var> perficiatur etiam quadratum <var>.h.n.</var> vnde in totali quadrato <var>.a.h.</var> habe bis <reg norm="duplum" type="context">duplÅ©</reg> <lb/>quadrati partis <var>.c.d.</var> nempe <var>.c.K.</var> et <var>.f.g.</var> & quadratum <var>.a.u.</var> cum gnomone <var>.u.g.h.k.</var> <lb/>cui deficit quadratum æquale <var>.d.i.</var> quadrato, vt Å¿int etiam duo quadrata partis <var>.a.c</var>.</s> - </p> - <p> - <s xml:id="s4406" xml:space="preserve">In decima <reg norm="autem" type="wordlist">aÅ©t</reg> propoÅ¿itione, <reg norm="quadratum" type="context">quadratÅ©</reg> totalis lineæ <var>.a.d.</var> Å¿it <var>.d.e.</var> & lineæ <var>.b.d.</var> Å¿it <var>.b.i.</var> et <lb/><var>c.d.</var> Å¿it <var>.d.n.</var> et <var>.a.c.</var> Å¿it <var>.c.f.</var> et <var>.f.e.</var> Å¿it <var>.e.u.</var> vnde <var>.n.u.</var> æquale erit quadrato <var>.b.i.</var> vnde in qua <lb/>drato totali <var>.a.h.</var> videbis duo quadrata æqualia <var>.f.c.</var> et <var>.g.k.</var> partis <var>.a.c.</var> & quadratum <var>.c.<lb/>K.</var> cum gnomone <var>.n.f.e.g.</var> cui addito quadrato <var>.b.i.</var> habebis duplum quadrati partis <var>.<lb/>c.d</var>.</s> - </p> - <figure position="here"> - <image file="0383-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0383-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="40"> - <div type="letter" level="4" n="1"> - <head xml:id="head552" xml:space="preserve">DE STELLA CASSIOPEIÆ.</head> - <head xml:id="head553" style="it" xml:space="preserve">Annibali Raymundo AÅ¿trologo Peritißimo.</head> - <p> - <s xml:id="s4407" xml:space="preserve"><emph style="sc">POstqvam</emph> tua doctiſſima Å¿cripta perlegi, conÅ¿ideraui, quod Å¿i à mul-<lb/>titudine exhalationum in regione elementari acciderit anno .1572. & <lb/>1573. vt totos Å¿ex menÅ¿es ab omnibus per vniuerÅ¿um terrarum orbem <lb/>viÅ¿a fuerit Å¿tella illa, quæ eÅ¿t in angulo Å¿eptentrionali quadrilateri Caſſio - <pb o="372" rhead="IO. BAPT. BENED." n="384" file="0384" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0384"/> - peiæ tam lucida, vt ipÅ¿o lucifero videretur rutilantior <reg norm="atque" type="simple">atq;</reg> cæterarum (abÅ¿que vlla <lb/>aÅ¿pectus diuerÅ¿itate) magis Å¿cintillans. </s> - <s xml:id="s4408" xml:space="preserve">Quî fieri poterat, vt Å¿tellæ quæ ab illa pa-<lb/>rum diÅ¿tant, alioqui multo maiores, non etiam illa clariores apparuerint? </s> - <s xml:id="s4409" xml:space="preserve">Å¿ed Å¿i ali-<lb/>quis diceret eam exhalationem non ita fortaſſe dilatari, vt inter nos, & aliam ali-<lb/>quam Å¿tellam interponeretur. </s> - <s xml:id="s4410" xml:space="preserve">Tunc ego reÅ¿ponderem neceſſariò Å¿equi debere ta-<lb/>lem exhalationem, tantam latitudinem occupare, quod aliquibus populis aliam <lb/><reg norm="aliquam" type="context">aliquã</reg> <reg norm="Å¿tellam" type="context">Å¿tellã</reg> circunuicinãhac ipÅ¿a de qua mentionem Å¿ecimus redderet lucidiorem. <lb/></s> - <s xml:id="s4411" xml:space="preserve">Sed cum hoc perÅ¿pectum fuerit nulli, Å¿equebatur lucem illam ab ipÅ¿is exhalatio-<lb/>nibus elementaribus haud poſſe oriri: </s> - <s xml:id="s4412" xml:space="preserve">quod nobis Å¿cintillatio illa maxima perma-<lb/>gno fuit inditio, Å¿i phas eÅ¿t credere, <reg norm="nam" type="context">nã</reg> quo magis aliquod coeleÅ¿te corpus Å¿cintillat, <lb/>eo longius à nobis diÅ¿tare.</s> - </p> - <p> - <s xml:id="s4413" xml:space="preserve">Verum quoniam efflagitaÅ¿ti à me vt aliquid circa huiuÅ¿ce rei Å¿peculationem tibi <lb/>Å¿cribam, idcirco tibi morem gerere volens paucis Å¿ubiungam.</s> - </p> - <p> - <s xml:id="s4414" xml:space="preserve">ConÅ¿idera primo hanc Å¿ubÅ¿criptam primam figuram, in qua <var>.c.a.e.</var> Å¿ignatur pro <lb/>Globo terreÅ¿tri cuius <var>.i.</var> centrum Å¿it et <var>.u.o.n.</var> pro conuexo ignis, Å¿ed <var>.K.x.s.</var> pro orbe <lb/>octauo <var>.x.</var> autem pro Å¿tella iam Å¿uperius dicta, quæ Å¿emper fuit, eÅ¿t, & erit, quamuis <lb/>cæteris tribus nunc obÅ¿curior Å¿it. </s> - <s xml:id="s4415" xml:space="preserve">Accipiantur deinde duo loca in Å¿uperÅ¿icie terræ, <lb/>quÄ™ Å¿int <var>.c.</var> et <var>.e.</var> diametraliter inuicem oppoÅ¿ita, ita quod circa eorum orizontes poÅ¿ <lb/>Å¿ibile Å¿it Å¿tellam <var>.x.</var> videre, radijs ipÅ¿ius Å¿tellæ mediantibus <var>.x.n.e.</var> et <var>.x.u.c.</var> <reg norm="quorum" type="context">quorÅ©</reg> par-<lb/>tes <var>.n.c.</var> et <var>.u.e.</var> ita breues Å¿int, reÅ¿pectu eorum <reg norm="totorum" type="context">totorÅ©</reg>, vt vix Å¿excenteÅ¿ima pars Å¿it vna <lb/><reg norm="quæque" type="simple">quæq;</reg> illarum, nec non <var>.c.e.</var> ita breuis reÅ¿pectu Å¿emidiametri octauæ Å¿phæræ, quod <lb/>vix Å¿it vna ex partibus decemmillibus, vt Å¿cis, Å¿equitur quod recta terminata ab <var>.u.</var> <lb/>et <var>.n.</var> minor Å¿enÅ¿ibiliter non Å¿it ipÅ¿o terræ diametro <var>.c.e.</var> cum duo hæc interualla ex <lb/>triangulorum Å¿imilitudine Å¿e habeant vt <var>.x.i.</var> ad <var>.x.o.</var> hoc eÅ¿t ferè vt .602. ad .601. vn-<lb/>de anguli <var>.n.e.c.</var> et <var>.u.c.e.</var> à rectis minime differre videbuntur, cum eorum differen-<lb/>tia certo modo minima Å¿it. </s> - <s xml:id="s4416" xml:space="preserve">ductÄ™ poÅ¿tea <reg norm="cum" type="context">cÅ©</reg> fuerint duæ diagonales <var>.e.u.</var> et <var>.n.c.</var> <reg norm="termi- nabunt" type="context">termi-<lb/>nabÅ©t</reg> angulos <var>.n.e.u.</var> et <var>.e.n.c.</var> <reg norm="inuicem" type="context">inuicẽ</reg> ferè Ä™quales, <reg norm="idem" type="context">idẽ</reg> aſſero de angulis <var>.u.c.n.</var> et <var>.e.u.c</var>.</s> - </p> - <p> - <s xml:id="s4417" xml:space="preserve">Supponatur nunc primò tuam exhalationem Å¿ublimatam eſſe ad Å¿upremas par-<lb/>tes elementaris regionis circum circa lineam <var>.o.i</var>: </s> - <s xml:id="s4418" xml:space="preserve">tunc clarum eſſet quod Å¿i ratione hu <lb/>iuÅ¿modi exhalationis Å¿tella <var>.x.</var> ita lucida viÅ¿a fuerit tam aÅ¿picientibus ab <var>.e.</var> quam ab <lb/>c. exhalatio minoris latitudinis quam <var>.u.n.</var> eſſe non poterat, hoc eÅ¿t, quam terræ dia-<lb/>meter, cum idem in longitudine ferè Å¿it, Å¿ed punctum <var>.u.</var> Å¿atis videri poteÅ¿t ab oculo <lb/>in <var>.e.</var> & punctum <var>.n.</var> ab oculo in <var>.c.</var> vt alias tibi probaui, ratione refractionis radiorum <lb/>per diuerÅ¿a diafana tranÅ¿euntium. </s> - <s xml:id="s4419" xml:space="preserve">Nunc producti cum fuerint ij duo radij <var>.e.u.</var> et <var>.c.<lb/>n.</var> vÅ¿que ad octauum orbem ad puncta <var>.s.</var> et <var>.K.</var> reliquum erit nos videre quantitates <lb/>graduum arcus <var>.s.x.</var> et <var>.k.x.</var> Å¿ed <var>.s.x.</var> Å¿ubiacet a ngulo <var>.s.e.x.</var> et <var>.k.x.</var> angulo <var>.k.c.x.</var> qui qui <lb/>quidem anguli nihil differunt Å¿enÅ¿ibiliter ac Å¿i eſſent in centro <var>.i</var>. </s> - <s xml:id="s4420" xml:space="preserve">Et cum Å¿uperius di-<lb/>xerimus angulos <var>.s.e.x.</var> et <var>.k.c.x.</var> Å¿enÅ¿ibiliter mi nime differre ab angulis <var>.c.n.e.</var> et <var>.e.<lb/>u.c.</var> Å¿i cognouerimus quantitatem iÅ¿torum, cognita etiam nobis erit quantitas <lb/>ill orum.</s> - </p> - <p> - <s xml:id="s4421" xml:space="preserve">Cum igitur Å¿emidiameter elementaris regionis maior Å¿it Å¿emidiametro terræ, vt <lb/>33. ad vnum, & cogitata <var>.c.n.</var> vt dicta Å¿emidiameter, quia Å¿enÅ¿ibiliter ab ea minime <lb/>diffe<unsure/>rt, nunc Å¿i Å¿upponatur dicta <var>.n.c.</var> vt <reg norm="baÅ¿istrianguli" type="context">baÅ¿istriãguli</reg> orthogonij eſſe <reg norm="partium" type="context">partiÅ©</reg> .100000 <lb/>& dixerimus Å¿i <var>.c.n.</var> vt partium .33. præbet nobis <var>.c.e.</var> duarum partium, quid nobis <lb/>pręſtabit eadem <var>.c.n.</var> vt partium .100000. vnde proueniet nobis <var>.c.e.</var> vt <reg norm="partium" type="context">partiÅ©</reg> .6060. <lb/>cuius angulus <var>.c.n.e.</var> erit graduum .3. & min .29. ita etiam erit angulus <var>.k.</var> e<unsure/> <var>.x.</var> cuius ar- - <pb o="373" rhead="EPISTOL AE." n="385" file="0385" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0385"/> - c<unsure/>us <var>.s.x.</var> eorundem graduum erit, <lb/> - <anchor type="figure" xlink:label="fig-0385-01a" xlink:href="fig-0385-01"/> - & minutorum. </s> - <s xml:id="s4422" xml:space="preserve">idem dico de arcu <var>.<lb/>x.k</var>. </s> - <s xml:id="s4423" xml:space="preserve">Sed circa dictam Å¿tellam om-<lb/>nes aliæ non diÅ¿tant huiuÅ¿modi in <lb/>teruallo.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0385-01" xlink:href="fig-0385-01a"> - <image file="0385-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0385-01"/> - </figure> - </div> - <p> - <s xml:id="s4424" xml:space="preserve">Nihilominus nec tu nec alij pe <lb/>ripatetici qui hanc Å¿equuti Å¿unt <lb/>opinionem exhalationum, ad Å¿er <lb/>uandam nullitatem diucrÅ¿itatis <lb/>aÅ¿pectus, affirmant poſſe tam lon <lb/>ge à terra <reg norm="aÅ¿cendere" type="context">aÅ¿cẽdere</reg> exhalationes, <lb/>imo nec attingere Å¿upremas tertiÄ™ <lb/>regionis a eris partes, ita ut non <reg norm="in- grediantur" type="context">in-<lb/>grediãtur</reg> Å¿uum <reg norm="igneum" type="context">igneÅ©</reg> orbem, qui <lb/>quidem orbis Å¿ecundum illorum <lb/>opinionem incipit non valde lon-<lb/>gè à ſuperÅ¿icie terræ, vt <reg norm="iam" type="context">iã</reg> in mea <lb/>conÅ¿ideratione contra Antonium <lb/>Bergam probaui, Å¿ed demus, <reg norm="quod" type="simple">ê™</reg> <lb/>dictæ exhalationes aÅ¿cenderint <lb/>per decem Å¿emidiametros terræ, <lb/>diÅ¿currendo poÅ¿tea Å¿ic, cum <var>.c.n.</var> ut <lb/>decem, nobis dat <var>.c.e.</var> vt duo, <reg norm="quid" type="simple">ê—d</reg> <lb/>dabit nobisipÅ¿a <var>.c.n.</var> vt .100000. <lb/>& proueniet nobis <var>.c.e.</var> vt .200000. <lb/>cuius Å¿inus angulus erit gra .11. mi <num value="32">.<lb/>32.</num> & ita erunt anguli <var>.s.e.x.</var> et <var>.k.<lb/>c.x.</var> & Å¿ic eorum arcus <var>.s.x.</var> et <var>.k.x.</var> <lb/>Å¿ed quis vnquam dubitabit <reg norm="quod" type="simple">ê™</reg> in <lb/>tanto interuallo à dicta Å¿tella non <lb/>fint aliæ multæ ipÅ¿a maiores? </s> - <s xml:id="s4425" xml:space="preserve">li-<lb/>neas vero <var>.e.o.r.</var> et <var>.c.o.t.</var> duxi, vt <lb/>videres effectum maioris aÅ¿pectus <lb/>diuerÅ¿itatis ab oculis <var>.e.</var> et <var>.c.</var> in cir-<lb/>culo altitudinis quando <var>.o.</var> fuiſſet <lb/>punctum illud lucidiſſimum, & <lb/>non <var>.x</var>.</s> - </p> - <pb o="374" rhead="IO. BAPT. BENED." n="386" file="0386" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0386"/> - <p> - <s xml:id="s4426" xml:space="preserve">At poterit aliquis mihi obijcere quod cum <var>.i.o.</var> fuiſſet longior <var>.i.e.</var> per decem vi-<lb/>ces tantummodo, exiÅ¿tente oculo in <var>.e.</var> uel <var>.c.</var> per gradus .90. ab <var>.a</var>. </s> - <s xml:id="s4427" xml:space="preserve">tunc punctus <var>.u.</var> vel <lb/>n. ab ipÅ¿o oculo non videretur ob terræ globoÅ¿itatem. </s> - <s xml:id="s4428" xml:space="preserve">Imaginemur igitur à puncto <lb/>u. recta <var>.u.b.</var> tangens quartam <var>.a.e.</var> in puncto <var>.b.</var> vt in Å¿ecunda figura videre eÅ¿t, in qua <lb/>ducantur <var>.c.b</var>: <var>i.b</var>: et <var>.i.u.</var> quæ <var>.i.u.</var> Å¿ecabit arcum <var>.c.</var> <lb/> - <anchor type="figure" xlink:label="fig-0386-01a" xlink:href="fig-0386-01"/> - b. in puncto <var>.p.</var> per æqualia et <var>.c.b.</var> Å¿imiliter in pun-<lb/>cto <var>.y.</var> quod nulli dubium eÅ¿t, cum <var>.c.u.</var> æqualis Å¿it <var>.<lb/>u.b.</var> ex .35. tertij Euclidis, </s> - <s xml:id="s4429" xml:space="preserve">unde ex octaua primi an-<lb/>gulus <var>.c.i.u.</var> æqualis erit angulo <var>.u.i.b.</var> & ideo arcus <var>.<lb/>c.p.</var> æquabitur arcui <var>.p.b.</var> Å¿ed ex .4. primi <var>.c.y.</var> æqua <lb/>lis erit ipÅ¿i <var>.y.b</var>.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0386-01" xlink:href="fig-0386-01a"> - <image file="0386-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0386-01"/> - </figure> - </div> - <p> - <s xml:id="s4430" xml:space="preserve">Nunc Å¿uppoÅ¿ita <var>.c.i.</var> decima parte ipÅ¿ius <var>.c.u.</var> nemi <lb/>ni dubium erit quod cum <var>.u.i.</var> Å¿ubtendatur angulo <lb/>recto <var>.u.c.i.</var> (iam Å¿upra diximus angulum <var>.c.</var> Å¿enÅ¿ibi-<lb/>liter minime differre à recto) ipÅ¿a vt Å¿inus totus erit <lb/>partium .100000. cuius quadratum cum diuiÅ¿um <lb/>fuerit in partes æquales centum & <reg norm="vnam" type="context">vnã</reg>, illarum vna <lb/>æqualis erit quadrto <var>.c.i.</var> reliquÄ™ vero quadrato ip-<lb/>Å¿ius <var>.u.c.</var> ex proportione duplicata quadratorum ad <lb/>eam quam continent eorum latera. </s> - <s xml:id="s4431" xml:space="preserve">Sed quadra-<lb/>tum ipÅ¿ius <var>.u.i.</var> eÅ¿t partium .10000000000. </s> - <s xml:id="s4432" xml:space="preserve">quare <lb/>quadratum <var>.c.i.</var> erit .99009900. cuius radix <var>.c.i.</var> <lb/>erit partium .9950. vnde quadratum ipÅ¿ius <var>.c.u.</var> erit <lb/>partium .9900990100. cuius radix <var>.u.c.</var> erit <reg norm="partium" type="context">partiÅ©</reg> <lb/>99500. vnde angulus <var>.c.i.u.</var> erit graduum .84. & mi <lb/>nu .17. & angulus <var>.c.u.i.</var> qui reÅ¿pondet Å¿inui <var>.c.i.</var> erit <lb/>gra .5. & min .43. cuius duplum, hoc eÅ¿t angulus <var>.c.<lb/>u.b.</var> erit grad .11. min .26. æqualis ferè angulo iam <lb/>Å¿upradicto. </s> - <s xml:id="s4433" xml:space="preserve">Å¿ed <var>.c.y.</var> Å¿inus anguli <var>.c.i.y.</var> erit Å¿imiliter <lb/>partium .99500. talium vt <var>.c.i.</var> Å¿unt .100000. Å¿ed vt <lb/><var>c.i.</var> eÅ¿t partium .9950. </s> - <s xml:id="s4434" xml:space="preserve">tunc <var>.c.y.</var> erit partium .9900 <lb/>hoc eÅ¿t quaÅ¿i decima pars ipÅ¿ius <var>.c.u.</var> </s> - <s xml:id="s4435" xml:space="preserve">quare Å¿i ocu-<lb/>lus in <var>.e.</var> non videbit punctum <var>.u.</var> hoc punctum be-<lb/>ne videbitur ab oculo in <var>.b.</var> abÅ¿que Å¿enÅ¿ibili dimi-<lb/>nutione anguli in puncto <var>.u.</var> vt probauimus.</s> - </p> - </div> - </div> - <div type="section" level="3" n="41"> - <div type="letter" level="4" n="1"> - <head xml:id="head554" xml:space="preserve">DE MAGNITVDINIBVS FIGVRARVM <lb/>iÅ¿operimetrarum.</head> - <head xml:id="head555" style="it" xml:space="preserve">Domino Ioanni Mariæ Agatio.</head> - <p> - <s xml:id="s4436" xml:space="preserve"><emph style="sc">QVamvis</emph> à Theone Å¿upra Ptolomei AlmageÅ¿tum Å¿ufficienter traditum <lb/>Å¿it de magnitudinibus figurarum iÅ¿opetimetrarum, nihilominus vt tibi <lb/>morem geram, ea nunc Å¿cribo, quæ mihi in mentem venerunt contra <lb/><reg norm="Alexandrum" type="context">Alexãdrum</reg> Piccolhomineum, <reg norm="antequam" type="context">antequã</reg> aliquid ipÅ¿ius Theonis vidiſſem - <pb o="375" rhead="EPISTOLAE." n="387" file="0387" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0387"/> - Alexander Piccolhomineus in libro primo de mundi Å¿phæra vbi tractat de <reg norm="caeliro- tunditate" type="simple">cÄ™liro-<lb/>tunditate</reg>, ita inquit.</s> - </p> - <quote xml:lang="it"> - <s xml:id="s4437" xml:space="preserve">Oltre di queÅ¿to, douendo il decimo cielo contenere & in Å¿e chiudere tutte le co-<lb/>Å¿e, è conueneuol coÅ¿a il penÅ¿are, che foſſe fatto di quella più capace figura che eÅ¿-<lb/>Å¿er poſſa, la qual è la figura rotunda, però che Å¿i può trar da molti luoghi d'Euclide <lb/>che Å¿i come Å¿e noi ciimmagineremo più figure Å¿uperficiali talmente che tutte le li-<lb/>nee de l'vna congionte inÅ¿ieme, Å¿ieno vguali à tutte le linee pur inÅ¿iememente com <lb/>poÅ¿te di qual Å¿i voglia de l'altre figure, ne Å¿eguirà , che quella figura Å¿arà più capa-<lb/>ce la qual haurà manco angoli, & quella capaciſſima che Å¿arà ſenza alcuno come è <lb/>la figura circolare, & c.</s> - </quote> - <p> - <s xml:id="s4438" xml:space="preserve">Cogitemus igitur primò de triangulo æquilate-<lb/>ro & quadrato iÅ¿operimetris, Å¿it enim triangulus æ-<lb/>quilaterus <var>.o.b.g.</var> quadratum verò <var>.b.l.</var> quorum pe-<lb/>riferiæ inuicem æquales Å¿int. </s> - <s xml:id="s4439" xml:space="preserve">Dico quadratum ma-<lb/> - <anchor type="figure" xlink:label="fig-0387-01a" xlink:href="fig-0387-01"/> - ioris Å¿uperficiei eſſe ipÅ¿o triangulo. </s> - <s xml:id="s4440" xml:space="preserve">Accipio pri-<lb/>mum lineam <var>.f.h.</var> eiuÅ¿dem longitudinis quæ vnius <lb/>periferiæ dictarum figurarum, quam punctis <var>.r.K.</var> <lb/>mediantibus diuido in tres Ä™quas partes, in quatuor <lb/>verò mediantibus punctis <var>.l.x.i.</var> vnde proportio to-<lb/>tius <var>.f.h.</var> ad <var>.K.h.</var> erit vt <var>.l.h.</var> ad <var>.i.h.</var> ideÅ¿t tripla, & per <lb/>16. quinti erit <var>.f.h.</var> ad <var>.l.h.</var> vt <var>.k.h.</var> ad <var>.i.h.</var> per .19. verò <lb/><var>f.h.</var> ad <var>.f.l.</var> vt <var>.K.h.</var> ad <var>.K.i.</var> Å¿ed <var>.f.l.</var> eÅ¿t quarta pars ip-<lb/>Å¿ius <var>.f.h.</var> ergo <var>.k.i.</var> erit quarta pars ipÅ¿ius <var>.k.h</var>. </s> - <s xml:id="s4441" xml:space="preserve"><reg norm="Conium" type="context">ConiÅ©</reg> <lb/>gantur enim ambo iÅ¿tæ figuræ vt hic inferius vides, <lb/>vnde <var>.a.g.</var> erit quarta pars ipÅ¿ius <var>.b.g.</var> diuiÅ¿a poÅ¿tea <var>.<lb/>b.g.</var> per æqualia in <var>.c.</var> erit <var>.a.c.</var> æqualis <var>.a.g</var>. </s> - <s xml:id="s4442" xml:space="preserve">Ducatur <lb/>deinde <var>.o.c.</var> quÄ™ per .8. primi, nec <reg norm="non" type="context">nõ</reg> ex definitione, <lb/>perpendicularis erit ipÅ¿i <var>.b.g.</var> ergo etiam <reg norm="quadratum" type="context">quadratÅ©</reg> <lb/>b q. Å¿upra <var>.b.g.</var> <reg norm="producoque" type="simple">producoqÌ;</reg> <var>.o.c.</var> vÅ¿que ad <var>.m.</var> nam nul <lb/>li dubium eÅ¿t quin <var>.o.c.</var> breuior Å¿it <var>.o.g.</var> ex .18. vel .48 <lb/>primi cui æquatur <var>.q.g.</var> diuido etiam <var>.c.m.</var> per æqua <lb/>lia in puncto <var>.e.</var> <reg norm="ducoque" type="simple">ducoqÌ;</reg> <var>t.e.p.</var> æquidiÅ¿tantem <var>.b.g.</var> <lb/>vnde habebimus duo quadrata <var>.e.g.</var> et <var>.e.b.</var> Å¿ed <lb/>quadratum <var>.b.l.</var> æquatur quadrato ipÅ¿ius <var>.c.a.</var> <lb/>cum duplo illius quod fit ex <var>.b.c.</var> in <var>.c.g.</var> vt patet <lb/>ex .9. Å¿ecundi, hoc eÅ¿t æquatur quadrato <var>.c.a.</var> & re-<lb/>ctangulo <var>.t.g</var>. </s> - <s xml:id="s4443" xml:space="preserve">Deinde vt Å¿e habet <var>.p.g.</var> ad <var>.o.e.</var> ita Å¿e habet <var>.u.p.</var> ad <var>.u.e.</var> ex Å¿imilitudine <lb/>triangulorum. </s> - <s xml:id="s4444" xml:space="preserve">Sed <var>.p.g.</var> maior eÅ¿t ipÅ¿a <var>.o.e.</var> cum <var>.p.g.</var> æqualis Å¿it <var>.e.m.</var> </s> - <s xml:id="s4445" xml:space="preserve">quare triangu-<lb/>lus <var>.u.g.p.</var> maior erit triangulo <var>.o.e.u.</var> ex .17. Å¿exti. </s> - <s xml:id="s4446" xml:space="preserve">Similiter dico maiorem eſſe trian <lb/>gulum <var>.b.d.t.</var> triangulo <var>.e.o.d.</var> vnde Å¿equitur rectangulum <var>.t.g.</var> maiorem eſſe triangu-<lb/>lo <var>.b.o.g.</var> Å¿ed quadratum <var>.b.l.</var> eÅ¿t etiam maior ipÅ¿o rectangulo <var>.t.g.</var> ex quadrato ipÅ¿ius <lb/><var>c.a.</var> vt diximus, tanto igitur maior erit triangulo <var>.b.o.g</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0387-01" xlink:href="fig-0387-01a"> - <image file="0387-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0387-01"/> - </figure> - </div> - <pb o="376" rhead="IO. BABPT. BENED." n="388" file="0388" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0388"/> - <p> - <s xml:id="s4447" xml:space="preserve">Poſſumus etiam probare quod periferia quadrati æqualis triangulo æquilatero <lb/>minor Å¿it periferia ipÅ¿ius trianguli æquilateri. </s> - <s xml:id="s4448" xml:space="preserve">Cogita triangulum æquilaterum hic <lb/>Å¿ubÅ¿criptum <var>.d.l.q.</var> cuius baÅ¿is <var>.l.q.</var> diuiÅ¿a Å¿it per æqualia à perpendiculari <var>.d.o.</var> <reg norm="deÅ¿cri- ptumque" type="context simple">deÅ¿criÂ<lb/>ptÅ©qÌ;</reg> Å¿it rectangulum <var>.o.g.</var> quod æquale erit triangulo <var>.d.l.q.</var> Å¿ed periferia trianguli <lb/>maior eÅ¿t periferia rectanguli, nam <var>.l.q.</var> æqualis eÅ¿t <var>.o.q.</var> cum <var>.d.g.</var> Å¿ed <var>.q.d.</var> maior eÅ¿t <var>.o.<lb/>d.</var> ex .18. primi, vnde <var>.l.d.</var> maior etiam <var>.q.g.</var> cum ex .34. dicti latera oppoÅ¿ita ipÅ¿ius re <lb/>ctanguli Å¿int inuicem æqualia, accipiamus poÅ¿tea <var>.e.c.</var> æqualem <var>.o.d.</var> et <var>.c.h.</var> indire-<lb/>ctum æqualem <var>.o.q.</var> circa quem diametrum <var>.e.h.</var> intelligatur circulus <var>.e.i.h.k.</var> et. à punÂ<lb/>cto <var>.c.</var> dirigatur perpendicularis <var>.k.i.</var> ad <var>.e.h.</var> vnde ex .3. tertij <var>.c.i.</var> æqualis erit <var>.c.k.</var> & ex <lb/>34. quod fit ex <var>.c.i.</var> in <var>.c.k.</var> hoc eÅ¿t quadratum ipÅ¿ius <var>.c.i.</var> æquale erit ei quod fit .ex <var>.e.c.</var> <lb/>in <var>.c.h.</var> hoc eÅ¿t rectangulo <var>.g.o.</var> hoc eÅ¿t triangulo <var>.d.l.q.</var> Å¿ed <var>.e.h.</var> eÅ¿t dimidium perife-<lb/>riÄ™ ipÅ¿ius rectanguli <var>.g.o.</var> quæ minor eÅ¿t di midio periferiæ trianguli <var>.d.l.q.</var> vt vidimus <lb/>et <var>.i.k.</var> eÅ¿t dimidium periferiÄ™ quadrati ipÅ¿ius <var>.i.c.</var> & minor etiam ipÅ¿a <var>.e.h.</var> ex .14. tertij <lb/></s> - <s xml:id="s4449" xml:space="preserve">quare verum eÅ¿t propoÅ¿itum.</s> - </p> - <figure position="here"> - <image file="0388-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0388-01"/> - </figure> - <p> - <s xml:id="s4450" xml:space="preserve">Sed quando periferiæ Å¿unt inuicem æquales, poſſumus etiam breuiter videre id <lb/>quod Å¿upradiximus, hoc eÅ¿t, quod quadratum, maius Å¿it triangulo æquilatero. </s> - <s xml:id="s4451" xml:space="preserve">Nam <lb/>cum <var>.b.g.</var> Å¿eÅ¿quitertia Å¿it ad <var>.b.a.</var> ergo <var>.b.g.</var> erit vt .4. et <var>.b.a.</var> ut .3. vnde <var>.b.q.</var> erit vt .16 <lb/>et <var>.b.l.</var> vt .9. et <var>.c.q.</var> vt .8. </s> - <s xml:id="s4452" xml:space="preserve">quare <var>.b.l.</var> maius erit ipÅ¿o <reg norm="rectangulo" type="context">rectãgulo</reg> <var>.c.q.</var> Å¿ed <var>.c.q.</var> maius eÅ¿t <reg norm="triam" type="context">triã</reg> <lb/>gulo <var>.b.o.g.</var> cum <var>.q.g.</var> quæ æqualis eÅ¿t <var>.o.g.</var> maior Å¿it <var>.o.c.</var> ex .18. vel penultima primi, <lb/>nam Å¿i <var>.q.g.</var> æqualis eſſet <var>.o.c.</var> </s> - <s xml:id="s4453" xml:space="preserve">tunc <var>.c.q.</var> æqualis eſſet triangulo <var>.b.o.g.</var> ex .41. primi.</s> - </p> - <p> - <s xml:id="s4454" xml:space="preserve">Alia etiam via maiores noÅ¿tri vÅ¿i Å¿unt quæ generalis eÅ¿t vt in Theone Å¿upra Al-<lb/>mageÅ¿tum videre eÅ¿t, medijs perpendicularibus à centris ad latera figurarum, Å¿ed <lb/>quia <reg norm="differentia" type="context">differẽtia</reg> longitudinum ipÅ¿arum perpendicularium alio medio inueniri poteÅ¿t, <lb/>eo quo ipÅ¿i vÅ¿i Å¿unt, prætermittere nolo quin tibi Å¿cribam.</s> - </p> - <p> - <s xml:id="s4455" xml:space="preserve">Ego enim ita diÅ¿curro.</s> - </p> - <p> - <s xml:id="s4456" xml:space="preserve">Sint duæ figuræ iÅ¿operimetrÄ™ æquilaterÄ™ & æquiangulæ, puta primò trian-<lb/>gulum & quadratum quorum centra Å¿int <var>.e.</var> et <var>.o.</var> à quibus centris ad latera Å¿int per-<lb/>pendiculares <var>.e.n.</var> et <var>.o.u.</var> vnde <var>.n.</var> et <var>.u.</var> diuident latera per æqualia vt Å¿cis, ducantur <lb/>poÅ¿tea <var>.e.t.</var> et <var>.o.a.</var> ad angulos dictorum laterum, vnde habebimus angulum <var>.o.a.u.</var> <reg norm="di- midium" type="context">di-<lb/>midiÅ©</reg> recti, et <var>.e.t.n.</var> tertia pars vnius recti, vt ex te ipÅ¿o videre potes, </s> - <s xml:id="s4457" xml:space="preserve">quare angulus - <pb o="377" rhead="EPISTOLAE." n="389" file="0389" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0389"/> - a. Å¿eÅ¿quialter erit angulo <var>.t.</var> quod vt clarius videas cogita lineam <var>.b.d.</var> cuius medietas <lb/>Å¿it <var>.c.d.</var> tertia verò pars illius Å¿it <var>.g.d.</var> </s> - <s xml:id="s4458" xml:space="preserve">tunc dico <var>.c.d.</var> Å¿eÅ¿quialteram eſſe ipÅ¿i <var>.g.d.</var> Å¿it enim <lb/><var>f.d.</var> duplum ipÅ¿ius <var>.g.d.</var> </s> - <s xml:id="s4459" xml:space="preserve">quare <var>.f.d.</var> erunt duæ tertiæ totius lineÄ™ <var>.b.d.</var> & quia eadem pro <lb/>portio eÅ¿t totius <var>.b.d.</var> ad <var>.c.d.</var> quæ <var>.f.d.</var> ad <var>.g.d.</var> ergo permutando eadem erit totius <var>.b.<lb/>d.</var> ad <var>.f.d.</var> quæ <var>.c.d.</var> ad <var>.g.d</var>. </s> - <s xml:id="s4460" xml:space="preserve">Sed <var>.b.d.</var> ad <var>.f.d.</var> Å¿eÅ¿quialtera eÅ¿t, verum igitur erit quod an-<lb/>gulus <var>.a.</var> Å¿eÅ¿quialter Å¿it ipÅ¿i <var>.t.</var> deinde <var>.t.n.</var> eÅ¿t Å¿eÅ¿quitertia ipÅ¿i <var>.a.u.</var> vt Å¿uperius vidimus .<lb/>in eorum duplis. </s> - <s xml:id="s4461" xml:space="preserve">Å¿cimus etiam <var>.n.e.</var> eſſe dimidium ipÅ¿ius <var>.t.e.</var> co quod cum <var>.e.t.n.</var> Å¿it <lb/>tertia pars vnius recti, angulus, <var>t.e.n.</var> erit duo tertia vnius recti, vnde <var>.e.n.</var> erit latus. <lb/>exagoni æquilateris inÅ¿criptibilis circulo cuius diameter Å¿it <var>.e.t.</var> </s> - <s xml:id="s4462" xml:space="preserve">quare <var>.e.t.</var> dupla erit <lb/>ipÅ¿i <var>.e.n.</var> in longitudine, Å¿ed quadrupla in potentia: </s> - <s xml:id="s4463" xml:space="preserve">t.n. vero tripla in potentia ipÅ¿i <var>.n.<lb/>e.</var> ex penultima primi, quæ omnia etiam ex .8. tertijdecimi. Eucli. elicere potes, Å¿ed <lb/><var>c.n.</var> erat Å¿exquitertia ipÅ¿i <var>.a.u.</var> in longitudine, hoc eÅ¿t ipÅ¿i <var>.o.u.</var> nam <var>.o.u.</var> æqualis eÅ¿t ipÅ¿i <lb/><var>a.u</var>. </s> - <s xml:id="s4464" xml:space="preserve">quare <var>.n.t.</var> erit minus quam dupla in potentia ipÅ¿i <var>.o.u.</var> hoc eÅ¿t, vt .16. ad .9. ergo <lb/>maior proportio erit ipÅ¿ius <var>.t.n.</var> in potentia ad <var>.n.e.</var> quam ad <var>.o.u.</var> </s> - <s xml:id="s4465" xml:space="preserve">quare etiam in lon <lb/>gitudine, maior proportio erit ipÅ¿ius <var>.t.n.</var> ad <var>.n.e.</var> quam ad <var>.o.u.</var> vnde <var>.o.u.</var> longior erit <lb/>ipÅ¿a <var>.n.e.</var> quod eÅ¿t propoÅ¿itum.</s> - </p> - <p> - <s xml:id="s4466" xml:space="preserve">Sed Å¿i <var>.o.a.u.</var> eſſet pentagonus æquilaterus & æquiangulus, Å¿imiliter probabo per-<lb/>pendicularem <var>.o.u.</var> longiorem eſſe <var>.n.e.</var> ipÅ¿ius trianguli æquilateri, dummodo Å¿int iÅ¿o-<lb/>perimetrÄ™. </s> - <s xml:id="s4467" xml:space="preserve">Sit enim <var>.a.u.</var> dimidium lateris pentagoni ex Å¿uppoÅ¿ito, cuius centrum Å¿it <lb/>o. </s> - <s xml:id="s4468" xml:space="preserve">tunc proportio <var>.t.n.</var> ad <var>.a.u.</var> erit Å¿uperbipartienstertias, vt ex ordine iam hic Å¿upradi <lb/>cto à te facillimè elicere potes, hoc eÅ¿t, vt .5. ad .3. et <var>.a.u.</var> minor erit <var>.o.u.</var> eo quod <lb/>angulus <var>.o.</var> minor erit angulo <var>.a.</var> nam angulus <var>.o.</var> erit quinta pars <reg norm="duorum" type="context">duorÅ©</reg> rectorum, hoc <lb/>eÅ¿t duæ quintæ vnius recti, vnde angulus <var>.a.</var> reÅ¿iduum vnius recti erit tres quin-<lb/>tæ vnius recti, </s> - <s xml:id="s4469" xml:space="preserve">quare angulus <var>.a.</var> maior ericangulo <var>.o.</var> & conÅ¿equenter latus <var>.o.u.</var> ma-<lb/>ius latere <var>.a.u.</var> Å¿ed <var>.t.n.</var> minor eÅ¿t quam tripla in potentia ad <var>.a.u.</var> eo quod erit vt .25. <lb/>ad .9. cum in longitudine Å¿it vt .5. ad .3. Å¿ed dicta <var>.t.n.</var> tripla eÅ¿t in potentia ad <var>.e.n.</var> qua-<lb/>re <var>.a.u.</var> maior erit ipÅ¿a <var>.e.n.</var> Å¿ed <var>.o.u.</var> maior eÅ¿t ipÅ¿a <var>.a.u.</var> vt diximus, igitur multo magis <var>.<lb/>o.u.</var> maior eÅ¿t ipÅ¿a <var>.a.u.</var> vt <reg norm="diximus" type="simple">diximê°</reg> & <reg norm="conÅ¿equenter" type="context context">cõſequẽter</reg> multo magis <var>.o.u.</var> maior erit ipÅ¿a <var>.n.e</var>.</s> - </p> - <p> - <s xml:id="s4470" xml:space="preserve">QuotieÅ¿cunque enim cognoÅ¿cimus proportionem anguli <var>.o.</var> ad angulum <var>.a.</var> quod <lb/>quidem facillimum eÅ¿t, nec non proportionem <var>.t.n.</var> ad <var>.a.u.</var> quod, etiam illico cogno-<lb/>Å¿citur, </s> - <s xml:id="s4471" xml:space="preserve">tunc exÅ¿cientia cordarum & arcuum omnia etiam facillimè innueniuntur. <lb/></s> - <s xml:id="s4472" xml:space="preserve">Verum circa <reg norm="triangulum" type="context">triangulÅ©</reg> æquilaterum, & pentagonum, alium <reg norm="modum" type="context">modÅ©</reg> inueni, Å¿ed aliquan <lb/>tulum prolixiorem.</s> - </p> - <figure position="here"> - <image file="0389-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0389-01"/> - </figure> - <pb o="378" rhead="IO. BAPT. BENED." n="390" file="0390" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0390"/> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head556" style="it" xml:space="preserve">De incommenÅ¿ur abilitate, in longitudine perpendicu-<lb/>laris trianguli æquilateri cum eiuÅ¿dem latere.</head> - <head xml:id="head557" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4473" xml:space="preserve">ID quod à me poÅ¿tulas eÅ¿t omnino impoſſibile, velles enim duos numeros inueni <lb/>re inter Å¿e ita Å¿e habentes, vt Å¿e habent perpendicularis in triangulo æquilatero <lb/>cum vno eius laterum, quod vero hoc fieri non poſſit, conÅ¿idera in figura præcedenti <lb/>triangulum æquilaterum <var>.d.l.q.</var> cuius perpendicularis Å¿it <var>.d.o.</var> quæ diuidit <var>.l.q.</var> per <lb/>æqualia in <var>.o.</var> vnde ex .4. Å¿ecundi Euclidis, quadratum <var>.l.q.</var> (ideÅ¿t <var>.d.q.</var>) quadruplum <lb/>erit quadrato <var>.o.q.</var> & ex penultima primi Ä™quale quadratis <var>.d.o.</var> et <var>.o.q.</var> </s> - <s xml:id="s4474" xml:space="preserve">quare erit Å¿eÅ¿-<lb/>quitertium quadrato ipÅ¿ius <var>.d.o.</var> & ita quadratum <var>.d.o.</var> erit triplum quadrato ipÅ¿ius <var>.<lb/>o.q.</var> hæe autem proportiones non Å¿unt vt numeri quadrati ad numerum quadratum <lb/>quod Å¿i ita fuiſſent, Å¿equeretur ternarium numerum eſſe quadratum ex .22. octaui. <lb/></s> - <s xml:id="s4475" xml:space="preserve">Cum igitur non Å¿int vt numeri quadrati ad numerum quadratum, Å¿equitur ex Å¿epti-<lb/>ma decimi <var>.d.o.</var> eſſe incommenÅ¿urabilem ipÅ¿i <var>.l.q.</var> Å¿eu <var>.d.q.</var> in longitudine.</s> - </p> - <p> - <s xml:id="s4476" xml:space="preserve">Vel dicamus ita, proportio quadrati ipÅ¿ius <var>.l.q.</var> ad quadratum ipÅ¿ius <var>.o.d.</var> eÅ¿t in ge <lb/>nere Å¿uperparticulari, cum Å¿it Å¿eÅ¿quitertia, vnde quadratum ipÅ¿ius <var>.d.o.</var> numeris da-<lb/>ri non poteÅ¿t, eo quod Å¿i dabilis fuiſſet, Å¿equeretur, quod inter quadratum ipÅ¿ius. l<unsure/> <var>.<lb/>q.</var> & ipÅ¿ius <var>.d.o.</var> eſſet aliquis numerus medius proportionalis ex .16. octaui, vnde ex <lb/>octaua eiuÅ¿dem vnitas diuiÅ¿ibilis eſſet, quod fieri non poteÅ¿t.</s> - </p> - <figure position="here"> - <image file="0390-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0390-01"/> - </figure> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head558" style="it" xml:space="preserve">De triangulo & Pentagono æquilatero</head> - <head xml:id="head559" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4477" xml:space="preserve">MOdum quem conÅ¿ideraui circa triangulum æquilaterum & pentagonum, vt <lb/>tibi Å¿ignificaui ita Å¿e habet.</s> - </p> - <p> - <s xml:id="s4478" xml:space="preserve">Probandum primò eÅ¿t pentagonum altiorem eſſe triangulo Å¿ibi iÅ¿operimetro. <lb/></s> - <s xml:id="s4479" xml:space="preserve">Iam tibi notam puto proportionem lateris trianguli ad latus pentagoni eſſe vt .5. <lb/>ad .3.</s> - </p> - <p> - <s xml:id="s4480" xml:space="preserve">Sit igitur pentagonus <var>.b.d.m.g.v.</var> cuius periferia diÅ¿tenta Å¿it <var>.K.z.</var> baÅ¿is autem <var>.m.<lb/>g.</var> bifariam diuiÅ¿a Å¿it in puncto <var>.a.</var> <reg norm="ductaque" type="simple">ductaqÌ;</reg> <var>.a.b</var>: <var>b.g.</var> et <var>.b.m.</var> clarum erit <var>.a.b.</var> perdicu-<lb/>larem eſſe ad <var>.m.g.</var> ex .8. primi Eucli. cum <var>.b.m.</var> et <var>.b.g.</var> (baÅ¿es triangulorum <var>.b.d.m.</var> - <pb o="379" rhead="EPISTOLAE." n="391" file="0391" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0391"/> - et <var>.b.u.g.</var>) Å¿int inuicem æquales ex .4. eiuÅ¿dem.</s> - </p> - <p> - <s xml:id="s4481" xml:space="preserve">Accipiatur deinde vel intelligatur <var>.g.p.</var> æqualis duabus te<unsure/>rtijs ipÅ¿ius <var>.a.g.</var> ducaturÂ<lb/>qÌue <var>.b.p.</var> quam probabo maiorem eſſe duplo ipÅ¿ius <var>.a.p.</var> vnde maior erit latere ipÅ¿ius <lb/>trigoni æquilateris, cuius dimidium eÅ¿t <var>.a.p.</var> Å¿cimus enim ipÅ¿um latus Å¿e habere ad <var>.m.<lb/>g.</var> vt quinque ad .3. ita etiam <var>.a.p.</var> ad <var>.a.g.</var> vt diximus.</s> - </p> - <p> - <s xml:id="s4482" xml:space="preserve">Cum <reg norm="autem" type="context">autẽ</reg> angulus <var>.a.b.g.</var> Å¿it quarta pars anguli <var>.b.g.a.</var> ex .10. quarti & quinta pars <lb/>vnius recti ex .32. primi, dictus angulus erit graduum .18. et <var>.a.g.</var> erit partium .30902. <lb/>et <var>.a.b.</var> partium .95015 et <var>.a.p.</var> 51503. vnde ex penultima primi latus <var>.b.p.</var> erit par-<lb/>tium .108075. duplum vero ipÅ¿ius <var>.a.p.</var> erit .103006. latus igitur dicti trigoni, quod <lb/>ab <var>.p.</var> erigitur, Å¿ecabit perpendicularem <var>.a.b.</var> Å¿ub <var>.b.</var> hoc eÅ¿t inter <var>.b.</var> et <var>.a.</var> ex penultima <lb/>primi. </s> - <s xml:id="s4483" xml:space="preserve">Finiatur enim triangulus æquicrurus <var>.b.q.p.</var> quem probaui maiorem eſſe æ-<lb/>quilatero iÅ¿operimetro pentagono propoÅ¿ito, <reg norm="ducaturque" type="simple">ducaturqÌ;</reg> <var>.u.p.</var> ducatur etiam <var>.u.n.</var> pa-<lb/>rallela ipÅ¿i <var>.b.g.</var> quæ concludet triangulum <var>.g.u.n.</var> Å¿imilem triangulo <var>.m.b.g.</var> eo quod <lb/>cum angulus <var>.m.b.g.</var> æqualis Å¿it angulo <var>.b.g.u.</var> ex .16. tertij, per .27. primi <var>.m.b.</var> et <var>.g.u.</var> <lb/>erunt inuicem <reg norm="æquidiÅ¿tantes" type="context">æquidiÅ¿tãtes</reg>, vnde angulus <var>.b.m.g.</var> æqualis erit angulo <var>.u.g.n.</var> et. ex .29. <lb/>angulus <var>.g.u.n.</var> æqualis erit angulo <var>.u.g.b</var>. </s> - <s xml:id="s4484" xml:space="preserve">quare etiam angulo <var>.g.b.m.</var> & angulus <var>.u.n.<lb/>g.</var> angulo <var>.b.g.m.</var> ex .32. eiuÅ¿dem, </s> - <s xml:id="s4485" xml:space="preserve">vnde ex .4. Å¿exti proportio <var>.g.n.</var> ad <var>.g.m.</var> erit .vt <var>.g.u.</var> <lb/>ad <var>.m.b.</var> Å¿ed cum <var>.g.u.</var> maior Å¿it dimidio ipÅ¿ius <var>.b.g.</var> ex .20. primi, hoc eÅ¿t maior dimi-<lb/>dio ipÅ¿ius <var>.b.m.</var> ergo <var>.g.n.</var> etiam maior erit ipÅ¿a <var>.g.a.</var> quapropter maior erit ipÅ¿a <var>.g.p.</var> <lb/>cum <var>.g.p.</var> minor Å¿it ipÅ¿a <var>.g.a.</var> ex hypotheÅ¿i, ducta deinde cum fuerit <var>.b.n.</var> habebimus <lb/>triangulum <var>.b.n.g.</var> <reg norm="æqualem" type="context">æqualẽ</reg> triangulo <var>.b.u.g.</var> & <reg norm="maiorem" type="context">maiorẽ</reg> <reg norm="triangulo" type="context">triãgulo</reg> <var>.b.p.g.</var> ex prima Å¿exti <lb/>vel quia totum maius eÅ¿t Å¿ua parte. </s> - <s xml:id="s4486" xml:space="preserve">Triangulus igitur <var>.b.u.g.</var> maior eÅ¿t triangu-<lb/>lo <var>.b.p.g</var>. </s> - <s xml:id="s4487" xml:space="preserve">quare triangulus <var>.b.u.o.</var> maior erit triangulo <var>.g.o.p.</var> ex communi conceptu, <lb/>idem infero ab alia parte dictarum figurarum. </s> - <s xml:id="s4488" xml:space="preserve">Quare pentagonus <var>.b.d.m.g.u.</var> maior <lb/>erit triangulo <var>.b.q.p.</var> quem probauimus maiorem eſſe triangulo æquilatero Å¿ibi iÅ¿o-<lb/>perimetro.</s> - </p> - <figure position="here"> - <image file="0391-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0391-01"/> - </figure> - </div> - <div type="letter" level="4" n="4"> - <head xml:id="head560" style="it" xml:space="preserve">Comparatio periferiarum quadrati & trianguli aquilateri circunÅ¿criptorum ab eodem circulo.</head> - <head xml:id="head561" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4489" xml:space="preserve">QVod autem periferia quadrati in eodem circulo inÅ¿cripti, in quo Å¿it triangu-<lb/>lus æquilaterus, longior Å¿it periferia ipÅ¿ius trianguli æquilateri, abÅ¿que vllo - <pb o="380" rhead="IO. BAPT. BENED." n="392" file="0392" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0392"/> - negotio cordarum & arcuum poſſumus geometricè demonÅ¿trare quod valde de-<lb/>Å¿ideras.</s> - </p> - <p> - <s xml:id="s4490" xml:space="preserve">Quapropter Å¿it circulus <var>.b.a.e.q.</var> in quo Å¿it <reg norm="triangulum" type="context">triangulÅ©</reg> æquilaterum <var>.b.e.n.</var> & quadra <lb/>tum <var>.b.a.q.u.</var> cuius periferiam probabo longiorem eſſe periferia trianguli. </s> - <s xml:id="s4491" xml:space="preserve">Sit enim <lb/>diameter circuli <var>.b.q.</var> qui etiam erit diameter quadrati, vt à te Å¿cire potes. </s> - <s xml:id="s4492" xml:space="preserve">Sit etiam <lb/><reg norm="punctum" type="context">punctÅ©</reg> <var>.b.</var> commune tam anguli quadrati quam trianguli. </s> - <s xml:id="s4493" xml:space="preserve">vnde Å¿equitur quod dictus <lb/>diameter Å¿ecabit latus <var>.n.e.</var> trianguli ad rectos & per æqualia in <var>.t</var>. </s> - <s xml:id="s4494" xml:space="preserve">Nam cum arcus <var>.b.<lb/>e.</var> æqualis Å¿it arcui <var>.b.n.</var> ex .27. tertij, remanet vt arcus <var>.q.e.</var> equalis Å¿it arcui <var>.q.n.</var> vnde <lb/>angulus <var>.q.b.e.</var> æqualis erit angulo <var>.q.b.n.</var> ex .26. eiuÅ¿dem. </s> - <s xml:id="s4495" xml:space="preserve">quare ex .4. primi anguli <lb/>ad <var>.t.</var> erunt recti, et <var>.n.t.</var> æqualis erit ipÅ¿i <var>.t.e.</var> vt diximus.</s> - </p> - <p> - <s xml:id="s4496" xml:space="preserve">Deinde <var>.b.e.</var> et <var>.q.a.</var> Å¿einuicem <reg norm="Å¿ecant" type="context">Å¿ecãt</reg> in puncto <var>.o.</var> vt ex Å¿e clarum patet, ducatur po <lb/>Å¿tea <var>.q.e.</var> vnde habebimus angulum <var>.b.e.q.</var> rectum ex .30. tertij, </s> - <s xml:id="s4497" xml:space="preserve">quare ex .18. primi <var>.q.<lb/>o.</var> longior erit ipÅ¿a <var>.q.e.</var> et <var>.q.e.</var> longior erit ipÅ¿a <var>.e.t</var>. </s> - <s xml:id="s4498" xml:space="preserve">quare <var>.q.o.</var> longior erit ipÅ¿a <var>.t.e</var>.</s> - </p> - <p> - <s xml:id="s4499" xml:space="preserve">Vt probemus poÅ¿tea <var>.b.a.o.</var> longiorem eſſe ipÅ¿a <var>.b.e.</var> producatur <var>.b.a.</var> ita quod <var>.a.<lb/>p.</var> æqualis Å¿it ipÅ¿i <var>.a.o.</var> <reg norm="ducaturque" type="simple">ducaturqÌ;</reg> <var>o.p.</var> et <var>.a.e.</var> cum autem ex iam dicta .30. tertij angulus <lb/><var>b.a.o.</var> <reg norm="rectus" type="simple">rectê°</reg> Å¿it, erit angulus <var>.o.a.p.</var> Å¿imiliter <reg norm="rectus" type="simple">rectê°</reg> ex .13. primi, vnde ex .5. et .32. <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> <lb/>angulus <var>.a.p.o.</var> erit dimidium recti, & Å¿imiliter, exijÅ¿dem, angulus <var>.b.q.a.</var> eÅ¿t dimidium <lb/>recti </s> - <s xml:id="s4500" xml:space="preserve">quare angulus <var>.a.p.o.</var> æqualis erit angulo <var>.a.q.b.</var> Å¿ed angulus <var>.a.e.b.</var> æqualis eÅ¿t an <lb/>gulo <var>.a.q.b.</var> ex .20. tertij, ergo angulus <var>.b.p.o.</var> æqualis erit angulo .b, <var>e.a.</var> angulus vero <lb/><var>a.b.e.</var> communis eÅ¿t ambobus triangulis <var>.a.b.e.</var> et <var>.o.b.p</var>. </s> - <s xml:id="s4501" xml:space="preserve">quare ex .32. primi anguli <var>.<lb/>b.a.e.</var> et <var>.b.o.p.</var> reliqui ex duobus rectis æqua <lb/> - <anchor type="figure" xlink:label="fig-0392-01a" xlink:href="fig-0392-01"/> - les inuicem erunt. </s> - <s xml:id="s4502" xml:space="preserve">Quare ex quarta Å¿exti, <lb/>et .18. quinti proportio <var>.b.o.</var> ad <var>.b.p.</var> erit, vt <lb/><var>b.a.</var> ad <var>.b.e.</var> Å¿ed ex .18. primi <var>.b.o.</var> maior eÅ¿t <lb/>ipÅ¿a <var>.b.a</var>. </s> - <s xml:id="s4503" xml:space="preserve">quare ex .14. quinti <var>.b.p.</var> maior erit <lb/>ipÅ¿a <var>.b.e.</var> Å¿ed <var>.b.p.</var> æquatur ipÅ¿is <var>.b.a.</var> cum <var>.a.</var> o <lb/>ex hypoteÅ¿i, ergo <var>.b.a.</var> cum <var>.a.o.</var> maior erit <lb/>ipÅ¿a <var>.b.e.</var> Å¿ed <var>.q.o.</var> maior erat ipÅ¿a <var>.t.e.</var> vt Å¿upe <lb/>rius vidimus, </s> - <s xml:id="s4504" xml:space="preserve">quare <var>.b.a.</var> cum <var>.a.o.</var> et <var>.o.q.</var> ma <lb/>ior eÅ¿t ipÅ¿a <var>.b.e.</var> cum <var>.e.t.</var> hoc eÅ¿t dimidium <lb/>periferiÄ™ ipÅ¿ius quadrati, <reg norm="maius" type="simple">maiê°</reg> erit dimidio <lb/>periferiÄ™ <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <reg norm="trianguli" type="context">triãguli</reg> propoÅ¿iti, </s> - <s xml:id="s4505" xml:space="preserve">quare ex 14. <lb/>dicta tota periferia dicti trianguli, Å¿imiliter <lb/>probarem de omnibus alijs figuris regulari <lb/>bus eodem circulo inÅ¿criptis.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0392-01" xlink:href="fig-0392-01a"> - <image file="0392-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0392-01"/> - </figure> - </div> - </div> - </div> - <div type="section" level="3" n="42"> - <div type="letter" level="4" n="1"> - <head xml:id="head562" xml:space="preserve">CONSIDERATIONES NONNVLLÆ IN <lb/>Archimedem.</head> - <head xml:id="head563" style="it" xml:space="preserve">Doct ßimo atque Reuerendo Domino Vincentio <lb/>Mercato.</head> - <p> - <s xml:id="s4506" xml:space="preserve"><emph style="sc">QVod</emph> tibi alià s dixi verum eÅ¿t, intellectum Å¿cilicet non omninò quieÅ¿cere cir <lb/>ca illas duas Archimedis propoÅ¿itiones, quæ in translatione Tartaleæ Å¿unt <lb/>Å¿ub numeris .4. et .5. & in impreſſione BaÅ¿ileæ Å¿ub numeris .6. et .7. vbi <lb/>tractat - <pb o="381" rhead="EPISTOLAE." n="393" file="0393" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0393"/> - tractat de centris libræ, Å¿eu Å¿tateræ: </s> - <s xml:id="s4507" xml:space="preserve">A Å¿pice igitur in .4. Å¿upradicta, quod cum appen-<lb/>ſæ fuerint omnes illæ partes ponderum, partibus longitudinis ipÅ¿ius <var>.l.K.</var> in qua volo <lb/>vt à punctis <var>.e.</var> et <var>.d.</var> imagineris duas lineas <var>.e.o.</var> et <var>.d.u.</var> inuicem æquales, & ferè per-<lb/>pendiculares ipÅ¿i <var>.l.K.</var> hoc eÅ¿t reÅ¿picientes mundi centrum; </s> - <s xml:id="s4508" xml:space="preserve">imagineris etiam <var>.o.u.</var> <lb/> - <anchor type="handwritten" xlink:label="hd-0393-01a" xlink:href="hd-0393-01"/> - quæ Å¿it paralle la ipÅ¿i <var>.l.k.</var> quæ diuiÅ¿a Å¿it in puncto <var>.i.</var> Å¿upra <var>.g</var>. </s> - <s xml:id="s4509" xml:space="preserve">Hinc nulli dubium erit, <lb/>cum <var>.g.</var> fuerit centrum totius ponderis appenÅ¿i ipÅ¿i <var>.l.K.</var> quod <var>.i.</var> Å¿imiliter erit centrum <lb/>cum directe locatum Å¿it Å¿upra <var>.g.</var> hoc eÅ¿t in eadem directionis linea, quod quidem <lb/>non indiget aliqua demonÅ¿tratione, cum per Å¿e Å¿atis pateat. </s> - <s xml:id="s4510" xml:space="preserve">Vnde ex communi <lb/>conceptu <var>.o.</var> erit centrum ponderis appenÅ¿i ipÅ¿i <var>.l.h.</var> et <var>.u.</var> erit centrum ponderis ap-<lb/>penÅ¿i. ipÅ¿i <var>h.K</var>. </s> - <s xml:id="s4511" xml:space="preserve">Scimus <reg norm="igitur" type="simple">igit̃</reg> <var>.i.</var> eſſe <reg norm="centrum" type="context">cẽtrum</reg> duorum, hoc eÅ¿t ipÅ¿ius <var>.l.h.</var> & ipÅ¿ius <var>.h.k.</var> con <lb/>tinuatorum per totam <var>.l.k</var>. </s> - <s xml:id="s4512" xml:space="preserve">Nunc ergo Å¿i conÅ¿ideremus <var>.l.k.</var> diuiÅ¿am eſſe, hoc eÅ¿t di-<lb/>Å¿iunctam in puncto <var>.h.</var> inueniemus nihilominus <var>.i.</var> centrum eſſe dictorum ponderum, <lb/>& quod tantum eÅ¿t, ipÅ¿am eſſe <reg norm="continuam" type="context">continuã</reg>, quantum diuiÅ¿am in dicto puncto <var>.h.</var> neque <lb/>ex hoc, punctum <var>.i.</var> erit magis vel minus centrum duorum ponderum <var>.l.h.</var> et <var>.h.k.</var> quo <lb/>rum vnum pendet totum ab <var>.o.</var> aliud verò totum ab <var>.u.</var> & hoc modo in longitudine <var>.<lb/>o.u.</var> diuiÅ¿a vt dictum eÅ¿t, habebimus propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="1"> - <handwritten xlink:label="hd-0393-01" xlink:href="hd-0393-01a"/> - </div> - <p> - <s xml:id="s4513" xml:space="preserve">Reliquam propoÅ¿itionem tibi relinquo.</s> - </p> - <p> - <s xml:id="s4514" xml:space="preserve">Illa verò propoÅ¿itio, quam tibi dixi Archimedem tacuiſſe in huiuÅ¿modi materia <lb/>eÅ¿t, quod Å¿i duo pondera æquilibrant ab extremis alicuius Å¿tateræ, in certis præfixis <lb/>diÅ¿tantijs à centro. </s> - <s xml:id="s4515" xml:space="preserve">Tunc dico Å¿i eorum vno manente alterum moueatur remotius <lb/>ab ipÅ¿o centro quod illud deÅ¿cendet, & Å¿i vicinius ipÅ¿i centro appenÅ¿um fuerit aÅ¿cen-<lb/>det. </s> - <s xml:id="s4516" xml:space="preserve">Hæc enim propoÅ¿itio quotidie omnibus in locis videtur, ipÅ¿am verÅ¿o4; </s> - <s xml:id="s4517" xml:space="preserve">puto Ar <lb/>chimedem prætermiÅ¿iſſe ob facilitatem, cum ab antedicta ferè dependeat.</s> - </p> - <p> - <s xml:id="s4518" xml:space="preserve">Sit exempli gratia Å¿tatera <var>.a.u.</var> cuius centr um Å¿it <var>.i.</var> & pondera <var>.u.a.</var> appenÅ¿a, Å¿ein-<lb/> - <anchor type="handwritten" xlink:label="hd-0393-02a" xlink:href="hd-0393-02"/> - uicem habeant vt <var>.i.u.</var> et <var>.i.a.</var> Å¿e inuicem habent. </s> - <s xml:id="s4519" xml:space="preserve">Nunc dico quod Å¿i pondus ipÅ¿ius <var>.u.</var> <lb/>poÅ¿itum fuerit vicinius centro vt puta in <var>.o.</var> inmoto exiÅ¿tente pondere, a. quod bra-<lb/>chium <var>.i.o.u.</var> aÅ¿cendet, & è conuerÅ¿o, Å¿i remotius poÅ¿itum fuerit, deÅ¿cendet.</s> - </p> - <div type="float" level="5" n="2"> - <handwritten xlink:label="hd-0393-02" xlink:href="hd-0393-02a"/> - </div> - <p> - <s xml:id="s4520" xml:space="preserve"><reg norm="Ponatur" type="simple">Ponat̃</reg> ergo vt <reg norm="dictum" type="context">dictÅ©</reg> eÅ¿t in <var>.o.</var> vicinius <reg norm="centro" type="context">cẽtro</reg>, quapropter brachium <var>.i.o.</var> <reg norm="breuius" type="simple">breuiê°</reg> erit <lb/>brachio <var>.i.u.</var> vnde minor proportio erit ipÅ¿ius <var>.i.o.</var> ad <var>.i.a.</var> quà m.i.u. ad eundem <var>.a.i.</var> & <lb/>conÅ¿equenter quam ponderis ipÅ¿ius <var>.a.</var> (quod Å¿it <var>.n.e.</var>) ad pondus ipÅ¿ius <var>.u</var>. </s> - <s xml:id="s4521" xml:space="preserve">Quare Å¿i cx <lb/>pondere <var>.n.e.</var> dempta fuerit <var>.e.</var> pars eius, ita quod reliqua pars <var>.n.</var> Å¿e habeat ad pondus <lb/>o. vt Å¿e habet. i<unsure/> <var>.o.</var> ad <var>.i.a.</var> tunc Å¿tatera non mouebitur; </s> - <s xml:id="s4522" xml:space="preserve">addita verò parte <var>.e.</var> ex com-<lb/>muni conceptu, a. deÅ¿cendet vnde <var>.o.</var> aÅ¿cenderet conuerÅ¿um verò ex Å¿imilibus ratio-<lb/>nibus per te concludes.</s> - </p> - <figure position="here"> - <image file="0393-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0393-01"/> - </figure> - <pb o="382" rhead="IO. BAPT. BENED." n="394" file="0394" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0394"/> - <p> - <s xml:id="s4523" xml:space="preserve">In eo quod à me petis, mittendo te ad Eutotium, tibi non Å¿atisfacerem, cum Eu-<lb/>totius citet Å¿extum librum Pergei, quem nunquam vidimus, <reg norm="Å¿upponatque" type="simple">Å¿upponatqÌ;</reg> ea, quæ nec <lb/>ipÅ¿e nec alius vnquam quod Å¿cimus probauit.</s> - </p> - <p> - <s xml:id="s4524" xml:space="preserve">DeÅ¿ideras enim demonÅ¿trationem illius quod Archimedes dicit inter primam, <lb/>& Å¿ecundam propoÅ¿itionem Å¿ecundi libri, vbi tractat de centris grauium, propte-<lb/>rea quod illud Å¿upponit pro manifeÅ¿to.</s> - </p> - <p> - <s xml:id="s4525" xml:space="preserve">Sit enim figura hic Å¿ubÅ¿cripta, ferè Å¿imilis parabolæ poÅ¿itæ in .2. propoÅ¿itione di <lb/>cti libri, vt in impreſſione BaÅ¿ileenÅ¿i habetur, <reg norm="Å¿intque" type="simple">Å¿intqÌ;</reg> diuiſæ duæ <var>.a.b.</var> et <var>.b.c.</var> per æqua <lb/>lia à punctis <var>.x.</var> et <var>.u.</var> <reg norm="protractisque" type="simple">protractisqÌ;</reg> <var>.f.x.</var> et <var>.u.i.</var> ad <var>.b.d.</var> quæ inuicem etiam erunt parallelÄ™ <lb/>ex .30. primi Eucli. </s> - <s xml:id="s4526" xml:space="preserve">vnde ipſæ etiam, diametri erunt ipÅ¿arum portionum: </s> - <s xml:id="s4527" xml:space="preserve">vt ex eo col <lb/>ligere eÅ¿t, quod in .49. primi lib. Pergei probatur. </s> - <s xml:id="s4528" xml:space="preserve">Imaginando poÅ¿tea ad puncta <var>.b.<lb/>f.</var> er<unsure/> <var>.i.</var> tres contingentes, manifeÅ¿tum erit punctum <var>.b.</var> illud eſſe quod terminat alti-<lb/>tudinem huiuÅ¿modi portionis, et <var>.f.</var> et <var>.i.</var> terminantia altitudines partialium, ex .5. Å¿eÂ<lb/>cundi ipÅ¿ius Pergei, eo quod dictæ contingentes paralellæ erunt ipÅ¿is baÅ¿ibus, vnde <lb/>trianguli inÅ¿cripti, eaÅ¿dem habebunt altitudines, quas portiones ipſæ, quod erit ex <lb/>mente Archimedis. </s> - <s xml:id="s4529" xml:space="preserve">Et Å¿ic deinceps poteris multiplicare angulos Å¿iguræ rectilineæ <lb/>in parabola, quæ deÅ¿ignata erit vt deÅ¿iderat Archimedes, qui quidem dicit, quod <lb/>protractæ cum fuerint aliæ deinceps poÅ¿t <var>.f.i.</var> ipſæ inuicem Ä™quidiÅ¿tantes <reg norm="erunt" type="context">erÅ©t</reg>, diuiſę-<lb/>qÌue peræqualia ab <var>.d.b.</var> quod <reg norm="quanuis" type="context">quãuis</reg> <reg norm="verum" type="context">verÅ©</reg> Å¿it, <reg norm="tantum" type="wordlist/context">tñ</reg> ab Eutotio non Å¿atis <reg norm="demonÅ¿tratum" type="context context">demõſtratÅ©</reg> <lb/>eÅ¿t, cum Å¿upponat <var>.a.f.b.</var> æqualem eſſe ipÅ¿i <var>.b.i.c.</var> probare volens eius diametros æqua <lb/>les eſſe abÅ¿que aliqua citata ratione, quæ quidem ratio eſſet conuerÅ¿um .4. propoÅ¿i-<lb/>tionis libri de conoidalibus. </s> - <s xml:id="s4530" xml:space="preserve">Sed oporteret nos <reg norm="etiam" type="context">etiã</reg> videre .6. librum ipÅ¿ius Pergei, <lb/>& propterea tibi non Å¿atisfacerem.</s> - </p> - <p> - <s xml:id="s4531" xml:space="preserve">EÅ¿to igitur, ut inuenta Å¿it linea <var>.K.</var> cuius productum in <var>.u.i.</var> æquale Å¿it qua drato ip <lb/>Å¿ius <var>.u.c.</var> inuenta etiam Å¿it linea <var>.h.</var> cuius productum cum <var>.f.x.</var> æquale Å¿it quadrato ip-<lb/>Å¿ius <var>.a.x.</var> vnde ex conuerÅ¿o .49. primi ipÅ¿ius Pergei, proportio ipÅ¿ius <var>.K.</var> ad <var>.b.c.</var> erit ut <lb/>ipÅ¿ius <var>.b.c.</var> ad <var>.b.d.</var> & ipÅ¿ius <var>.h.</var> ad <var>.a.b.</var> vt ipÅ¿ius <var>.a.b.</var> ad <var>.b.d</var>. </s> - <s xml:id="s4532" xml:space="preserve">Erit igitur ex .16. Å¿exti Eucl. <lb/>quadratum <var>.b.c.</var> æquale producto ipÅ¿ius <var>.K.</var> in <var>.b.d.</var> & quadratum <var>.a.b.</var> æquale produ-<lb/>cto ipÅ¿ius <var>.h.</var> in <var>.b.d.</var> & ex prima Å¿exti, ita erit ipÅ¿ius <var>.K.</var> ad <var>.h.</var> vt producti quod fit ex <var>.K.</var> <lb/>in <var>.b.d.</var> ad productum ipÅ¿ius <var>.h.</var> in <var>.b.d.</var> hoc eÅ¿t vt quadrati ipÅ¿ius <var>.b.c.</var> ad quadratum ip <lb/>Å¿ius <var>.b.a.</var> ex .16. et .11. quinti, hoc eÅ¿t vt quadrati ipÅ¿ius <var>.u.c.</var> ad quadratum ipÅ¿ius <var>.a.x.</var> <lb/>hoc eÅ¿t ut productum ipÅ¿ius <var>.k.</var> in <var>.u.i.</var> ad productnm ipÅ¿ius <var>.h.</var> in <var>.x.f</var>. </s> - <s xml:id="s4533" xml:space="preserve">Nunc Å¿i ipÅ¿ius <var>.k.</var> <lb/>ad <var>.h.</var> c<unsure/>Å¿t vt producti ipÅ¿ius <var>.K.</var> in <var>.u.i.</var> ad productum ipÅ¿ius <var>.h.</var> in <var>.f.x.</var> ergo ex .24. Å¿exti, <lb/>& communi conceptu, proportio ipÅ¿ius <var>.k.</var> ad <var>.h.</var> compoÅ¿ita erit ex ea quæ ipÅ¿ius <var>.u.i.</var> <lb/>ad <var>.f.x.</var> & ex ea quæ ipÅ¿ius <var>.k.</var> ad <var>.h</var>. </s> - <s xml:id="s4534" xml:space="preserve">Cum ergo dempta fuerit proportio ipÅ¿ius <var>.k.</var> ad <var>.h.</var> <lb/>(vt Å¿implex) à proportione ipÅ¿ius <var>.k.</var> ad <var>.h.</var> (vt compoÅ¿ita) reliquum nihil erit. </s> - <s xml:id="s4535" xml:space="preserve">Qua-<lb/>re <var>.f.x.</var> æqualis erit ipÅ¿i <var>.u.i</var>.</s> - </p> - <p> - <s xml:id="s4536" xml:space="preserve">Sed quod <var>.f.m.</var> æqualis Å¿it ipÅ¿i <var>.m.i</var>. </s> - <s xml:id="s4537" xml:space="preserve">Videto in Eutotio, quia hoc Å¿atis Å¿ui natura <lb/>facile eÅ¿t.</s> - </p> - <p> - <s xml:id="s4538" xml:space="preserve">Sed accipe alium modum breuiorem ad probandum <var>.f.x.</var> eſſe æqualem ipÅ¿i <var>.u.i</var>.</s> - </p> - <p> - <s xml:id="s4539" xml:space="preserve">Finge lineam <var>.e.b.g.</var> conting entem in puncto <var>.b.</var> prolungatisqÌue diametris <var>f.<lb/>x.</var> et <var>.u.i.</var> vÅ¿que ad contingentem ipÅ¿am, habebis <var>.f.e.</var> æqualem ipÅ¿i <var>.f.x.</var> et <var>.g.i.</var> ipÅ¿i <var>.u.i.</var> <lb/>Ex .35. primi Pergei, producta poÅ¿tea <var>.x.u.</var> habeb is ex .2. Å¿exti Eucli <var>.x.u.</var> parallelam <lb/>ipÅ¿i <var>.a.c.</var> Å¿ed <var>.e.g.</var> parallela eÅ¿t ipÅ¿imet <var>.a.c.</var> ex quinta Å¿ecundi ipÅ¿ius Pergei, </s> - <s xml:id="s4540" xml:space="preserve">quare ex .30 <lb/>primi Euclid <var>.e.g.</var> parallela erit ipÅ¿i <var>.u.x.</var> & ex .34. eiuÅ¿dem æqualis erit <var>.e.x.</var> ipÅ¿i <var>.u.g.</var> <lb/>vnde <var>.f.x.</var> etiam æqualis erit <var>.u.i.</var> ex communi conceptu.</s> - </p> - <p> - <s xml:id="s4541" xml:space="preserve">Sed ne quid deÅ¿ideres probabo <var>.f.m.</var> æqualem eſſe ipÅ¿i <var>.m.i</var>. </s> - <s xml:id="s4542" xml:space="preserve">Iam igitur Å¿cis quod - <pb o="383" rhead="EPISTOLAE." n="395" file="0395" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0395"/> - cum Å¿it <var>.f.x.</var> æqualis ipÅ¿i <var>.u.i.</var> vt tibi probaui, & inuicem parallelæ ideo <var>.f.i.</var> parallela <lb/>erit ipÅ¿i <var>.x.u.</var> ex .33. primi Euclidis. </s> - <s xml:id="s4543" xml:space="preserve">Vnde ex .30. eiuÅ¿dem, parallela erit etiam ipÅ¿i <var>.a.<lb/>c.</var> Å¿ed cum <var>.x.u.</var> diuiÅ¿a Å¿it ab <var>.d.b.</var> per æqualia, eo quod diuidit <var>.a.c.</var> eodem modo, quÄ™ <lb/>ipÅ¿i parallela eÅ¿t ex .2. Å¿exti. </s> - <s xml:id="s4544" xml:space="preserve">Reliqua tibi conÅ¿ideranda relinquo. </s> - <s xml:id="s4545" xml:space="preserve">cum verò ambæ <var>.f.<lb/>x.</var> et <var>.u.i.</var> parallelæ Å¿int ipÅ¿i <var>.b.d.</var> Å¿equitur quod cum ex .34. primi <reg norm="vnaquæque" type="simple">vnaquæq;</reg> <var>.f.m.</var> et <var>.m.<lb/>i.</var> æqualis Å¿it medietati ipÅ¿ius <var>.x.u.</var> erunt inuicem æquales.</s> - </p> - <figure position="here"> - <image file="0395-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0395-01"/> - </figure> - <p> - <s xml:id="s4546" xml:space="preserve">Minime dubitabam tibi non Å¿atisfacere Eutocium in .3. propoÅ¿itione Å¿ecundi <lb/>lib. de centris Grauium Archimedis, cum citet .6. librum de elementis conicis, ad-<lb/>de quod Å¿i aliud in ipÅ¿o .6. libro ab eo citato non eſſet magis ad propoÅ¿itum, quà m <lb/>ca quæ ab ipÅ¿o citata Å¿unt, nihilominus adhuc irreÅ¿olutus maneres.</s> - </p> - <p> - <s xml:id="s4547" xml:space="preserve">ConÅ¿idera igitur eandem ipÅ¿am figuram præcedentem; </s> - <s xml:id="s4548" xml:space="preserve">pro alia verò parabola Å¿i <lb/>mili dictæ, accipe Å¿ecundam figuram ipÅ¿ius tertiæ dictæ propoÅ¿itionis. </s> - <s xml:id="s4549" xml:space="preserve">Deinde ima <lb/>ginabis duo latera <var>.o.x.</var> et <var>.o.p.</var> diuiÅ¿a eſſe per æqualia in punct is <var>.g.</var> et <var>.K.</var> <reg norm="protractisque" type="simple">protractisqÌ;</reg> <lb/>diametris <var>.g.y.</var> et <var>.K.u.</var> quæ, vt in præcedenti probaui, Å¿unt inuicem æquales, Å¿cire <lb/>debes quod Å¿imiles parabolæ inuicem aliæ non poſſunt eſſe, niÅ¿i eæ quæ diametros <lb/>proportionales Å¿uis baÅ¿ibus habeant, <reg norm="Å¿imiliterque" type="simple">Å¿imiliterqÌ;</reg> poÅ¿itæ, hoc eÅ¿t, ut proportio ipÅ¿ius <lb/><var>b.d.</var> ad <var>.a.c.</var> Å¿it eadem quæ ipÅ¿ius <var>.o.r.</var> ad <var>.x.p.</var> & quod anguli ad <var>.r.</var> Å¿int æquales angulis <lb/>circa <var>.d</var>. </s> - <s xml:id="s4550" xml:space="preserve">Notentur ergo primum puncta communia ip Å¿ius <var>.o.g.</var> cum <var>.y.t.</var> & ipÅ¿ius <var>.b.</var> x <lb/>cum <var>.f.m.</var> characteribus. ω<unsure/>. et <var>.n</var>. </s> - <s xml:id="s4551" xml:space="preserve">Nunc igitur Å¿cimus <var>.f.m.</var> æqualem eſſe <var>.m.i.</var> tota <reg norm="mque" type="simple">mqÌ;</reg> <var>.f.<lb/>i.</var> parallelam eſſe ipÅ¿i <var>.a.c</var>. </s> - <s xml:id="s4552" xml:space="preserve">Idem dico de <var>.y.t.u.</var> <reg norm="triangulique" type="simple">trianguliqÌ;</reg> <var>.x.f.n.</var> et <var>.g.y.</var> ω<unsure/>. eſſe Å¿imiles <lb/>triangulis <var>.n.m.b.</var> et. ω<unsure/> <var>.t.o.</var> quod ita probatur, nam ex .15. primi Euclid. anguli ad <var>.n.</var> <lb/>Å¿unt inuicem æquales, ex .29. verò eiuÅ¿dem anguli <var>.f.x.n.</var> et <var>.n.b.m.</var> Å¿imiliter æquales <lb/>ita etiam <var>.n.f.x.</var> et <var>.n.m.b</var>.</s> - </p> - <p> - <s xml:id="s4553" xml:space="preserve">Idem dico in Å¿ecunda figura, vnde ex .4. Å¿exti Eucli. proportio <var>.n.f.</var> ad <var>.m.n.</var> erit ea <lb/>dem quÄ™ <var>.f.x.</var> ad <var>.b.m.</var> & ipÅ¿ius <var>.n.f.</var> ad <var>.x.f.</var> vt <var>.n.m.</var> ad <var>.m.b.</var> ex .16. quinti. </s> - <s xml:id="s4554" xml:space="preserve">Quare ex .11. <lb/> - <anchor type="figure" xlink:label="fig-0395-02a" xlink:href="fig-0395-02"/> - <pb o="384" rhead="IO. BAPT. BENED." n="396" file="0396" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0396"/> - eiuÅ¿dem erit vt <var>.a.d.</var> ad <var>.d.b</var>. </s> - <s xml:id="s4555" xml:space="preserve">Idem etiam dico in Å¿ecunda parabola, Å¿ed ipÅ¿ius <var>.x.o.</var> ad <lb/><var>o.r.</var> eÅ¿t vt <var>.a.b.</var> ad <var>.b.d.</var> ex .6. Å¿exti Eucli. </s> - <s xml:id="s4556" xml:space="preserve">vnde ex .11. quinti <var>.n.f.</var> ad <var>.f.x.</var> erit vt <var>.ω.y.</var> <lb/>ad <var>.y.g</var>. </s> - <s xml:id="s4557" xml:space="preserve">Sed in precedenti iam tibi dixi <var>.a.b.</var> mediam proportionalem eſſe inter <var>.h.</var> <lb/>et <var>.b.d</var>. </s> - <s xml:id="s4558" xml:space="preserve">Sit nunc <var>.z.</var> pro Å¿ecunda parabola, ita ut <var>.h.</var> eÅ¿t pro prima, vnde <var>.o.x.</var> crit media <lb/>proportionalis inter <var>.z.</var> et <var>.o.r.</var> & ex .11. quinti ita erit <var>.h.</var> ad <var>.a.b.</var> vt <var>.z.</var> ad <var>.x.o.</var> & ex .22. <lb/>h. ad <var>.a.x.</var> ut z. ad <var>.x.g.</var> & quia ex .16. Å¿exti <var>.a.x.</var> media proportionalis eÅ¿t inter <var>.h.</var> et <var>.f.<lb/>x.</var> cum Å¿upponatur productum <var>.h.</var> in <var>.f.x.</var> æquale eſſe quadrato <var>.a.x</var>. </s> - <s xml:id="s4559" xml:space="preserve">Idem dico <var>.x.g.</var> <lb/>mediam eſſe proportionalem inter <var>.z.</var> et <var>.g.y.</var> </s> - <s xml:id="s4560" xml:space="preserve">quare ex .11. iam dicta, ita erit <var>.a.x.</var> ad <var>.f.<lb/>x.</var> vt <var>.y.g.</var> ad <var>.x.o.</var> & ex eadem, ita erit ipÅ¿ius <var>.f.n.</var> ad <var>.a.b.</var> ut <var>.y.ω.</var> ad <var>.x.o.</var> & Å¿ic <var>.f.n.</var> ad <var>.d.a.</var> <lb/>vt <var>.y.ω.</var> ad <var>.x.r.</var> Å¿ed <var>.f.m.</var> ad <var>f.n.</var> eÅ¿t vt <var>.y.t.</var> ad <var>.y.ω.</var> ex .18. quinti vnde <var>.f.m.</var> ad <var>.a.d.</var> erit vt <lb/><var>y.t.</var> ad <var>.x.r</var>. </s> - <s xml:id="s4561" xml:space="preserve">Idem dico de eorum duplis.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0395-02" xlink:href="fig-0395-02a"> - <image file="0395-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0395-02"/> - </figure> - </div> - <p> - <s xml:id="s4562" xml:space="preserve">Ex ijÅ¿dem rationibus dico ita eſſe <var>.b.d.</var> ad <var>.b.m.</var> vt <var>.o.r.</var> ad <var>.o.t.</var> & ex .17. quinti <var>.d.m.</var> <lb/>ad <var>.b.m.</var> vt <var>.r.t.</var> ad <var>.t.o</var>. </s> - <s xml:id="s4563" xml:space="preserve">Reliqua tibi conÅ¿ideranda relinquo.</s> - </p> - <figure position="here"> - <image file="0396-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0396-01"/> - </figure> - <p> - <s xml:id="s4564" xml:space="preserve">In reliquis verò propoÅ¿itionibus illius lib. nullo pacto poteris dubitare: </s> - <s xml:id="s4565" xml:space="preserve">Verum ne <lb/>in .4. aliquid tibi noui exurgat, te Å¿cire volo <ref id="ref-0025">corollarium .20. in libr. de quadratuÂ<lb/>ra parabolÄ™</ref> docere poſſibile eſſe inÅ¿criptionem rectilineæ, ea tamen conditione <reg norm="quam" type="context">quã</reg> <lb/>dicit Archimedes.</s> - </p> - <p> - <s xml:id="s4566" xml:space="preserve">In quinta poÅ¿tea animaduertendum eÅ¿t, quod prima pars, probat tantummodo de <lb/>centro trianguli, et .2. pars probat de centro pentagoni, à te ipÅ¿o deinde potes pro-<lb/>bare de centro nonanguli: </s> - <s xml:id="s4567" xml:space="preserve">& Å¿ic de cæteris: </s> - <s xml:id="s4568" xml:space="preserve">eo quod cum probatum fuerit de centro <lb/>figuræ in medio locatæ Å¿i conÅ¿titutæ poÅ¿tea fuerint Å¿imiles figuræ in portionibus la-<lb/>teralibus habebitur propoÅ¿itum in infinitum.</s> - </p> - <p> - <s xml:id="s4569" xml:space="preserve">Idem intelligendum eÅ¿t in .3. propoÅ¿itione quamuis exemplum vlterius non ex-<lb/>tendatur quam ad pentagonos.</s> - </p> - <p> - <s xml:id="s4570" xml:space="preserve">Sexta verò <reg norm="propoÅ¿itio" type="simple">ê“poÅ¿itio</reg> tibi Å¿acilis erit, quæ nihilominus <reg norm="pont" type="context">põt</reg> <reg norm="demonÅ¿trari" type="context">demõſtrari</reg> hoc <reg norm="mon" type="context">mõ</reg> Å¿ciliÂ<lb/>cet. </s> - <s xml:id="s4571" xml:space="preserve">Sint .4. <reg norm="quantitates" type="context">quãtitates</reg> <var>.a.b.c.d.</var> ipÅ¿ius Archimedis <reg norm="Å¿upponendo" type="context">Å¿upponẽdo</reg> <var>.a.</var> pro figura rectilinea <lb/>inÅ¿cripta in parabola, et <var>.b.</var> pro reÅ¿iduo ipÅ¿ius parabolÄ™ et <var>.c.</var> pro triangulo <var>.a.b.c.</var> in me <lb/>dio ipÅ¿ius parabolÄ™ et <var>.d.</var> pro triangulo <var>.r</var>. </s> - <s xml:id="s4572" xml:space="preserve">Nunc cum <var>.a.</var> maior Å¿it <var>.c.</var> prout totum ma-<lb/>ius eÅ¿t Å¿ua parte, ideo ex .8. quinti maior proportio habebit <var>.a.</var> ad <var>.b.</var> quam <var>.c.</var> ad <var>.b.</var> <lb/>Cum autem <var>.b.</var> minor Å¿it <var>.d.</var> ex Å¿uppoÅ¿ito, ideo ex eadem dicta, maior proportio habe <lb/>bit <var>.a.</var> ad <var>.b.</var> quam <var>.c.</var> ad <var>.d.</var> cum verò centrum cuiuÅ¿uis figuræ plenæ neceſſariò Å¿it intra <lb/>ipÅ¿am figuram, idcirco centrum reÅ¿idui ipÅ¿ius parabolÄ™ intra ipÅ¿am reperietur. </s> - <s xml:id="s4573" xml:space="preserve">quod <lb/>ita <reg norm="clarum" type="context">clarÅ©</reg> <reg norm="per" type="simple">ê‘</reg> Å¿e eÅ¿t, <reg norm="quemadmodum" type="wordlist">quẽadmodÅ©</reg> quoduis aliud axioma, & quia <reg norm="dictum" type="context">dictÅ©</reg> <reg norm="centrum" type="context">centrÅ©</reg> ex .8. primi <lb/>de centris, neceſſariò eÅ¿t in linea <var>.b.h.</var> inter <var>.b.</var> et <var>.h</var>. </s> - <s xml:id="s4574" xml:space="preserve">Sit igitur <var>.g.</var> vnde ex eadem .8. ita <lb/>erit <var>.g.h.</var> ad <var>.h.e.</var> vt <var>.a.</var> ad <var>.b.</var> ergo <var>.g.h.</var> ad <var>.h.e.</var> maior proportio erit <reg norm="quam" type="context">quã</reg> <var>.c.</var> ad <var>.d.</var> hoc eÅ¿t <lb/>quam <var>.b.h.</var> ad <var>.f.</var> ex .12. quinti. </s> - <s xml:id="s4575" xml:space="preserve">Sed <reg norm="cum" type="context">cÅ©</reg> <var>.h.b.</var> maior Å¿it ipÅ¿a <var>.h.g.</var> prout omne totum ma-<lb/>ius eÅ¿t Å¿ua parte, ideo maior proportio habebit <var>.h.b.</var> ad <var>.h.e.</var> quam <var>.h.g.</var> ad <var>.h.e.</var> vnde <lb/>multo <reg norm="maiorem" type="context">maiorẽ</reg> <reg norm="quam" type="context">quã</reg> <var>.h.b.</var> ad <var>.f.</var> ex <reg norm="coni" type="context">cõi</reg> <reg norm="conceptu" type="context">cõceptu</reg>, </s> - <s xml:id="s4576" xml:space="preserve">quare <var>.h.e.</var> erit minor ipÅ¿a <var>.f.</var> ex .10. <reg norm="quinti" type="context">quÄ©ti</reg>.</s> - </p> - <p> - <s xml:id="s4577" xml:space="preserve">Septima verò et .8. propoÅ¿itio nullius tibi erit difficultatis.</s> - </p> - <pb o="385" rhead="EPISTOL AE." n="397" file="0397" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0397"/> - <p> - <s xml:id="s4578" xml:space="preserve">Quamuis Eutotius Å¿cribat Å¿uper duas vltimas lib. Å¿ecundi de centris <reg norm="grauium" type="context">grauiÅ©</reg>, nihil <lb/>miror ipÅ¿um tibi non Å¿atisfacere. </s> - <s xml:id="s4579" xml:space="preserve">Accipe igitur quod ego nunc tibi mitto.</s> - </p> - <p> - <s xml:id="s4580" xml:space="preserve">Archimedes eo in loco <reg norm="primum" type="context">primÅ©</reg> Å¿upponit in penultima dicti libri quatuor lineas <lb/>proportionales <var>.a.b</var>: <var>c.b</var>: <var>d.b</var>: et <var>.e.b</var>: Å¿upponit etiam quod proportio quæ eÅ¿t ipÅ¿ius <var>.<lb/>e.b.</var> ad <var>.e.a.</var> <reg norm="eadem" type="context">eadẽ</reg> Å¿it quæ ipÅ¿ius <var>.f.g.</var> ad tres quintas ipÅ¿ius <var>.a.d.</var> & quod proportio com <lb/>poÅ¿iti dupli ipÅ¿ius <var>.a.b.</var> cum quadruplo ipÅ¿ius <var>.b.c.</var> cum Å¿excuplo ipÅ¿ius <var>.b.d.</var> cum triplo <lb/> - <anchor type="figure" xlink:label="fig-0397-01a" xlink:href="fig-0397-01"/> - <pb o="386" rhead="IO. BAPT. BENED." n="398" file="0398" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0398"/> - ipÅ¿ius <var>.b.e.</var> ad compoÅ¿itum quintupli ipÅ¿ius <var>.a.b.</var> cum decuplo ipÅ¿ius <var>.c.b.</var> cum decuplo <lb/>ipÅ¿ius <var>.b.d.</var> cum quintuplo ipÅ¿ius <var>.b.e.</var> eadem Å¿it quæ ipÅ¿ius <var>.g.h.</var> ad <var>.a.d.</var> & vult proba-<lb/>re <var>.f.h.</var> eſſe duas quintas ipÅ¿ius <var>.a.b</var>.</s> - </p> - <div type="float" level="5" n="4"> - <figure xlink:label="fig-0397-01" xlink:href="fig-0397-01a"> - <image file="0397-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0397-01"/> - </figure> - </div> - <p> - <s xml:id="s4581" xml:space="preserve">Cum autem dicit proportionem ipÅ¿ius <var>.a.c.</var> ad <var>.c.d.</var> & ipÅ¿ius <var>.c.d.</var> ad <var>.d.e.</var> eſſe vt ipÅ¿ius <lb/><var>a.b.</var> a<unsure/>d.b.c. & cÄ™tera, verum dicit ex .19. quinti Eucli. eo quod cum ex hypotheÅ¿i Å¿it <lb/>ipÅ¿ius, <var>a.b.</var> totalis ad <var>.c.b.</var> totalom vt ipÅ¿ius <var>.c.b.</var> partialis (Å¿umptæ vt pars abÅ¿ciÅ¿a ab <var>.a.<lb/>b.</var> pro nunc) ad <var>.d.b.</var> partialem (abÅ¿ciÅ¿am ab <var>.c.b.</var>) erit ex .19. dicta ipÅ¿ius <var>.a.c.</var> (reÅ¿idui <lb/>ex <var>.a.b.</var>) ad <var>.c.d.</var> (reÅ¿iduum ex <var>.c.b.</var>) vt ipÅ¿ius <var>.a.b.</var> ad <var>.c.b.</var> & ita probabitur de pro-<lb/>portione ipÅ¿iuas <var>.c.d.</var> ad <var>.d.e.</var> eadem ratione.</s> - </p> - <p> - <s xml:id="s4582" xml:space="preserve">Cum verò ex .18. quinti Å¿it ipÅ¿ius <var>.a.b.</var> cum <var>.c.b.</var> ad <var>.c.b.</var> vt ipÅ¿ius <var>.a.d.</var> ad <var>.d.e.</var> ergo ex <lb/>22. eiuÅ¿dem, ita erit ipÅ¿ius <var>.a.b.</var> cum <var>.c.b.</var> ad <var>.d.b.</var> vt <var>.a.d.</var> ad <var>.d.e.</var> & exijÅ¿dem rationibus <lb/>eadem proportio erit ipÅ¿ius <var>.c.b.</var> cum <var>.d.b.</var> ad <var>.b.e.</var> vt <var>.a.d.</var> ad <var>.d.e.</var> quod inquit Archi. <lb/></s> - <s xml:id="s4583" xml:space="preserve">Verum etiam erit (ex .13. quinti) cum dicit eandem proportionem eſſe ipÅ¿ius <var>.a.d.</var> <lb/>ad <var>.d.e.</var> quÄ™ dupli primi antecedentis cum Å¿implo Å¿ecundi antecedentis ad duplum <lb/>primi conÅ¿equentis cum Å¿implo Å¿ecundi conÅ¿equentis, hoc eÅ¿t dupli ipÅ¿ius <var>.a.b.c.</var> <reg norm="cum" type="context">cÅ©</reg> <lb/>Å¿implo <var>.c.b.d.</var> ad duplum ipÅ¿ius <var>.d.b.</var> cum Å¿implo <var>.e.b.</var> hoc eft dupli <var>.a.b.</var> cum triplo ip-<lb/>Å¿ius <var>.b.c.</var> cum Å¿implo <var>.d.b.</var> ad duplum ipÅ¿ius <var>.d.b.</var> cum Å¿implo <var>.e.b</var>. </s> - <s xml:id="s4584" xml:space="preserve">Nunc duplum <var>.a.b.</var> <lb/>cum triplo <var>.b.c.</var> cum Å¿implo <var>.b.d.</var> Å¿ignatum Å¿it charactere <var>.D.</var> Å¿uum verò conÅ¿equens, <lb/>hoc eÅ¿t duplum <var>.d.b.</var> <reg norm="cum" type="context">cÅ©</reg> Å¿implo <var>.e.b.</var> Å¿ignificetur à charactere <var>.B.</var> hinc proportio ipÅ¿ius <lb/> - <anchor type="note" xlink:label="note-0398-01a" xlink:href="note-0398-01"/> - <var>a.d.</var> ad <var>.d.e.</var> erit vt <var>.D.</var> ad <var>.B</var>.</s> - </p> - <div type="float" level="5" n="5"> - <note xlink:label="note-0398-01" xlink:href="note-0398-01a" position="left" xml:space="preserve">M</note> - </div> - <p> - <s xml:id="s4585" xml:space="preserve">Inquit nunc Archimedes, Å¿i quis Å¿umeret aliquod maius antecedens æquale Å¿ci-<lb/>licet duplo ipÅ¿ius <var>.a.b.</var> cum quadruplo ipÅ¿ius <var>.b.e.</var> cum quadruplo ipÅ¿ius <var>.b.d.</var> cum du-<lb/>plo ipÅ¿ius <var>.b.e.</var> <reg norm="compararetque" type="simple">compararetqÌ;</reg> illud cum <reg norm="conÅ¿equente" type="context">cõſequente</reg> <var>.B.</var> clarum eſſet ex .8. quinti quod <lb/>tale antecedens maiorem proportionem haberet ad <var>.B.</var> quam ad <var>.D.</var> hoc eÅ¿t maiorem <lb/>quà m ipÅ¿ius <var>.a.d.</var> ad <var>.d.e.</var> ex .12. quinti.</s> - </p> - <p> - <s xml:id="s4586" xml:space="preserve">Nunc Å¿i Å¿umpta fuerit aliqua linea, puta <var>.d.o.</var> cui <var>.a.d.</var> <reg norm="dictam" type="context">dictã</reg> habeat proportionem <lb/>maiorem, larum erit ex Å¿ecunda parte decimÄ™ quinti quod <var>.d.o.</var> minor erit ipÅ¿a <var>.d.e.</var> <lb/>Corrige igitur impreſſionem BaÅ¿ileÄ™ locando characterem <var>.o.</var> inter <var>.d.</var> et <var>.e.</var> eo quod <lb/>ibi poÅ¿itum non fuit.</s> - </p> - <p> - <s xml:id="s4587" xml:space="preserve">Volo nunc quod dictum maius antecedens æquale Å¿cilicet duplo ipÅ¿ius <var>.a.b.</var> cum <lb/>quadruplo ipÅ¿ius <var>.b.c.</var> cum quadruplo ipÅ¿ius <var>.b.d.</var> cum duplo ipÅ¿ius <var>.b.c.</var> Å¿ignificetur à <lb/>charactere <var>.A</var>. </s> - <s xml:id="s4588" xml:space="preserve">Hinc habebimus proportionem ipÅ¿ius <var>.a.d.</var> ad <var>.d.o.</var> ut <var>.A.</var> ad <var>.B</var>.</s> - </p> - <note position="left" xml:space="preserve">β</note> - <!-- note-0398-02 --> - <p> - <s xml:id="s4589" xml:space="preserve">Ex .18. quinti poÅ¿tea habebimus <var>.A.B.</var> ad <var>.B.</var> vt <var>.a.o.</var> ad <var>.d.o.</var> & proportionalitate <lb/>euerÅ¿a in .19. dicti ita erit <var>.A.B.</var> ad <var>.A.</var> vt <var>.a.o.</var> ad <var>.a.d</var>. </s> - <s xml:id="s4590" xml:space="preserve">Sed hoc vltimum antecedens in <lb/> - <anchor type="note" xlink:label="note-0398-03a" xlink:href="note-0398-03"/> - Å¿e continet id quod Archimedes Å¿cribit, hoc eÅ¿t duplum ipÅ¿ius <var>.a.b.</var> <reg norm="quadruplum" type="context">quadruplÅ©</reg> ipÅ¿ius <lb/><var>b.c.</var> Å¿excuplum ipÅ¿ius <var>.b.d.</var> & triplum ipÅ¿ius <var>.b.e</var>. </s> - <s xml:id="s4591" xml:space="preserve">ConÅ¿equens verò <var>.A.</var> continet du <lb/>plum ipÅ¿ius <var>.a.b.</var> quadruplum ipÅ¿ius <var>.b.c.</var> quadruplum ipÅ¿ius <var>.b.d.</var> & duplum ipÅ¿ius <var>.b.e</var>.</s> - </p> - <div type="float" level="5" n="6"> - <note xlink:label="note-0398-03" xlink:href="note-0398-03a" position="left" xml:space="preserve">T</note> - </div> - <p> - <s xml:id="s4592" xml:space="preserve">Ex Å¿uppoÅ¿ito deinde ipÅ¿ius Archimedis & ex conuerÅ¿a proportionalitate in .19. <lb/>dicta, verum eÅ¿t id quod dicit Archimedes, videlicet quod eadem proportio eÅ¿t <lb/>ipÅ¿ius <var>.a.d.</var> ad <var>.g.h.</var> quod quintupli ipÅ¿ius <var>.a.b.</var> cum quintuplo ipÅ¿ius <var>.b.e.</var> cum decuplo <lb/>ipÅ¿ius <var>.b.c.</var> cum decuplo ipÅ¿ius <var>.b.d.</var> (quod quidem antecedens Å¿ignificetur per <var>.V.</var>) <lb/>ad duplum ipÅ¿ius <var>.a.b.</var> cum quadruplo ipÅ¿ius <var>.b.c.</var> cum Å¿excuplo ipÅ¿ius <var>.b.d.</var> cum triplo <lb/>ipÅ¿ius <var>.b.e.</var> hoc eÅ¿t ad <var>.A.B</var>.</s> - </p> - <p> - <s xml:id="s4593" xml:space="preserve">Erit igitur <var>.V.</var> ad <var>.A.B.</var> vt ipÅ¿ius <var>.a.d.</var> ad <var>.g.h.</var> Å¿ed Å¿uperius vbi Å¿ignatum eÅ¿t <var>.T.</var> iam <lb/>probatum fuit ita eſſe <var>.A.B.</var> ad <var>.A.</var> vt ipÅ¿ius <var>.a.o.</var> ad <var>.a.d</var>. </s> - <s xml:id="s4594" xml:space="preserve">Ergò ex .23. quinti Archime <lb/>des verum Å¿cribit, hoc eÅ¿t quod ita erit ipÅ¿ius <var>.V.</var> ad <var>.A.</var> vt ipÅ¿ius <var>.a.o.</var> ad <var>.g.h</var>.</s> - </p> - <p> - <s xml:id="s4595" xml:space="preserve">Clarum per Å¿e etiam eÅ¿t, id quod Archimed. dicit hoc eÅ¿t quod <var>.V.</var> ad <var>.A.</var> eÅ¿t vt <lb/> - <anchor type="note" xlink:label="note-0398-04a" xlink:href="note-0398-04"/> - <pb o="387" rhead="EPISTOL AE." n="399" file="0399" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0399"/> - quinque ad duo, cum quodlibet ingredientium in compoÅ¿ito <var>.V.</var> ad quodlibet in-<lb/>gredientium in compoÅ¿ito <var>.A.</var> Å¿it vt quinque ad duo. </s> - <s xml:id="s4596" xml:space="preserve">Quare ex .13. quinti verum <lb/>dicit. </s> - <s xml:id="s4597" xml:space="preserve">Vnde <var>.a.o.</var> ad <var>.g.h.</var> erit vt <reg norm="quinque" type="simple">quinq;</reg> ad duo ex .11. <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> vt inquit Archimedes.</s> - </p> - <div type="float" level="5" n="7"> - <note xlink:label="note-0398-04" xlink:href="note-0398-04a" position="left" xml:space="preserve">ω</note> - </div> - <p> - <s xml:id="s4598" xml:space="preserve">Corrige impreſſionem vbi Å¿criptum eÅ¿t, rurÅ¿us quoniam <var>.o.a.</var> quia oportet dicere <lb/>RurÅ¿us quoniam <var>.o.d</var>.</s> - </p> - <p> - <s xml:id="s4599" xml:space="preserve">Archimedes igitur verum dicit, quod ipÅ¿ius <var>.o.d.</var> ad <var>.d.a.</var> eÅ¿t vt ipÅ¿ius <var>.B.</var> ad <var>.A.</var> ex <lb/> - <anchor type="figure" xlink:label="fig-0399-01a" xlink:href="fig-0399-01"/> - <pb o="388" rhead="IO. BABPT. BENED." n="400" file="0400" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0400"/> - conuerÅ¿a proportionalitate in .19. quinti, cum <var>.a.d.</var> ad <var>.d.o.</var> iam probatum fuit (vbi <lb/>B.) ita eſſe ut <var>.A.</var> ad <var>.B</var>.</s> - </p> - <div type="float" level="5" n="8"> - <figure xlink:label="fig-0399-01" xlink:href="fig-0399-01a"> - <image file="0399-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0399-01"/> - </figure> - </div> - <p> - <s xml:id="s4600" xml:space="preserve">Sed in principio huius Å¿peculationis probatum iam fuit ita eſſe ipÅ¿ius <var>.d.a.</var> ad <var>.d.e.</var> <lb/>vt ipÅ¿ius <var>.D.</var> ad <var>.B.</var> vbi notatum eÅ¿t <var>.M</var>. </s> - <s xml:id="s4601" xml:space="preserve">quare ex .23. quinti, Archimedes verum dicit, <lb/>qu od <var>.d.o.</var> ad <var>.d.e.</var> erit vt <var>.D.</var> ad <var>.A</var>.</s> - </p> - <p> - <s xml:id="s4602" xml:space="preserve">Sed cum <var>.d.o.</var> ad <var>.d.e.</var> Å¿e habeat ut <var>.D.</var> ad <var>.A.</var> erit ex conuerÅ¿a proportionalitate iam <lb/>dicta <var>.d.e.</var> ad <var>.d.o.</var> vt <var>.A.</var> ad <var>.D.</var> per euerÅ¿am vero erit <var>.d.e.</var> ad <var>.a.o.</var> vt <var>.A.</var> ad Å¿uum reÅ¿i-<lb/> - <anchor type="note" xlink:label="note-0400-01a" xlink:href="note-0400-01"/> - duum. </s> - <s xml:id="s4603" xml:space="preserve">quod reÅ¿iduum componitur ex Å¿implo <var>.b.c.</var> cum triplo <var>.b.</var> cum duplo <var>.b.o.</var> quod <lb/>à te ipÅ¿o videre poteris detrahendo numeros ipÅ¿arum quantitatum quæ in <var>.D.</var> <lb/>reperiuntur, ex numeris earundem, quæ in <var>.A.</var> quod quidem reÅ¿iduum Å¿igniÅ¿icetur <lb/>à charactere <var>.E</var>. </s> - <s xml:id="s4604" xml:space="preserve">Vnde ex conuerÅ¿a proportionalitate verum dicit Archime. hoc eÅ¿t <lb/>quod ita Å¿e hab ebit <var>.o.e.</var> ad <var>.d.e.</var> vt <var>.E.</var> ad <var>.A</var>.</s> - </p> - <div type="float" level="5" n="9"> - <note xlink:label="note-0400-01" xlink:href="note-0400-01a" position="left" xml:space="preserve">λ</note> - </div> - <p> - <s xml:id="s4605" xml:space="preserve">Cum autem Å¿it <var>.a.b.</var> ad <var>.c.b.</var> vt <var>.c.b.</var> ad <var>.d.b.</var> & ita <var>.d.b.</var> ad <var>.e.b.</var> ex Å¿uppoÅ¿ito, ideo ex <lb/>17. quinti verum dicit Archim. </s> - <s xml:id="s4606" xml:space="preserve">hoc eÅ¿t quod ita erit ipÅ¿ius <var>.d.e.</var> ad <var>.e.b.</var> vt <var>.a.c.</var> ad <var>.c.b.</var> <lb/>& vt <var>.c.d.</var> ad <var>.d.b.</var> & ex .13. eiuÅ¿dem eadem proportio erit tripli ipÅ¿ius <var>.c.d.</var> ad triplum <lb/>ipÅ¿ius <var>.d.b.</var> quæ dupli ipÅ¿ius <var>.d.e.</var> ad. duplum ipÅ¿ius <var>.e.b.</var> vt inquit Archi.</s> - </p> - <p> - <s xml:id="s4607" xml:space="preserve">Ex qua .13. compoÅ¿itum ex <var>.a.c.</var> cum triplo ipÅ¿ius <var>.c.d.</var> cum duplo ipÅ¿ius <var>.d.e.</var> ean-<lb/>dem proportionem habebit ad <reg norm="compoÅ¿itum" type="context">compoÅ¿itÅ©</reg> ipÅ¿ius <var>.c.b.</var> cum triplo <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <var>.d.b.</var> cum duplo <lb/>ipÅ¿ius <var>.e.b.</var> quam ipÅ¿ius <var>.d.e.</var> ad <var>.e.b</var>. </s> - <s xml:id="s4608" xml:space="preserve">Sed horum compoÅ¿itorum primum Å¿ignificetur <lb/>per <var>.H.</var> Å¿ecundum verò Å¿ignificatum fuit per <var>.E.</var> vnde <var>.H.</var> ad <var>.E.</var> Å¿e habebit vt <var>.d.e.</var> ad <lb/><var>e.b.</var> Å¿ed <var>.E.</var> ad <var>.A.</var> iam dictum eÅ¿t eſſe vt <var>.o.e.</var> ad <var>.d.e.</var> vbi Å¿ignatum eÅ¿t <var>.λ</var>. </s> - <s xml:id="s4609" xml:space="preserve">quare ex .23. <lb/>quinti eadem proportio erit ipÅ¿ius <var>.o.e.</var> ad <var>.e.b.</var> quæ <var>.H.</var> ad <var>.A.</var> vt ipÅ¿e inquit.</s> - </p> - <note position="left" xml:space="preserve">X</note> - <!-- note-0400-02 --> - <p> - <s xml:id="s4610" xml:space="preserve">Ex .18. poÅ¿tea eiuÅ¿dem ita erit <var>.o.b.</var> ad <var>.e.b.</var> vt <var>.H.A.</var> ad <var>.A</var>.</s> - </p> - <p> - <s xml:id="s4611" xml:space="preserve">Notandum etiam eÅ¿t quod Å¿i collectæ fuerint omnes partes compoÅ¿iti <var>.H.A.</var> hoc <lb/>eÅ¿t duplum <var>.a.b.</var> cum duplo <var>.b.e.</var> cum quadruplo <var>.b.c.</var> cum quadruplo <var>.b.d.</var> cum Å¿implo <lb/><var>a.c.</var> cum triplo <var>.c.d.</var> cum duplo <var>.d.e.</var> habebitur triplum <var>.a.b.</var> triplum <var>.b.d.</var> & Å¿excuplum <lb/><var>b.c.</var> vt ipÅ¿e dixit. </s> - <s xml:id="s4612" xml:space="preserve">Quod autem hoc verum Å¿it, cum diÅ¿tinctæ fuerint omnes partes, <lb/>vt in Å¿ubÅ¿criptis his lineis videre eÅ¿t, videbis quod Å¿i ex <var>.H.</var> detracta fuerit Å¿implex <var>.a.<lb/>c.</var> quæ quidem poÅ¿tea iuncta vni ex partibus quadrupli <var>.b.c.</var> ipÅ¿ius <var>.A.</var> reÅ¿ultabit nobis <lb/>vna inte gra <var>.a.b</var>. </s> - <s xml:id="s4613" xml:space="preserve">Vnde habebimus triplum ipÅ¿ius <var>.a.b.</var> & in <var>.A.</var> remanebit triplum ip <lb/>Å¿ius <var>.c.b</var>. </s> - <s xml:id="s4614" xml:space="preserve">Deinde Å¿i ex <var>.H.</var> auferatur triplum ipÅ¿ius <var>.c.d.</var> & ipÅ¿um addatur tribus parti-<lb/>bus quadrupli <var>.b.d.</var> ipÅ¿ius <var>.A.</var> habebimus tres vices <var>.b.c.</var> quæ Å¿i iungantur tribus, quæ <lb/>remanebant in <var>.A.</var> vt dixi, habebimus Å¿excuplum ipÅ¿ius <var>.b.c.</var> & in <var>.A.</var> remanebit Å¿im <lb/>plum <var>.b.d.</var> cum duplo ipÅ¿ius <var>.b.e</var>. </s> - <s xml:id="s4615" xml:space="preserve">Vnde Å¿i ex <var>.H.</var> demptum fuerit duplum ipÅ¿ius <var>.d.<lb/>e.</var> quod quidem iungatur cum duplo ipÅ¿ius <var>.b.e.</var> habebimus duplum ipÅ¿ius <var>.b.d.</var> quod <lb/>coniunctum cum Å¿implo <var>.b.d.</var> quod in <var>.A.</var> relictum fuerat, habebimus triplum ipÅ¿ius <lb/><var>d.b</var>. </s> - <s xml:id="s4616" xml:space="preserve">Verum igitur eÅ¿t quod inquit Archimedes, hoc eÅ¿t, quod <var>.H.A.</var> eÅ¿t triplum ip-<lb/>Å¿ius <var>.a.b.</var> Å¿excuplum ipÅ¿ius <var>.b.c.</var> & triplum ipÅ¿ius <var>.b.d</var>.</s> - </p> - <p> - <s xml:id="s4617" xml:space="preserve">Verum etiam dicit ex eo (vt Å¿upra probatum eÅ¿t) quod <var>.a.c</var>: <var>c.d</var>: et <var>.d.e.</var> Å¿e <reg norm="habebant" type="context">habebãt</reg> <lb/>in continua proportionalitate, </s> - <s xml:id="s4618" xml:space="preserve">quare ex conuerÅ¿a proportionalitate erunt Å¿ibi inui-<lb/>cem continuæ proportionales.</s> - </p> - <p> - <s xml:id="s4619" xml:space="preserve">Nunc autem cum <var>.a.c</var>: <var>c.d.</var> et <var>.d.e.</var> Å¿int continuæ proportionales in ea proportione <lb/>in qua Å¿unt <var>.a.b</var>: <var>c.b</var>: <var>d.b</var>: et <var>.e.b.</var> vt in principio diximus, erit ex .22. quinti <var>.a.c.</var> ad <var>.d.<lb/>e.</var> vt <var>.a.b.</var> ad <var>.d.b.</var> & Å¿ic etiam <var>.c.b.</var> ad <var>.e.b</var>. </s> - <s xml:id="s4620" xml:space="preserve">Vnde ex .24. eiuÅ¿dem <var>.a.d.</var> ad <var>.d.e.</var> erit vt <var>.a.<lb/>b.</var> cum <var>.b.e.</var> ad <var>.d.b.</var> & vt <var>.c.b.</var> cum <var>.b.d.</var> ad <var>.e.b.</var> & ex .13. dicti vt <var>.a.b.</var> cum <var>.b.e.</var> bis <lb/>Å¿umpto, & cum <var>.b.d.</var> ad <var>.e.b</var>. </s> - <s xml:id="s4621" xml:space="preserve">Quare ex conuerÅ¿a proportionalitate, vt Å¿e habet <var>.e.d.</var> <lb/>ad <var>.d.a.</var> ita Å¿e habebit <var>.e.b.</var> <reg norm="cum" type="context">cÅ©</reg> <var>.d.b.</var> ad <var>d.b.</var> <reg norm="cum" type="context">cÅ©</reg> <var>.b.c.</var> duplicato & <reg norm="cum" type="context">cÅ©</reg> <var>.b.a.</var> vt inquit Archi <lb/>medes. </s> - <s xml:id="s4622" xml:space="preserve">Nunc antecedens vocetur <var>.M.</var> hoc eÅ¿t <var>.b.e.</var> cum <var>.d.b.</var> conÅ¿equens verò, hoc - <pb o="389" rhead="EPISTOLAE." n="401" file="0401" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0401"/> - eÅ¿t <var>.d.b.</var> cum duplo <var>.b.c.</var> cum Å¿implo <var>.b.a.</var> vocetur <var>.N</var>.</s> - </p> - <p> - <s xml:id="s4623" xml:space="preserve">Animaduertendum tamen eÅ¿t quod impreſſio mendoÅ¿a eÅ¿t ubi dicit. <lb/></s> - <s xml:id="s4624" xml:space="preserve"><q open="„" close="">vnaquæque <var>.c.b</var>: <var>b.d.</var> & cætera,</q> <lb/>propterea quod dicendum eÅ¿t ita <lb/><reg norm="vnaquæque" type="simple">vnaquæq;</reg> <var>e.b</var>: <var>b.d</var>.</s> - </p> - <p> - <s xml:id="s4625" xml:space="preserve">Nunc ex .18. quinti, quemadmodum Å¿e habet <var>.a.e.</var> ad <var>.d.a.</var> ita Å¿e habebit <var>.M.N.</var> ad <var>.N.</var> <lb/> - <anchor type="figure" xlink:label="fig-0401-01a" xlink:href="fig-0401-01"/> - </s> - <pb o="390" rhead="IO. BAPT. BENED." n="402" file="0402" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0402"/> - <s xml:id="s4626" xml:space="preserve">Vbiautem Å¿criptum eÅ¿t <lb/><q open="„" close="">ad vtrunque Å¿imul <var>.b.d</var>: <var>d.a.</var> cum dupla <var>.b.c.</var></q> <lb/>dicendum eÅ¿t ita, <lb/>ad vtranque Å¿imul <var>.b.d.b.a.</var> cum dupla <var>.b.c</var>.</s> - </p> - <div type="float" level="5" n="10"> - <figure xlink:label="fig-0401-01" xlink:href="fig-0401-01a"> - <image file="0401-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0401-01"/> - </figure> - </div> - <p> - <s xml:id="s4627" xml:space="preserve">Inquit deinde Archi. quod Å¿icut Å¿e haber <var>.e.a.</var> ad <var>.d.a.</var> ita Å¿e habebit duplum <var>.M.N.</var> <lb/>ad duplum <var>.N</var>. </s> - <s xml:id="s4628" xml:space="preserve">Quod quidem verum eÅ¿t ex .13. quinti, huiuÅ¿modi verò antecedons <lb/>& conÅ¿equens, Archi. manifeÅ¿tat ex Å¿uis partibus, Å¿umendo duplum <var>.e.b.</var> cum duplo <lb/><var>b.d.</var> pro duplo <var>.M.</var> & duplum <var>.b.d.</var> cum duplo <var>.a.b.</var> cum quadruplo <var>.b.c.</var> pro duplo <var>.N.</var> <lb/>quÄ™ Å¿imul iuncta æquantur duplo <var>.e.b.</var> cum duplo <var>.a.b.</var> cum quadruplo <var>.b.d.</var> cum qua-<lb/>druplo <var>.b.c.</var> ex quo æquabuntur <var>.A.</var> vocentur igitur hæc omnia <var>.A.</var> potius quà m du-<lb/>plum ipÅ¿ius <var>.M.N</var>.</s> - </p> - <p> - <s xml:id="s4629" xml:space="preserve">Verum etiam Å¿cribit, vbi dicit, quod proportio <var>.e.a.</var> ad tres quintas ipÅ¿ius <var>.a.d.</var> erit <lb/>vt <var>.A.</var> ad tres quintas dupli <var>.N.</var> ex .22. quinti. </s> - <s xml:id="s4630" xml:space="preserve">Sed cum ex Å¿uppoÅ¿ito ita Å¿e habeat <var>.f.<lb/>g.</var> ad tres quintas ipÅ¿ius <var>.a.d.</var> quemadmodum <var>.b.e.</var> ad <var>.e.a.</var> erit ex .16. quinti verum <reg norm="quod" type="simple">ê™</reg> <lb/>dicit Archimed. </s> - <s xml:id="s4631" xml:space="preserve">hoc eÅ¿t, ita Å¿e habere <var>.b.e.</var> ad <var>.f.g.</var> vt <var>.e.a.</var> ad tres quintas ipÅ¿ius <var>.a.d</var>.</s> - </p> - <p> - <s xml:id="s4632" xml:space="preserve">Et per .11. eiuÅ¿dem verum etiam erit quod Å¿icut Å¿e habet <var>.e.b.</var> ad <var>.f.g.</var> ita Å¿e habe-<lb/>bit <var>.A.</var> ad tres quintas dupli <var>.N.</var> quod quidem duplum <var>.N.</var> Å¿ignificetur per <var>.Q</var>.</s> - </p> - <p> - <s xml:id="s4633" xml:space="preserve">Sed Å¿uperius iam demonÅ¿tratum fuit (vbi <var>.X.</var>) quod <var>.o.b.</var> ad <var>.b.e.</var> ita Å¿e habebat vt <lb/><var>H.A.</var> ad <var>.A.</var> & <reg norm="nunc" type="context">nÅ©c</reg> demum probatum fuit ita eſſe <var>.A.</var> ad tres quintas ipÅ¿ius <var>.Q.</var> vt <var>.e.b.</var> <lb/>ad <var>.f.g</var>. </s> - <s xml:id="s4634" xml:space="preserve">Quare ex .22. quinti ita erit <var>.H.A.</var> ad tres quintas ipÅ¿ius <var>.Q.</var> vt <var>.o.b.</var> ad <var>.f.g.</var> vt <lb/> - <anchor type="note" xlink:label="note-0402-01a" xlink:href="note-0402-01"/> - idem inquit.</s> - </p> - <div type="float" level="5" n="11"> - <note xlink:label="note-0402-01" xlink:href="note-0402-01a" position="left" xml:space="preserve">Y</note> - </div> - <p> - <s xml:id="s4635" xml:space="preserve">Sed <var>.H.A.</var> ad <var>.Q.</var> (vt ex Å¿uis partibus videre eÅ¿t) ita Å¿e habet vt tres ad duo ex .13. <lb/>quinti, vt inquit Archimedes.</s> - </p> - <p> - <s xml:id="s4636" xml:space="preserve">IpÅ¿e etiam dicit proportionem <var>.H.A.</var> ad tres quintas ipÅ¿ius <var>.Q.</var> eſſe vt quinque <lb/>ad duo. </s> - <s xml:id="s4637" xml:space="preserve">Pro cuius rei euidentia imaginemur tam <var>.H.A.</var> quam <var>.Q.</var> diuiÅ¿a per <reg norm="quinque" type="simple">quinq;</reg> <lb/>partes æquales, vnde ex .16. quinti habebimus quamlibet quintam <reg norm="partem" type="context">partẽ</reg> ipÅ¿ius <var>.Q.</var> <lb/><reg norm="æqualem" type="context">æqualẽ</reg> eſſe duabus tertijs vniuÅ¿cuiuÅ¿que quintæ partis <var>.H.A.</var> vnde tres quintæ ipÅ¿ius <lb/>Q. erunt, ex communi conceptu, Å¿ex tertiæ vnius quintæ ipÅ¿ius <var>.H.A.</var> hoc eÅ¿t duæ <lb/>quintæ. ipÅ¿ius <var>.H.A</var>. </s> - <s xml:id="s4638" xml:space="preserve">Quare <var>.o.b.</var> ita Å¿e habebit ad <var>.f.g.</var> vt quinque ad duo ex commu <lb/>ni <reg norm="conceptu" type="context">cõceptu</reg>, cum <var>.o.b.</var> ad <var>.f.g.</var> probatum fuerit Å¿e habere vt <var>.H.A.</var> ad tres quintas ipÅ¿ius <lb/>Q. (vbi <var>.Y.</var>) Å¿ed iam probatum fuit (vbi. ω) quod <var>.o.a.</var> ad <var>.h.g.</var> erat etiam vt <lb/>quinque ad duo, hoc eÅ¿t quod <var>.f.h.</var> erit duæ quintÄ™ ipÅ¿ius <var>.a.b</var>. </s> - <s xml:id="s4639" xml:space="preserve">Quod eÅ¿t propoÅ¿itum.</s> - </p> - <pb o="391" rhead="EPISTOL AE." n="403" file="0403" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0403"/> - <figure position="here"> - <image file="0403-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0403-01"/> - </figure> - <pb o="392" rhead="IO. BAPT. BENED." n="404" file="0404" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0404"/> - <p> - <s xml:id="s4640" xml:space="preserve">In vltima verò propoÅ¿itione Å¿ecundi lib. de ponderibus Archi. hoc modo intelliÂ<lb/>gendus eÅ¿t, vt Å¿i diceret, <lb/>Sit paraboles <var>.a.</var> cuius baÅ¿is Å¿it <var>.a.c.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.d.e.</var> recta parallela dictæ baÅ¿i <var>.a.c.</var> <reg norm="diameterque" type="simple">diameterqÌ;</reg> <lb/><var>b.f</var>. <lb/></s> - <s xml:id="s4641" xml:space="preserve">Inquit deinde quod linea contingens in <var>.b.</var> parallela erit ipÅ¿i <var>.a.c.</var> et <var>.e.d.</var> quod proba <lb/>bimus hoc modo. <lb/></s> - <s xml:id="s4642" xml:space="preserve">Cum <var>.b.f.</var> diameter Å¿it et <var>.a.c.</var> baÅ¿is, clarum erit ex definitione quod <var>.b.f.</var> diuidet <var>.a.c.</var> <lb/>per æqualia in <var>.g</var>. </s> - <s xml:id="s4643" xml:space="preserve">Vnde ex .7. vel etiam ex .46. primi Pergei <var>.d.e.</var> diuiÅ¿a erit per æqua <lb/>lia à diametro <var>.b.f</var>. </s> - <s xml:id="s4644" xml:space="preserve">Quare verum dicit ex quinta Å¿ecundi ipÅ¿ius Pergei hoc eÅ¿t quod <lb/>dicta contingens in puncto. b parallela erit ambobus <var>.a.c.</var> et <var>.e.d</var>.</s> - </p> - <p> - <s xml:id="s4645" xml:space="preserve">Inquit poÅ¿tea quod diuiÅ¿a cum fuerit pars diametri quÄ™ inter <var>.d.e.</var> et <var>.a.c.</var> poÅ¿ita eÅ¿t <lb/>(hoc eÅ¿t <var>.g.f.</var>) per quinque partes æquales, <reg norm="quarum" type="context">quarÅ©</reg> partium media Å¿it <var>.h.k.</var> diuiÅ¿a etiam <lb/>imaginatione Å¿it in puncto <var>.i.</var> ita quod proportio ipÅ¿ius <var>.h.i.</var> ad <var>.i.K.</var> eadem Å¿it quæ in-<lb/>ter duo Å¿olida quorum vnum (illud Å¿cilicet à quo relatio incipit, hoc eÅ¿t antecedens) <lb/>pro Å¿ua baÅ¿i teneat quadratum ipÅ¿ius <var>.a.f.</var> cuius etiam Å¿olidi altitudo compoÅ¿ita Å¿it ex <lb/> - <anchor type="note" xlink:label="note-0404-01a" xlink:href="note-0404-01"/> - duplo ipÅ¿ius <var>.d.g.</var> cum Å¿implo <var>.a.f</var>. </s> - <s xml:id="s4646" xml:space="preserve">Aliud verò Å¿olidum habeat pro Å¿ua baÅ¿i quadra-<lb/>tum ipÅ¿ius <var>.d.g.</var> eius verò altitudo compoÅ¿ita Å¿it ex duplo ipÅ¿ius <var>.a.f.</var> cum Å¿implo <var>.d.g</var>.</s> - </p> - <div type="float" level="5" n="12"> - <note xlink:label="note-0404-01" xlink:href="note-0404-01a" position="left" xml:space="preserve">R</note> - </div> - <p> - <s xml:id="s4647" xml:space="preserve">Inquit nunc Archi. quod cum ita factum fuerit, oÅ¿tendet punctum <var>.i.</var> centrum eſſe <lb/>portionis abÅ¿ciſſę à tota Å¿ectione, quod <reg norm="fruÅ¿tum" type="context">fruÅ¿tÅ©</reg> <reg norm="nominatur" type="simple">nominat̃</reg> <reg norm="Å¿ignatum" type="context">Å¿ignatÅ©</reg> characteribus <var>.a.d.e.c</var>.</s> - </p> - <p> - <s xml:id="s4648" xml:space="preserve">Sit igitur num@. <var>m.n.</var> inquit, æqualis diametro <var>.b.f.</var> et <var>.n.o.</var> æqualis <var>.b.g.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.x.n.</var> me <lb/>dia proportionalis inter <var>.n.m.</var> et <var>.n.o.</var> et <var>.t.n.</var> in continua proportionalitate poÅ¿t <var>.o.n.</var> <lb/>hoc eÅ¿t quod ea proportio quæ eÅ¿t ipÅ¿ius <var>.o.n.</var> ad <var>.n.t.</var> eadem Å¿it ipÅ¿ius <var>.x.n.</var> ad <var>.n.o</var>. </s> - <s xml:id="s4649" xml:space="preserve">Hinc <lb/>habebimus .4. lineas in continua proportionalitate Å¿ibi inuicem coniunctas <var>.m.n</var>: <var>x.<lb/>n</var>: <var>o.n.</var> et <var>.t.n</var>.</s> - </p> - <p> - <s xml:id="s4650" xml:space="preserve">Vult etiam quod à linea <var>.i.b.</var> incipiens ab <var>.i.</var> verÅ¿us <var>.g.</var> alia linea abÅ¿ciſſa Å¿it, cui li-<lb/> - <anchor type="note" xlink:label="note-0404-02a" xlink:href="note-0404-02"/> - neæ, ita proportionata Å¿it <var>.f.h.</var> vt <var>.t.m.</var> eÅ¿t ad <var>.t.n.</var> quæ quidem linea Å¿ignata Å¿it <var>.i.r</var>.</s> - </p> - <div type="float" level="5" n="13"> - <note xlink:label="note-0404-02" xlink:href="note-0404-02a" position="left" xml:space="preserve">A</note> - </div> - <p> - <s xml:id="s4651" xml:space="preserve">Dicit poÅ¿tea quod diameter <var>.b.f.</var> erit fortaſſe a xis vel aliqua reliquarum diame-<lb/>trorum, quod quidem in .46. primi Pergei videre eÅ¿t, cum omnes diametri Å¿int in-<lb/>uicem paralleli ipÅ¿i axi.</s> - </p> - <p> - <s xml:id="s4652" xml:space="preserve">Cum poÅ¿tea dicit, quod <var>.a.f.</var> et <var>.d.g.</var> Å¿unt intentæ ductæqÌue, ibi vult id em infer-<lb/>re, quod Pergeus vocat ordinatè, vt ex .11. et .49. primi ipÅ¿ius Pergei videre li-<lb/>cet, vnde ex .20. eiuÅ¿dem proportio <var>.b.f.</var> ad <var>.b.g.</var> erit vt quadrati <var>.a.f.</var> ad quadratum <lb/>ipÅ¿ius <var>.d.g.</var> vt ipÅ¿e dicit.</s> - </p> - <p> - <s xml:id="s4653" xml:space="preserve">Sed ita erit quadrati <var>.m.n.</var> ad qua <reg norm="dratum" type="context">dratÅ©</reg> <var>.x.n.</var> ex .18. Å¿exti Eucli. </s> - <s xml:id="s4654" xml:space="preserve">Quare ex .11. quin-<lb/> - <anchor type="note" xlink:label="note-0404-03a" xlink:href="note-0404-03"/> - ti quadratum ipÅ¿ius <var>.m.n.</var> ad quadratum ipÅ¿ius <var>.n.x.</var> eandem habebit proportionem, <lb/>quam quadratum ipÅ¿ius <var>.a.f.</var> ad quadratum ipÅ¿ius <var>.d.g</var>. </s> - <s xml:id="s4655" xml:space="preserve">Vnde ex .18. & ex communi <lb/><reg norm="Å¿cientia" type="context">Å¿ciẽtia</reg>, eadem proportio erit ipÅ¿ius <var>.m.n.</var> ad <var>.n.x.</var> quÄ™ ipÅ¿ius <var>.a.f.</var> ad <var>.d.g.</var> vt inquit Arch.</s> - </p> - <div type="float" level="5" n="14"> - <note xlink:label="note-0404-03" xlink:href="note-0404-03a" position="left" xml:space="preserve">α</note> - </div> - <p> - <s xml:id="s4656" xml:space="preserve">Quaptopter proportio cubi ipÅ¿ius <var>.m.n.</var> ad cubum ipÅ¿ius <var>.n.x.</var> erit vt cubi ipÅ¿ius <var>.a.<lb/>f.</var> ad cubum ipÅ¿ius <var>.d.g.</var> vt etiam dicit ex communi Å¿cientia, nec non ex .36. vndecimi.</s> - </p> - <p> - <s xml:id="s4657" xml:space="preserve">Inquit poÅ¿tea quod proportio totius Å¿ectionis <var>.a.b.c.</var> ad portionem <var>.d.b.e.</var> eadem <lb/>eÅ¿t quæ cubi ipÅ¿ius <var>.a.f.</var> ad cubum ipÅ¿ius <var>.d.g.</var> quod verum eÅ¿t, vt alià s tibi monÅ¿traui in <lb/>diuiÅ¿ione parabolæ Å¿ecundum aliquam propoÅ¿itam proportionem.</s> - </p> - <p> - <s xml:id="s4658" xml:space="preserve">Quando autem dicit quod proportio cubi ipÅ¿ius <var>.m.n.</var> ad cubum ipÅ¿ius <var>.n.x.</var> eadem <lb/> - <anchor type="note" xlink:label="note-0404-04a" xlink:href="note-0404-04"/> - eÅ¿t quæ ipÅ¿ius <var>.m.n.</var> ad <var>.n.t.</var> verum dicit ex .36. vndecimi. </s> - <s xml:id="s4659" xml:space="preserve">Vnde ex .11. quinti ita Å¿e <lb/>habebit totalis Å¿ectio <var>.a.b.c.</var> ad portionem <var>.d.b.c.</var> vt <var>.m.n.</var> ad <var>.n.t.</var> & ex .17. eiuÅ¿dem ita <lb/>erit ipÅ¿ius <var>.m.t.</var> ad <var>.t.n.</var> vt fruÅ¿ti <var>.a.d.e.c.</var> ad Å¿ectionem <var>.d.b.e.</var> quemadmodum ipÅ¿e di-<lb/>cit. </s> - <s xml:id="s4660" xml:space="preserve">Sed quia Å¿uperius, vbi <var>.A.</var> ipÅ¿a <var>.f.h.</var> (quæ eÅ¿t tres quintæ ipÅ¿ius <var>.f.g.</var>) ad <var>.i.r.</var> ita rela- - <pb o="393" rhead="EPISTOL AE." n="405" file="0405" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0405"/> - ta fuit vt <var>.m.t.</var> ad <var>.t.n.</var> idcirco ex .11. quinti ita erit ipÅ¿ius fruÅ¿ti <var>.a.e.</var> ad Å¿ectionem <var>.d.b.<lb/>e.</var> vt tres quintÄ™ ipÅ¿ius <var>.f.g.</var> ad <var>.i.r</var>.</s> - </p> - <div type="float" level="5" n="15"> - <note xlink:label="note-0404-04" xlink:href="note-0404-04a" position="left" xml:space="preserve">β</note> - </div> - <p> - <s xml:id="s4661" xml:space="preserve">Inquit deinde quod proportio corporis iam Å¿upradicti, quod pro Å¿ua baÅ¿i habeat <lb/>quadratum ipÅ¿ius <var>.a.f.</var> altitudinem verò compoÅ¿itam ex duplo ipÅ¿ius <var>.d.g.</var> cum Å¿implo <lb/><var>a.f.</var> ad cubum ipÅ¿ius <var>.a.f.</var> eadem erit quæ dupli ipÅ¿ius <var>.d.g.</var> cum Å¿implo <var>.a.f.</var> ad <var>.a.f</var>. </s> - <s xml:id="s4662" xml:space="preserve">Quod <lb/>quidem verum eÅ¿t ex .33. vndecimi & ex prima Å¿exti.</s> - </p> - <p> - <s xml:id="s4663" xml:space="preserve">Sed Å¿uperius (vbi. α.) iam probauimus eandem proportio nem eſſe inter <var>.m.n.</var> & <lb/><var>n.x.</var> quæ inter <var>.a.f.</var> et <var>.d.g.</var> ideo ex conuerÅ¿a pro portion alitate ita erit ipÅ¿ius <var>.x.n.</var> ad <var>.n.<lb/>m.</var> vt ipÅ¿ius <var>.d.g.</var> ad <var>.a.f.</var> Å¿ed dupli <var>.x.n.</var> ad Å¿implum <var>.x.n.</var> eÅ¿t vt dupli <var>.d.g.</var> ad <var>.d.g</var>. </s> - <s xml:id="s4664" xml:space="preserve">Qua <lb/>re ex .22. quinti dupli <var>.x.n.</var> ad <var>.m.n.</var> erit vt dupli <var>.d.g.</var> ad <var>.a.f.</var> & ex .18. eiuÅ¿dem ita erit <lb/>dupli <var>.x.n.</var> cum Å¿implo <var>.m.n.</var> ad <var>.m.n.</var> vt dupli <var>.d.g.</var> cum Å¿implo <var>.a.f.</var> ad <var>.a.f</var>. </s> - <s xml:id="s4665" xml:space="preserve">Quare Å¿olidi <lb/> - <anchor type="figure" xlink:label="fig-0405-01a" xlink:href="fig-0405-01"/> - <pb o="394" rhead="IO. BAPT. BENED." n="406" file="0406" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0406"/> - iam dicti ad cubum inÅ¿ius <var>.a.f.</var> ex .11. quinti erit vt dupli <var>.x.n.</var> <reg norm="cum" type="context">cÅ©</reg> Å¿implo <var>.m.n.</var> ad <var>.m.n</var>.</s> - </p> - <div type="float" level="5" n="16"> - <figure xlink:label="fig-0405-01" xlink:href="fig-0405-01a"> - <image file="0405-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0405-01"/> - </figure> - </div> - <note position="left" xml:space="preserve">δ</note> - <!-- note-0406-01 --> - <p> - <s xml:id="s4666" xml:space="preserve">Superius autem vbi. β. demonÅ¿tratum fuit ita eſſe ipÅ¿ius <var>.m.n.</var> ad <var>.n.t.</var> vt cubi <var>.m.n.</var> <lb/>ad cubum <var>.x.n.</var> & inter. α et. β probatum fuit ita eſſe cubi <var>.a.f.</var> ad cubum <var>.d.g.</var> vt <lb/>cubi <var>.m.n.</var> ad cubum <var>.x.n</var>. </s> - <s xml:id="s4667" xml:space="preserve">Vnde ex .11. quinti <var>.m.n.</var> ad <var>.n.t.</var> erit vt cubi <var>.a.f.</var> ad cubum <lb/><var>d.g</var>.</s> - </p> - <p> - <s xml:id="s4668" xml:space="preserve">Dicit poÅ¿tea quod eadem proportio erit inter cubum <var>.d.g.</var> & corpus illud quod <lb/>pro baÅ¿i habeat quadratum inÅ¿ius <var>.d.g.</var> altitudinem verò vt dictum eÅ¿t, quæ eÅ¿t inter <lb/><var>d.g.</var> & compoÅ¿itum ex duplo <var>.a.f.</var> cum Å¿implo <var>.d.g.</var> quod compoÅ¿itum eÅ¿t altitudo di <lb/>cta, & <reg norm="verum" type="context">verÅ©</reg> dicit ex ratione Å¿uperius allegata pro reliquo corpore & cubo ipÅ¿ius <var>.a.f</var>. <lb/></s> - <s xml:id="s4669" xml:space="preserve">Quare etiam quemadmodum <var>.t.n.</var> Å¿e habet ad duplum ipÅ¿ius <var>.o.n.</var> cum Å¿implo <var>.t.n.</var> <lb/>ex ijÅ¿dem rationibus Å¿upradictis, vbiloquuti Å¿umus de <var>.x.n.</var> cum <var>.m.n</var>.</s> - </p> - <p> - <s xml:id="s4670" xml:space="preserve">DiÅ¿ponantur <reg norm="nunc" type="context">nÅ©c</reg> omnia tali ordine, ita vt <var>.u.</var> primum Å¿it corpus quod pro Å¿ua ba <lb/>Å¿i habeat quadratum ipÅ¿ius <var>.a.f.</var> & c.</s> - </p> - <p> - <s xml:id="s4671" xml:space="preserve">Et <var>.y.</var> Å¿it cubus ipÅ¿ius <var>.a.f.</var> et <var>.s.</var> Å¿it cubus ipÅ¿ius <var>.d.g.</var> et <var>.z.</var> Å¿it corpus quod baÅ¿im ha-<lb/>bet quadratum ipÅ¿ius <var>.d.g.</var> altitudinem verò vt Å¿upradictum eÅ¿t, et <var>.p.</var> Å¿it compoÅ¿itum <lb/>dupli <var>.n.x.</var> cum Å¿implo <var>.m.n.</var> et <var>.l.</var> Å¿it compoÅ¿itum dupli ipÅ¿ius <var>.n.o.</var> cum Å¿implo <var>.t.n.</var> <lb/>Sed <var>.u.</var> locata Å¿it è regione <var>.p.</var> et <var>.y.</var> è regione <var>.m.n.</var> et <var>.s.</var> è regione <var>.n.t.</var> et <var>.z.</var> è regione <var>.l.</var> <lb/>& habebimus proportionem ipÅ¿ius <var>.u.</var> ad <var>.y.</var> vt <var>.y.</var> ad <var>.m.n.</var> & ipÅ¿ius <var>.y.</var> ad <var>.s.</var> vt <var>.m.n.</var> ad <var>.<lb/>n.t.</var> quod Å¿uperius iam demonÅ¿tratum fuit, vbi, δ. et <var>.s.</var> ad <var>.z.</var> ita Å¿e habebit vt <var>.n.t.</var> ad <var>.<lb/>l.</var> vt vltimò probatum fuit. </s> - <s xml:id="s4672" xml:space="preserve">Quare ex .22. quinti ita Å¿e habebit <var>.u.</var> ad <var>.z.</var> vt <var>.p.</var> ad <var>.l.</var> <lb/>quemadmodum dicit Archi.</s> - </p> - <p> - <s xml:id="s4673" xml:space="preserve">Et quia vt Å¿e habet <var>.u.</var> ad <var>.z.</var> ita facta fuit <var>.h.i.</var> ad <var>.i.K.</var> vbi <var>.R.</var> ideo ex .11. quinti vt Å¿e <lb/>habet <var>.h.i.</var> ad <var>.i.K.</var> ita Å¿e habebit <var>.p.</var> ad <var>.l.</var> vt ipÅ¿e dicit: </s> - <s xml:id="s4674" xml:space="preserve">Et ex .18. quinti ita erit <var>.h.K.</var> <lb/>ad <var>.K.i.</var> vt <var>.p.l.</var> ad <var>.l.</var> & ex communi conceptu <var>.g.f.</var> Å¿e habebit ad <var>.h.K.</var> vt quintuplum <lb/>ipÅ¿ius <var>.p.l.</var> ad <var>.p.l.</var> & ex .22. eiuÅ¿dem ita Å¿e habebit <var>.f.g.</var> ad <var>.i.k.</var> vt quintuplum ipÅ¿ius <var>.p.<lb/>l.</var> ad <var>.l.</var> quintuplum autem ipÅ¿ius <var>.p.l.</var> compoÅ¿itum eÅ¿t ex quintuplo ipÅ¿ius <var>.n.m.</var> cum <lb/>decuplo ipÅ¿ius <var>.n.x.</var> cum quintuplo ipÅ¿ius <var>.n.t.</var> cum decuplo ipÅ¿ius <var>.n.o.</var> vt à te facilè <lb/>computare potes.</s> - </p> - <p> - <s xml:id="s4675" xml:space="preserve">Verum etiam erit ex communi Å¿cientia quod <var>.g.f.</var> ad <var>.f.k.</var> eÅ¿t ut quintuplum ipÅ¿ius <lb/><var>p.l.</var> ad duplum ipÅ¿ius <var>.p.l.</var> eo quod Å¿uperius Å¿uppoÅ¿itum fuit <var>.h.K.</var> eſſe <reg norm="quintam" type="context">quintã</reg> mediam, <lb/>vnde <var>.k.f.</var> relinquebatur pro duabus quintis inferioribus, duplum autem <var>.p.l.</var> com-<lb/>poÅ¿itum eÅ¿t ex duplo ipÅ¿ius <var>.m.n.</var> cum duplo ipÅ¿ius <var>.n.t.</var> cum quadruplo ipÅ¿ius <var>.n.x.</var> & <lb/>cum quadruplo ipÅ¿ius <var>.x.o</var>.</s> - </p> - <p> - <s xml:id="s4676" xml:space="preserve">Ex conuerÅ¿a proportionalitate deinde ita Å¿e habet, <var>i.K.</var> ad <var>.i.k.</var> ad <var>.f.g.</var> vt <var>.l.</var> ad quin-<lb/>tuplum ipÅ¿ius <var>.p.l.</var> et <var>.k.f.</var> ad <var>.f.g.</var> vt duplum ipÅ¿ius <var>.p.l.</var> ad quintuplum ipÅ¿ius <var>.p.l</var>. </s> - <s xml:id="s4677" xml:space="preserve">Vnde <lb/>ex .24. quinti <var>.i.f.</var> Å¿e habebit ad <var>.f.g.</var> vt <reg norm="duplum" type="context">duplÅ©</reg> ipÅ¿ius <var>.p.l.</var> cum Å¿implo <var>.l.</var> ad quintuplum <lb/>ipÅ¿ius <var>.p.l</var>. </s> - <s xml:id="s4678" xml:space="preserve">Deinde ex conuerÅ¿a proportionalitate quintuplum ipÅ¿ius <var>.p.l.</var> Å¿e habebit <lb/> - <anchor type="note" xlink:label="note-0406-02a" xlink:href="note-0406-02"/> - ad duplum ipÅ¿ius <var>.p.l.</var> cum Å¿implo <var>.l.</var> vt <var>.f.g.</var> ad <var>.f.i</var>. </s> - <s xml:id="s4679" xml:space="preserve">Sed compoÅ¿itum dupli ipÅ¿ius <var>.p.l.</var> <lb/>cum Å¿implo <var>.l.</var> æquale eÅ¿t duplo ipÅ¿ius <var>.m.n.</var> cum quadruplo ipÅ¿ius <var>.x.n.</var> cum Å¿excuplo <lb/>ipÅ¿ius <var>.o.n.</var> cum triplo ipÅ¿ius <var>.n.t.</var> vt per te computare potes.</s> - </p> - <div type="float" level="5" n="17"> - <note xlink:label="note-0406-02" xlink:href="note-0406-02a" position="left" xml:space="preserve">θ</note> - </div> - <p> - <s xml:id="s4680" xml:space="preserve">Superius enim Å¿umpta fuit <var>.i.r.</var> ad quam ita Å¿e haberet <var>.f.h.</var> hoc eÅ¿t tres quintæ ip-<lb/>Å¿ius <var>.f.g.</var> vt <var>.m.t.</var> ad <var>.t.n</var>. </s> - <s xml:id="s4681" xml:space="preserve">Quare ex conuerÅ¿a proportionalitate ita Å¿e habebit <var>.i.r.</var> ad tres <lb/>quintas ipÅ¿ius <var>.f.g.</var> vt <var>.t.n.</var> ad <var>.t.m</var>. </s> - <s xml:id="s4682" xml:space="preserve">Et quia <var>.o.n.</var> Å¿umpta fuit æqualis ipÅ¿i <var>.b.g.</var> et <var>.m.n.</var> ipÅ¿i <lb/><var>b.f.</var> ideo <var>.m.o.</var> ex communi Å¿cientia æ qualis erit ipÅ¿i <var>.g.f</var>. </s> - <s xml:id="s4683" xml:space="preserve">Vnde proportio <var>.r.i.</var> ad tres <lb/>quintas ipÅ¿ius <var>.m.o.</var> erit vt <var>.n.t.</var> ad <var>.t.m.</var> vt inquit Archi.</s> - </p> - <p> - <s xml:id="s4684" xml:space="preserve">Sed vbi. θ. iam probauimus ita Å¿e habere <var>.i.f.</var> ad <var>.f.g.</var> vt duplum <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <var>.p.l.</var> cum Å¿im-<lb/>plo <var>.l.</var> Å¿e habet ad quintuplum ipÅ¿ius <var>.p.l.</var> hoc eÅ¿t <var>.i.f.</var> ad <var>.m.o.</var> vt duplum ipÅ¿ius <var>.p.l.</var> cum <lb/>Å¿implo <var>.l.</var> ad quintuplum ipÅ¿ius <var>.p.l</var>.</s> - </p> - <pb o="395" rhead="EPISTOL AE." n="407" file="0407" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0407"/> - <p> - <s xml:id="s4685" xml:space="preserve">Habemus igitur <reg norm="nuncomnem" type="context">nuncomnẽ</reg><unsure/>s illas conditiones quas Archimedes in præcedenti <lb/>propoÅ¿itione Å¿upponit. </s> - <s xml:id="s4686" xml:space="preserve">Vnde ex rationibus ibi allegatis Å¿equitur <var>.f.r.</var> eſſe duas quin-<lb/>tas ipÅ¿ius <var>.m.n.</var> hoc eÅ¿t ipÅ¿ius <var>.f.b</var>. </s> - <s xml:id="s4687" xml:space="preserve">Quapropter punctum <var>.r.</var> centrum erit ponderis to-<lb/>tius Å¿ectionis parabolæ ex .8. Å¿ecundi lib. de ponderibus eiuÅ¿dem Archimedis.</s> - </p> - <p> - <s xml:id="s4688" xml:space="preserve">Inquit nunc Archimedes, quod exiÅ¿tente <var>.q.</var> centro ponderis ipÅ¿ius parabolæ <var>.d.<lb/>b.e.</var> partialis, centrum fruÅ¿ti erit in linea recta <var>.q.r.f.</var> ita remotum à centro <var>.r.</var> quod <lb/>proportio <var>.q.r.</var> ad partem illam ipÅ¿ius <var>.r.f.</var> quæ reperitur inter centrum <var>.r.</var> & centrum <lb/>huius fruÅ¿ti æqualis eÅ¿t proportioni totius parabolæ ad partialem. </s> - <s xml:id="s4689" xml:space="preserve">Quod quidem ve <lb/>rum eÅ¿t ex .8. primi libri eiuÅ¿dem.</s> - </p> - <p> - <s xml:id="s4690" xml:space="preserve">Inquit etiam punctum <var>.i.</var> illud eſſe, eo quod cum probatum Å¿it <var>.f.r.</var> duas quintas eÅ¿-<lb/>Å¿e ipÅ¿ius <var>.f.b.</var> ideo <var>.b.r.</var> tres quintas erit ipÅ¿ius <var>.b.f.</var> vt ipÅ¿e dicit.</s> - </p> - <figure position="here"> - <image file="0407-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0407-01"/> - </figure> - <pb o="396" rhead="IO, BAPT. BENED." n="408" file="0408" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0408"/> - <p> - <s xml:id="s4691" xml:space="preserve">Sed <var>.q.b.</var> Å¿imiliter tres quintæ eÅ¿t ipÅ¿ius <var>.d.b.</var> ex .8. prædicta. </s> - <s xml:id="s4692" xml:space="preserve">Quare <var>.q.r.</var> tres quintæ <lb/>erit ipÅ¿ius <var>.f.g.</var> ex .19. quinti.</s> - </p> - <p> - <s xml:id="s4693" xml:space="preserve">Dicamus igitur hoc modo cum <var>.f.b.</var> totum ad totum <var>.b.r.</var> ita Å¿e habeat vt abÅ¿ciÅ¿-<lb/>Å¿um <var>.b.g.</var> ad abÅ¿ciſſum <var>.q.b.</var> ex .7. et .8. dicti primi libri eiuÅ¿dem ideo reÅ¿iduum <var>.f.g.</var> ex <lb/><var>f.b.</var> ad reÅ¿iduum <var>.r.q.</var> ex <var>.r.b.</var> erit vt totum <var>.f.b.</var> ad. totum <var>.r.b.</var> ex .19. quinti Eucli.</s> - </p> - <p> - <s xml:id="s4694" xml:space="preserve">Sed iam Å¿ub. β. probauimus ita Å¿e habere fruÅ¿tum <var>.a.d.e.c.</var> ad parabolam <var>.d.b.e.</var> vt <lb/><var>m.t.</var> ad <var>.t.n.</var> Å¿ed vt <var>.m.t.</var> ad <var>.t.n.</var> ita aſſ umpta fuit (vbi <var>.A.</var>). <var>i.r.</var> ad quam Å¿ic Å¿e haberet <var>.f.<lb/>h.</var> hoc eÅ¿t tres quintæ ipÅ¿ius <var>.f.g.</var> hoc eÅ¿t <var>.q.r</var>. </s> - <s xml:id="s4695" xml:space="preserve">quare ex .11. quinti prop ortio fruÅ¿ti <var>.a.<lb/>d.e.c.</var> ad parabolam partialem erit vt <var>.q.r.</var> ad <var>.r.i</var>. </s> - <s xml:id="s4696" xml:space="preserve">ExiÅ¿tente igitur <var>.r.</var> centro totius pa <lb/>rabolæ et <var>.q.</var> centro partialis, ergo <var>.i.</var> centrum erit fruÅ¿ti propoÅ¿iti.</s> - </p> - <p> - <s xml:id="s4697" xml:space="preserve">Sed Å¿i nullo Å¿olido intercedente, voluerimus centrum <var>.i.</var> fruÅ¿ti <var>.a.e.</var> citius inuenire, <lb/>inueniemus primò centrum <var>.r.</var> totius figuræ ex .8. Å¿ecundi eiuÅ¿dem conÅ¿tituendo <var>.b.r.</var> <lb/>tres quintas totius axis <var>.b.f.</var> & centrum <var>.q.</var> parabolæ <var>.d.b.e.</var> partialis Å¿imiliter.</s> - </p> - <p> - <s xml:id="s4698" xml:space="preserve">Nunc igitur manifeÅ¿tum eÅ¿t nobis, eandem proportionem fore ipÅ¿ius <var>.q.r.</var> <lb/>ad <var>.r.i.</var> quæ fruÅ¿ti <var>.a.e.</var> ad portionem <var>.d.b.e.</var> ex .8. dicta. </s> - <s xml:id="s4699" xml:space="preserve">Vnde ex coniuncta pro-<lb/>portionalitate ita Å¿e habebit <var>.q.i.</var> ad <var>.i.r.</var> vt <var>.a.b.c.</var> ad <var>.d.b.e.</var> Å¿ed vt <var>.a.b.c.</var> ad <var>.d.b.e.</var> ita Å¿e <lb/>habet <var>.m.n.</var> ad <var>.n.t.</var> eo quod vnaquæque harum duarum proportionum Å¿eÅ¿quialtera <lb/>eÅ¿t proportioni <var>.f.b.</var> ad <var>.b.g.</var> eo. quod <var>.f.b.</var> ad <var>.b.g.</var> ita Å¿e habet. vt <var>.m.n.</var> ad <var>.o.n.</var> </s> - <s xml:id="s4700" xml:space="preserve">quare <lb/><var>m.n.</var> ad <var>.t.n.</var> ita Å¿e habebit vt <var>.g.i.</var> ad <var>.r.i.</var> vnde diÅ¿iunctim <var>.m.t.</var> ad <var>.t.n.</var> ita Å¿e habebit vt <lb/><var>q.r.</var> ad <var>.r.i</var>. </s> - <s xml:id="s4701" xml:space="preserve">Iungatur igitur <var>.r.i.</var> quæ quidem <var>.r.i.</var> ita Å¿e habeat ad <var>.r.q.</var> vt <var>.t.n.</var> ad <var>.t.m.</var> vt <lb/>habeatur centrum fruÅ¿ti.</s> - </p> - <pb o="397" rhead="EPISTOL AE." n="409" file="0409" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0409"/> - <figure position="here"> - <image file="0409-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0409-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="43"> - <div type="letter" level="4" n="1"> - <head xml:id="head564" xml:space="preserve">DEFENSIO NOSTRA CONTRA ANTONIVM <lb/>Bergam, & Alexandrum Piccolhomineum.</head> - <head xml:id="head565" style="it" xml:space="preserve">IlluÅ¿tri Domino Horatio Muto.</head> - <p> - <s xml:id="s4702" xml:space="preserve"><emph style="sc">INter</emph> ea quæ olim contra Antonium Bergam, Å¿ermone Italico Å¿cripÅ¿i, <lb/>hoc vnum erat, quod ip Å¿e Berga non viderat quendam notatu dignum <lb/>errorem ipÅ¿ius Pi ccolhominei, vbi ipÅ¿e Alexander arguit quendam au-<lb/>thorem in tractatu de magnitudine terræ & aquæ pag .37. linea .26. ita di <lb/>cens, & erit maior aqua.</s> - </p> - <pb o="398" rhead="IO. BAPT. BENED." n="410" file="0410" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0410"/> - <p> - <s xml:id="s4703" xml:space="preserve">Quo in loco clare videtur ipÅ¿um putare eandem proportionem inter diametros, <lb/>quæ inter Å¿phæras ipÅ¿as eſſe, nec amplius recordari eius quod Å¿cripÅ¿erat pag .24.</s> - </p> - <p> - <s xml:id="s4704" xml:space="preserve">Piccolhom. igitur ibi <reg norm="Å¿upponens" type="context">Å¿upponẽs</reg> centrum <var>.D.</var> eſſe magnitudinis aquæ, & intra Å¿phæ <lb/>ram terreÅ¿trem, putat omnino cauÅ¿am eſſe vt terra Å¿uperet aquam magnitudine, qua-<lb/>Å¿i quod Å¿i punctum <var>.D.</var> vt centrum Å¿phæræ aquæ, vnum <reg norm="idemque" type="simple">idemqÌ;</reg> eſſet cum puncto <var>.E.</var> <lb/>extremo diametri ipÅ¿ius terræ, Å¿phæra <var>.A.G.H.</var> Å¿phæræ <var>.A.B.E.</var> dupla eſſe deberet, <lb/>quod quidem nullo pacto fieri poteÅ¿t, quamuis etiam proportio <var>.A.H.</var> ad <reg norm="diametrum" type="context">diametrÅ©</reg> <lb/><var>A.E.</var> Å¿uperbipartiensÅ¿eptimas exiÅ¿teret, quæ minor eſſet quam Å¿eÅ¿quitertia, ita quod <lb/>quando etiam <var>.D.E.</var> maior medietate ipÅ¿ius <var>.D.H.</var> fuiſſet, nihilominus tamen <lb/>terra minor eſſet aqua, eo quod proportio dupla minor eÅ¿t, quam tripla ad <reg norm="propor- tionem" type="context">propor-<lb/>tionẽ</reg> Å¿uperbipartientenÅ¿eptimas, & maior <reg norm="quam" type="context">quã</reg> tripla ad proportionem Å¿eÅ¿quiquar-<lb/>tam. </s> - <s xml:id="s4705" xml:space="preserve">Vnde Å¿i Piccolhom. Å¿uppòſuiſſet proportionem ipÅ¿ius <var>.D.H.</var> ad <var>.C.E.</var> eſſe <lb/>Å¿eÅ¿quiquartam, rectè profectò dixiſſet, Å¿ed dicere quod ubicunque exiÅ¿tat <reg norm="punctum" type="context">punctÅ©</reg> <var>.<lb/>D.</var> intra Å¿phæram terreÅ¿trem, Å¿equitur ipÅ¿am eſſe maiorem aquea, verum non eÅ¿t.</s> - </p> - <p> - <s xml:id="s4706" xml:space="preserve">ScripÅ¿i etiam quod Piccoloho. decipiebatur vbi loquitur de diaphaneitate aquæ <lb/>pag .40. ita dicens.</s> - </p> - <quote> - <s xml:id="s4707" xml:space="preserve">Et cum rationabiliter aliquis exiÅ¿timare non poteÅ¿t, quod vmbra quæ facit ori-<lb/>ri e cclipÅ¿es Lunæ, producta Å¿it à terra, & ab aqua Å¿imul, vt ab vno corpore aggre-<lb/>gato exijs duobus elementis, & ad vnam communem Å¿phæreceitatem reductis, pro <lb/>pterea quod cum vmbra produci debeat à corporibus opacis, quorum opacitas effi-<lb/>cit illa corpora vmbroÅ¿a, aqua autem, Å¿it corpus diaphanum, & tranÅ¿parens, nullam <lb/>vmbram poterit à ſe eminus producere.</s> - </quote> - <p> - <s xml:id="s4708" xml:space="preserve">Hic enim decipitur Piccolhom. duabus rationibus, quarum prima eÅ¿t, quod ra-<lb/>dius luminoÅ¿us non poteÅ¿t multum in profundum mergi, vt probaui in .8. epiÅ¿tola ad <lb/>Vimercatum, altera verò eÅ¿t, quod cum Å¿phærica Å¿it aqua maris, Å¿upponatur etiam <lb/>quod Å¿ub ea nulla terræ portio eſſet, & quod radij Å¿olares ipÅ¿am, non Å¿ecus ac pilam <lb/>ex criÅ¿tallo fabrÄ™factam penetrarent, cum autem ipÅ¿i radij, tam ab una, quam ab alia <lb/>parte Å¿up erficiei huiuÅ¿modi globi <reg norm="frangantur" type="context">frãgantur</reg>, ob diſſimilem diaphaneitatem inter ae <lb/>rem & aquam, ipÅ¿i Å¿einuicem interÅ¿ecarent, vt poÅ¿t pilam criÅ¿tallinam videre eÅ¿t, de-<lb/>inde procedentes, diÅ¿gregarentur, <reg norm="diÅ¿ciparenturque" type="simple">diÅ¿ciparenturqÌ;</reg> quouÅ¿que nullam vim illuminatio <lb/>nis haberent, quod quilibet experiri poterit mediante aliquo vaÅ¿e uitreo Å¿phærico, <lb/>aqua pleno, cuiuÅ¿uis magnitudinis, Å¿oli expoÅ¿ito.</s> - </p> - <p> - <s xml:id="s4709" xml:space="preserve">Rationes etiam quas eodem loco Piccolho. adducit ad probandum quod Å¿i quis <lb/>in fundo maris exiÅ¿teret, nullum uideret lumen, nihil ualent. </s> - <s xml:id="s4710" xml:space="preserve">Quarum prima eÅ¿t, <lb/>ubi ita dicit.</s> - </p> - <quote> - <s xml:id="s4711" xml:space="preserve">Ille qui Å¿e in aquam mergit, cum maiorem lucem, quæ Å¿upra aquam eÅ¿t, relin-<lb/>quat, iudicat pro magno temporis Å¿patio locum illum obÅ¿curum, quemadmodum <lb/>accidit quando per multum temporis Å¿patium fixis oculis in corpore Solis intuiti Å¿u <lb/>mus, ab eodem poÅ¿tea eoÅ¿dem amouentes, omnia obÅ¿cura nobis videntur.</s> - </quote> - <p> - <s xml:id="s4712" xml:space="preserve">IpÅ¿e autem non conÅ¿iderat quod talis obÅ¿curitas quæ Å¿equitur viÅ¿ionem maioris <lb/>luminis, parum durat, immo cito euaneÅ¿cit, Å¿ed in aqua nunquam reuertimur ad vi-<lb/>dendum, ne que veÅ¿tigium aliquod luminis ibi videtur, in fundo maris dico, quem-<lb/>admodum nobis nuntiauerunt hi qui margaritas expiÅ¿cantur in imis partibus ingen <lb/>tium æquorum indicorum.</s> - </p> - <p> - <s xml:id="s4713" xml:space="preserve">Secunda uerò ratio ipÅ¿ius Piccolhom. eÅ¿t ubi ita dicit.</s> - </p> - <quote> - <s xml:id="s4714" xml:space="preserve">Altera cauÅ¿a quod nobis obÅ¿curus appareat locus Å¿ub aqua, eſſe poteÅ¿t obÅ¿tacu-<lb/>lum quod aquæ habent ab opacitate terræ Å¿ub eorum fundo, etenim Å¿icut <reg norm="chriÅ¿tallum" type="context">chriÅ¿tallÅ©</reg> - <pb o="399" rhead="EPISTOLAE." n="411" file="0411" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0411"/> - quamuis <reg norm="perÅ¿picuum" type="context">perÅ¿picuÅ©</reg> Å¿iue <reg norm="tranÅ¿parens" type="context">tranÅ¿parẽs</reg> Å¿it, nihilominus propter obÅ¿taculum plumbi Å¿ub <lb/>ipÅ¿o poÅ¿iti, efficit vt radij viÅ¿uales repercuſſi reuertantur. ita etiam quamuis aqua Å¿it <lb/>corpus tranÅ¿parens, nihilominus propter obÅ¿taculum terræ opacæ, quæ Å¿ubÅ¿idet in <lb/>fundo maris efficere poteÅ¿t obÅ¿curas partes illas Å¿ub aqua, illis hominibus qui in <lb/>ipÅ¿a aqua mèrguntur.</s> - </quote> - <p> - <s xml:id="s4715" xml:space="preserve">In hac Å¿ecunda ratione decipitur Piccolhom. </s> - <s xml:id="s4716" xml:space="preserve">Primum quia Å¿i vÅ¿que ad imam par <lb/>tem maris, Solis radius ferri poſſet, ille qui ibi eſſet, attollens oculos Å¿urÅ¿um Solem <lb/>cerneret, </s> - <s xml:id="s4717" xml:space="preserve">deinde aÅ¿piciendo ipÅ¿um fundum Maris, videret illum, ratione reflexio-<lb/>nis luminis ab ipÅ¿o fundo, & ex eadem ratione Å¿peculi ab ipÅ¿o adducta, quæ contra <lb/>ipÅ¿um eÅ¿t.</s> - </p> - <p> - <s xml:id="s4718" xml:space="preserve">Decipitur etiam cum dicatradios viÅ¿uales à ſpeculo Å¿eu plumbo repercuti, eo <reg norm="quod" type="simple">ê™</reg> <lb/>non radij viÅ¿uales Å¿unt hi qui reflectuntur, Å¿ed Å¿unt radij luminoÅ¿i primarij, Å¿eu Å¿ecun <lb/>darij qui non ab oculis exeunt Å¿ed à corpore lucido.</s> - </p> - <p> - <s xml:id="s4719" xml:space="preserve">ScripÅ¿i etiam quod Å¿i verum eſſet proportionalitatem continuam <reg norm="quantitatum" type="context">quãtitatum</reg> ele-<lb/>mentorum ex proportione decupla conÅ¿tare, ignem pro maximo, terram verò pro <lb/>minimo terminorum Å¿umentes, totum aggregatum ex terra, aqua, aere, & igne, ita <lb/>eſſet maius terra, quemadmodum mille centum & vndecim ad vnum, vnde Å¿emidia <lb/>meter regionis elementaris eſſet quaÅ¿i aut paulo maior decuplo Å¿olum Å¿emidiame-<lb/>tro terræ, vnde inter conuexum ignis, & concauum minimi, Å¿eu inferioris orbis lu-<lb/>naris, relinqueretur quidam orbis vacuus Å¿piſſitudinis vnius interualli plus quam vi-<lb/>ginti terræ Å¿emidiametrorum, quod Å¿patium vacuum orbiculariter, maius exi-<lb/>Å¿teret ipÅ¿a totali regione elementari plus quam trigeÅ¿ies millies, immo Å¿i Å¿emidia <lb/>meter dicti primi orbislunaris maior eſſet terreÅ¿tri vt trigintanouem ad unum, <reg norm="dictus" type="simple">dictê°</reg> <lb/>orbis vacuus maior eſſet elementari regione plus quam .58208. ad vnum, proportio <lb/>nalitatem igitur continuam quæ ex decupla proportionalitate reÅ¿ultat in elementis <lb/>eſſe putare eÅ¿t maximus error.</s> - </p> - <p> - <s xml:id="s4720" xml:space="preserve">Subdit deinde Berga, hoc voluiſſe Platonem neceſſario requiri, vt extrema ele-<lb/>menta, nempeignis & terra cum duobus medijs aere, & aqua coniungerentur, cum in <lb/>corporibus Å¿olidis (quaÅ¿i BergÄ™ Å¿int quædam corpora quæ Å¿olida non extent) poſſit <lb/>dari medium æquale in geometrica proportione.</s> - </p> - <p> - <s xml:id="s4721" xml:space="preserve">Sed vbi Plato ad Å¿ermonem de numero elementorum Å¿e confert, poÅ¿tquam ra-<lb/>tione creationis ignis, & terrÄ™ Å¿e propoÅ¿uiſſe putat, vt <reg norm="idem" type="context">idẽ</reg> de alijs duobus corporibus <lb/>medijs probet, comparatione proportionalitatis continuæ geometricæ in tribus ter-<lb/>minis, ratione rerum Å¿uperficialium primò, deinde in quatuor, ratione corporearum <lb/>vtitur, ita dicens.</s> - </p> - <quote> - <s xml:id="s4722" xml:space="preserve">Vinculorum verò ideÅ¿t aptiſſimum atque pulcherrimum quod exÅ¿e, & ex ijs quÄ™ <lb/>aÅ¿tringunt, quam maximè vnum efficit, <reg norm="&c." type="unresolved">&c.</reg></s> - </quote> - <p> - <s xml:id="s4723" xml:space="preserve">Quo in loco Plato inÅ¿erre vult de proportionalitate geometrica trium termino-<lb/>rum, in qua ijdem ita Å¿e habent, vt medius, primi, <reg norm="vltimique" type="simple">vltimiqÌ;</reg> vice fungatur, ita vt vtriuÅ¿-<lb/>que ipÅ¿orum extremorum particeps fiat, cum productum quod à medio termino in <lb/>Å¿eipÅ¿o progignitur idem Å¿it ei quod ab extremis fuit, vnde medius, potentia idem eÅ¿t <lb/>quod productum ab extremis.</s> - </p> - <p> - <s xml:id="s4724" xml:space="preserve">Subdit deinde Plato dicens.</s> - </p> - <quote> - <s xml:id="s4725" xml:space="preserve">Quando enim in tribus numeris, aut molibus, aut viribus, medium ita Å¿e habet <lb/>ad poÅ¿tremum vt primum ad medium, <reg norm="viciſſimque" type="simple">viciſſimqÌ;</reg> vt poÅ¿tremum cum medio, ita me-<lb/>dium cum primo congruit, </s> - <s xml:id="s4726" xml:space="preserve">tunc quod medium eÅ¿t, & primum fit & <reg norm="poÅ¿tremum" type="context">poÅ¿tremÅ©</reg>, po-<lb/>Å¿tremum quoque, & primum & media fiunt.</s> - </quote> - <pb o="400" rhead="IO. BAPT. BENED." n="412" file="0412" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0412"/> - <p> - <s xml:id="s4727" xml:space="preserve">Hic <reg norm="animaduertendum" type="context">animaduertendÅ©</reg> eÅ¿t omnes interpretes falli, qui hoc loco Platonem de omni-<lb/>bus proportionalitatibus continuis quæ ternario numero (alia enim Arithmetica, <lb/>alia geometrica, alia harmonica dicitur) continentur, intelligendum eſſe cenÅ¿ent, <lb/>quia de numeris, magnitudinibus, <reg norm="viribusque" type="simple">viribusqÌ;</reg>, aut ut dici Å¿olet, virtutibus mentionem <lb/>fecerit. </s> - <s xml:id="s4728" xml:space="preserve">Plato enim nihil aliud inferre voluit, quam eandem paſſionem (ut ipÅ¿e reci-<lb/>tat) inter medium <reg norm="extremaque" type="simple">extremaqÌ;</reg> vnius proportionalitatis continuæ geometricæ, tam <lb/>in quantitate, quam in qualitate <reg norm="reÅ¿ultaturam" type="context">reÅ¿ultaturã</reg>, cum tres termini eiuÅ¿dem eſſent Å¿pe-<lb/>ciei, & quia quantitas in duas principes <reg norm="primariasque" type="simple">primariasqÌ;</reg> partes, ideÅ¿t in continuam, & <lb/>diÅ¿cretam diuiditur, hanc ob cauÅ¿am Plato hoc præcipuè Å¿ignificat numerorum ma-<lb/><reg norm="gnitudinisque" type="simple">gnitudinisqÌ;</reg> vocabulis vtens, quibus vniuerÅ¿um quantitatis genus complectitur.</s> - </p> - <p> - <s xml:id="s4729" xml:space="preserve">Cum verò ait vires, uniuerÅ¿um qualitatis genus inferre uult. </s> - <s xml:id="s4730" xml:space="preserve">Quia proportio & <lb/>proportionalitas tam continua quam diÅ¿creta, non Å¿olum interterminos quanti, Å¿ed <lb/>inter eos etiam qui quali attribuuntur elucet.</s> - </p> - <p> - <s xml:id="s4731" xml:space="preserve">Sed quod eo loco de harmonica proportionalitate quæ <reg norm="cum" type="context">cÅ©</reg> geometrica magis Å¿im <lb/>bola eÅ¿t quam cum Arithmetica Plato minime intelligat, ex eiuÅ¿dem uerbis cum ita <lb/>Å¿cribit manifeÅ¿tè patet.</s> - </p> - <quote> - <s xml:id="s4732" xml:space="preserve">Quando enim medium ita Å¿e habet ad poÅ¿tremum ut primum ad medium, <reg norm="uiciÅ¿- Å¿imque" type="simple">uiciÅ¿-<lb/>Å¿imqÌ;</reg> ut poÅ¿tremum cum medio ita medium cum primo congruit.</s> - </quote> - <p> - <s xml:id="s4733" xml:space="preserve">Id enim in harmonica proportionalitate non cernitur in qua primus terminus ad <lb/>poÅ¿tremum, & non ad medium, ita Å¿e habet geometricè ut differentia inter primum <lb/>& medium ad differentiam inter medium & ultimum.</s> - </p> - <p> - <s xml:id="s4734" xml:space="preserve">Quod Å¿i clarum eÅ¿t ipÅ¿um de harmonica proportionalitate nullo modo intellige-<lb/>re, quanto minus de Arithmetica, quæ cum geometrica nihil habet commune.</s> - </p> - <p> - <s xml:id="s4735" xml:space="preserve">Cum uerò Plato ait.</s> - </p> - <quote> - <s xml:id="s4736" xml:space="preserve">Tunc quod medium eÅ¿t & primum fit & poÅ¿tremum, poÅ¿tremum quoque, & pri-<lb/>mum media fiunt, <reg norm="&c." type="unresolved">&c.</reg></s> - </quote> - <p> - <s xml:id="s4737" xml:space="preserve">Nihil aliud oÅ¿tendere uult, quam Å¿imilitudinem quæ inter huiuÅ¿modi medium & <lb/>extrema intercedit, cum ipÅ¿um medium ad poÅ¿tremum, quem primus ad Å¿eipÅ¿um, <lb/>eundem reÅ¿pectum habeat, in quo eÅ¿t Å¿imilis primo, & contra ad primum <reg norm="terminum" type="context">terminÅ©</reg>, <lb/>eundem reÅ¿pectum, quem poÅ¿tremum ad Å¿eipÅ¿um habet, </s> - <s xml:id="s4738" xml:space="preserve">unde hac ratione ultimum <lb/>repręſentat, uolens Plato inferre de conuenientia quÄ™ inter media elementa, & ex-<lb/>trema intercedit, ut aquæ inter aerem, & terram, cum aqua, ratione Å¿uæ frigiditatis, <lb/>terrÄ™, ratione uero Å¿uæ humiditatis aeri Å¿imilis euadat. </s> - <s xml:id="s4739" xml:space="preserve">Aer uero qui inter ignem, <lb/><reg norm="aquamque" type="simple">aquamqÌ;</reg> ponitur quod ad caliditatem attinet cum igne, quod uero ad humidita-<lb/>tem Å¿pectat cum aqua communicet.</s> - </p> - <p> - <s xml:id="s4740" xml:space="preserve">Sed quia Plato multis in rebus doctrinam Pythagoricam Å¿equutus eÅ¿t, Pythago-<lb/>rici aut em omnia numeris metiebantur, & de omnire Å¿ecundum numerorum ratio <lb/>nem diſſerebant, <reg norm="uidensque" type="simple">uidensqÌ;</reg> Plato quod inter duos numeros Å¿uperficiales, <reg norm="inuicemque" type="simple">inuicemqÌ;</reg> <lb/>Å¿imiles exiÅ¿tentes, unum tantum numerum medium in proportionalitate continua <lb/>geometrica cadere poteÅ¿t, ideo Å¿ubiungit.</s> - </p> - <quote> - <s xml:id="s4741" xml:space="preserve">Quod Å¿i uniuerÅ¿i corpus latitudinem habere debuiſſet, nullam uerò profundita-<lb/>tem, unum Å¿anè, tum ad Å¿eipÅ¿um, tum ad extrema uincienda interiectum medium <lb/>Å¿uffeciſſet.</s> - </quote> - <p> - <s xml:id="s4742" xml:space="preserve">Sequitur poÅ¿tea Å¿ic.</s> - </p> - <quote> - <s xml:id="s4743" xml:space="preserve">Sed cum Å¿oliditatem mundus requireret, Å¿olida uerò non uno, Å¿ed duobus Å¿em-<lb/>per modis copulentur, inter ignem, & terram, Deus, Aerem, AquamqÌue loca-<lb/>uit, <reg norm="&c." type="unresolved">&c.</reg></s> - </quote> - <pb o="401" rhead="EPISTOLAE." n="413" file="0413" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0413"/> - <p> - <s xml:id="s4744" xml:space="preserve">Volens inferre, quod quemadmodum inter duos numeros Å¿olidos, & inuicem <lb/>Å¿imiles, <reg norm="vnus" type="simple">vnê°</reg> <reg norm="tantum" type="context context">tãtÅ©</reg> medius proportionalis intercedere <reg norm="non" type="context">nõ</reg> poteÅ¿t, Å¿ed duo neceſſariò re <lb/>quiruntur (vt exijs quæ Euclid .8. lib. 16. 17. 18. et .19. propoÅ¿itione proponit videnÂ<lb/>tur) ita <reg norm="dictante" type="context">dictãte</reg> ratione inter igneum, <reg norm="terreumque" type="simple">terreumqÌ;</reg> corpus duo corpora interiecta <reg norm="eÅ¿sent" type="context">eÅ¿sẽt</reg>, <lb/>non ratione proportionalitatis continuæ in quantitate <reg norm="eorundem" type="context">eorÅ©dem</reg> corporum, Å¿ed pro <lb/>pter Å¿imilitudinem connexionis, cum productum ex duobus medijs proportionali-<lb/>bus æquale Å¿it producto ab extremis, & idem reÅ¿pectus, quem primum ipÅ¿orum qua <lb/>tuor ad Å¿ecundum habet, Å¿ecundi ad tertium extet, vnde Å¿ecundum primo Å¿imile <lb/>euadit, & contra, reÅ¿pectus qui eÅ¿t quarti ad tertium, Å¿it etiam tertij ad Å¿ecundum, vn-<lb/>de ipÅ¿um tertium, ratione vltimi Å¿ubit, & eius imaginem induit, & hanc ob cauÅ¿am <lb/>Å¿ic Å¿cribit Plato.</s> - </p> - <quote> - <s xml:id="s4745" xml:space="preserve">Propterea ex huiuÅ¿modi rebus numero quaternario concluÅ¿is, mundi corpus con <lb/>flatum eÅ¿t, ea connexum comparatione qua dixi. </s> - <s xml:id="s4746" xml:space="preserve">Ex quo Å¿eipÅ¿um amicitia concor-<lb/>di complectitur, <reg norm="&c." type="unresolved">&c.</reg></s> - </quote> - <p> - <s xml:id="s4747" xml:space="preserve">Vbi Platonem, elementa maiora, <reg norm="minoranue" type="context">minorãue</reg> in proportionalitate continua, nec <lb/>geometrica, nec alterius cuiuÅ¿uis generis eſſe noluiſſe, clarè perÅ¿picitur, Å¿ed huiuÅ¿mo <lb/>di Å¿imilitudine, in eo quod media elementa cum extremis conueniunt eÅ¿t vÅ¿us, quæ <lb/>quidem conuenientia, nullibi maior, quam in proportionalitate continua geome-<lb/>trica reperitur. </s> - <s xml:id="s4748" xml:space="preserve">Sed etiam Å¿i Plato de huiuÅ¿modi corporea elementorum magnitu-<lb/>dine Å¿eipÅ¿um intelligi voluiſſet, Å¿i Å¿emidiameter regionis elementaris ex Ä™quo vt .39 <lb/>ad vnum, reÅ¿pectu Å¿emidiametri terræ fuiſſet, aqua, ipÅ¿am terram, magis quam tri-<lb/>geÅ¿ies, & octies, non Å¿olum decies, & aer quoque eandem magis quam .1500. & <lb/>ignis magis quam .55000. partibus magnitudine Å¿uperaret.</s> - </p> - <p> - <s xml:id="s4749" xml:space="preserve">SubÅ¿tantia vero rerum quas Å¿cripÅ¿eram circa finem illius conÅ¿iderationis talis fuit.</s> - </p> - <p> - <s xml:id="s4750" xml:space="preserve">Nunc autem tempus eſſe videtur, vt ego etiam, ne tantum deÅ¿truxiſſe, Å¿ed etiam <lb/>conÅ¿truxiſſe videar aliquid pro veritate diſſeram.</s> - </p> - <p> - <s xml:id="s4751" xml:space="preserve">Non eÅ¿t igitur dubium, Å¿olidæ doctrinæ viris, quin præſtantiſſimus Piccolo. Å¿e-<lb/>cutus Å¿it tutam viam ad explorandum, quod terra maior Å¿it quam aqua, metiendo <lb/>vtriuÅ¿que horum corporum Å¿uperficiem detectam. </s> - <s xml:id="s4752" xml:space="preserve">Omittamus autem compenÅ¿a-<lb/>tionem illam curuitatis, & concauitatis vallium, & montium, <reg norm="&c." type="unresolved">&c.</reg> quam ipÅ¿e Piccolo. <lb/>propè finem Å¿exti cap. vellet dare fluminibus, Å¿tagnis, fontibus, & eiuÅ¿modi aquis. <lb/></s> - <s xml:id="s4753" xml:space="preserve">eo enim in loco labitur Piccolo. vbi non conÅ¿iderat, quod eiuÅ¿modi obliquis Å¿uper-<lb/>ficiebus non reÅ¿pondent anguli Å¿olidi centri Å¿phæræ, qui reÅ¿piciunt eorum baÅ¿im ad <lb/>rectos angulos. </s> - <s xml:id="s4754" xml:space="preserve">Sed poÅ¿tquam Piccolo. comperit Å¿uperficiem terrÄ™ detectam, eſſe <lb/>maiorem apparente Å¿uperficie Å¿phærica aquæ, proculdubio poterat concludere ter-<lb/>ram eſſe maiorem aqua, Å¿icuti fecit, <reg norm="etiam" type="context">etiã</reg> Å¿i aqua profunda eſſet pyramidaliter <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> <lb/>ad mundi centrum, ideÅ¿t .3500. milliaria, Å¿upponendo tantum eſſe huius globi Å¿emi <lb/>diametrum.</s> - </p> - <p> - <s xml:id="s4755" xml:space="preserve">Verum quia poſſet aliquis dubitare circa diligentiam Piccolo. in hiÅ¿cæ duabus Å¿u <lb/>perficiebus dimetiendis, viÅ¿um eÅ¿t mihi non alienum Å¿equi aliam viam pro hac veri <lb/>tate probanda, Å¿upponendo verum eſſe, quod non vnus Å¿olus metitus fuerit, Å¿ed mul <lb/>ti, ideÅ¿t Å¿upponendo <reg norm="verum" type="context">verÅ©</reg> eſſe quod maris profunditas menÅ¿urari poſſit, & præterea, <lb/>quod non modo ipÅ¿ius maris maxima profunditas non perueniat ad quingentos paÅ¿ <lb/>Å¿us, Å¿icuti refert Piccolo. in fine Å¿ui tractatus, & mihi aſſeruerunt HiÅ¿pani multi, & <lb/>LuÅ¿itani præſtantiſſimi nautæ, tum Venetijs, tum Parmæ, in Aula Sereniſſimæ quon <lb/>dam Principis, inter quos, Venetijs fuit IlluÅ¿tris Rodericus Guzmanus, Dominus <lb/>FranciÅ¿cus Lopes, Dominus Garzias de Seuilia, <reg norm="multique" type="simple">multiqÌ;</reg> alij. </s> - <s xml:id="s4756" xml:space="preserve">Parmæ autem varij - <pb o="402" rhead="IO. BAPT. BENED." n="414" file="0414" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0414"/> - quos omnes <reg norm="recenÅ¿erem" type="context">recenÅ¿erẽ</reg> moleÅ¿tum eſſet. </s> - <s xml:id="s4757" xml:space="preserve">Sed etiam Å¿upponendo quod maxima <lb/>pelagi profunditas Å¿it, non modo .500. paſſuum, Å¿ed etiam .500. millium paſſuum, vt <lb/>dixi, & quod mare Å¿it huius profunditatis, non vno in loco tantum, aut multis, Å¿ed <lb/>quod Å¿upra totam etiam faciem terræ, mare tantÄ™ profunditatis ipÅ¿am terram vn-<lb/>dique operiret, ideÅ¿t, quod vbicunque nunc terra detecta eÅ¿t, eſſet aqua, Å¿piſſitudi-<lb/>nis .500. millium paſſuum. </s> - <s xml:id="s4758" xml:space="preserve">Atque vt planius intelligar Å¿upponendo quod Å¿icuti to-<lb/>tus huius globi Å¿emidiameter eÅ¿t <reg norm="milliarium" type="context">milliariÅ©</reg> .3500. </s> - <s xml:id="s4759" xml:space="preserve">TerreÅ¿tris partis Å¿emidiameter <lb/>eſſet <reg norm="tantum" type="wordlist">tm̃</reg> .3000. & reliquum Å¿emidiametri, id eÅ¿t quingenta milliaria eſſet craſſitudo <lb/>Å¿iue profunditas orbis aquei, in quo nihil neceſſe eſſet laborare in dimetiendis fon-<lb/>tibus, fluminibus, lacubus, Å¿tagnis, paludibus, & huiuÅ¿modi particulis nullius momen <lb/>ti apud peritos, nec curare Å¿ubterraneas aquas cauernarum, aut aliorum terræ cauo-<lb/>rum, Å¿eu terræ porroÅ¿itatum, quæ omnia Å¿unt circa ipÅ¿ius terræ Å¿uperficiem. </s> - <s xml:id="s4760" xml:space="preserve">Quia ve <lb/>riÅ¿imile non eÅ¿t naturam eiuÅ¿modi caua Å¿iue Å¿pong oÅ¿itates produxiſſe demiſſius li-<lb/>bramenti maris. </s> - <s xml:id="s4761" xml:space="preserve">Supponendo igitur ea quæ nunc dicta Å¿unt, terra tamen eſſet ferè <lb/>duplo maior aqua, hoc eÅ¿t, vt .12. ad .7. </s> - <s xml:id="s4762" xml:space="preserve">Quod quidem, cuiuis mathematicæ philoÅ¿o-<lb/>phiæ mediocriter perito, Å¿upputatu facillnnum eÅ¿t. </s> - <s xml:id="s4763" xml:space="preserve">Cum proportio diametrorum, <lb/>Å¿eu Å¿emidiametrorum, tertia pars exiÅ¿tat proportionis eorundem Å¿phærarum. </s> - <s xml:id="s4764" xml:space="preserve">Sed <lb/>vt parum periti minore labore Å¿upputare poſſint.</s> - </p> - <p> - <s xml:id="s4765" xml:space="preserve">Primum Å¿ciendum eÅ¿t, quod Å¿upponendo diametrum globi, ex terra, & aqua com <lb/>poÅ¿iti, eſſe .3500. milliarium, & Å¿emidiametrum puræ terreÅ¿tris partis eſſe .3000. tan <lb/>tum, eiuÅ¿modi proportio erit ut .7. ad .6. quia communis maior numerator horum <lb/>duum Å¿emidiametrorum erit .500. qui in maiorem ingredietur Å¿epties, in minorem <lb/>a utem Å¿exies. </s> - <s xml:id="s4766" xml:space="preserve">Et eiuÅ¿modi proportio Å¿uperparticularis, vocatur Å¿eÅ¿quiÅ¿exta, cuius <lb/>triplum erit vt .57. cum Å¿exta parte ad .36. & idem erit inter dictum globum compo-<lb/>Å¿itum, & partem terreÅ¿trem Å¿implicem. </s> - <s xml:id="s4767" xml:space="preserve">Quare Å¿ubtrahendo puram, Å¿eu Å¿implicem <lb/>partem terreÅ¿trem, ex compoÅ¿ito, reliqua pars erit, vt .21. cum Å¿exta, pro quantitate <lb/>aquei orbis, ad quam, terreÅ¿tris quantitas .36. erit ferè in <reg norm="eadem" type="context">eadẽ</reg> proportione, quæ .12. <lb/>ad .7.</s> - </p> - <p> - <s xml:id="s4768" xml:space="preserve">Nunc fortaſſe alienum non erit videre quanto ferè maior eſſet terra, quam tota <lb/>aqua, non dico <reg norm="autem" type="context">autẽ</reg> Å¿olum de parte illa maximæ e ius profunditatis, quæ nuÅ¿quam <lb/>ad quingentos paſſus peruenit, Å¿ed de ficto illo orbe aqueo, profunditatis .500. paÅ¿-<lb/>Å¿uum, qui totum terreÅ¿trem orbem circundaret, & tegeret, Å¿upponendo quod per <lb/>quingentos paſſus profunditatis, quidquid eÅ¿t terra, eſſet aqua, ideÅ¿t Å¿uppoÅ¿ito quod <lb/>ex totius orbis compoÅ¿iti Å¿emidiametro exiÅ¿tente .3500. milliarium, purÄ™ terræ Å¿emi <lb/>diameter eſſet milliarium .3499. cum dimidio. </s> - <s xml:id="s4769" xml:space="preserve">Supponendo igitur, vt Å¿upradixi. <lb/></s> - <s xml:id="s4770" xml:space="preserve">Comperietur quod terra eſſet maior aqua amplius quam .2333. vicibus. </s> - <s xml:id="s4771" xml:space="preserve">Sed quia <lb/>partes terræ detectæ rumpunt eiuÅ¿modi fictum orbem aqueum, quæ quidem partes, <lb/>Å¿unt ampliores Å¿uperficiÄ™ aquæ, vt obÅ¿eruauit Piccolo. atque alij <reg norm="præſtantes" type="context">præſtãtes</reg> viri, ideo <lb/>Å¿equetur, vt terra Å¿it maior aqua amplius .4666. vicibus imo amplius quinquies mil-<lb/>lecuplo. </s> - <s xml:id="s4772" xml:space="preserve">Si autem quis diceret, in quantitate aquæ computari etiam illam, quæ gi-<lb/>gnatur ex vaporibus, qui globum hunc compoÅ¿itum circundant: </s> - <s xml:id="s4773" xml:space="preserve">reÅ¿pondeo quod <lb/>non modò ei concedo computari eiuÅ¿modi aquam, Å¿ed Å¿upponendo etiam quodto <lb/>tus locus à vaporibus occupatus, qui attolluntur .52. milliaria Å¿upra Å¿uperficiem <lb/>huius globi, vt iam Å¿upradictum eÅ¿t, totus eſſet aqueus, & amplius, Å¿upponendo quod <lb/>orbis hic aqueus eſſet Å¿piſſitudinis, Å¿iue altitudinis quingentorum milliarium Å¿upra <lb/>totum ipÅ¿um globum compoÅ¿itum. </s> - <s xml:id="s4774" xml:space="preserve">Tamen terra eſſet maior ipÅ¿a aqua ferè duplo; <lb/></s> - <s xml:id="s4775" xml:space="preserve">qua dere, quiÅ¿que eiuÅ¿modi Å¿upputationum peritus certior fieri poterit. </s> - <s xml:id="s4776" xml:space="preserve">Vnde iti- - <pb o="403" rhead="EPISTOLAE." n="415" file="0415" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0415"/> - dem affirmare poſſemus, terram non Å¿olum maiorem eſſe aqua, Å¿ed aqua & præte-<lb/>rea aere, Å¿i aer non tam altè pertingit, quam multi alij præter Piccolo. Å¿entiunt, qui <lb/>dicuntinde euenire quod aerea humiditas non tam altè aÅ¿cendere poteÅ¿t, quoniam <lb/>humiditas ipÅ¿a grauitatem Å¿ecum affert, præterquam quod nubium Å¿itus oÅ¿tendit Å¿u <lb/>pra eas materiam eſſerariorem quam Å¿int ipſę nubes, infra vero denÅ¿iorem. </s> - <s xml:id="s4777" xml:space="preserve">Corpo-<lb/>ra enim eouÅ¿que aÅ¿cendunt donec inueniunt conÅ¿titutionem mediam formæ æqua-<lb/>lis (vt ita dicam) Å¿uis. </s> - <s xml:id="s4778" xml:space="preserve">Quare materia illa quæ impropriè ignis vocatur (non enim <lb/>eÅ¿t ignis) incipit carere humiditate (qua mediante aer definitur) circa quinquage-<lb/>Å¿imum Å¿ecundum milliarium Å¿upra Å¿uperficiem terræ, vt iam Å¿upradixi à Vitellione <lb/>demonÅ¿tratum fuiſſe. </s> - <s xml:id="s4779" xml:space="preserve">AriÅ¿to. autem affert <reg norm="rationem" type="context">rationẽ</reg> quare nubes altius <reg norm="non" type="context">nõ</reg> <reg norm="tranÅ¿cendant" type="context context">tranÅ¿cẽdãt</reg>. <lb/></s> - <s xml:id="s4780" xml:space="preserve">Vnde apparet tertiam aeris regionem impropriè aerem appellari, Å¿i humiditate ca-<lb/>ret, vt ait AriÅ¿t. qua mediante aer definitur, immo potius retinet ignis naturam, vt <lb/>etiam aſſerunt interpretes AriÅ¿totelis in primum Meteororum. </s> - <s xml:id="s4781" xml:space="preserve">Qui AriÅ¿to. in locis <lb/>Å¿upra citatis itidem oÅ¿tendit Å¿e etiam huius modi eſſe opinionis.</s> - </p> - <p> - <s xml:id="s4782" xml:space="preserve">Quod autem attinet ad probandum quod Å¿uperficies terrÄ™ detecta Å¿it altior quam <lb/>Å¿uperficies detecta aquæ, id tam clarum eÅ¿t Å¿ua Å¿ponte philoÅ¿ophis, qui Å¿ciunt quid Å¿it <lb/>altum, quidue demiſſum, quod Å¿uperfluum eſſet quidquid Å¿uper hoc dicerem præ <lb/>terquam, quòd conÅ¿tat ex demonÅ¿tratione ab AriÅ¿to. Å¿acta textu 31. li .2. de cÅ“lo, in <lb/>quo agit de corporibus in aqua poÅ¿itis, vnde eiuÅ¿modi veritas planiſſimè aperitur. <lb/></s> - <s xml:id="s4783" xml:space="preserve">Omittimus etiam quod præſtantes Moderni omnes, eam pro manifeÅ¿tiſſima <reg norm="ponunt" type="context">ponÅ©t</reg>, <lb/>Å¿icutiapud omnes Å¿ani iudicij homines reuera exiÅ¿timatur.</s> - </p> - <p> - <s xml:id="s4784" xml:space="preserve">Hæc enim Å¿unt quæ in fine illius conÅ¿iderationis Å¿cripÅ¿eram.</s> - </p> - <p> - <s xml:id="s4785" xml:space="preserve">Anno autem præterito editus in lucem fuit tractatus quidam Pulcherrimus, ab Ex <lb/>cellentiſſimo, nec non Doctiſſimo viro AuguÅ¿tino Michele, Patritio Veneto, ad cor <lb/>roborandam opinionem antiquorum, vbi tot authoritates, <reg norm="totque" type="simple">totqÌ;</reg> rationes adducit, <lb/>vt nil amplius dici poſſit. </s> - <s xml:id="s4786" xml:space="preserve">Atego Å¿enÅ¿um, <reg norm="rationemque" type="simple">rationemqÌ;</reg>, & non authoritatem <reg norm="aliquam" type="context">aliquã</reg> <lb/>Å¿equutus Å¿um: </s> - <s xml:id="s4787" xml:space="preserve">cum verò dico Å¿enÅ¿um, de Å¿enÅ¿u illorum intelligo, qui profunditatem <lb/>maris metiti Å¿unt, vt non mihi Å¿olum, Å¿ed, & Piccolo. & alijs permultis retulerunt, <lb/>de ratione vero à me adducta, aliorum Å¿it iudicium.</s> - </p> - <quote> - <s xml:id="s4788" xml:space="preserve">SediÅ¿te mirabilis & Excellentiſſimus vir, verba mea non accepit in eo Å¿enÅ¿u, vt <lb/>ego Å¿cripÅ¿i, ita vt omnino alienas conÅ¿equentias Å¿ibi confingat, <reg norm="quemadmodum" type="context">quemadmodÅ©</reg> pag. <lb/>3. Å¿ui tractatus inquit, me non concedere naturam produxiſſe in magna quantitate, <lb/>atque immenÅ¿a, id totum, quod bonum, & neceſſarium eÅ¿t. </s> - <s xml:id="s4789" xml:space="preserve">Hanc enim conÅ¿equen <lb/>tiam ipÅ¿e colligit ex eo, quod ego pag .19. meæ conÅ¿iderationis contra Antonium <lb/>Bergam Å¿cripÅ¿eram, quod videntur multa corpora alijs nobiliora, nihilominus mi-<lb/>nora, eo quod quantitas non Å¿equitur nobilitatem, neque ab ea pender, ita vt res <lb/>illa quæ nobilior eÅ¿t, neceſſarium Å¿it vt etiam maior exiÅ¿tat. </s> - <s xml:id="s4790" xml:space="preserve">Sed Excellentiſſimus <lb/>iÅ¿tæ vir Å¿cribit ita me dixiſſe.</s> - </quote> - <p> - <s xml:id="s4791" xml:space="preserve">Multa immo infinita corpora Å¿unt nobilia, & neceſſaria, nihilominus Å¿unt paruÄ™ <lb/>molis.</s> - </p> - <p> - <s xml:id="s4792" xml:space="preserve">Vide igitur quantum hoc diÅ¿tat ab illo.</s> - </p> - <p> - <s xml:id="s4793" xml:space="preserve">Præterea cap .12. aliam conÅ¿equentiam facit, quam ego non tam amplam facio. <lb/></s> - <s xml:id="s4794" xml:space="preserve">„ IpÅ¿e enim me inferre vult in alijs terrÄ™ partibus cauernas non reperiri, eo quod Mon <lb/>„ tes Å¿int cauernoÅ¿i. </s> - <s xml:id="s4795" xml:space="preserve">AÅ¿pice quæſo. pag .29. meæ conÅ¿iderationis, & clarè videbis me <lb/>nullo modo negare illas concauitates Å¿eu porroÅ¿itates terræ extra montana loca, <lb/>circa Å¿uperficiem terræ, vÅ¿que ad æquilibrium, orbiculariter, infimæ profunditatis <lb/>maris.</s> - </p> - <p> - <s xml:id="s4796" xml:space="preserve">Sed putare inferius has porroÅ¿itates reperiri, cum nulla ratio nobis perÅ¿uaÅ¿ibilis - <pb o="404" rhead="IO. BAPT. BENED." n="416" file="0416" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0416"/> - adhuc ab aliquo prodita Å¿it, idoneum nullo pacto eſſet. </s> - <s xml:id="s4797" xml:space="preserve">Rationes autem ab ipÅ¿o Ex <lb/>cellentiſſimo AuguÅ¿tino adductas circa huiuÅ¿modi rem, alij dijudicent, de authori-<lb/>tatibus verò, nihil dicam, quia ab illis petendæ Å¿unt, qui profitentur tales facultates, <lb/>quorum vnius tantummodo authoritas præualere deberet, contra omnes alias <reg norm="eorum" type="context">eorÅ©</reg> <lb/>qui nunquam attigerunt Å¿ummis labris orificia harum <reg norm="Å¿cientiarum" type="context">Å¿cientiarÅ©</reg>. </s> - <s xml:id="s4798" xml:space="preserve">Vt Å¿i exempli gra-<lb/>tia non Å¿olum authoritas illorum virorum, quos ipÅ¿e recenÅ¿uit, Å¿ufficiens eſſet vt pu <lb/>ta Pioccolo. </s> - <s xml:id="s4799" xml:space="preserve">Naibodæ, Bordini, Clauij, <reg norm="reliquorumque" type="simple">reliquorumqÌ;</reg> fautorum verÄ™ opinionis, <lb/>Å¿ed FranciÅ¿ci Maurolici tantummodo, qui in primo Dialogo Å¿uæ coÅ¿mographiæ ita <lb/>inquit.</s> - </p> - <quote> - <s xml:id="s4800" xml:space="preserve">ExiÅ¿timo autem totum terræ corpus rigidum eſſe Å¿axum, nam Å¿i arena eſſet, aue <lb/>gleba fragilis, ita humorem imbiberet, vt cum eo quaÅ¿i confunderetur; </s> - <s xml:id="s4801" xml:space="preserve">huc ac-<lb/>cedit, quod Å¿i mineræ, ac rupes, quæ Å¿unt grauiſſimæ partes in ipÅ¿a plerunque Å¿uper-<lb/>ficie comperiuntur, multo magis apud centrum eſſe debent. </s> - <s xml:id="s4802" xml:space="preserve">Videtur ita ratio exi-<lb/>gere, vt grauiora centro quoque Å¿int propinquiora.</s> - </quote> - <p> - <s xml:id="s4803" xml:space="preserve">Hæcigitur Å¿ola authoritas, inÅ¿tar reliquarum omnium Å¿ufficere poſſet. </s> - <s xml:id="s4804" xml:space="preserve">Verum <lb/>de authoritatibus minime curandum eÅ¿t, vbi Å¿enÅ¿us, <reg norm="ratioque" type="simple">ratioqÌ;</reg> vera illis opponuntur.</s> - </p> - <p> - <s xml:id="s4805" xml:space="preserve">Quod autem numerus animalium aquatilium maior exiÅ¿tat numero terreÅ¿trium, <lb/>Å¿atis reÅ¿pondimus pag .41. noÅ¿træ conÅ¿iderationis.</s> - </p> - <p> - <s xml:id="s4806" xml:space="preserve">Sed in cap .14. Excellentiſſimus AuguÅ¿tinus ita inquit (vt etiam Å¿uperius dixerat) <lb/></s> - <quote> - <s xml:id="s4807" xml:space="preserve">quod certiorem cognitionem homo non habet illa, quæ à ſenÅ¿u prouenit. </s> - <s xml:id="s4808" xml:space="preserve">Et quod <lb/>nemo eÅ¿t qui aÅ¿piciat terram, & aquam, quod hanc maiorem illa non iudicet, & <reg norm="non" type="context">nõ</reg> <lb/>exiÅ¿timet.</s> - </quote> - </p> - <p> - <s xml:id="s4809" xml:space="preserve">Quod autem certiorem cognitionem homo non habeat illa, quæ à ſenÅ¿u proue-<lb/>nit, concedendum non cenÅ¿eo. </s> - <s xml:id="s4810" xml:space="preserve">Nam omnis cognitio mathematica (cum primum <lb/>gradum certitudinis obtineat) ab ipÅ¿o Å¿enÅ¿u fieret, quod omnino alienum <lb/>eÅ¿t à veritate. </s> - <s xml:id="s4811" xml:space="preserve">SenÅ¿us enim nunquam vidit incommenÅ¿urabilitates magnitudinum, <lb/>vel incoincidentias linearum non tangentium cum curuitate hyperbolica, aut angu <lb/>lum contingentiæ aliquem, nec (vt vno verbo dicam) aliquam concluÅ¿ionem ma-<lb/>thematicam, quam volueris. </s> - <s xml:id="s4812" xml:space="preserve">Neque per Å¿enÅ¿um eÅ¿t Å¿cire, inquit AriÅ¿toteles. </s> - <s xml:id="s4813" xml:space="preserve">Co-<lb/>gnitio igitur Å¿enÅ¿itiua, certior non eÅ¿t illa, quæ per habitum Å¿cientiÅ¿icum acquiritur.</s> - </p> - <p> - <s xml:id="s4814" xml:space="preserve">Ad reliqua verò, Å¿upponamus nos tunc fuiſſe in Arca Noe, <reg norm="cum" type="context">cÅ©</reg> aquæ cooperiebant <lb/>omnia cacumina montium, vbi nullum terræ veÅ¿tigium videbatur, </s> - <s xml:id="s4815" xml:space="preserve">quare proculdu <lb/>bio aquam iudicaremus, atque exiÅ¿timaremus maiorem terra, <reg norm="dum" type="context">dÅ©</reg> nulla aliare vtere-<lb/>mur niÅ¿i Å¿enÅ¿u abÅ¿que alio diÅ¿curÅ¿u intellectuali, ut reliqua illa animalia irrationalia, <lb/>quæ nobiÅ¿cum erant in dicta arca. </s> - <s xml:id="s4816" xml:space="preserve"><reg norm="Non" type="context">Nõ</reg> Å¿ufficit igitur Å¿uperficiem aquæ tantummodo <lb/>aÅ¿picere, quia neque tunc temporis, aqua erat maior terra, etiam Å¿i non Å¿olum tot <lb/>cubitis attolleretur Å¿upra cacumina montium, Å¿ed quingenta milliaria, vt Å¿upradi-<lb/>ximus.</s> - </p> - <p> - <s xml:id="s4817" xml:space="preserve">Ratio autem illa, ex infinitis, ab ipÅ¿o, eo in loco adducta, talis eÅ¿t.</s> - </p> - <quote> - <s xml:id="s4818" xml:space="preserve">Aqua eÅ¿t eccentrica ad terram, & pro <reg norm="centro" type="context">cẽtro</reg> habet centrum grauitatis terræ, aqua <lb/>igitur maioris eÅ¿t amplitudinis ipÅ¿a terra.</s> - </quote> - <quote> - <s xml:id="s4819" xml:space="preserve">Hanc etiam conÅ¿equentiam alijs relinquo PhiloÅ¿ophis dijudicandam.</s> - </quote> - <p> - <s xml:id="s4820" xml:space="preserve">SubÅ¿equitur poÅ¿tea dicens.</s> - </p> - <quote> - <s xml:id="s4821" xml:space="preserve">Præterea proprius locus terræ, eÅ¿t Å¿uperÅ¿icies aquæ, igitur terram oportet ab <lb/>aqua tegi.</s> - </quote> - <p> - <s xml:id="s4822" xml:space="preserve">Ad hoc etiam aliquis poſſet quærere, quis nam erit locus illius partis terræ de-<lb/>tectæ ab aqua? </s> - <s xml:id="s4823" xml:space="preserve">nulli dubium erit quin Å¿uperficies aeris, & non aquæ exiÅ¿tet.</s> - </p> - <pb o="405" rhead="EPISTOL AE." n="417" file="0417" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0417"/> - <p> - <s xml:id="s4824" xml:space="preserve">Nune autem Å¿i locus terræ eÅ¿t Å¿ub aqua, ergo locus aquæ proprius eÅ¿t Å¿ub aere, & <lb/>non Å¿ub terra, vnde non erit rationabile putare maiorem copiam aquarum exiÅ¿tere <lb/>in cauernis Å¿ubterraneis, quam Å¿upra Å¿uperficiem terræ. </s> - <s xml:id="s4825" xml:space="preserve">Adde quod locus illarum <lb/>aquarum non eſſet Å¿uperficies aeris, Å¿ed terræ, vnde non minus locus aquÄ™ eſſet ter-<lb/>ra, quam locus terræ, aqua. </s> - <s xml:id="s4826" xml:space="preserve">Sed miſſa faciamus hæc.</s> - </p> - <p> - <s xml:id="s4827" xml:space="preserve">Cap. verò .20. ita inquit.</s> - </p> - <p> - <s xml:id="s4828" xml:space="preserve">Materia elementorum æqualis eÅ¿t. </s> - <s xml:id="s4829" xml:space="preserve">Ergo aqua maior eÅ¿t terra.</s> - </p> - <quote> - <s xml:id="s4830" xml:space="preserve">Hæc enim conÅ¿equentia veriſſima eſſet. </s> - <s xml:id="s4831" xml:space="preserve">Sed nullus vnquam PhiloÅ¿ophus (vt Phi-<lb/>loÅ¿ophus dico) concedet totam materiam elementarem, in quatuor æquales partes <lb/>eſſe diuiÅ¿am.</s> - </quote> - <p> - <s xml:id="s4832" xml:space="preserve">Cap. verò .21. inquit me dixiſſe non Å¿uffecturam paucam Å¿piſſitudinem. </s> - <s xml:id="s4833" xml:space="preserve">Eo enim <lb/>in loco pag .26. mei tractatus contradicens ipÅ¿i Bergæ, dixi, quod Å¿ecundum ipÅ¿um <lb/>Bergam non Å¿ufficeret pauca Å¿piſſitudo.</s> - </p> - <p> - <s xml:id="s4834" xml:space="preserve">Similiter etiam dixi, quod Å¿ecundum ipÅ¿um, quanto remotius diffunditur lumen <lb/>fortaſſe tantò magis illuminat. </s> - <s xml:id="s4835" xml:space="preserve">Putans ipÅ¿e Berga quod in propinquo debilius exi-<lb/>Å¿teret dictum lumen. </s> - <s xml:id="s4836" xml:space="preserve">Et propter ea dixi, quod apud ipÅ¿um fortaſſe nihil valet illa <lb/>propoÅ¿icio, quæ dicit. </s> - <s xml:id="s4837" xml:space="preserve">Agens in propinquo, fortius agit quam in remoto.</s> - </p> - <p> - <s xml:id="s4838" xml:space="preserve">Cap. autem .22. vbi Excellentiſſimus AuguÅ¿tinus inquit, vnum tantummodo ele <lb/>mentum non Å¿ufficere ad generationem miÅ¿torum. </s> - <s xml:id="s4839" xml:space="preserve">Hoc enim concedo, Å¿ed hoc ni-<lb/>hil ad me Å¿pectat, eo quod meum reÅ¿ponÅ¿um ad Bergam, erat circa tranÅ¿itum lumi-<lb/>nis, & non circa generationem elementorum.</s> - </p> - <p> - <s xml:id="s4840" xml:space="preserve">Cap. demum .23. pag .20. linea .10. vbi Å¿cribit me dixiſſe, iudicare, oportebat <lb/>Å¿cribere, dubitare.</s> - </p> - <p> - <s xml:id="s4841" xml:space="preserve">Puto tamen hoc vocabulum eſſe errorem Thypographi, quamuis in correctione <lb/>illud non inuenerim, quia vt ego multoties expertusÅ¿um, difficillimum omnes Thy <lb/>pographi errores corrigere, neque (vt fertur) Argi oculi Å¿ufficerent.</s> - </p> - <p> - <s xml:id="s4842" xml:space="preserve">Hactenus enim in mei defenÅ¿ionem hæc Å¿ubiungere volui.</s> - </p> - <p> - <s xml:id="s4843" xml:space="preserve">Ad defenÅ¿ionem autem Piccolo. <reg norm="aliorumque" type="simple">aliorumqÌ;</reg> virorum meæ opinionis, nec non de <lb/>proportione duplicata profunditatis maris ad Å¿uam amplitudinem, ex conÅ¿equentia <lb/>pyramidali: </s> - <s xml:id="s4844" xml:space="preserve"><reg norm="alijsque" type="simple">alijsqÌ;</reg> Å¿imilibus rationibus, prodeant alij. </s> - <s xml:id="s4845" xml:space="preserve">HuiuÅ¿modi tamen Doctiſſi-<lb/>mi viriingenium, memoriam, nec non doctrinam valde admiror, atque obÅ¿eruo.</s> - </p> - </div> - </div> - <div type="section" level="3" n="44"> - <div type="letter" level="4" n="1"> - <head xml:id="head566" xml:space="preserve">DE METHODO PRODVCTIONIS FRACTORVM <lb/>qua vtuntur Pedemontani AgrimenÅ¿ores.</head> - <head xml:id="head567" style="it" xml:space="preserve">AnÅ¿elmo RoÅ¿emburg AgrimenÅ¿ori CaÅ¿areo.</head> - <p> - <s xml:id="s4846" xml:space="preserve"><emph style="sc">MEthodvs</emph> quà m mihi Å¿cribis in Prouincia tua maximè in vÅ¿u eſſe, nimis <lb/>longa atque prolixa eÅ¿t, Pedemontani verò AgrimenÅ¿ores in produ-<lb/>ctione fractorum, valde breui methodo vti Å¿olent, quam libenter tibi <lb/>Å¿cribo, eo maxime, vt videas quam rationabiliter operentur.</s> - </p> - <p> - <s xml:id="s4847" xml:space="preserve">Scire igitur primum te oportet illos, maximam eorum communem menÅ¿uram <lb/>vocare Trabucum, cuius Å¿extam partem vocant Pedem, duodecimam verò pe-<lb/>dis, Vnciam, <reg norm="duodecimam" type="context">duodecimã</reg> <reg norm="autem" type="context">autẽ</reg> vnciæ <reg norm="punctum" type="context">punctÅ©</reg>, <reg norm="duodecimam" type="context">duodecimã</reg> demum puncti; </s> - <s xml:id="s4848" xml:space="preserve">Attomum.</s> - </p> - <p> - <s xml:id="s4849" xml:space="preserve">QuotieÅ¿cunque igitur multiplicant trabucum, per trabucum nulli dubium eÅ¿t <lb/>quin producant trabucum Å¿uperficialem Å¿cilicet.</s> - </p> - <pb o="406" rhead="IO. BAPT. BENED." n="418" file="0418" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0418"/> - <p> - <s xml:id="s4850" xml:space="preserve">Similiter multiplicando pedes, vncias, puncta, & attoma per trabucum, produ-<lb/>cunt pedes, vncias, puncta, & attoma Å¿uperficialia rectangula oblonga, quorum lon <lb/>gitudo eÅ¿t ipÅ¿ius trabuchi, latitudo vero lineæ dictarum Å¿pecierum.</s> - </p> - <p> - <s xml:id="s4851" xml:space="preserve">Dum vero multiplicant pedem per pedem, nulli dubium eÅ¿t quin producant pe-<lb/>dem quadratum, Å¿ed apud ipÅ¿os non vocatur quadratum, quamuis reuera ita Å¿it, Å¿ed <lb/>illud vocant duas vncias, quæ quidem Å¿unt rectangula oblonga iam hic Å¿upradicta, <lb/>quarum vniuÅ¿cuiuÅ¿que longitudo Å¿it vnius trabuchi, latitudo vero vnius duodecimæ <lb/>partis ipÅ¿ius pedis linearis.</s> - </p> - <p> - <s xml:id="s4852" xml:space="preserve">Productum autem pedis per vnciam, vocant duo puncta, quæ etiam Å¿unt duo re-<lb/>ctangula oblonga, vt Å¿upra.</s> - </p> - <p> - <s xml:id="s4853" xml:space="preserve">Productum deinde vnciæ per vnciam, vocant duos attomos, qui <reg norm="etiam" type="context">etiã</reg> Å¿unt duo re-<lb/>ctangula oblonga, vt dictum eÅ¿t, quæ omnia Å¿cientificè videbimus.</s> - </p> - <p> - <s xml:id="s4854" xml:space="preserve">Pro cuius rei cognitione, Å¿it, exempli gratia <var>.a.e.</var> vnus Trabuchus linearis <var>.e.i.</var> ve-<lb/>ro vnus pes <var>.i.o.</var> autem vna vncia, <var>o.u.</var> poÅ¿tea vnum punctum, et <var>.u.t.</var> vnus At-<lb/>tomus.</s> - </p> - <p> - <s xml:id="s4855" xml:space="preserve">Vnde <var>.e.i.</var> erit Å¿exta pars ipÅ¿ius <var>.a.e.</var> et <var>.i.o.</var> duodecima ipÅ¿ius <var>.e.i.</var> et <var>.o.u.</var> duodecima <lb/>ipÅ¿ius <var>.i.o.</var> et <var>.u.t.</var> duodecima ipÅ¿ius <var>.o.u</var>. </s> - <s xml:id="s4856" xml:space="preserve">Sit etiam <var>.a.b.</var> æqualis <var>.a.e.</var> lineæ & Å¿ic <var>.e.d</var>: <var>i.<lb/>f</var>: <var>o.g</var>: <var>o.n.</var> & c. <reg norm="terminenturque" type="simple">terminenturqÌ;</reg> parallelogramma <var>.b.e</var>: <var>d.i</var>: <var>f.o</var>: <var>g.u.</var> et <var>.c.t.</var> vnde <var>.b.e.</var> erit <lb/>trabuchum quadratum, et <var>.d.i.</var> pes rectangulus oblongus vt Å¿upra, et <var>.f.o.</var> vncia rectan <lb/>gula oblonga, et <var>.g.u.</var> punctum rectangulum oblongum, et <var>.c.t.</var> attomus rectangu-<lb/>lus oblongus.</s> - </p> - <p> - <s xml:id="s4857" xml:space="preserve">De producto igitur trabuchi per <reg norm="trabuchum" type="context">trabuchÅ©</reg>, nulli dubium eÅ¿t quin Å¿it quadratum <var>.<lb/>a.d.</var> vt Å¿uperius diximus.</s> - </p> - <p> - <s xml:id="s4858" xml:space="preserve">Productum autem trabuchi cum pede erit <var>.d.i.</var> Å¿exta pars ipÅ¿ius <var>.a.d.</var> cum <var>.e.i.</var> Å¿it Å¿ex <lb/>ta ipÅ¿ius <var>.a.e.</var> ex prima Å¿exti vel .18. aut .19. Å¿eptimi, Å¿iue etiam ex .15. quinti Eucli.</s> - </p> - <p> - <s xml:id="s4859" xml:space="preserve">Productum autem pedis cum pede erit <var>.e.K.</var> quadratum, quod probandum eÅ¿t <lb/> - <anchor type="figure" xlink:label="fig-0418-01a" xlink:href="fig-0418-01"/> - <pb o="407" rhead="EPISTOL AE." n="419" file="0419" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0419"/> - duplum eſſe rectangulo <var>.f.o.</var> <reg norm="Nam" type="context">Nã</reg> <var>.K.i.</var> Å¿exta pars eÅ¿t ipÅ¿ius <var>.f.i.</var> ex Å¿uppoÅ¿ito, et <var>.i.o.</var> duo-<lb/>decima ipÅ¿ius <var>.e.i.</var> proportio igitur <var>.e.i.</var> ad <var>.o.i.</var> dupla eÅ¿t proportioni ipÅ¿ius <var>.f.i.</var> ad <var>.K.i</var>. <lb/></s> - <s xml:id="s4860" xml:space="preserve">quare <var>.K.e.</var> duplo maius eÅ¿t ipÅ¿ius <var>.f.o.</var> eo quod Å¿i <var>.i.o.</var> vel <var>.f.g.</var> (quod idem eÅ¿t) duplo <lb/>maius eſſet ipÅ¿o latere pręſenti <var>.o.i.</var> vel <var>.f.g.</var> </s> - <s xml:id="s4861" xml:space="preserve">tunc <var>.f.o.</var> æquale eſſet ipÅ¿i <var>.K.e.</var> ex .15. Å¿exti <lb/>vel .20. Å¿eptimi quod quidem <var>.f.o.</var> duplo maius eſſet ipÅ¿o præſenti <var>.f.o</var>. </s> - <s xml:id="s4862" xml:space="preserve">Rectè igitur <lb/>inquiunt dicentes productum pedis cum pede eſſe duas vncias, vel Å¿i mauis, ita dicas <lb/><var>e.K.</var> Å¿exta pars eÅ¿t ipÅ¿ius <var>.d.i.</var> ex iam dictis propoÅ¿itionibus <var>.f.o.</var> autem eÅ¿t duodecima <lb/>ipÅ¿ius <var>.d.i.</var> ex ijÅ¿dem, cum exÅ¿uppoÅ¿ito <var>.i.o.</var> duodecima Å¿it ipÅ¿ius <var>.e.i.</var> </s> - <s xml:id="s4863" xml:space="preserve">quare <var>.e.K.</var> <reg norm="duplum" type="context">duplÅ©</reg> <lb/>erit ipÅ¿ius <var>.f.o.</var> ex commu ni notione.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0418-01" xlink:href="fig-0418-01a"> - <image file="0418-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0418-01"/> - </figure> - </div> - <p> - <s xml:id="s4864" xml:space="preserve">Productum verò pedis cum vncia. Å¿it <var>.K.o.</var> quod probabimus ex ijÅ¿dem rationibus <lb/>duplum eſſe ipÅ¿ius <var>.g.u.</var> puncti rectanguli oblongi. </s> - <s xml:id="s4865" xml:space="preserve">Nam <var>.l.o.</var> Å¿exta pars Å¿imiliter eÅ¿t <lb/>ipÅ¿ius <var>.g.o.</var> et <var>.o.u.</var> duodecima ipÅ¿ius <var>.o.i.</var> </s> - <s xml:id="s4866" xml:space="preserve">quare proportio <var>.i.o.</var> ad <var>.o.n.</var> dupla eÅ¿t propor <lb/>tioni <var>.g.o.</var> ad <var>.o.l.</var> Å¿equitur ergo ex prædictis rationibus <var>.k.o.</var> duplum eſſe ipÅ¿ius <var>.g.u.</var> <lb/>vel Å¿ic, vtlin præcedenti, cum <var>.K.o.</var> Å¿it Å¿exta pars ipÅ¿ius <var>.f.o.</var> ex dictis propoÅ¿itionibus <var>.<lb/>g.u.</var> verò duodecima eiuÅ¿dem <var>.f.o.</var> ex ijÅ¿dem, nam <var>.o.u.</var> duodecima eÅ¿t ipÅ¿ius <var>.o.i.</var> ergo <lb/><var>K.o.</var> duplo maius eÅ¿t ipÅ¿o <var>.g.u</var>.</s> - </p> - <p> - <s xml:id="s4867" xml:space="preserve">Ex ijÅ¿demmet rationibus productum <var>.l.u.</var> pedis cum puncto duplum eÅ¿t ipÅ¿ius <var>.c.t.</var> <lb/>attomi rectanguli oblongi.</s> - </p> - <p> - <s xml:id="s4868" xml:space="preserve">Probandum nunc relinquitur productum <var>.o.n.</var> vnciæ cum vncia, quod eÅ¿t quadra-<lb/>tum, duplum eſſe ipÅ¿ius <var>.c.t.</var> attomi rectanguli oblongi. </s> - <s xml:id="s4869" xml:space="preserve">Nam <var>.i.n.</var> eÅ¿t pars vna ex .72. <lb/>ipÅ¿ius <var>.c.u.</var> et <var>.u.t.</var> pars vna ex .144. ipÅ¿ius <var>.o.i.</var> ex Å¿uppoÅ¿ito, </s> - <s xml:id="s4870" xml:space="preserve">quare proportio <var>.i.o.</var> ad <var>.u.t.</var> <lb/>dupla eÅ¿t proportioni ipÅ¿ius <var>.c.u.</var> ad <var>.n.i.</var> ex dictis igitur rationibus <var>.o.n.</var> duplo maius <lb/>eÅ¿t ipÅ¿o <var>.c.t</var>. </s> - <s xml:id="s4871" xml:space="preserve">Vel Å¿i placet dicas <var>.n.o.</var> eÅ¿t vna pars ex .72. ipÅ¿ius <var>.f.o.</var> exÅ¿upradictis, eo <lb/>quod <var>.n.i.</var> ita Å¿e habet ad <var>.f.i.</var> vt vnitas ad .72. Å¿ed ex ijÅ¿dem rationibus <var>.c.t.</var> pars vna ex <lb/>144. eÅ¿t ipÅ¿ius <var>.f.o.</var> eo quod ita Å¿e habet <var>.u.t.</var> ad <var>.o.i.</var> </s> - <s xml:id="s4872" xml:space="preserve">quare <var>.o.n.</var> duplo maius erit ipÅ¿o <var>.<lb/>c.t</var>.</s> - </p> - <p> - <s xml:id="s4873" xml:space="preserve">PropoÅ¿itum Å¿it nobis nunc, exercitij gratia, quærere Å¿uperficiem alicuius rectan <lb/>guli, cuius vnum latus Å¿it <reg norm="trabuchorum" type="context">trabuchorÅ©</reg> .3. pedum .2. & vnciarum .3. aliud vero latus Å¿it <lb/>trabuchorum .2. pedum .3. vnciarum vero .2.</s> - </p> - <p> - <s xml:id="s4874" xml:space="preserve">HuiuÅ¿modi autem methodo mediante, multiplicando primum latus <reg norm="dictum" type="context">dictÅ©</reg> .3. 2. 3. <lb/>per numerum trabucorum Å¿ecundi lateris .2. Å¿cilicet producentur nobis primò trabu <lb/>cha Å¿uperficialia .6. pedes .4. & vnciæ .6. omnia rectagula, vt dictum eÅ¿t. </s> - <s xml:id="s4875" xml:space="preserve">Multiplican-<lb/>do deinde idem primum latus .3. 2. 3. per pedes .3. Å¿ecundi lateris. </s> - <s xml:id="s4876" xml:space="preserve">Ex trabuchis .3. <lb/>primi lateris cum .3. pedibus Å¿ecundi, producentur .9. pedes rectanguli, hoc eÅ¿t <lb/>vnus trabuchus cum tribus pedibus rectangulis. </s> - <s xml:id="s4877" xml:space="preserve">Ex pedibus autem huius .2. cum <lb/>ijÅ¿dem alterius lateris .3. producentur .12. vnciæ rectangulæ ideÅ¿t vnus pes rectangu-<lb/>lus. </s> - <s xml:id="s4878" xml:space="preserve">ExijÅ¿dem pedibus .3. Å¿ecundi lateris, cum .3. vncijs primi lateris producentur <num value="18">.<lb/> - <anchor type="table" xlink:label="table-0419-01a" xlink:href="table-0419-01"/> - <pb o="408" rhead="IO. BAPT. BENED." n="420" file="0420" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0420"/> - 18.</num> puncta rectangula, hoc eÅ¿t vna vncia cum .6. punctis rectangulis. </s> - <s xml:id="s4879" xml:space="preserve">Deinde ex <lb/>multiplicatione vnciarum .2. Å¿ecundi lateris, cum .3. trabuchis primi lateris, produ-<lb/>centur .6. vnciæ. </s> - <s xml:id="s4880" xml:space="preserve">Ex multiplicatione poÅ¿tea dictarum .2. vnciarum Å¿ecundi lateris <lb/>cum .2. pedibus primi, producentur .8. puncta.</s> - </p> - <div type="float" level="5" n="2"> - <xhtml:table xlink:label="table-0419-01" xlink:href="table-0419-01a"> - <xhtml:tr> - <xhtml:th xml:space="preserve">Trabucha.</xhtml:th> - <xhtml:th xml:space="preserve">pedes.</xhtml:th> - <xhtml:th xml:space="preserve">vnciæ.</xhtml:th> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">3.</xhtml:td> - <xhtml:td xml:space="preserve">2.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">2.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - <xhtml:td xml:space="preserve">2.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">6.</xhtml:td> - <xhtml:td xml:space="preserve">4.</xhtml:td> - <xhtml:td xml:space="preserve">6.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">1.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - <xhtml:td xml:space="preserve">1.</xhtml:td> - <xhtml:td xml:space="preserve">6.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td xml:space="preserve">1.</xhtml:td> - <xhtml:td xml:space="preserve">6.</xhtml:td> - <xhtml:td xml:space="preserve">8.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td xml:space="preserve">1.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">8.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - <xhtml:td xml:space="preserve">2.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - </xhtml:tr> - </xhtml:table> - </div> - <p> - <s xml:id="s4881" xml:space="preserve">Demum ex ijÅ¿dem .2. vncijs Å¿ecundi lateris cum .3. primi, producentur .12. atto-<lb/>mi, ideÅ¿t vnum punctum. </s> - <s xml:id="s4882" xml:space="preserve">Quæ omnia collecta facient trabucha .8. pedes .3. uncias <lb/>2. & attomi .3. omnes rectanguli oblongi. </s> - <s xml:id="s4883" xml:space="preserve">Pulcherrima profecto operatio.</s> - </p> - <xhtml:table> - <xhtml:tr> - <xhtml:th xml:space="preserve">Trabucha.</xhtml:th> - <xhtml:th xml:space="preserve">pedes.</xhtml:th> - <xhtml:th xml:space="preserve">vnciæ.</xhtml:th> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">3.</xhtml:td> - <xhtml:td xml:space="preserve">2.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">2.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - <xhtml:td xml:space="preserve">2.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">6.</xhtml:td> - <xhtml:td xml:space="preserve">4.</xhtml:td> - <xhtml:td xml:space="preserve">6.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">1.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - <xhtml:td xml:space="preserve">1.</xhtml:td> - <xhtml:td xml:space="preserve">6.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td xml:space="preserve">1.</xhtml:td> - <xhtml:td xml:space="preserve">6.</xhtml:td> - <xhtml:td xml:space="preserve">8.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td xml:space="preserve">1.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">8.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - <xhtml:td xml:space="preserve">2.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - </xhtml:tr> - </xhtml:table> - <p> - <s xml:id="s4884" xml:space="preserve">Videamus nunc exercitij cauÅ¿a, vt dixi, quomodo conueniat calculus iÅ¿te cum <lb/>calculo ordinario communi?</s> - </p> - <p> - <s xml:id="s4885" xml:space="preserve">Nam quotieÅ¿cunque dicta latera, fracta fuerint in vncias, primum latus erit <lb/>vnciarum .243. Å¿ecundum autem .182. productum vero vnius in alterum erit vn-<lb/>ciarum quadratarum .44 226. quod quidem productum cum diuiÅ¿um fuerit per <num value="5184">.<lb/>5184.</num> vncias quadratas vnius trabuchi quadrati, prouentus erit .8. trabucho-<lb/>rum, reliquus verò numerus, Å¿iue fractus, erit vnciarum quadratarum .2754. qui <lb/>cum diuiÅ¿us fuerit per numerum .144. vnciarum vnius pedis quadrati, prouenient <lb/>pedes .19. quadrati cum vncijs .18. Å¿uperabundantibus, dicti autem pedes .19. Å¿ignifi-<lb/>cant tres pedes rectangulos oblongos cum vno pede quadrato, hoc eÅ¿t cum duabus <lb/>vncijs rectangulis oblongis, vt Å¿upra.</s> - </p> - <p> - <s xml:id="s4886" xml:space="preserve">Videndum nunc eÅ¿t, vtrum illæ .18. vnciæ æquipolleant tribus punctis rectangu-<lb/>lis oblongis: </s> - <s xml:id="s4887" xml:space="preserve">Å¿ed hoc manifeÅ¿tè videre eÅ¿t, ex hoc, quia quÄ™libet vncia rectangula <lb/>oblonga componitur ex .72. quadratis, punctum autem rectangulum oblongum, <reg norm="cum" type="context">cÅ©</reg> <lb/>Å¿it duodecima pars ipÅ¿ius vnciæ rectangulæ oblongæ, ipÅ¿um componetur ex .6. vn-<lb/>cijs quadratis .18. igitur vncijs quadratis, triplum erit ipÅ¿ius puncti rectanguli dicti. <lb/></s> - <s xml:id="s4888" xml:space="preserve">Vnde clarè patet, quod, quotieÅ¿cunque voluerimus Å¿cire proportionem ipÅ¿arum vn <lb/>ciarum quadratarum Å¿uperabundantium, ad punctum rectangulum oblongum, Å¿i <lb/>dixerimus ex regula de tribus, Å¿i .72. (vncia rectangula oblonga) dat .18. quid <reg norm="dabunt" type="context">dabÅ©t</reg> <lb/>12? </s> - <s xml:id="s4889" xml:space="preserve">puncta rectangula oblonga, quarum vnaquæque eÅ¿t duodecima pars ipÅ¿ius vn-<lb/>ciæ rectangulæ oblongæ, in præſenti autem caÅ¿u prouenient .3. pro quarto termino <lb/>quæſito, & habebimus propÅ¿itum.</s> - </p> - <pb o="409" rhead="EPISTOL AE." n="421" file="0421" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0421"/> - </div> - </div> - <div type="section" level="3" n="45"> - <div type="letter" level="4" n="1"> - <head xml:id="head568" xml:space="preserve">SOLVTIO CVIVSDAM QVÆSITI.</head> - <head xml:id="head569" style="it" xml:space="preserve">Magnifico Ludouico Fauzzoni amico cariβimo.</head> - <p> - <s xml:id="s4890" xml:space="preserve">TVI quæſiti Å¿olutio quam neÅ¿cio quis te docuit, valde diuerÅ¿a eÅ¿t à vera. <lb/></s> - <s xml:id="s4891" xml:space="preserve">quæſitum enim tale fuit.</s> - </p> - <p> - <s xml:id="s4892" xml:space="preserve">Reperiuntur quatuor Å¿ocij, Ludouicus, Hieronymus, FranciÅ¿cus, & Lau <lb/>rentius quorum primus, Ludouicus Å¿cilicet, poÅ¿uit aureos .6000. Hierony <lb/>mus verò aureos .5000. FranciÅ¿cus autem .2000. & Laurentius .1000. quorum Å¿um-<lb/>ma faciebat aureos .14000. </s> - <s xml:id="s4893" xml:space="preserve">interim tamen de tali Å¿umma Ludouicus recepit aureos <lb/>2000. Hieronymus verò .1000. FranciÅ¿cus autem .900. & Laurentius .800. quapro-<lb/>pter in Å¿umma reÅ¿idua Ludouicus non habebat niÅ¿i aureos .4000. Hieronymus <reg norm="etiam" type="context">etiã</reg> <lb/>4000. FranciÅ¿cus .1100. & Laurentius .200. quorum Å¿umma erat .9300. </s> - <s xml:id="s4894" xml:space="preserve">Nunc au-<lb/>tem iÅ¿ti Å¿ocij cupiunt augere hanc Å¿ummam per aureos .20000. tali tamen conditio-<lb/>ne quod quilibet tantum tribuat vt in totali Å¿umma, tantam partem unus habeat, <lb/>quantam alter.</s> - </p> - <p> - <s xml:id="s4895" xml:space="preserve">Hoc autem problema tam Å¿acile eÅ¿t, & cum Å¿uo theoremate ita coniunctum, quod <lb/>miror amicum noÅ¿trum illud illico non vidiſſe.</s> - </p> - <p> - <s xml:id="s4896" xml:space="preserve">Accipe igitur illos aureos .20000. & eos collige cum Å¿umma .9300. vnde habebis <lb/>aureos .29300. pro <reg norm="summa" type="context">sÅ©ma</reg> totali, cuius quarta pars erit .7325. <reg norm="quam" type="context">quã</reg> <reg norm="vnuÅ¿quisque" type="simple">vnuÅ¿quisq;</reg> poÅ¿tea <lb/>habebit in dicta Å¿umma. </s> - <s xml:id="s4897" xml:space="preserve">Sed ut reperias quantitatem aureorum quam quilibet <lb/>prius debet contribuere, vt poÅ¿tea habeat aureos .7325. in dicta Å¿ocietate. </s> - <s xml:id="s4898" xml:space="preserve">Iubeo, <lb/>vt Ludouicus demat illos aureos .4000. quos demum habebat, ex .7325. reliquum <lb/>autem erit .3325. qui quidem numerus erit aureorum nunc contribuendorum ipÅ¿ius <lb/>Ludouici. </s> - <s xml:id="s4899" xml:space="preserve">Demptis Å¿imiliter aureis .4000. ex dictis .7325. <reg norm="remanebunt" type="context">remanebÅ©t</reg> .3325. pro con <lb/>tributione ipÅ¿ius Hieronymi. </s> - <s xml:id="s4900" xml:space="preserve">Deinde Å¿i ex .7325. extracti fuerint aurei .1100. relin-<lb/>quent .6225. pro contributione FranciÅ¿ci. </s> - <s xml:id="s4901" xml:space="preserve">Demptis demum .200. ex .7325. reÅ¿idui <lb/>erunt .7125. pro contributione Laurentij, & Å¿ic quilibet habebit æqualem portio-<lb/>nem in totaliÅ¿umma.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head570" style="it" xml:space="preserve">Speculatio cuiuÅ¿dam Methodire ductionis numiÅ¿matum <lb/>unius Å¿peciei in aliam.</head> - <head xml:id="head571" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4902" xml:space="preserve">MIrum tibi videtur quo pacto verum Å¿it, quod Å¿umma <reg norm="mendietatis" type="context">mẽdietatis</reg> cuiuÅ¿uis <lb/>numeri illorum numiÅ¿matum, quæ hic vocantur Blanci, cum Å¿exta parte eiuÅ¿ <lb/>dem medietatis, Å¿emper Å¿it numerus florenorum huius prouinciæ. </s> - <s xml:id="s4903" xml:space="preserve">Vt exempli gra <lb/>tia, quotieÅ¿cunque reducere voluerimus .48. Blancos in Florenos, Å¿i medietati ip-<lb/>Å¿ius .48. hoc eÅ¿t .24. adiecta fuerit Å¿exta pars ipÅ¿ius medietatis, quæ eÅ¿t .4. </s> - <s xml:id="s4904" xml:space="preserve">tunc habebi <lb/>mus .28. & ita dicemus quod .48. Blanci conÅ¿tituunt Florenos .28. quod quidem <lb/>verum eÅ¿t.</s> - </p> - <p> - <s xml:id="s4905" xml:space="preserve">HuiuÅ¿modi autem rei Å¿peculatio ita Å¿e habet. </s> - <s xml:id="s4906" xml:space="preserve">Nam vnuÅ¿quiÅ¿que Blancus diuidi-<lb/>tur in .7. æquales partes, quarum .12. conÅ¿tituunt vnum Florenum, horum verò nu-<lb/>miÅ¿matum communis menÅ¿ura, vocatur Groſſus, vt Å¿cis, ex quo Å¿equitur, quod Å¿i - <pb o="410" rhead="IO. BAPT. BENED." n="422" file="0422" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0422"/> - 28. Floreni æquantur Blancis .48. tot Groſſi erunt in .28. Florenis quot in .48. Blan-<lb/>cis. </s> - <s xml:id="s4907" xml:space="preserve">Fingamus igitur, mente, noÅ¿tram figuram .79. Theorematis Arithmetici <var>.x.u.o.<lb/>e.n.</var> Å¿upponendo ambo producta <var>.u.x.</var> et <var>.n.e.</var> inuicem Ä™qualia exiÅ¿tere, & vnumquod-<lb/>que eſſe groſſorum .336. Å¿it etiam <var>.o.x.</var> vnus Florenus .12. groſſorum <var>.o.n.</var> verò Blan-<lb/>cus .7. eorundem groſſorum <var>.o.e.</var> autem Blancorum .48. </s> - <s xml:id="s4908" xml:space="preserve">Nunc certi erimus ex .15. <lb/>Å¿exti vel .20. Å¿eptimi Euclidis eandem fore proportionem <var>.o.u.</var> ad <var>.o.e.</var> quæ <var>.o.n.</var> ad <var>.o.<lb/>x.</var> Å¿ed <var>.o.n.</var> eÅ¿t Å¿umma medietatis ipÅ¿ius <var>.o.x.</var> cum Å¿exta parte dictæ medietatis, ita igi-<lb/>tur erit <var>.o.u.</var> ipÅ¿ius <var>.o.e.</var> hoc eÅ¿t Å¿umma medietatis <var>.o.e.</var> <reg norm="cum" type="context">cÅ©</reg> Å¿exta parte medietatis eiuÅ¿-<lb/>dem, quæ Å¿umma in præſenti exemplo erit .28.</s> - </p> - <p> - <s xml:id="s4909" xml:space="preserve">Hac enim Å¿peculatione mediante, poteris methodum inuenire conuertendi Flo-<lb/>renos in Blancos. </s> - <s xml:id="s4910" xml:space="preserve">Vt Å¿i nobis propoÅ¿iti fuerint Floreni .28. </s> - <s xml:id="s4911" xml:space="preserve">VoluerimusqÌue inuenire <lb/>quot Blancos faciant, Å¿uppoÅ¿ita menÅ¿ura communi, iam Å¿upradicta. </s> - <s xml:id="s4912" xml:space="preserve">Nam duplica-<lb/>bimus numerum Florenorum, à quo duplo detrahemus Å¿eptimam partem, <reg norm="reliquum" type="context">reliquÅ©</reg> <lb/>verò erit numerus quæſitus.</s> - </p> - <p> - <s xml:id="s4913" xml:space="preserve">HuiuÅ¿modi autem rei ratio eÅ¿t, quia, cum in Å¿upradicta figura, proportio <var>.o.e.</var> ad <lb/><var>o.u.</var> Ä™qualis exiÅ¿tat ei, quæ <var>.o.x.</var> ad <var>.o.n.</var> atque etiam <var>.o.x.</var> Å¿it minor duplo ipÅ¿ius <var>.o.n.</var> <lb/>per Å¿eptimam partem ipÅ¿ius dupli <var>.o.n.</var> minor erit <var>.o.e.</var> duplo ipÅ¿ius <var>.o.u.</var> per <reg norm="Å¿eptimam" type="context">Å¿eptimã</reg> <lb/>partem eiuÅ¿dem dupli ipÅ¿ius <var>.o.u</var>.</s> - </p> - <p> - <s xml:id="s4914" xml:space="preserve">Idem affirmo de quauis conuerÅ¿ione aliorum numiÅ¿matum, quorum Å¿emper <var>.o.x.</var> <lb/>maior Å¿it <var>.o.n.</var> verò minor. </s> - <s xml:id="s4915" xml:space="preserve">Vt Å¿i <var>.o.x.</var> æquiualeret .7: et <var>.o.n.</var> valeret .4. et <var>.o.e.</var> valeret <lb/>42. quæ quidem <var>.o.e.</var> menÅ¿uraretur ab <var>.o.n</var>.</s> - </p> - <p> - <s xml:id="s4916" xml:space="preserve">Si cuperemus Å¿cire quot <var>.o.x.</var> Å¿int in <var>.o.n</var>. </s> - <s xml:id="s4917" xml:space="preserve">Primo dicemus in <var>.o.n.</var> reperiri Å¿ummam <lb/>medietatis Å¿ex Å¿eptimorum ipÅ¿ius <var>.o.x.</var> collectæ cum vna Å¿eptima parte ipÅ¿ius <var>.o.x.</var> Å¿eu <lb/>(vt ita dicam) cum tertia ipÅ¿ius medietatis. </s> - <s xml:id="s4918" xml:space="preserve">Vnde dempta Å¿eptima parte ipÅ¿ius .42. <lb/>quæ eÅ¿t .6. <reg norm="collectaque" type="simple">collectaqÌ</reg> cum medietate reÅ¿idui, quæ eÅ¿t .18. habebimus .24. res, quarum <lb/>vnaquæque æqualis erit ipÅ¿i <var>.o.x</var>.</s> - </p> - <p> - <s xml:id="s4919" xml:space="preserve">Sed Å¿i quis cupiat reperire <var>.o.e.</var> dato <var>.o.u.</var> duplicet <var>.o.u.</var> à quo demat quartam <reg norm="par- tem" type="context">par-<lb/>tẽ</reg> ipÅ¿ius <var>.o.u.</var> & habebit propoÅ¿itum. </s> - <s xml:id="s4920" xml:space="preserve">Nam ita Å¿e habere oportet <var>.o.e.</var> ad <var>.o.u.</var> quemad <lb/>modum <var>.o.x.</var> ad <var>.o.n</var>.</s> - </p> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head572" style="it" xml:space="preserve">De lucro mercantili.</head> - <head xml:id="head573" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4921" xml:space="preserve"><emph style="sc">QVod</emph> demum Å¿cire à me deÅ¿ideras, eÅ¿t, quod cum vendideris libram vnam <lb/>mercis pro .4. Å¿olidis, & lucratus fueris .2. cum quarta parte vnius pro Å¿ingu-<lb/>lis decem libris, Å¿cire velles quantum lucri facturus eſſes in libris <reg norm="decem" type="context">decẽ</reg> dan-<lb/>do Å¿ingulam libram pro .6. Å¿olidis.</s> - </p> - <p> - <s xml:id="s4922" xml:space="preserve">Nulli dubium eÅ¿t quin decima pars de .2. cum quarta vnius Å¿it lucrum libræ vnius. <lb/></s> - <s xml:id="s4923" xml:space="preserve">Quæ decima pars Å¿unt <reg norm="nouem" type="context">nouẽ</reg> quadrageÅ¿imæ partes, & hæc Å¿ubducta à ſolidis .4. reli-<lb/>qui erunt Å¿olidi .3. cum .31. quadrageÅ¿imis partibus pro Å¿orte vnius libræ. </s> - <s xml:id="s4924" xml:space="preserve">QuÄ™ Å¿ors <lb/>Å¿ubtracta à ſolidis .6. remanebunt Å¿ol .2. cum .9. quadrageÅ¿imis lucri pro libra, quod <lb/>multiplicatum per .10. proueniunt Å¿ol .22. cum quarta parte vnius, & tantum aÅ¿cen-<lb/>deret lucrum, quod fieri poſſet in libris decem Å¿i quamlibet, Å¿ol .3. cum .31. quadra <lb/>geÅ¿imis nobis conÅ¿taret.</s> - </p> - <p> - <s xml:id="s4925" xml:space="preserve">Vel Å¿ic multiplicemus Å¿ortem vnius libræ per .10. productum erit .37. cum tribus - <pb o="411" rhead="EPISTOL AE." n="423" file="0423" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0423"/> - quartis, iterum multiplicemus per .10. Å¿ortem cum lucro vnius librÄ™ quod eÅ¿t .4. pro-<lb/>ductum erit .40. differens à primo Å¿ol .2. cum quarta parte, multiplicemus pariter <lb/>per .10. precium .6. Å¿olidorum proueniens erit .60. à quo deducendo productum Å¿or-<lb/>tis librarum .10. quod erat Å¿ol .37. cum tribus quartis Å¿upererunt Å¿ol .22. cum quar-<lb/>ta parte, vt Å¿upra.</s> - </p> - </div> - </div> - <div type="section" level="3" n="46"> - <div type="letter" level="4" n="1"> - <head xml:id="head574" xml:space="preserve">DE DIGNITATIBVS PLANETARVM.</head> - <head xml:id="head575" style="it" xml:space="preserve">Adriano Panetio.</head> - <p> - <s xml:id="s4926" xml:space="preserve"><emph style="sc">QVod</emph> eam diÅ¿tinctionem orbium, quæiam inualuit, nonteneas, Å¿ed putes <lb/>totum eſſe quoddam continuum excipiens corpora Å¿tellarum, nouum <reg norm="non" type="context">nõ</reg> <lb/>eÅ¿t, nam nonnulli Å¿olidæ doctrinæ PhiloÅ¿ophi idem cenÅ¿uerunt. </s> - <s xml:id="s4927" xml:space="preserve">Sed <lb/>quod attinet ad dignitates planetarum in Å¿ignis zodiaci, Å¿cias huiuÅ¿mo-<lb/>di ordinem me comprÄ™hendere eſſe deÅ¿umptum ab ordine antiquo orbium <reg norm="ipÅ¿orum" type="context">ipÅ¿orÅ©</reg> <lb/>planetarum, quiquidem ordo erat, vt Å¿tatim poÅ¿t Lunam Å¿uccederet Sol, poÅ¿t So-<lb/>lem Mercurius, tum Venus deinde Mars, poÅ¿tea Iupiter, & tandem Saturnus per <lb/><reg norm="eoÅ¿demque" type="simple">eoÅ¿demqÌ;</reg> orbes, retro redibant, atque hoc cognoÅ¿citur conÅ¿tituendo Cancrum do <lb/>micilium Lunæ, Leonem, Solis, Virginem, Mercurij, Libram, Veneris, Scorpio-<lb/>nem, Martis, Sagittarium, Iouis, Capricornum, Saturni, Incipientes deinde ab <lb/>Aquario, quiad nos propius accedit <reg norm="eundemque" type="simple">eundemqÌ;</reg> tribuentes Saturno, PiÅ¿ces, Ioui, <lb/>Arietem, Marti, Taurum, Veneri, & Gemellos, Mercurio, Å¿eptem Planetas cum <lb/>duodecim Å¿ignis zodiaci concordes reddebant.</s> - </p> - <p> - <s xml:id="s4928" xml:space="preserve">Quod deinde AriÅ¿toteles in libris de Å¿enÅ¿u & ijs quæ Å¿enÅ¿ibus percipiuntur, dicit <lb/>pupillam oculi eſſe nigram, non ita Å¿e habet, nam idem eÅ¿t, ac Å¿i quis diceret <reg norm="nigrum" type="context">nigrÅ©</reg> <lb/>eſſe illud medium, quod permitteret tranÅ¿itum lumini per Å¿uam diaphaneitatem, nul <lb/>lum lumen à ſeipÅ¿o reflectens, & etiam ac Å¿i quis diceret nigrum eſſe aerem alicuius <lb/>cubiculi vndequaque clauÅ¿i tenebroÅ¿i.</s> - </p> - <p> - <s xml:id="s4929" xml:space="preserve">Quod etiam idem AriÅ¿toteles volens adducere cauÅ¿am, cur oculus magis mate-<lb/>riam aquæ, quam aeris participet, dicensidea ratione fieri, quod aqua magis quam <lb/>aer Å¿eruari poſſit, eodem libro Å¿cribit, eÅ¿t reuera admirandum. </s> - <s xml:id="s4930" xml:space="preserve">Ibi enim clarè de-<lb/>monÅ¿trat Å¿e planè ignorare, & conÅ¿tructionem oculi, & cauÅ¿am diuerÅ¿itatis eorum <lb/>humorum tam in Å¿ubÅ¿tantia, quam in figura, quæ non aliunde dependet quam quod <lb/>diuerÅ¿am refractionem radiorum luminoÅ¿orum producat, qui per pupillam ingre-<lb/>diuntur, vt ad proprios <reg norm="Å¿ibique" type="simple">Å¿ibiqÌ;</reg> deÅ¿tinatos locos dirigantur radij, vt à virtute viÅ¿iua per <lb/>fectius Å¿en tiantur.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head576" style="it" xml:space="preserve">De ratione Frigiditatis locorum umbroÅ¿orum.</head> - <head xml:id="head577" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4931" xml:space="preserve">VEra ratio vnde fiat, vt quanto magis Å¿entitur calor in locis expoÅ¿itis Soli, tan-<lb/>to minus Å¿entiatur in vmbra, vbi Solis radius non reflectitur, eÅ¿t quia cum ra <lb/>refactus eÅ¿t aer à vehementi calore radij Å¿olaris, Å¿eipÅ¿um colligit, & condenÅ¿atur in <lb/>locis, à quibus à calore, ratione rarefactionis, non expellitur, & quia naturaliter ca-<lb/>lor Å¿equitur rarum, rarum calorem, & frigidum <reg norm="densum" type="context">densÅ©</reg>, & <reg norm="densum" type="context">densÅ©</reg> frigidum, vt vnicui <lb/>que Å¿anÄ™ mentis patet, hanc ob cauÅ¿am Å¿equitur rem ita Å¿e habere vt diximus. </s> - <s xml:id="s4932" xml:space="preserve">Poſſu <lb/>mus etiam abÅ¿que dubio credere huiuÅ¿modi ratione fieri, vt frigus matutini tempo <lb/>ris, in crepuÅ¿culo maius eſſe eo, quod noctu viguit. </s> - <s xml:id="s4933" xml:space="preserve">Nam materia conÅ¿iÅ¿tens in co-<lb/>no vmbræ terræ, Å¿emper denÅ¿ior eÅ¿t ea, quæ extra reperitur, imo noua materia con <lb/>tinuo condenÅ¿atur, propter motum vmbrÄ™, quæ Å¿emper corpori Å¿olari opponitur. </s> - <s xml:id="s4934" xml:space="preserve">hÄ™c - <pb o="412" rhead="IO. BAPT. BENED." n="424" file="0424" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0424"/> - autem noua condenÅ¿atio dico Å¿emper fit in crepuÅ¿culo matutino, hoc eÅ¿t in parte co <lb/>ni à Sole pulÅ¿a, in parte vero contrari a ipÅ¿ius coni hoc eÅ¿t in parte crepuÅ¿culi ve-<lb/>Å¿pertini, contrarium accidit, quia potius aliquantulum in hac parte materia coni ra <lb/>rificatur, quia extrinÅ¿eca condenÅ¿atur, in parte vero matutina extrinÅ¿eca rarificatur; <lb/></s> - <s xml:id="s4935" xml:space="preserve">& propterea intrinÅ¿eca conde nÅ¿atur.</s> - </p> - </div> - </div> - <div type="section" level="3" n="47"> - <div type="letter" level="4" n="1"> - <head xml:id="head578" xml:space="preserve">QVOD RECTE ARIST. SENSERIT COELVM <lb/>caÅ¿u non eſſe productum.</head> - <head xml:id="head579" style="it" xml:space="preserve">Hieronymo Condrumerio.</head> - <p> - <s xml:id="s4936" xml:space="preserve">FErunt <reg norm="AriÅ¿tippum" type="context">AriÅ¿tippÅ©</reg> tempeÅ¿tate maris ad incognita littora delatum, cum in are-<lb/>na vidiſſet <reg norm="quaÅ¿dam" type="context">quaÅ¿dã</reg> figuras geometricas delineatas <reg norm="exultantem" type="context">exultantẽ</reg> lÄ™titia dixiſſe: </s> - <s xml:id="s4937" xml:space="preserve">Hæc <lb/>Å¿unt hominum veÅ¿tigia. </s> - <s xml:id="s4938" xml:space="preserve">Nam conÅ¿onum rationi non erat, vt huiuÅ¿modi figuræ ca-<lb/>Å¿u eſſent impreſſæ: </s> - <s xml:id="s4939" xml:space="preserve">neque etiam credendum eÅ¿t ingentem hanc ma chinam tanto or <lb/>dine conÅ¿tantem fortuitò eſſe productam, cum nulla quantumuis minima eiuÅ¿dem <lb/>particula, dummodo nitatur ordine, aliquo modo caÅ¿u effecta fuerit; </s> - <s xml:id="s4940" xml:space="preserve">cum caÅ¿us ni-<lb/>hil producat, quod regulam & ordinem Å¿eruet. </s> - <s xml:id="s4941" xml:space="preserve">Non eÅ¿t igitur producta caÅ¿u admi <lb/>randa correÅ¿pondentia, quæ eÅ¿t obiectorum cum potentijs, luminis cum oculo, Å¿o-<lb/>ni cum auditu, Å¿aporis cum guÅ¿tatu, odoris cum odoratu, qualitatum tangibilium <reg norm="cum" type="context">cÅ©</reg> <lb/>tactu. </s> - <s xml:id="s4942" xml:space="preserve">Si diligenter deinde cuiuÅ¿libet rei naturalis operationem conÅ¿iderabimus, <lb/>eas tanta arte conÅ¿tructas videbimus, vt cogamur fateri aliquam prudentiſſimam, <lb/>& Å¿agaciſſimam mentem eas formaſſe, Å¿i ergo quælibet <reg norm="mundi" type="context">mÅ©di</reg> pars tanta cum ratione <lb/>& ordine eÅ¿t conÅ¿tructa: </s> - <s xml:id="s4943" xml:space="preserve">quomodo fieri poterit, vt de toto ipÅ¿o mundo id in dubium <lb/>vocemus, <reg norm="certiſſimeque" type="simple">certiſſimeqÌ;</reg> non credamus diuiniſſimam aliquam <reg norm="mentem" type="context">mẽtem</reg> eſſe à qua exqui-<lb/>Å¿itiſſima huius vniuerÅ¿i harmonia, quæ ex tot <reg norm="tantisque" type="simple">tantisqÌ;</reg> partibus, maximo ordine ni-<lb/>tentibus conficitur, non dependeat?</s> - </p> - </div> - </div> - <div type="section" level="3" n="48"> - <div type="letter" level="4" n="1"> - <head xml:id="head580" xml:space="preserve">VARIA RESPONSA.</head> - <head xml:id="head581" style="it" xml:space="preserve">Nicolao Petreio.</head> - <p> - <s xml:id="s4944" xml:space="preserve">AD ea quæ mihi Å¿cribis dico, quod excrementa quæ ex corpore Å¿ano prodeunt <lb/>in Å¿ua <reg norm="ipÅ¿orum" type="context">ipÅ¿orÅ©</reg> qualitate Å¿enÅ¿ibili ita Å¿e <reg norm="habent" type="context">habẽt</reg> ad <reg norm="facultatem" type="context">facultatẽ</reg> illius partis eiuÅ¿dem <lb/>corporis, ut <reg norm="eam" type="context">eã</reg> non lÄ™dant, <reg norm="quemadmodum" type="wordlist">quẽadmodÅ©</reg> efficeret <reg norm="Å¿putum" type="context">Å¿putÅ©</reg>, Å¿i eſſet <reg norm="amarum" type="context">amarÅ©</reg>, aut quod ex <lb/>cernitur naÅ¿o <reg norm="fÄ™tidum" type="context">fÄ™tidÅ©</reg> eſſet. </s> - <s xml:id="s4945" xml:space="preserve">Imagineris igitur <reg norm="quemadmodum" type="wordlist">quẽadmodÅ©</reg> poſſit eſſe <reg norm="verum" type="context">verÅ©</reg> id quod <reg norm="idem" type="context">idẽ</reg> <lb/><reg norm="amicus" type="simple">amicê°</reg> noÅ¿ter ait. </s> - <s xml:id="s4946" xml:space="preserve">PrÄ™terea Å¿i aliquid tibi in <reg norm="oculum" type="context">oculÅ©</reg> inciderit, an neÅ¿cis quomodo Å¿tatim <lb/>affatim affluat humor, vt id foras <reg norm="propellat" type="simple">ê“pellat</reg>, vel abducat? </s> - <s xml:id="s4947" xml:space="preserve">(mirabile opus naturæ.) </s> - <s xml:id="s4948" xml:space="preserve">Dic <lb/><reg norm="etiam" type="context">etiã</reg> eidem non abÅ¿que myÅ¿terio naturam in tot miÅ¿erijs Å¿enectutem poÅ¿uiſſe, cum <lb/>ſæpiſſimè Å¿enex mori deÅ¿ideret, ut huius vitæ calamitatibus liberetur, vnde fit, vt <lb/>cum eius aduentum Å¿entiat, minus affligatur. </s> - <s xml:id="s4949" xml:space="preserve">Dicito etiam eidem, naturam non <lb/>fuiſſe tam Å¿olicitam de quibuÅ¿dam partibus quemadmodum eÅ¿t de toto, vnde ma-<lb/>gis rotunda, & polita poterat eſſe Å¿uperficies terræ, quam nunc eÅ¿t, quia natura ma <lb/>gis reÅ¿picit totum, quam partes, & magis maiores, quam minores.</s> - </p> - <p> - <s xml:id="s4950" xml:space="preserve">Dum tuas legerem, me continere non potui quin riÅ¿erim, id quod Å¿cribis te inter-<lb/>rogaſſe eum PhiloÅ¿ophum naturalem, vnde fit, vt ventus Å¿it frigidus, <reg norm="eumque" type="simple">eumqÌ;</reg> tibi re <lb/>Å¿pondiſſe, quod à remotiſſimis partibus veniat, <reg norm="genereturque" type="simple">genereturqÌ;</reg> à vaporibus terræ frigi-<lb/>dis. </s> - <s xml:id="s4951" xml:space="preserve">( cum ipÅ¿a Å¿it frigida.) </s> - <s xml:id="s4952" xml:space="preserve">Cæterum miror quod ab eo non quæſieris, vnde oriatur <lb/>frigiditas, quæ percipitur ab agitatione aeris, qui quidem à vaporibus terræ non <lb/>proÅ¿ilit, nec à remotiſſimis partibus ad nos accedit. </s> - <s xml:id="s4953" xml:space="preserve">Sed quia de eadem re me in- - <pb o="413" rhead="EPISTOL AE." n="425" file="0425" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0425"/> - rerrogas, Å¿cito <reg norm="naturam" type="context">naturã</reg> coniunxiſſe <reg norm="frigiditatem" type="context">frigiditatẽ</reg> <reg norm="cum" type="context">cÅ©</reg> denÅ¿itate, & <reg norm="caliditatem" type="context">caliditatẽ</reg> <reg norm="cum" type="context">cÅ©</reg> raritate, <lb/>vt Å¿up. diximus, ita vt cum aliquod corpus <reg norm="denÅ¿atur" type="simple">denÅ¿at̃</reg>, <reg norm="frigidum" type="context">frigidÅ©</reg> <reg norm="reddatur" type="simple">reddat̃</reg>, & dum rarefit ma-<lb/>iorem caliditatem acquirat, & Å¿ic econtra fit, vt quanto magis aliquod corpus refri <lb/>geratur, tanto denÅ¿ius reddatur, & quanto calidius fit tanto rarius efficiatur. </s> - <s xml:id="s4954" xml:space="preserve">Quo-<lb/>ties igitur agitabitur aer, aut aliud corpus, quod ratione Å¿uæ Å¿ubtilitatis, velociter <lb/>condenÅ¿ari, & rarefieri poſſit, eius partes denÅ¿iores Å¿emper erunt frigidæ, & hanc <lb/>obrem quilibet ventus, qui per calida loca non tranÅ¿eat, natura Å¿ua frigidus, calidus <lb/>autem per accidens erit. </s> - <s xml:id="s4955" xml:space="preserve">Hinc fit vt vaÅ¿a vitrea, & terrea tam in vehementi frigore, <lb/>quam in magno æſtu frangantur, quia horum vnum fit, ne aliquis locus vacuus rema <lb/>neat, & aliud ob loci neceſſitatem, Å¿ed hoc non Å¿equeretur, Å¿i in materia, qua huiuÅ¿ <lb/>modi vas conÅ¿tat, aliqua aeris portio non contineretur.</s> - </p> - </div> - </div> - <div type="section" level="3" n="49"> - <div type="letter" level="4" n="1"> - <head xml:id="head582" xml:space="preserve">DE LVMINE LVNÆ, DE FINE LVMINIS, <lb/>de fine motus corporum cÄ™leÅ¿tium, de albedine, <lb/>de Å¿phæra.</head> - <head xml:id="head583" style="it" xml:space="preserve">Clariβimo Antonio Nauaiero.</head> - <p> - <s xml:id="s4956" xml:space="preserve"><emph style="sc">LVmen</emph> Lunæ etiam Å¿i Å¿it lumen reflexum Solis ab ipÅ¿a Luna, ab ea tamen <lb/>non ita reflectitur, vt à ſuperficie polita Å¿peculi, <reg norm="cum" type="context">cÅ©</reg> eius luminis <reg norm="tantam" type="context">tantã</reg> <reg norm="quan titatem" type="context">quãÂ<lb/>titatem</reg> Å¿uper ipÅ¿um corpus lunare videamus, & eo modo <reg norm="terminatam" type="context">terminatã</reg> quo <lb/>conÅ¿picimus. per Å¿e lumen, cauÅ¿a oculi eÅ¿t effectum, per accidens autem <lb/>puta quod vis. </s> - <s xml:id="s4957" xml:space="preserve">Terra deinde nunquam lunari lumine (<reg norm="quanuis" type="context">quãuis</reg> Å¿olaris reflexio exiÅ¿tat) <lb/>omnino deÅ¿tituta eÅ¿t, dico etiam, neque in ipÅ¿is ecclipÅ¿ibus Å¿olaribus vel lunaribus, <lb/>in Å¿olaribus enim cum Soltot millia vices maior Å¿it Luna, Luna verò minor terra, Å¿e <lb/>quitur, vt terra non omnino priuata remancat lumine Lunæ, in ecclipÅ¿ibus ve-<lb/>rò lunaribus Luna Å¿emper videtur, gratia luminis Å¿olaris, quamuis refracti. </s> - <s xml:id="s4958" xml:space="preserve">Mo-<lb/>tus corporum cÅ“leÅ¿tium fit ratione Å¿itus, & varietatis virtutis Å¿tellæ in diuerÅ¿is locis, <lb/>hæc autem varietas abÅ¿que diuerÅ¿o Å¿itu eiuÅ¿dem Å¿tellæ, nec diuerÅ¿us hic Å¿itus abÅ¿que <lb/>motu fieri poſſet, ita vt motus Å¿tellarum Å¿it ratione diuerÅ¿itatis Å¿ituum ipÅ¿arum, er-<lb/>go motus, & diuerÅ¿itas Å¿ituum, fit, ob diuerÅ¿am influentiam. </s> - <s xml:id="s4959" xml:space="preserve">Quæ autem de albe-<lb/>dine fratri tuo dixeram, erant, quod inter <reg norm="oens" type="context">oẽs</reg> colores albedo, certo quodam modò, <lb/><reg norm="maiorem" type="context">maiorẽ</reg> Å¿imilitudinem habet cum lumine. </s> - <s xml:id="s4960" xml:space="preserve">Primò quia magis coniungitur cum lumi-<lb/>ne. </s> - <s xml:id="s4961" xml:space="preserve">Secundo quia magis afficit Å¿enÅ¿um. </s> - <s xml:id="s4962" xml:space="preserve">Tertiò quia abÅ¿que reÅ¿iÅ¿tentia magis reci-<lb/>pit qualitatem aliorum colorum, quam alij colores. </s> - <s xml:id="s4963" xml:space="preserve">Quartò quia maximus <lb/>eÅ¿t omnium colorum. </s> - <s xml:id="s4964" xml:space="preserve">Quintò quia Å¿implicior eÅ¿t reliquis. </s> - <s xml:id="s4965" xml:space="preserve">Sextò quia diÅ¿gregat vi-<lb/>Å¿um. </s> - <s xml:id="s4966" xml:space="preserve">Septimò quia qualitas quæ in niue alba eſſe videtur, nihil aliud eÅ¿t quam mul-<lb/>titudo quædam luminum reflexorum, & non albedo, Å¿imilis ei, quæ eÅ¿t lactis, aut <lb/>panni, quæ quidem Å¿eptima cauÅ¿a effecit, vt ipÅ¿am albedinem, magis quam alium <lb/>quemuis colorem cum ipÅ¿o lumine compararem, cum nihil Å¿it, quod eſſe Å¿uum <reg norm="tranÅ¿" type="context">trãſ</reg> <lb/>mutans, aut apparenter, aut eſſentialiter, illud ipÅ¿um prius non tranÅ¿mutet in for-<lb/>mam Å¿ibi propin quiorem, vt manifeÅ¿tè patet. </s> - <s xml:id="s4967" xml:space="preserve">EÅ¿t etiam huius rei octaua ratio, <lb/>magni ponderis, quia Å¿cilicet nullus Å¿it color, qui magis reÅ¿iÅ¿tat lumini, aut in quem <lb/>lumen minorem impreſſionem faciat, quam albedo. </s> - <s xml:id="s4968" xml:space="preserve">Vnde Å¿equitur, obiecta alba, <lb/>minus eſſe combuÅ¿tibilia quam alia, cum quælibetres in Å¿uum contrarium quam in - <pb o="414" rhead="IO. BAPT. BENED." n="426" file="0426" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0426"/> - Å¿ibi Å¿imile valentius agat, vtrectè vidit AriÅ¿toteles cum dixit, omne contrarium @ <lb/>Å¿uo contrario patinatum eÅ¿t.</s> - </p> - <p> - <s xml:id="s4969" xml:space="preserve">Inter corpora, multum Å¿implicitatisretinet Å¿phæra.</s> - </p> - <p> - <s xml:id="s4970" xml:space="preserve">Circa quod, præter rationes adductas ab AriÅ¿totele in libris de CÅ“lo, poſſumus <lb/>etiam ratiocinarià facilitate motus <reg norm="vndique" type="simple">vndiqÌ;</reg> ab eo quod violentiæ non reÅ¿iÅ¿tar, ab eo, <lb/>quod apta <reg norm="nataque" type="simple">nataqÌ;</reg> Å¿it quieÅ¿cere Å¿upra quoduis punctum Å¿uÄ™ Å¿uperficiei, ab eo quod ab <lb/>aliqua Å¿uperficie alterius corporis Å¿eÅ¿e tangi non permittat, quæ curuitate concaua <lb/>non adæquetur, niÅ¿i medio vnius puncti. </s> - <s xml:id="s4971" xml:space="preserve">Verum eÅ¿t, quod licet hæc vltima ratio <reg norm="non" type="context">nõ</reg> <lb/>Å¿it propria Å¿phæræ, eÅ¿t tamen cauÅ¿a Å¿implicitatis in eo, in quo reperitur, Å¿ed propriÄ™ <lb/>paſſiones Å¿phæræ Å¿unt Å¿upradictæ, præter quam quod alia eiuÅ¿dem Å¿phæræ eÅ¿t pro-<lb/>prijſſima, quæ eÅ¿t diÅ¿tantia eiustermini ab vno tantummodo puncto ideÅ¿t ab <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> <lb/>centro, & etiam poſſe diuidere corpus aliquod medium, cum æquali reÅ¿iÅ¿tentia circa <lb/>punctum, quod prius in motu reperitur.</s> - </p> - <p> - <s xml:id="s4972" xml:space="preserve">Aequalitas autem rerum, eÅ¿t etiam valde Å¿imilis Å¿implicitati, & vnitati.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head584" style="it" xml:space="preserve">Comparatio uiÅ¿us, & auditus.</head> - <head xml:id="head585" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s4973" xml:space="preserve">QVodad viÅ¿um & auditum attinet, magis neceſſarium eſſe viÅ¿um, & nobilio-<lb/>rem quam auditum exiÅ¿timo, primò quia Å¿i quis viÅ¿u orbatus eſſet, contra <lb/>frigus, & calorem, contra famen, & Å¿itim nil prouidere poſſet, neque aliud quic-<lb/>quam hoc vocabulum prouidere Å¿ignificat, neque abÅ¿que periculo vitæ ab vno loco <lb/>ad alium ferri poſſet, neque aliquid arte facere.</s> - </p> - <p> - <s xml:id="s4974" xml:space="preserve">Sed Å¿i quis deÅ¿titutus eſſet facultate audiendi, Å¿upradictas tamen operationes <reg norm="prae- Å¿tare" type="simple">prÄ™-<lb/>Å¿tare</reg> poſſet, neque modo careret, quo animi Å¿ui Å¿enÅ¿a abÅ¿que beneficio Å¿oni, Å¿ed <lb/>ope figurarum & characterum alteri aperiret: </s> - <s xml:id="s4975" xml:space="preserve">neque etiam munere Å¿peculandi Å¿cien <lb/>tias (excepta muÅ¿ica) deÅ¿titueretur. </s> - <s xml:id="s4976" xml:space="preserve">Ad Å¿cientiam comparandam, longè magis ne <lb/>ceſſarius eÅ¿t viÅ¿us, quam auditus præterquam, quod viÅ¿us maiorem numerum obie-<lb/>ctorum, & differentiarum rerum percipit, & inter reliquos Å¿enÅ¿us velociſſimè imò <lb/>in inÅ¿tanti operatur, magis remotè quam alij, & exactius Å¿entit, <reg norm="minusque" type="simple">minusqÌ;</reg> quam reli-<lb/>qui afficitur, præterquam quod Å¿emperagit, dummodò non dormiat animal. </s> - <s xml:id="s4977" xml:space="preserve">Præ-<lb/>terea Å¿eÅ¿e magis patefacit, & prodit anima per oculos, quam per aliud, cuiuslibet <lb/>Å¿enÅ¿us, inÅ¿trumentum. </s> - <s xml:id="s4978" xml:space="preserve">Oculo magis quam alia corporis parte, hominis natura co-<lb/>gnoÅ¿citur: </s> - <s xml:id="s4979" xml:space="preserve">& Å¿i aliquid Å¿peculari volumus, quod Å¿ine imaginatiua fieri non poteÅ¿t, <lb/>Å¿tatim imaginamur nos videre huiuÅ¿modirem, ac Å¿i oculo fuiſſet compræhenÅ¿a, & <lb/>ab imagine quæ eÅ¿t vnum ex obiectis oculi, imaginatiua nuncupatur. </s> - <s xml:id="s4980" xml:space="preserve">Beneficio <lb/>oculorum omnes ferè Å¿cientiæ Å¿unt adinuentæ. </s> - <s xml:id="s4981" xml:space="preserve">Auditus nil aliud quam Å¿onum ca-<lb/>pit, auditus nunquam detulit intellectui figuram, corpus Å¿uperficiem, aut lineam, <lb/>materiam, formam, locum, dimenÅ¿ionem, plenum inane, nec innumera alia acci-<lb/>dentia, quæ ab oculo compræhenduntur. </s> - <s xml:id="s4982" xml:space="preserve">Quæ verò viÅ¿ui, & auditui Å¿unt commu-<lb/>nia, Å¿unt etiam tactui communia, vt numerus, motus, maius, & minus, Å¿unt tamen ali <lb/>qua oculo & tactui communia, quæ auditus non poteÅ¿t capere, vt durum, molle, acu <lb/>tum, obtuÅ¿um, aÅ¿perum, lene, planum, curuum, concauum, conuexum, magnum, <lb/>paruum, & Å¿upradicta, ideÅ¿t figura corpus & cÄ™tera, vt ctiam rectum, obliquum, & <lb/>Å¿imilia.</s> - </p> - <pb o="415" rhead="EPISTOL AE." n="427" file="0427" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0427"/> - <p> - <s xml:id="s4983" xml:space="preserve">AriÅ¿toteles circa finem primi capitis libri de Å¿enÅ¿u ait mediante viÅ¿u, magis <reg norm="quam" type="context">quã</reg> <lb/>quolibet alio Å¿enÅ¿u, nos percipere Å¿enÅ¿ibilia communia. </s> - <s xml:id="s4984" xml:space="preserve">Vbi eundem per Å¿e, & <lb/>non per accidens magis neceſſarium eſſe quam auditum, tam in ijs quæ ad victum, <lb/>quam in ijs quæ ad Å¿cientiam pertinent eſſe aſſerit, quia auditus intellectui confert <lb/>per accidens. </s> - <s xml:id="s4985" xml:space="preserve">Vide etiam quod idem Å¿cribit primo metaphyÅ¿icorum. </s> - <s xml:id="s4986" xml:space="preserve">Et Å¿i ad ali-<lb/>quid perfectè cognoÅ¿cendum, oculo Å¿eÅ¿e nobis offerrent ea omnia obiecta, quorum <lb/>Å¿pecies in imaginatiua formamus, ipÅ¿a imaginatiua non egeremus. </s> - <s xml:id="s4987" xml:space="preserve">Sed quia hoc <lb/>fieri non poteÅ¿t, hunc <reg norm="theÅ¿aurum" type="context">theÅ¿aurÅ©</reg> imaginatiuè, Å¿eu memoriæ ad conÅ¿eruandam imagi <lb/>nem omnium obiectorum Å¿enÅ¿ibilium nobis dedit natura, vt ope diÅ¿curÅ¿us <reg norm="intellectus" type="simple">intellectê°</reg> <lb/>circa dictas imagines, rerum veritatem venari poſſimus. </s> - <s xml:id="s4988" xml:space="preserve">Sed vt ad propoÅ¿itum re-<lb/>deamus, beneficio oculi animal liberum eÅ¿t, cum Å¿ine ipÅ¿o locum mutare nequeat, <lb/>vt Å¿it tutum. tenebræ, <reg norm="priuatioque" type="simple">priuatioqÌ;</reg> viÅ¿us Å¿unt ferè vnum, & idem. </s> - <s xml:id="s4989" xml:space="preserve">Neque vllus eÅ¿t@Å¿en-<lb/>Å¿us, qui Å¿it magis Å¿imilis intellectui quam viÅ¿us: </s> - <s xml:id="s4990" xml:space="preserve">neque alij Å¿enÅ¿us habent obiecta vi-<lb/>ciſſim communia, quæ non Å¿int etiam oculo communia, Å¿ed inter oculum, & quem <lb/>libet alium ex Å¿enÅ¿ibus, inuenientur quidem obiecta communia, quæ cum alijs non <lb/>communicabunt, vt inter oculum & tactum, figura, acutum, obtuÅ¿um, & Å¿imilia, <lb/>quæ alijs Å¿enÅ¿ibus non percipiuntur. </s> - <s xml:id="s4991" xml:space="preserve">Mediante viÅ¿u, & auditu etiam, <reg norm="compræhendum" type="context">compræhendÅ©</reg> <lb/>tur variæ diÅ¿tantiæ, <reg norm="Å¿itusque" type="simple">Å¿itusqÌ;</reg> obiectorum, nec non proportiones, & alia quÄ™ ab alijs Å¿en-<lb/>Å¿ibus non compræhenduntur. </s> - <s xml:id="s4992" xml:space="preserve">Multa obiecta deinde Å¿unt Å¿ubiecta guÅ¿tatui, quÄ™ alijs <lb/>accidentibus prÄ™dita Å¿unt, vnde cum fuerint Å¿emel deguÅ¿tata, talia, qualia Å¿unt ab o-<lb/>culo percipiuntur, quod nullus ex alijs Å¿enÅ¿ibus præſtabit. </s> - <s xml:id="s4993" xml:space="preserve">Idem de obiectis odora-<lb/>tus dico. </s> - <s xml:id="s4994" xml:space="preserve">SenÅ¿uum nullus eÅ¿t qui maiorem Å¿imilitudinem gerat cum vigilia & cum <lb/>vita, quam viÅ¿us, neque aliquid eÅ¿t, quod magis repræſentet imaginem Å¿omni, & <lb/>mortis, quà m cÄ™citas.</s> - </p> - <p> - <s xml:id="s4995" xml:space="preserve">Qui Å¿ibi oculos eruit vt melius Å¿pecularetur maxima Å¿tultitia prius obcÄ™catus fuit <lb/>quia Å¿oni magis impediunt Å¿peculationem quà m lumina, imò qui commodè vult <lb/>contemplari, quantum plus poteÅ¿t nititur longius eſſe ab omni Å¿trepitu, magis quà m <lb/>à locis luminoÅ¿is, & animal magis lætatur lumine quam Å¿ono: </s> - <s xml:id="s4996" xml:space="preserve">& ad Å¿peculationem <lb/>nos magis inuitat harmonia luminum variorum colorum & figurarum, quà m har-<lb/>monia Å¿onorum, præterquam quod inÅ¿trumentum viÅ¿us totius corporis partium eÅ¿t <lb/>pulcherrima, & in eminentiori loco locata, Å¿i de inÅ¿trumentis Å¿enÅ¿uum loquamur, & <lb/>veluti fineÅ¿træ animæ. </s> - <s xml:id="s4997" xml:space="preserve">Et Å¿i AriÅ¿toteles dicat oculos & aures in vno <reg norm="eodemque" type="simple">eodemqÌ;</reg> orbe <lb/>exiÅ¿tere, volens inferre quod in eodem æquilibrio Å¿int æqualiter alta non ita Å¿e ha-<lb/>bet, quia (Å¿i de homine loquamur) oculus eÅ¿t altior aure. </s> - <s xml:id="s4998" xml:space="preserve">Beneficio huius Å¿enÅ¿us, eo <lb/>rum quæ abÅ¿unt, & longo iam tempore Å¿unt mortui, animi Å¿enÅ¿a, & conceptus intel-<lb/>ligimus, neque alia ratione rerum omnium memoria Å¿eruari poteÅ¿t. </s> - <s xml:id="s4999" xml:space="preserve">Si cabala un-<lb/>quam vera fuit, nulla alia ratione eÅ¿t deleta, quam quia alicuius Å¿igni viÅ¿ibilis medio <lb/>conÅ¿eruata non fuerit, & quæcunque non Å¿cribuntur, ideÅ¿t oculo non <reg norm="commendantur" type="context">cõmendantur</reg> <lb/>parum durant cito obliuioni <reg norm="traduntur" type="simple">tradunt̃</reg>. </s> - <s xml:id="s5000" xml:space="preserve">In maiori Å¿emper pretio fuit pictura <reg norm="quam" type="context">quã</reg> muÅ¿i-<lb/>ca: </s> - <s xml:id="s5001" xml:space="preserve">obiectis viÅ¿ibilibus magis quam ijs quæ Å¿ub auditu cadunt, affectus animi, <reg norm="atque" type="simple">atq;</reg> <lb/>alia quælibet res naturalis exprimi poſſunt. </s> - <s xml:id="s5002" xml:space="preserve">Aegyptij volentes Å¿ignificare Deum, <lb/>oculi medio id præſtabant.</s> - </p> - <p> - <s xml:id="s5003" xml:space="preserve">Oculus, reÅ¿pectu aliorum inÅ¿trumentorum Å¿enÅ¿uum, eÅ¿t quaÅ¿i epicyclus animæ, <lb/>neque defuit qui crederet oculum eſſe principem animi partem.</s> - </p> - <p> - <s xml:id="s5004" xml:space="preserve">Oculus à Sole, & à Luna ita dependet, vt qui tempore defectus cuiuslibet lumi-<lb/>naris naÅ¿citur, Å¿tatim cæcus euadat, neque aliqua eÅ¿t corporis pars in qua magis ap- - <pb o="416" rhead="IO. BABPT. BENED." n="428" file="0428" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0428"/> - pareat differentia vitæ à morte; </s> - <s xml:id="s5005" xml:space="preserve">quam in oculo. </s> - <s xml:id="s5006" xml:space="preserve">AriÅ¿toteles ad finem cap .15. lib. pri <lb/>mi poÅ¿teriorum ait, clarum eſſe quod Å¿i aliquis Å¿enÅ¿us deficiat, futurum vt aliqua <lb/>quoque Å¿cientia deÅ¿it. </s> - <s xml:id="s5007" xml:space="preserve">ConÅ¿idera, quot Å¿cientijs careret homo, Å¿i viÅ¿u orbaretur.</s> - </p> - <p> - <s xml:id="s5008" xml:space="preserve">Et in tertio de anima ait, eum qui non Å¿entit, nihil intelligere poſſe; </s> - <s xml:id="s5009" xml:space="preserve">id quod in-<lb/>de confirmat, quia nihil Å¿it in intellectu, quod prius non fuerit in Å¿enÅ¿u. </s> - <s xml:id="s5010" xml:space="preserve">Plato in ti <lb/>meo ait, oculos nobis attuliſſe rerum optimarum notitiam, & Å¿i oculus non fuiſſet ni <lb/>hil eorum, quæ ad cÅ“lum Å¿pectant inueniri potuiſſe, & <reg norm="cognitionem" type="context">cognitionẽ</reg> diei ac noctis ab <lb/>oculis ortum duxiſſe, vt reuolutiones menÅ¿ium, & annorum metiri, & tempus co-<lb/>gnoÅ¿cere, & inueÅ¿tigare ordinem naturæ vniuerÅ¿alis poſſemus; </s> - <s xml:id="s5011" xml:space="preserve">quibus <reg norm="philoÅ¿ophiam" type="context">philoÅ¿ophiã</reg> <lb/>nobis comparauimus, ut alia multa omittam, quæ ibi à Platone dicuntur. </s> - <s xml:id="s5012" xml:space="preserve">Addam <lb/>hic & aliam Å¿pecialem differentiam inter auditum & viÅ¿um, quæ eÅ¿t, vt obiectum vi <lb/>Å¿us Å¿it permanens, & obiectum auditus tranÅ¿itorium Å¿iue Å¿ucceſſiuum aut, vt alio mo <lb/>do idem dicamus, obiectum viÅ¿us particpes Å¿it æternitatis, illud autem quod eÅ¿t au-<lb/>ditus non item, nam auditus tempori Å¿ubiectus eÅ¿t, viÅ¿us autem minimè. </s> - <s xml:id="s5013" xml:space="preserve">Vel Å¿i di-<lb/>camus operationem auditus abÅ¿que tempore fieri non poſſe cum Å¿it motio, operatio <lb/>verò viÅ¿us, nullo indiget tempore, cum ip Å¿a Å¿it momentanea, & propterea inÅ¿tan-<lb/>tanea. </s> - <s xml:id="s5014" xml:space="preserve">Nam momentum non eÅ¿t motus, nec inÅ¿tans tempus.</s> - </p> - </div> - </div> - <div type="section" level="3" n="50"> - <div type="letter" level="4" n="1"> - <head xml:id="head586" xml:space="preserve">QVARE HYEME VIDEATVR HALITVS <lb/>animalium non autem æſtate, & de vento.</head> - <head xml:id="head587" style="it" xml:space="preserve">Pancratio Mellano.</head> - <p> - <s xml:id="s5015" xml:space="preserve"><emph style="sc">VNde</emph> fiat vt hyeme halitum noÅ¿trum videamus, & non æſtate, ratio eÅ¿t ab <lb/>eiuÅ¿dem halitus congelatione, quæ ab extrinÅ¿eco frigore fit. </s> - <s xml:id="s5016" xml:space="preserve">Prius enim <lb/>Å¿cire debes aerem <reg norm="attractum" type="context">attractÅ©</reg> in pulmone, foras deinde erumpere cum alio <lb/>vapore aliquantulum craſſiore humido, & excrementitio expulÅ¿o à natu-<lb/>ra, quæ continuò noÅ¿trum corpus euaporare facit, vnde Å¿equitur dum aer foras à pul <lb/>mone pellitur, maiorem Å¿emper materiæ portionem, ea quæ intus attracta eÅ¿t exire: <lb/></s> - <s xml:id="s5017" xml:space="preserve">vnde Å¿tatim vt dicta materia foras expulÅ¿a, frigidum aerem offendit, cum conÅ¿tet ex <lb/>partibus craſſis, & obnoxiis congelationi, condenÅ¿atur in formam vaporis, ad dif-<lb/>ferentiam aeris ambientis qui in Å¿e eas partes craſſas non habet, à quibus <reg norm="quidem" type="context">quidẽ</reg> par-<lb/>tibus condenÅ¿atis, & redditis opacis reflectitur lumen, atque hanc ob cauÅ¿am æſtate <lb/>hoc non fit, quia calor vim condenÅ¿andi non habet.</s> - </p> - <p> - <s xml:id="s5018" xml:space="preserve">Ventus nihil aliud eÅ¿t quam quidam aeris motus, cum condenÅ¿atur, ob defectum <lb/>caloris, neque (pace AriÅ¿totelis dicam) eÅ¿t exhalatio Å¿icca. </s> - <s xml:id="s5019" xml:space="preserve">Exemplum à Vitruuio <lb/>allatum nil planè valet, quantum Å¿pectatad venti naturam, cuius rationem à mere-<lb/>quiris. </s> - <s xml:id="s5020" xml:space="preserve">Exemplum etiam ventilabri quo tempore æſtate vtimur negligendum pe-<lb/>nitus non eÅ¿t, quia eius beneficio, non Å¿olum arcemus à nobis aerem ambientem <lb/>calidum, Å¿ed alium etiam aerem circa nos condenÅ¿amus: </s> - <s xml:id="s5021" xml:space="preserve">& quia ordo naturæ eÅ¿t hu <lb/>iuÅ¿modi quod quemadmodum calor Å¿equitur raritatem <reg norm="corporum" type="context">corporÅ©</reg>, Å¿ic etiam frigus <lb/>eorundem denÅ¿itatem Å¿equatur. </s> - <s xml:id="s5022" xml:space="preserve">Quod Å¿i vis vt exemplo illuÅ¿trem, diligenter ob-<lb/>Å¿eruato tempore æſtatis cum aliqua nubes nobis Solem adimit, vbiaer qui in eius - <pb o="417" rhead="EPISTOL AE." n="429" file="0429" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0429"/> - vmbra reperitur, tantum quantum defectus caloris radij Å¿olaris fert, qui per vim, <lb/>dictum aerem rarefactum conÅ¿eruabat, Å¿tatim dictum aerem condenÅ¿ari cognoÅ¿ces: <lb/></s> - <s xml:id="s5023" xml:space="preserve">& quia ea condenÅ¿atio homogenea non eÅ¿t, ob diuerÅ¿as rationes, hanc ob cauÅ¿am <lb/>percipimus eam aeris impulÅ¿ionem, & inæqualiter, dum verò eadem vmbra diÅ¿ce-<lb/>dit, ventus, ferè, Å¿tatim ceſſat, & ſæpe ante quam dicta vmbra diÅ¿cedat; </s> - <s xml:id="s5024" xml:space="preserve">cuius rei cau <lb/>Å¿a eÅ¿t longa mora quam trahi vmbra, ita vt prius abÅ¿oluatur reditus aeris ad <reg norm="formam" type="context">formã</reg>, <lb/>quæ ei conuenit in huiuÅ¿modi vmbra, quam faciet nubes dum Sol deregitur.</s> - </p> - <p> - <s xml:id="s5025" xml:space="preserve">Vera non Å¿unt ea, quæ tibi Arnoldus dixit, vt mihi tuis literis Å¿ignificaÅ¿ti. </s> - <s xml:id="s5026" xml:space="preserve">Nam ego <lb/>ita dixi, videlicet, quod quoti eÅ¿cunque aliquis aÅ¿pexerit aliquod punctum in Å¿uper-<lb/>ficie Å¿peculi, </s> - <s xml:id="s5027" xml:space="preserve">tunc imaginem ipÅ¿ius poÅ¿t dictam Å¿uperficie<unsure/>m videbit duplicatam, Å¿i <lb/>verò aÅ¿pexerit imaginem intra Å¿peculum, </s> - <s xml:id="s5028" xml:space="preserve">tunc illud punctum videbit duplicatum, <lb/>huiuÅ¿modi autem rei ratio pendet ab hijs quÄ™ ad FranciÅ¿cum Vimercatum Å¿cri-<lb/>pÅ¿i, quæ Å¿i memoria tenes, nullum tibi dubium remanebit. </s> - <s xml:id="s5029" xml:space="preserve">Nam ea tibi omnia <lb/>oÅ¿tendi.</s> - </p> - <p> - <s xml:id="s5030" xml:space="preserve">Dum verò dicis omnem proportionem rationalem diuidi poſſe duobus numeris <lb/>mediantibus in tres æquas partes, mihi ad memoriam reuocas id quod quidam Vitru <lb/>uij commentator aſſerit Å¿uper primum cap. noni lib. eiuÅ¿dem Authoris, ita dicens.</s> - </p> - <quote> - <s xml:id="s5031" xml:space="preserve">Benè eſſe poteÅ¿t vt diagonalis (quadrati Å¿cilicet) numerorum via reperiatur, Å¿ed <lb/>fortaſſe intercedent fracta.</s> - </quote> - <p> - <s xml:id="s5032" xml:space="preserve">Miror te non memoria tenere quid Å¿int numeri rationales quidúe Å¿urdi, <reg norm="neque" type="simple">neq;</reg> con <lb/>Å¿ideras, non Å¿olum non eſſe diuiÅ¿ibilem in tres æquas partes omnem proportionem <lb/>rationabilem, Å¿ed neque in duas, vt Å¿unt Å¿uperparticulares proportiones, necnon <lb/>aliæ innumeræ, Å¿ed cum talia Å¿cribis te nimis parum verÅ¿atum in iÅ¿tis rebus oÅ¿tendis.</s> - </p> - <p> - <s xml:id="s5033" xml:space="preserve">Id verò quod tibi dicere volebam nudiustertius de Mercurio erat, quod nullo pa <lb/>cto confidendum eÅ¿t calculis qui fiunt de curÅ¿u Mercurij, eo quod eius Å¿itus nullo mo <lb/>do obÅ¿eruabilis eÅ¿t, nam ipÅ¿e nunquam nec vbiuis locorum orbis terrarum viÅ¿ibilis <lb/>eÅ¿t altior .18. gradibus Å¿upra orizontem, Å¿ed neque confidendum eſſet Å¿i <reg norm="etiam" type="context">etiã</reg> ipÅ¿um <lb/>videremus altum .20. gradibus, </s> - <s xml:id="s5034" xml:space="preserve">propterea quod magna refractio <reg norm="radiorum" type="context">radiorÅ©</reg> infra hos <lb/>gradus nos valde fallit, quæ quidem refractio, nec <reg norm="vbique" type="simple">vbiq;</reg>, nec omni tempore vnifor-<lb/>mis eÅ¿t, propter diformem Å¿eu inæqualem craſſiciem vaporum quæ continuò muta <lb/>tur. </s> - <s xml:id="s5035" xml:space="preserve">Imo multoties eum. videre putabimus Å¿upra orizontem, exiÅ¿tente ipÅ¿o Å¿ub <lb/>orizonte.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:id="head588" style="it" xml:space="preserve">Quod Ouidius tr anÅ¿currit à motu diurno, ad motum annuum <lb/>prater rem.</head> - <head xml:id="head589" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s5036" xml:space="preserve">TVus etiam Ouidius ceÅ¿pitauit, cum pro itinere vnius diei efficiat, vt Phaeton à <lb/>patre edoctus Å¿it etiam de itinere annuali.</s> - </p> - <p> - <s xml:id="s5037" xml:space="preserve">Nam, quod Phaeton petat pro curÅ¿u vnius diei, clarè patet <reg norm="ex" type="conjecture">ẽx</reg> diuerÅ¿is locis, & pri-<lb/>mò vbi ita Å¿cribit Ouidius.</s> - </p> - <quote> - <s xml:id="s5038" xml:space="preserve">Currus petit ille paternos. <lb/></s> - <s xml:id="s5039" xml:space="preserve"><reg norm="Inque" type="simple">Inq;</reg> diem alipedum ius & moderamen equorum.</s> - </quote> - <p> - <s xml:id="s5040" xml:space="preserve">Deinde vbi Pater ita loquitur.</s> - </p> - <quote> - <s xml:id="s5041" xml:space="preserve">Ardua prima via eÅ¿t, & qua vix <reg norm="manem" type="context">manẽ</reg><unsure/> recentes. </s> - <pb o="418" rhead="IO. BAPT. BENED." n="430" file="0430" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0430"/> - <s xml:id="s5042" xml:space="preserve">Enituntur equi medio eÅ¿t altiſſima cÄ™lo. <lb/></s> - <s xml:id="s5043" xml:space="preserve">Vnde mare, & terras ipÅ¿i mihi ſæpe videre. <lb/></s> - <s xml:id="s5044" xml:space="preserve">Fit timor & pauida trepidat formidine pectus. <lb/></s> - <s xml:id="s5045" xml:space="preserve">Vltima prona via eÅ¿t & eget moderamine certo.</s> - </quote> - <p> - <s xml:id="s5046" xml:space="preserve">Etiam vbi dicit.</s> - </p> - <quote> - <s xml:id="s5047" xml:space="preserve"><reg norm="Dumque" type="simple">DumqÌ;</reg> ea magnanimus <reg norm="Phaenton" type="context">Phaẽton</reg> miratur, <reg norm="opusque" type="simple">opusqÌ;</reg> <lb/>PerÅ¿picit, ecce vigil nitido patefecit ab ortu. <lb/></s> - <s xml:id="s5048" xml:space="preserve">Purpureas aurora fores, & plena roÅ¿arum. <lb/></s> - <s xml:id="s5049" xml:space="preserve">Atria, diffugiunt Å¿tellæ, quarum agmina cogit. <lb/></s> - <s xml:id="s5050" xml:space="preserve">Lucifer, & coeli Å¿tatione nouiſſimus exit.</s> - </quote> - <p> - <s xml:id="s5051" xml:space="preserve">Necnon vbi ita inquit.</s> - </p> - <quote> - <s xml:id="s5052" xml:space="preserve">Et Å¿i (modo credimus) vnum <lb/>Iſſe diem Å¿ine Sole ferunt, incendia lumen Præbebant.</s> - </quote> - <p> - <s xml:id="s5053" xml:space="preserve">Quod autem à Patre inÅ¿truatur etiam de curÅ¿u annuali, <lb/>videbitur vbi ita dicit.</s> - </p> - <quote> - <s xml:id="s5054" xml:space="preserve">Nitor in aduerÅ¿um, nec me, qui cætera vincit. <lb/></s> - <s xml:id="s5055" xml:space="preserve">Impetus, & rapido contrarius euehor orbi.</s> - </quote> - <p> - <s xml:id="s5056" xml:space="preserve">Et vbi ita loquitur.</s> - </p> - <quote> - <s xml:id="s5057" xml:space="preserve">ForÅ¿itan & lucos illic, <reg norm="vrbesque" type="simple">vrbesqÌ;</reg> Deorum. <lb/></s> - <s xml:id="s5058" xml:space="preserve">Concipias animo <reg norm="delubraque" type="simple">delubraqÌ;</reg> ditia donis <lb/>Eſſe per inÅ¿idias iter eÅ¿t, <reg norm="formasque" type="simple">formasqÌ;</reg> ferarum. <lb/></s> - <s xml:id="s5059" xml:space="preserve"><reg norm="Vtque" type="simple">VtqÌ;</reg> viam teneas, <reg norm="nulloque" type="simple">nulloqÌ;</reg> errore traharis. <lb/></s> - <s xml:id="s5060" xml:space="preserve">Per tamen aduerÅ¿i, gradieris cornua Tauri. <lb/><reg norm="Aemoniosque" type="simple">AemoniosqÌ;</reg> arcus, <reg norm="violentique" type="simple">violentiqÌ;</reg> ora Leonis. <lb/><reg norm="Sæuaque" type="simple">SæuaqÌ;</reg> circuitu curuantem brachia longo. <lb/>Scorpion atque aliter curuantem brachia cancrum. <lb/></s> - <s xml:id="s5061" xml:space="preserve">Nec tibi quadrupedes animoÅ¿os ignibus illis. <lb/></s> - <s xml:id="s5062" xml:space="preserve">Quos in pectore habent quos ore & naribus efflant, & c.</s> - </quote> - <p> - <s xml:id="s5063" xml:space="preserve">Sed lucidius etiam hoc videre eÅ¿t inferius vbi ita loquitur.</s> - </p> - <quote> - <s xml:id="s5064" xml:space="preserve">Nec tibi directos placeat via quinque per arcus. <lb/></s> - <s xml:id="s5065" xml:space="preserve">Sectus in obliquum eÅ¿t lato curuamine limes. <lb/></s> - <s xml:id="s5066" xml:space="preserve"><reg norm="Zonarumque" type="simple">ZonarumqÌ;</reg> trium contentus fine, <reg norm="polumque" type="simple">polumqÌ;</reg> <lb/>Effugit auÅ¿tralem <reg norm="iunctamque" type="simple">iunctamqÌ;</reg> aquilonibus arcton. <lb/></s> - <s xml:id="s5067" xml:space="preserve">Hac Å¿it iter, manifeÅ¿ta rotæ veÅ¿tigia cernes.</s> - </quote> - <p> - <s xml:id="s5068" xml:space="preserve">Et vbi etiam dicit.</s> - </p> - <quote> - <s xml:id="s5069" xml:space="preserve">Neute dexterior tortum declinet ad anguem. <lb/></s> - <s xml:id="s5070" xml:space="preserve">Ne ve Å¿iniÅ¿terior preſſam rota ducat ad aram.</s> - </quote> - </div> - <div type="letter" level="4" n="3"> - <head xml:id="head590" style="it" xml:space="preserve">De Å¿upputatione quinque corporum regularium. <lb/>De aliquibus etiam eorum Å¿ympathijs.</head> - <head xml:id="head591" xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:id="s5071" xml:space="preserve">ID quod à me deÅ¿ideras, ab alijs etiam factum eÅ¿t, Å¿ed ne me putes laborem euita <lb/>re, non præter mittam aliquid tibi Å¿cribere, earum rerum quæ ab Euclide colle - <pb o="419" rhead="EPISTOLAE." n="431" file="0431" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0431"/> - gi, methodo etiam qua vtebar dum in iÅ¿tisrebus me aliquo modo exercebam.</s> - </p> - <p> - <s xml:id="s5072" xml:space="preserve">QuotieÅ¿cunque igitur Å¿cire volueris quantitatem corpulentiæ <reg norm="cuiuÅ¿que" type="simple">cuiuÅ¿q;</reg> <reg norm="quinque" type="simple">quinq;</reg> cor-<lb/>porum regularium ab vna <reg norm="eademque" type="simple">eademqÌ;</reg> Å¿phæra terminatorum Å¿eu <reg norm="circunÅ¿criptibilium" type="context">circunÅ¿criptibiliÅ©</reg> cu-<lb/>rabis primum, cognoÅ¿cere quantitatem lateris <reg norm="cuiusque" type="simple">cuiusqÌ;</reg> eorum, talium partium, qua-<lb/>lium Å¿emidiameter dictæ Å¿phæræ Å¿it .100000. extabulis Å¿inuum Nicolai Copernici. <lb/></s> - <s xml:id="s5073" xml:space="preserve">Propone igitur tibiante oculos figuram Å¿emicircularem vltimæ propoÅ¿itionis .13. <lb/>lib. Eucli. & inuenies <var>.c.d.</var> tertiam partem Å¿emidiametri <var>.d.b.</var> eſſe partium .33333. æ-<lb/>qualem Å¿inui arcus <var>.f.e.</var> graduum .19. mi .28. qui quidem arcus <reg norm="demptus" type="context">dẽptus</reg> <reg norm="cum" type="context">cÅ©</reg> fuerit à tota <lb/>quarta <var>.b.f.</var> remanebitarcus <var>.e.b.</var> gra .70. mi .32. cuius corda erit latus exaedri, quod<unsure/> <lb/>latus ita cognoÅ¿ces, Å¿umendo Å¿cilicet Å¿inum medietatis <var>.b.e.</var> hoc eÅ¿t Å¿inum gra .35. mi <num value="16">.<lb/>16.</num> qui erit partium .57738. cuius duplum erit partium .115476. pro latere cubi.</s> - </p> - <p> - <s xml:id="s5074" xml:space="preserve">Dempto poÅ¿tea quadrato lateris exaedri, & quadrato totius diametri <var>.a.b.</var> reÅ¿i-<lb/>dui radix quadrata, erit <var>.a.e.</var> latus Tetraedri. </s> - <s xml:id="s5075" xml:space="preserve">Vel Å¿i duplicaueris Å¿inum dimidij ar-<lb/>cus <var>.a.e.</var> qui quidem arcus, componitur ex quarta <var>.a.f.</var> & ex arcu <var>.f.e.</var> iam inuento, Å¿iue, <lb/>vt reÅ¿iduus totius dimidij circuli, dempto <var>.b.e.</var> iam Å¿upra inuento, habebimus idem <lb/>latus <var>.a.e.</var> partium .163294.</s> - </p> - <p> - <s xml:id="s5076" xml:space="preserve">Pro latere verò Octaedri accipere potes radicem quadratam dupli quadrati ip-<lb/>Å¿ius <var>.d.b.</var> & habebis <var>.f.b.</var> latus quæſitum. </s> - <s xml:id="s5077" xml:space="preserve">Vel Å¿i malis accipe duplum Å¿inus medietatis <lb/>arcus <var>.b.f.</var> quod duplum erit <var>.f.b.</var> partium .14142.</s> - </p> - <p> - <s xml:id="s5078" xml:space="preserve">Pro latere verò Duodecaedri, diuide latus Exaedri ex methodo .11. Å¿ecundi <lb/>Eucli. cuius maior pars erit latus quæſitum, partium .71368.</s> - </p> - <p> - <s xml:id="s5079" xml:space="preserve">Sed pro latere IcoÅ¿aedri, te primum oportebit inuenire quantitatem anguli <var>g.d.<lb/>a.</var> hoc eÅ¿t ipÅ¿ius arcus <var>.b.n.</var> qui tali angulo Å¿ubiacet, quod cum pluribus modis inue-<lb/>niri poſſit, nihilominus, hunc Å¿eruabis, inuenies primò quantitatem <var>.d.g.</var> quæ eÅ¿t ra <lb/>dix quadrata Å¿ummæ duorum quadratorum hoc eÅ¿t <var>.d.a.</var> et <var>.a.g.</var> quæ <var>.a.g.</var> æqualis eÅ¿t <lb/>diametro <var>.a.b.</var> vt Å¿cis, dices poÅ¿tea, Å¿i <var>.d.g.</var> correÅ¿pondet ipÅ¿i <var>.g.a.</var> cui correÅ¿pondet <var>.d.<lb/>h.</var> Å¿emidiametro Å¿phæræ? </s> - <s xml:id="s5080" xml:space="preserve">tibi veniet <var>.h.k.</var> Å¿inus arcus <var>.a.h.</var> hoc eÅ¿t <var>.b.n.</var> graduum .63 -<lb/>min .26. cuius medietas gra .31. mi .43. pro Å¿inu Å¿uo habet partes .52571. cuius Å¿inus du <lb/>plum eÅ¿t partium .105142. pro latere IcoÅ¿aedri.</s> - </p> - <p> - <s xml:id="s5081" xml:space="preserve">Incipiendo nunc à Tetraedro, Å¿cire debes, quod pars <var>.a.c.</var> totius diametri <var>.a.b.</var> æ-<lb/>qualis eÅ¿t axi ipÅ¿ius Tetraedri, quæ quidem <var>.a.c.</var> vt Å¿ubÅ¿eÅ¿quialtera ipÅ¿ius <var>.a.b.</var> erit par <lb/>tium .13333.</s> - </p> - <p> - <s xml:id="s5082" xml:space="preserve">Quæres poÅ¿tea quantitatem Å¿uperficialem vnius faciei ipÅ¿ius Tetraedri, hac me-<lb/>thodo, inueniendo primum radicem quadratam trium quartarum quadrati <lb/>ipÅ¿ius <var>.a.e.</var> lateris Tetraedri, eo quod latus hoc, Å¿eÅ¿quitertium in potentia eÅ¿t ipÅ¿i perÂ<lb/>pendiculari terminatÄ™ ab vno angulorum trianguli æquilateris & à latere ei oppoÅ¿i-<lb/>to ex .11. tertijdecimi ipÅ¿ius Eucli. quÄ™ quidem perpendicularis, erit <reg norm="partium" type="context">partiÅ©</reg> .141416. <lb/>& hæc multiplicata cum medietate lateris trianguli, hoc eÅ¿t cum .81647. tibi dabit <lb/>Å¿uperficiem quæſitam, hoc eÅ¿t baÅ¿im Tetraedri <reg norm="partium" type="context">partiÅ©</reg> <reg norm="Å¿uperficialium" type="context">Å¿uperficialiÅ©</reg> .11546192152. <lb/><reg norm="Hanc" type="context">Hãc</reg> demum baÅ¿im multiplicando cum tertia parte axis Tetraedri habebis corpu-<lb/>lentiam totius Tetraedri, quæ erit .513158964003488.</s> - </p> - <p> - <s xml:id="s5083" xml:space="preserve">Neque tibi hoc loco occultare volo quandam meam animaduerÅ¿ionem, quæ eÅ¿t, <lb/>quod diameter Å¿eu perpendicularis (Å¿upradicta) faciei ipÅ¿ius Tetraedri, Å¿emper æ-<lb/>qualis eÅ¿t lateri ipÅ¿ius Octaedri circunÅ¿criptibilis ab eadem Å¿phæra, hoc eÅ¿t ipÅ¿i <var>.b.f.</var> <lb/>quapropter quotieÅ¿cunque ipÅ¿am perpendicularem habere voluerimus accipiendo <lb/><var>b.f.</var> habebimus intentum. </s> - <s xml:id="s5084" xml:space="preserve">Et quod hoc verum Å¿it poſſumus ita demonÅ¿trare.</s> - </p> - <p> - <s xml:id="s5085" xml:space="preserve">Primum, notum nobis eÅ¿t, ipÅ¿am perpendicularem, triplam eſſe eius parti, quæ - <pb o="420" rhead="IO. BAPT. BENED." n="432" file="0432" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0432"/> - à centro circuli, ipÅ¿um triangulum circunÅ¿cribentis, terminatur, & à baÅ¿i, vt in tertio <lb/>propoÅ¿ito decimæſeptimæ quartidecimi Eucli. probatur, ex quo Å¿equitur proportioÂ<lb/>nem huiuÅ¿modi perpendicularis ad axem Tetraedri, hoc eÅ¿t ad <var>.a.c.</var> Å¿eÅ¿quioctauam <lb/>eſſe in potentia, ex penultima primi Eucli. </s> - <s xml:id="s5086" xml:space="preserve">Sed cum <var>.d.c.</var> tertia pars Å¿it ipÅ¿ius <var>.d.a.</var> vt <lb/>etiam ex .2. propoÅ¿ito, Å¿eu corollario decimæſeptimæ .14. lib. diÅ¿currere licet, cum ex <lb/>dicto corollario <var>.d.c.</var> Å¿it Å¿exta pars ipÅ¿ius <var>.a.b</var>. </s> - <s xml:id="s5087" xml:space="preserve">Quare <var>.d.c.</var> quarta pars erit ipÅ¿ius <var>.a.c.</var> vn <lb/>de <var>.a.c.</var> Å¿eÅ¿quitertia erit ipÅ¿i <var>.a.d.</var> in longitudine, <reg norm="ideoque" type="simple">ideoqÌ;</reg> quadratum ipÅ¿ius <var>.a.d.</var> ad qua-<lb/>dratum ipÅ¿ius <var>.a.c.</var> erit vt .9. ad .16: </s> - <s xml:id="s5088" xml:space="preserve">& ita duplum quadrati ipÅ¿ius <var>.a.d.</var> hoc eÅ¿t quadra-<lb/>tum ipÅ¿ius <var>.b.f.</var> ad quadratum ipÅ¿ius <var>.a.c.</var> erit, vt .18. ad .16. hoc eÅ¿t Å¿eÅ¿quioctauum, er-<lb/>go <var>.b.f.</var> æqualis erit dictæ perpendiculari, ex .9. quinti.</s> - </p> - <p> - <s xml:id="s5089" xml:space="preserve">Cubus poÅ¿tea ipÅ¿ius <var>.b.e.</var> erit partium .1539838576570176.</s> - </p> - <p> - <s xml:id="s5090" xml:space="preserve">Pro Octaedro deinde, accipies productum diametri in Å¿emidiametrum, quod <lb/>productum, æquale erit quadrato diuidenti per æqualia Octaedron, hocigitur pro-<lb/>ductum, multiplicando per .100000. Å¿emidiametrum Å¿phæræ, tibi dabit columnam <lb/>quadrilateram cuius tertia pars, erit partium .666666666666666. cuius duplum <lb/>erit ipÅ¿um Octaedron partium .1333333333333.</s> - </p> - <p> - <s xml:id="s5091" xml:space="preserve">Pro IcoÅ¿aedro autem, oportet prius quantitatem perpendicularis inuenire, quæ <lb/>perpendicularis, per æqualia diuidit baÅ¿im ipÅ¿ius IcoÅ¿aedri, quæ vt radix quadrata <lb/>trium quartarum quadrati lateris ipÅ¿ius baÅ¿is, erit partium .91055. talium, qualium <lb/>dictum latus erit partium .105142. cuius medietas eÅ¿t .52571. quæ medietas Å¿i mul-<lb/>tiplicata fuerit cum dicta perpendiculari, dabit totam baÅ¿im Å¿uperficialem, hoc eÅ¿t <lb/>Å¿uperficiem vnius trianguli æquilateris partium Å¿uperficialium .4786852405. quo <lb/>facto, accipe quadratum duarum tertiarum ipÅ¿ius, hic Å¿upra dictæ perpendicularis, <lb/><reg norm="ipÅ¿umque" type="simple">ipÅ¿umqÌ;</reg> deme ex quadrato Å¿emidiametri Å¿phæræ, hoc eÅ¿t, ex quadrato <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> .100000 <lb/>radix poÅ¿tea quadrata reÅ¿idui, erit partium .79468. & hæc erit perpendicularis à cen <lb/>tro Å¿phærÄ™ ad vnam baÅ¿im ipÅ¿ius IcoÅ¿aedri, quam volueris, quam perpendicularem <lb/>Å¿i multiplicaueris cum quantitate Å¿uperficiali, hic Å¿uperius reperta, vnius baÅ¿is, con-<lb/>Å¿equeris columnam trilateram partium .380401586920540. cuius tertia pars, erit <lb/>partium .126800528973513. pro vna ex .20. </s> - <s xml:id="s5092" xml:space="preserve">Pyramidibus ipÅ¿um corpus compo-<lb/>nentibus. </s> - <s xml:id="s5093" xml:space="preserve">Breuius tamen hoc efficiens, Å¿i multiplicaueris baÅ¿im dictam, cum tertia <lb/>parte ipÅ¿ius perpendicularis, hanc poÅ¿tea pyramidem multiplicando per .20. habebis <lb/>totam corpulentiam ipÅ¿ius IcoÅ¿aedri partium .2536010579470260.</s> - </p> - <p> - <s xml:id="s5094" xml:space="preserve">Pro Duodecaedro demum, accipe Å¿inum gra .36. qui <reg norm="gradus" type="simple">gradê°</reg> Å¿unt pro dimidio quin <lb/>tæ partis totius gyri circularis, <reg norm="qui" type="simple">ê—</reg> <reg norm="quidem" type="context">quidẽ</reg> Å¿inus, erit partium .58778. cuius quadratum <lb/>Å¿i <reg norm="dem" type="context">dẽ</reg> pÅ¿eris ex quadrato <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> .100000. Å¿emidiametri circuli <reg norm="circunÅ¿cribentis" type="context">circũſcribentis</reg> <reg norm="aliquem" type="context">aliquẽ</reg> <reg norm="pen- tago" type="context">pẽ-<lb/>tago</reg> num æquilaterum, & æquiangulum, </s> - <s xml:id="s5095" xml:space="preserve">tunc radix reÅ¿idui, erit perpendicularis du-<lb/>cta à centro dicti circuli ad medium vnius lateris ipÅ¿ius pentagoni, quæ perp endicu <lb/>laris, erit partium .80902. talium qualium medietas lateris dicti fuerit .58778. <lb/>Nunc verò dicendo Å¿i .58778. dat .80902. quid nobis dabit .35684? </s> - <s xml:id="s5096" xml:space="preserve">medietas lateris <lb/>ipÅ¿ius Duodecaedri, vnde da bit .49116. pro perpendiculari, à centro ipÅ¿ius penta-<lb/>goni, ad latus ipÅ¿ius Duodecaedri, quæ multiplicata cum me dietate Å¿upradicta ip-<lb/>Å¿ius lat eris, hoc eÅ¿t cum .35684. producet vnum ex quinque triangulis componenti-<lb/>bus vn um pentagonum, Å¿eu vnam baÅ¿im ipÅ¿ius Duodecaedri, quod quidem triangu <lb/>lum, erit partium .1752655344. Å¿u perficialium, quas Å¿i per quinque multiplicaueris <lb/>habeb is vnam baÅ¿im pentagonam dicti corporis partium .8763276720. </s> - <s xml:id="s5097" xml:space="preserve">Dicendum <lb/>poÅ¿tea eÅ¿t, Å¿i ad .80901. conuenit Å¿emidiameter circularis partium .100000. quid <reg norm="con" type="context">cõ</reg> <lb/>ueniet partibus .49116. dabit .60711. pro tali Å¿emidiametro circulari, cuius quadra- - <pb o="421" rhead="EPISTOL AE." n="433" file="0433" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0433"/> - tum, Å¿i dempÅ¿eris ex quadrato ipÅ¿ius .100000. Å¿emidiametro Å¿phÄ™ræ, </s> - <s xml:id="s5098" xml:space="preserve">tuncradix qua-<lb/>drata reÅ¿idui, erit perpendicularis à centro Å¿phæræ ad centrum pentagoni partium, <lb/>79461. cuius tertia pars, Å¿i multiplicata fuerit cum pentagono Å¿upra reperto dicti cor <lb/>poris producet vnam ex .12. pyramidibus componentibus dictum Duodecaedron, <lb/>quæ pyramis, demum, multiplicata per .12. dabit totam corpulentiam ipÅ¿ius Duo <lb/>decaedri partium .2785354925791680.</s> - </p> - <p> - <s xml:id="s5099" xml:space="preserve">Nunc verò Å¿i experiri voluerimus vtrum iÅ¿ti calculi duorum corporum maiorum <lb/>Å¿int rectè Å¿upputati, <reg norm="dicemus" type="simple">dicemê°</reg> Å¿i ad <reg norm="corpus" type="simple">corpê°</reg> .12. <reg norm="baÅ¿ium" type="context">baÅ¿iÅ©</reg>, <reg norm="quod" type="wordlist">qÄ‘</reg> eÅ¿t <reg norm="partium" type="context">partiÅ©</reg> .2785354925791680 <lb/>conuenit numerus partium .2536010579470260. ipÅ¿ius IcoÅ¿aedri, quid conueniet <lb/>lateri cubi partium .115476. & inueniemus conuenire latus ipÅ¿ius IcoÅ¿aedri partium <lb/>105138. eo quod probatum Å¿it in <ref id="ref-0026">.10. propoÅ¿itione .14. li. Eucl.</ref> eandem <reg norm="proportionem" type="context">proportionẽ</reg> <lb/>eſſe corpulentiæ ipÅ¿ius Duodecaedri ad corpulentiam ipÅ¿ius IcoÅ¿aedri, quæ lateris <lb/>cubi ad latus IcoÅ¿aedri.</s> - </p> - <p> - <s xml:id="s5100" xml:space="preserve">Hæc autem corpora, ita Å¿ibi inuicem, & cum eorum Å¿phæra harmonicè <reg norm="conueniunt" type="context">conueniÅ©t</reg> <lb/>quemadmodum antiqui philoÅ¿ophi inuenerunt, vt <reg norm="mirandum" type="context">mirandÅ©</reg> non Å¿it, ipÅ¿os credidiÅ¿-<lb/>Å¿e omnia quæ natura conÅ¿tant, aliquo pacto exiÅ¿tis corporibus fieri. </s> - <s xml:id="s5101" xml:space="preserve">ConÅ¿idera quæ-<lb/>Å¿o quomodo conueniant inuicem Tetraedron, Octaedron, & IcoÅ¿aedron, cum uniuÅ¿-<lb/>cuiuÅ¿que baÅ¿es Å¿int triangulares æquilateræ intelli gendo Å¿emper hæc corpora ab ea-<lb/>dem Å¿phæra circunÅ¿criptibilia.</s> - </p> - <p> - <s xml:id="s5102" xml:space="preserve">Octaedron, cum Tetraedro etiam in hoc conuenit, quod latus Octaedri æquale <lb/>Å¿it ei perpendiculari, quæ diuidit baÅ¿im Tetraedri per æqualia, vtÅ¿upra demonÅ¿tra-<lb/>uimus.</s> - </p> - <p> - <s xml:id="s5103" xml:space="preserve">Harmonicis etiam interua llis hæc duo corpora inuicem concordantur, cum baÅ¿is <lb/>Tetraedri ad baÅ¿im Octaedri Å¿eruet proportionem Å¿eÅ¿quitertiam, conÅ¿onantiæ dia-<lb/>teſſaron. </s> - <s xml:id="s5104" xml:space="preserve">Et proportio omnium Å¿uperficierum Å¿iue baÅ¿ium Octaedri Å¿imul Å¿umpta-<lb/>rum, ad omnes baÅ¿es ipÅ¿ius Tetraedri Å¿imul Å¿umptas Å¿it Å¿eÅ¿quialtera, conÅ¿onantiæ dia <lb/>pentis. </s> - <s xml:id="s5105" xml:space="preserve">Neque omittendum eÅ¿t, quod proportio Octaedriad triplum Tetraedri Å¿it, <lb/>vt latus Octaedri ad latus Tetraedri.</s> - </p> - <p> - <s xml:id="s5106" xml:space="preserve">Proportio verò lateris Octaedri, ad axem Tetraedri, potentia eÅ¿t Å¿eÅ¿quioctaua, <lb/>vt Å¿upra vidimus interuallum Å¿cilicet harmonicum toni maioris.</s> - </p> - <p> - <s xml:id="s5107" xml:space="preserve">Harmonia verò Tetraedri, & Exaedri <reg norm="cum" type="context">cÅ©</reg> eorum Å¿phæra, talis eÅ¿t, vt proportio dia <lb/>metriÅ¿phæræ, potentia, tripla Å¿it lateri Exaedri, & Å¿eÅ¿quialtera lateri Tetraedri, ex <lb/>quo Å¿equitur latus Tetraedri potentia duplum exiÅ¿tere lateri Exaedri. </s> - <s xml:id="s5108" xml:space="preserve">Interuallum <lb/>enim triplum in harmonicis, componitur ex diapaÅ¿on, & diapente, & Å¿onat Å¿peciem <lb/>diapentis. </s> - <s xml:id="s5109" xml:space="preserve">Duplum verò eÅ¿t diapaÅ¿on, Å¿eÅ¿quialterum autem eÅ¿t di apente, quÄ™ con-<lb/>Å¿onantiæ perfectiſſimæ Å¿unt.</s> - </p> - <p> - <s xml:id="s5110" xml:space="preserve">Proportio verò diametri Å¿phæræ, potentia dupla eÅ¿t lat eri Octaedri, conÅ¿onantiÄ™ <lb/>diapaÅ¿on. </s> - <s xml:id="s5111" xml:space="preserve">Ex quo Å¿equitur proportionem lateris Tetraedri ad latus Octaedri, po-<lb/>tentia, Å¿eÅ¿quitertiam eſſe, hoc eÅ¿t conÅ¿onantiæ diateſſaron, & proportionem lateris <lb/>Octaedri ad latus Exaedri, potentia, Å¿eÅ¿quialteram eſſe, ita quod quatuor iÅ¿tæ poten <lb/>tiæ, ideÅ¿t diametri Å¿phæræ, lateris Tetraedri, lateris Octaedri, & lateris Exaedri con-<lb/>Å¿tituunt harmoniam ferè perfectiſſimam, ijs terminis comprehenÅ¿am .6. 4. 3. 2. (dixi <lb/>ferè, quia ditonus Å¿upra terminum .3. vel Å¿emiditonus Å¿ub termino .2. hoc loco non <lb/>reperitur, cuius quidem terminus eſſet .2. cum duabus quintis.)</s> - </p> - <p> - <s xml:id="s5112" xml:space="preserve">Adde quod diameter Å¿phæræ triplus eÅ¿t longitudine ad <reg norm="perpendicularem" type="context">perpendicularẽ</reg> ductam <lb/>à centro Å¿phæræ ad baÅ¿im Octaedri, quæ proportio, vt Å¿upra dictum eÅ¿t, dicitur dia-<lb/>paÅ¿ondiapente, practici verò eam vocant duodecimam.</s> - </p> - <pb o="422" rhead="IO. BAPT. BENED." n="434" file="0434" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0434"/> - <p> - <s xml:id="s5113" xml:space="preserve">Diameter verò Å¿phæræ Å¿eÅ¿quialter eÅ¿t longitudine axi Tetraedri, conÅ¿onantiæ <lb/>diapentis. </s> - <s xml:id="s5114" xml:space="preserve">Axis autem Tetraedri Å¿eÅ¿quitertius eÅ¿t longitudinis Å¿emidiametro Å¿phæ-<lb/>ræ conÅ¿onantiæ diateſſaron. </s> - <s xml:id="s5115" xml:space="preserve">Ita quod iÅ¿ti tres termini, qui Å¿unt, diameter Å¿phæræ, <lb/>axis Tetraedri, & Å¿emidiameter Å¿phæræ conÅ¿tituunt etiam valde perfectam harmo-<lb/>niam huiuÅ¿modi numeris contentam .6. 4. 3. corpulentia verò Exaedri ad corpu-<lb/>lentiam Tetraedri tripla eÅ¿t, conÅ¿onantiæ iam Å¿upradictæ diapaÅ¿ondiapente. </s> - <s xml:id="s5116" xml:space="preserve">Si ve-<lb/>rò de vniÅ¿ono aliquid videre deÅ¿ideras, conÅ¿idera æqualitatem dupli quadrati dia-<lb/>metri ipÅ¿ius Å¿phæræ, cum omnibus baÅ¿ibus Exaedri, vel potentia diametri Å¿phæræ <lb/>cum duabus potentijs Å¿imul Å¿umptis, quarum vna eÅ¿t lateris Tetraedri, reliqua verò <lb/>lateris Exaedri, vel æqualitatem numerorum laterum Tetraedri, cum baÅ¿ibus Exae <lb/>dri. </s> - <s xml:id="s5117" xml:space="preserve">Nec mihi videtur Å¿ilentio inuoluendum eſſe, antequam vlterius progrediar noÂ<lb/>tabilem Å¿ympatiam inter triangulum æquilaterum, & Tetraedron (<reg norm="quanuis" type="context">quãuis</reg> <reg norm="triangulum" type="context">triangulÅ©</reg> <lb/>corpus non Å¿it) non Å¿olum ob <reg norm="inalterabilitatem" type="context">inalterabilitatẽ</reg> harum duarum figurarum. </s> - <s xml:id="s5118" xml:space="preserve">(nam omnes <lb/>aliæ alterabiles eſſe poſſunt, ijÅ¿dem lateribns exiÅ¿tentibus, cum ex quadrato rom-<lb/>bus, vel ex pentagono Ä™quiangulo, pentagonum non æquiangulum & c. efficiatur) <lb/></s> - <s xml:id="s5119" xml:space="preserve">Å¿ed quod quemadmodum latus trianguli æquilateri Å¿eÅ¿quitertium potentia eÅ¿t per-<lb/>pendiculari ipÅ¿um per æqualia diuidenti, ita latus Tetraedri, Å¿eÅ¿quialterum eÅ¿t po-<lb/>tentia axi ipÅ¿ius Tetraedri, vnde cum dempta fuerit illa proportio Å¿eÅ¿quitertia, ex <lb/>hac Å¿eÅ¿quialtera relinquetur nobis proportio Å¿eÅ¿quioctaua, inter perpendicularem <lb/>trianguli, & axem Tetraedri (quod etiam Å¿upra demonÅ¿trauimus.) </s> - <s xml:id="s5120" xml:space="preserve">TranÅ¿eamus nunc <lb/>hÄ™c, nec omittamus tamen Å¿ympatias quaÅ¿dam inter Exaedron, Octaedron, & Tetra <lb/>edron, hoc eÅ¿t quod eadem proportio Å¿it inter corpulentias Exaedri, & Octaedri, <lb/>quæinter eorum Å¿uperficies, nec non, vt latus Exaedri ad Å¿emidiametrum Å¿phæræ. <lb/></s> - <s xml:id="s5121" xml:space="preserve">Proportio verò baÅ¿is Exaedri ad baÅ¿im Tetraedri, vtlatus Tetraedri ad perpendicu <lb/>larem diuidentem per æqualia eius baÅ¿im.</s> - </p> - <p> - <s xml:id="s5122" xml:space="preserve">Hactenus Å¿atis dictum Å¿it de Tetraedro, Exaedro, & Octaedro cum Å¿phæra. </s> - <s xml:id="s5123" xml:space="preserve"><reg norm="Dicem" type="context">Dicẽ</reg> <lb/>dum nunc cenÅ¿eo aliquid de reliquis duobus mirabilibus corporibus, quamuis ferè <lb/>omnia hæc ab antiquis philoÅ¿ophis inuenta Å¿int, quorum primum eÅ¿t, quod tam ba-<lb/>Å¿is Duodecaedri, quam IcoÅ¿aedri, ab vno <reg norm="eodemque" type="simple">eodemqÌ;</reg> circulo circunÅ¿criptibiles Å¿unt, ve <lb/>rùm, talis paſſio accidit etiam baÅ¿ibus Exaedri & Octaedri. </s> - <s xml:id="s5124" xml:space="preserve">Præterea quemadmo-<lb/>dum in Duodecaedro, quilibet angulus Å¿olidus terminatur tribus angulis pentago-<lb/>norum æquiangulorum ita in IcoÅ¿aedro, quilibet angulus Å¿olidus viceuerÅ¿a termi-<lb/>natur quinque angulis triangulorum æquiangulorum. </s> - <s xml:id="s5125" xml:space="preserve">Et tam vnum, quam alte-<lb/>rum horum corporum, triginta lateribus continetur. </s> - <s xml:id="s5126" xml:space="preserve">Et tot Å¿olidos angulos trian-<lb/>gulares, habet Duodecaedron, quot baÅ¿es triangulares continet IcoÅ¿aedron.</s> - </p> - <p> - <s xml:id="s5127" xml:space="preserve">Et IcoÅ¿aedron, tot Å¿olidos angulos <reg norm="pentagonos" type="context">pẽtagonos</reg>, quot baÅ¿es <reg norm="pentagonas" type="context">pẽtagonas</reg> habet Duo <lb/>decaedron. </s> - <s xml:id="s5128" xml:space="preserve">Et tam vnum quam alterum habet .60. angulos Å¿uperficiales. </s> - <s xml:id="s5129" xml:space="preserve"><reg norm="Eademque" type="context simple">EadẽqÌ;</reg> <lb/>proportio eÅ¿t omnium baÅ¿ium Å¿imul <reg norm="Å¿umptarum" type="context">Å¿umptarÅ©</reg> Duodecaedri ad omnes baÅ¿es Å¿imul <lb/>Å¿umptas ipÅ¿ius IcoÅ¿aedri, quæ corpulentiæ ipÅ¿ius Duodecaedri ad corpulentiam <lb/>IcoÅ¿aedri (quamuis hÄ™c paſſio accidat Exaedro cum Octaedro, vt Å¿pra diximus) quÄ™ <lb/>quidem proportio, eadem etiam eÅ¿t, quÄ™ lateris Exaedri ad latus IcoÅ¿aedri, vt Å¿u-<lb/>pra iam dictum fuit.</s> - </p> - <pb o="423" rhead="EPISTOL AE." n="435" file="0435" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0435"/> - </div> - </div> - <div type="section" level="3" n="51"> - <div type="letter" level="4" n="1"> - <head xml:id="head592" xml:space="preserve">NOVA INVENTIO COMPONENDI ASTROLABIA <lb/>cum Horologijs artificialibus.</head> - <head xml:id="head593" style="it" xml:space="preserve">Facobo Mayeto IngenioÅ¿ißimo Horologiorum Serenißimi <lb/>Sabaudiæ Ducis Artifici.</head> - <p> - <s xml:id="s5130" xml:space="preserve"><emph style="sc">NOnnvnqvam</emph> conÅ¿ideraui mirabilem pulchritudinem, Å¿imul cum vtili-<lb/>tate coniunctam, illorum horologiorum, quæin Germania conÅ¿truuntur <lb/><reg norm="cum" type="context">cÅ©</reg> mobili Rete, Å¿eu Aranea AÅ¿trolabij <reg norm="Å¿uper" type="simple">Å¿uê‘</reg> <reg norm="Tabulam" type="context">Tabulã</reg> regionis, in <reg norm="quibus" type="simple">ê—bus</reg> <reg norm="conti" type="context">cõti</reg> <lb/>nuo <reg norm="videntur" type="simple">vident̃</reg> oriri, <reg norm="occidereque" type="simple">occidereqÌ;</reg> cæleÅ¿tia Å¿igna, cælum mediare Å¿upra orizon <lb/><reg norm="tem" type="context">tẽ</reg><unsure/>, necnon Å¿ub eo, & vt vno verbo dicam, continuo erecta videtur tota coelifigura. <lb/></s> - <s xml:id="s5131" xml:space="preserve">Sed quia talia horologia omnia eorum limbum diÅ¿tinctum habent in .24. horas, qua <lb/>propter diametrum limbi, minorem duobus palmis, Å¿eu Å¿emipede eſſe non oportet <lb/>neinterÅ¿titia horarum iuÅ¿tò breuiora Å¿eu anguÅ¿tiora efficiantur, etiam ne interualla <lb/>dentium rotæ indicis nimis anguÅ¿ta Å¿int. </s> - <s xml:id="s5132" xml:space="preserve">Sed quia talis magnitudo vt plurimum in-<lb/>commoda exiÅ¿tit. </s> - <s xml:id="s5133" xml:space="preserve">Ideo non inutile fore cogitaui, Å¿i modus aliquis inuentus fuerit, <lb/>vt ea omnia efficiantur in limbo diuiÅ¿o tantummodo in .12. horas æquales, <reg norm="ipÅ¿umque" type="simple">ipÅ¿umqÌ;</reg> <lb/>inueni, qui quidem erit, efficiendo vt Tabula (in qua deÅ¿ignantur cÄ™leÅ¿tes domus, <lb/>cum almicantarat, atque azimut) Reti Å¿ubiectæ, mobilis Å¿it, tardior tamen ipÅ¿o Re-<lb/>te cum indice, pro duplo temporis, hoc eÅ¿t, quod eo tempore, quo Aranea cum in <lb/>dice circunuoluetur Å¿pacio .12. horarum vno gyro perfecto, ipÅ¿a Tabula efficiat tan <lb/>tummodo Å¿exinterÅ¿titia horarum. </s> - <s xml:id="s5134" xml:space="preserve">IdeÅ¿t dum Tabula dicta eſſicit vnam integram re <lb/>uolutionem, Aranea, Å¿eu Zodiacus cum indice, duas efficiat reuolutiones. </s> - <s xml:id="s5135" xml:space="preserve">Ita quod <lb/>Aranea cum indice perficiet vnam reuolutionem Å¿paci o temporis .12. horarum, Ta-<lb/>bula verò perficiet eam Å¿pacio temporis .24. horarum. </s> - <s xml:id="s5136" xml:space="preserve">Vnde Å¿equetur quod Ara-<lb/>nea Å¿eu Zodiacus cum indice, Å¿pacio .24. horarum perfectè circunuoluetur Å¿upra Ta-<lb/>bulam, & ita huiuÅ¿modi horologia, in hoc nihil differrent ab illis Å¿upradictis. </s> - <s xml:id="s5137" xml:space="preserve">Vt au <lb/>tem facias dictam tabulam tardiorem duplo temporis Araneæ cum indice, quamuis <lb/>diuerÅ¿is modis hoc fieri poſſit, pręſtantiorem tamen iudico, Å¿i cum Rota indicis, <reg norm="aliam" type="context">aliã</reg> <lb/>Rotam <reg norm="concentricam" type="context">concentricã</reg> coniunxeris, ita tamen, vt <reg norm="vnaquÄ™que" type="simple">vnaquÄ™q;</reg> liberè poſſit volui, Å¿imiliter <lb/>Å¿i cum ea horologii particula (quÄ™ <reg norm="circumagit" type="context">circÅ©agit</reg> Rotam indicis, quæ Italicè <emph style="sc">rochetto</emph> <lb/>Germanicè verò <emph style="sc">trib</emph> vocatur, Latinè <reg norm="autem" type="wordlist">aÅ©t</reg> ipÅ¿um vocabo, <emph style="sc">colinvm</emph>, qui Å¿ubro-<lb/>ta fuÅ¿i reperitur) coniunxeris alium colinum quem, Å¿ecundum vocabo, <reg norm="concentricum" type="context">concentricÅ©</reg> <lb/>verò cum primo, cum <reg norm="eoque" type="simple">eoqÌ;</reg> conÅ¿olidato, numerum verò dentium, tam RotÄ™ adiunctÄ™ <lb/>quam Å¿ecundi colini, varijs modis poteris inuenire, quorum primus erit, vt numerus <lb/>dentium Å¿ecundæ RotÄ™ duplus exiÅ¿tat numero dentium primÄ™, efficiendo Å¿ecundum <lb/>colinum eiuÅ¿dem numeri dentium quo primum, Å¿ed quia interualla dentium huiuÅ¿-<lb/>modi RotÄ™, nimis anguÅ¿ta fortaſſe reÅ¿ultabunt, </s> - <s xml:id="s5138" xml:space="preserve">propterea alios etiam modos inue-<lb/>ni, quorum vnus erit (dum numerus dentium primi colini par fuerit) <reg norm="efficiendo" type="context">efficiẽdo</reg> Å¿ecun <lb/>dam <reg norm="Rotam" type="context">Rotã</reg> <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> numeri <reg norm="dentium" type="context">dentiÅ©</reg> cuius eÅ¿t prima. <reg norm="Å¿ecundum" type="context context">Å¿ecÅ©dÅ©</reg> vero colinum, medietatis <lb/>numeri dentium cuius erit primus. </s> - <s xml:id="s5139" xml:space="preserve">Attamen Å¿i primus colinus eſſet .4. dentium, Å¿ecun <lb/>dum oporteret eſſe duorum dentium, vnde motus Å¿ecundÄ™ RotÄ™ non eſſet ita conti-<lb/>nuus. </s> - <s xml:id="s5140" xml:space="preserve">Quapropter alium etiam mòdum excogitaui, hoc eÅ¿t, cupiendo vt Å¿ecundus <lb/>colinus, extribus dentibus exiÅ¿tat, Å¿i primus ex .4. repertus fuerit, oportebit prius ex <lb/>regula de tribus, numerum quendam inuenire quo inuento ipÅ¿um duplicare, & hunc <lb/>duplicatum numerum conueniet Å¿ecundam Rotam habere, vt ipÅ¿a poſſit ab illo co-<lb/>lino <reg norm="trium" type="context">triÅ©</reg> <reg norm="dentium" type="context">dentiÅ©</reg> circunuolui in duplo temporis, quo prima à ſuo colino quatuor den - <pb o="424" rhead="IO. BAPT. BENED." n="436" file="0436" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0436"/> - tium. </s> - <s xml:id="s5141" xml:space="preserve">Exempli gratia, Å¿i prima Rota conÅ¿taret ex .36. dentibus, dicendum eſſet, Å¿i <lb/>4. conuenit cum .36. cum quibus conuenient .3. & inueniemus .27. cum quo numero <lb/>dicta Å¿ecunda Rota circunuolueretur eodem tempore à ſuo colino trium dentium, <lb/>quo prima à ſuo quatuor dentium, </s> - <s xml:id="s5142" xml:space="preserve">quare duplicando .27. haberemus .54. pro nume-<lb/>ro dentium dictÄ™ Å¿ecundæ Rotæ, vt duplo temporis circunuoluatur quo prima. </s> - <s xml:id="s5143" xml:space="preserve">Sed <lb/>Å¿i primus colinus conÅ¿taret ex .6. dentibus, exiÅ¿tente Å¿ua Rota ex .36. <reg norm="vellemusque" type="simple">vellemusqÌ;</reg> <reg norm="quod" type="simple">ê™</reg> <lb/>Å¿ecundus exiÅ¿teret ex .4. </s> - <s xml:id="s5144" xml:space="preserve">tunc Å¿uam Rotam oporteret habere dentes .48. ex dicta re-<lb/>gula. </s> - <s xml:id="s5145" xml:space="preserve">Si autem primus colinus conÅ¿taret ex numero impari, nihil referret, dummo-<lb/>do huiuÅ¿modi numerus impar, Å¿eu par, exiÅ¿teret pars propria numeri dentium, vel <lb/>ipÅ¿ius dupli primæ RotÄ™, hoc eÅ¿t, eſſet pars aliquota numeri dentium ipÅ¿ius primæ <lb/>Rotæ vel ipÅ¿ius dupli. </s> - <s xml:id="s5146" xml:space="preserve">In ijs verò horologiis in quibus duplum numeri dentium di-<lb/>ctÄ™ primÄ™ RotÄ™ non erit multiplex numero dentium primi colini, hoc fieri non pote <lb/>rit. </s> - <s xml:id="s5147" xml:space="preserve">Ratio enim tam clarè, tibi conÅ¿ideranti, patebit, vt nullis verbis indigeat cum <lb/>Å¿emper numerus dentium Å¿ecundÄ™ RotÄ™ multiplex eſſe debeat numero dentium Å¿e-<lb/>cundi colini. </s> - <s xml:id="s5148" xml:space="preserve">Idem autem non dico de prima Rota cum Å¿uo colino, hoc eÅ¿t, vt nu-<lb/>merus primÄ™ multiplex Å¿it numero Å¿ui colini, nam hoc neceſſarium non eÅ¿t. </s> - <s xml:id="s5149" xml:space="preserve">Pona-<lb/>mus exempli gratia primum colinum conÅ¿tare Å¿ex dentibus, Å¿uam vero Rotam den-<lb/>tibus .21. cuius quidem numeri, 6. non eÅ¿t pars aliquota, Å¿ed dupli ipÅ¿ius .21. ipÅ¿e .6. <lb/>eÅ¿t pars aliquota. </s> - <s xml:id="s5150" xml:space="preserve">Nunc verò Å¿i voluerimus numerum dentium Å¿ecundæ RotÄ™ inue-<lb/>nire, cuius colinus ex quinque dentibus exiÅ¿tat (Å¿uppoÅ¿ito primo ex .6. conÅ¿tare) </s> - <s xml:id="s5151" xml:space="preserve">tunc <lb/>ex regula de tribus, diuiÅ¿o producto, quod fit ex .21. in .5. per .6. exibit .17. cum di-<lb/>midio, cuius duplum eſſet .35. qui multiplex eÅ¿t ipÅ¿i quinque. </s> - <s xml:id="s5152" xml:space="preserve">Reperto igitur nume <lb/>ro Å¿ecundÄ™ RotÄ™, cum numero ipÅ¿ius colini, oportet nunc Å¿cire modum compoÅ¿itio-<lb/>nis, Å¿eu coniunctionis harum rerum, hoc eÅ¿t duorum colinorum <reg norm="concentricorum" type="context">concentricorÅ©</reg> (Å¿ed <lb/>de ijs Å¿atis iam Å¿uperius dictum fuit) duarum Rotarum concentricarum cum Tabula, <lb/>cum Zodiaco, & cum indice, Å¿eu OÅ¿tenÅ¿ore, cuius quidem OÅ¿tenÅ¿oris medietas tan <lb/>tummodo nobis Å¿ufficiet. </s> - <s xml:id="s5153" xml:space="preserve">Sciendum igitur nunc eÅ¿t quod cum primus colinus re-<lb/>uoluat totam primam Rotam, Å¿pacio temporis .12. horarum, oportet vt eius axis, Å¿eu <lb/>arbor voluat oÅ¿tenÅ¿orem, <reg norm="Zodiacumque" type="simple">ZodiacumqÌ;</reg>, eodem temporis Å¿pacio, & quia Rota hÄ™c <lb/>inalterabis eÅ¿t, propter eius coniunctionem cum Å¿uo colino, & nos oporteat indicem <lb/><reg norm="Zodiacumque" type="simple">ZodiacumqÌ;</reg>, quotidie ferè, dirigere, <reg norm="Å¿uisque" type="simple">Å¿uisqÌ;</reg> locis collocare, ideo nos oportet, indi <lb/>cem, Zodiacum, & primam Rotam, ita cum axe, Å¿eu arbore coniungere, vt poſſimus <lb/>dicta omnia efficere. </s> - <s xml:id="s5154" xml:space="preserve">Pars igitur Arboris, Å¿eu axis dicti, quæ ingredi debet in prima <lb/>Rota, Å¿it rotunda, & contigua ipÅ¿i Rotæ, non autem continua, vel cum Rota conÅ¿oli-<lb/>data. </s> - <s xml:id="s5155" xml:space="preserve">Pars verò quÄ™ per foramen Zodiaci, Å¿eu AraneÄ™ tranÅ¿ibit, Å¿it quadrata vÅ¿que <lb/>ad <reg norm="Å¿ummitatem" type="context">Å¿ummitatẽ</reg> ipÅ¿ius axis (tali Å¿piſſitudine, vt in claui ipÅ¿ius horologij ingredi poÅ¿-<lb/>Å¿it) & ita foramen ipÅ¿ius AraneÄ™, quadratum Å¿it, OÅ¿tenÅ¿or autem circa axem, com <lb/>poÅ¿itus Å¿it tali ordine, vt circa paruum circulum volui poſſit, qui paruus circulus ha-<lb/>beat quadratum foramen, per quod tranÅ¿eat axis, qui axis aliquantulum emineat <lb/>Å¿upra <reg norm="oÅ¿tenÅ¿orem" type="context">oÅ¿tenÅ¿orẽ</reg>. </s> - <s xml:id="s5156" xml:space="preserve">Sub Aranea vero vel Zodiaco, locata erit Tabula, vt <reg norm="nunc" type="context">nÅ©c</reg> dicemus, <lb/>Å¿ed Å¿ciendum eÅ¿t prius, quod inter Tabulam, & Å¿uam Å¿ecundam Rotam, aliam lami-<lb/>nam immobilem interpoÅ¿itam eſſe oportet, quÄ™ circulare foramen habeat, per <reg norm="quod" type="wordlist">qÄ‘</reg> <lb/>quÄ™dam breuis fiÅ¿tula tranÅ¿eat circundans axem & coniungens <reg norm="Tabulam" type="context">Tabulã</reg> cum Å¿ua Ro-<lb/>ta, cuius quidem fiÅ¿tulæ Å¿uperficies concaua, rotunda Å¿it, Å¿uperficies verò extrinÅ¿e-<lb/>ca, nontota, niÅ¿i ea pars, quÄ™ Å¿ecundam Rotam ingreditur, vt in rotundo foramine <lb/>ipÅ¿ius RotÄ™, dicta fiÅ¿tula volui poſſit, pars vero extrinÅ¿eca quÄ™ Tabulam ingredi de-<lb/>bet, Å¿it quadrata. </s> - <s xml:id="s5157" xml:space="preserve">Tabula vero quatuor paruiſſima foramina habeat in extremitati- - <pb o="425" rhead="EPISTOL AE." n="437" file="0437" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0437"/> - bus linearum, meridianÄ™, & verticalis, vt acu mediante volui poſſit, prout oportebit.</s> - </p> - <p> - <s xml:id="s5158" xml:space="preserve">Perfectum igitur cum fuerit op us hoc, te oportet Å¿cire modum ipÅ¿o vtendi. </s> - <s xml:id="s5159" xml:space="preserve">Qua-<lb/>propter quotieÅ¿cunque volueris, aÅ¿pice Solis locum in Zodiaco, Ephemeridibus me <lb/>diantibus, idem dico de vnoquoque reliquorum planetarum. </s> - <s xml:id="s5160" xml:space="preserve">Inuento poÅ¿tea So-<lb/>lis loco in noÅ¿tro Zodiaco horologij, manu mediante, volue oÅ¿tenÅ¿orem, ita, vt li-<lb/>nea fiduciæ tranÅ¿eat per gradum Solis, deinde, claui ipÅ¿ius horologij mediante, vol-<lb/>ue indicem, ita cum Zodiaco coniunctum, vt linea fiduciÄ™, punctum, Å¿eu partem ho-<lb/>rÄ™ oÅ¿tendat in limbo horologii, quÄ™ quidem hora notanda eÅ¿t Å¿i fuerit ex ijs quÄ™ in-<lb/>cipiunt à meridie vÅ¿que ad mediam noctem, vel à media nocte vÅ¿que ad meri-<lb/>diem, </s> - <s xml:id="s5161" xml:space="preserve">tunc acu Å¿upradicta mediante, poÅ¿ita in aliquo illorum quatuor foraminum, <lb/>circunuoluenda eÅ¿t Tabula, ita, vt extremitas lineÄ™ meridianÄ™ Å¿upra orizontem, ex <lb/>Ä™quo incidat inter duodecimam horam, & lineam fiduciÄ™, computum incipiendo à <lb/>duodecima hora, Å¿i vero dicta indicis hora fuerit ex ijs quÄ™ <reg norm="incipiunt" type="context">incipiÅ©t</reg> à media nocte & <lb/>deÅ¿inunt poÅ¿tea in meridie, oportebit, acu mediante, circunuoluere Tabulam, quo-<lb/>uÅ¿que punctum extremum meridianæ Å¿ub terra, medio loco exiÅ¿tat inter <reg norm="duodecimam" type="context">duodecimã</reg> <lb/>horam, & horam oÅ¿tenÅ¿am à linea fiduciÄ™. </s> - <s xml:id="s5162" xml:space="preserve">Quo facto continuo videbis erectam <reg norm="cae- li" type="simple">cÄ™-<lb/>li</reg> figuram. </s> - <s xml:id="s5163" xml:space="preserve">& quia vidiÅ¿ti loca planetarum in Ephemeridibus, videbis etiam <lb/>eorum loca accidentalia in domibus Å¿cilicetaccidentalibus, Å¿i aliquas fixarum in <lb/>Aranea deÅ¿iderabis, accipere poteris Ocu. ♉, cor. ♌, Å¿pi. â™, Liram, Aquilam, & <lb/>Arcturum, dum locus fuerit capax. </s> - <s xml:id="s5164" xml:space="preserve">Nec te moueat, quod oportebit lineam fiduciÄ™ <lb/>Å¿upra gra. Solis quotidie collocare, quod nihil refert. </s> - <s xml:id="s5165" xml:space="preserve">Nam oportet etiam quoti-<lb/>die cordam fuÅ¿o circunuoluere.</s> - </p> - </div> - </div> - <div type="section" level="3" n="52"> - <div type="letter" level="4" n="1"> - <head xml:id="head594" xml:space="preserve">DE DEMONSTRATIONIBVS PROPOSITIONVM <lb/>Mathematicarum, nec non de AÅ¿trologia Iudiciaria.</head> - <head xml:id="head595" style="it" xml:space="preserve">Fu<unsure/>llustriſſ <var>.D.</var> Volfardo AiÅ¿eÅ¿tain.</head> - <p> - <s xml:id="s5166" xml:space="preserve"><emph style="sc">NIhil</emph> mihi gratius & iucundius afferri potuit tuis literis, quibus te cupi-<lb/>dum oÅ¿tendis Å¿ciendi rationem, quare ego non vna methodo ad omnes <lb/>propoÅ¿itiones demonÅ¿trandas vÅ¿us Å¿im, hoc eÅ¿t, </s> - <s xml:id="s5167" xml:space="preserve">quare non omnia ea Eucl. <lb/>Theoremata citem in vnaquaque propoÅ¿itione, quÄ™ ad <reg norm="eam" type="context">eã</reg> demonÅ¿trandam <reg norm="faciunt" type="context">faciÅ©t</reg>, <lb/>quemadmodum in mea Gnomonica vidiÅ¿ti me aliquando omiÅ¿iſſe. </s> - <s xml:id="s5168" xml:space="preserve">ReÅ¿pondeo <reg norm="quod" type="simple">ê™</reg> <lb/>mathematicÄ™ demonÅ¿trationes, hominibus Euclidis Elementa poſſidentibus, non in <lb/>digent aliqua citatione numerorum Theorematum ipÅ¿ius Euclidis, & Å¿i aliquando <lb/>vÅ¿us Å¿um aliqua citatione eorundem, hoc feci propter conÅ¿uetudinem noÅ¿tri tempo <lb/>ris, vel etiam ad faciliorem intelligentiam illorum, quibus Å¿cribebam. </s> - <s xml:id="s5169" xml:space="preserve">Sed omnia <lb/>quamuis minima citare, vt <reg norm="faciunt" type="context">faciÅ©t</reg> nonnulli, mihi, nimis laborioÅ¿um, <reg norm="Å¿uperfluumque" type="simple">Å¿uperfluumqÌ;</reg> <lb/>videtur, preÅ¿ertim ijs (vt dixi) qui memoria tenent prima Elementa. </s> - <s xml:id="s5170" xml:space="preserve">HÄ™c igitur <lb/>eÅ¿t vna ratio. </s> - <s xml:id="s5171" xml:space="preserve">Alia, quia multoties, ita coniuncta eÅ¿t Å¿peculatio cum ipÅ¿a concluÅ¿io <lb/>ne, vt mihi ſępius viÅ¿um Å¿it Å¿uperfluum, aliquid de ipÅ¿a theoria Å¿cribere. </s> - <s xml:id="s5172" xml:space="preserve">In iis <lb/>enim, quÄ™ dum puer eramÅ¿cripÅ¿i, videbis Å¿crupuloÅ¿am illam methodum, Å¿ed po-<lb/>Å¿tea, non niÅ¿i in arduis propoÅ¿itionibus me nihil eſſentiale prÄ™termittere.</s> - </p> - <p> - <s xml:id="s5173" xml:space="preserve">Circa vero id de quo me interrogas, Å¿cilicet, vtrum putem omnia vera eſſe, ea <lb/>quÄ™ Å¿cripta reperiuntur in libris AÅ¿trologæ iudiciariÄ™. </s> - <s xml:id="s5174" xml:space="preserve">ReÅ¿pondeo quod non, imo - <pb o="426" rhead="IO. PAPT. BENED." n="438" file="0438" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0438"/> - puto plurima falÅ¿a eſſe. </s> - <s xml:id="s5175" xml:space="preserve">Nam illa multitudo partium, vt pars vitÄ™, pars Hylech, pars <lb/>futurorum, & reliquÄ™ omnium domorum cÅ“leÅ¿tium, Å¿alua parte fortunÄ™, Å¿unt merÄ™ <lb/>nugÄ™. </s> - <s xml:id="s5176" xml:space="preserve">Idem dico de faciebus, Å¿iue decanis, de terminis, & de gradibus ipÅ¿is, vt pu-<lb/>ta azemenis, puteis, vacuis, fumoÅ¿is, & de reliquis. </s> - <s xml:id="s5177" xml:space="preserve">De Domibus vero, Exaltationi <lb/>bus, nec non triplicitatibus, experientia <reg norm="confirmat" type="context">cõfirmat</reg> ea vera eſſe. </s> - <s xml:id="s5178" xml:space="preserve"><reg norm="Idem" type="context">Idẽ</reg> affirmo de Domi <lb/>bus accidentalibus, rationalibus tamen, non <reg norm="autem" type="context">autẽ</reg> de Domibus Campani, & Gazuli. <lb/></s> - <s xml:id="s5179" xml:space="preserve">ObÅ¿eruationes etiam complexionum Å¿eu inÅ¿luentiarum ipÅ¿orum Planetarum rectè <lb/>factæ Å¿unt, quÄ™ etiam à coloribus ipÅ¿orum Planetarum ferè iudicari poſſunt. </s> - <s xml:id="s5180" xml:space="preserve">Con-<lb/>iunctiones <reg norm="aÅ¿pectusque" type="simple">aÅ¿pectusqÌ;</reg> ipÅ¿orum inuicem, Å¿imiliter mirabilia faciunt, & ex maiori par <lb/>te, ea, quÄ™ de iÅ¿tis Å¿cribuntur vera Å¿unt. </s> - <s xml:id="s5181" xml:space="preserve">Reuolutiones annuÄ™ Å¿imiliter, cum Domino <lb/>anni. </s> - <s xml:id="s5182" xml:space="preserve">Dominum verò orbis <reg norm="DiuiÅ¿oremque" type="simple">DiuiÅ¿oremqÌ;</reg> non approbo, nam hic pendet à termino, <lb/>ille verò ab hora. </s> - <s xml:id="s5183" xml:space="preserve">Nouenarias autem Dodecathemoria, Alfridarias, & multa iis Å¿i <lb/>milia omnia nego. </s> - <s xml:id="s5184" xml:space="preserve">AntiÅ¿cia, vera Å¿unt, ideÅ¿t influunt, malos tamen effectus, alia <lb/>plus alia verò minus, prout aliqua eorum Å¿unt tetragona, alia verò trigona, alia ma-<lb/>gna, alia parua, magna Å¿unt, vt Arietis cum Virgine, & LibrÄ™ cum PiÅ¿cibus, parua ve <lb/>rò, <reg norm="debiliaque" type="simple">debiliaqÌ;</reg> Geminorum cum Cancro, & Sagittarij <reg norm="cum" type="context">cÅ©</reg> Capricorno. </s> - <s xml:id="s5185" xml:space="preserve">Sed difuÅ¿ius <lb/>hÄ™c <reg norm="omnia" type="wordlist">oÄ©a</reg> videbis in meo illo particulari tractatu, de quo tibi alià s dixi, in quo multa <lb/>videbis, quÄ™ omnia ab experientia, ex multis à me obÅ¿eruatis, comprobata Å¿unt, <lb/>quem quidem tractatum cum quibuÅ¿dam alijs meis Å¿peculationibus in lucem prode <lb/>re cupio, Å¿i fieri poterit, antequam ad directionem mei HoroÅ¿copi cum corpore <lb/>Martis AnÄ™retÄ™ perueniam, quÄ™ quidem directio circa annum milleÅ¿imum quin-<lb/>genteÅ¿imum nonageÅ¿imum Å¿ecundum eueniet.</s> - </p> - <head xml:id="head596" type="footer" xml:space="preserve">FINIS.</head> - </div> - </div> - </div> - </div> - <div type="back" level="1" n="1"> - <div type="errata" level="2" n="1"> - <pb n="439" file="0439" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0439"/> - <head xml:id="head597" xml:space="preserve">ERRATA CORRIGITO IVXTA INFRASCRIPTAM TABVLAM,</head> - <head xml:id="head598" xml:space="preserve">Reliquos verò errores, qui orthographiam reÅ¿piciunt, benignus lector corrigat, quiÅ¿ciat multos errores in <lb/>editione irrepÅ¿iſſe, quod non paucos dies morbo fuerim detentus dum præſens opus excuderetur.</head> - <xhtml:table class="it"> - <xhtml:tr> - <xhtml:th xml:space="preserve">Pag.</xhtml:th> - <xhtml:th xml:space="preserve">Lin.</xhtml:th> - <xhtml:th xml:space="preserve">Errata</xhtml:th> - <xhtml:th xml:space="preserve">Correcta</xhtml:th> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">3</xhtml:td> - <xhtml:td xml:space="preserve">29</xhtml:td> - <xhtml:td xml:space="preserve">æqualis</xhtml:td> - <xhtml:td xml:space="preserve">æquali</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">8</xhtml:td> - <xhtml:td xml:space="preserve">35</xhtml:td> - <xhtml:td xml:space="preserve">maius</xhtml:td> - <xhtml:td xml:space="preserve">maior</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">9</xhtml:td> - <xhtml:td xml:space="preserve">15</xhtml:td> - <xhtml:td xml:space="preserve">in vnitate Å¿uperficialis, erit ac</xhtml:td> - <xhtml:td xml:space="preserve">in vnitate, Å¿upreficialis erit, ac</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">11</xhtml:td> - <xhtml:td xml:space="preserve">1</xhtml:td> - <xhtml:td xml:space="preserve">proueuiens</xhtml:td> - <xhtml:td xml:space="preserve">prouenientem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">11</xhtml:td> - <xhtml:td xml:space="preserve">8</xhtml:td> - <xhtml:td xml:space="preserve">futurum</xhtml:td> - <xhtml:td xml:space="preserve">futurus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">11</xhtml:td> - <xhtml:td xml:space="preserve">31</xhtml:td> - <xhtml:td xml:space="preserve">illæ nihil aliud Å¿unt</xhtml:td> - <xhtml:td xml:space="preserve">illud nihil aliud est</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">11</xhtml:td> - <xhtml:td xml:space="preserve">38</xhtml:td> - <xhtml:td xml:space="preserve">diuidemus</xhtml:td> - <xhtml:td xml:space="preserve">diuidamus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">12</xhtml:td> - <xhtml:td xml:space="preserve">1</xhtml:td> - <xhtml:td xml:space="preserve">tertiæ, Å¿int</xhtml:td> - <xhtml:td xml:space="preserve">tertiæ Å¿int</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">12</xhtml:td> - <xhtml:td xml:space="preserve">2</xhtml:td> - <xhtml:td xml:space="preserve">producturn</xhtml:td> - <xhtml:td xml:space="preserve">productus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">19</xhtml:td> - <xhtml:td xml:space="preserve">30</xhtml:td> - <xhtml:td xml:space="preserve">proueniens</xhtml:td> - <xhtml:td xml:space="preserve">prouenientem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">19</xhtml:td> - <xhtml:td xml:space="preserve">30</xhtml:td> - <xhtml:td xml:space="preserve">productum æquale</xhtml:td> - <xhtml:td xml:space="preserve">productus æqualis</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">21</xhtml:td> - <xhtml:td xml:space="preserve">27</xhtml:td> - <xhtml:td xml:space="preserve">eadem</xhtml:td> - <xhtml:td xml:space="preserve">eædcm</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">24</xhtml:td> - <xhtml:td xml:space="preserve">18</xhtml:td> - <xhtml:td xml:space="preserve">eÅ¿t</xhtml:td> - <xhtml:td xml:space="preserve">eſſet</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">26</xhtml:td> - <xhtml:td xml:space="preserve">39</xhtml:td> - <xhtml:td xml:space="preserve">est</xhtml:td> - <xhtml:td xml:space="preserve">eſſet</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">41</xhtml:td> - <xhtml:td xml:space="preserve">24</xhtml:td> - <xhtml:td xml:space="preserve">quæ</xhtml:td> - <xhtml:td xml:space="preserve">quæ</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">41</xhtml:td> - <xhtml:td xml:space="preserve">31</xhtml:td> - <xhtml:td xml:space="preserve">distinguendæ</xhtml:td> - <xhtml:td xml:space="preserve">diÅ¿tinguendo</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">59</xhtml:td> - <xhtml:td xml:space="preserve">5</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ubÅ¿equens</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ubÅ¿equentem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">61</xhtml:td> - <xhtml:td xml:space="preserve">3</xhtml:td> - <xhtml:td xml:space="preserve">hæc via tenendæ</xhtml:td> - <xhtml:td xml:space="preserve">hanc viamtenere</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">61</xhtml:td> - <xhtml:td xml:space="preserve">3</xhtml:td> - <xhtml:td xml:space="preserve">fuit</xhtml:td> - <xhtml:td xml:space="preserve">fuerit</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">61</xhtml:td> - <xhtml:td xml:space="preserve">45</xhtml:td> - <xhtml:td xml:space="preserve">numerum quæſitum</xhtml:td> - <xhtml:td xml:space="preserve">numerus quæſitus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">62</xhtml:td> - <xhtml:td xml:space="preserve">30</xhtml:td> - <xhtml:td xml:space="preserve">quantum est <reg norm="dimidiam" type="context">dimidiã</reg> occupatam</xhtml:td> - <xhtml:td xml:space="preserve">quanta est dimidia occupata</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">64</xhtml:td> - <xhtml:td xml:space="preserve">4</xhtml:td> - <xhtml:td xml:space="preserve">Å¿peculari</xhtml:td> - <xhtml:td xml:space="preserve">conÅ¿iderari</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">64</xhtml:td> - <xhtml:td xml:space="preserve">15</xhtml:td> - <xhtml:td xml:space="preserve"><reg norm="totque" type="simple">totq;</reg> Å¿unt termini</xhtml:td> - <xhtml:td xml:space="preserve"><reg norm="totque" type="simple">totq;</reg> eße terminos</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">64</xhtml:td> - <xhtml:td xml:space="preserve">18</xhtml:td> - <xhtml:td xml:space="preserve">hoc est numerum</xhtml:td> - <xhtml:td xml:space="preserve">hoc est per numerum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">64</xhtml:td> - <xhtml:td xml:space="preserve">19</xhtml:td> - <xhtml:td xml:space="preserve">primo quod vnus est</xhtml:td> - <xhtml:td xml:space="preserve">primo qui vnus est</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">66</xhtml:td> - <xhtml:td xml:space="preserve">18</xhtml:td> - <xhtml:td xml:space="preserve">minimum</xhtml:td> - <xhtml:td xml:space="preserve">minimns</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">67</xhtml:td> - <xhtml:td xml:space="preserve">13</xhtml:td> - <xhtml:td xml:space="preserve">maximum terminum <reg norm="addendum" type="context">addendÅ©</reg></xhtml:td> - <xhtml:td xml:space="preserve">maximus terminus addendus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">72</xhtml:td> - <xhtml:td xml:space="preserve">1</xhtml:td> - <xhtml:td xml:space="preserve">numerum</xhtml:td> - <xhtml:td xml:space="preserve">numerus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">72</xhtml:td> - <xhtml:td xml:space="preserve">8</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ingulos itinere</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ingulos in itinere</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">74</xhtml:td> - <xhtml:td xml:space="preserve">7</xhtml:td> - <xhtml:td xml:space="preserve">itinerarium</xhtml:td> - <xhtml:td xml:space="preserve">itinerant ium</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">78</xhtml:td> - <xhtml:td xml:space="preserve">21</xhtml:td> - <xhtml:td xml:space="preserve">morum</xhtml:td> - <xhtml:td xml:space="preserve">modum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">78</xhtml:td> - <xhtml:td xml:space="preserve">24</xhtml:td> - <xhtml:td xml:space="preserve">noueratius</xhtml:td> - <xhtml:td xml:space="preserve">nouenarius</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">81</xhtml:td> - <xhtml:td xml:space="preserve">10</xhtml:td> - <xhtml:td xml:space="preserve">iuncta</xhtml:td> - <xhtml:td xml:space="preserve">iunctæ</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">88</xhtml:td> - <xhtml:td xml:space="preserve">35</xhtml:td> - <xhtml:td xml:space="preserve">armonicæ</xhtml:td> - <xhtml:td xml:space="preserve">harmonicæ</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">91</xhtml:td> - <xhtml:td xml:space="preserve">6</xhtml:td> - <xhtml:td xml:space="preserve">Quare <reg norm="argumentande" type="context context">argumẽtãde</reg> permut ando</xhtml:td> - <xhtml:td xml:space="preserve">Quare permutando</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">95</xhtml:td> - <xhtml:td xml:space="preserve">27</xhtml:td> - <xhtml:td xml:space="preserve">vitium</xhtml:td> - <xhtml:td xml:space="preserve">vicium</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">97</xhtml:td> - <xhtml:td xml:space="preserve">14</xhtml:td> - <xhtml:td xml:space="preserve">Å¿umma</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ummam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">97</xhtml:td> - <xhtml:td xml:space="preserve">30</xhtml:td> - <xhtml:td xml:space="preserve">illud verò quod con</xhtml:td> - <xhtml:td xml:space="preserve">illud verò con</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">98</xhtml:td> - <xhtml:td xml:space="preserve">22</xhtml:td> - <xhtml:td xml:space="preserve">deÅ¿ideraremus</xhtml:td> - <xhtml:td xml:space="preserve">deÅ¿iderauerimus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">103</xhtml:td> - <xhtml:td xml:space="preserve">23</xhtml:td> - <xhtml:td xml:space="preserve">diÅ¿poÅ¿itis facit</xhtml:td> - <xhtml:td xml:space="preserve">diÅ¿poÅ¿itis, tantum facit</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">104</xhtml:td> - <xhtml:td xml:space="preserve">39</xhtml:td> - <xhtml:td xml:space="preserve">diÅ¿poÅ¿itis, facit</xhtml:td> - <xhtml:td xml:space="preserve">diÅ¿poÅ¿itis, tantum facit</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">107</xhtml:td> - <xhtml:td xml:space="preserve">21</xhtml:td> - <xhtml:td xml:space="preserve">eccedit</xhtml:td> - <xhtml:td xml:space="preserve">excedit</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">110</xhtml:td> - <xhtml:td xml:space="preserve">41</xhtml:td> - <xhtml:td xml:space="preserve">habuerit eius cerebrum</xhtml:td> - <xhtml:td xml:space="preserve">habuerit cerebrum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">113</xhtml:td> - <xhtml:td xml:space="preserve">46</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ufficeret</xhtml:td> - <xhtml:td xml:space="preserve">Å¿uffecißet</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">116</xhtml:td> - <xhtml:td xml:space="preserve">8</xhtml:td> - <xhtml:td xml:space="preserve">quarum Å¿ecundam</xhtml:td> - <xhtml:td xml:space="preserve">quorum Å¿ecundum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">116</xhtml:td> - <xhtml:td xml:space="preserve">8</xhtml:td> - <xhtml:td xml:space="preserve">primæ, tertiam</xhtml:td> - <xhtml:td xml:space="preserve">primi, tertium</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">116</xhtml:td> - <xhtml:td xml:space="preserve">9</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ecundæ</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ecundi</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">130</xhtml:td> - <xhtml:td xml:space="preserve">23</xhtml:td> - <xhtml:td xml:space="preserve">ducenda</xhtml:td> - <xhtml:td xml:space="preserve">ducendus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">133</xhtml:td> - <xhtml:td xml:space="preserve">10</xhtml:td> - <xhtml:td xml:space="preserve">lineam qua</xhtml:td> - <xhtml:td xml:space="preserve">lineam, qua</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">133</xhtml:td> - <xhtml:td xml:space="preserve">11</xhtml:td> - <xhtml:td xml:space="preserve">dratam</xhtml:td> - <xhtml:td xml:space="preserve">dratum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">134</xhtml:td> - <xhtml:td xml:space="preserve">16</xhtml:td> - <xhtml:td xml:space="preserve">inueniet</xhtml:td> - <xhtml:td xml:space="preserve">inuenimus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">137</xhtml:td> - <xhtml:td xml:space="preserve">22</xhtml:td> - <xhtml:td xml:space="preserve">distans</xhtml:td> - <xhtml:td xml:space="preserve">diÅ¿tantius</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">137</xhtml:td> - <xhtml:td xml:space="preserve">45</xhtml:td> - <xhtml:td xml:space="preserve">Notißimum igitur primum</xhtml:td> - <xhtml:td xml:space="preserve">Notißimum primum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">139</xhtml:td> - <xhtml:td xml:space="preserve">15</xhtml:td> - <xhtml:td xml:space="preserve">est</xhtml:td> - <xhtml:td xml:space="preserve">Å¿it</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">139</xhtml:td> - <xhtml:td xml:space="preserve">28</xhtml:td> - <xhtml:td xml:space="preserve">duas</xhtml:td> - <xhtml:td xml:space="preserve">duos</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">141</xhtml:td> - <xhtml:td xml:space="preserve">5</xhtml:td> - <xhtml:td xml:space="preserve">comperuiſſe</xhtml:td> - <xhtml:td xml:space="preserve">comperiſſe</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">142</xhtml:td> - <xhtml:td xml:space="preserve">19</xhtml:td> - <xhtml:td xml:space="preserve"><reg norm="quam" type="context">quã</reg></xhtml:td> - <xhtml:td xml:space="preserve">ac</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">143</xhtml:td> - <xhtml:td xml:space="preserve">17</xhtml:td> - <xhtml:td xml:space="preserve">linea</xhtml:td> - <xhtml:td xml:space="preserve">lineam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">144</xhtml:td> - <xhtml:td xml:space="preserve">23</xhtml:td> - <xhtml:td xml:space="preserve">patebit, Å¿i quis</xhtml:td> - <xhtml:td xml:space="preserve">patebit, quòd Å¿i quis</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">145</xhtml:td> - <xhtml:td xml:space="preserve">18</xhtml:td> - <xhtml:td xml:space="preserve">constante</xhtml:td> - <xhtml:td xml:space="preserve">constantem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">146</xhtml:td> - <xhtml:td xml:space="preserve">20</xhtml:td> - <xhtml:td xml:space="preserve">paÅ¿tam</xhtml:td> - <xhtml:td xml:space="preserve">maſſam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">146</xhtml:td> - <xhtml:td xml:space="preserve">22</xhtml:td> - <xhtml:td xml:space="preserve">pastam</xhtml:td> - <xhtml:td xml:space="preserve">maſſam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">148</xhtml:td> - <xhtml:td xml:space="preserve">13</xhtml:td> - <xhtml:td xml:space="preserve"><reg norm="proponit" type="simple">ê“ponit</reg> <reg norm="non" type="context">nõ</reg> <reg norm="concludit" type="context">cõcludit</reg> melius <reg norm="autem" type="wordlist">aÅ©t</reg></xhtml:td> - <xhtml:td xml:space="preserve">proponit, <reg norm="non" type="context">nõ</reg> concludit, melius <reg norm="autem" type="wordlist">aÅ©t</reg></xhtml:td> - 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<xhtml:td xml:space="preserve">153</xhtml:td> - <xhtml:td xml:space="preserve">24</xhtml:td> - <xhtml:td xml:space="preserve">existente. vnde aliqua</xhtml:td> - <xhtml:td xml:space="preserve">exiÅ¿tente, aliqua</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">154</xhtml:td> - <xhtml:td xml:space="preserve">23</xhtml:td> - <xhtml:td xml:space="preserve">futurum</xhtml:td> - <xhtml:td xml:space="preserve">effecturam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">155</xhtml:td> - <xhtml:td xml:space="preserve">25</xhtml:td> - <xhtml:td xml:space="preserve">carta</xhtml:td> - <xhtml:td xml:space="preserve">charta</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">156</xhtml:td> - <xhtml:td xml:space="preserve">23</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ufficere</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ufficeret</xhtml:td> - </xhtml:tr> - </xhtml:table> - <xhtml:table class="it"> - <xhtml:tr> - <xhtml:th xml:space="preserve">Pag.</xhtml:th> - <xhtml:th xml:space="preserve">Lin.</xhtml:th> - <xhtml:th xml:space="preserve">Errata</xhtml:th> - <xhtml:th xml:space="preserve">Correcta</xhtml:th> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">158</xhtml:td> - <xhtml:td xml:space="preserve">26</xhtml:td> - <xhtml:td xml:space="preserve">verÅ¿a</xhtml:td> - <xhtml:td xml:space="preserve">verÅ¿am</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">158</xhtml:td> - <xhtml:td xml:space="preserve">26</xhtml:td> - <xhtml:td xml:space="preserve">Å¿it</xhtml:td> - <xhtml:td xml:space="preserve">Å¿int</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">162</xhtml:td> - <xhtml:td xml:space="preserve">22</xhtml:td> - <xhtml:td xml:space="preserve">cindenda</xhtml:td> - <xhtml:td xml:space="preserve">Å¿cindenda</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">163</xhtml:td> - <xhtml:td xml:space="preserve">7</xhtml:td> - <xhtml:td xml:space="preserve">oppeÅ¿itus</xhtml:td> - <xhtml:td xml:space="preserve">oppoÅ¿itum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">164</xhtml:td> - <xhtml:td xml:space="preserve">24</xhtml:td> - <xhtml:td xml:space="preserve">tanta</xhtml:td> - <xhtml:td xml:space="preserve">tantæ</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">164</xhtml:td> - <xhtml:td xml:space="preserve">37</xhtml:td> - <xhtml:td xml:space="preserve">adiunctæ nobis eſſent duæ aliæ</xhtml:td> - <xhtml:td xml:space="preserve">adiuncti nobis eßent duo alii</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">164</xhtml:td> - <xhtml:td xml:space="preserve">39</xhtml:td> - <xhtml:td xml:space="preserve">Å¿estineret</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ustineret</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">164</xhtml:td> - <xhtml:td xml:space="preserve">40</xhtml:td> - <xhtml:td xml:space="preserve">dictæ</xhtml:td> - <xhtml:td xml:space="preserve">dicti</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">164</xhtml:td> - <xhtml:td xml:space="preserve">41</xhtml:td> - <xhtml:td xml:space="preserve">aliquam</xhtml:td> - <xhtml:td xml:space="preserve">aliquem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">165</xhtml:td> - <xhtml:td xml:space="preserve">3</xhtml:td> - <xhtml:td xml:space="preserve">Å¿uffieiant</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ufficiat</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">165</xhtml:td> - <xhtml:td xml:space="preserve">4</xhtml:td> - <xhtml:td xml:space="preserve"> Å¿ubÅ¿eÅ¿quialter<unsure/> - </xhtml:td> - <xhtml:td xml:space="preserve">Å¿ubÅ¿eÅ¿quialterum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">186</xhtml:td> - <xhtml:td xml:space="preserve">32</xhtml:td> - <xhtml:td xml:space="preserve">finit</xhtml:td> - <xhtml:td xml:space="preserve">finitam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">187</xhtml:td> - <xhtml:td xml:space="preserve">29</xhtml:td> - <xhtml:td xml:space="preserve">propinqua</xhtml:td> - <xhtml:td xml:space="preserve">propinqui</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">187<unsure/> - </xhtml:td> - <xhtml:td xml:space="preserve">30</xhtml:td> - <xhtml:td xml:space="preserve">philoÅ¿ophi Å¿upra</xhtml:td> - <xhtml:td xml:space="preserve">philoÅ¿ophi, proximè Å¿upra</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">187<unsure/> - </xhtml:td> - <xhtml:td xml:space="preserve">31</xhtml:td> - <xhtml:td xml:space="preserve">contingeret menÅ¿e</xhtml:td> - <xhtml:td xml:space="preserve">contingeret vt menÅ¿e</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">187</xhtml:td> - <xhtml:td xml:space="preserve">34</xhtml:td> - <xhtml:td xml:space="preserve">regionis) non conÅ¿iderans</xhtml:td> - <xhtml:td xml:space="preserve">regionis) in quo non conÅ¿iderauit</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">190</xhtml:td> - <xhtml:td xml:space="preserve">11</xhtml:td> - <xhtml:td xml:space="preserve">lenas</xhtml:td> - <xhtml:td xml:space="preserve">lenæs</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">190</xhtml:td> - <xhtml:td xml:space="preserve">23</xhtml:td> - <xhtml:td xml:space="preserve">lenis</xhtml:td> - <xhtml:td xml:space="preserve">leuis</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">191</xhtml:td> - <xhtml:td xml:space="preserve">10</xhtml:td> - <xhtml:td xml:space="preserve">aliud quadratum</xhtml:td> - <xhtml:td xml:space="preserve">alius quadratus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">191</xhtml:td> - <xhtml:td xml:space="preserve">15</xhtml:td> - <xhtml:td xml:space="preserve">qualis</xhtml:td> - <xhtml:td xml:space="preserve">qui</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">193</xhtml:td> - <xhtml:td xml:space="preserve">17</xhtml:td> - <xhtml:td xml:space="preserve">cum</xhtml:td> - <xhtml:td xml:space="preserve">tum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">193</xhtml:td> - <xhtml:td xml:space="preserve">36</xhtml:td> - <xhtml:td xml:space="preserve">fit</xhtml:td> - <xhtml:td xml:space="preserve">facit</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">193</xhtml:td> - <xhtml:td xml:space="preserve">40</xhtml:td> - <xhtml:td xml:space="preserve">pleno vbi</xhtml:td> - <xhtml:td xml:space="preserve">pleno poÅ¿uerimus, vbi</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">195</xhtml:td> - <xhtml:td xml:space="preserve">10</xhtml:td> - <xhtml:td xml:space="preserve">valent</xhtml:td> - <xhtml:td xml:space="preserve">valent apud ipÅ¿os</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">196</xhtml:td> - <xhtml:td xml:space="preserve">19</xhtml:td> - <xhtml:td xml:space="preserve">phyloÅ¿ophiæ</xhtml:td> - <xhtml:td xml:space="preserve">philoÅ¿ophiæ</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">196</xhtml:td> - <xhtml:td xml:space="preserve">20</xhtml:td> - <xhtml:td xml:space="preserve">pellendo</xhtml:td> - <xhtml:td xml:space="preserve">plendo</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">198</xhtml:td> - <xhtml:td xml:space="preserve">18</xhtml:td> - <xhtml:td xml:space="preserve">Å¿unt etiam</xhtml:td> - <xhtml:td xml:space="preserve">eße etiam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">204</xhtml:td> - <xhtml:td xml:space="preserve">12</xhtml:td> - <xhtml:td xml:space="preserve">priñcipium</xhtml:td> - <xhtml:td xml:space="preserve">principum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">204</xhtml:td> - <xhtml:td xml:space="preserve">18</xhtml:td> - <xhtml:td xml:space="preserve">indigna</xhtml:td> - <xhtml:td xml:space="preserve">indignas</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">204</xhtml:td> - <xhtml:td xml:space="preserve">21</xhtml:td> - <xhtml:td xml:space="preserve">diffundetur, creÅ¿cat</xhtml:td> - <xhtml:td xml:space="preserve">diffundatur, & creÅ¿cat</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">205</xhtml:td> - <xhtml:td xml:space="preserve">16</xhtml:td> - <xhtml:td xml:space="preserve">aliquando Å¿er</xhtml:td> - <xhtml:td xml:space="preserve">aliquando me Å¿er</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">205</xhtml:td> - <xhtml:td xml:space="preserve">19</xhtml:td> - <xhtml:td xml:space="preserve">anno præter, <reg norm="neceßitatem" type="context">neceßitatẽ</reg>, gignitur</xhtml:td> - <xhtml:td xml:space="preserve">anno, præter <reg norm="neceßitatem" type="context">neceßitatẽ</reg> gignitur</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">206</xhtml:td> - <xhtml:td xml:space="preserve">21</xhtml:td> - <xhtml:td xml:space="preserve">plenilunium, quod</xhtml:td> - <xhtml:td xml:space="preserve">plenilunium fieri, quod</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">207</xhtml:td> - <xhtml:td xml:space="preserve">26</xhtml:td> - <xhtml:td xml:space="preserve">inchoet annus</xhtml:td> - <xhtml:td xml:space="preserve">inchoetur annus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">209</xhtml:td> - <xhtml:td xml:space="preserve">9</xhtml:td> - <xhtml:td xml:space="preserve">Å¿oli, quod punctum</xhtml:td> - <xhtml:td xml:space="preserve">Å¿olis, cuius punctum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">210</xhtml:td> - <xhtml:td xml:space="preserve">10</xhtml:td> - <xhtml:td xml:space="preserve">cælebrandi</xhtml:td> - <xhtml:td xml:space="preserve">celebrandis</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">212</xhtml:td> - <xhtml:td xml:space="preserve">33</xhtml:td> - <xhtml:td xml:space="preserve">Inuentæ</xhtml:td> - <xhtml:td xml:space="preserve">Inuenti</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">212</xhtml:td> - <xhtml:td xml:space="preserve">33</xhtml:td> - <xhtml:td xml:space="preserve">duæ</xhtml:td> - <xhtml:td xml:space="preserve">duo</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">214</xhtml:td> - <xhtml:td xml:space="preserve">9</xhtml:td> - <xhtml:td xml:space="preserve">claßi</xhtml:td> - <xhtml:td xml:space="preserve">claßis</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">214</xhtml:td> - <xhtml:td xml:space="preserve">10</xhtml:td> - <xhtml:td xml:space="preserve">Inter Eximias</xhtml:td> - <xhtml:td xml:space="preserve">Quia inter Eximias</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">214</xhtml:td> - <xhtml:td xml:space="preserve">26</xhtml:td> - <xhtml:td xml:space="preserve">reperiretur</xhtml:td> - <xhtml:td xml:space="preserve">reperietur</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">214</xhtml:td> - <xhtml:td xml:space="preserve">34</xhtml:td> - <xhtml:td xml:space="preserve">neceſſario Å¿it futurum, vt</xhtml:td> - <xhtml:td xml:space="preserve">neceßarium Å¿it, vt</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">215</xhtml:td> - <xhtml:td xml:space="preserve">37</xhtml:td> - <xhtml:td xml:space="preserve">quod Å¿i velimus</xhtml:td> - <xhtml:td xml:space="preserve">Å¿i velimus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">215</xhtml:td> - <xhtml:td xml:space="preserve">39</xhtml:td> - <xhtml:td xml:space="preserve"> poteÅ¿t vno <reg norm="eodemque" type="simple">eodemq;</reg> - </xhtml:td> - <xhtml:td xml:space="preserve">vno <reg norm="eodemque" type="simple">eodemq;</reg></xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">215</xhtml:td> - <xhtml:td xml:space="preserve">45</xhtml:td> - <xhtml:td xml:space="preserve">Å¿it circulus</xhtml:td> - <xhtml:td xml:space="preserve">Å¿it verè circulus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">217</xhtml:td> - <xhtml:td xml:space="preserve">2</xhtml:td> - <xhtml:td xml:space="preserve">Quod cum verum</xhtml:td> - <xhtml:td xml:space="preserve">Quod Å¿i verum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">217</xhtml:td> - <xhtml:td xml:space="preserve">4</xhtml:td> - <xhtml:td xml:space="preserve">nulla estratio</xhtml:td> - <xhtml:td xml:space="preserve">nulla eſſet ratio</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">221</xhtml:td> - <xhtml:td xml:space="preserve">31</xhtml:td> - <xhtml:td xml:space="preserve">inueniemus</xhtml:td> - <xhtml:td xml:space="preserve">deÅ¿ignabimus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">222</xhtml:td> - <xhtml:td xml:space="preserve">20</xhtml:td> - <xhtml:td xml:space="preserve">falli</xhtml:td> - <xhtml:td xml:space="preserve">fallat</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">225</xhtml:td> - <xhtml:td xml:space="preserve">20</xhtml:td> - <xhtml:td xml:space="preserve">Å¿i quæ</xhtml:td> - <xhtml:td xml:space="preserve"><reg norm="Å¿ique" type="simple">Å¿iq;</reg> - </xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">225</xhtml:td> - <xhtml:td xml:space="preserve">25</xhtml:td> - <xhtml:td xml:space="preserve">chnum</xhtml:td> - <xhtml:td xml:space="preserve">chnium</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">226</xhtml:td> - <xhtml:td xml:space="preserve">14</xhtml:td> - <xhtml:td xml:space="preserve">oleum effundebat</xhtml:td> - <xhtml:td xml:space="preserve">effundebatur</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">227</xhtml:td> - <xhtml:td xml:space="preserve">5</xhtml:td> - <xhtml:td xml:space="preserve">euadit</xhtml:td> - <xhtml:td xml:space="preserve">euadet</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">231</xhtml:td> - <xhtml:td xml:space="preserve">36</xhtml:td> - <xhtml:td xml:space="preserve">Å¿upradicta, minuta</xhtml:td> - <xhtml:td xml:space="preserve">Å¿upradicta minuta</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">232</xhtml:td> - <xhtml:td xml:space="preserve">21</xhtml:td> - <xhtml:td xml:space="preserve">progrediuntur</xhtml:td> - <xhtml:td xml:space="preserve">progredi</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">234</xhtml:td> - <xhtml:td xml:space="preserve">15</xhtml:td> - <xhtml:td xml:space="preserve">aliiori cÅ“lo</xhtml:td> - <xhtml:td xml:space="preserve">altius cælum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">239</xhtml:td> - <xhtml:td xml:space="preserve">12</xhtml:td> - <xhtml:td xml:space="preserve">dum</xhtml:td> - <xhtml:td xml:space="preserve">cum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">241</xhtml:td> - <xhtml:td xml:space="preserve">41</xhtml:td> - <xhtml:td xml:space="preserve">afferens</xhtml:td> - <xhtml:td xml:space="preserve">cogitans</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">245</xhtml:td> - <xhtml:td xml:space="preserve">25</xhtml:td> - <xhtml:td xml:space="preserve">diametri</xhtml:td> - <xhtml:td xml:space="preserve">diametros</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">248</xhtml:td> - <xhtml:td xml:space="preserve">42</xhtml:td> - <xhtml:td xml:space="preserve">Å¿it & ob id</xhtml:td> - <xhtml:td xml:space="preserve">Å¿it, ob id</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">249</xhtml:td> - <xhtml:td xml:space="preserve">42</xhtml:td> - <xhtml:td xml:space="preserve">alium numerum</xhtml:td> - <xhtml:td xml:space="preserve">alio numero</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">250</xhtml:td> - <xhtml:td xml:space="preserve">13</xhtml:td> - <xhtml:td xml:space="preserve">um voluerimus</xhtml:td> - <xhtml:td xml:space="preserve">cum inuestigare voluerimus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">251</xhtml:td> - <xhtml:td xml:space="preserve">1</xhtml:td> - <xhtml:td xml:space="preserve">inuenerimus</xhtml:td> - <xhtml:td xml:space="preserve">inuenimus</xhtml:td> - 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Å¿it</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">263</xhtml:td> - <xhtml:td xml:space="preserve">22</xhtml:td> - <xhtml:td xml:space="preserve">Å¿emidiameter eße vnius</xhtml:td> - <xhtml:td xml:space="preserve">Å¿emidiameter vnius</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">265</xhtml:td> - <xhtml:td xml:space="preserve">4</xhtml:td> - <xhtml:td xml:space="preserve">cognitæ</xhtml:td> - <xhtml:td xml:space="preserve">cogniti</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">272</xhtml:td> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td xml:space="preserve">infigura vbieÅ¿t <var>.P.</var></xhtml:td> - <xhtml:td xml:space="preserve">ponatur. R</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">272</xhtml:td> - <xhtml:td xml:space="preserve">18</xhtml:td> - <xhtml:td xml:space="preserve">E.i. ad <var>.E.i.</var></xhtml:td> - <xhtml:td xml:space="preserve">E.I. ad <var>.E.i.</var></xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">273</xhtml:td> - <xhtml:td xml:space="preserve">3</xhtml:td> - <xhtml:td xml:space="preserve">licet</xhtml:td> - <xhtml:td xml:space="preserve">liceret</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">274</xhtml:td> - <xhtml:td xml:space="preserve">7</xhtml:td> - <xhtml:td xml:space="preserve">te</xhtml:td> - <xhtml:td xml:space="preserve">ibi</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">275</xhtml:td> - <xhtml:td xml:space="preserve">10</xhtml:td> - <xhtml:td xml:space="preserve">dividere</xhtml:td> - <xhtml:td xml:space="preserve">diuidendo</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">276</xhtml:td> - <xhtml:td xml:space="preserve">47</xhtml:td> - <xhtml:td xml:space="preserve">accipiemus</xhtml:td> - <xhtml:td xml:space="preserve">accipiamus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">278</xhtml:td> - <xhtml:td xml:space="preserve">8</xhtml:td> - <xhtml:td xml:space="preserve">Moreta</xhtml:td> - <xhtml:td xml:space="preserve">Moreta</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">280</xhtml:td> - <xhtml:td xml:space="preserve">25</xhtml:td> - <xhtml:td xml:space="preserve">diminutæ, quartæ</xhtml:td> - <xhtml:td xml:space="preserve">diminutæ Å¿eu defectiuæ, quarta</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">281</xhtml:td> - <xhtml:td xml:space="preserve">36</xhtml:td> - <xhtml:td xml:space="preserve">grauißimum</xhtml:td> - <xhtml:td xml:space="preserve">grauißimo</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">281</xhtml:td> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td colspan="2" xml:space="preserve">in fine pag. vnamquanque Zifram accommoda Å¿ub vnoquoque al-<lb/>phabeti characterem ad priores Zifras prius addendo .1. 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<xhtml:tr> - <xhtml:td xml:space="preserve">290</xhtml:td> - <xhtml:td xml:space="preserve">10</xhtml:td> - <xhtml:td xml:space="preserve">Å¿atile</xhtml:td> - <xhtml:td xml:space="preserve">Å¿atilis</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">290</xhtml:td> - <xhtml:td xml:space="preserve">19</xhtml:td> - <xhtml:td xml:space="preserve">æqualis</xhtml:td> - <xhtml:td xml:space="preserve">æquale</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">290</xhtml:td> - <xhtml:td xml:space="preserve">43</xhtml:td> - <xhtml:td xml:space="preserve">circunÅ¿criptibilis</xhtml:td> - <xhtml:td xml:space="preserve">circunÅ¿cribentis</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">291</xhtml:td> - <xhtml:td xml:space="preserve">1</xhtml:td> - <xhtml:td xml:space="preserve">od</xhtml:td> - <xhtml:td xml:space="preserve">Quod</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">294</xhtml:td> - <xhtml:td xml:space="preserve">33</xhtml:td> - <xhtml:td xml:space="preserve">detractis</xhtml:td> - <xhtml:td xml:space="preserve">detracti</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">294</xhtml:td> - <xhtml:td xml:space="preserve">34</xhtml:td> - <xhtml:td xml:space="preserve"> angulis contingentiæ <reg norm="Å¿olidiÅ¿que" type="simple">Å¿olidiÅ¿q;</reg> - </xhtml:td> - <xhtml:td xml:space="preserve">anguli contingentiæ <reg norm="Å¿olidique" type="simple">Å¿olidiq;</reg></xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">295</xhtml:td> - <xhtml:td xml:space="preserve">24</xhtml:td> - <xhtml:td xml:space="preserve">chorda</xhtml:td> - <xhtml:td xml:space="preserve">chordam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">295</xhtml:td> - <xhtml:td xml:space="preserve">32</xhtml:td> - <xhtml:td xml:space="preserve">lixum</xhtml:td> - <xhtml:td xml:space="preserve">lixus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">297</xhtml:td> - <xhtml:td xml:space="preserve">1</xhtml:td> - 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</xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">305</xhtml:td> - <xhtml:td xml:space="preserve">21</xhtml:td> - <xhtml:td xml:space="preserve">Idem facere</xhtml:td> - <xhtml:td xml:space="preserve">Idem poßumus facere</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">312</xhtml:td> - <xhtml:td xml:space="preserve">37</xhtml:td> - <xhtml:td xml:space="preserve">duplæ</xhtml:td> - <xhtml:td xml:space="preserve">duplo</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">313</xhtml:td> - <xhtml:td xml:space="preserve">6</xhtml:td> - <xhtml:td xml:space="preserve">quotieÅ¿cunque</xhtml:td> - <xhtml:td xml:space="preserve">qui</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">315</xhtml:td> - <xhtml:td xml:space="preserve">23</xhtml:td> - <xhtml:td xml:space="preserve">retrogradandum</xhtml:td> - <xhtml:td xml:space="preserve">rotrogradiendum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">316</xhtml:td> - <xhtml:td xml:space="preserve">22</xhtml:td> - <xhtml:td xml:space="preserve">vbi</xhtml:td> - <xhtml:td xml:space="preserve">tibi</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">324</xhtml:td> - <xhtml:td xml:space="preserve">41</xhtml:td> - <xhtml:td xml:space="preserve">Å¿i conÅ¿titueremus</xhtml:td> - <xhtml:td xml:space="preserve">Å¿i nos conÅ¿titueremus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">325</xhtml:td> - <xhtml:td xml:space="preserve">12</xhtml:td> - <xhtml:td xml:space="preserve">Å¿imili ad</xhtml:td> - <xhtml:td xml:space="preserve">Å¿imili. Ad</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">326</xhtml:td> - <xhtml:td xml:space="preserve">9</xhtml:td> - <xhtml:td xml:space="preserve">longare</xhtml:td> - <xhtml:td xml:space="preserve">ducere</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">329</xhtml:td> - <xhtml:td xml:space="preserve">5</xhtml:td> - <xhtml:td xml:space="preserve">ipÅ¿i ellipÅ¿is</xhtml:td> - <xhtml:td xml:space="preserve">ipÅ¿ius ellipÅ¿is</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">333</xhtml:td> - <xhtml:td xml:space="preserve">15</xhtml:td> - <xhtml:td xml:space="preserve">cogitetur</xhtml:td> - <xhtml:td xml:space="preserve">cogitemus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">333</xhtml:td> - <xhtml:td xml:space="preserve">17</xhtml:td> - <xhtml:td xml:space="preserve">vnde pro genera</xhtml:td> - <xhtml:td xml:space="preserve">vnde ex genera</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">333</xhtml:td> - <xhtml:td xml:space="preserve">20</xhtml:td> - <xhtml:td xml:space="preserve">æqualem eſſe longitudini</xhtml:td> - <xhtml:td xml:space="preserve">æqualem longitudini</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">333</xhtml:td> - <xhtml:td xml:space="preserve">21</xhtml:td> - <xhtml:td xml:space="preserve">ei</xhtml:td> - <xhtml:td xml:space="preserve">eam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">333</xhtml:td> - <xhtml:td xml:space="preserve">21</xhtml:td> - <xhtml:td xml:space="preserve">minor</xhtml:td> - <xhtml:td xml:space="preserve">minorem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">333</xhtml:td> - <xhtml:td xml:space="preserve">23</xhtml:td> - <xhtml:td xml:space="preserve">nor</xhtml:td> - <xhtml:td xml:space="preserve">norem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">333</xhtml:td> - <xhtml:td xml:space="preserve">24</xhtml:td> - <xhtml:td xml:space="preserve">maior</xhtml:td> - <xhtml:td xml:space="preserve">maiorem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">333</xhtml:td> - <xhtml:td xml:space="preserve">25</xhtml:td> - <xhtml:td xml:space="preserve">maior</xhtml:td> - <xhtml:td xml:space="preserve">maiorem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">333</xhtml:td> - <xhtml:td xml:space="preserve">37</xhtml:td> - <xhtml:td xml:space="preserve"><reg norm="communeque" type="simple">communeq;</reg> - </xhtml:td> - <xhtml:td xml:space="preserve"><reg norm="communique" type="simple">communiq;</reg> - </xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">335</xhtml:td> - <xhtml:td xml:space="preserve">26</xhtml:td> - <xhtml:td xml:space="preserve">reflectit</xhtml:td> - <xhtml:td xml:space="preserve">reflectitnr</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">335</xhtml:td> - <xhtml:td xml:space="preserve">47</xhtml:td> - <xhtml:td xml:space="preserve"> & remotiori - </xhtml:td> - 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DefenÅ¿io nostra <lb/>contra Antonium Bergam, & <lb/>Alexandrum Piccol.</xhtml:td> - <xhtml:td xml:space="preserve">dicatur. contra Anto. Bergam & <lb/>Alex Piccol atq, defenÅ¿io nostra <lb/>contra Excell. August. Michelem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">399</xhtml:td> - <xhtml:td xml:space="preserve">32</xhtml:td> - <xhtml:td xml:space="preserve">tione</xhtml:td> - <xhtml:td xml:space="preserve">tionem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">400</xhtml:td> - <xhtml:td xml:space="preserve">39</xhtml:td> - <xhtml:td xml:space="preserve">vnum tantum numerum medium</xhtml:td> - <xhtml:td xml:space="preserve">unus tantum numerus medius</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">402</xhtml:td> - <xhtml:td xml:space="preserve">19</xhtml:td> - <xhtml:td xml:space="preserve">diametrum</xhtml:td> - <xhtml:td xml:space="preserve">Å¿emidiametrum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">403</xhtml:td> - <xhtml:td xml:space="preserve">7</xhtml:td> - <xhtml:td xml:space="preserve">Å¿uis</xhtml:td> - <xhtml:td xml:space="preserve">Å¿uæ</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">403</xhtml:td> - <xhtml:td xml:space="preserve">9</xhtml:td> - <xhtml:td xml:space="preserve">iam Å¿upræ</xhtml:td> - <xhtml:td xml:space="preserve">eo tractatu</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">403</xhtml:td> - <xhtml:td xml:space="preserve">17</xhtml:td> - <xhtml:td xml:space="preserve">quidquid</xhtml:td> - <xhtml:td xml:space="preserve">quidquam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">404</xhtml:td> - <xhtml:td xml:space="preserve">37</xhtml:td> - <xhtml:td xml:space="preserve">milliaria</xhtml:td> - <xhtml:td xml:space="preserve">milliaribus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">406</xhtml:td> - <xhtml:td xml:space="preserve">19</xhtml:td> - <xhtml:td xml:space="preserve">trabuchum quadratum</xhtml:td> - <xhtml:td xml:space="preserve">trabuchus quadratus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">410</xhtml:td> - <xhtml:td xml:space="preserve">38</xhtml:td> - <xhtml:td xml:space="preserve">quæ Å¿ors</xhtml:td> - <xhtml:td xml:space="preserve">qua Å¿orte</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">410</xhtml:td> - <xhtml:td xml:space="preserve">41</xhtml:td> - <xhtml:td xml:space="preserve">quamlibet</xhtml:td> - <xhtml:td xml:space="preserve">quælibet</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">411</xhtml:td> - <xhtml:td xml:space="preserve">27</xhtml:td> - <xhtml:td xml:space="preserve">dicens</xhtml:td> - <xhtml:td xml:space="preserve">dicat</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">411</xhtml:td> - <xhtml:td xml:space="preserve">28</xhtml:td> - <xhtml:td xml:space="preserve">eodem libro</xhtml:td> - <xhtml:td xml:space="preserve">vt eodem libre</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">412</xhtml:td> - <xhtml:td xml:space="preserve">24</xhtml:td> - <xhtml:td xml:space="preserve">non dependeat?</xhtml:td> - <xhtml:td xml:space="preserve">dependeat?</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">413</xhtml:td> - <xhtml:td xml:space="preserve">2</xhtml:td> - <xhtml:td xml:space="preserve">vt Å¿upra</xhtml:td> - <xhtml:td xml:space="preserve">vt alias</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">413</xhtml:td> - <xhtml:td xml:space="preserve">22</xhtml:td> - <xhtml:td xml:space="preserve">millia vices</xhtml:td> - <xhtml:td xml:space="preserve">millibus viciu<unsure/>m</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">415</xhtml:td> - <xhtml:td xml:space="preserve">32</xhtml:td> - <xhtml:td xml:space="preserve">fineÅ¿træ</xhtml:td> - <xhtml:td xml:space="preserve">fenestræ</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">416</xhtml:td> - <xhtml:td xml:space="preserve">15</xhtml:td> - <xhtml:td xml:space="preserve">operatio</xhtml:td> - <xhtml:td xml:space="preserve">operationem</xhtml:td> - </xhtml:tr> - </xhtml:table> - <head xml:id="head599" type="footer" xml:space="preserve">FINIS.</head> - <pb n="441" file="0441" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0441"/> - </div> - <div type="cover" level="2" n="1"> - <handwritten/> - <pb n="442" file="0442" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0442"/> - <pb n="443" file="0443" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0443"/> - <pb n="444" file="0444" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0444"/> - <p> - <s xml:id="s5186" xml:space="preserve">Test: Dieser Satz enthält ein Zeichen mit Unicode-Codepoint über FFFF, nämlich ð†‘ (U+10191; D800+DD91). </s> - <s xml:id="s5187" xml:space="preserve">Das gleiche Zeichen innerhalb eines Wortes: vorherð†‘nachher. </s> - <s xml:id="s5188" xml:space="preserve">Das Zeichen wird testweise zu X normalisiert. </s> - <s xml:id="s5189" xml:space="preserve">Das Zeichen ð†’ (U+10192; D800+DD92) wird dagegen nicht normalisiert: vorherð†’nachher. </s> - </p> - </div> - </div> - </text> -</echo> \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-xml/examples/Benedetti_1585.xml Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,15419 +0,0 @@ -<?xml version="1.0" encoding="UTF-8"?><echo xmlns="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" xmlns:dcterms="http://purl.org/dc/terms" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:xhtml="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.0"> - <metadata> - <dcterms:identifier>ECHO:163127KK.xml</dcterms:identifier> - <dcterms:creator identifier="GND:118859676 LCCN:nr89-8408 VIAF:88720955">Benedetti, Giovanni Battista de</dcterms:creator> - <dcterms:title xml:lang="la">Diversarum Speculationum mathematicum, & physicarum liber</dcterms:title> - <dcterms:language xsi:type="dcterms:ISO639-3">lat</dcterms:language> - <dcterms:date xsi:type="dcterms:W3CDTF">1585</dcterms:date> - <dcterms:rights>open access</dcterms:rights> - <dcterms:license>http://echo.mpiwg-berlin.mpg.de/policy/oa_basics/declaration</dcterms:license> - <dcterms:accessRights>free</dcterms:accessRights> - <echolink>echo.mpiwg-berlin.mpg.de?titleID=163127KK</echolink> - </metadata> - <text type="book" xml:lang="la"> - <div type="front" level="1" n="1"> - <div type="cover" level="2" n="1"> - <pb n="1" file="0001" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0001"/> - <pb n="2" file="0002" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0002"/> - <pb n="3" file="0003" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0003"/> - <pb n="4" file="0004" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0004"/> - <pb n="5" file="0005" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0005"/> - </div> - <div type="title" level="2" n="1"> - <head xml:space="preserve">IO. BAPTISTAE <lb/>BENEDICTI <lb/>Patritij Veneti PhiloÅ¿ophi.</head> - <head xml:space="preserve"><emph style="it">DIVERSARVM SPECVLATIONVM <lb/>Mathematicarum, & PhyÅ¿icarum</emph> <lb/>Liber.</head> - <head xml:space="preserve">Quarum Å¿eriem Å¿equens pagina indicabit.</head> - <head xml:space="preserve">AD SERENISSIMVM CAROLVM EMANVELEM <lb/>ALLOBROGVM, ET SVBALPINORVM <lb/>DVCEM INVICTISSIMVM.</head> - <figure position="here"> - <image file="0005-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0005-01"/> - </figure> - <head xml:space="preserve"><emph style="sc">Tavrini</emph>, Apud Hæredem Nicolai Beuilaquæ, <num value="1585" style="sc">mdlxxxv</num>. <lb/><emph style="it">Superioribus permiſſum</emph>.</head> - <pb n="6" file="0006" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0006"/> - </div> - <div type="toc" level="2" n="1"> - <head style="it" xml:space="preserve">TRACTATVS QVI IN HOC <lb/>volumine continentur.</head> - <head xml:space="preserve">Theoremata Arithmetica.</head> - <head xml:space="preserve">Derationibus operationum perÅ¿pectiuæ.</head> - <head xml:space="preserve">De Mechanicis.</head> - <head xml:space="preserve">DiÅ¿putationes de quibuÅ¿dam placitis AriÅ¿t.</head> - <head xml:space="preserve">In quintum Euclidis librum.</head> - <head xml:space="preserve">PhyÅ¿ica, & Mathematica reÅ¿ponÅ¿a per EpiÅ¿tolas.</head> - <pb n="7" file="0007" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0007"/> - </div> - <div type="dedication" level="2" n="1"> - <head xml:space="preserve">SERENISSIMO <lb/>CAROLO EMANVELI <lb/>Sabaudiæ Duci, <reg norm="&c." type="unresolved">&c.</reg></head> - <p style="it"> - <s xml:space="preserve"><emph style="sc it">AGitvr</emph> nonusdecimus annus ex quo litte-<lb/>ris Serenißimi patris tuæ CelÅ¿itudinis, ac-<lb/>cerÅ¿itus ex vrbe ParmenÅ¿i in banc me ciui-<lb/>tatem contuli. </s> - <s xml:space="preserve">Is aduenientem tam bumanè <lb/>excepit, tanta deinde liberalitate fuit com-<lb/>plexus ego vicißim ei deÅ¿eruiendi, tam vebe-<lb/>menti cupiditate fui accenÅ¿us, vt Å¿ub eius ditione quodÅ¿uper-<lb/>eßet vitæ agere conÅ¿tituerem. </s> - <s xml:space="preserve">Cuius in me benignitas, mea <lb/>in illum obÅ¿eruantia mirum in modum mutuo vÅ¿u, & conÅ¿ue-<lb/>tudine eÅ¿t adaucta, vt idem Dux me Å¿ecum dum ruÅ¿ticaretur <lb/>eße vellet, ſæpè etiam Å¿ecum pernoctare; </s> - <s xml:space="preserve">quo quidem tempo-<lb/>re de Matbematicis Å¿cientijs mecum agebat, in quibus perdi-<lb/>Å¿cendis mea opera vtebatur, quæſtiones, Arithmeticam, GeoÂ<lb/>metriam, Opticen, MuÅ¿icam, aut Astrologiam Å¿pectantes <lb/>proponens. </s> - <s xml:space="preserve">Cui vt quod in me eßet Å¿atisfacerem, acrius <lb/>quà m anteainea studia (adquætamen Å¿emper fui propenÅ¿ißi-<lb/>mus) incubui. </s> - <s xml:space="preserve"><reg norm="Illiusque" type="simple">IlliusqÌê«</reg> imitatione (vt ferècæteri Principum <lb/>studiaimitantur) non pauci aut præſentes, aut per litter as me <lb/>de his, atque illis Mathematicis quæstionibus conÅ¿uluerunt. <lb/></s> - <s xml:space="preserve">Cùmque ego nunquam laborem amicorum cauÅ¿a defugerim, <lb/>euenit vt post tot annorum curricula, mea Å¿crinia Å¿crutatus, <lb/>inuenerim tot abÅ¿olutas quæſtiones, vt ex eis corpus mediocre <lb/>effici poſſe videretur. </s> - <s xml:space="preserve">Quas, cùm rationibus in epiÅ¿tola Å¿ub-<lb/>Å¿equenti allatis edere constituiſſem, non Å¿ub cuiuÅ¿que alte-<lb/>rius nomine, & auÅ¿picijs quam tuæ CelÅ¿itudinis volui apparere; <lb/></s> - <s xml:space="preserve">tum quòd patri debitum libellum filio reddere par erat, tum - <pb n="8" file="0008" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0008"/> - quòd in tuæ CelÅ¿itudine paternam in me fouendo, & augendo<unsure/> <lb/>benignit atem ineße Å¿emper Å¿um expertus, tum quòd tuæ CelÅ¿i-<lb/>tudinis interrog ationibus excitatus non pauca quæ hoc volu-<lb/>mine continentur, elucubraui. </s> - <s xml:space="preserve">Acceßit, quod ego Å¿emper in <lb/>his dedic ationibus Å¿pectandum put aui, tuam CelÅ¿itudinem t<unsure/>an-<lb/>tos progreßus in Mathematicis feciſſe, vt vel idonea æſtima-<lb/>trix mearum vigiliarum eſſe poßit. </s> - <s xml:space="preserve">Quare, & veterum Per-<lb/>Å¿arum Regum gloriam æquauit, & nos veluti in Å¿pem certam <lb/>fælicitatis buius ſæculi induxit, Å¿i verum eÅ¿t Platonis va-<lb/>ticinium, beat am eam futuram Rempublic am in qua <lb/>Principes PhiloÅ¿ophentur. </s> - <s xml:space="preserve">Tua igitur celÅ¿i-<lb/>tudo libellum tot ei nominibus debitum, <lb/>ea qua Å¿olet bumanitate accipe-<lb/>re nè grauetur. </s> - <s xml:space="preserve">Deus tuas <lb/>omnes cogitationes, <lb/>& conatus ad <lb/>fÅ“licißi-<lb/>mos <lb/>Å¿emper exitus perducat, <lb/><reg norm="teque" type="simple">teqÌê«</reg> diutißimè Å¿er-<lb/>uet incolu-<lb/>mem.</s> - </p> - <pb n="9" file="0009" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0009"/> - </div> - <div type="preface" level="2" n="1"> - <head xml:space="preserve">AD LECTOREM</head> - <p> - <s xml:space="preserve"><emph style="sc">CVm</emph> Varijs temporibus permulta in diuerÅ¿is <lb/>diÅ¿ciplinis contemplatus Å¿im, partim à præ-<lb/>Å¿tantibus viris patronis ac amicis meis exci-<lb/>tatus, quiÅ¿uper eis Å¿ententiam meam exquire-<lb/>bant, partim, abingenito mihi deÅ¿iderio, ra-<lb/>tionem, & cauÅ¿am eorum percipiendi, com-<lb/>mittendum non putaui, quin qualiacunque <lb/>meaÅ¿cripta in illis Å¿cientijs, Å¿tudioÅ¿is impartirer, <lb/>non dubitans quin illis aliquid commodi atque vtilitatis allatura Å¿int, prÄ™ <lb/>Å¿ertim cum in eiuÅ¿modi quæſtionibus inueÅ¿tigandis atque perpendendis, <lb/>nemo ( quod Å¿ciam ) hactenus elaborauerit. </s> - <s xml:space="preserve">Nihil enim his libris à me <lb/>traditum eÅ¿t, quod aut legiſſe, aut ab alijs audiuiſſe meminerim, nam Å¿i <lb/>aliena attigi, ea, aut cum aliqua differentia demonÅ¿trationis, aut diluci-<lb/>dius Å¿cripÅ¿i, quod Å¿i forte alius eadem tradidit, aut eius lucubrationes ad <lb/>me non peruenerunt, aut earum perlectionis memoria excidit. </s> - <s xml:space="preserve">Vtenim <lb/>etiam AriÅ¿toteles ipÅ¿e Å¿enÅ¿it facilè fieri poteÅ¿t, vt pluribus, eædem opinio-<lb/>nes in mentem veniant. </s> - <s xml:space="preserve">Immo multa Å¿cribenti euenire poteÅ¿t, vt cum <lb/>iamdiu aliquid Å¿cripÅ¿erit, iam oblitus, idem repetat, quod mihi etiam <lb/>nonnunquam accidit. </s> - <s xml:space="preserve">In his autemlibris non Å¿uÅ¿cepi munus integræ ali <lb/>cuius Å¿cientiæ tradendæ, ne, quæ abalijs iam tradita Å¿unt, ipÅ¿e inutiliter re <lb/>peterem, mihiqÌue viderer exalienis laboribus laudem voluiſſe comparare. <lb/></s> - <s xml:space="preserve">Singularum enim Å¿cientiarum volumina, iam ab alijs collecta, at-<lb/>que in ordinemÅ¿unt digeÅ¿ta, & Å¿i pauciſſimi Å¿int libri quorum omnes <lb/>Å¿ententiæ, omniaqÌue inuenta vnius Å¿int authoris, excipio Archime-<lb/>dis volumina. </s> - <s xml:space="preserve">Cumque multi Å¿int, qui vel vnam rem à ſe inuentam <lb/>in publicum proferre non dubitent, multo magis mihi qui multa ex-<lb/>cogitaui, & Å¿i inter Å¿e hætereogenea, atque vtcunque expreſſa, idem <lb/>licere Å¿um arbitratus. </s> - <s xml:space="preserve">In his autem meditandis, ex Arithmeticis autho-<lb/>ribus quos inÅ¿pexi, præcipuus fuit Nicolaus Tartalea, quippe quem fe-<lb/>rè omnia ab alijs Å¿cripta collegiſſe conÅ¿tat, nec alios ex præcipuis, quos le-<lb/>gere potui omittendos duxi, inter quos Å¿unt Hieronymus Cardanus, Mi-<lb/>chael Stifelius, Gemma FriÅ¿ius, Ioannes Nouiomagus, Cuthebertus <lb/>TonÅ¿tallus, <reg norm="cæterique" type="simple">cæteriqÌ;</reg> huiuÅ¿inodi. </s> - <s xml:space="preserve">Quorundam tamen volumina illorum <lb/>qui à Tartalea citantur, vt Leonardi PiÅ¿ani, ProÅ¿docimi, Ioannis Infor-<lb/>tunati, Fratris Lucæ, Petri Borgi, aliorumqÌue aliquot inÅ¿piciendorum, - <pb n="10" file="0010" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0010"/> - facultas mihi non fuit. </s> - <s xml:space="preserve">Præterea, licet in his libris nonnullÄ™ inueniantur <lb/>propoÅ¿itiones, quæ diÅ¿iunctam ab alijs habeant rationem, eæ non Å¿per-<lb/>nendæ tamen Å¿unt, viam fortaſſe alicui aperient vlterius progrediendi. <lb/></s> - <s xml:space="preserve">Quemadmodum enim, exempli gratia, ex Å¿ub contraria coni Å¿ectione, <lb/>Å¿umpta poÅ¿tea fuit diuina illa Planisferijdelineation, quæ Å¿ub Ptolomæi no-<lb/>mine legitur, & Å¿icuti ex <ref id="ref-0001">penultima primi Euclidis</ref>, quam Pythagoras <lb/>excogitauit propè innumeræ pulchræ conÅ¿equentiæ in AÅ¿tronomia, in <lb/>Architectura, in <reg norm="multisque" type="simple">multisqÌ;</reg> alijs Å¿cientijs deÅ¿umptæ Å¿unt, immo quemad-<lb/>modum ex Å¿ingulis propoÅ¿itionibus à noÅ¿tris maioribus excogitatis mul-<lb/>ta egregia Å¿unt deducta, ita fortaſſe continget, vt ex mearum muentio-<lb/>num aliqua, <reg norm="nonnihil" type="context">nõnihil</reg> in poÅ¿terum vtilitatis deÅ¿umatur. </s> - <s xml:space="preserve">Si quid verò, hic in-<lb/>ueneris, quod tuo genio non arrideat, illa prudentiſſimi hominis Å¿en-<lb/>tentia in mentem veniat. </s> - <s xml:space="preserve"><emph style="it">Quot capita, tot Å¿ententiæ</emph>, ac per raro con-<lb/>tingere, vt idem omnibus probari, atque placere queat, & perdifficulter <lb/>inueniri hominem cui placeant omnia quæ alteri Å¿atisfaciunt. </s> - <s xml:space="preserve">Nec te mo <lb/>ueat, quodhÄ™c Theoremata Å¿iue excogitationes non videas ordine illo di-<lb/>Å¿poÅ¿itas, quo collocari debere exiÅ¿timaueris, tum in Arithmeticis, tum in <lb/>cæteris. </s> - <s xml:space="preserve">Cum enim in huiuÅ¿modi rebus ordo non Å¿it neceſſarr<unsure/>us, vi-<lb/>Å¿um eÅ¿t mihi poſſe me, Å¿ine repræhenÅ¿ione, illum negligere, cum Å¿pe-<lb/>culationi, Å¿iue inuentioni preÄ™cipuè adeo mihi incumbendum decreuerim <lb/>vtin collocatione operam ponere, & tempus abÅ¿umere operæpretium <lb/>non duxerim, quod idem in epiÅ¿tolarum collocatione feci, in quibus per-<lb/>Å¿onarum ad quas Å¿cribo nullus ferè graduum ordo Å¿eruatus eÅ¿t, nec tem-<lb/>poris, quo Å¿unt Å¿criptæ, quæſitorum tantummodo ratione habita. </s> - <s xml:space="preserve">Nec <lb/>admirari quenquam velim, quod in Å¿peculandis numerorum paſſioni-<lb/>bus, figuris vtar geometricis, ita enim in <ref id="ref-0002">.2. libr.</ref> fecit Euclides, qui mo-<lb/>dus, eo magis mihi arridet, quo minus eÅ¿t abÅ¿tractus, <emph style="it">quoniam oportet in-<lb/>telligentem phantaÅ¿mata Å¿peculari</emph>, cum prÄ™terea perÅ¿picuum Å¿it, diÅ¿cretum <lb/>omne, ex continui diuiÅ¿ione aliquo modo oriri, Å¿iue actu, Å¿iue potentia. <lb/></s> - <s xml:space="preserve">Deinde Å¿i forte meis in deinonÅ¿trationibus tibi videbor aliquando bre-<lb/>uior, illud in cauÅ¿a fuiſſe Å¿cias, quod ibi ad viros Å¿cribebam in his diÅ¿cipli-<lb/>nis exercitatos, quibus Å¿atis fuit rem Å¿ignificare. </s> - <s xml:space="preserve">Libuit autem mihi om-<lb/>nes voluminis Arithmetici propoÅ¿itiones potius vocabulo theorema-<lb/>tum appellare, quam problematum, quia pars earum Å¿peculatiua tan-<lb/>tum mea eÅ¿t, & Å¿i ex varijs eiuÅ¿modi propoÅ¿itionibus etiam operatiuam <lb/>adinuenerim. </s> - <s xml:space="preserve">Quoniam verò multis in locis accidit, vt veritatis iudi-<lb/>candæ cauÅ¿a neceſſe mihi fuerit quorundam Å¿ententijs aduerÅ¿ari nolim te - <pb n="11" file="0011" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0011"/> - hoc mihi vitio tribuere, <reg norm="meque" type="simple">meqÌ;</reg> hoc nomine carptorem <reg norm="maledicumque" type="simple">maledicumqÌ;</reg> ha-<lb/>bere quod alienos errores aperiam, cum potius habenda Å¿it mihi gratia, <lb/>quod in ijs interdum laborans (quÄ™ AntiÅ¿thenes in diÅ¿ciplinis magis ne-<lb/>ceſſaria eſſe dixit, <emph style="it">vt mala Å¿cilicet prius dediÅ¿cantur<sic comment="closing parenthesis in italics">)</sic></emph> falÅ¿as opiniones euel-<lb/>lere Å¿tudeam, <reg norm="veritatemque" type="simple">veritatemqÌ;</reg> oÅ¿tendere, quam omnis philoÅ¿ophus, AriÅ¿to-<lb/>telis exemplo, pluris quam cuiuÅ¿uis hominis authoritatem, aut gratiam <lb/>facere debet. </s> - <s xml:space="preserve">CumqÌue in hoc volumine aliquid eiuÅ¿modi legeris <lb/>te oratum volo, vt in iudicando, affectum omnem exuas, <lb/>SalluÅ¿tianum illud præ oculis habens. </s> - <s style="it" xml:space="preserve">Omnes qui dere-<lb/>bus <reg norm="dubijs" type="lig">dubijs</reg> conÅ¿ultant, ab odio amicitia, ira, atque <lb/>miÅ¿ericordia vacuos eſſe decet. </s> - <s xml:space="preserve">Hinc fiet, vt <lb/>non perÅ¿onæ (vt multiÅ¿olent) Å¿ed <lb/>veritati, quÄ™ Å¿ummo Å¿tudio di-<lb/>gniſſima eÅ¿t, Å¿emper po <lb/>tius faueas. </s> - <s xml:space="preserve">Vale <lb/>noÅ¿trisqÌue <lb/>labo-<lb/>ribus vtere, Å¿i quem inde fructum, <lb/>Å¿icuti Å¿pero tuleris, illi præ-<lb/>cipuè habeas gratiam à <lb/>quo omnes fluunt <lb/>Å¿cientiæ.</s> - </p> - <pb n="12" file="0012" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0012"/> - <pb n="13" file="0013" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0013"/> - </div> - </div> - <div type="body" level="1" n="1"> - <div type="chapter" level="2" n="1"> - <head xml:space="preserve">IO. BAPTISTAE <lb/>BENEDICTI <lb/>PATRITII VENETI <lb/>SERENISS. CAR. EM. <lb/>ALLOBROGVM DVCIS <lb/>PHILOSOPHI.</head> - <head style="it" xml:space="preserve">Theoremata Arithmetica.</head> - <p> - <s xml:space="preserve"><emph style="sc">PRaeclare</emph> multa veteres mathematici philoÅ¿ophi de nuÂ<lb/>meris eorumqÌue effectibus excogitata poÅ¿teris tradide-<lb/>runt, quorum cum vix vllam rationem reddiderint, aut <lb/>certè per exiguam, occaÅ¿ione diuerÅ¿orum problematum <lb/>mihi à Sereniſſimo Sabaudiæ Duce propoÅ¿itorum præbi-<lb/>ta, de ijs quæ ab antiquis propoÅ¿ita fuerunt contemplanda <lb/>nonnulla occurrerunt, quæ poÅ¿teritati comendare non <lb/>inutile arbitratus fum, ne hæ meæ cogitationes intercide-<lb/>rent, & occaÅ¿ionem præberem quamplurimis abÅ¿truÅ¿a hÄ™c <lb/>indagandi, quæ problematibus & thæorematibus inuoluta, vix aliquem qui euol-<lb/>ueret nacta funt.</s> - </p> - <p> - <s xml:space="preserve">Inter cætera vero à me queÅ¿ita, hoc fuit theorema.</s> - </p> - <div type="math:theorem" level="3" n="1"> - <head xml:space="preserve">THEOREMA PRIMVM.</head> - <p> - <s xml:space="preserve"><emph style="sc">INterrogavit</emph> me Sereniſſimus Dux Sabaudiæ, qua ratione cognoÅ¿ci poÅ¿-<lb/>Å¿et Å¿cientificè & Å¿peculatiue (vt dicitur) productum ex duobus fractis numeris, <lb/>quolibet producentium minus eſſe. </s> - <s xml:space="preserve">Cui reÅ¿pondi, mente & cogitatione conci-<lb/>piendum eſſe fractos producentes cum fractis productis, non vnius eiuÅ¿demqÌue na-<lb/>turæ eſſe, imò longè diuerfæ.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, fractis numeris propofitis <var>.a.i.</var> et <var>.a.c.</var> quorum integri Å¿int <var>.a.<lb/>b.</var> et <var>.a.d.</var> qui tanquam lineæ cogitentur, apertum fanè eſſet productum <var>.c.i.</var> fu-<lb/>perficiale futurum, quod nomen caperet à producto Å¿uperficiali <var>.d.b.</var> generato ex <lb/>vno in aliud totorum linearium, nam Å¿i conÅ¿titueretur <var>.a.i.</var> octauum ipÅ¿ius <var>.a.b.</var> et <var>.a.<lb/>c.</var> dimidium <var>.a.d.</var> multiplicato <var>.a.i.</var> cum <var>.a.c.</var> produceretur fextumdecimum ipÅ¿ius <var>.<lb/>d.b</var>. </s> - <s xml:space="preserve">Quare <var>.d.b.</var> eſſet totum <reg norm="relatiuum" type="context">relatiuÅ©</reg> ipÅ¿ius <var>.c.i.</var> non aliquod totum producentium. <lb/></s> - <s xml:space="preserve">Mirum itaque non eÅ¿t Å¿i productum <var>.c.i.</var> minus videatur fuis producentibus, cum <lb/>toto, diuerſæ naturæ à primis conferatur, fractum fiquidem ab integro eiuÅ¿dem <lb/>naturæ, linearis, Å¿uperficialis, aut corporeæ denominatur.</s> - </p> - <p> - <s xml:space="preserve">Quòd Å¿i amplioris cognitionis gratia ex Å¿cientiæ præceptis Å¿peculari voluerit a@ - <pb o="2" rhead="IO. BAPT. BENED." n="14" file="0014" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0014"/> - quis, qua ratione fractus numerus <var>.c.i.</var> minor Å¿it in Å¿uo integro <var>.d.b.</var> fracto <var>.a.i.</var> in <lb/>Å¿uo integro <var>.a.b.</var> aut fracto <var>.a.c.</var> in Å¿uo integro <var>.a.d.</var> conÅ¿ideret is quo pacto pro-<lb/>portio <var>.c.i.</var> ad <var>.d.b.</var> minor Å¿it proportione <var>.a.i.</var> ad <var>.a.b.</var> et <var>.a.c.</var> ad <var>.a.d.</var> hac ratione. </s> - <s xml:space="preserve">Ma-<lb/>nifeÅ¿tum eÅ¿t ex <ref id="ref-0003">prima Å¿exti de quantitate <lb/>continua</ref>, aut <ref id="ref-0004">.18. Å¿eptimi Euclidis</ref> de diÅ¿cre <lb/> - <anchor type="figure" xlink:label="fig-0014-01a" xlink:href="fig-0014-01"/> - ta, proportionem ipÅ¿ius <var>.d.i.</var> ad <var>.d.b.</var> eſſe Å¿i-<lb/>cut <var>.a.i.</var> ad <var>.a.b.</var> & cum <var>.c.i.</var> minor Å¿it <var>.d.i.</var> <lb/>velut pars Å¿uo toto, proportio, <var>c.i.</var> ad <var>.d.b.</var> <lb/>minor erit proportione <var>.d.i.</var> ad <var>.d.b.</var> ex .8. <lb/>quinti, </s> - <s xml:space="preserve">quare minor erit pariter proportio-<lb/>ne <var>.a.i.</var> ad <var>.a.b.</var> ex <ref id="ref-0005">.12. <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg></ref> vnà etiam pro-<lb/>portio <var>.c.i.</var> ad <var>.d.b.</var> minor erit <var>.a.c.</var> ad <var>.a.d.</var> <lb/>ex eiÅ¿dem cauÅ¿is, medio <var>.c.b</var>. </s> - <s xml:space="preserve">Ex quibus pa-<lb/>tet ratio, cur fracti diuerÅ¿arum denomina-<lb/>tionum ad vnicam reducantur. </s> - <s xml:space="preserve">Cur etiam <lb/>numeros integros in partes fractis Å¿imiles <lb/>frangere liceat, quæ omnia ex Å¿ubÅ¿equenti <lb/>figura facilè cognoÅ¿ci poſſunt.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0014-01" xlink:href="fig-0014-01a"> - <image file="0014-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0014-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="2"> - <head xml:space="preserve">THEOREMA <num value="2">II</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">QVae</emph> Å¿it ratio, cur hi, qui numeros, fractos diuerÅ¿arum denominationum col-<lb/>ligere volunt, & in Å¿ummam redigere, multiplicent vnum ex numerantibus <lb/>per denominatorem alterius, & poÅ¿tmodum denominatores adinuicem, quorum <lb/>vltimum productum, commune eÅ¿t denominans duorum priorum productorum, <lb/>quæ collecta in Å¿ummam efficiunt quod quærebatur.</s> - </p> - <p> - <s xml:space="preserve">Qua in re Å¿ciendum eÅ¿t, denominantes conÅ¿iderari tanquam partes vnius <reg norm="eiuÅ¿dem- qÌue" type="context">eiuÅ¿dẽ-<lb/>qÌue</reg> magnitudinis quantitatis continuæ, linearum (verbigratia) <var>a.b.</var> et <var>.a.d.</var> <reg norm="æqualium" type="context">æqualiÅ©</reg> <lb/>in longitudine, <reg norm="quarum" type="context">quarÅ©</reg> <var>.a.b.</var> in quatuor partes diuidatur, et <var>.a.d.</var> in tres. </s> - <s xml:space="preserve">Quare Å¿i colli-<lb/>gere voluerimus duo tertia cum tribus quartis, multiplicabimus <var>.a.c.</var> duo tertia, <lb/>cum <var>.a.b.</var> diuiÅ¿a in 4. partes, produceturqÌue <var>.c.b.</var> octo partium Å¿uperficialium, de-<lb/>hinc multiplicando <var>.a.i.</var> tres quartas cum <var>.a.d.</var> diuiÅ¿a in .3. partes producetur <var>.i.d.</var> pri <lb/>mis Å¿ingulis æqualis, nouem partium Å¿uper <lb/>ficialium, multiplicata deinde <var>a.b.</var> diui-<lb/> - <anchor type="figure" xlink:label="fig-0014-02a" xlink:href="fig-0014-02"/> - Å¿a in .4. partes per <var>.a.d.</var> in .3. diuiÅ¿a, produ-<lb/>cetur quadratum <var>.d.b.</var> in continuo, in 12. <lb/>partes diuiÅ¿um, quod erit totum commune <lb/>Å¿ingulis productis, quorum primum erat <var>.c.<lb/>b</var>. </s> - <s xml:space="preserve">Quare <var>.c.b.</var> ita Å¿e habet ad totum <var>.d.b.</var> Å¿i-<lb/>cut <var>.a.c.</var> ad <var>.a.d.</var> ex prima Å¿exti in continuis, <lb/>aut .18. Å¿eptimi in diÅ¿cretis quantitatibus, <lb/>et <var>.d.i.</var> ad <var>.d.b.</var> Å¿icut <var>.a.i.</var> ad <var>.a.b.</var> ex eiÅ¿dem <lb/>propoÅ¿itionibus. </s> - <s xml:space="preserve">Collectis deinde parti-<lb/>bus producti <var>.c.b.</var> cum partibus producti <var>.<lb/>d.i.</var> manifeÅ¿tè depræhendetur eiuÅ¿modi <lb/>Å¿ummam componi ex partibus vnius totius <lb/>communis Å¿ingulis earum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0014-02" xlink:href="fig-0014-02a"> - <image file="0014-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0014-02"/> - </figure> - </div> - <pb o="3" rhead="THEOR. ARITH." n="15" file="0015" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0015"/> - </div> - <div type="math:theorem" level="3" n="3"> - <head xml:space="preserve">THEOREMA <num value="3">III</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">CVr</emph> reperturi qualis Å¿it fractus aliquis numerus reÅ¿pectu alterius; </s> - <s xml:space="preserve">multiplicare <lb/>debeant numeratores adinuicem & ita etiam denominatores, ex quo produ-<lb/>ctum ex numeratoribus nomen capiat à producto denominatorum.</s> - </p> - <p> - <s xml:space="preserve">Huius Å¿i cauÅ¿am noÅ¿ce vis, Å¿ume <var>.o.i.</var> & <var>.o.u.</var> pro totis denominatoribus, tum <var>.o.e.</var> <lb/>& <var>.o.a.</var> pro numeratoribus (exempli cauÅ¿a) Å¿it <var>.o.i.</var> Å¿enarius <var>.o.u.</var> quaternarius <var>.o.e.</var> <lb/>quinarius <var>.o.a.</var> ternarius. </s> - <s xml:space="preserve">Si noÅ¿ce vis quæ Å¿int tres quartÄ™ partes quinque Å¿extarum, <lb/>patet ex regulis practicis oriri quindecim vigeÅ¿imaÅ¿quartas. </s> - <s xml:space="preserve">Id quomodo fiat, ex <lb/>Å¿ubÅ¿cripta Å¿igura depræhendetur, memores tamen eſſe oportet, quodlibet <reg norm="productum" type="context">productÅ©</reg> <lb/>conÅ¿iderari <reg norm="tanquam" type="context">tanquã</reg> Å¿uperficiem, producentia <reg norm="autem" type="context">autẽ</reg> tan-<lb/>quam lineas. </s> - <s xml:space="preserve">In hac igitur Å¿igura productum ex totis <lb/> - <anchor type="figure" xlink:label="fig-0015-01a" xlink:href="fig-0015-01"/> - linearibus eÅ¿t <var>.u.i.</var> aggregatum ex .24. partibus, & <var>.u.e.</var> <lb/>productum aggregatum ex .20. </s> - <s xml:space="preserve">Quodita Å¿e habebit <lb/>ad productum totale <var>.u.i.</var> Å¿icut <var>.o.e.</var> ad <var>o.i.</var> ex prima <lb/>Å¿exti aut .18. Å¿eptimi, ita <var>.u.e.</var> erunt quinque Å¿extæ par <lb/>tes <var>.u.i.</var> quarum in propoÅ¿ito exemplo, tres quartæ <lb/><reg norm="quæruntur" type="context">quærÅ©tur</reg>. </s> - <s xml:space="preserve">Si <reg norm="itaque" type="simple">itaq;</reg> multiplicabitur <var>.o.e.</var> <reg norm="cum" type="context">cÅ©</reg> <var>.o.a.</var> orietur <lb/>productum <var>.a.e.</var> ita <reg norm="proportionatum" type="context">proportionatÅ©</reg> ad <var>.u.e.</var> Å¿icut <var>.o.a.</var> ad <lb/><var>o.u.</var> reperitur, ex prædictis rationibus. </s> - <s xml:space="preserve">Quòd Å¿i <reg norm="Å¿tatutum" type="context">Å¿tatutÅ©</reg> <lb/>eÅ¿t <var>.o.a.</var> tres quartas partes eſſe ipÅ¿ius <var>.u.o.</var> <reg norm="etiam" type="context">etiã</reg> <var>.a.e.</var> tres <lb/>quartæ partes <reg norm="erunt" type="context">erÅ©t</reg> <var>.u.e.</var> Å¿ed <var>.u.e.</var> quinque Å¿extæ Å¿unt ip-<lb/>Å¿ius <var>.u.i.</var> ex quo Å¿equitur bonum eſſe huiuÅ¿modi opus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0015-01" xlink:href="fig-0015-01a"> - <image file="0015-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0015-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="4"> - <head xml:space="preserve">THEOREMA <num value="4">IIII</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">CVr</emph> multiplicaturi fractos cum integris, rectè multiplicent numerantem fra-<lb/>cti per numerum integrorum, partianturqÌue productum per <reg norm="denominantem" type="context">denominantẽ</reg> <lb/>fracti, ex quo numerus quæſitus colligitur.</s> - </p> - <p> - <s xml:space="preserve">Propter quod mente concipiamus in Å¿ubÅ¿equenti figura, numerum integrorum <lb/>tanquam lineam <var>.a.e.</var> qui, verbigratia, Å¿it denarius, quorum vnuÅ¿quiÅ¿que Å¿it æqualis <lb/><var>a.i.</var> cogiteturqÌue productum ipÅ¿ius <var>.a.e.</var> in <var>.a.i.</var> Å¿itqÌue <var>.u.e.</var> quod quidem erit dena-<lb/>rius Å¿uperficialis, conÅ¿tituta prius <var>.a.u.</var> æqualis <var>.a.i.</var> & <var>.a.o.</var> Å¿int duæ tertiæ <var>.a.u.</var> <reg norm="quarum" type="context">quarÅ©</reg> <lb/>duarum tertiarum productum in numerum <var>.a.e.</var> Å¿it <var>.o.e.</var> pariter <var>.u.i.</var> vnitas Å¿it Å¿uper-<lb/>ficialis prout <var>.a.i.</var> vnitas eÅ¿t linearis, quam <var>.u.i.</var> reÅ¿picere debet productum <var>.o.e.</var> ex <lb/>quo integer Å¿uperficialis <var>.u.i.</var> erit tanquam ternarius, & productum <var>.o.i.</var> tanquam bi <lb/>narius, & quia quælibet pars è viginti ipÅ¿ius <var>.o.e.</var> æqualis eÅ¿t tertiæ parti <var>.u.i.</var> vnita-<lb/>tis Å¿uperficialis; </s> - <s xml:space="preserve">Å¿i cupiamus Å¿cire quot integræ vnitates Å¿int in partibus <var>.o.e.</var> conÅ¿ul-<lb/>tum eÅ¿t eaÅ¿dem diuidere per denominantem <var>.u.i.</var> compoÅ¿itum ex tribus partibus Å¿u <lb/>perficialibus, & cum tam linea <var>u.a.</var> quam Å¿uperficies <var>.u.i.</var> diuidatur in 3. partes <reg norm="aequa les" type="simple">Ä™quaÂ<lb/>les</reg> noÅ¿ce peroportunum eÅ¿t eiuÅ¿modi partitionem numeri <var>.o.e.</var> fieri per numerum <lb/>ipÅ¿ius <var>.u.i.</var> non <var>.u.a.</var> ex prædictis cauÅ¿is.</s> - </p> - <figure position="here"> - <image file="0015-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0015-02"/> - </figure> - <pb o="4" rhead="IO. BAPT. BENED." n="16" file="0016" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0016"/> - </div> - <div type="math:theorem" level="3" n="5"> - <head xml:space="preserve">THEOREMA <num value="5">V</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">ALia</emph> quoque via prædicti effe <lb/> - <anchor type="figure" xlink:label="fig-0016-01a" xlink:href="fig-0016-01"/> - ctus cauÅ¿a, Å¿peculando inno-<lb/>teÅ¿cere poteÅ¿t, cuius rei gratia for-<lb/>metur Å¿equens figura <var>.e.o.a.u.n.</var> <lb/>eiuÅ¿modi, vt <var>a.e.</var> Å¿it numerus li-<lb/>linearis integrorum, & <var>o.e.</var> produ-<lb/>ctum numerantis ipÅ¿orum <reg norm="fractorum" type="context">fractorÅ©</reg> <lb/>in integris, ex quo <var>.a.o.</var> erunt duæ <lb/>tertiæ, verbigratia, <var>a.i.</var> aut <var>a.u.</var> qua-<lb/>rum <reg norm="linearum" type="context">linearÅ©</reg> Å¿ingulÄ™ Å¿tatuuntur æqua <lb/>les vnitati lineari, Å¿uperficies autem <lb/>parallelogramma <var>.u.n.</var> conÅ¿tituatur <lb/>æqualis magnitudinis Å¿uperficiei <var>.o.<lb/>e.</var> ex quo <var>.u.n.</var> erit nobis cognita Å¿u-<lb/>perficies. </s> - <s xml:space="preserve">CognoÅ¿cetur pariter quan <lb/>titas partium <var>.a.u.</var> quam in propoÅ¿i-<lb/>to exemplo diximus eſſe trium par-<lb/>tium. </s> - <s xml:space="preserve">ex regula igitur de tribus, di-<lb/>cemus Å¿i <var>.u.a.</var> dat <var>.a.e.</var> Å¿ine dubio <var>.o.<lb/>a.</var> dabit <var>.a.n.</var> numerum linearem. <lb/>quæ regula ex 15. Å¿exti in continuis, <lb/>& ex 20. Å¿eptimi in diÅ¿cretis, depro-<lb/>mitur. </s> - <s xml:space="preserve">rectè igitur <reg norm="multiplicantur" type="context">multiplicãtur</reg> fra-<lb/>cti numerantes cum integris, & productum diuiditur per <reg norm="denominantem" type="context">denominantẽ</reg> fractorum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0016-01" xlink:href="fig-0016-01a"> - <image file="0016-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0016-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="6"> - <head xml:space="preserve">THEOREMA <num value="6">VI</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">ITem</emph> & alia Å¿peculatione cognoÅ¿ci poteÅ¿t hoc rectè fieri, mul-<lb/>tiplicantes enim has duas tertias per decem, debemus conÅ¿ide-<lb/> - <anchor type="figure" xlink:label="fig-0016-02a" xlink:href="fig-0016-02"/> - rare quantitatem duarum <reg norm="tertiarum" type="context">tertiarÅ©</reg> decies produci, ex quo oriuntur <lb/>20. tertia, quandoquidem Å¿ingulæ vnitates, </s> - <s xml:space="preserve">tunc pro duobus ter-<lb/>tijs Å¿umuntur, Å¿ed c<unsure/>um quilibet integer tria fragmenta contineat, <lb/>ideo ex ratione partiendi quoties ternarius ingrediatur viginti, <lb/>Å¿tatim cognoÅ¿cemus quod optabamus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0016-02" xlink:href="fig-0016-02a"> - <image file="0016-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0016-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Id ipÅ¿um accideret Å¿i integri in eiuÅ¿modi Å¿pecie fractorum diui-<lb/>derentur. quo facto hi multiplicandi eſſent cum numerante propo <lb/>Å¿ito, & <reg norm="partiendum" type="context">partiendÅ©</reg> productum per quadratum denominantis.</s> - </p> - <p> - <s xml:space="preserve">Cuius rei hæc eÅ¿t Å¿peculatio. </s> - <s xml:space="preserve">Sit linea <var>.a.e.</var> conÅ¿tans ex <reg norm="quinque" type="simple">quinq;</reg> <lb/>integris numeris, quorum <reg norm="vnuÅ¿quiÅ¿que" type="simple">vnuÅ¿quiÅ¿q;</reg> æqualis Å¿it <var>.a.u.</var> vel <var>.a.i.</var> & <var>.a.o.</var> <lb/>Å¿int duo tertia vnitatis integræ linearis. </s> - <s xml:space="preserve">cogitemus nunc hos <reg norm="quinque" type="simple">quinq;</reg> <lb/>integros diuidi in Å¿ua <reg norm="fragmenta" type="context">fragmẽta</reg> linearia, quÄ™ in propoÅ¿ito exemplo <lb/>erunt 15. multiplicatis iam 15. cum propoÅ¿itis, videlicet <var>a.o.</var> orie-<lb/>tur productum <var>.o.e.</var> triginta fragmentorum Å¿uperficialium, <reg norm="quorum" type="context">quorÅ©</reg> <lb/>in Å¿ingulos integros Å¿uperficiales <reg norm="cadunt" type="context">cadÅ©t</reg> <reg norm="nouem" type="context">nouẽ</reg> in hoc <reg norm="exemplo" type="context">exẽplo</reg>, & cum <lb/>notauerimus quoties <reg norm="nouem" type="context">nouẽ</reg> ingrediatur triginta, propoÅ¿itum con-<lb/>Å¿equemur.</s> - </p> - <pb o="5" rhead="THEOR. ARITH." n="17" file="0017" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0017"/> - </div> - <div type="math:theorem" level="3" n="7"> - <head xml:space="preserve">THEOREMA <num value="7">VII</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">CVr</emph> multiplicaturi integros numeros & fractos, cum integris & fractis, de-<lb/>beant integros reducere ad Å¿pecies fractorum, eos colligendo cum fractis: <lb/></s> - <s xml:space="preserve">deinde multiplicare hos vltimos numerantes adinuicem & productum partiri <lb/>per productum denominantium.</s> - </p> - <p> - <s xml:space="preserve">Vt (exempli cauÅ¿a) Å¿i volumus multiplicare vnum & duo tertia, per duo & tria <lb/>quarta, reducentur omnia in fractos, ex quo vna ex parte eſſent quinque ter-<lb/>tia, multiplicanda cum vndecim quartis ex altera, quo facto oriretur productum <lb/>quinquagintaquinque fractorum, quod diuiÅ¿um per <lb/>productum ternarijin quaternarium, videlicet per duode <lb/>cim, quatuor integri proferentur cum Å¿eptem duodeci-<lb/>mis fractis vnius integri.</s> - </p> - <figure position="here"> - <image file="0017-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0017-01"/> - </figure> - <p> - <s xml:space="preserve">Detur Å¿ubÅ¿equens figura in qua linea <var>a.i.</var> æqualis Å¿it li-<lb/>neæ <var>.u.a.</var> quarum <reg norm="vnaquæque" type="simple">vnaquæq;</reg> <reg norm="conÅ¿ideretur" type="context">cõſideretur</reg> pro integro nume <lb/>ro: </s> - <s xml:space="preserve"><reg norm="cogiteturque" type="simple">cogiteturqÌ;</reg> <var>.a.i.</var> valere quatuor in pręſenti <reg norm="exemplo" type="context">exẽplo</reg>, & <var>.a.<lb/>u.</var> tria: </s> - <s xml:space="preserve">detur deinde linea <var>.a.o.</var> æquipollens vni integro <reg norm="cum" type="context">cÅ©</reg> <lb/>duobus tertijs, & <var>a.e.</var> æquipollens duobus integris & tri-<lb/>bus quartis. </s> - <s xml:space="preserve">Iam Å¿i hæ duæ lineæ in Å¿uos fractos redu-<lb/>cantur, multiplicata (vt in Å¿equenti figura apparet.) <var>a.o.</var> <reg norm="cum" type="context">cÅ©</reg> <lb/><var>a.e.</var> orietur productum <var>o.e.</var> fractorum Å¿uperficialium <lb/><reg norm="quinquagintaquinque" type="simple">quinquagintaquinq;</reg>, quorum integer Å¿uperficialis va-<lb/>let duodecim, Å¿cilicet <var>.u.i.</var> vt cuique manifeÅ¿tum eÅ¿t, ex <lb/>quo, quærenti media partitione, quoties duodecim in-<lb/>grediatur quinquagintaquinque, citra errorem, quæſitum <lb/>occurret.</s> - </p> - </div> - <div type="math:theorem" level="3" n="8"> - <head xml:space="preserve">THEOREMA <num value="8">VIII</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">ID ipsvm</emph> accideret Å¿i fractiad vnam eandemqÌue denominationem reduceren-<lb/>tur, qui poÅ¿tmodum Å¿imul multiplicarentur, productumqÌue partiremur per qua-<lb/>dratum denominantis communis.</s> - </p> - <p> - <s xml:space="preserve">Exempli cauÅ¿a, Å¿int eadem quinque tertia, & vndecim quarta adinuicem multi-<lb/>plicanda, quæ Å¿i reducantur ad vnam & eandem denominationem quinarius <lb/>numerans vnius, multiplicabitur cum quaternario deno-<lb/>minante alterius, & vndenarius Å¿ecundi cum ternario de-<lb/>nominante primi. ex quo vna ex parte eſſent viginti, ex <lb/> - <anchor type="figure" xlink:label="fig-0017-02a" xlink:href="fig-0017-02"/> - altera 33. numerantia vnius <reg norm="communis" type="context">cõmunis</reg> denominantis, quod <lb/>eſſet productum ternarij in quaternarium, videlicet duo-<lb/>decim, vt ex veteri regula patet. </s> - <s xml:space="preserve">Iam Å¿i multiplicentur vi <lb/>ginti cum trigintatribus, dabuntur 660. fracti, quorum in-<lb/>teger erit quadratum duodenarij, nempe 144. quibus qui-<lb/>dem 660. diuiÅ¿is per 144. proferentur quatuor integri & <lb/>Å¿eptem duodecimi.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0017-02" xlink:href="fig-0017-02a"> - <image file="0017-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0017-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Cuius rei gratia Å¿it in Å¿ubÅ¿cripta figura linea <var>.a.i.</var> & ei <lb/>æqualis <var>.a.u.</var> pro integro lineari, quæ <var>.a.i.</var> diuidatur in qua-<lb/>tuor partes, & <var>.a.u.</var> in tres, & linea <var>.a.e.</var> Å¿it vndecim <reg norm="partium" type="context">partiÅ©</reg> <lb/>talium qualium <var>.a.i.</var> eÅ¿t quatuor, & <var>.a.o.</var> Å¿it quinque pro-<lb/>ut <var>.a.u.</var> eÅ¿t trium. </s> - <s xml:space="preserve">nunc multiplicato <var>.a.o.</var> & <var>.a.i.</var> orietur pro-<lb/>ductum <var>.o.i.</var> viginti partium Å¿uperficialium. </s> - <s xml:space="preserve">tum multipli- - <pb o="6" rhead="IO. BAPT. BENED." n="18" file="0018" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0018"/> - cato <var>.a.e.</var> per <var>.a.u.</var> dabitur productum <var>.u.e.</var> <reg norm="trigintatrium" type="context">trigintatriÅ©</reg> <lb/> - <anchor type="figure" xlink:label="fig-0018-01a" xlink:href="fig-0018-01"/> - partium. </s> - <s xml:space="preserve">ad hæc quadratum <var>.u.i.</var> conÅ¿tabit ex duode-<lb/>cim partibus eiuÅ¿dem rationis cum reliquis duobus <lb/>productis, quod quadratum <var>.u.i.</var> vnitas eÅ¿t Å¿uperficia-<lb/>lis, & communis denominans duorum productorum. <lb/></s> - <s xml:space="preserve">quod Å¿i in præſentiarum cogitabimus lineam <var>.c.d.</var> tri-<lb/>gintatrium partium æqualium, et <var>.c.t.</var> duodecim Å¿imi-<lb/>lium, et <var>.c.f.</var> viginti <var>.c.n.</var> duodecim, multiplicato <var>.c.<lb/>d.</var> cum <var>.c.f.</var> dabitur Å¿uperficies <var>.f.d.</var> 660. fractorum <lb/>Å¿uperficialium, quorum vnitas integra Å¿uperficialis <lb/>erit quadratum <var>.n.t.</var> 144. partium cuiuÅ¿modi <var>.f.d.</var> <lb/>partes habet .660. diuiÅ¿o itaque <var>.f.d.</var> per <var>.n.t.</var> pro-<lb/>poÅ¿itum conÅ¿equetur. </s> - <s xml:space="preserve">eo quòd eadem proportio erit <lb/> - <anchor type="figure" xlink:label="fig-0018-02a" xlink:href="fig-0018-02"/> - producti <var>.f.d.</var> ad <var>.n.t.</var> quæ producti eius quòd fit ex <var>.<lb/>a.e.</var> in <var>.a.o.</var> ad <var>.u.i.</var> nam proportio <var>.c.d.</var> ad <var>.c.t.</var> ea-<lb/>dem eÅ¿t quæ <var>.a.e.</var> ad <var>.a.i.</var> & <var>c.f.</var> ad <var>.c.n.</var> vt <var>.a.o.</var> ad <var>.a.<lb/>u.</var> ex prima Å¿exti vel 18. Å¿eptimi, Å¿ed vt <var>.f.d.</var> ad id <reg norm="quod" type="simple">ê™</reg> <lb/>fit ex <var>.f.c.</var> in <var>.c.t.</var> eÅ¿t vt <var>.c.d.</var> ad <var>.c.t.</var> & vt eius <reg norm="quod" type="simple">ê™</reg> fit ex <lb/><var>f.c.</var> in <var>.c.t.</var> ad <var>.n.t.</var> eÅ¿t vt <var>.f.c.</var> ad <var>.c.n.</var> ex dictis pro-<lb/>poÅ¿itionibus </s> - <s xml:space="preserve">quare ex æqua proportionalitate, eodem <lb/>modo diÅ¿currendo in figura <var>.o.a.e.</var> ita Å¿e habebit <var>.f.d.</var> <lb/>ad <var>.n.t.</var> vt <var>.o.e.</var> ad <var>.u.i</var>. </s> - <s xml:space="preserve">Porrò ex ijs, quæ hactenus de <lb/>fractorum multiplicatione conÅ¿iderata fuerunt, apertè <lb/>ratio deprehenditur, cur productum, Å¿ingulis producen <lb/>tibus Å¿emper minus Å¿it, cum producta Å¿int Å¿uperficialia <lb/>producentia verò Å¿emper linearia, omiſſis productis <lb/>corporeis, quæ omnia ad Å¿uperficialia reducuntur.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0018-01" xlink:href="fig-0018-01a"> - <image file="0018-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0018-01"/> - </figure> - <figure xlink:label="fig-0018-02" xlink:href="fig-0018-02a"> - <image file="0018-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0018-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="9"> - <head xml:space="preserve">THEOREMA <num value="9">IX</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">IN Ipsa</emph> fractorum diuiÅ¿ione, animaduertendum eÅ¿t, denominantes numeros <lb/>Å¿emper æquales inuicem eſſe debere, vnius Å¿cilicet Å¿peciei, quòd Å¿i æquales non <lb/>fuerint, neceſſe eÅ¿t via multiplicationis ipÅ¿orum denominantium adinuicem effice-<lb/>re æquales vt Å¿int, ex quo productum oritur eiuÅ¿modi, vt aptum Å¿it habere partes <lb/>fractorum, quæ deÅ¿iderabantur.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proponerentur diuidenda Å¿eptem octaua per tria quarta præ-<lb/>cipit antiquorum regula, vt ad vnam tantum denominationem reducantur. </s> - <s xml:space="preserve">quare <lb/>multiplicant denominantes inuicem. </s> - <s xml:space="preserve">ex quo productum in materia propoÅ¿ita ori-<lb/>tur triginta duarum partium commune denominans, cuius duo numerantes Å¿unt vi-<lb/>gintiquatuor & vigintiocto, producti ex multiplicatione vnius numerantis in deno <lb/>minantem alterius, ex quo dantur vigintiquatuor tamquam tria quarta trigintaduo <lb/>rum, & vigintiocto tanquam Å¿eptem octaua particularum vniformium, prout ope <lb/>primæ Å¿exti aut decimæoctauæ Å¿eptimi in Å¿ubÅ¿cripta figura cognoÅ¿ci poteÅ¿t.</s> - </p> - <pb o="7" rhead="THEOR. ARITH." n="19" file="0019" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0019"/> - <p> - <s xml:space="preserve">Sit itaque linea <var>.a.i.</var> diuifa in partes octo, & ei æqualis in longitudine <var>.a.u.</var> in qua-<lb/>tuor, productum verò vnius in alteram <lb/> - <anchor type="figure" xlink:label="fig-0019-01a" xlink:href="fig-0019-01"/> - Å¿it <var>.u.i.</var> trigintaduarum particularum <lb/>fuperficialium fimilium & <reg norm="æqualium" type="context">æqualiÅ©</reg> ad-<lb/>inuicem. </s> - <s xml:space="preserve">fit deinde <var>.a.e.</var> Å¿eptem <reg norm="partium" type="context">partiÅ©</reg> <lb/>lineæ <var>.a.i.</var> & <var>.a.o.</var> trium partium <var>.a.u</var>. <lb/></s> - <s xml:space="preserve">tunc productum <var>.a.e.</var> in <var>.a.u.</var> erit <var>.u.e.</var> <lb/>particularum Å¿uperficialium vigintiocto <lb/>& productum <var>.a.o.</var> in <var>.a.i.</var> erit <var>.o.i.</var> par <lb/>ticularum <reg norm="Å¿uperficialium" type="context">Å¿uperficialiÅ©</reg> vigintiquatuor <lb/>eiuÅ¿dem naturæ cum partibus triginta-<lb/>duabus totius denominantis communis. <lb/></s> - <s xml:space="preserve">vnde diuifo numerante vigintiocto per-<lb/>numerantem vigintiquatuor, dabitur <lb/>vnum cum fexta parte illius vnius.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0019-01" xlink:href="fig-0019-01a"> - <image file="0019-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0019-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="10"> - <head xml:space="preserve">THEOREMA <num value="10">X</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">PArtiri</emph> Å¿eu diuidere vno numero alium numerum, eÅ¿t etiam quodammodo <lb/>eiuÅ¿modi partem numeri diuifibilis inuenire refpectu totius numeri diuifibilis, <lb/>cuiuÅ¿modi eÅ¿t vnitas in diuidente refpectu totius diuidentis, partem inquam numeri <lb/>diuiÅ¿ibilis Å¿ic Å¿e habentem ad totum numerum diuiÅ¿ibilem Å¿icut vnitas ad totum di-<lb/>uidentem, quod Å¿imiliter ex regula de tribus præſtamus dicentes, Å¿i tantus numerus <lb/>diuidens dat <reg norm="vnitatem" type="context">vnitatẽ</reg>, quid dabit numerus diuifibilis, quemadmodum ex <ref id="ref-0006">.15. Å¿exti</ref> <lb/>Å¿eu <ref id="ref-0007">.20. Å¿eptimi</ref> licet Å¿peculari, Idcircò quotieÅ¿cunque minorem numerum per <lb/>maiorem diuidimus, Å¿emper qui prouenit fractus eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i cogitaremus lineam <var>.a.e.</var> diuiÅ¿am in octo partes æquales, qua <lb/>rum vna Å¿cilicet vnitas effet <var>.a.i.</var> & cupere-<lb/>mus eam diuidere in nouem partes, ac Å¿cire <lb/> - <anchor type="figure" xlink:label="fig-0019-02a" xlink:href="fig-0019-02"/> - quan a Å¿it nona illius pars; </s> - <s xml:space="preserve">manifeÅ¿tum eſſet, <lb/>nonam partem ipÅ¿ius <var>.a.e.</var> minorem futuram <lb/>ipÅ¿a <var>.a.i.</var> cum <var>.a.i.</var> diminui debeat à ſua inte-<lb/>gritate eadem proportione, qua <var>.a.e.</var> minor <lb/>reperitur vna linea nouem partium æqualium <lb/>fingularum <var>.a.i</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0019-02" xlink:href="fig-0019-02a"> - <image file="0019-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0019-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Quod vt dilucidè cuiuis innoteÅ¿cat, hoc <lb/>etiam modo licebit videre Å¿itlinea <var>.n.c.</var> no-<lb/>nupla ad <var>.a.i.</var> & parallela ad <var>.a.e.</var> dubium non <lb/>eÅ¿t quin <var>.n.c.</var> maior futura Å¿it ipÅ¿a <var>.a.e.</var> iam Å¿i <lb/>earum extrema congiungantur medijs duabus <lb/>lineis <var>.n.a.</var> et <var>.c.e.</var> quæ Å¿imul concurrant in <lb/>puncto <var>.o.</var> (quod eÅ¿t probatu facillimum) da-<lb/>buntur certe duo trianguli fimiles <var>.a.o.e.</var> et <var>.n.o.c</var>. </s> - <s xml:space="preserve">Sit deinde <var>.n.t.</var> vna è partibus <lb/>ipÅ¿ius <var>.n.c.</var> quæ <var>.n.t.</var> æqualis erit <var>.a.i.</var> ex præſuppoÅ¿ito. </s> - <s xml:space="preserve">ducatur deinde <var>.o.t.</var> quÄ™ <lb/>interÅ¿ecet <var>.a.e.</var> in puncto <var>.x.</var> dico <var>.a.x.</var> tanto minorem futuram <var>.a.i.</var> quanto <var>.a.e.</var> <lb/>minor eÅ¿t <var>.n.c.</var> neque enim dubium eſſe poteÅ¿t quin proportiones <var>.n.t.</var> ad <var>.a.x.</var> et <var>. - <pb o="8" rhead="IO. BAPT. BENED." n="20" file="0020" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0020"/> - n.c.</var> ad <var>.a.e.</var> Å¿int æquales inuicem quandoqui-<lb/> - <anchor type="figure" xlink:label="fig-0020-01a" xlink:href="fig-0020-01"/> - dem vnaquæque earum ex triangulorum Å¿imi <lb/>litudine æqualis eÅ¿t proportioni <var>.o.n.</var> ad <var>.o.a</var>. <lb/></s> - <s xml:space="preserve">itaque <var>.n.t.</var> hoc eÅ¿t <var>.a.i.</var> tanto maior erit <var>.a.x.</var> <lb/>quanto <var>.n.c.</var> maior eÅ¿t <var>.a.e.</var> vnde ficut <var>.a.e.</var> con-<lb/>Å¿tat octo nonis ipÅ¿ius <var>.n.c.</var> ita pars <var>.a.x.</var> ipÅ¿ius <var>.<lb/>a.e.</var> octo nonis conÅ¿tabit ipÅ¿ius <var>.a.i</var>.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0020-01" xlink:href="fig-0020-01a"> - <image file="0020-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0020-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Hinc patet ratio cur partituri numerum mino <lb/>rem per maiorem collocent minorem fupra <lb/>virgulam & maiorem infra & zerum ad <reg norm="læuam" type="context">læuã</reg>.</s> - </p> - <p> - <s xml:space="preserve">Sciendum eÅ¿t præterea diuidere numerum <lb/>per numerum: </s> - <s xml:space="preserve">eſſe inuenire <reg norm="alterum" type="context">alterÅ©</reg> latus à quo <lb/>producitur, Å¿uppoÅ¿ito Å¿emper quòd numerus <lb/>diuifibilis Å¿uperÅ¿icialis Å¿it, & rectangulus.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proponantur triginta diuidenda per quinarium, nihil aliud erit <lb/>hæc diuiÅ¿io, quam inuentio alterius numeri, qui multiplicatus per quinarium produ-<lb/>cat triginta Å¿uperficies rectangulas, huiuÅ¿modi verò eÅ¿t Å¿enarius, cuius Å¿ingulæ vnita-<lb/>tes Å¿uperficiales erunt.</s> - </p> - <p> - <s xml:space="preserve">Cuius rei gratia Å¿it Å¿ubÅ¿criptum rectangulum <var>.a.e.</var> triginta vnitatum <reg norm="Å¿uperÅ¿icialium" type="context">Å¿uperÅ¿icialiÅ©</reg>, <lb/>cuius latus <var>.e.n.</var> Å¿it quinque vnitatum. </s> - <s xml:space="preserve">hinc latus <var>.a.n.</var> erit Å¿ex vnitatum; </s> - <s xml:space="preserve">ita diuiden-<lb/>tes rectangulum <var>.e.a.</var> nihil a iud faciemus, quam vt inue-<lb/> - <anchor type="figure" xlink:label="fig-0020-02a" xlink:href="fig-0020-02"/> - nia mus quantum valeat latus <var>.a.n.</var> quod erit Å¿ex vnitatum. <lb/></s> - <s xml:space="preserve">Sin verò diuiÅ¿erimus per latus <var>.a.n.</var> quæremus latus <var>.e.n.</var> <lb/>quinque vnitatum. </s> - <s xml:space="preserve">ex quo, proportio totius numeri diuifi-<lb/>bilis ad numerum qui oritur, erit Å¿icut diuidentis ad vnita-<lb/>tem, ex prima Å¿exti, aut .18. vel .19. Å¿eptimi, & permutatim <lb/>ita Å¿e habebit diuiÅ¿ibile ad diuidentem, Å¿icut numerus qui <lb/>oritur ad vnitatem.</s> - </p> - <div type="float" level="4" n="3"> - <figure xlink:label="fig-0020-02" xlink:href="fig-0020-02a"> - <image file="0020-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0020-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Partiri igitur nihil aliud eÅ¿t, quam inuenire latus rectanguli, quod productum in <lb/>diuidente, numerum diuiÅ¿ibilem compl at, ex quo numerus diuiÅ¿ibilis Å¿uperficialis <lb/>eÅ¿t, diuidens autem, & qui oritur, numeri lineares & latera producentia huiuÅ¿cemodi <lb/>numerum diuiÅ¿ibilem. </s> - <s xml:space="preserve">nam multiplicare & diuidere opponuntur inuicem, cum au-<lb/>tem ex multiplicatione laterum Å¿iue linearum generatur Å¿uperficies, ex diuiÅ¿ione po-<lb/>Å¿tea ipÅ¿ius Å¿uperficiei inuenitur alterum latus. </s> - <s xml:space="preserve">quare mirum non eÅ¿t, Å¿i proueniens ex <lb/>vna diuiÅ¿ione (via fractorum) Å¿it Å¿emper maius numero diuiÅ¿ibili.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, diuidendo dimidium per tertiam partem, reÅ¿ultat vnus integer nu <lb/>merus cum dimidio pro numero qui oritur. </s> - <s xml:space="preserve">Sit itaque dimidium Å¿uperÅ¿iciale diuiÅ¿i-<lb/>bile <var>.b.c.</var> cuius totum Å¿it <var>.b.p.</var> quadratum. </s> - <s xml:space="preserve">tertium verò lineare diuidens, <var>b.n.</var> cuius to-<lb/>tum lineare Å¿it <var>.b.d.</var> quærendum nobis eÅ¿t latus <var>.b.s.</var> quod cum latere <var>.b.n.</var> producat re <lb/>ctangulum <var>.n.s.</var> æquale dimidio Å¿uperÅ¿iciali propoÅ¿ito <var>.b.c.</var> quod Å¿i Å¿iat, ex .15. Å¿exti, <lb/>aut .20. Å¿eptimi. erit eadem proportio <var>.b.n.</var> ad <var>.b.q.</var> quæ eÅ¿t <var>.q.c.</var> ad <var>.b.s.</var> dicemus itaque <lb/>Å¿i <var>.n.b.</var> dat <var>.b.q.</var> quid dabit <var>.q.c</var>? </s> - <s xml:space="preserve">certè <var>.b.s.</var> Å¿ed <var>.n.b.</var> eÅ¿t tertium lineare et <var>.b.q.</var> lineare <reg norm="in- tegrum" type="context">in-<lb/>tegrÅ©</reg>, & <var>b.s.</var> proueniens lineare. </s> - <s xml:space="preserve">& quia <var>.b.c.</var> dimidium Å¿uperficiale, producitur à <var>.q.c.</var> <lb/>dimidio lineari in <var>.q.b.</var> integro lineari. </s> - <s xml:space="preserve">quare cum <var>.n.s.</var> Å¿it Ä™qualis <var>.b.c.</var> & productum ex <var>.<lb/>b.n.</var> minori <var>.q.c.</var> neceſſe eÅ¿t, vt producatur in <var>.b.s.</var> maiore <var>.q.b.</var> quod <var>.q.b.</var> maius eÅ¿t <var>.q.c.</var> <lb/>quod quidem <var>.q.c.</var> ita appellatur Å¿icut <var>.b.c</var>. </s> - <s xml:space="preserve">quare mirum non eÅ¿t Å¿i proueniens per fra-<lb/>ctos numeros ex diuiÅ¿ione, maior Å¿it numero diuiÅ¿ibili.</s> - </p> - <pb o="9" rhead="THEOR. ARITH." n="21" file="0021" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0021"/> - <p> - <s xml:space="preserve">Hinc manifeÅ¿te patet quamlibet <reg norm="diuiÅ¿ionem" type="context">diuiÅ¿ionẽ</reg> aut partitionem oriri ex regula de tri-<lb/>bus, quandoquidem Å¿inguli diuidentes æquipollent vni integro, & loco illius Å¿u-<lb/>muntur. </s> - <s xml:space="preserve">Perinde enim eÅ¿t diuidere centum per viginti, ac <reg norm="regulam" type="context">regulã</reg> obÅ¿eruare de tri-<lb/>bus <reg norm="dicentes" type="context">dicẽtes</reg>, Å¿i viginti æquipollent vni, quibus <reg norm="Ä™quiualebunt" type="context">Ä™quiualebÅ©t</reg> <reg norm="centum" type="context">cẽtum</reg>? </s> - <s xml:space="preserve">Hoc autem ex Å¿ub <lb/>Å¿equenti figura facile deprehendetur, in qua linea <var>.a.b.</var> Å¿ignificat viginti, et <var>.a.o.</var> <reg norm="vni- tatem" type="context">vni-<lb/>tatẽ</reg> <reg norm="linearem" type="context">linearẽ</reg>, et <var>.a.c.</var> vnitates lineares <reg norm="centum" type="context">centÅ©</reg>: </s> - <s xml:space="preserve">o.c. verò centum vnitates Å¿uperficiales, <lb/>et <var>.a.d.</var> <reg norm="quinque" type="simple">quinq;</reg> vnitates lineares, et <var>.d.b.</var> centum vnitates Å¿uperficiales, ex quo manife-<lb/>ftè deprehenditur quòd quemadmodum multiplicare, nihil aliud eÅ¿t, quam inueni <lb/>re <reg norm="productum" type="context">productÅ©</reg> ex duobus lateribus propoÅ¿itis, it a partiri nihil aliud eÅ¿t, quam da-<lb/>to vno latere inuenire aliud latus producti propoÅ¿iti.</s> - </p> - <figure position="here"> - <image file="0021-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0021-01"/> - </figure> - <figure position="here"> - <image file="0021-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0021-02"/> - </figure> - <p> - <s xml:space="preserve">Nam <reg norm="quotieÅ¿cunque" type="simple">quotieÅ¿cunq;</reg> <reg norm="ratiocinantes" type="context">ratiocinãtes</reg> dicimus tantundem numeri, immediate produci <lb/>mus Å¿uperficiem, <reg norm="mediante" type="context">mediãte</reg> vnitate in huiuÅ¿modi numero, qui numerus <reg norm="antequam" type="context">antequã</reg> pro-<lb/>ducatur in vnitatem, mente concipiendus eÅ¿t tanqua m linearis, tanquam linea in-<lb/>quam diuiÅ¿a in totidem particulas lineares, Å¿ingulas continuas & æquales vnitati <lb/>propoÅ¿itæ. </s> - <s xml:space="preserve"><reg norm="Cum" type="context">CÅ©</reg> verò productus fuerit numerus in vnitate Å¿uperficialis, erit ac Å¿i tot eÅ¿-<lb/>Å¿ent vnitates quadratæ, quod Å¿i ita non eſſet, nulla mentio facienda eſſet quo-<lb/>rumuis <reg norm="fractorum" type="context">fractorÅ©</reg>. </s> - <s xml:space="preserve">Ex <reg norm="eadem" type="context">eadẽ</reg> regula de tribus reduci poteÅ¿tad praxim <reg norm="tertium" type="context">tertiÅ©</reg> theorema.</s> - </p> - <p> - <s xml:space="preserve">Quare cupientes Å¿cire quæ Å¿int illæ partes, quæ Å¿unt tres quartÄ™, ipÅ¿arum quin-<lb/>que Å¿extarum, dicemus Å¿i quatuor dant tria, quid dabunt <reg norm="quinque" type="simple">quinq;</reg> Å¿extæ? </s> - <s xml:space="preserve">dabunt .15. <lb/>vigeÅ¿imas quartas, quæ quindecim Å¿unt tres quartæ ipÅ¿ius .20. viginti <reg norm="autem" type="context">autẽ</reg> <reg norm="quinque" type="simple">quinq;</reg> Å¿ex <lb/>tæ vigintiquatuor, quandoquidem nos numerum quÄ™rimus, cui ita proportionentur <lb/><reg norm="quinque" type="simple">quinq;</reg> Å¿extæ alterius numeri, Å¿icut quatuor ad tria, vnde Å¿ic Å¿e habent .20. ad .15. Å¿i-<lb/>cut .4. ad .3. ipÅ¿e autem .20. <reg norm="quinque" type="simple">quinq;</reg> Å¿extÄ™ partes Å¿unt vigintiquatuor, vt per Å¿e <reg norm="notum" type="context">notÅ©</reg> eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Ex eadem regula de tribus, huiuÅ¿modi quęſito reÅ¿ponderi poteÅ¿t, Å¿i conÅ¿tituamus <lb/>prædictas <reg norm="quinque" type="simple">quinq;</reg> Å¿extas eſſe numerum, cuius tres quartæ quÄ™rantur, dicentes, Å¿i vnus <lb/>integer dat tres quartas, quid dabunt <reg norm="quinque" type="simple">quinq;</reg> Å¿extæ? </s> - <s xml:space="preserve">quare Å¿equentes regulam de <lb/>tribus, dabuntur quindecim vigeÅ¿imæ quartæ. </s> - <s xml:space="preserve">Valet eadem regula de tribus; </s> - <s xml:space="preserve">vt quis <lb/>Å¿cire poſſit, quæ pars aut partes numeri propoÅ¿iti Å¿it aliquis numerus.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿cire cupienti, quæ pars aut partes ipÅ¿ius vigintiquatuor Å¿int Å¿ex-<lb/>decim, conÅ¿tituentur .24. tanquam vnum totum, cuius pars aut partes Å¿int Å¿exdecim, <lb/>dicemus igitur Å¿i .24. dant Å¿exdecim, quid dabit vnum? </s> - <s xml:space="preserve">Å¿exdecim videlicet vigeÅ¿i-<lb/>masquartas, quæ cum ad primos numeros reductæ fuerint, erunt duæ tertiæ. <lb/></s> - <s xml:space="preserve">Eadem ratione qui Å¿cire uellet, quæ partes aut pars eſſent tres quartæ, octo no-<lb/>narum, diceret, Å¿i octo nonæ danttres quartas, quid dabit vnum? </s> - <s xml:space="preserve">prouenient .27. <lb/>trigeÅ¿imęſecundæ.</s> - </p> - <p> - <s xml:space="preserve">SubÅ¿eruit pariter ad <reg norm="Å¿ciendum" type="context">Å¿ciendÅ©</reg> <reg norm="naturam" type="context">naturã</reg> <reg norm="partium" type="context">partiÅ©</reg> numeri propoÅ¿iti. </s> - <s xml:space="preserve">Exempli cauÅ¿a, Å¿i quis <lb/>quærat, cuius numeri, duodecim Å¿int duæ tertiæ partes. </s> - <s xml:space="preserve">Dicet Å¿i duo dant tria, quid - <pb o="10" rhead="IO. BAPT. BENED." n="22" file="0022" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0022"/> - dabunt duodecim? </s> - <s xml:space="preserve">nempe dabunt decemocto, numerum quæſitum Å¿cilicet, <lb/></s> - <s xml:space="preserve">Tunc autem nil aliud pręſtamus quam quòd quærimus numerum ad quem ita Å¿e <lb/>habeant duodecim, Å¿icut duo ad tria. </s> - <s xml:space="preserve">Ita etiam Å¿i quis quærat, cuius numeri duo <lb/>tertia Å¿int tres quintÄ™, dicet, Å¿i tria dant <reg norm="quinque" type="simple">quinq;</reg>, quid dabunt duo tertia? </s> - <s xml:space="preserve">nempe da-<lb/>bunt integrum cum fracto nono. </s> - <s xml:space="preserve">Hoc erit <reg norm="itaque" type="simple">itaq;</reg> quÄ™rere numerum ad quem Å¿ic Å¿e <lb/>habeant duo tertia Å¿icut tria ad <reg norm="quinque" type="simple">quinq;</reg>, quod manifeÅ¿tum eÅ¿t per Å¿e.</s> - </p> - <p> - <s xml:space="preserve">Eadem ratione qui Å¿cire vellet, cuius numeri duæ Å¿eptimæ, eſſent octo integra-<lb/>rum cum duabus quintis, diceret, Å¿i duo dant Å¿eptem quid dabunt octo integra cum <lb/>duabus quintis? </s> - <s xml:space="preserve">nempe dabunt .29. integra cum duabus quintis numerum quæſi-<lb/>tum. </s> - <s xml:space="preserve">Sic etiam qui transferre uellet fractum numerum in fractum, id perficeret <lb/>ex regula de tribus.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia Å¿i proponerentur vnde cim tertiædecimæ vnius totius, toto diui-<lb/>Å¿o in .13. partes, <reg norm="deÅ¿ideraremusque" type="simple">deÅ¿ideraremusqÌ;</reg> Å¿cire, quot partes totius <reg norm="eÅ¿sent" type="context">eÅ¿sẽt</reg> vndecim <reg norm="tertiaedeci- mæ" type="simple">tertiÄ™deci-<lb/>mæ</reg>, toto in .4. partes diuiÅ¿o, diceremus Å¿i .13. dant .11. quid dabunt quatuor? </s> - <s xml:space="preserve">nem <lb/>pe <reg norm="dabunt" type="context">dabÅ©t</reg> tres quartas <reg norm="cum" type="context">cÅ©</reg> <reg norm="quinque" type="simple">quinq;</reg> tertijsdecimis unius quartæ, hoc verò nihil aliud eÅ¿t <lb/>quam querere numerum, ad quem Å¿ic Å¿e habeat totum in 4. partes diuiÅ¿um, Å¿icut <lb/>idem totum diuiÅ¿um in tredecim Å¿e habet ad undecim tertiasdecimas, Porrò ad <lb/>alia etiam multa hæc regula accommodata eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Hæc enim <reg norm="non" type="context">nõ</reg> Å¿ine propoÅ¿ito dicta Å¿unt, Å¿ed ut <reg norm="quiÅ¿que" type="simple">quiÅ¿q;</reg> videat cauÅ¿am Å¿imilium ope-<lb/>rationum, quæ à practicis circa fractos numeros Å¿criptæ Å¿unt, omnem à diuina illa <lb/>regula de tribus originem trahere ut etiam in Å¿equentibus videbimus.</s> - </p> - </div> - <div type="math:theorem" level="3" n="11"> - <head xml:space="preserve">THEOREMA <num value="11">XI</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">CVr</emph> productum ex eo quod oritur in diuidente, Å¿emper æquale eÅ¿t numero <lb/>diuiÅ¿ibili Å¿i queras ita accipe.</s> - </p> - <p> - <s xml:space="preserve">Sit numerus diuiÅ¿ibilis <var>.b.</var> quod oritur Å¿it <var>.c.</var> diuidens <var>.d.</var> & vnitas diuidentis <var>.t.</var> cum <lb/>igitur, vt in præcedenti theoremate dictum <lb/>fuit, eadem Å¿it proportio <var>.b.</var> ad <var>.c.</var> quæ eÅ¿t <var>.d.</var> <lb/> - <anchor type="figure" xlink:label="fig-0022-01a" xlink:href="fig-0022-01"/> - ad <var>.t.</var> manifeÅ¿te deprehenditur ex .20. Å¿epti <lb/>mi, productum ex <var>.b.</var> in <var>.t.</var> æquale eſſe pro-<lb/>ducto <var>.c.</var> in <var>d</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0022-01" xlink:href="fig-0022-01a"> - <image file="0022-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0022-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="12"> - <head xml:space="preserve">THEOREMA <num value="12">XII</num>.</head> - <p> - <s xml:space="preserve">ID ipÅ¿um alia ratione contemplari licet.</s> - </p> - <p> - <s xml:space="preserve">Numerus diuiÅ¿ibilis Å¿ignificetur per lineam <var>.n.e.</var> diuidens verò per lineam <var>.a.e.</var> <lb/>quod oritur linea <var>.u.e.</var> vnitas diuidentis <var>.o.e.</var> <reg norm="quam" type="context">quã</reg> cogitamus eſſe vnitatem linearem; <lb/></s> - <s xml:space="preserve">ad hæc productum ex <var>.u.e.</var> in <var>.a.e.</var> Å¿it Å¿uperficies <var>.u.a</var>. </s> - <s xml:space="preserve">Dico Å¿uperficiem <var>.u.a.</var> componi <lb/>ex tot vnitatibus Å¿uperficialibus quot linearibus conÅ¿tat linea <var>.n.e.</var> nam ex ijs quæ <lb/>diuidendi ratione notauimus, <reg norm="conÅ¿tituitur" type="context">cõſtituitur</reg> <lb/>eandem proportionem eſſe <var>.n.e.</var> ad <var>.u.e.</var> <lb/> - <anchor type="figure" xlink:label="fig-0022-02a" xlink:href="fig-0022-02"/> - quÄ™ eÅ¿t <var>.a.e.</var> ad <var>.o.e</var>. </s> - <s xml:space="preserve">At ex prima Å¿exti aut <lb/>18. Å¿eptimi Å¿ic Å¿e habet totale <reg norm="productum" type="context">productÅ©</reg> <var>.<lb/>u.a.</var> ad partiale <var>.u.o.</var> Å¿icut <var>.a.e.</var> ad <var>.o.e</var>. <lb/></s> - <s xml:space="preserve">quare Å¿ic Å¿e habebit <var>.u.a.</var> ad <var>.u.o.</var> Å¿icut <var>.n.<lb/>e.</var> ad <var>.u.e.</var> Å¿ed <var>.u.e.</var> et <var>.u.o.</var> numero non differunt, cum Å¿int vnius & eiuÅ¿dem Å¿peciei, (ta-<lb/>met Å¿i numerus <var>.u.o.</var> Å¿it Å¿uperficialis et <var>.u.e.</var> linearis). </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> ex nona quinti numerus <var>.<lb/>u.a.</var> æqualis erit numero <var>.n.e</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0022-02" xlink:href="fig-0022-02a"> - <image file="0022-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0022-02"/> - </figure> - </div> - <pb o="11" rhead="THEOREM. ARITH." n="23" file="0023" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0023"/> - </div> - <div type="math:theorem" level="3" n="13"> - <head xml:space="preserve">THEOREMA. <num value="13">XIII</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">CVr</emph> diuidentibus numerum diuiÅ¿ibilem per proueniens, oritur numerus diui-<lb/>dens?</s> - </p> - <p> - <s xml:space="preserve">Sit Å¿ubÅ¿criptus rectangulus <var>.o.e.</var> numerus diuiÅ¿i <lb/> - <anchor type="figure" xlink:label="fig-0023-01a" xlink:href="fig-0023-01"/> - bilis, qui producitur, tam ex <var>.a.o.</var> in <var>.a.e.</var> quám ex <var>.a.<lb/>e.</var> in <var>.a.o</var>. </s> - <s xml:space="preserve">quare Å¿i <var>.a.o.</var> diuidens fuerit <var>.a.e.</var> proue-<lb/>niens erit, Å¿i veró <var>.a.e.</var> diuidens extiterit, <var>a.o.</var> pro-<lb/>ueniens erit futurum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0023-01" xlink:href="fig-0023-01a"> - <image file="0023-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0023-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="14"> - <head xml:space="preserve">THEOREMA. <num value="14">XIIII</num>.</head> - <p> - <s xml:space="preserve">HOcipÅ¿um, alia <reg norm="quoque" type="simple">quoq;</reg> uia licebit Å¿peculari.</s> - </p> - <p> - <s xml:space="preserve">Sit linea <var>.a.</var> <reg norm="denotans" type="context">denotãs</reg> numerum diuiÅ¿ibilem, et <var>.o.</var> primi prouenientis linea <var>.e.</var> pri <lb/>mi diuidentis <var>.u.</var> Å¿ecundi prouenientis ideÅ¿t cum <var>.o.</var> pro diuidente Å¿umetur. </s> - <s xml:space="preserve">Iam ex <lb/>indicata definitione diuiÅ¿ionis nono theoremate huius libri, dabitur proportio <var>.a.</var> <lb/>ad <var>.o.</var> prout datur <var>.e.</var> ad vnitatem Å¿ignificatam li-<lb/>nea <var>.i.</var> & permutatim <var>.a.</var> ad <var>.e.</var> Å¿icut <var>.o.</var> ad <var>.i.</var> Å¿ed <var>.a.</var> <lb/> - <anchor type="figure" xlink:label="fig-0023-02a" xlink:href="fig-0023-02"/> - ad <var>.u.</var> Å¿ic Å¿e habet prout <var>.o.</var> ad <var>.i.</var> ex eadem definitio-<lb/>ne diuiÅ¿ionis, <reg norm="itaque" type="simple">itaq;</reg> Å¿ic Å¿e habebit <var>.a.</var> ad <var>.u.</var> Å¿icut <var>.a.</var> ad <var>.<lb/>e.</var> vnde <var>.u.</var> æqualis erit <var>.e.</var> ex .9. quinti.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0023-02" xlink:href="fig-0023-02a"> - <image file="0023-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0023-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="15"> - <head xml:space="preserve">THEOREMA. <num value="15">XV</num>.</head> - <p> - <s xml:space="preserve">VNde prouenit, vt qui velit cognoÅ¿cere cuius numeri quatuor quintæ par-<lb/>tes, Å¿int duæ tertiÄ™, aut quid Å¿imile, <reg norm="conÅ¿ultiſſime" type="context">cõſultiſſime</reg> faciat, Å¿i ad unam <reg norm="eandemque" type="simple">eandemq;</reg> <lb/>denominationem reduxerit.</s> - </p> - <p> - <s xml:space="preserve">Prout in propoÅ¿ito exemplo, <reg norm="cum" type="context">cÅ©</reg> <reg norm="denominans" type="context">denominãs</reg> <reg norm="communis" type="context">cõmunis</reg> Å¿it quindecim, cuius duæ ter <lb/>tiæ Å¿unt <reg norm="decem" type="context">decẽ</reg>, & quatuor quintæ duodecim, <reg norm="communis" type="context">cõmunis</reg> <reg norm="autem" type="context">autẽ</reg> denominans .15. multipli <lb/>candus Å¿it per quatuor quintas, Å¿cilicet duodecim, & productum diuidendum per <lb/>duas tertias, hoc eÅ¿t decem, ex quo oriantur decemocto quęſitus numerus?</s> - </p> - <p> - <s xml:space="preserve">Quod ad <reg norm="reductionem" type="context">reductionẽ</reg> <reg norm="numeratorum" type="context">numeratorÅ©</reg> ad vnam & eandem denominationem attinet, <lb/>ea de cauÅ¿a fit quo uti poſſimus regula de tribus, quæ tribus tantummodo notis ter-<lb/>minis indiget, quo quartus à prÄ™dictis dependens, inueniri poſſit, quandoquidem <lb/>bini illi reÅ¿pectus, tribus terminis comprehendi <reg norm="poÅ¿sunt" type="context">poÅ¿sÅ©t</reg>. </s> - <s xml:space="preserve">At quod ad multiplicatio-<lb/>nem Å¿pectat denominantis <reg norm="communis" type="context">cõmunis</reg> <reg norm="cum" type="context">cÅ©</reg> numerante denominantis in cogniti & diui-<lb/>Å¿ionem producti per numerantem <reg norm="cognitum" type="context">cognitÅ©</reg> illÄ™ nihil aliud Å¿unt, quam <reg norm="quartum" type="context">quartÅ©</reg> <reg norm="terminum" type="context">terminÅ©</reg> <lb/>inuenire, ita proportionatum tertio, vt Å¿ecundus primo.</s> - </p> - <p> - <s xml:space="preserve">Excmpli gratia, Å¿it <var>.a.</var> <reg norm="denotans" type="context">denotãs</reg> nume-<lb/>rantem denominantis cogniti, qui Å¿igni <lb/> - <anchor type="figure" xlink:label="fig-0023-03a" xlink:href="fig-0023-03"/> - ficetur linea <var>.o.</var> et <var>.e.</var> Å¿it denominantis in-<lb/>cogniti numerans, denotati linea <var>.u.</var> imò <lb/>verò & cogniti <var>.o.</var> nempe quatuor <lb/>quintæ, Iam Å¿i <var>.o.</var> cum <var>.e.</var> multiplicemus, & productum per <var>.a.</var> diuidemus dabitur <var>.u.</var> <lb/>Å¿ic Å¿e habens ad <var>.e.</var> Å¿icut <var>.o.</var> ad <var>.a.</var> ex .20. Å¿eptimi.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0023-03" xlink:href="fig-0023-03a"> - <image file="0023-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0023-03"/> - </figure> - </div> - <pb o="12" rhead="IO. BAPT. BENED." n="24" file="0024" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0024"/> - </div> - <div type="math:theorem" level="3" n="16"> - <head xml:space="preserve">THEOREMA <num value="16">XVI</num>.</head> - <p> - <s xml:space="preserve">INuenire autem cupienti cuius numeri, duæ tertiæ, Å¿int quatuor quintÄ™ partes, mulÂ<lb/>tiplicandÄ™ eſſent duæ tertiæ per denominantem communem, & productum diui-<lb/>dendum per quatuor quintas ipÅ¿ius de-<lb/>nominantis. </s> - <s xml:space="preserve">Ac Å¿i quis diceret Å¿i <var>.e.</var> dat <var>.<lb/> - <anchor type="figure" xlink:label="fig-0024-01a" xlink:href="fig-0024-01"/> - o.</var> quid dabit <var>.a</var>? </s> - <s xml:space="preserve">nempe dabit <var>.u.</var> nam in <lb/>propoÅ¿ito exemplo, terminus <var>.a.</var> loco <var>.e.</var> <lb/>duos Å¿ortietur denominantes, cognitum <lb/>videlicet <var>.o.</var> et <var>.u.</var> incognitum quod po-<lb/>Å¿tea cognitum oritur ex regula de tribus, vt dictum eÅ¿t.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0024-01" xlink:href="fig-0024-01a"> - <image file="0024-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0024-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="17"> - <head xml:space="preserve">THEOREMA <num value="17">XVII</num>.</head> - <p> - <s xml:space="preserve">QVA ratione cognoÅ¿ci poterit proportionem quantitatis cenÅ¿icæ cenÅ¿icæ ad <lb/>Å¿imilem quantitatem quadruplam eſſe ad eam, quæ eÅ¿t Å¿uarum radicum; </s> - <s xml:space="preserve">pro-<lb/>portionem <reg norm="autem" type="context">autẽ</reg> primarum relatarum eſſe quintuplam, <reg norm="atque" type="simple">atq;</reg> ita deinceps?</s> - </p> - <p> - <s xml:space="preserve">Cuiusrei gratia, <reg norm="Å¿ciendus" type="context">Å¿ciẽdus</reg> eÅ¿t modus <reg norm="productionis" type="simple">ê“ductionis</reg> <reg norm="harum" type="context">harÅ©</reg> <reg norm="dignitatum" type="context">dignitatÅ©</reg> qui oritur ex produ-<lb/>ctione primæ radicis in Å¿eipÅ¿am, prout qui <reg norm="cubum" type="context">cubÅ©</reg> requirit, ducat radicé in Å¿uo quadra-<lb/>to, & orietur cubus, hæc poÅ¿tea ducta in cubum, <reg norm="quantitatem" type="context">quantitatẽ</reg> cenÅ¿icam <reg norm="cenÅ¿icam" type="context">cenÅ¿icã</reg>, et in <lb/>hanc, prædictam radicem, dabit quantitatem primam relatam. </s> - <s xml:space="preserve">Quod vbi Å¿ciueri-<lb/>mus, meminiſſe oportet Euclidem decimaoctaua Å¿exti aut .11. octaui docere, pro-<lb/>portionem quadrati ad <reg norm="quadratum" type="context">quadratÅ©</reg>, duplam eſſe proportioni Å¿uarum radicum, & .36. <lb/>vndecimi aut .11. octaui, cubi ad <reg norm="cubum" type="context">cubÅ©</reg> triplam eſſe, ego verò nunc aſſero, cenÅ¿ici cen <lb/>Å¿ici ad radicum proportionem quadruplam eſſe, primi verò relati ad primum re-<lb/>latum quintuplam <reg norm="atque" type="simple">atq;</reg> ita gradatim.</s> - </p> - <p> - <s xml:space="preserve">Cuius Å¿peculationis gratia, detur linea <var>.d.</var> quæ cubum maiorem Å¿ignificet. et <var>.b.</var> <lb/>minorem <var>.c.</var> verò Å¿it radixipÅ¿ius <var>.d.</var> et <var>.e.</var> ipÅ¿ius <var>.b.</var> ita ordinate adinuicem, vt in Å¿ub-<lb/>Å¿cripta figura cernitur. </s> - <s xml:space="preserve">Iam <var>.c.</var> cum <var>.d.</var> producatur <reg norm="proueniatque" type="simple">proueniatqÌ;</reg> <var>.q.</var> cenÅ¿icum cenÅ¿i-<lb/>cum, tum producatur <var>.e.</var> cum <var>.b.</var> et dabitur <var>.p.</var> alterum cenÅ¿icum cenÅ¿icum. </s> - <s xml:space="preserve">Dico <lb/>igitur proportionem <var>.q.</var> ad <var>.p.</var> quadruplam eſſe proportioni <var>.c.</var> ad <var>.e.</var> hac de <lb/>cauÅ¿a quòd proportio <var>.q.</var> ad <var>.p.</var> compo-<lb/>natur ex proportione <var>.d.</var> ad <var>.b.</var> et <var>.c.</var> ad <var>.e.</var> <lb/> - <anchor type="figure" xlink:label="fig-0024-02a" xlink:href="fig-0024-02"/> - prout facile ex .24. Å¿exti, aut quinta octaui <lb/>deprÄ™henditur. </s> - <s xml:space="preserve">Quare <reg norm="cum" type="context">cÅ©</reg> proportio <var>.d.</var> ad <var>.<lb/>b.</var> proportioni <var>.c.</var> ad <var>.e.</var> tripla Å¿it, patet pro-<lb/>portionem <var>.q.</var> ad <var>.p.</var> quadruplam eſſe pro-<lb/>portioni <var>.c.</var> ad <var>.e</var>. </s> - <s xml:space="preserve">Idem de cæteris dignitati <lb/>bus dico, Å¿umptis Å¿emper <var>.d</var> et <var>.b.</var> pro duo-<lb/>bus cenÅ¿ibus cenÅ¿uum, aut duobus primis relatis, aut alio quouis axiomate.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0024-02" xlink:href="fig-0024-02a"> - <image file="0024-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0024-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="18"> - <head xml:space="preserve">THEOREMA. <num value="18">XVIII</num>.</head> - <p> - <s xml:space="preserve">CVR diuidentibus nobis dignitatem, per dignitatem, radix prouenientis: </s> - <s xml:space="preserve">pro <lb/>ueniens Å¿it diuiÅ¿ionis vnius radicis per alteram?</s> - </p> - <p> - <s xml:space="preserve">Sint exempli gratia duÄ™ lineæ <var>.b.q.</var> et <var>.f.g.</var> quæ Å¿ignificent duas radices cuiuÅ¿uis <lb/>dignitatis; </s> - <s xml:space="preserve"><reg norm="demusque" type="simple">demusqÌ;</reg> eſſe radices duorum quadratorum, <reg norm="quadratumque" type="simple">quadratumqÌ;</reg> ipÅ¿ius <var>b.q.</var> <lb/>per quadratum ipÅ¿ius <var>.f.g.</var> diuidatur; </s> - <s xml:space="preserve">quadrataqÌue radix prouenientis Å¿it <var>.d.q.</var> <lb/>vnitas verò linearis Å¿it <var>.i.g</var>. </s> - <s xml:space="preserve">Dico ipÅ¿am <var>.d.q.</var> eſſe proueniens ex diuiÅ¿ione <var>.b.q.</var> <lb/>per <var>.f.g</var>. </s> - <s xml:space="preserve">Patet enim ex definitione diuiÅ¿ionis nono theoremate tradita quadra- - <pb o="13" rhead="THEOR. ARITH." n="25" file="0025" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0025"/> - tum ipÅ¿ius <var>.d.q.</var> talem eſſe partem quadrati ipÅ¿ius <var>.b.q.</var> qualis quadratum ipÅ¿ius <var>.g.i.</var> <lb/>eÅ¿t quadrati ipÅ¿ius <var>.f.g</var>. </s> - <s xml:space="preserve">Scimus prÄ™terea ex .19. Å¿exti, aut vndecima octaui, propor-<lb/>tioné quadrati ipÅ¿ius <var>.b.q.</var> ad <reg norm="quadratum" type="context">quadratÅ©</reg> ipÅ¿ius <var>.d.q.</var> duplam eſſe proportioni <var>.b.q.</var> ad <var>.<lb/>d.q.</var> Å¿uarum radicum (cuborum enim tripla eſſet & cenÅ¿uum cenÅ¿uum, quadrupla, <lb/><reg norm="atque" type="simple">atq;</reg> ita deinceps ex præcedenti theoremate) Id ipÅ¿um dico de dignitatibus ipÅ¿ius <var>.<lb/>f.g.</var> et <var>.i.g.</var> reÅ¿pectu radicum <var>.f.g.</var> et <var>.i.g</var>. </s> - <s xml:space="preserve">Vnde <lb/>cum proportio dignitatis ipÅ¿ius <var>.b.q.</var> ad il-<lb/>lam <var>.d.q.</var> Ä™qualis Å¿it proportioni dignitatis <lb/> - <anchor type="figure" xlink:label="fig-0025-01a" xlink:href="fig-0025-01"/> - ipÅ¿ius <var>.f.g.</var> ad illam <var>.g.i.</var> ex communi Å¿cien-<lb/>tia apertè cognoÅ¿cemus Å¿implices propor-<lb/>tiones eſſe interÅ¿e æquales, nempe eam quÄ™ <lb/>eÅ¿t <var>.b.q.</var> ad <var>.d.q.</var> æqualem eſſe ei, quæ eÅ¿t <var>.f.<lb/>g.</var> ad <var>.i.g.</var> <reg norm="itaque" type="simple">itaq;</reg> Å¿equitur ex definitione diuiÅ¿ionis <var>.d.q.</var> eſſe proueniens ex diuiÅ¿ione <var>.<lb/>b.q.</var> per <var>.f.g</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0025-01" xlink:href="fig-0025-01a"> - <image file="0025-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0025-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="19"> - <head xml:space="preserve">THEOREMA <num value="19">XVIIII</num>.</head> - <p> - <s xml:space="preserve">CVR productum ex duabus radicibus quadratis, eÅ¿t quadrata radix, producti <lb/>Å¿uorum quadratorum Å¿imul?</s> - </p> - <p> - <s xml:space="preserve">In cuius rei gratiam, Å¿int duo quadrata <var>.d.a.</var> et <var>n.o.</var> coniuncta Å¿imul, prout in Å¿ub-<lb/>Å¿cripta figura apparet, ita tamen vtangulus <var>.a.n.u.</var> Å¿itre <lb/>ctus, </s> - <s xml:space="preserve">quare ex quartadecima primi, duo latera <var>.n.c.</var> et <var>.<lb/> - <anchor type="figure" xlink:label="fig-0025-02a" xlink:href="fig-0025-02"/> - n.a.</var> directe <reg norm="coniungentur" type="context">coniÅ©gentur</reg> adinuicem, prout etiam reli-<lb/>qua duo latera <var>.n.u.</var> et <var>.n.d</var>. </s> - <s xml:space="preserve">Cogitato deinde <var>.a.u.</var> pro <lb/>ducto ipÅ¿ius <var>.a.n.</var> in <var>.n.u.</var> duarum videlicet radicum <lb/>quadratarum Å¿imul, dabitur ex prima Å¿exti, aut de-<lb/>cimaottaua Å¿eptimi, productum <var>.a.u.</var> medium propor <lb/>tionale inter quadratum <var>.a.d.</var> et <var>.u.c.</var> quod Å¿i cogi-<lb/>temus has tres Å¿uperficies, tres numeros eſſe, pate-<lb/>bit ex vigeÅ¿imaprima Å¿eptimi productum <var>.a.u.</var> in Å¿e-<lb/>ipÅ¿um, quadratum Å¿cilicet <var>.a.u.</var> æquale eſſe producto <var>.<lb/>a.d.</var> in <var>.u.c.</var> ex quo propoÅ¿iti euidentia conÅ¿equetur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0025-02" xlink:href="fig-0025-02a"> - <image file="0025-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0025-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="20"> - <head xml:space="preserve">THEOREMA <num value="20">XX</num>.</head> - <p> - <s xml:space="preserve">QVA ratione id ipÅ¿um in cubis cognoÅ¿ci poterit. <lb/></s> - <s xml:space="preserve">Sit cubus <var>.l.b.</var> & cubus <var>.o.p.</var> quorum productum Å¿it <var>.u.g.</var> quod aſſero eÅ¿le <lb/> - <anchor type="figure" xlink:label="fig-0025-03a" xlink:href="fig-0025-03"/> - cubum, quamuis Eucli. idem probet <lb/>in <ref id="ref-0008">.4. noni.</ref> cuius radicem demonÅ¿tra-<lb/>bo eſſe numeri æqualis numero <var>.m.q.</var> <lb/>qui <var>.m.q.</var> productum eÅ¿t ipÅ¿ius <var>.m.e.</var> in <var>.e.<lb/>q.</var> radicum propoÅ¿itorum cuborum. </s> - <s xml:space="preserve">Pa-<lb/>tet enim ex præcedenti theoremate <var>.m.<lb/> - <anchor type="figure" xlink:label="fig-0025-04a" xlink:href="fig-0025-04"/> - <pb o="14" rhead="IO. BAPT. BENED." n="26" file="0026" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0026"/> - q.</var> radicem eſſe quadratam producti <var>.l.e.</var> in <var>.e.p.</var> quod <reg norm="productum" type="context">productÅ©</reg> Å¿it quadratuni<unsure/> <lb/>corporeum <var>.c.g.</var> cogitemus pariter duo quadrata <var>.l.e.</var> et <var>.e.p.</var> eſſe pariter corpo-<lb/>rea, tantÄ™ profunditatis, quantam, vnitas linearis radicum <var>.m.e.</var> et <var>.e.q.</var> requirit. <lb/></s> - <s xml:space="preserve">Hæc duo corpora producentur à ſuperficie in vnitatem, <reg norm="vocenturque" type="simple">vocenturqÌ;</reg> <var>.l.x.</var> et <var>.x.p.</var> quo <lb/>facto, cogitemus corpus <var>.a.g.</var> tamquam productum cubi <var>.l.b.</var> in quadratum <var>.e.p</var>. </s> - <s xml:space="preserve">Vn-<lb/>de ex decimaoctaua, aut decimanona Å¿eptimi, eadem erit proportio <var>.a.g.</var> ad <var>.c.g.</var> <lb/>quæ eÅ¿t <var>.l.b.</var> ad <var>.l.x.</var> corporeum, Å¿ed ex .25. vndecimi & prima Å¿exti, ita Å¿e habet <var>.a.K.</var> <lb/>ad <var>.K.c.</var> vnitatem linearé Å¿icut <var>.a.g.</var> ad <var>.c.g.</var> & ex <reg norm="eiÅ¿dem" type="context">eiÅ¿dẽ</reg> ita Å¿e habebit <var>.b.e.</var> ad <var>.e.x.</var> vnita-<lb/>tem linearem, Å¿icut <var>.l.b.</var> ad quadratum <var>.l.x.</var> corporeum. </s> - <s xml:space="preserve">Itaque Å¿ic Å¿e habebit <var>.b.e.</var> ad <lb/>vnitatem linearem <var>.e.x.</var> videlicet <var>.K.c.</var> Å¿icut <var>.a.K.</var> ad ipÅ¿am <var>.K.c</var>. </s> - <s xml:space="preserve">Vnde ex nona quinti <var>.<lb/>a.K.</var> æqualis erit <var>.e.b.</var> & conÅ¿equenter æqualis <var>.m.e</var>. </s> - <s xml:space="preserve">Iam verò Å¿it <var>.u.g.</var> productum <var>.l.b.</var> <lb/>cubi, in cubum <var>.o.p.</var> vt Å¿upra dictum eÅ¿t, Hinc patebit ex quauis duarum propoÅ¿itio-<lb/>num, decimaoctaua, aut decimanona Å¿eptimi, eandem futuram proportionem <var>.u.g.</var> <lb/>ad <var>.a.g.</var> quæ eÅ¿t <var>.o.p.</var> ad <var>.x.p.</var> quadratum corporeum. </s> - <s xml:space="preserve">Quare ex poÅ¿tremis, dictis ratio-<lb/>nibus, eadem erit proportio <var>.u.K.</var> ad <var>.a.K.</var> quæ eÅ¿t <var>.o.e.</var> ad vnitatem linearem <var>.e.x.</var> at <lb/>ex dictis decimaoctaua & decimanona Å¿eptimi, ita Å¿e habet <reg norm="numerus" type="simple">numerê°</reg> <var>.m.q.</var> ad <reg norm="numerum" type="context">numerÅ©</reg> <lb/><reg norm="Å¿uperficialem" type="context">Å¿uperficialẽ</reg> <var>.m.e.</var> qui <reg norm="producitur" type="simple">ê“ducitur</reg> à lineari <var>.m.e.</var> in vnitaté <reg norm="linearem" type="context">linearẽ</reg> ipÅ¿ius <var>.e.q.</var> Å¿icut nume <lb/>rus <var>.q.e.</var> ad Å¿uam vnitaté, Å¿ed <reg norm="cum" type="context">cÅ©</reg> numerus <var>.a.K.</var> æqualis Å¿it numero <var>.m.e.</var> vt <reg norm="probatum" type="context">probatÅ©</reg> eÅ¿t <lb/>erit ergo ex vndecima & nona quinti, numerus <var>.u.K.</var> æqualis numero <var>.m.q</var>. </s> - <s xml:space="preserve">At <var>.f.g.</var> <lb/>pariter æqualis eÅ¿t numero <var>.m.q.</var> ex præcedenti theoremate, vnde <var>.K.u.</var> pariter æqua <lb/>lis erit <var>.f.g</var>. </s> - <s xml:space="preserve">Itaque Å¿equitur <var>.u.g.</var> cubum eſſe, & <var>f.g.</var> radicem ipÅ¿ius, æqualem numero <var>.<lb/>m.q.</var> quod quærebatur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0025-03" xlink:href="fig-0025-03a"> - <image file="0025-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0025-03"/> - </figure> - <figure xlink:label="fig-0025-04" xlink:href="fig-0025-04a"> - <image file="0025-04" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0025-04"/> - </figure> - </div> - <figure position="here"> - <image file="0026-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0026-01"/> - </figure> - <figure position="here"> - <image file="0026-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0026-02"/> - </figure> - </div> - <div type="math:theorem" level="3" n="21"> - <head xml:space="preserve">THEOREMA <num value="21">XXI</num>.</head> - <p> - <s xml:space="preserve">VT autem in uniuerÅ¿um Å¿ciri poſſit totum <reg norm="infinitum" type="context">infinitÅ©</reg> dignitatum, hoc eÅ¿t radicem <lb/>producti duarum dignitatum Å¿imilium, productum eſſe duarum radicum ea-<lb/>rundem dignitatum.</s> - </p> - <p> - <s xml:space="preserve">Ponamus, exempli gratia, duas radices quadratas <var>.q.p.</var> et <var>.g.K.</var> incognitas, quas <lb/>qui velit adinuicem multiplicare, cogatur earum quadrata cognita <var>.n.</var> cum <var>.i.</var> multi-<lb/>plicare, quorum productum Å¿it quadratum <var>.m.</var> radix cuius Å¿it <var>.b.d.</var> quam dico æqualé - <pb o="15" rhead="THEOREM. ARIT." n="27" file="0027" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0027"/> - eſſe <reg norm="producto" type="simple">ê“ducto</reg> <var>.q.p.</var> in <var>.g.k.</var> <reg norm="quod" type="wordlist">qÄ‘</reg> <reg norm="autem" type="context">autẽ</reg> Å¿it <var>.o</var>. </s> - <s xml:space="preserve">Patet enim <reg norm="proportionem" type="context">proportionẽ</reg> <var>.o.</var> ad <var>.q.p.</var> <reg norm="eandem" type="context">eandẽ</reg> eſſe <lb/>cum proportione <var>.g.k.</var> ad Å¿uam vnitatem linearem, ex decimaoctaua, aut decima-<lb/>nona Å¿eptimi, hæc vero vnitas linearis Å¿it <var>.t.</var> cuius Å¿uperficialis Å¿it <var>.u.</var> vnitas Å¿cilicet to-<lb/>ties in Å¿eipÅ¿am multiplicata quoties propoÅ¿ita dignitas patitur, tametÅ¿i in præſen <lb/>ti exemplo quadrata dignitas Å¿umatur. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> ex eiÅ¿dem propoÅ¿itionibus decimaocta <lb/>ua aut decimanona, Å¿ic Å¿e habet <var>.m.</var> ad <var>.n.</var> Å¿icut <var>.i.</var> ad <var>.u</var>. </s> - <s xml:space="preserve">Scimus prÄ™terea <reg norm="proportionem" type="context">proportionẽ</reg> <var>.<lb/>m.</var> ad <var>.n.</var> (eo quod in propoÅ¿ito exemplo Å¿int quadrata) duplam eſſe proportioni <var>.b.<lb/>d.</var> ad <var>.q.p.</var> et ipÅ¿ius <var>.i.</var> ad <var>.u.</var> pariter duplam proportioni <var>.g.k.</var> ad <var>.t.</var> iam autem dictum <lb/>fuit Å¿ic Å¿e habere <var>.m.</var> ad <var>.n.</var> Å¿icut <var>.i.</var> ad <var>.u</var>. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> <var>.<lb/>b.d.</var> Å¿ic Å¿e habebit ad <var>.q.p.</var> Å¿icut <var>.g.k.</var> ad <var>.t.</var> <lb/> - <anchor type="figure" xlink:label="fig-0027-01a" xlink:href="fig-0027-01"/> - quandoquidem Å¿ic Å¿e habeattotum ad <reg norm="to- tum" type="context">to-<lb/>tÅ©</reg>, Å¿icut pars ad <reg norm="partem" type="context">partẽ</reg>, <reg norm="dum" type="context">dÅ©</reg> Å¿imiles Å¿int, proba <lb/><reg norm="tum" type="context">tÅ©</reg> <reg norm="autem" type="context">autẽ</reg> eÅ¿t Å¿uperius ita Å¿e habere <var>.o.</var> ad <var>.q.p.</var> <lb/>Å¿icut <var>.g.k.</var> ad <var>.t.</var> <reg norm="itaque" type="simple">itaq;</reg> <var>.o.</var> Å¿ic Å¿e habebit ad <var>.q.p.</var> <lb/>Å¿icut <var>.b.d.</var> ad <var>.q.p.</var> vnde <var>.o.</var> æqualis erit <var>.b.d.</var> <lb/>HocipÅ¿um cÄ™teris dignitatibus conueniet, <lb/>mutatis tantummodo proportionibus <var>.m.<lb/>n.</var> ad proportionem <var>.b.d</var>: <var>q.p.</var> Å¿ic propor-<lb/>tionibus duarum dignitatum <var>.i.u.</var> ad pro-<lb/>portionem Å¿uarum radicum <var>.g.k.t</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0027-01" xlink:href="fig-0027-01a"> - <image file="0027-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0027-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="22"> - <head xml:space="preserve">THEOREMA <num value="22">XXII</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">DOcent</emph> veteres, quòd Å¿i quilibet numerus in duas partes inæquales diuiÅ¿us <lb/>fuerit, <reg norm="totumque" type="simple">totumqÌ</reg> diuiÅ¿um per <reg norm="vnam" type="context">vnã</reg> partium, & per eandem pars altera diuiÅ¿a fue-<lb/>rit: </s> - <s xml:space="preserve">differentia prouenientium Å¿emper vnitas erit. </s> - <s xml:space="preserve">quodquidem veriſſimum eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Detur enim <var>.b.d.</var> propoÅ¿itus numerus in duas partes inæquales diuiÅ¿us <var>.b.c.</var> et <var>.c.d.</var> <lb/>& in primis <reg norm="totum" type="context">totÅ©</reg> <var>.b.d.</var> per <var>.c.d.</var> diuidatur, ex quo oriatur <var>e.o.</var> vnitas autem <reg norm="per" type="punctuation simple">.ê‘</reg> <var>.i.o.</var> Å¿igni-<lb/>ficetur, tum pars ipÅ¿a <var>.b.c.</var> <reg norm="per" type="simple punctuation">ê‘.</reg> <reg norm="eandem" type="context">eãdem</reg> <var>.c.d.</var> diuidatur, <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <reg norm="proueniens" type="context">proueniẽs</reg> <var>.a</var>. </s> - <s xml:space="preserve">Sanè ex defini-<lb/>tione diuiÅ¿ionis, eadem erit proportio <var>.b.d.</var> ad <var>.e.o.</var> quæ eÅ¿t <var>.c.d.</var> ad <var>.i.o.</var> et ita <var>.b.c.</var> ad <var>.a.</var> <lb/>Å¿icut <var>.c.d.</var> ad <var>.i.o</var>. </s> - <s xml:space="preserve">Ex <ref id="ref-0009">.19. autem quinti</ref>, ita Å¿e habet <var>.b.c.</var> ad <var>.e.i.</var> Å¿icut <var>.b.d.</var> ad <var>.e.o.</var> at <var>.b.d.</var> <lb/>ad <var>.e.o.</var> Å¿ic Å¿e habet Å¿icut <var>.c.d.</var> ad <var>.i.o.</var> hoc eÅ¿t Å¿icut <var>.b.c.</var> ad <var>.a</var>. </s> - <s xml:space="preserve">Quare ex .11. quinti Å¿ic Å¿e <lb/>habebit <var>.b.c.</var> ad <var>.e.i.</var> Å¿icut .ad <var>.a.</var> ex quo ex .9. <reg norm="praedi cti" type="simple">prÄ™diÂ<lb/>cti</reg> <var>.a.</var> æqualis erit <var>.e.i.</var> Å¿ed <var>.e.i.</var> minor eÅ¿t <var>.e.o.</var> <lb/> - <anchor type="figure" xlink:label="fig-0027-02a" xlink:href="fig-0027-02"/> - per <var>.i.o</var>. </s> - <s xml:space="preserve">Quare Å¿equitur propoÅ¿itum verum eÅ¿Â<lb/>Å¿e. </s> - <s xml:space="preserve">Quod ipÅ¿um pauciſſimis verbis Å¿ic definiri <lb/>poteÅ¿t, Å¿i dixerimus, eiuÅ¿modi diuidens .in par-<lb/>te diuiÅ¿ibili, <reg norm="quam" type="context">quã</reg> in toto, Å¿emel minus ingredi, <lb/>quandoquidem altera pars eÅ¿t, ex qua totum integrum perficitur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0027-02" xlink:href="fig-0027-02a"> - <image file="0027-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0027-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="23"> - <head xml:space="preserve">THEOREMA <num value="23">XXIII</num>.</head> - <p> - <s xml:space="preserve">HOcipÅ¿um alia ratione contemplari poÂ<lb/> - <anchor type="figure" xlink:label="fig-0027-03a" xlink:href="fig-0027-03"/> - terimus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0027-03" xlink:href="fig-0027-03a"> - <image file="0027-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0027-03"/> - </figure> - </div> - <p> - <s xml:space="preserve">Significetur enim totalis numerus per <var>.a.e.</var> <lb/>in duas partes diuiÅ¿us <var>.a.u.</var> et <var>.u.e.</var> totius autem diuidens Å¿it <var>.u.e.</var> & partis alterius <var>.a.u.</var> <lb/>totius verò <reg norm="proueniens" type="context">proueniẽs</reg> Å¿it <var>.a.c.</var> partis <reg norm="autem" type="context">autẽ</reg>, Å¿it <reg norm="proueniens" type="context">proueniẽs</reg> <var>.a.n.</var> tum differentia Å¿it <var>.n.c.</var> vni - <pb o="16" rhead="IO. BAPT. BENED." n="28" file="0028" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0028"/> - tas vero cui <reg norm="differentiam" type="context">differentiã</reg> <var>.n.c.</var> æquari dico, Å¿it <var>.a.i</var>. </s> - <s xml:space="preserve">Patet enim in primis, eandem propor <lb/>tionem eſſe <var>.a.e.</var> ad <var>.a.c.</var> quæ eÅ¿t <var>.u.e.</var> ad <var>.a.i.</var> ex definitione diuiÅ¿ionis, et eandem <lb/>eſſe <var>.a.u.</var> ad <var>.a.n.</var> quæ eÅ¿t <var>.u.e.</var> ad <var>.a.i.</var> vnde ex .<lb/>11. quinti Å¿ic Å¿e habebit <var>.a.e.</var> ad <var>.a.c.</var> Å¿icut <var>.a.<lb/> - <anchor type="figure" xlink:label="fig-0028-01a" xlink:href="fig-0028-01"/> - u.</var> ad <var>.a.n.</var> et ex .19. eiuÅ¿dem Å¿ic Å¿e habe-<lb/>bit <var>.u.e.</var> ad <var>.n.c.</var> Å¿icut <var>.a.e.</var> ad <var>.a.c.</var> Å¿ed. Å¿ic Å¿e <lb/>habebat <var>.u.e.</var> ad <var>.a.i</var>. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> ex prædicta .11. quinti, Å¿ic Å¿e habebit <var>.u.e.</var> ad <var>.n.c.</var> Å¿icut ad <var>.a.<lb/>i</var>. </s> - <s xml:space="preserve">Quare ex .9. eiuÅ¿dem <var>.n.c.</var> æqualis erit <var>.a.i.</var> etidcirco <var>.n.c.</var> pariter vnitas erit.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0028-01" xlink:href="fig-0028-01a"> - <image file="0028-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0028-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="24"> - <head xml:space="preserve">THEOREMA <num value="24">XXIIII</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">CVr</emph> quibuslibet duobus numeris diuiÅ¿is adinuicem, <reg norm="multiplicatisque" type="simple">multiplicatisqÌ</reg> prouenien <lb/>tibus Å¿imul, productum, Å¿emper eÅ¿t vnitas Å¿uperficialis? </s> - <s xml:space="preserve">Nempe ex .20. Å¿eptimi, <lb/>quoniam vnitas linearis Å¿emper media proportionalis eÅ¿t inter bina prouenientia. <lb/></s> - <s xml:space="preserve">Quodita Å¿pecularilicet.</s> - </p> - <p> - <s xml:space="preserve"><reg norm="Significentur" type="context">Significẽtur</reg> duo propoÅ¿iti numeri per <var>.b.p.</var> et <var>.b.d.</var> mutuo diuiÅ¿i, proueniens au-<lb/>tem <var>.b.p.</var> per <var>.b.d.</var> diuiÅ¿um Å¿it <var>.b.n.</var> tum proueniens <var>.b.d.</var> diuiÅ¿um per <var>.b.p.</var> Å¿it <var>.b.a.</var> <lb/>et <var>.b.t.</var> Å¿it vnitas <var>.b.p.</var> et <var>.b.e.</var> vnitas <var>.b.d.</var> ex quo <var>.b.t.</var> æqualis erit <var>.b.e</var>.</s> - </p> - <p> - <s xml:space="preserve">Iam ex definitio ne diuiÅ¿ionis, dabitur eadem proportio <var>.b.p.</var> ad <var>.b.n.</var> quæ eÅ¿t <var>.b.d.</var> <lb/>ad <var>.b.e.</var> et proportio <var>.b.d.</var> ad <var>.b.a.</var> quæ eÅ¿t <var>.b.p.</var> ad <var>.b.t</var>. </s> - <s xml:space="preserve">Sed cum Å¿ic Å¿e habeat <var>.b.<lb/>p.</var> ad <var>.b.n.</var> Å¿icut <var>.b.d.</var> ad <var>.b.e.</var> permutando Å¿ic Å¿e habebit <var>.b.p.</var> ad <var>.b.d.</var> Å¿icut <var>.b.n.</var> ad <var>.b.<lb/>e.</var> hoc eÅ¿t ad <var>.b.t.</var> et cum Å¿ic Å¿e habeat <var>.b.d.</var> ad <var>.b.a.</var> Å¿icut <var>.b.p.</var> ad <var>.b.t</var>: permutando Å¿ic Å¿e <lb/>habebit <var>.b.d.</var> ad <var>.b.p.</var> Å¿icut <var>.b.a.</var> ad <var>.b.t</var>. <lb/></s> - <s xml:space="preserve">Quare euerÅ¿im Å¿ic Å¿e habebit <var>.b.p.</var> ad <var>.<lb/> - <anchor type="figure" xlink:label="fig-0028-02a" xlink:href="fig-0028-02"/> - <lb/>b.d.</var> Å¿icut <var>.b.t.</var> ad <var>.b.a.</var> Å¿ed <var>.b.n.</var> ad <var>.b.t.</var> Å¿ic <lb/>Å¿e habebat vt <var>.b.p.</var> ad <var>.b.d</var>. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> ex .11. <lb/>quintiÅ¿ic Å¿e habebit <var>.b.n.</var> ad <var>.b.t.</var> Å¿icut <var>.b.<lb/> - <anchor type="figure" xlink:label="fig-0028-03a" xlink:href="fig-0028-03"/> - e.</var> ad <var>.b.a</var>. </s> - <s xml:space="preserve">Dictum autem eÅ¿t <var>.b.e.</var> et <var>.b.t.</var> idem omnino eſſe. </s> - <s xml:space="preserve">Quare ex .20. Å¿eptimi pro-<lb/>poÅ¿iti veritas innoteÅ¿cet.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0028-02" xlink:href="fig-0028-02a"> - <image file="0028-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0028-02"/> - </figure> - <figure xlink:label="fig-0028-03" xlink:href="fig-0028-03a"> - <image file="0028-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0028-03"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="25"> - <head xml:space="preserve">THEOREMA <num value="25">XXV</num>.</head> - <p> - <s xml:space="preserve">IDipÅ¿um & hac altera uia patebit.</s> - </p> - <p> - <s xml:space="preserve">Duo illi numeri per <var>.o.</var> et <var>.u.</var> Å¿ignificentur mutuo diuiÅ¿i, proueniens <reg norm="autem" type="context">autẽ</reg> <var>.o.</var> per <var>.<lb/>u.</var> Å¿it <var>.e.</var> et proueniens <var>.u.</var> per <var>.o.</var> Å¿it <var>.x.</var> vnitas uerò per <var>.i.</var> Å¿ignificetur, quas tamen quanti-<lb/>tates Å¿ubÅ¿cripto modo ad inuicem diÅ¿poni-<lb/>to. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> ex definitione diuiÅ¿ionis, eadem erit <lb/> - <anchor type="figure" xlink:label="fig-0028-04a" xlink:href="fig-0028-04"/> - proportio <var>.o.</var> ad <var>.e.</var> quÄ™ eÅ¿t <var>.u.</var> ad <var>.i.</var> et <var>.o.</var> ad <var>.i.</var> quÄ™ <lb/>eÅ¿t <var>.u.</var> ad <var>.x</var>. </s> - <s xml:space="preserve">Quare ex æqualitate <reg norm="proportionum" type="context">proportionÅ©</reg> <var>.<lb/>c.</var> ad <var>.i.</var> Å¿ic Å¿e habebit Å¿icut <var>.i.</var> ad <var>.x.</var> erit enim <var>.i.</var> <lb/>media proportionalis inter <var>.e.</var> et <var>.x.</var> ex .20. <reg norm="autem" type="context">autẽ</reg> <lb/>Å¿eptimi propoÅ¿itum concludetur. </s> - <s xml:space="preserve">HuiuÅ¿modi rei cauÅ¿a etiam eÅ¿t, quod proueniens <lb/>diuiÅ¿ionis vnius eÅ¿t numerator æqualis denominatori diuiÅ¿ionis alterius.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0028-04" xlink:href="fig-0028-04a"> - <image file="0028-04" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0028-04"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="26"> - <head xml:space="preserve">THEOREMA <num value="26">XXVI</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">CVr</emph> duobus numeris mutuo diuiÅ¿is, <reg norm="sumptis" type="context">sÅ©ptis</reg> deinde prouenientibus Å¿imul et adinui <lb/>cem, & per hanc Å¿ummam, diuiÅ¿a Å¿umma quadratorum dictorum <reg norm="propoÅ¿itorum" type="context">propoÅ¿itorÅ©</reg> - <pb o="17" rhead="THEOREM. ARITH." n="29" file="0029" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0029"/> - numerorum, proueniat numerus æqualis numero producti duorum primorum nu-<lb/>m erorum Å¿imul.</s> - </p> - <p> - <s xml:space="preserve">Sint exempli gratia propoÅ¿iti numeri .2. et .8. qui mutuo diuiÅ¿i in primis dent pro <lb/>uenientia quatuor integra, tum quartam partem pro altero proueniente, hæc colle-<lb/>cta dabunt Å¿ummam quatuor integrorum et quartæ partis vnius, Å¿umma autem qua <lb/>dratorum binarij & octonarij erit .68. qui quidem numerus per quatuor & quar <lb/>tam partem vnius diuiÅ¿us dabit .16. pro proueniente, quæ .16. æqualia erunt pro <lb/>ducto binarii in octonarium.</s> - </p> - <p> - <s xml:space="preserve">Cuius rei hæc erit Å¿peculatio, Å¿int duæ lineæ <var>.o.e.</var> et <var>.o.n.</var> quæ duos numeros pro-<lb/>poÅ¿itos Å¿ignificent, inuicem ad angulum rectum <var>.o.</var> coniunctæ, quarum quadrata <lb/>Å¿int <var>.o.a.</var> et <var>.o.p.</var> ipÅ¿orum productum Å¿it <var>.n.e.</var> tum <var>.o.t.</var> Å¿it proueniens ex diuiÅ¿ione <var>.o.e.</var> <lb/>per <var>.o.n</var>. </s> - <s xml:space="preserve">HÄ™c Å¿ingulatim conÅ¿ideremus (<reg norm="nam" type="context">nã</reg> Å¿i in partibus Å¿implicibus quod dicimus ac <lb/>ciderit, id ipÅ¿um in compoÅ¿itis conÅ¿equenter eueniet) quamobrem ex definitione di <lb/>uiÅ¿ionis dabitur eadem proportio <var>.o.e.</var> ad <var>.o.t.</var> quæ eft <var>.o.n.</var> ad vnitatem, quæ Å¿it <var>.o.<lb/>x</var>. </s> - <s xml:space="preserve">Nunc cogitemus <reg norm="Å¿uperficiem" type="context">Å¿uperficiẽ</reg> <reg norm="rectangulam" type="context">rectangulã</reg> <var>.o.c.</var> <reg norm="æqualem" type="context">æqualẽ</reg> quadrato <var>.o.a</var>. </s> - <s xml:space="preserve">tunc numerus <var>.<lb/>c.t.</var> proueniens erit, ut patet, ex diuiÅ¿ione numeri quadrati <var>.o.a.</var> per <reg norm="numerum" type="context">numerÅ©</reg> <var>.o.t.</var> <reg norm="eritque" type="simple">eritqÌ</reg> <lb/><reg norm="eadem" type="context">eadẽ</reg> proportio <var>.c.t.</var> ad <var>.o.e.</var> quæ eÅ¿t <var>.o.e.</var> ad <var>.o.t.</var> ex Å¿ecunda parte quintæ decimæ Å¿exti, <lb/>aut .20. Å¿eptimi. </s> - <s xml:space="preserve"><reg norm="Iam" type="context">Iã</reg> <reg norm="autem" type="context">autẽ</reg> dictum eÅ¿t <var>.o.e.</var> ad <var>.o.t.</var> Å¿ic Å¿e habere Å¿icut <var>.o.n.</var> ad <var>.o.x</var>. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> ex .<lb/>11. quinti Å¿ic Å¿e habebit <var>.c.t.</var> ad <var>.o.e.</var> Å¿icut <var>.o.n.</var> ad <var>.o.x</var>. </s> - <s xml:space="preserve">Sed ex prima Å¿exti, aut .18. vel .<lb/>19. Å¿eptimi, Å¿ic Å¿e habet <reg norm="productum" type="simple">ê“ductum</reg> <var>.n.e.</var> ad <var>.e.x.</var> Å¿icut <var>.o.n.</var> ad <var>.o.x</var>. </s> - <s xml:space="preserve">quare denuo Å¿ic Å¿e ha-<lb/>bebit numerus <var>.c.t.</var> ad numerum <var>.o.e.</var> Å¿icut nume-<lb/>rus <var>.n.e.</var> ad numerum <var>.x.e</var>. </s> - <s xml:space="preserve">Sed numerus <var>.o.e.</var> cum <lb/> - <anchor type="figure" xlink:label="fig-0029-01a" xlink:href="fig-0029-01"/> - numero <var>.x.e.</var> Å¿pecie idem eÅ¿t, igitur ex .9. quinti nu <lb/>merus <var>.c.t.</var> numero <var>.n.e.</var> æqualis erit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0029-01" xlink:href="fig-0029-01a"> - <image file="0029-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0029-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Id ipÅ¿um de quadrato ipÅ¿ius <var>.o.n.</var> videlicet <var>.p.o.</var> <lb/>dico. </s> - <s xml:space="preserve">Nam Å¿i proueniens <var>.o.n.</var> diuiÅ¿o per <var>.o.e.</var> ideÅ¿t <var>.<lb/>o.i.</var> proportionale reÅ¿pondens ad <var>.o.t.</var> cum <var>.o.t.</var> <lb/><reg norm="coniunctum" type="context">coniunctÅ©</reg> fuerit, et per <reg norm="hanc" type="context">hãc</reg> Å¿ummam diuiÅ¿a Å¿umma <lb/>quadratorum <var>.o.a.</var> et <var>.o.p.</var> patet per Å¿e proueniens <lb/>futurum eiuÅ¿dem numeri <var>.c.t.</var> <reg norm="ipÅ¿umque" type="simple">ipÅ¿umqÌ</reg> <var>.c.t.</var> proue-<lb/>niens Å¿emper Å¿uturum.</s> - </p> - <p> - <s xml:space="preserve">Quo autem lucidius res hæc innoteÅ¿cat. </s> - <s xml:space="preserve">Cogi <lb/>temus proueniens quadrati <var>.o.p.</var> diuiÅ¿i ab <var>.o.i.</var> re-<lb/><reg norm="Å¿pondentisque" type="simple">Å¿pondentisq;</reg> <var>.o.t.</var> eſſe <var>.i.u.</var> quod via prædicta inue-<lb/>nitur æqualis eſſe numero <var>.n.e.</var> ex quo conÅ¿e-<lb/>quenter æquale <var>.c.t</var>: cogitato deinde rectangu-<lb/>lo <var>.o.u.</var> æquali <var>.o.p.</var> coniuncto <var>.o.c</var>:totum <var>.t.u.</var> æqua-<lb/>le erit compoÅ¿ito duorum quadratorum <var>.o.a.</var> et <var>.o.<lb/>p.</var> cum in nullo numerus <var>.c.t.</var> mutetur, tam ex com-<lb/>poÅ¿ito <var>.t.u.</var> <reg norm="quam" type="context">quã</reg> ex Å¿implici <var>.o.c.</var> ex quo propoÅ¿iti Å¿e <lb/>Å¿e ueritas profert.</s> - </p> - </div> - <div type="math:theorem" level="3" n="27"> - <head xml:space="preserve">THEOREMA <num value="27">XXVII</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">PRoposvervnt</emph> veteres nobile quidem problema, Å¿ed quod tamen citra al-<lb/>gebraticam effectionem, aut neÅ¿cierunt, aut noluerunt diſſoluere, quod nihi-<lb/>lominus facillimum eÅ¿t.</s> - </p> - <pb o="18" rhead="IO. BAPT. BENED." n="30" file="0030" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0030"/> - <p> - <s xml:space="preserve">Proponunt hi numerum in binas eiuÅ¿modi partes diuidendum, vt Å¿umma qua-<lb/>dratorum dictarum partium, alteri numero poÅ¿sibili propoÅ¿ito æqualis Å¿it, poſſi-<lb/>bili inquam, etenim Å¿i eiuÅ¿modi numerus propoÅ¿itus, minor eſſet producto totius <lb/>primi in Å¿uum dimidium, eſſet huiuÅ¿modi factum impoſſibile. </s> - <s xml:space="preserve">Quod nos exequi <lb/>cupientes, Å¿umamus primum <reg norm="numerum" type="context">numerÅ©</reg> propoÅ¿itum, quem in Å¿e ipÅ¿um multiplice-<lb/>mus. </s> - <s xml:space="preserve">ab hoc quadrato deducamus Å¿ecundum numerum propoÅ¿itum, tum quod re-<lb/>manÅ¿erit duplicemus, quod duplum denuo iubeo ex eodem primo quadrato detra-<lb/>hi, accepta poÅ¿tea radice quadrata reÅ¿idui & dempta ex priori numero propoÅ¿ito, <lb/></s> - <s xml:space="preserve">tunc dimidium reÅ¿idui vna pars erit ex duabus primi numeri quæſita.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia proponantur .20. diuidenda in duas eiuÅ¿modi partes, vt Å¿umma <lb/>quadratorum ipÅ¿arum partium æqualis Å¿it .272. qui numerus maior eÅ¿t .200. maior <lb/>inquam dimidio quadrati .400. ipÅ¿orum .20. hic autem numerus .272. è quadra-<lb/>to .400. deducatur, <reg norm="remanebunt" type="context">remanebÅ©t</reg> enim .128. quod duplicari iubeo, <reg norm="producentur" type="context">producẽtur</reg> <reg norm="Å¿iquidem" type="context">Å¿iquidẽ</reg> <num value="256">.<lb/>256.</num> quæ pariter deducta è quadrato totali, remanebunt .144. cuius radicem Å¿umi <lb/>volo, quæ erit .12. & dempta ex .20. priori numero dato remanebit .8. cuius di-<lb/>midium erit .4: pars vna ex quæſitis, quæ ex primo numero propoÅ¿ito .20. detra-<lb/>hetur, <reg norm="remanebitque" type="simple">remanebitqÌ</reg> .16. pro altera parte.</s> - </p> - <p> - <s xml:space="preserve">Cuius demonÅ¿trationis cauÅ¿a, in primis cogitemus quadratum <var>.a.c.</var> cognitum nu-<lb/>meri <var>.a.b.</var> primò propoÅ¿iti, qui cogitetur diuiÅ¿us in duo quadrata <var>.d.e.</var> et <var>.e.b.</var> <reg norm="duo- que" type="simple">duo-<lb/>qÌ</reg> Å¿upplementa <var>.a.e.</var> et <var>.e.c.</var> numerus autem Å¿ummæ duorum quadratorum <var>.d.e.<lb/>b.</var> pro Å¿ecundo propoÅ¿ito datur; </s> - <s xml:space="preserve">ex quo, Å¿umma duorum Å¿upplementorum <var>.a.e.c.</var> <lb/>conÅ¿equenter erit cognita, quÄ™ cum duplicata fuerit, & quatuor hæc Å¿upplementa<unsure/> <lb/>cogitatione accommodata, prout in <lb/>quadrato <var>.f.g.</var> apparet (<reg norm="quanuis" type="context">quãuis</reg> idipÅ¿um <lb/> - <anchor type="figure" xlink:label="fig-0030-01a" xlink:href="fig-0030-01"/> - proueniret Å¿i modo Eucl. octaua <reg norm="Å¿ecundi" type="context">Å¿ecÅ©di</reg> <lb/>aptaretur) æquali quadrato <var>.a.c.</var> ita vt <lb/>cogitatis quatuor Å¿upplementis numeri <lb/>cogniti in quadrato <var>.f.g.</var> ex conÅ¿equen-<lb/>ti cognoÅ¿cetur numerus quadrati partia <lb/>lis <var>.h.i.</var> & vna etiam eius radix qua de-<lb/>tracta ex numero <var>.a.b.</var> aut <var>.f.n.</var> (quod <lb/>idem eÅ¿t) primo propoÅ¿iti, relinquetur numerus cognitus duplum <var>.x.k.n.</var> aut <var>.t.b.</var> <lb/>pars vna totius <var>.a.b.</var> ex quo uerum erit hoc meum problema.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0030-01" xlink:href="fig-0030-01a"> - <image file="0030-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0030-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="28"> - <head xml:space="preserve">THEOREMA <num value="28">XXVIII</num>.</head> - <p> - <s xml:space="preserve">SI quis & aliam rationem perficiendæ <lb/> - <anchor type="figure" xlink:label="fig-0030-02a" xlink:href="fig-0030-02"/> - huius rei quærat, hoc præſtet inuen-<lb/>to numero huius Å¿upplementi, cum in <lb/>præcedenti theoremate dictum fuerit, <lb/>qua ratione manifeÅ¿tetur duplum Å¿upple-<lb/>menti ipÅ¿ius.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0030-02" xlink:href="fig-0030-02a"> - <image file="0030-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0030-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Cogitemus in Å¿ubÅ¿cripta figura lineam <var>.<lb/>a.b.</var> tanquam primum numerum propoÅ¿i-<lb/>tum, & productum <var>.a.e.</var> Å¿upplemento <var>.a.e.</var> primæ præcedentis figuræ æquale Å¿it, <lb/>ac deinde ordine ab antiquis tradito procedatur, ad quadratum reducto dimidio <var>.<lb/>a.b.</var> videlicet <var>.b.c.</var> quod erit <var>.b.d.</var> ex quo detrahatur deinde <var>.a.e</var>. </s> - <s xml:space="preserve">quare remane- - <pb o="19" rhead="THEOREM. ARIT." n="31" file="0031" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0031"/> - bit quadratum <var>.e.d.</var> cognitum, cuius radix æqualis erit <var>.c.t.</var> qua coniuncta dimi-<lb/>dio <var>.c.a.</var> ex quinta Å¿ecundi Eucli. dabit quod propoÅ¿itum erat.</s> - </p> - </div> - <div type="math:theorem" level="3" n="29"> - <head xml:space="preserve">THEOREMA <num value="29">XXIX</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">QVid</emph> cauſæ eÅ¿t, cur Å¿ubtracto duplo producti duorum numerorum ad inui-<lb/>cem <reg norm="multiplicatorum" type="context">multiplicatorÅ©</reg> ex Å¿umma Å¿uorum quadratorum, Å¿emper quod Å¿uper <lb/>eÅ¿t duorum numerorum quadratum differentiæ Å¿it?</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia Å¿i proponerentur duo numeri .16. et .4. duplum producti eorum <lb/>eſſet .128. quò detracto ex Å¿umma Å¿uorum quadratorum, nempè ex .272. rema-<lb/>neret .144. cuius quadrati radix eſſet .12. tanquam differentia inter .4. et .16.</s> - </p> - <p> - <s xml:space="preserve">Id vtÅ¿ciamus, duo numeri propoÅ¿iti, duabus lineis Å¿ignificentur, maiore <var>.q.g.</var> <lb/>et minore <var>.g.p.</var> directè coniunctis, Å¿uper quas, totale quadratum extruatur <var>.a.p.</var> <lb/>in quo cogitetur diameter <var>.a.p.</var> et à puncto <var>.g.</var> ducatur parallela <var>.g.n.c.</var> et à pun-<lb/>cto <var>.n.</var> parallela <var>.n.s.r.</var> ex quo duo producta <reg norm="dabuntur" type="context">dabÅ©tur</reg> <var>.q.n.</var> et <var>.n.u.</var> Å¿ingula æqualia pro-<lb/>ducto <var>.q.g.</var> in <var>g.p.</var> et <var>.a.n.</var> et <var>.n.p.</var> duo quadrata dictorum numerorum propoÅ¿i-<lb/>torum, quod Å¿atis <reg norm="Å¿uperque" type="simple">Å¿uperqÌ</reg> , probatur quarta Å¿ecundi Eucli. </s> - <s xml:space="preserve">Cogitemus deinde <var>.n.<lb/>o.</var> æqualem <var>.n.p.</var> et à puncto <var>.o.</var> ducatur <var>.o.m.t.</var> parallela <var>.r.s.</var> et <var>.o.e.</var> ad <var>.n.<lb/>c</var>. </s> - <s xml:space="preserve">quare ex allatis ab Eucli. octaua Å¿ecundi, dabi-<lb/>tur quantitas <var>.m.n.</var> æqualis <var>.q.n.</var> producto <var>.q.g.</var> in <lb/> - <anchor type="figure" xlink:label="fig-0031-01a" xlink:href="fig-0031-01"/> - <var>g.p.</var> et quantitas <var>.o.c.</var> minor ipÅ¿o producto, ex <lb/>quantitate quadrati <var>.n.p.</var> ex quo quantitas <var>.m.n.e.</var> <lb/>vna cum quadrato <var>.n.p.</var> æqualis erit duplo produ-<lb/>cti <var>.q.g.</var> in <var>.g.p.</var> Å¿ed hæ duæ quantitates, Å¿unt par-<lb/>tes duorum quadratorum dictorum, & quæ Å¿uper <lb/>eÅ¿t <var>.m.e.</var> quadratum differentiæ vnius numeri pro-<lb/>poÅ¿iti ab altero, prout in Å¿ubÅ¿cripta figura licebit cui <lb/>libet conÅ¿iderare. </s> - <s xml:space="preserve">Itaque veritas hæc manifeÅ¿ta <lb/>erit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0031-01" xlink:href="fig-0031-01a"> - <image file="0031-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0031-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="30"> - <head xml:space="preserve">THEOREMA <num value="30">XXX</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">CVr</emph> ij qui ex duobus numeris propoÅ¿itis maiorem per minorem diuidunt, Å¿i <lb/>proueniens per maiorem numerum multiplicauerint, productum æquale <lb/>erit prouenienti ex diuiÅ¿ione quadrati maioris numeri per minorem?</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia Å¿i proponantur duo numeri .20. et .4. <reg norm="ipÅ¿eque" type="simple">ipÅ¿eqÌ</reg> .20. per .4. diui-<lb/>datur, dabit quinque, tum .400. quadrato .20. diuiÅ¿o per prioré .4. dabit .100. <lb/>quod proueniens, producto ex .20. in .5. primo prouenienti adæquatur.</s> - </p> - <p> - <s xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, Å¿int duo numeri, qui lineis <var>.x.u.</var> et <var>.x.s.</var> maiore <reg norm="atque" type="simple">atq;</reg> mi-<lb/>nore Å¿ignificétur, tum <var>.u.x.</var> numerus per <var>.s.x.</var> di-<lb/>uidatur, Å¿itqÌue proueniens <var>.x.n.</var> poÅ¿tmodum qua-<lb/> - <anchor type="figure" xlink:label="fig-0031-02a" xlink:href="fig-0031-02"/> - dratum <var>.u.x.</var> Å¿it <var>.x.o.</var> et productum ex <var>.n.x.</var> in <var>.u.<lb/>x.</var> Å¿it <var>.x.e.</var> quod æquale eſſe dico prouenienti ex <lb/>diuiÅ¿ione quadrati <var>.o.x.</var> per <var>.s.x.</var> quod Å¿it <var>.m</var>. </s> - <s xml:space="preserve">Patet <lb/>enim ex definitione diuiÅ¿ionis, talem futuram pro-<lb/>portionem <var>.u.x.</var> ad <var>.n.x.</var> qualis eÅ¿t <var>.s.x.</var> ad vnitatem, <lb/>& quadratum <var>.o.x.</var> ad rectangulum <var>.e.x.</var> ita Å¿e ha- - <pb o="20" rhead="I O. BAPT. BENED." n="32" file="0032" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0032"/> - biturum, Å¿icut <var>.u.x.</var> ad <var>.n.x.</var> ex prima Å¿exti aut .18. vel .19. Å¿eptimi, </s> - <s xml:space="preserve">quare ex 11. <lb/>quinti ita Å¿e habebit <var>.o.x.</var> ad <var>.e.x.</var> Å¿icut <var>.s.x.</var> ad vnitatem; </s> - <s xml:space="preserve">Å¿ed Å¿icut Å¿e habet <var>.s.x.</var> ad. <lb/>vnitatem, ita Å¿e habet pariter <var>.o.x.</var> ad <var>.m</var>. </s> - <s xml:space="preserve">vnde ex .11. prædicta ita Å¿e habebit <var>.o.<lb/>x.</var> ad <var>.m.</var> Å¿icut idipÅ¿um <var>.o.x.</var> ad <var>.e.x.</var> itaqÌue ex .9. prædicti quinti <var>.m.</var> æqualis erit <var>.o.x</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0031-02" xlink:href="fig-0031-02a"> - <image file="0031-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0031-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="31"> - <head xml:space="preserve">THEOREMA <num value="31">XXXI</num>.</head> - <p> - <s xml:space="preserve">CVR propoÅ¿ito aliquo numero in duas partes inæquales diuiÅ¿o, Å¿i rurÅ¿us per <lb/>quamlibet ipÅ¿arum diuidatur, prouenientia tantumdem coniuncta quantum <lb/>multiplicata efficiant.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿it denarius prop oÅ¿itus numerus, per binarium & octonarium <lb/>diuiÅ¿us, prouenientia erunt quinque & vnum cum quarta parte, quæ coniuncta <lb/>crunt .6. cum quarta parte lineari, quæ Å¿i mul multiplicata, pariter erunt .6. cum <lb/>quarta parte Å¿uperficiali.</s> - </p> - <p> - <s xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, totalis numerns, linea <var>.q.p.</var> Å¿ignificetur, eius duæ <lb/>partes, per <var>.k.</var> maiorem et <var>.u.</var> minorem, ipÅ¿a vnitas per .t: proueniens ex diuiÅ¿io-<lb/>ne <var>.q.p.</var> per <var>.k.</var> Å¿it <var>.q.i.</var> proueniens autem ipÅ¿ius <var>.q.p.</var> per <var>.u.</var> Å¿it <var>.q.f.</var> </s> - <s xml:space="preserve">quare ex defini-<lb/>tione diuiÅ¿ionis ita Å¿e habebit <var>.q.p.</var> ad <var>.q.i.</var> Å¿icut <var>.k.</var> ad <var>.t.</var> et <var>.q.p.</var> ad <var>.q.f.</var> Å¿icut <var>.u.</var> ad <var>.t.</var> <lb/>hoc eÅ¿t <var>.q.f.</var> ad <var>.q.p.</var> Å¿icut <var>.t.</var> ad <var>.u.</var> vnde ex æqualitate <reg norm="proportionum" type="context">proportionÅ©</reg> Å¿ic Å¿e habebit <var>.q.f.</var> <lb/>ad <var>.q.i.</var> Å¿icut <var>.k.</var> ad <var>.u.</var> et conuerÅ¿im. </s> - <s xml:space="preserve">Ad hæc in linea <var>.q.p.</var> vnitas, per lineam <var>.q.o.</var> Å¿igni-<lb/>ficetur, quo facto, dicamus, Å¿i <var>.q.p.</var> ad <var>.q.i.</var> Å¿ic Å¿e habet vt <var>.k.</var> ad <var>.q.o.</var> itaque permu-<lb/>tando, Å¿ic Å¿e habebit <var>.q.p.</var> ad <var>.k.</var> Å¿icut <var>.q.i.</var> ad <var>.q.o.</var> hoc eÅ¿t <var>.k.u.</var> ad <var>.k.</var> Å¿icut <var>.i.q.f.</var> ad <var>.<lb/>q.f.</var> (nam <var>.k.u.</var> partes Å¿unt integrales totius <var>.q.p.</var> et <var>.k.u.</var> ad <var>.k.</var> eÅ¿t Å¿icut <var>.i.q.f.</var> ad <var>.q.f.</var> <lb/>ex .18. quinti) </s> - <s xml:space="preserve">Quare ita erit <var>.i.q.f.</var> ad <var>.q.f.</var> Å¿icut <var>.q.i.</var> ad vnitatem <var>.q.o.</var> ex .11. quinti <lb/>Addatur deinde <var>.q.i.</var> ad <var>.q.f.</var> et <var>.q.i.</var> per <var>.<lb/>q.f.</var> multiplicetur, cuius multiplicatio-<lb/> - <anchor type="figure" xlink:label="fig-0032-01a" xlink:href="fig-0032-01"/> - nis productum, Å¿it <var>.x.f.</var> quod probabo <lb/>æquale eſſe Å¿ummæ <var>.f.q.</var> cum <var>.q.i</var>. </s> - <s xml:space="preserve">Sece-<lb/>tur enim linea <var>.q.x.</var> in puncto <var>.s.</var> ita. vt <var>.<lb/>q.s.</var> æqualis Å¿it <var>.q.o.</var> Å¿igneturqÌue pro-<lb/>ductum <var>.s.f.</var> </s> - <s xml:space="preserve">quare <reg norm="eadem" type="context">eadẽ</reg> erit propor-<lb/>tio quantitatis <var>.x.f.</var> ad <var>.s.f.</var> quæ eÅ¿t <var>.q.x.</var> <lb/>ad <var>.q.s.</var> ex prima Å¿exti, aut .18. vel 19. <lb/>Å¿eptimi, hoc eÅ¿t, Å¿icut <var>.q.i.</var> ad <var>.q.o.</var> et <lb/>ex .11. quinti (vt dictum eÅ¿t) Å¿icut <var>.i.q.<lb/>f.</var> ad <var>.q.f.</var> Å¿ed numerus <var>.s.f.</var> fuperficia-<lb/>lis tantus eÅ¿t, quantus linearis <var>.q.f</var>. <lb/></s> - <s xml:space="preserve">quare ex .9. quinti tantus erit (Å¿u-<lb/>perficialiter) numerus <var>.x.f.</var> quantus <lb/>(lineariter). <var>f.q.i.</var> quod erat pro-<lb/>poÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0032-01" xlink:href="fig-0032-01a"> - <image file="0032-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0032-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="32"> - <head xml:space="preserve">THEOREMA. <num value="32">XXXII</num>.</head> - <p> - <s xml:space="preserve">CVR numero aliquo in duas partes inæquales diuiÅ¿o, Å¿i rurÅ¿us diuidatur per <lb/>Å¿ingulas partes, Å¿umma duorum prouenientium per binarium, Å¿emper ma-<lb/>ior Å¿it Å¿umma prouenientium ex diuiÅ¿ione vnius partis per alteram.</s> - </p> - <p> - <s xml:space="preserve"><reg norm="Exempli" type="context">Exẽpli</reg> gratia, Å¿i proponeretur numerus .24. qui in duas partes inæquales diuide - <pb o="21" rhead="THEOREM. ARIT." n="33" file="0033" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0033"/> - retur .20. Å¿cilicet et .4. certè .24. perÅ¿ingulas partes diuiÅ¿o, daretur vnum proue-<lb/>niens Å¿ex integra, & alterum vnum & quinta pars, quorum Å¿umma eſſet Å¿eptem in-<lb/>tegra cum quinta parte, tum altera parte per alteram diuiÅ¿a, daretur vnum proue-<lb/>niens quinque integrorum & alterum vnius quinti tantum, quorum Å¿umma eſſet <lb/>quinque integra, & vna quinta pars, minor prima reliquorum duorum prouenien-<lb/>tium per binarium.</s> - </p> - <p> - <s xml:space="preserve">Cuius conÅ¿iderationis cauÅ¿a, propoÅ¿itus numerus linea <var>.q.p.</var> Å¿ignificetur, eius duÄ™ <lb/>partes lineis <var>.q.x.</var> et <var>.x.p.</var> <reg norm="tum" type="context">tÅ©</reg> <var>.q.f.</var> Å¿it proueniens ex diuiÅ¿ione totius <var>.q.p.</var> per <var>.x.p.</var> et <var>.<lb/>q.i.</var> Å¿it proueniens ex diuiÅ¿ione eiuÅ¿dem <var>.q.p.</var> per <var>.q.x.</var> adhæc <var>.h.m.</var> Å¿it proueniens, <lb/>ex diuiÅ¿ione <var>.q.x.</var> per <var>x.p.</var> et <var>.h.k.</var> proue-<lb/>niensex diuiÅ¿ione <var>.p.x.</var> per <var>.q.x.</var> patet igi-<lb/> - <anchor type="figure" xlink:label="fig-0033-01a" xlink:href="fig-0033-01"/> - tur ex .22. theoremate huiuslibri proue-<lb/>niés.h.m. minus eſſe proueniente <var>.q.f.</var> per <lb/>vnitaté, & proueniens <var>.h.k.</var> minus proue-<lb/>niente <var>.q.i.</var> per alteram vnitatem. </s> - <s xml:space="preserve">Itaque <var>.<lb/>f.q.i.</var> maior erit <var>.m.h.k.</var> per numerum binarium, quoderat propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0033-01" xlink:href="fig-0033-01a"> - <image file="0033-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0033-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="33"> - <head xml:space="preserve">THEOREMA. <num value="33">XXXIII</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">QVilibet</emph> numerus, medius eÅ¿t <lb/>proportionalis inter numerum <lb/> - <anchor type="figure" xlink:label="fig-0033-02a" xlink:href="fig-0033-02"/> - Å¿ui quadrati & vnitatem.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0033-02" xlink:href="fig-0033-02a"> - <image file="0033-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0033-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Detur enim numerus propoÅ¿itus, <lb/>qui linea <var>.a.u.</var> Å¿ignificetur, cuiusqua-<lb/>dratum Å¿it <var>.u.n.</var> vnitas linearis Å¿it <var>.i.a.</var> <lb/>et Å¿uperficialis <var>.o.</var> patebit ex .18. Å¿exti <lb/>aut 11. octaui proportionem <var>.u.n.</var> ad <var>.<lb/>o.</var> futuram duplam proportioni <var>.u.a.</var> <lb/>ad <var>.i.a.</var> Å¿ed <var>.i.a.</var> e<unsure/>t.o. eadem (Å¿pecie) <lb/>res <reg norm="sunt" type="context">sÅ©t</reg>, tanta Å¿cilicet <var>.a.i.</var> quanta <var>.o.</var> vni <lb/> - <anchor type="figure" xlink:label="fig-0033-03a" xlink:href="fig-0033-03"/> - tas eÅ¿t, Itaque proportio numeri <var>.u.n.</var> <lb/>ad <var>.u.a.</var> æqualis erit proportioni <var>.u.a.</var> <lb/>ad <var>.i.a</var>. </s> - <s xml:space="preserve">Quare numerus <var>.u.a.</var> inter nu-<lb/>merum <var>.u.n.</var> & vnitatem, medius erit <lb/>proportionalis.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0033-03" xlink:href="fig-0033-03a"> - <image file="0033-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0033-03"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="34"> - <head xml:space="preserve">THEOREMA <num value="34">XXXIIII</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">HOc</emph> ipÅ¿um quod diximus & alia ratione Å¿peculari licebit.</s> - </p> - <p> - <s xml:space="preserve">PropoÅ¿itus numerus, nunc etiam per <var>.a.u.</var> Å¿ignificetur, eius quadratum per <var>.<lb/>u.n.</var> vnitas linearis per <var>.a.i.</var> <reg norm="productumque" type="simple">productumqÌ;</reg> <var>.a.u.</var> in <var>.a.i.</var> terminetur, <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.n.i</var>. </s> - <s xml:space="preserve">quare <lb/><var>n.i.</var> conÅ¿tabit numero Ãuperficiali æquali numero lineari <var>.a.u.</var> & ex prima fexti aut .<lb/>18. vel .19. Å¿eptimi, eadem erit proportio <var>.u.n.</var> ad <var>.i.n.</var> quæ eÅ¿t <var>.a.u.</var> ad <var>.a.i.</var> Å¿ed nu-<lb/>merus <var>.a.u.</var> cum numero <var>.n.i.</var> idem Å¿pecie eÅ¿t. </s> - <s xml:space="preserve">Itaque medius eÅ¿t proportiona-<lb/>lis inter <var>.u.n.</var> & vnitatem.</s> - </p> - <pb o="22" rhead="IO. BAPT. BENED." n="34" file="0034" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0034"/> - </div> - <div type="math:theorem" level="3" n="35"> - <head xml:space="preserve">THEOREMA <num value="35">XXXV</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">QVivis</emph> numerus per alterum multiplicatus, & diuiÅ¿us, medius eÅ¿t propor-<lb/>tionalis inter productum multiplicationis, & proueniens diaiÅ¿ionis.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i .20. <reg norm="multiplicentur" type="context">multiplicẽtur</reg> per quinque & inde per quinque diuidantur <lb/>productum erit .100. proueniens .4. inter quos numeros .20. medius eÅ¿t propor-<lb/>tionalis.</s> - </p> - <p> - <s xml:space="preserve">Hoc vt Å¿peculemur, proponatur numerus multiplicandus & diuidendus, qui Å¿i-<lb/>gnificetur linea <var>.u.e.</var> multiplicans autem & diuidens linea <var>.a.u.</var> multiplicationis <lb/>productum Å¿it <var>.e.a.</var> proueniens ex diuiÅ¿ione Å¿it <var>.o.e</var>. </s> - <s xml:space="preserve">Nunc proueniens <var>.e.o.</var> per <reg norm="nu- merum" type="context">nu-<lb/>merÅ©</reg> <var>.a.u.</var> diuidentem multiplicetur, cuius multiplicationis productum Å¿it <var>.e.i.</var> <lb/>quare, eadem erit proportio numeri <var>.a.e.</var> <lb/>ad numerum <var>.e.i.</var> quæ eÅ¿t numeri <var>.u.e.</var> ad <lb/> - <anchor type="figure" xlink:label="fig-0034-01a" xlink:href="fig-0034-01"/> - numerum <var>.e.o.</var> ex prima Å¿extiaut .18. vel <lb/>19. Å¿eptimi. </s> - <s xml:space="preserve">Sed cum numerus <var>.u.e.</var> ex <ref id="ref-0010">.11. theoremate præſentis libri</ref>, numero <var>.e.<lb/>i.</var> æqualis Å¿it. </s> - <s xml:space="preserve">verum eſſe, quod propoÅ¿i-<lb/>tum fuit conÅ¿equetur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0034-01" xlink:href="fig-0034-01a"> - <image file="0034-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0034-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="36"> - <head xml:space="preserve">THEOREMA <num value="36">XXXVI</num>.</head> - <p> - <s xml:space="preserve">CVR ij, qui propoÅ¿itum numerum ita multiplicare & diuidere cupiunt, vt pro <lb/>ductum multiplicationis, tam Å¿it multiplex prouenienti ex diuiÅ¿ione, quam <lb/>quæritur, rectè Å¿umant aliquem numerum pro multiplicante & diuidente, qui Å¿it ra <lb/>dix quadrata denominantis quęſitÄ™ multiplicitatis.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, proponuntur .20. multiplicanda atque diuidenda, ita vt pro-<lb/>ductum multiplicationis nonuplum Å¿it prouenienti ex diuiÅ¿ione, nempè, vt pro-<lb/>ueniens, nona pars Å¿it eiuÅ¿modi producti, </s> - <s xml:space="preserve">quare quadratam radicem ipÅ¿orum no-<lb/>uem, ideÅ¿t denominantis Å¿umunt, tria Å¿cilicet, multiplicant igitur & diuidunt <lb/>data .20. ex quo productum erit .60. proueniens autem .6. cum duabus tertijs. </s> - <s xml:space="preserve">& <lb/>propoÅ¿itum Å¿equitur.</s> - </p> - <p> - <s xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, Å¿ignificetur numerus propoÅ¿itus linea <var>.u.e.</var> multipli-<lb/>cans autem & diuidens linea <var>.u.a.</var> productum Å¿it <var>.e.a.</var> proueniens <var>.e.o.</var> quadratum <lb/>verò <var>.a.u.</var> Å¿it <var>.x.a.</var> erit igitur proportio <var>.a.e.</var> ad <var>.e.o.</var> dupla proportioni <var>.a.e.</var> ad nume <lb/>rum <var>.u.e.</var> ex præcedenti theoremate: </s> - <s xml:space="preserve">Adhæc, cogitemus in linea <var>.u.a.</var> vnitatem <var>.<lb/>u.i.</var> <reg norm="terminenturque" type="simple">terminenturqÌ;</reg> duo producta <var>.e.i.</var> et <var>.x.i.</var> </s> - <s xml:space="preserve">quare eadem erit proportio <var>.a.e.</var> ad <var>.e.i.</var> <lb/>quæ eÅ¿t <var>.a.e.</var> ad <var>.u.e.</var> numerus enim <var>.e.i.</var> (quamuis Å¿uperficialis) idem eÅ¿t cum nume-<lb/>ro lineari <var>.u.e.</var> Å¿ed <var>.a.e.</var> ad <var>.e.i.</var> Å¿ic Å¿e habet Å¿icut <var>.a.u.</var> ad <var>.u.i.</var> ex prima Å¿exti aut .18. <lb/>vel .19. Å¿eptimi, (quod ipÅ¿um dico de <var>.a.x.</var> ad <var>.x.i.</var>) </s> - <s xml:space="preserve">quare proportio <var>.a.x.</var> ad <var>.x.i.</var> hoc <lb/>eÅ¿t <var>.x.u.</var> Ä™qualis erit <reg norm="proportioni" type="simple">ê“portioni</reg> <var>.a.e.</var> ad <var>.u.e.</var> at trigeÅ¿imotertio & trigeÅ¿imoquarto theo <lb/>remate probatum eÅ¿t proportionem numeri <var>.a.x.</var> ad vnitatem, duplam eſſe propor-<lb/>tioni eiuÅ¿dem numeri <var>.a.x.</var> ad <var>.u.x.</var> Å¿equitur <lb/>igitur cum dimidia Å¿int æqualia, tota etiam <lb/>æqualia eſſe: </s> - <s xml:space="preserve">hoc eÅ¿t proportionem numeri <var>.<lb/> - <anchor type="figure" xlink:label="fig-0034-02a" xlink:href="fig-0034-02"/> - a.e.</var> ad numerum <var>.e.o.</var> æqualem eſſe propor <lb/>tioni numeri <var>.a.x.</var> ad vnitatem. </s> - <s xml:space="preserve">Itaque rectè <lb/>Å¿umitur numerus <var>.a.u.</var> eiuÅ¿modi vt <reg norm="quadratum" type="context">quadratÅ©</reg> - <pb o="23" rhead="THEOR. ARITH." n="35" file="0035" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0035"/> - ipÅ¿ius <var>.a.x.</var> tam Å¿it multiplex ad vnitatem, quam cupimus numerum <var>.a.e.</var> numero <var>.<lb/>e.o.</var> multiplicem eſſe.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0034-02" xlink:href="fig-0034-02a"> - <image file="0034-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0034-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="37"> - <head xml:space="preserve">THEOREMA <num value="37">XXXVII</num>.</head> - <p> - <s xml:space="preserve">CVR inuenire cupientes duos numeros, quorum quadrata in Å¿ummam colle-<lb/>cta, æqualia Å¿int numero propoÅ¿ito, & ijÅ¿dem numeris multiplicatis ad-<lb/>inuicem, productum alteri numero propoÅ¿ito Å¿it æquale, rectè Å¿umant dimidium <lb/>primi numeri propoÅ¿iti, cui Å¿umma quadratorum æquari debet, <reg norm="hocque" type="simple">hocqÌ;</reg> dimidium <lb/>in Å¿eipÅ¿um multiplicent, vnà etiam alterum numerum propoÅ¿itum in Å¿eipÅ¿um <lb/>multiplicent, quod quadratum detrahunt de primo, & reÅ¿idui quadratam radicem, <lb/>dimidio primi numeri propoÅ¿iti coniungunt, ex qua Å¿umma, quadratam radicem <lb/><reg norm="eruunt" type="context">eruÅ©t</reg>, quæ duobus quæſitis numeris maior erit, cuius quadrato de primo numero <lb/>detracto, & exreliquo erutaradice quadrata, detur minor numerus, duorum <reg norm="quae- Å¿itorum" type="simple">quÄ™-<lb/>Å¿itorum</reg>.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proponerentur .34. pro primo numero cui æquari de-<lb/>beret Å¿umma duorum quadratorum, quorum radicum productum æquale eſſe de-<lb/>beret alteri numero, verbi gratia .15. iubet antiquorum regula, dimidium primi <lb/>numeri in Å¿eipÅ¿um multiplicari, cuius dimidij quadratum erit .289. è quo Å¿i detra-<lb/>has quadratum Å¿ecundi numeri, nempe .225. remanebit .64. <reg norm="atque" type="simple">atq;</reg> huius Å¿i quadra-<lb/>tam radicem Å¿umas nempe .8. quam dimidio primi numeri, nempe .17. coniun-<lb/>gas, dabitur duorum quadratorum numerorum quęſitorum maior numerus .25. hac <lb/>deinde radice è dimidio detracta, minus quadratum dabitur .9. Å¿cilicet, quorum <lb/>radices .5. et .3. eſſent ij numeri, qui quæruntur.</s> - </p> - <p> - <s xml:space="preserve">Cuius Å¿peculationis gratia, cogitemus primum numerum, cui quadratorum fum <lb/>ma æquari debet, Å¿ignificari linea <var>.a.n.</var> tum concipiamus quæſita quadrata Å¿ignifi-<lb/>cari, <reg norm="coniungique" type="simple">coniungiqÌ</reg> modo Å¿ubÅ¿cripto <var>.t.b.k.</var> Å¿ecundum porrò numerum propoÅ¿itum, <lb/>Å¿ignificari producto <var>.d.b</var>. </s> - <s xml:space="preserve">Iam nil Å¿upereÅ¿t aliud quam vt quantitates <var>.d.p.</var> et <var>.b.p.</var> <lb/>quæramus.</s> - </p> - <p> - <s xml:space="preserve">Itaque cum in linea <var>.a.n.</var> Å¿ummæ quadratorum numerus detur, quadratum di-<lb/>midij <var>.o.a.</var> Å¿it <var>.s.a.</var> quod nobis erit cognitum; </s> - <s xml:space="preserve">Å¿it etiam <var>.a.u.</var> numerus quadrati ma <lb/>ioris, et <var>.u.n.</var> minoris, et <var>.a.z.</var> productum vnius in alterum; </s> - <s xml:space="preserve">qui quidem numerus <var>.a.<lb/>z.</var> æqualis erit <lb/>quadrato nume <lb/> - <anchor type="figure" xlink:label="fig-0035-01a" xlink:href="fig-0035-01"/> - ri <var>.d.b.</var> ex .19. <lb/>theoremate hu-<lb/>ius libri. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> <lb/><var>a.z.</var> cognitum <lb/>erit, cum eius <lb/>radix <var>.d.b.</var> Å¿it <reg norm="Å¿e- cundus" type="context">Å¿e-<lb/>cÅ©dus</reg> numerus <lb/>propoÅ¿itus, quæ <lb/>minor erit <var>.a.s.</var> ex quinta Å¿ecundi, aut Å¿eptima conÅ¿equentia poÅ¿t .16. noni Eucli-<lb/>dis. </s> - <s xml:space="preserve">Iam Å¿ubtracta quantitate <var>.z.a.</var> è quadrato <var>.a.s.</var> cognoÅ¿cetur quadratum <var>.t.x.</var> <lb/>cuius radix æqualis erit <var>.o.u.</var> ex poÅ¿tremo adductis, Itaque cognoÅ¿cemus <var>.o.u.</var> qui <lb/>numerus coniunctus dimidio <var>.o.a.</var> cognito, dabit quadratum <var>.a.u.</var> cognitum, at-<lb/>queita <var>.u.n.</var> pariter cognoÅ¿cetur, & eorum radices conÅ¿equenter.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0035-01" xlink:href="fig-0035-01a"> - <image file="0035-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0035-01"/> - </figure> - </div> - <pb o="24" rhead="IO. BAPT. BENED." n="36" file="0036" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0036"/> - <p> - <s xml:space="preserve">Hoc ipÅ¿um & alia ratione perfici poteÅ¿t, nempe, iuncta Å¿umma <var>.k.b</var>: <var>b.d</var>: ec<unsure/> <var>.<lb/>b.t.</var> alteri rectangulo æquali <var>.b.d.</var> quod Å¿it <var>.b.c.</var> ex quo totum quadratum lineæ <var>.d.k.</var> <lb/>cognitum erit, <reg norm="atque" type="simple">atq;</reg> ita etiam conÅ¿equenter eius radicem <var>.d.k.</var> cognoÅ¿cemus, cuius <lb/>ope ac producti <var>.d.b.</var> cognoÅ¿cemus <var>.d.p.</var> et <var>.p.k.</var> prout ex theoremate quadrageÅ¿i-<lb/>moquinto huius libri patebit.</s> - </p> - <p> - <s xml:space="preserve">Michael Stifelius, vndecimo cap. tertij libri, problema eiuÅ¿modi proponit, <lb/>quod tamen ipÅ¿e via algebræ diÅ¿soluit.</s> - </p> - <figure position="here"> - <image file="0036-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0036-01"/> - </figure> - </div> - <div type="math:theorem" level="3" n="38"> - <head xml:space="preserve">THEOREMA <num value="38">XXXVIII</num>.</head> - <p> - <s xml:space="preserve">CVR ij, qui duos numeros inuenire volunt, quorum productum alicui nu-<lb/>mero propoÅ¿ito æquetur, & quadratorum eorundem differentia alteri nu-<lb/>mero propoÅ¿ito æqualis Å¿ir. </s> - <s xml:space="preserve">Rectè dimidium Å¿ecundi numeri propoÅ¿iti in Å¿eipÅ¿um <lb/>multiplicent, cui quidem numero differentia quadratorum æquari debet; </s> - <s xml:space="preserve">porrò <lb/>huic quadrato primi propoÅ¿iti numeri, cui æquandum eÅ¿t productum numerorum <lb/>quæſitorum, quadratum adiungant; </s> - <s xml:space="preserve">tum radicem quadratam huius Å¿ummæ co-<lb/>pulet dimidio Å¿ecundi numeri propoÅ¿iti, ei inquam, cui differentia quadratorum <lb/>æqualis eſſe debet, ex quo quadratum maius conÅ¿urgit, à quo, detracto Å¿ecundo <lb/>numero, Å¿upereÅ¿t quadratum minus.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proponeretur primo loco numerus .8. cui æquandum eÅ¿t <lb/>productum numerorum quæſitorum, tum proponeretur numerus .12. cui, detra-<lb/>cto minore à maiore, differentia quadratorum vtriuÅ¿que quæſiti numeri æqualis <lb/>eſſe debet, oportet huius vltimi numeri .12. dimidium in Å¿eipÅ¿um multiplicare, <reg norm="fient- qÌue" type="context">fiẽt-<lb/>qÌue</reg> .36. quadratum dimidij, vnde in Å¿ummam colligeremus quadratum primi <lb/>numeri .8. quod eſſet .64. quæ cum .36. efficerent .100. cuius centenarij radice, nem <lb/>pe .10. collecta in Å¿ummam cum dimidio Å¿ecundi numeri, nempe .6. daretur qua-<lb/>dratum maius, nempe .16. ex quo, detracto Å¿ecundo numero, nempe .12. rema-<lb/>neret quadratum minus .4.</s> - </p> - <p> - <s xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, maius quadratum <lb/> - <anchor type="figure" xlink:label="fig-0036-02a" xlink:href="fig-0036-02"/> - incognitum Å¿ignificetur linea <var>.q.g.</var> minus verò <lb/>pariter incognitum linea <var>.g.i.</var> </s> - <s xml:space="preserve">quare <var>.q.i.</var> eorum <lb/>differentia, tanquam data remanebit cognita, <lb/>vnà etiam <var>.b.i.</var> et <var>.q.b.</var> Å¿ua dimidia; </s> - <s xml:space="preserve">tunc cogite-<lb/>tur quadratum <var>.y.g.</var> Å¿uper <var>.b.g.</var> et <reg norm="parallelogram- mum" type="context">parallelogrã-<lb/>mum</reg> rectangu<unsure/>lum <var>.g.r.</var> deÅ¿ignatum, & ita etiam <lb/>gnomon <var>.u.g.t.</var> prout Å¿exta Å¿ecundi Euclidis pro <lb/>ponitur, ex quo quadratum <var>.b.i.</var> nempe <var>.u.t.</var> co-<lb/>gnitum erit, Å¿ed gnomon æqualis eÅ¿t rectangulo <var>.g.r.</var> ex prædicta, aut ex .8. poÅ¿t .16. - <pb o="25" rhead="THEOREM. ARIT." n="37" file="0037" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0037"/> - noni, <reg norm="hocque" type="simple">hocqÌ;</reg> rectangulum <var>.g.r.</var> quadratum eÅ¿t primi numeri propoÅ¿iti ex .19. theo-<lb/>remate huius libri, <reg norm="itaque" type="simple">itaq;</reg> cognitum erit. </s> - <s xml:space="preserve">vnà etiam gnomon <var>.u.g.t.</var> cognoÅ¿cetur, <lb/>quare totum quadratum <var>.g.y.</var> <reg norm="eiusque" type="simple">eiusqÌ;</reg> radix <var>.b.g.</var> manifęſta erit, cui coniuncta <var>.q.b.</var> <lb/>data, maius quadratum <var>.q.g.</var> cognoÅ¿cetur, ex qua <var>.b.g.</var> detracta <var>.b.i.</var> data, cogno-<lb/>Å¿cetur <var>.i.g.</var> quadratum minus conÅ¿equenter, etiam eorum radices notæ erunt.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0036-02" xlink:href="fig-0036-02a"> - <image file="0036-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0036-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="39"> - <head xml:space="preserve">THEOREMA <num value="39">XXXIX</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">ALia</emph> etiam ratione idipÅ¿um definiri poteÅ¿t, prætermiſſa antiquorum via, <lb/>nempe multiplicatis in Å¿emetipÅ¿is primo & Å¿ecundo, numeris propoÅ¿itis, qua-<lb/><reg norm="druplicatoque" type="simple">druplicatoqÌ;</reg> quadrato primi, qua Å¿umma coniuncta cum quadrato Å¿ecundi nume-<lb/>ri, & ex hac altera Å¿umma eruta radice quadrata, ex qua detracto Å¿ecundo nume-<lb/>ro, & è reliquo Å¿umpto dimidio, quod erit <reg norm="quadratum" type="context">quadratÅ©</reg> minus, quo detracto ex radi-<lb/>ce poÅ¿tremo iuncta, Å¿upererit quadrarum maius.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proponeretur numerus .8. cui productum duorum numerorum <lb/>quæſitorum æquandum eÅ¿t, proponeretur idem .12. cui differentia quadratorum <lb/>duorum numerorum æqualis eſſe debet. </s> - <s xml:space="preserve">Iubeo primum numerum, nempe .8. in Å¿e <lb/>ipÅ¿um multiplicari, ex quo exurget .64. pro numero Å¿ui quadrati, quod quadru-<lb/>plicari volo, <reg norm="eritque" type="simple">eritqÌ;</reg> productum .256. quod cenÅ¿eo <reg norm="coniungendum" type="context">coniÅ©gendum</reg> cum quadrato Å¿e-<lb/>cundi numeri propoÅ¿iti, nempe .144. <reg norm="eritque" type="simple">eritqÌ;</reg> Å¿umma .400. ex quaÅ¿umetur radix, Å¿ci <lb/>licet .20. & ex hac detrahetur Å¿ecundus numerus .12. <reg norm="reÅ¿iduique" type="simple">reÅ¿iduiqÌ;</reg> dimidium, nempe <num value="4">.<lb/>4.</num> pro quadrato minore, quo in Å¿ummam collecto cum, 12. dabit quadratum <lb/>maius .16.</s> - </p> - <p> - <s xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, quadratum maius per lineam <var>.q.g.</var> minus per <var>.g.p.</var> Å¿i-<lb/>gnificetur: </s> - <s xml:space="preserve">Å¿uper integram autem <var>.q.p.</var> erigatur quadratum integrum <var>.d.p.</var> diuiÅ¿um, <lb/>vt quadratum <var>.f.g.</var> vigeÅ¿imiÅ¿eptimi theorematis huius libri, (idipÅ¿um accideret di-<lb/>uiÅ¿o quadrato modo octauæ Å¿ecundi Euclidis) quæ quidem diuiÅ¿io, eÅ¿t via quatuor <lb/>productorum <var>.q.g.</var> in <var>.g.p.</var> è quibus vnum Å¿it <var>.g.r.</var> quod erit cognitum ex .19. theore <lb/>mate cum Å¿it <reg norm="quadratum" type="context">quadratÅ©</reg> primi numeri ppoÅ¿iti, ex quo illa quatuor cognita <reg norm="erunt" type="context">erÅ©t</reg>. </s> - <s xml:space="preserve">Iam <lb/>verò Å¿i cogitemus <var>.q.p.</var> Å¿ectam in puncto <var>.t.</var> ita vt <var>.q.t.</var> æqualis Å¿it <var>.p.g.</var> dabitur differen <lb/>tia <var>.t.g.</var> cognita, vt radix quadrati <var>.e.o.</var> cum ex præſup-<lb/>poÅ¿ito <var>.r.n.</var> æqualis Å¿it <var>.q.g.</var> et <var>.r.e</var>: <var>g.p.</var> ex quo etiam <var>.q.t.</var> <lb/> - <anchor type="figure" xlink:label="fig-0037-01a" xlink:href="fig-0037-01"/> - ita pariter <var>.e.n.t.g.</var> æqualis erit. </s> - <s xml:space="preserve">Collecto <reg norm="itaque" type="simple">itaq;</reg> quadra <lb/>to <var>.e.o.</var> ipÅ¿ius <var>.t.g.</var> cum quadruplo <var>.g.r</var>: cognitum erit <lb/>quadratum <var>.d.p.</var> ipÅ¿ius <var>.q.p.</var> </s> - <s xml:space="preserve">quare cognoÅ¿cetur <var>.q.p.</var> de <lb/>quo numero detracta differétia quadratorum cognita <var>.<lb/>t.g.</var> Å¿upererit aggregatum <var>.p.g.</var> et <var>.q.t.</var> cognitum. </s> - <s xml:space="preserve">Qua-<lb/>re ex conÅ¿equenti, dimidium aggregati, nempe <var>.g.p.</var> <lb/>cognoÅ¿cetur, tanquam minus duorum quadratorum. <lb/></s> - <s xml:space="preserve">cui iuncta <var>.g.t.</var> aut detracta <var>.p.g.</var> ex <var>.p.q.</var> quadratum <var>.q.<lb/>g.</var> maius cognitum remanebit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0037-01" xlink:href="fig-0037-01a"> - <image file="0037-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0037-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="40"> - <head xml:space="preserve">THEOREMA <num value="40">XL</num>.</head> - <p> - <s xml:space="preserve">CVR ijs, qui volunt duos eiuÅ¿modi numeros inuenire, vt eorum maior mi-<lb/>norem, numero propoÅ¿ito Å¿uperet, & productum vnius in alterum, alteri nu-<lb/>mero propoÅ¿ito adÄ™quetur, conÅ¿ultiÅ¿simum Å¿it dimidium primi numeri propoÅ¿iti, - <pb o="26" rhead="IO. BAPT. BENED." n="38" file="0038" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0038"/> - numerum inquam, cui differentia duorum quæſitorum æquanda eÅ¿t, in Å¿eipÅ¿um <lb/>multiplicare, atque huic quadrato, Å¿ecundum numerum propoÅ¿itum iungere, cui, <lb/>productum numerorum quæſitorum æquale eſſe debet, & ex hac Å¿umma eruere qua <lb/>dratam radicem, quæ coniuncta dimidio primi numeri propoÅ¿iti, dabit maiorem <lb/>duorum numerorum & ex eadem radice detracto dimidio primi numeri, minorem <lb/>numerum duorum quæſitorum.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proponeretur .12. cui differentia vnius numeri ab altero æqua-<lb/>ri deberet, tum proponeretur .64. cui productum multiplicationis duorum quæſi-<lb/>torum Å¿imul <reg norm="æquandum" type="context">æquãdum</reg> eſſet. </s> - <s xml:space="preserve">Dimidium primi numeri in Å¿eipÅ¿um multiplicaremus, <lb/><reg norm="proueniretque" type="simple">proueniretqÌ;</reg> <reg norm="quadratum" type="context">quadratÅ©</reg> .36. cui coniuncto Å¿ecundo, nempe .64. totum eſſet .100. <lb/>ex quo detracta quadrata radice .10. etipÅ¿i coniuncto Å¿enario, dimidio primi nume <lb/>ri, & ex eadem detracto eodem dimidio .6. pro maiore numero proueniret .16. & <lb/>pro minore .4.</s> - </p> - <p> - <s xml:space="preserve">Cuius rei Å¿peculatio hæc eÅ¿t. </s> - <s xml:space="preserve">Sit <var>.e.o.</var> differentia cognita duorum incognitorum <lb/>numerorum <var>.a.o.</var> et <var>.a.e.</var> quorum productum datum Å¿iue cognitum Å¿it <var>.a.s</var>: conÅ¿ide-<lb/>remus nunc <var>.e.i.</var> dimidium <var>.e.o.</var> datæ differentiæ, & ex compoÅ¿ito <var>.a.i.</var> imaginetur <lb/>quadratum <var>.a.x.</var> in quo protracta Å¿it <var>.t.u.</var> æquidiÅ¿tans lateri <var>.a.i.</var> & tam ab ipÅ¿a <var>.a.i.</var> re <lb/>mota, quam <var>.x.i.</var> ab <var>.s.e.</var> vnde <var>.t.e.</var> quadratum erit <var>.e.i.</var> <lb/>dimidiæ Å¿cilicet differentiæ datæ <var>.e.o.</var> et <var>.t.n.</var> rectan-<lb/> - <anchor type="figure" xlink:label="fig-0038-01a" xlink:href="fig-0038-01"/> - gulum æquale erit rectangulo <var>.n.c.</var> vt cuilibet licet <lb/>per Å¿e conÅ¿iderare, vnde Å¿equitur gnomonem <var>.e.r.t.</var> <lb/>æqualem eſſe producto <var>.a.s.</var> ideo cognitus, qui <reg norm="quidem" type="context">quidẽ</reg> <lb/>gnomon, Å¿i coniunctus fuerit quadrato <var>.e.t.</var> cognito <lb/>ex radice <var>.e.i.</var> cognita (vt dimidia toralis differentiÄ™ <var>.<lb/>e.o.</var> datæ) habebimus quadratum totale <var>.a.x.</var> cogni-<lb/>tum, & ita eius radicem <var>.a.i.</var> cognitam & reliqua om <lb/>nia conÅ¿equenter quæ quidem Å¿peculatio eadem eÅ¿t <lb/>quæ .6. Å¿ecundi Å¿eu .8. noni Euclidis.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0038-01" xlink:href="fig-0038-01a"> - <image file="0038-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0038-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Poteris tamen ex modo & rationibus præceden-<lb/>ti theoremate allatis, hocipÅ¿um concludere.</s> - </p> - </div> - <div type="math:theorem" level="3" n="41"> - <head xml:space="preserve">THEOREMA <num value="41">XLI</num>.</head> - <p> - <s xml:space="preserve">CVR ij, qui aliquo propoÅ¿ito numero, inuenturi Å¿unt duos numeros inter Å¿e <lb/>differentes, quorum quadratorum Å¿umma altero numero propoÅ¿ito æqualis <lb/>Å¿it, rectè primum numerum propoÅ¿itum in Å¿eipÅ¿um multiplicant, quod quadratum <lb/>exÅ¿ecundo numero <reg norm="detrahunt" type="context">detrahÅ©t</reg>, & dimidium reÅ¿idui Å¿umunt, quod productum erit <lb/>multiplicationis duorum numerorum interÅ¿e, in reliquis præcedentis theorematis <lb/>ordinem Å¿equuntur.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proponeretur .12. tanquam numerus, cui differentia duorum <lb/>numerorum quæſitorum æquanda eÅ¿t, proponerentur præterea .272. quibus Å¿um-<lb/>ma quadratorum duorum numerorum quæſitorum æquari deberet, oporteret Å¿anè <lb/>primum numerum, nempe .12. in Å¿eipÅ¿um multiplicare, cuius <reg norm="quadratum" type="context">quadratÅ©</reg> hoc loco <lb/>eſſet .144. atque hoc detrahere ex Å¿ecundo numero, Å¿upereſſet .128. Å¿umpto <lb/>deinde dimidio huiuÅ¿ce numeri, népe .64. producto in quam duorum numerorum <lb/><reg norm="quæſitorum" type="context">quæſitorÅ©</reg>. </s> - <s xml:space="preserve">Cum hoc .64. proÅ¿tea et duodenario primo propoÅ¿ito numero, præceden <lb/>tis theorematis ordinem Å¿equeremur.</s> - </p> - <pb o="27" rhead="THEOREM. ARIT." n="39" file="0039" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0039"/> - <p> - <s xml:space="preserve">Quod vt Å¿peculemus, conÅ¿ideremus Å¿ubÅ¿criptam figuram, vigefiminoni theore-<lb/>matis figuræ Å¿imilem, in qua numeri quæſiti duabus <lb/>lineis directè coniunctis <var>.q.g.</var> et <var>.g.p.</var> fignificentur, ho <lb/> - <anchor type="figure" xlink:label="fig-0039-01a" xlink:href="fig-0039-01"/> - <reg norm="rum" type="context">rÅ©</reg> quadrata <reg norm="erunt" type="context">erÅ©t</reg> <var>.r.c.</var> et <var>.g.s.</var> <reg norm="quorum" type="context">quorÅ©</reg> <reg norm="summa" type="context">sÅ©ma</reg> <reg norm="iterum" type="context">iterÅ©</reg> propo <lb/>nitur, quare etiam cognita. </s> - <s xml:space="preserve"><reg norm="Differentia" type="context">Differẽtia</reg> autem <reg norm="duorum" type="context">duorÅ©</reg> <lb/>numerorum primo propofita fit <var>.q.i.</var> eius verò qua-<lb/>dratum <var>.m.e.</var> quod cognitum eÅ¿t ex Å¿ua radice <var>.q.i</var>. <lb/></s> - <s xml:space="preserve">quare gnomon <var>.e.n.m.</var> Å¿imul cum quadrato minori <var>.<lb/>g.s.</var> cognitus erit, quæ Å¿umma æqualis eÅ¿t duplo <var>.g.r.</var> <lb/>producto datorum numerorum. </s> - <s xml:space="preserve">Itaque & ipÅ¿a <var>.g.<lb/>r.</var> cognoÅ¿cetur, nunc Å¿i præcedentis theorematis Å¿pe-<lb/>culationem in reliquis conÅ¿uluerimus propoÅ¿itum <lb/>conÅ¿equemur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0039-01" xlink:href="fig-0039-01a"> - <image file="0039-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0039-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="42"> - <head xml:space="preserve">THEOREMA <num value="42">XLII</num>.</head> - <p> - <s xml:space="preserve">ADhuc etiam & alia ratione idipÅ¿um conÅ¿equi poſſemus, non conÅ¿ulto qua-<lb/>drageÅ¿imo theoremate. </s> - <s xml:space="preserve">Nam Å¿ubtracto quadrato differentiæ, numeri primi <lb/>(<reg norm="inquam" type="context">inquã</reg>) propoÅ¿iti, ex <reg norm="summa" type="context">sÅ©ma</reg> duorum quadratorum, nempe ex Å¿ecundo numero pro-<lb/>poÅ¿ito colligendum eſſet reÅ¿iduum in Å¿ummam cum prædicto Å¿ecundo numero, & <lb/>ex Å¿umma hac deÅ¿umenda quadrata radix, quæ duorum numerorum Å¿umma erit, <lb/>de qua detracto primo numero, remanebit duplum minoris numeri quæſiti, cuius <lb/>dimidio addito primo numero propoÅ¿ito, aut detracto minore inuento ex radice <lb/>poÅ¿tremo inuenta, dabitur numerus maior, qui quæritur.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, cum Å¿uperfuerint .128. hæc Å¿i cum Å¿ecundo numero <reg norm="nempe" type="context">nẽpe</reg> .272. <lb/>iunxerimus, dabunt .400. quorum radix erit .20. de quo numero detracto primo <lb/>propoÅ¿ito, nempe .12. Å¿upererunt .8. quorum <reg norm="dimidium" type="context">dimidiÅ©</reg> erit .4. quo ex .20. detracto <lb/>aut coniuncto .12. maior numerus orietur.</s> - </p> - <p> - <s xml:space="preserve">Cuius rei contemplatio, præcedenti figura aperitur. </s> - <s xml:space="preserve">Nam reÅ¿iduum detractionis <lb/>quadrati <var>.m.e.</var> ex Å¿umma <reg norm="duorum" type="context">duorÅ©</reg> quadratorum <var>.r.c.</var> et <var>.g.s.</var> numerum præbet æqua-<lb/>lem duobus Å¿upplementis <var>.q.n.</var> et <var>.n.u.</var> ex .8. Å¿ecundi Euclidis. qui coniunctus duo-<lb/>bus quadratis (quorum Å¿umma Å¿ecundo propoÅ¿ita fuit) cognitionem profert qua-<lb/>drati <var>.q.u.</var> & eius radicis <var>.q.p.</var> de qua, detracto primo dato numero, Å¿cilicet <var>.q.i.</var> Å¿u-<lb/>pereÅ¿t <var>.i.p.</var> cuius dimidium nempe <var>.g.p.</var> minor eÅ¿t numerus qui quÄ™ritur; </s> - <s xml:space="preserve">reÅ¿iduum <lb/>verò totius <var>.g.q.</var> maior Å¿cilicet.</s> - </p> - </div> - <div type="math:theorem" level="3" n="43"> - <head xml:space="preserve">THEOREMA <num value="43">XLIII</num>.</head> - <p> - <s xml:space="preserve">CVR ij, qui volunt duos numeros inuenire, quorum Å¿umma æqualis propo-<lb/>fito alicui numero futura Å¿it, & Å¿umma quadratorum maior eorum produ-<lb/>cto per quantitatem alterius propoÅ¿iti numeri, rectè dimidium primi dati numeri in <lb/>Å¿eipÅ¿um multiplicant, quod quadratum ex <reg norm="Å¿ecundo" type="context">Å¿ecÅ©do</reg> dato numero detrahunt, Å¿umuntÂ<lb/>qÌue tertiÄ™ partis refidui quadratam radicem, quam dimidio primi numeri coniun-<lb/>gunt, ex quo maior numerus <reg norm="duorum" type="context">duorÅ©</reg> <reg norm="quæſitorum" type="context">quæſitorÅ©</reg> datur, quo ex toto primo detracto, Å¿u-<lb/>pererit minor.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, propoÅ¿ito numero .20. cui æquanda eÅ¿t Å¿umma duorum nume-<lb/>rorum quæſitorum, <reg norm="datoque" type="simple">datoqÌ;</reg> Å¿ecundo numero .208. qui Å¿emper maior eſſe debet - <pb o="28" rhead="IO. BAPT. BENED." n="40" file="0040" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0040"/> - quadrato dimidij, prout ex Å¿peculatione huiuÅ¿modi operis cognoÅ¿cetur, <reg norm="cuiæquanda" type="context">cuiæquãda</reg> <lb/>eÅ¿t <reg norm="differentia" type="context">differẽtia</reg> inter <reg norm="Å¿ummam" type="context">Å¿ummã</reg> <reg norm="quadratorum" type="context">quadratorÅ©</reg> <reg norm="duorum" type="context">duorÅ©</reg> qui <reg norm="quæruntur" type="context">quærÅ©tur</reg> <reg norm="numerorum" type="context">numerorÅ©</reg>, Å¿imul <reg norm="cum" type="context">cÅ©</reg> pro <lb/>ducto <reg norm="eorum" type="context">eorÅ©</reg> radicum. </s> - <s xml:space="preserve">Dimidium numeri .20. in Å¿eipÅ¿um multiplicandum eſſet, qua-<lb/><reg norm="dratumque" type="simple">dratumqÌ;</reg> detrahendum ex .208. vtremanerent .108. quorum .108. tertiæ partis qua <lb/>drata radix eſſet .6. quæ Å¿i iuncta fuerit dimidio .20. nempe .10. daretur maior nu-<lb/>merus quæſitus .16. quo detracto è .20. darentur .4.</s> - </p> - <p> - <s xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, datus primus numerus Å¿ignificetur linea <var>.g.h.</var> in qua <lb/>maior numerus incognitus Å¿it <var>.g.h.</var> minor verò <var>.b.h.</var> quorum quadrata Å¿int <var>.y.t.</var> et <var>.<lb/>b.l.</var> in quadrato maximo <var>.g.p.</var> tum productum <var>.g.b.</var> in <var>.b.h.</var> Å¿it <var>.g.c.</var> <reg norm="cogitenturque" type="simple">cogitenturqÌ;</reg> duo <lb/>diametri <var>.q.h.</var> et <var>.g.p.</var> diuiÅ¿i per medium in puncto <var>.o.</var> per quod duÄ™ lineæ ducan-<lb/>tur <var>.f.d.</var> et <var>.k.m.</var> parallelæ lateribus maximi quadrati. </s> - <s xml:space="preserve">Hæ dictum quadratum in <lb/>quatuor quadrata æqualia diuident, quorum <reg norm="vnumquodque" type="simple">vnumquodqÌ;</reg>, æquale erit quadrato <var>.<lb/>g.f.</var> dimidij ipÅ¿ius <var>.g.h.</var> datÄ™, </s> - <s xml:space="preserve">quare eorum <reg norm="vnumquodque" type="simple">vnumquodqÌ;</reg> cognitum erit. </s> - <s xml:space="preserve">Iterum co <lb/>gitemus <var>.s.x.</var> per <var>.e.</var> <reg norm="parallelam" type="context">parallelã</reg> <var>.g.k.</var> tantum diÅ¿tan-<lb/>tem à <var>.g.k.</var> quantum <var>.y.l.</var> ab <var>.g.h.</var> diÅ¿tare inueni-<lb/> - <anchor type="figure" xlink:label="fig-0040-01a" xlink:href="fig-0040-01"/> - tur. </s> - <s xml:space="preserve">Cogitetur pariter <var>.z.i.a.</var> per punctum <var>.i.</var> <lb/>parallela <var>.d.p.</var> </s> - <s xml:space="preserve">quare <var>.a.t.</var> æqualis erit <var>.f.c.</var> et <var>.y.x.</var> <lb/>æqualis <var>.f.e.</var> et <var>.y.s</var>: <var>b.l.</var> æqualis. </s> - <s xml:space="preserve">Ita Å¿ubtractis è <lb/>duobus quadratis Å¿uperius dictis <var>.a.t.y.x.</var> et <var>.b.l.</var> <lb/>producto <var>.y.b.</var> æqualibus, Å¿upererunt <var>.k.d.</var> et <var>.a.c.<lb/>x.</var> cognita, tanquam æqualia dato Å¿ecundo nu-<lb/>mero, Å¿ed <var>.k.d.</var> quadratum eÅ¿t medietatis <var>.g.f.</var> <lb/>cognitæ, cognoÅ¿cetur igitur reÅ¿iduum <var>.a.c.x.</var> vnà <lb/>etiam Å¿ingulæ tertiæ partes nempe quadrata <var>.o.<lb/>i.o.c.</var> et <var>.o.e.</var> & radix <var>.b.f.</var> vel <var>.f.s.</var> Å¿ingularum, <lb/>qua coniuncta dimidio <var>.g.f.</var> <reg norm="rurfusque" type="simple">rurfusqÌ;</reg> ab <reg norm="eodem" type="context">eodẽ</reg> de-<lb/>tracta, propoÅ¿itum conÅ¿equemur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0040-01" xlink:href="fig-0040-01a"> - <image file="0040-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0040-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="44"> - <head xml:space="preserve">THEOREMA <num value="44">XLIIII</num>.</head> - <p> - <s xml:space="preserve">CVR Å¿i quis cupiat numerum propoÅ¿itum in duas eiuÅ¿modi partes diuidere, vt <lb/>quadratum maioris, quadratum minoris Å¿uperet quantitate alterius numeri <lb/>propoÅ¿iti, rectè primum numerum in Å¿eipÅ¿um multiplicabit, & ab eodem Å¿ecun-<lb/>dum numerum detrahet, reÅ¿iduum verò per duplum primi diuidet, ex quo proue-<lb/>niens primi pars minor erit, quæ ex illo primo detracta, partem maiorem <lb/>proferet.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proponantur .20. diuiÅ¿a in duas eiuÅ¿modi partes, vt <reg norm="quadratum" type="context">quadratÅ©</reg> <lb/>maioris Å¿uperet quadratum minoris numero æquali ipÅ¿i .240. oportebit primum <lb/>numerum, qui quadratus cum fuerit, erit .400. in Å¿eipÅ¿um multiplicare, & ex hoc <lb/>quadrato Å¿ecundum numerum nempe .240. detrahere, </s> - <s xml:space="preserve">tunc remanebunt .160. quÄ™ <lb/>diuiÅ¿a per .40. <reg norm="numerum" type="context">numerÅ©</reg> <reg norm="duplum" type="context">duplÅ©</reg> primo, dabuntur quatuor pro minori numero, à reÅ¿i-<lb/>duo verò .20. detractis quatuor, erunt .16. pro maiorinumero.</s> - </p> - <p> - <s xml:space="preserve">Quod vt exactè conÅ¿ideremus, primus numerus propoÅ¿itus Å¿ignificetur linea <var>.q.<lb/>h.</var> diuidendus in duas partes <var>.q.p.</var> et <var>.p.h.</var> tales quales quærimus. </s> - <s xml:space="preserve">PoÅ¿tmodum eriga <lb/><gap extent="2"/>r quadratum <var>.q.e.</var> diuiÅ¿um diametro <var>.f.h.</var> <reg norm="ductisque" type="simple">ductisqÌ;</reg> <var>.p.o.t.</var> et <var>.a.o.c.</var> parallelis lateri-<lb/>bus quadrati, dabuntur imaginaria quadrata <var>.c.t.</var> et <var>.p.a.</var> duarum partium <var>.q.p.</var> et <var>.p.<lb/>h.</var> incognitarum. </s> - <s xml:space="preserve">Ad hæc cogitemus quadratum <var>.u.n.</var> æquale quadrato <var>.p.a.</var> è quadra - <pb o="29" rhead="THEOR. ARITH." n="41" file="0041" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0041"/> - to maiore <var>.c.t.</var> extractum quare reÅ¿iduum qua-<lb/> - <anchor type="figure" xlink:label="fig-0041-01a" xlink:href="fig-0041-01"/> - drati <var>.c.p.</var> cognitum erit, quam quantitatem co-<lb/>gnitam, cum Å¿it Å¿ecundo loco data, cogitemus <lb/>detrahi è toto quadrato cognito <var>.q.e.</var> ex quo <lb/>Å¿umma duorum Å¿upplementorum <var>.q.o.</var> et <var>.o.e.</var> <lb/>cognoÅ¿cetur, vnà cum quadratis <var>.u.n.</var> et <var>.p.a.</var> du <lb/>plo Å¿cilicet <var>.q.a.</var> quo diuiÅ¿o per duplum <var>.q.h.</var> aut <lb/>Å¿implex <var>.q.a.</var> per <var>.q.h.</var> Å¿implicem, dabitur <var>.a.h.</var> <lb/>nempe <var>.p.h.</var> minor numerus quæſitus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0041-01" xlink:href="fig-0041-01a"> - <image file="0041-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0041-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="45"> - <head xml:space="preserve">THEOREMA <num value="45">XLV</num>.</head> - <p> - <s xml:space="preserve">CVR volentes diuidere numerum propoÅ¿itum in duas eiuÅ¿modi partes, vt pro <lb/>ductum vnius in alteram, alteri numero propoÅ¿ito æquetur, rectè dimidium <lb/>primi dati numeri in Å¿eipÅ¿um multiplicant, ex quo quadrato Å¿ecundum datum nu-<lb/>merum detrahunt, <reg norm="reÅ¿iduique" type="simple">reÅ¿iduiqÌ;</reg> radicem Å¿umunt, qua coniuncta vni dimidio primi nu-<lb/>meri, pars maior datur, ex altero verò dimidio detracta, minorem manifeÅ¿tabit.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i numerus partiendus eſſet .34. alter verò numerus eſſet .64. cui <lb/>productum vnius partis in alteram æquale eſſe deberet. </s> - <s xml:space="preserve">Dimidium primi numeri, in <lb/>Å¿eipÅ¿um multiplicaremus, cuius quadratum eſſet .289. de quo detracto Å¿ecundo nu-<lb/>mero nempe .64. remaneret .225. cuius quadrata radix nempe .15. coniuncta .17. <lb/>dimidio .34. proferet .32. maiorem partem, <reg norm="detractoque" type="simple">detractoqÌ;</reg> ex .17. Å¿upereſſet .2. pars <lb/>inquam minor.</s> - </p> - <p> - <s xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, primus numerus propoÅ¿itus Å¿ignificetur linea <var>.a.d.</var> cu-<lb/>ius dimidium <var>.c.d.</var> cognitum erit, vnà etiam eius quadratum <var>.c.f.</var> quo diuiÅ¿o per dia <lb/>metrum <var>.e.d.</var> Å¿upponantur partes ignotæ <lb/> - <anchor type="figure" xlink:label="fig-0041-02a" xlink:href="fig-0041-02"/> - ipÅ¿ius <var>.a.d.</var> eſſe <var>.a.b.</var> et <var>.b.d.</var> & à puncto <var>.b.</var> <lb/>duci lineam <var>.b.h.g.</var> parallelam <var>.d.f.</var> et <var>.m.<lb/>h.k.</var> parallelam <var>.d.a.</var> extructa figura Å¿imi <lb/>li figuræ quintæ Å¿ecundi Eucli. </s> - <s xml:space="preserve">quare da <lb/>bitur <reg norm="gnomon" type="context">gnomõ</reg> <var>.l.d.g.</var> æqualis producto <var>.b.<lb/>k.</var> & proinde cognitus, quo detracto è <lb/>quadrato, <var>c.f.</var> remanebit quadratum <var>.g.l.</var> <lb/>cuius radice æquali <var>.c.b.</var> coniuncta <var>.a.c.</var> <lb/>& detracta ex <var>.c.d.</var> partes <var>.a.b.</var> et <var>.b.d.</var> quæſitæ dabuntur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0041-02" xlink:href="fig-0041-02a"> - <image file="0041-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0041-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="46"> - <head xml:space="preserve">THEOREMA <num value="46">XLVI</num>.</head> - <p> - <s xml:space="preserve">CVR propoÅ¿itis tribus numeris, quorum prior in duas eiuÅ¿modi partes diui-<lb/>dendus Å¿it, ut mutuò diuiſæ, & per Å¿ummam prouenientium diuiÅ¿o Å¿ecundo <lb/>numero, proueniens vltimum Å¿it æquale tertio numerorum propoÅ¿itorum. </s> - <s xml:space="preserve">ConÅ¿ul <lb/>tiÅ¿simum Å¿it Å¿ecundum numerum per tertium diuidere, ex quo proueniens Å¿it Å¿um-<lb/>ma prouenientium è duabus partibus mutuò diuiÅ¿is, quam Å¿ummam Å¿i quis velit di-<lb/>Å¿tinguere, rectè poſſit medio operationis <reg norm="prÄ™cedentis" type="context">prÄ™cedẽtis</reg> theorematis <reg norm="sumpta" type="context">sÅ©pta</reg> vnitate Å¿uper <lb/>ficiali pro Å¿ecundo numero diÅ¿tinctis poÅ¿tmodum prouenientibus, rectè meo iudi-<lb/>cio operabimur per <reg norm="regulam" type="context">regulã</reg> de tribus (quod fuit ab antiquis prætermiſſum) Si dixe- - <pb o="30" rhead="IO. BAPT. BENED." n="42" file="0042" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0042"/> - rimus, Å¿i Å¿umma vnius dictorum prouenientium cum vnitate dat primum numerum, <lb/>quid ipÅ¿a eadem vnitas dabit? </s> - <s xml:space="preserve">ex quo propoÅ¿itum oriatur.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, proponuntur tres numeri, primus .20. Å¿ecundus .34. tertius .8. <lb/>Iam quærimus diuidere primum .20. in duas partes quæ mutuò diuiſæ prÄ™beant duo <lb/>prouenientia, quorum Å¿umma tanta Å¿it vt per eam diuiÅ¿o .34. proueniat numerus <lb/>æqualis tertio numero .8. </s> - <s xml:space="preserve">Quod vt præſtemus iubet regula Å¿ecundum .34. per <reg norm="tertium" type="context">tertiÅ©</reg> <num value="8">.<lb/>8.</num> diuidi, vnde proueniet .4. cum vna quarta parte, quod proueniens erit Å¿umma pro <lb/>uenientium ex diuiÅ¿ione duarum partium quæſitarum, quæ Å¿i diÅ¿tinguere volueri-<lb/>mus, præcedentis theorematis methodum Å¿equemur, vnitate Å¿uperficiali pro Å¿ecun <lb/>do numero propoÅ¿ito Å¿umpta, ac Å¿i diceremus, diuidatur .4. cum vna quarta parte <lb/>in duas eiuÅ¿modi partes, vt productum vnius in alteram Å¿it vnitas Å¿uperficialis, cer-<lb/>tè fractis integris cum quarta parte coniungendis, darentur vnitatis decemÅ¿eptem <lb/>quartæ lineares, verum cum neceſſe Å¿it, ex præcedenti theoremate, dimidium in <lb/>Å¿eipÅ¿um multiplicare, <reg norm="eſſetque" type="simple">eſſetqÌ;</reg> dimidium .8. quartarum partium cum octaua, com-<lb/>modius totum conÅ¿tituetur .34. octauarum, quarum dimidium, nempe decemÅ¿ep-<lb/>tem octauæ, in Å¿eipÅ¿um multiplicatum erunt .289. Å¿exageÅ¿imæ quartæ vnius integri <lb/>Å¿uperficialis, quandoquidem <reg norm="integrum" type="context">integrÅ©</reg> Å¿uperficiale, cuius vnitas linearis in .8. partes <lb/>diuiditur eÅ¿t .64. vt ex primo theoremate huius libri depræhendi poteÅ¿t. </s> - <s xml:space="preserve">Nunc vni-<lb/>tate hac Å¿uperficiali, nempe .64. ex .289. detracta, Å¿upererit .225. cuius radix qua-<lb/>drata, Å¿cilicet .15. coniuncta dimidio dictorum prouenientium, nempe .17. dabit <lb/>maius proueniens .32. <reg norm="detractaque" type="simple">detractaqÌ;</reg> ex altero dimidio, dabit proueniens minus .2. hoc <lb/>eÅ¿t pro maiore proueniente .32. octauas, & pro minore duas, quatuor Å¿cilicet inte-<lb/>gros pro maiore, & quartam partem vnius integri pro minore. </s> - <s xml:space="preserve">Nunc Å¿i ex regula <lb/>de tribus dixerimus, Å¿i .4. iuncta vni, nempe .5. dant .20. primum numerum, quid <lb/>dabunt .4. integra (proueniens inquam maius) <reg norm="dabunt" type="context">dabÅ©t</reg> certè .16. partem maiorem. <lb/></s> - <s xml:space="preserve">Tum Å¿i dixerimus, Å¿i quarta pars coniuncta vnitati dat .20: </s> - <s xml:space="preserve">quid dabit quarta illa <lb/>pars (hoc eÅ¿t proueniens minus) dabit <reg norm="profectò" type="simple">ê“fectò</reg> quatuor Å¿cilicet <reg norm="minorem" type="context">minorẽ</reg> partem, quod <lb/>ab antiquis certè ignoratum fuit, qui, inuentis prouenientibus quieuerunt, ne-<lb/>Å¿cientes ijs vti ad inueniendas duas primi numeri partes.</s> - </p> - <p> - <s xml:space="preserve">Cuius Å¿peculationis gratia, demus primum numerum Å¿ignificari linea <var>.e.u.</var> cuius <lb/>partes <var>.e.a.</var> & <var>a.u.</var> Å¿int quæ quæruntur, alter verò numerus Å¿ignificetur linea <var>.b.<lb/>d.</var> tertius linea <var>.g.f.</var> proueniens <reg norm="autem" type="wordlist">aÅ©t</reg> diuiÅ¿ionis <var>.e.a.</var> per <var>.a.u.</var> Å¿it <var>.n.t.</var> diuiÅ¿ionis <reg norm="autem" type="wordlist">aÅ©t</reg> <var>.a.u.</var> <lb/>per <var>.a.e.</var> Å¿it <var>.t.o.</var> Å¿umma erit <var>.n.t.o.</var> vnitas verò <var>.n.i.</var> et <var>.o.i</var>. </s> - <s xml:space="preserve">Iam Å¿i numerus <var>.f.g.</var> tertiò <lb/>propoÅ¿itus ex diuiÅ¿ione Å¿ecundi per <var>.o.t.n.</var> proferri debet. </s> - <s xml:space="preserve">Ex .13. theoremate patet, <lb/>quòd Å¿i <var>.b.d.</var> per <var>.g.f.</var> diuiÅ¿erimus, proferetur <var>.o.t.n.</var> qui cum fuerit inuentus, <reg norm="Å¿ummam" type="context">Å¿ummã</reg> <lb/>eſſe oportet <reg norm="duorum" type="context">duorÅ©</reg> <reg norm="prouenientium" type="context">prouenientiÅ©</reg>, ex diuiÅ¿ione mutua <reg norm="duorum" type="context">duorÅ©</reg> numerorum, nempe <var>.<lb/>a.e.</var> per <var>.a.u.</var> et <var>.a.u.</var> per <var>.a.e.</var> deinde manifeÅ¿tum eÅ¿t ex .24. aut .25. theoremate <reg norm="eorum" type="context">eorÅ©</reg> <lb/>productum (multiplicatis prouenientibus adinuicem) vnitatem Å¿uperficialem futu <lb/>ram eſſe. </s> - <s xml:space="preserve">Hactenus igitur, totum <var>.o.n.</var> ex doctrina præcedentis theorematis diui-<lb/>ditur in puncto <var>.t.</var> ita vt productum <var>.o.t.</var> in <var>.t.n.</var> <lb/>Å¿olam vnitatem Å¿uperficialem <reg norm="contineat" type="context">cõtineat</reg>, quo <lb/> - <anchor type="figure" xlink:label="fig-0042-01a" xlink:href="fig-0042-01"/> - facto, Å¿i, vt antedictum eÅ¿t, cogitauerimus <var>.n.<lb/>t.</var> <reg norm="proueniens" type="context">proueniẽs</reg> eſſe ex diuiÅ¿ione <var>.e.a.</var> per <var>.a.u.</var> et <var>.<lb/>t.o.</var> proueniens ex diuiÅ¿ione <var>.a.u.</var> per <var>.a.e.</var> pa-<lb/>tebit ex definitione diuiÅ¿ionis, quod eadem <lb/>erit proportio <var>.a.e.</var> ad <var>.n.t.</var> quæ eÅ¿t <var>.a.u.</var> ad vni-<lb/>tatem <var>.n.i.</var> et <var>.a.u.</var> ad <var>.o.t.</var> eadem quæ eÅ¿t <var>.e.a.</var> - <pb o="31" rhead="THEOREM. ARITH." n="43" file="0043" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0043"/> - ad vnitatem <var>.o.i.</var> <reg norm="permutandoque" type="simple">permutandoqÌ;</reg> <var>.e.a.</var> ad <var>.a.u.</var> Å¿icut <var>.t.n.</var> ad <var>.n.i.</var> & componendo <var>.e.a.u.</var> <lb/>ad <var>a.u.</var> Å¿icut <var>.t.n.i.</var> ad <var>.n.i</var>: & euerÅ¿im <var>.e.a.u.</var> ad <var>.e.a.</var> vt <var>.t.n.i.</var> ad <var>.t.n</var>. </s> - <s xml:space="preserve">Quare, ex .20. Å¿epti <lb/>mi, recte vtimur regula de tribus. </s> - <s xml:space="preserve">Idem & de altera parte dico, quamuis qui vnam <lb/>teneat, alteram quo que habiturus Å¿it. </s> - <s xml:space="preserve">Non mirum tamen Å¿i huiuÅ¿modi problema <lb/>ab antiquis definitum non fuerit, qui hanc vltimam partem non cognouerunt.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0042-01" xlink:href="fig-0042-01a"> - <image file="0042-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0042-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="47"> - <head xml:space="preserve">THEOREMA <num value="47">XLVII</num>.</head> - <p> - <s xml:space="preserve">CVR duobus numeris mutuó diuiÅ¿is, Å¿i per Å¿ummam prouenientium, produ-<lb/>ctum vnius in alterum multiplicetur, vltimum productum, Å¿ummæ quadra-<lb/>tn<gap extent="2"/>m duorum numerorum æquale futurum Å¿it.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, propoÅ¿itis .16. et .4. mutuò diuiÅ¿is, Å¿umma prouenientium erit <num value="4">.<lb/>4.</num> integrorum cum quarta parte, qua Å¿umma multiplicata cum producto <reg norm="primorum" type="context">primorÅ©</reg> <lb/>numerorum, nempe .64. dabuntur .272. integri Å¿uperficiales, qui Å¿ummæ quadra-<lb/>torum duorum numerorum æquantur.</s> - </p> - <p> - <s xml:space="preserve">Hoc vt conÅ¿ideremus, duo numeri partibus <var>.a.e.</var> et <var>.e.i.</var> in linea <var>.a.i.</var> Å¿ignificentur, <lb/>quorum productum Å¿it <var>.e.d.</var> & <reg norm="quadratum" type="context">quadratÅ©</reg> ipÅ¿ius <var>.a.e.</var> Å¿it <var>.e.p</var>: ipÅ¿ius verò <var>.e.i.</var> Å¿it <var>.e.q.</var> pro-<lb/>ueniens <reg norm="autem" type="wordlist">aÅ©t</reg> ex diuiÅ¿ione <var>.e.i.</var> per <var>.a.e.</var> Å¿it <var>.o.u.</var> proueniens <reg norm="autem" type="wordlist">aÅ©t</reg> <var>.a.e.</var> per <var>.e.i.</var> Å¿it <var>.o.t.</var> quo-<lb/>rum Å¿umma Å¿it <var>.o.u.t.</var> tum productum <var>.e.d</var>: linea <var>.u.n.</var> Å¿ignificetur ad angulum <reg norm="rectum" type="context">rectÅ©</reg> <lb/>coniuncta in puncto <var>.u.</var> extremo ipÅ¿ius <var>.o.u.t.</var> productum <reg norm="autem" type="wordlist">aÅ©t</reg> <var>.u.o.t.</var> in <var>.u.n.</var> Å¿it <var>.n.t</var>. </s> - <s xml:space="preserve">Iam <lb/>probandum nobis eÅ¿t <var>.n.t.</var> æqualem eſſe Å¿ummæ duorum quadratorum <var>.q.e.p</var>. </s> - <s xml:space="preserve">Quod <lb/>Å¿ingillatim probo, & aſſero productum <var>.o.n.</var> æquale eſſe quadrato <var>.q.e.</var> & <reg norm="productum" type="context">productÅ©</reg> <var>.<lb/>s.t.</var> quadrato <var>.e.p</var>. </s> - <s xml:space="preserve">Nam ex .35. theoremate patet numerum <var>.e.i.</var> medium eſſe <reg norm="pro- portionalem" type="context">pro-<lb/>portionalẽ</reg> inter <var>.e.d.</var> et <var>.o.u</var>: cum numerus <var>.e.i.</var> ex præſuppoÅ¿ito ab <var>.e.a.</var> multiplicetur <lb/>& diuidatur, cuius multiplicationis produ-<lb/>ctum eÅ¿t <var>.d.e</var>: nempe <var>.u.n.</var> & proueniens ex <lb/> - <anchor type="figure" xlink:label="fig-0043-01a" xlink:href="fig-0043-01"/> - diuiÅ¿ione eÅ¿t <var>.o.u</var>: </s> - <s xml:space="preserve">quare ex dicto theorema-<lb/>te <var>.e.i.</var> media proportionalis eÅ¿t inter <var>.u.n.</var> et <var>.<lb/>u.o</var>. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> productum <var>.o.n.</var> æquale eÅ¿t qua-<lb/>drato <var>.e.q.</var> ex .16. Å¿exti vel .20. Å¿eptimi. </s> - <s xml:space="preserve">Idem <lb/>dico de producto <var>.s.t.</var> <reg norm="nempe" type="context">nẽpe</reg> æquale eſſe qua-<lb/>drato <var>.e.p.</var> quandoquidem numerus <var>.a.e.</var> ab <lb/><var>e.i.</var> multiplicatur ac diuiditur, cuius multi-<lb/>plicationis productum eÅ¿t <var>.d.e.</var> nempe <var>o.s.</var> & <lb/>proueniens ex diuiÅ¿ione <var>.o.t</var>: </s> - <s xml:space="preserve">inter quæ ex .<lb/>35. theoremate <var>.a.e.</var> media proportionalis <lb/>eÅ¿t. </s> - <s xml:space="preserve">Quare ex allatis propoÅ¿itionibus <reg norm="productum" type="context">productÅ©</reg> <var>.s.t.</var> æquale eÅ¿t quadrato <var>.e.p.</var> Å¿ed <reg norm="totum" type="context">totÅ©</reg> <lb/>productum <var>.n.t.</var> Å¿umma eÅ¿t duorum productorum <var>.o.n.</var> et <var>.s.t.</var> ex prima Å¿ecundi Eucli. <lb/></s> - <s xml:space="preserve">Itaque verum eſſe quod dictum eÅ¿t, conÅ¿equitur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0043-01" xlink:href="fig-0043-01a"> - <image file="0043-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0043-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="48"> - <head xml:space="preserve">THEOREMA <num value="48">XLVIII</num>.</head> - <p> - <s xml:space="preserve">CVR Å¿i quis maiorem duorum numerorum Å¿ola vnitate inter Å¿e differentium, <lb/>per minorem diuidat, <reg norm="maioremque" type="simple">maioremqÌ;</reg> per proueniens multiplicet, productum, <lb/><reg norm="summæ" type="context">sÅ©mæ</reg> ipÅ¿ius maioris cum eodem proueniente æquale erit.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia .10 per .9. diuiÅ¿o, datur vnum cum nona parte, quo multiplica-<lb/>to per proueniens, ipÅ¿o nempe .10: </s> - <s xml:space="preserve">datur productum .11. cum nona parte, tantum Å¿ci - <pb o="32" rhead="I.O. BAPT. BENED." n="44" file="0044" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0044"/> - licet quanta Å¿umma eÅ¿t maioris cum proueniente.</s> - </p> - <p> - <s xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, maior numerus Å¿ignificetur <var>.a.i.</var> et minor linea <var>.a.o.</var> ex <lb/>quo ex præſupoÅ¿ito <var>.o.i.</var> vnitas erit. </s> - <s xml:space="preserve">Sit autem proueniens ex diuiÅ¿ione <var>.a.i.</var> per <var>.a.o.<lb/>a.e</var>: </s> - <s xml:space="preserve">quod <var>.e.a.</var> directè coniungatur ipÅ¿i <var>.a.i.</var> et productum <var>.a.i.</var> in <var>.a.e.</var> Å¿it <var>.u.i</var>. </s> - <s xml:space="preserve">Probabo <lb/>numerum Å¿uperficialem <var>.u.i.</var> æqualem eſſe lineari <var>.i.a.e</var>. </s> - <s xml:space="preserve">quare meminiſſe oportet, <lb/>decimotertio theoremate probatum fuiſſe, quod Å¿i numerus diuiÅ¿ibilis per pro-<lb/>ueniens diuidatur, proueniens futurus Å¿it numerus diuidens, </s> - <s xml:space="preserve">quare <var>.a.o.</var> erit pro-<lb/>ueniens ex diuiÅ¿ione <var>.a.i.</var> per <var>.a.e.</var> & ex deÅ¿initione diuiÅ¿ionis ita Å¿e habebit <var>.e.a.</var> ad <var>.<lb/>a.i.</var> Å¿icut <var>.o.i.</var> ad <var>.o.a.</var> & componondo ita <var>.e.i.</var> ad <var>.a.i.</var> Å¿icut <var>.i.a.</var> ad <var>.o.a.</var> </s> - <s xml:space="preserve">quare <var>.a.i.</var> erit me-<lb/>dia pportionalis inter <var>.e.i.</var> et <var>.a.o.</var> Å¿ed <var>.a.i.</var> non modò diuiÅ¿a <reg norm="nunc" type="context">nÅ©c</reg> cogitatur ab <var>.e.a.</var> ex <lb/>quo Å¿it proueniens <var>.a.o.</var> Å¿ed etiam per eandem <var>.e.a.</var> multiplicata, ex quo produ-<lb/>ctum oriatur <var>.u.i</var>. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> ex .25. theobema-<lb/>te <var>.a.i.</var> media eÅ¿t proportionalis inter <var>.u.</var> <lb/> - <anchor type="figure" xlink:label="fig-0044-01a" xlink:href="fig-0044-01"/> - i. et <var>.a.o</var>. </s> - <s xml:space="preserve">Quare. ex .11. quinti. eadem erit <lb/>proportio <var>.u.i.</var> ad <var>.a.i.</var> Å¿icut <var>.e.i.</var> ad eandem <var>.<lb/>a.i</var>. </s> - <s xml:space="preserve">Igitur ex .9. prædicti numerus <var>.u.i.</var> <lb/>æqualis erit numero <var>.e.i.</var> quod erat propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0044-01" xlink:href="fig-0044-01a"> - <image file="0044-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0044-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="49"> - <head xml:space="preserve">THEOREMA <num value="49">XLIX</num>.</head> - <p> - <s xml:space="preserve">IDipÅ¿tim etiam alia ratione conÅ¿iderari poteÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Linea <var>.u.a.</var> Å¿ecetur in puncto <var>.t.</var> ita vt <var>.a.t.</var> æqualis Å¿it vnitati <var>.o.i.</var> & media paral <lb/>lela <var>.t.n.</var> terminetur productum <var>.t.i.</var> quod conÅ¿tabit æquali numero, quamuis Å¿uperfi-<lb/>ciali, numero <var>.a.i.</var> tametÅ¿i lineari. </s> - <s xml:space="preserve">Tumparallela ducatur à puncto <var>.o.</var> ipÅ¿i <var>.a.u.</var> termi <lb/><reg norm="neturque" type="simple">neturqÌ;</reg> productum <var>.o.u.</var> ex quo bina producta dabuntur <var>.u.o.</var> et <var>.t.i.</var> inter Å¿e æqualia <lb/>ex .15. Å¿exti aut .20. Å¿eptimi cum ita Å¿e habeat <var>.a.i.</var> ad <var>.a.u.</var> Å¿icut <var>.a.o.</var> ad <var>.a.t.</var> Å¿ed <var>.a.i.</var> ad <var>.<lb/>a.o.</var> permutando Å¿ic Å¿e habet Å¿icut <var>.a.u.</var> ad <var>.a.t.</var> & ex prima Å¿exti aut .18. vel .19. Å¿epti-<lb/>mi Å¿ic Å¿e habet <var>.u.i.</var> ad <var>.u.o.</var> Å¿icut <var>.a.i.</var> ad <var>.a.</var> <lb/> - <anchor type="figure" xlink:label="fig-0044-02a" xlink:href="fig-0044-02"/> - o. hoc eÅ¿t <var>.u.i.</var> ad <var>.t.i.</var> ope .11. quinti. </s> - <s xml:space="preserve">Iam <lb/>ex definitione diuiÅ¿ionis ita Å¿e habet <var>.a.e.</var> <lb/>ad <var>.a.i.</var> Å¿icut <var>.o.i.</var> ad <var>.o.a.</var> & componendo <var>.<lb/>e.i.</var> ad <var>.a.i.</var> Å¿icut <var>.i.a.</var> ad <var>.o.a</var>. </s> - <s xml:space="preserve">Itaque ex præ-<lb/>dicta .11. Å¿ic Å¿e habebit <var>.e.i.</var> ad <var>.i.a.</var> Å¿icut <var>.u.<lb/>i.</var> ad <var>.t.i.</var> Å¿ed <var>.t.i.</var> numero conÅ¿tat æquali <var>.a.<lb/>i</var>. </s> - <s xml:space="preserve">quare ex .9. quinti numerus <var>.u.i.</var> numero <var>.e.i.</var> æqualis erit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0044-02" xlink:href="fig-0044-02a"> - <image file="0044-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0044-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="50"> - <head xml:space="preserve">THEOREMA <num value="50">L</num>.</head> - <p> - <s xml:space="preserve">CVR diuidentes numerum propoÅ¿itum in duas eiuÅ¿modi partes, vt <reg norm="productum" type="context">productÅ©</reg> <lb/>vnius in alteram cum i pÅ¿arum differentia in Å¿ummam collectum, æquale Å¿it <lb/>alicui alteri numero maiori primo. </s> - <s xml:space="preserve">Rectè primum ex Å¿ecundo detrahunt, reÅ¿iduum <lb/>verò conÅ¿eruant, tum ex primo Å¿emper binarium deÅ¿umunt, <reg norm="dimidiumque" type="simple">dimidiumqÌ;</reg> conÅ¿er-<lb/>uant, alterum verò dimidium in Å¿eipÅ¿o multiplicant, & ex quadrato numerum con <lb/>Å¿eruatum eruunt, <reg norm="reÅ¿iduique" type="simple">reÅ¿iduiqÌ;</reg> radicem ex dimidio conÅ¿eruato, quod vltimum reÅ¿i-<lb/>duum propoÅ¿iti numeri quæſita pars minor eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proponatur numerus .20. ita <reg norm="diuidendus" type="context">diuidẽdus</reg>, vt <reg norm="productum" type="context">productÅ©</reg> vnius partis <lb/>in alteram, cum partium differentia collectum in Å¿ummam, æquale Å¿it propoÅ¿ito - <pb o="33" rhead="THEOR. ARITH." n="45" file="0045" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0045"/> - numero, verbi gratia .92. præcepit regula detrahi primum numerum ex Å¿ecundo, <lb/>nempe .20. ex .92. cuius reÅ¿iduum, Å¿cilicet .72. conÅ¿eruetur, tum detrahi iubet bi <lb/>narium ex primo, Å¿ic in propoÅ¿ito exemplo remanebunt .18. huius autem .18. dimi <lb/>dium in Å¿eipÅ¿um multiplicari iubet, quod cum Å¿it .9. datur numerus .81. ex quo .81. <lb/>primum numerum conÅ¿eruatum, nempe .72. vult regula detrahi, Å¿ic remanebit .9. <lb/>tum huius .9. quadrata radix detrahenda eÅ¿t ex dimidio ipÅ¿ius .18. quod fuit ante qua <lb/>dratum, Å¿ic Å¿upererit .6. hoc eÅ¿t .9. excepta radice quadrata, qui .6. erit minor pars <lb/>quæſita, maior verò .14. quarum productum .84. coniunctum cum partium differen <lb/>tia præbet exactè .92.</s> - </p> - <p> - <s xml:space="preserve">Cuius rei hæc eÅ¿t Å¿peculatio. </s> - <s xml:space="preserve">Primus numerus minor, qui proponitur diuiÅ¿ibilis <lb/>Å¿ignificetur linea <var>.q.g.</var> maior vero linea <var>.x.</var> tum cogitemus <var>.q.g.</var> diuiÅ¿am, cuius maior <lb/>pars Å¿it <var>.q.o.</var> minor <var>.o.g.</var> differentia <var>.q.p.</var> ex quo <var>.p.o.</var> æqualis erit <var>.o.g.</var> Å¿it autem produ-<lb/>ctum <var>.b.o</var>. </s> - <s xml:space="preserve">Oportet igitur, ut <var>.b.o.</var> Å¿imul cum differentia <var>.q.p.</var> æquale Å¿it numero <var>.x.</var> Å¿e-<lb/>cundò propoÅ¿ito, qui notus eÅ¿t, </s> - <s xml:space="preserve">quare etiam Å¿umma producti <var>.b.o.</var> cum differentia <lb/><var>q.p.</var> cognita erit, ex qua detracto primo numero <var>.q.g.</var> reÅ¿iduum cognitum erit, nunc <lb/>igitur quodnam erit hoc reÅ¿iduum? </s> - <s xml:space="preserve">attendamus qua ratione ex Å¿umma <var>.b.o.</var> et <var>.q.p.</var> <lb/>detrahenda Å¿it <var>.q.g</var>. </s> - <s xml:space="preserve">In primis Å¿i Å¿ubtraxerimus ex dicta Å¿umma <var>.q.p.</var> quÄ™ pars eÅ¿t <var>.q.g.</var> <lb/>Å¿upererit detrahenda <var>.p.g.</var> ex <var>.b.o.</var> pars inquam ipÅ¿ius <var>.q.g.</var> quod fiet quotieÅ¿cunque <lb/>cogitauerimus <var>.q.o.</var> duabus vnitatibus diminutam, et per <var>.o.g.</var> multiplicatam, Å¿it au-<lb/>tem productum <var>.b.e.</var> nam cum <var>.o.g.</var> toties <var>.b.o.</var> ingrediatur, quot Å¿unt in <var>.q.o.</var> vnitates <lb/>ex prima Å¿exti aut .18. vel .19. Å¿eptimi, <reg norm="detrahendaque" type="simple">detrahendaqÌ;</reg> Å¿it <var>.p.g.</var> ex <var>.b.o.</var> quæ <var>.p.g.</var> dupla <lb/>eÅ¿t <var>.o.g.</var> patebit <var>.o.c.</var> æqualem eſſe <var>.p.g.</var> fu-<lb/>pererit ita que <var>.b.e.</var> productum <var>.q.e.</var> in <var>.e.</var> <lb/> - <anchor type="figure" xlink:label="fig-0045-01a" xlink:href="fig-0045-01"/> - i. cognitum, erutis autem ex <var>.q.g.</var> ijÅ¿dem <lb/>duabus vnitatibus, remanebit <var>.q.i.</var> nobis <lb/>nota, ex quo <var>.e.i.</var> æqualis erit <var>.e.c</var>. </s> - <s xml:space="preserve">Cum <lb/>igitur productum <var>.q.e.</var> in <var>.e.i.</var> cognoÅ¿camus <lb/>Å¿imul cum <var>.q.i</var>: Sivoluerimus partes <var>.q.e.</var> <lb/>et <var>.e.i.</var> cognoÅ¿cere, vtemur .45. theorema-<lb/>te huius libri, & propoÅ¿itum obtinebimus, nam cognoÅ¿cemus <var>.e.i.</var> & ex conÅ¿equen-<lb/>ti <var>.o.g.</var> eius æqualem.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0045-01" xlink:href="fig-0045-01a"> - <image file="0045-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0045-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="51"> - <head xml:space="preserve">THEOREMA <num value="51">LI</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">DIvidere</emph> numerum in duas eiuÅ¿modi partes, quæ pro medio proportionali <lb/>alterum numerum propoÅ¿itum recipiant, primi dimidio minorem, aliud ni <lb/>hil eÅ¿t, quà m binas primi numeri partes inuenire, quæ inter Å¿e multiplicatæ quadra <lb/>to Å¿ecundi numeri numerum æqualem proferant, ex .16. Å¿exti aut .20. Å¿eptimi, quod <lb/>tamen .45. theoremate fuit à nobis Å¿peculatum.</s> - </p> - </div> - <div type="math:theorem" level="3" n="52"> - <head xml:space="preserve">THEOREMA <num value="52">LII</num>.</head> - <p> - <s xml:space="preserve">CVR pro poÅ¿itis tribus numeris quibuÅ¿cunque, Å¿i productum primi in Å¿ecun-<lb/>dum per tertium multiplicetur, atque Å¿ecundum hoc productum <reg norm="corporeum" type="context">corporeÅ©</reg>, <lb/>per primum numerum diuidatur, proueniens erit numerus æqualis producto Å¿e-<lb/>cundi in tertium.</s> - </p> - <p> - <s xml:space="preserve">Exempli cauÅ¿a, proponantur hi tres numeri .10. 11. 12. <reg norm="multiplicenturque" type="simple">multiplicenturqÌ;</reg> .10. <reg norm="cum" type="context">cÅ©</reg>.</s> - <pb o="34" rhead="IO. BAPT. BENED." n="46" file="0046" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0046"/> - <s xml:space="preserve">11. dabuntur .110. quo producto multiplicato cum .12. dabuntur .1320. hoc pro <lb/>ueniens per primum nempe .10. diuiÅ¿um dabit .132. numerum æqualem producto <lb/>Å¿ecundi in tertium numerorum propoÅ¿itorum, Å¿cilicet .132.</s> - </p> - <p> - <s xml:space="preserve">Hoc vt Å¿peculemur, primus numerus Å¿ignificetur line <var>a.o.u.</var> Å¿ecundus <var>.e.o.</var> tertius <var>.<lb/>e.a.</var> productum verò <var>.o.u.</var> in <var>.o.e.</var> Å¿it <var>.o.i.</var> ipÅ¿ius ve <lb/>rò <var>.o.i.</var> per <var>.e.a.</var> <reg norm="productum" type="context">productÅ©</reg> <reg norm="corporeum" type="context">corporeÅ©</reg> Å¿it <var>.i.c.</var> tum <lb/> - <anchor type="figure" xlink:label="fig-0046-01a" xlink:href="fig-0046-01"/> - <reg norm="productum" type="context">productÅ©</reg> <var>.e.o.</var> in <var>.e.a.</var> Å¿it <var>.e.c</var>. </s> - <s xml:space="preserve">Dico <reg norm="nunc" type="context">nÅ©c</reg> quod di-<lb/>uiÅ¿o numero corporeo <var>.i.c.</var> per <reg norm="primum" type="context">primÅ©</reg> <var>.o.u.</var> <reg norm="proue" type="simple">ê“ue</reg> <lb/>niens æquale erit numero producti <var>.e.c</var>. </s> - <s xml:space="preserve">Qua-<lb/>re in primis cogitandum eÅ¿t, quod cum produ-<lb/>ctum <var>.i.c.</var> ortum fuerit ex multiplicatione <var>.o.i.</var> <lb/>in <var>.e.a</var>: dictum <var>.o.i.</var> toties ingredietur <var>.i.c.</var> quo-<lb/>ties vnitas reperitur in <var>.e.a.</var> eadem ratione, to-<lb/>ties <var>.e.c.</var> in <var>.i.c.</var> quot vnitates erunt in <var>.o.u</var>. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> <lb/>Å¿equitur quòd diuiÅ¿o <var>.i.c.</var> per <var>o.u.</var> proueniens Å¿it <lb/><var>e.c.</var> corporeum, æquale nihilominus producto <var>.e.c.</var> Å¿uperficiali.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0046-01" xlink:href="fig-0046-01a"> - <image file="0046-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0046-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="53"> - <head xml:space="preserve">THEOREMA <num value="53">LIII</num>.</head> - <p> - <s xml:space="preserve">CVR diuidens propoÅ¿itum numerum in tres partes Å¿ic Å¿e habentes vt produ-<lb/>ctum primi in Å¿ecundam, in tertia <reg norm="multiplicatum" type="context">multiplicatÅ©</reg>, præbeat numerum alteri nu-<lb/>mero propoÅ¿ito æqualem. </s> - <s xml:space="preserve">Rectè Å¿ecundum numerum per quemcunque alium mino <lb/>rem primo diuidit, qui diuidens vna erit ex tribus partibus quæſitis, proueniens <lb/>autem erit productum vnius in alteram reliquarum duarum, quarum Å¿umma cogni <lb/>ta erit, detracto numero diuidente ex primo dato, quam quidem Å¿i diÅ¿tinguere <lb/>quis voluerit, vtetur theoremate .45.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, proponitur numerus .20. in tres partes diuidendus, quæ Å¿ic Å¿e <lb/>habeant, ut productum primæ in Å¿ecundam in tertia multiplicatum det .90. itaque <lb/>Å¿umenda erit pro prima vna pars ipÅ¿ius .20. quæcunque illa Å¿it, verbi gratia .2. qua <lb/>Å¿ecundus numerus, nempe .90. diuidatur, dabitur igitur .45. quod erit productum <lb/>cæterarum partium inter Å¿e, quarum Å¿umma eÅ¿t .18. quam Å¿ummam Å¿i diÅ¿tinguere <lb/>volueris in cÄ™teris duabus partibus Å¿eparatis, vteris .45. theoremate, vt quà m citiÅ¿-<lb/>Å¿imè quod cupis exequaris, erunt autem partes .3. et .15.</s> - </p> - <p> - <s xml:space="preserve">In cuius Å¿peculationis gratiam nihil aliud occurrit, quà m quod præcedenti theo-<lb/>remate, & Å¿uperiore .45. allatum eÅ¿t.</s> - </p> - </div> - <div type="math:theorem" level="3" n="54"> - <head xml:space="preserve">THEOREMA <num value="54">LIIII</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">DIvidere</emph> numerum in .3. eiuÅ¿modi partes, vt quadratum vnius Å¿it æquale <lb/>producto reliquarum duarum inter Å¿e, idem omnino eÅ¿t cum 51. theoremate. <lb/></s> - <s xml:space="preserve">Nam qui Å¿umet quamlibet partem propoÅ¿iti numeri, quæ tertia parte maior tamen <lb/>non Å¿it, <reg norm="reÅ¿iduumque" type="simple">reÅ¿iduumqÌ</reg> in duas tales partes diuiÅ¿erit, vt prima Å¿umpta, media proportio <lb/>nalis Å¿it ex probatione .51. theoremate allata, propoÅ¿itum conÅ¿equetur.</s> - </p> - </div> - <div type="math:theorem" level="3" n="55"> - <head xml:space="preserve">THEOREMA <num value="55">LV</num>.</head> - <p> - <s xml:space="preserve">ID ipÅ¿um alia ratione ab ea diuerÅ¿a <reg norm="quam" type="context">quã</reg> .51. theoremate adduximus, <reg norm="profici" type="simple">ê“fici</reg> poteÅ¿t.</s> - </p> - <pb o="35" rhead="THEOREM. ARIT." n="47" file="0047" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0047"/> - <p> - <s xml:space="preserve">Sumantur enimtres numeri continui proportionales, cuiuÅ¿cunque denique pro <lb/>portionalitatis, qui in Å¿ummam colligantur, ac poÅ¿tmodum, regula de trib. dica-<lb/>mus. </s> - <s xml:space="preserve">Si Å¿umma hæc primo numero propoÅ¿ito in tres partes diuidendo reÅ¿pondet, <lb/>cuireÅ¿pondebit vna ex tribus partibus huiuÅ¿cÄ™ <reg norm="summæ" type="context">sÅ©mæ</reg>? </s> - <s xml:space="preserve">idem dereliquis duabus pa<unsure/>rti <lb/>bus dico.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proponatur numerus .57. diuidendus in tres continuas partes <lb/>proportionales proportione Å¿eÅ¿quialtera, tres numeros in eiuÅ¿modi proportio-<lb/>nalitate diÅ¿tinctos Å¿umemus, vt potè .4. 6. 9. qui in Å¿ummam collecti dabunt <reg norm="Å¿um- mam" type="context">Å¿um-<lb/>mã</reg> .19. <reg norm="dicemusque" type="simple">dicemusqÌ;</reg> Å¿i .19. dant .4. quid <reg norm="dabunt" type="context">dabÅ©t</reg> .57? </s> - <s xml:space="preserve">vnde proueniens vnius partis erit <num value="12">.<lb/>12</num>. </s> - <s xml:space="preserve">Tum Å¿i dicamus, Å¿i .19. dat .6. quid dabit .57? </s> - <s xml:space="preserve">nempe dabit .18. </s> - <s xml:space="preserve">PoÅ¿tremò, Å¿i <num value="19">.<lb/>19.</num> dat .9. quid dabit .57? </s> - <s xml:space="preserve">nempe .26. atque ita dabitur .18. cuius quadratum æqua-<lb/>bitur producto reliquarum duarum partium inter Å¿e.</s> - </p> - <p> - <s xml:space="preserve">Quod vt Å¿ciamus, numerus propoÅ¿itus in tres quaÅ¿libet partes diuidendus Å¿i-<lb/>gnificetur linea <var>.a.d.</var> tres autem numeri dictæ proportionalitatis, lineis <var>.e.f</var>: <var>f.g.</var> <lb/>et <var>.g.h.</var> directè inter Å¿e coniunctis denotentur. </s> - <s xml:space="preserve">Cogitemus pariter lineam <var>.d.a.</var> in <lb/>tres partes diuiÅ¿am <var>.a.b</var>: <var>b.c.</var> et <var>.c.d.</var> eadem cum cæteris proportionalitate, </s> - <s xml:space="preserve">tunc ea-<lb/>dem erit proportio <var>.a.d.</var> ad quamlibet Å¿uarum partium, quæ eÅ¿t <var>.e.h.</var> ad reÅ¿ponden <lb/>tem ipÅ¿ius in <var>.a.d</var>: Verbi gratia reÅ¿pondentem <var>.a.b.</var> ipÅ¿i <var>.e.f.</var> et <var>.b.c</var>: <var>f.g.</var> et <var>.c.d</var>: <var>g.h</var>. </s> - <s xml:space="preserve">Di <lb/>co enim quòd ita Å¿e habebit <var>.a.d.</var> ad <var>.c.d.</var> Å¿icut <var>.e.h.</var> ad <var>.g.h</var>. </s> - <s xml:space="preserve">Nam cum Å¿ic Å¿e habeat <var>.a.<lb/>b.</var> ad <var>.b.c.</var> Å¿icut <var>.e.f.</var> ad <var>.f.g.</var> ex præſuppoÅ¿ito, permutando Å¿ic Å¿e habebit <var>.a.b.</var> ad <var>.e.f.</var> Å¿i-<lb/>cut <var>.b.c.</var> ad <var>.f.g.</var> & eadem ratione Å¿ic Å¿e habe-<lb/>bit <var>.c.d.</var> ad <var>.g.h.</var> Å¿icut <var>.b.c.</var> ad <var>.f.g.</var> & <reg norm="conÅ¿equen- ter" type="context">cõſequen-<lb/> - <anchor type="figure" xlink:label="fig-0047-01a" xlink:href="fig-0047-01"/> - ter</reg> Å¿icut <var>.a.b.</var> ad <var>.e.f.</var> ex quo ex .13. quinti Å¿ic <lb/>Å¿e habebit tota <var>.a.d.</var> ad totam <var>.e.h.</var> Å¿icut <var>.c.d.</var> <lb/>ad <var>.g.h.</var> aut <var>.b.c.</var> ad <var>.f.g.</var> aut <var>.a.b.</var> ad <var>.e.f.</var> per-<lb/>mutando itaque propoÅ¿itum manifeÅ¿tum erit, ipÅ¿um autem productum <var>.a.b.</var> in <var>.c.b.</var> <lb/>æquale erit quadrato <var>.b.c.</var> ex .15. fexti aut .20. Å¿eptimi.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0047-01" xlink:href="fig-0047-01a"> - <image file="0047-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0047-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="56"> - <head xml:space="preserve">THEOREMA <num value="56">LVI</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">VEteres</emph> aliud quoque problema indeterminatum propoÅ¿uerunt, quod ex <lb/>more ratione à me definietur, eÅ¿t autem eiuÅ¿modi.</s> - </p> - <p> - <s xml:space="preserve">Quomodo propoÅ¿itus numerus in tres eiuÅ¿modi partes diuidatur, vt <reg norm="quadratum" type="context">quadratÅ©</reg> <lb/>vnius æquale fit fummæ quadratorum reliquarum duarum partium.</s> - </p> - <p> - <s xml:space="preserve">Hoc vt efficiamus tria quadrata Å¿eparata Å¿umamus, <reg norm="quorum" type="context">quorÅ©</reg> <reg norm="vnum" type="context">vnÅ©</reg> æquale Å¿it reliquis <lb/>duobus; </s> - <s xml:space="preserve"><reg norm="eorum" type="context">eorÅ©</reg> <reg norm="autem" type="context">autẽ</reg> radices in Å¿ummam Å¿imul colligantur, tum regulam de tribus Å¿e <lb/>quemur, ratione præcedenti theoremate demonÅ¿trata, & rectè vt infra docebimus, <lb/>quod autem dico de quadratis, etiam de cubis, & quibuÅ¿uis dignitatibus aſſero.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i numerus diuiÅ¿ibilis proponatur .30. in tres eiuÅ¿modi partes di <lb/>uidendus, vt quadratum vnius æquale Å¿it Å¿ummæ quadratorum reliquarum duarum <lb/>partium, in primis radices trium quadratorum Å¿umemus, Å¿ic quomodocunque Å¿e <lb/>habentes, vt maius ipÅ¿orum æquale Å¿it Å¿ummæ reliquorum duorum, verbi gratia .25. <lb/>16. et .9. nempe .5. 4. et .3. quæ Å¿i colligantur in Å¿ummam efficiunt .12. </s> - <s xml:space="preserve">Tum ex regu-<lb/>la de tribus dicemus, Å¿i .12. reÅ¿pondet .30: </s> - <s xml:space="preserve">cui, 5. radix maior reÅ¿pondebit? </s> - <s xml:space="preserve">nem-<lb/>pe .12. cum dimidio.</s> - </p> - <p> - <s xml:space="preserve">Deinde Å¿i dixerimus Å¿i .12. valet .30. quid valebit .4. radix media? </s> - <s xml:space="preserve">nempe vale-<lb/>bit .10. tertia autem minor .7. cum dimidio. </s> - <s xml:space="preserve">Itaquetota Å¿umma erit .30. & quadra- - <pb o="36" rhead="IO. BAPT. BENED." n="48" file="0048" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0048"/> - tum .12. cum dimidio erit .155. quod æquale erit Å¿ummæ quadratorum duarum par <lb/>tium, nempe .100. cum .55.</s> - </p> - <p> - <s xml:space="preserve">Hoc vt <reg norm="demonÅ¿tremus" type="context">demõſtremus</reg>, numerus diuiÅ¿ibilis propoÅ¿itus Å¿ignificetur linea <var>.a.d.</var> & Å¿um <lb/>ma radicum, noÅ¿tro modo Å¿umptarum, linea <var>.e.h.</var> quarum prima & maior Å¿it <var>.e.f.</var> Å¿e-<lb/>cunda <var>.f.g.</var> tertia <var>.g.h.</var> cogitemus etiam lineam <var>.a.d.</var> ea ratione diuiÅ¿am eſſe qua <var>.e.h.</var> <lb/>patebit cnim ex modo præcedentis theorematis vnamquanque partium <var>.a.d.</var> ita Å¿e <lb/>habituram ad Å¿uum totum Å¿icut Å¿e habent Å¿ingulæ <var>.e.h.</var> ad Å¿uum. </s> - <s xml:space="preserve">Quod ideo dico, vt <lb/>intelligamus rectè nos dicere. </s> - <s xml:space="preserve">Si <var>.e.h.</var> dat <var>.a.d.</var> ergo <var>.e.f.</var> dabit <var>.a.b.</var> <reg norm="atque" type="simple">atq;</reg> ita de cæteris. <lb/></s> - <s xml:space="preserve">Quare permutando Å¿ic Å¿e habebit <var>.a.b.</var> ad <var>.b.c.</var> Å¿icut <var>.e.f.</var> ad <var>.f.g.</var> idem dico de reliquis. <lb/></s> - <s xml:space="preserve">Igitur ex .18. Å¿exti aut .11. octaui, eadem erit proportio quadrati <var>.a.b.</var> ad <reg norm="quadratum" type="context">quadratÅ©</reg> <var>.<lb/>b.c.</var> quæ quadrati <var>.e.f.</var> ad quadratum <var>.f.g.</var> tota enim Å¿unt æqualia, cum eorum partes <lb/>Å¿imiles inter Å¿e Å¿unt æquales. </s> - <s xml:space="preserve">Idem dico de proportione qu@drati <var>.a.b.</var> nempe ita <lb/>Å¿e habere ad <var>.c.d.</var> Å¿icut quadratum <var>.e.f.</var> ad quadratum <var>.g.h.</var> ex quo ex .24. quinti pro-<lb/>portio quadrati <var>.a.b.</var> ad Å¿ummam quadratorum duarum partium <var>.b.c.</var> et <var>.c.d.</var> Å¿ic Å¿e ha <lb/>bebit ut quadrati <var>.e.f.</var> ad Å¿ummam quadra-<lb/>torum <var>.f.g.</var> et <var>.g.h</var>. </s> - <s xml:space="preserve">At quadratum <var>.e.f.</var> æquale <lb/> - <anchor type="figure" xlink:label="fig-0048-01a" xlink:href="fig-0048-01"/> - eÅ¿t Å¿ummæ quadratorum <var>.f.g.</var> et <var>.g.h.</var> igitur <lb/>Å¿ic etiam Å¿e habebit quadratum <var>.a.b.</var> nempe <lb/>æquale quadratis <var>.b.c.</var> et <var>.c.g</var>. </s> - <s xml:space="preserve">IdipÅ¿um de cæ <lb/>teris dignitatibus dices, <reg norm="vterisque" type="simple">vterisqÌ;</reg> .21. theoremate huius libri.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0048-01" xlink:href="fig-0048-01a"> - <image file="0048-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0048-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="57"> - <head xml:space="preserve">THEOREMA <num value="57">LVII</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">SImile</emph> quoque problema ab antiquis indeterminatum proponitur, quod eiuÅ¿-<lb/>modi eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">An numerus aliquis in tres eiuÅ¿modi partes di@idi poſſit, vt quadratum vnius æ-<lb/>quale Å¿it Å¿ummæ quadratorum cæterarum duarum partium Å¿imul cum producto <lb/>vnius in alteram.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proponatur numerus .50. vt iam dictum eÅ¿t diuidendus, repe <lb/>riendus erit alius quilibet numerus, qui tamen Å¿umma Å¿it trium radicum Å¿ic Å¿e ha-<lb/>bentium, vt quadratum vnius æquale Å¿it Å¿ummæ quadratorum duarum partium Å¿i-<lb/>mul cum producto vnius in alteram, eum autem qui primò occurrit Å¿umamus, utpo <lb/>tè .30. qui Å¿umma eÅ¿t numerorum .6. 10. 14. partium Å¿ic Å¿e habentium, vt quadratum <lb/>ipÅ¿ius .14. æquale Å¿it Å¿ummæ quadratorum cæterarum partium Å¿imul cum produ-<lb/>cto vnius in alteram, agamusqÌue regula de tribus, ac dicamus, Å¿i .30. valet <num value="50">.<lb/>50.</num> quid valebit .14. nempe .23. cum tertia parte. </s> - <s xml:space="preserve">Idem efficiemus in cæte-<lb/>ris partibus, quarum vna erit .16. cum duabus tertijs, altera verò .10. abÅ¿que @ractis, <lb/>ex quo quadratum primæ erit .544. cum .4. nonis, Å¿ecundæ .277. cum Å¿eptem nonis, <lb/>tertiæ .100. & productum Å¿ecundæ in tertiam .166. cum .6. nonis, quod productum, <lb/>cum quadratis Å¿ecundæ & tertiæ collectum erit .544. cum .4. nonis.</s> - </p> - <p> - <s xml:space="preserve">Huius rei Å¿peculatio eadem eÅ¿t, quÄ™ fuit præcedentis theorematis vÅ¿quequo no-<lb/>ueris eandem proportionem eſſe quadrati <var>.a.b.</var> ad Å¿ummam quadratorum <var>.b.c.</var> et <var>.c.<lb/>d.</var> quæ quadrati <var>.e.f.</var> ad Å¿ummam quadratorum <var>.f.g.</var> et <var>.g.h</var>. </s> - <s xml:space="preserve">Sed cum hic non demus <lb/>quadratum <var>.e.f.</var> æquale Å¿ummæ quadratorum <var>.f.g.</var> et <var>.g.h.</var> fed maius ex producto <var>.g.h.</var> <lb/>in <var>.f.g.</var> aut quod idem eÅ¿t, è contrario, Å¿ubÅ¿equentes figuræ cogitandæ erunt, qua-<lb/>rum <var>.i.</var> Å¿it quadratum <var>.a.b</var>: l. Å¿it quadratum <var>.e.f</var>: x. quadratum <var>.b.c</var>: y. quadratum <var>.f.g</var>: p. <lb/>quadratum <var>.c.d</var>: q. quadratum <var>.g.h</var>: k. Å¿it productum <var>.b.c.</var> in <var>.c.d</var>: m. Å¿it productum <var>.f.</var> - <pb o="37" rhead="THEOREM. ARITH." n="49" file="0049" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0049"/> - g. in <var>.g.h</var>. </s> - <s xml:space="preserve">Nunc ex Å¿peculatione præcedentis theorematis, eadem erit proportio <var>.n.<lb/>t.</var> ad <var>.o.u.</var> quæ eÅ¿t <var>.n.s.</var> ad <var>.o.r.</var> </s> - <s xml:space="preserve">quare pro-<lb/>ductum <var>.k.</var> ex definitione Å¿imile erit <lb/> - <anchor type="figure" xlink:label="fig-0049-01a" xlink:href="fig-0049-01"/> - producto <var>.m.</var> cum vtraque Å¿int rectan-<lb/>gula, vnde proportio <var>.k.</var> ad <var>.m.</var> ad pro-<lb/>portionem <var>.n.t.</var> ad <var>.o.u.</var> ex .18. Å¿exti du-<lb/>pla erit. </s> - <s xml:space="preserve">Igitur proportio <var>.k.</var> ad <var>.m.</var> æ-<lb/>qualis erit proportioni <var>.x.</var> ad <var>.y.</var> et <var>.p.</var> <lb/>ad <var>.q.</var> et <var>.i.</var> ad <var>.l.</var> & permutando Å¿ic Å¿e ha-<lb/>bebit <var>.k.</var> ad <var>.i.</var> Å¿icut <var>.m.</var> ad <var>.l.</var> Å¿ed <var>.x.p.</var> ad <var>.i.</var> <lb/>Å¿icÅ¿e habere probatum eÅ¿t vt <var>.y.q.</var> ad <var>.l</var>. <lb/></s> - <s xml:space="preserve">Quare ex eadem .24. quinti Å¿ic Å¿e habe <lb/>bit <var>.x.p.k.</var> ad <var>.i.</var> Å¿icut <var>.y.q.m.</var> ad <var>.l.</var> Å¿ed <var>.y.q.<lb/>m.</var> æqualis eÅ¿t <var>.l</var>. </s> - <s xml:space="preserve">Itaque <var>.x.p.k.</var> pariter <var>.i.</var> <lb/>æqualis erit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0049-01" xlink:href="fig-0049-01a"> - <image file="0049-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0049-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="58"> - <head xml:space="preserve">THEOREMA <num value="58">LVIII</num>.</head> - <p> - <s xml:space="preserve">ALIVD quoque problema, nec tamen definitum, veteres propoÅ¿uerunt, <lb/>nempe an aliquis numerus in .4. eiuÅ¿modi partes diuidi poſſit, vt Å¿umma qua-<lb/>dratorum duarum partium dupla Å¿it Å¿ummæ quadratorum reliquarum duarum.</s> - </p> - <p> - <s xml:space="preserve">Verum huius effectio & Å¿peculatio non erit difficilis, <reg norm="cum" type="context">cÅ©</reg> Å¿it eadem quæ præmiÅ¿sis <lb/>proximè duobus theorematibus allata fuit, Å¿umpta nempe Å¿umma radicum quarun <lb/>cunque Å¿ic Å¿e habentium, prout dictum fuit. </s> - <s xml:space="preserve">Verbigratia .44. cuius partes erunt. <lb/>16. 12. 14. 2. <reg norm="tunc" type="context">tÅ©c</reg> progrediemur regula de tribus dicentes. </s> - <s xml:space="preserve">Si .44 numerum propoÅ¿i-<lb/>tum valet, quid .16. pars maior? </s> - <s xml:space="preserve">nempe valebit partem maiorem numeri propoÅ¿i-<lb/>ti reÅ¿pondentem .16. idem de cæteris dico.</s> - </p> - <p> - <s xml:space="preserve">Porrò Å¿peculatio eadem eÅ¿t cum Å¿uperioribus.</s> - </p> - </div> - <div type="math:theorem" level="3" n="59"> - <head xml:space="preserve">THEOREMA <num value="59">LIX</num>.</head> - <p> - <s xml:space="preserve">CVR diuidens propoÅ¿itum numerum in duas eiuÅ¿modi partes, vt productum <lb/>radicum quadratarum ipÅ¿arum partium æquale Å¿it alteri numero propoÅ¿ito, <lb/>cuius <reg norm="tamen" type="context">tamẽ</reg> quadratum maius <reg norm="non" type="context">nõ</reg> Å¿it quadrato dimidij primi numeri propoÅ¿iti. </s> - <s xml:space="preserve">Rectè <lb/>Å¿ecundum numerum propoÅ¿itum in Å¿eipÅ¿um multiplicat, & <reg norm="eundem" type="context">eundẽ</reg> ex quadrato di-<lb/>midij primi detrahit, <reg norm="reÅ¿iduique" type="simple">reÅ¿iduiqÌ;</reg> quadratam radicem Å¿ubtrahit ex dimidio ipÅ¿ius pri-<lb/>mi, ex quo datur minor pars quæſita, quaipÅ¿i dimidio coniuncta, maior pars ha-<lb/>betur.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proponatur numerus, 20. propoÅ¿ito modo, in duas partes <lb/>eiuÅ¿modi diuidendus, vt productum radicum æquale Å¿it (verbigratia) 8. </s> - <s xml:space="preserve">Dimi-<lb/>dium priminumeri in Å¿eipÅ¿um multiplicabimus, cuius quadratum erit .100. ex <lb/>quo quadratum Å¿ecundi numeri, nempe .64. detrahemus, <reg norm="remanebitque" type="simple">remanebitqÌ;</reg> .36. cuius radi <lb/>ce quadrata coniuncta .10. dimidio inquam primi numeri propoÅ¿iti, dabitur nume <lb/>rus .16. pars maior, & Å¿ubtracta à dimidio, dabitur minor pars, nempe .4.</s> - </p> - <pb o="38" rhead="IO. BAPT. BENED." n="50" file="0050" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0050"/> - <p> - <s xml:space="preserve">Hoc vt demonÅ¿tremus, primus nu-<lb/> - <anchor type="figure" xlink:label="fig-0050-01a" xlink:href="fig-0050-01"/> - merus linea <var>.a.b.</var> Å¿ignificetur, quam di-<lb/>uiÅ¿am cogitemus in puncto <var>.c.</var> in partes <lb/>quæſitas, ex quo præſupponitur duas li-<lb/>neas <var>.a.c.</var> et <var>.c.b.</var> duo quadrata eſſe, quæ <lb/>in altera figura Å¿ignificetur per <var>.d.</var> et <var>.e.</var> <lb/>productum autem radicum cognitum <var>.<lb/>f.</var> quandoquidem datum eÅ¿t, cuius qua-<lb/>dratum æquale erit producto quadra-<lb/>torum <var>.d.e.</var> adinuicem, nempe <var>.b.c.</var> in <var>.a.c.</var> ex .19. theoremate huius. </s> - <s xml:space="preserve">Quod verbi <lb/>gratia Å¿it <var>.x.</var> <reg norm="itaque" type="simple">itaq;</reg> cognitum, quo facto, doctrinam .45. theorematis libri huius Å¿ecuti, <lb/>propoÅ¿itum conÅ¿equemur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0050-01" xlink:href="fig-0050-01a"> - <image file="0050-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0050-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="60"> - <head xml:space="preserve">THEOREMA <num value="60">LX</num>.</head> - <p> - <s xml:space="preserve">CVR productum differentiæ duarum radicum in Å¿ummam ipÅ¿arum, Å¿emper <lb/>differentia Å¿it quadratorum ipÅ¿arum radicum.</s> - </p> - <p> - <s xml:space="preserve"><reg norm="Exempli" type="context">Exẽpli</reg> gratia, quoslibet duos numeros pro radicibus Å¿umpÅ¿erimus, vt potè .3. et <num value="5">.<lb/>5.</num> quorum differentia eÅ¿t .2. certè Å¿i differentiam hanc per Å¿ummam radicum Å¿cili-<lb/>cet .8. multiplicauerimus, dabitur numerus .16. quod productum differentia eÅ¿t <lb/>Å¿uorum quadratorum, nempeinter .9. et .25.</s> - </p> - <p> - <s xml:space="preserve">Hoc vt Å¿peculemur, duæ radices in linea <var>.n.i.</var> Å¿ignificentur, quarum vna Å¿it <var>.n.c.</var> & <lb/>altera <var>.c.i.</var> ipÅ¿arum autem differentia <var>.n.t.</var> ex quo <var>.t.<lb/>c.</var> æqualis erit <var>.c.i</var>. </s> - <s xml:space="preserve">Tum cogitato toto quadrato <var>.d.i.</var> <lb/> - <anchor type="figure" xlink:label="fig-0050-02a" xlink:href="fig-0050-02"/> - cum diametro <var>.d.i.</var> <reg norm="ductaque" type="simple">ductaqÌ</reg> parallela lateri <var>.n.d.</var> à <lb/>puncto <var>.c.</var> & altera à puncto <var>.t.</var> & à puncto <var>.o.</var> tertia <lb/>ipÅ¿i <var>.n.i.</var> & à puncto <var>.a.</var> quarta <var>.x.a.e.</var> parallela ipÅ¿i <var>.<lb/>o.</var> inueniemus <var>.b.n.</var> productum eſſe differentiæ <var>.n.<lb/>t.</var> in Å¿umma radicum <var>.n.i.</var> & cum <var>.d.o.</var> et <var>.a.o.</var> Å¿int <lb/>quadrata radicum prædictarum: </s> - <s xml:space="preserve">b.e. æquale erit <var>.<lb/>n.u.</var> cum vtrunque horum productorum æquale Å¿it <var>.<lb/>x.u.</var> ex quo gnomon <var>.e.d.u.</var> æqualis erit producto <var>.<lb/>b.n.</var> quod Å¿cire cupiebamus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0050-02" xlink:href="fig-0050-02a"> - <image file="0050-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0050-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="61"> - <head xml:space="preserve">THEOREMA <num value="61">LXI</num>.</head> - <p> - <s xml:space="preserve">CVR propoÅ¿itum aliquem numerum diuiÅ¿uri in duas eiuÅ¿modi partes, vt diffe-<lb/>rentia radicum quadratarum æqualis Å¿it alteri numero propoÅ¿ito, cuius ta-<lb/>men quadratum dimidij primi quadratum non excedat. </s> - <s xml:space="preserve">Rectè Å¿ecundum numerum <lb/>in Å¿eipÅ¿um multiplicant, productum verò ex primo numero detrahunt, <reg norm="rurÅ¿usque" type="simple">rurÅ¿usqÌ;</reg> di <lb/>midium reÅ¿idui quadrant, & quadratum hoc ex quadrato dimidij primi Å¿ubtrahunt, <lb/>atque ita radice quadrata reÅ¿idui, dimidio primi coniuncta, pars maior datur, qua <lb/>ex ipÅ¿o dimidio detracta, pars minor relinquitur.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, propoÅ¿ito numero .20. ita ut propoÅ¿itum eÅ¿t, diuidendo, nem-<lb/>pe vt differentia radicum quadratarum dictarum partium æqualis Å¿it binario, bina-<lb/>rium hocin Å¿eipÅ¿um multiplicabimus, cuius quadratum .4. è primo numero .20. de - <pb o="39" rhead="THEOREM. AR IT." n="51" file="0051" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0051"/> - trahemus, <reg norm="Å¿upereritque" type="simple">Å¿upereritqÌ;</reg> numerus .16. cuius dimidium Å¿cilicet .8. in Å¿eipÅ¿um multipli-<lb/>cabimus, <reg norm="dabiturque" type="simple">dabiturqÌ;</reg> numerus .64. qui cum ex quadrato dimidij primi detractus fue-<lb/>rit, nempe ex .100. & reÅ¿iduo .36. radix quadrata nempe .6. coniuncta denario, di-<lb/>midio primi, dabit .16. partem maiorem, & ex denario detracta, partem minorem.</s> - </p> - <p> - <s xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, primus numerus <lb/>propoÅ¿itus Å¿igniÅ¿icetur linea <var>.x.y.</var> pro voto diui-<lb/> - <anchor type="figure" xlink:label="fig-0051-01a" xlink:href="fig-0051-01"/> - Å¿a in puncto <var>.c.</var> et <var>.x.t.</var> productum Å¿it ipÅ¿ius <var>.x.<lb/>c.</var> in <var>.c.y.</var> pariter etiam <var>.q.p.</var> Å¿it Å¿umma radicum <lb/>quadratarum, nempe <var>.q.g.</var> ipÅ¿ius <var>.t.c.</var> et <var>.g.p.</var> ip-<lb/>Å¿ius <var>.c.y</var>. </s> - <s xml:space="preserve">Tum Å¿uper <var>.q.p.</var> extruatur & diuidatur <lb/>quadratum <var>.q.u.</var> ea ratione qua .41. theoremate <lb/>aut .29. diuiÅ¿imus, in quo Å¿anè quadrato, quadra <lb/>tum ipÅ¿ius <var>.q.i.</var> cernemus datæ differentiæ, & in <lb/>eo collocata quadrata <var>.x.c.</var> et <var>.c.y.</var> ita etiam & <lb/>rationem, qua cognoÅ¿cimus productum <var>.g.r.</var> (vÅ¿i <lb/>modo .29. theorematis) cuius quidem <var>.g.r.</var> qua-<lb/>dratum, ex .19. theoremate æquale erit produ-<lb/>cto <var>.x.t.</var> ideo etiam <reg norm="cognitum" type="context">cognitÅ©</reg>, ac proinde cum no <lb/>uerimus <var>.x.y.</var> Å¿i rationem Å¿equemur .45. theore <lb/>mate cognoÅ¿cemus non Å¿olum ratione .41. theoremate allata hocrectè perfici, Å¿ed <lb/>hac etiam alia ratione.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0051-01" xlink:href="fig-0051-01a"> - <image file="0051-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0051-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="62"> - <head xml:space="preserve">THEOREMA <num value="62">LXII</num>.</head> - <p> - <s xml:space="preserve">CVR propoÅ¿itum numerum diuiÅ¿uri in duas eiuÅ¿modi partes, vt differentia <lb/><reg norm="Å¿uarum" type="context">Å¿uarÅ©</reg> <reg norm="radicum" type="context">radicÅ©</reg> <reg norm="quadratarum" type="context">quadratarÅ©</reg> æqualis Å¿it alteri numero propoÅ¿ito. </s> - <s xml:space="preserve">Cuius <reg norm="tamen" type="wordlist">tamẽ</reg> <reg norm="qua- dratum" type="context">qua-<lb/>dratÅ©</reg> maius non Å¿it quadrato medietatis ipÅ¿ius primi propoÅ¿iti numeri. </s> - <s xml:space="preserve">Rectè <reg norm="etiam" type="context">etiã</reg> <lb/><reg norm="quadratum" type="context">quadratÅ©</reg> dimidij Å¿ecundi numeri ex dimidio primi <reg norm="detrahunt" type="context">detrahÅ©t</reg>, <reg norm="reÅ¿iduique" type="simple">reÅ¿iduiqÌ;</reg> radicem per <lb/>Å¿ecundum multiplicant, & productum ex dimidio primi detrahunt, vt reÅ¿iduum <lb/>pars quæſita minor Å¿it, & illud alterum totius reÅ¿iduum, pars maior.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i numerus .50. in <lb/>prædictas duas partes diuidendus pro-<lb/> - <anchor type="figure" xlink:label="fig-0051-02a" xlink:href="fig-0051-02"/> - poneretur, & alter etiam .6. quadratum <lb/>dimidij Å¿ecundi numeri eſſet .9. eo detra <lb/>cto ex dimidio primi, remaneret .16. cu <lb/>ius radix .4. Å¿cilicet per totum Å¿ecundum <lb/>nempe .6. multiplicata, proferet .24. <lb/>quo producto ex dimidio primi detra-<lb/>cto, nempe .25. dabitur .1. pars minor, <lb/>maior <reg norm="autem" type="context">autẽ</reg> erit <reg norm="reÅ¿idum" type="context">reÅ¿idÅ©</reg> .50. hoc eÅ¿t .49. <lb/>radices autem erunt .1. et .7. differentes <lb/>inter Å¿e, numero Å¿enario.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0051-02" xlink:href="fig-0051-02a"> - <image file="0051-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0051-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Hocvt Å¿ciamus, duo numeri lineis <reg norm="Å¿i- gnificentur" type="context">Å¿i-<lb/>gnificẽtur</reg>, primus linea .b: <reg norm="Å¿ecundus" type="context">Å¿ecÅ©dus</reg> linea <var>.<lb/>c.</var> duæ autem partes <var>.b.</var> duobus quadra-<lb/>tis <var>.q.i.</var> et <var>.i.d.</var> notentur, eorum verò radi-<lb/>ces lineis <var>.a.g.</var> et <var>.g.d.</var> differentia porrò ip <lb/>Å¿i <var>.c.</var> æqualis & co gnita Å¿it <var>.a.h.</var> ex quo <var>.h.</var> - <pb o="40" rhead="IO. BAPT. BENED." n="52" file="0052" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0052"/> - g. æqualis erit <var>.g.d.</var> tum productum <var>.a.g.</var> in <var>.g.d.</var> Å¿it <var>.a.i.</var> et <var>.t.i.</var> æqualis <var>.a.i.</var> et <var>.l.i.</var> pariter <lb/>Å¿ecetur æqualis <var>.t.i.</var> quæ omnia ex diametro <var>.q.d.</var> cogitari poſſunt: </s> - <s xml:space="preserve">erit igitur <var>.u.i.</var> æ-<lb/>qualis <var>.i.d.</var> <reg norm="Å¿upereritque" type="simple">Å¿upereritqÌ;</reg> quadratum <var>.q.u.</var> differentiæ <var>.a.h.</var> cognitum, hoc verò cogi-<lb/>temus diuiÅ¿um eſſe in .4. partes æquales medijs diametris <var>.p.r.</var> et <var>.n.e.</var> </s> - <s xml:space="preserve">quare <reg norm="vnaquæque" type="simple">vnaquæq;</reg> <lb/>partium cognoÅ¿cetur, & <reg norm="quadratum" type="context">quadratÅ©</reg> erit ipÅ¿ius <var>.a.K.</var> aut ipÅ¿ius <var>.K.h.</var> dimidij <var>.a.h</var>. </s> - <s xml:space="preserve">Quòd <lb/>Å¿i aliquod iÅ¿torum quadratorum detrahere voluerimus, nempe <var>.n.r.</var> ex dimidio Å¿um <lb/>mæ <var>.b.</var> duorum quadratorum <var>.q.i.</var> et <var>.i.d.</var> cognitæ, hac via procedemus, primum con <lb/>Å¿iderabimus <var>.t.r.</var> coniunctam <var>.t.i.</var> quæ quantitates erunt Å¿umma dimidij <reg norm="duorum" type="context">duorÅ©</reg> qua-<lb/>dratorum <var>.q.i.</var> et <var>.i.d.</var> quando quidem <var>.t.r.</var> <lb/><reg norm="dimidium" type="context">dimidiÅ©</reg> eÅ¿t quadrati <var>.t.l.</var> et <var>.t.i.</var> <reg norm="dimidium" type="context">dimidiÅ©</reg> <lb/> - <anchor type="figure" xlink:label="fig-0052-01a" xlink:href="fig-0052-01"/> - gnomonis <var>.t.i.l.</var> coniunctum dimidio <lb/>quadrati <var>.i.d.</var> ex quo <var>.i.t.r.</var> dimidium erit <var>.<lb/>b.</var> ex qua quantitate <var>.i.t.r.</var> cogitare debe <lb/>mus detrahi quadratum ipÅ¿ius <var>.K.h.</var> nem <lb/>pe <var>.n.r</var>: </s> - <s xml:space="preserve">quare quod Å¿upereÅ¿t cognitum <lb/>erit nempe <var>.y.s.</var> cum <var>.n.i.</var> Å¿ed <var>.y.m.</var> æqualis <lb/>eÅ¿t <var>.n.i.</var> et <var>.y.m.</var> cum <var>.y.s.</var> conÅ¿tituunt qua-<lb/>dratum <var>.p.m</var>. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> <var>.p.m.</var> quadratum & <lb/>conÅ¿equenter <var>.p.s.</var> eius radix cognoÅ¿ce-<lb/>tur, ita etiam & productum huius <var>.p.s.</var> in <var>.<lb/>s.x.</var> æqualis <var>.c.</var> nempe <var>.p.x</var>: <reg norm="eÅ¿tque" type="simple">eÅ¿tqÌ;</reg> produ-<lb/>ctum huiuÅ¿modi Å¿emper minus quantita <lb/>te <var>.r.t.i</var>: per <var>.u.i.</var> æquale quadrato minori <var>.<lb/>i.d</var>. </s> - <s xml:space="preserve">quare <var>.i.d.</var> cognoÅ¿cetur, conÅ¿equen-<lb/>ter <var>.i.</var> @q. tanquam reÅ¿iduum ex <var>.b.</var> & eo-<lb/>rum radices quadratæ cognoÅ¿centur <var>.a.<lb/>g.</var> et <var>.g.d</var>.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0052-01" xlink:href="fig-0052-01a"> - <image file="0052-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0052-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="63"> - <head xml:space="preserve">THEOREMA <num value="63">LXIII</num>.</head> - <p> - <s xml:space="preserve">IDEM præſtari hac alia via, meo iudicio poteÅ¿t. </s> - <s xml:space="preserve">Secundus numerus in <reg norm="Å¿uum" type="context">Å¿uÅ©</reg> dimi <lb/><reg norm="dium" type="context">diÅ©</reg> multiplicetur, <reg norm="productum" type="context">productÅ©</reg> autem ex dimidio primi detrahatur, ex quo re-<lb/>manens erit productum vnius quadratæ radicis in alteram partium primi numeri <lb/>quæſitarum, deinde productum hoc duplicetur, & primo numero dato coniunga-<lb/>tur, <reg norm="Å¿icque" type="simple">Å¿icqÌ;</reg> huius Å¿ummæ quadrata radix erit Å¿umma radicum quadratarum dictarum <lb/>partium, cui iuncto producto ex quadrageÅ¿imoquinto theoremate Å¿ingulæ radices <lb/>proferentur.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, primus numerus diuiÅ¿ibilis erat .50. alter verò .6. </s> - <s xml:space="preserve">Iam Å¿i multi-<lb/>plicemus .6. per .3. nempe dimidium proferetur numerus .18. quo ex dimidio pri-<lb/>mi, nempe .25. detracto, Å¿upererit .7. productum vnius radicis in alteram, quod du <lb/>plicatum dabit .14. quo coniuncto cum primo numero .50. dabitur numerus .64. <lb/>cuius quadrata radix Å¿cilicet .8. erit Å¿umma radicum duarum partium quæſitarum, <lb/>qua & producto .7. ex quadrag eÅ¿imoquinto theoremate dictæ radices diÅ¿tinguen, <lb/>tur, quarum vna erit .7. & altera <var>.I</var>.</s> - </p> - <p> - <s xml:space="preserve">Vtautem hocÅ¿peculemur, præcedenti figura vti poterimus, in qua patet <var>.t.r.</var> pro <lb/>ductum eſſe Å¿ecundi numeri <var>.c.</var> nempe <var>.a.h.</var> hoc eÅ¿t <var>.t.u.</var> in dimidio <var>.a.e.</var> Å¿cilicet <var>.p.t.</var> re-<lb/>Å¿iduum autem dimidij primi <var>.b.</var> eſſe <var>.t.i.</var> nempe <var>.a.i.</var> productum radicum, quod Å¿upple - <pb o="41" rhead="THEOREM. ARITH." n="53" file="0053" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0053"/> - mentum eÅ¿t quadrati <var>.q.d.</var> totalis. </s> - <s xml:space="preserve">Quare duplicato <var>.a.i.</var> & coniuncto <var>.b.</var> cognoÅ¿ci-<lb/>mustotum <var>.q.d.</var> & conÅ¿equenter <var>.a.d.</var> Å¿uam radicem, hoc eÅ¿t Å¿ummam duarum radi <lb/>cum <var>.a.g.</var> et <var>.g.d.</var> quæ medio <var>.a.i.</var> cognito, & quadrageÅ¿imoquinto theoremate Å¿ingu-<lb/>læ cognoÅ¿cuntur.</s> - </p> - </div> - <div type="math:theorem" level="3" n="64"> - <head xml:space="preserve">THEOREMA <num value="64">LXIIII</num>.</head> - <p> - <s xml:space="preserve">CVR propoÅ¿itum aliquem num erum in duas eiuÅ¿modi partes diuiÅ¿uri, vt Å¿um-<lb/>ma radicum dictarum partium æqualis Å¿it alteri numero propoÅ¿ito. </s> - <s xml:space="preserve">Rectè Å¿e-<lb/>cundum numerum in Å¿eipÅ¿um multiplicant, ex quo quadrato, primum datum nu-<lb/>merum detrahunt, <reg norm="rurÅ¿usque" type="simple">rurÅ¿usqÌ;</reg> reÅ¿iduum in Å¿eipÅ¿um multiplicant, & ex eo quadrato <lb/>quartam partem deÅ¿umunt, <reg norm="quam" type="context">quã</reg> ex quadrato dimidij primi numeri detrahunt, radi-<lb/>cemqÌue qua dratam reÅ¿idui cum iunxerint, & ex dimidio primi numeri detraxerint, <lb/>partes quæſitæ proferuntur.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proponeretur primus numerus .20. diuidendus et .6. Å¿ecundus <lb/>pro Å¿umma radicum, hunc Å¿ecundum .6. in Å¿eipÅ¿um multiplicabimus, <reg norm="dabiturque" type="simple">dabiturqÌ;</reg> nu-<lb/>merus .36. ex quo quadrato primus numerus detrahetur, <reg norm="Å¿upereritque" type="simple">Å¿upereritqÌ;</reg> numerus .16. <lb/>qui quadratus dabit .256. cuius numeri quarta pars Å¿umetur, nempe .64. quæ ex qua <lb/>drato dimidij primi numeri detrahetur, nempe .100. <reg norm="Å¿upereritque" type="simple">Å¿upereritqÌ;</reg> .36. cuius radix qua <lb/>drata .6. coniuncta & detracta ex .10. dabit .16. partem maiorem et .4. minorem.</s> - </p> - <p> - <s xml:space="preserve">Cuius rei hæc Å¿peculatio, primus numerus diuiÅ¿ibilis Å¿ignificetur linea <var>.a.b.</var> diui-<lb/>Å¿a in puncto <var>.e.</var> in partes adhuc incognitas, et <var>.a.c.</var> Å¿it productum <var>.a.e.</var> in <var>.e.b.</var> item <var>.q.<lb/>p.</var> Å¿ecundum numerum Å¿ignificet, æqualem Å¿ummæ radicum, quæ puncto <var>.n.</var> diÅ¿tin-<lb/>guantur. </s> - <s xml:space="preserve">PoÅ¿tmodum totum quadratum <var>.p.d.</var> erigatur (quod nobis eÅ¿t cognitum), <lb/>in duo quadrata diuiÅ¿um <var>.o.p.</var> et <var>.o.d.</var> quorum Å¿umma <var>.a.b.</var> cum detur, cognita rema-<lb/>net Å¿umma <reg norm="duorum" type="context">duorÅ©</reg> <reg norm="Å¿upplementorum" type="context">Å¿upplementorÅ©</reg> <var>.o.u.</var> et <var>.o.q.</var> qua quadrata <reg norm="cum" type="context">cÅ©</reg> fuerit dabit quadru <lb/><reg norm="plum" type="context">plÅ©</reg> quadrati <reg norm="Å¿upplementi" type="context">Å¿upplemẽti</reg> <var>.o.q.</var> <reg norm="nempe" type="context">nẽpe</reg> <reg norm="quadruplum" type="context">quadruplÅ©</reg> producti <var>.a.c.</var> etenim <var>.a.c.</var> ex .19. theo <lb/>remate huius libri quadratum eft ipÅ¿ius <var>.q.o.</var> <reg norm="Å¿icque" type="simple">Å¿icqÌ;</reg> poterant etiam veteres quadrare <lb/>dimidium differentiæ <var>.a.b.</var> ab <var>.p.d.</var> nempe quadrato tantummodo Å¿upplemento <var>.q.<lb/>o</var>. </s> - <s xml:space="preserve">Tunc habito <var>.a.c.</var> eius ope tanquam producti <var>.a.e.</var> in <var>.e.b.</var> ex .45. theoremate Å¿ingu <lb/>læ partes cognoÅ¿centur.</s> - </p> - <p> - <s xml:space="preserve">Quod alia etiam ratione præſtari poterat, nempe cognito Å¿upplemento <var>.<lb/>q.o.</var> diÅ¿tinguendæ radices <var>q.n.</var> et <var>.n.p.</var> ex .45. theoremate, quibus cognitis, eorum <lb/>etiam quadrata cognoÅ¿cuntur.</s> - </p> - <figure position="here"> - <image file="0053-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0053-01"/> - </figure> - <figure position="here"> - <image file="0053-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0053-02"/> - </figure> - <pb o="42" rhead="IO. BAPT. BENED." n="54" file="0054" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0054"/> - </div> - <div type="math:theorem" level="3" n="65"> - <head xml:space="preserve">THEOREMA <num value="65">LXV</num>.</head> - <p> - <s xml:space="preserve">CVR propoÅ¿ito numero in tres qualeÅ¿cunque partes diuiÅ¿o, Å¿i prima in <lb/>tertiam multiplicetur, & huic producto, Å¿ecundæ in primam productum <lb/>coniungatur, <reg norm="itemque" type="simple">itemqÌ;</reg> Å¿ecundæ in tertiam, hæc Å¿umma duplicata æqualis Å¿it Å¿ummæ <lb/>productorum Å¿ingularum in cæteras duas.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proponatur .20. diuiÅ¿us in tres partes nempe .12. 5. 3. multipli-<lb/>cato primo .12. per .3. tertiam partem dabitur .36. Å¿ecunda verò multiplicata per re <lb/>liquas duas, hoc eÅ¿t .5. per .12. et .3. in primis dabitur .60. poÅ¿tea .15. <reg norm="quorum" type="context">quorÅ©</reg> <reg norm="trium" type="context">triÅ©</reg> pro <lb/>ductorum Å¿umma erit .111. quæ duplicata dabit .222. qui numerus æqualis eſſe di-<lb/>citur Å¿ummæ productorum Å¿ingularum partium in reliquas duas, nempe Å¿ummæ .60. <lb/>36. 60. 15. 36. 15. hoc eÅ¿t ipÅ¿is .222.</s> - </p> - <p> - <s xml:space="preserve">Cuius rei per Å¿e patet Å¿peculatio, cum in his Å¿ex vltimis productis, Å¿ingula tria <lb/>prima duplicentur.</s> - </p> - </div> - <div type="math:theorem" level="3" n="66"> - <head xml:space="preserve">THEOREMA <num value="66">LXVI</num>.</head> - <p> - <s xml:space="preserve">CVR propoÅ¿ito numero in .3. qualeÅ¿cunque partes diuiÅ¿o, Å¿i in reliquas duas Å¿in-<lb/>gulæ multiplicentur, & hæc producta cum Å¿umma Å¿uorum quadratorum con-<lb/>iungantur, tota Å¿umma hæc vltima æqualis erit quadrato totali propoÅ¿iti numeri.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i fuerit idem numerus .20. in .3. partes diuiÅ¿us .12. 5. 3. </s> - <s xml:space="preserve">Si .12. in <lb/>5. et .3. producatur, Å¿umma productorum erit .96. at .5. in .12. et .3. erit .75. poÅ¿tmo-<lb/>dum .3. in .12. et .5. erit .51. nempe in vniuerÅ¿um .222. quadratorum porrò Å¿umma <lb/>erit .178 quæ coniuncta .222. dabit .400. quadratum ipÅ¿ius .20.</s> - </p> - <p> - <s xml:space="preserve">Erit autem huiuÅ¿ce rei facillima Å¿peculatio, Å¿i Å¿equentem figuram mente conce-<lb/>perimus, in qua <var>.a.b.</var> propoÅ¿itum numerum Å¿ignificet, cuius partes diÅ¿tinctæ Å¿int me-<lb/>dio <var>.e.</var> et <var>.c</var>. </s> - <s xml:space="preserve">Ip Å¿um autem <var>.q.b.</var> Å¿it quadratum <lb/>totale parallelis <var>.e.s.</var> et <var>.c.x.</var> diuiÅ¿um, quæ qua <lb/> - <anchor type="figure" xlink:label="fig-0054-01a" xlink:href="fig-0054-01"/> - dratum in triarectangula diuident, quorum <lb/>primum erit <var>.q.e.</var> compoÅ¿itum ex producto <var>.a.<lb/>e.</var> in Å¿emetipÅ¿am, nempe quadratum <var>.o.e.</var> & <lb/>ex producto eiuÅ¿dem <var>.a.e.</var> in <var>.e.b.</var> quod erit re <lb/>ctangulum <var>.o.s.</var> ex quo tria rectangula <var>.o.s.</var> et <var>.<lb/>n.x.</var> et <var>.t.u.</var> tria producta erunt Å¿ingularum par <lb/>tium in cæteras duas, et <var>.e.o</var>: <var>c.n</var>: <var>b.t.</var> tria qua-<lb/>drata erunt: </s> - <s xml:space="preserve">quibus Å¿ex quantitatibus quadra <lb/>tum totale <var>.q.b.</var> completur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0054-01" xlink:href="fig-0054-01a"> - <image file="0054-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0054-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="67"> - <head xml:space="preserve">THEOREMA <num value="67">LXVII</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">VEteres</emph> aliud quoque problema indefinitum propoÅ¿uerunt, quod tamen à <lb/>nobis determinabitur.</s> - </p> - <p> - <s xml:space="preserve">Cur diuiÅ¿uri propoÅ¿itum numerum in duas eiuÅ¿modi partes, vt mutuò diuiÅ¿is, & <lb/>per Å¿ummam prouenientium diuiÅ¿a Å¿umma qua dratorum partium, oriatur proue-<lb/>niens alter numerus propoÅ¿itus.</s> - </p> - <p> - <s xml:space="preserve">PropoÅ¿ito deinde tertio quolibet numero diuidendo per Å¿ingulas partes primi, - <pb o="43" rhead="THEOREM. ARIT." n="55" file="0055" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0055"/> - ita vt Å¿imul prouenientibus in Å¿ummam collectis huius fummæ ad primum nume-<lb/>rum propoÅ¿itum proportio futura Å¿it ea quæ eÅ¿t tertij ad Å¿ecundum. </s> - <s xml:space="preserve">Rectè dimidium <lb/>primi numeri in Å¿eipÅ¿um multiplicant, ex quo quadrato Å¿ecundum numerum detra <lb/>hunt, tum reÅ¿idui radicem Å¿umunt, quam iungentes, & detrahentes ex dimidio <lb/>primi, partes quæſitas habent, cætera ex neceÅ¿sitate Å¿ubÅ¿equuntur, prout nunc a <lb/>me docebitur.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, proponitur numerus .20. in duas partes diuidendus, quibus po <lb/>Å¿tea mutuò diuiÅ¿is, & per Å¿ummam prouenientium diuiÅ¿a Å¿umma quadratorum, <lb/>dent <reg norm="Å¿ecundum" type="context">Å¿ecundÅ©</reg> numerum propoÅ¿itum .36. nam reliqua conÅ¿equuntur. </s> - <s xml:space="preserve">Itaque .10. <lb/>dimidium primi in Å¿eipÅ¿um multiplicatur, & ex quadrato .100. eruitur numerus .36. <lb/>nempe Å¿ecundus propoÅ¿itus reÅ¿idui porrò .64. quadrata radix .8. fumitur, quam con <lb/>iungimus & detrahimus ex dimidio primi Å¿cilicet .10. ex quo partes quæſitæ dabun <lb/>tur .18. et .2. quæ mutuo diuiſæ dabunt Å¿uorum prouenientium Å¿ummam .9. cum no-<lb/>na parte, per quam diuidentes .328. Å¿ummam quadratorum ipÅ¿arum partium, <lb/>exactè dabitur numerus .36. qui fuit Å¿ecundò propoÅ¿itus. </s> - <s xml:space="preserve">Tum Å¿i per Å¿ingu-<lb/>las iam inuentas partes quilibet numerus diuiÅ¿us fuerit, verbi gratia .72. Å¿umma pro <lb/>uenientium erit .40. qui num@rus eandem proportionem cum primo nempe .20. Å¿er <lb/>uabit, quam tertius propoÅ¿itus .72. cum Å¿ecundo .36.</s> - </p> - <p> - <s xml:space="preserve">Quod vt Å¿peculemur, primus numerus Å¿ignificetur linea <var>.n.e.</var> ita diuidendus à <lb/>puncto <var>.o.</var> vt diuiÅ¿a parte <var>.n.o.</var> per <var>.o.e.</var> et <var>.o.e.</var> per <var>.n.o.</var> & per Å¿ummam prouenien-<lb/>tium diuiÅ¿a Å¿umma quadratorum <var>.n.o.</var> et <var>.o.e.</var> detur Å¿ecundus numerus notatus linea <var type="line">.<lb/>q.K</var>. </s> - <s xml:space="preserve">Porrò meminiſſe oportet quòd .26. theoremate probatum fuit vltimum hoc <lb/>proueniens æquale producto partium inter Å¿e futurum, nempe producto <var>.n.o.</var> in <var>.o.<lb/>e.</var> quod Å¿ignificetur rectangulo <var>.n.e</var>. </s> - <s xml:space="preserve">Itaque datis <var>.n.e.</var> et <var>.q.K.</var> Å¿i .45. theorema conÅ¿u-<lb/>luerimus, partes <var>.n.o.</var> et <var>.o.e.</var> cognoÅ¿cemus.</s> - </p> - <p> - <s xml:space="preserve">Proponitur deinde tertius quilibetnumerus, verbi gratia <var>.x.</var> diuidendus per <var>.o.e.</var> <lb/>et <var>.o.n.</var> qui Å¿i diuidatur per <var>.o.e.</var> dabit pro <lb/>ueniens <var>.b.o</var>. </s> - <s xml:space="preserve">Si verò per <var>.n.o.</var> proueniens <lb/> - <anchor type="figure" xlink:label="fig-0055-01a" xlink:href="fig-0055-01"/> - erit <var>.d.n.</var> nunc aſſerimus <reg norm="Å¿ummam" type="context">Å¿ummã</reg> duorum <lb/>horum prouenientium, Å¿ic primo nume-<lb/>ro <var>.n.e.</var> dato proportionatam eſſe, Å¿icut <lb/>tertius <var>.x.</var> <reg norm="Å¿ecundo" type="context">Å¿ecÅ©do</reg> <var>.q.K</var>. </s> - <s xml:space="preserve">Producatur enim li-<lb/>nea <var>.d.n.</var> donec <var>.n.q.</var> æqualis Å¿it <var>.o.b.</var> ex <lb/>quo <var>.q.d.</var> erit Å¿umma vltimò prouenien-<lb/>tium: </s> - <s xml:space="preserve">item producatur <var>.e.n.</var> donec <var>.n.u.</var> æ-<lb/>qualis Å¿it <var>.o.e.</var> <reg norm="termineturque" type="simple">termineturqÌ</reg> rectangulum <var>.<lb/>q.u.</var> quod tertio numero propoÅ¿ito <var>.x.</var> vt <lb/>patet, æquale erit, </s> - <s xml:space="preserve">quare ex .15. Å¿exti aut .<lb/>20. Å¿eptimi eadem erit proportio <var>.d.n.</var> ad <lb/><var>n.q.</var> quæ <var>.u.n.</var> nempe <var>.o.e.</var> ad <var>.o.n.</var> & com-<lb/>ponendo <var>.d.q.</var> ad <var>.q.n.</var> Å¿icut <var>.e.n.</var> ad <var>.n.o.</var> & <lb/>permutando <var>.d.q.</var> ad <var>.e.n.</var> quæ <var>.q.n.</var> hoc eÅ¿t <var>.<lb/>b.o.</var> ad <var>.o.n.</var> nempe Å¿icut <var>.b.e.</var> ad <var>.e.n.</var> Å¿uperficialem, ex prima Å¿exti aut .18. vel .19. <lb/>Å¿eptimi, Å¿ed rectangulum <var>.e.n.</var> conÅ¿titutum fuit æquale numero <var>.q.K</var>. </s> - <s xml:space="preserve">itaque verum <lb/>eÅ¿t propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0055-01" xlink:href="fig-0055-01a"> - <image file="0055-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0055-01"/> - </figure> - </div> - <pb o="44" rhead="IO. BAPT. BENED." n="56" file="0056" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0056"/> - </div> - <div type="math:theorem" level="3" n="68"> - <head xml:space="preserve">THEOREMA <num value="68">LXVIII</num>.</head> - <p> - <s xml:space="preserve">CVR numero per numerum diuiÅ¿o, <reg norm="productoque" type="simple">productoqÌ;</reg> duorum numerorum per pro-<lb/>ueniens multiplicato, quod vltimò productum eÅ¿t, diuiÅ¿i numeri Å¿emper qua <lb/>dratum exiÅ¿tat.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i diuidamus .10. per .2. proueniens erit .5. quo producto ex duo <lb/>bus numeris multiplicato, nempe .20. habe <lb/>bimus .100. quadratum numeri diuiÅ¿i.</s> - </p> - <figure position="here"> - <image file="0056-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0056-01"/> - </figure> - <p> - <s xml:space="preserve">Cuius gratia duo numeri Å¿int <var>.a.</var> et <var>.e.</var> por <lb/>rò <var>.a.</var> per <var>.e.</var> diuiÅ¿o detur <var>.u.</var> tum <var>.o.</var> produ-<lb/>ctum <var>.a.</var> in <var>.e.</var> eſſe conÅ¿tituatur, quo per <var>.u.</var> <lb/>multiplicato dabitur <var>.x.</var> quadratum <var>.a.</var> pro-<lb/>ptereà quòd <var>.a.</var> medium eÅ¿t proportionale <lb/>inter <var>.o.</var> et <var>.u.</var> ex .35. theoremate. </s> - <s xml:space="preserve">itaque <lb/>ex .16. Å¿exti aut .20. Å¿eptimi, propoÅ¿iti veri-<lb/>tas eluceÅ¿cet.</s> - </p> - </div> - <div type="math:theorem" level="3" n="69"> - <head xml:space="preserve">THEOREMA <num value="69">LXIX</num>.</head> - <p> - <s xml:space="preserve">CVR numero aliquo per duos alios multiplicato & diuiÅ¿o, Å¿i per horum duo-<lb/>rum productum, Å¿umma duorum primorum productorum diuiÅ¿a fuerit, vl-<lb/>timum proueniens, Å¿ummæ duorum primorum prouenientium æquale Å¿it.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, proponitur numerus .24. per .8. et .6. multiplicandus & diuiden <lb/>dus Å¿umma productorum crit .336. prouenientium autem .7. Å¿i igitur Å¿ummam .336. <lb/>productorum per productum duorum Å¿ecundorum numerorum nempe .48. diuiÅ¿e-<lb/>rimus, proueniens pariter erit .7.</s> - </p> - <p> - <s xml:space="preserve">In cuius <reg norm="gratiam" type="context">gratiã</reg> primus numerus Å¿ignificetur linea <var>.q.b.</var> multiplicandus & diuiden-<lb/>dus numeris deÅ¿ignatis per <var>.k.m.</var> et <var>.y.m.</var> productorum Å¿umma Å¿it <var>.k.z.</var> prouenien-<lb/>tium autem <var>.a.e</var>: et <var>.a.o.</var> ex <var>.k.m.</var> et <var>.o.e.</var> ex <var>.y.m</var>: tum productum <var>.k.m.</var> in <var>.m.y.</var> Å¿it <var>.f.<lb/>m</var>. </s> - <s xml:space="preserve">Dico quòd Å¿i <var>.k.z.</var> per <var>.f.m.</var> diuiÅ¿erimus proueni et <var>.a.e</var>. </s> - <s xml:space="preserve">Quod cum Å¿ic fuerit, erit <lb/>quoque verum quòd diuiÅ¿a <var>.k.z.</var> per <var>.a.e.</var> proueniet <var>.f.m.</var> numerus Å¿cilicet æqualis <lb/>numero <var>.f.m.</var> ex .13. theoremate huius. </s> - <s xml:space="preserve">Itaque quotieÅ¿cunque probauero quòd di-<lb/>uiÅ¿a <var>.k.z.</var> per <var>.a.e.</var> proueniat numerus æqualis ipÅ¿i <var>.f.m.</var> propoÅ¿itum verum eſſe con <lb/>Å¿equetur. ex .13. theoremate. </s> - <s xml:space="preserve">Quòd Å¿i proueniens ex diuiÅ¿ione <var>.k.z.</var> per <var>.a.e.</var> æqua <lb/>le fuerit <var>.f.m.</var> patet ex .7. quinti quòd <reg norm="eadem" type="context">eadẽ</reg> erit proportio numeri <var>.k.m.y.</var> ad ipÅ¿um <lb/>proueniens, quæ ad numerum <var>.f.m</var>. </s> - <s xml:space="preserve">Cogitemus <reg norm="itaque" type="simple">itaq;</reg> <var>.k.u.</var> æqualem <var>.a.e.</var> Å¿uper quam <lb/>mente concipiamus rectangulum <var>.u.p.</var> æqualem <var>.k.z.</var> ex quo eadem erit proportio <var>.<lb/>k.p.</var> ad <var>.k.y.</var> quæ <var>.g.k.</var> ad <var>.k.u.</var> ex .15. Å¿exti, aut, 20. Å¿eptimi, numerus autem <var>.k.p.</var> erit <lb/>proueniens, quod probandum eÅ¿t æquale eſſe <var>.f.m</var>.</s> - </p> - <p> - <s xml:space="preserve">Probabitur autem Å¿ic, ex .9. quinti, nempe demonÅ¿trato quòd numerus <var>.k.p.</var> ean <lb/>dem proportionem habeat ad numerum <var>.k.y.</var> quam habet numerus <var>.f.m.</var> ad eundem <lb/><var>k.y</var>. </s> - <s xml:space="preserve">Sed probatum eÅ¿t Å¿ic Å¿e habere <var>.k.g.</var> ad <var>.k.u.</var> Å¿icut <var>.k.p.</var> ad <var>.k.y.</var> Å¿ufficiet igitur pro-<lb/>bare Å¿ic Å¿e habere <var>.k.g.</var> ad <var>.k.u.</var> Å¿icut <var>.f.m.</var> ad <var>.k.y</var>. </s> - <s xml:space="preserve">Sed <var>.k.g.</var> dicitur æqualis eſſe <var>.q.b</var>: et <var>.k.</var> <lb/>u; </s> - <s xml:space="preserve">a.e. Å¿atis erit igitur probare ita Å¿e habere <var>.q.b.</var> ad <var>.a.e.</var> Å¿icut <var>.f.m.</var> ad <var>.k.y</var>. </s> - <s xml:space="preserve">Scimus au-<lb/>tem quòd eadem eÅ¿t proportio <var>.q.b.</var> ad <var>.a.o.</var> quæ <var>.m.k.</var> ad vnitatem, quæ Å¿it <var>.x.</var> & quod <lb/>proportio <var>.o.e.</var> ad <var>.q.b.</var> eadem eÅ¿t, quæ <var>.x.</var> ad <var>.m.y.</var> ex definitione diuiÅ¿ionis. </s> - <s xml:space="preserve">Quare <lb/>ex æqualitate proportionum eadem erit proportio <var>.k.m.</var> ad <var>.m.y.</var> quæ <var>.e.o.</var> ad <var>.o.a.</var> & - <pb o="45" rhead="THEOREM. ARIT." n="57" file="0057" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0057"/> - componendo Å¿ic Å¿e habebit <var>.k.y.</var> ad <var>.m.y.</var> Å¿icut <var>.e.a.</var> ad <var>.o.a.</var> & permutando <var>.k.y.</var> ad <var>.e.<lb/>a.</var> Å¿icut <var>.m.y.</var> ad <var>.o.a.</var> & ex .19. quinti ita <var>.k.m.</var> ad <var>.e.o.</var> Å¿icut <var>.k.y.</var> ad <var>.e.a.</var> & permutando <var>.<lb/>k.m.</var> ad <var>.k.y.</var> Å¿icut <var>.e.o.</var> ad <var>.e.a</var>. </s> - <s xml:space="preserve">Nunc producatur <var>.f.t.</var> donec <var>.t.i.</var> æqualis Å¿it <var>.k.y.</var> <reg norm="produ- ctaque" type="simple">produ-<lb/>ctaqÌ;</reg> <var>.m.t.</var> done <var>c.t.s.</var> æqualis Å¿it vnitati <var>.x.</var> <reg norm="termineturque" type="simple">termineturqÌ;</reg> rectangulum <var>.s.i.</var> ex quo da-<lb/>bitur proportio numeri <var>.f.m.</var> ad numerum <var>.s.i.</var> compoÅ¿ita ex <var>.m.t.</var> ad <var>.t.s.</var> et <var>.f.t.</var> ad <var>.t.i.</var> <lb/>ex .24. Å¿exti, aut quinta octaui, Å¿ed ita etiam proportio <var>.q.b.</var> ad <var>.a.e.</var> componitur ex <lb/>eiÅ¿dem proportionibus, nempe ex <var>.q.b.</var> ad <var>.o.e.</var> æquali <var>.m.t.</var> ad <var>.t.s.</var> & ex proportione <var>.<lb/>o.e.</var> ad <var>.a.e.</var> æquali <var>.f.t.</var> ad <var>.t.i.</var> ita que proportio numeri <var>.f.m.</var> ad <var>.s.i.</var> hoc eÅ¿t ad <reg norm="numerum" type="context">numerÅ©</reg> <lb/>ipÅ¿ius <var>.k.y.</var> Ä™qualis eÅ¿t proportioni numeri <var>.q.b.</var> ad <var>.a.e.</var> <reg norm="nempe" type="context">nẽpe</reg> <var>.k.g.</var> ad <var>.k.u.</var> hoc eÅ¿t <var>.k.p.</var> ad <lb/><var>x.y.</var> ex quo Å¿equitur <var>.k.p.</var> conÅ¿tare numero Ä™quali <var>.f.m.</var> proueniens igitur ex diuiÅ¿ione <lb/>numeri <var>.k.z.</var> per <var>.f.m.</var> æquale eÅ¿t numero ipÅ¿ius <var>.a.e</var>.</s> - </p> - <figure position="here"> - <image file="0057-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0057-01"/> - </figure> - </div> - <div type="math:theorem" level="3" n="70"> - <head xml:space="preserve">THEOREMA <num value="70">LXX</num>.</head> - <p> - <s xml:space="preserve">HAEC porrò concluÅ¿io alia etiam via demonÅ¿trari poteÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Significetur numerus diuidendus atque multiplicandus linea <var>.b.a</var>. </s> - <s xml:space="preserve">Deinde <lb/>diuidentes & <reg norm="multiplicantes" type="context">multiplicãtes</reg> Å¿int <var>.k.m.</var> et <var>.m.y.</var> prouenientia ex diuiÅ¿ione Å¿int <var>.a.o.</var> et <var>.o.<lb/>e.</var> atque <var>.a.o.</var> ex <var>.m.y</var>: <var>o.e.</var> verò ex <var>.k.m.</var> proueniat, quorum Å¿umma Å¿it <var>.a.e</var>: productum <lb/>autem <var>.b.a.</var> in <var>.k.m.</var> Å¿it <var>.b.p.</var> et <var>.p.s.</var> productum <var>.b.a.</var> in <var>.m.y.</var> ad hæc rectangulum <var>.k.y.</var> Å¿it <lb/>productum <var>.k.m.</var> in <var>.m.y</var>: quo to-<lb/>tum productum <var>.a.s.</var> diuidatur, pro <lb/> - <anchor type="figure" xlink:label="fig-0057-02a" xlink:href="fig-0057-02"/> - <reg norm="ueniensque" type="simple">ueniensqÌ;</reg> Å¿it <var>.a.c.</var> cui, <var>a.c</var>: <reg norm="productum" type="context">productÅ©</reg> <var>.<lb/>a.s.</var> <reg norm="eandem" type="context context">eãdẽ</reg> <reg norm="proportionem" type="context">proportionẽ</reg> Å¿eruabit, <reg norm="quam" type="context">quã</reg> <lb/><var>k.y.</var> rectangulum ad vnitatem ex <lb/>definitione diuiÅ¿ionis, hoc autem <lb/>proueniens <var>.a.c.</var> <reg norm="conÅ¿tare" type="context">cõſtare</reg> numero æ-<lb/>quali aſſero Å¿ummæ <var>.a.e</var>. </s> - <s xml:space="preserve">Primum <lb/>enim ex dicta definitione diuiÅ¿io-<lb/>nis habemus eandem eſſe propor-<lb/>tionem <var>.b.a.</var> ad <var>.a.o.</var> quæ <var>.m.y.</var> ad <lb/>vnitatem, & quod Å¿ic Å¿e habet <var>.b.a.</var> <lb/>ad <var>.o.e.</var> Å¿icut <var>.k.m.</var> ad eandem vnita <lb/>tem. </s> - <s xml:space="preserve">Itaque vnitas hæc linearis Å¿i-<lb/>gnificetur per <var>.m.x.</var> in Å¿ingulis late-<lb/>ribus <var>.k.m.</var> et <var>.m.y.</var> producentibus rectangulum <var>.k.y</var>: Å¿uperficialis autem vnitas Å¿it. - <pb o="46" rhead="IO. BAPT. BENED." n="58" file="0058" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0058"/> - <var>g.m.</var> <reg norm="cogiteturque" type="simple">cogiteturqÌ;</reg> rectangulum <var>.y.x.</var> & rectangulum <var>.k.x</var>. </s> - <s xml:space="preserve">Itaque dabitur eadem pro <lb/>portio <var>.k.m.</var> ad <var>.m.x.</var> nempe <var>.k.x.</var> rectanguli ad <var>.m.g.</var> quæ eÅ¿t <var>.b.a.</var> ad <var>.o.e.</var> et <var>.y.x.</var> ad <var>.m.<lb/>g.</var> quæ <var>.b.a.</var> ad <var>.a.o.</var> Å¿ed ex prima Å¿exti aut .18. vel .19. Å¿eptimi, Å¿ic Å¿e habet rectangu-<lb/>lum <var>.k.y.</var> ad <var>.x.y.</var> Å¿icut <var>.k.m.</var> ad <var>.m.x.</var> </s> - <s xml:space="preserve">quare Å¿icut <var>.b.a.</var> ad <var>.o.e.</var> ex .11. quinti, & eiuÅ¿dem <lb/>rectanguli <var>.k.y.</var> ad rectangulum <var>.k.x.</var> Å¿icut <var>.y.m.</var> ad <var>.x.m.</var> nempe <var>.b.a.</var> ad <var>.a.o</var>. </s> - <s xml:space="preserve">Quare <lb/>ex communi Å¿cientia, Å¿ic Å¿e habebit duplum rectanguli <var>.k.y.</var> ad Å¿ummam <var>.y.x.</var> cum <var>.<lb/>k.x.</var> rectangulorum, Å¿icut duplum <var>.b.a.</var> ad Å¿ummam <var>.a.o.e.</var> et proportio Å¿ummæ re-<lb/>ctangulorum <var>.y.x.</var> et <var>.k.x.</var> duplo <var>.g.m.</var> Å¿icut duplum <var>.b.a.</var> ad <var>.a.o.e</var>. </s> - <s xml:space="preserve">Igitur Å¿umma duo-<lb/>rum rectangulorum <var>.y.x.</var> et <var>.x.k.</var> media proportionalis erit inter duplum rectanguli <var>.<lb/>k.y.</var> & duplum vnitatis Å¿uperÅ¿icialis <var>.g.m</var>. </s> - <s xml:space="preserve">Nunc terminetur rectangulum <var>.a.r.</var> ex quo <lb/>dabitur eadem proportio dupli <var>.a.s.</var> ad <var>.a.r.</var> Å¿icut dupli <var>.b.a.</var> ad <var>.a.e.</var> ex propoÅ¿itioni-<lb/>bus notatis, Å¿exti aut Å¿eptimi. </s> - <s xml:space="preserve">Quare etiam Å¿icut dupli rectanguli <var>.k.y.</var> ad <reg norm="Å¿ummam" type="context">Å¿ummã</reg> <lb/>rectangulorum <var>.y.x.</var> et <var>.k.x</var>. </s> - <s xml:space="preserve">Iam verò Å¿i conÅ¿tituatur <var>.e.c.</var> pro vnitate lineari ipÅ¿ius <var>.<lb/>e.r.</var> certi erimus numerum <var>.a.c.</var> æqualem eſſe <var>.a.e.</var> & proportionem <var>.r.e.</var> ad <var>.e.c.</var> hoc <lb/>eÅ¿t <var>.a.r.</var> ad <var>.a.c.</var> eandem quæ <var>.y.x.</var> et <var>.x.k.</var> rectangulorum ad <var>.m.g.</var> ex prædictis rationi-<lb/>bus, & ex hypotheÅ¿i, nempe quòd <var>.<lb/>e.r.</var> æqualis Å¿it numero <var>.k.m.y.</var> <lb/> - <anchor type="figure" xlink:label="fig-0058-01a" xlink:href="fig-0058-01"/> - hoc eÅ¿t rectangulorum <var>.y.x.</var> et <var>.x.<lb/>k</var>. </s> - <s xml:space="preserve">Quamobrem <var>.a.r.</var> ex communi <lb/>Å¿cientia <reg norm="medium" type="context">mediÅ©</reg> proportionale erit <lb/>inter duplum <var>.a.s.</var> & duplum <var>.a.c.</var> <reg norm="ea demque" type="context simple">eaÂ<lb/>dẽqÌ;</reg> <reg norm="proportio" type="simple">ê“portio</reg> dupli prÄ™dicti <var>.a.s.</var> ad <lb/>duplum <var>.a.c.</var> ex æqualitate propor-<lb/>tionum Å¿imul collectarum, eadem <lb/>erit qùæ proportio dupli rectangu-<lb/>li <var>.k.y.</var> ad duplum <var>.m.g.</var> hoc eÅ¿t <var>.a.s.</var> <lb/>Å¿implicis ad Å¿implicem <var>.a.c.</var> quæ Å¿im <lb/>plicis rectanguli <var>.k.y.</var> ad Å¿implicem <lb/>vnitatem <var>.g.m.</var> Å¿ic enim Å¿e habet Å¿im <lb/>plex ad Å¿implex, Å¿icut duplum ad <lb/>duplum. </s> - <s xml:space="preserve">Sed pariter ita Å¿e habet <var>.a.s.</var> ad <var>.a.</var> c<unsure/>. cogitato <var>.a.c.</var> tamquam proueniente <lb/>ex diuiÅ¿ione <var>.a.s.</var> per rectangulum <var>.k.y.</var> vt conÅ¿titutum eÅ¿t, Å¿icut <var>.k.y.</var> ad <var>.m.g.</var> ex defi-<lb/>nitione diuiÅ¿ionis vt iam dictum eÅ¿t, </s> - <s xml:space="preserve">quare numerus <var>.a.c.</var> æqualis erit numero <var>.a.o.e</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0057-02" xlink:href="fig-0057-02a"> - <image file="0057-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0057-02"/> - </figure> - <figure xlink:label="fig-0058-01" xlink:href="fig-0058-01a"> - <image file="0058-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0058-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="71"> - <head xml:space="preserve">THEOREMA <num value="71">LXXI</num>.</head> - <p> - <s xml:space="preserve">CVR propoÅ¿itis .4. numeris, duobus nempe diuidentibus ac duobus diuiden-<lb/>dis, Å¿i <reg norm="adinuicem" type="context">adinuicẽ</reg> diuiÅ¿i fuerint, <reg norm="duoque" type="simple">duoqÌ;</reg> <reg norm="prouenientia" type="context">proueniẽtia</reg> <reg norm="inuicem" type="context">inuicẽ</reg> multiplicata <reg norm="quenuis" type="context">quẽuis</reg> nu <lb/>merum producant, qui Å¿eruetur, Å¿i deinde ijdem numeri verÅ¿a vice mutuo diuiÅ¿i fue <lb/>rint, & inter Å¿e multiplicata prouenientia, <reg norm="productum" type="context">productÅ©</reg> hoc, primo Å¿eruato numero <lb/>æquale erit.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia propoÅ¿itis his .4. numeris .20. 30. 5. 10. duo autem .20. Å¿cilicet <lb/>et .30. Å¿int numeri diuidendi, porrò .5. et .10. numeri diuidentes, <reg norm="nempe" type="context">nẽpe</reg> vt primo .20 <lb/>per .5. diuidatur, tum .30. per .10. producetur .4. et .3. qui Å¿imul multiplicati <reg norm="proferent" type="context">proferẽt</reg> <num value="12">.<lb/>12.</num> tum .20. per .10. d iuiÅ¿o et .30. per .5. prouenientia erunt .2. 6. quæ inter Å¿e multi-<lb/>plicata producent etiam .12.</s> - </p> - <pb o="47" rhead="THEOR. ARITH." n="59" file="0059" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0059"/> - <p> - <s xml:space="preserve">Cuius rationem Å¿i quæris, Å¿ignificentur .4. numeri lineis, <var>a.e.o.u.</var> <reg norm="diuidaturque" type="simple">diuidaturqÌ;</reg> .2. <lb/>per <var>.o.</var> & <reg norm="oriatur" type="simple">oriat̃</reg>. s. & per <var>.u.</var> <reg norm="oriatur" type="simple">oriat̃</reg> <var>.y.</var> et <var>.<lb/> - <anchor type="figure" xlink:label="fig-0059-01a" xlink:href="fig-0059-01"/> - e.</var> diuiÅ¿o per <var>.o.</var> oriatur <var>.z.</var> & per <var>.u.</var> <lb/>proueniat <var>.f.</var> tum <var>.n.</var> Å¿it productum <var>.z.</var> <lb/>in <var>.y.</var> et <var>.m.</var> productum <var>.s.</var> in <var>.f</var>. </s> - <s xml:space="preserve">Dico <lb/>n. futurum æquale <var>.m</var>. </s> - <s xml:space="preserve">Sit deinde <var>.<lb/>x.</var> vnitas, quare ex definitione diui-<lb/>Å¿ionis eadem erit proportio <var>.s.</var> ad <var>.a.</var> <lb/>et <var>.z.</var> ad <var>.e.</var> quæ <var>.x.</var> ad <var>.o</var>. </s> - <s xml:space="preserve">Sed ita Å¿e ha-<lb/>bet <var>.a.</var> ad <var>.y.</var> et <var>.e.</var> ad <var>.f.</var> Å¿icut <var>.u.</var> ad <var>.x.</var> ex <lb/>quo Å¿ic Å¿e habebit <var>.s.</var> ad <var>.a.</var> Å¿icut <var>.z.</var> ad <lb/>e. et <var>.a.</var> ad. y, Å¿icut <var>.e.</var> ad <var>.f</var>. </s> - <s xml:space="preserve">Itaque ex <lb/>æqualitate proportionum Å¿ic Å¿e ha-<lb/>bebit s. ad <var>.y.</var> Å¿icut <var>.z.</var> ad <var>.f</var>. </s> - <s xml:space="preserve">Igitur ex <lb/>15. Å¿exti aut .20. Å¿eptimi productum <var>.<lb/>n.</var> producto <var>.m.</var> æquale erit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0059-01" xlink:href="fig-0059-01a"> - <image file="0059-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0059-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="72"> - <head xml:space="preserve">THEOREMA <num value="72">LXXII</num>.</head> - <p> - <s xml:space="preserve">ALIVD quoque problema à me inuentum eÅ¿t, nempe vt proponantur .4. <lb/>numeri qualeÅ¿cunque tandem, quorum duo diuiÅ¿ibiles Å¿int, tertius diuiÅ¿or <lb/>vnius è duobus pro libito, <reg norm="quæramusque" type="simple">quæramusqÌ;</reg> alterius diuidentem, qui Å¿ic Å¿e habeat vt pro <lb/>ductum duorum prouenientium quarto numero propoÅ¿ito Å¿it æquale.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, proponuntur .4. numeri .20. 48. 5. 12. porrò .20. et .48. numeri <lb/>Å¿int diuiÅ¿ibiles et .5. <reg norm="diuidens" type="context">diuidẽs</reg> vnius, ut potè .20. </s> - <s xml:space="preserve"><reg norm="Quærendus" type="context">Quærẽdus</reg> nunc erit diuidens alterius <lb/>nempe .48. eiuÅ¿modi vt productum prouenientium æquale Å¿it .12. </s> - <s xml:space="preserve">Diuidam itaque <num value="20">.<lb/>20.</num> per .5. <reg norm="prouenietque" type="simple">prouenietqÌ;</reg> 4. quem per .48. multiplicabo, nempe per alterum diuiÅ¿ibi-<lb/>lem, <reg norm="Å¿icque" type="simple">Å¿icqÌ;</reg> proueniet .192. quod productum per quartum numerum nempe .12. diui-<lb/>fum dabit .16. qui erit diuidens quæſitus, quo diuiÅ¿o .48. proueniet .3. Å¿ecundum Å¿ci <lb/>licet proueniens, quo per alterum hoc eÅ¿t .4. multiplicato producetur quartus nu-<lb/>merus .12.</s> - </p> - <p> - <s xml:space="preserve">Quod vt Å¿ciamus, primus nume-<lb/>rus diuiÅ¿ibilis Å¿ignificetur <reg norm="rectangulo" type="context">rectãgulo</reg> <var>.<lb/> - <anchor type="figure" xlink:label="fig-0059-02a" xlink:href="fig-0059-02"/> - a.i.</var> Å¿ecundus rectangulo <var>.o.u.</var> primus <lb/>diuidens latere <var>.a.e.</var> quartum nume-<lb/>rum rectangulo <var>.i.o.</var> primum proue-<lb/>niens latere <var>.e.i.</var> Å¿ecundus diuidens la <lb/>tere <var>.e.u.</var> (hic autem eÅ¿t quem quæri-<lb/>mus) tum alterum proueniens Å¿igni <lb/>ficetur latere <var>.e.o</var>. </s> - <s xml:space="preserve">Iam <reg norm="eadem" type="context">eadẽ</reg> erit pro-<lb/>portio <var>.e.i.</var> ad <var>.e.u.</var> quæ <var>.o.i.</var> ad <var>.o.u.</var> <lb/>Sed cum cognitæ Å¿int tres quantita-<lb/>tes <var>.e.i</var>: <var>i.o</var>: et <var>.o.u.</var> quarta quoque. e<unsure/> <var>.u.</var> exregula de tribus immediatè cognoÅ¿cetur, <lb/>cætera in Å¿ubÅ¿cripta figura facillimè patebunt.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0059-02" xlink:href="fig-0059-02a"> - <image file="0059-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0059-02"/> - </figure> - </div> - <pb o="48" rhead="IO. BAPT. BENED." n="60" file="0060" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0060"/> - </div> - <div type="math:theorem" level="3" n="73"> - <head xml:space="preserve">THEOREMA <num value="73">LXXIII</num>.</head> - <p> - <s xml:space="preserve">HOC etiam problema à me inuentum eÅ¿t, nempe Å¿i duæ radices quadratæ in <lb/>Å¿ummam collectæ fuerint, & ex dimidio eiuÅ¿modi Å¿ummæ detracta fuerit mi <lb/>nor radix, <reg norm="reÅ¿iduique" type="simple">reÅ¿iduiqÌ;</reg> quadratum duplicatum <reg norm="eique" type="simple">eiqÌ;</reg> Å¿ummæ coniungatur du-<lb/>plum producti ipÅ¿ius reÅ¿idui in dimidium Å¿ummæ radicum, atque huic Å¿ummæ du-<lb/>plum producti eiuÅ¿dem reÅ¿idui in radicem minorem coniunctum fuerit; </s> - <s xml:space="preserve">vltima hæc <lb/>Å¿umma differentia erit duorum quadratorum propoÅ¿itorum.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia duæ radices quadraræ Å¿int .5. et .11. harum Å¿umma erit .16. & dimi <lb/>dium .8. differentia minoris ab ipÅ¿o dimidio erit .3: duplum quadrati huius differen <lb/>tiæ erit .18: </s> - <s xml:space="preserve">duplum producti huius differentiÄ™ in dimidium Å¿ummÄ™ radicum erit .48. <lb/>item & huius differentiæ duplum in minorem radicem erit .30. quarum omnium <lb/>Å¿umma erit .96. tantaqÌue erit differentia Å¿uorum quadratorum, quorum vnum <lb/>erit .25. alterum verò .121.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius rei Å¿cientia, duæ quadratæ radices Å¿int <var>.h.o.</var> et <var>.o.d.</var> directæ inter Å¿e con-<lb/>iunctæ, quæ Å¿umma per medium in puncto <var>.e.</var> diuidatur, tum cogitetur <var>.e.b.</var> æqualis <lb/><var>o.e.</var> perpendicularis <var>.h.d.</var> <reg norm="ducanturque" type="simple">ducanturqÌ;</reg> lineæ <var>.b.h</var>: <var>b.o.</var> et <var>.b.d</var>. </s> - <s xml:space="preserve">Iam ex .4. primi <var>.b.h.</var> æqua <lb/>lis erit <var>.b.d.</var> & quadratum <var>.b.h.</var> æquale quadrato <var>.h.o.</var> & quadrato <var>.o.b.</var> Å¿imul cum du <lb/>plo producti <var>.o.e.</var> in <var>.o.h.</var> ex .12. Å¿ecundi Eucli. </s> - <s xml:space="preserve">Sed ex .13. <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> quadratum <var>.b.d.</var> <lb/>minus eÅ¿t quadrato <var>.o.d.</var> cum quadrato <var>.o.b.</var> ex duplo producti <var>.o.e.</var> in <var>.o.d.</var> at duplum <lb/>eiuÅ¿modi producti æquale eÅ¿t duplo qua-<lb/>drati <var>.o.e.</var> & duplo producti <var>.o.e.</var> in <var>.e.d.</var> ex <lb/> - <anchor type="figure" xlink:label="fig-0060-01a" xlink:href="fig-0060-01"/> - tertia eiuÅ¿dem, itaque duo quadrata Å¿cili-<lb/>cet <var>.o.b.</var> et <var>.o.d.</var> maiora erunt duobus qua-<lb/>dratis, nempe <var>.o.b.</var> et <var>.o.h.</var> collectis cum du <lb/>plo producti <var>.o.e.</var> in <var>.o.h.</var> ex duplo quadrati <lb/><var>o.e.</var> vna <reg norm="cum" type="context">cÅ©</reg> duplo producti <var>.o.e.</var> in <var>.e.d</var>. </s> - <s xml:space="preserve">Qua <lb/>re <reg norm="differentia" type="context">differẽtia</reg> Å¿ummæ duorum quadratorum <lb/><var>o.b.</var> et <var>.o.d.</var> à ſumma duorum <var>o.b.</var> et <var>.o.h.</var> du <lb/>plum erit quadrati <var>.o.e.</var> cum duplo produ-<lb/>cti <var>.o.e.</var> in <var>.e.d.</var> & duplo producti <var>.o.e.</var> in <var>.o.h.</var> <lb/>Quòd Å¿i ex Å¿ingulis duabus Å¿ummis quadratorum demptum fuerit quadratum <var>.o.b.</var> <lb/>eadem producta & quadrata ipÅ¿ius <var>.o.e.</var> remanebunt, tanquam differentia duorum <lb/>quadratorum <var>.o.u.</var> et <var>.h.c</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0060-01" xlink:href="fig-0060-01a"> - <image file="0060-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0060-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="74"> - <head xml:space="preserve">THEOREMA <num value="74">LXXIIII</num>.</head> - <p> - <s xml:space="preserve">CVR Å¿umma duorum <reg norm="extremorum" type="context">extremorÅ©</reg> quatuor terminorum <reg norm="proportionalium" type="context">proportionaliÅ©</reg> arith-<lb/>meticè, æqualis eÅ¿t Å¿ummæ duorum mediorum, vbi nota hac in re neceſſa-<lb/>rium non eſſe proportionalitatem continuam exiÅ¿tere.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i darentur hi quatuor termini .20. 17. 9. 6. quorum proportio ea <lb/>dem eſſet primi ad Å¿ecundum quæ tertij ad quartum, Å¿umma primi cum quarto eſſet <lb/>26. <reg norm="tantaque" type="simple">tantaqÌ;</reg> Å¿ecundi cum tertio.</s> - </p> - <p> - <s xml:space="preserve">Cuius Å¿peculationis cauÅ¿a, primus <reg norm="maiorque" type="simple">maiorqÌ;</reg> numerus Å¿ignificetur linea <var>.e.o.</var> Å¿ecun-<lb/>dus <var>.s.q.</var> tertius <var>.u.c.</var> quartus <var>.g.t.</var> differentia porrò inter <var>.e.o.</var> et <var>.s.q.</var> Å¿it <var>.i.o.</var> quæ æqualis <lb/>erit differentiæ <var>.r.c.</var> qua quartus à tertio Å¿uperatur ex hypotheÅ¿i. </s> - <s xml:space="preserve">Itaque aſſero Å¿um <lb/>mam <var>.e.o.</var> cum <var>.g.t.</var> nempe <var>.a.o.</var> æqualem eſſe Å¿ummÄ™ <var>.q.s.</var> et <var>.u.c.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.q.p</var>. </s> - <s xml:space="preserve">Nam in <var>.a.o.</var> - <pb o="49" rhead="THEOREM. ARIT." n="61" file="0061" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0061"/> - Secundus tertiusqÌue terminus reperiuntur, eÅ¿t <lb/> - <anchor type="figure" xlink:label="fig-0061-01a" xlink:href="fig-0061-01"/> - enim Å¿ecundus <var>.e.i.</var> tertius <var>.i.o.</var> et <var>.e.a.</var> quando-<lb/>quidem ex præſuppoÅ¿ito <var>.e.i.</var> æqualis eÅ¿t <var>.s.q.</var> et <lb/><var>i.o.</var> æqualis <var>.r.c.</var> et <var>.a.e.</var> cum Å¿it æqualis <var>.g.t.</var> cui <lb/>pariter æqualis eÅ¿t <var>.r.u.</var> ex quo <var>.a.e.</var> æqualis <lb/>eÅ¿t <var>.u.r</var>. </s> - <s xml:space="preserve">Itaque illud Å¿equitur <var>.a.o.</var> ipÅ¿i <var>.q.p.</var> <lb/>æqualem eſſe.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0061-01" xlink:href="fig-0061-01a"> - <image file="0061-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0061-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="75"> - <head xml:space="preserve">THEOREMA <num value="75">LXXV</num>.</head> - <p> - <s xml:space="preserve">CVR Å¿umma duorum terminorum extremorum imparium arithmeticæ pro-<lb/>portionalitatis Å¿emper duplo medij termini æqualis eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿unt hitres termini proportionalitatis arithmeticæ .20. 15. 10 <lb/>Å¿umma duorum extremorum erit .30. quæ duplum eÅ¿t medij termini .15.</s> - </p> - <p> - <s xml:space="preserve">Quod vt Å¿peculemur, tres termini, tribus lineis <var>.b.d</var>: <var>n.u.</var> et <var>.q.p.</var> <reg norm="Å¿ignificentur" type="context">Å¿ignificẽtur</reg>. </s> - <s xml:space="preserve">Di-<lb/>co nunc quòd Å¿umma <var>.b.d.</var> cum <var>.q.p.</var> nempe <var>.<lb/>h.d.</var> Å¿emper duplo <var>.n.u.</var> Å¿cilicet <var>.g.u.</var> æqualis <lb/> - <anchor type="figure" xlink:label="fig-0061-02a" xlink:href="fig-0061-02"/> - erit. </s> - <s xml:space="preserve">Tum differentia <var>.b.d.</var> ad <var>.n.u.</var> Å¿it <var>.c.d.</var> quæ <lb/>æqualis erit <var>.e.u.</var> differentiæ inter <var>n.u.</var> et <var>.q.p.</var> <lb/>patet enim in linea <var>.h.d</var>: <var>b.c.</var> æqualem eſſe <var>.n.<lb/>u.</var> Å¿ed <var>.n.u.</var> ex <var>.n.e.</var> componitur æquali <var>.q.p.</var> et <lb/>ex <var>.e.u.</var> æquali <var>.c.d.</var> cum <reg norm="itaque" type="simple">itaq;</reg> in <var>.h.d.</var> partem <var>.<lb/>h.b.</var> reperiamus æqualem <var>.n.e.</var> gratia <var>.q.p.</var> & <lb/>partem <var>.c.d.</var> æquale <var>m.e.u.</var> manifeÅ¿tum erit <lb/><var>h.d.</var> æqualem eſſe <var>.g.u</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0061-02" xlink:href="fig-0061-02a"> - <image file="0061-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0061-02"/> - </figure> - </div> - <head style="it" xml:space="preserve">BINA PROBLEMAT A EX DVOBVS PRAEDICTIS <lb/>THEOREMATIBVS DEPENDENTIA.</head> - <p> - <s xml:space="preserve">EX duobus prædictis theorematibus duo problemata oriuntur, <reg norm="quorum" type="context">quorÅ©</reg> primum <lb/>eÅ¿t. </s> - <s xml:space="preserve">Datis tribus quantitatibus cognitis, Å¿i quis quartam inuenire voluerit, <lb/>quæ eiuÅ¿modi Å¿it reÅ¿pectu tertiæ, qualis eÅ¿t Å¿ecunda reÅ¿pectu primæ, Å¿ecunda cum <lb/>tertia in Å¿ummam colligenda erit, ex qua detracta prima, Å¿upererit quarta.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, cognitis tribus quantitatibus .20. 17. 9. Å¿i quartam inuenire vo <lb/>luerimus eiuÅ¿modi proportionem cum tertia arithmeticè Å¿eruantem, quam Å¿ecunda <lb/>cum prima, Å¿ecundam cum tertia in Å¿ummam colligemus, <reg norm="dabiturque" type="simple">dabiturqÌ;</reg> Å¿umma .26. ex <lb/>qua detracta prima quantitate, quarta relinquetur nempe .6. quod ex .74. theore-<lb/>mate dependet.</s> - </p> - <p> - <s xml:space="preserve">IdipÅ¿um tamen proueniret Å¿i quis ex tertio termino differentiam primi atque Å¿e-<lb/>cundi detraheret; </s> - <s xml:space="preserve">hæc tamen via non tam vniuerÅ¿alis eÅ¿tqu à m illa. </s> - <s xml:space="preserve">N Å¿i quartus ter <lb/>minus incognitus tertio maior eſſe deberet, dictam differentiam cum tertio termi-<lb/>mino in Å¿ummam colligere oporteret.</s> - </p> - <p> - <s xml:space="preserve">Alterum problema eÅ¿t, quòd inuentis duobus terminis, Å¿i tertius requiratur, Å¿e-<lb/>cundus duplicandus erit, ex qua Å¿umma detracto primo, Å¿tatim tertius proferetur, <lb/>quod problema ex præcedenti theoremate dependet.</s> - </p> - <pb o="50" rhead="IO. BAPT. BENED." n="62" file="0062" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0062"/> - <p> - <s xml:space="preserve">Progredi nihilominus etiam hac in re poſſemus per differentiam primi & Å¿ecun-<lb/>di termini, eam detrahendo aut in Å¿ummam cum Å¿ecunda colligendo, attamen prior <lb/>ratio magis latè patet, ideÅ¿t vniuerÅ¿alior eÅ¿t.</s> - </p> - </div> - <div type="math:theorem" level="3" n="76"> - <head xml:space="preserve">THEOREMA <num value="76">LXXVI</num>.</head> - <p> - <s xml:space="preserve">CVR Å¿i quis cupiat Å¿ecundum terminum inuenire, quatuor terminorum arith-<lb/>meticè proportionalis continuæ, quorum nobis duo extrema proponantur. <lb/></s> - <s xml:space="preserve">Rectè primum duplicabit <reg norm="coniungetque" type="simple">coniungetqÌ;</reg> vltimo termino, nempe quarto, ex qua Å¿um-<lb/>ma tertiam partem deÅ¿umet, quæ erit Å¿ecundus terminus quęſitus.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i horum quatuor terminorum .12. 9. 6. 3. duo nobis extrema <lb/>proponantur. </s> - <s xml:space="preserve">nempe .12. et .3. quorum Å¿ecundus inueniendus Å¿it, Å¿umpto quolibet <lb/>pro primo, Å¿it autem .3. primus numerus, quartus verò .12. </s> - <s xml:space="preserve">quare duplicato 3. vtpo <lb/>tè primo, & coniuncto .12. quarto, Å¿umma erit .18. cuius eÅ¿t tertia pars .6. Å¿ecundus <lb/>numerus Å¿cilicet Å¿umpto principio à minimo. </s> - <s xml:space="preserve">IdipÅ¿um euenit Å¿umpto principio à <lb/>maximo. </s> - <s xml:space="preserve">Nam Å¿i datur Å¿ecundus à minimo aut à maximo, illico tertius datur diffe-<lb/>rentia inter hunc & primum, Å¿ecundo coniuncta, aut ex eodem detracta.</s> - </p> - <p> - <s xml:space="preserve">Cuius ratio Å¿ic demonÅ¿tratur, quatuor termini quatuor lineis <var>.m.g</var>: <var>q.p</var>: <var>u.n</var>: <var>c.t.</var> <lb/>Å¿ignificentur, quorum <var>.m.g.</var> et <var>.c.t.</var> tantummodo cognoÅ¿cantur. </s> - <s xml:space="preserve"><reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.m.g.</var> primus ac <lb/>maior terminus: </s> - <s xml:space="preserve">k.g. verò Å¿it duplum primi <var>.m.g</var>: cui coniungatur <var>.b.k.</var> æqualis <var>.c.t.</var> <lb/>Dico tertiam partem <var>.b.g.</var> quæ Å¿umma totalis eÅ¿t, æqualem eſſe <var>.q.p</var>. </s> - <s xml:space="preserve">In primis enim <lb/>certi Å¿umus <var>.m.f.</var> in <var>.m.g.</var> reperiri æqualem <var>.q.p.</var> <reg norm="Å¿upereÅ¿tque" type="simple">Å¿upereÅ¿tqÌ;</reg> <var>.f.g.</var> differentia inter <var>.m.g.</var> <lb/>et <var>.q.p.</var> æqualis <var>.e.p.</var> differentiæ inter <var>.q.p.</var> et <var>.u.n.</var> & æqualis <var>.o.n.</var> differen-<lb/>tiæ inter <var>.u.n.</var> et <var>.c.t</var>: Å¿imul etiam in <var>.k.m.</var> habemus <var>.d.m.</var> æqualem <var>.m.f.</var> </s> - <s xml:space="preserve">quare etiam <var>.q.<lb/>p.</var> et <var>.k.d.</var> æqualem <var>.f.g.</var> nempe <var>.e.p.</var> aut <var>.o.n</var>: Hactenus in <var>.k.g.</var> reperimus duplum <var>.q.<lb/>p.</var> Å¿imul cum <var>.f.g.</var> et <var>.k.d.</var> æqualibus <var>.e.p.</var> et <var>.o.n.</var> & quia <var>.b.K.</var> æqualis <var>.c.t.</var> fuit coniuncta. <lb/></s> - <s xml:space="preserve">conÅ¿iderandum eÅ¿t an hætres quantitates <var>.f.g</var>: <var>K.d.</var> et <var>.b.K.</var> Å¿imul æquales Å¿int <var>.q.p.</var> <lb/>quod tamen per Å¿e manifeÅ¿tum eÅ¿t. </s> - <s xml:space="preserve">nam <var>.q.p.</var> Å¿uperat <var>.u.n.</var> per <var>.e.p.</var> et <var>.u.n.</var> ex-<lb/>cedit <var>.c.t.</var> per <var>.o.n.</var> æqualem <var>.e.p.</var> </s> - <s xml:space="preserve">quare <var>.q.p.</var> per duplum differentiÄ™ <var>.f.g.</var> Å¿uperat <var>.c.t.</var> ita <lb/>que <var>.f.g</var>: <var>k.d.</var> et <var>.K.b.</var> ipÅ¿i <var>.q.p.</var> Å¿unt <reg norm="ae- quales" type="simple">Ä™-<lb/>quales</reg>, ex quo Å¿equitur <var>.q.p.</var> <reg norm="tertiam" type="context">tertiã</reg> <lb/> - <anchor type="figure" xlink:label="fig-0062-01a" xlink:href="fig-0062-01"/> - partem eſſe <var>.b.g.</var> Hæc quæ hacte-<lb/>nus dicta fuerunt, in genere maio-<lb/>ris inæqualitatis probata fuerunt. <lb/></s> - <s xml:space="preserve">At in genere minoris, Å¿umpto or-<lb/>dinis principio à minimo termino <lb/>rum, duplicetur <var>.c.t.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> duplum <lb/>hoc <var>.K.t.</var> cui <var>.k.b.</var> æqualis <var>.m.g.</var> con-<lb/>iungatur, quæſumma Å¿it <var>.b.t</var>. </s> - <s xml:space="preserve">Di-<lb/>co <var>.u.n.</var> tertiam eſſe partem ipÅ¿ius. <lb/></s> - <s xml:space="preserve">Nam in primis in <var>.b.t.</var> datur termi <lb/>nus <var>.b.K.</var> æqualis vltimo <var>.m.g.</var> in <lb/>quo Å¿emel reperitur <var>.u.n.</var> vnà cum <lb/>duabus differentijs, nempe <var>.i.g.</var> in <lb/>ipÅ¿a autem <var>.b.t</var>: <var>u.n.</var> Å¿ignificetur pri <lb/>mo loco per <var>.r.K.</var> ex quo Å¿upererit <var>.b.r.</var> duabus differentijs prædictis æqualis, Å¿ed ex <lb/>præſuppoÅ¿ito <var>.u.n.</var> componitur ex <var>.o.u.</var> æquali <var>.c.t.</var> et <var>.o.n.</var> Ä™quali vni differentiæ. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> - <pb o="51" rhead="THEOREM. ARIT." n="63" file="0063" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0063"/> - cum in <var>.b.t.</var> præter <var>.r.K.</var> bis detur <var>.c.t.K.t.</var> et <var>.b.r.</var> duabus differentijs æquipol-<lb/>lens, illud efficitur <var>.u.n.</var> pariter ipÅ¿ius <var>.b.t.</var> eſſe tertiam partem, quod erat <reg norm="propoÅ¿itum" type="context">propoÅ¿itÅ©</reg>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0062-01" xlink:href="fig-0062-01a"> - <image file="0062-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0062-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="77"> - <head xml:space="preserve">THEOREMA <num value="77">LXXVII</num>.</head> - <p> - <s xml:space="preserve">CVR Å¿i quis velit Å¿ecundum quinque continuorum proportionalium termi-<lb/>num inuenire, Å¿olis extremis cognitis. </s> - <s xml:space="preserve">Rectè <reg norm="vltimum" type="context">vltimÅ©</reg> triplo primi coniunget, <lb/>ex qua Å¿umma quartam partem detraher, quæ erit Å¿ecundus terminus quæſitus. <lb/></s> - <s xml:space="preserve">Quod ipÅ¿um faciet qui inuenire vult Å¿ecundum terminum Å¿enarij Å¿eptenarij, octo-<lb/>narij aut alterius cuiuÅ¿cunque, creÅ¿cente tamen multiplicatione primi, <reg norm="vltimoque" type="simple">vltimoqÌ;</reg> <lb/>coniuncto.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, dantur duo extremi termini, horum quinque numerorum .18. <lb/>16. 14. 12. 10. nempe .18. et .10. Å¿i .18. primus erit, hoc eÅ¿t, Å¿i à genere maioris inæ-<lb/>qualitatis progrediemur, triplicabimus terminum .18. <reg norm="dabunturque" type="simple">dabunturqÌ;</reg> .54. cui numero <lb/>coniuncto quinto termino .10. dabitur numerus .64. cuius quarta pars erit .16. vtpo <lb/>tè Å¿ecundus terminus gratia, aut Å¿ecundi Å¿ex terminorum, quadruplicandus eſſet pri <lb/>mus .18. deinde adiuncto vltimo, quinta pars Å¿ummæ eſſet Å¿ecundus terminus, <reg norm="atque" type="simple">atq;</reg> <lb/>ita deinceps.</s> - </p> - <p> - <s xml:space="preserve">Cuius Å¿peculationis gratia, dicti termini lineis <var>.z.h</var>: <var>f.s</var>: <var>u.p</var>: <var>e.g.</var> et <var>.r.x.</var> <reg norm="Å¿igniÅ¿icentur" type="context">Å¿igniÅ¿icẽtur</reg>. <lb/></s> - <s xml:space="preserve">In primis ex genere maioris inæqualitatis, triplicabimus <var>.z.h.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> triplum hoc <var>.k.<lb/>h.</var> <reg norm="cuiconiungatur" type="context">cuicõiungatur</reg> <var>.b.k.</var> Ä™qualis vltimo termino <var>.r.x</var>. </s> - <s xml:space="preserve">Dico <var>.f.s.</var> <reg norm="quartam" type="context">quartã</reg> partem eſſe Å¿um-<lb/>mÄ™ <var>.b.h</var>. </s> - <s xml:space="preserve">Nam in <var>.k.h.</var> Å¿ecundus terminus <var>.f.s.</var> ter cum tribus differentijs æqualibus <var>.n.h.</var> <lb/>reperitur. </s> - <s xml:space="preserve">Probandum nunc eÅ¿t tres has differentias <var>.n.h</var>: <var>a.c.</var> et <var>.d.k.</var> Å¿imul cum <var>.b.<lb/>K.</var> Ä™quales eſſe <var>.f.s.</var> <lb/> - <anchor type="figure" xlink:label="fig-0063-01a" xlink:href="fig-0063-01"/> - quod in <reg norm="dubium" type="context">dubiÅ©</reg> re <lb/>uocari <reg norm="non" type="context">nõ</reg> poteÅ¿t, <lb/>cum <var>.f.s.</var> Å¿uperet <var>.<lb/>r.x.</var> per <var>.o.s</var>: <var>t.p.</var> et <var>.<lb/>i.g</var>. </s> - <s xml:space="preserve">At in genere <lb/>minoris inæquali <lb/>tatis, triplum <var>.r.x.</var> <lb/>Å¿it <var>.x.a.</var> et <var>.a.b.</var> Å¿it <lb/>æqualis <var>.z.h.</var> & <reg norm="cum" type="context">cÅ©</reg> <lb/><var>z.h.</var> tribus <reg norm="differem" type="context">differẽ</reg> <lb/>tijs <var>.n.h</var>: <var>o.s</var>: <var>t.p.</var> Å¿u-<lb/>peret <var>.e.g.</var> quæ in <var>.<lb/>a.b.</var> Å¿int <var>.b.K</var>: <var>K.d</var>: <lb/><var>d.c.</var> ex quo <var>.a.c.</var> <lb/>æqualis erit <var>.e.g.</var> <lb/>et <var>.a.x.</var> cum <var>.b.c.</var> tripla <var>.e.g</var>. </s> - <s xml:space="preserve">Itaque tota Å¿umma <var>.b.x.</var> qua drupla erit <var>.e.g</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0063-01" xlink:href="fig-0063-01a"> - <image file="0063-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0063-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="78"> - <head xml:space="preserve">THEOREMA <num value="78">LXXVIII</num>.</head> - <p> - <s xml:space="preserve">QVantitates quæ fuerint inuicem in proportionalitate arithmetica proportio-<lb/>nales, permutan do quoque proportionales erunt.</s> - </p> - <pb o="52" rhead="IO. BAPT. BENED." n="64" file="0064" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0064"/> - <p> - <s xml:space="preserve">Sint exempli gratia .4. quantitates <var>.a.b</var>: <var>c.d</var>: <var>e.f</var>: et <var>.g.h</var>: inuicem proportionales in <lb/>proportionalitate arithmetica. </s> - <s xml:space="preserve">Hoc eÅ¿t vt quæ proportio (licet impropriè dicta) <lb/>eÅ¿t ipÅ¿ius <var>.a.b.</var> ad <var>.c.d.</var> <reg norm="eadem" type="context">eadẽ</reg> Å¿it ipÅ¿ius <var>.e.f.</var> ad <var>.g.h</var>. </s> - <s xml:space="preserve">Tunc permutando dico eandem pro <lb/>portionem fore ipÅ¿ius <var>.a.b.</var> ad <var>.e.f.</var> quæ ipÅ¿ius <var>.c.d.</var> ad <var>.g.h</var>.</s> - </p> - <p> - <s xml:space="preserve">Nam, ex hypotheÅ¿i, differentia qua <var>.a.b.</var> Å¿uperat <var>.c.d.</var> (quæ Å¿it <var>.m.b.</var>) æqualis eÅ¿t <lb/>differentiæ qua <var>.e.f.</var> Å¿uperat <var>.g.h.</var> (quæ Å¿it <var>.i.f.</var>) vnde <var>.a.m.</var> reÅ¿iduum ex <var>.a.b.</var> æquale erit <lb/><var>c.d.</var> & reÅ¿iduum <var>.e.i.</var> æquale <var>.g.h</var>. </s> - <s xml:space="preserve">Sit igitur exempli gratia <var>.c.d.</var> maior <var>.g.h.</var> per <var>.c.n.</var> <lb/>vnde <var>.n.d.</var> æqualis erit <var>.g.h.</var> </s> - <s xml:space="preserve">quare <var>.a.m.</var> maior erit <var>.e.i.</var> per <var>.a.K.</var> æqualem <var>.c.n.</var> ex com-<lb/>muni Å¿cientia. </s> - <s xml:space="preserve">Vnde <var>.K.m.</var> æqualis erit <var>.n.d.</var> hoc eÅ¿t ipÅ¿i <var>.g.h.</var> hoc eÅ¿t ipÅ¿i <var>e.i</var>. </s> - <s xml:space="preserve">Quare ex <lb/>communi conceptu <var>.b.K.</var> æqualis erit ipÅ¿i <var>.f.e.</var> Å¿ed <var>.n.d.</var> æqualis eÅ¿t <var>.g.h.</var> vt dictum eÅ¿t. <lb/></s> - <s xml:space="preserve">Cum ergo <var>.b.K.</var> æqualis Å¿it <var>.e.f.</var> et <var>.d.n.</var> ipÅ¿i <var>.g.h.</var> et <var>.a.b.</var> maior Å¿it ipÅ¿a <var>.K.b.</var> per <var>.a.K.</var> æqua-<lb/>lem ipÅ¿i <var>.c.n.</var> per quam <var>c.n</var>: <var>d.c.</var> maior eÅ¿t ipÅ¿a <var>.d.n.</var> Å¿equitur verum eſſe <reg norm="propoÅ¿itum" type="context">propoÅ¿itÅ©</reg> hoc <lb/>eÅ¿t, quod eadem proportio Å¿it ipÅ¿ius <var>.a.b.</var> ad <var>.e.f.</var> quæ <var>.c.d.</var> ad <var>.g.h.</var> arithmetice Å¿cilicet.</s> - </p> - <figure position="here"> - <image file="0064-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0064-01"/> - </figure> - </div> - <div type="math:theorem" level="3" n="79"> - <head xml:space="preserve">THEOREMA <num value="79">LXXIX</num>.</head> - <p> - <s xml:space="preserve">CVR prouenientia duorum numerorum diuidentium eiuÅ¿dem numeri diuiÅ¿i-<lb/>bilis, geometricè <reg norm="eandem" type="context">eandẽ</reg> inter Å¿e <reg norm="proportionem" type="context">proportionẽ</reg> Å¿eruant, <reg norm="quam" type="context">quã</reg> ipÅ¿imet <reg norm="diuidentes" type="context">diuidẽtes</reg>.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia Å¿i per Å¿enarium & octonarium numerus vigintiquatuor diuida-<lb/>tur, prouenientia erunt .4. et .3. eadem proportione, qua diuidentes.</s> - </p> - <p> - <s xml:space="preserve">Cuius eÅ¿t ratio numerus diuiÅ¿ibilis Å¿ignificetur rectangulis <var>.u.x.</var> et <var>.n.e.</var> diuidentes <lb/>autem Å¿int <var>.u.o.</var> et <var>.e.o.</var> </s> - <s xml:space="preserve">quare ex ijs, quæ .10. <lb/> - <anchor type="figure" xlink:label="fig-0064-02a" xlink:href="fig-0064-02"/> - theoremate dicta fuerunt <var>.u.x.</var> per <var>.u.o.</var> diui-<lb/>Å¿o dabit <var>.x.o.</var> & diuiÅ¿o <var>.n.e.</var> per <var>.e.o.</var> dabit <var>.o.<lb/>n</var>. </s> - <s xml:space="preserve">Dicimus itaque <reg norm="eandem" type="context">eandẽ</reg> eſſe <reg norm="proportionem" type="context">proportionẽ</reg> <lb/><var>o.x.</var> ad <var>.o.n.</var> quæ <var>.e.o.</var> ad <var>.o.u.</var> quod patet Å¿ub <lb/>Å¿criptam figuram conÅ¿iderantibus, in qua, <lb/>ex .15. Å¿exti aut .20. Å¿eptimi, eadem propor-<lb/>tio cernitur <var>.o.x.</var> ad <var>.o.n.</var> quæ <var>.o.e.</var> ad <var>.o.u</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0064-02" xlink:href="fig-0064-02a"> - <image file="0064-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0064-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="80"> - <head xml:space="preserve">THEOREMA <num value="80">LXXX</num>.</head> - <p> - <s xml:space="preserve">CVR quauis quantitate, tribus <lb/> - <anchor type="figure" xlink:label="fig-0064-03a" xlink:href="fig-0064-03"/> - aut quatuor aut etiam pro libi-<lb/>to pluribus diuidentibus numeris di-<lb/>uifa, prouenientia eandem prorÅ¿us <lb/>inter Å¿e proportionem Å¿eruabunt, <lb/>quam ipÅ¿i diuidentes habere compe <lb/>riuntur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0064-03" xlink:href="fig-0064-03a"> - <image file="0064-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0064-03"/> - </figure> - </div> - <p> - <s xml:space="preserve">Exempli gratia, proponitur nu-<lb/>merus .60. quinque numeris diuiden <lb/>dus, vtpotè .30. 20. 15. 12. 10. pro-<lb/>uenientia erunt .2. 3. 4. 5. 6. eadem - <pb o="53" rhead="THEOREM. ARITH." n="65" file="0065" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0065"/> - proportione diuidentium, quamuis ex aduerÅ¿o.</s> - </p> - <p> - <s xml:space="preserve">Cuius ratio ex .15. Å¿exti aut .20. Å¿eptimi dependet. </s> - <s xml:space="preserve">prout in Å¿ubÅ¿cripto ordine fa-<lb/>cillimè deprehendi poteÅ¿t.</s> - </p> - </div> - <div type="math:theorem" level="3" n="81"> - <head xml:space="preserve">THEOREMA <num value="81">LXXXI</num>.</head> - <p> - <s xml:space="preserve">CVR quantitate in tres continuas partes proportionales Å¿ecta, & per Å¿ingulas <lb/>ipÅ¿arum diuiÅ¿a, Å¿umma trium prouenientium quadrato medij prouenientis <lb/>æqualis eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, proponitur .14. diuidendus in tres continuas partes proportio-<lb/>nales, nempe .8. 4. 2. <reg norm="ipÅ¿eque" type="simple">ipÅ¿eqÌ;</reg> numerus .14. per Å¿ingulas diuiditur, ex quo tria proue-<lb/>nientia oriuntur, nempe ex prima parte .8. <reg norm="proueniens" type="context">proueniẽs</reg> erit .1. cum tribus quartis par <lb/>tibus ex Å¿ecunda .4. datur proueniens .3. cum dimidio vnius, & ex tertia .2. proue-<lb/>nient .7. integri, qui in Å¿ummam collecti dant .12. integros & vnam quartam par-<lb/>tem tantumdem, videlicet quantum quadratum prouenientis medij, nempe .3. <lb/>cum dimidio.</s> - </p> - <p> - <s xml:space="preserve">Cuius Å¿peculationis gratia, totalis numerus Å¿ignificetur linea <var>.n.c.</var> qui in tres par-<lb/>tes diuidatur <var>.n.a</var>: <var>a.e.</var> et <var>.e.c.</var> quæ Å¿int continuæ proportionales, quarum Å¿ingulis, <lb/>numerum <var>.n.c.</var> diuiÅ¿um eſſe cogitemus, proueniens autem ex diuiÅ¿ione <var>.n.c.</var> per <var>.n.<lb/>a.</var> Å¿it <var>.i.d.</var> quod verò prouenit ex diuiÅ¿ione <var>.n.c.</var> per <var>.a.e.</var> Å¿it <var>.d.u.</var> proueniens quoque ex <lb/>diuiÅ¿ione <var>.n.c.</var> per <var>.e.c.</var> Å¿it <var>.u.o.</var> quorum Å¿umma Å¿it <var>.i.o.</var> quæ aſſeritur eſſe numeri æqua-<lb/>lis numero quadrati <var>.d.u</var>. </s> - <s xml:space="preserve">Quod hac ratione probabo, producatur linea <var>.i.o.</var> donec <var>.<lb/>o.p.</var> æqualis Å¿it <var>.o.u.</var> <reg norm="erigaturque" type="simple">erigaturqÌ;</reg> <var>.o.m.</var> æqualis <var>.d.i.</var> perpendiculariter <var>.o.p.</var> in puncto <var>.o.</var> <lb/>quæ producatur donec <var>.o.q.</var> vnitati Å¿it æqualis, <reg norm="terminenturque" type="simple">terminenturqÌ;</reg> duo rectangula <var>.m.p.</var> <lb/>et <var>.q.i.</var> ex quo habebimus rectangulum, aut productum <var>.m.p.</var> æquale quadrato <var>.d.u.</var> <lb/>ex .16 Å¿exti aut .20. Å¿eptimi, quandoquidem tria prouenientia <var>.o.u</var>: <var>u.d.</var> et <var>.d.i.</var> ex <lb/>prÄ™cedenti theoremate Å¿unt inter Å¿e continua proportionalia, proportionalitate qua <lb/>partes <var>.n.c</var>. </s> - <s xml:space="preserve">Iam verò Å¿i probauero <var>.q.i.</var> productum, producto <var>.m.p.</var> æquale eſſe, pro-<lb/>poÅ¿itum quoque probatum erit. </s> - <s xml:space="preserve">Numerus enim producti <var>.q.i.</var> æqualis eÅ¿t numero. <lb/></s> - <s xml:space="preserve">Å¿ummæ <var>.i.o</var>. </s> - <s xml:space="preserve">Habemus autem ex definitione diuiÅ¿ionis ita Å¿e habere <var>.n.c.</var> ad <var>.i.d.</var> Å¿icut <var>.<lb/>n.a.</var> ad <var>.o.q</var>. </s> - <s xml:space="preserve">Itaque permutando Å¿ic Å¿e habebit <var>.n.c.</var> ad <var>.n.a.</var> Å¿icut <var>.d.i.</var> hoc eÅ¿t <var>.m.o.</var> ad <var>.<lb/>o.q.</var> Å¿ed Å¿icut Å¿e habet <var>.n.c.</var> ad <var>.n.a.</var> ita pariter Å¿e habet <var>.i.o.</var> ad <var>.o.u.</var> hoc eÅ¿t ad <var>.o.p</var>. </s> - <s xml:space="preserve">Ita-<lb/>que <var>.i.o.</var> ad <var>.o.p.</var> Å¿ic Å¿e habebit Å¿icut <var>.m.o.</var> ad <var>.o.q.</var> ex quo ex .15. Å¿exti aut .20. Å¿eptimi <var>.<lb/>q.i.</var> æqualis erit <var>.m.p.</var> & conÅ¿equenter quadrato <var>.d.u</var>. </s> - <s xml:space="preserve">Vt autem lector minori labo-<lb/>re cognoÅ¿cere queat <var>.i.o.</var> ad <var>.o.u.</var> Å¿ic Å¿e habere, vt <var>.n.c.</var> ad <var>.n.a.</var> Å¿ciendum eÅ¿t quòd, Å¿ic <lb/>Å¿e habet <var>.i.d.</var> ad <var>.d.u.</var> ut <var>.c.e.</var> ad <var>.e.a.</var> ex quo componendo Å¿ic Å¿e habebit <var>.i.u.</var> ad <var>.d.u.</var> Å¿i-<lb/>cut <var>.c.a.</var> ad <var>.a.e.</var> & permutando ita <var>.i.u.</var> <lb/> - <anchor type="figure" xlink:label="fig-0065-01a" xlink:href="fig-0065-01"/> - ad <var>.c.a.</var> vt <var>.d.u.</var> ad <var>.e.a.</var> Å¿ed cum ex <reg norm="præ- cedenti" type="context">præ-<lb/>cedẽti</reg> theoremate Å¿ic Å¿e habeat <var>.d.u.</var> <lb/>ad <var>.u.o.</var> Å¿icut <var>.e.a.</var> ad <var>.a.n.</var> permutando <lb/>Å¿ic Å¿e habebit <var>.d.u.</var> ad <var>.a.e.</var> Å¿icut <var>.u.o.</var> ad <lb/><var>a.n.</var> ex quo ex .11. quinti Å¿ic Å¿e habe-<lb/>bit <var>.i.u.</var> ad <var>.c.a.</var> prout <var>.o.u.</var> ad <var>.a.n.</var> per-<lb/>mutandoqÌue <var>.i.u.</var> ad <var>.u.o.</var> vt <var>.c.a.</var> ad <var>.a.n.</var> & componendo, ita <var>.i.o.</var> ad <var>.u.o.</var> Å¿icut <var>.c.n.</var> <lb/>ad <var>.a.n</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0065-01" xlink:href="fig-0065-01a"> - <image file="0065-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0065-01"/> - </figure> - </div> - <pb o="54" rhead="IO. BAPT. BENED." n="66" file="0066" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0066"/> - </div> - <div type="math:theorem" level="3" n="82"> - <head xml:space="preserve">THEOREMA <num value="82">LXXXII</num>.</head> - <p> - <s xml:space="preserve">CVR quantitate aliqua in quatuor partes <reg norm="continuas" type="context">cõtinuas</reg> proportionales Å¿ecta per-<lb/>qÌue Å¿ingulas diuiÅ¿a, Å¿umma quatuor prouenientium æqualis Å¿it producto Å¿e-<lb/>cundi in tertium.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i triginta in quatuor partes proportionales Å¿ecetur, hoc eÅ¿t. <lb/>16. 8. 4. 2. <reg norm="perque" type="simple">perqÌ;</reg> harum Å¿ingulas idem numerus .30. diuidatur, primum proueniens <lb/>erit .1. cum Å¿eptem octauis partibus. </s> - <s xml:space="preserve">Secundum .3. cum tribus quartis, tertium .7. <lb/>cum dimidio, quartum .15. integri, quorum Å¿umma erit .28. cum octaua parte, tan <lb/><reg norm="tumque" type="simple">tumqÌ;</reg> erit productum Å¿ecundi prouenientis in tertium.</s> - </p> - <p> - <s xml:space="preserve">Quod vt Å¿ciamus, quantitas <var>.n.c.</var> in partes continuas proportionales quatuor Å¿e-<lb/>cetur <var>.n.a</var>: <var>a.t</var>: <var>t.e.</var> et <var>.e.c.</var> <reg norm="rurÅ¿usque" type="simple">rurÅ¿usqÌ;</reg> per Å¿ingulas partes illa ipÅ¿a diuiÅ¿a, prouenientia <lb/>Å¿int <var>.i.d</var>: <var>d.x</var>: <var>x.u</var>: <var>u.o.</var> <reg norm="quorum" type="context">quorÅ©</reg> Å¿umma Å¿it <var>.i.o.</var> hanc <reg norm="Å¿ummam" type="context">Å¿ummã</reg> dicimus æqualem eſſe nume-<lb/>ro producti <var>.d.x.</var> in <var>.x.u</var>.</s> - </p> - <p> - <s xml:space="preserve">Quod hac ratione probo, cogito productam eſſe lineam <var>.i.o.</var> <reg norm="quousque" type="simple">quousqÌ;</reg> <var>.o.p.</var> æqua <lb/>lis Å¿it <var>.o.u.</var> <reg norm="erectamque" type="simple">erectamqÌ;</reg> <var>.m.o.</var> æqualem <var>.i.d.</var> perpendiculariter <var>.o.p.</var> & productam donec <var>.<lb/>o.q.</var> vnitati Å¿it æqualis. </s> - <s xml:space="preserve">Iam terminatis rectangulis <var>.m.p.</var> et <var>.i.q.</var> patebit ex .15. Å¿exti <lb/>aut .20. Å¿eptimi, productum <var>.m.p.</var> producto <var>.d.x.</var> in <var>.x.u.</var> æquale eſſe. </s> - <s xml:space="preserve">Ita quòd Å¿i pro-<lb/>bauero productum <var>.i.q.</var> producto <var>.m.p.</var> æquale eſſe, facile patebit propoÅ¿itum. </s> - <s xml:space="preserve">Cuius <lb/>gratia, Å¿equuti præcedentis theorematis ordinem, primum ex <reg norm="definitionem" type="context">definitionẽ</reg> diuiÅ¿ionis, <lb/>eadem proportio erit <var>.n.c.</var> ad <var>.i.d.</var> quæ <var>.n.a.</var> ad <var>.o.q.</var> ex quo permutando <var>.n.c.</var> ad <var>.n.a.</var> Å¿ic <lb/>Å¿e habebit vt <var>.i.d.</var> hoc eÅ¿t <var>.m.o.</var> ad <var>.o.q.</var> & Å¿i progrediamur eodem ordine, quo præ-<lb/>cedenti theoremate, Å¿umpto principio ab <var>.i.d.</var> et <var>.e.c.</var> verÅ¿us <var>.d.x.</var> et <var>.e.t.</var> gradatimqÌue <lb/>permutando ac coniungendo, inue-<lb/> - <anchor type="figure" xlink:label="fig-0066-01a" xlink:href="fig-0066-01"/> - niemus eandem proportionem eſſe <lb/><var>c.n.</var> ad <var>.n.a.</var> quæ <var>.i.o.</var> ad <var>.o.u.</var> nempe <var>.<lb/>o.p.</var> ex quo ex .11 quinti, ita Å¿e habe <lb/>bit <var>.i.o.</var> ad <var>.o.p.</var> vt <var>.m.o.</var> ad <var>.o.q.</var> </s> - <s xml:space="preserve">quare <lb/>ex .15. Å¿extiaut .20. Å¿eptimi <reg norm="produ- ctum" type="context">produ-<lb/>ctÅ©</reg> <var>.i.q.</var> erit producto<unsure/> <var>.m.p.</var> æquale, <lb/>ex quo etiam æquale erit producto <var>.<lb/>d.x.</var> in <var>.x.u</var>. </s> - <s xml:space="preserve">Idem ordo in qualibet <lb/>quantitate in quantaÅ¿uis partes diuiÅ¿a Å¿eruari poterit, cum huiuÅ¿modi <reg norm="Å¿cientia" type="context">Å¿ciẽtia</reg> in vni <lb/>uerÅ¿um pateat.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0066-01" xlink:href="fig-0066-01a"> - <image file="0066-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0066-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="83"> - <head xml:space="preserve">THEOREMA <num value="83">LXXXIII</num>.</head> - <p> - <s xml:space="preserve">CVR termini medij cubus, trium continuè proportionalium, Å¿emper producto <lb/>rectanguli compræhenÅ¿i à maximo & medio in minimo termino æqualis Å¿it.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, datis his tribus terminis continuis proportionalibus .9. 6. 4. Å¿i <lb/>Å¿umpÅ¿erimus productum maximi in medium nempe .54. quod per <reg norm="minimum" type="context">minimÅ©</reg> .4. multi-<lb/>plicemus, dabitur numerus .216. cubo medij .6. æqualis.</s> - </p> - <p> - <s xml:space="preserve">In cuius gratiam tres numeri continui proportionales tribus lineis <var>.a.e.i.</var> <reg norm="Å¿ignifi- centur" type="context">Å¿ignifi-<lb/>cẽtur</reg>, cubus autem <var>.e.</var> Å¿ignificetur figura <var>.d.n.</var> <reg norm="productumque" type="simple">productumqÌ</reg> <var>.a.</var> in <var>.e.</var> Å¿it <var>.b.n.</var> ipÅ¿ius <reg norm="au- temmet" type="context">au-<lb/>tẽmet</reg> in <var>.i.</var> Å¿it <var>.p.o.</var> ita quod <var>.q.p.</var> aut <var>.b.o.</var> cum Å¿int <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> Å¿peciei, æqualis erit .a: et <var>.o.n.</var> - <pb o="55" rhead="THEOR. ARITH." n="67" file="0067" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0067"/> - æqualis .e: et <var>.q.n.</var> æqualis <var>.i</var>. </s> - <s xml:space="preserve">Nunc co-<lb/> - <anchor type="figure" xlink:label="fig-0067-01a" xlink:href="fig-0067-01"/> - gitemus abÅ¿olui corpus <var>.n.h.</var> ita ut <var>.b.<lb/>o.c.</var>Å¿it vnica recta linea, ex quo ex .25. <lb/>vndecimi proportio <var>.n.h.</var> ad <var>.n.k.</var> ea-<lb/>dem eÅ¿t quæ <var>.o.h.</var> ad <var>o.k.</var> Å¿ed Å¿ic Å¿e ha-<lb/>bet <var>.o.h.</var> ad <var>.o.k.</var> vt <var>.h.b.</var> ad <var>.b.k.</var> <lb/>ex prima Å¿exti aut .18. vel .19. Å¿e-<lb/>ptimi itaque <var>.n.h.</var> ad <var>.n.k.</var> ex .11. <lb/>quinti Å¿ic Å¿e habebit. vt <var>.h.b.</var> ad <var>.b.k.</var> <lb/>Å¿ed <var>.n.h.</var> ad <var>.n.d.</var> ex eiÅ¿dem Å¿ic Å¿e habet <lb/>ut <var>.h.u.</var> ad <var>.d.u.</var> et <var>.h.u.</var> ad <var>.u.d.</var> ita ut <var>.h.<lb/>b.</var> ad <var>.b.k.</var> ex præſuppoÅ¿ito. </s> - <s xml:space="preserve">Itaque ex <lb/>11. prædicta <var>.n.h.</var> ad <var>.n.k.</var> eadem erit <lb/>proportio quæ <var>.n.h.</var> ad <var>.n.d</var>. </s> - <s xml:space="preserve">Quare <lb/>ex .9. quinti <var>.n.k.</var> æqualis erit <var>.n.d.</var> <lb/>Quod erat propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0067-01" xlink:href="fig-0067-01a"> - <image file="0067-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0067-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="84"> - <head xml:space="preserve">THEOREMA <num value="84">LXXXIIII</num>.</head> - <p> - <s xml:space="preserve">CVR quadrato vnius quantitatis radice proportionalis, per Å¿ingulos tres termi <lb/>nos diuiÅ¿o, prouenientia, Å¿ingulis dictis terminis Å¿int æqualia.</s> - </p> - <p> - <s xml:space="preserve"><reg norm="Exempli" type="context">Exẽpli</reg> gratia, datis tribus terminis continuis proportionalibus .9. 6. 4. qua <lb/>dratum medij erit .36. quod per .9. diuiÅ¿um dabit .4: per .6: 6. per .4: 9.</s> - </p> - <p> - <s xml:space="preserve">Cuius gratia, Å¿int tres termini <reg norm="continui" type="context">cõtinui</reg> <reg norm="proportionales" type="simple">ê“portionales</reg> <var>.a.o</var>: <var>o.c.</var> et <var>.c.q.</var> <reg norm="quadratum" type="context">quadratÅ©</reg> <reg norm="autem" type="context">autẽ</reg> <lb/>medij Å¿it <var>.e.c</var>. </s> - <s xml:space="preserve">Iam Å¿i applicetur <reg norm="rectangulum" type="context">rectangulÅ©</reg> <var>.a.d.</var> æquale quadrato <var>.e.c.</var> ipÅ¿i <var>.a.o.</var> & re-<lb/>ctangulum <var>.q.p.</var> æquale eidem quadrato <var>.e.c.</var> ipÅ¿i <var>.c.q.</var> Å¿i quadratum <var>.e.c.</var> per <var>.a.o.</var> diui <lb/>Å¿erimus, proueniens erit <var>.o.d.</var> <reg norm="diuiÅ¿oque" type="simple">diuiÅ¿oqÌ</reg> per <var>.c.q.</var> proueniens erit <var>.c.p.</var> quod Å¿i per Å¿uam <lb/>radicem <var>.o.c.</var> diuidatur, proueniens erit <var>.o.</var> <lb/> - <anchor type="figure" xlink:label="fig-0067-02a" xlink:href="fig-0067-02"/> - e. quod Å¿ine dubio æquale eÅ¿t <var>.o.c.</var> Å¿ed dico <var>.<lb/>o.d.</var> æqualem eſſe <var>.c.q</var>. </s> - <s xml:space="preserve">Nam ex .16. Å¿exti aut <lb/>20. Å¿eptimi eadem eÅ¿t proportio <var>.a.o.</var> ad <var>.o.<lb/>c.</var> quÄ™ <var>.o.e.</var> ad <var>.o.d.</var> nempe <var>.o.c.</var> ad <var>.o.d.</var> itaque <lb/><var>o.d.</var> ex .9. quinti æqualis eÅ¿t <var>.c.q.</var> quandoqui <lb/>dem ex .11. Å¿ic Å¿e habet <var>.o.c.</var> ad <var>.o.d.</var> Å¿icut <var>.o.<lb/>c.</var> ad <var>.c.q</var>. </s> - <s xml:space="preserve">Applicatis ijÅ¿dem rationibus ipÅ¿i <var>.<lb/>p.c.</var> probabimus <var>.c.p.</var> æqualem eſſe <var>.a.o.</var> cum <lb/><var>o.c.</var> media Å¿it proportionalis, <reg norm="tam" type="context">tã</reg> inter <var>.c.p.</var> et <lb/><var>c.q.</var> quam inter <var>.a.o.</var> et <var>.c.q.</var> itaque <var>.c.p.</var> æqua-<lb/>lis eÅ¿t <var>.a.o</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0067-02" xlink:href="fig-0067-02a"> - <image file="0067-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0067-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="85"> - <head xml:space="preserve">THEOREMA <num value="85">LXXXV</num>.</head> - <p> - <s xml:space="preserve">CVR propoÅ¿itis tribus quantitatibus continuis proportionalibus proportione <lb/>aliarum duarum nobis datarum, multiplicata maiori poÅ¿tremarum dua-<lb/>rum in Å¿ummam mediæ cum minima trium primarum, productum æqua-<lb/>le Å¿it producto minoris duarum in Å¿ummam maximæ cum media trium.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia proponuntur quantitates .9. 6. 4. proportione numerorum pro- - <pb o="56" rhead="IO. BAPT. BENED." n="68" file="0068" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0068"/> - poÅ¿itorum .3. et .2. multiplicato .3. per .10. <reg norm="Å¿ummam" type="context">Å¿ummã</reg> .6. cum .4. dantur .30. quod pro-<lb/>ductum æquale erit producto .2. per .15. nempe per Å¿ummam 9. et .6.</s> - </p> - <p> - <s xml:space="preserve">Quod vt cognoÅ¿camus, tres quan <lb/> - <anchor type="figure" xlink:label="fig-0068-01a" xlink:href="fig-0068-01"/> - titates continuæ proportionales Å¿int <lb/><var>b.a.p.</var> proportione <var>.d.q.</var> productum <lb/>autem <var>.d.</var> in Å¿ummam <var>.a.</var> cum <var>.p.</var> Å¿it <var>.f.t.</var> <lb/>& productum <var>.q.</var> in Å¿ummam <var>.b.a.</var> Å¿it <var>.<lb/>K.h.</var> et <var>.K.n.</var> Å¿it æqualis <var>.b.</var> et <var>.n.o.</var> æqua <lb/>lis <var>.a.</var> & ita etiam <var>.o.u.</var> eidem <var>.a.</var> et <var>.u.t.</var> <lb/>æqualis <var>.p.</var> et <var>.h.o.</var> ipÅ¿i <var>.q.</var> et <var>.f.o.</var> ipÅ¿i <var>.d</var>. <lb/></s> - <s xml:space="preserve">quare ita Å¿e habebit <var>.K.n.</var> ad <var>.n.o.</var> Å¿icut <lb/><var>o.u.</var> ad <var>.u.t.</var> & componendo <var>.K.o.</var> ad <var>.<lb/>n.o.</var> vt <var>.o.t.</var> ad <var>.u.t.</var> & permutando <var>.K.<lb/>o.</var> ad <var>.o.t.</var> vt <var>.n.o.</var> hoc eÅ¿t <var>.o.u.</var> ad <var>.u.t.</var> & <lb/>pariter <var>.f.o.</var> ad <var>.o.h.</var> vt <var>.o.u.</var> ad <var>.u.t</var>. </s> - <s xml:space="preserve">Ita-<lb/>que Å¿icut <var>.k.o.</var> ad <var>.o.t.</var> ex quo ex .15. Å¿exti aut .20. Å¿eptimi <var>.K.h.</var> æqualis erit <var>.f.t</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0068-01" xlink:href="fig-0068-01a"> - <image file="0068-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0068-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="86"> - <head xml:space="preserve">THEOREMA <num value="86">LXXXVI</num>.</head> - <p> - <s xml:space="preserve">CVR multiplicatis Å¿ingulis tribus quantitatibus continuis proportionalibus in <lb/>reliquas duas, Å¿ex producta æqualia Å¿int producto dupli Å¿ummæ ipÅ¿arum trium <lb/>in mediam proportionalem.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, proponuntur hitres termini continui proportionales .9. 6. 4. pro <lb/>ductum .9. in .6. erit .54. at .9. in .4. erit .36. et .6. in .9: 54. et .6. in .4: 24. et .4. in .9: 36. et <num value="4">.<lb/>4.</num> in .6: 24. quæ producta Å¿imul collecta efficiunt numerum .228 Å¿ed <reg norm="tantum" type="context">tantÅ©</reg> eÅ¿t pro-<lb/>ductum dupli Å¿ummæ trium terminorum in Å¿ecundum nempe .38 in .6.</s> - </p> - <p> - <s xml:space="preserve">Cuius <reg norm="intelligentiæ" type="context">intelligẽtiæ</reg> cauÅ¿a, tres termini <reg norm="continui" type="context">cõtinui</reg> proportionales Å¿ignificentur linea <var>.<lb/>b.e.</var> nempe <var>.b.d</var>: <var>d.c</var>: <var>c.e.</var> cuius duplum Å¿it <var>.u.e.</var> et <var>.b.f.</var> æqualis Å¿it <var>.b.d.</var> et <var>.f.n</var>: <var>d.c.</var> et <var>.n.u</var>: <lb/>c. e productum verò <var>.u.e.</var> in <var>.d.c.</var>Å¿it <var>.u.s.</var> cui dico æqualem eſſe Å¿ummam productorum <lb/>Å¿ingulorum trium terminorum in reliquos duos. </s> - <s xml:space="preserve">Quamobrem ducantur perpendi-<lb/>culares <var>.c.g</var>: <var>d.o</var>: <var>b.i</var>: <var>f.a.</var> et <var>.n.p.</var> inter <var>.u.e.</var> et <var>.q.s.</var> ex quo pro producto <var>.c.e.</var> in <var>.c.d.</var> ha-<lb/>bebimus rectangulum <var>.c.s.</var> & rectan-<lb/> - <anchor type="figure" xlink:label="fig-0068-02a" xlink:href="fig-0068-02"/> - gulum <var>.d.g.</var> pro producto <var>.c.e.</var> in <var>.d.b.</var> <lb/>ex .16. Å¿exti aut .20. Å¿eptimi itemqÌue <lb/>rectangulum <var>.q.n.</var> pro producto <var>.d.c.</var> <lb/>in <var>.c.e.</var> & rectangulum <var>.b.o.</var> ex <var>.d.c.</var> in <var>.<lb/>b.d.</var> & rectangulum <var>.b.a.</var> ex <var>.b.d.</var> in <var>.d.<lb/>c.</var> et <var>.p.f.</var> ex <var>.d.b.</var> in <var>.c.e.</var> ex .16. aut .20. <lb/>prÄ™dictas. </s> - <s xml:space="preserve">Quare Å¿ex producta æquantur inter Å¿e, <reg norm="replentque" type="simple">replentqÌ</reg> productum <var>.u.s.</var> ex quo <lb/>verum eÅ¿t propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0068-02" xlink:href="fig-0068-02a"> - <image file="0068-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0068-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="87"> - <head xml:space="preserve">THEOREMA <num value="87">LXXXVII</num>.</head> - <p> - <s xml:space="preserve">QVA ratione cognoÅ¿ci poſſit <reg norm="verum" type="context">verÅ©</reg> eſſe proportionem Å¿ummæ quatuor quan-<lb/>titatum continuarum proportionalium ad Å¿ummam Å¿ecundæ & tertiæ, ean-<lb/>dem eſſe, quæ Å¿ummæ primæ & tertiæ ad Å¿ecundam Å¿implicem.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i inue nirentur hæ quatuor quantitates continuæ proportiona-<lb/>es .16. 8. 4. 2. earum Å¿umma erit .30. Å¿unima verò Å¿ecundæ & tertiæ .12. tum Å¿umma - <pb o="57" rhead="THEOREM. ARITH." n="69" file="0069" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0069"/> - primæ cum tertia .20. ex quo Å¿icÅ¿e habet .20. ad .8. nempe ad Å¿ecundam, vt .30. <lb/>ad .12.</s> - </p> - <p> - <s xml:space="preserve">Quod vt Å¿ciamus, quatuor prædictæ quantitates Å¿ignificentur linea <var>.a.e.i.o.</var> pro-<lb/>babo ita Å¿e habere <var>.a.e.i.o.</var> ad <var>.e.i.</var> vt <var>.a.i.</var> ad <var>.e</var>. </s> - <s xml:space="preserve">Nam cum Å¿ic Å¿e habeat <var>.a.</var> ad <var>.e.</var> ut <var>.e.</var> <lb/>ad <var>.i.</var> & vt <var>.i.</var> ad .o: ex æqualitate proportionum vel permutando ita Å¿e habebit <var>.a.</var> ad <var>.i.</var> <lb/>vt <var>.e.</var> ad <var>.o.</var> & è conuerÅ¿o ita <var>.o.</var> ad <var>.e.</var> vt <var>.i.</var> ad <var>.a.</var> & <reg norm="componendo" type="context">cõponendo</reg> ita <var>.o.e.</var> ad e. vt <var>.i.a.</var> ad <var>.a.</var> <lb/><reg norm="permutandoque" type="simple">permutandoqÌ</reg> <var>.o.e.</var> ad <var>.i.a.</var> vt <var>.e.</var> ad <var>.a.</var> nempe <var>.i.</var> ad <var>.e.</var> & componendo ita <var>.o.i.e.a.</var> ad <var>.<lb/>i.a.</var> vt <var>.i.e.</var> ad <var>.e.</var> & permutando ita <var>.o.i.e.a.</var> ad <var>.i.e.</var> vt <var>.i.a.</var> ad <var>.e.</var> quod erat propoÅ¿itum. <lb/></s> - <s xml:space="preserve">Ex quo patet error antiquorum quiidipÅ¿um, accidere arbitrati Å¿unt in quantitatibus <lb/>diÅ¿cretæ proportionalitatis, quod tamen falÅ¿um eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proponantur .12. 6. 4. 2. proportio .12. ad .6. eadem eÅ¿t quæ .4. <lb/>ad .2. </s> - <s xml:space="preserve">Sed à proportione .6. ad .4. frangitur, cum non Å¿it eadem quæ .12. ad .6. harum <lb/>autem Å¿umma erit .24. & Å¿umma Å¿ecundæ cum tertia .10. Å¿ed primæ cum tertia erit <lb/>16. ex quo .16. ad .6. non Å¿ic Å¿e habebit vt .24. ad .10. <lb/>At in Å¿peculatione quatuor quantitatum <var>.a.</var> <lb/> - <anchor type="figure" xlink:label="fig-0069-01a" xlink:href="fig-0069-01"/> - <var>e.i.o.</var> Å¿i proportio <var>.e.</var> ad <var>.i.</var> non eſſet eadem <lb/>quæ <var>.a.</var> ad <var>.e.</var> minimè licuiſſet dicere ita Å¿e <lb/>habere <var>.i.</var> ad <var>.e.</var> vt <var>.e.</var> ad <var>.a</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0069-01" xlink:href="fig-0069-01a"> - <image file="0069-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0069-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="88"> - <head xml:space="preserve">THEOREMA <num value="88">LXXXVIII</num>.</head> - <p> - <s xml:space="preserve">CVR extribus quantitatibus quibuÅ¿libet, productum duarum in tertiam, vna <lb/>Å¿emper <reg norm="eademque" type="simple">eademqÌ;</reg> Å¿it quantitas.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, proponuntur .15. 8. 2. Å¿i multiplicauerimus .15. per .8. tum produ <lb/>ctum per .2. tantum erit quantum Å¿i quis multiplicaret .8. per .2. & hoc per .15. et .15. <lb/>per .2. <reg norm="rurÅ¿usque" type="simple">rurÅ¿usqÌ;</reg> per .8.</s> - </p> - <p> - <s xml:space="preserve">Quod ut pateat, tres quantitates tri-<lb/> - <anchor type="figure" xlink:label="fig-0069-02a" xlink:href="fig-0069-02"/> - bus lineis Å¿ignificentur <var>.m.f</var>: a. et <var>.o</var>. </s> - <s xml:space="preserve">Dico <lb/>productum <var>.m.f.</var> in <var>.a.</var> multiplicatum. <lb/></s> - <s xml:space="preserve">per <var>.o.</var> æquale eſſe producto <var>.a.</var> in <var>.o.</var> mul-<lb/>tiplicato per <var>.m.f.</var> aut producto <var>.m.f.</var> in <var>.<lb/>o.</var> multiplicato per <var>.a</var>. </s> - <s xml:space="preserve">Sit enim corpus <var>.d.<lb/>u.</var> <reg norm="rectangulum" type="context">rectãgulum</reg>, cuius latus <var>.n.u.</var> Å¿it æquale <lb/><var>m.f.</var> et <var>.u.t</var>: a: et <var>.u.c</var>: o. patebit manifeÅ¿tè <lb/><var>n.t.</var> eſſe productum <var>.m.f.</var> in <var>.a.</var> quod <var>.n.t.</var> <lb/>multiplicatum in <var>.u.c.</var> æquali <var>.o.</var> producit <lb/>corpus <var>.d.u.</var> Å¿ed idipÅ¿um corpus <var>.d.u.</var> ex <lb/>multiplicatione producti <var>.c.t.</var> in latus <var>.n.<lb/>u.</var> æquale <var>.m.f.</var> oritur, & idipÅ¿um <var>.d.u.</var> ex <lb/>multiplicatione <var>.n.c.</var> in latus <var>.u.t.</var> æquale <var>.a.</var> profertur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0069-02" xlink:href="fig-0069-02a"> - <image file="0069-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0069-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="89"> - <head xml:space="preserve">THEOREMA <num value="89">LXXXIX</num>.</head> - <p> - <s xml:space="preserve">CVR quarumcunque quatuor quantitatum, Å¿i prima in Å¿ecundam multiplice-<lb/>tur & hoc productum in tertiam, <reg norm="rurÅ¿usque" type="simple">rurÅ¿usqÌ</reg> hoc alterum in quartam, vltimum <lb/>productum æquale Å¿it producto producti Å¿ecundæ in tertiam, in productum primæ <lb/>in quartam.</s> - </p> - <pb o="58" rhead="IO. BAPT. BENED." n="70" file="0070" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0070"/> - <p> - <s xml:space="preserve">Exempli gratia, caÅ¿u Å¿eÅ¿e offerunt hi quatuor numeri .8. 5. 3. 2. multiplicato .8. <lb/>per .5. & hoc .40. per .3. rurÅ¿us hoc .120. per .2. vltimum productum eſſet .240. æqua <lb/>le producto .15. (quod ex .5. in .3. oritur) in productum .16. quod ex .8. in .2. pro-<lb/>fertur.</s> - </p> - <p> - <s xml:space="preserve">Cuius Å¿peculationis gratia, cogitemus quatuor numeros quatuor lineis <var>.a.e.i.o.</var> <lb/>Å¿ignifi cari, productum autem <var>.e.</var> in <var>.i.</var> eſſe <var>.m.f.</var> et <var>.r.s.</var> Å¿imiliter & productum <var>.a.</var> in <var>.o.</var> eÅ¿-<lb/>Å¿e <var>.m.z</var>: et <var>.z.f.</var> productum eſſe <var>.m.f.</var> in <var>.m.z.</var> cui productum <var>.a.</var> in <var>.e.</var> multiplicatum per <lb/>i. & hoc tandem per <var>.o.</var> æquari debet.</s> - </p> - <p> - <s xml:space="preserve">Sit itaque <var>.u.y.</var> productum <var>.a.</var> in <var>.e.</var> quod <var>.u.y.</var> per <var>.i.</var> multiplicatum proferat <var>.u.s.</var> <lb/>hocqÌue <var>.u.s.</var> multiplicatum per <var>.o</var>. </s> - <s xml:space="preserve">Dico quod dabit numerum æqualem numero <var>.f.z.</var> <lb/>Quamobrem <var>.r.s.</var> aut <var>.m.f.</var> quod idem eÅ¿t, in figura præcedentis theore matis Å¿igni-<lb/>ficetur linea <var>.n.u.</var> & linea <var>.r.u.</var> hu-<lb/>ius, nempe <var>.a.</var> Å¿ignificetur per <var>.u.t.</var> <lb/> - <anchor type="figure" xlink:label="fig-0070-01a" xlink:href="fig-0070-01"/> - præcedentis, ex quo numerus pro <lb/>ducti <var>.u.s.</var> præſentis, in præcedenti <lb/>Å¿ignificabitur producto <var>.n.t.</var> quod <lb/><reg norm="productum" type="simple context">ê“ductÅ©</reg> <var>.u.s.</var> <reg norm="prÄ™sens" type="context">prÄ™sẽs</reg> <reg norm="per" type="simple">ê‘</reg> <reg norm="præsens" type="context">præsẽs</reg> <var>.o.</var> mulÂ<lb/>tiplicatum, quod erat in præceden <lb/>ti <var>.u.c.</var> Å¿ignificabitur per <var>.d.u.</var> præce <lb/>dentis, quod non modo ex multi-<lb/>plicatione <var>.n.t.</var> præcedentis, nempe <var>.u.s.</var> præſentis. in <var>.u.c.</var> præcedentis æquali <var>.o.</var> præ-<lb/>Å¿entis oritur, Å¿ed etiam ex <var>.c.t.</var> præcedentis æquali <var>.m.z.</var> præſentis in <var>.n.u.</var> præceden <lb/>tis æquali <var>.m.f.</var> præſentis. </s> - <s xml:space="preserve">Itaque verum eÅ¿t propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0070-01" xlink:href="fig-0070-01a"> - <image file="0070-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0070-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="90"> - <head xml:space="preserve">THEOREMA <num value="90">XC</num>.</head> - <p> - <s xml:space="preserve">CVR quibuÅ¿libet & quantiÅ¿uis numeris in Å¿ummam collectis, Å¿i ab vnitate in Å¿e-<lb/>cunda Å¿pecie progreſſionis arithmeticÄ™ imparium numerorum progreſſi fue-<lb/>rimus, eiuÅ¿modi Å¿umma Å¿emper eÅ¿t quadratus numerus.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i horum quatuor diÅ¿parium numerorum <reg norm="Å¿ummam" type="context">Å¿ummã</reg>, in dicta pro-<lb/>greſſione arithmetica quis Å¿umat, principio ab vnitate Å¿umpto, nempe .1. 3. 5. 7. Å¿um-<lb/>ma erit .16. numerus quadratus inquam. </s> - <s xml:space="preserve">Idem de cæteris.</s> - </p> - <p> - <s xml:space="preserve">Quamobrem animaduertendum eÅ¿t, vnitatem, tam Å¿umi pro Å¿ui ipÅ¿ius radicem, <lb/>quam pro quadrato, cubo, cenÅ¿o cenÅ¿i, primo relato, & alia quauis dignitate. <lb/></s> - <s xml:space="preserve">Nunc autem pro quadrato Å¿umamus per <var>.o.</var> Å¿ignificato, <reg norm="cogitemusque" type="simple">cogitemusqÌ</reg> quadratum <var>.o.</var> <lb/>includi quadrato vnitatem Å¿equenti, quod, vt patet, eÅ¿t quatuor vnitatum, ac pro-<lb/>priè primum quadratum numerorum, ex quo etiam nomen accepit, vnde ex Å¿imi-<lb/>litudine quam cætera quadrata cum hoc primo retinent, ex quaternario denomina-<lb/>tionem acceperunt. </s> - <s xml:space="preserve"><reg norm="Hocitaque" type="simple">Hocitaq;</reg> Å¿it <var>.o.u.c.e.</var> ita ex communi Å¿cientia quadrato <var>.o.</var> iun-<lb/>gitur gnomon <var>.e.c.u.</var> conÅ¿tans tribus vnitatibus, quare primus gnomon, numero im-<lb/>pari conÅ¿tat. </s> - <s xml:space="preserve">Scimus etiam ex additione numeri binarij ad imparem, numeris di-<lb/>Å¿paribus Å¿ummam excreÅ¿cere, cum propius accedere <reg norm="quam" type="context">quã</reg> binario nequeant, ex quo <lb/>medio binario, Å¿ibi inuicem Å¿uccedunt. </s> - <s xml:space="preserve">Dico igitur quòd quinario ternarium Å¿ub <lb/>Å¿equente, coniuncto quadrato <var>.o.u.c.e.</var> profertur quadratum, quod in numeris, bi-<lb/>narij quadratum Å¿equitur, <reg norm="eritque" type="simple">eritqÌ;</reg> ternarij, <reg norm="quodque" type="simple">quodqÌ;</reg> Å¿ignificetur per <var>.o.f.</var> patet enim pri <lb/>mo non differre ab <var>.o.c.</var> præter quam gnomone <var>.b.f.d.</var> qui coniungitur quadrato <var>.o.<lb/>c.</var> quique duabus vnitatibus maior eÅ¿t <var>.e.c.u</var>. </s> - <s xml:space="preserve"><reg norm="Iam" type="context">Iã</reg> Å¿cimus gnomonem <var>.e.o.u.</var> æqualem - <pb o="59" rhead="THEOREM. ARIT." n="71" file="0071" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0071"/> - eſſe gnomoni <var>.e.c.u.</var> <reg norm="itemque" type="simple">itemqÌ;</reg> gnomonem <var>.b.f.d.</var> æqualem gnomoni <var>.b.o.d.</var> at hic gno-<lb/>mon <var>.b.o.d.</var> ex præſuppoÅ¿ito, maior eÅ¿t gnomone <var>.e.o.u.</var> duabus vnitatibus <var>.b.</var> et <var>.d.</var> <lb/>Itaque etiam gnomon <var>.b.f.d.</var> duabus vnitatibus gnomonem <var>.e.c.u.</var> Å¿uperabit. </s> - <s xml:space="preserve">Qua-<lb/>re <var>.b.f.d.</var> erit impar immediatè Å¿equens ternarium, qui coniunctus quadrato <var>.o.c.</var> <lb/>quadratum Å¿ubÅ¿equens componet. </s> - <s xml:space="preserve">Eadem ratione probabitur de quadrato <var>.o.n.</var> Å¿e <lb/>quenti <var>.o.f.</var> & gnomone <var>.i.n.a.</var> cum hic ordo Å¿peculationis Å¿it vniuerÅ¿alis. </s> - <s xml:space="preserve">In <lb/>quo cernitur quemlibet gnomonem Å¿ibi <reg norm="contiguum" type="context">contiguÅ©</reg> inferiorem Å¿emper duabus vni-<lb/>tat ibus excedere, cumque quadrata non niÅ¿i gnomonibus Å¿ibi inuicem Å¿uccedant. <lb/></s> - <s xml:space="preserve">Sed <reg norm="cum" type="context">cÅ©</reg> primus <var>.e.c.u.</var> diÅ¿par fuerit, <reg norm="proculdubio" type="simple">ê“culdubio</reg> <reg norm="etiam" type="context">etiã</reg> <reg norm="neceſſarioque" type="simple">neceſſarioqÌ;</reg> cæteri diÅ¿pares <reg norm="erunt" type="context">erÅ©t</reg>. <lb/></s> - <s xml:space="preserve">Ex qua Å¿peculatione, oritur regula ab antiquis tradita <lb/>inueniendi vltimi numeri diÅ¿paris <reg norm="concurrentis" type="context">cõcurrentis</reg> ad <reg norm="compo ſitionem" type="context">cõpoÂ<lb/> - <anchor type="figure" xlink:label="fig-0071-01a" xlink:href="fig-0071-01"/> - Å¿itionem</reg> alicuius quadrati. </s> - <s xml:space="preserve">Vt Å¿i quis Å¿eire deÅ¿ideret nu-<lb/>merum vltimum diÅ¿parem, quo mediante quadratum <var>.<lb/>o.n.</var> conÅ¿titutum fuit, quod aliud non eÅ¿t quam Å¿cire <lb/>quantus Å¿it numerus vltimi gnomonis <var>.i.n.a.</var> æqualis gno <lb/>moni <var>.i.o.a</var>. </s> - <s xml:space="preserve">Itaque vt Å¿ciamus hunc gnomonem <var>.i.o.a.</var> <lb/>patet duplicandam eſſe radicem <var>.o.e.b.i.</var> <reg norm="dabiturque" type="simple punctuation">dabiturqÌ,</reg> <var>.o.e.<lb/>b.i.</var> et <var>.o.u.d.a.</var> vbi bis reperitur <var>.o.</var> nos autem tantummo <lb/>do quærimus Å¿cire gnomonem .i.b.e.o.u.d.a. </s> - <s xml:space="preserve">Itaque <lb/>minor eÅ¿t vnitate duplo radicis, cum unitas <var>.o.</var> bis repe-<lb/>tatur, quæ tamen in gnomone Å¿emel tantum Å¿umebatur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0071-01" xlink:href="fig-0071-01a"> - <image file="0071-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0071-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="91"> - <head xml:space="preserve">THEOREMA <num value="91">XCI</num>.</head> - <p> - <s xml:space="preserve">CVR Å¿umma quadratorum, quorum radices Å¿unt in proportione Å¿eÅ¿quitertia <lb/>nempe .4. ad .3. quadrata Å¿it.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿umemus quadratum .3. Å¿cilicet 9. quod in Å¿ummam cum qua-<lb/>drato .4. colligemus, nempè .16. <reg norm="eritque" type="simple">eritqÌ;</reg> quadratum .25. & ita quadratum .6. hoc eÅ¿t <num value="36">.<lb/>36.</num> collectum cum quadrato .8. nempè .64. efficiet quadratum .100. ita etiam qua-<lb/>dratum .9. hoceÅ¿t .81. coniunctum quadrato .12. nempè .144. producet quadra-<lb/>tum .225.</s> - </p> - <p> - <s xml:space="preserve">In cuius gratiam Å¿int duo quadrata Å¿ubÅ¿cripta <var>.q.o.</var> et <var>.q.a.</var> quorum radices Å¿int <var>.q.</var> <lb/> - <anchor type="figure" xlink:label="fig-0071-02a" xlink:href="fig-0071-02"/> - g. et <var>.q.p.</var> hoc eÅ¿t <var>.q.g.</var> quatuor vnitatum, et <var>.q.<lb/>p.</var> trium, ex quo <var>.q.a.</var> erit .16. vnitatum et <var>.q.o.</var> <lb/>nouem. </s> - <s xml:space="preserve">Ad hæc cogitemus applicari quadra-<lb/>to <var>.q.a.</var> gnomonem <var>.f.s.h.</var> tam amplum Å¿iue la-<lb/>tum <reg norm="quam" type="context">quã</reg> gnomon <var>.b.a.g.</var> nempè vt <var>.h.</var> Å¿it æqua <lb/>lis .g: g. verò differentia Å¿it qua <var>.q.g.</var> maior eÅ¿t <var>.<lb/>q.p.</var> <reg norm="huncque" type="simple">huncqÌ;</reg> gnomonem <var>.f.s.h.</var> dico Ä™qualem eÅ¿ <lb/>Å¿e quadrato <var>.q.o.</var> nam ex preÅ¿uppoÅ¿ito <var>.g.</var> terra <lb/>dicem <var>.q.p.</var> ingreditur, & quater <var>.q.g.</var> ex quo, <lb/>tres partes <var>.q.k.p.</var> inter Å¿e æquales Å¿unt vnde <lb/>etiam quadratum <var>.q.o.</var> nouem partibus Å¿uper-<lb/>ficialibus quadratis conÅ¿tabit, quarum Å¿ingula <lb/>rum radix æqualis erit <var>.g.</var> cumque præcedenti <lb/>theoremate didicerimus quemlibet gnomo-<lb/>nem quadrati immediatè Å¿equentis æquÄ™ amplitudinis cum gnomone præcedentis, - <pb o="60" rhead="IO. BAPT. BENED." n="72" file="0072" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0072"/> - per duab. vnitatibus Å¿uperficialibus creÅ¿cere, <reg norm="quarum" type="context">quarÅ©</reg> <reg norm="Å¿ingularum" type="context">Å¿ingularÅ©</reg> radix æqualis eÅ¿t <var>.g.</var> ne <lb/>ceſſariò Å¿equitur gnomonem <var>.b.a.g.</var> duabus partibus aut vnitatibus gnomonem <var>.d.<lb/>o.p.</var> Å¿uperare, ita vt gnomon <var>.b.a.g.</var> Å¿eptem vnitatibus, aut partibus Å¿uperficialibus <lb/>quadratis conÅ¿tet. </s> - <s xml:space="preserve">Quare eadem ratione gnomon <var>.f.s.h.</var> conÅ¿tabit nouem Å¿imilibus. <lb/></s> - <s xml:space="preserve">Itaque æqualis erit quadrato <var>.q.o</var>. </s> - <s xml:space="preserve">Quamobrem verum eÅ¿t, quòd quadrato <var>.q.o.</var> <lb/>coniuncto quadrato <var>.q.a.</var> proueniet quadratum <var>.q.s.</var> cuius radix ita differet à <var>.q.g.</var> vt <var>.<lb/>q.g.</var> à <var>.q.p</var>: ex quo tres radices arithmeticè inter Å¿e continuæ proportionales erunt. <lb/></s> - <s xml:space="preserve">IdipÅ¿um dico Å¿i <var>.q.p.</var> fuerit .6. et <var>.q.g</var>: 8: </s> - <s xml:space="preserve">tunc enim Å¿ingulæ partes <var>.q.k.p.g.h.</var> æquipol <lb/>lebunt duabus vnitatibus, quæ cogitabuntur <lb/> - <anchor type="figure" xlink:label="fig-0072-01a" xlink:href="fig-0072-01"/> - in Å¿ummam collectæ, ut cum patribus <var>.q.k.p.<lb/>g.h.</var> integris contemplari liceat. </s> - <s xml:space="preserve">Idem acci-<lb/>det fi <var>.q.p.</var> erit .9. et <var>.q.g.</var> 12. fingulæ enim par-<lb/>tes <var>.q.K.p.g.h.</var> tripartitæ erunt. </s> - <s xml:space="preserve">Idcircò dixi <lb/>gnomonem <var>.f.s.h.</var> tam amplum cogitari de-<lb/>bere, quam gnomon <var>.b.a.g.</var> nempè ut <var>.h.</var> æqua <lb/>lis Å¿it <var>.g</var>. </s> - <s xml:space="preserve">Idem occurret Å¿i <var>.q.g.</var> erit .12. et <var>.q.p.</var> <lb/>quinque, quod cum fuerit patebitex præce-<lb/>dentis theorematis Å¿peculatione, gnomonem <lb/><var>f.s.h</var>: 25. vnitatibus conÅ¿tare, cogitatum am-<lb/>plitudinis Å¿implicis vnitatis denominatæ in <var>.q.<lb/>p.</var> aut <var>.q.g.</var> non amplitudinis gnomonis <var>.b.a.g.</var> <lb/>qui Å¿eptem vnitatibus latus eſſet. </s> - <s xml:space="preserve">Cum igitur <var>.<lb/>q.p.</var> quinque vnitatibus linearibus conÅ¿tet Å¿cimus <var>.q.o</var>: 25. Å¿uperficialibus conÅ¿tare, <lb/>collecto itaque in Å¿ummam quadrato <var>.q.o.</var> cum quadrato <var>.q.a.</var> cognoÅ¿cetur quadra-<lb/>tum <var>.q.s.</var> vnà etiam eius radix. </s> - <s xml:space="preserve">Eadem ratione, alia multa quadrata Å¿imilia contem-<lb/>plari licebit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0071-02" xlink:href="fig-0071-02a"> - <image file="0071-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0071-02"/> - </figure> - <figure xlink:label="fig-0072-01" xlink:href="fig-0072-01a"> - <image file="0072-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0072-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="92"> - <head xml:space="preserve">THEOREMA <num value="92">XCII</num>.</head> - <p> - <s xml:space="preserve">CVR propoÅ¿ito numero pari maiori binario, qui detrahi & in Å¿ummam colli-<lb/>gi debeat ex altero numero quærendo, vt tam reÅ¿iduum quam Å¿umma Å¿int <lb/>quadrata numerorum integrornm. </s> - <s xml:space="preserve">Rectè dimidium propoÅ¿iti numeri in Å¿eipÅ¿um <lb/>multiplicamus, & quadrato huic addimus vnitatem, <reg norm="eritque" type="simple">eritqÌ;</reg> numerus quæfitus.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia proponitur .12. numerus detrahendus, & coniungendus nume-<lb/>ro inueÅ¿tigando, ut reÅ¿iduum detractionis, & Å¿umma Å¿int quadrati numeri. </s> - <s xml:space="preserve">Addi-<lb/>ta vnitate ipÅ¿i .36. quadrato dimidij, dabitur .37. numerus quæſitus.</s> - </p> - <p> - <s xml:space="preserve">Cuius Å¿peculationis gratia, Å¿ubÅ¿cripta quatuor quadrata cogitemus <var>.g.p</var>: <var>u.i</var>: <var>t.c</var>: <var>n.<lb/>K.</var> <reg norm="cogitemusque" type="simple">cogitemusqÌ;</reg> quadratum <var>.g.p.</var> eſſe quadratum Å¿ummæ, <var>K.n.</var> verò reÅ¿idui Å¿ubtractio-<lb/>nis: <var>u.i.</var> <reg norm="autem" type="wordlist">aÅ©t</reg> numerum <reg norm="inueÅ¿tigandum" type="context context">inueÅ¿tigãdÅ©</reg>, ex quo gnomon <var>.u.d.i.</var> cognoÅ¿cetur ita etiam et <var>.n.<lb/>o.K.</var> qui inter Å¿e Å¿unt æquales. </s> - <s xml:space="preserve">Iam certi erimus <var>.e.i.</var> eſſe plus quam dimidium gno-<lb/>monis <var>.n.o.K</var>. </s> - <s xml:space="preserve">Itaque cogitemus rectangulum <var>.r.c.</var> exactum <reg norm="dimidium" type="context">dimidiÅ©</reg> eſſe gnomonis <var>.<lb/>n.o.K.</var> ex unitatibus Å¿uperficialibus quarum una erit <var>.m.a</var>.</s> - </p> - <p> - <s xml:space="preserve">Cuius numeri quadratum Å¿it <var>.t.c.</var> vnde etiam cognitum & cum <var>.K.c.</var> ex communi <lb/>Å¿cientia Å¿it vnitas linearis, </s> - <s xml:space="preserve">propterea quod <var>.m.a.</var> eÅ¿t Å¿uperficialis hoc eÅ¿t quadrata, <lb/>quæ detracta ex <var>.q.c.</var> dimidio gnomonis <var>.n.o.K.</var> (quamuis lineari) Å¿upererit <var>.K.q.</var> co <lb/>gnita, numerorum integrorum (nota <var>q.K.i.</var> Å¿emper minor erit duabus vnitatibus li-<lb/>nearibus & maior vna ex dictis vnitatibus, ut ex te ipÅ¿o contemplari potes) </s> - <s xml:space="preserve">quare <var>. - <pb o="61" rhead="THEOR. ARITH." n="73" file="0073" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0073"/> - n.k.</var> ipÅ¿ius quadratum numerorum integrorum cognoÅ¿cetur, cui addito gnomone <var type="gnomon">.<lb/>n.o.K.</var> cognoÅ¿cemus numerum <var>.u.i.</var> quæſitum.</s> - </p> - <p> - <s xml:space="preserve">Sed cum nobis hæc via, tenenda propoÅ¿itum non fuit, hoc eÅ¿t primo loco inue <lb/>niendi quadrati minoris <var>.n.K.</var> ideo Å¿upereÅ¿t probandum gnomonem <var>.t.o.c.</var> vnitati <reg norm="ae- qualem" type="simple">Ä™-<lb/>qualem</reg> eſſe, nempe quadratulo <var>.m.a.</var> quod patebit, Å¿i conÅ¿ideremus nos Å¿umpÅ¿iſſe <lb/>rectangulum <var>.r.c.</var> pro dimidio gnomonis <var>.n.o.K</var>. </s> - <s xml:space="preserve">etenim Å¿i Å¿upplemento etiam <var>.n.r.</var> qua <lb/>dratulum æquale <var>.m.a.</var> adderetur, pateret gnomonem <var>.n.a.K.</var> cum dicto quadratulo <lb/>collectum, æqualem eſſe gnomoni <var>.n.o.K</var>: cum duo Å¿upplementa <var>.m.t.</var> et <var>.m.c.</var> inter Å¿e <lb/>fint æqualia. </s> - <s xml:space="preserve">Quamobrem inuento quadrato <var>.t.c.</var> ex dimidio gnomonis cognito, <lb/>additur vnitas, gnomon Å¿cilicet <var>.t.o.c.</var> ex quo cognoÅ¿citur numerus <var>.u.i.</var> quæſitus. <lb/></s> - <s xml:space="preserve">Quod autem quadratum <var>.g.p.</var> numeris integris conÅ¿tet, hac ratione probatur viÅ¿um <lb/>enim fuit Å¿upra quadratum <var>.n.K.</var> verè quadratum eſſe, & numeris integris conÅ¿tare, <lb/>pariter etiam <var>.t.c.</var> <reg norm="Å¿eque" type="simple">Å¿eqÌ;</reg> mutuo conÅ¿equi (nam <var>.K.c.</var> eÅ¿t vnitas linearis) ex quo gnomon <lb/><var>n.a.K.</var> numero diÅ¿pari conÅ¿tabit, ex ijs quæ .90. theoremate probata fuerunt. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> <lb/>ex eodem theoremate neceſſe eÅ¿t gnomonem <var>.t.d.c.</var> etiam numero diÅ¿pari conÅ¿tare, <lb/>ita vt à numero <var>.n.a.K.</var> non niÅ¿i duabus vnitatibus differat, nempe vt <var>.c.p.</var> Å¿it vnitas li-<lb/>nearis, Å¿ed ita reuera eÅ¿t, numerus enim <var>.u.d.i.</var> ex præſuppoÅ¿ito par eÅ¿t, </s> - <s xml:space="preserve">quare nume <lb/>rus <var>.t.d.c.</var> diÅ¿par erit, cum alterum vnitate Å¿uperet, videlicet gnomone <var>.t.o.c.</var> vnita <lb/>ri æquali, tum <var>.n.a.K.</var> minor eÅ¿t <var>.n.o.K.</var> ex eodem gnomone <var>.t.o.c.</var> unitati æquali. </s> - <s xml:space="preserve">Ita <lb/>que <var>.n.a.K.</var> minor erit <var>.u.d.i.</var> per vnitatem, & minor <var>.t.d.c.</var> per duas unitates, ex quo Å¿e-<lb/>quitur <var>.g.p.</var> eſſe quadratum <reg norm="integrorum" type="context">integrorÅ©</reg> ex dicto theoremate ac con Å¿equens quadrato <lb/><var>t.c</var>. </s> - <s xml:space="preserve">quare <var>.c.p.</var> vnitas erit, & radices <var>.q.K.</var> et <var>.q.p.</var> horum quadratorum numero bina-<lb/>rio inter Å¿e different. </s> - <s xml:space="preserve">Vnà etiam Å¿cienda eÅ¿t cauÅ¿a, cur numerus propoÅ¿itus neceſſa <lb/> - <anchor type="figure" xlink:label="fig-0073-01a" xlink:href="fig-0073-01"/> - riò binario maior eſſe debeat. </s> - <s xml:space="preserve">Etenim <reg norm="cum" type="context">cÅ©</reg> ipÅ¿e <lb/>Å¿it futurus gnomon <var>.n.o.K.</var> nec poſſit minor eſſe <lb/>numero ternario, vt patet ex .90. theoremate, <lb/>idcirco Å¿equitur neceſſariò maiorem eſſe bina-<lb/>rio debere. </s> - <s xml:space="preserve">Quòd Å¿i diÅ¿par numerus propone-<lb/>retur, nec forma operis nec Å¿peculationis <reg norm="mutan- da" type="context">mutã-<lb/>da</reg> eſſet. </s> - <s xml:space="preserve">Non erit tamen neceſſarium vt ipÅ¿a <lb/>quadrata <var>.n.K.</var> et <var>.g.p.</var> numeris integris conÅ¿ta-<lb/>rent. </s> - <s xml:space="preserve">Sæpius enim fractis <reg norm="componerentur" type="context">cõponerentur</reg>, quod <lb/>ex .90. theoremate facile erit Å¿peculari nihilo-<lb/>minus fractis integris, <reg norm="ipÅ¿isque" type="simple">ipÅ¿isqÌ;</reg> collectis cum Å¿uis <lb/>fractis Å¿ummæ eſſent quadratæ.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0073-01" xlink:href="fig-0073-01a"> - <image file="0073-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0073-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="93"> - <head xml:space="preserve">THEOREMA <num value="93">XCIII</num>.</head> - <p> - <s xml:space="preserve">CVR propoÅ¿itis duobus numeris altero pari, altero verò diÅ¿pari, duplo primi <lb/>minore per vnitatem, Å¿i alium inuenire numerum voluerimus, cui alterum iÅ¿to <lb/>rum coniunctum proferat quadratum, & altero detracto, quadratum Å¿uperÅ¿it. </s> - <s xml:space="preserve">Re-<lb/>ctè datos numeros in Å¿ummam colligemus, quam Å¿ummam in duas quam maximas <lb/>poterimus partes diuidemus, quarum vna pari, altera diÅ¿pari conÅ¿tet, tum vtran-<lb/>que in Å¿eipÅ¿am multiplicabimus, & quadrato minori, duorum numerorum propo-<lb/>Å¿itorum quemuis ademus, ex quo cupimus nobis quadratum minus Å¿upereſſe, & pro <lb/>ueniet nobis numerum quæſitum.</s> - </p> - <p> - <s xml:space="preserve">Exempli gtatia, proponuntur numeri .11. et .6. quorum alter alicui numero ad- - <pb o="62" rhead="IO. BAPT. BENED." n="74" file="0074" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0074"/> - dendus, alter ex eodem detrahendus Å¿it, ex quo proferri debeant bina qua-<lb/>drata. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> numeri illi in Å¿ummam collecti dabunt .17. differentiam minoris quadra <lb/>ti & maioris. </s> - <s xml:space="preserve">I am Å¿i ex hoc .17. binas partes fecerimus, altera erit .8. altera .9. qui <lb/>bus in Å¿eipÅ¿is multiplicatis alterum quadratum erit .64. alterum .81. addito <reg norm="itaque" type="simple">itaq;</reg> ipÅ¿i. <lb/>64. 11. aut .6. pro libito, propoÅ¿itum numerum conÅ¿equemur. </s> - <s xml:space="preserve">cui addito .6. vel .11. <lb/>dabit nobis .81. vel ex ipÅ¿o detracto .11. vel .6. relinquet nobis 64. in pręſenti autem <lb/>exemplo talis numerus erit, aut .70. vel .75. </s> - <s xml:space="preserve">Huius autem theorematis Å¿peculatio <lb/>ex .90. dependet, quo demonÅ¿tratum fuit gnomonem proximè quadratum Å¿equen <lb/>tem, vnitate duplo radicis minorem eſſe.</s> - </p> - </div> - <div type="math:theorem" level="3" n="94"> - <head xml:space="preserve">THEOREMA <num value="94">XCIIII</num>.</head> - <p> - <s xml:space="preserve">CVR Å¿i quis cupiat Å¿ummam progreſſionis arithmeticæ quam citiſſimè cogno <lb/>Å¿cere. </s> - <s xml:space="preserve">Rectè coniunget vltimo termino vnitatem primum terminum, huius <lb/>poÅ¿tea vltimi termini dimidium cum numero terminorum multiplicabit, ex <lb/>quo multiplicationis productum, erit omnium propoÅ¿itorum terminorum Å¿umma, <lb/>aut eundem vltimum terminum iunctum primo, per dimidium numeri terminorum <lb/>multiplicabit. </s> - <s xml:space="preserve">Nam idipÅ¿um eueniet.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proponerentur .17. termini in prima progreſſione arithmeti-<lb/>ca naturali, vltimus eſſet .17. cui coniuncta vnitate primo termino Å¿umma erit .18. <lb/>cuius dimidium cum numero terminorum, nempe .17. multiplicatum cum fuerit, <lb/>oritur productum .153. </s> - <s xml:space="preserve">IdpÅ¿um eueniet, multiplicato dimidio numeri <reg norm="terminorum" type="context">terminorÅ©</reg> <lb/>per vltimum coniunctum vnitati primo termino.</s> - </p> - <p> - <s xml:space="preserve">Quod vt Å¿ciamus, cogitemus terminos progreſſionis collocari, vt in figura Å¿ub-<lb/>Å¿cripta <var>.a.o.n.</var> collocantur, <reg norm="tanquam" type="context">tanquã</reg> per gradus, Å¿umpto principio ab vnitate <var>.n.</var> tum <var>.<lb/>u.t.</var> atque ita gradatim. </s> - <s xml:space="preserve">Sic cogitato abÅ¿oluto parallelogrammo <var>.q.o.</var> Å¿ciemus aper-<lb/>tè Å¿ummam progreſſionis tanto maiorem eſſe dimidio totius <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>, quan <lb/>tum dimidium numeri diametri <var>.a.e.i.c.u.n.</var> requirit. </s> - <s xml:space="preserve">Nam cum parallelogram-<lb/>mum diuidatur à dl<unsure/>ametro in tres partes, diameter vnam occupat, reliquæ verò duÄ™ <lb/>ambientes diametrum inter Å¿e Å¿unt æquales. </s> - <s xml:space="preserve">Sumpto <reg norm="itaque" type="simple">itaq;</reg> diametro cum altera di <lb/>ctarum duarum partium, patet Å¿umi pluÅ¿quam <reg norm="dimidium" type="context">dimidiÅ©</reg> totius <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>. </s> - <s xml:space="preserve">pro <lb/>tanta portione, quantum eÅ¿t dimidiam occupatam à diametro, qui <reg norm="cam" type="context">cã</reg><unsure/> ex diÅ¿cretis <lb/>reÅ¿pondentibus numero terminorum componatur, conÅ¿tat numero æquali eſſe di-<lb/>cto numero terminorum <var>.o.n</var>. </s> - <s xml:space="preserve">Iam Å¿i quis multiplicet <var>.a.o.</var> per dimidium <var>.o.n.</var> procul <lb/>dubio, ex prima Å¿exti aut .18. Å¿eptimi, orietur <reg norm="dimidium" type="context">dimidiÅ©</reg> numeri <reg norm="parallelogrammi" type="context">parallelogrãmi</reg> <var>.q.o.</var> <lb/>quod minus erit Å¿umma progreſſionis dimidio numeri diametri, aut quod idem eÅ¿t <lb/>dimidio <var>.o.n.</var> Å¿ed hoc dimidium <var>.o.n.</var> æquale eÅ¿t producto dimidij vnitatis <var>.n.</var> in <var>.o.n.</var> <lb/>ex .20. Å¿eptimi, cum dimidium <var>.o.n.</var> Å¿it eius productum in <reg norm="vnitatem" type="context">vnitatẽ</reg>. </s> - <s xml:space="preserve">Itaque multipli-<lb/>cato <var>.n.o.</var> per dimidium <var>.o.a.</var> coniunctum dimidio vnitatis <var>.n.</var> oritur Å¿umma quæſita <lb/>propoÅ¿itæ progreſſionis. </s> - <s xml:space="preserve">IdipÅ¿um accidet multiplicata Å¿umma <var>.o.a.</var> & vnitate <var>.n.</var> <reg norm="per" type="simple">ê‘</reg> <lb/>dimidium <var>.o.n.</var> ex .20. Å¿eptimi, cum proportio totius ad totum eadem Å¿it, quæ dimi <lb/>dijad dimidium, ex cauÅ¿a permutationalitatis. </s> - <s xml:space="preserve">Patet etiam in progreſſionibus, <lb/>quæ ab vnitate initium ducunt, Å¿i fiat aÅ¿cenÅ¿us per binarium Å¿umma vltimi termini <lb/>cum primo Å¿emper duplam futuram eſſe numero terminorum, quod Å¿equentes figu - <pb o="63" rhead="THEOREM. ARIT." n="75" file="0075" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0075"/> - ras confideranti Å¿peculari licebit, Diametros harum figurarum notaui literis Å¿iue <lb/>characteribus <var>.a.e.i.c.u.n</var>.</s> - </p> - <figure position="here"> - <image file="0075-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0075-01"/> - </figure> - <figure position="here"> - <image file="0075-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0075-02"/> - </figure> - </div> - <div type="math:theorem" level="3" n="95"> - <head xml:space="preserve">THEOREMA <num value="95">XCV</num>.</head> - <p> - <s xml:space="preserve">IN progreſſionibus, quæ ab alio termino quam vnitate incohantur, idipÅ¿um vt <lb/>monuimus accidit, hoc tamen notato, quòd ex conÅ¿equenti quælibet pars dia-<lb/>metri <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>, minimo termino æqualis erit, prout in progreſſionibus quæ <lb/>ab vnitate originem ducunt, Å¿ingulæ partes diametri, vnitati Å¿ui primi termini æ-<lb/>quales Å¿unt. </s> - <s xml:space="preserve">At in reliquis progreſſionibus, vt in figura patet, eadem eÅ¿t propor-<lb/>tio totius diametri ad <var>.o.n.</var> quæ minimi termini ad vnitatem ex .13. quinti, nempe <var>.<lb/>a.o.</var> ad <var>.o.n.</var> vt <var>.n.n.n.n.</var> ad <var>.n</var>. </s> - <s xml:space="preserve">In eiuÅ¿modi progreſſionibus accidit quoque <reg norm="parallelo- grammum" type="context">parallelo-<lb/>grãmum</reg> à diametro in tres partes diuidi, quarum vnam ipÅ¿e occupat, reliquæ ve-<lb/>ro inter Å¿e æquales ipÅ¿um ambiunt. </s> - <s xml:space="preserve">Ex quo illud etiam Å¿equitur, productum <var>.a.o.</var> in <lb/>dimidium <var>.o.n.</var> æquale eſſe dimidio <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>, quod minus eÅ¿t Å¿umma progreÅ¿-<lb/>Å¿ionis dimidio diametri, quod dimidum Å¿i inuenire voluerimus, minimum <reg norm="terminum" type="context">terminÅ©</reg> <var>.<lb/>n.n.n.n.</var> per dimidium <var>.o.n.</var> multiplicabimus, & ex .18. aut .19. Å¿eptimi ipÅ¿um habe-<lb/>bimus, <reg norm="quandoquidem" type="context">quandoquidẽ</reg> minimo termino per totum <var>.o.n.</var> multiplicato profertur integer <lb/>diameter ex .20. prædicti. </s> - <s xml:space="preserve">Etenim vt diximus, eadem eÅ¿t proportio totius diame-<lb/>tri ad <var>.o.n.</var> quæ minimi termini ad vnitatem. </s> - <s xml:space="preserve">Ita etiam dico ex dicta .20. Å¿eptimi. <lb/></s> - <s xml:space="preserve">idem dimidium diametri oriri, Å¿i quis dimidium minimi termini nempè <var>.n.n.</var> per to <lb/>tum <var>.o.n.</var> multiplicauerit. </s> - <s xml:space="preserve">Quamobrem qui Å¿tatim Å¿ummam propoÅ¿itæ progreſſionis <lb/>cognoÅ¿cere voluerit, <lb/> - <anchor type="figure" xlink:label="fig-0075-03a" xlink:href="fig-0075-03"/> - Å¿emper primum termi <lb/>num <var>.n.n.n.n.</var> cum <var>.a.o.</var> <lb/>coniunget, qua Å¿umma <lb/>per <reg norm="dimidium" type="context">dimidiÅ©</reg> <var>.o.n.</var> mul-<lb/>tiplicata, aut <var>.o.n.</var> per <lb/>dimidium dictæ Å¿um-<lb/>mæ, ex prædictis rationibus propofitum conÅ¿equemur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0075-03" xlink:href="fig-0075-03a"> - <image file="0075-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0075-03"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="96"> - <head xml:space="preserve">THEOREMA <num value="96">XCVI</num>.</head> - <p> - <s xml:space="preserve">CVR Å¿i quis numerum terminorum inuenire velit, cognitis tantummodo pri <lb/>mo atque vltimo, rectè vltimum per primum diuidet, ex quo proueniens - <pb o="64" rhead="IO. BAPT. BENED." n="76" file="0076" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0076"/> - numerus quæſitus erit.</s> - </p> - <p> - <s xml:space="preserve">Quod intelligendum eÅ¿ttamen quoties primus terminus differentia <reg norm="terminorum" type="context">terminorÅ©</reg> <lb/>eÅ¿t, nempe aÅ¿cendens ipÅ¿orum ter minorum.</s> - </p> - <p> - <s xml:space="preserve">Cuius ratio manifeÅ¿tè Å¿peculari poteÅ¿t in figura præcedentis theorematis. </s> - <s xml:space="preserve">Nam <lb/>diuiÅ¿a <var>.a.o.</var> per <var>.n.n.n.n.</var> eadem proportio erit <var>.a.o.</var> ad proueniens, quæ. n <var>.n.n.<lb/>n.</var> ad vnitatem <var>.n.</var> ex definitione diuiÅ¿ionis. </s> - <s xml:space="preserve">At Å¿uperius dictum fuit ita Å¿e ha bere <var>.a.<lb/>o.</var> ad <var>.o.n.</var> vt <var>.n.n.n.n.</var> ad <var>.n.</var> ex quo Å¿equitur ex .11. et .9. quinti pr oueniens eſſe nume-<lb/>rum quæſitum <var>.o.n</var>.</s> - </p> - </div> - <div type="math:theorem" level="3" n="97"> - <head xml:space="preserve">THEOREMA <num value="97">XCVII</num>.</head> - <p> - <s xml:space="preserve">VBI verò primus terminus, reliquorum non erit differentia. </s> - <s xml:space="preserve">Hac de caufa ne-<lb/>ceſſe eÅ¿t detrahere primum ex vltimo, <reg norm="reÅ¿iduumque" type="simple">reÅ¿iduumqÌ;</reg> per numerum aÅ¿cenden-<lb/>tem differentiam Å¿cilicet, partiri, <reg norm="proueniensque" type="simple">proueniensqÌ;</reg> vnitati coniungere, quò numerum <lb/>terminorum habere poſſimus. </s> - <s xml:space="preserve">Scimus etenim tam multas vnitates eſſe in vltimo <lb/>terminorum quot in omnibus interuallis aut differentijs in Å¿ummam collectis Å¿imul <lb/>cum vnitatibus primi termini, <reg norm="totque" type="simple">totqÌ;</reg> funt termini, quot interualla Å¿imul cum pri-<lb/>motermino. </s> - <s xml:space="preserve">Quare fi minimus terminus interuallo æqualis fuerit. </s> - <s xml:space="preserve">Vltimo per pri-<lb/>mum diuiÅ¿o, ex a dductis præcedenti theoremate propofitum confequemur. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> <lb/>primo termino ex vltimo detracto <reg norm="refiduoque" type="simple">refiduoqÌ;</reg> per interuallum, hoc eft numerum dif-<lb/>ferentiæ diuifo, proueniens erit numerus terminorum abÅ¿que primo quod vnus eft, <lb/>coni uncto quoque dicto prouenienti propoÅ¿itum conÅ¿equemur.</s> - </p> - </div> - <div type="math:theorem" level="3" n="98"> - <head xml:space="preserve">THEOREMA <num value="98">XCVIII</num>.</head> - <p> - <s xml:space="preserve">CVR fi quis arithmeticæ progreſſionis dato primo & vltimo fimul cum nume <lb/>ro terminorum, afcendentem numerum cognofcere voluerit. </s> - <s xml:space="preserve">Rectè primuin <lb/>ex vltimo detrahet, <reg norm="refiduumque" type="simple">refiduumqÌ;</reg> per numerum terminorum excepto vno diuidet.</s> - </p> - <p> - <s xml:space="preserve">Huius theorematis Å¿peculatio ex .13. theoremate manifeÅ¿ta crit, nam in præce-<lb/>denti cap. numerus terminorum erat proueniens diuiÅ¿ionis reÅ¿idui Å¿ubtractionis pri-<lb/>mi termini ex vltimo.</s> - </p> - </div> - <div type="math:theorem" level="3" n="99"> - <head xml:space="preserve">THEOREMA <num value="99">XCIX</num>.</head> - <p> - <s xml:space="preserve">CVR Å¿i quis maximum omnium terminorum dictæ progreffionis cognofcere <lb/>voluerit, dato primo vnà cum numero aÅ¿cendenti, <reg norm="numeroque" type="simple">numeroqÌ;</reg> terminorum. </s> - <s xml:space="preserve">Re-<lb/>ctè numerum afcendentem cum numero terminorum excepto vno multiplicabit, <lb/><reg norm="productoque" type="simple">productoqÌ;</reg> primum terminum coniunget.</s> - </p> - <p> - <s xml:space="preserve">Cuius quidem theorematis tum ex vndecimo, tum ex ijs quæ præcedentibus ca-<lb/>pitibus dicta fuerunt, aperta eÅ¿t ratio.</s> - </p> - </div> - <div type="math:theorem" level="3" n="100"> - <head xml:space="preserve">THEOREMA <num value="100">C</num>.</head> - <p> - <s xml:space="preserve">CVR veteres cupientes obtinere Å¿ummam pr<unsure/>ogreffionis continuæ naturalis, <lb/>quæab vnitate initium ducit, dato vltimo termino tantummodo. </s> - <s xml:space="preserve">Dimidium <lb/>vltimi-termini <reg norm="cum" type="context">cÅ©</reg> toto fequente multiplicabant, <reg norm="productumque" type="simple">productumqÌ;</reg> Å¿umma quæſita erat.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i vltimus terminus eiuÅ¿modi progreſſionis fuerit .7. multiplica- - <pb o="65" rhead="THEOR. ARITH." n="77" file="0077" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0077"/> - to dimidio ipÅ¿ius nempe .3. & dimidio, cum numero ipÅ¿um terminum <reg norm="Å¿equenti" type="context">Å¿equẽti</reg>, nem <lb/>pè .8. Å¿umma dictorum terminorum erit .28.</s> - </p> - <p> - <s xml:space="preserve">Huius autem Å¿peculatio ex .94. theoremate dependet, in quo facilè depræhen-<lb/>dere licet ex figura continuæ progreſſionis naturalis, numerum terminorum maxi-<lb/>mo termino Å¿emper æqualem eſſe; </s> - <s xml:space="preserve">ex quo <reg norm="tantum" type="context">tãtum</reg> eÅ¿t dimidium numeriterminorum, <lb/>quantum maximi dimidium, <reg norm="tantusque" type="simple">tantusqÌ;</reg> eÅ¿t vltimus terminus vnitati coniunctus, quan <lb/>tus numerus is, qui vltimum terminum conÅ¿equitur.</s> - </p> - </div> - <div type="math:theorem" level="3" n="101"> - <head xml:space="preserve">THEOREMA <num value="101">CI</num>.</head> - <p> - <s xml:space="preserve">CVR antiqui idip fum, quod iam dictum eft, in ea progreſſione, cuius vltimus ter <lb/>minus diÅ¿par eÅ¿t Å¿cire cupientes, numerum integrorum proximè dimidium <lb/>maximi Å¿equentem Å¿umebant, quem per maximum multiplicabant, ex quo <lb/>Å¿umma quæſita oriebatur.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i dimidium maximi fuiſſet .3. cum dimidio, fumebant quatuor, <lb/>& per maximum .7. multiplicabant, ex quo pariter proferebatur Å¿umma .28.</s> - </p> - <p> - <s xml:space="preserve">Cuius ratio ex .20. Å¿eptimi Euclidis oritur, cum eadem Å¿it proportio numeri fe-<lb/>quentis ma ximum ad numerum dimidium maximi Å¿equentem; </s> - <s xml:space="preserve">quæ maximi ad <reg norm="fuum" type="context">fuÅ©</reg> <lb/>dimidium, eÅ¿t enim dupla.</s> - </p> - </div> - <div type="math:theorem" level="3" n="102"> - <head xml:space="preserve">THEOREMA <num value="102">CII</num>.</head> - <p> - <s xml:space="preserve">TRaditum eÅ¿t à nonnullis, à veteribus obÅ¿eruatam fuiſſe hancregulam, qua Å¿ci-<lb/>re poſſent Å¿ummam alicuius progreſſionis arithmeticæ diÅ¿continuæ aut inter <lb/>cifæ, quæ numero pari terminetur. </s> - <s xml:space="preserve"><reg norm="Multiplicabant" type="context">Multiplicabãt</reg> enim <reg norm="dimidium" type="context">dimidiÅ©</reg> vltimi termini per <lb/>pro ximum numerum dimidio dicto maiorem, ex quo <reg norm="inquiebant" type="context">inquiebãt</reg> Å¿emper productum <lb/>Å¿ummæ quæſitæ æquale eſſe, <reg norm="Å¿ubijciuntque" type="simple">Å¿ubijciuntqÌ;</reg> exemplum progreſſionis, quæ à binario in-<lb/>choata crefcit per binarium. </s> - <s xml:space="preserve">In qua quidem progreſſione non per fe, fed per acci-<lb/>dens regula vera eft. </s> - <s xml:space="preserve">Hoc eÅ¿t, non quia ex Å¿e vnus ex producentibus numeris dimi-<lb/>dium termini maioris futurus Å¿it, alter uerò proximè Å¿equens dimidium, fed quia <lb/>vt dictum eÅ¿t .95. theoremate, eadem eÅ¿t proportio maximi termini ad numerum <lb/>terminorum, quæ minimi ad vnitatem. </s> - <s xml:space="preserve">CumqÌue in præfenti exemplo minimum <lb/>Å¿it duplum vnitati in eiuÅ¿modi caÅ¿u, numerus terminorum, dimidio maximi termini <lb/>æqualis eÅ¿t, qui terminorum numerus ex Å¿e, vt patet, vnus eÅ¿t ex producentibus, al-<lb/>ter verò producens numerus, eÅ¿t proximè dimidium Å¿equens, non exÅ¿e, fed quia nu <lb/>merus Å¿equens, dimidium eÅ¿t Å¿ummæ maximi, & minimi, quæ per fe alter eſſe de-<lb/>bet producens numerus. </s> - <s xml:space="preserve">In cæteris enim progreſſionibus, quæ binario non creÅ¿cút <lb/>regulafal<unsure/>fa eÅ¿t, prout facilè patere poteÅ¿t ei, qui ex Å¿cientiæ legibus ope Å¿peculatio-<lb/>nis .95. theorematis Å¿peculatus fuerit.</s> - </p> - </div> - <div type="math:theorem" level="3" n="103"> - <head xml:space="preserve">THEOREMA <num value="103">CIII</num>.</head> - <p> - <s xml:space="preserve">ALIAM quoque tradunt regulam, qua veteres vÅ¿os fuiſſe dicunt, quo Å¿um-<lb/>mam Å¿cire poſſent progreſſionis diÅ¿continuæ, quænumero diÅ¿pari abÅ¿olui-<lb/>tur. </s> - <s xml:space="preserve">Ea autem eÅ¿t eiuÅ¿modi. </s> - <s xml:space="preserve">Vltimum terminum in duas quam maximè poterant ma-<lb/>ximas partes diuidebant, quarum vna Å¿emper altera maior erat, banc autem maio-<lb/>rem in Å¿eipÅ¿am multiplicabant, at que quadratum hoc, Å¿ummam progreffionis effe - <pb o="66" rhead="IO. BAPT. BENED." n="78" file="0078" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0078"/> - affirmabant. </s> - <s xml:space="preserve">Quæ Å¿anè regula, non Å¿emper, etÅ¿i interdum vera Å¿it.</s> - </p> - <p> - <s xml:space="preserve">Sumebant hi exemplum progreſſionis, quæ ab vnitate incohata creÅ¿cit per bina <lb/>rium, in qua per accidens euenit vt numerus dimidium vltimi termini proximè Å¿e-<lb/>quens, nempe è duabus partibus vltimi termini maior, æqualis Å¿it numero termino <lb/>rum, qui per Å¿e vnus è producentibus, ex ijs que .94. theoremate diximus, eſſe debet; <lb/></s> - <s xml:space="preserve">alter vero producens, qui per Å¿e dimidium Å¿ummæ primi & vltimi eſſe debet, per <lb/>accidens pars maior eÅ¿t duarum vltimi termini, & alteri producenti æqualis.</s> - </p> - <p> - <s xml:space="preserve">Aut alio modo ratiocinemur, dicentes, in huiuÅ¿modi progreſſione dimidium <lb/>Å¿ummæ vltimi termini cum primo, Å¿emper medium proportionale eÅ¿t inter eam <lb/>Å¿ummam & dimidium numeri terminorum, etenim huiuÅ¿modi Å¿umma numero ter-<lb/>minorum Å¿emper dupla eÅ¿t, prout .94. theoremate tradimus. </s> - <s xml:space="preserve">Itaque ex .20. Å¿eptimi, <lb/>quadratum partis maioris, producto Å¿ummæ dictæ in numerum dimidij <reg norm="terminorum" type="context">terminorÅ©</reg> <lb/>æquale erit, quod productum per Å¿e Å¿ummæ progreſſionis eÅ¿t æquale. </s> - <s xml:space="preserve">At in cæte-<lb/>ris eiuÅ¿modi progreſſionibus fallit regula, vt ex Å¿upradictis facilè demonÅ¿tratur.</s> - </p> - </div> - <div type="math:theorem" level="3" n="104"> - <head xml:space="preserve">THEOREMA <num value="104">CIIII</num>.</head> - <p> - <s xml:space="preserve">PErmultis terminis ad libitum propoÅ¿itis, diÅ¿poÅ¿itis nihilominus progreſſio-<lb/>ne, aut proportionalitate geometrica continua, Å¿i minimus ex maximo & exfe-<lb/>quenti minimum detrahatur, reÅ¿iduum maximi, eam proportionem ad fum-<lb/>mam reliquorum omnium terminorum retinebit, quam reÅ¿iduum Å¿ecundi ad pri-<lb/>mum.</s> - </p> - <p> - <s xml:space="preserve">Proponuntur, exempli gratia, quatuor termini .3. 12. 48. 192. continui geome-<lb/>tricè proportionales, Å¿i primum, hoc eÅ¿t minimum, ex Å¿ecundo, & maximo detra <lb/>has, exÅ¿ecundo Å¿upererit .9. ex maximo .189. quod Å¿i minimum per reÅ¿iduum maxi <lb/>mi multiplicaueris, hoc eÅ¿t .189. orietur .567. tum Å¿i huiuÅ¿modi productum per .9. <lb/>( refiduum Å¿ecundi ) diuiÅ¿eris, proueniet .63. quod proueniens æquale erit Å¿ummæ <lb/>reliquorum omnium terminorum, maximo excepto. </s> - <s xml:space="preserve">Ex quo inferre licet ex .20. Å¿e <lb/>ptimi eandem proportionem eſſe .189. ad .63. quæ .9. ad .3. aut Å¿i reÅ¿iduum Å¿ecundi <lb/>per Å¿ummam dictorum terminorum multiplicaueris produceturidem .567. </s> - <s xml:space="preserve">quare <lb/>ex .20. Å¿eptimi & cætera.</s> - </p> - <p> - <s xml:space="preserve">Quod vt <reg norm="Å¿cientificè" type="context">Å¿ciẽtificè</reg> poſſimus, & in vniuerÅ¿um Å¿peculari. </s> - <s xml:space="preserve">Quatuor termini propo-<lb/>Å¿iti, quatuor Å¿ubÅ¿criptis lineis <reg norm="Å¿ignificentur" type="context">Å¿ignificẽtur</reg> <var>.b.i</var>: <var>c.a</var>: <var>f.r</var>: <var>m.s.</var> (quod <reg norm="autem" type="wordlist">aÅ©t</reg> de his quatuor di <lb/>co de <reg norm="centummillibus" type="context">centÅ©millibus</reg>, & eo amplius dicere poſſum.) </s> - <s xml:space="preserve">Nunc minimus terminus <var>.m.s.</var> ex <lb/>maximo <var>.b.i.</var> detrahatur, <reg norm="Å¿uperÅ¿itque" type="simple">Å¿uperÅ¿itqÌ;</reg> <var>.n.i.</var> <reg norm="idemque" type="simple">idemqÌ;</reg> <var>.m.s.</var> ex Å¿ecundo termino <var>.f.r.</var> Å¿ubtra-<lb/>hatur, <reg norm="Å¿uperÅ¿itque" type="simple">Å¿uperÅ¿itqÌ;</reg> <var>.o.r</var>. </s> - <s xml:space="preserve">Dico proportionem <var>.n.i.</var> ad Å¿ummam reliquorum omnium ter-<lb/>minorum <var>.c.a</var>: <var>f.r</var>: <var>m.s.</var> eandem effe, quæ <var>.o.r.</var> ad <var>.m.s</var>. </s> - <s xml:space="preserve">Quamobrem ex tertio & quar-<lb/>to Å¿ecundus <var>.f.r.</var> <reg norm="detrahatur" type="simple">detrahat̃</reg>, ex <reg norm="tertioque" type="simple">tertioqÌ;</reg> Å¿uperÅ¿it <var>.t.a.</var> & ex quarto <var>.e.i.</var> ita etiam tertius <var>.<lb/>c.a.</var> ex quarto <var>.b.i.</var> <reg norm="Å¿uperÅ¿itque" type="simple">Å¿uperÅ¿itqÌ;</reg> <var>.d.i.</var> Å¿anè <lb/> - <anchor type="figure" xlink:label="fig-0078-01a" xlink:href="fig-0078-01"/> - Å¿ic Å¿e habebit <var>.c.a.</var> ad <var>.f.r.</var> vt <var>.c.t.</var> ad <var>.f.o.</var> <lb/>vt <reg norm="quisque" type="simple">quisq;</reg> per Å¿e Å¿cire poteÅ¿t. </s> - <s xml:space="preserve">Quare ex <lb/>19. quinti Å¿ic Å¿e habebit <var>.a.t.</var> ad <var>.r.o.</var> vt <var>.<lb/>c.a.</var> ad <var>.f.r.</var> & permutando ita <var>.a.t.</var> ad <var>.a.<lb/>c.</var> vt <var>.o.r.</var> ad <var>.r.f.</var> & Å¿eparando Å¿ic <var>.a.t.</var> ad <var>.<lb/>a.c.</var> (hoc eÅ¿t <var>.f.r.</var>) vt <var>.r.o.</var> ad <var>.o.f.</var> vide-<lb/>licet <var>.m.s</var>. </s> - <s xml:space="preserve"><reg norm="Idem" type="context">Idẽ</reg> dico de <var>.d.i.</var> ad <var>.a.c.</var> nem-<lb/>pe Å¿ic Å¿e habebit <var>.d.i.</var> ad <var>.a.c.</var> vt <var>.a.t.</var> ad <var>. - <pb o="67" rhead="THEOREM. ARIT." n="79" file="0079" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0079"/> - r.f.</var> hoc eÅ¿t <var>.o.r.</var> ad <var>.m.s.</var> ex .11. quinti. </s> - <s xml:space="preserve">Itaque ex communi Å¿cientia Å¿ic Å¿e habe-<lb/>bit <var>.d.i.</var> ad <var>.d.b.</var> vt <var>.e.d.</var> ad <var>.e.b</var>: cum <var>.e.d.</var> æqualis Å¿it <var>.t.a</var>. </s> - <s xml:space="preserve">Ita etiam vt <var>.e.n.</var> ad <var>.n.b</var>: cum <var>.n.<lb/>e.</var> æqualis Å¿it <var>.o.r</var>. </s> - <s xml:space="preserve">Iam Å¿i Å¿ic Å¿e habeat <var>.d.i.</var> ad <var>.d.b.</var> vt <var>.d.e.</var> ad <var>.e.b.</var> permutando <reg norm="quoque" type="simple">quoq;</reg> Å¿ic <lb/>Å¿e habebit <var>.d.i.</var> ad <var>.d.e.</var> vt <var>.d.b.</var> ad <var>.b.e.</var> & compon endo ita <var>.i.d.e.</var> ad <var>.e.d.</var> vt <var>.d.b.e.</var> ad <var>.e.<lb/>b.</var> & permutando Å¿ic <var>.i.d.e.</var> ad <var>.d.b.e.</var> vt. de <var>.a.d.e.b.</var> nempe vt <var>.e.n.</var> ad <var>.n.b.</var> & permutan <lb/>do ita <var>.i.d.e.</var> ad <var>.e.n.</var> vt <var>.d.b.e.</var> ad <var>.b.n.</var> & componendo ita <var>.i.d.e.n.</var> ad <var>.n.e.</var> vt <var>.d.b.e.</var> et <var>.b.<lb/>n.</var> ad <var>.b.n.</var> & permutando Å¿ic <var>.i.d.e.n.</var> ad <var>.d.b.e.</var> et <var>.b.n.</var> nempe ad <var>.a.c</var>: <var>f.r</var>: <var>m.s</var>: vt <var>.e.n.</var> ad <var>.<lb/>n.b.</var> hoc eÅ¿t. ut <var>.o.r.</var> ad <var>.m.s.</var> quod erat propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0078-01" xlink:href="fig-0078-01a"> - <image file="0078-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0078-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="105"> - <head xml:space="preserve">THEOREMA <num value="105">CV</num>.</head> - <p> - <s xml:space="preserve">CVR deÅ¿ideranti Å¿ummam quorumcunque terminorum progreſſionis conti-<lb/>nuæ geometricæ cognoÅ¿cere. </s> - <s xml:space="preserve">Rectè minimus terminus ex maximo detrahen <lb/>dus eÅ¿t, <reg norm="reÅ¿iduumque" type="simple">reÅ¿iduumqÌ;</reg> per denominantem progreſſionis dempta vnitate diuidendum, <lb/><reg norm="prouenientique" type="simple">prouenientiqÌ;</reg> maximum terminum addendum, ex quo oritur Å¿umma quæſita.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i darentur quatuor termini continui proportionales .8. 12. 18. <lb/>27. primum hoc eÅ¿t minimum .8. ex vltimo .27. detraheremus: </s> - <s xml:space="preserve"><reg norm="remaneretque" type="simple">remaneretqÌ;</reg> .19. qui <lb/>per denominantem progreſſionis, dempta vnitate, diuideretur. </s> - <s xml:space="preserve">Quo loco animad <lb/>uertendum eÅ¿t, quamlibet <reg norm="denominationem" type="context">denominationẽ</reg> cuiuÅ¿cunque proportionis numerorum <lb/>Å¿upra vnitatem fieri, nam de proportionibus multiplicibus dubitandum non eÅ¿t, & <lb/>idipÅ¿um de Å¿uperparticularibus, & Å¿uperpartientibus eÅ¿t intelligendum, vt in præ-<lb/>Å¿enti proportio Å¿eÅ¿quialtera inter duos terminos cogitanda eÅ¿t, nempe inter vnum <lb/>& dimidium, atque vnum. </s> - <s xml:space="preserve">SeÅ¿quitertia autem inter vnum & tertiam partem, <lb/>& vnum. </s> - <s xml:space="preserve">SeÅ¿quiquinta inter vnum cum quinta parte, & vnum. </s> - <s xml:space="preserve">De Å¿uperpartien <lb/>tibus idem aſſero quod de proportione <reg norm="Å¿uperbipartiente" type="context">Å¿uperbipartiẽte</reg> tertias appellata, vt .5. <lb/>ad .3. quæ cogitanda eſſet inter vnum duas tertias, & vnum, Å¿uperbipartiens quar-<lb/>tas inter vnum tres quartas, & vnum, ita vt minor terminus, numerans Å¿cilicet, Å¿em <lb/>per Å¿it vnitas, alter verò denominans. </s> - <s xml:space="preserve">Idem de cæteris. </s> - <s xml:space="preserve">Quare in præſenti exem <lb/>plo, detracta vnitate ex denominante progreſſionis, Å¿upererit tantummodo dimi-<lb/>dium, quo diuiÅ¿o .19. proueniet .38. qui numerus æqualis erit Å¿ummæ <reg norm="reliquorum" type="context">reliquorÅ©</reg> <lb/>omnium terminorum, cui coniuncto vltimo termino .27. dabitur Å¿umma quæſita .65.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius Å¿peculatione, quatuor termini Å¿ignificentur, quatuor lineis <var>.m.s</var>: <var>f.r</var>: <var>c.a.<lb/>b.i.</var> primus autem terminus <var>.m.s.</var> ex vltimo <var>.b.i.</var> detrahatur, <reg norm="reÅ¿iduumque" type="simple">reÅ¿iduumqÌ;</reg> Å¿it <var>.n.i.</var> & ex <lb/>Å¿ecundo <var>.f.r.</var> cuius reÅ¿iduum Å¿it <var>.o.r.</var> proportio verò progreſſionis ea Å¿it, quæ <var>.g.h.</var> ad <var>.<lb/>y.</var> quo vnitas repræſentatur (ex quo Å¿ic Å¿e habebit <var>.g.h.</var> ad <var>.y.</var> vt <var>.f.r.</var> ad <var>.m.s.</var>) qua <var>.y.</var> de <lb/>tracta ex <var>.g.h.</var> Å¿uperÅ¿it <var>.h</var>. </s> - <s xml:space="preserve">Tum erecta <lb/>cogitetur linea <var>.n.u.x.</var> indefinita per <lb/> - <anchor type="figure" xlink:label="fig-0079-01a" xlink:href="fig-0079-01"/> - pendicularis <var>.b.i.</var> à puncto <var>.n.</var> quÄ™ diui <lb/>datur in puncto <var>.x.</var> ita vt <var>.n.x.</var> æqualis <lb/>Å¿it vnitati <var>.y.</var> & in puncto <var>.u.</var> ita. vt <var>.n.<lb/>u.</var> æqualis Å¿it <var>.h.</var> ex quo eadem erit <lb/>proportio <var>.n.u.</var> ad <var>.n.x.</var> vt <var>.h.</var> ad <var>.y.</var> <reg norm="nem- pe" type="context">nẽ-<lb/>pe</reg> <var>.o.r.</var> ad <var>.m.s</var>. </s> - <s xml:space="preserve">Nam cú Å¿ic Å¿e habeat <var>.<lb/>f.r.</var> ad <var>.m.s.</var> hoc eÅ¿t ad <var>.f.o.</var> vt <var>.g.h.</var> ad <var>.y</var> <lb/>hoc eÅ¿t ad <var>.g.</var> permutando <reg norm="quoque" type="simple">quoq;</reg> Å¿ic <lb/>Å¿e habebit <var>.f.r.</var> ad <var>.g.h.</var> vt <var>.f.o.</var> ad <var>.g</var>. </s> - <s xml:space="preserve">Ita <lb/>que ex .19. quinti <var>.o.r.</var> ad <var>.h.</var> vt <var>.f.r.</var> ad <var>.g.h.</var> ex quo ex .11. eiuÅ¿dem <var>.o.r.</var> ad <var>.h.</var> vt <var>.f.o.</var> ad - <pb o="70" rhead="IO. BAPT. BENED." n="80" file="0080" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0080"/> - g. & permutando <var>.o.r.</var> ad <var>.f.o.</var> hoc eÅ¿t ad <var>.m.s.</var> vt <var>.h.</var> ad <var>.g.</var> hoc eÅ¿t <var>.y</var>. </s> - <s xml:space="preserve">Quamobrem ea-<lb/>dem erit proportio <var>.o.r.</var> ad <var>.m.s.</var> quæ <var>.n.u.</var> ad <var>.n.x</var>. </s> - <s xml:space="preserve">Abfoluantur itaque duo rectangu <lb/>la <var>.x.i.</var> et <var>.u.z.</var> ita tamen vt <reg norm="rectangulum" type="context">rectangulÅ©</reg> <var>.u.z.</var> cogitetur Ä™quale rectangulo <var>.x.i.</var> cuius <var>.x.i.</var> <lb/>Å¿uperficialis numerus ex communi conceptione lineari <var>.n.i.</var> æqualis erit, </s> - <s xml:space="preserve">quare ex <lb/><reg norm="eadem" type="context">eadẽ</reg> communi conceptione, numerus Å¿uperficialis <var>.u.z.</var> lineari <var>.n.i.</var> æqualis erit, qui <lb/>quidem numerus in figura rectangu-<lb/>la Å¿uperficialis cognitandus erit, cum <lb/> - <anchor type="figure" xlink:label="fig-0080-01a" xlink:href="fig-0080-01"/> - diuidendus Å¿it per <var>.h.</var> hoc eÅ¿t per <var>.n.u.</var> <lb/>ex quo proueniens ex huiuÅ¿modi di <lb/>uiÅ¿ione erit numerus <var>.n.z.</var> ex ijs <lb/>quæ .10. theoremate dicta fuerunt. <lb/></s> - <s xml:space="preserve">Sed ex .15. fexti aut .20. Å¿epti-<lb/>mi eadem eÅ¿t proportio <var>.n.i.</var> ad <var>.n.z.</var> <lb/>quæ <var>.n.u.</var> ad <var>.n.x.</var> hoc eÅ¿t <var>.o.r.</var> ad <var>.m.s.</var> <lb/>videlicet vt <var>.n.i.</var> ad aggregatum reli-<lb/>quorum omnium terminorum <var>.c.a</var>: <var>f.<lb/>r</var>: </s> - <s xml:space="preserve">m.s. ex præcedenti theoremate, & ex .11. quinti Euclidis. </s> - <s xml:space="preserve">Itaque ex .9. eiuf-<lb/>dem numerus <var>.n.z.</var> æqualis erit Å¿ummæ trium terminorum <var>.c.a</var>: f. num <var>.s.</var> cui coniuncto <lb/>quarto termino <var>.b.i.</var> propoÅ¿itum obtinetur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0079-01" xlink:href="fig-0079-01a"> - <image file="0079-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0079-01"/> - </figure> - <figure xlink:label="fig-0080-01" xlink:href="fig-0080-01a"> - <image file="0080-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0080-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="106"> - <head xml:space="preserve">THEOREMA <num value="106">CVI</num>.</head> - <p> - <s xml:space="preserve">PRopoÅ¿uere veteres quæſita nonnulla de itineribus <reg norm="interque" type="simple">interqÌ;</reg> hoc vnum fuit. </s> - <s xml:space="preserve">Po-<lb/>namus duos iter agere per eandem viam quorum alter quatuor milliaria Å¿in-<lb/>gulis diebus conficiat, alter verò prima die milliare vnum, Å¿ecunda duo, tertia tria, <lb/>atque ita Å¿ingulis diebus milliare addit; </s> - <s xml:space="preserve">quærimus quot dierum Å¿pacio Å¿ocium con <lb/>Å¿equetur.</s> - </p> - <p> - <s xml:space="preserve">Quamobrem numerus milliarium primi viatoris duplicatur, Å¿ic Å¿unt .8. milliaria. <lb/></s> - <s xml:space="preserve">ex quo Å¿emper vnitas detrahicur, quæ in præſenti exemplo erit .7. <reg norm="totque" type="simple">totqÌ;</reg> dies erunt <lb/>quibus Å¿ocius Å¿ocium conÅ¿equetur, & milliarium numerum æqualem abÅ¿oluerit.</s> - </p> - <p> - <s xml:space="preserve">Cuius rei facilis erit Å¿peculatio, Å¿i Å¿ubÅ¿cripta figura diligenter conÅ¿ideretur, in <lb/>qua primus viator, die prima, quatuor milliaria linea <var>.q.d.</var> Å¿ignificata conficit, at-<lb/>que illa ipÅ¿a die alter vnum tantum defignatum per <var>.d.</var> perficit, ita vt primus via-<lb/>tor tribus milliaribus Å¿ocium anteceſſerit, altera verò die fecundus uiator cum duo <lb/>milliaria <reg norm="conficiat" type="context">cõficiat</reg>, excedètur à primo duobus milliaribus tantummodo, quÄ™ cum tri-<lb/>bus primæ diei quinque erunt; </s> - <s xml:space="preserve">tertia die ijÅ¿dem de cauÅ¿is primus Å¿ex tantum millia-<lb/>ribus à ſecundo diÅ¿tabit, cum verò quarta die tot Å¿ecundus quot primus milliaria <lb/>conficiat, primus à ſecundo amplius quam antea non diÅ¿tabit; </s> - <s xml:space="preserve">quinta verò cum Å¿e <lb/>cundus vnum milliare amplius quam primus conficiat. </s> - <s xml:space="preserve">propius accedit ad primum <lb/>vno ex Å¿ex milliaribus, quibus anteà diÅ¿tabat, tum Å¿exta cum duobus primum Å¿upe-<lb/>ret, detrahet ex Å¿ex milliaribus præteritæ diÅ¿tantiæ tria, Å¿eptima tandem illa Å¿ex <lb/>detraxerit. </s> - <s xml:space="preserve">In quo conÅ¿iderandum eÅ¿t Å¿ecundum viatorem iter agere progreſſio-<lb/>ne arithmetica continua naturali <var>.d.c.f.</var> primum autem per rectangulum <var>.q.f.</var> <reg norm="quarum" type="context">quarÅ©</reg> <lb/>duarum figurarum <var>.d.o.p.f.</var> pars <reg norm="communis" type="context">cõmunis</reg> eſſe reperitur, quæ quantitates Å¿i inuicem <lb/>æquales eſſe debent, neceſſe eÅ¿t Å¿eparatas partes <var>.u.q.n.</var> et <var>.t.i.c.</var> inter Å¿e æquales eſſe, <lb/>& quoniam quarta die (hoc eÅ¿t die Å¿ic diÅ¿tante à primo, nempè numero milliarium - <pb o="71" rhead="THEOREM. ARITH." n="81" file="0081" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0081"/> - primi viatoris) tot milliaria abÅ¿oluat vnus <lb/> - <anchor type="figure" xlink:label="fig-0081-01a" xlink:href="fig-0081-01"/> - quot alter abÅ¿que vlla differentia, quæ Å¿igni-<lb/>ficetur per <var>.o.s.</var> neceſſe eÅ¿titaque ex communi <lb/>conceptione tot dies eſſe poÅ¿t <var>.o.s.</var> quotante-<lb/>ceſſerant, vt exceſſus æqualis Å¿it defectui, qui <lb/>Å¿imul collecti, iuncta etiam <var>.o.s.</var> duplum erunt <lb/><var>d.s.</var> dempta vnitate, prout facilè in Å¿ubÅ¿cripta <lb/>figura qui Å¿que per Å¿e Å¿cientificè poterit Å¿pecu <lb/>lari. </s> - <s xml:space="preserve">Quamobrem conÅ¿ultum erit duplicare <lb/>numerum <var>.o.s.</var> & exduplo vnitatem detrahe-<lb/>re, quandoquidem dies Å¿upra <reg norm="infraque" type="simple">infraqÌ;</reg> <var>.o.s.</var> cum <lb/>die <var>.o.s.</var> minores Å¿unt duplo numeri <var>.d.s.</var> aut <var>.o.<lb/>s.</var> (quodidem eÅ¿t) vnitate.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0081-01" xlink:href="fig-0081-01a"> - <image file="0081-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0081-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="107"> - <head xml:space="preserve">THEOREMA <num value="107">CVII</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">QVod</emph> <reg norm="Å¿iÅ¿ecundus" type="context">Å¿iÅ¿ecÅ©dus</reg> viator <reg norm="ordinem" type="context">ordinẽ</reg> <reg norm="Å¿ecundæ" type="context">Å¿ecÅ©dæ</reg> progreſſionis arithmeticæ <reg norm="Å¿eruansiter" type="context">Å¿eruãsiter</reg> <lb/>agat, nempe ea quæ ab vno per binarium aÅ¿cendit, Å¿emper numerus dierum <lb/>æqualis erit numero milliarium diurnorum primi viatoris.</s> - </p> - <p> - <s xml:space="preserve">In cuius gratiam animaduertendum eÅ¿t numerus ne milliarium diurnorum pri-<lb/>mi viatoris par an impar Å¿it. </s> - <s xml:space="preserve">Etenim Å¿i par eÅ¿t, primus viator in fine Å¿ingulorum die-<lb/>rum primæ medietatis numeri omnium dierum Å¿ecundum antecedet numero diÅ¿pa <lb/>ri milliarium; </s> - <s xml:space="preserve">altero verò dimidio numero dierum, à ſecundo numero etiam diÅ¿pa <lb/>ri præteribitur, vt in Å¿equenti figura patet. </s> - <s xml:space="preserve">Nam prima die, Å¿ecundus ex primo <lb/>milliare vnum ex numero pari, qui à primo conficitur detrahit; </s> - <s xml:space="preserve">Å¿ecunda verò die <lb/>idem Å¿ecundus, duo Å¿ubtrahit milliaria exdiÅ¿pari, qui primo reliquus fuerat, <reg norm="Å¿icque" type="simple">Å¿icqÌ;</reg> <lb/>perpetuò diÅ¿par remanet vÅ¿que ad vnitatem, ad quam cum peruenerint, nempe ad <lb/>illius diei exitum, quo primus Å¿ecundum vnitate tantummodò Å¿uperat, manifeÅ¿tè <lb/>depræhendetur Å¿ubÅ¿equente dieÅ¿ecundum vnitate primum Å¿uperaturum, altera ve <lb/>rò tribus vnitatibus, prout penultima die Å¿ecundus à primo tribus vnitatibus Å¿upera <lb/>batur. </s> - <s xml:space="preserve">Quare neceſſe erit, tot diebus Å¿ecundum cum primo iter agere, inchoan-<lb/>do ab ea die, qua fecundus primam Å¿uperabit, quot egerat dum à primo Å¿uperare-<lb/>tur, vt ex communi conceptione, media figura <var>.A.</var> depræhendi poteÅ¿t. </s> - <s xml:space="preserve">Quod au-<lb/>tem Å¿ingula dimidia dierum, dimi-<lb/> - <anchor type="figure" xlink:label="fig-0081-02a" xlink:href="fig-0081-02"/> - dia Å¿int numeri milliarium diurno-<lb/>rum primi; </s> - <s xml:space="preserve">patebit exÅ¿equenti fi-<lb/>gura, cogitato termino <var>.u.n.</var> vlti-<lb/>mo progreſſionis Å¿uperatÄ™ à primo <lb/>vÅ¿que ad vnitatem <var>.e.</var> quiterminus <lb/><var>u.n.</var> coniunctus primo <var>.o.</var> nempe <var>.e.</var> <lb/>Å¿emper <reg norm="duplum" type="context">duplÅ©</reg><unsure/> eÅ¿t numeritermino-<lb/>rum <var>.o.n.</var> vt .94. theoremate circa <lb/>finem dictum fuit. </s> - <s xml:space="preserve">Sed <var>.u.n.</var> cum <var>.e.</var> <lb/>numero æquali conÅ¿tat numero <lb/>milliarium diurnorum primi viatoris, ex quo Å¿equitur totum numerum dierum, quo <lb/>rum <var>.o.n.</var> dimidium eÅ¿t, æqualem eſſe numero milliarium diuruorum primi via-<lb/>toris.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0081-02" xlink:href="fig-0081-02a"> - <image file="0081-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0081-02"/> - </figure> - </div> - <pb o="70" rhead="IO. BAPT. BENED." n="82" file="0082" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0082"/> - </div> - <div type="math:theorem" level="3" n="108"> - <head xml:space="preserve">THEOREMA <num value="108">CVIII</num>.</head> - <p> - <s xml:space="preserve">AT Å¿i numerus milliarium primi viatoris diÅ¿par fuerit, Å¿ecundum numero pari <lb/>Å¿emper Å¿uperabit, vt facile erit Å¿equentem figuram conÅ¿ideranti intelligere, <lb/>ex quo illud Å¿equetur, futuram quandam diem, qua paria milliaria conficient. </s> - <s xml:space="preserve">Sit-<lb/>qÌue illa dies <var>.u.n.</var> Å¿equitur <reg norm="etiam" type="context">etiã</reg> tranÅ¿acta ea die, tot diebus vtrique ambulandum eſſe <lb/>quot iter egere anteaquam ad diem <var>.u.n.</var> <lb/>peruenirent, vt tanto numero primus à ſe-<lb/> - <anchor type="figure" xlink:label="fig-0082-01a" xlink:href="fig-0082-01"/> - cundo Å¿uperetur, <reg norm="quanto" type="context">quãto</reg> Å¿ecundum primus <lb/>Å¿uperauerat, vnde totalis numerus <var>.o.f.</var> mi <lb/>nor erit duplo <var>.o.n.</var> vnitate ex communi <lb/>conceptione, Å¿ed ita etiam Å¿e habet termi-<lb/>nus <var>.u.n.</var> hoc eÅ¿t minor duplo <var>.o.n.</var> per <var>.o.</var> vt <lb/>94. theoremate dictum fuit, itaque <var>.o.f.</var> æ-<lb/>qualis erit <var>.u.n.</var> quoderat propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0082-01" xlink:href="fig-0082-01a"> - <image file="0082-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0082-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="109"> - <head xml:space="preserve">THEOREMA <num value="109">CIX</num>.</head> - <p> - <s xml:space="preserve">SIN verò progreſſio Å¿ecundi viatoris, non ab vnitate Å¿ed à binario inchoata, <lb/>per binarium quoque aÅ¿cenderet, <reg norm="numerusque" type="simple">numerusqÌ;</reg> milliarium diurnorum primi via <lb/>toris par eſſet, abÅ¿que dubio quadam die paria milliaria <reg norm="vterque" type="simple">vterq;</reg> conficeret, quÄ™ Å¿igni <lb/>ficetur <var>.u.n.</var> qua tranÅ¿acta, tot diebus vtrique <reg norm="ambulandum" type="context">ambulãdum</reg> erit, quot <reg norm="fuerunt" type="context">fuerÅ©t</reg> <reg norm="dum" type="context">dÅ©</reg> primus <lb/>Å¿ecundum Å¿uperaret, vt totidem alijs pri-<lb/>mus à ſecundo Å¿uperetur, in qua tamen <lb/> - <anchor type="figure" xlink:label="fig-0082-02a" xlink:href="fig-0082-02"/> - progreſſione terminus <var>.u.n.</var> Å¿emper duplus <lb/>eÅ¿t numero terminorum <var>.o.n.</var> ex .95. theo-<lb/>remate, <reg norm="totque" type="simple">totqÌ;</reg> Å¿unt infra <var>.u.n.</var> termini vÅ¿que <lb/>ad <var>.f.</var> quot Å¿upra. </s> - <s xml:space="preserve">ex quo illud Å¿equitur om <lb/>nesterminosaut dies <var>.o.n.f.</var> pauciores eſſe <lb/><var>u.n.</var> vnitate, atque ita præcipit, regula de-<lb/>trahendam eſſe vnitatem ex numero mil-<lb/>liarium diurnorum primi viatoris, Å¿i dierum numerum habere voluerimus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0082-02" xlink:href="fig-0082-02a"> - <image file="0082-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0082-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="110"> - <head xml:space="preserve">THEOREMA <num value="110">CX</num>.</head> - <p> - <s xml:space="preserve">SED Å¿i in eiuÅ¿modi progreſſione numerus milliarium diurnorum primi viato-<lb/>ris diÅ¿par fuerit, patet quòd primus Å¿ecundum numero diÅ¿pari Å¿uperabit, do-<lb/>necad vnitatem perueniatur <reg norm="vi- ciſſimque" type="simple">vi-<lb/> - <anchor type="figure" xlink:label="fig-0082-03a" xlink:href="fig-0082-03"/> - ciſſimqÌ;</reg> primum Å¿ecundus, in-<lb/>choando ab vnitate, </s> - <s xml:space="preserve">quare nul-<lb/>la <reg norm="vnquam" type="context">vnquã</reg> die paria milliaria vter-<lb/>que conficiet, Å¿it itaque vltima <lb/>dies, qua primus Å¿ecundum vnita <lb/>tem antecedit <var>.u.n.</var> qui terminus <lb/>duplus eÅ¿t numero terminorum <lb/><var>o.n.</var> & cum illa die primus Å¿ecun-<lb/>dum milliario antecedat, Å¿equen - <pb o="71" rhead="THEOREM. ARIT." n="83" file="0083" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0083"/> - te verò à ſecundo milliario vno primus antecedatur, ex communi Å¿cientia neceſſe <lb/>eÅ¿t Å¿ecundum tot diebus <reg norm="cum" type="context">cÅ©</reg> primo iter agere quot Å¿unt <var>.o.n.</var> qui Å¿imul æquales erunt <var>.<lb/>u.n.</var> Å¿ed <var>.u.n.</var> minor eft numero milliarium diurnorum primi vnitate <var>.e</var>. </s> - <s xml:space="preserve">Itaque rectè <lb/>Å¿equemur regulam, quæ iubet ex numero milliarium vnitatem demere, quo nu <lb/>merum dierum habere poſſimus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0082-03" xlink:href="fig-0082-03a"> - <image file="0082-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0082-03"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="111"> - <head xml:space="preserve">THEOREMA <num value="111">CXI</num>.</head> - <p> - <s xml:space="preserve">SI verò Å¿ecundi viatoris progreſſio per ternarium aÅ¿cenderet, Å¿umpto initio ab <lb/>ipÅ¿o ternario, animaduertendum eÅ¿t an numerus milliarium diurnorum primi, <lb/>ternario menÅ¿uretur necne, etenim Å¿i menÅ¿uretur, tandem aliquando paria millia-<lb/>ria conficient, quæ dies Å¿it <var>.u.n.</var> </s> - <s xml:space="preserve">quare Å¿ub <lb/> - <anchor type="figure" xlink:label="fig-0083-01a" xlink:href="fig-0083-01"/> - <var>u.n.</var> totidem quot Å¿upra termini <reg norm="erunt" type="context">erÅ©t</reg>, <lb/>& <reg norm="cum" type="context">cÅ©</reg> <var>.o.n.</var> tertia Å¿it pars <var>.u.n.</var> ex .95. theo-<lb/>remate. </s> - <s xml:space="preserve">Itaque tota <var>.o.f.</var> minor erit <lb/>duabus tertijs <var>.u.n.</var> vnitate, vtiam re-<lb/>ctè Å¿umendæ Å¿int duæ tertiæ partes <var>.u.n.</var> <lb/>ex quibus vnitas detrahatur Å¿uperÅ¿itqÌue <lb/>numerus <var>.o.f.</var> dierum quæſitorum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0083-01" xlink:href="fig-0083-01a"> - <image file="0083-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0083-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="112"> - <head xml:space="preserve">THEOREMA <num value="112">CXII</num>.</head> - <p> - <s xml:space="preserve">CVM verò milliarium numerus p rimi viatoris metirinon poterit à numero <lb/>aÅ¿cendente Å¿ecundi, patet nullam futuram diem qua pari milliaria conficient, <lb/></s> - <s xml:space="preserve">quare illa vltima qua primus Å¿ecundum antecedet, vno aut duobus milliaribus an-<lb/>tecedet in præſenti caÅ¿u. </s> - <s xml:space="preserve">Antecedat itaque duobus milliaribus, <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> dies <var>.u.n.</var> & alte <lb/>ra <var>.t.i.</var> Å¿ecundus primum vno milliari Å¿uperabit, ita quod Å¿ub <var>.t.i.</var> non poterunt plu-<lb/>res integros dies iter agere, quam ambulauerunt ante diem <var>.u.n.</var> hoc eÅ¿t vÅ¿quequo <lb/>Å¿ecundtis iunctus Å¿it primo, qui numerus dierum, tertia parte <var>.o.n.</var> ipÅ¿ius <var>.u.n.</var> vnitate <lb/>minor erit, cum ex .95. theoremate <var>.o.n.</var> Å¿it tertia pars <var>.u.n.</var> ex quo numerus <var>.o.f.</var> ter-<lb/>minorum aut dierum intergrorum cognitus erit, qui Å¿i cum numero alcendente <lb/>cognoÅ¿cetur, Å¿tatim ex .99. theoremate deueniemus in cognitionem vltimi diei in <lb/>tegri <var>.s.f.</var> atque ita etiam totius Å¿ummæ progreſſionis ex .95. theoremate. </s> - <s xml:space="preserve">Iam verò <lb/>cognito numero milliarium diurnorum primi, Å¿imul cum numero terminorum, aut <lb/>dierum conÅ¿equenter nouerimus rectanguli Å¿ummam, hoc eÅ¿t productum à primo <lb/>viatore formatum, quarum duarum Å¿ummarum in præſenti caÅ¿u Å¿emper ea, quæ <lb/>huiuÅ¿modi producti eÅ¿t, maior erit, cum conÅ¿titutum fuerit Å¿ecundum viatorem à <lb/>primo Å¿uperari ipÅ¿a die <var>.u.n.</var> vno milliari amplius quam Å¿equente die <var>.t.i.</var> primus à ſe <lb/>cundo Å¿uperatur, tum pari gradu iter egerunt Å¿ub <var>.t.i.</var> quo Å¿upra <var>.u.n.</var> ambulauerant. <lb/></s> - <s xml:space="preserve">Hoc animaduertendo, quòd Å¿i Å¿umma progreſſionis maior eſſet rectangulo, ex ea <lb/>Å¿umma neceſſe eſſet <reg norm="numerum" type="context">numerÅ©</reg> mil <lb/>liarium vltimi termini in Å¿umma <lb/> - <anchor type="figure" xlink:label="fig-0083-02a" xlink:href="fig-0083-02"/> - incluÅ¿i detrahere, & reÅ¿iduo ope-<lb/>rari. </s> - <s xml:space="preserve">Nunc verò Å¿ummam pro-<lb/>greſſionis exÅ¿umma rectanguli à <lb/>primo viatore facti Å¿ubtrahi de-<lb/>bet, <reg norm="reÅ¿iduumque" type="simple">reÅ¿iduumqÌ;</reg> Å¿eruari <reg norm="voceturque" type="simple">voceturqÌ;</reg> - <pb o="72" rhead="IO. BAPT. BENED." n="84" file="0084" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0084"/> - <reg norm="primum" type="context">primÅ©</reg> <reg norm="reÅ¿iduum" type="context">reÅ¿iduÅ©</reg>. </s> - <s xml:space="preserve">Ad hæc <reg norm="numerum" type="context">numerÅ©</reg> <reg norm="milliariorum" type="context">milliariorÅ©</reg>, quæ <reg norm="Å¿ecundus" type="context">Å¿ecÅ©dus</reg> viator die <reg norm="Å¿eqennti" type="context">Å¿eqẽnti</reg> <var>.s.f.</var> confi <lb/>ciet <reg norm="Å¿umatur" type="simple">Å¿umat̃</reg>, ex quo numerus <reg norm="milliarium" type="context">milliariÅ©</reg> <reg norm="diurnorum" type="context">diurnorÅ©</reg> primi detrahatur, <reg norm="residuumque" type="context simple">reÅ¿iduÅ©qÌ;</reg> pariter re <lb/>Å¿eruetur, <reg norm="voceturque" type="simple">voceturqÌ;</reg> Å¿ecundum reÅ¿iduum, poÅ¿tmodum numerum milliarium primi <lb/>vnius diei multiplicetur per primum reÅ¿iduum Å¿eruatum, <reg norm="productumque" type="simple">productumqÌ;</reg> per <reg norm="Å¿ecundum" type="context">Å¿ecundÅ©</reg> <lb/>reÅ¿iduum diuidatur <var>.a.c.</var> <reg norm="proueniens" type="context">proueniẽs</reg> (<reg norm="quod" type="simple">ê™</reg> erit iter primi in Å¿equenti die) iungatur reÅ¿i-<lb/>Å¿iduo primo, tot enim erunt milliaria conficienda a Å¿ecundo Å¿equenti die, vt <lb/>Å¿eÅ¿e conÅ¿equantur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0083-02" xlink:href="fig-0083-02a"> - <image file="0083-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0083-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Vtautem Å¿ciamus quantam partem diei <reg norm="Å¿eqenntis" type="context">Å¿eqẽntis</reg>, Å¿ingulos itinere agere oporteat, <lb/><reg norm="proueniens" type="context">proueniẽs</reg> per .24. horas multiplicetur (Å¿uppoÅ¿ito quod <reg norm="ambulantes" type="context">ambulãtes</reg> <reg norm="nullam" type="context">nullã</reg> <reg norm="requiem" type="context">requiẽ</reg> nec <lb/>die nec nocte <reg norm="capiant" type="context">capiãt</reg>), <reg norm="productumque" type="simple context simple">ê“ductÅ©qÌ;</reg> <reg norm="per" type="simple">ê‘</reg> <reg norm="numerum" type="context">numerÅ©</reg> milliariorum vuius diei primi viatoris di <lb/>uidatur, ex quo dabitur <reg norm="quantitas" type="context">quãtitas</reg> <reg norm="horarum" type="context">horarÅ©</reg>, & pars horæ, qua <reg norm="cuique" type="simple">cuiq;</reg> illa die <reg norm="ambulandum" type="context">ambulandÅ©</reg> <lb/>eÅ¿t. </s> - <s xml:space="preserve">Idem accideret Å¿i primum reÅ¿iduum reÅ¿eruatum cum proueniente in Å¿ummam <lb/>colligeretur, <reg norm="eaque" type="simple">eaqÌ;</reg> Å¿umma per .24. horas multiplicaretur, <reg norm="productumque" type="simple">productumqÌ;</reg> per <reg norm="nume- rum" type="context">nume-<lb/>rÅ©</reg> milliariorum Å¿equenti die à ſecundo conficiendorum diuideretur. </s> - <s xml:space="preserve">I dipÅ¿um quo-<lb/>que eueniret multiplicato primo reÅ¿iduo per .24. & producto per Å¿ecundum reÅ¿i-<lb/>duum diuiÅ¿o.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, primus viator diurna milliaria vndecim conficit, Å¿ecundus, pri-<lb/>ma die tria, Å¿ecunda .6. tertia .9. <reg norm="atque" type="simple">atq;</reg> ita deinceps, diuidatur ergo .11. per .3. vnde <lb/>pro numero <var>.o.n.</var> dabitur .3. <reg norm="Å¿upereritque" type="simple">Å¿upereritqÌ;</reg> .2. </s> - <s xml:space="preserve">quare: <var>u.n.</var> ab <var>.e.n.</var> duobus milliaribus Å¿u-<lb/>perabitur, et <var>.i.t.</var> dictum <var>.e.n.</var> vno milliario, ex quo ante diem <var>.e.u.n.</var> duobus diebus <lb/>iter egerunt, <reg norm="totque" type="simple">totqÌ;</reg> diebus ambulandum erit poÅ¿t <var>.t.i.</var> hoceÅ¿t .6. in vniuerÅ¿um inte-<lb/>gris. </s> - <s xml:space="preserve">Ad hÄ™c multiplicato <var>.o.f.</var> hoc eÅ¿t .6. per <var>.x.o.</var> hoc eÅ¿t .3. habebimus <var>.s.f.</var> <reg norm="milliariorum" type="context">milliariorÅ©</reg> <lb/>18. <reg norm="tum" type="context">tÅ©</reg> <reg norm="coniuncto" type="context context">cõiÅ©cto</reg> <var>.x.o.</var> primo termino hoc eÅ¿t .3. <reg norm="cum" type="context">cÅ©</reg> <var>.s.f.</var> hoc eÅ¿t .18. vltimo termino, habe <lb/>bimus .21. quo multiplicato <reg norm="cum" type="context">cÅ©</reg> dimidio <var>.o.f.</var> hoc eÅ¿t .3. habebimus totam Å¿ummam <lb/>progreſſionis .63. Å¿ex dierum integrorum ex .94. theoremate, tum multiplicato .11. <lb/>nempe numero <reg norm="milliariorum" type="context">milliariorÅ©</reg> diurnorum primi cum .6. hoc eÅ¿t cum <var>.o.f.</var> habebimus pa <lb/>rallelogrammum à primo Å¿ex diebus integris confectum milliariorum .66. ex quo <lb/>detracta .63. Å¿umma inquam progreſſionis, Å¿upererit pro primo reÅ¿iduo .3. Å¿umptis <lb/>poÅ¿tea milliaribus .21. pro itinere, quod Å¿ecundus die Å¿equenti <var>.s.f.</var> conficeret, & ex <lb/>ijs detracto numero <reg norm="milliariorum" type="context">milliariorÅ©</reg> diurnorum primi, nempe .11. Å¿ecundum reÅ¿iduum <lb/>erit 10. quod pro diuidenti Å¿eruabitur. </s> - <s xml:space="preserve">Iam multiplicato .11. cum primo reÅ¿iduo <num value="3">.<lb/>3.</num> dabitur .33. qui diuiÅ¿us per .10. Å¿ecundum reÅ¿iduum profert .3. cum tribus decimis, <lb/><reg norm="eritque" type="simple">eritqÌ</reg> iter à primo viatore Å¿equenti die conficiendum, hoc etiam ipÅ¿um proueniens <lb/>cum primo reÅ¿iduo .3. coniunctum, dat .6. cum tribus decimis, quod eÅ¿t iter Å¿ecundi <lb/>viatoris illa Å¿equenti die. </s> - <s xml:space="preserve">Ad inueniendam autem quantitatem diei, qua vtrique <lb/>ambulandum eÅ¿t, perinde erit multiplicare proueniens .3. & tres decimas per .24. ho <lb/>ras, & productum per .11. dimidium iter primi viatoris partiri, ac multiplicare Å¿um <lb/>mam .6. & tres decimas cum .24. horis, <reg norm="productumque" type="simple">productumqÌ</reg> diuidere per .21. hoc eÅ¿t periter <lb/>Å¿ecundi viatoris Å¿equentis diei, vtrinque enim Å¿emper Å¿eptem horæcum .12. minu <lb/>tis prouenient. </s> - <s xml:space="preserve">IdipÅ¿um accidet multiplicato per .24. horas primo reÅ¿iduo .3. pro-<lb/><reg norm="ductoque" type="simple">ductoqÌ;</reg> diuiÅ¿o per Å¿ecundum reÅ¿iduum .10.</s> - </p> - <p> - <s xml:space="preserve">Quarum Å¿peculationum gratia, totum iter parallelogrammi primi viatoris die-<lb/>rum integrorum fignificetur linea <var>.n.e.</var> Å¿umma verò progreſſionis Å¿ecundi linea <var>.f.m.</var> <lb/>parallela <var>.n.e.</var> <reg norm="eritque" type="simple">eritqÌ;</reg> <var>.f.m.</var> minor <var>.n.e</var>. </s> - <s xml:space="preserve">ConÅ¿tituamus deinde à termino <var>.f.n.</var> (majoris <lb/><reg norm="intelligentiÄ™" type="context">intelligẽtiÄ™</reg> gratia) vtranque <reg norm="perpendiculariter" type="context">perpẽdiculariter</reg> duci, <reg norm="producatur" type="simple">ê“ducatur</reg> deinde <var>.n.e.</var> donec <var>.e.<lb/>d.</var> æqualis Å¿it itineri diurno primi viatoris, item etiam producatur <var>.f.m.</var> donec <var>.m.K.</var> <lb/>æqualis Å¿it itineri à ſecundo confecto Å¿equenti die vltimum integrum progreſſio- - <pb o="73" rhead="THEOREM. ARITH." n="85" file="0085" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0085"/> - nis, ex quo <var>.m.k.</var> prolixior erit <var>.e.d.</var> ex præſup poÅ¿ito. </s> - <s xml:space="preserve">PoÅ¿tmodum <var>.m.e.</var> et <var>.k.d.</var> dua-<lb/>bus lineis rectis coniungantur, quæ productæ concurrentin puncto <var>.b.</var> ducatur pari-<lb/>ter <var>.e.g.</var> à puncto <var>.e.</var> parallela <var>.b.k.</var> et <var>.m.a</var>: <var>e.h.</var> et <var>.b.q.</var> parallelæ <var>.f.n.</var> ex quo <var>.f.m.</var> æqua-<lb/>lis erit <var>.n.a.</var> et <var>.m.h</var>: <var>a.e.</var> et <var>.h.q</var>: <var>e.o.</var> et <var>.g.k</var>: <var>e.d.</var> et <var>.f.q</var>: <var>n.o.</var> ex .34. primi Eucli. </s> - <s xml:space="preserve">vnde pro <lb/>portio <var>.m.h.</var> ad <var>.h.q.</var> erit vt <var>.m.g.</var> ad <var>.g.k.</var> quandoquidem vtraque æqualis eÅ¿t propor-<lb/>tioni <var>.m.e.</var> ad <var>.e.b.</var> ex .2. Å¿exti, Å¿ed cum <var>.m.k.</var> et <var>.g.k.</var> notæ Å¿int, pariter cognoÅ¿cetur <var>.m.<lb/>g.</var> Å¿ecundum reÅ¿iduum, cum etiam notæ Å¿int <var>.n.e.</var> et <var>.n.a</var>. </s> - <s xml:space="preserve">Itaque cognoÅ¿cemus <var>.a.e.</var> hoc <lb/>eÅ¿t <var>.m.h.</var> cognitis verò <var>.m.g</var>: <var>g.k.</var> et <var>.m.h.</var> ex .15. Å¿exti aut .20. Å¿eptimi cognoÅ¿cetur <var>.h.<lb/>q.</var> erit igitur <var>.a.e.</var> aut quod idem eÅ¿t <var>.m.</var> hprimum reÅ¿iduum, et <var>.m.g.</var> Å¿ecundum, et <var>.h.<lb/>q.</var> aut <var>.e.o.</var> proueniens, et <var>.n.o.</var> et <var>.f.q.</var> itinera vtriuÅ¿que viatoris inter Å¿e æqualia. <lb/></s> - <s xml:space="preserve">Nec verò prætermittenda eÅ¿t Å¿peculatio vltimæ rationis inueniendæ quantitatis <lb/>diei, quæ conÅ¿tat ope diuiÅ¿ionis producti <var>.m.h.</var> in .24. per <var>.m.g</var>. </s> - <s xml:space="preserve">Ea autem eiuÅ¿modi <lb/>eÅ¿t. </s> - <s xml:space="preserve">Probatum fuit Å¿ic Å¿e habere <var>.m.h.</var> ad <var>.h.q.</var> ut <var>.m.g.</var> ad <var>.g.k</var>. </s> - <s xml:space="preserve">Itaque componendo <lb/>Å¿ic Å¿e habebit <var>.m.q.</var> ad <var>.h.q.</var> vt <var>.m.k.</var> ad <var>.g.k.</var> & permutando <var>.m.q.</var> ad <var>.m.k.</var> vt <var>.h.q.</var> ad <var>.g.<lb/>k</var>. </s> - <s xml:space="preserve">Sed cum Å¿ic Å¿e habeat <var>.m.h.</var> ad <var>.h.q.</var> vt <var>.m.g.</var> ad <var>.g.k.</var> permutando Å¿ic Å¿e habebit <var>.m.<lb/>h.</var> ad <var>.m.g.</var> vt <var>.h.q.</var> ad <var>.g.k.</var> itaque <lb/>ex .11. quinti ita <var>.m.h.</var> ad <var>.m.g.</var> vt <var>.<lb/> - <anchor type="figure" xlink:label="fig-0085-01a" xlink:href="fig-0085-01"/> - m.q.</var> ad <var>.m.k.</var> ex quo permutando <lb/><var>m.h.</var> ad <var>.m.q.</var> vt <var>.m.g.</var> ad <var>.m.k.</var> Å¿ed <lb/><reg norm="cum" type="context">cÅ©</reg> <var>.m.k.</var> Å¿it motus toti diei reÅ¿pon <lb/>dens, Å¿ecurè dicere poterimus, Å¿i <lb/><var>m.g.</var> talis eÅ¿t reÅ¿pectu horarum <num value="24">.<lb/>24.</num> Å¿ignificatarum per <var>.m.k.</var> qualis <lb/> - <anchor type="figure" xlink:label="fig-0085-02a" xlink:href="fig-0085-02"/> - erit <var>.m.h.</var> & quo <lb/>tæ parti dieire-<lb/>Å¿pondens: </s> - <s xml:space="preserve">quæ <lb/><reg norm="poÅ¿tmodum" type="context">poÅ¿tmodÅ©</reg> erit <var>.<lb/>m.q.</var> quæ, vt <reg norm="di- ctum" type="context">di-<lb/>ctÅ©</reg> fuit, talis eÅ¿t <lb/>reÅ¿pectu <var>.m.k.</var> <lb/>qualis <var>.m.h.</var> re-<lb/>Å¿pectu <var>.m.g</var>. </s> - <s xml:space="preserve">Reli <lb/>quÄ™ duæ Å¿pecula <lb/>tiones priorum <lb/><reg norm="modorum" type="context">modorÅ©</reg>, vna & <lb/>eadem eÅ¿t, <reg norm="facilisque" type="simple">facilisqÌ;</reg> per Å¿e mediocriter intelligenti. </s> - <s xml:space="preserve">Eodem modo reliquæ omnes <lb/>progreſſiones Å¿ecundi viatoris <reg norm="cum" type="context">cÅ©</reg> rectangulo primi conferri ex hoc theoremate <lb/>poterunt.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0085-01" xlink:href="fig-0085-01a"> - <image file="0085-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0085-01"/> - </figure> - <figure xlink:label="fig-0085-02" xlink:href="fig-0085-02a"> - <image file="0085-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0085-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="113"> - <head xml:space="preserve">THEOREMA <num value="113">CXIII</num>.</head> - <p> - <s xml:space="preserve">PRoponitur & aliud, primum Å¿cilicet viatorem iter incipere diebus aliquot an-<lb/>tè Å¿ecundum, primum tamen lentius, quà m Å¿ecundum ambulare, & utrunque <lb/>eorum certa quædam milliaria conficere. </s> - <s xml:space="preserve">Iam Å¿iÅ¿cire voluerimus in quot diebus <lb/>Å¿eÅ¿e conÅ¿equentur, uulgaris regula iubet, inÅ¿pici quot milliaria primus Å¿olus iter a-<lb/>gens confecerit, tum animaduerti differentiam diurnam motus vnius ab altero, <reg norm="atque" type="simple">atq;</reg> <lb/>milliarium numerum primi viatoris Å¿oli abundantis per hanc <reg norm="differentiam" type="context">differentiã</reg> diuidi, pro <lb/>ueniens autem erit numerus dierum quæſitus.</s> - </p> - <pb o="74" rhead="IO. BAPT. BENED." n="86" file="0086" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0086"/> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i primus octo diebus antequam Å¿ecundus iter arripuiſſet, con-<lb/><reg norm="feciſſetque" type="simple">feciſſetqÌ;</reg> fingulis diebus .20. milliaria, tum Å¿ecundus .25. quotidie perfeciſſet, mul <lb/>tiplicandus eſſet numerus .8. cum .20. ex quo darentur .160. milliaria à primo Å¿olo <lb/><reg norm="ambulante" type="context">ambulãte</reg> confecta, quibus diuiÅ¿is per .5. differentiam motuum diurnorum, daretur <num value="32">.<lb/>32.</num> numerus quæſitus dierum.</s> - </p> - <p> - <s xml:space="preserve">Cuius ratio apertiſſima eÅ¿t. </s> - <s xml:space="preserve">Sint enim duo rectanguli <var>.a.n.</var> et <var>.u.i.</var> æquales inter <lb/>Å¿e, quibus motus itinerarium Å¿ignificentur, quorum <var>.a.n.</var> Å¿it primi, et <var>.u.i.</var> Å¿ecundi, præ <lb/>tereà . <var>a.c.</var> numerum milliarium diurnorum primi .et <var>.u.e.</var> Å¿ecundi, ex quo <var>.a.c.</var> minor <lb/>erit <var>.u.e.</var> per <var>.o.e.</var> atque ita <var>.o.e.</var> co-<lb/>gnoÅ¿cetur. </s> - <s xml:space="preserve">Tum <var>.c.o.</var> numerum <reg norm="dierum" type="context">dierÅ©</reg> <lb/> - <anchor type="figure" xlink:label="fig-0086-01a" xlink:href="fig-0086-01"/> - primi Å¿oli iter agentis denotet, <reg norm="cumque" type="context simple">cÅ©q;</reg> <lb/>conÅ¿tituamus <var>.a.n.</var> æqualem eſſe <var>.u.i.<lb/>o.i.</var> Ä™qualis erit <var>.o.a.</var> atque <var>.o.a.</var> cogni <lb/>tus ex Å¿uis producentibus <var>.a.c.</var> et <var>.c.o.</var> <lb/>itaque <var>.o.i.</var> etiam cognitus, qui diui-<lb/>Å¿us per latus cognitum <var>.o.e.</var> dabit <var>.e.<lb/>i.</var> cognitum numerum Å¿cilicet dierum, quibus Å¿ecundo ambulandum eÅ¿t, vt primum <lb/>conÅ¿equatur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0086-01" xlink:href="fig-0086-01a"> - <image file="0086-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0086-01"/> - </figure> - </div> - <head xml:space="preserve">APPENDIX THEOREM. CXIII.</head> - <p> - <s xml:space="preserve">AB hoc theoremate Å¿umpÅ¿i ordinem illius operationis, numeris mediantibus, ad <lb/>inueniendam exactam temporis quantitatem, Å¿eu interuallum, <reg norm="quod" type="simple">ê™</reg> tranÅ¿it, vel in <lb/>tercedit inter vnam mediocrem coniunctionem & aliam proximam <reg norm="Å¿equentem" type="context">Å¿equentẽ</reg> duo <lb/>rum planetarum, vt patet in epiÅ¿tola noÅ¿tra ad IlluÅ¿trem Bernardum Trottum con-<lb/>tra Benedictum Altauillam repræhenÅ¿orem Ephemeridum. </s> - <s xml:space="preserve">Verum tamen eÅ¿t <reg norm="quod" type="simple">ê™</reg> <lb/>cum praxis huiuÅ¿modi theorematis Å¿it multiplex, viÅ¿um fuit vnam proponere, quÄ™ <lb/>non ita perÅ¿picua Å¿it, Å¿ed Å¿ubobÅ¿cura, non quòd aliquid voluerim latere illum ami <lb/>cum mihi dilectiſſimum, cui priuatim omnes modos prius oÅ¿tenderam, Å¿ed vt cere-<lb/>brum illius mei aduerÅ¿arij in laberintum conijcerem inextricabilem vt feci, quam-<lb/>uis modus ille egregius etiam Å¿it, vt nunc oÅ¿tendam.</s> - </p> - <p> - <s xml:space="preserve">In dicta epiÅ¿tola igitur mente cogitaui medium motum tardioris planetæ, pu-<lb/>ta Å¿aturni, illius temporis quo velocior planeta, Å¿cilicet Iupiter, percurrit Å¿uo medio <lb/>motu totum zodiacum, incipiendo ambo eodem temporis puncto, nec non ab vna <lb/>eorum media coniunctione, hoc eÅ¿t ab eodem zodiaci puncto, in quo coniunctÄ™ fue <lb/>runt eorum lineæ mediorum motuum, vbi inueni vi regulæ de tribus, quòd Satur <lb/>nus Å¿pacio dierum vnius mediocris reuolutionis Iouis, qui Å¿unt .4328. progreditur <lb/>medio motu gra .145. min .4. hoc eÅ¿t min .8704. pofito quòd ipÅ¿e Saturnus perficiat <lb/>vnam mediam reuolutionem Å¿pacio dierum .10740. vt dixi. </s> - <s xml:space="preserve">Incipiendo igitur ite <lb/>rum Iupiter aliam reuolutionem percurrere, reperto Saturno per min .8704. ante <lb/>ipÅ¿um Å¿pacio .4328. dierum, certus eram hos dies Å¿ignificatos eſſe à linea <var>.a.u.</var> vel <var>.c.<lb/>o.</var> (æquales enim inuicem Å¿unt) in figura huiuÅ¿modi theorematis, & quòd rectangu <lb/>lum <var>.a.o.</var> præbebat Å¿ummam graduum .145. min .4. hoc eÅ¿t min .8704. et quòd <var>.a.c.</var> <lb/>vel <var>.o.u.</var> Å¿ignificabat iter vnius diei ipÅ¿ius Saturni, et <var>.u.e.</var> iter vnius diei Iouis. </s> - <s xml:space="preserve">Cogi-<lb/>temus nunc <var>.u.x.</var> Å¿ignificari dies .30. & à puncto <var>.x.</var> productam eſſe <var>.x.f.</var> parallelam ipÅ¿i <lb/><var>u.o.e.</var> vnde certi erimus rectangulum <var>.e.x.</var> Å¿ignificare iter Ionis Å¿pacio temporis die-<lb/>rum .30. rectangulum verò <var>.o.x.</var> iter Saturni eodem temporis interuallo, vnde rectan - <pb o="75" rhead="THEOREM. ARIT." n="87" file="0087" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0087"/> - gulum <var>.e.x.</var> erit minutorum .149. & Å¿ecundorum .43. et <var>.o.x.</var> minutorum .60. & Å¿ecun. <lb/>20. vt in dicta epiÅ¿tola, vnde rectangulum <var>.o.f.</var> erit min .89. & Å¿ecun .23. & quia re-<lb/>ctangulum <var>.o.i.</var> æquale eÅ¿t rectangulo <var>.a.o.</var> ergo <var>.o.i.</var> Å¿imiliter continebit min .8704. <lb/>Nunc quia <var>.a.c.</var> vel <var>.o.u.</var> denotat iter vnius diei Saturni et <var>.u.e.</var> vnius diei Iouis vt di-<lb/>ximus ergo <var>.u.o.</var> erit minutorum .2. Å¿ecun <var>.o.</var> & tertiarum .40. videlicet tertiarum <num value="7240">.<lb/>7240.</num> Å¿uppoÅ¿ito periodo totaliipÅ¿ius Saturni dierum .10740. et <var>.u.e.</var> erit <reg norm="minutorum" type="context">minutorÅ©</reg>. <lb/>4. Å¿ecun .59. & ter .27. vel circa hoc eÅ¿t tertiarum .17967. vnde <var>.o.e.</var> erit tertiarum <num value="10727">.<lb/>10727.</num> </s> - <s xml:space="preserve">NuncÅ¿i dixerimus cum <var>.o.e.</var> tertiarum .10727. dat <var>.o.u.</var> vel <var>.a.c.</var> (nam tam <lb/>vna quam altera eÅ¿t tertiarum .7240.) quid dabit <var>.a.u.</var> vel <var>.o.c.</var> (quia tam vna quam <lb/>altera eÅ¿t partium .4328.) clarum erit quòd dabit <var>.o.n.</var> vel <var>.u.t.</var> uel <var>.e.i.</var> quia tam vna <lb/>quam altera erit partium .2921. quæ partes coniunctæ cum fuerint cum partibus ip-<lb/>Å¿ius <var>.a.u.</var> dabunt totam <var>.a.t.</var> <reg norm="partium" type="context">partiÅ©</reg> .7249. quæ erunt tot dies, hoc eÅ¿t periodus quæſita.</s> - </p> - <p> - <s xml:space="preserve">Alia methodo Å¿imiliter poſſumus idem cognoÅ¿cere, Å¿cilicet dicendo Å¿i rectangu <lb/>lum <var>.f.o.</var> quod eÅ¿t minutorum .89. & Å¿ecun .23. hoc eÅ¿t Å¿ecundorum .5363. dat rectan <lb/>gulum <var>.o.x.</var> minutorum .60. & Å¿ecun .20. hoc eÅ¿t Å¿ecun .3620. quid dabit <var>.a.u.</var> partium <lb/>4328. vnde veniet <var>.u.t.</var> partium .2921. Å¿imiliter, eo quod eadem proportio eÅ¿t rectan <lb/>guli <var>.f.o.</var> ad <var>.o.x.</var> quæ <var>.e.o.</var> ad <var>.o.u.</var> ex prima Å¿exti, vel .18. 19. Å¿eptimi Å¿eu .15. quinti.</s> - </p> - <p> - <s xml:space="preserve">Poſſet etiam aliquis dicere Å¿i <var>.f.o.</var> dat <var>.o.x.</var> quid dabit <var>.o.a.</var> vnde veniet <var>.o.t.</var> quo <lb/>diuiÅ¿o per <var>.o.u.</var> daret <var>.u.<lb/>t.</var> quia ita Å¿e habet <var>.a.o.</var> <lb/> - <anchor type="figure" xlink:label="fig-0087-01a" xlink:href="fig-0087-01"/> - ad <var>.o.t.</var> vt <var>.a.u.</var> ad <var>.u.t.</var> ex <lb/>Å¿upra hic iam citatis.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0087-01" xlink:href="fig-0087-01a"> - <image file="0087-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0087-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Sed ego, in dicta epi-<lb/>Å¿tola, aliam methodum <lb/>obÅ¿eruaui, quæ eÅ¿t multi <lb/>plicando minuta .8704. <lb/>per .30. <reg norm="productumque" type="simple">productumqÌ;</reg> di <lb/>uiÅ¿i per min .5363. quaÅ¿i <lb/>dicens. </s> - <s xml:space="preserve">Si <var>.o.f.</var> dat <var>.o.i.</var> <lb/>quid dabit <var>.e.f</var>. </s> - <s xml:space="preserve">Vnde exiam Å¿upradictis propoÅ¿itionibus veniet <var>.e.i.</var> & quia permu-<lb/>tando ita Å¿e habet <var>.o.f.</var> ad <var>.e.f.</var> vt <var>.o.i.</var> ad <var>.e.i.</var> ideo dixi, Å¿i min .89. cum Å¿ecun .23. dat .30 <lb/>quid dabit min .8704.</s> - </p> - </div> - <div type="math:theorem" level="3" n="114"> - <head xml:space="preserve">THEOREMA <num value="114">CXIIII</num>.</head> - <p> - <s xml:space="preserve">PRoponunt veteres & quærunt aliud, nempe Å¿i duo iter agentes, eodem in-<lb/>Å¿tanti diuerÅ¿is è locis proficiÅ¿cantur, ita vt vnus locum vnde alter profectus <lb/>eÅ¿t petat, <reg norm="alterque" type="simple">alterqÌ;</reg> altero velocior Å¿it, quo loco <reg norm="quanue" type="context">quãue</reg> die Å¿ibi inuicem occurrent.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Patauio profectus quidam Taurinum petit, eodem inÅ¿tanti al-<lb/>ter Taurino Patauium, <reg norm="eÅ¿tque" type="simple">eÅ¿tqÌ;</reg> iter .400. milliarium, ille tamen vndecim diebus, hic <lb/>9. motu regulari & vniformi appellit. </s> - <s xml:space="preserve">Quærimus quot milliaria quiÅ¿que confece-<lb/>rit, <reg norm="quotque" type="simple">quotqÌ;</reg> diebus iter egerit, priuÅ¿quam Å¿ibi occurrant.</s> - </p> - <p> - <s xml:space="preserve">Iubent nos veteres dies vtriuÅ¿que inuicem inter Å¿e multiplicare, <reg norm="eritque" type="simple">eritqÌ;</reg> produ-<lb/>ctum .99. item etiam in Å¿ummam colligere, <reg norm="eritque" type="simple">eritqÌ;</reg> Å¿umma .20. per quam <reg norm="productum" type="context">productÅ©</reg>. <lb/>99.@diuiÅ¿erimus dabuntur dies .4. cum .19. vigeÅ¿imis vnius diei. </s> - <s xml:space="preserve">At pro milliaribus <lb/>vtriuÅ¿que, pro eo qui .11. diebus iter conficit, multiplicatis .400. per .4. et .19. vigeÅ¿i <lb/>mis, tum diuiÅ¿o per .11. dabitur numerus .180. à Patauio Taurinum & è contra, qui - <pb o="76" rhead="IO. BAPT. BENED." n="88" file="0088" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0088"/> - Taurino Patauium .220. quæ quiÅ¿que confecerit.</s> - </p> - <p> - <s xml:space="preserve">Dum autem hæc Å¿pecularer attentius, occurrit alius Å¿oluendi modus, quamuis pro <lb/>lixior. </s> - <s xml:space="preserve">Is <reg norm="autem" type="wordlist">aÅ©t</reg> eÅ¿t eiuÅ¿modi. </s> - <s xml:space="preserve"><reg norm="Accipiatur" type="simple">Accipiat̃</reg> medietas minoris numeri <reg norm="dierum" type="context">dierÅ©</reg>, <reg norm="nempe" type="context">nẽpe</reg> .4. <reg norm="cum" type="context">cÅ©</reg> dimi <lb/>dio, & per .400. multiplicetur, <reg norm="productumque" type="context simple">productÅ©qÌ;</reg> per <reg norm="maiorem" type="context">maiorẽ</reg> numerum diuidemus Å¿cilicet <lb/>11. ex quo dabuntur .163. cum .7. vndecimis, quo proueniente è dimidio <reg norm="millia- riorum" type="context">millia-<lb/>riorÅ©</reg> itineris .200. detracto, & <reg norm="preÅ¿iduum" type="context">preÅ¿iduÅ©</reg> <reg norm="nempe" type="context">nẽpe</reg> .36. <reg norm="cum" type="context">cÅ©</reg> .4. vndecimis multiplicato pro <lb/><reg norm="ductoque" type="simple">ductoqÌ;</reg> diuiÅ¿o <reg norm="per" type="simple">ê‘</reg> <reg norm="Å¿ummam" type="context">Å¿ummã</reg> dimidij itineris .200. <reg norm="cum" type="context">cÅ©</reg> primo <reg norm="prouentu" type="context">prouẽtu</reg> .163. et .7. vndecimis <lb/><reg norm="nempe" type="context">nẽpe</reg> <reg norm="per" type="simple">ê‘</reg> .363. ct .7. vndecimas partes <reg norm="proueniet" type="simple">ê“ueniet</reg> .16. <reg norm="cum" type="context">cÅ©</reg> .4. vndecimis, quo <reg norm="coniuncto" type="context">cõiuncto</reg> pri <lb/>mo <reg norm="prouenienti" type="simple context">ê“ueniẽti</reg>, primus .180. milliaria <reg norm="confecerit" type="context">cõfecerit</reg>, quæ è .400. detracta Å¿upererunt .220. <lb/>pro itinere Å¿ecundi, qui .9. diebus iter abÅ¿oluit. </s> - <s xml:space="preserve">Ad hæc Å¿i tempus Å¿cire velimus <lb/>eius, qui .11. diebus appellit, multiplicabimus .11. cum .180. <reg norm="productumque" type="simple">productumqÌ;</reg> per .400. <lb/>partiemur, <reg norm="prouenientque" type="simple">prouenientqÌ;</reg> paulominus, quam quinque dies, nempe .4. cum .22. horis <lb/>et .48. minutis, quod tempus vtrique viatori inÅ¿eruiet, quandoquidem idipÅ¿um pro <lb/>uenit multiplicato .220. per .9. <reg norm="productoque" type="simple">productoqÌ;</reg> per .400. diuiÅ¿o.</s> - </p> - <p> - <s xml:space="preserve">Huius autem, qui à me pręſcribitur modi, Å¿peculatio talis eÅ¿t. </s> - <s xml:space="preserve">Duo termini duabus <lb/>rectis lineis æqualibus, & parallelis inter Å¿e <var>.b.p.</var> et <var>.d.q.</var> Å¿ignificentur, quæ alijs dua-<lb/>bus <var>.b.d.</var> et <var>.q.p.</var> <reg norm="coniungantur" type="simple">coniungant̃</reg>, quÄ™ parallelæ & æquales erunt ex .33. primi, quibus Å¿igni <lb/>ficentur duo itinera. </s> - <s xml:space="preserve">Viator primus quidem lentior à . b in <var>.d.</var> velocior à <var>.q.</var> in <var>.p</var>. </s> - <s xml:space="preserve">Iam <lb/>Å¿umatur <reg norm="punctum" type="context">punctÅ©</reg> medium <var>.q.p.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.k.</var> & ab ipÅ¿o ad <var>.b.d.</var> ducatur <var>.k.i.</var> parallela <var>.d.q.</var> aut <lb/><var>b.p.</var> quod idem eÅ¿t, ex quo <var>.b.i.</var> æqualis erit <var>.p.k.</var> ex .34. primi, hoc eÅ¿t <var>.q.k.</var> <reg norm="certique" type="simple">certiqÌ;</reg> eri-<lb/>mus primum viatorem <var>.q.p.</var> in dimidio itineris <var>.q.k.</var> occurrere non potuiſſe viatori ip <lb/>Å¿ius <var>.b.i.</var> quandoquidem eo tempore, quo is, qui ipÅ¿ius <var>.q.p.</var> mouetur per <var>.q.k.</var> (cum Å¿it <lb/>altero velocior) qui per <var>.b.d.</var> nondum peruenerit ad .i: Sit itaque punctum <var>.c.</var> in quo <lb/>lentior reperitur, dum velocior eÅ¿t in <var>.k.</var> ex quo certi erimus eos inter <var>.c.</var> et <var>.i.</var> Å¿ibi in-<lb/>uicem obuiaturos eſſe. </s> - <s xml:space="preserve">Cogito deinde rectam lineam ductam <var>.k.c.</var> & ut Å¿e habet <var>.i.<lb/>c.</var> ad <var>.c.b.</var> ita cogito Å¿e habere. u<unsure/> <var>.k.</var> ad <var>.k.q.</var> & à puncto <var>.u.</var> ad <var>.i.</var> duco <var>.u.i.</var> quæ, vt manife <lb/>Å¿tum eÅ¿t, lineam <var>.k.c.</var> in puncto <var>.e.</var> interÅ¿ecabit, à quo cum fuerit ducta <var>.e.o.n.</var> parallela <lb/><var>k.i.</var> habebimus <var>.o.n.</var> ea Å¿cilicet puncta, quibus occurrunt Å¿ibijpÅ¿is, nam cum Å¿ic Å¿e ha <lb/>beat <var>.q.k.</var> ad <var>.k.u.</var> vt <var>.b.c.</var> ad <var>.c.i.</var> et <var>.k.u.</var> ad <var>.k.n.</var> vt <var>.c.i.</var> ad <var>.c.o.</var> ex Å¿imilitudine manifeÅ¿ta <lb/>triangulorum, ex æqualitate proportionum Å¿ic Å¿e habebit <var>.q.k.</var> ad <var>.k.n.</var> vt <var>.b.c.</var> ad <var>.c.o.</var> <lb/>& permutando ita <var>.k.q.</var> ad <var>.b.c.</var> vt <var>.k.n.</var> ad <var>.c.o.</var> & cum <var>.q.k.</var> et <var>.b.c.</var> Å¿patia Å¿int tempori-<lb/>bus æqualibus confecta, itaque Å¿patia <var>.k.n.</var> et <var>.c.o.</var> ex communi Å¿cientia temporibus <lb/>æqualibus conficientur.</s> - </p> - <p> - <s xml:space="preserve">Quare rectè dicimus, Å¿i tot diebus à <var>.b.</var> in <var>.d.</var> aliquis peruenit, quot milliaria in di <lb/>midio temporis alterius viatoris idem conficiet? </s> - <s xml:space="preserve">ex quo ex regula de tribus quam <lb/>primum iter <var>.b.c.</var> cognoÅ¿citur, quo ex dimidio itineris detracto, remanet <var>.c.i.</var> cogni <lb/>tus, Å¿ed cum probauerimus <var>.q.k.</var> ad <var>.k.n.</var> hoc eÅ¿t <var>.i.o.</var> (cum Å¿int æquales inter Å¿e, ex .34 <lb/>primi) ita Å¿e habere. vt <var>.b.c.</var> ad <var>.c.o.</var> permutando Å¿ic Å¿e habebit <var>.q.k.</var> ad <var>.b.c.</var> vt <var>.i.o.</var> ad <var>.<lb/>o.c.</var> & <reg norm="componendo" type="context">cõponendo</reg> <var>.q.k.</var> et <var>.b.c.</var> ad <var>.b.c.</var> vt <var>.i.c.</var> ad <var>.c.o.</var> </s> - <s xml:space="preserve">quare rectè dicimus Å¿i Å¿umma <var>.q.<lb/>k.</var> cum <var>.b.c.</var> dat <var>.b.c.</var> quid dabit <var>.i.c</var>? </s> - <s xml:space="preserve">nempe dabit <var>.c.o.</var> quo coniuncto cum <var>.b.c.</var> cogno-<lb/>Å¿citur <var>.b.o.</var> quo <var>.b.o.</var> detracto ex <var>.b.d.</var> remanet cognitus <var>.o.d.</var> nempe <var>.q.n.</var> illi æqualis <lb/>ex .34. prædicta. </s> - <s xml:space="preserve">Gratia verò <reg norm="temporis" type="context">tẽporis</reg> patet nos rectè dicere Å¿i <var>.b.d.</var> tot diebus abÅ¿olui <lb/>tur, aut etiam <var>.q.p</var>: quo <var>.b.o.</var> aut <var>.q.n.</var> abÅ¿oluetur.</s> - </p> - <p> - <s xml:space="preserve">Vt autem ad Å¿peculationem regulæ antiquorum deueniamus, cogitemus pri-<lb/>mum viatorem ipÅ¿ius <var>.q.p.</var> velociorem eo, qui per <var>.b.d.</var> iter agit, tanto tempore præ <lb/>tergredi <var>.p.</var> quanto alter <var>.b.d.</var> abÅ¿oluit. </s> - <s xml:space="preserve">Is autem ad <var>.g.</var> pertingat, ex quo eadem pro-<lb/>portio Å¿pacij <var>.q.g.</var> ad <var>.q.p.</var> hoc eÅ¿t <var>.b.d.</var> dabitur, quæ temporis quo <var>.b.d.</var> abÅ¿oluitur ab - <pb o="77" rhead="THEOREM. ARIT." n="89" file="0089" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0089"/> - eo qui per <var>.b.d.</var> ad tempus quo <var>.q.p.</var> Å¿olum, qui per <var>.q.p.</var> mouetur (mo-<lb/>tus enim continui regulares & vniformes conÅ¿tituuntur) eadem ratione ita-<lb/>que ea erit proportio <var>.q.k.</var> ad <var>.b.c.</var> quæ <var>.q.g.</var> ad <var>.q.p.</var> & cum probatum <lb/>fuerit ita Å¿e habere <var>.k.n.</var> ad <var>.c.o.</var> vt <var>.q.k.</var> ad <var>.b.c.</var> itaque Å¿ic Å¿e habebit <var>.k.n.</var> ad <var>.c.<lb/>o.</var> ut <var>.q.g.</var> ad <var>.q.p.</var> probatum etiam fuit ita Å¿e habere <var>.q.k.</var> ad <var>.k.n.</var> vt <var>.b.c.</var> ad <var>.c.o.</var> ex quo <lb/>componendo Å¿ic Å¿e habebit <var>.q.n.</var> ad <var>.n.k.</var> vt <var>.b.o.</var> ad <var>.o.c.</var> & permutando ita <var>.q.n.</var> ad <var>.b.<lb/>o.</var> vt <var>.k.n.</var> ad <var>.c.o.</var> hoc eÅ¿t <var>.q.g.</var> ad <var>.q.p.</var> nempe vt tempus lenti ad tempus velocis itine-<lb/>rantis, & componendo ita <var>.q.n.</var> cum <var>.o.b.</var> hoc eÅ¿t <var>.b.d.</var> ad <var>.b.o.</var> vt Å¿umma <reg norm="dierum" type="context">dierÅ©</reg> vnius & <lb/>alterius viatoris ad <reg norm="minorem" type="context">minorẽ</reg> <reg norm="numerum" type="context">numerÅ©</reg> <reg norm="dierum" type="context">dierÅ©</reg> velocioris. </s> - <s xml:space="preserve">Breuiter <reg norm="itaque" type="simple">itaq;</reg> obtineremus in <lb/><reg norm="tentum" type="context">tentÅ©</reg> <reg norm="quando" type="wordlist">qñ</reg> diceremus Å¿i Å¿umma dierum, quibus iter agitur à viatoribus talis eÅ¿t (20) re-<lb/>Å¿pectu numeri dierum velocioris(9) qualis & cui <reg norm="reÅ¿pondebit" type="context">reÅ¿põdebit</reg> totum Å¿pacium <var>.b.d</var>? </s> - <s xml:space="preserve">vn-<lb/>de dabitur Å¿pacium <var>.b.o.</var> vnde reliqua omnia nobis cognita emergent.</s> - </p> - <p> - <s xml:space="preserve">Cum autem antiquorum regula iubeat numerum dierum vnius, cum numero die-<lb/>rum alterius multiplicari, ac poÅ¿tmodum diuidi productum per Å¿ummam omnium <lb/>dierum, rectèid quidem fit. </s> - <s xml:space="preserve">Nam cum Å¿ic Å¿e habeat <var>.b.d.</var> ad <var>.b.o.</var> vt Å¿umma omnium <lb/>dierum ad minorem quantitatem dierum velocioris Å¿cilicet. </s> - <s xml:space="preserve">Ideo temporis propor <lb/>tio à mobili per <var>.b.d.</var> abÅ¿umpti ad tempus mobilis per <var>.b.o.</var> eadem erit, quæ Å¿ummæ <lb/>omnium dierum ad namerum dierum velocioris. </s> - <s xml:space="preserve">Quarerectè dicemus, Å¿i eiuÅ¿modi <lb/>Å¿umma talem reÅ¿pectum habet ad minorem numerum dierum, quem numerum re-<lb/>Å¿piciet dies ipÅ¿ius <var>.b.d</var>? </s> - <s xml:space="preserve">ex quo proferentur dies reÅ¿pondentes ipÅ¿i <var>.b.o.</var> cætera iam <lb/>dicta fuerunt.</s> - </p> - <p> - <s xml:space="preserve">HuiuÅ¿modi verò Å¿peculationis am-<lb/> - <anchor type="figure" xlink:label="fig-0089-01a" xlink:href="fig-0089-01"/> - plitudo ad pauciſſima verba reduci <lb/>poteft, in cuius <reg norm="gratiam" type="context">gratiã</reg> Å¿it Å¿ubÅ¿cripta <lb/>figura pars <reg norm="inquam" type="context">inquã</reg> prÄ™cedentis, in qua <lb/><reg norm="conÅ¿tituamus" type="context simple">cõſtituamê°</reg> <var>.o.n.</var> <reg norm="locum" type="context">locÅ©</reg> <reg norm="eum" type="context">eÅ©</reg> eſſe quo Å¿ibi <lb/>viatores obuient, ex quo Å¿pacium <var>.q.<lb/>n.</var> à ſuo viatore conficietur, eo ipÅ¿o <lb/>tempore, quo à ſuo Å¿pacium <var>.b.o.</var> ita <lb/>que eadem erit proportio <var>.q.n.</var> ad <var>.b.</var> <lb/> - <anchor type="figure" xlink:label="fig-0089-02a" xlink:href="fig-0089-02"/> - o. quæ <var>.q.g.</var> ad <var>.b.d.</var> eadem erit <reg norm="inquam" type="context">inquã</reg> <lb/>proportio <var>.d.o.</var> ad <var>.o.b.</var> quæ numeri <lb/>dierum eius, qui à <var>.b.</var> pergit in <var>.d.</var> ad <lb/>numerum dierum alterius qui à <var>.q.</var> in <lb/>p. proficiÅ¿citur, & componendo <reg norm="eadem" type="context">eadẽ</reg> <lb/>erit proportio <var>.d.b.</var> ad <var>.b.o.</var> quæ Å¿um-<lb/>mæ dierum ad minorem numerum ipÅ¿orum, & eadem quæ dierum <var>.b.d.</var> ad dies <lb/>ipÅ¿ius <var>.b.o</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0089-01" xlink:href="fig-0089-01a"> - <image file="0089-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0089-01"/> - </figure> - <figure xlink:label="fig-0089-02" xlink:href="fig-0089-02a"> - <image file="0089-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0089-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="115"> - <head xml:space="preserve">THEOREMA <num value="115">CXV</num>.</head> - <p> - <s xml:space="preserve">CIRCA hæc ipÅ¿a itinera aliud quæritur peruenuÅ¿tè, in quo quæſito illud conÂ<lb/>Å¿tituitur cognitum eſſe, nempe interuallum inter duo diuerÅ¿a loca, è quibus <lb/>duo viatores eodem inÅ¿tanti vt Å¿ibi occurrant proficiÅ¿cuntur, <reg norm="certaque" type="simple">certaqÌ;</reg> milliaria Å¿in-<lb/>gulis diebus conficiant, ita tamen, ut unus ordinatè plura altero ambulet, quæritur <lb/>deinde quoto die Å¿ibi occurrent. </s> - <s xml:space="preserve">Hoc autem fit diuiÅ¿o toto interuallo locorum per <lb/>Å¿ummam milliariorum quam vterque quotidie abÅ¿oluit.</s> - </p> - <pb o="78" rhead="IO. BAPT. BENED." n="90" file="0090" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0090"/> - <p> - <s xml:space="preserve">Exempli gratia, diÅ¿tant loca .100. milliaribus à ſe inuicem; </s> - <s xml:space="preserve">vnus autem viator <lb/>Å¿ingulis diebus .15 milliaria, alter .10. conficit Å¿i ita que .15. cum .10. <reg norm="coniungamus" type="context">coniÅ©gamus</reg>, <lb/>Å¿umma erit .25. per <reg norm="quam" type="context">quã</reg> diuiÅ¿is milliaribus .100. totius interualli proferetur .4. nume <lb/>rus quæſitus dierum quo viatoribus iter agendum erit prius quam Å¿ibi obuient.</s> - </p> - <p> - <s xml:space="preserve">In cuius Å¿peculationis gratiam totum iter Å¿ignificetur linea <var>.a.u</var>: primi autem via-<lb/>toris iter diurnum fit <var>.a.e.</var> & alterius <var>.u.o</var>: terminus verò <var>.i.</var> Å¿it occurÅ¿us ita vt eodem <lb/>tempore, alter Å¿pacium <var>.a.i.</var> alter <var>.u.i.</var> confecerit, Å¿pacij autem <var>.a.e.</var> tempus <lb/>per <var>.b.</var> Å¿ignificetur & tempus Å¿pacij <var>.u.o.</var> per <var>.c.</var> quæ tempora erunt inter Å¿e <lb/>æqualia, porrò Å¿pacij <var>.a.i.</var> tempus per <var>.d.</var> & Å¿pacij <var>.u.i.</var> per <var>.f.</var> denotetur, æquali <lb/>bus inquam, ex quo eadem proportio erit <var>.a.e.</var> ad <var>.a.i.</var> quæ <var>.b.</var> ad <var>.d.</var> et <var>.o.u.</var> ad <var>.u.i.</var> quæ <lb/>c. ad <var>.f.</var> vnde permutando eadem erit proportio itineris ipÅ¿ius <var>.b.</var> ad iter ipÅ¿ius <var>.c.</var> quæ <lb/>itineris <var>.d.</var> ad iter ipÅ¿ius <var>.f.</var> & componendo itinerum ipÅ¿ius <var>.b.c.</var> ad iter <var>.c.</var> vt itinerum <var>.<lb/>d.f.</var> ad iter <var>.f.</var> & permutando itinerum <lb/><var>b.c.</var> ad itinera <var>.d.f.</var> vt itineris <var>.c.</var> ad. iter <lb/> - <anchor type="figure" xlink:label="fig-0090-01a" xlink:href="fig-0090-01"/> - ipÅ¿ius <var>.f.</var> meritò itaque quÄ™ritur Å¿i itine <lb/>ra <var>.b.c.</var> dat itinera <var>.d.f.</var> quid dabit tem-<lb/>pus <var>.c.</var> nempe dabit tempus <var>.f.</var> Å¿ed <var>.c.</var> <lb/>Å¿ignatum eÅ¿t pro vna die, </s> - <s xml:space="preserve">quare in pro <lb/>poÅ¿ito exemplo <var>.f.</var> Å¿ignificabit 4: dies.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0090-01" xlink:href="fig-0090-01a"> - <image file="0090-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0090-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="116"> - <head xml:space="preserve">THEOREMA <num value="116">CXVI</num>.</head> - <p> - <s xml:space="preserve">ANtiquorum monumentis traditum motum reperimus diuinandi numeri quem <lb/>quis mente conceperit, quo iubemus eum qui numerum cogitauerit, dimi-<lb/>dium cogitari numeri addere cogitato, atque huic Å¿ummæ, rurÅ¿us eiuÅ¿dem Å¿ummÄ™ <lb/>dimidium adiungere, tum quærimus, quoties noueratius totam eam Å¿ummam ingre <lb/>diatur patefactis fractis Å¿i qui occurrant.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i quis cogitaſſet numerum .12. iubebant huic dimidium addi, <lb/>nempe .6. ex quo Å¿umma erat .18. iubebant, præterea dimidium huius Å¿ummæ nem-<lb/>pe .9. toti Å¿ummæ adiungi, quæ fuiſſet .27. adhæc quærebant Å¿ibi patefieri quoties <num value="9">.<lb/>9.</num> Å¿ummam prædictam ingrederetur, & Å¿i in prima aut Å¿ecunda diuiÅ¿ione aut <reg norm="etiam" type="context">etiã</reg> <lb/>vtraque, fracti reperirentur, ac quoties nouem vltimam Å¿ummam ingrediebatur, <lb/>toties .4. multiplicabant. </s> - <s xml:space="preserve">Quod Å¿i in prima diuiÅ¿ione fracti erant, vltimo produ-<lb/>cto addebant vnitatem; </s> - <s xml:space="preserve">Å¿in verò in Å¿ecunda, binarium adiungebant, ex quo exa-<lb/>ctus numerus quæſitus proferebatur.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius rei ratione Å¿it <var>.a.</var> numerus cogitatione compræhenÅ¿us et <var>.e.</var> ipÅ¿ius <var>.a.</var> cum <lb/>eiuÅ¿dem medietate Å¿umma et <var>.i.</var> ipÅ¿ius <var>.e.</var> cum eiuÅ¿dem medietate itidem Å¿umma, vn <lb/>de <var>.i.e.a.</var> tres numeri continui proportionales, in Å¿eÅ¿quialtera proportione euadent. <lb/></s> - <s xml:space="preserve">Sumantur nunc tres numeri .4. 6. 9. in eadem proportionalitate. </s> - <s xml:space="preserve">Vnde ratione Ä™qua <lb/>litatis proportionum ita Å¿e habebit <var>.i.</var> ad <var>.a.</var> <reg norm="quemadmodum" type="wordlist">quẽadmodum</reg> .9. ad .4. & permutando <var>.i.</var> <lb/>ad .9. quemadmodum <var>.a.</var> ad .4. & ob id .4. toties ingredietur <var>.a.</var> quoties .9. ipÅ¿am <var>.i</var>. </s> - <s xml:space="preserve">Sed <lb/>quia ſępe contingit, vt in Å¿ecunda diuiÅ¿ione, aut in ambabus etiam diuiÅ¿ionibus re <lb/>periantur numeri fracti, anima duertendum eÅ¿t numerum animo compræhenÅ¿um <var>.a.</var> <lb/>Å¿cilicet aut parem aut imparem Å¿emper futurum. </s> - <s xml:space="preserve">Si par eÅ¿t, aut multiplex erit ad <num value="4">.<lb/>4.</num> aut non. </s> - <s xml:space="preserve">Si priori modo Å¿e habebit in duabus diuiÅ¿ionibus, nullus numerus fra-<lb/>ctus admittetur, Å¿ed Å¿i ad .4. multiplex non erit, à multiplicibus per duo Å¿emper dif <lb/>feret, & Å¿i per medium diuidatur, eiuÅ¿dem medietas impar Å¿emper erit, vnde prior - <pb o="79" rhead="THEOR. ARITH." n="91" file="0091" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0091"/> - quoque Å¿umma par nunquam exiÅ¿tet, cuius medietatem aliquod medium Å¿emper <lb/>ingredietur, & hanc ob cauÅ¿am poÅ¿terior Å¿umma cum fracto Å¿emper erit, & nume-<lb/>rum deÅ¿umptum maiorem eſſe multiplici ad quatuor per duo Å¿ignificabit.</s> - </p> - <p> - <s xml:space="preserve">At verò Å¿i inter impares reponatur, aut eorum erit qui Å¿uperant multiplicem <lb/>ipÅ¿ius quatuor per vnum, Å¿eu per tria, quod hinc innoteÅ¿cet, nempe, quia Å¿i eorum <lb/>erit qui dictum multiplicem per vnum tantum vincunt, Å¿ua medietate ipÅ¿i numero <lb/>addita, & præter hanc medietatem medio etiam integro adiuncto, tota hæc prior <lb/>Å¿umma in numerum parem Å¿emper euadet, vnde in poÅ¿teriori Å¿umma nullus nume-<lb/>rus fractus conÅ¿picietur, & hanc ob <reg norm="causam" type="context">causã</reg> multiplici ipÅ¿ius .4. vnitas Å¿emper addetur.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i numerus deÅ¿umptus, in Å¿erie eorum, qui multiplicem ipÅ¿ius .4. pertria Å¿u-<lb/>perant, collocabitur, hinc compræhendetur, quia primæ Å¿ummæ numerus cum <lb/>media vnitate Å¿emper impar erit, vnde Å¿ecunda Å¿umma præter integras cum me-<lb/>dia vnitate nobis Å¿emper occur ret.</s> - </p> - <p> - <s xml:space="preserve">Quod autem nobis prodere faciamus an in prima diuiÅ¿ione, & Å¿ecunda numerus <lb/>aliquis fractus conÅ¿iÅ¿tat, eò tantum nobis inÅ¿eruit, quò deueniamus in cognitionem <lb/>an numerus animo conceptus multiplicem ipÅ¿ius .4. per vnum, per duo, aut tria Å¿upe <lb/>ret. </s> - <s xml:space="preserve">Quòd etiam medias eas vnitates ad integros reducere faciamus, eò tantum re <lb/>fertur, vt minori labore eum, qui numerum imaginatione compræhendit, onere-<lb/>mus, quia reuera numerus impar nunquam mente concipi poteÅ¿t, quin aliquis fra-<lb/>ctus in prima diuiÅ¿ione, aut in Å¿ecunda Å¿equatur: </s> - <s xml:space="preserve">vnde à numeris imparibus, qui mul <lb/>tiplicem ipÅ¿ius .4. unitatis tantum exceſſu <reg norm="Å¿uperant" type="context">Å¿uperãt</reg>, poÅ¿terior Å¿umma <reg norm="cum" type="context">cÅ©</reg> quarta parte <lb/>vnitatis, præter integros numeros, & ab imparibus qui dictum multiplicem ipÅ¿ius <num value="4">.<lb/>4.</num> per tria vincunt, cum tribus quartis vnius integri præter integras vnitates ; </s> - <s xml:space="preserve">& à <lb/>numeris paribus, qui multiplicem ipÅ¿ius .4. per duo cum medietate vnitatis præter <lb/>integros Å¿emper procedit. </s> - <s xml:space="preserve">Ita cum is qui numerum Å¿ecum conÅ¿iderat, Å¿i in nume-<lb/> - <anchor type="figure" xlink:label="fig-0091-01a" xlink:href="fig-0091-01"/> - ris fractis verÅ¿atus eſſet, qui eum in-<lb/>terrogat prudenter Å¿e gereret, Å¿i Å¿ibi <lb/>declarari curaret, quis nam ex fractis <lb/>Å¿u per integros <reg norm="Å¿ecundæ" type="context">Å¿ecÅ©dæ</reg> <reg norm="summæ" type="context">sÅ©mæ</reg> remane <lb/>ret, quia <reg norm="per" type="simple">ê‘</reg> quot quarta integros <reg norm="Å¿ecun- dæ" type="context">Å¿ecÅ©-<lb/>dæ</reg> Å¿ummæ Å¿uperaret, per <reg norm="totidem" type="context">totidẽ</reg> inte <lb/>gros numerus mente conceptus multiplicem ipÅ¿ius .4. Å¿uperaret.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0091-01" xlink:href="fig-0091-01a"> - <image file="0091-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0091-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="117"> - <head xml:space="preserve">THEOREMA <num value="117">CXVII</num>.</head> - <p> - <s xml:space="preserve">VNDE fiat, vt Å¿i ali quis quemuis numerum animo compræhendat, eique <lb/>numero alium etiam quemlibet numerum propoÅ¿itum addat, & à tertia par <lb/>te huius Å¿ummæ tertiam partem numeri imaginati detrah et, reÅ¿iduum Å¿ecundi nu-<lb/>meri adiuncti, ideÅ¿t propoÅ¿iti, tertia pars erit.</s> - </p> - <p> - <s xml:space="preserve">Vt exempli gratia, Å¿i aliquis de numero denario cogitaſſet, <reg norm="huicque" type="simple">huicqÌ;</reg> .24. adderet, <lb/>vnde triginta quatuor efficerent, detra hendo nunc tertiam partem numeri de na-<lb/>rij cogitatione concepti, ideÅ¿t .3. cum tertia parte vnius, à tertia parte huius Å¿um mæ <lb/>ideÅ¿t ab vndecim & vna tertia parte remanerent .8. ideÅ¿t tertia pars numeri additi. <lb/></s> - <s xml:space="preserve">Id quod mihi inter iocos in honeÅ¿torum hominum cætu in mentem venit.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius ratione, prior numerus ima <lb/> - <anchor type="figure" xlink:label="fig-0091-02a" xlink:href="fig-0091-02"/> - ginatus mediante linea <var>.a.b.</var> et is, qui ad-<lb/>ditus eÅ¿t <reg norm="intercedente" type="context">intercedẽte</reg> linea <var>.b.d.</var> è directo - <pb o="80" rhead="IO. BAPT. BENED." n="92" file="0092" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0092"/> - coniunctis denotetur, et <var>.b.e.</var> Å¿it tertia pars ipÅ¿ius <var>.a.b.</var> prioris numeri im aginati, et. b <lb/>c. tertia pars ipÅ¿ius, <var>b.d.</var> Å¿ecundi numeri propoÅ¿iti, vnde coniunctum vnius harum ter <lb/>tiarum <reg norm="partium" type="context">partiÅ©</reg> <reg norm="cum" type="context">cÅ©</reg> alia Å¿it <var>.e.c.</var> quod quidem <var>.e.c.</var> eſſe tertiam partem Å¿ummæ duorum <lb/>primorum ideÅ¿t <var>.a.d.</var> aſſero. </s> - <s xml:space="preserve">Iam manifeÅ¿tum eÅ¿t ipÅ¿ius <var>.d.b.</var> ad <var>.b.c.</var> eſſe quemadmo <lb/>dum ipÅ¿ius <var>.a.b.</var> ad <var>.b.e.</var> vnde viciſſim ipÅ¿ius <var>.d.b.</var> ad <var>.b.a.</var> erit quemadmodum ipÅ¿ius <var>.b.<lb/>c.</var> ad <var>.b.e.</var> & coniunctim ipÅ¿ius <var>.d.a.</var> ad <var>.a.b.</var> quemadmodum ipÅ¿ius <var>.c.e.</var> ad <var>.e.b.</var> & viciÅ¿-<lb/>Å¿im ipÅ¿ius <var>.d.a.</var> ad <var>.c.e.</var> quemadmodum ipÅ¿ius <var>.b.a.</var> ad <var>.b.e.</var> Å¿ed proportio ipÅ¿ius <var>.b.a.</var> ad <var>.<lb/>b.e.</var> eÅ¿t tripla, ergo ea quæ eÅ¿t ipÅ¿ius <var>.a.d.</var> ad <var>.e.c.</var> erit quoque tripla; </s> - <s xml:space="preserve">vnde Å¿umendo <var>.e.<lb/>c.</var> pro tertia parte ipÅ¿ius <var>.a.d.</var> & ab ipÅ¿a <var>.e.c.</var> Å¿ubtrahendo tertiam partem ipÅ¿ius <var>.a.b.</var> <lb/>tertia pars ipÅ¿ius <var>.b.d.</var> remanebit <var>.b.c</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0091-02" xlink:href="fig-0091-02a"> - <image file="0091-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0091-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Aut alio hoc modo, Å¿upponendo <var>.e.c.</var> tertiam partem ipÅ¿ius <var>.a.d.</var> et <var>.e.b.</var> ipÅ¿ius <var>.a.<lb/>b.</var> exiÅ¿ter. </s> - <s xml:space="preserve">Dico <var>.b.c.</var> tertiam partem ipÅ¿ius <var>.b.d.</var> futuram:</s> - <s xml:space="preserve">quia Å¿i totius <var>.a.d.</var> ad totum <lb/><var>e.c.</var> ita Å¿e habet, quemadmodum <var>.a.b.</var> à toto <var>.a.d.</var> diffecti atque diuulÅ¿i ad <var>.e.b.</var> à toto <var>.<lb/>e.c.</var> diÅ¿ractum, ergo ex <ref id="ref-0011">.19. lib. quinti Eu-<lb/> - <anchor type="figure" xlink:label="fig-0092-01a" xlink:href="fig-0092-01"/> - clid.</ref> reÅ¿idui <var>.b.d.</var> totius <var>.a.d.</var> ad reÅ¿iduum <var>.b.c.</var> <lb/>totius <var>.e.c.</var> erit, vt totius <var>.a.d.</var> ad <reg norm="totum" type="context">totÅ©</reg> <var>.e.c.</var> at-<lb/>que hic quidem modus rem <reg norm="propoÅ¿itam" type="context">propoÅ¿itã</reg> Å¿pe-<lb/>culandi mihi aptior & commodior eſſe videtur.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0092-01" xlink:href="fig-0092-01a"> - <image file="0092-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0092-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="118"> - <head xml:space="preserve">THEOREMA <num value="118">CXVIII</num>.</head> - <p> - <s xml:space="preserve">PErmulta ac varia problemata inuenerunt antiqui, longioribus verò vijs reÅ¿olu-<lb/>ta, proptereà quòd <reg norm="non" type="context">nõ</reg> Å¿emper nobis Å¿uccurrit breuiſſima in vnaquaque re ex-<lb/>plicatio. </s> - <s xml:space="preserve">Vt exempli gratia, proponitur numerus .50. diuidendus in tres tales par-<lb/>tes, quod Å¿ecunda dupla Å¿it primÄ™, & adhuc eam Å¿uperet tribus vnitatibus, tertia ve <lb/>rò æqualis Å¿it aggregato primæ cum Å¿ecunda, & amplius ipÅ¿um aggregatum Å¿uperet <lb/>quinque vnitatibus.</s> - </p> - <p> - <s xml:space="preserve">Ad hoc autem quæſitum Å¿oluendum antiqui vtebantur regula falÅ¿i, quod reuera <lb/>breuiori modo poteÅ¿t Å¿olui, videlicet detra hendo illud Å¿ecundum exceſſum, quin-<lb/>que Å¿cilicet ex .50. ita vt nobis .45. remaneret, cui medietati hoc eÅ¿t .22. cum dimidia <lb/>vnitate, Å¿i addiderimus illud quinque habebimus .27. cum dimidia vnitate pro ter-<lb/>tia parte quæſita ipÅ¿ius numeri .50. deinde Å¿i ab eodem numero .22. cum dimidia <lb/>vnitate detractum fuerit illud .3. primus exceſſus datus, remanebit .19. cum dimi-<lb/>dia vnitate, cuius tertia pars, hoc eÅ¿t .6. cum dimidia vnitate, prima pars, ex tri-<lb/>bus quæſita erit, quæ quidem Å¿i detraxerimus ex .19. cum dimidia vnitate, reli-<lb/>quum erit .13. cui <reg norm="cum" type="context">cÅ©</reg> additus fuerit primus exceſſus ideÅ¿t .3. </s> - <s xml:space="preserve">Iam propoÅ¿itum re-<lb/>Å¿ultabit nobis .16. pro Å¿ecunda parte quæſita.</s> - </p> - <p> - <s xml:space="preserve">Ratio verò huiuÅ¿modi operationis talis eÅ¿t, Å¿it verbi gratia totalis numerus pro-<lb/>poÅ¿itus Å¿ignificatus per lineam <var>.a.b.</var> cuius Å¿ecundæ partis numerus datus Å¿ignificetur <lb/>per lineam <var>.g.</var> & numerus tertiæ partis propoÅ¿itus per lineam <var>.h</var>. </s> - <s xml:space="preserve">Nunc dempta <var>.h.</var> ex <lb/><var>a.b.</var> nobis cognita, remanebit <var>.f.a.</var> qua <reg norm="quidem" type="context">quidẽ</reg> per æqualia imaginatione diuiÅ¿a in pun <lb/>cto <var>.e.</var> & ipÅ¿i <var>.e.f.</var> addita <var>.f.b.</var> tota <var>.e.b.</var> nobis cognita erit, quæ quidem tertia pars <lb/>quæſita ipÅ¿ius <var>.a.b.</var> erit, proptereà quòd <var>.a.e.</var> (quæ æqualis eÅ¿t ipÅ¿i <var>.e.f.</var>) erit Å¿umma <lb/>primæ, & Å¿ecundæ partis. </s> - <s xml:space="preserve">Detrahatur poÅ¿teà . g. ex <var>.e.a.</var> & remanebit <var>.d.a.</var> cuius ter <lb/>tia pars Å¿it <var>.a.c.</var> quæ quidem prima pars quæſita erit, & nunc cognita, & ita <var>.c.d.</var> <lb/>cognita, cui cum addita fuerit <var>.d.e.</var> habebimus Å¿ecundam partem quæſitam, quæ <lb/>compo- - <pb o="81" rhead="THEOREM. ARIT." n="93" file="0093" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0093"/> - componitur ex <var>.d.c.</var> dupla .ad <var>.a.c.</var> pri-<lb/> - <anchor type="figure" xlink:label="fig-0093-01a" xlink:href="fig-0093-01"/> - mam partem, & ex <var>.d.e.</var> numero dato. <lb/></s> - <s xml:space="preserve">tertia verò pars <var>.e.b.</var> compoÅ¿ita eÅ¿t ex <var>.<lb/>e.f.</var> æquali <var>.a.e.</var> hoc eÅ¿t æquali compoÅ¿i-<lb/>to ex prima, & Å¿e cunda parte, & ex <var>.f.<lb/>b.</var> numero dato vt proponebatur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0093-01" xlink:href="fig-0093-01a"> - <image file="0093-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0093-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="119"> - <head xml:space="preserve">THEOREMA <num value="119">CXIX</num>.</head> - <p> - <s xml:space="preserve">INter alia problemata ab antiquis inuenta, hoc etiam ponitur. </s> - <s xml:space="preserve">Aliquis inter-<lb/>rogat quot Å¿int horæ, alius verò reÅ¿pondit tot eſſe, quot duæ tertiæ præteriti <lb/>temporis Å¿imul iuncta cum tribus quintis futuri temporis totius dieri naturalis effi-<lb/>ciunt. </s> - <s xml:space="preserve">Nunc quæritur quot Å¿int horÄ™.</s> - </p> - <p> - <s xml:space="preserve">Antiqui, hoc etiam problema Å¿oluebant mediante regula falÅ¿i, Å¿ed mihi alio mo <lb/>do Å¿oluendum eſſe dictum problema videtur. </s> - <s xml:space="preserve">Accipio enim ex quinque, tres vni-<lb/>tates, pro parte futuri temporis, quas quidem in tres vnitates præteriti temporis <lb/>duco, vnde proueniunt mihi nouem vnitates, quod productum coniungo <reg norm="cum" type="context">cÅ©</reg> quin-<lb/>que futuri temporis, vnde veniunt .14. vnitates, ex regula poftea de tribus ita dico <lb/>Å¿i ex .14. mihi prouenit .9. quid reÅ¿ultabit ex .24. & prouenient mihi horæ .15. cum <lb/>tribus Å¿eptimis vnius horæ, hoc eÅ¿t minuta ferè .26.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius ratione, quinque vnitates, feu partes temporis futuri Å¿ignificentur à <lb/>linea <var>.e.u.</var> quarum trium Å¿igniÅ¿icentur a linea <var>.e.i.</var> Å¿umpta deinde Å¿it linea <var>.e.o.</var> æqualis <lb/>lineæ <var>.e.i.</var> et <var>.e.a.</var> tripla Å¿it ad <var>.o.e.</var> vel ad <var>.e.i.</var> quod idem eÅ¿t, vnde <var>.a.e.</var> compoÅ¿ita erit <lb/>ex <var>.a.o.</var> (hoc eÅ¿t ex duabus tertijs ip Å¿ius <var>.a.e.</var>) & ex <var>o.e.</var> (hoc eÅ¿t ex. tribus quintis ip-<lb/>Å¿ius <var>.e.u.</var>) vnde <var>.a.u.</var> ad <var>.a.e.</var> eandem rationem obtinebit, quæ .14. ad .9. </s> - <s xml:space="preserve">propterea igi <lb/>tur poſſumus recte ratiotinari <lb/> - <anchor type="figure" xlink:label="fig-0093-02a" xlink:href="fig-0093-02"/> - fi .14. nobis dat .9. quid dabit .24. <lb/>qui quidem .24. nobis dabit .15. <lb/>cum min .26. quod rectè factum <lb/>erit ex <ref id="ref-0012">.20. Å¿eptimi Euclidis</ref>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0093-02" xlink:href="fig-0093-02a"> - <image file="0093-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0093-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="120"> - <head xml:space="preserve">THEOREMA <num value="120">CXX</num>.</head> - <p> - <s xml:space="preserve">SVpponunt etiam antiqui tres Å¿ocios nummos habere, quorum Å¿umma primi & <lb/>Å¿ecundi cognita Å¿it, item Å¿umma primi & tertij cognita & Å¿umma Å¿ecundi & <lb/>tertij item cognita, at que ex huiuÅ¿modi tribus aggregatis veniunt in cognitionem <lb/>particularem vniuÅ¿cuiuÅ¿que illorum.</s> - </p> - <p> - <s xml:space="preserve">GemafriÅ¿ius Å¿oluit hoc problema ex regula Å¿alÅ¿i. </s> - <s xml:space="preserve">At ego tali ordine progredior. <lb/></s> - <s xml:space="preserve">Sit verbi gratia, Å¿umma primi cum Å¿ecundo .50. & Å¿ecundi cum tertio .70. & primi <lb/>cum tertio .60. harum trium Å¿ummarum accipiantur duæ quæuis, vt puta .50. & .70 <lb/>quæ coniunctæ Å¿imul dabunt .120. à qua Å¿umma detrahatur reliqua, ideÅ¿t .60. & <lb/>reÅ¿tabit nobis .60. cuius medietas erit .30. hoc eÅ¿t numerus nummorum Å¿ecundi <lb/>Å¿ocij quo numero detracto à .70. hoc eÅ¿t à ſumma Å¿ecundi cum tertio remanebit .40. <lb/>hoc eÅ¿t numerus tertij Å¿ocij, & hic numerus deÅ¿umptus à .60. reÅ¿iduus erit nume-<lb/>rus primi Å¿ocij.</s> - </p> - <pb o="82" rhead="IO. BAPT. BENED." n="94" file="0094" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0094"/> - <p> - <s xml:space="preserve">Pro cuius ratione conÅ¿ideremus triangulum hic Å¿ubnotatum <var>.a.b.c.</var> cuius <lb/>unumquodque latus Å¿ignificet Å¿ummam duorum Å¿ociorum, vtputa latus <var>.a.b.</var> Å¿ignifi-<lb/>cet Å¿ummam primi cum Å¿ecundo, latus verò <var>.b.c.</var> Å¿ummam Å¿ecundi cum tertio, la-<lb/>rus autem <var>.a.c.</var> Å¿ummam primi cum tertio, et <var>.a.e.</var> Å¿eu <var>.a.o.</var> Å¿it numerus primi Å¿ocij, et <var>.<lb/>e.b.</var> vel <var>.b.u.</var> Å¿it Å¿ecundi Å¿ocij, et <var>.c.u.</var> Å¿eu <var>.c.o.</var> Å¿it tertij, cum autem <var>.a.e.</var> æqualis Å¿it <var>.a.o.</var> <lb/> - <anchor type="figure" xlink:label="fig-0094-01a" xlink:href="fig-0094-01"/> - et <var>.b.e</var>:æ qualis <var>.b.u.</var> et <var>.c.u.</var> æqualis <var>.c.o.</var> <lb/>ex Å¿uppoÅ¿ito Å¿i <reg norm="dempta" type="context">dẽpta</reg> fuerit Å¿umma Å¿eu <lb/>latus <var>.a.c.</var> datum ex aggregato laterum <var>.<lb/>a.b.</var> cum <var>.b.c.</var> reliquarum Å¿ummarum, re <lb/>linquet nobis cognitum aggregatum <lb/>ex <var>.b.e.</var> cum <var>.b.u</var>. </s> - <s xml:space="preserve">Quare & eius medic-<lb/>tas <var>.b.e.</var> Å¿iue <var>.b.u.</var> nobis cognita erit, qua <lb/>detracta exÅ¿umma <var>.b.a.</var> relinquetur no <lb/>bis cognitus numerus <var>.a.e.</var> detracto ve-<lb/>ro numero <var>.a.e.</var> hoc eÅ¿t <var>.a.o.</var> ex <var>.a.c.</var> Å¿um-<lb/>ma, Å¿eu latus, aut <var>.b.u.</var> ex <var>.b.c.</var> remanebit <lb/><var>o.c.</var> Å¿eu <var>.c.u.</var> cognitus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0094-01" xlink:href="fig-0094-01a"> - <image file="0094-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0094-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="121"> - <head xml:space="preserve">THEOREMA <num value="121">CXXI</num>.</head> - <p> - <s xml:space="preserve">HAC etiam methodo hoc facere poſſumus non <reg norm="Å¿olum" type="context">Å¿olÅ©</reg> de tribus Å¿ocijs, Å¿ed <reg norm="etiam" type="context">etiã</reg> <lb/>de omnibus quotquot volueris, vt exempli gratia, <lb/> - <anchor type="figure" xlink:label="fig-0094-02a" xlink:href="fig-0094-02"/> - Å¿int Å¿ex Å¿ocij <var>.a.b.c.d.e.f.</var> quorum Å¿umma per binos co-<lb/>gnita, vtputà ſumma numeri <var>.a.</var> cum <var>.b.</var> cognita nobis Å¿it, <lb/>& Å¿umma numeri <var>.b.</var> cum <var>.c.</var> & Å¿umma <var>.c.</var> cum <var>.d.</var> & Å¿um-<lb/>ma <var>.d.</var> cum <var>.e.</var> & Å¿umma <var>.e.</var> cum <var>.f.</var> neceÅ¿le eft etiam Å¿cire <lb/>Å¿ummam duorum vno relicto, vtputa Å¿ummam <var>.a.</var> cum <lb/>c. vt poſſimus triangulum <var>.a.b.c.</var> conÅ¿tituere. </s> - <s xml:space="preserve">Vnde ex <lb/>præmiffa, cognitus numerus nobis erit vniuÅ¿cuiuÅ¿que <var>.a.<lb/>b.c</var>. </s> - <s xml:space="preserve">Quapropter dempto numero <var>.c.</var> ex Å¿umma <var>.c.</var> cum <lb/>d. & numero <var>.d.</var> ex Å¿umma <var>.d.</var> cum <var>.e.</var> & numero <var>.e.</var> ex Å¿um <lb/>ma <var>.e.</var> cum <var>.f.</var> habebimus intentum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0094-02" xlink:href="fig-0094-02a"> - <image file="0094-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0094-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="122"> - <head xml:space="preserve">THEOREMA <num value="122">CXXII</num>,</head> - <p> - <s xml:space="preserve">CVM aliquando, illud quod Archimedes inuenit, vt furtum Regiab aurifa-<lb/>bro in regia corona factum, quemadmodum Å¿cribit Vitruuius, proderet, con-<lb/>templarer, mihi etiam viÅ¿um eÅ¿t, vt aliquem modum Å¿cientiÅ¿icum inueÅ¿tigarem, quo <lb/>proportio auri ad argentum, quod in aliquo propoÅ¿ito corpore exipÅ¿is miÅ¿to cogni <lb/>ti ponderis cognoÅ¿ci poſſet. </s> - <s xml:space="preserve">Et cum multos diuerÅ¿is temporibus excogitarim offi-<lb/>cio meo deeſſe nolui in ijÅ¿dem literarum monumentis mandandis, quorum hic <lb/>vnus erit: </s> - <s xml:space="preserve">propoÅ¿ita nobis Å¿int tria corpora <var>.A.M.V.</var> æqualia inter Å¿e, Å¿ed diuer-<lb/>Å¿arum Å¿pecierum materiei, vtputa quod <var>.A.</var> Å¿it argenteum, & omogeneum <var>.V.</var> ve-<lb/>rò aureum omogeneum, & M. mixtum exauro, & argento, ideÅ¿t heterogeneum, <lb/>cupimusergo Å¿cire <reg norm="iuÅ¿tam" type="context">iuÅ¿tã</reg> quantitatem auri & argenti, quæ eÅ¿t in ipÅ¿o corpore <var>.M.</var> <lb/>miÅ¿to. </s> - <s xml:space="preserve">Ita igitur faciamus. </s> - <s xml:space="preserve">Videamus primum quantum Å¿it pondus vniuÅ¿cuiuÅ¿que <lb/>ipÅ¿orum corporum, ponamus autem pondus corporis <var>.V.</var> auri eſſe vt .234. pondus - <pb o="89" rhead="THEOR. ARITH." n="95" file="0095" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0095"/> - autem corporis <var>.M.</var> miÅ¿ti <reg norm="vt" type="punctuation">.vt</reg> .216. argentei verò <var>.A.</var> vt .156. detrahatur nunc pon-<lb/>dus <var>.A.</var> ex pondere <var>.V</var>. </s> - <s xml:space="preserve">Reliquum erit .78. quod vocetur prima differentia Å¿eruan-<lb/>da, dematur etiam pondus <var>.M.</var> ex pondere <var>.V.</var> reliquum erit .18. pro Å¿ecunda diffe-<lb/>rentia, etiam Å¿eruanda, multiplicetur poÅ¿teà pondus <var>.A.</var> per Å¿ecundam differen-<lb/>tiam, productum verò diuidatur per primam differentiam. </s> - <s xml:space="preserve">Vnde in præſenti exem <lb/>plo proueniet nobis .36. quiquidem prouentus erit quantitas argenti ipÅ¿ius corpo-<lb/>ris miÅ¿ti <var>.M.</var> quo etiam detracto ex pondere totali ipÅ¿ius <var>.M.</var> reliquum erit quanti-<lb/>tas auri eius corporis, hoc eÅ¿t .180.</s> - </p> - <p> - <s xml:space="preserve">In cuius operationis Å¿peculatione, aliquid natura Å¿ua prius cognitum præcedere <lb/>oportet hoc eÅ¿t, quod omnia corpora omogenea eandem proportionem obtinent <lb/>inter quantitates, quam inter pondera. </s> - <s xml:space="preserve">Quo Å¿uppoÅ¿ito denotetur corpus <var>.A.</var> li-<lb/>nea <var>.o.a.</var> corpus autem <var>.V.</var> linea <var>.o.c.</var> & corpus <var>.M.</var> linea <var>.e.u</var>: Å¿ed <var>.e.o.</var> Å¿ignificet par-<lb/>tem argenti, et <var>.o.u.</var> partem auri in corpore miÅ¿to <var>.M.</var> vnde ex communi conceptu <lb/>habebimus <var>.o.e.</var> æqualem <var>.u.c.</var> cum ex hypotheÅ¿i <var>.e.u.</var> æqualis Å¿it <var>.o.c.</var> et <var>.a.o.</var> Å¿imiliter. <lb/></s> - <s xml:space="preserve">Significetur poÅ¿teà pondus <var>.a.o.</var> ab <var>.f.</var> & pondus <var>.e.u.</var> ab <var>.b.x.</var> & pondus <var>.o.c.</var> ab <var>.f.g.</var> pon <lb/>dus verò <var>.o.e.</var> ab <var>.b.</var> pondus autem <var>.o.u.</var> ab <var>.x.</var> pondus enim <var>.u.c.</var> ab <var>.b.d.</var> et <var>.g.</var> Å¿it diffe-<lb/>rentia, qua <var>.f.g.</var> maior eÅ¿t .f: et <var>.d.</var> <lb/> - <anchor type="figure" xlink:label="fig-0095-01a" xlink:href="fig-0095-01"/> - differentia qua <var>.b.d.</var> maior eÅ¿t <var>.b</var>. <lb/></s> - <s xml:space="preserve">Vnde ex ratione omogeneitatis ea <lb/>dem proportio erit <var>.a.o.</var> ad <var>.e.o.</var> vt <var>.<lb/>f.</var> ad <var>.b.</var> et <var>.o.c.</var> ad <var>.u.c.</var> quæ <var>.x.b.d.</var> Å¿eu <lb/><var>f.g.</var> (quodidem eÅ¿t) ad <var>.b.d</var>. </s> - <s xml:space="preserve">Quare <lb/>ex <ref id="ref-0013">.11. quinti <reg norm="eadem" type="context">eadẽ</reg></ref> erit proportio <var>.<lb/>f.</var> ad <var>.b.</var> vt <var>.f.g.</var> ad <var>.b.d.</var> & permutan-<lb/>do ita erit <var>.f.</var> ad <var>.f.g.</var> vt <var>.b.</var> ad <var>.b.d.</var> & <lb/>Å¿eparando ita <var>.f.</var> ad <var>.g.</var> vt <var>.b.</var> ad <var>.d</var>. </s> - <s xml:space="preserve">Sed <var>.g.</var> cognita nobis eÅ¿t, vt differentia in <lb/>ter <var>.f.</var> g, et <var>.f</var>: cognita nobis eÅ¿t etiam <var>.f</var>: cognoÅ¿cimus itidem <var>.d.</var> vt differentiam inter <var>.<lb/>x.b.d.</var> et <var>.b.x.</var> quapropter cognoÅ¿cemus <var>.b.</var> ex .20. Å¿eptimi Eucli. & Å¿ic <var>.x.</var> reÅ¿iduum. <lb/></s> - <s xml:space="preserve">ex <var>.b.x</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0095-01" xlink:href="fig-0095-01a"> - <image file="0095-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0095-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="123"> - <head xml:space="preserve">THEOREMA <num value="123">CXXIII</num>.</head> - <p> - <s xml:space="preserve">NVNC ex methodo præcedentis propoÅ¿iti deuenire poſſumus in cognitio-<lb/>nem veræ quantitatis auri, & argenti confuÅ¿i in corona Hieronis conÅ¿tituen-<lb/>do primum duo corpora Å¿implicia æqualia inter Å¿e, & coronæ hoc modo videlicet, <lb/>immergendo coronam, Å¿eu corpus miÅ¿tum in aliquod vas aqua plenum, & diligen-<lb/>ter colligere aquam, quæ ex eo effundetur, poÅ¿teà verò oportet, aliud vas inuenire <lb/>præciſæ capax illius a quæ collectæ, in quod demum infundatur tantum auri, & po-<lb/>Å¿teà tantum argenti, quantum Å¿ieri poteÅ¿t, vnde vnumquodque horum duorum cor <lb/>porum Å¿implicium æquale erit mixto, Å¿eu coronæ, & Å¿ic quod dictum eÅ¿t in præce-<lb/>cedenti theoremate exequemur.</s> - </p> - </div> - <div type="math:theorem" level="3" n="124"> - <head xml:space="preserve">THEOREMA <num value="124">CXXIIII</num>.</head> - <p> - <s xml:space="preserve">SED vt breuiori methodo idem præſtemus, quod in antecedenti propoÅ¿ito di-<lb/>ctum eÅ¿t, quædam theoremata præmittenda Å¿unt, videlicet quòd quotÃeÅ¿cunque <lb/>fuerint tria corpora, quorum duo inuicem æqualia Å¿int in quantitate, Å¿ed diuerÅ¿a- - <pb o="84" rhead="IO. BAPT. BENED." n="96" file="0096" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0096"/> - rum Å¿pecierum materiæ, tertium verò corpus maius, vel minus Å¿it in quantitate vtro-<lb/>que illorum, Å¿ed eiuÅ¿dem materiæ vnius quod vis illorum, ponderis verò alterius, <lb/><reg norm="semper" type="context">sẽper</reg> eadem proportio erit inter pondera æqualium corporum, quæ inter <reg norm="quantita- tem" type="context">quãtita-<lb/>tem</reg> corporis inæqualis, & eam quæ vnius cuiuÅ¿uis æqualium.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿it <var>.b.</var> corpus aliquod aureum æquale corpori <var>.u.</var> argenteo, Å¿it <lb/>etiam corpus <var>.a.</var> argenteum maius corpore <var>.b.</var> vel <var>.u.</var> Å¿ed ponderis eiuÅ¿dem, quod au-<lb/>ri <var>.b</var>. </s> - <s xml:space="preserve">Tunc dico eandem eſſe proportionem ponde-<lb/> - <anchor type="figure" xlink:label="fig-0096-01a" xlink:href="fig-0096-01"/> - ris <var>.b.</var> ad pondus <var>.u.</var> quæ eÅ¿t magnitudinis <var>.a.</var> ad ma-<lb/>gnitudinem <var>.u</var>. </s> - <s xml:space="preserve">Quod ratiocinemur hoc modo, nam <lb/>cum proportio corporeitatis <var>.a.</var> ad corporeitatem <var>.u.</var> <lb/>eadem Å¿it, quæ ponderis <var>.a.</var> ad pondus <var>.u.</var> ex ratione <lb/>omogeneitatis, ponderis verò <var>.b.</var> ad pondus <var>.u.</var> ex .7. <lb/>quinti, eadem quæ ponderis <var>.a.</var> ad pondus <var>.u.</var> ideo ex <lb/>11. eiuÅ¿dem proportio ponderis <var>.b.</var> ad pondus <var>.u.</var> eadem erit, quæ corporeitatis <var>.a.</var> <lb/>ad corporeitatem <var>.u.</var> vel ad corporeitatem <var>.b.</var> quæ æqualis eÅ¿t alteri.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0096-01" xlink:href="fig-0096-01a"> - <image file="0096-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0096-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="125"> - <head xml:space="preserve">THEOREMA <num value="125">CXXV</num>.</head> - <p> - <s xml:space="preserve">QVotieÅ¿cunque nobis propoÅ¿ita fuerint duo corpora cuiuÅ¿uis magnitudinis æ-<lb/>que ponderantia, Å¿ed diuerÅ¿arum Å¿pecierum materiæ, cum Å¿cire volueri-<lb/>mus proportionem ponderum illarum Å¿pecierum inter ipÅ¿as hoc modo faciemus.</s> - </p> - <p> - <s xml:space="preserve">Sint exempli gratia, duo nobis propoÅ¿ita corpora <var>.a.</var> et <var>.b.</var> (vt dictum eÅ¿t) quæ Å¿i <lb/>fuerint æqualium magnitudinum inter Å¿e, clarum erit quod quæritur, Å¿ed inæqua-<lb/>lia erunt, immergatur <reg norm="unumquodque" type="simple">unumquodq;</reg> eorum in vas aqua plenum, & collecta Å¿it aqua <lb/>effuÅ¿a ab vnoquoque illorum, </s> - <s xml:space="preserve">tunc <reg norm="vnaquæque" type="simple">vnaquæq;</reg> iÅ¿tarum aquarum æqualis magnitudi-<lb/>nis erit Å¿ui corporis impellentis, & proportio ponderoÅ¿itatis illarum eadem erit, <lb/>quæ earum magnitudinum ex omogeneitate, quapropter Å¿i vnamquamque illarum <lb/>ponderabimus, habebimus propoÅ¿itum ex præcedenti theoremate.</s> - </p> - </div> - <div type="math:theorem" level="3" n="126"> - <head xml:space="preserve">THEOREMA <num value="126">CXXVI</num>.</head> - <p> - <s xml:space="preserve">SED cum Å¿cire voluerimus pondus alicuius magnitudinis aquæ æqualis alicui <lb/>corpori ponderoÅ¿o, breuiſſimus modus erit ponderando ipÅ¿um corpus tam in ae-<lb/>re, quà m in aqua, & quia Å¿emper leuius erit in aqua, </s> - <s xml:space="preserve">tunc differentia ponderum ip-<lb/>Å¿ius corporis, erit pondus quæſitum, hoc eÅ¿t vnius corporis aquei æqualis magnitu-<lb/>dinis magnitudini corporis propoÅ¿iti ex <ref id="ref-0014">.7. propoÅ¿itione lib. Archimedis de inÅ¿i-<lb/>dentibus aquæ</ref>. </s> - </p> - <p> - <s xml:space="preserve">Quare ex præmiſſis quotieÅ¿cunque immerÅ¿a fuerint in aquam dicti vaÅ¿is duo cor <lb/>pora æquè ponderantia, Å¿ed diuerÅ¿arum Å¿pecierum, vt dictum eÅ¿t, proportio pon-<lb/>deris aquæ maioris ad pondus aquæ minoris magnitudinis eadem Å¿emper erit, quæ <lb/>ponderis minoris corporis ad pondus alicuius corporis eidem æqualis, Å¿peciei verò <lb/>maioris, vel eadem proportio ponderis alicuius corporis æqualis maiori, Å¿peciei ve <lb/>rò minoris ad pondus ipÅ¿ius maioris.</s> - </p> - <p> - <s xml:space="preserve">Vt puta Å¿it corpus <var>.a.</var> argenteum æqualis ponderis corpori <var>.b.</var> aurei, & corpus <var>.u.</var> <lb/>argenteum æqualis magnitudinis corpori <var>.b.</var> aurei, corpus verò <var>.n.</var> aureum æqualis <lb/>magnitudinis corpori <var>.a.</var> argentei, corpus verò <var>.f.</var> aqueum æqualis magnitudinis cor- - <pb o="85" rhead="THEOREM. ARIT." n="97" file="0097" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0097"/> - pori <var>.a.</var> argentei, corpus autem <var>.e.</var> <reg norm="aqueum" type="context">aqueÅ©</reg> æqualis ma-<lb/> - <anchor type="figure" xlink:label="fig-0097-01a" xlink:href="fig-0097-01"/> - gnitudinis corpori <var>.b.</var> aurei. </s> - <s xml:space="preserve">Tunc dico proportio-<lb/>nem ponderis <var>.f.</var> ad pondus <var>.e.</var> eadem eſſe, quæ pon-<lb/>deris <var>.b.</var> ad pondus <var>.u.</var> vt in præcedenti theoremate <lb/>iam dictum eÅ¿t, vel quæ ponderis <var>.n.</var> ad pondus <var>.a.</var> ex <lb/>11. quinti Euclidis. </s> - <s xml:space="preserve">Proptereà quòe ponderis <var>.<lb/>n.</var> ad pondus <var>.a.</var> eft vt poderis <var>.b.</var> ad pondus <var>.u.</var> eo <lb/>quòd permutando ponderis <var>.n.</var> ad pondus <var>.b.</var> eÅ¿t vt <lb/>ponderis <var>.a.</var> ad pondus <var>.u.</var> ex corporum omogenei-<lb/>tate, & ex æqualitate magnitudinum corporum antecedentium & conÅ¿equentium.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0097-01" xlink:href="fig-0097-01a"> - <image file="0097-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0097-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="127"> - <head xml:space="preserve">THEOREMA <num value="127">CXXVII</num>.</head> - <p> - <s xml:space="preserve">SCire etiam nos oportet, quòd quotieÅ¿cumque fuerint duo corpora aquea, quo-<lb/>rum vnum æqualis magnitudinis Å¿it alicui miÅ¿to, quod quidem miÅ¿tum graue <lb/>Å¿it tam in aere, quà m in aqua, alterum verò corpus aquem æqualis Å¿it magnitudi-<lb/>nis alicui corpoli Å¿implici, quod quidem corpus Å¿implex æqualis ponderis Å¿it dicto <lb/>corpori miÅ¿to. </s> - <s xml:space="preserve">Tunc proportio ponderis aquei, cuius magnitudo æquatur magni <lb/>tudini corporis miÅ¿ti, ad pondus corporis aquei, cuius magnitudo æqualis eÅ¿t ma-<lb/>gnitudini corporis Å¿implicis, eadem erit, quæ proportio ponderis alicuius corpo-<lb/>ris Å¿implicis, cuius magnitudo æqualis Å¿it magnitudini corporis miÅ¿ti Å¿uperius dicti, <lb/>Å¿ed Å¿peciei corporis Å¿implicis iam dicti, ad pondus dicti miÅ¿ti.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿it corpus aqueum <var>.e.</var> magnitudinis æqualis corpori <var>.m.</var> mixto, <lb/>corpus verò aqueum <var>.i.</var> æqualis magnitudinis Å¿it corpori Å¿implici <var>.a.</var> quod quidem <lb/>corpus <var>.a.</var> æqualis ponderis Å¿it cum corpore <var>.m.</var> & corpus <var>.u.</var> Å¿it æqualis magnitudinis <lb/>cum corpore <var>.m.</var> Å¿ed Å¿peciei corporis <var>.a</var>. </s> - <s xml:space="preserve">Tunc dico proportionem ponderis <var>.e.</var> ad <lb/>pondus <var>.i.</var> <reg norm="eandem" type="context">eãdem</reg> eſſe, quæ ponderis <var>.u.</var> ad pondus <var>.m.</var> primum nulli dubium eÅ¿t, quin <lb/>eadem proportio Å¿it magnitudinis <var>.e.</var> ad magnitudinem <var>.i.</var> quæ magnitudinis <var>.m.</var> ad <lb/>a. Å¿ed <var>.m.</var> ad <var>.a.</var> eÅ¿t vt <var>.u.</var> ad <var>.a.</var> ex .7. quinti </s> - <s xml:space="preserve">quare <lb/>ex .11. eiuÅ¿dem proportio <var>.e.</var> ad <var>.i.</var> erit vt <var>.u.</var> ad <var>.a.</var> <lb/> - <anchor type="figure" xlink:label="fig-0097-02a" xlink:href="fig-0097-02"/> - de ipÅ¿ius magnitudinibus loquendo, Å¿ed propor-<lb/>tio ponderis <var>.u.</var> ad pondus <var>.a.</var> eadem eÅ¿t, quæ ma <lb/>gnitudinis <var>.u.</var> ad magnitudinem <var>.a.</var> ex omogenei-<lb/>tate. </s> - <s xml:space="preserve">Idem dico de pondere <var>.e.</var> ad pondus .1. </s> - <s xml:space="preserve">Qua-<lb/>re proportio ponderis <var>.e.</var> ad pondus <var>.i.</var> eadem erit <lb/>quæ ponderis <var>.u.</var> ad pondus <var>.a</var>. </s> - <s xml:space="preserve">Sed ponderis <var>.u.</var> <lb/>ad pondus <var>.m.</var> eadem eÅ¿t quæ ponderis <var>.u.</var> ad pondus <var>.a.</var> ex .7. quinti, ergò ex .11. <lb/>eiuÅ¿dem proportio ponderis <var>.e.</var> ad pondus <var>.i.</var> eadem erit, quæ ponderis <var>.u.</var> ad pon-<lb/>dus <var>.m.</var> quod eÅ¿t propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0097-02" xlink:href="fig-0097-02a"> - <image file="0097-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0097-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="128"> - <head xml:space="preserve">THEOREMA <num value="128">CXXVIII</num>.</head> - <p> - <s xml:space="preserve">NVNC ad cognoÅ¿cendam proportionem duarum diuerÅ¿arum Å¿pecierum in <lb/>corpore miÅ¿to propoÅ¿ito, tribus corporibus aqueis mediantibus, quæ <reg norm="quidem" type="context">quidẽ</reg> <lb/>corpora æqualium magnitudinum Å¿int alijs tribus corporibus vnius & eiuÅ¿dem pon <lb/>deris, quorum vnum Å¿it mixtum, reliqua verò duo Å¿implicia, Å¿ed Å¿pecierum mixti, <lb/>hoc ordine procedemus.</s> - </p> - <pb o="96" rhead="IO. BAPT. BENED." n="98" file="0098" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0098"/> - <p> - <s xml:space="preserve">Sint exempli gratia, tria corpora æquè ponderantia, & vnumquodque illorum <lb/>Å¿itquinque librarum, quorum vnum Å¿it aureum, aliud argenteum, reliquum verò <lb/>mixtum ex ijs duobus metallis, vnde corpus aureum Å¿implex minus erit, & argen <lb/>teum maius corpore mixto, quod nulli dubium eÅ¿t, Å¿it nunc pondus corporis aquei <lb/>Ä™qualis corpori aureo, <reg norm="librarum" type="context">librarÅ©</reg> .3. aquei verò Ä™qualis miÅ¿to, Å¿it <reg norm="librarum" type="context">librarÅ©</reg> 3. <reg norm="cum" type="context">cÅ©</reg> quarta par <lb/>te, aquei demum æqualisargenteo, librarum .4. cum dimidia, vnde exijs, quæ in præ <lb/>cedenti theoremate, & in .126. theoremate diximus, Å¿i imaginatione concipiemus <lb/>alia duo corpora Å¿implicia, auri, & argenti, Å¿ed æqualium magnitudinum mixto, <lb/>habebimus proportionem ponderis aurei ad pondus corporis mixti vt <reg norm="trium" type="context">triÅ©</reg> librarum <lb/>cum quarta vnius ad .3. libras, & proportio ponderis mixti ad pondus argentei erit, <lb/>vt proportio librarum .4. cum dimidia ad tres libras cum quarta parte vnius libræ, <lb/>& proportio ponderis aurei ad pondus argentei vt librarum .4. cum dimidia ad li-<lb/>bras .3: hoc eÅ¿t aurei ad mixtum, vt .13. ad .12. & mixti ad argenteum, vt .18. ad .13. <lb/>& aurei ad argenteum, vt .3. ad .2. ideÅ¿t, vt .18. ad .12.</s> - </p> - <p> - <s xml:space="preserve">Nunc inueniantur duo numeri ita inter Å¿e proportionati, vt .3. ad .2. habentes ta-<lb/>men inter ipÅ¿os numerum ita proportionatum ad maximum, vt .12. Å¿e habet ad <num value="13">.<lb/>13.</num> & ita proportionatum ad minimum, vt Å¿e habet .18. ad .13. quod hoc modo in-<lb/>ueniemus, multiplicabimus .18. per .12. & proueniet nobis .216. pro numero me-<lb/>dio, poÅ¿teà multiplicabimus .18. per .13. & proueniet .234. pro maximo, <reg norm="demum" type="context">demÅ©</reg> multi <lb/>plicando .12. per .13. proueniet .156. pro minimo, ita quod .234. correÅ¿pondebit <lb/>ponderi corporis aurei: </s> - <s xml:space="preserve">216. verò ponderi mixti, et .156. ponderi argentei æqua-<lb/>lium magnitudinum.</s> - </p> - <p> - <s xml:space="preserve">Cum autem proportiones horum trium corporum inuenerimus, Å¿i ordinem theo <lb/>rematis .122. Å¿equemur, habebimus quod quærebamus, & inueniemus in præſenti <lb/>exemplo proportionem ponderis auri ad pondus argenti in corpore mixto eſſe, vt <num value="180">.<lb/>180.</num> ad .36. Å¿ed quia Å¿uppoÅ¿itum fuit corpus mixtum eſſe quinque librarum, propte-<lb/>reà dicemus. </s> - <s xml:space="preserve">Si .216. hoc eÅ¿t toti corpori mixto correÅ¿pondent quinque libræ tunc <lb/>parti .180. hoc eÅ¿t auro in ipÅ¿o corpore mixto, correÅ¿pondent libræ .4. cum duabus <lb/>vncijs, ex regula detribus, reÅ¿iduum verò quinque librarum, ideÅ¿t vnciæ decem, <lb/>correÅ¿pondent parti .36. hoc eÅ¿t argento in dicto corpore mixto.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i tria corpora dicta fuiſſent inuicem ita proportionata, vt .40. 47. 60. </s> - <s xml:space="preserve">tunc <lb/>proportio auri ad argentum in corpore mixto eſſet vt .13. ad .7. quapropter <reg norm="cum" type="context">cÅ©</reg> pon <lb/>dus mixti fuiſſet .120. librarum, </s> - <s xml:space="preserve">tunc aurum ipÅ¿ius eſſet librarum .78. argentum ve-<lb/>rò librarum .42. ex eadem regula.</s> - </p> - <p> - <s xml:space="preserve">Pro quarum rerum Å¿peculatione nil aliud oportet nunc dicere cum Å¿atis dictum à no <lb/>bis Å¿uperius fuerit, vno excepto, hoc eÅ¿t rationem reddere, qua motus fui ad inue <lb/>niendos illos .3. numeros ita inter Å¿e diÅ¿poÅ¿itos, vt dictum eÅ¿t, quæ quidem ratio fuit, <lb/>vt haberemus .3. numeros ita inter ipÅ¿os ordinatè diÅ¿poÅ¿itos, vt Å¿unt pondera trium <lb/>illorum corporum æqualium magnitudinum. </s> - <s xml:space="preserve">Proptereà quòd quamuis inter pri-<lb/>mos .3. numeros ponderum corporum aqueorum eædem fuerint proportiones pon <lb/>derum corporum metallicorum, nihilominus medius numerus extra proprium lo-<lb/>cum, & inordinatè inueniebatur, reÅ¿pectu extremorum, vnde medius numerus in <lb/>Å¿uo vero Å¿itu inter .18. et .12. fuiſſent .16. <reg norm="cum" type="context">cÅ©</reg> .8. tertijs decimis, Å¿ed vt <reg norm="fractorum" type="context">fractorÅ©</reg> incom <lb/>moditatem euitemus, præcepi, vt multiplicarentur extrema per .13. vnde produ-<lb/>cti fuerunt numeri .234. et .156. in <reg norm="eadem" type="context">eadẽ</reg> proportione, quæ eÅ¿t .18. ad .12. ex .18. Å¿epti <lb/>mi, iuſſi etiam multiplicari .18. per .12. vt nobis prodiret .216. ad quem numerum, <lb/>numerus .234. ita Å¿e haberet, ut .13. ad .12. ex .19. Å¿eptimi, quod autem ita Å¿it propor - <pb o="87" rhead="THEOREM. ARITH." n="99" file="0099" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0099"/> - tionatus .216. ad .156. vt .18. ad .13. maniteÅ¿tum eÅ¿t exijÅ¿dem, nam tam .18. quam <num value="13">.<lb/>13.</num> multiplicatus fuit per .12.</s> - </p> - </div> - <div type="math:theorem" level="3" n="129"> - <head xml:space="preserve">THEOREMA <num value="129">CXXIX</num>.</head> - <p> - <s xml:space="preserve">ALIVD proponitur problema hoc modo: </s> - <s xml:space="preserve">Å¿upponitur obÅ¿idio alicuius loci, vbi <lb/>alimento ad nutriendos .10000. homines Å¿ufficiunt pro quinque menÅ¿ibus tan-<lb/>tum, Å¿ed quia eum locum obÅ¿idione non liberari putatur niÅ¿i .18. menÅ¿ibus exactis, <lb/>quæritur, quot homines eo tempore illis alimentis nutriri poſſint, hoc eÅ¿t .18. <lb/>menÅ¿ibus.</s> - </p> - <p> - <s xml:space="preserve">Præcipitregula, vt multiplicetur primus numerus, hoc eÅ¿t hominum .10000. cum <lb/>Å¿ecundo, hoc eÅ¿t menÅ¿ium quinque, productum verò diuidatur per .18. hoc eÅ¿t men-<lb/>Å¿ium, </s> - <s xml:space="preserve">tunc proueniet .2777. cum .7. nonis.</s> - </p> - <p> - <s xml:space="preserve">Cuius operationis ratio eÅ¿t hæc, Å¿int exempli gratia duo hic Å¿ubÅ¿cripta producta <lb/>Å¿uperficialia <var>.a.n.</var> et <var>.o.u.</var> inuicem æqualia, Å¿ed tal@ figura delineata, vt proportio <var>.u.<lb/>x.</var> ad <var>.x.o.</var> Å¿it, vt .10000. ad quinque, & proportio <var>a.x.</var> ad <var>.x.o.</var> Å¿it vt .18. ad quinque, <lb/>ct <var>.x.n.</var> Å¿it nobis ignota, quæ quidem eÅ¿t illa, quæ indagatur, ita <reg norm="quod" type="simple">ê™</reg> vnumquodque <lb/>iÅ¿torum productorum Å¿ignificabit alimentum, et <var>.u.x.</var> Å¿ignificabit numerum homi-<lb/>num .10000. qui quidem homines comederent totum alimentum <var>.u.o.</var> Å¿pacio tem-<lb/>poris <var>.x.o.</var> quinque menÅ¿ium, proptereà quòd <var>u.o.</var> Å¿upponitur productum eſſe ab <var>.<lb/>u.x.</var> in <var>.x.o</var>. </s> - <s xml:space="preserve">Deinde <reg norm="Å¿upponendo" type="context">Å¿upponẽdo</reg> <var>.a.x.</var> tem <lb/> - <anchor type="figure" xlink:label="fig-0099-01a" xlink:href="fig-0099-01"/> - pus eſſe .18. menÅ¿ium, ergo <var>.x.n.</var> Å¿ignifi-<lb/>cabit numerum hominum, qui eo tem-<lb/>poris Å¿pacio ali poſſunt, hoc eÅ¿t <var>.x.a.</var> ali-<lb/>mento <var>.n.a.</var> eo quòd <var>.a.n.</var> producitur ex <var>.<lb/>n.x.</var> in <var>.a.x.</var> vnde ex .15. Å¿exti, Å¿eu ex, 20. <lb/>Å¿eptimi proportio <var>.x.u.</var> ad <var>.x.n.</var> <reg norm="eadem" type="context">eadẽ</reg> erit, <lb/>quÄ™ <var>.a.x.</var> ad <var>.x.o.</var> quapropter rectè factum <lb/>erit accipere <reg norm="productum" type="context">productÅ©</reg> <var>.u.o.</var> quodidem <lb/>eÅ¿t in quantitate, quod productum .2. n. & ipÅ¿um diuidere per <var>.a.x.</var> vnde nobis <lb/>proueniat <var>.n.x</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0099-01" xlink:href="fig-0099-01a"> - <image file="0099-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0099-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="130"> - <head xml:space="preserve">THEOREMA <num value="130">CXXX</num>.</head> - <p> - <s xml:space="preserve">QVotieÅ¿cunque nobis propoÅ¿itum fuerit inuenire tertium terminum, trium ter <lb/>minorum continuè proportionalium armonicæ proportionalitatis, quo-<lb/>tum duo nobis cogniti Å¿int, ita agemus.</s> - </p> - <p> - <s xml:space="preserve">Sint, exempli gratia, tres termini <var>.q.p</var>: <var>a.g.</var> et <var>.e.c.</var> continuæ proportionalium at <lb/>monicæ proportionalitatis, quorum <var>.q.p.</var> maior et <var>.a.g.</var> medius Å¿int nobis cogniti, <lb/>cum ergo voluerimus tertium <var>.e.<lb/>c.</var> cognitum nobis eſſe: </s> - <s xml:space="preserve">a.g. detra-<lb/> - <anchor type="figure" xlink:label="fig-0099-02a" xlink:href="fig-0099-02"/> - hatur ex <var>.q.p.</var> differentia verò <var>.d.<lb/>p.</var> addatur <var>.q.p.</var> quorum Å¿umma <lb/>erit <var>.q.o.</var> cognita, qua mediante <lb/>diuidatur productum, quod ex <var>.a.<lb/>g.</var> in <var>.d.p.</var> exurgit, & proueniet no <lb/>bis <var>.n.g.</var> hoc e@t minor differentia, eo quòd productum <var>.q.o.</var> in <var>.n.g.</var> æquale eÅ¿t pro- - <pb o="88" rhead="IO. BAPT. BENED." n="100" file="0100" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0100"/> - ducto .2. g. in <var>.d.p.</var> ex .20. Å¿eptimi, proptereà quòd proportio <var>.q.o.</var> ad <var>.o.p.</var> hoc eÅ¿t ad <var>.<lb/>d.p.</var> eÅ¿t vt <var>.a.g.</var> ad <var>.g.n.</var> coniunctim cum diÅ¿iunctim it a Å¿it <var>.q.p.</var> ad <var>.p.o.</var> vt <var>.a.n.</var> ad <var>.n.g.</var> <lb/><reg norm="permutando" type="context">permutãdo</reg> eo quòd <var>.q.p.</var> ad <var>.a.n.</var> (ideÅ¿t ad <var>.e.c.</var>) ita Å¿e <reg norm="hent" type="context">hẽt</reg> ut <var>.p.o.</var> (hoc eÅ¿t <var>.d.p.</var>) ad <var>.n.g.</var> <lb/>ex <reg norm="conditionibus" type="context">cõditionibus</reg> armonicæ proportio nalitatis. </s> - <s xml:space="preserve">Deinde Å¿i detraxerimus <var>.n.g.</var> ex <var>.a.g.</var> <lb/>remanebit <var>.e.c.</var> minor terminus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0099-02" xlink:href="fig-0099-02a"> - <image file="0099-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0099-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Sed Å¿i <var>.e.c.</var> tertius terminus nobis propoÅ¿itus eſſet Å¿imul cum <var>.a.g.</var> medio, & volue <lb/>rimus maiorem inuenire <var>.q.p.</var> Å¿cilicet, oportebit <var>.e.c.</var> ex <var>.a.g.</var> detrahere, differentiam <lb/>verò <var>.n.g.</var> Å¿imiliter demeremus <lb/>ex <var>.e.c.</var> unde remaneret nobis <var>.e.t.</var> <lb/> - <anchor type="figure" xlink:label="fig-0100-01a" xlink:href="fig-0100-01"/> - cognitum, quo reÅ¿iduo <var>.c.t.</var> me-<lb/>diante diuidemus productum, <reg norm="quod" type="simple">ê™</reg> <lb/>furgit ex <var>.a.g.</var> in <var>.t.c.</var> & prouentus <var>.<lb/>d.p.</var> erit differentia maior, eo <reg norm="quod" type="simple">ê™</reg> <lb/><reg norm="productum" type="context">productÅ©</reg> quod Å¿it ex <var>.e.t.</var> in <var>.d.p.</var> <lb/>æquale eÅ¿t producto quòd fit ex <var>.a.g.</var> in <var>.t.c.</var> per 20. Å¿eptimi Eucli. eo quòd <var>.a.g.</var> (id-<lb/>eÅ¿t <var>.q.d.</var>) ad <var>.d.p.</var> eÅ¿t ut <var>.e.t.</var> ad <var>.t.c.</var> diÅ¿iunctim, cum coniunctim ita Å¿it <var>.q.p.</var> ad <var>.d.p.</var> vt <var>.e.<lb/>c.</var> ad <var>.t.c.</var> permutando, quia <var>.q.p.</var> ad <var>.e.c.</var> eÅ¿t vt <var>.d.p.</var> ad <var>.t.c.</var> hoc eÅ¿t ad <var>.n.g.</var> ex legibus <lb/>dictis.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0100-01" xlink:href="fig-0100-01a"> - <image file="0100-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0100-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="131"> - <head xml:space="preserve">THEOREMA <num value="131">CXXXI</num>.</head> - <p> - <s xml:space="preserve">ALIA etiam methodo hoc perfici poſſe comperi. </s> - <s xml:space="preserve">PropoÅ¿iti enim cum nobis fue <lb/>rint duo termini <var>.c.e.</var> minimus et <var>.g.a.</var> medius, maximus verò quærendus Å¿it, de <lb/>trahatur differentia <var>.g.n.</var> ex <var>.e.c.</var> & per reÅ¿iduum <var>.e.t.</var> diuidatur productum <reg norm="quod" type="simple">ê™</reg> fit ex <var>.a.<lb/>g.</var> in <var>.e.c.</var> prouentus quæ erit <var>.q.p.</var> terminus quæſitus.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius ratione, ponamus in eſſe terminum <var>.q.p.</var> </s> - <s xml:space="preserve">tunc ex forma huius proportio <lb/>nalitatis nulli dubium erit quin <var>.q.p.</var> ad <var>.e.c.</var> fit vt <var>.d.p.</var> ad <var>.n.g.</var> hoc eft ad <var>.t.c.</var> vnde ex <lb/>19. quinti vel .12. Å¿eptimi ita eſſet <var>.q.d.</var> ad <var>.e.t.</var> vt <var>.q.p.</var> ad <var>.e.c.</var> </s> - <s xml:space="preserve">quare ex .20. @cptimi pro <lb/>ductum <reg norm="quod" type="simple">ê™</reg> naÅ¿citur ex <var>.p.d.</var> (hoc eÅ¿t <var>.a.g.</var>) in <var>.e.c.</var> æquale eric producto <var>.e.t.</var> in <var>.q.p.</var> qua-<lb/>propter Å¿i diuiÅ¿erimus id per <var>.e.t.</var> proueniet nobis <var>.q.p</var>.</s> - </p> - <p> - <s xml:space="preserve">Sed <reg norm="cum" type="context">cÅ©</reg> nobis propoÅ¿iti fuerint duo termini <var>.q.p.</var> maximus, et <var>.a.g.</var> medius, Å¿i <reg norm="mini- mum" type="context">mini-<lb/>mÅ©</reg> <var>.e.c.</var> <reg norm="voluerimus" type="simple">voluerimê°</reg> inuenire. </s> - <s xml:space="preserve">Termino <var>.q.p.</var> maximo, <reg norm="iungatur" type="context simple">iÅ©gat̃</reg>. <var>p.o.</var> Ä™qualis, <var>p.d.</var> <reg norm="differentiÄ™" type="context">differẽtiÄ™</reg> <lb/>propoÅ¿itæ, diuidatur poÅ¿tea productum <reg norm="quod" type="simple">ê™</reg> ex <var>.q.p.</var> in <var>.a.g.</var> generatur per <var>.q.o.</var> prouen <lb/>tus autem Å¿it <var>.e.c.</var> qui quidem erit terminus quæſitus.</s> - </p> - <p> - <s xml:space="preserve">Cuius operationis Å¿peculutio hæc erit, Å¿upponatur terminum <var>.e.c.</var> inuentum eſſe <lb/>vnde <var>.n.g.</var> differentia Å¿it inter <var>.e.c.</var> <lb/>et <var>.a.g.</var> ex forma igitur armonicæ <lb/> - <anchor type="figure" xlink:label="fig-0100-02a" xlink:href="fig-0100-02"/> - proportionalitis ita erit <var>.q.p.</var> ad <var>.a.<lb/>n.</var> vt <var>.p.o.</var> ad <var>.n.g.</var> vnde ex .13. quin-<lb/>ti. </s> - <s xml:space="preserve">Ita erit <var>.q.o.</var> ad <var>.a.g.</var> vt <var>.q.p.</var> ad <var>.a.<lb/>n.</var> ergo <reg norm="productum" type="context">productÅ©</reg> quòd fit ex <var>.a.g.</var> <lb/>in <var>.q.p.</var> (ex .20. Å¿eptimi) æquale erit <lb/>producto <var>.q.o.</var> in <var>.a.n</var>. </s> - <s xml:space="preserve">Quare Å¿i diuiÅ¿um fuerit tale productum per <var>.q.o.</var> proueniet no-<lb/>bis <var>.e.c.</var> quòd querebamus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0100-02" xlink:href="fig-0100-02a"> - <image file="0100-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0100-02"/> - </figure> - </div> - <pb o="89" rhead="THEOR. ARITH." n="101" file="0101" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0101"/> - </div> - <div type="math:theorem" level="3" n="132"> - <head xml:space="preserve">THEOREMA <num value="132">CXXXII</num>.</head> - <p> - <s xml:space="preserve">SED quia aliquis poſſet in dubium reuocare, an poſſibile Å¿it inuenire tertium <lb/>terminum rationalem, Å¿eu communicantem duobus datis terminis inter Å¿e com <lb/>municantibus in tali proportionalitate, hoc eÅ¿t harmonica. </s> - <s xml:space="preserve">Vthoc oÅ¿tendatur.</s> - </p> - <p> - <s xml:space="preserve">Sint duo termini dati <var>.a.o.</var> et <var>.a.e.</var> inter Å¿e communicantes, tertius verò inuentus <lb/>Å¿it <var>.a.c.</var> qui maximus, primò, Å¿it in ea proportionalitate, quem dico communicantem <lb/>eſſe cum primis datis.</s> - </p> - <p> - <s xml:space="preserve">Nam ex conditionibus huiuÅ¿modi proportionalitatis, habebimus primum ean-<lb/>dem proportionem eſſe <var>.a.c.</var> ad <var>.a.o.</var> quæ eÅ¿t <var>.e.c.</var> ad <var>.e.o.</var> vnde permutando ita erit <var>.a.<lb/>c.</var> ad <var>.e.c.</var> vt <var>.a.o.</var> ad <var>.o.e.</var> & quia ex .9. decimi Euclid <var>.a.o.</var> communicat cum <var>.o.e.</var> </s> - <s xml:space="preserve">quare <lb/>ex .10. eiuÅ¿dem <var>.a.c.</var> communicabit cum <var>.e.c.</var> & per .9. cum <var>.a.e.</var> et per .8. cum <var>.a.o.</var> <lb/>quod<unsure/>eÅ¿t propoÅ¿itum.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i datus fuerit maximus <var>.a.c.</var> cum medio <var>.a.e.</var> interÅ¿e communicantes mini-<lb/>mum verò <var>.a.o.</var> probabo <reg norm="communicantem" type="context">cõmunicantem</reg> cum illis eſſe. </s> - <s xml:space="preserve">Cogitemus ergo <var>.c.f.</var> æqua-<lb/>jem eſſe differentiæ <var>.c.e.</var> cognitæ, vnde habebimus proportionem, <var>a.c.</var> ad <var>.c.f.</var> vt <var>.a.o.</var> <lb/>ad <var>.o.e.</var> & componendo <var>.a.f.</var> ad <var>.f.c.</var> vt <var>.a.e.</var> ad <var>.e.o.</var> & quia (ex Å¿uppoÅ¿ito). <var>a.c.</var> commu-<lb/>nicat cum <var>.e.c.</var> hoc eÅ¿t cum <var>.c.f.</var> </s> - <s xml:space="preserve">quare <lb/>ex eadem .9. dicti decimi <var>.a.f.</var> et <var>.f.c.</var> <reg norm="erunt" type="context">erÅ©t</reg> <lb/> - <anchor type="figure" xlink:label="fig-0101-01a" xlink:href="fig-0101-01"/> - inter Å¿e communicantes, & per .10. <var>a.e.</var> <lb/>communicabit cum <var>.o.e.</var> & per .9. <var>a.e.</var> cò <lb/>municabit cum <var>.a.o.</var> vnde per .8. <var>a.o.</var> communicabit cum <var>.a.c.</var> Å¿imiliter.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0101-01" xlink:href="fig-0101-01a"> - <image file="0101-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0101-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="133"> - <head xml:space="preserve">THEOREMA <num value="133">CXXXIII</num>.</head> - <p> - <s xml:space="preserve">SED Å¿i nobis duo extremi termini propoÅ¿iti fuerint, & medium inuenire deÅ¿ide <lb/>remus in dicta proportionalitate, ita faciendum erit.</s> - </p> - <p> - <s xml:space="preserve">Sint, exempli gratia, duo termini dati <var>.q.b.</var> et <var>.b.r.</var> minor <var>.b.r.</var> ex maiori <var>.b.q.</var> de-<lb/>trahatur, reÅ¿iduum verò <var>.q.x.</var> multiplicetur per <var>.b.r.</var> productum poÅ¿teà diuidatur per <lb/><var>q.r.</var> vnde proueniet nobis <var>.x.l.</var> pro differentia minori, quæ addita cum <var>.b.x.</var> minimo <lb/>termino, dabit nobis <var>.b.l.</var> mcdium terminum harmonicum.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius ratione cogitemus dictum medium terminum <var>.b.l.</var> iam inuentum eſſe, <lb/>vnde ita erit proportio <var>.q.l.</var> ad <var>.l.x.</var> vt <var>.q.b.</var> ad <var>.b.r.</var> ex forma huius proportionalitatis, <lb/></s> - <s xml:space="preserve">quare coniunctim ita erit <var>.q.r.</var> ad <var>.r.b.</var> vt <lb/><var>q.x.</var> ad <var>.x.l.</var> & proptereà ex .20. Å¿eptimi <lb/> - <anchor type="figure" xlink:label="fig-0101-02a" xlink:href="fig-0101-02"/> - productum, quod fit ex <var>.q.r.</var> in <var>.x.l.</var> æqua-<lb/>le erit producto <var>.q.x.</var> in <var>.b.r</var>. </s> - <s xml:space="preserve">Rectè igitur <lb/>fit cum diuiditur hoc productum per <var>.q.r.</var> vt proueniat nobis <var>.x.l.</var> differentia minor.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0101-02" xlink:href="fig-0101-02a"> - <image file="0101-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0101-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="134"> - <head xml:space="preserve">THEOREMA <num value="134">CXXXIIII</num>.</head> - <p> - <s xml:space="preserve">POſſumus etiam harmonicè diuidere vnam datam proportionem abÅ¿que aliqua <lb/>diuiÅ¿ione productorum, ne nobis fractiones proueniant, hoc modo videlicet. <lb/></s> - <s xml:space="preserve">Nobis propoÅ¿itum Å¿it diuidere harmonicè Å¿eÅ¿quialteram <reg norm="proportionem" type="context">proportionẽ</reg> inuenian-<lb/>tur primo minimi termini huius proportionis ut putà .3. et .2. quarum Å¿umma, hoc <lb/>eÅ¿t quinque, multiplicetur per minorem ideÅ¿t .2. vnde proueniet nobis .10. qui qui-<lb/>dem erit minor terminus trium quæſitorum, quorum maximus erit productum Å¿um - <pb o="90" rhead="IO. BAPT. BENED." n="102" file="0102" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0102"/> - mæ iam dictæ in maiorem eorum, hoc eÅ¿t quod fit ex quinque in .3. quod erit .15. </s> - <s xml:space="preserve">Vt <lb/>autem medium terminum harmonicum inter iÅ¿tos habeamus, accipiatur <reg norm="duplum" type="context">duplÅ©</reg> pro-<lb/>ducti, quod fit ex primis minimis terminis, quod erit .12.</s> - </p> - <p> - <s xml:space="preserve">Cuius rei Å¿peculatio eÅ¿t iÅ¿ta: </s> - <s xml:space="preserve">Å¿ignificentur duo termini datæ proportionis ab <var>.q.b.</var> <lb/>et <var>.b.r.</var> quorum Å¿umma erit <var>.q.r.</var> cuius quadratum Å¿it <var>.q.o.</var> Å¿it etiam imaginata <var>.b.e.</var> <lb/>parallela ad <var>.o.r</var>. </s> - <s xml:space="preserve"><reg norm="Sitque" type="simple">SitqÌ;</reg> <var>.b.x.</var> æqualis <var>.b.r.</var> et <var>.q.u.</var> Å¿imiliter, & ducatur <var>.x.y.</var> parallela ad <lb/><var>r.o.</var> et <var>.u.l.</var> ad <var>.q.x</var>. </s> - <s xml:space="preserve">Tunc habebimus <var>.b.o.</var> æquale ei producto, quod fit ex <var>.q.r.</var> in <var>.b.r.</var> <lb/>et <var>.b.y.</var> eidem etiam æquale, et <var>.q.e.</var> pro producto, quod fit ex <var>.q.r.</var> in <var>.q.b.</var> et <var>.q.l.</var> pro <lb/>eo, quod fit ex <var>.q.x.</var> in <var>.b.r</var>. </s> - <s xml:space="preserve">Vnde <var>.q.l.</var> cum <var>.b.y.</var> æquale fiet duplo ei, quod fit ex <var>.q.b.</var> <lb/>in <var>.b.r</var>. </s> - <s xml:space="preserve">Dico nunc <var>.b.o.</var> eſſe minimum terminum eorum, quos quærimus, et <var>.y.b.</var> cum <var>.<lb/>x.u.</var> medium <var>.q.e.</var> verò maximum huiuÅ¿modi proportionalitatis.</s> - </p> - <p> - <s xml:space="preserve">Primum ergo certi Å¿cimus ex prima Å¿exti vel .18. Å¿eptimi eandem exiÅ¿tere pro-<lb/>portionem <var>.q.e.</var> ad <var>.b.o.</var> Å¿eu ad <var>.b.y.</var> quæ <var>.q.b.</var> ad <var>.b.r</var>: Å¿ed <var>.u.y.</var> ad <var>.u.x.</var> eÅ¿t vt <var>.y.l.</var> ad <var>.l.x.</var> <lb/>hoc eÅ¿t vt <var>.q.b.</var> ad <var>.b.r.</var> ideÅ¿t vt <var>.q.e.</var> ad <var>.b.o.</var> & Å¿umma <var>.u.y.</var> cum <var>.u.x.</var> ideÅ¿t <var>.q.y.</var> minor eÅ¿t <lb/>quam <var>.q.e.</var> maximus terminus per <var>.b.y.</var> minimum ter-<lb/>minum. </s> - <s xml:space="preserve">& <reg norm="coniunctim" type="context">cõiunctim</reg> <var>.q.y.</var> ad <var>.q.l.</var> vt <var>.y.x.</var> ad <var>.x.l.</var> hoc eÅ¿t <lb/> - <anchor type="figure" xlink:label="fig-0102-01a" xlink:href="fig-0102-01"/> - vt <var>.q.r.</var> ad <var>.r.b</var>. </s> - <s xml:space="preserve">Vnde ex Å¿peculatione <reg norm="præcedentis" type="context">præcedẽtis</reg> theo <lb/>rematis, Å¿equitur <var>.u.y.</var> eſſe differentiam inter <reg norm="maximum" type="context">maximÅ©</reg> <lb/>& medium terminum, et <var>.u.x.</var> eſſe differentiam inter <lb/>medium & minimum dictæ proportionalitatis. </s> - <s xml:space="preserve">Nam <lb/>eadem proportio eÅ¿t <var>.q.e.</var> maximi termini ad <var>.b.o.</var> mi-<lb/>nimi. quæ <var>.u.y.</var> (differentia inter <var>.q.e.</var> & gnomonem <var>.<lb/>u.b.y.</var>) ad <var>.u.x.</var> (differentia inter dictum <var>.u.b.y.</var> et <var>.b.y.</var> <lb/>minimum terminum, quia Å¿unt ambæ ut <var>.q.b.</var> ad <var>.b.r.</var> <lb/>vt diximus. </s> - <s xml:space="preserve">Quare <var>.b.y.</var> <reg norm="coniunctum" type="context">coniunctÅ©</reg> cum <var>.x.u.</var> medius <lb/>terminus erit, qui quidem (vt dictum eÅ¿t) duplus eÅ¿t ei <lb/>quod fit ex <var>.q.b.</var> in <var>.b.r</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0102-01" xlink:href="fig-0102-01a"> - <image file="0102-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0102-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="135"> - <head xml:space="preserve">THEOREMA <num value="135">CXXXV</num>.</head> - <p> - <s xml:space="preserve">ALIVM etiam modum ab antiquis traditum ad hoc problema perficiendum <lb/>inueni, qui talis eÅ¿t. </s> - <s xml:space="preserve">Inueniatur primo inter datos terminos extremos, me-<lb/>dius terminus in arithmetica proportione, per <reg norm="quem" type="context">quẽ</reg> <lb/> - <anchor type="figure" xlink:label="fig-0102-02a" xlink:href="fig-0102-02"/> - multiplicetur vnuÅ¿quiÅ¿que dictorum extremorum, <lb/>deinde multiplicentur ipÅ¿i extremi interÅ¿e, vnde <lb/>habebimus tria producta eadem proportione inui <lb/>cem exiÅ¿tentia, vt quærebatur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0102-02" xlink:href="fig-0102-02a"> - <image file="0102-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0102-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Exempli gratia, ponamus duos propoÅ¿itos ter-<lb/>minos eſſe .3. et .2. quorum medius arithmeticè <lb/>eſſet .2. cum dimidia vnitate, per quem cum vnum <lb/>quemque priorum multiplicauerimus, <reg norm="emergent" type="context">emergẽt</reg> no-<lb/>bis duo producta, quorum primum ideÅ¿t maius eſſet <lb/>7. cum dimidia vnitate, reliquum verò eſſet <lb/>quinque, productum poÅ¿teà quod ex ipÅ¿is extremis <lb/>prouenit, erit .6. quod quidem eÅ¿t harmonicè collo <lb/>catum inter .7. cum dimidia vnitate, & quinque.</s> - </p> - <p> - <s xml:space="preserve">Cuius rei Å¿peculatio omnis à præcedenti theore-<lb/>mate dependet. </s> - <s xml:space="preserve">Sint exempli gratia, duo termini - <pb o="91" rhead="THEOREM. ARIT." n="103" file="0103" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0103"/> - propoÅ¿iti <var>.a.e.</var> maior, et <var>.e.o.</var> minor, <reg norm="Sitque" type="simple">SitqÌ;</reg> <var>.o.k.</var> medius arithmeticus inter dictos, vn-<lb/>de clarè patebit <var>.o.k.</var> eſſe dimidium Å¿ummæ dictorum terminorum ex .75. theorema <lb/>te huius libri. </s> - <s xml:space="preserve">Sit ergo productum <var>a.t.</var> id quod fit ex <var>.a.e.</var> in <var>.o.k.</var> et <var>.o.t.</var> Å¿it <reg norm="productum" type="context">productÅ©</reg> <lb/>quod fit ex <var>.e.o.</var> in <var>.o.k.</var> et <var>.n.m.</var> Å¿it productum quod Å¿it ex <var>.a.e.</var> in <var>.e.o.</var> quorum vnum-<lb/>quodque erit dimid ium vniuÅ¿cuiuÅ¿que producti præcedentis theorematis, <lb/>ex .18. et .19. Å¿eptimi Eucli. vnumquodque Å¿ui relatiui. </s> - <s xml:space="preserve">Quare argumentando per <lb/>mutando à concluÅ¿ionibus præcedentis theorematis ad has præſentis, habebimus <lb/>productum.</s> - </p> - </div> - <div type="math:theorem" level="3" n="136"> - <head xml:space="preserve">THEOREMA <num value="136">CXXXVI</num>.</head> - <p> - <s xml:space="preserve">MEDIVM autem contra <reg norm="harmonicum" type="context">harmonicÅ©</reg> inuenire cum quis voluesit inter duos <lb/>propoÅ¿itos terminos, ita faciendum erit, hoc eÅ¿t per Å¿ummam datorum ex <lb/>tremorum diuidatur productum quod fit ex minimo termino in <reg norm="differentiam" type="context">differẽtiam</reg> dato-<lb/>rum, prouentus poÅ¿tea erit differentia inter maximum & med<unsure/>um quæſitum.</s> - </p> - <p> - <s xml:space="preserve">Vt exempli gratia, Å¿i nobis propoÅ¿iti fuerint hi duo termini .3. et .2. Å¿umma eo-<lb/>rum erit quinque, per quam cum diuiÅ¿erimus productum, quod naÅ¿citur ex mini-<lb/>mo .2. in differentiam eorum, quæ eÅ¿t vnum, quod quidem erit .2. </s> - <s xml:space="preserve">tunc duæ quintæ <lb/>partes prouenient, quæ Å¿i demptæ fuerint ex maximo termino, reliquum erit .2. <reg norm="cum" type="context">cÅ©</reg> <lb/>3. quintis, hoc eÅ¿t medius terminus contta harmonicus.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius ratione cogitemus <var>.u.d.</var> et <var>.x.c.</var> eſſe duosterminosnobis propoÅ¿itos, in-<lb/>ter quos deÅ¿ideremus inuenire <var>.o.s.</var> medium ita illis <reg norm="relatum" type="context">relatÅ©</reg>, vt proportio exceſſus ip-<lb/>Å¿ius Å¿upra <var>.x.c.</var> (qui Å¿it <var>.e.n.</var>) ad exceÅ¿-<lb/>Å¿um <var>.u.d.</var> Å¿upra <var>.o.s.</var> (qui Å¿it <var>.n.d.</var>) ea-<lb/> - <anchor type="figure" xlink:label="fig-0103-01a" xlink:href="fig-0103-01"/> - dem Å¿it quæ <var>.u.d.</var> ad <var>.x.c</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0103-01" xlink:href="fig-0103-01a"> - <image file="0103-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0103-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Cogitemus igitur <var>.x.c.</var> coniunctum <lb/>eſſe cum <var>.u.d.</var> & hæcÅ¿umma vocetur <var>.<lb/>b.d.</var> vnde habebimus proportionem <var>.<lb/>u.d.</var> ad <var>.u.b.</var> vt <var>.e.n.</var> ad <var>.n.d</var>. </s> - <s xml:space="preserve">Quare <reg norm="com- ponendo" type="context">cõ-<lb/>ponendo</reg> ita erit <var>.d.b.</var> ad <var>.u.b.</var> ut <var>.e.d.</var> 3d.n.d. Å¿ed quia <var>.d.b</var>: <var>u.b.</var> et <var>.e.d.</var> quantitates no-<lb/>bis cognitÄ™ Å¿unt, ideò <var>.d.n.</var> ex .20. Å¿eptimi cognita nobis erit.</s> - </p> - </div> - <div type="math:theorem" level="3" n="137"> - <head xml:space="preserve">THEOREMA <num value="137">CXXXVII</num>.</head> - <p> - <s xml:space="preserve">SVpponunt antiqui aliquot mercatores dantes pecunias lucro in diuerÅ¿is vnius <lb/>anni temporibus, </s> - <s xml:space="preserve">tunc in fine anni Å¿umma torius lucri datur cognita, Å¿ed quæ-<lb/>ritur quantuni<unsure/> vnicuique illorum exipÅ¿a Å¿umma debeatur.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, primus in principio anni poÅ¿uit .100. aurcos, Å¿ecundus verò .100 <lb/>diebus poÅ¿t primum poÅ¿uit .50. aureos tertius autem .200. diebus poÅ¿t primum po-<lb/>Å¿uit .25. aureos Å¿umma lucri poÅ¿tea in fine anni fuit aureorum .60.</s> - </p> - <p> - <s xml:space="preserve">Nunc vt Å¿ciamus quantum huius Å¿ummæ vniduique illorum proueniat, præcipit <lb/>regula, vt faciamus tria producta, quorum primum Å¿it ex numero dierum totius an-<lb/>ni in numerum aureorum primi, vnde tale productum in præſenti caÅ¿u erit .36500. <lb/>Å¿ecundum verò Å¿it ex numero dierum à primo die in quo ipÅ¿e Å¿ecundus poÅ¿uit uÅ¿que <lb/>ad finem anni, in numerum ipÅ¿orum nummorum, quod erit .13250. tertium autem <lb/>productum ex diebus tertij in numerum Å¿uorum aureorum, quod <reg norm="quidem" type="context">quidẽ</reg> erit .4125. <lb/>quæ producta Å¿imul collecta faciunt .53875. deinde multiplicetur vnumquodque - <pb o="92" rhead="IO. BAPT. BENED." n="104" file="0104" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0104"/> - ipÅ¿orum prochictorum per Å¿ummam lucri hoc eÅ¿t per .60. vnde multiplicatio primi <lb/>producti erit .2190000. multiplicatio verò Å¿ecundi producti erit .795000. tertij po <lb/>Å¿tca erit .247500. quarum multiplicationum vnaquæque diuidatur per Å¿ummam <lb/>53875. productori<unsure/>t, & proueniet ex prima diuiÅ¿ione .40. <reg norm="cum" type="context">cÅ©</reg> fractis .35000. vnius in-<lb/>tegri diuiÅ¿i in partes .53875. quod erit lucrum primi, prouentus autem Å¿ecundæ di-<lb/>uiÅ¿ionis erit .14. cum fractis .41050. vnius integri diuiÅ¿i in partes .53875. lucrum <reg norm="Å¿ecu di." type="context">Å¿ecÅ©Â<lb/>di</reg> </s> - <s xml:space="preserve">prouentus verò quartæ diuiÅ¿ionis erit .4. cum fractis .32000. vnius integri, vt Å¿u <lb/>pra diuiÅ¿i in partes .53875. hoc eÅ¿t lucrum tertij.</s> - </p> - <p> - <s xml:space="preserve">Cuius rei Å¿peculatio ex Å¿e in Å¿ub Å¿cripta figura patet, vbi <var>.a.q.</var> Å¿ignificat numerum <lb/>dierum totius anni pro primo mercatore <var>.q.n.</var> autem Å¿ignificat numerum dierum Å¿e <lb/>cundi mercatoris <var>.e.q.</var> poÅ¿teà ſignificat numerum dierum tertij Å¿it etiam <var>.s.a.</var> pro nu-<lb/>mero denariorum primi, et <var>.o.n.</var> pro numero Å¿ecundi, et <var>.e.t.</var> pro numero <lb/>tertij, productum autem <var>.q.s.</var> Å¿ignificet valorem primi lucri, et <var>.q.o.</var> Å¿ecundi, <lb/>et <var>.q.t.</var> tertij <var>.x.y.</var> autem Å¿ignificet Å¿ummam lucri omnium, et <var>.x.i.</var> Å¿ignificet <lb/>partem primi, et <var>.i.p.</var> Å¿ecundi, et <var>.p.y.</var> tertij. </s> - <s xml:space="preserve">vnde clarè patebit ex communi <lb/>Å¿cientia quòd eadem proportio erit <var>.x.y.</var> ad <var>.x.i.</var> quæ aggregati omnium producto-<lb/>rum <var>.q.s</var>: <var>q.o.</var> et <var>.q.t.</var> ad <var>.q.s.</var> & ita <var>.x.y.</var> ad <var>.i.p.</var> vt aggregati dictiad <var>.q.o.</var> et <var>.x.y.</var> ad <var>.p.y.</var> <lb/>vt dicti aggregati ad <var>.q.t</var>. </s> - <s xml:space="preserve">Rectè igitur ex regula de tribus multiplicatio <var>.q.s.</var> in <var>.x.y.</var> <lb/>diuiditur per aggregatum omnium <lb/> - <anchor type="figure" xlink:label="fig-0104-01a" xlink:href="fig-0104-01"/> - productorum, ita vt Å¿i aliquis dice-<lb/>ret, Å¿i ex dicto aggregato, prouenit <lb/><var>x.y.</var> quid proueniet vnicuique <reg norm="illo- rum" type="context">illo-<lb/>rÅ©</reg> <reg norm="productorum" type="context">productorÅ©</reg>. </s> - <s xml:space="preserve"><reg norm="Nam" type="context">Nã</reg> Å¿i numerus dena-<lb/>riorum <reg norm="Å¿ecundi" type="context">Å¿ecÅ©di</reg> æqualis eſſet numero <lb/><var>a.s.</var> primi vt putà . <var>n.b</var>. </s> - <s xml:space="preserve">tunc eius <reg norm="lucrum" type="context">lucrÅ©</reg> <lb/>Å¿ignificaretur à rectangulo <var>.q.b.</var> & ita <lb/>de tertio dico <reg norm="quod" type="simple">ê™</reg> Å¿ignificaretur à <reg norm="re- ctangulo" type="context">re-<lb/>ctãgulo</reg> <var>.q.c.</var> vel Å¿i Å¿i<unsure/>antibus <reg norm="ijÅ¿dem" type="context">ijÅ¿dẽ</reg> <reg norm="denariorum" type="context">denariorÅ©</reg> quantitatibus <var>.n.o.</var> et <var>.e.t.</var> omnes Å¿uas pe-<lb/>cunias eodem tempore poÅ¿uiſſent, </s> - <s xml:space="preserve">tunc rectangula Å¿ignificantia eorum lucra eÅ¿lent <lb/><var>q.s.q.d.</var> et <var>.q.f.</var> Å¿ed cum nec eodem tempore, nec eandem quantitatem poÅ¿ueruntr<unsure/>e <lb/>ctè eorum lucra Å¿ignificantur à rectangulis <var>.q.s.q.o.</var> et <var>.q.t.</var> <reg norm="quod" type="simple">ê™</reg> ex prima .6. vel .18. aut <num value="19">.<lb/>19.</num> Å¿eptimi ratiocinando clarè patebit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0104-01" xlink:href="fig-0104-01a"> - <image file="0104-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0104-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="138"> - <head xml:space="preserve">THEOREMA <num value="138">CXXXVIII</num>.</head> - <p> - <s xml:space="preserve">NIcolaus Tartalea in primo libro vltimæ partis numerorum ad .35. quæſitum <lb/>docet inuenire quantitatem laterum vnius propoÅ¿iti trianguli, cuius la-<lb/>r<unsure/>erum proportio nobis data Å¿it Å¿imul cum area Å¿uperÅ¿iciali ipÅ¿ius trianguli, Å¿ed quia <lb/>ipÅ¿e Tartalea vtiturregula algebræ, mihi viÅ¿um eÅ¿t breuiori methodo hoc idein fa <lb/>cere, & etiam vniuerÅ¿aliori via.</s> - </p> - <p> - <s xml:space="preserve">Supp onamus igitur duo triangula, quorum vnum <var>.u.n.i.</var> Å¿it nobis <reg norm="propoÅ¿itum" type="context">propoÅ¿itÅ©</reg>, & <lb/>cognitæ Å¿uperficiei, proportiones Å¿imiliter laterum <var>.i.n.</var> ad <var>.n.u</var>: et <var>.u.n.</var> ad <var>.u.i.</var> Å¿int no <lb/>bis datæ, <reg norm="alterum" type="context">alterÅ©</reg> verò <reg norm="triangulum" type="context">triangulÅ©</reg> Å¿it <var>.a.o.u.</var> à nobis tamen ita <reg norm="confectum" type="context">confectÅ©</reg>, v<unsure/>latera Å¿int inÂ<lb/>er Å¿e proportionata eodem modo, quo latera prioris trianguli, Å¿ed hæc nobis <reg norm="etiam" type="context">etiã</reg> <lb/>cognita Å¿int, <reg norm="quod" type="simple">ê™</reg> facillimum eÅ¿t. </s> - <s xml:space="preserve">Nunc vero Å¿i <reg norm="demptum" type="context">demptÅ©</reg> fuerit <reg norm="quadratum" type="context">quadratÅ©</reg> <var>.a.o.</var> minimi <lb/>lateris, ex quadrato <var>.o.u.</var> maximi, relinquet nobis duplum producti <var>.o.u.</var> in <var>.u.e.</var> per <lb/><reg norm="penultimam" type="context">penultimã</reg> .2. Eucli. <reg norm="Å¿upponendo" type="context">Å¿upponẽdo</reg> <var>.a.e.</var> perpendicularem ad <var>.o.u.</var> vnde tale productum <lb/>quòd fit ex <var>.o.u.</var> in <var>.u.e.</var> conÅ¿equenter nobis cognitum erit, & quia <var>.o.u.</var> nobis cogni- - <pb o="93" rhead="THEOREM. ARIT." n="105" file="0105" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0105"/> - tum eÅ¿t, ideo cognoÅ¿cemus <var>.e.u.</var> Å¿ed <reg norm="cum" type="context">cÅ©</reg> <var>.e.u.</var> minor Å¿it <var>.a.u.</var> ex .18. & penultima primi, <lb/>Å¿i <reg norm="demptum" type="context">demptÅ©</reg> fuerit quadratum <var>.e.u.</var> ex quadrato <var>.a.u.</var> remanebit nobis <reg norm="cognitum" type="context">cognitÅ©</reg> <reg norm="quadra- tum" type="context">quadra-<lb/>tÅ©</reg> <var>.a.e.</var> & Å¿ic nota erit nobis perpendicularis <var>.a.e.</var> ex penultima primi, quæ quidem <var>.<lb/>a.e.</var> Å¿i multiplicata fuerit in dimidium <var>.o.u.</var> dabit nobis <reg norm="Å¿uperficiem" type="context">Å¿uperficiẽ</reg> trianguli <var>.a.o.u.</var> ex <lb/>41. dicti libri. </s> - <s xml:space="preserve">Et quia proportio trianguli <var>.a.o.u.</var> ad triangulum <var>.u.i.n.</var> (propter Å¿imi <lb/>litudinem) eÅ¿t vt quadrati <var>.o.u.</var> ad quadratum <var>.n.i.</var> ex communi Å¿cientia cum vna-<lb/>quæque iÅ¿tarum proportionum dupla Å¿it proportioni <var>.o.u.</var> ad <var>.n.i.</var> ex .17. et .18. Å¿exti, <lb/>deinde cum nobis cognitæ Å¿int tres iÅ¿tarum quatuor quantitatum hoc eÅ¿t Å¿uperficies <lb/>trianguli <var>.a.o.u.</var> Å¿uperficies trianguli <var>.u.n.i.</var> & quadrati <var>.o.u.</var> </s> - <s xml:space="preserve">quare ex regula de tribus <lb/>cognoÅ¿cemus etiam quadratum <var>.n.i.</var> & Å¿ic <var>.n.i.</var> latus primi trianguli, vnde reliqua la <lb/>tera illicò nobis innoteÅ¿cent exipÅ¿a regula de tribus, cum dixerimus, Å¿i <var>.o.u.</var> dat nobis <lb/><var>u.a</var>. </s> - <s xml:space="preserve">tunc <var>.i.n.</var> dabit <var>.u.n.</var> quòd etiam infero de <var>.u.i</var>.</s> - </p> - <p> - <s xml:space="preserve">Poſſemus etiam ita hoc perficere, <lb/>Å¿cilicet inuenire <var>.x.</var> quantitatem me-<lb/> - <anchor type="figure" xlink:label="fig-0105-01a" xlink:href="fig-0105-01"/> - diam proportionalem inter duas Å¿u-<lb/>perficies triangulorum, vnde Å¿uper-<lb/>ficies trianguli <var>.i.a.u.o.</var> ad <var>.x.</var> Å¿e ha-<lb/>beret ut <var>.o.u.</var> ad <var>.i.n.</var> & ita ex regula <lb/>detribus cognoÅ¿cemus <var>.i.n</var>. </s> - <s xml:space="preserve">Multo <reg norm="tem" type="context">tẽ</reg> <lb/>pore poÅ¿tquà m hoc theorema conÅ¿truxi, ipÅ¿um conÅ¿criptum inueni in decimo <lb/>Å¿ecundi libri Ioannis de monte Regio, Å¿atis tamen obÅ¿curè expreſſum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0105-01" xlink:href="fig-0105-01a"> - <image file="0105-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0105-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="139"> - <head xml:space="preserve">THEOREMA <num value="139">CXXXIX</num>.</head> - <p> - <s xml:space="preserve">IN eodem primo libro vltimæ partis numerorum, Tartalea probat, via algebrÄ™ <lb/>quòd quælibet duo latera trianguli orthogonij, angulumrectum continentia, <lb/>Å¿int tertio longiora per diame-<lb/> - <anchor type="figure" xlink:label="fig-0105-02a" xlink:href="fig-0105-02"/> - trum circuli inÅ¿criptibilis in ip-<lb/>Å¿o triangulo. </s> - <s xml:space="preserve">Å¿ed hoc breuius <lb/>geometricè poteÅ¿t <reg norm="demonÅ¿trari" type="context">demõſtrari</reg>, <lb/>quemadmodum in Å¿ubÅ¿cripta <lb/>hic figura videre eÅ¿t, proptereà <lb/>quòd cum anguli <var>.A.o.u.</var> et <var>.n.</var> <lb/>omnes Å¿int recti et <var>.A.u.</var> æqualis <lb/><var>o.n.</var> et <var>.A.n.</var> Ä™qualis <var>.u.o.</var> ipſæ <var>.A.<lb/>u.</var> et <var>.A.n.</var> æquales erunt diame-<lb/>tro ipÅ¿ius circuli. </s> - <s xml:space="preserve">Sed eædem <var>.<lb/>A.u.</var> et <var>.A.n.</var> Å¿unt Å¿uperfluum, quo <var>.A.B.</var> et <var>.A.C.</var> Å¿unt maiores <var>.B.C.</var> cum <var>.B.u.</var> et <var>.C.n.</var> <lb/>Å¿int æquales <var>.B.C.</var> ex penultima tertij Eucli.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0105-02" xlink:href="fig-0105-02a"> - <image file="0105-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0105-02"/> - </figure> - </div> - <head xml:space="preserve">THEO. SEQVENS THEO. CXXXIX.</head> - <p> - <s xml:space="preserve">SImiliter in nono capite Å¿ecundi libri nouæ Å¿cientiæ poterat ipÅ¿e Tartalea breuio <lb/>ri methodo abÅ¿que vlla operatione ipÅ¿ius Algebræ inuenire <var>.A.H.</var> reÅ¿pectu <var>.A.<lb/>E.</var> eſſe vt .4. <reg norm="cum" type="context">cÅ©</reg> vno Å¿eptimo ad <reg norm="vnum" type="context">vnÅ©</reg>. </s> - <s xml:space="preserve"><reg norm="Nam" type="context">Nã</reg> ipÅ¿e Å¿upponit <var>.A.E.</var> <reg norm="decimam" type="context">decimã</reg> <reg norm="partem" type="context">partẽ</reg> eſſe ipÅ¿ius - <pb o="94" rhead="IO. BAPT. BENED." n="106" file="0106" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0106"/> - <var>A.I.</var> vnde quadratum lineæ <var>.A.I.</var> erit .100. idem dico de quadrato lineæ <var>.I.L</var>. </s> - <s xml:space="preserve">quare <lb/>ex penultima primi <var>.A.L.</var> erit radix quadrata quadrati .200. ideÅ¿t .14. cum vno Å¿epti-<lb/>mo ferè. </s> - <s xml:space="preserve">quare <var>.A.L.</var> iuncta <var>.A.O.</var> erit .28. cum duobus Å¿eptimis. </s> - <s xml:space="preserve">Å¿ed <var>.L.O.</var> ex Å¿uppoÅ¿i-<lb/>to erit .20. eo quòd <var>.L.I.</var> Ä™quatur ipÅ¿i <var>.A.I.</var> Å¿imiliter et <var>.I.O.</var> vt ipÅ¿e etiam probauit. </s> - <s xml:space="preserve">qua <lb/>dempta ex <var>.L.A.O.</var> relinquetur <var>.H.A.M.</var> (nam <var>.L.H.</var> cum <var>.O.M.</var> æquatur ipÅ¿i <var>.L.O.</var> ex .<lb/>35. tertij ipÅ¿ius Eucli. partium .8. <reg norm="cum" type="context">cÅ©</reg> duabus Å¿eptimis. cuius <reg norm="dimidium" type="context">dimidiÅ©</reg> hoc eÅ¿t <var>.A.H.</var> erit <lb/>4. cum una Å¿eptima, quod eÅ¿t propoÅ¿itum. </s> - <s xml:space="preserve">ReÅ¿pice figuram ipÅ¿ius Tartaleæ.</s> - </p> - </div> - <div type="math:theorem" level="3" n="140"> - <head xml:space="preserve">THEOREMA <num value="140">CXL</num>.</head> - <p> - <s xml:space="preserve">QVadrageÅ¿imum nonum quæſitum Å¿imiliter poſſumus alio modo Å¿oluere, vt <lb/>putà cum vnumquodque latus rhombi Å¿imul cum area cognitum, Å¿eu datum <lb/>nobis Å¿it <reg norm="cognitum" type="context">cognitÅ©</reg> Å¿imiliter nobis erit quadratum lateris <var>.a.d.</var> hoc eÅ¿t Å¿umma duorum <lb/>quadratorum <var>.a.o.</var> et <var>.o.d.</var> ex penultima primi Euclid. </s> - <s xml:space="preserve">cúmque nobis cognita etiam <lb/>Å¿it totalis Å¿uperficies rhombi, cognita etiam nobis erit eius medietas, hoc eÅ¿t produ-<lb/>ctum <var>.o.d.</var> in <var>.o.a.</var> vnde ex methodo .37. Theorematis cognoÅ¿cemus .a<unsure/><var>.o.</var> et <var>.o.d.</var> & Å¿ic <lb/>etiam eorum dupla, quod quærebatur.</s> - </p> - <figure position="here"> - <image file="0106-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0106-01"/> - </figure> - </div> - <div type="math:theorem" level="3" n="141"> - <head xml:space="preserve">THEOREMA <num value="141">CXLI</num>.</head> - <p> - <s xml:space="preserve">PVlchrum quæſitum fuit id, quod Tartalea ponit pro .18. noni libri in quarto fo-<lb/>lio, quod huiuÅ¿modi eÅ¿t. </s> - <s xml:space="preserve">Aliquis habet dolium mero plenum, ex quo <lb/>duas vrnas extrahit ipÅ¿ius vini, Å¿ed loco ipÅ¿ius vini infundit duas vrnas aquæ. </s> - <s xml:space="preserve">Dein <lb/>de poÅ¿t aliquot dies extrahit iterum alias duas vrnas illius miÅ¿ti, & iterum infundit <lb/>duas vrnas aquæ, & poÅ¿t alios aliquot dies idem facit, & hac vltima tertia vice in-<lb/>uenit aquam tantam eſſe, quantum vinum. </s> - <s xml:space="preserve">Quæritur nunc quot vrnas capiat il-<lb/>lud dolium.</s> - </p> - <p> - <s xml:space="preserve">Solutio ipÅ¿ius Tartaleæ bona eÅ¿t, cum Å¿upponat illas quatuor quantitates vini eſſe <lb/>inuicem continuas proportionales, vt putà primò totum vinum merum, poÅ¿teà re-<lb/>Å¿iduum pro Å¿ecunda quantitate, deinde pro tertia in Å¿ecunda, & pro quarta in ter-<lb/>tia extractione, hoc eÅ¿t quòd proportio totius vini meri ad vinum in prima Å¿it, vt hu <lb/>ius ad vinum in Å¿ecunda, & vt huius ad vinum in tertia miÅ¿tione. </s> - <s xml:space="preserve">Sed quia ipÅ¿e <lb/>non probat hanc continuam proportionalitatem ex methodo Å¿cientifica, mihi <reg norm="visum" type="context">visÅ©</reg> <lb/>eÅ¿t hoc loco illam deÅ¿cribere.</s> - </p> - <p> - <s xml:space="preserve">Cogitemus igitur <var>a.u.</var> pro capacitate dolij, et <var>.a.i.</var> pro quantitate duarum vrna-<lb/>rum. </s> - <s xml:space="preserve">Nunc uerò Å¿upponamus quamlibet partem huius miÅ¿ti omogeneam eſſe Å¿uo <lb/>toto, quapropter Å¿equetur eandem proportionem eſſe vini ad aquam in qualibet <lb/>parte, quæ erit in toto, & ideò imaginemur <var>.e.o.</var> æqualem <var>.a.i</var>. </s> - <s xml:space="preserve">Sed in puncto <var>.i.</var> tali <lb/>modo diuiÅ¿am, vt proportio <var>.i.e.</var> ad <var>.i.o.</var> eadem Å¿it quæ <var>.i.a.</var> ad <var>.i.u</var>. </s> - <s xml:space="preserve">Supponamus <reg norm="etiam" type="context">etiã</reg> - <pb o="95" rhead="THEOREM. ARITH." n="107" file="0107" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0107"/> - <var>e.o.</var> eſſe duas primas vrnas vini miÅ¿ti hoc eÅ¿t primæ miÅ¿tionis, vnde cum eadem pro <lb/>portio Å¿it <var>.a.i.</var> ad <var>.i.u.</var> vt <var>.e.i.</var> ad <var>.i.o.</var> ita erit (ex .19. quinti). <var>a.e.</var> ad <var>.o.u.</var> ut <var>.a.i.</var> ad <var>.i.u.</var> & <lb/><reg norm="componendo" type="context">componẽdo</reg> ita erit <var>.a.e.</var> cum <var>.o.u.</var> hoc eÅ¿t <var>.i.o.u.</var> (proptereà quòd <var>.i.o.</var> æqualis eÅ¿t <var>.a.e.</var> <lb/>vt reÅ¿idua totorum æqualium) ad <var>.o.u.</var> quemadmodum <var>.a.i.u.</var> ad <var>.i.u</var>. </s> - <s xml:space="preserve">Quare <var>.i.u.</var> erit <lb/>media proportionalis inter <var>.a.u.</var> et <var>.o.u.</var> vnde proportio <var>.a.u.</var> ad <var>.o.u.</var> dupla erit pro <lb/>portioni <var>.i.u.</var> ad <var>.o.u</var>. </s> - <s xml:space="preserve">Nunc autem cum extracta fuerit quantitas <var>.e.o.</var> ex primo mi-<lb/>Å¿to, & poÅ¿teà infuÅ¿a aqua vÅ¿que ad plenitudinem dolij, proportio ingredientium <lb/>huius Å¿ecundi miÅ¿ti erit ea, quæ eÅ¿t inter <var>.o.u.</var> et <var>.o.a.</var> eo quòd in prima miÅ¿tione pro-<lb/>proportio ingredientium erat ea, quæ eÅ¿t inter <var>.o.u.</var> et <var>.a.e.</var> vel inter <var>.a.e.</var> et <var>.o.u.</var> <lb/>vt demonÅ¿trauimus. </s> - <s xml:space="preserve">Accipiamus ergo <var>.t.m.</var> huiuÅ¿modi Å¿ecundi mifti, magnitudi-<lb/>nis <var>.a.i.</var> vel <var>.e.o.</var> Å¿ignificantis duas vrnas, & permutemus eum in tantam aquam, <lb/><reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> punctum <var>.o.</var> quod nobis diuidat <var>t.m.</var> in <var>.o.m.</var> et, <var>o.t.</var> partes Å¿implices, tali propor <lb/>tione inuicem relatas, vt Å¿unt <var>.o.u.</var> et <var>.o.a.</var> vnde habebimus ex Å¿upradictis rationibus <lb/>eandem proportionem ipÅ¿ius <var>.a.t.</var> ad <var>.m.u.</var> vt <var>.a.o.</var> ad <var>.o.u.</var> & componendo <var>.a.t.</var> cum <var>.m.<lb/>u.</var> hoc eÅ¿t <var>.i.m.u.</var> (eo quod cum <var>.t.m.</var> æqualis Å¿it <var>.a.i.</var> per conÅ¿equens <var>.i.m.</var> æqualis erit <var>.<lb/>a.t.</var>) ad <var>.m.u.</var> vt <var>.a.o.u.</var> ad <var>.o.u.</var> Å¿ed proportio <var>.a.o.u.</var> ad <var>.o.u.</var> dupla erat proportioni <var>.i.o.<lb/>u.</var> ad <var>.o.u.</var> quemadmodum Å¿upra diximus. </s> - <s xml:space="preserve">Ergo proportio <var>.i.m.u.</var> ad <var>.m.u.</var> erit dupla <lb/>Å¿imiliter proportioni <var>.i.o.u.</var> ad <var>.o.<lb/>u.</var> quapropter <var>.o.u.</var> erit media proÂ<lb/> - <anchor type="figure" xlink:label="fig-0107-01a" xlink:href="fig-0107-01"/> - portionalis inter <var>.i.u.</var> et <var>.m.u</var>. </s> - <s xml:space="preserve">Ec-<lb/>ce igitur quomodo eadem eÅ¿t pro <lb/>portio <var>.a.u.</var> ad <var>.i.u.</var> quæ <var>.i.u.</var> ad <var>.o.u.</var> & quæ <var>.o.u.</var> ad <var>.m.u.</var> qui quidem modus neceſſarius <lb/>eÅ¿t vt intellectus acquieÅ¿cat, id quod experientia non facit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0107-01" xlink:href="fig-0107-01a"> - <image file="0107-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0107-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="142"> - <head xml:space="preserve">THEOREMA <num value="142">CXLII</num>.</head> - <p> - <s xml:space="preserve">PRæcedens Tartaleæ quæſitum elegans quidem eÅ¿t, Å¿ed pulchrum etiam vide-<lb/>tur quærere proportionem ingredientium in ultima miÅ¿tione, cum cognita fue <lb/>rit nobis proportio continentiæ dolij ad capacitatis vrnæ Å¿imul <reg norm="cum" type="context">cÅ©</reg> numero vitium <lb/>extractionum & impletionum.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proportio <var>.a.u.</var> ad <var>.a.i.</var> cognita nobis fuerit, cognoÅ¿cemus etiam <lb/><var>e.i.</var> ex regula de tribus & per conÅ¿equens etiam <var>.i.o.</var> reÅ¿iduum ex <var>.e.o.</var> & Å¿imiliter ag-<lb/>gregatum <var>.a.i.</var> cum <var>.i.o.</var> & Å¿ic <var>.o.u.</var> reÅ¿iduum totius, et <var>.o.t.</var> Å¿imiliter, eo quòd <var>.a.u.</var> ad <var>.a.<lb/>o.</var> eÅ¿t ut <var>.t.m.</var> ad <var>.o.t.</var> vnde cognoÅ¿cemus etiam <var>.o.m.</var> vt reÅ¿iduum <var>.t.m.</var> & Å¿imiliter ag-<lb/>gregatum <var>.a.o.</var> cum <var>.o.m.</var> hoc eÅ¿t <var>.a.m.</var> & etiam <var>.m.u.</var> reÅ¿iduum totius.</s> - </p> - <p> - <s xml:space="preserve">CognoÅ¿cere autem proportionem totius dolij ad vrnam, vel ècontrà , cum cogni <lb/>ta nobis fuerit proportio ingredientium in vltima miÅ¿tione Å¿imul cum numero vi-<lb/>tium extractionum, & repletionum, quod Å¿cribit Tartalea, hoc etiam modo <lb/>poſſumus.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proportio <var>.m.u.</var> ad <var>.m.a.</var> cognita nobis fuerit, illicò Å¿cie-<lb/>mus proportionem <var>.a.u.</var> ad <var>.m.u.</var> & cum Å¿ciuerimus numerum vitium extractionum, <lb/>& impletionum illicò cognoÅ¿ci-<lb/>mus multiplicitatem proportio-<lb/>nis <var>.a.u.</var> ad <var>.m.u.</var> ad proportionem <var>.<lb/> - <anchor type="figure" xlink:label="fig-0107-02a" xlink:href="fig-0107-02"/> - o.u.</var> ad <var>.m.u.</var> quapropter propor-<lb/>tio <var>.o.u.</var> ad <var>.m.u.</var> nobis cognita erit <lb/>hoc eÅ¿t <var>.a.u.</var> ad <var>.i.u.</var> & Å¿imiliter ea, quæ eÅ¿t <var>.a.u.</var> ad <var>.a.i.</var> & è conuerÅ¿o Å¿imiliter.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0107-02" xlink:href="fig-0107-02a"> - <image file="0107-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0107-02"/> - </figure> - </div> - <pb o="96" rhead="IO. BAPT. BENED." n="108" file="0108" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0108"/> - <p> - <s xml:space="preserve">Vnde cum aliquis diceret priori modo, dolium habeo vrnarum .400. vini, & per <lb/>vices .25. extraxi & impleui ipÅ¿um, vt dictum eÅ¿t. </s> - <s xml:space="preserve">Nunc verò velim Å¿cire proportio-<lb/>nem vini ad a quam hac vltima vice. </s> - <s xml:space="preserve">Nunc igitur Å¿i procedemus iuxta doctrinam <lb/>primi exempli huius theorematis, obtinebimus quod quærebamus.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i diceret iuxta Tartaleæ quæſitum, hoc eÅ¿t dolium habeo, quod ignoro quot <lb/><reg norm="nam" type="context">nã</reg> urnas contineat, volo tamen per .25. vices extrahere, & implere vt <reg norm="Å¿upradictum" type="context">Å¿upradictÅ©</reg> <lb/>eÅ¿t, ita vt vltima vice proportio vini ad aquam Å¿it Å¿eÅ¿quialtera. </s> - <s xml:space="preserve">Tunc Å¿i iuxta mo-<lb/>dum Å¿ecundi exempli huius theorematis procedemus habebimus quod cupimus.</s> - </p> - <p> - <s xml:space="preserve">Alio etiam modo aliquis quærere poſſet, hoc eÅ¿t, habeo <reg norm="dolium" type="context">doliÅ©</reg> quod capit .400. <lb/>vrnas. </s> - <s xml:space="preserve">Habeo etiam vas trium vrnarum, quo mediante me oportet extrahere, & <lb/>implere. </s> - <s xml:space="preserve">Velim tamen Å¿cire quoties me hoc facere oporteat, ita vt poÅ¿trema vi-<lb/>ce vinum Å¿e habeat ad aquam in proportione Å¿eÅ¿quialtera, vnde multoties accidet <lb/>vltimam extractionem, & impletionem mutilatam, Å¿eu imperfectam, euadere.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿i proportio vini ad aquam in vltima miÅ¿tione deberet eſſe vt <var>.n.<lb/>u.</var> ad <var>.n.a.</var> ita vt extrema vice fuiſſet <var>.t.m.</var> quæ quidem <var>.t.m.</var> excederet terminum per <var>.<lb/>n.m.</var> quæ <var>.n.m.</var> reuera eſſet nobis cognita, eò quòd ex priori modo hic Å¿upra dicto <lb/>proportio <var>.a.m.</var> ad <var>.m.u.</var> nobis in-<lb/>noteÅ¿ceret, & proportio <var>.n.a.</var> ad <var>.<lb/>n.u.</var> nobis data eÅ¿t Å¿imul cum <reg norm="quan titate" type="context">quãÂ<lb/> - <anchor type="figure" xlink:label="fig-0108-01a" xlink:href="fig-0108-01"/> - titate</reg> <var>.a.u.</var> </s> - <s xml:space="preserve">quare quantitas <var>.n.u.</var> & <lb/><var>m.u.</var> nobis cognita, remanebit, et <lb/><var>n.m.</var> eorum differentia Å¿imiliter, etiam, et <var>.t.n.</var> reÅ¿iduum vaÅ¿is, quo metimur, vnde <lb/>neceſſe erit, quo<unsure/>d vltima vice vas contineret Å¿olum <var>.t.n.</var> reliqua uerò per Å¿e patent.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0108-01" xlink:href="fig-0108-01a"> - <image file="0108-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0108-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="143"> - <head xml:space="preserve">THEOREMA <num value="143">CXLIII</num>.</head> - <p> - <s xml:space="preserve">HIeronymus Cardanus in <ref id="ref-0015">lib. Å¿uæ arithmeticæ cap .66. quæſtione .56.</ref> quam CarÂ<lb/>danicam vocat, ita inquit.</s> - </p> - <p> - <s xml:space="preserve">Quidam perambulauit prima die certam quantitatem Å¿patij, & Å¿ecunda die, <reg norm="tan tò" type="context">tãÂ<lb/>tò</reg> plus proportionaliter, quantò diameter eÅ¿t maior coÅ¿ta, & tertia die tantò plus <lb/>Å¿ecunda, quantò proportionaliter portio lineæ diuiſæ Å¿ecundum proportionem ha <lb/>bentem medium, & duo extrema excedit minorem portionem, & quarta die in <lb/>proportione ad tertiam vt Å¿ecunda ad primam, & quinta die proportionaliter tan-<lb/>tò plus quarta, quantò in tertia plus Å¿ecunda, & ita alternatis vicibus in diebus no-<lb/>uem peregit nouem milliaria. </s> - <s xml:space="preserve">Quæritur igitur quantum ambulauit die prima.</s> - </p> - <p> - <s xml:space="preserve">Hoc autem nihil aliud eÅ¿t, quà m Å¿i aliquis diceret, propono tibi, exempli gratia, <lb/>lineam <var>.a.l.</var> nouem partibus inuicem non æqualibus ita diuiÅ¿am <var>.a.c</var>: <var>c.d</var>: <var>d.e</var>: & cæte-<lb/>ris, quarum partium proportiones tibi etiam do, vt putà . <var>a.c.</var> ad <var>.c.d.</var> et <var>.c.d.</var> ad <var>.d.e.</var> et <var>.<lb/>d.e.</var> ad <var>.e.f.</var> & Å¿ic de cæteris vÅ¿que ad poÅ¿tremam <var>.k.l.</var> quæ quidem proportiones Å¿int <lb/>etiam inuicem diſſimiles, Å¿eu inæquales, do tibi etiam <reg norm="proportionem" type="context">proportionẽ</reg> totius lineæ <var>.a.l.</var> <lb/>ad <var>.a.b.</var> Å¿uam partem, quæ vt in propoÅ¿ito exemplo nonupla eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Quæro nunc quam proportionem habebit <var>.a.c.</var> ad <var>.a.b.</var> & Å¿ic de cæteris partibus <lb/>eiuÅ¿dem ad eandem <var>.a.b</var>.</s> - </p> - <p> - <s xml:space="preserve">Quod quidem facillimum erit Å¿peculari, nec non operari vnicuique, qui omnino <lb/>practicæ numerorum ignarus non fuerit, dum ab ordine Å¿cientifico non diÅ¿cedat.</s> - </p> - <p> - <s xml:space="preserve">Cum enim cognoÅ¿cimus proportionem <var>.a.c.</var> ad <var>.c.d.</var> conÅ¿equenter cognoÅ¿cemus <lb/>ctiam proportion em aggregati <var>.a.c.d.</var> ad <var>.c.d.</var> cum autem cognouerimus proportio- - <pb o="97" rhead="THEOREM. ARIT." n="109" file="0109" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0109"/> - nem <var>.c.d.</var> ad <var>.d.e.</var> Å¿i <var>.c.d.</var> accipiemus, vt medium inter <var>.a.d.</var> et <var>.d.e.</var> cognoÅ¿cemus etiam <lb/>proportionem <var>.a.d.</var> ad <var>.d.e.</var> </s> - <s xml:space="preserve">quare etiam eam quæ <var>.a.e.</var> ad <var>.d.e.</var> collocando poÅ¿teà . <lb/><var>d.e.</var> inter <var>.e.f.</var> et <var>.a.e.</var> innoteÅ¿cet ea, quæ eÅ¿t <var>.a.e.</var> ad <var>.e.f.</var> & ita gradatim accedenrus ad <lb/>perfectam cognitionem proportionis totius <var>.a.l.</var> ad <var>.k.l</var>. </s> - <s xml:space="preserve">Nunc autem mediante <var>.k.l.</var> <lb/>cognoÅ¿cemus proportionem totius <var>.a.l.</var> ad <var>.i.k.</var> & hac mediante, cam cognoÅ¿cemus, <lb/>quæ totius <var>.a.l.</var> ad <var>.g.h.</var> & hac mediante eam quæ totius <var>.a.l.</var> ad <var>.f.g.</var> & Å¿ic gradatim, co <lb/>gnita nobis erit proportio totius <lb/>lineæ <var>.a.l.</var> ad Å¿uam partem <var>.a.c.</var> be-<lb/> - <anchor type="figure" xlink:label="fig-0109-01a" xlink:href="fig-0109-01"/> - neficio poÅ¿teà totius lineæ <var>.a.l.</var> co <lb/>gnoÅ¿cemus proportionem <var>a.c.</var> ad <lb/><var>a.b.</var> & Å¿ic aliarum reÅ¿pectu lineæ <var>.a.b.</var> vt quærebatur, quæ quidem propoÅ¿itio, etÅ¿i car <lb/>danica uocetur leuiſſima tamen eÅ¿t.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0109-01" xlink:href="fig-0109-01a"> - <image file="0109-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0109-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="144"> - <head xml:space="preserve">THEOREMA <num value="144">CXLIIII</num>.</head> - <p> - <s xml:space="preserve">QVamuis multi de modo in Å¿umma colligendi, Å¿ubtrahendi, <reg norm="multiplicandi" type="context">multiplicãdi</reg>, & di <lb/>uidendi proportiones Å¿cripÅ¿erint, nullus tamen (quod Å¿ciam) perfectè, ac <lb/>Å¿cientificè Å¿peculatus eÅ¿t has operationes, quapropter hanc rem cum Å¿ilentio tranÅ¿i <lb/>re nolui, quin aliquid de ipÅ¿a conÅ¿cribam à ſumma dictarum proportionum in-<lb/>cohando.</s> - </p> - <p> - <s xml:space="preserve">QuotieÅ¿cunque igitur volunt duas proportiones inuicem aggregare, Å¿imul ea-<lb/>rum antecedentia multiplicant, & Å¿imiliter earum conÅ¿equentia. </s> - <s xml:space="preserve">Tunc proportio <lb/>terminata ab illis productis euadit in Å¿ummam illarum duarum propoÅ¿itarum <lb/>proportionum.</s> - </p> - <p> - <s xml:space="preserve">Vt exempli gratia, Å¿i voluerimus colligere proportionem Å¿eÅ¿quialteram cum Å¿eÅ¿-<lb/>quitertia, multiplicando .3. cum .4. antecedentia Å¿cilicet, pro ductum erit .12. poÅ¿teà <lb/>multiplicando .2. cum .3. conÅ¿equentia, tunc productum erit .6. </s> - <s xml:space="preserve">Proportio igitur, <lb/>quæ inter .12. et .6. reperitur. (quæ dupla eÅ¿t) eÅ¿t Å¿umma propoÅ¿itarum <reg norm="proportionum" type="context">proportionÅ©</reg>.</s> - </p> - <p> - <s xml:space="preserve">Cuius rei Å¿peculatio erit huiuÅ¿modi Å¿int <var>.x.</var> et <var>.u.</var> <lb/>duo antecedentia quarunruis proportionum <var>.t.</var> <lb/> - <anchor type="figure" xlink:label="fig-0109-02a" xlink:href="fig-0109-02"/> - verò et. n Å¿int eorum conÅ¿equentia, productum <lb/>autem antecedentium Å¿it <var>.a.g.</var> illud verò quod <reg norm="con" type="context">cõ</reg> <lb/>Å¿equentium Å¿it <var>.d.a.</var> vnde proportio <var>.a.g.</var> ad <var>.a.d.</var> <lb/>compoÅ¿ita erit ex proportione <var>.x.</var> ad <var>.t.</var> & ex ea, <lb/>quæ eÅ¿t <var>.u.</var> ad <var>.n.</var> per .24. Å¿exti vel quintam octaui. <lb/></s> - <s xml:space="preserve">Patet igitur ratio rectè faciendi, vt Å¿uprà dictum <lb/>eÅ¿t.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0109-02" xlink:href="fig-0109-02a"> - <image file="0109-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0109-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="145"> - <head xml:space="preserve">THEOREMA <num value="145">CXLV</num>.</head> - <p> - <s xml:space="preserve">QVotieÅ¿cunque deinde detrahere volunt vnam proportionem ex altera mul-<lb/>tiplicant antecedens vnius cum conÅ¿equenti alterius. </s> - <s xml:space="preserve">Tunc proportio, quÄ™ <lb/>inter talia duo producta incluÅ¿a reperitur, eÅ¿t reÅ¿iduum, Å¿eu differentia illarum dua-<lb/>rum proportionum datarum.</s> - </p> - <p> - <s xml:space="preserve">Vt exempli gratia, Å¿i aliquis vellet ex proportione dupla detrahere Å¿eÅ¿quialte-<lb/>ram, multiplicaret .2. antecedens duplæ cum .2. conÅ¿equenti Å¿eÅ¿quialteræ, quorum <lb/>productum eſſet .4. pro antecedenti reÅ¿iduÄ™ proportionis. </s> - <s xml:space="preserve">Deinde multiplicaret .3 <lb/>antecedens Å¿eÅ¿quialteræ cum .1. conÅ¿equenti duplæ, & productum eſſet .3. pro <reg norm="con- Å¿equenti" type="context">cõ- - <pb o="98" rhead="IO. BAPT. BENED." n="110" file="0110" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0110"/> - Å¿equenti</reg> reÅ¿iduÄ™ proportionis; </s> - <s xml:space="preserve">quæ quidem reÅ¿idua proportio eſſet vt .4. ad .3. hoc <lb/>eÅ¿t Å¿eÅ¿quitertia, & Å¿ic de cæteris.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius ratione, Å¿it proportio <var>.x.</var> ad <var>.n.</var> ea quæ (exempli gratia) maior Å¿it, à <lb/>qua volumus demere proportionem <var>.t.</var> ad <var>.u.</var> minorem Å¿cilicet. </s> - <s xml:space="preserve">Nunc autem <lb/>productum <var>.x.</var> in <var>.u.</var> Å¿it <var>.a.g.</var> illud verò <var>.t.</var> in <var>.<lb/>n.</var> Å¿it <var>.a.d</var>. </s> - <s xml:space="preserve">Tunc dico proportionem <var>.a.g.</var> ad <var>.a.</var> <lb/> - <anchor type="figure" xlink:label="fig-0110-01a" xlink:href="fig-0110-01"/> - d. eſſe reÅ¿iduam quæſitam. </s> - <s xml:space="preserve">Sit <var>.b.a.</var> productum <lb/>u. in <var>.n.</var> vnde eadem proportio erit producti <var>.a.<lb/>g.</var> ad productum <var>.a.b.</var> quÄ™ <var>.x.</var> ad <var>.n.</var> et <var>.a.d.</var> ad <var>a.b.</var> <lb/>quæ <var>.t.</var> ad <var>.u.</var> ex prima Å¿exti, Å¿eu .18. vel .19. Å¿e-<lb/>ptimi, Å¿ed proportio <var>.a.g.</var> ad <var>.a.b.</var> hoc eÅ¿t <var>.x.</var> ad <var>.<lb/>n.</var> componitur ex ea, quæ eÅ¿t <var>.a.g.</var> ad <var>.a.d.</var> & ea, <lb/>quæ eÅ¿t <var>.a.d.</var> ad <var>.a.b.</var> hoc eÅ¿t <var>.t.</var> ad <var>.u.</var> ergò ea, quÄ™ <lb/>eÅ¿t <var>.a.g.</var> ad <var>.a.d.</var> erit quà m quærebamus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0110-01" xlink:href="fig-0110-01a"> - <image file="0110-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0110-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="146"> - <head xml:space="preserve">THEOREMA <num value="146">CXLVI</num>.</head> - <p> - <s xml:space="preserve">RATIO verò, quòd rectè fiat, quotieÅ¿cunque aliquam proportionem dupli-<lb/>care volentes, quadramus terminos ipÅ¿ius proportionis, vel Å¿i eam triplicare <lb/>voluerimus, cubamus ipÅ¿os terminos, vel Å¿i eam quadruplicare voluerimus <lb/>inuenimus cenÅ¿icos cenÅ¿icos terminorum ipÅ¿ius proportionis, & Å¿ic de Å¿ingulis, in <ref id="ref-0016">.17 <lb/>Theo. huiuÅ¿modi tractatus</ref> manifeÅ¿ta eÅ¿t.</s> - </p> - </div> - <div type="math:theorem" level="3" n="147"> - <head xml:space="preserve">THEOREMA <num value="147">CXLVII</num>.</head> - <p> - <s xml:space="preserve">QVotieÅ¿cunque nobis propoÅ¿iti fuerint duo numeri ad libitum, deÅ¿ideraremusÂ<lb/>qÌue duas proportiones tali relatione inuicem refertas, quali Å¿unt hi duo pro <lb/>poÅ¿iti numeri inter Å¿e, ita faciendum erit.</s> - </p> - <p> - <s xml:space="preserve">Sciendum primo eÅ¿t proportionem maioris numeri propoÅ¿iti ad minorem Å¿em-<lb/>per eſſe alicuius ex quinque generum, hoc eÅ¿t aut erit generis multiplicis, aut Å¿u-<lb/>perparticularis, aut multiplicis Å¿uperparticularis, aut Å¿uper partientis, aut multi-<lb/>plicis Å¿uperpartientis.</s> - </p> - <p> - <s xml:space="preserve">Nunc autem Å¿i erit ex genere multiplici, iam ab antiquis traditus eÅ¿t modus, <reg norm="quem" type="context">quẽ</reg> <lb/>Å¿equi debemus. </s> - <s xml:space="preserve">Cuius Å¿peculatio à me inuenta patet .in .17. Theo. huius libri, vt <lb/>in præcedenti dixi.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i talis proportio datorum numerorum erit alicuius aliorum generum, ita <lb/>agemus, Å¿i fuerit Å¿uperparticularis.</s> - </p> - <p> - <s xml:space="preserve">Sit exempli gratia, Å¿eÅ¿quialtera, tunc Å¿umantur duo numeri inuicem inæquales, <lb/>quos à caÅ¿u volueris <var>.o.</var> et <var>.c.</var> qui quidem cubentur, & eorum cubi Å¿int <var>.a.</var> et <var>.e</var>. </s> - <s xml:space="preserve">Inuenia <lb/>tur poÅ¿teà . u. ita proportionatus ad <var>.o.</var> vt <var>.o.</var> eÅ¿t ad <var>.c.</var> ex regula de tribus, hoc eÅ¿t diui-<lb/>dendo quadratum ipÅ¿ius <var>.o.</var> per <var>.c.</var> vnde nobis proueniat <var>.u.</var> & quia proportio <var>.a.</var> ad <var>.e.</var> <lb/>tripla eÅ¿t proportioni <var>.o.</var> ad <var>.c.</var> & proportio <var>.u.</var> ad <var>.c.</var> dupla eÅ¿t <reg norm="eidem" type="context">eidẽ</reg>, quæ <var>.o.</var> ad <var>.c.</var> ideo <lb/>proportio <var>.a.</var> ad <var>.e.</var> Å¿eÅ¿quialtera erit proportioni <var>.u.</var> ad <var>.c</var>.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i proportio numerorum propoÅ¿itorum fuerit Å¿eÅ¿quitertia, faciemus <var>.a.</var> et <var>.e.</var> <lb/>eſſe cenÅ¿ica cenÅ¿ica ipÅ¿ius <var>.o.</var> et <var>.c</var>. </s> - <s xml:space="preserve">tunc Å¿umemus <var>.u.</var> conÅ¿equentem ad <var>.o.</var> vt dictum eÅ¿t, <lb/>deinde inueniremus <var>.i.</var> conÅ¿equens ad <var>.u.</var> ita ut <var>.u.</var> conÅ¿equens ipÅ¿ius <var>.o</var>. </s> - <s xml:space="preserve">tunc habebi-<lb/>mus proportionem <var>.i.</var> ad <var>.c.</var> triplam, & eam quæ eÅ¿t <var>.a.</var> ad <var>.e.</var> quadruplam proportio- - <pb o="99" rhead="THEOREM. ARIT." n="111" file="0111" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0111"/> - ni <var>.o.</var> ad <var>.c</var>. </s> - <s xml:space="preserve">Idem dico de reliquis proportionibus Å¿uperparticularibus.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i data proportio numerorum fuerit ex Å¿uper partientibus, vt exempli gra-<lb/>tia de quinque ad tria, efficiemus, vt <var>.a.</var> et <var>.e.</var> Å¿int prima relata ipÅ¿ius <var>.o.</var> et <var>.c.</var> vnde <lb/>proportio <var>.a.</var> ad <var>.e.</var> ita Å¿e habe-<lb/>bit ad proportionem <var>.o.</var> ad <var>.c.</var> <lb/> - <anchor type="figure" xlink:label="fig-0111-01a" xlink:href="fig-0111-01"/> - vt quinque ad <reg norm="vnum" type="context">vnÅ©</reg> & propor-<lb/>tio <var>.i.</var> ad <var>.c.</var> ut tria ad <reg norm="vnum" type="context">vnÅ©</reg>. </s> - <s xml:space="preserve">Qua-<lb/>re proportio <var>.a.</var> ad <var>.e.</var> ad pro-<lb/>portionem <var>.i.</var> ad <var>.c.</var> Å¿e habebit, <lb/>vt quinque ad tria, & Å¿ic de reliquis.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0111-01" xlink:href="fig-0111-01a"> - <image file="0111-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0111-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Pro alijs, eundem ordinem Å¿eruando, obtinebimus quod volumus.</s> - </p> - </div> - <div type="math:theorem" level="3" n="148"> - <head xml:space="preserve">THEOREMA <num value="148">CXLVIII</num>.</head> - <p> - <s xml:space="preserve">QVamuis in .16. Å¿exti et .20. Å¿eptimi manifeÅ¿tè pateat ratio, quare rectè fiatac <lb/>cipiendam radicem quadratam illius producti, quod fit ex duobus datis <lb/>terminis, vt medium proportionale geometricè inter ipÅ¿os habeamus: </s> - <s xml:space="preserve">nihilomi-<lb/>nus, quia per aliam methodum hoc idem Å¿cire poſſumus, inconueniens non erit a-<lb/>liquid circa hoc dicere.</s> - </p> - <p> - <s xml:space="preserve">Cogitemus igitur exempli gratia, tres numeros continuè proportionales geo-<lb/>metricè <var>.a.b</var>: <var>c.d.</var> et <var>.e.f.</var> quorum <var>.a.b.</var> et <var>.e.f.</var> tantummodo nobis cogniti Å¿int, imagine-<lb/>mur etiam <var>.g.a.</var> eſſe productum quod fit ex <var>.a.b.</var> in <var>.e.f.</var> et <var>.d.k.</var> quadratum <var>.c.d.</var> et <var>.a.h.</var> <lb/>id quod fit ex <var>.a.b.</var> vnde eandem proportionem habebimus <var>.a.h.</var> ad <var>.a.g.</var> quæ eÅ¿t <var>.h.b.</var> <lb/>ad <var>.b.g.</var> ex prima .6. aut .18. vel .19. Å¿epti-<lb/>mi, Å¿ed per .11. octaui ita eÅ¿t quadrati <var>.a.</var> <lb/> - <anchor type="figure" xlink:label="fig-0111-02a" xlink:href="fig-0111-02"/> - h. ad quadratum <var>.k.d.</var> vt <var>.a.b.</var> ad <var>.e.f.</var> hoc <lb/>eÅ¿t vt <var>.h.b.</var> ad <var>.b.g.</var> ergo per .11. quinti ita <lb/>erit <var>.a.h.</var> ad <var>.a.g.</var> vt ad <var>.k.d.</var> vnde <var>.a.g.</var> æqua <lb/>le erit <var>.k.d.</var> per .9. quinti. </s> - <s xml:space="preserve">Rectè ergo erit <lb/>accipere radicem quadratam <var>.a.g.</var> pro <var>.c.<lb/>d.</var> quod etiam eÅ¿t diuidere vnam datam <lb/><reg norm="proportionem" type="context">proportionẽ</reg> per æqualia, hoc eÅ¿t in duas <lb/>æquales partes, non dubito quin poſſer aliquis dicere non oportere vti poÅ¿teriori-<lb/>bus Theorematibus ad demonÅ¿trandum priora illis, Å¿ed hoc .148. dictum Å¿it luden <lb/>di loco.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0111-02" xlink:href="fig-0111-02a"> - <image file="0111-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0111-02"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="149"> - <head xml:space="preserve">THEOREMA <num value="149">CXLIX</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">Vnde</emph> fiat <reg norm="quod" type="simple">ê™</reg> Å¿i quis inuenire voluerit Å¿ecundum terminum ex quatuor nume <lb/>ris continuè, & geometricè proportionalibus, quorum duo extremi tantum-<lb/>modo nobis cogniti Å¿int, rectè factum Å¿it quadrare primum eorum, & hoc quadra-<lb/>tum poÅ¿teà per alium terminum cognitum multiplicare, cuius producti demum ac-<lb/>cipere radicem cubam pro Å¿ecundo termino quæſito, hocloco videbimus.</s> - </p> - <p> - <s xml:space="preserve">Imaginemur quatuor terminos continuè proportionales, vt dictum eÅ¿t, eſſe.</s> - <pb o="100" rhead="IO. BAPT. BENED." n="112" file="0112" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0112"/> - <s xml:space="preserve">a.b: <var>c.d</var>: <var>e.f.</var> et <var>.g.h.</var> quorum <var>.a.b.</var> et <var>.g.h.</var> nobis tantummodo cogniti Å¿int, <reg norm="Å¿itque" type="simple">Å¿itqÌ</reg> imagina <lb/>tione deÅ¿criptus cubus <var>.a.q.</var> primi termini, <reg norm="cubusque" type="simple">cubusqÌ</reg> <var>.d.k.</var> Å¿ecundi rermini, conÅ¿idere-<lb/>mus etiam baÅ¿im <var>.a.i.</var> quadratam ipÅ¿ius cubi <var>.a.q.</var> hoc eÅ¿t præcedentem dignitatem ip <lb/>Å¿ius cubi eiuÅ¿dem radicis, quæ quidem baÅ¿is <var>.a.i.</var> multiplicetur per quartum <reg norm="terminum" type="context">terminÅ©</reg> <lb/><var>g.h.</var> productum autem Å¿it <var>.g.a.</var> vnde eadem proportio erit <var>.a.q.</var> ad <var>.a.g.</var> quæ <var>.b.q.</var> ad <var>.b.<lb/>g.</var> per .25. vndecimi, Å¿ed per primam Å¿exti, vel .18. aut .19. Å¿eptimi ita eÅ¿t <var>.q.i.</var> ad <var>.i.g.</var> <lb/>vt <var>.b.q.</var> ad <var>.b.g.</var> </s> - <s xml:space="preserve">quare per .11. quinti <lb/>ita erit <var>.a.q.</var> ad <var>.a.g.</var> vt <var>.q.i.</var> ad <var>.i.g.</var> ideÅ¿t <lb/> - <anchor type="figure" xlink:label="fig-0112-01a" xlink:href="fig-0112-01"/> - vt <var>.a.b.</var> ad <var>.g.h.</var> Å¿ed vt eÅ¿t <var>.a.b.</var> ad <var>.g.h.</var> <lb/>Å¿ic eÅ¿t <var>.a.q.</var> ad <var>.k.d.</var> per .36. vndecimi, <lb/>Å¿eu per .11. octaui, vnde per .11. quin <lb/>ti Å¿ic erit <var>.a.q.</var> ad <var>.a.g.</var> vt ad <var>.k.d</var>. </s> - <s xml:space="preserve">Qua-<lb/>re per .9. eiuÅ¿dem <var>.a.g.</var> Ä™qualis erit <var>.k.<lb/>d</var>. </s> - <s xml:space="preserve">Vnde rectè erit accipere radicem <lb/>cubam <var>.a.g.</var> pro <reg norm="Å¿ecundo" type="context">Å¿ecÅ©do</reg> termino <var>.c.d.</var> <lb/>id, quod nobis inÅ¿eruit ad inueniendam tertiam partem vnius propoÅ¿itæ propor-<lb/>tionis.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0112-01" xlink:href="fig-0112-01a"> - <image file="0112-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0112-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="150"> - <head xml:space="preserve">THEOREMA <num value="150">CL</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">Sed</emph> vt Å¿peculatio iÅ¿ta ita vniuerÅ¿alis fiat vt ad <reg norm="oens" type="context">oẽs</reg> dignitates applicari poſſit; <lb/></s> - <s xml:space="preserve">Supponamus <var>.a.q.</var> et <var>.k.d.</var> eſſe duas dignitates quas volueris vnius, Å¿ed eiuÅ¿dem <lb/>Å¿peciei, et <var>.a.i.</var> dignitas præcedens dignitatem <var>.a.q.a.</var> cuius multiplicatione in <var>.a.b.</var> <lb/>eius radix producitur dignitas <var>.a.q.</var> & ab ipÅ¿ius <var>.a.i.</var> multiplicatione in <var>.g.h.</var> reÅ¿ultet <var>.a.<lb/>g.</var> vnde ex .18. vel .19. Å¿eptimi eadem proportio erit <var>.a.q.</var> ad <var>.a.g.</var> quæ <var>.a.b.</var> ad <var>.g.h.</var> Å¿ed <lb/>eadem etiam eÅ¿t <var>.a.q.</var> ad <var>.k.d.</var> ex ijs, quæ in .17. theoremare dixi, vnde ex .11. quinti, <lb/>ita erit <var>.a.q.</var> ad <var>.a.g.</var> vt ad <var>.k.d</var>. </s> - <s xml:space="preserve">Quapropter <var>.a.g.</var> æqualis erit <var>.k.d.</var> & ideo cum inuenta <lb/>fuerit radix huiuÅ¿modi dignitatis ex quantitate <var>.a.g.</var> habebimus <var>.c.d.</var> Å¿ecundum ter-<lb/>minum quæſitum.</s> - </p> - </div> - <div type="math:theorem" level="3" n="151"> - <head xml:space="preserve">THEOREMA <num value="151">CLI</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">Vnde</emph> verò fiat, quòd cum quis voluerit dimidium alicuius datæ proportio-<lb/>nis inuenire, rectè faciat, Å¿i accipiat radices quadratas illorum datorum rer-<lb/>minorum, etÅ¿i voluerit tertiam partem, accipiat radices cubas: </s> - <s xml:space="preserve">Å¿i autem quartam, <lb/>accipereradices cenÅ¿icas cenÅ¿icas ipÅ¿orum, & Å¿ic de Å¿ingulis in .17. </s> - <s xml:space="preserve">Theoremate om-<lb/>nia patent.</s> - </p> - </div> - <div type="math:theorem" level="3" n="152"> - <head xml:space="preserve">THEOREMA <num value="152">CLII</num>.</head> - <p> - <s xml:space="preserve"><emph style="sc">Vnde</emph> autem fiat, vt cum quis voluerit multiplicare aliquam proportionem <lb/>per fractos, rectè faciat prius multiplicando eam per numeratorem, dein-<lb/>de productum diuiÅ¿erit per denominationem ipÅ¿orum fractorum.</s> - </p> - <p> - <s xml:space="preserve">Vt exempli gratia, cum aliquis voluerit multiplicare proportionem Å¿eÅ¿quiquar-<lb/>tam per duo tertia, multiplicabit prius ipÅ¿am proportionem per numeratorem .2. <lb/>& productum, erit proportio .25. ad .16. qua poÅ¿tea diuiÅ¿a per .3. denominatorem, <lb/>prouentus erit proportio radicis cubæ .25. ad radicem cubam .16. vel vt proportio.</s> - <pb o="101" rhead="THEOREM. ARIT." n="113" file="0113" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0113"/> - <s xml:space="preserve">25. ad radicem cubam .10000. quæ quidem proportiones æquales inuicem Å¿unt, cu <lb/>tam vna, quà m alia, Å¿it tertia pars totius.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius ratione cogitem is <var>.a.b.</var> eſſe aliquod totum, quod multiplicare cupimus <lb/>per duas tertias, quod <reg norm="quidem" type="context">quidẽ</reg> nihil aliud eÅ¿t, quà m accipere duas tertias partes vnius <lb/>totius Å¿uperficialis, imaginemur igitur hoc totum <var>.a.b.</var> lineare diuiÅ¿um eſſe in tertias <lb/>partes mediantibus <var>.e.</var> et <var>.d</var>. </s> - <s xml:space="preserve">& tunc multiplicando ipÅ¿um per 2. tertias lineares produ-<lb/>ctum erit <var>.a.c.</var> Å¿ex vnitatum Å¿uperficialium, quod quidem productum poÅ¿teà diuiÅ¿um <lb/>per .3. dabit <var>.d.c.</var> hoc eÅ¿t duas tertias Å¿uperficiales (quæ eÅ¿t tertia pars ipÅ¿ius <var>.a.c.</var>) & <lb/>Ä™quales numero <var>.c.b.</var> duabus vnitatibus linearibus, ideÅ¿t duabus tertijs ipÅ¿ius <var>.a.b</var>. </s> - <s xml:space="preserve">No <lb/>tandum etiam eÅ¿t, quòd cum ferè omnia reducantur ad regulam de tribus, proptereà <lb/>etiam multiplicatio alicuius quantitatis per aliam quantitatem, nihil aliud eÅ¿t quà m <lb/>quædam operatio ipÅ¿ius regulæ de tribus, vt eyempli gratia volo multiplicare .25. <lb/>per 20. hoc nihil aliud eÅ¿t niÅ¿i quærere alium numerum ita proportionatum ad .25. <lb/>vt 20. Å¿e habetad vnum, vnde multiplicando .25. cum .20. & productum diuidendo <lb/>per vnum exregula de tribus, prouentus eÅ¿t idem numerus ipÅ¿ius producti, & propte <lb/>rea cum volumus multiplicare aliquem numerum per fractos hoc nihil aliud eÅ¿t <lb/>quà m quærere aliquem numerum ita proportionatum ad ipÅ¿um numerum datum, <lb/>vt Å¿e habet numerator ad denominatorem, exempli gratia Å¿i .24. aliquis voluerit mul <lb/>tiplicare per duo tertia hoc idem eÅ¿t vt Å¿i quæreret numerum ad quem .24. ita Å¿e <lb/>habeat, vt .3. ad .2. & idem dico de proportionibus, hoc eÅ¿t quod aliud non eÅ¿t mulri-<lb/>plicare aliquam proportionem per fractos, quà m aliam proportionem quærere ad <lb/><reg norm="quam" type="context">quã</reg> data Å¿e habeat, vt denominator Å¿e <reg norm="hent" type="context">hẽt</reg> ad <reg norm="numeratorem" type="context">numeratorẽ</reg>; </s> - <s xml:space="preserve">& hoc exregula de tribus <lb/>perficitur, <reg norm="conÅ¿tituendo" type="context context">cõſtituẽdo</reg> <reg norm="denominatorem" type="context">denominatorẽ</reg> in primo loco, quilocus eÅ¿t diuiÅ¿oris, numerato <lb/><reg norm="rem" type="context">rẽ</reg> verò in <reg norm="Å¿ecundo" type="context">Å¿ecÅ©do</reg> loco, <reg norm="multiplicando" type="context">multiplicãdo</reg> poÅ¿teà pro <lb/>portionem per <reg norm="numeratorem" type="context">numeratorẽ</reg>, & <reg norm="productum" type="context">productÅ©</reg> <reg norm="diuidem" type="context">diuidẽ</reg> <lb/> - <anchor type="figure" xlink:label="fig-0113-01a" xlink:href="fig-0113-01"/> - do per denominatorem, prouentus demum erit <lb/>proportio, ad quam data Å¿e habebit, vt denomi-<lb/>nator Å¿e <reg norm="hent" type="context">hẽt</reg> ad numeratorem ex ratione ipÅ¿ius re <lb/>gulÄ™ de tribus. </s> - <s xml:space="preserve">Ratio verò methodi <reg norm="diuidendi" type="context">diuidẽdi</reg> <reg norm="vnam" type="context">vnã</reg> <lb/>datam <reg norm="proportionem" type="context">proportionẽ</reg> per fractos, ex Å¿e Å¿atis patet, <lb/>cum idem Å¿it modus diuidendi quemhbet nume <lb/>rum integrum per fractos. </s> - <s xml:space="preserve">Quare, quæ vnius, <lb/>& alterius eÅ¿t ratio.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0113-01" xlink:href="fig-0113-01a"> - <image file="0113-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0113-01"/> - </figure> - </div> - </div> - <div type="math:theorem" level="3" n="153"> - <head xml:space="preserve">THEOREMA <num value="153">CLIII</num>.</head> - <p> - <s xml:space="preserve">NIcolaus Tartalea in .3. lib. quintæ partis numerorum Å¿oluit .24. quæſitum Å¿i-<lb/>bi propoÅ¿itum à Hieronymo Cardano, via particulari & non generali. </s> - <s xml:space="preserve">Quæ-<lb/>Å¿itum autem tale eÅ¿t quamlibet propoÅ¿itam rectam lineam in duas partes ita diuide <lb/>re via Euclidis, ut cubus totius lineæ ad cubos partium Å¿e habeat in proportione <lb/>tripla.</s> - </p> - <p> - <s xml:space="preserve">Tartalea igitur inquit quòd vt Å¿atisfiat Å¿peculatiuis ingenijs Å¿oluendum Å¿it huiuÅ¿-<lb/>modi quæſitum, Å¿ecando lineam propoÅ¿itam <var>.a.b.</var> in tres æquales partes, quarum vna <lb/>fit <var>.c.b.</var> vnde problema Å¿olutum erit.</s> - </p> - <p> - <s xml:space="preserve">Verum dicit, Å¿ed hæc non eÅ¿t methodus generalis, proptereà , quod cum tale <lb/>problema alterius fuiÅ¿let proportionis quam triplæ, talis methodus nihil valeret.</s> - <pb o="102" rhead="IO. BAPT. BENED." n="114" file="0114" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0114"/> - <s xml:space="preserve">Quapropter non tacebo quod mihi in mentem venit circa hoc problema.</s> - </p> - <p> - <s xml:space="preserve">Sit ergo linea <var>.a.b.</var> diuiÅ¿ibilis in puncto <var>.c.</var> ita vt cubum totius dictæ <var>.a.b.</var> lineæ ad <lb/>Å¿ummam cuborum <reg norm="Å¿uarum" type="context">Å¿uarÅ©</reg> partium <var>.a.c.</var> et <var>.c.b.</var> oporteat eam proportionem <reg norm="habere" type="conjecture">habére</reg>, <lb/>exempli gratia, vt .125. ad .65. vt vitemus fracta pro nunc, notantes talem propor-<lb/>tionem quadrupla nunquam maiorem eſſe poſſe, vt quilibet ex Å¿e contemplari po-<lb/>teÅ¿t, conÅ¿tituendo punctum <var>.c.</var> in medio loco inter <var>.a.</var> et <var>.b.</var> vnde proportio totalis <lb/>cubi ad Å¿ummam partialium eſſet omnium maxima quæ poſſint eſſe, collocando <var>.c.</var> <lb/>vbi volueris in dicta linea <var>.a.b.</var> & hæc eſſet quadrupla.</s> - </p> - <p> - <s xml:space="preserve">Sed vt ad propoÅ¿itum reuertamur, conÅ¿iderabimus cubum totalem ipÅ¿ius <var>.a.b.</var> <lb/>eſſe vt .125. & Å¿ummam partialium vt .65. quam detrahemus ex cubo totali & nobis <lb/>remanebit .60. pro Å¿umma trium Å¿olidorum inuicem æqualium, quorum longitu-<lb/>do vniuÅ¿cuiuÅ¿que erit tota linea <var>.a.b.</var> nobis cognita vt radix dati cubi totalis, quæ erit <lb/>in hoc exemplo quinque partium, latitudo verò vniuÅ¿cuiuÅ¿que dictorum <reg norm="Å¿olidorum" type="context">Å¿olidorÅ©</reg> <lb/>erit <var>.a.c.</var> pars maior ipÅ¿ius <var>.a.b.</var> quæ quidem <var>.a.c.</var> adhuc nobis ignota eÅ¿t, profunditas <lb/>Å¿eu altitudo vniuÅ¿cuiuÅ¿que illorum Å¿olidorum, erit <var>.c.b.</var> pars reliqua ipÅ¿ius <var>.a.b.</var> & <reg norm="etiam" type="context">etiã</reg> <lb/>nobis incognita, Å¿ed quia Å¿umma horum trium Å¿olidorum nobis manifeÅ¿ta Å¿uperius <lb/>fuit, quæ erat .60. propterà nobis cognita erit quantitas vniuÅ¿cuiuÅ¿que illorum Å¿oli-<lb/>dorum, vt tertia pars totius Å¿ummæ ipÅ¿orum quæ erit .20. in propoÅ¿ito <reg norm="exemplo" type="context">exẽplo</reg>, dein <lb/>de cum vnumquodque illorum Å¿olidorum producatur à ſuperficie contenta Å¿eu pro<lb/> ducta ab <var>.c.a.</var> in <var>.c.b.</var> in tota linea <var>.a.b.</var> Å¿equitur quòd Å¿i diuiÅ¿erimus hoc Å¿olidum .20. <lb/>per lineam <var>.a.b.</var> quinque partium proueniet nobis cognita Å¿uperficies producta ab <var>.<lb/>a.c.</var> in <var>.c.b.</var> quatuor partium, Å¿ed cum quadratum totius <var>.a.b.</var> nobis cognitum Å¿it, eo <lb/>quod <var>.a.b.</var> vt eius latus etiam cognitum eÅ¿t. </s> - <s xml:space="preserve">Tunc dictum quadratum erit .25. quod <lb/>quidem æquale eÅ¿t quadruplo illius quod fit ex <var>.a.c.</var> in <var>.c.b.</var> Å¿imul cum quadrato diffe <lb/>rentiæ inter <var>.a.c.</var> et <var>.c.b.</var> per .8. Å¿ecundi Eucli. </s> - <s xml:space="preserve">Vnde quia quadruplum illius quod fit <lb/>ex <var>.a.c.</var> in <var>.c.b.</var> nobis cognitum eÅ¿t, vt <lb/>16. eo quod Å¿implum quod eÅ¿t .4. <reg norm="iam" type="context">iã</reg> <lb/> - <anchor type="figure" xlink:label="fig-0114-01a" xlink:href="fig-0114-01"/> - inuentum fuit, ideo Å¿i hoc quadru-<lb/>plum .16. demptum fuerit ex totali <lb/>quadrato .25. reliquum erit .9. qua <lb/><reg norm="dratum" type="context">dratÅ©</reg> Å¿cilicet vnius partis <var>.a.c.</var> ipÅ¿ius <lb/>hoc eÅ¿t illius partis, quæ differentia <lb/>eÅ¿t inter <var>a.c.</var> et <var>.c.b.</var> quæ quidem erit <num value="3">.<lb/>3.</num> partium quæ differentia cum Å¿ub-<lb/>tracta fuerit ex <var>.a.b.</var> reliquum erit du <lb/>plum ipÅ¿ius <var>.c.b.</var> duo Å¿cilicet. </s> - <s xml:space="preserve">Quare <var>.<lb/>c.b.</var> erit vt <var>.I.</var> et <var>.a.c.</var> vt .4. & productum <var>.a.c.</var> in <var>.c.b.</var> erit .4. vnitatum Å¿uperficialium. <lb/></s> - <s xml:space="preserve">& c.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0114-01" xlink:href="fig-0114-01a"> - <image file="0114-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0114-01"/> - </figure> - </div> - <pb o="103" n="115" file="0115" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0115"/> - </div> - <div type="appendix" level="3" n="1"> - <head xml:space="preserve">APPENDIX</head> - <head xml:space="preserve">DE SPECVLATIONE <lb/>REGVLAE FALSI.</head> - <p> - <s xml:space="preserve"><emph style="sc">Nvnc</emph> idem ferè mihi accidit, quod & Michaeli Stifelio, à quo <lb/>cum Petreius Tipographus nuper totam Å¿uam Arithmeticam re <lb/>cepiſſet, mox poÅ¿teà per literas petijt <reg norm="explicationem" type="context">explicationẽ</reg> regulæ falÅ¿i.</s> - </p> - <p> - <s xml:space="preserve">Similiter poÅ¿t inciÅ¿as omnes Å¿uperiorum Theorematum figu-<lb/>ras, <reg norm="opereque" type="simple">opereqÌ;</reg> Typographo commiſſo, amicus quidam omnium <lb/>Å¿cientiarum ornatiſſimus maxima neceſſitudine mecum coniun-<lb/>ctus monuit me, vt aliquid de regula falÅ¿i Å¿cribere vellem, cuius <lb/>Å¿uaÅ¿u hæc, quæ Å¿equuntur appendicis vice ponere libuit, nelector, quidpiam quod <lb/>ad hancrem pertinet iure merito à nobis deÅ¿iderare poſſet; </s> - <s xml:space="preserve">vt autem ad ipÅ¿am <reg norm="re- gulam" type="context">re-<lb/>gulã</reg> accedamus Ego Å¿icut, & in alijs multis, ita & in huiuÅ¿cæ regulÄ™ inuentione cum <lb/>ipÅ¿o Stifelio maximè conuenio, putans regulam falÅ¿i, Å¿eu falÅ¿arum poÅ¿itionum in-<lb/>uentam fuiſſe per paruos numeros in quæſtionibus facillimis & cognitis, eodem fer <lb/>mè modo, quo ipÅ¿e monÅ¿trat illis duobus exemplis, quæ quamuis ipÅ¿e appellet theo <lb/>remata, nihilominus the oremata ego illa non vocarem, niÅ¿i adiuncta fuerit Å¿pecu-<lb/>latio ab ipÅ¿o præterita, & non experientia tantummodo, vt ipÅ¿e fecit. </s> - <s xml:space="preserve">Primum eius <lb/>exemplum eÅ¿t, quòd.</s> - </p> - <p> - <s xml:space="preserve">Quorumcumque <reg norm="duorum" type="context">duorÅ©</reg> numerorum differentia, Å¿i fuerit multiplicata in aggre <lb/>gatum eorum, producit ipÅ¿am differentiam, quæ eÅ¿t inter quadrata eorum.</s> - </p> - <p> - <s xml:space="preserve">Secundum verò exemplum eÅ¿t, quod.</s> - </p> - <p> - <s xml:space="preserve">Datis tribus numeris Å¿ecundum progreſſionem arithmeticam diÅ¿poÅ¿itis, facit mul <lb/>tiplicatio medij in Å¿e, <reg norm="quantum" type="context">quãtum</reg> multiplicatio extremorum inter Å¿e cum multiplicatio <lb/>ne differentiarum inter Å¿e.</s> - </p> - <p> - <s xml:space="preserve">Talia enim exempla ipÅ¿e aliter non probat niÅ¿i experientia in aliquibus numeris, <lb/>arbitratus ex eo inuentam eſſe regulam falÅ¿i, experientia tantummodo confirma-<lb/>tam, quod quidem etiam & ego credo. </s> - <s xml:space="preserve">At experientia in philoÅ¿ophia mathema-<lb/>tica, aut <reg norm="nullam" type="context">nullã</reg> prorÅ¿us facit <reg norm="Å¿cientiam" type="context">Å¿cientiã</reg>, aut omnino Å¿uperfluus fuit Euclides in multis <lb/>Å¿uis propoÅ¿itionibus, & præcipuè in eius Å¿ecundo libro, Å¿i Å¿ufficeret experientia. </s> - <s xml:space="preserve">Id-<lb/>circo quo magis ad euidentiam ipÅ¿ius veritatis, quam profiteor, deuenire poſſim, <lb/><reg norm="accipiam" type="context">accipiã</reg> primò primum exemplum <lb/>ipÅ¿ius Stifelij hic Å¿uperius citatum, <lb/>& pro numero maiori, in prima hic <lb/> - <anchor type="figure" xlink:label="fig-0115-01a" xlink:href="fig-0115-01"/> - Å¿ubÅ¿cripta figura .AE. accipio <var>.a.i.</var> <lb/>cuius quadratum Å¿it <var>.a.c</var>: pro minori <lb/>vero numero capio <var>.a.e.</var> <reg norm="partem" type="context">partẽ</reg> ipÅ¿ius <lb/><var>a.i.</var> cuius quadratum fit <var>.a.t.</var> differen <lb/>tia autem horum numerorum erit <var>.<lb/>e.i.</var> reliqua pars ipÅ¿ius <var>.a.i</var>: & differen <lb/>tia ipÅ¿orum quadratorum erit gno-<lb/>mon <var>.e.c.o</var>: Nunc autem protraho <var>.<lb/>i.c.</var> latus quadrati maioris quouÅ¿que <lb/><var>c.n.</var> æqualis Å¿it <var>.a.e.</var> numero minori, <lb/><reg norm="perficioque" type="simple">perficioqÌ;</reg> rectangulum <var>.e.n.</var> quod - <pb o="104" rhead="IO. BAPT. BENED." n="116" file="0116" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0116"/> - producitur ex <var>.i.e.</var> differentia in <var>.i.n.</var> aggregatum amborum numerorum, Å¿ed hoc pro <lb/>ductum excedit productum <var>e.c</var>: partem gnomonis dicti per <var>.u.n.</var> quod quidem <var>.u.<lb/>n.</var> æquatur ipÅ¿i <var>.u.o.</var> reliquæ Å¿cilicet parti ipÅ¿ius gnomonis, <reg norm="nam" type="context">nã</reg> <var>.e.u.</var> æqualis eft <var>.i.c.</var> qua <lb/>re et <var>.a.i.</var> Å¿ed <var>.e.t.</var> Ä™quatur <var>.e.a.</var> vnde <var>.t.u.</var> æqualis erit <var>.e.i.</var> </s> - <s xml:space="preserve">quare et <var>.u.c</var>: at cum <var>.c.n.</var> æqua <lb/>lis Å¿it ipÅ¿i <var>.a.e.</var> erit etiam æqualis ipÅ¿i <var>.<lb/>o.t</var>. </s> - <s xml:space="preserve">quare <var>.u.n.</var> æqualis erit ipÅ¿i <var>.u.o.</var> <lb/>& tunc intellectus quieÅ¿cit, & <reg norm="abÅ¿que" type="simple">abÅ¿q;</reg> <lb/> - <anchor type="figure" xlink:label="fig-0116-01a" xlink:href="fig-0116-01"/> - aliqua alia experientia verè Å¿cientifi <lb/><reg norm="ceque" type="simple">ceqÌ;</reg> dicere poteft, quòd.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0115-01" xlink:href="fig-0115-01a"> - <image file="0115-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0115-01"/> - </figure> - <figure xlink:label="fig-0116-01" xlink:href="fig-0116-01a"> - <image file="0116-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0116-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Quorumcumque duorum nume-<lb/>rorum differentia, fi fuerit multipli-<lb/>cata in aggregatum eorum, producit <lb/>ipÅ¿am <reg norm="differentiam" type="context">differentiã</reg>, quæ eftinter qua-<lb/>drata eorum.</s> - </p> - <p> - <s xml:space="preserve">Hæcautem propoÅ¿itio à me ipÅ¿o <lb/>etiam in .60. </s> - <s xml:space="preserve">Theoremate huius libri <lb/>aliter demonftrata fuit.</s> - </p> - <p> - <s xml:space="preserve">DE Å¿peculatione autem, etÅ¿cientia Å¿ecundi exempli, in Å¿ecunda hic Å¿ubÅ¿cripta <lb/>figura <var>.ω.</var> cogitemus lineam <var>.u.a.</var> tribusin partibus arithmeticè diuiÅ¿am, qua <lb/>rum maxima Å¿it <var>.u.o.</var> media. Å¿it <var>.o.e.</var> minima verò Å¿it <var>.e.a.</var> multiplicatio autem mediæ <var>.<lb/>o.e.</var> in Å¿e Å¿it quadratum <var>.o.t.</var> abÅ¿cindatur deinde ex <var>.o.e</var>: <var>e.i.</var> æqualis <var>.e.a.</var> </s> - <s xml:space="preserve">tunc <var>.o.i.</var> erit <lb/>differentia inter <var>.o.e.</var> et <var>.e.a.</var> & æqualis differentiæ inter <var>.o.e.</var> et <var>.o.u.</var> ex hypotefi, quæ <lb/>quidem <var>.o.i.</var> in Å¿e ducta procreabit quadratum <var>.o.c.</var> quod erit productum ex differen <lb/>tijs ipÅ¿arum partium, & erit pars quadrati <var>.o.t.</var> Å¿uperius dicti, vt exÅ¿e patet. </s> - <s xml:space="preserve">Nunc <lb/>autem dico gnomonem <var>.i.t.n.</var> æqualem eſſe ei quod fit ex <var>.a.e.</var> in <var>.o.u</var>. </s> - <s xml:space="preserve">Producatur igi <lb/>tur <var>.e.t.</var> quouÅ¿que <var>.t.r.</var> æqualis Å¿it ipÅ¿i <var>.o.i</var>. </s> - <s xml:space="preserve">tunc <var>.e.r.</var> erit æqualis <var>.o.u.</var> quod etiam clarum <lb/>eÅ¿t. </s> - <s xml:space="preserve">Claudatur ergo rectangulum <var>.i.r.</var> quod erit æquale producto ipÅ¿ius <var>.e.a.</var> in <var>.o.u.</var> <lb/>Nam <var>.e.i.</var> Å¿umpta fuit <lb/>æqualis <var>.e.a.</var> Å¿ed ex ra <lb/> - <anchor type="figure" xlink:label="fig-0116-02a" xlink:href="fig-0116-02"/> - tionibus in priori <reg norm="exem" type="context">exẽ</reg> <lb/>plo allatis, <reg norm="productum" type="simple">ê“ductum</reg> <var>.<lb/>i.r.</var> æquale erit gno-<lb/>moni <var>.i.t.n</var>. </s> - <s xml:space="preserve">Nuncau <lb/>tem verè, Å¿cientifice-<lb/>qÌue poſſumus affirma <lb/>re, quòd. </s> - <s xml:space="preserve">Datis tribus <lb/>numeris <reg norm="Å¿ecundum" type="context">Å¿ecundÅ©</reg> pro <lb/>greffionem arithme-<lb/>ticam diÅ¿pofitis, fa-<lb/>cit multiplicatio me-<lb/>dij in Å¿e quantum mul <lb/>tiplicatio extremorum inter Å¿e, cum multiplicatione differentiarum inter Å¿e.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0116-02" xlink:href="fig-0116-02a"> - <image file="0116-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0116-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Et Å¿ic de alijs huiuÅ¿modi inuentionibus infero.</s> - </p> - <p> - <s xml:space="preserve">DIcturus igitur aliquid circa <reg norm="regulam" type="context">regulã</reg> falÅ¿i, videtur mihi nullam oportere facere <lb/>mentionem de origine huiuÅ¿cæ regulæ, cum in hoc Stifelius Å¿atisfecerit, Å¿ed - <pb o="105" rhead="THEOREM. ARITH." n="117" file="0117" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0117"/> - potius veras rationes <reg norm="propriaque" type="simple">propriaqÌ;</reg> fundamenta huiuÅ¿modi operationis oftendere, fu-<lb/>mendo eadem exempla propoÅ¿ita abipÅ¿is practicis, & maximè à Nicolao Tartalea <lb/>viro accuratiffimo, qui vbicunque potuit Å¿peculatus eÅ¿t cauiÅ¿as <reg norm="ipÅ¿arum" type="context">ipÅ¿arÅ©</reg> operationum, <lb/>etÅ¿i de huiuÅ¿modi falÅ¿i regula circa finem cap .8. lib. 17. promittat poÅ¿tea loqui, nub-<lb/>libi tamen loquutus eft. </s> - <s xml:space="preserve">Monendum etiam cenÅ¿eo, me nihil de rationibus regulæ <lb/>falÅ¿i Å¿implicis dicturum, cum ex Å¿eipÅ¿is Å¿atis appareant, quod non ita eÅ¿t de poÅ¿itio-<lb/>nibus duplis. </s> - <s xml:space="preserve">Incipiam ergo à primo problemate lib. 17. ipÅ¿ius Tartaleæ, quo <reg norm="etiam" type="context">etiã</reg> <lb/>ipÅ¿e vtitur pro exemplo docendi gratia, ipÅ¿am regulam duplæ poÅ¿itionis, quod qui <lb/>dem problema aliter à me <reg norm="Å¿olutum" type="context">Å¿olutÅ©</reg> fuit in .118. </s> - <s xml:space="preserve">Theoremate huius mei lib. quod Å¿imi <lb/>liter ob hanc demum occaÅ¿ionem mihi oblatam, alia etiam via, Å¿peculatus Å¿umidem <lb/>poſſe fieri, quæ quidem via Å¿eu methodus generalis erit, & ita Å¿e habet.</s> - </p> - <p> - <s xml:space="preserve">Accipio enim propoÅ¿itum numerum diuiÅ¿ibilem, à quo detraho Å¿ummam <lb/>datorum numerorum, primo duplicato, eo quòd tam in Å¿ecunda quam in <lb/>tertia parte reperitur, vt in propofito exemplo, datus numerus eft, 50. à <lb/>quo detraho Å¿ummam dictorum numerorum, quæ eÅ¿t .11. nam tres, & tres, & <lb/>quinque Å¿unt vndecim, eo quòd primus ingreditur in Å¿ecunda, & in tertia parte, <lb/>dempto igitur hoc numero .11. ex .50. remanet .39. qui quidem numerus intelligen-<lb/>dus eÅ¿t pro Å¿umma trium partium Å¿implicium adhuc incognitarum, à quo extrahen <lb/>da eÅ¿t prima, eo modo quo nunc proponam exregula de tribus, hoc eÅ¿t aggregan <lb/>do dictas partes Å¿implices Å¿ine aliqua additione vtcunque volueris (Å¿ed commodius <lb/>erit in minimis numeris) iuxta propoÅ¿itum, quod quidem propoÅ¿itum eÅ¿t, vt Å¿ecun <lb/>da pars dupla Å¿it primæ, tertia verò æqualis fit primæ & Å¿ecundæ, quæ partes in di-<lb/>ctis minimis numeris, ita diÅ¿poÅ¿itæ erunt .1. 2. 3. quarum Å¿umma erit .6. </s> - <s xml:space="preserve">Nunc Å¿i ex <lb/>regula de tribus dixerimus, cum hæc Å¿umma proueniat nobis ab vno, à quo proue-<lb/>niet .39. et veniet nobis .6. cum dimidio pro prima parte quæfita in propoÅ¿ito nume-<lb/>ro .39. cum ergo habuerimus primam <reg norm="partem" type="context">partẽ</reg>, reliquas poÅ¿teà illicò cognoÅ¿cemus.</s> - </p> - <p> - <s xml:space="preserve">HuiuÅ¿modi verò operationis ratio ex Å¿e manifeÅ¿ta patet, eo quòd proportio Å¿um <lb/>mæ partium in minimis numeris ad primam eorum partem eadem eſſe debet, quæ <lb/>ipÅ¿ius .39. ad primam partem quæſitam huiuÅ¿modi aggregati partium <reg norm="Å¿implicium" type="context">Å¿impliciÅ©</reg>, Å¿ed <lb/>quia nemo adhuc, quod Å¿ciam, Å¿atis animaduertit rationem modorum, qui ab anti-<lb/>quis obÅ¿eruati Å¿unt, qui quidem modi duo Å¿unt circa hoc Helcataym duplæ falſæ <lb/>pofitionis, igitur non prætermittam aliquid de hacreÅ¿peculari, & primo de pri-<lb/>mo modo.</s> - </p> - <p> - <s xml:space="preserve">In primis igitur <reg norm="Å¿ciendum" type="context">Å¿ciendÅ©</reg> eft, <reg norm="quod" type="simple">ê™</reg> <lb/> - <anchor type="figure" xlink:label="fig-0117-01a" xlink:href="fig-0117-01"/> - veritas ita inueniri poterit eo-<lb/>rum modo, me diantibus Å¿impliÂ<lb/>cibus partibus, vt <reg norm="etiam" type="context">etiã</reg> <reg norm="median- tibus" type="simple">median-<lb/>tibê°</reg> <reg norm="compoÅ¿itis" type="context">cõpoÅ¿itis</reg>, ut in pręſenti <reg norm="exem" type="context">exẽ</reg> <lb/>plo pro primis pofitionibus ac-<lb/>ceperunt .10. et .8. pro Å¿ecundis <lb/>verò compoÅ¿itis <reg norm="cum" type="context">cÅ©</reg> numero .3. <lb/><reg norm="inuenerunt" type="context">inuenerÅ©t</reg> .23. et .19. pro tertijs <lb/><reg norm="autem" type="wordlist">aÅ©t</reg> <reg norm="compoÅ¿itis" type="context">cõpoÅ¿itis</reg> <reg norm="cum" type="context">cÅ©</reg> <reg norm="quinque" type="simple">quinq;</reg>, notaue <lb/>runt .38. et .32. vnde prima Å¿um <lb/>marefultauit .71. Å¿ecunda verò <lb/>59. ita <reg norm="quod" type="simple">ê™</reg> <reg norm="primus" type="simple">primê°</reg> error remanebat <lb/>21. <reg norm="Å¿ecundus" type="context simple">Å¿ecÅ©dê°</reg> <reg norm="autem" type="wordlist">aÅ©t</reg> .9. vt in figura <var>.A</var>.</s> - </p> - <div type="float" level="4" n="3"> - <figure xlink:label="fig-0117-01" xlink:href="fig-0117-01a"> - <image file="0117-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0117-01"/> - <caption xml:space="preserve">Compositorum</caption> - </figure> - </div> - <pb o="106" rhead="IO. BAPT. BENED." n="118" file="0118" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0118"/> - <p> - <s xml:space="preserve">SEDijdem errores proueniunt exÅ¿ummis partium Å¿implicium.</s> - </p> - <p> - <s xml:space="preserve">Vtexempli gratia, in figura <var>.B.</var> Å¿umma propoÅ¿ita partium Å¿implicium eÅ¿t .39. <lb/>vt diximus, eo quòd ab ipÅ¿o .50. detraxerimus .11. Å¿umma Å¿cilicet numerorum adij <lb/>ciendorum ad efficiendas partes compofitas, Å¿umma poÅ¿teà fimplicium partium <lb/>primæ poÅ¿itionis, erit .60. eo quòd prima pars erat .10. Å¿ecunda autem Å¿implex 20. <lb/>tertia verò fimplex .30. iuxta ordinem propoÅ¿iti. </s> - <s xml:space="preserve">Summa deinde Å¿implicium <reg norm="partium" type="context">partiÅ©</reg> <lb/>fecundæ poÅ¿itionis effet .48. quia prima eius pars erat .8. Å¿ecunda verò Å¿implex .16. <lb/>tertia autem Å¿implex .24. vnde prima Å¿umma excederet datam .39. per .21. differen-<lb/>tiæ, Å¿ecunda verò per .9. vt Å¿upra vidimus de Å¿ummis compoÅ¿itis à dato .50. compo-<lb/>fito, & hoc quidem mirandum non eft, quod Å¿cilicet tres Å¿ummæ fimplicium par-<lb/>tium Å¿intinuicem inæqua-<lb/>les, ijÅ¿dem differentijs me-<lb/> - <anchor type="figure" xlink:label="fig-0118-01a" xlink:href="fig-0118-01"/> - diantibus, quibus <reg norm="differunt" type="context">differÅ©t</reg> <lb/>dictæ tres Å¿ummæ compofi <lb/>tæ, cum ab vnaquaque <reg norm="con" type="context">cõ</reg> <lb/><reg norm="poÅ¿itarum" type="context">poÅ¿itarÅ©</reg> ablatus fit nume-<lb/>rus .11. æqualiter, vnde ex <lb/>neceſſitate, permutando, <lb/><reg norm="earum" type="context">earÅ©</reg> differentiæ <reg norm="relinquem" type="context">relinquẽ</reg> <lb/>dæ erant æquales inuicem <lb/>ex <ref id="ref-0017">.78. theoremate hu-<lb/>ius noÅ¿tri lib.</ref> Å¿ummæ enim <lb/>compofitæ erant .71. 59. et <lb/>50. fimplices verò .60. 48. <lb/>et .39. differentes à primis <lb/>per .11. vt dictum eft, qua <lb/>re veritas ita manabit à compofitis, quemadmodum à fimplicibus, Å¿ed à fimplici-<lb/>bus per Å¿e, & a compofitis per accidens vtiam iam videbimus.</s> - </p> - <div type="float" level="4" n="4"> - <figure xlink:label="fig-0118-01" xlink:href="fig-0118-01a"> - <image file="0118-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0118-01"/> - <caption xml:space="preserve">Simpricium</caption> - </figure> - </div> - <p> - <s xml:space="preserve">ANtiquorumigitur primus m odus vtitur regula detribus, hocordine, multi-<lb/>plicando Å¿cilicet Å¿ecundum errorem, qui eft .9. cum differentia primarum par <lb/>tium pofitarum, quæ eft .2. & productum diuidendo per differentiam errorum, quæ <lb/>eft .12. proueniens poftea quod eft .1. cum dimidio additur hoc loco primæ parti Å¿e-<lb/>cundæ poÅ¿itionis. <reg norm="&c." type="unresolved">&c.</reg> quòd benè Å¿e habet. </s> - <s xml:space="preserve">Vbi animaduertendum eÅ¿t, quod ille <lb/>numerus .12. non eft accipiendus per Å¿e vt differentia errorum hoc eft .21. et .9. nifi <lb/>perà ccidens, fed benè perfe, vt <reg norm="differentia" type="context">differẽtia</reg> inter .60. er .48. Å¿implices Å¿ummas, quem <lb/>admodum .9. in hoc propoÅ¿ito eft differentia per Å¿e inter .48. et .39 per accidens ve-<lb/>ro inter .59. et .50.</s> - </p> - <p> - <s xml:space="preserve">CognoÅ¿cendum igitur eft mediante .24. quinti Eucli. quod eadem proportio <lb/>eft primæ Å¿ummæ (Å¿implicium dico) ad Å¿uam primam partem, quæ Å¿ecundæ Å¿um-<lb/>mæ ad Å¿uam, & tertiæ Å¿ummæ ad fuam fimiliter (vbi rectè etiam feciffent hoc in lo-<lb/>co antiqui Å¿i multiplicauiffent tertiam Å¿ummam fim plicem cum prima parte prioris <lb/>fummæ fimplicis, & productum diuififfent per primam Å¿ummam, vnde prima pars <lb/>quæſita tertiæ Å¿ummæ orta fuiffet, abÅ¿que ullo negotio ipfius plus velminus) </s> - <s xml:space="preserve">Quare <lb/>habebimus tres terminos antecedentes ab vna parte, & tres terminos conÅ¿equen-<lb/>tesab alia parte continentes vnam <reg norm="eandemque" type="simple">eandemqÌ;</reg> proportionem, vnde ex .19. quinti, <lb/>vel .12. Å¿eptimi eorum differentiæ proportionales erunt, hoc eft, <reg norm="quod" type="simple">ê™</reg> eadem propor - <pb o="107" rhead="THEOREM. ARIT." n="119" file="0119" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0119"/> - tio erit eius differentiæ, quæ eÅ¿t inter primam & fecundam Å¿ummam, ad differen-<lb/>tiam quæ eÅ¿t inter primas earum partes, quæ illius differentiæ, quæ eÅ¿t inter Å¿ecun-<lb/>dam & tertiam Å¿ummam, ad differentiam, quæ eft inter primas illarum partes, Å¿ed <lb/>harum .4. differentiarum, tres nobis cognitæ Å¿unt, ideft .12. 2. et .9. ergo ex regula de <lb/>tribus ab Eucli. in .20. Å¿eptì<unsure/>mi Å¿peculata inueniebatur quarta differentia, quæ eft .1. <lb/>cum dimidio.</s> - </p> - <p> - <s xml:space="preserve">A compofitis Å¿ummis idem etiam proueniet, Å¿ed non vt ex proprijs caufis, & per <lb/>Å¿e, Å¿edper accidens. </s> - <s xml:space="preserve">Nam quamuis eadem differentia fit inter 71. et .59. quæ in-<lb/>ter .60. et .48. & <reg norm="eadem" type="context">eadẽ</reg> inter .59. et .50. quæ inter .48. et .39. </s> - <s xml:space="preserve">Nihilominus non eft <reg norm="eadem" type="context">eadẽ</reg> <lb/>proportio (propriè) ipÅ¿ius .71. ad .59. quæ ipÅ¿ius .60. ad .48. nec ea quæ ipÅ¿ius .59. ad <num value="50">.<lb/>50.</num> eft quæ ipÅ¿ius .48. ad .39: </s> - <s xml:space="preserve">Vnde non erit eadem proportio ipÅ¿ius .71. ad .59. quæ <lb/>ipfius .10. ad .8. ne@ea quæ eft ipfius .59. ad .50. quæ ipÅ¿ius .8. ad .6. cum dimidio. </s> - <s xml:space="preserve">Sed <lb/>minores illis. </s> - <s xml:space="preserve">Nam ex æqualibus additamentis diminuuntur proportiones maio-<lb/>ris inÄ™qualitatis.</s> - </p> - <p> - <s xml:space="preserve">A fimplicibus igitur Å¿ummis pendet ratio huiuÅ¿modi effectus.</s> - </p> - <p> - <s xml:space="preserve">Si vero prima pars fecundæ poÅ¿itionis effet .4. tunc Å¿ecunda eius pars effet .8. & ter-<lb/>tia .12. quarum Å¿umma effet .24. (harum fimplicium partium Å¿eilicet) & minor vera <lb/>(39.) per .15. & differens à ſumma primarum. (60.) per .36. & differentia primarum <lb/>partium effet .6. differentia vero primæpartis Å¿ecundæ poÅ¿itionis, a prima parte quÄ™ <lb/>fita effet .2. cum dimidio. </s> - <s xml:space="preserve">Vnde in huiuÅ¿modi exemplo videre eft quare colligan-<lb/>tur errores inuicem, quando alter eorum eccedit, reliquus vero deficit à numero pro <lb/>pofito. </s> - <s xml:space="preserve">Quod quidem ob aliam caufam non fit, nifi vt cognoÅ¿catur differentia .36. <lb/>differentia Å¿cilicet Å¿implicium Å¿ummarum ipÅ¿arum poÅ¿itionum.</s> - </p> - <p> - <s xml:space="preserve">Secundus autem modus ab antiquis magis exercitatus eÅ¿t, quod multiplicabant <lb/>diametraliter errores cum primis partibus, hoc eÅ¿t primum errorem cum prima par <lb/>te, hoc eÅ¿t cum numero Å¿ecundæ poÅ¿itionis, Å¿ecundum vero errorem cum prima <lb/>parte, hoc eÅ¿t cum numero primæ poÅ¿itionis, differentiam poÅ¿teà vel aggregatum <lb/>horum duorum productorum diuidebant per differentiam vel aggregatum dicto-<lb/>rum errorum, proueniens poÅ¿teà erat prima pars quæſita numeri propoÅ¿iti. </s> - <s xml:space="preserve">Vn-<lb/>de oriebantur tria producta, quorum <reg norm="tertium" type="context">tertiÅ©</reg>, hoc eÅ¿t differentia, Å¿eu aggregatum il-<lb/>lorum conÅ¿tituebatur ex differentia feuaggregato errorum, & ex numero quæ-<lb/>fito.</s> - </p> - <p> - <s xml:space="preserve">Vtin præfenti exemplo, primus error eÅ¿t .21. qui multiplicatus cum prima par-<lb/>te Å¿ecundæ poÅ¿itionis, quæ eÅ¿t .8. producit .168. <reg norm="Å¿ecundus" type="context">Å¿ecÅ©dus</reg> verò error eÅ¿t .9. qui multi-<lb/>plicatus cum prima parte primÄ™ poÅ¿itionis producit .90. differentia autem horum <lb/>productorum eÅ¿t .78. quæ diuifa per differentiam errorum, quæ eÅ¿t 12. dabit .6. <reg norm="cum" type="context">cÅ©</reg> di <lb/>midio, pro prima parte quæſita dati numeri diuiÅ¿ibilis, qui erat .50.</s> - </p> - <p> - <s xml:space="preserve">Hæc omnia rectè Å¿e habent. </s> - <s xml:space="preserve">Sed, vt Å¿upra dixi diuiÅ¿or non eft per Å¿e differentia <lb/>errorum, neque etiam differentia per Å¿e Å¿ummarum compoÅ¿itarum, fed bene fim-<lb/>plicium.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius rei Å¿peculatione, accipiendæ Å¿unt Å¿ummæ Å¿implices, quarum differen-<lb/>tiæ per Å¿e vtiles Å¿unt in huiuÅ¿modi operatione; </s> - <s xml:space="preserve">& quia etiam rationes veritatis ex <lb/>iÅ¿tis, & non ex illis fluunt; </s> - <s xml:space="preserve">quamuis tam vnæ, quam aliæ Å¿int eædem in quantitate, <lb/>ideÅ¿t æquales.</s> - </p> - <pb o="108" rhead="IO. BAPT. BENED." n="120" file="0120" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0120"/> - <p> - <s xml:space="preserve">DiÅ¿ponantur igitur huiuÅ¿mo-<lb/> - <anchor type="figure" xlink:label="fig-0120-01a" xlink:href="fig-0120-01"/> - di numeri tali ordine, vt fim-<lb/>plex Å¿umma, quæ ab vna reli-<lb/>quarum Å¿uperatur, & aliam Å¿upe-<lb/>rat, medium locum teneat; </s> - <s xml:space="preserve">@t <lb/>in propoÅ¿ito exemplo Å¿umma <lb/>mediocris eft .48. quÄ™ à ſumma <num value="60">.<lb/>60.</num> Å¿uperatur, & Å¿uperat Å¿um-<lb/>mam .39. locata igitur fit hÄ™c .48. <lb/>inter illas, Å¿uæ verò primæ partes <lb/>fimiliter conftitutæ Å¿int Å¿upra di-<lb/>ctas Å¿ummas, cum Å¿uis <reg norm="differentijs" type="context">differẽtijs</reg>, <lb/>& tria producta iam dicta, vt in fi <lb/>guris <var>.C.</var> et <var>.D.</var> arithmeticis <lb/>clarè patet: </s> - <s xml:space="preserve">figura enim <var>.C.</var> eft <lb/>pro exemplo ipÅ¿ius plus Å¿impli-<lb/>citer: </s> - <s xml:space="preserve">figura verò <var>.D.</var> pro exem-<lb/>plo ipÅ¿ius plus, & minus. </s> - <s xml:space="preserve">Et fic <lb/> - <anchor type="figure" xlink:label="fig-0120-02a" xlink:href="fig-0120-02"/> - in figura <var>.C.</var> habebimus tres <lb/>numeros confequentes .60. 48. <lb/>39. & tres antecedentes .10. 8. <lb/>6. cum dimidio, vnam, & ean-<lb/>dem proportionem terminantes, <lb/>ex .24. quinti, vt diximus; </s> - <s xml:space="preserve">qua-<lb/>re eorum differentiæ fimiliter <lb/>proportionales erunt, quod etiam <lb/>vidimus. </s> - <s xml:space="preserve">Supponamus nunc nos <lb/>ignorare æqualitatem maximi <lb/>producti cum reliquis duobus, <lb/>accipiendo Å¿olum pro hypoteÅ¿i, <lb/>quòd dicta producta oriantur <lb/>ex lateribus iam dictis.</s> - </p> - <div type="float" level="4" n="5"> - <figure xlink:label="fig-0120-01" xlink:href="fig-0120-01a"> - <image file="0120-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0120-01"/> - </figure> - <figure xlink:label="fig-0120-02" xlink:href="fig-0120-02a"> - <image file="0120-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0120-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">DemonÅ¿trandum nobis nunc relinquetur, maximum productum æquale effere-<lb/>liquis duobus; </s> - <s xml:space="preserve">hoc eÅ¿t productum .168. æquale effe productis .90. et .78. quorum <lb/>duorum productorum alterum .90. Å¿cilicet, generatur à differentia .9. quæ eft Å¿e-<lb/>cundÄ™, & tertiÄ™ Å¿ummæ, in primum numerum antecedentem, qui eÅ¿t .10. alterum vc-<lb/>ro productum .78. Å¿cilicet, generatur à differentia .12. quÄ™ eÅ¿t primÄ™, & Å¿ecundÄ™, Å¿um <lb/>mÄ™ in tertium numerum antecedentem, qui eÅ¿t .6. cum dimidio, maximum vero <lb/>productum .168. Å¿cilicet generatur à differentia maxima .21. quÄ™ eft primÄ™, & tertiÄ™ <lb/>Å¿ummÄ™ (& Å¿emper Ä™qualis prioribus duabus differentijs .12. et .9.) in Å¿ecundum nu-<lb/>merum antecedentem, qui eÅ¿t .8.</s> - </p> - <p> - <s xml:space="preserve">ConÅ¿tituantur igitur duo producta fimul iuncta Ä™qualia duobus .90. et .78. <lb/>lateralibus Å¿upra vnam aliquam rectam lineam <var>.q.p.</var> <reg norm="fitque" type="simple">fitqÌ;</reg> productum <var>.f.g.</var> Ä™quale <num value="90">.<lb/>90.</num> productum verò <var>.g.n.</var> Ä™quale .78. fit etiam baÅ¿is <var>.g.p.</var> vt .9. et <var>.g.q.</var> vt .12. vnde <var>.g.i.</var> <lb/>vel <var>.q.n.</var> erit vt .6. cum dimidio .et <var>.g.d.</var> vel <var>.p.f.</var> vt .10. & ideo <var>.i.d.</var> differentia erit .3. - <pb o="109" rhead="THEOREM. ARIT." n="121" file="0121" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0121"/> - cum dimidio, ut in figura <var>.C.</var> geometrica hic Å¿ubÅ¿cripta videre licet, et <var>.q.p.</var> erit .21. <lb/>Cogitemus nunc differentiam <var>.d.i.</var> diuiÅ¿am eſſe in puncto <var>.e.</var> ita vt eadem proportio <lb/>Å¿it ipÅ¿ius <var>.d.e.</var> ad <var>.e.i.</var> quæ ipÅ¿ius <var>.q.g.</var> ad <var>.g.p.</var> hoc eÅ¿t vt .1 2. ad .9. quapropter <var>.d.e.</var> erit <num value="2">.<lb/>2.</num> et <var>.e.i.</var> erit .1. cum dimidio, vt in dicta figura <var>.C.</var> arithmetica reperiuntur eſſe dif-<lb/>ferentiæ ipÅ¿orum antecedentium numerorum, deinde à puncto <var>.e.</var> ducatur imagina-<lb/>tione <var>.u.e.o.</var> æ quidiÅ¿tans ipÅ¿i <var>.q.p.</var> & producatur <var>.q.n.</var> vÅ¿que ad <var>.u.</var> vnde ita Å¿e habebit <lb/><var>u.e.</var> ad <var>.e.o.</var> ut <var>.q.g.</var> ad <var>g.p</var>. </s> - <s xml:space="preserve">quare vt <var>.d.e.</var> ad <var>.e.i.</var> ideo ex .15. Å¿exti vel .20. Å¿eptimi <var>.n.e.</var> <lb/>rectangulum æquale crit ipÅ¿i <var>.e.f.</var> qua propter rectang ulum <var>.q.o.</var> æquale erit duobus<lb/> rectangulis <var>.f.g.</var> et <var>.g.n</var>: Å¿ed cum <var>.g.i.</var> Å¿it vt .6. cum dimidio, et <var>.i.e.</var> vt .1. cum dimidio, er <lb/>go <var>.g.e.</var> erit ut .8. qui quidem numerus multiplicatus cum <var>.q.p.</var> 21. producit .168. ve <lb/>rum eÅ¿t igitur quod dictum fuit, hoc eÅ¿t <reg norm="quod" type="simple">ê™</reg> maximum productum Ä™quale Å¿it reliquis <lb/>duobus.</s> - </p> - <figure position="here"> - <image file="0121-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0121-01"/> - </figure> - <figure position="here"> - <image file="0121-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0121-02"/> - </figure> - <pb o="110" rhead="IO. BAPT. BENED." n="122" file="0122" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0122"/> - <p> - <s xml:space="preserve">DEmpto poÅ¿teà quo volueris horum altero productorum ex maximo, <reg norm="diuiÅ¿oque" type="simple">diuiÅ¿oqÌ;</reg> <lb/>reliquo per differentiam conÅ¿equentium, ipÅ¿i diametraliter oppoÅ¿itam, pro <lb/>ueniet tibi numerus antecedens <reg norm="correÅ¿pondensque" type="simple">correÅ¿pondensqÌ;</reg> illi.</s> - </p> - <p> - <s xml:space="preserve">Animaduertendum tamen eÅ¿t, quòd Å¿i in figura à me ita ordinata, Å¿umma Å¿im-<lb/>plex propoÅ¿ita medium locum occuparet, vt in figura <var>.D.</var> arithmetica videri poteÅ¿t; <lb/></s> - <s xml:space="preserve">tunc vt habeatur eius productum, addenda Å¿imul erunt circunÅ¿tantia producta .eo <lb/><reg norm="quod" type="simple">ê™</reg> eius Å¿ecundum latus eſſet antecedens medio loco conÅ¿titutum, & prima pars <reg norm="quae- Å¿ita" type="simple">quÄ™-<lb/>Å¿ita</reg> numeri propoÅ¿iti: </s> - <s xml:space="preserve">in qua figura <var>.D.</var> manifeÅ¿tè patet ratio, quare colligendi Å¿int <lb/>tam errores, quam producta, dum eorum alterum eÅ¿t plus, reliquum verò minus.</s> - </p> - <p> - <s xml:space="preserve">Speculatio figurÄ™ <var>.D.</var> arithmeticÄ™ videbitur in figura <var>.D.</var> geometrica, eodem fe <lb/>rè modo quo fecimus in figuris <var>.C.</var> mutatis mutandis, reÅ¿pectu ipÅ¿ius plus, & minus.</s> - </p> - <p> - <s xml:space="preserve">Collectio namque <reg norm="errorum" type="context">errorÅ©</reg> Å¿imiliter accidentalis eÅ¿t, eo quod eſſentialis numerus <lb/>diuiÅ¿or per Å¿e, eÅ¿t maxima differentia Å¿ummarum Å¿implicium, vt in dicta figura <var>.D.</var> <lb/>cerni poteÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Sed vt Å¿uperius dixi, nunc etiam repeto, quòd rectè hoc loco multiplicabatur <lb/>Å¿umma Å¿implex propoÅ¿ita, cum prima par <lb/>te primÄ™ poÅ¿itionis, vt productum diuide <lb/>retur per primam Å¿implicem Å¿ummam, <lb/> - <anchor type="figure" xlink:label="fig-0122-01a" xlink:href="fig-0122-01"/> - vnde proueniret nobis pars prima <reg norm="quaeÅ¿i- ta" type="simple">quęſi-<lb/>ta</reg> noÅ¿tri numeri propoÅ¿iti, ex regula de <lb/>tribus, vnica poÅ¿itione.</s> - </p> - <div type="float" level="4" n="6"> - <figure xlink:label="fig-0122-01" xlink:href="fig-0122-01a"> - <image file="0122-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0122-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Vt exempli gratia, datus numerus diui <lb/>dendus Å¿it .100. in quinque partes, tales <lb/>verò, <reg norm="quod" type="simple">ê™</reg> Å¿ecunda duplo maior Å¿it prima <lb/>cum .2. Å¿imul, tertia autem æqualis Å¿it pri-<lb/>mæ & Å¿ecundæ cum .3. vnitatibus iunctis, <lb/>quarta poÅ¿teà maior Å¿it prima Å¿ecunda, & <lb/>tertia per .4. vnitates, quinta demum Å¿u-<lb/>peret reliquas omnes per quinque vnita <lb/>tes, vt in figura <var>.E.</var> videre eÅ¿t, quæ quidem <lb/>partes compoÅ¿itæ (Å¿umpta vnitate pro <lb/>prima) ita diÅ¿poÅ¿itæ erunt .1. 4. 8. 17. 35. <lb/>quarum Å¿umma erit .65. Å¿implices autem <lb/>cum diÅ¿poÅ¿itæ fuerint erunt .1. 2. 3. 6. 12. <lb/>quarum Å¿umma erit .24. dempta igitur <lb/>cum fuerit hæc Å¿implex Å¿umma .24. à com <lb/>poÅ¿ita .65. reÅ¿iduum erit .41. hoc eÅ¿t Å¿um-<lb/>ma numerorum propoÅ¿itorum cum Å¿uis <lb/>iterationibus in ipÅ¿is partibus, quod cum <lb/>per Å¿e clariſſimum Å¿it, Å¿uperÅ¿luum eÅ¿t <reg norm="ipsam" type="context">ipsã</reg> <lb/>Å¿ummam annatomizare per Å¿ingulas par-<lb/>tes, niÅ¿i quis habuerit eius cerebrum à fi-<lb/>gura Omega <reg norm="terminatum" type="context">terminatÅ©</reg>, cui tamen poÅ¿-<lb/>Å¿emus dicere dictam Å¿ummam .41. in .4. <lb/>partes diuidi, cuius prima eſſet .2. pro ad <lb/>ditione ad <reg norm="Å¿ecundam" type="context">Å¿ecÅ©dam</reg> partem Å¿implicium, - <pb o="111" rhead="THEOREM. ARIT." n="123" file="0123" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0123"/> - Å¿ecunda verò eſſet .5. pro additione ad tertiam partem Å¿implicium, tertia autem eÅ¿-<lb/>Å¿et .11. pro additione ad quartam partem Å¿implicium, quarta demum eſſet .23. pro <lb/>additione quintæ partis Å¿implicium, quarum partium .2. 5. 11. 23. Å¿umma eÅ¿t .41. vt <lb/>diximus. </s> - <s xml:space="preserve">Hæc igitur Å¿umma .41. Å¿ubducenda eÅ¿t à numero .100. propoÅ¿ito, vnde re-<lb/>linquetur .59. pro Å¿umma partium Å¿implicium numeri propoÅ¿iti, quarum prima erit <lb/>2. cum vndecim vigeÅ¿imisquartis ex diuiÅ¿ione huiuÅ¿modi .59. per .24. Å¿ummam par-<lb/>tium Å¿implicium ex viregulæ de tribus, dicendo Å¿i .24. prouenit nobis ab .1. prima <lb/>partium Å¿implicium, à quo proueniet nobis .59? </s> - <s xml:space="preserve">vnde proueniet à .2. cum vndecim <lb/>vigeÅ¿imisquartis pro prima parte quæſita, Å¿ecunda verò iuxta propoÅ¿itum, erit .6. <lb/>cum .22. vigeÅ¿imisquartis, tertia autem .12. cum nouem vigeÅ¿imisquartis, quarta poÂ<lb/>Å¿teà .25. cum .18. vigeÅ¿imisquartis, quinta demum erit .52. cum .12. vigeÅ¿imisquartis, <lb/>quarum omnium Å¿umma erit .100.</s> - </p> - <p> - <s xml:space="preserve">STifelius in primo exemplo regulæ falÅ¿i, ita inquit.</s> - </p> - <p> - <s xml:space="preserve">Quæratur numerus, à cuius dimidio Å¿ubtractæ partes tertia, & quarta relin-<lb/>quatur .300.</s> - </p> - <p> - <s xml:space="preserve">IpÅ¿e enim Å¿upponit .300. pro reÅ¿iduo cognito alterius numeri incogniti, deinde <lb/>accipit .24. pro prima poÅ¿itione numeri cogniti, à cuius medietate abÅ¿cindit tertiam <lb/>& quartam partem ipÅ¿ius medietatis, vnde remanet .5. qui quidem numerus .5. ex <num value="22">.<lb/>22.</num> quinti vel .15. Å¿eptimiÅ¿e ha-<lb/>bebit ad .24. vt .300. ad <reg norm="numerum" type="context">numerÅ©</reg> <lb/> - <anchor type="figure" xlink:label="fig-0123-01a" xlink:href="fig-0123-01"/> - quæſitum, </s> - <s xml:space="preserve">quare cum quis multi <lb/>plicauerit .300. per .24. & produ-<lb/>ctum diuiÅ¿erit per .5. proueniet <num value="1440">.<lb/>1440.</num> numerus quæſitus, ex vi <lb/>regulæ de tribus.</s> - </p> - <div type="float" level="4" n="7"> - <figure xlink:label="fig-0123-01" xlink:href="fig-0123-01a"> - <image file="0123-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0123-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">ConÅ¿ideremus igitur <reg norm="meam" type="context">meã</reg> di-<lb/>Å¿poÅ¿itionem numerorum huiuÅ¿-<lb/>modi exempli, in figura hic Å¿up-<lb/>poÅ¿ita <var>.F.</var> in qua videre licebit <lb/>quo pacto ipÅ¿e etiam Stifelius ac <lb/>cipiat diuiÅ¿orem .5. vt <reg norm="differentiam" type="context">differentiã</reg> <lb/>errorum & non ut differentiam <lb/>duorum conÅ¿equentium .5. et .10 <lb/>Å¿icuti eÅ¿t re uera, ut diuiÅ¿or dico, <lb/>ex rationibus à me hic Å¿upra ad-<lb/>ductis, quamuis vna & eadem Å¿it <lb/>quantitas neceſſariò ut patet.</s> - </p> - <p> - <s xml:space="preserve">ACcipiamus adhuc aliud exemplum à Tartalea propoÅ¿itione .9. <reg norm="datum" type="context">datÅ©</reg>, & <reg norm="oppoÅ¿itum" type="context">oppoÅ¿itÅ©</reg> <lb/>priori; </s> - <s xml:space="preserve">nam Å¿icut in illo numerus Å¿implex habebatur per Å¿ubtractionem Å¿um-<lb/>mæ numerorum adijciendorum, in hoc fitèconuerÅ¿o, hoc eÅ¿t per additionem nu-<lb/>merorum Å¿ubtrahendorum.</s> - </p> - <p> - <s xml:space="preserve">Problema igitur ita Å¿e habet. </s> - <s xml:space="preserve">Fuit quidam mercator qui habebat aliquot au-<lb/>reos, cuius quantitas poÅ¿teà quærenda erit, hic enim fecit duo itinera, ut aliquod <lb/>dictis aureis mediantibus lucrum faceret, in primo autem itinere duplicauit nume-<lb/>rum Å¿uorum aureorum, ex quibus poÅ¿teà conÅ¿umpÅ¿it .4. pro aliquibus expenÅ¿is, in - <pb o="112" rhead="IO. BAPT. BENED." n="124" file="0124" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0124"/> - Å¿ecundo itinere iterum duplicauit Å¿uos aureos, ex quibus etiam poÅ¿tea conÅ¿umpÅ¿it <num value="8">.<lb/>8.</num> numeratis poÅ¿teà pecunijs reperit tantummodo .24. aureos in eius marÅ¿upio, <reg norm="quae ritur" type="simple">quÄ™Â<lb/>ritur</reg> nunc quot habebat aureos in principio primi itineris.</s> - </p> - <p> - <s xml:space="preserve">Intali caÅ¿u, cum ipÅ¿e quolibet itinere duplicabat eius pecuniam, nulli dubium <lb/>eÅ¿t quòd in fine Å¿ecundi itineris ipÅ¿e habuiſſet pecuniam Å¿uam quadruplicatam, Å¿i <lb/>ex ipÅ¿a nihil detractum fuiſſet, Å¿ed quia in fine primi itineris conÅ¿umpÅ¿it .4. aureos, <lb/>quibus alios .4. lucratus eſſet in Å¿ecundo itinere, poÅ¿teà conÅ¿umpÅ¿it iterum <num value="8">.<lb/>8.</num> aureos, ita <reg norm="quod" type="simple">ê™</reg> ex quadruplo Å¿uæ primæ pecuniæ, rectè dici poteÅ¿t, quod conÅ¿um-<lb/>pÅ¿erit .16. aureos; </s> - <s xml:space="preserve">qui quidem numerus ex communi conceptu erit differentia in-<lb/>ter .24. & quadruplum prioris pecuniæ, cum qua profectus fuit in principio eius iti-<lb/>neris; </s> - <s xml:space="preserve">quapropter Å¿i addiderimus .16. ipÅ¿i .24. habebimus .40. pro quadruplo eius <lb/>prioris pecuniæ. </s> - <s xml:space="preserve">Rectè igitur dici poteÅ¿t, Å¿i .4. prouenit ab vno, à quo numero pro <lb/>ueniet .40.</s> - </p> - <p> - <s xml:space="preserve">Videamus igitur nunc quo pacto hoc reÅ¿pondeat cum methodo antiquorum. <lb/></s> - <s xml:space="preserve">Ego enim inueni duas poÅ¿itiones Å¿criptas à Tartalea pro prima pecunia hoc eÅ¿t .12. <lb/>et .14. Å¿ed à .12. pro primo errore reperi .8. more antiquo à .14. verò pro Å¿ecundo er-<lb/>rore proueniebat .16. producta autem horum numerorum diametraliter, Å¿unt .112. <lb/>et .192. quorum differentia eÅ¿t .80. pro tertio producto, quo diuiÅ¿o per differen-<lb/>tiam <reg norm="errorum" type="context">errorÅ©</reg> .8. Å¿cilicet, præbetnobis .10. pro pecunia quæſita, vt etiam ego inueni.</s> - </p> - <p> - <s xml:space="preserve">Sed hoc mihi viÅ¿um eÅ¿t Å¿ubtilius examinare mea methodo mediante, vtin figu-<lb/>ra <var>.G.</var> videre eÅ¿t, prius enim Å¿uo loco poÅ¿uitria producta dicta, deinde duas poÅ¿itio <lb/>nes .12. et .14. & quia Å¿ciebam productum .112. oriri à multiplicatione .14. cum .8. <lb/>ideo poÅ¿ui talem numerum .8. Å¿uo loco diametraliter oppoÅ¿ito ei producto .112. <lb/>& quia Å¿ciebam etiam productum .192. naÅ¿ci ex .12. et .16. ideo Å¿uo loco poÅ¿ui hunc <lb/>numerum .16. qui eÅ¿t maxima differentia inter duos conÅ¿equentes ( ita à me Å¿upra <lb/>nominatos) à qua differentia dempta priori .8. iam inuenta, reliqua .8. mihi daba-<lb/>tur, <reg norm="quam" type="context">quã</reg> Å¿uo loco notaui, Å¿uo <reg norm="etiam" type="context">etiã</reg> <lb/>loco Å¿cripÅ¿i .2. differentiam inter <lb/>12. et .14. antecedentium. </s> - <s xml:space="preserve">Å¿ed <lb/> - <anchor type="figure" xlink:label="fig-0124-01a" xlink:href="fig-0124-01"/> - quia Å¿ciebam eandem proportio <lb/>nem eſſe inter hanc differentiam <lb/>& <reg norm="differentiam" type="context">differentiã</reg> .8. huic <reg norm="Å¿uppoÅ¿itam" type="context">Å¿uppoÅ¿itã</reg>, <lb/>quæ reperitur inter .12. <reg norm="antecedem" type="context">antecedẽ</reg> <lb/>tem, & <reg norm="Å¿uum" type="context">Å¿uÅ©</reg> conÅ¿equentem; </s> - <s xml:space="preserve">ideo <lb/>poÅ¿ui .48. pro dicto conÅ¿equenti, <lb/>diuiÅ¿i poÅ¿tea productum .80. per <num value="8">.<lb/>8.</num> differentiam ei diametraliter <lb/>oppoÅ¿itam, vnde prouenit mihi <lb/>10. cui ita proportionatus eÅ¿t <reg norm="Å¿uus" type="simple">Å¿uê°</reg> <lb/>numerus conÅ¿equens .40. vt .48. <lb/>ad .12. et .56. ad .14. exijÅ¿dem ra-<lb/>tionibus à meÅ¿upra dictis. <lb/></s> - <s xml:space="preserve">In tali igitur figura videntur nu-<lb/>merinaturaliter <reg norm="correÅ¿pondentes" type="context">correÅ¿pondẽtes</reg> <lb/>ipÅ¿is poÅ¿itionibus, & hac metho-<lb/>do poſſumus inuenire tales numeros conÅ¿equentes in omnibus alijs exemplis à no-<lb/>Å¿tris maioribus Å¿criptis.</s> - </p> - <div type="float" level="4" n="8"> - <figure xlink:label="fig-0124-01" xlink:href="fig-0124-01a"> - <image file="0124-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0124-01"/> - </figure> - </div> - <pb o="113" rhead="THEOREM. ARIT." n="125" file="0125" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0125"/> - <p> - <s xml:space="preserve">PRoponitur etiam quoddam vas, cuius pes Å¿it quarta pars totius vaÅ¿is cum oper <lb/>culo, pars autem media Å¿ine operculo, Å¿it quinta pars ipÅ¿ius pedis, operculum <lb/>verò .18. libras pendeat. </s> - <s xml:space="preserve">quæritur nunc quantitas dicti pedis.</s> - </p> - <p> - <s xml:space="preserve">Ex methodo enim antiquorum inuentus eÅ¿t pes .4. cum .14. decimisnonis ta-<lb/>lium partium, Å¿eu librarum, qualium operculus eÅ¿t .18. </s> - <s xml:space="preserve">Videamus igitur & nos ex <lb/>noÅ¿tra figura, quo pacto hoc reÅ¿pondeat veritati.</s> - </p> - <p> - <s xml:space="preserve">Inuenta enim Å¿unt tria producta, <reg norm="iam" type="context">iã</reg> orta ex dicta methodo .10. 100. 90. quæ Å¿uis <lb/>locis notaui, vt in Å¿igura <var>.H.</var> Å¿ubÅ¿cripÅ¿i etiam duas illorum poÅ¿itiones .5. et .10. cum <lb/>Å¿ua differentia .5. & cum productum .10. oriretur ab vno latere .10. reliquum erat <num value="1">.<lb/>1.</num> quod Å¿uo loco notaui, Å¿imiliter quia .100. productum, pro vno eius laterum erat <num value="5">.<lb/>5.</num> reliquum autem .20. Å¿uo loco poÅ¿ui, & quia differentia inter .20. et .1. duo latera, <lb/>quÄ™ eÅ¿t .19. æqualis eÅ¿t ei, quæ inter duo conÅ¿equentia duarum poÅ¿itionum, etiam <lb/>Å¿uo loco ipÅ¿am conÅ¿titui, Å¿ed quia hæc differentia eÅ¿t vnum laterum producti .90. er <lb/>go reliquum latus <reg norm="quæſitum" type="context">quæſitÅ©</reg> erit .4. cum .14. decimisnonis, rectè igitur operatur. <lb/></s> - <s xml:space="preserve">Å¿ed cum eadem proportio Å¿it inter differentiam .5. Å¿uperiorem, et .19. inferiorem, <lb/>quæ eÅ¿t vnius <reg norm="antecedentis" type="context">antecedẽtis</reg> ad Å¿uum <lb/>conÅ¿equens, </s> - <s xml:space="preserve">quare .10. <reg norm="antecedens" type="context">antecedẽs</reg> <lb/>habebit pro Å¿uo conÅ¿equenti .38. <lb/> - <anchor type="figure" xlink:label="fig-0125-01a" xlink:href="fig-0125-01"/> - et .5. habebit .19. et .4. cum .14. de-<lb/>cimisnonis habebit .18. rectè <reg norm="igitur" type="simple">igit̃</reg> <lb/>dictum fuiſſet Å¿i .19. prouenit .à .5. <lb/>à quo proueniet .18?</s> - </p> - <div type="float" level="4" n="9"> - <figure xlink:label="fig-0125-01" xlink:href="fig-0125-01a"> - <image file="0125-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0125-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">HuiuÅ¿modi autem rei ratio ita <lb/>Å¿e <reg norm="hent" type="context">hẽt</reg>, eÅ¿to linea <var>.a.e.u.</var> cuius pars <lb/>a. Å¿it quarta reliquarum <var>.e.u.</var> iuncta <lb/>rum, Å¿ed <var>.e.</var> Å¿it quinta ipÅ¿ius <var>.a</var>. </s> - <s xml:space="preserve">Tunc <lb/>clarum erit quod <var>.e.</var> erit vigeÅ¿ima <lb/>dictarum <var>.e.u.</var> </s> - <s xml:space="preserve">quare erit decima-<lb/>nona ipÅ¿ius <var>.u.</var> Å¿ed <reg norm="cum" type="context">cÅ©</reg> u. <reg norm="sumpta" type="context">sÅ©pta</reg> Å¿it vt <num value="18">.<lb/>18.</num> rectè igitur dici poteÅ¿t, Å¿i .u: ut <num value="19">.<lb/>19.</num> prouenit ab <var>.a.</var> ut quinque, à <lb/>quot ipÅ¿ius <var>.a.</var> proueniet <var>.u.</var> ut .18.</s> - </p> - <p> - <s xml:space="preserve">Quis enim non uidet quod diui <lb/>Å¿a cum fuerit <var>.u.</var> in partes .19. quod <lb/>quinque illarum æquabuntur ipÅ¿i <var>.<lb/>a.</var> cum quælibet fuerit æqualis <var>.e.</var> <lb/>quintæ parti ipÅ¿ius <var>.a</var>.</s> - </p> - <p> - <s xml:space="preserve">HAc igitur mea numerorum diÅ¿poÅ¿itione mediante reperiuntur ipÅ¿i numeri in <lb/>feriores naturaliter conÅ¿equentes, correÅ¿pondentesqÌue ipÅ¿is Å¿uperioribus an <lb/>tecedentibus; </s> - <s xml:space="preserve">quamuis multoties <reg norm="contingere" type="context">cõtingere</reg> poſſit, ut generationes Å¿eu com-<lb/>poÅ¿itiones ipÅ¿orum ignorentur: </s> - <s xml:space="preserve">& quia tam à differentijs errorum, quam ab illis, <lb/>quÄ™ Å¿unt inter ueros conÅ¿equentes numeros ( propter eorum æqualitatem ) elicitur <lb/>ipÅ¿a ueritas, proptereà rectè antiqui illis vÅ¿i Å¿unt, quamuis Å¿int potius Å¿enÅ¿um <lb/>Å¿equuti, uel experientiam, quam rationem: </s> - <s xml:space="preserve">quæ quidem ratio pendet ab ipÅ¿is na-<lb/>turalibus numeris conÅ¿equentibus ( ut Å¿upra uidimus ) etÅ¿i incognitis ut plurimum, <lb/>quod Å¿i ipÅ¿os inuenire primò nobis datum fuiſſet, unica <reg norm="tantummodo" type="context">tantÅ©modo</reg> poÅ¿itio Å¿uffice- - <pb o="114" rhead="IO. BAPT. BENED." n="126" file="0126" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0126"/> - ret, mediante ipÅ¿a regula de tribus, vt <reg norm="iam" type="context">iã</reg> ſępius <reg norm="dictum" type="context">dictÅ©</reg> eÅ¿t, quod <reg norm="etiam" type="context">etiã</reg> clarè patet ex di-<lb/>uerÅ¿is problematibus .17. lib. ipÅ¿ius Tartaleæ, vt ex primo, quod aſſumpÅ¿imus pro <lb/>noÅ¿tro etiam primo exemplo, ex .9. 15. 16. 17. 18. 19. 20. 27. 28. 29. 30. 33. & ex <lb/>alijs multis, vbi facillimè inue nitur conÅ¿equens ipÅ¿ius poÅ¿itionis, qui quidem nume-<lb/>rus eÅ¿t diuiÅ¿or producti ipÅ¿ius numeri propoÅ¿iti in numerum poÅ¿itionis, vnde poÅ¿teà <lb/>prouenit <reg norm="Å¿ecundum" type="context">Å¿ecundÅ©</reg> latus huiuÅ¿modi producti, hoc eÅ¿t numerus quæſitus, per <reg norm="regulam" type="context">regulã</reg> <lb/>de tribus, vt dixi.</s> - </p> - <p> - <s xml:space="preserve">Alia verò multa problemata inueniuntur, pro quorum re@olutione poſſumus ali <lb/>qua methodo vti, in qua manifeÅ¿tè pateant <reg norm="eorum" type="context">eorÅ©</reg> rationes abÅ¿que regula falÅ¿i, cuius <lb/>regulæ rationes non ita promptè ipÅ¿i intellectui Å¿e offerunt, vt Å¿upra vidimus.</s> - </p> - <p> - <s xml:space="preserve">Accipiamus pro exemplo .21. problema ipÅ¿ius Tartalæ in dicto .17. libr. vbi Å¿up-<lb/>ponit vnum hædum diuiÅ¿um in .4. partes, quarum quælibet vendebatur eodem pre <lb/>cio, interiora vero .6. denarijs minus quam quælibet dictarum partium, Å¿umma <lb/>autem omnium iÅ¿torum denariorum fuit .127. quæritur nunc precium cuiuÅ¿que <lb/>partis.</s> - </p> - <p> - <s xml:space="preserve">Tale enim problema hoc etiam alio breuiori modo poteÅ¿t Å¿olui, vt rationes ma-<lb/>gis pateant, quam ex regula falÅ¿i.</s> - </p> - <p> - <s xml:space="preserve">Nam Å¿i illi numero .127. denariorum, additus fuerit numerus .6. Å¿umma erit .133. <lb/>qua diuiÅ¿a per quinque, illico proueniet .26. cum tribusquintis pro precio vniuÅ¿cu-<lb/>iuÅ¿que quatuor partium, à quo .26. cum tribusquintis dempto .6. remanebit .20. cum <lb/>tribusquintis pro precio interiorum.</s> - </p> - <p> - <s xml:space="preserve">Simili modo in .24. problemate inquit.</s> - </p> - <p> - <s xml:space="preserve">Duodecim pyra cum .28. pomis venduntur .36. denarijs, et .20. pyra. cum .200 po <lb/>mis <reg norm="venduntur" type="context">vẽduntur</reg> .44. denarijs, <reg norm="quæritur" type="simple">quærit̃</reg> nunc, quod <reg norm="nam" type="context">nã</reg> fuerit <reg norm="precium" type="context">preciÅ©</reg> <reg norm="vniuÅ¿cuiuÅ¿que" type="simple">vniuÅ¿cuiuÅ¿q;</reg> illorum.</s> - </p> - <p> - <s xml:space="preserve">Hoc etiam problema, hac alia methodo Å¿olui poteÅ¿t, dicendo exregula de tribus, <lb/>Å¿i ex .20. vtrorunque qui ea vendit, vult .44. quid volet ex .12? </s> - <s xml:space="preserve"><reg norm="manifeÅ¿tum" type="context">manifeÅ¿tÅ©</reg> erit quod <lb/>volet .26. cum duobus quintis, </s> - <s xml:space="preserve">quare .12. pyra cum .12. pomis valebunt .26. cum duo <lb/>bus quintis, Å¿ed 12. cum .28. pomis valebant .36. ergo .16. poma Å¿ola valebunt .9. <lb/>cum tribus quintis, hoc enim clarè ex Å¿e patet; </s> - <s xml:space="preserve">quare cum dixerimus, Å¿i .16. poma Å¿o <lb/>la valent .9. cum tribusquintis, vnum valebit <var>.o.</var> cum tribusquintis, Å¿ed quemadmo-<lb/>dum .20. pyra cum .20. pomis valent .44. vnum pyrum, cum vno pomo valebunt .2. <lb/>cum quinta parte, à quo numero detractus cum fuerit <var>.o.</var> cum tribus quintis, precio <lb/>Å¿cilicet vnius pomi, reliquum .1. cum tribusquintis, erit precium vnius pyri.</s> - </p> - <p> - <s xml:space="preserve">Idem etiam dico de .28. problemate, vbi Å¿upponit quod quidam comparaſſet <lb/>quatuor petias, vt vulgo dicitur, panni pro ducatis .96. quarum primæ precium ob-<lb/>litus Å¿it, Å¿ed memoria tenet pro Å¿ecunda Å¿oluiſſe .6. plusquam pro prima, & pro ter-<lb/>tia Å¿oluiſſe .8. plus quam pro Å¿ecunda, & pro quarta Å¿oluiſſe .10. plus quam pro ter-<lb/>tia, quæritur nunc quantum fuerit precium vniuÅ¿cuiuÅ¿que illarum.</s> - </p> - <p> - <s xml:space="preserve">Quod <reg norm="quidem" type="context">quidẽ</reg> problema <lb/> - <anchor type="figure" xlink:label="fig-0126-01a" xlink:href="fig-0126-01"/> - breuius eſſetita Å¿olui, vt in <lb/>Å¿ubÅ¿cripta figura <var>.I.</var> videri <lb/>poteÅ¿t, <reg norm="addendo" type="context">addẽdo</reg> Å¿imul omnes <lb/>exceſſus. </s> - <s xml:space="preserve">Nam exceſſus <reg norm="Å¿ecum" type="context">Å¿ecÅ©</reg> <lb/>dæ Å¿upra primam eÅ¿t .6. Å¿ed <lb/>cum exceſſus tertiæ Å¿upra Å¿e <lb/>cundam Å¿it .8. ergo exceſſus <lb/>tertiæ Å¿upra primam erit .14 - <pb o="115" rhead="THEOREM. ARIT." n="127" file="0127" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0127"/> - Å¿ed exceſſus quartæ Å¿upra tertiam eÅ¿t .10. vnde Å¿upra Å¿ecundam erit .18. & Å¿upra pri-<lb/>mam erit .24. quæ omnia Å¿imul addita erunt .44. & in qualibet harum trium remane-<lb/>bit una pars æqualis primæ quantitati, </s> - <s xml:space="preserve">quare Å¿i ex .96. detractus fuerit numerus .44. <lb/>reliquus 52. erit quadruplus primæ, </s> - <s xml:space="preserve">quare prima pars valebit .13. Å¿ecunda .19. ter-<lb/>tia .27. & quarta .37. quarum omnium Å¿umma eÅ¿t .96.</s> - </p> - <div type="float" level="4" n="10"> - <figure xlink:label="fig-0126-01" xlink:href="fig-0126-01a"> - <image file="0126-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0126-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">EX poÅ¿itionibus autem Tartaleæ in noÅ¿tra figura <var>.K.</var> digeÅ¿tis, videre poſſumus <lb/>quo pacto <reg norm="colligantur" type="context">colligãtur</reg> huiuÅ¿ <lb/>modi <reg norm="conÅ¿equentes" type="context">conÅ¿equẽtes</reg> numeri Å¿impli-<lb/>ces .36. et .52. more figuræ <var>.E.</var> quia <lb/> - <anchor type="figure" xlink:label="fig-0127-01a" xlink:href="fig-0127-01"/> - colliguntur primò partes compoÅ¿i <lb/>tæ .9. 15. 23. 33. ex quarum Å¿umma <lb/>80. Å¿ubtrahitur .36. Å¿umma Å¿im-<lb/>plex ex Å¿implicibus partibus .9. 9. <lb/>9. 9. & <reg norm="reÅ¿iduum" type="context">reÅ¿iduÅ©</reg> quod eÅ¿t .44. Å¿ubdu <lb/>citur ex .96. Å¿umma compoÅ¿ita & <lb/>propoÅ¿ita, vnde remanet .52. pro <lb/>Å¿umma Å¿implici, ex numero dato, <lb/>cuius proportio ad .13. eadem eÅ¿t <lb/>quæ .36 ad .9. & proptereà ſuper-<lb/>flua eÅ¿t Å¿ecunda poÅ¿itio, <reg norm="quando" type="context">quãdo</reg> Å¿ci <lb/>mus inuenire tales duos numeros <lb/>conÅ¿equentes, vt in hoc exemplo <lb/>Å¿unt .36. et .52. quia ex regula de <lb/>tribus poÅ¿teà elicitur veritas quæ-<lb/>Å¿ita. </s> - <s xml:space="preserve"><reg norm="Idem" type="context">Idẽ</reg> dico de 33. problemate.</s> - </p> - <div type="float" level="4" n="11"> - <figure xlink:label="fig-0127-01" xlink:href="fig-0127-01a"> - <image file="0127-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0127-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">PRO quo .33. problemate acci <lb/>piantur poÅ¿itiones primi <reg norm="exem" type="context">exẽ</reg> <lb/>pli TonÅ¿talli hoc eÅ¿t .33. et .31. vt in figuris hic Å¿ubiectis <var>.P.Q.</var> facile quis poteÅ¿t vi-<lb/>dere, vbi in figura P. videbit nume-<lb/>ros compoÅ¿itos, in figura verò <var>.Q.</var> cer <lb/> - <anchor type="figure" xlink:label="fig-0127-02a" xlink:href="fig-0127-02"/> - net numeros Å¿implices, à quibus pro <lb/>ueniunt rationes per Å¿e huiaÅ¿modi <lb/>operationis, in figura autem <var>.R.</var> vide <lb/>bitur meus ordo, & iÅ¿tæ tres figuræ Å¿i <lb/>miles <reg norm="erunt" type="context">erÅ©t</reg> tribus illis primis <var>.A.B.C.</var> <lb/>ita quòd cum quis illas intellexerit, il <lb/>lico etiam iÅ¿tas cognoÅ¿cet, vbi <reg norm="etiam" type="context">etiã</reg> <lb/>videbit quam confusè <reg norm="ratiocinentur" type="context">ratiocinẽtur</reg> ij <lb/>qui ignorant hunc meum ordinem <lb/>Å¿implicium <reg norm="numerorum" type="context">numerorÅ©</reg>, à quibus fluit <lb/>tota ratio (vt Å¿upra dixi) huiuÅ¿cemo <lb/>di operationis.</s> - </p> - <div type="float" level="4" n="12"> - <figure xlink:label="fig-0127-02" xlink:href="fig-0127-02a"> - <image file="0127-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0127-02"/> - </figure> - </div> - <pb o="116" rhead="IO. BAPT. BENED." n="128" file="0128" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0128"/> - <figure position="here"> - <image file="0128-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0128-01"/> - </figure> - <figure position="here"> - <image file="0128-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0128-02"/> - </figure> - <p> - <s xml:space="preserve">I Dem etiam poteÅ¿t dici de .15. problemate (Å¿icut de alijs multis) vbi ponit tres <lb/>homines habentes .40. aureos quorum primus habet duas quintas partes Å¿ecun-<lb/>di, <reg norm="Å¿ecundus" type="context">Å¿ecÅ©dus</reg> verò <reg norm="quinque" type="simple">quinq;</reg> octauas tertij, <reg norm="quæritur" type="simple">quærit̃</reg> <reg norm="nunc" type="context">nÅ©c</reg> quot ducatos habeat vnuÅ¿quiÅ¿que.</s> - </p> - <p> - <s xml:space="preserve">Quis non videt quæſo, <reg norm="quod" type="simple">ê™</reg> omnes partes erunt .15. quare cum dixerimus Å¿i .15. dat <lb/>nobis .2. (pro prima portione primi hominis) quid dabit .40? </s> - <s xml:space="preserve">vnde nobis proueniet <lb/>5. cum tertia parte.</s> - </p> - <p> - <s xml:space="preserve">Et de .29. Å¿imiliter aſſero, vbi ponit <reg norm="alium" type="context">aliÅ©</reg> emiſſe tria fruÅ¿ta panni pro ducatis .48. <lb/>quarum Å¿ecundam habuit pro dimidio precio primæ, tertiam autem pro quarta <lb/>parte ipÅ¿ius Å¿ecundæ, </s> - <s xml:space="preserve">quare omnes partes erunt .13. quapropter precium tertiæ pe-<lb/>tiæ erit tertiadecima pars ipÅ¿ius .48. hoc eÅ¿t .3. cum .9. tertijs decimis.</s> - </p> - <p> - <s xml:space="preserve">Adhuc duo exempla videtur mihi proponere, quorum primum eÅ¿t .38. eiuÅ¿dem <lb/>lib. vbi Å¿upponitur operarium quendam velle perficere opus quoddam Å¿pacio die-<lb/>rum .36. tali pacto, quod qualibet die, in qua ipÅ¿e operaturus Å¿it lucretur Å¿olidos <num value="16">.<lb/>16.</num> qualibet verò die, in qua nihil agat perdat Å¿olidos .24. </s> - <s xml:space="preserve">Tunc accidit, vt exacto <lb/>termino <reg norm="perfectoque" type="simple">perfectoqÌ;</reg> opere, tantum lucratus Å¿it, quantum perdiderit. </s> - <s xml:space="preserve">Quæritur <reg norm="nunc" type="context">nÅ©c</reg> <lb/>quot fuerint dies lucri, quotúe perditionis.</s> - </p> - <p> - <s xml:space="preserve">HuiuÅ¿modi problematis operatio breuiſſima abÅ¿que vlla falÅ¿a poÅ¿itione ita erit, <lb/>hoc eÅ¿t diuidendo productum .36. in .24. per .40. ideÅ¿t per aggregatum ipÅ¿ius .24. <reg norm="cum" type="context">cÅ©</reg> <lb/>16. & prouentus erit .21. cum tribus quintis pro diebus lucri, vnde reliquum ex .36. <lb/>erit .14. cum duabus quintis pro diebus perditionis.</s> - </p> - <p> - <s xml:space="preserve">Cuius operationis ratio ex Å¿e Å¿atis patet, cum duo producta, vnum lucri, alterum <lb/>vero perditio-<lb/>nis æqualia eſſe <lb/> - <anchor type="figure" xlink:label="fig-0128-03a" xlink:href="fig-0128-03"/> - debeant, vnde <lb/>ex duodecima, <lb/>& vigeÅ¿imaÅ¿epti <lb/>mi ex regula de <lb/>tribus reperiun <lb/>tur partes ipÅ¿ius <lb/>36. eodem mo- - <pb o="117" rhead="THEOR. ARITH." n="129" file="0129" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0129"/> - do Å¿e inuicem habentes, vt .24. et .16. quæ Å¿unt .21. cum tribus quintis, et .14. <reg norm="cum" type="context">cÅ©</reg> dua <lb/>bus quintis, ex quo Å¿equitur, vt quod fit ex .21. cum tribus quintis, in 16. Ä™quale Å¿it ei <lb/>quod fit ex .14 cum duabus quintis, in .24. & ita reperiuntur duo producta æqualia, <lb/>vnum lucri, reliquum vero perditionis, vt in figura <var>.M.</var> clarè videtur.</s> - </p> - <div type="float" level="4" n="13"> - <figure xlink:label="fig-0128-03" xlink:href="fig-0128-03a"> - <image file="0128-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0128-03"/> - </figure> - </div> - <p> - <s xml:space="preserve">A Liud verò exemplum eÅ¿t .39. quod quidem à ſuperiori non differt, niÅ¿i quod <lb/>in fine operationis, operarius dictus lucratus eÅ¿t Å¿olidos .60: </s> - <s xml:space="preserve">quÄ™ritur <reg norm="nunc" type="context">nÅ©c</reg> vt Å¿u-<lb/>pra, quot fuerunt dies lucri, & quot perditionis.</s> - </p> - <p> - <s xml:space="preserve">Hoc etiam abÅ¿que vlla falÅ¿a poÅ¿itione dicto citius poteÅ¿t Å¿olui, hoc modo, <reg norm="diuidem" type="context">diuidẽ</reg> <lb/>do Å¿cilicet illos .60 Å¿olidos per .40. ideÅ¿t per aggregatum .24. cum .16. proueniens <lb/>autem, quod erit .1. cum dimidio, adde ad latus Å¿uperius inuentum, hoc eÅ¿t .21. cum <lb/>tribus quintis, & Å¿unima erit .23. cum decima parte pro numero dierum lucri, dein-<lb/>de idem prouentum deme ex alio latere Å¿uperius reperto .14. cum duabus quintis, & <lb/>refiduum erit .12. cum nouem decimis, vnde habebis numerum dierum perdi-<lb/>tionis.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius rei Å¿peculatione cogitemus in figura <var>.N.</var> duo dicta producta inuicem <lb/>æqualia <var>.o.b.</var> et <var>.n.c.</var> exiÅ¿tente latere <var>.u.c.</var> vt .24. <var>u.o.</var> ut .16: </s> - <s xml:space="preserve">b.u. vt .21. cum tribus quin <lb/>tis, et <var>.u.n.</var> vt .14. cum duabus quintis. </s> - <s xml:space="preserve">Nunc verò Å¿i mente concepta fuerit recta <var>.e.<lb/>a.t.</var> æquidiÅ¿tans <var>.o.c.</var> ita vt rectangulum <var>.o.e.</var> Å¿it .60. </s> - <s xml:space="preserve">tunc rectangulum, Å¿eu productum <lb/><var>b.t.</var> Å¿uperabit rectangulum Å¿eu productum <var>.n.e.</var> per idem .60. ex communi conceptu, <lb/>eo quòd ex producto <var>.n.c.</var> Å¿ublatum eÅ¿t productum <var>.a.c.</var> 24. & producto <var>.o.b.</var> additum <lb/>eÅ¿t productum <var>e.a.</var> 16. rectè igitur feci cum diuiÅ¿erim .60 per .40. vnde prouenit mi <lb/>hi <var>.u.a.</var> ideÅ¿t .1. cum dimidio, quod additum ipÅ¿i <var>.b.u.</var> compoÅ¿uit <var>.b.a.</var> & dempto ex <var>.u.<lb/>n.</var> relinquit <var>.a.n.</var> pro lateribus duorum productorum <var>.b.t.</var> et <var>.n.e</var>.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i idem operator perdidiſſet .60. tunc cogitaremus parallelam dictam <var>.e.a.</var> t <lb/>Å¿uperius ductam eſſe ita vt Å¿ecaret <var>.b.u.</var> & non <var>.u.n.</var> vnde adderet .24. ipÅ¿i producto <var>.n.<lb/>c.</var> & d@meret .16. à producto <var>.b.o</var>.</s> - </p> - <figure position="here"> - <image file="0129-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0129-01"/> - </figure> - <p> - <s xml:space="preserve">CIRCA verò talia quæſita videtur mihi non inutile fore Å¿i aliquid notatu di-<lb/>gnum aduerterim, hoc eÅ¿t quod ſæpe accidere poterit ut caÅ¿us impoſſibiles <lb/>proponantur. </s> - <s xml:space="preserve">Quemadmodum Å¿i aliquis diceret, cupio mihi ueÅ¿timentum con-<lb/>ficere ex duobus pannis colore & pretio differentibus, quorum unus exempli gra- - <pb o="118" rhead="IO. BAPT. BENED." n="130" file="0130" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0130"/> - tia Å¿it albus, rubeus uerò alter, deinde albus Å¿it pretij .40. Å¿olidorum uniuÅ¿cuiuÅ¿-<lb/>que cubiti, rubeusuerò precij .50. uellemqÌue omnes cubitos eſſe .8. nec plus <lb/>nec minus. </s> - <s xml:space="preserve">Vellem etiam Å¿oluere Å¿olidos 450. neque minus.</s> - </p> - <p> - <s xml:space="preserve">Hic igitur caÅ¿us impoſſibilis eÅ¿t, eo quòd .8. cubiti totius rubei eſſent precij Å¿o-<lb/>lidorum .400. tantummodo, unde ex alio panno albo minoris precij Å¿umere ali-<lb/>quid non poſſumus.</s> - </p> - <p> - <s xml:space="preserve">Idem etiam eueniret Å¿i uoluiſſet Å¿oluere Å¿olidos .320. neque plus, eo quòd .8. cu-<lb/>biti illius minoris precij, hoc eÅ¿t .40. Å¿olidorum, eÅ¿sent ualoris .320. Å¿olidorum tan <lb/>tummodo, quare pro alio panno nullus eÅ¿set locus. </s> - <s xml:space="preserve">Animaduertendum igitur erit <lb/>quod numerus poſſibilis ad Å¿oluendum tale quæſitum erit inter .400. et .320. & non <lb/>extra iÅ¿tos terminos, vt vnicuique patere poteÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Similiter idem in hoc alio caÅ¿u accidere poterit, ut Å¿i quis diceret.</s> - </p> - <p> - <s xml:space="preserve">Emi quinque petias panni pro aureis .55. pretium tamen primæ oblitus Å¿um, Å¿ed <lb/>memoria teneo, quòd Å¿ecunda altioris pretij erat quam ipÅ¿a prima per .4. & ter-<lb/>tia precioÅ¿ior Å¿ecunda per .7. et quarta carior tertia per .9. quinta verò Å¿uperabat <lb/>quartam per .2.</s> - </p> - <p> - <s xml:space="preserve">Hic etiam reperitur impoſſibilitas quædam, eo quòd aggregatum omnium ha-<lb/>rum rerum, dato etiam quòd pro prima nihil Å¿olutum eſſet, Å¿uperat aureos .55 quòd <lb/>quidem nullo pacto fieri poteÅ¿t, vt veri Å¿int Å¿upra dicti exceÅ¿sus, Å¿i verus eÅ¿t numerus <lb/>totalis aureorum .55. </s> - <s xml:space="preserve">Nam .4. cum .7. faciunt .11. qui quidem .11. cum .9. efficiunt <num value="20">.<lb/>20.</num> & hic cum .2. facit .22. Å¿ed .22. cum .20. et .11. et .4. dant .57. qui numerus maior <lb/>eÅ¿t quam .55.</s> - </p> - <head type="footer" xml:space="preserve">FINIS THEOREM. ARIT.</head> - <pb o="119" n="131" file="0131" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0131"/> - </div> - </div> - <div type="chapter" level="2" n="2"> - <head xml:space="preserve">DE RATIONIBVS <lb/>OPERATIONVM <lb/>PERSPECTIVAE.</head> - <div type="section" level="3" n="1"> - <head xml:space="preserve">CAP.I.</head> - <p> - <s xml:space="preserve">CVM nullus adhuc (quod Å¿ciam) veras <reg norm="internasque" type="simple">internasqÌ</reg> cauÅ¿as <lb/>operationis perÅ¿pectiuæ perÅ¿ectè docuerit, operæpre-<lb/>cium exiÅ¿timaui <reg norm="aliquam" type="context">aliquã</reg> de ijs diÅ¿putationem Å¿uÅ¿cipere.</s> - </p> - <p> - <s xml:space="preserve">Multi enim <reg norm="eorum" type="context">eorÅ©</reg>, qui huiuÅ¿modi operationis regulas <lb/>præſcribunt, cum eius effectuum veras cauÅ¿as igno-<lb/>rent, varios <reg norm="diuerÅ¿osque" type="simple">diuerÅ¿osqÌ;</reg> errores committunt, vt exempli <lb/>gratia in Å¿ubÅ¿cripta figura Å¿uperficiali <var>.A.</var> volentes degra <lb/>dare (vt dicunt) rectangulum <var>.q.a.</var> in triangulo <var>.i.d.q.</var> du-<lb/>cunt <reg norm="parallelam" type="context">parallelã</reg> ipÅ¿i <var>.q.d.</var> à puncto <var>.B.</var> interÅ¿ecationis lineæ-<lb/>o.l. cum latere <var>.i.d.</var> trianguli, & (idem) indifferenter, ean-<lb/>dem quoque à puncto <var>.Z.</var> interÅ¿ecationis ipÅ¿ius <var>.o.l.</var> cum perpendiculari <var>.x.i.</var> ducunt. <lb/></s> - <s xml:space="preserve">neÅ¿cientes hunc Å¿olum eſſe verum modum, n onitem alium, quia Å¿i alius, talis eſſet, <lb/>hic, verus non exiÅ¿teret, nam Å¿i vellent Å¿eÅ¿e excuÅ¿ are, quòd ducendo dictam paralle-<lb/>lam à puncto <var>.B.</var> hoc fiat præſupponendo planum ipÅ¿ius <var>.i.d.q.</var> verÅ¿us rectangulum <var>.<lb/>q.a.</var> orizontale inclinatum, Å¿ecundum angulum <var>.i.d.q.</var> hæc excuÅ¿atio accipien-<lb/>da non eſſet, quia horum conÅ¿enÅ¿u, præſupponendo planum <var>.i.d.q.</var> inclinatum, <lb/>anguli inferiores rectanguli degradati, non tam acuti, quam Å¿unt duo <var>.i.d.q.</var> et <var>.i.q.<lb/>d.</var> eſſe deberent, quod facilè eorum ratione innoteÅ¿cet, quæ de figura corporea <var>.<lb/>A.</var> hîc Å¿ubÅ¿cripta mox proponam, præter id, quòd volentes deinde aÅ¿picere qua-<lb/>dratum degradatum, oporteret huiuÅ¿modi planum reÅ¿pectu oculi ita collocare, <lb/>quemadmodum Å¿e habet linea <var>.i.d.</var> reÅ¿pectu <var>.o.</var> quod factu nimis arduum eſſet.</s> - </p> - <p> - <s xml:space="preserve">Vera igitur ratio erit ducere parallelam <var>.e.r.</var> ad <var>.q.d.</var> à puncto <var>.Z.</var> communi ip-<lb/>Å¿is <var>.o.l.</var> et <var>.x.i.</var> perpendiculari ipÅ¿i <var>.l.p</var>.</s> - </p> - <figure position="here"> - <image file="0131-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0131-01"/> - </figure> - <pb o="120" rhead="IO. BAPT. BENED." n="132" file="0132" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0132"/> - <p> - <s xml:space="preserve">Pro cuius rei Å¿peculatione imaginemur in figura corporea .A: <var>q.a.</var> eſſe figuram re-<lb/>ctangulam <reg norm="orizontalemque" type="simple">orizontalemqÌ;</reg> ad degradandam Å¿uper aliquod planum perpendiculare <lb/>orizonti, & cum eo primum coniunctam in linea <var>.q.d.</var> cuius plani triangulum <var>.i.q.d.</var> <lb/>pars erit, Å¿it autem oculus reÅ¿picientis <var>.o.</var> cuius altitudo <var>.o.p.</var> ab orizonte, qui <reg norm="quidem" type="context">quidẽ</reg> <lb/>conÅ¿picit rectangulum dictum orizontale <var>.q.a.</var> in pyramide <var>.o.q</var>: <var>o.u</var>: <var>o.a.</var> et <var>.o.d.</var> <lb/>terminata quatuor triangulis <var>.o.q.u</var>: <var>o.u.a</var>: <var>o.a.d.</var> et <var>.o.d.q.</var> Å¿it verò primum ita <lb/>collocatus pes <var>.p.</var> eius qui reÅ¿picit, vt linea <var>.p.l.</var> perpendicularis ipÅ¿i <var>.u.a.</var> lateri re-<lb/>ctanguli, medio loco poÅ¿ita Å¿it, inter <var>.a.n.</var> et <var>.u.s</var>. </s> - <s xml:space="preserve"><reg norm="Idque" type="simple">IdqÌ;</reg> primum nobis erit exem-<lb/>plum.</s> - </p> - <p> - <s xml:space="preserve">Imaginemur nunc lineas <var>.u.q.</var> et <var>.a.d.</var> indefinitè productas eſſe, quæ in Å¿uperficie-<lb/>bus duorum triangulorum <var>.o.u.q.</var> et <var>.o.a.d.</var> & rectanguli orizontalis <var>.q.a.</var> ex <ref id="ref-0018">prima <lb/>vndecimi Euclid.</ref> poÅ¿itæ erunt. </s> - <s xml:space="preserve">Imaginemur etiam lineam <var>.p.s.n.</var> perpendicula-<lb/>rem ipÅ¿i <var>.p.l.</var> quæ etiam cum duabus <var>.u.q.s.</var> et <var>.a.d.n.</var> ex .34. primi Euclid. angulos <lb/>rectos conÅ¿tituet, cum ex .28. duæ <var>.u.q.s.</var> et <var>.a.d.n.</var> Å¿int parallelæ ipÅ¿i <var>.p.l.</var> et <var>.s.n.</var> ipÅ¿i <var>.u.<lb/>a.</var> & quia Å¿upponitur <var>.o.p.</var> perpendicularis plano orizontali, Angulus ergò <var>.o.p.l.</var> re-<lb/>ctus erit ex Å¿ecunda definitione .11. Euclid. </s> - <s xml:space="preserve">Imaginemur quoque ductas eſſe <lb/>duas <var>.o.s.</var> et <var>.o.n.</var> vnde <var>.l.p.</var> ei Å¿uperficiei, in qua Å¿unt duæ lineæ <var>.o.p.</var> et <var>.s.n.</var> ex .4. <lb/>11. perpendicularis erit, & Å¿uperficies orizontalis <var>.a.s.</var> perpendicularis erit cum dicta <lb/><var>o.s.n.</var> ex .18. eiuÅ¿dem lib. vnde ex dicta definitione <var>.o.s.u.</var> et <var>.o.n.a.</var> erunt anguli recti <lb/>et <var>.o.s.</var> et <var>.o.n.</var> ex communi Å¿cientia, in Å¿uperficiebus duorum triangulorum <var>.o.u.q.</var> et <var>.<lb/>o.a.d.</var> erunt, Å¿i noluerimus cogere aduerÅ¿arium ad confitendum duas lineas rectas in-<lb/>cludere Å¿uperficiem, quemadmodum cogere-<lb/> - <anchor type="figure" xlink:label="fig-0132-01a" xlink:href="fig-0132-01"/> - tur facere, Å¿i opinaretur duas alias rectas per <lb/>eadem puncta <var>.o.s.n.</var> tranÅ¿ire, quæſunt in di-<lb/>ctis Å¿uperficiebus. </s> - <s xml:space="preserve">Vnde <var>.o.s.</var> et <var>.o.n.</var> communes <lb/>erunt Å¿ectiones duarum dictarum <reg norm="Å¿uperficierum" type="context">Å¿uperficierÅ©</reg> <lb/>cum Å¿uperficie <var>.o.s.n</var>. </s> - <s xml:space="preserve">Imaginemur nunc has <lb/>duas Å¿uperficies <var>.o.u.</var> et <var>.o.a.</var> quarum commu-<lb/>nis Å¿ectio Å¿it <var>.o.t.</var> (quæ erit linea recta ex .3. lib. <lb/>II.) quæ erunt perpendiculares Å¿uperficiei <var>.o.s.<lb/>n.</var> ex .4. et .14. iam dictis. </s> - <s xml:space="preserve">& ex .19. eiuÅ¿dem <lb/><var>o.t.</var> perpendicularis eidem Å¿uperficiei <var>.o.s.n.</var> <lb/>erit, & ex .6. eiuÅ¿dem hæc linea <var>.o.t.</var> duabus <var>.u.<lb/>q.s.</var> et <var>.a.d.n.</var> parallela exiÅ¿ter, & ex .9. eiuÅ¿dem <lb/>hæc linea <var>.o.t.</var> duabus <var>.u.q.s.</var> et <var>.a.d.n.</var> parallela <lb/>exiÅ¿tet, & ex eadem .9. erit parallela ipÅ¿i <var>.p.l.</var> <lb/>Imaginemur nunc planum, Å¿uper quod deÅ¿ide <lb/>remus videre quadrangulum orizontale, quod <lb/>planum, exempli gratia, Å¿it primo, vt iam dixi-<lb/>mus, locatum in linea <var>.q.d.</var> ad angulos rectos <lb/>cum plano orizontali, cuius communes Å¿ectio <lb/>nes cum Å¿uperficiebus <var>.s.t.</var> et <var>.n.t.</var> viÅ¿ionis la-<lb/>terum <var>.u.q.</var> et <var>.a.d.</var> Å¿int <var>.i.q.</var> et <var>.i.d.</var> & com-<lb/>munis Å¿ectio trianguli <var>.o.u.a.</var> ideÅ¿t viÅ¿ionis <lb/>lateris <var>.a.u.</var> cum dicto plano, Å¿it <var>.r.e</var>. </s> - <s xml:space="preserve">Vnde ex <lb/>communi Å¿cientia rectangulum orizontale, <lb/>oculo <var>.o.</var> Å¿eipÅ¿um patefaciet in plano <var>.i.q.d.</var> Å¿e- - <pb o="121" rhead="DE PERSPECT." n="133" file="0133" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0133"/> - cundum figuram quadrilateram <var>.q.d.r.e</var>. </s> - <s xml:space="preserve">Communis autem Å¿ectio Å¿uperficiei <var>.p.t.</var> <lb/>cum dicto plano, Å¿it <var>.i.x.</var> quæ <var>.i.x.</var> perpendicularis erit <var>.s.a.</var> Å¿uperficiei orizontali ex <lb/>19. lib. 11. quia <var>.p.t.</var> eÅ¿t etiam orizonti perpendicularis ex .18. eiuÅ¿dem, cum <var>.o.p.</var> ei-<lb/>dem perpendicularis exiÅ¿tat. </s> - <s xml:space="preserve">Vnde <var>.i.x.</var> erit altitudo trianguli <var>.i.q.d.</var> & æqualis ipÅ¿i <var>.<lb/>o.p.</var> ex <ref id="ref-0019">.34. primi</ref>. </s> - <s xml:space="preserve">Sit deinde <var>.o.l.</var> <reg norm="communis" type="context">cõmunis</reg> Å¿ectio Å¿uperficiei triangularis <var>.o.a.u.</var> <reg norm="cum" type="context">cÅ©</reg> <lb/>Å¿uperficie <var>.p.t.</var> quæ <var>.o.l.</var> Å¿ecando lineam <var>.e.r.</var> in puncto <var>.Z.</var> nobis oÅ¿tendet quantum di-<lb/>Å¿tare Å¿eu eminens eſſe debeat latus <var>.e.r.</var> in plano ab <var>.q.d.</var> medio ipÅ¿ius <var>.z.x</var>. </s> - <s xml:space="preserve">Et quia <lb/>præſuppoÅ¿uimus <var>.p.l.</var> in eodem medio, inter <var>.u.s.</var> et <var>.a.n.</var> ideo <var>.x.q.</var> Ä™qualis erit <var>.x.d.</var> <lb/>& ex .4. lib. primi <var>.i.q.</var> ipÅ¿i <var>.i.d.</var> et <var>.e.r.</var> parallela ipÅ¿i <var>.q.d.</var> ex .6. lib. 11. cum ipÅ¿a quoque <lb/>Å¿it perpendicularis Å¿uperficiei <var>.p.t.</var> ex <ref id="ref-0020">.19. eiuÅ¿dem</ref>. </s> - <s xml:space="preserve">HucuÅ¿que igitur in figura cor-<lb/>porea <var>.A.</var> prodeunt in lucem omnes cauſæ effectuum figuræ Å¿uperficialis <var>.A.</var> ideÅ¿t vn <lb/>de fiat, vt in ipÅ¿a figura Å¿uperficiali, triangulum <var>.o.p.l.</var> tale conÅ¿urgat, & quid Å¿ignifi-<lb/>cet <var>.o.</var> et <var>.o.p.</var> et <var>.p.l.</var> et <var>.o.l.</var> & quam ob cauÅ¿am tale quoque formetur triangulum <var>.i.<lb/>q.d.</var> atque in tantam altitudinem, quantam obtinet <var>.o.p.</var> & quid Å¿int latera <var>.i.q.</var> et <var>.i.<lb/>d.</var> & quare erigatur <var>.x.i.</var> parallela ipÅ¿i <var>.p.o.</var> ab eadem <var>.p.o.</var> tanto Å¿patio diÅ¿tans, & qua <lb/>ratione producatur à puncto <var>.Z.</var> ipÅ¿a <var>.Z.r.e.</var> parallela ipÅ¿i <var>.q.d</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0132-01" xlink:href="fig-0132-01a"> - <image file="0132-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0132-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Nunc obÅ¿eruandum eÅ¿t, quòd Å¿i planum ipÅ¿ius <var>.i.q.d.</var> in figura corporea aliquan-<lb/>tulum inclinatum eſſet orizontem verÅ¿us, anguli <var>.i.q.d.</var> et <var>.i.d.q.</var> maiores exiÅ¿terent, <lb/>quà m cum idem eÅ¿t ipÅ¿i orizonti perpendiculare, quemadmodum clarè demonÅ¿tra-<lb/>tum fuit in .39. primi Vitelionis.</s> - </p> - <p> - <s xml:space="preserve">Non igitur rectè fit Å¿i in figura Å¿uperficiali ducatur à puncto <var>.B.</var> parallela ipÅ¿i <var>.q.d.</var> <lb/>abÅ¿que maiori apertura angulorum <var>.i.q.d.</var> et <var>.i.d.q</var>.</s> - </p> - <figure position="here"> - <image file="0133-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0133-01"/> - <caption xml:space="preserve">SVPERFICIALIS.</caption> - </figure> - </div> - <div type="section" level="3" n="2"> - <head xml:space="preserve">CAP. II.</head> - <p> - <s xml:space="preserve">CVM verò duæ præcedentes figuræ intellectæ erunt, facilè quoque erit intel-<lb/>ligere duas Å¿ubÅ¿equentes <var>.B.B.</var> in corporea quarum <var>.p.l.</var> extra lineas <var>.u.s.</var> et <var>.a.n.</var> <lb/>reperitur, vbi enim aduertendum erit oportere Å¿umere Å¿emper <var>.p.x.</var> figuræ Å¿uperfi-<lb/>cialis æqualem ei, quæ eÅ¿t corporeæ, & eidem Å¿uperficiali, adiungere <var>.x.d.</var> æqualem <lb/>ei, quæ eÅ¿t corporeæ, & compoÅ¿ito <var>.p.d.</var> ex dictis duabus lineis, in figura Å¿uperficiali, <lb/>addere <var>.d.q.</var> æqualem ei, quæ eÅ¿t figuræ corporeæ, deinde accipere punctum <var>.l.</var> in fu-<lb/>perficiali - <pb o="122" rhead="IO. BAPT. BENED." n="134" file="0134" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0134"/> - perficiali, ita diÅ¿tans à <var>.p.</var> vt in corporea reperitur. </s> - <s xml:space="preserve">Po-<lb/> - <anchor type="figure" xlink:label="fig-0134-01a" xlink:href="fig-0134-01"/> - Å¿teà . <var>x.i.</var> erigetur æqualis lineæ <var>.p.o.</var> & Å¿uis terminis <lb/>concludetur triangulum <var>.i.q.d.</var> & id quod remanet. <lb/></s> - <s xml:space="preserve">Vnde Å¿i longius diÅ¿erendo progrediare, patebit ex <num value="4">.<lb/>4.</num> primi <var>.i.d.</var> Å¿uperficialem, futuram æqualem <var>.i.d.</var> <lb/>corporeæ. </s> - <s xml:space="preserve">Idem dico de <var>.i.q.</var> & de reliquis.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0134-01" xlink:href="fig-0134-01a"> - <image file="0134-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0134-01"/> - <caption xml:space="preserve">CORPOREA.</caption> - </figure> - </div> - <figure position="here"> - <image file="0134-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0134-02"/> - <caption xml:space="preserve">SVPERFICIALIS.</caption> - </figure> - </div> - <div type="section" level="3" n="3"> - <head xml:space="preserve">CAP. III.</head> - <p> - <s xml:space="preserve"><emph style="sc">MOdvs</emph> hic, proprius eÅ¿t, & vniuerÅ¿alis, licet in figura Å¿uperficiali <var>.A.</var> Å¿upe-<lb/>rius poÅ¿ita, Å¿ecundum communem antiquorum conÅ¿uetudinem exemplum <lb/>dederim, effectus enim idem eÅ¿t. </s> - <s xml:space="preserve">Sed Å¿i quis vellet conÅ¿iderare dictam figuram Å¿u-<lb/>perficialem <var>.A.</var> Å¿ecundum eum modum, quem de figura <var>.B.</var> Å¿uperficiali pręſcripÅ¿i, id <lb/>poterit in Å¿ubÅ¿cripta figura <var>.C.</var> Å¿peculari in hunc modum. </s> - <s xml:space="preserve">Accipiet enim <var>.p.x.</var> Å¿uper <lb/>ficialem, æqualem corporeæ, & quia in ipÅ¿a corporea <var>.A.</var> Å¿uppoÅ¿ita fuit linea <var>.p.l.</var> <lb/>ideÅ¿t punctum <var>.x.</var> inter duas <var>.u.s.</var> et <var>.a.n.</var> Å¿ecabimus <var>.p.x.</var> Å¿uperficialem in puncto <var>.d.</var> ita, <lb/>vt <var>.d.x.</var> Å¿uperficialis, æqualis Å¿it corporeæ, & ipÅ¿i Å¿uperficiali <var>.p.x.</var> addetur <var>.x.q.</var> æqualis <lb/>corporeæ. </s> - <s xml:space="preserve">vnde <var>.q.d.</var> Å¿uperfi-<lb/>cialis æqualis erit corporeÄ™,<lb/> - <anchor type="figure" xlink:label="fig-0134-03a" xlink:href="fig-0134-03"/> - et <var>.p.x.</var> Å¿uperficiali addetur <var>.<lb/>x.l.</var> æqualis corporeæ. </s> - <s xml:space="preserve">De <lb/>ijs poÅ¿tea quæ dicenda Å¿uper-<lb/>Å¿unt, iam Å¿atis Å¿uperqÌue di-<lb/>ximus.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0134-03" xlink:href="fig-0134-03a"> - <image file="0134-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0134-03"/> - <caption xml:space="preserve">SVPERFICIALIS.</caption> - </figure> - </div> - <p> - <s xml:space="preserve">Quamobrem, punctum <var>.<lb/>x.</var> aut intra, aut extra lineam <var>.<lb/>q.d.</var> veniat, hunc modum Å¿e-<lb/>quentes, in errorem non in-<lb/>cidemus, imò efficietur qua -<lb/>drilaterum <var>.q.r.</var> Å¿uperficiale, <lb/>Å¿imile, & æquale corporeo.</s> - </p> - </div> - <div type="section" level="3" n="4"> - <head xml:space="preserve">CAP. IIII.</head> - <p> - <s xml:space="preserve"><emph style="sc">PVnctvm</emph> verò <var>.i.</var> (quod verum eÅ¿t punctum perÅ¿pectiuæ, vt practici dicere Å¿o <lb/>lent) quid Å¿it, hac via & ratione Å¿ub noÅ¿tram cognitionem cadit: </s> - <s xml:space="preserve">quòd nihil - <pb o="123" rhead="DE PERSPECT." n="135" file="0135" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0135"/> - aliud eÅ¿t, quà m punctum, varijs Å¿ectionibus commune, & huiuÅ¿modi punctum, ocu <lb/>lus non eÅ¿t, quemadmodum multi Pictores, Sculptores, Architecti, & PerÅ¿pectiui <lb/>ignari, ipÅ¿um punctum, oculum appellando, falsò crediderunt, quaÅ¿i <reg norm="punctum" type="context">punctÅ©</reg> <var>.i.</var> per-<lb/>Å¿pectiuæ oculus eſſet.</s> - </p> - <p> - <s xml:space="preserve">In Å¿upradictis igitur figuris manifeÅ¿te eluceÅ¿cit cauÅ¿a diminutionis obiectorum, <lb/>& altitudinis trianguli æqualis ei, quæ eÅ¿t oculi à plano orizontali, vt etiam diÅ¿tantiÄ™ <var>.<lb/>p.l.p.x.</var> & cuiuÅ¿uis tandem rei. </s> - <s xml:space="preserve">Sed vt huius effectus Å¿cientia magis in vniuerÅ¿um pa-<lb/>retur. </s> - <s xml:space="preserve">Volo duas hic Å¿ubÅ¿criptas figuras <var>.D.</var> corpoream, & <var>.D.</var> Å¿uperficialem à vo-<lb/>bis conÅ¿iderari, in quarum corporea, linea <var>.p.l.</var> Å¿it extra duas <var>.u.s.</var> et <var>.a.n.</var> vt in figu-<lb/>ra <var>.B.</var> locata, ita tamen vt planum trianguli <var>.i.q.d.</var> diÅ¿iunctum Å¿it à rectangulo Å¿uper-<lb/>ficiali, ideÅ¿t, vt Å¿eparatum exiÅ¿tat à linea <var>.q.d.</var> latere ipÅ¿ius rectanguli, & Å¿it etiam obli <lb/>quum, reÅ¿pectu ipÅ¿ius rectanguli, ideÅ¿t vt communis Å¿ectio dicti plani cum Å¿uperficie <lb/><var>a.s.</var> orizontalis ipÅ¿i <var>.u.a.</var> parallela non Å¿it, Å¿ed Å¿it obliqua, Å¿i tamen idem planuni per-<lb/>pendiculare dictæ Å¿uperficiei orizontali <var>.a.s.</var> erit: </s> - <s xml:space="preserve">& dicta communis Å¿ectio exprimaÂ<lb/>tur characteribus .q.ω.α.d.x. nunc in figura corporea habebimus figuram <var>.e.r.c.m.</var> <lb/>in plano, quod viÅ¿ualem pyramidem Å¿ecat, medio cuius figuræ <var>.e.r.c.m.</var> oculus po-<lb/>Å¿itus in <var>.o.</var> rectangulum orizontale conÅ¿picit. </s> - <s xml:space="preserve">Volentes vero nunc in figura <var>.D.</var> Å¿uper-<lb/>ficiali eam deÅ¿cribere, faciem us <var>.p.x.</var> Å¿uperficialem, æqualem corporeæ, eiqÌue <lb/>addemus <var>.x.l.</var> æqualem corporeæ, aut Å¿umemus <var>.p.l.</var> eidem corporeæ <reg norm="aequa- lem" type="simple">Ä™qua-<lb/>lem</reg>, quam Å¿ecabimus in puncto <var>.x.</var> eodem planè modo, quo corporea reperi-<lb/>tur diuiÅ¿a; </s> - <s xml:space="preserve">erigemus deinde <var>.p.o.</var> et <var>.x.i.</var> æquales corporeis. </s> - <s xml:space="preserve">Secabimus deinde <var>.x.q.</var> <lb/>æqualem corporeæ, & ducemus <var>.q.i.</var> et <var>.l.o.</var> vnde habebimus triangulos <var>.o.p.l.</var> et <var>.i.x.<lb/>q.</var> Å¿imiles & æquales corporeis ex .4. primi Eucli. </s> - <s xml:space="preserve">Secabimus deinde <var>.q.x.</var> in pun-<lb/>cto <var>.d.</var> eadem ratione, qua Å¿ecta fuit corporea, & ducemus lineam <var>.d.i.</var> vnde habebi-<lb/>mus triangulos <var>.i.d.q.</var> et <var>.i.d.x.</var> Å¿imiles corporeis. </s> - <s xml:space="preserve">& mediante triangulo <var>.i.q.d.</var> hu-<lb/> - <anchor type="figure" xlink:label="fig-0135-01a" xlink:href="fig-0135-01"/> - cuÅ¿que habebimus Å¿itus duorum laterum figurÄ™ <lb/>rectanguli degradati, ideÅ¿t Å¿itus ipÅ¿ius <var>.e.m.</var> et <var>.r.<lb/>c.</var> etiam Å¿i adhuc neÅ¿ciatur in qua parte ipÅ¿ius <var>.i.<lb/>q.</var> & ipÅ¿ius <var>.i.d.</var> eſſe <reg norm="debeant" type="context">debeãt</reg>. </s> - <s xml:space="preserve">Quod Å¿i Å¿cire volue <lb/>rimus <reg norm="Å¿ecabitur" type="simple">Å¿ecabit̃</reg>. <var>p.l.</var> in <reg norm="puncto" type="context">pÅ©cto</reg> <var>.g.</var> Å¿imilis corporeæ, <lb/>Å¿i in ipÅ¿a tamen corporea prius protraxerimus <lb/>lineam <var>.q.d.</var> latus rectanguli vÅ¿que ad <var>.p.l.</var> in pun <lb/>cto <var>.g</var>. </s> - <s xml:space="preserve">Ducetur deinde linea <var>.o.g.</var> Å¿uperficialis, <lb/>quæ Å¿ecabit lineam <var>.i.x.</var> in puncto <var>.f.</var> linea vero <var>.<lb/>o.l.</var> in puncto <var>.z.</var> punctis Å¿itis in <var>.i.x.</var> Å¿uperficiali, <lb/>prÄ™cisè vt in corporea, <reg norm="quemadmodum" type="context">quemadmodÅ©</reg> quilibet <lb/>ex Å¿e facilè cognoÅ¿cere poteÅ¿t. </s> - <s xml:space="preserve">Deinde in cor <lb/>porea, in Å¿uperficie orizontali ducatur <var>.p.q.</var> et <lb/> - <anchor type="figure" xlink:label="fig-0135-02a" xlink:href="fig-0135-02"/> - <pb o="124" rhead="IO. BAPT. BENED." n="136" file="0136" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0136"/> - <var>.p.u.</var> & imaginemur <var>.o.q.</var> in Å¿uperficie <var>.t.s.</var> vnde trianguli <var>.o.p.q.</var> et <var>.o.p.u.</var> erunt perpen <lb/>diculares orizonti ex <ref id="ref-0021">.18. lib. 11.</ref> et <var>.ω.m.</var> et. α. e. communes Å¿ectiones dictorum <reg norm="duorum" type="context">duorÅ©</reg> <lb/>triangulorum cum plano trianguli <var>.i.q.x.</var> ipÅ¿i quoque plano ex .19. eiuÅ¿dem lib. erunt <lb/>perpendiculares. </s> - <s xml:space="preserve">Nuncautem Å¿ecetur <var>.q.x.</var> Å¿uperficialis in punctis <var>.ω.</var> et .α. eadem ra-<lb/>tione; </s> - <s xml:space="preserve">qua corporea Å¿ecta Å¿uit à duabus <var>.p.q.</var> et <var>.p.u.</var> à quibus punctis <var>.ω.</var> et. α. Å¿uperficia <lb/>libus ductæ Å¿int duæ <var>ω.m.</var> et .α.e. perpendiculares vÅ¿que ad latus <var>.i.q.</var> in punctis <var>.m.</var> et <var>.<lb/>e.</var> quÄ™ Å¿itum habebunt in <var>.i.q.</var> Å¿uperficiali prÄ™cisè, vt in corporea, ex .26. primi, du-<lb/>cendo deinde in Å¿uperficiali duas <var>.m.f.</var> et <var>.e.Z.</var> eæ æquales erunt corporeis ex .4. pri-<lb/>mi, & Å¿ic anguli <var>.i.e.z.</var> et <var>.i.m.f.</var> & eÄ™ duÄ™ lineÄ™ <var>.e.z.</var> et <var>.m.f.</var> fectÄ™ erunt à linea <var>.i.d.</var> in duo<lb/> - <anchor type="figure" xlink:label="fig-0136-01a" xlink:href="fig-0136-01"/> - bus punctis <var>.r.</var> et <var>.c.</var> vnde <var>.e.r.</var> et <var>.m.c.</var> æquales <lb/>erunt corporeis ex .26. primi, Å¿ed ita quo-<lb/>que Å¿e habent duÄ™ <var>.e.m.</var> et <var>.r.c.</var> Å¿i verum eÅ¿t <reg norm="quod" type="simple">ê™</reg> dif <lb/>ferentiÄ™ rerum æqualium Å¿int adinuicem etiam <lb/>æquales. </s> - <s xml:space="preserve">Hac ratione igitur habebimus figu-<lb/>ram quadrilateram <var>.m.e.r.c.</var> Å¿uperficialem om <lb/>ninò Å¿inlilem, & Ä™qualem corporeæ. </s> - <s xml:space="preserve">Is tamen <lb/>modus prolixus eÅ¿t, & arduus, quam ob cau-<lb/>Å¿am neque ego vnquam <reg norm="eum" type="context">eÅ©</reg> viui accommo-<lb/>darem, neque alijs, vt eodem vterentur Å¿ua-<lb/>derem.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0135-01" xlink:href="fig-0135-01a"> - <image file="0135-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0135-01"/> - <caption xml:space="preserve">CORPOREA.</caption> - </figure> - <figure xlink:label="fig-0135-02" xlink:href="fig-0135-02a"> - <image file="0135-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0135-02"/> - <caption xml:space="preserve">SVPERFICIALIS.</caption> - </figure> - <figure xlink:label="fig-0136-01" xlink:href="fig-0136-01a"> - <image file="0136-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0136-01"/> - <caption xml:space="preserve">CORPOREA.</caption> - </figure> - </div> - <figure position="here"> - <image file="0136-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0136-02"/> - <caption xml:space="preserve">SVPERFICIALIS.</caption> - </figure> - </div> - <div type="section" level="3" n="5"> - <head xml:space="preserve">CAP.V.</head> - <p> - <s xml:space="preserve"><emph style="sc">ESt</emph> igitur Å¿ciendum, quòd qui Å¿ciuerit vnum Å¿olum punctum locare in perÅ¿pe <lb/>ctiua, eo modo quem nunc proponam, facilè quoque Å¿ciet Å¿upra quoduis <reg norm="planum" type="context">planÅ©</reg> <lb/>(quod tamen Å¿it perpendiculare orizonti) quamlibet rem locare. </s> - <s xml:space="preserve">Quam ob cauÅ¿am <lb/>imaginemur hic Å¿ubÅ¿criptas duas figuras <var>.E.</var> <reg norm="corpoream" type="context">corporeã</reg>, & E. Å¿uperficialem, & in qua-<lb/>drilatero rectangulo orizontali <var>.a.u.q.d.</var> imaginemur eſſe punctum <var>.b.</var> quodlibet col-<lb/>locandum in aliquo plano perpendiculari orizonti locato, quemadmodum Å¿uppo-<lb/>nebatur in figura <var>.A.</var> corporea. </s> - <s xml:space="preserve">Imaginemur ergo in ipÅ¿a figura <var>.E.</var> corporea radi-<lb/>um viÅ¿ualem <var>.o.b.</var> qui Å¿ectus Å¿it à noÅ¿tro plano in <var>.k.</var> quod quidem <var>.k.</var> quærendum eÅ¿t <lb/>in triangulo <var>.i.q.d.</var> ipÅ¿ius plani. </s> - <s xml:space="preserve">Volo ob hanc igitur rem, vt à puncto <var>.b.</var> in figura <var>.E.</var> <lb/>Å¿uperficiali ducatur <var>.b.c.</var> ad rectos <reg norm="cum" type="context">cÅ©</reg> <var>.q.d.</var> & à puncto <var>.c.</var> ad <var>.i.</var> ducatur linea <var>.c.i.</var> et <var>.b.m.</var> <lb/>parallela ipÅ¿i <var>.q.d.</var> quÄ™ ab ipÅ¿a <var>.x.l.</var> in puncto <var>.m.</var> erit diuiÅ¿a, & hÄ™c <var>.x.m.</var> è directo con-<lb/>iuncta cum <var>.p.x.</var> ducatur <var>.o.m.</var> quæ ab <var>.i.x.</var> Å¿ecta erit in puncto <var>.f.</var> à quo ducendo dein-<lb/>de <var>.f.g.h.</var> parallela <var>.q.d.</var> ab <var>.i.c.</var> in puncto <var>.K.</var> erit diuiÅ¿a. </s> - <s xml:space="preserve">Atque id erit quod nobis <lb/>inquirendum propoÅ¿ueramus.</s> - </p> - <pb o="125" rhead="DE PERSPECT." n="137" file="0137" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0137"/> - <p> - <s xml:space="preserve">Ad cuius rei <reg norm="Å¿peculationem" type="context">Å¿peculationẽ</reg>, imaginatione con<lb/> - <anchor type="figure" xlink:label="fig-0137-01a" xlink:href="fig-0137-01"/> - cipiamus lineam <var>.b.c.</var> corpoream, protractam eÅ¿ <lb/>Å¿e vÅ¿que ad <var>.y.</var> lineæ <var>.s.n.</var> & imaginatione Å¿it com <lb/><reg norm="præhenÅ¿a" type="context">præhẽſa</reg> linea <var>.y.o.</var> et <var>.b.</var> R<unsure/>. parallela eidem, ideo <lb/>ob rationes iam dictas de figura <var>.A.</var> hæ tres li-<lb/>neæ <var>.o.y</var>: <var>i.c</var>: et. </s> - <s xml:space="preserve">R<unsure/> <var>.b.</var> Å¿imul cum linea <var>.o.b.</var> erunt <lb/>in vna eademqÌue Å¿uperficie plana, quam cha-<lb/>racteribus <var>.y.</var> R<unsure/>. notemus .et <var>.i.c.</var> eius erit Å¿e-<lb/>ctio communis cum plano, in quo quæritur <reg norm="pun- ctum" type="context">pÅ©-<lb/>ctum</reg>, et <var>.f.k.</var> ipÅ¿ius plani cum triangulo <var>.o.b.m.</var> <lb/>erit Å¿ectio communis, & parallela ipÅ¿i <var>.q.d.</var> ex <ref id="ref-0022">.<lb/>6. lib. 11.</ref> quia <var>.k.f.</var> perpendicularis eÅ¿t Å¿uperfi-<lb/>ciei <var>.p.t.</var> ex .19. eiuÅ¿dem cum triangulus <var>.o.<lb/>b.m.</var> eidem Å¿uperficiei <var>.p.t.</var> ex .18. eiuÅ¿dem <lb/>perpendicularis exiÅ¿tat. </s> - <s xml:space="preserve">Vnde perÅ¿picuè pa-<lb/>tet ratio quare protracta<unsure/> Å¿it parallela <var>.b.c.</var> et <lb/>quare ducta Å¿it <var>.i.c.</var> et coniuncta <var>.x.m.</var> cum <var>.x.<lb/>p.</var> directè, & quare ducta Å¿it <var>.o.m.</var> et <var>.f.k</var>. </s> - <s xml:space="preserve">Lau-<lb/>do igitur vt Å¿emper præſupponatur <var>.p.x.</var> perpen <lb/>dicularis baÅ¿i ipÅ¿ius plani & præſupponatur, (vt <lb/>rem totam vnò verbo complectar) Å¿uperficies <var>.<lb/>p.t.</var> perpendicularis plano, & orizonti. </s> - <s xml:space="preserve">Quod <lb/>reliquum eÅ¿t, neceſſariv<unsure/>m non eÅ¿t, niÅ¿i ad Å¿pe-<lb/>culandum. </s> - <s xml:space="preserve">Neceſſariæ ergo non Å¿unt aliæli-<lb/>neæ, quà m.p.x: <var>p.o.x.i</var>: <var>b.c</var>: et <var>.x.m.</var> è dire-<lb/>cto coniuncta cum <var>.p.x.</var> (quæ <var>.x.m.</var> coniuncta <lb/> - <anchor type="figure" xlink:label="fig-0137-02a" xlink:href="fig-0137-02"/> - <pb o="126" rhead="IO. BAPT. BENED." n="138" file="0138" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0138"/> - æqualis Å¿it ipÅ¿i <var>.b.c.</var>) <lb/> - <anchor type="figure" xlink:label="fig-0138-01a" xlink:href="fig-0138-01"/> - <var>o.m.</var> etiam <var>.i.c</var>: et <var>.f.<lb/>k.</var> vt in figura <var>.F.</var> cla <lb/>riſſimè patet. </s> - <s xml:space="preserve">Alias <lb/><reg norm="autem" type="context">autẽ</reg> multas lineas in <lb/>alijs figuris non <reg norm="aliam" type="context">aliã</reg> <lb/>ob <reg norm="camm" type="context">cãm</reg> duxi, <reg norm="quam" type="context">quã</reg> ad <lb/>facilius <reg norm="eruendas" type="context">eruẽdas</reg> è te-<lb/>nebris ignorantiæ, & <lb/>in cognitionis lucem <lb/>proferendas horum <lb/>effectuum cauÅ¿as, vt <lb/>dixi.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0137-01" xlink:href="fig-0137-01a"> - <image file="0137-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0137-01"/> - <caption xml:space="preserve">CORPOREA.</caption> - </figure> - <figure xlink:label="fig-0137-02" xlink:href="fig-0137-02a"> - <image file="0137-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0137-02"/> - <caption xml:space="preserve">SVPERFICIALIS.</caption> - </figure> - <figure xlink:label="fig-0138-01" xlink:href="fig-0138-01a"> - <image file="0138-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0138-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="6"> - <head xml:space="preserve">CAP. VI.</head> - <p> - <s xml:space="preserve"><emph style="sc">SEd</emph> vtlocum altitudinis, in noÅ¿tro plano perpendiculari orizonti, & ita <reg norm="locatum" type="context">locatÅ©</reg><unsure/>, <lb/>vt poÅ¿tremo diximus, inueniamus; </s> - <s xml:space="preserve">duas hîc Å¿ubÅ¿criptas figuras conÅ¿iderabimus <var>.<lb/>G.</var> corpoream, & G. Å¿uperficialem, Å¿imiles duabus <var>.E.E.</var> proximè præcedentibus, <lb/>in quarum corporea Å¿it linea <var>.b.M.</var> altitudinis perpendicularis orizonti. </s> - <s xml:space="preserve">Quare Å¿i <lb/>deÅ¿iderabis inuenire in noÅ¿tro plano Å¿itum puncti <var>.M.</var> ideÅ¿t punctum radij <var>.o.M.</var> vi-<lb/>Å¿ualis in quo ipÅ¿e radius à plano eÅ¿t diuiÅ¿us, quod Å¿it <var>.R.</var> quamuis extra <reg norm="triangulum" type="context">triangulÅ©</reg> <lb/><var>i.q.d.</var> tibi imaginatione confige ductam eſſe lineam <var>.p.b.</var> quæ erit Å¿ectio commu-<lb/>nis orizontis cum Å¿uperficie <var>.o.p.b.M.</var> quæ Å¿uperficies erit perpendicularis ipÅ¿i ori-<lb/>zonti ex .18. lib 11. </s> - <s xml:space="preserve">Quòd autemnon minus <var>.o.p.</var> quà m.M.b. Å¿it in vna eademqÌue <lb/>Å¿uperficie dubitandum non eÅ¿t, quia Å¿i imaginabimur ductam eſſe lineam <var>.p.M.</var> ha <lb/>bebimus triangulum <var>.o.p.b.</var> cum triangulo <var>.M.b.p.</var> communibus partibus in vna ea-<lb/>demqÌue Å¿uperficie conÅ¿tantem, vt triangulum quoque <var>.o.p.M.</var> cum triangulo <var>M.b.</var> <lb/>o & triangulum <var>.o.p.b.</var> cum triangulo <var>.o.p.M.</var> & triangulum <var>.M.b.p.</var> cum triangulo <var>.<lb/>M.b.o</var>. </s> - <s xml:space="preserve">Vnde cum quilibet triangulus in vnica tantum Å¿uperficie Å¿it ex .2. lib. 11. Å¿e-<lb/>quetur Å¿uperficiem <var>.o.p.b.M.</var> planam eſſe, & vnicam, cuius communis Å¿ectio cum no-<lb/>Å¿tro plano Å¿it. θ.K.R. quæ perpendicularis orizonti exiÅ¿tet ex .19. lib. 11. eritqÌue pa-<lb/>rallela ipÅ¿i <var>.i.x.</var> ex .6. eiuÅ¿dem. </s> - <s xml:space="preserve">Imaginare nunc erectam eſſe <var>.m.T.</var> æqualem ipÅ¿i <var>.<lb/>b.M.</var> orizonti perpendicularem, quæ extenÅ¿a erit in Å¿uperficie <var>.p.t.</var> quod ex Å¿e ad <lb/>conÅ¿iderandum admodum facilè, clarumqÌue exiÅ¿tit, reducendo ad impoſſibilia <lb/>quemlibet hæc negare volentem. </s> - <s xml:space="preserve">Imaginemur quoque ductam eſſe lineam <var>.M.<lb/>T.</var> quæ <var>.b.m.</var> ex .33. primi erit parallela, quia <var>.m.T.</var> Ä™qualis <var>.b.M.</var> parallela eÅ¿t <lb/>ipÅ¿i <var>.b.M.</var> ex .6. lib. 11. præter hæc <var>.b.m.</var> parallela eÅ¿t ipÅ¿i <var>.q.d.</var> quia Å¿ic fuit ducta <lb/>Å¿uperius, vnde <var>.M.T.</var> parallela erit ipÅ¿i <var>.q.d.</var> ex .9. vndecimi, & obid perpendi-<lb/>cularis erit Å¿uperficiei <var>.b.t.</var> ex .8. eiuÅ¿dem. </s> - <s xml:space="preserve">Nunc Å¿it <var>.R.V.</var> communis Å¿ectio trian-<lb/>guli <var>.o.M.T.</var> cum noÅ¿tro plano, vnde <var>.R.V.</var> perpendicularis erit Å¿uperficiei <var>.p.t.</var> <lb/>ex .19. lib. 11. quam ob cauÅ¿am parallela erit ipÅ¿i <var>.q.d.</var> ex .6. aut ex .9. eiuÅ¿dem <lb/>quia ex .6. dicta, parallela eÅ¿t ipÅ¿i <var>.M.T</var>. </s> - <s xml:space="preserve">AtÅ¿i <var>.R.V.</var> parallela eÅ¿t ipÅ¿i <var>.q.d.</var> <lb/>etiam <var>.f.K.</var> probatum iam fuit parallelam eſſe eidem, ergo <var>.R.V.</var> parallela erit <lb/>ipÅ¿i <var>.K.f.</var> ex .30. primi, </s> - <s xml:space="preserve">Vnde ex .34. æqualis erit ipÅ¿i <var>.K.f</var>. </s> - <s xml:space="preserve">Accedamus nunc <lb/>ad <reg norm="figuram" type="context">figurã</reg> <var>.G.</var> <reg norm="extructam" type="context">extructã</reg> Å¿upra figuram <var>.E.</var> Å¿uperficialem, & erigamus <var>.m.T.</var> perpendi-<lb/>cularem ipÅ¿i <var>.m.p.</var> Å¿ed æqualem perfectæ altitudini, & ducamus <var>.T.o.</var> vt Å¿ecet li-<lb/>neam <var>.i.x.</var> in puncto <var>.V.</var> ab ipÅ¿o ducentes <var>.V.R.</var> parallelam ipÅ¿i <var>.q.d.</var> ducendo de- - <pb o="127" rhead="DE PERSPECT." n="139" file="0139" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0139"/> - inde <var>.k.R.</var> parallelam ipÅ¿i <var>.i.a.</var> habebimus altitudinem <var>.k.R.</var> quam quærebamus in <lb/>noÅ¿tro plano. </s> - <s xml:space="preserve">Quod cum Å¿ui natura clarum euadat, laborem ratiocinandi de eo, <lb/> - <anchor type="figure" xlink:label="fig-0139-01a" xlink:href="fig-0139-01"/> - cuilibet vel mediocriter in præclariſſima hac Å¿cien-<lb/>tia erudito relinquo. </s> - <s xml:space="preserve">ideÅ¿t, vt probetur <var>.k.R.</var> <reg norm="Å¿u- perficialem" type="context">Å¿u-<lb/>perficialẽ</reg>, <reg norm="æqualem" type="context">æqualẽ</reg> eſſe corporeæ. </s> - <s xml:space="preserve">Sed <reg norm="tollendo" type="context">tollẽdo</reg> <reg norm="Å¿uper- fluitatem" type="context">Å¿uper-<lb/>fluitatẽ</reg> linearum, & hoc <reg norm="accommodantes" type="context">accõmodantes</reg> vt in figura <var>.<lb/>F.</var> diligenter conÅ¿ideretur figura <var>.H</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0139-01" xlink:href="fig-0139-01a"> - <image file="0139-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0139-01"/> - <caption xml:space="preserve">CORPOREA.</caption> - </figure> - </div> - <figure position="here"> - <image file="0139-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0139-02"/> - <caption xml:space="preserve">SVPERFICIALIS.</caption> - </figure> - <figure position="here"> - <image file="0139-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0139-03"/> - <caption xml:space="preserve">SVPERFICIALIS</caption> - </figure> - <pb o="128" rhead="IO. BAPT. BENED." n="140" file="0140" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0140"/> - </div> - <div type="section" level="3" n="7"> - <head xml:space="preserve">CAP. VII.</head> - <p> - <s xml:space="preserve">ALiarn tamen inueni viam breuiorem vt in figura <var>.H.H.</var> in qua Å¿it punctus <var>.<lb/>b.</var> perfecti, & <var>.k.</var> degradati plani. </s> - <s xml:space="preserve">Nunc ducatur <var>.b.c.s.</var> ad rectos cum <var>.<lb/>p.m.</var> indefinitè, quæ quidem abÅ¿cindatur in puncto <var>.s.</var> ita quòd <var>.c.s.</var> æqualis Å¿it alti <lb/>tudini perfectæ, deinde coniungatur rectà . s. cum <var>.i</var>. </s> - <s xml:space="preserve">Tunc Å¿i ab <var>.k.</var> vÅ¿que ad <reg norm="protractam" type="context">protractã</reg> <lb/><var>i.s.</var> ducta fuerit <var>.k.R.</var> parallela li-<lb/> - <anchor type="figure" xlink:label="fig-0140-01a" xlink:href="fig-0140-01"/> - neÄ™ <var>.c.s.</var> hæc <var>.R.k.</var> erit altitudo <lb/>quæſita Å¿eu degradata.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0140-01" xlink:href="fig-0140-01a"> - <image file="0140-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0140-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Quod ita probo. </s> - <s xml:space="preserve">Iam nulli du <lb/>bium eÅ¿t quin <var>.f.V.</var> Å¿it æqualis alti-<lb/>tudini quęſitæ Å¿eu degradatÄ™, quo <lb/><reg norm="tieÅ¿cunque" type="simple">tieÅ¿cunq;</reg> ergo <reg norm="probauerimus" type="simple">ê“bauerimus</reg> <var>.k.R.</var> <lb/>æqualem eſſe lineæ <var>.f.V.</var> habebi-<lb/>mus propoÅ¿itum. </s> - <s xml:space="preserve">Quare certum <lb/>nobis erit eandem proportionem <lb/>eſſe lineæ <var>.c.s.</var> ad <var>.k.R.</var> quam <var>.c.i.</var> ad <lb/><var>k.i.</var> et <var>.c.i.</var> ad <var>.k.i.</var> vt <var>.x.i.</var> ad <var>.f.i.</var> et <var>.x.<lb/>i.</var> ad <var>.f.i.</var> vt <var>.m.o.</var> ad <var>.f.o.</var> et <var>.m.o.</var> ad <var>.<lb/>f.o.</var> vt <var>.m.T.</var> ad <var>.f.V.</var> ex Å¿imilitudine <lb/>triangulorum. </s> - <s xml:space="preserve">Ergo <var>.m.T.</var> ad <var>.f.V.</var> <lb/>erit vt <var>.c.s.</var> ad <var>.k.R.</var> ex .11. quinti, <lb/>Å¿ed <var>.c.s.</var> Å¿umpta fuit æqualis <var>.m.T</var>. <lb/></s> - <s xml:space="preserve">quare <var>.c.s.</var> ad <var>.f.V.</var> erit, vt <var>.m.T.</var> ad <lb/><reg norm="eandem" type="context context">eãdẽ</reg> <var>.f.V.</var> ex .7. <reg norm="quinti" type="simple">ê—nti</reg>, & ex .11. eiuÅ¿-<lb/>dem <var>.c.s.</var> ad <var>.f.V.</var> erit vt <var>.c.s.</var> ad <var>.k.R.</var> <lb/>quapropter ex .9. eiuÅ¿dem <var>.k.R.</var> æqualis erit <var>.f.V</var>.</s> - </p> - </div> - <div type="section" level="3" n="8"> - <head xml:space="preserve">CAP. VIII.</head> - <p> - <s xml:space="preserve">MOdus ab antiquis philoÅ¿ophis obÅ¿eruatus, eÅ¿t etiam vtilis, <reg norm="compendioÅ¿aque" type="simple">compendioÅ¿aqÌ;</reg> via <lb/>progreditur, cuius Å¿peculationem, in Å¿ubÅ¿cripta figura, quadam ex parte <reg norm="Å¿ecun- dum" type="context">Å¿ecÅ©-<lb/>dum</reg> morem antiquum, quadam etiam ex parte Å¿ecundum ingenij mei vires <reg norm="conÅ¿tru- cta" type="context">cõſtru-<lb/>cta</reg>, cognoÅ¿cemus. </s> - <s xml:space="preserve">In qua ego diuiÅ¿i <var>.x.i.</var> in puncto <var>.s.</var> ab <var>.x.</var> ita eleuato, quanta eÅ¿t <lb/> - <anchor type="figure" xlink:label="fig-0140-02a" xlink:href="fig-0140-02"/> - <pb o="129" rhead="DE PERSPECT." n="141" file="0141" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0141"/> - vera altitudo ipÅ¿ius <var>.M.T.</var> et <var>I.s.</var> duxi Å¿upponendo eſſe <var>.I.</var> <reg norm="punctum" type="context">punctÅ©</reg> pÅ¿pectiuæ <reg norm="Å¿ecundum" type="context">Å¿ecundÅ©</reg> <lb/>antiquos, ideÅ¿t angulum Å¿upremum trianguli antiquorum à punctoqÌue <var>.k.</var> meo duxi <lb/><var>k.f.</var> parallelam ipÅ¿i <var>.c.m.p.</var> vÅ¿que ad <var>.i.x.</var> in puncto <var>.f.</var> & à puncto à communi ipÅ¿is <var>.k.f.</var> <lb/>et <var>.i.x.</var> vÅ¿que ad <var>.I.s.</var> duxi quoque <var>.A.B.</var> parallelam ipÅ¿i <var>.i.x.</var> atque hæc omnia ex more <lb/>antiquo præſtiti.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0140-02" xlink:href="fig-0140-02a"> - <image file="0140-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0140-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Nunc verò eum conÅ¿iderans modum, quem ego de figuris <var>.G.H.</var> antecedentibus <lb/>præſcripÅ¿i, videndum eÅ¿t, an punctum <var>.B.</var> tribus lineis <var>.A.B.I.s.</var> et <var>.R.V.</var> quarum hÄ™c vl <lb/>tima à me iam ducta fuit, commune exiÅ¿tat, ideÅ¿t vtrum <var>.A.B.</var> æqualis exiÅ¿tat ipÅ¿i <var>.K.<lb/>R.</var> quam Å¿ecundum modum à me adinuentum, reuera Å¿cimus eſſe deÅ¿ideratam altitu <lb/>dinem in perÅ¿pectiua. </s> - <s xml:space="preserve">Quod tunc à nobis probatum erit, quando rationibus clarè <lb/>patebit ipÅ¿am <var>.A.B.</var> æqualem eſſe ipÅ¿i <var>.f.V</var>. </s> - <s xml:space="preserve">Quamobrem ducamus <var>.I.f.</var> vÅ¿que ad <var>.ω.</var> <lb/>lineæ <var>.c.p.</var> vnde ratione Å¿imilitudinis triangulorum manifeÅ¿tè intelligemus, eandem <lb/>proportionem eſſe ipÅ¿ius <var>.m.T.</var> ad <var>.f.V.</var> quæ eÅ¿t <var>.m.o.</var> ad <var>.f.o.</var> & eius, quæ eÅ¿t <var>.m.o.</var> ad <var>.f.<lb/>o.</var> quæ eÅ¿t <var>.ω.I.</var> ad <var>.f.I.</var> & eius, quæ eÅ¿t <var>.ω.I.</var> ad <var>.f.I.</var> quæ eÅ¿t <var>.x.I.</var> ad <var>.A.I.</var> & eius, quæ eÅ¿t <var>.x.<lb/>I.</var> ad <var>.A.I.</var> quæ eÅ¿t <var>.x.s.</var> ad <var>.A.B.</var> ideÅ¿t vt eius, quæ eÅ¿t <var>.m.T.</var> ad <var>.A.B.</var> Å¿ed idem <reg norm="quoque" type="simple">quoq;</reg> erat <lb/>de <var>.m.T.</var> ad <var>.f.V</var>. </s> - <s xml:space="preserve">Vnde Å¿equitur <var>.A.B.</var> æqualem eſſe <var>.f.V.</var> ex .9. quinti Eucli. <reg norm="atque" type="simple">atq;</reg> etiam <lb/>ipÅ¿i <var>.k.R.</var> quod à nobis propoÅ¿itum eÅ¿t inquirendum.</s> - </p> - </div> - <div type="section" level="3" n="9"> - <head xml:space="preserve">CAP. IX.</head> - <figure position="here"> - <image file="0141-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0141-01"/> - </figure> - <p> - <s xml:space="preserve"><emph style="sc">INstitvens</emph> etiam Å¿ermonem de figuris Å¿u-<lb/>perficialibus orizontalibus, Å¿eu de plantis, <lb/>pulcherrimum quendam modum, quem ego ad <lb/>locandum quodlibet punctum in perÅ¿pectiua, <lb/>(degradatum cum fuerit <reg norm="parallelogrammum" type="context">parallelogrãmum</reg> quod <lb/>dam rectangulum, in noÅ¿tro plano perpendicula <lb/>ri orizonti, quemadmodum in Å¿uperioribus figu-<lb/>ris <var>.A.</var> demonÅ¿trauimus) conÅ¿ideraui, Å¿ilentio <lb/>haud prætereundum eſſe.</s> - </p> - <p> - <s xml:space="preserve">Sit igitur in Å¿ubÅ¿cripta hîc figura <var>.K.</var> in paralle <lb/><reg norm="logrammo" type="context">logrãmo</reg> perfecto <reg norm="punctum" type="context">pÅ©ctum</reg> <var>.b.</var> quod locari debeat <lb/>in degradato <var>.e.q.d.r</var>. </s> - <s xml:space="preserve">Nunc à duobus quorumli-<lb/>bet quatuor angulorum <var>.q.u.a.d.</var> ducuntur duæ li-<lb/>neæ occultæ <var>.q.g.</var> et <var>.u.f.</var> per punctum <var>.b.</var> vÅ¿que ad <lb/>latera <var>.q.d.</var> et <var>.u.a.</var> ita tamen vt eorum extremita-<lb/>tes <var>.g.</var> et <var>.f.</var> intus cadant inter <var>.q.d.</var> et <var>.u.a.</var> ipÅ¿orum <lb/>laterum, ideÅ¿t vt non Å¿ecent duo latera <var>.q.u.</var> aut <var>.d.<lb/>a</var>. </s> - <s xml:space="preserve">Deinde punctum <var>.f.</var> inter <var>.q.</var> et <var>.d.</var> <reg norm="coniungatur" type="context">cõiungatur</reg> oc-<lb/>cultè cum angulo degradato <var>.e.</var> qui <reg norm="correÅ¿pondet" type="context">correÅ¿põdet</reg> <var>.<lb/>u.</var> perfecti, mediante linea <var>.e.f.</var> quæ erit <var>.u.f.</var> degra <lb/>dita in noÅ¿tro plano. </s> - <s xml:space="preserve">Deinde Å¿umatur punctum <var>.<lb/>n.</var> in linea <var>.q.d.</var> tam diÅ¿tans à <var>.q.</var> quà m.g. diÅ¿tat ab <var>.<lb/>u.</var> ducaturqÌue linea <var>.i.n.</var> quæ lineam <var>.e.r.</var> in puncto <lb/>c. diuidet, quod exijs, quæ Å¿uperius iam diximus <lb/>ad ipÅ¿um <var>.g.</var> referetur. </s> - <s xml:space="preserve">Ducendo poÅ¿tea lineam oc <lb/>cultam <var>.q.c.</var> patebit eam correÅ¿pondere lineæ <var>.q.g.</var> quæ Å¿ecans lineam <var>.e.f.</var> in puncto <var>.<lb/>t.</var> hoc, communi Å¿cientiæ ratione, reÅ¿pondebit ipÅ¿i <var>.b.</var> vt omnes cognoÅ¿cent.</s> - </p> - <pb o="130" rhead="IO. BAPT. BENED." n="142" file="0142" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0142"/> - <p> - <s xml:space="preserve">Sed Å¿i fortè punctum <var>.b.</var> eſſet in aliquo <lb/> - <anchor type="figure" xlink:label="fig-0142-01a" xlink:href="fig-0142-01"/> - laterum, puta <var>.q.u.</var> volo vt in rectangulo per-<lb/>fecto <var>.q.d.a.u.</var> ducta Å¿it vna diagonalis quam <lb/>volueris puta <var>.q.a.</var> deinde à puncto <var>.b.</var> ad reli-<lb/>quum angulum oppoÅ¿iti lateris ducta Å¿it recta <var>.<lb/>b.d.</var> ita quod à diagonali Å¿ecetur in puncto <var>.ω.</var> <lb/>per quod punctum demum à reliquo angulo la <lb/>teris <var>.q.u.</var> ducta Å¿it <var>.u.ω.</var> vÅ¿que ad latus <var>.q.d.</var> in <reg norm="pum" type="context">pÅ©</reg> <lb/>cto <var>.f.</var> quo facto, ita faciendum erit in rectangu <lb/>lo degradato, hoc eÅ¿t ducenda erit diagonalis <var>.<lb/>q.r.</var> quÄ™ correÅ¿pondet diagonali <var>.q.a.</var> perfecti <lb/>deinde <var>.f.e.</var> quæ correÅ¿pondet rectæ <var>.f.u.</var> perfe-<lb/>cti, quæ etiam interÅ¿ecabitur à diagonali <var>.q.r.</var> <lb/>in puncto <var>.o.</var> correÅ¿pondens <var>.ω.</var> perfecti, per <reg norm="quem" type="context">quẽ</reg> <var>.<lb/>o.</var> à puncto <var>.d.</var> cum ducta fuerit <var>.d.o.</var> vÅ¿que ad <var>.t.</var> <lb/>in latere <var>.q.e.</var> hoc <reg norm="punctum" type="context">punctÅ©</reg> <var>.t.</var> correÅ¿pondebit pun <lb/>cto <var>.b.</var> perfecti.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0142-01" xlink:href="fig-0142-01a"> - <image file="0142-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0142-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Idem eueniet Å¿i loco diametri <var>.q.a.</var> Å¿umpta <lb/>fuerit diameter <var>.u.d.</var> & loco <var>.b.d.</var> protracta fue <lb/>rit <var>.b.a.</var> deinde loco <var>.u.ω.f.</var> ducta fuerit <var>.q.ω.f.</var> vn-<lb/>de punctum correÅ¿pondens ipÅ¿i <var>.f.</var> in figura de-<lb/>gradata erit in latere Å¿upremo <var>.e.r.</var> correÅ¿pon-<lb/>dens lateri <var>.u.a.</var> & ita ducenda erit diameter <var>.d.<lb/>e.</var> correÅ¿pondens diametro <var>.d.u.</var> et <var>.q.f.</var> Å¿urÅ¿um <lb/>verÅ¿us correÅ¿pondens <var>.q.f.</var> imum verÅ¿us deinde <var>.<lb/>r.o.</var> reÅ¿pondens <var>.a.ω.</var> quæ terminabicur ab <reg norm="eodem- met" type="context">eodẽ-<lb/>met</reg> puncto <var>.t.</var> vt prius.</s> - </p> - </div> - <div type="section" level="3" n="10"> - <head xml:space="preserve">CAP.X.</head> - <p> - <s xml:space="preserve"><emph style="sc">Ex</emph> mea figura <var>.F.</var> Å¿uperficiali perÅ¿pectiuæ facillimum modum locandi quoduis <lb/>punctum in perÅ¿pectiua elicui. </s> - <s xml:space="preserve">Iuſſi enim vt aptaretur tabula quædam rectan-<lb/>gula exactè plana, triplo aut quadruplo, aut quanto volueris maioris longitu-<lb/>dinis, quà m latitudinis protenÅ¿a, quæ quidem latitudo erat ad duos circiter pedes <lb/>deÅ¿ignata ab <var>.A.B.C.D.</var> cuius duobus lateribus <var>.A.B.</var> et <var>.B.C.</var> iuſſi, vt duæ regulæ affi <lb/>gerentur, quæ Å¿uperficiem eiuÅ¿dem tabulæ excederent, vt <reg norm="vnum" type="context">vnÅ©</reg> ex lateribus alicuius <lb/>anguli recti materialis, qui <reg norm="appellatur" type="simple">appellat̃</reg> norma (vt inferius dicam) ei adherere poſſit, cu-<lb/>raui poÅ¿tea, ut iuxta angulum <var>.D.</var> in puncto <var>.o.</var> fixo mobilis regula <var>.o.Q.</var> affigeretur <lb/>tantæ longitudinis, aut paulò minoris, quantam occupabat latus <var>.D.A.</var> quæ circum <var>.<lb/>o.</var> volueretur, in rectitudine poÅ¿teà . <var>o.i.</var> parallela ipÅ¿i <var>.D.A.</var> in puncto <var>.i.</var> duobus pedi-<lb/>bus longè à latere <var>.A.B.</var> aliam quoque mobilem appendere feci <var>.i.M.</var> in tantam ferè <lb/>longitudinem extenÅ¿am, quanta conÅ¿tat <var>.A.B.</var> <reg norm="conÅ¿titui" type="context">cõſtitui</reg> etiam, vt <reg norm="quoddam" type="context">quoddã</reg> angulum re <lb/>ctum materiale tantæ magnitudinis, quanta nobis vÅ¿ui eſſe poterat Å¿uper eadem ta-<lb/>bula; </s> - <s xml:space="preserve">necnon regula quædam materialis neceſſariæ longitudinis <reg norm="Å¿tatuerentur" type="simple">Å¿tatuerent̃</reg>, <reg norm="atque" type="simple">atq;</reg> hæc <lb/>omnia tenuiſſima, vt fierent curaui. </s> - <s xml:space="preserve">Quandam deinde lineam ad <var>.o.i.</var> parallelam, <lb/>ideÅ¿t <var>.p.E.</var> Å¿uper eadem tabula adeò diÅ¿tantem ab <var>.o.i.</var> vt inter <var>.E.p.</var> et <var>.B.c.</var> perfe-<lb/>ctæ res, quæ degradari debebant, locari poſſent, Å¿ignaui. </s> - <s xml:space="preserve">Hæc autem diÅ¿tantia, quæ - <pb o="131" rhead="DE PERSPECT." n="143" file="0143" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0143"/> - inter <var>.o.i.</var> et <var>.p.E.</var> intercedebat, altitudinem oculi ab orizonte Å¿ignificabat. </s> - <s xml:space="preserve">Signa-<lb/>ui etiam lineam <var>.i.G.</var> perpendicularem lineæ <var>.E.p.</var> cui affigi poſſet non nihil chartæ <lb/>quotieÅ¿cunque volebam in perÅ¿pectiua aliquid delineare. </s> - <s xml:space="preserve">Quod cum facere deÅ¿i-<lb/>derabam, ponebam perfectum optimè affixum in quadrangulo <var>.E.G</var>: & in quadran <lb/>gulo <var>.E.i.</var> aliquod folium papyri affigebam. </s> - <s xml:space="preserve">Ponamus nunc, me voluiſſe conÅ¿titaere <lb/><reg norm="punctum" type="context">punctÅ©</reg> <var>.b.</var> <reg norm="Å¿umebam" type="context">Å¿umebã</reg> <reg norm="angulum" type="context">angulÅ©</reg> <reg norm="rectum" type="context">rectÅ©</reg> <reg norm="materialem" type="context">materialẽ</reg>, Å¿eu <reg norm="normam" type="context">normã</reg>, & eius vnum latus, iuxta latus <var>.<lb/>B.c.</var> ponebam, atque aliud per punctum <var>.b.</var> tranÅ¿ire faciebam, & vbi hoc latus <reg norm="lineam" type="context">lineã</reg> <lb/><var>E.p.</var> diuidebat, punctum .c Å¿ignabam per quod efficiebatur, vt regula <var>.i.M.</var> tranÅ¿iret, <lb/>quieÅ¿ceretqÌue aliquantulum aliquo modo in huiuÅ¿modi Å¿itu, opera deinde circini <lb/>interuallum <var>.b.c.</var> Å¿umebam, & in <var>.p.E.</var> à puncto <var>.x.</var> verÅ¿us <var>.E.</var> punctum <var>.m.</var> <reg norm="Å¿ignabam" type="context">Å¿ignabã</reg>: </s> - <s xml:space="preserve">per <lb/>quod faciebam, vt tranÅ¿iret regula <var>.o.Q.</var> quæ lineam <var>.x.i.</var> in puncto <var>.f.</var> diuidebat. <lb/></s> - <s xml:space="preserve">Angulum deinderectum materialem accipiebam, cuius vnum latus <var>.A.B.</var> ponebam, <lb/>aliud verò per punctum <var>.f.</var> tranÅ¿ibat, quod quidem latus regulam <var>.i.M.</var> in puncto <var>.k.</var> <lb/>(quod Å¿tatim Å¿uper folio papyri Å¿ignabatur) interÅ¿ecabat, atque hoc erat <reg norm="punctum" type="context">punctÅ©</reg>, <lb/>quod quærebam, puncto <var>.b.</var> correÅ¿pondens. </s> - <s xml:space="preserve">HuiuÅ¿modi effectus rationes ab ijs, quæ <lb/>Å¿uperius dixi eliciuntur. </s> - <s xml:space="preserve">Atque hæc ad baÅ¿es rerum, vt in Å¿ubÅ¿cripta figura eluceÅ¿cit, <lb/>Å¿pectabant.</s> - </p> - <figure position="here"> - <image file="0143-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0143-01"/> - </figure> - </div> - <div type="section" level="3" n="11"> - <head xml:space="preserve">CAP. XI.</head> - <p> - <s xml:space="preserve"><emph style="sc">Ad</emph> degradandas deinde altitudines, vÅ¿us Å¿um mea figura tam <var>.H.</var> <reg norm="quam" type="context">quã</reg> <reg norm="ent" type="context">ẽt</reg> <var>.H.H.</var> <lb/>vt milii Å¿eÅ¿e offerebat occaÅ¿io. </s> - <s xml:space="preserve">In primis ratione modi figuræ <var>.H.</var> <reg norm="curabam" type="context">curabã</reg>, vt - <pb o="132" rhead="IO. BAPT. BENED." n="144" file="0144" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0144"/> - vnum ex lateribus anguli recti, Å¿eu normæ regulæ <var>.B.C.</var> anniteretur, aliud verò <lb/>per <var>.m.</var> in rectitudine cuius Å¿ignabam <var>m.T.</var> interuallum æquale altitudini perfecti, <lb/>ideÅ¿t punctum <var>.T.</var> æqualiter diÅ¿tans ab <var>.m.</var> tranÅ¿ire faciebam, deinde regulam <var>.o.Q.</var> <lb/>per punctum <var>.T.</var> tranÅ¿ire quoque faciebam: </s> - <s xml:space="preserve">& notabam interÅ¿ectionem ipÅ¿ius cum <lb/>linea <var>.i.x.</var> in puncto <var>.V.</var> efficiebam deinde vt vnum ex lateribus anguli recti, lateri ta-<lb/>bulæ <var>.B.C.</var> anniteretur, aliudqÌue per punctum <var>.k.</var> tranÅ¿ire faciebam, & in huiuÅ¿modi <lb/>rectitudine à puncto k. Å¿ignabam quandam menÅ¿uram æ qualem lineæ <var>.f.V.</var> quÄ™ erat <lb/><var>k.R.</var> pro altitudine degradata.</s> - </p> - <head xml:space="preserve">ALITER IDEM.</head> - <p> - <s xml:space="preserve"><emph style="sc">MEdiante</emph> deinde figura <var>.H.H.</var> vnum ex lateribus anguli recti, lateri tabu-<lb/>læ <var>.B.C.</var> vt anniteretur faciebam; </s> - <s xml:space="preserve">aliud verò per punctum <var>.b.</var> perfecti, ideÅ¿t ba <lb/>Å¿is eiuÅ¿dem perfecti tranÅ¿ire faciebam. </s> - <s xml:space="preserve">Et in huiuÅ¿modi rectitudine Å¿ignabam <var>.c.s.</var> <lb/>æquale interuallum altitudini perfecti, ideÅ¿t punctum <var>.s.</var> ita diÅ¿tans à <var>.c.</var> efficiendo de <lb/>inde, vt latus anguli recti, lateri <var>.B.C.</var> tabulæ anniteretur, aliudqÌue per punctum <var>.k.</var> <lb/>tranÅ¿ire faciens Å¿ignabam <var>.k.R.</var> indeterminatam. </s> - <s xml:space="preserve">Faciens deinde tranÅ¿ire regulam <lb/><var>i.M.</var> per punctum <var>.s.</var> notabam punctum <var>.R.</var> interÅ¿ectionis eiuÅ¿dem cum linea <var>.k.R.</var> <reg norm="iam" type="context">iã</reg> <lb/>ducta. </s> - <s xml:space="preserve">Itaque altitudinem <var>.k.R.</var> degradatam habebam. </s> - <s xml:space="preserve">Hæc autem via aliquan-<lb/>tulum breuior, expeditiorqÌue altera.</s> - </p> - <figure position="here"> - <image file="0144-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0144-01"/> - </figure> - <pb o="133" rhead="DE PERSPECT." n="145" file="0145" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0145"/> - </div> - <div type="section" level="3" n="12"> - <head style="it" xml:space="preserve">JACOBO SOLDATO MEDIOLANENSI <lb/>Serenißimi Ducis Sabaudiæ Architecto peritißimo.</head> - <head xml:space="preserve">CAP. VII.</head> - <p> - <s xml:space="preserve"><emph style="sc">SVperioribvs</emph> diebus non diu poÅ¿tquam de perÅ¿pectiuis inter nos Å¿ermonem <lb/>habuimus, dum animus totus adhuc in his eſſet. </s> - <s xml:space="preserve">Illud in mentem venit quòd exi <lb/>mius ille vir, & profundiſſimæ doctrinæ, nec vnquam Å¿atis laudatus Daniel Barba-<lb/>rus Å¿e accepiſſe profitetur à Ioanne Zamberto patritio Veneto, qui ad verbum om <lb/>nia deÅ¿umpÅ¿erat a Ioanne CuÅ¿ino PariÅ¿ienÅ¿e. </s> - <s xml:space="preserve">Nec parum mirabar peritiſſimum il-<lb/>lum CuÅ¿inum, quod in capite quarto Å¿ecundæ partis perÅ¿pectiuæ, vt quodpiam pla-<lb/>num quadrilaterum in quadratam figuram redigeret, Å¿uper vnam datam <reg norm="lineam" type="context">lineã</reg> qua-<lb/>dratam compoÅ¿uiſſe. </s> - <s xml:space="preserve">Non animaduertens diÅ¿tantiam aut interuallum <var>.b.c.</var> degra-<lb/>datum Ãn linea <var>.b.f.</var> (quod eÅ¿t <var>.b.E.</var>) ita eſſe poſſe latus parallelogrammi rectanguli <lb/>magis longi quam lati, aut magis lati quam longi, vt etiam latus quadrati, quod be-<lb/>neficio Å¿ubÅ¿criptæ hic figuræ facilè depræhendi poteÅ¿t. </s> - <s xml:space="preserve">Vbi <var>.b.c.</var> latitudo eſſe po-<lb/>teÅ¿t, tam perfecti degradati in triangulo <var>.b.n.m.</var> aut in triangulo <var>.b.q.t.</var> quam in trian <lb/>gulo <var>.b.a.c</var>. </s> - <s xml:space="preserve">Sed perfectum degradati in triangulo <var>.b.n.m.</var> magis longum quam <reg norm="latum" type="context">latÅ©</reg> <lb/>& perfectum degradatum in triangulo <var>.b.q.t.</var> magis latum quam longum, & perfe-<lb/>ctum degradati in triangulo <var>.b.a.c.</var> quadratum erit quemadmodum à meis etiam fi-<lb/>guris <var>.A.</var> Å¿cientificè intelligi poteÅ¿t. </s> - <s xml:space="preserve">Hinc, ad inueniendum perfectum alicuius pla-<lb/>ni degradati, non Å¿ufficere degradationem Å¿olum interualli inter duos terminos <lb/>Å¿olos, ideÅ¿t <var>.b.E.</var> aſſignare, apertè patet, quia non omnia parallelogramma. perfecta <lb/>ab vno <reg norm="tantum" type="wordlist">tm̃</reg> interuallo producuntur, eo <reg norm="quod" type="simple">ê™</reg> non Å¿unt omnia quadrata. </s> - <s xml:space="preserve">Ad <reg norm="inquirendum" type="context">inquirendÅ©</reg> <lb/>igitur perfectum alicuius plani <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>, alicuius propoÅ¿iti degradati, oportet <lb/>vniuerÅ¿am degradationem tam latitudinis, <reg norm="quam" type="simple">ê—̃</reg> longitudinis, & <reg norm="non" type="context">nõ</reg> folius longitudinis <lb/>aſſignare; </s> - <s xml:space="preserve">Vt <reg norm="exempli" type="context">exẽpli</reg> gratia, in Å¿ubÅ¿cripta hic figura, <reg norm="volendo" type="context">volẽdo</reg> inuenire perfectum paral <lb/>lelogrammum degradati <var>.b.h.l.m.</var> dando diÅ¿tantiam orizontalem <var>.b.d.</var> à pede <var>.d.</var> ho-<lb/> - <anchor type="figure" xlink:label="fig-0145-01a" xlink:href="fig-0145-01"/> - <pb o="134" rhead="IO. BAPT. BENED." n="146" file="0146" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0146"/> - minis vÅ¿que ad planiÅ¿itum in quo degradatio facta Å¿it: </s> - <s xml:space="preserve">Å¿tatim altitudo <var>.A.</var> oculi à pe <lb/>de, quæ tanta Å¿emper eſſe debet quanta eÅ¿t altitudo trianguli <var>.b.n.m.</var> qui clauditur, <lb/>protrahendo <var>.m.l.</var> et <var>.b.h.</var> vÅ¿que ad concurÅ¿um in <var>.n.</var> in lucem prodibit. </s> - <s xml:space="preserve">Oporter <lb/>deinde erigere lineam <var>.b.o.</var> perpendicularem lineæ <var>.d.b.m.</var> & vÅ¿que ad eandem pro <lb/>ducere lineam <var>.l.h.</var> in puncto <var>.E.</var> et à puncto <var>.A.</var> per <var>.E.</var> vÅ¿que ad <var>.c.</var> ipÅ¿ius <var>.d.b.m.</var> produ <lb/>ctæ ducere <var>.A.E.c.</var> atque deinde protrahere lineam <var>.o.b.</var> vÅ¿que ad <var>.T.</var> ita vt <var>.b.T.</var> Ä™qua <lb/>lis Å¿it ipÅ¿i <var>.b.c.</var> & ad ipÅ¿am à puncto <var>.m.</var> ducere parallelam <var>.m.R.</var> & à puncto <var>.T.</var> ducere <var>.<lb/>T.R.</var> parallelam ipÅ¿i <var>.b.m</var>. </s> - <s xml:space="preserve">Vnde ex .34. primi Eucli <var>.m.R.</var> æqualis erit ipÅ¿i <var>.b.T.</var> et <var>.R.<lb/>T.</var> ipÅ¿i <var>.m.b.</var> & anguli in rectos euadent, atque hoc parallelo grammum rectangulum <lb/>erit verum perfectum degradati <var>.b.m.l.h.</var> obrationes à me circa <reg norm="figuram" type="context">figurã</reg> <var>.A.</var> adductas. <lb/> - <anchor type="figure" xlink:label="fig-0146-01a" xlink:href="fig-0146-01"/> - </s> - <s xml:space="preserve">Sed eÅ¿t hic quod magis nos commoueat, quia cum ex linea <var>.b.c.</var> quadratum <var>.b.g.</var> pro <lb/>duxerit, vult eum poÅ¿tea degradare. </s> - <s xml:space="preserve">Quod vt faciat (hanc figuram videbis in cap <num value="4">.<lb/>4.</num> Å¿ecundæ partis Danielis Barbari) oculum <var>.A.</var> in eadem Å¿uperficie extenÅ¿a, <lb/>quadrati <var>.b.g.</var> collocat. </s> - <s xml:space="preserve">Quod rectè fieti non poteÅ¿t, quia oculum hoc mo-<lb/>do locantes, viÅ¿ualesqÌue radios beneficio vnius plani Å¿ituati in <var>.b.f.</var> Å¿ecantes in ipÅ¿o <lb/>plano, nihil aliud quam <reg norm="dictam" type="context">dictã</reg> <reg norm="lineam" type="context">lineã</reg> <var>.b.f.</var> & <reg norm="nullam" type="context">nullã</reg> <reg norm="degradationem" type="context">degradationẽ</reg> <reg norm="inuenient" type="context">inueniẽt</reg>. </s> - <s xml:space="preserve">Id quod, & Å¿i <lb/>natura Å¿ua Å¿it omnibus notum, ponit <reg norm="tantum" type="wordlist/context">tñ</reg> id ipÅ¿um Vitelio pro quinta propoÅ¿itione <lb/>quarti libri de perÅ¿pectiua. </s> - <s xml:space="preserve">Præter hæc, credit latera <var>.b.d.</var> et <var>.c.e.</var> quadrati degra-<lb/>dati Å¿emper videri mediantibus angulis <var>.b.A.c.</var> et <var>.f.A.g.</var> quod fieri <reg norm="non" type="context">nõ</reg> poteÅ¿t, quem-<lb/>a dmodum ex mea figura corporea <var>.A.</var> facilè cognoÅ¿cere poſſumus, </s> - <s xml:space="preserve">propterea quòd <lb/>latera <var>.d.r.</var> et <var>.q.e.</var> meæ figuræ, mediantibus angulis <var>.d.o.r.</var> et <var>.q.o.e.</var> qui extra Å¿uperfi-<lb/>ciem <var>.s.a.</var> exiÅ¿tunt videntur, vnde Å¿i quis imaginaretur in puncto <var>.p.</var> oculum eſſe, & <lb/>ab ipÅ¿o ad <var>.u.</var> et <var>.q.</var> duas lineas duceret, angulus <var>.q.o.u.</var> nunc maior, nunc minor eſſet <lb/>angulo <var>.q.p.u.</var> aliquando etiam æqualis, quamuis rariſſime; </s> - <s xml:space="preserve">Sub diuerÅ¿is igitur an-<lb/>gulis, pro maiori parte, deteguntur latera, à partibus quadrati tam degradati, quà m <lb/>perfecti, quæ non Å¿unt anguli <var>.b.A.c.</var> et <var>.f.A.G</var>. </s> - <s xml:space="preserve">Quod vero idem poÅ¿tea dicat eam <lb/>proportionem eſſe ab <var>.b.E.</var> ad <var>.f.h.</var> & Å¿imul ad <var>.c.g.</var> quæ eÅ¿t ab <var>.a.g.</var> ad <var>.h.g.</var> id tuo relin - <pb o="135" rhead="DE PERSPECT." n="147" file="0147" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0147"/> - quam iudicio. </s> - <s xml:space="preserve">Tibi quoque conÅ¿iderandum relinquo; </s> - <s xml:space="preserve">cum rationabilis degrada-<lb/>tio eſſe debeat, qua ratione neceſſarium Å¿it, vt diÅ¿tantiæ reÅ¿qÌue, in vna & <reg norm="eadem" type="context">eadẽ</reg> pro-<lb/>portione cum altitudine oculi ad rem degradatam exiÅ¿tant? </s> - <s xml:space="preserve">Cum poÅ¿tea degrada-<lb/>uerit <reg norm="quadratum" type="context">quadratÅ©</reg>, is Å¿criptor, in figura <var>.d.b.c.e.</var> eum bene & ex perÅ¿pectiuæ optimis <lb/>legibus degradatum fuiſſe probare nititur; </s> - <s xml:space="preserve">Å¿olum probans <var>.d.e.</var> æqualem eſſe ipÅ¿i <var>.<lb/>E.h.</var> <reg norm="qua" type="wordlist">q̃</reg> <var>.E.h.</var> <reg norm="Å¿ecundum" type="context">Å¿ecundÅ©</reg> ipÅ¿um eÅ¿t degra datio lateris <var>.c.g.</var> & <reg norm="cum" type="context">cÅ©</reg> Å¿uperius dixerit, Å¿etria <lb/>quadrati plana degradauiſſe, quia <var>.b.E.</var> degradat <var>.b.c.</var> et <var>.E.h.</var> degradat <var>.c.g.</var> et <var>.f.<lb/>h.</var> degradat <var>.f.g.</var> nec quidem de lateribus <var>.b.d.</var> et <var>.c.e.</var> loquitur, quia Å¿i <var>.c.g.</var> <lb/>perfecti, degradatum eÅ¿t in <var>.E.h</var>: et <var>.d.e.</var> rectè protracta exiÅ¿tit, cum Å¿it æqua-<lb/>lis ipÅ¿i <var>.E.h.</var> cum etiam <var>.b.d.</var> et <var>.c.e.</var> rectè protractæ eſſe debeant: </s> - <s xml:space="preserve">qua de cau-<lb/>Å¿a ipÅ¿is <var>.b.E.</var> et <var>.f.h.</var> quæ, ex ipÅ¿o, Å¿unt degradationes <var>.b.c.</var> et <var>.f.g.</var> æquales eſſe non de-<lb/>bent? </s> - <s xml:space="preserve">Poſſet is mihi quidem reÅ¿pondere, <reg norm="quod" type="simple">ê™</reg> hoc pacto nulla Å¿uperficies clauderetur. <lb/></s> - <s xml:space="preserve">Ergo tria latera <var>.b.c</var>: <var>c.g.</var> et <var>.g.f.</var> <reg norm="non" type="context">nõ</reg> benè Å¿unt degradata, <reg norm="eiusque" type="simple">eiusqÌ;</reg> <reg norm="proportionalitates" type="simple">ê“portionalitates</reg> ma <lb/>lè intellectæ nil probant. </s> - <s xml:space="preserve">quia Å¿i dictæ proportionalitates, nobis tutò promitterent <lb/>degradationes, ab eo primum effectas, in linea <var>.b.f.</var> eſſe bonas, ergo duæ <var>.b.d.</var> et <var>.e.c.</var> <lb/>falſæ exiÅ¿terent, quarum quælibet maior eÅ¿t <var>.b.E.</var> et <var>.f.h.</var> ex .18. primi Eucli. </s> - <s xml:space="preserve">Omitta-<lb/>mus etiam quod vbi is Å¿cribit eam eſſe rationem, aut comparationem ab <var>.A.d.</var> ad <var>.b.<lb/>E.</var> quæ eÅ¿t ab <var>.d.c.</var> ad <var>.b.c.</var> eandemqÌue eſſe ab <var>.E.h.</var> ad <var>.c.g.</var> quæ eÅ¿t ab <var>.A.E.</var> ad <var>.A.c.</var> nil <lb/>probet; </s> - <s xml:space="preserve">nec Å¿imilitudinem triangulorum, nec aliquam propoÅ¿itionem Eucli. citans. <lb/></s> - <s xml:space="preserve">In quo excuÅ¿ari non poteÅ¿t, quòd non Å¿oleat Euclidem, aut alium quemuis autorem <lb/>citare, cum vel in ipÅ¿o operis principio capite .3. primæ partis, A pollonium <reg norm="Pergeum" type="context">PergeÅ©</reg> <lb/><reg norm="Euclidemque" type="simple">EuclidemqÌ;</reg>, & Å¿i etiam præter rem, citet. </s> - <s xml:space="preserve">Deinde <reg norm="quum" type="context">quÅ©</reg> idem probare vult <var>.d.e.</var> æqua <lb/>lem eſſe ipÅ¿i <var>.E.h.</var> eandem inquit eſſe proportionem <var>.a.b.</var> ad <var>.a.d.</var> quæ eÅ¿t ipÅ¿ius <var>.A.c.</var> <lb/>ad <var>.A.E.</var> quod & Å¿i verum Å¿it, hic tamen modus ratiocinandi nullo ordine nititur, <lb/>quia rectius dixiſſet pro clariori intelligentia ipÅ¿ius <var>.a.c.</var> ad <var>.a.e.</var> eandem proportio-<lb/>nem eſſe, quæ eÅ¿t <var>.A.c.</var> ad <var>.A.E.</var> propter Å¿imilitudinem, quæ inter duos triangulos <var>.A.<lb/>c.a.</var> et <var>.E.c.e.</var> intercedit, cum <var>.E.e.</var> Å¿upponatur parallela ipÅ¿i <var>.A.a.</var> quod etiam vt de-<lb/>monÅ¿traretur longiori oratione ei opus fuiſſet Å¿i voluiſſet intellectum eorum, qui pa <lb/>rum Å¿unt exercitati, perduci ad <reg norm="cognoÅ¿cendum" type="context">cognoÅ¿cendÅ©</reg> idem planè futurum de <var>.a.c.</var> ad <var>.a.e.</var> vt <lb/>eÅ¿t ipÅ¿ius <var>.A.c.</var> ad <var>.A.E.</var> in hunc modum, ideÅ¿t probando primùm duos triangulos <var>.A.<lb/>c.a.</var> et <var>.E.c.e.</var> æquiangulos eſſe, mediante .29. primi Eucli. cum <var>.A.a.</var> et <var>.E.e.</var> inuicem <lb/>Å¿int parallelæ. </s> - <s xml:space="preserve">Vnde ex .4. Å¿exti. idem extitiſſet de <var>.A.c.</var> ad <var>.E.c.</var> vt <var>.a.c.</var> ad <var>.e.c.</var> et. ex <lb/>16. quinti idem de <var>.A.c.</var> ad <var>.a.c.</var> vt ipÅ¿ius <var>.E.c.</var> ad <var>.e.c.</var> & ex .19. eiuÅ¿dem de <var>.A.E.</var> ad <var>.a.<lb/>e.</var> vt ipÅ¿ius <var>.A.c.</var> ad <var>.a.c.</var> & ex .16. iam dicta de <var>.A.E.</var> ad <var>.A.c.</var> vt ipÅ¿ius <var>.a.e.</var> ad <var>.a.c.</var> ideÅ¿t <lb/>ipÅ¿ius <var>.A.c.</var> ad <var>.A.E.</var> vt eÅ¿t ipÅ¿ius <var>.a.c.</var> ad <var>.a.e</var>: Aut hoc alio modo, qui breuior eÅ¿t pro-<lb/>cedendum, incipiendo Å¿cilicet à ſecunda Å¿exti Eucli. dicendo <reg norm="quod" type="simple">ê™</reg> exiÅ¿tente <var>.E.e.</var> paral <lb/>lela ipÅ¿i <var>.A.a</var>: ex dicta .2. lib. 6. erit idem de <var>.c.E.</var> ad <var>.E.A.</var> vt de <var>.c.e.</var> ad <var>.e.a.</var> vnde ex <num value="18">.<lb/>18.</num> quinti innotuiſſet Å¿tatim quod de <var>.c.A.</var> ad <var>.E.A.</var> vt de <var>.c.a.</var> ad <var>.e.a.</var> extitiſſet. </s> - <s xml:space="preserve">Nunc <lb/>mediantibus Å¿upradictis duabus propoÅ¿itionibus ideÅ¿t .29. primi, & 4. Å¿exti, cogno-<lb/>Å¿citur idem planè eſſe de <var>.b.c.</var> ad <var>.d.e.</var> quod ipÅ¿ius <var>.a.c.</var> ad <var>.a.e.</var> & ex eiÅ¿dem idem eſſe <lb/>de <var>.c.g.</var> ad <var>.E.h.</var> quod ipÅ¿ius <var>.A.c.</var> ad <var>.A.E.</var> vnde ex .11. quinti bis repetita idem erit de <lb/><var>b.c.</var> ad <var>.d.e.</var> quod de <var>.c.g.</var> ad <var>.E.h.</var> Å¿ed cum ex Å¿uppoÅ¿ito <var>.c.g.</var> Å¿it æqualis ipÅ¿i <var>.c.b.</var> idem <lb/>erit de <var>.c.g.</var> ad <var>.e.d.</var> quod ipÅ¿ius <var>.c.b.</var> ad eandem ex .7. quinti, vnde ex .11. idem erit de <lb/><var>c.g.</var> ad <var>.E.h.</var> quod eiuÅ¿dem <var>.c.g.</var> ad <var>.e.d.</var> ex .9. igitur eiuÅ¿dem <var>.d.e.</var> æqualis erit ipÅ¿i <var>.E.<lb/>h.</var> atque hic verus eÅ¿t modus ducendi intellectum parum exercitatum in cognicio-<lb/>nis campum. </s> - <s xml:space="preserve">quem quidem mihi obÅ¿eruandum proponerem Å¿i onus Å¿cribendi Å¿u-<lb/>Å¿ciperem ijs, qui in Å¿cientijs parum verÅ¿ati Å¿unt, quos tanquam puerulos manu du- - <pb o="136" rhead="IO. BAPT. BENED." n="148" file="0148" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0148"/> - cere oportet. </s> - <s xml:space="preserve">Ratio verò ab ipÅ¿o adducta propter quam <var>.E.</var> repreÅ¿entatur oculo al-<lb/>tius quam <var>.b.</var> nempe eo quod <var>.A.</var> Å¿uperÅ¿tet ipÅ¿i <var>.E.</var> nihil valet, quia Å¿i inferius eſſet, <lb/>idem contingeret, Å¿ed hoc euenit eo quod <var>.E.</var> altius eÅ¿t ipÅ¿o <var>.b</var>. </s> - <s xml:space="preserve">Idem dico de <var>.h.</var> <lb/>vbi Å¿imiliter decipitur. </s> - <s xml:space="preserve">Idem etiam in .7. cap. fallitur in Å¿ecundo modo, quem oÅ¿ten <lb/>dit pro Å¿ecundo quadrato aliquo degradato à parallelogrammo degradato magis <lb/>longo quà m lato, cum ducat parallelam <var>.l.m.</var> ad <var>.b.c.</var> à puncto <var>.l.</var> interÅ¿ection is ipÅ¿ius <var>.<lb/>o.c.</var> id, quod non rectè efficitur quemadmodum ex rationibus à me allegatis circa <lb/>meas figuras <var>.A.A.</var> facilè innoteÅ¿cit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0145-01" xlink:href="fig-0145-01a"> - <image file="0145-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0145-01"/> - </figure> - <figure xlink:label="fig-0146-01" xlink:href="fig-0146-01a"> - <image file="0146-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0146-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Nono deinde cap. contrario planè ordine, quam oporteret proceÅ¿sit, quia <reg norm="cum" type="context">cÅ©</reg> <lb/>angulus .2. trianguli perfecti magis diÅ¿tet à plano Å¿uper quod degradari debet <lb/>triangulum, quà m latus .1. 3. oppoÅ¿itum dicto angulo .2. & per confequens longère <lb/>motior Å¿it ab oculo, ipÅ¿e in degradato, <reg norm="eum" type="context">eÅ©</reg> magis propinquum eſſe facit, è con-<lb/>tra eap .10. rectè fecit contra id, quod capite .9. tradiderat.</s> - </p> - <p> - <s xml:space="preserve">Quod autem deinceps in prima parte .11. & vltimi capitis aſſerit eÅ¿t, <reg norm="admittendum" type="context">admittendÅ©</reg>. <lb/></s> - <s xml:space="preserve">Quod verò in Å¿ecunda parte ab eo traditur, ideÅ¿t alius quidam modus quem de <reg norm="tranÅ¿" type="context">trãſ</reg> <lb/>ferendis punctis à perfecto in degradato proponit, non eÅ¿t modus vniuerÅ¿alis; </s> - <s xml:space="preserve">quia <lb/>Å¿i altitudo <var>.T.Q.</var> oculi à plano orizontali, non eſſet æqualis medietati lateris <var>.B.D.</var> <lb/>perfecti, interualla <var>.a.b.c.d.e.</var> lateris <var>B.D.</var> admittenda non eſſent.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius rei intelligentia Å¿it in Å¿ubÅ¿cripta hic figura corporea <var>.ω.</var> parallelogram-<lb/>mum rectangulum <var>A.B.C.D.</var> in plano orizontali, & linea <var>.Q.H.</var> illud per medium <lb/>diuidat, quæ Å¿it parallela duobus lateribus <var>.A.B.</var> et <var>.C.D.</var> in cuius quolibet puncto <var>.<lb/>Q.</var> Å¿it infimus terminus altitudinis oculi, & in <var>.<lb/> - <anchor type="figure" xlink:label="fig-0148-01a" xlink:href="fig-0148-01"/> - T.</var> ad perpendiculum ipÅ¿ius <var>.Q.</var> Å¿it verus Å¿itus <lb/>eiuÅ¿dem, tantum eleuatus à <var>.Q.</var> quanta eÅ¿t <lb/>medietas ipÅ¿ius <var>.D.B.</var> Å¿itqÌue figura corpo-<lb/>rea finita Å¿imilis meæ <var>.A.</var> vnde <var>.Q.T.</var> æqualis <lb/>erit ipÅ¿i <var>.Q.æ.</var> & planum perpendiculare <reg norm="orizon- ti" type="context">orizõ-<lb/>ti</reg>, Å¿uper quod punctum <var>.k.</var> perfecti duci debet <lb/>Å¿it <var>.R.D.B.</var> Å¿intqÌue ductæ per imaginationem <lb/>lineæ <var>.T.K</var>: <var>Q.K.</var> et Å¿it <var>.K.N.</var> perpendicularis la-<lb/>teri <var>.C.D.</var> à quo puncto <var>.N.</var> imaginatione Å¿it <reg norm="con" type="context">cõ</reg> <lb/>præhenÅ¿a linea <var>.N.Q.</var> at que hæ tres lineæ Å¿ectæ <lb/>Å¿int à plano in punctis <var>.c.i.</var> et .2. quorum <reg norm="punctum" type="context">punctÅ©</reg>. <lb/>2. erit quæſitum plani. </s> - <s xml:space="preserve">Imaginemur nunc duos <lb/>triangulos <var>.K.T.Q.</var> et <var>.N.Q.æ.</var> qui Å¿ecti <reg norm="erunt" type="context">erÅ©t</reg> <lb/>à plano <var>.R.B.D.</var> quorum communes Å¿ectiones <lb/>erunt .1. 2. et <var>.D.c.</var> & quia <var>.N.K.D.i.</var> et <var>.æ.Q.</var> <lb/>inuicem Å¿unt parallelæ, Å¿equitur eandem pro-<lb/>portionem futuram ipÅ¿ius <var>.Q.K.</var> ad <var>.K.i.</var> quæ eÅ¿t <lb/>ipÅ¿ius <var>.æ.N.</var> ad <var>.N.D.</var> imaginatione concipien <lb/>do a puncto <var>.K.</var> vÅ¿que ad <var>.æ.Q.</var> quandam paral-<lb/>lelam ipÅ¿i <var>.N.æ.</var> quemadmo dum ex te ipÅ¿o intel <lb/>ligere potes. </s> - <s xml:space="preserve">Sed ratione Å¿imilitudinis trian-<lb/>gulorum ita Å¿e res habet de <var>.æ.Q.</var> ad <var>.D.c.</var> vt de <var>.<lb/>æ.N.</var> ad <var>.N.D.</var> vt quoque de <var>.T.Q.</var> ad .2. 1. quemadmodum ipÅ¿ius <var>.Q.K.</var> ad <var>.K.i.</var> vn-<lb/>de ex .11. quinti, idem erit de <var>.Q.T.</var> ad .1. 2. quod de <var>.Q.æ.</var> ad <var>.c.D.</var> & ex .16. eiuÅ¿dem <lb/>de <var>.Q.T.</var> ad <var>.Q.æ.</var> quod de .1. 2. ad <var>.c.D.</var> & exiÅ¿tente <var>.æ.Q.</var> ex Å¿uppoÅ¿ito æquali ipÅ¿i.</s> - <pb o="137" rhead="DE PERSPECT." n="149" file="0149" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0149"/> - <s xml:space="preserve">1. 2. </s> - <s xml:space="preserve">Vnde huiuÅ¿modi regula tunc bona redditur, quando <var>T.Q.</var> æqualis eÅ¿t ipÅ¿i .Å“.<lb/>Q. ideÅ¿t medietati ipÅ¿ius <var>.D.B.</var> at verò Å¿i æqualis non eſſet hoc minime Å¿equeretur, <lb/>vt facilè patet. </s> - <s xml:space="preserve">Quòd verò .2. <var>R.Z.</var> &. Å¿int benè diÅ¿poÅ¿ita, dubitandum non eÅ¿t, quia <lb/>punctum <var>.i.</var> meæ hic Å¿ubÅ¿criptæ figuræ, quod coreÅ¿pondet K. eius Å¿iguræ adeò diÅ¿tat <lb/>a medio <var>.R.X.</var> trianguli <var>.R.B.D.</var> vt .2. cum .1. 2. dicto medio <var>.R.X.</var> ex .6. </s> - <s xml:space="preserve">Vndecimi fit <lb/>parallela. </s> - <s xml:space="preserve">Idem de reliquis dico. quod manifeÅ¿tè cognoÅ¿ci poteÅ¿t, ab eo, quod in <lb/>Å¿uperius poÅ¿itis figuris corporeis dixi. </s> - <s xml:space="preserve">HuiuÅ¿modi modus ducendi res in perÅ¿pectiua, <lb/>non Å¿olum à Gallis, Å¿ed à Germanis etiam in vÅ¿um reducitur. </s> - <s xml:space="preserve">Sed quia ad hæc <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> <lb/>tempora eiuÅ¿dem perfectionis ratio, quam ego Å¿uperius propoÅ¿ui, <reg norm="nondum" type="context">nõdum</reg> in lucem <lb/>emerÅ¿it, factum fuit, vt <reg norm="errorum" type="context">errorÅ©</reg> laqueis irretirentur, Å¿umentes <var>.T.Q.</var> modo maiorem, <lb/>modo minorem medietate lateris <var>.D.B</var>. </s> - <s xml:space="preserve">Cum hunc igitur modum hic Autor <lb/>vniuerÅ¿alem eſſe putet, labitur in errorem, cum debuiſſet longitudinem ipÅ¿ius <var>.T.Q.</var> <lb/>debere eſſe æqualem medietati ipÅ¿ius <var>.D.B.</var> proferre. </s> - <s xml:space="preserve">Aſſerit deinde diÅ¿tantiam ip-<lb/>Å¿ius <var>.T.Q.</var> à latere <var>.B.D.</var> æ qualem eſſe debere lateri <var>.C.D.</var> quod neceſſarium non eÅ¿t, <lb/>quia in quibuslibet diÅ¿tantijs, iuÅ¿ta operatio fieri poteÅ¿t, quemadmodum in Å¿ubÅ¿cri-<lb/>pta hîc figura facile patet, ideÅ¿t, quòd quibuÅ¿cunque modis <var>.c.D.</var> æqualis remaneat <lb/>ipÅ¿i .1. 2. & Å¿ic interualla, quæ <reg norm="per" type="simple">ê‘</reg> tranÅ¿uerÅ¿um aguntur <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> ad <reg norm="medium" type="context">mediÅ©</reg> trianguli <var>.D.R.B.</var> <lb/>Neque etiam probandus eÅ¿t auctor ille, cum pro oculo, Å¿uum <var>.T.</var> loco <var>.Q.</var> à me poÅ¿i-<unsure/><lb/>ti, ponit, cum is locus Å¿it verus Å¿itus pedis eius quireÅ¿picit, & non oculi. </s> - <s xml:space="preserve">Quòd <reg norm="autem" type="context">autẽ</reg> <lb/>Auctor iÅ¿te, modo vniuerÅ¿ali intelligat, vt iam diximus, <reg norm="conÅ¿ideretur" type="context">cõſideretur</reg> figura tertij mo <lb/>di primi cap. tertiæ partis, in qua Å¿uum oculum (vt ita dicam) ponit in <var>.o.</var> altius Å¿eu <lb/>diÅ¿tans à rectitudine lateris <var>.c.d.</var> plus quam Å¿it totum latus <var>.d.b</var>.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0148-01" xlink:href="fig-0148-01a"> - <image file="0148-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0148-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="13"> - <head xml:space="preserve">AD EVNDEM IACOBVM. <lb/>CAP. XIII.</head> - <p> - <s xml:space="preserve"><emph style="sc">TVas</emph> accepiliteras omnis humanitatis & officij plenas, in quibus requiris cau-<lb/>Å¿am, quæ me in alijs meis literis impulit ad <reg norm="dicendum" type="context">dicendÅ©</reg>, <reg norm="angulum" type="context">angulÅ©</reg> <var>.q.o.u.</var> modo ma-<lb/>iorem, modo verò minorem futurum angulo <var>.q.p.u.</var> meæ figuræ corporeæ <var>.A.</var> hanc <lb/>igitur ob cauÅ¿am imagineris in Å¿ubÅ¿cripta hîc figura duo triangula <var>.q.o.u.</var> et <var>.q.p.u.</var> <lb/>quorum <var>.q.p.u.</var> perpendiculariter Å¿it Å¿uper Å¿uperficie trianguli <var>.q.o.p.</var> collocatum, <lb/>præcisè vt in mea figura corporea <var>.A.</var> Å¿uperficies verò trianguli <var>.q.o.p.</var> Å¿it exempli-<lb/>gratia <var>.V.M.</var> & trian-<lb/>guli <var>.u.o.p.</var> Å¿it <var>.V.D.</var> <lb/> - <anchor type="figure" xlink:label="fig-0149-01a" xlink:href="fig-0149-01"/> - quarum <reg norm="communis" type="context">cõmunis</reg> Å¿e-<lb/>ctio Å¿it <var>.V.p.o.x.</var> non <lb/>eÅ¿t enim <reg norm="dubitandum" type="context">dubitãdum</reg> <lb/>quin triangulum <var>.q.<lb/>p.u.</var> Å¿it perpendicula-<lb/>re triangulo <var>.q.o.p.</var> <lb/><reg norm="cum" type="context">cÅ©</reg> hoc ex .18. lib. 11. <lb/>Eucli. perpendicula-<lb/>re Å¿it Å¿uperficiei <var>.a.s.</var> <lb/>in qua reperitur <reg norm="trian- gulum" type="context">triã-<lb/>gulum</reg> <var>.q.p.u.</var> & hoc <lb/>ex linea <var>.o.p.</var> perpendiculari dictæ Å¿uperficiei <var>.a.s</var>. </s> - <s xml:space="preserve">Nunc dico angulum <var>.q.o.u.</var> modo <lb/>maiorem, modo minorem eſſe angulo <var>.q.p.u</var>. </s> - <s xml:space="preserve">Notiſſimum igitur primum nobis - <pb o="138" rhead="IO. BAPT. BENED." n="150" file="0150" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0150"/> - eÅ¿t angulum <var>.p.q.u.</var> obtuÅ¿um eſſe; </s> - <s xml:space="preserve">Imaginemur ergo circa triang ulum <var>.p.q.u.</var> circun-<lb/>Å¿criptum eſſe circulum, cuius portio <var>.p.q.u.</var> minor erit medietate eiuÅ¿dem medij cir-<lb/>culi, vt iam ex <ref id="ref-0023">30. Eucli. lib. tertij</ref> nouiÅ¿ti. </s> - <s xml:space="preserve">nunc imaginemur dictum circulum circum <lb/>lineam <var>.q.u.</var> loco axis verÅ¿us <var>.x.</var> moueri, vnde girus eiuÅ¿dem, per quem tranÅ¿ibat linea <lb/><var>V.x.</var> remouebitur ab eadem linea non nihil cum motus erit à primo Å¿itu vÅ¿quequò <lb/>ad Å¿ecandam dictam lineam <var>.V.x.</var> in alio quodam puncto inter <var>.p.</var> et <var>.x.</var> redibit; </s> - <s xml:space="preserve">quod <lb/>quidem punctum Å¿i <lb/>erit inter <var>.o.</var> et <var>.x.</var> angu <lb/> - <anchor type="figure" xlink:label="fig-0150-01a" xlink:href="fig-0150-01"/> - lus <var>.q.o.u.</var> maior erit <lb/>angulo <var>.q.p.u</var>. </s> - <s xml:space="preserve">Sed Å¿i <lb/>idem <reg norm="punctum" type="context">punctÅ©</reg> erit in-<lb/>ter <var>.p.</var> et <var>.o.</var> dictus an-<lb/>gulus <var>.q.o.u.</var> minor <lb/>erit <var>.q.p.u.</var> de qua <reg norm="qui- dem" type="simple context">ê—-<lb/>dẽ</reg> re tu ipÅ¿e median-<lb/>te .20. lib. 3. et .16. lib. <lb/>primi certior fieri po-<lb/>tes. </s> - <s xml:space="preserve">Valde miror <reg norm="quod" type="simple">ê™</reg> <lb/>hæc Ioannis CuÅ¿ini di <lb/>cta ad hæc vÅ¿que tempora tanto in prætio Å¿int habita, vt ab excellentibus Å¿criptori-<lb/>bus quaÅ¿i Å¿i proprij eorum ingenij partus eſſent, de verboad verbum vt theÅ¿auros, in <lb/>fuis <reg norm="ipÅ¿orummet" type="context">ipÅ¿orÅ©met</reg> libris reÅ¿cripta fuerint, quemadmodum iam omnes admonui in mea <lb/>gnomonica Orontium, MunÅ¿terum, <reg norm="aliosque" type="simple">aliosqÌ;</reg> permultos feciſſe.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0149-01" xlink:href="fig-0149-01a"> - <image file="0149-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0149-01"/> - </figure> - <figure xlink:label="fig-0150-01" xlink:href="fig-0150-01a"> - <image file="0150-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0150-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="14"> - <head xml:space="preserve">CAP. XIIII.</head> - <p> - <s xml:space="preserve"><emph style="sc">Ex</emph> ijs, qu æ de nonnullis effectibus ducendo in perÅ¿pectiua tertÃum corpus regu <lb/>lare, <reg norm="quod" type="simple">ê™</reg> octo triangulis æquilateribus eÅ¿t term inatum, Å¿cire deÅ¿ideras, hoc <reg norm="vnum" type="context">vnÅ©</reg> <lb/>eÅ¿t caput: </s> - <s xml:space="preserve">vnde fiat, aut quomodo probetur quaÅ¿libet duas facies oppoÅ¿itas eiuÅ¿-<lb/>dem corporis octoaedri <reg norm="inuicem" type="context">inuicẽ</reg> æquidiÅ¿tantes eſſe. </s> - <s xml:space="preserve">Quamobrem Å¿it hîc <reg norm="Å¿ubÅ¿criptum" type="context">Å¿ubÅ¿criptÅ©</reg> <lb/><reg norm="octoaedrum" type="context">octoaedrÅ©</reg>, cuius diameter vna Å¿it <var>.b.q.</var> et <var>.b.p.<lb/>l.</var> vna ex faciebus, cui opponatur facies <var>.q.k.</var> <lb/> - <anchor type="figure" xlink:label="fig-0150-02a" xlink:href="fig-0150-02"/> - d. quas <reg norm="adinuicem" type="context">adinuicẽ</reg> æquidiÅ¿tantes eſſe contendo <lb/>Å¿int aliæ duæ facies, quæ inter has ponuntur <var>.<lb/>b.d.k.</var> et <var>.q.p.l.</var> & à punctis extremis <var>.b.q.</var> dia-<lb/>metri. ductæ Å¿int quatuor lineæ <var>.b.a</var>: <var>b.u</var>: <var>q.a</var>: <var>q.<lb/>u.</var> ad puncta <var>.a.</var> et <var>.u.</var> diuidentia <var>.k.d.</var> et <var>.l.p.</var> per <lb/>medium, vnde ex 4. primi Eucli. quatuor hæ <lb/>lineæ adinuicem Ä™quales erunt <reg norm="Å¿umendo" type="context">Å¿umẽdo</reg> eas vt <lb/>baÅ¿es <reg norm="triangulorum" type="context">triangulorÅ©</reg> <var>.a.d.b</var>: <var>u.l.b</var>: <var>a.d.q.</var> et <var>.u.l.q.</var> <lb/><reg norm="adinuicem" type="context">adinuicẽ</reg> <reg norm="quoque" type="simple">quoq;</reg> <reg norm="æquidiÅ¿tabunt" type="simple context">æê—diÅ¿tabÅ©t</reg> <var>.a.b.</var> ab <var>.u.q.</var> et <var>.b.<lb/>u.</var> ab <var>.q.a.</var> ex .27. primi; </s> - <s xml:space="preserve"><reg norm="quia" type="simple">ê—a</reg> Å¿i imaginabimur dia <lb/>metrum <var>.b.q.</var> tunc ex .4. aut ex .8. eiuÅ¿dem lib. <lb/>habebimus angulos <var>.a.b.q.</var> et <var>.u.q.b.</var> æquales <lb/>inuicem; </s> - <s xml:space="preserve">Å¿ed ob eaÅ¿dem rationes <var>.p.l.</var> paralle-<lb/>la eÅ¿t ipÅ¿i <var>.d.k.</var> vnde ex 15. lib. 11. facies <var>.b.p.l.</var> <lb/>parallela fit, aut æquidiÅ¿tans ipÅ¿i <var>.q.d.k.</var> ideÅ¿t <lb/>primum propoÅ¿itum.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0150-02" xlink:href="fig-0150-02a"> - <image file="0150-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0150-02"/> - </figure> - </div> - <pb o="139" rhead="DE PERSPECT." n="151" file="0151" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0151"/> - <p> - <s xml:space="preserve">Ad habendam deinde quantitatem diÅ¿tantiæ, aut interualli Å¿imul cum Å¿itu, in fa-<lb/>cie <var>.q.d.k.</var> quem latus <var>.p.l.</var> perpendiculariter reÅ¿picit. </s> - <s xml:space="preserve">Imaginemur à puncto <var>.u.</var> Å¿uper <lb/><var>q.a.</var> cad ere lineam perpendicularem <var>.u.o.</var> quæ illico reperitur cum triangulum <var>.a.<lb/>u.q.</var> ex lateribus datis & cognitis conÅ¿tet, <reg norm="quodquidem" type="context">quodquidẽ</reg> triangulum, medietas eÅ¿t qua-<lb/>drilateri, Å¿eu. rumbi <var>.q.a.b.u.</var> cui vnaquæque dictarum quatuor facierum perpendi-<lb/>cularis exiÅ¿tit ex .4. ct .18. lib. 11. & ob id linea <var>.u.o.</var> extenÅ¿a in Å¿uperficie dicti quadri-<lb/>lateri, & perpendicularis lineæ <var>.q.a.</var> perpendicularis erit faciei <var>.q.d.k.</var> & ex .29. <lb/>primi, angulus <var>.b.u.o.</var> rectus erit, ut <reg norm="etiam" type="context">etiã</reg> angulus <var>.o.u.l.</var> ex .2. definitione lib. 11. vnde <lb/>ex .4. eiuÅ¿dem lib <var>.o.u.</var> perpendicularis erit faciei <var>.b.p.l</var>. </s> - <s xml:space="preserve">Ha bebimus ergo Å¿itum in fa-<lb/>cie <var>.q.d.k.</var> qui reÅ¿picietur ad angulos rectos à linea <var>.p.l.</var> quiquidem erit in perpendi-<lb/>culari à puncto <var>.o.</var> ad <var>.q.a.</var> ducta.</s> - </p> - <p> - <s xml:space="preserve">Quòd autem <var>.a.o.</var> Å¿it latus exagoni æquilateris circumÅ¿crip tibilis ab eodem circu <lb/>lo, qui vnam ex faciebus triangularibus æquilateribus propoÅ¿iti corporis circunÅ¿cri-<lb/>bere pot eÅ¿t, ita oÅ¿tenditur. Å¿it <reg norm="comprehenÅ¿um" type="context">cõprehenÅ¿um</reg> imaginatione, triangulum <var>.a.q.u.</var> Å¿epara <lb/>tim, cuius latus <var>.a.u.</var> æquale eÅ¿t vni ex lateribus <reg norm="triangulorum" type="context">triangulorÅ©</reg> eiuÅ¿dem corporis ex .33. <lb/>primi, quo dlibet verò aliorum duorum æquale perpendicularibus dictorum trian-<lb/>gulorum, in quo triangulo <var>.a.u.q.</var> ducta Å¿it perpendicularis <var>.u.o.</var> ab vna <reg norm="extremitatum" type="context">extremitatÅ©</reg> <lb/>lateris maioris, ad vnum ex minoribus lateribus, quÄ™ perpendicularis intra triangu-<lb/>lum cadet, quia dictum triangulum oxigonium eÅ¿t. </s> - <s xml:space="preserve">quod autem attinet ad duos angu <lb/>los <var>.a.</var> et <var>.u.</var> cum æquales Å¿int ex quinta lib. primi; </s> - <s xml:space="preserve">17. nos certiores facit; </s> - <s xml:space="preserve">quod verò anÂ<lb/>gulus <var>.q.</var> Å¿it <reg norm="etiam" type="context">etiã</reg> acutus: </s> - <s xml:space="preserve">30. lib. tertii nos cer-<lb/>tos reddit, <reg norm="quia" type="simple">ê—a</reg><var>.a.u.</var> minor eÅ¿t diametro <reg norm="Å¿phae ræ" type="simple">Å¿phÄ™Â<lb/> - <anchor type="figure" xlink:label="fig-0151-01a" xlink:href="fig-0151-01"/> - ræ</reg> datum corpus circumÅ¿cribentis, cum <var>.q.</var> <lb/>dictæ Å¿phÄ™rÄ™ Å¿uperficiem tangat.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0151-01" xlink:href="fig-0151-01a"> - <image file="0151-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0151-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Ad probandum <var>.a.o.</var> Ä™qualem eſſe lateri <lb/>exagoni dicti, Å¿atis erit probare <var>.a.q.</var> Å¿eÅ¿qui <lb/>alteram eſſe ad <var>.a.o.</var> quia Å¿i in Å¿ubÅ¿cripto <lb/>hîc circulo ducemus duas Å¿emidiametros <var>.<lb/>n.p.</var> et <var>.n.l.</var> ad. angulos <reg norm="trianguli" type="context">triãguli</reg> Ä™quilateri <var>.p.</var> <lb/>et <var>.l.</var> & cum quodlibet laterum ipÅ¿ius exago <lb/>ni, Ä™quale Å¿it Å¿emidiametro circuli ex .15. <lb/>lib. 4. habebimus ex .8. primi, angulum <var>.n.<lb/>p.l.</var> æqualem angulo <var>.q.p.l</var>. </s> - <s xml:space="preserve">Vnde ex .4. eiuÅ¿ <lb/>dem <var>.o.n.</var> Ä™qualiserit ipÅ¿i <var>.o.q.</var> ideÅ¿t <var>.q.a.</var> Å¿eÅ¿ <lb/>quialtera erit ad <var>.a.o</var>.</s> - </p> - <p> - <s xml:space="preserve">Ad probandum nunc in triangulo <var>.a.q.<lb/>u</var>: <var>a.q.</var> Å¿eÅ¿quialteram eſſe ad <var>.a.o.</var> eÅ¿t <reg norm="quoque" type="simple">quoq;</reg> <lb/>Å¿ciendum primò omne latus trianguli Ä™quilateri in potentia Å¿eÅ¿quitertium eſſe ad <lb/>perpendicularem eiuÅ¿dem trianguli, quod vndecima lib. 14. Eucli. breuiter demon <lb/>Å¿tratum eÅ¿t.</s> - </p> - <pb o="140" rhead="IO. BAPT. BENED." n="152" file="0152" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0152"/> - <p> - <s xml:space="preserve">Ponamus nunc quadratum lateris <var>.a.u.</var> eſſe .12. clarum erit quodlibet quadratum <lb/>aliorum duorum laterum <var>.a.q.</var> et <var>.u.q.</var> futurum nouem, ex ijs quæ poÅ¿teriore loco dixi <lb/>mus, & quia quadratum ipÅ¿ius <var>.q.a.</var> eÅ¿t tantò minus aliorum duorum quadratorum <lb/>Å¿umma, quantum eÅ¿t duplum producti ipÅ¿ius <var>.q.a.</var> in <var>.a.o.</var> ex .13. Å¿ecundi, Å¿ed alia duo <lb/>quadrata Å¿imul collecta faciunt .21. à quo numero Å¿ubtrahendo quadratum ipÅ¿ius <var>.a.<lb/>q.</var> ideÅ¿t nouem, remanebit numerus .12. pro duplo producti ipÅ¿ius <var>.q.a.</var> in <var>.a.o.</var> cuius <lb/>dupli me-<lb/> - <anchor type="figure" xlink:label="fig-0152-01a" xlink:href="fig-0152-01"/> - dia pars, id-<lb/>eÅ¿t Å¿implex <lb/>productum <lb/>ipÅ¿ius <var>.q.a.</var> <lb/><reg norm="in" type="wordlist">Ä©</reg> <var>a.o.</var> erit 6. <lb/>Sed <reg norm="quia" type="simple">ê—a</reg> qua <lb/>dratum ip-<lb/>Å¿ius <var>.q.a.</var> eÅ¿t <lb/>nouem, <lb/>eius radix <var>.<lb/>q.a.</var> crit .3. <lb/>per <reg norm="quam" type="context">quã</reg> di-<lb/>uidendo .6. <lb/>productum <lb/>ipÅ¿ius <var>.q.a.</var> <lb/>in <var>.a.o.</var> pro <lb/>latere <var>.a.o.</var> <lb/>conÅ¿urgent <lb/>duo, cum er <lb/>go <var>.a.o.</var> Å¿int <lb/>duo tertia <lb/>ipÅ¿ius <var>.a.q.</var> <lb/>certi <reg norm="erimus" type="simple">erimê°</reg> <lb/><var>a.o.</var> eſſe latus dicti exagoni.</s> - </p> - <div type="float" level="4" n="3"> - <figure xlink:label="fig-0152-01" xlink:href="fig-0152-01a"> - <image file="0152-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0152-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="15"> - <head xml:space="preserve">CAP. XV.</head> - <p> - <s xml:space="preserve"><reg norm="DEÅ¿iderantes" type="context">DEÅ¿iderãtes</reg> Å¿cire deinde <var>.l.k.</var> in figura <var>.M.</var> quar <lb/> - <anchor type="figure" xlink:label="fig-0152-02a" xlink:href="fig-0152-02"/> - ti cap. tertiæ partis perÅ¿pectiuÄ™ Danielis <lb/>Barbari, Å¿eu Zamberti, eſſe veram <reg norm="altitudinem" type="context">altitudinẽ</reg> cor-<lb/>poris octoaedri, <reg norm="primum" type="context">primÅ©</reg> Å¿cire debemus <reg norm="quod" type="simple">ê™</reg> <reg norm="exiÅ¿tente" type="context">exiÅ¿tẽte</reg> <var>.b.<lb/>h.</var> vt <reg norm="etiam" type="context">etiã</reg> <var>.b.l.</var> tripla ad <var>.b.k.</var> vt ex ijs, quÄ™ Å¿uperius <reg norm="iam" type="context">iã</reg> <lb/>diximus, facile percipi poteÅ¿t; </s> - <s xml:space="preserve">ex penultima primi <var>.<lb/>b.l.</var> in potentia, Å¿eÅ¿quioctaua erit ad <var>.k.l.</var> ipÅ¿a et <var>.k.<lb/>l.</var> dupla <reg norm="inpotentia" type="context">inpotẽtia</reg> ad <var>.h.k.</var> & ob id ducta <reg norm="cum" type="context">cÅ©</reg> eſſet <var>.h.<lb/>l.</var> exiÅ¿teret in potentia tripla ad <var>.h.k.</var> & Å¿eÅ¿quialtera <lb/>ad <var>.l.k.</var> & Å¿eÅ¿quitertia ad <var>.l.b.</var> & Å¿ic ad <var>.h.b.</var> vnde <var>.l.h.</var> <lb/>æqualis eſſet vni ex lateribus <reg norm="trianguli" type="context">triãguli</reg> Ä™quilateri di-<lb/>cti corporis. </s> - <s xml:space="preserve">Ex rationibus igitur Å¿uperius hîc poÅ¿i-<lb/>tis <var>.l.k.</var> erit altitudo dicta, id eÅ¿t diÅ¿tantia inter duas <lb/>facies inuicem oppoÅ¿itas, octoaedri.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0152-02" xlink:href="fig-0152-02a"> - <image file="0152-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0152-02"/> - </figure> - </div> - <p> - <s xml:space="preserve"><reg norm="Neque" type="simple">Neq;</reg> volo te ignorare <reg norm="alium" type="context">aliÅ©</reg> <reg norm="non" type="context">nõ</reg> <reg norm="paruum" type="context">paruÅ©</reg> fuiſſe <reg norm="errorem" type="context">errorẽ</reg> <lb/>illius Zamberti: </s> - <s xml:space="preserve">cum <reg norm="eodem" type="context">eodẽ</reg> capite affirmet angulos <lb/>octoacdri rectos eſſe <reg norm="cum" type="context">cÅ©</reg> Å¿int acuti, <reg norm="nam" type="context">nã</reg> <reg norm="vnuÅ¿quiÅ¿que" type="simple">vnuÅ¿quiÅ¿q;</reg> minor eÅ¿t angulo cubi Å¿olido.</s> - </p> - <pb o="141" n="153" file="0153" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0153"/> - </div> - </div> - <div type="chapter" level="2" n="3"> - <head xml:space="preserve">DE MECHANICIS.</head> - <p style="it"> - <s xml:space="preserve"><emph style="sc">SCripservnt</emph> multi multa, & quidem Å¿citißimè, de mechn<unsure/>-<lb/>nicis, at cum natura <reg norm="vÅ¿usque" type="simple">vÅ¿usq;</reg> aliquid Å¿emper vel nouum, vel <lb/>Latens in apertum emittere Å¿oleant, nec ingenui aut grati Å¿it <lb/>animi, posteris inuidere, Å¿i quid ei contigerit comperuiße prius <lb/>tenebris inuolutum: </s> - <s xml:space="preserve">cum tam multa ipÅ¿e ex aliorum diligentia <lb/>Å¿it conÅ¿equut us. </s> - <s xml:space="preserve">Paucula <reg norm="quædam" type="context">quædã</reg> futùra, vt reor, non ingrata his <lb/>qui in biÅ¿ce mechanicis verÅ¿antur, nuÅ¿quam ante bac tentata, <lb/>aut Å¿atis exastè explicata in medium proferre volui: </s> - <s xml:space="preserve">quo vel iuuandi deÅ¿iderium, vel <lb/>Å¿altem non ocioÅ¿i ingenioli argumentum aliquod exbiberem: </s> - <s xml:space="preserve">at que vel boc vno modo me <lb/>inter bumanos vixiſſe testatum relinquerem.</s> - </p> - <div type="section" level="3" n="1"> - <head style="it" xml:space="preserve">De differentia Å¿itus brachiorum libra.</head> - <head xml:space="preserve">CAP.I.</head> - <p> - <s xml:space="preserve"><emph style="sc">OMne</emph> pondus poÅ¿itum in extremitate alicuius brachij libræ maiorem, aut mi-<lb/> - <anchor type="handwritten" xlink:label="hd-0153-01a" xlink:href="hd-0153-01"/> - norem grauitatem habet, pro diuerÅ¿a ratione Å¿itus ipÅ¿ius brachij. </s> - <s xml:space="preserve">Å¿it exe<unsure/>mpli <lb/>gratia <var>.B.</var> centrum, aut, quod diuidit brachia alicuius libræ, & <var>.A.B.Q.</var> vertica-<lb/>lis linea, aut, vt rectius dicam, axis orizontis, & <var>.B.C.</var> vnum brachium dictæ li-<lb/>bræ, & in <var>.C.</var> Å¿it pondus, & <var>.C.O.</var> linea inclinationis, Å¿euicineris<unsure/> <var>.C.</var> verÅ¿us cen-<lb/>trum mundi, cum qua <var>.B.C.</var> angulum rectum conÅ¿tituat in puncto <var>.C</var>. </s> - <s xml:space="preserve">ExiÅ¿tente <lb/>igitur in huiuÅ¿modi Å¿itu brachio <var>.B.C.</var> dico pondus <var>.C.</var> grauius futurum, quam <lb/>in alio quolibet Å¿itu. </s> - <s xml:space="preserve">quia Å¿upra centrum <var>.B.</var> omninò non quieÅ¿cet, quemadmodum <lb/>in quouis alio Å¿itu faceret. </s> - <s xml:space="preserve">Ad quod intelligendum, Å¿it dictum brachium, in Å¿itu <var>.B.<lb/>F.</var> cum eodem pondere in puncto <var>.F.</var> & linea itineris Å¿eu inclinationis dicti ponderis <lb/>Å¿it <var>.F.u.M.</var> per quam lineam dictum pondus progredi non poteÅ¿t, niÅ¿i brachium <var>.B.F.</var> <lb/>breuius redderetur. </s> - <s xml:space="preserve">Vnde clarum erit <lb/> - <anchor type="figure" xlink:label="fig-0153-01a" xlink:href="fig-0153-01"/> - quòd pondus <var>.F.</var> aliquantulum Å¿upra cen <lb/>trum <var>.B.</var> mediante brachio <var>.B.F.</var> nititur. <lb/></s> - <s xml:space="preserve">EÅ¿t quidem verum, quòd pondus <var>.C.</var> nec <lb/>ipÅ¿um etiam per lineam <var>.C.O.</var> proficiÅ¿ce-<lb/>tur, quia iter extremitatis brachij eÅ¿t cir-<lb/>cularis, & <var>.C.O.</var> in vno <reg norm="quodam" type="context">quodã</reg> puncto eÅ¿t <lb/>contingens. </s> - <s xml:space="preserve">Sit hociter <var>.A.C.Q</var>. </s> - <s xml:space="preserve">Opor-<lb/>tet nunc præſupponere pondus extremi-<lb/>tatis brachij deberetanto magis <reg norm="centro" type="context">cẽtro</reg> <var>.B.</var> <lb/>inniti, quanto magis linea Å¿uæ inclinatio-<lb/>nis (ponamus <var>.F.u.M.</var>) propinqua erit di <lb/>cto centro <var>.B.</var> quod Å¿equenti cap. proba-<lb/>bo, vt exempli gratia, Å¿it <var>.F.</var> Å¿uper <var>.u.</var> pun-<lb/>ctum medij ex æquo inter <var>.C.</var> et <var>.B.</var> qua-<lb/>propter <var>.u.B.</var> æqualis erit <var>.u.C.</var> vndeÅ¿e- - <pb o="142" rhead="IO. BAPT. BENED." n="154" file="0154" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0154"/> - quetur dictum pondus grauius futurum pro parte <var>.F.C.</var> quam pro ea, quæ eÅ¿t <var>.A.F.</var> & <lb/>minus Å¿upra centrum <var>.B.</var> pro dicta parte <var>.F.C.</var> quam pro parte <var>.A.F.</var> quieturum; </s> - <s xml:space="preserve">& <lb/>dictum brachium quanto magis orizontale erit à ſitu <var>.B.F.</var> tantò minus-Å¿upra dictum <lb/>centrum <var>.B.</var> quieÅ¿cet, & hac ratione grauius quoque erit, & quanto magis vicinum <lb/>erit ipÅ¿i <var>.A.</var> à dicto <var>.F.</var> tantò magis Å¿uper centrum <var>.B.</var> quoque quieÅ¿cet, vnde <reg norm="tantò" type="context">tãtò</reg> quo-<lb/>que leuius exiÅ¿tet. </s> - <s xml:space="preserve">Idem dico de omni Å¿itu brachij per girum inferiorem <var>.C.Q.</var> vbi <lb/>pondus pendebit à centro <var>.B.</var> dictum centrum attrahendo, quemadmodum Å¿uperius <lb/>illud impellebat. </s> - <s xml:space="preserve">Hæc verò omnia cap. Å¿equenti melius percipientur.</s> - </p> - <div type="float" level="4" n="1"> - <handwritten xlink:label="hd-0153-01" xlink:href="hd-0153-01a"/> - <figure xlink:label="fig-0153-01" xlink:href="fig-0153-01a"> - <image file="0153-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0153-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="2"> - <head style="it" xml:space="preserve">De proportione ponderis extremitatis brachij libr & <lb/>in diuerÅ¿o Å¿itu ab orizontali.</head> - <head xml:space="preserve">CAP. II.</head> - <p> - <s xml:space="preserve"><emph style="sc">PRoportio</emph> ponderis in <var>.C.</var> ad idem pondus in F. erit quemadmodum totius <lb/>brachij <var>.B.C.</var> ad partem <var>.B.u.</var> poÅ¿itam inter centrum & lineam <var>.F.u.M.</var> inclinatio-<lb/>nis, quam pondus ab extremitate <var>.F.</var> liberum verÅ¿us mundi <reg norm="centrum" type="context">centrÅ©</reg> conficeret. </s> - <s xml:space="preserve">Quod <lb/>vt facilius intelligamus imaginemur <reg norm="alterum" type="context">alterÅ©</reg> brachium libræ <var>.B.D.</var> & in extremo <var>.D.</var> <lb/>locatum aliquod pondus minus pondere <var>.C.</var> vt <var>.B.u.</var> pars <var>.B.C.m.</var> nor eÅ¿t <var>.B.D.</var> cla-<lb/>rè cognoÅ¿cetur ex .6. lib. primi de ponderibus Archimedis, quòd Å¿i in puncto <var>.u.</var> col-<lb/>locatum erit pondus ipÅ¿ius <var>.C.</var> libra nihil penitus à ſitu orizontali dimouebitur. </s> - <s xml:space="preserve">Sed <lb/>perinde eÅ¿t quòd pondus <var>.F.</var> æquale <var>.C.</var> Å¿it in extremo <var>.F.</var> in Å¿itu brachij <var>.B.F.</var> <reg norm="quam" type="context">quã</reg> vt Å¿it <lb/>in puncto <var>.u.</var> in Å¿itu ipÅ¿ius <var>.B.u.</var> orizontali. </s> - <s xml:space="preserve">Ad cuius rei euidentiam imaginemur <reg norm="filum" type="context">filÅ©</reg> <var>.<lb/>F.u.</var> perpendiculare, & in cuius extremo <var>.u.</var> pendere pondus, quod erat in <var>.F.</var> vnde cla <lb/>rum erit quòd eundem effectum gignet, ac Å¿i fuiſſet in <var>.F.</var> quod, vt iam diximus re-<lb/>manens affixum puncto <var>.u.</var> brachij <var>.B.u.</var> tantò minus graue eÅ¿t Å¿itu ipÅ¿ius <var>.C.</var> quantò <var>.u.<lb/>B.</var> minus eÅ¿t ipÅ¿o <var>.B.C</var>. </s> - <s xml:space="preserve">Idem aſſero Å¿i brachium eſſet in Å¿itu <var>.e.B.</var> quod facilè cogno-<lb/>Å¿cere poterimus, Å¿i imaginemur filum appenÅ¿um ipÅ¿i <var>.u.</var> brachij <var>.B.C.</var> & vÅ¿que ad <var>.e.</var> <lb/><reg norm="perpendicularem" type="context">perpendicularẽ</reg>, in quo extremo <reg norm="appensum" type="context">appensÅ©</reg> eſſet pondus æquale ponderi <var>.C.</var> & <reg norm="liberum" type="context">liberÅ©</reg> <lb/>ab <var>.e.</var> brachij <var>.B.e.</var> vnde libra orizontalis manebit. </s> - <s xml:space="preserve">Sed Å¿i brachium <var>.B.e.</var> conÅ¿olida-<lb/>tum fuiſſet in tali Å¿itu cum orizontali <var>.B.D.</var> <lb/> - <anchor type="figure" xlink:label="fig-0154-01a" xlink:href="fig-0154-01"/> - & <reg norm="appenÅ¿o" type="context">appẽſo</reg> <reg norm="pondere" type="context">põdere</reg> <var>.C.</var> in <var>.e.</var> libero à filo, nec <lb/><reg norm="aÅ¿cenderet" type="context">aÅ¿cẽderet</reg>, <reg norm="neque" type="simple">neq;</reg> deÅ¿cenderet. </s> - <s xml:space="preserve">quia tantum <lb/>eÅ¿t quod ipÅ¿um Å¿it appenÅ¿um filo, <reg norm="quod" type="simple">ê™</reg> pendet <lb/>ab <var>.u.</var> quantum quòd ab ipÅ¿o liberum <reg norm="appem" type="context">appẽ</reg> <lb/>nÅ¿um fuiſſet <var>.e.</var> brachij <var>.B.e.</var> & hoc procede <lb/>ret ab eo quòd partim pendereta centro <var>.<lb/>B.</var> & Å¿i <reg norm="brachium" type="context">brachiÅ©</reg> eſſet in Å¿itu <var>.B.Q.</var> totum <reg norm="pon" type="context">põ</reg> <lb/>dus centro <var>.B.</var> remaneret appenÅ¿um, <reg norm="quem- admodum" type="context">quem-<lb/>admodÅ©</reg> in Å¿itu <var>.B.A.</var> <reg norm="totum" type="context">totÅ©</reg> dicto centro an-<lb/>niteretur. </s> - <s xml:space="preserve">vnde fit vt hoc modo pondus <lb/>magis aut minus Å¿it graue, quò magis <lb/>aut minus à centro pendet, aut eidem niti-<lb/>tur: </s> - <s xml:space="preserve"><reg norm="atque" type="simple">atq;</reg> hæc eÅ¿t cauÅ¿a proxima, & per Å¿e, <lb/> - <anchor type="handwritten" xlink:label="hd-0154-01a" xlink:href="hd-0154-01"/> - qua fit vt vnum <reg norm="idemque" type="simple">idemq;</reg> pondus in vno eo-<lb/><reg norm="demque" type="simple">demqÌ;</reg> medio magis aut minus graue exi- - <pb o="143" rhead="DE MECHAN." n="155" file="0155" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0155"/> - Å¿tat. </s> - <s xml:space="preserve">Et quamuis appellem latus <var>.B.C.</var> orizontale, Å¿upponens illud angulum rectum <lb/>cum <var>.C.O.</var> facere, vnde angulus <var>.C.B.Q.</var> fit vt minor Å¿it recto, ob quantitatem vnius <lb/>anguli Ä™qualis ei, quem duæ <var>.C.O.</var> et <var>.B.Q.</var> in centro regionis <reg norm="elementaris" type="context">elemẽtaris</reg> <reg norm="conÅ¿tituunt" type="context">conÅ¿tituÅ©t</reg>, <lb/>hoc tamen nihil refert, cum dictus angulus inÅ¿enÅ¿ibilis Å¿it magnitudinis. </s> - <s xml:space="preserve">Ab iÅ¿tis au-<lb/>tem rationibus elicere poſſumus, quod Å¿i punctus <var>.u.</var> erit ex æquo medius inter cen-<lb/>trum <var>.B.</var> & extremum <var>.C.</var> pondus <var>.F.</var> aut <var>.M.</var> pendebit, aut nitetur pro medietate dicto <lb/>centro <var>.B.</var> & Å¿i dictum <var>.u.</var> erit propius <var>.B.</var> quam puncto <var>.C.</var> pendebit ab ipÅ¿o, aut nitetur <lb/>ipÅ¿i amplius <reg norm="quam" type="context">quã</reg> exmedietate, & Å¿i magis verÅ¿us <var>.C.</var> minus <reg norm="quam" type="context">quã</reg> ex medietate <reg norm="nitetur" type="simple">nitet̃</reg>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0154-01" xlink:href="fig-0154-01a"> - <image file="0154-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0154-01"/> - </figure> - <handwritten xlink:label="hd-0154-01" xlink:href="hd-0154-01a"/> - </div> - </div> - <div type="section" level="3" n="3"> - <head style="it" xml:space="preserve">Quòd quantit as cuiuÅ¿libet ponderis, aut uirtus mouens re-<lb/>Å¿pectu alterius quantitatis cognoÅ¿catur beneficio <lb/>perpendicularium ductarum à centro <lb/>libr & ad line am inclinationis.</head> - <head xml:space="preserve">CAP. III.</head> - <p> - <s xml:space="preserve">EX ijs, quæ à nobis hucuÅ¿que Å¿unt dicta, facilè intelligi poteÅ¿t, <reg norm="quod" type="simple">ê™</reg> quantitas <var>.B.u.</var> <lb/>quæ ferè perpendicularis eÅ¿t à centro <var>.B.</var> ad lineam <var>.F.u.</var> inclinationis, ea eÅ¿t, <lb/> - <anchor type="handwritten" xlink:label="hd-0155-01a" xlink:href="hd-0155-01"/> - quæ nos ducit in cognitionem quantitatis virtutis ipÅ¿ius <var>.F.</var> in huiuÅ¿modi Å¿itu, conÅ¿ti <lb/>tuens videlicet linea <var>.F.u.</var> cum brachio <var>.F.B.</var> angulum acutum <var>.B.F.u</var>. </s> - <s xml:space="preserve">Vt hoc tamen <lb/>melius intelligamus, imaginemur libram <var>.b.o.a.</var> fixam in centro <var>.o.</var> ad. cuius etrema <lb/>Å¿int appenÅ¿a duo pondera, aut duæ virtutes mouentes <var>.e.</var> et <var>.c.</var> ita tamen <reg norm="quod" type="simple">ê™</reg> linea incli-<lb/>nationis <var>.e.</var> ideÅ¿t <var>.b.e.</var> faciat angulum rectum cum <var>.o.b.</var> in puncto <var>.b.</var> linea verò inclina <lb/>tionis <var>.c.</var> ideÅ¿t <var>.a.c.</var> faciat angulum acutum, aut obtuÅ¿um cum <var>.o.a.</var> in puncto <var>.a</var>. </s> - <s xml:space="preserve">Imagi-<lb/>nemur ergo lineam <var>.o.t.</var> perpendicularem lineæ <var>.c.a.</var> inclinationis, vnde <var>.o.t.</var> minor <lb/>erit <var>.o.a.</var> ex .18. primi Euclidis. Å¿ecetur deinde imaginatione <var>o.a.</var> in puncto <var>.i.</var> ita ut <lb/><var>o.i.</var> æqualis. </s> - <s xml:space="preserve">Å¿it <var>.o.t.</var> & puncto <var>.i.</var> appenÅ¿um Å¿it pondus æquale ipÅ¿i <var>.c.</var> cuius inclinationis <lb/>linea parallela Å¿it lineæ inclinationis ponderis <var>.e.</var> Å¿upponendo tamen pondus aut vir <lb/>tutem <var>.c.</var> ea ratione maiorem eſſe ea, quæ eÅ¿t <var>.e.</var> qua <var>.b.o.</var> maior eÅ¿t <var>.o.t.</var> abÅ¿que dubio <lb/>ex .6. lib. primi Archi. de ponderibus <var>.b.o.i.</var> non mouebitur Å¿itu, Å¿ed Å¿i loco <var>.o.i.</var> imagi <lb/>nabimur <var>.o.t.</var> conÅ¿olidatam cum <var>.o.b.</var> & per lineam <var>.t.c.</var> attractam virtute <var>.c.</var> Å¿imiliter <lb/>quoque continget ut <var>b.o.</var> t; </s> - <s xml:space="preserve">communi quadam Å¿cientia, non moueatur Å¿i tu. </s> - <s xml:space="preserve">EÅ¿t ergo <lb/> - <anchor type="handwritten" xlink:label="hd-0155-02a" xlink:href="hd-0155-02"/> - quod propoÅ¿uimus verum quantitatem alicuius ponderis reÅ¿pectu ad eam, quæ eÅ¿t <lb/>alterius debere depræhendi à perpendicularibus, quæ à centro libræ ad lineas incli <lb/>nationis exiliunt. </s> - <s xml:space="preserve">Hinc autem innoteÅ¿cit facillimè, quantum vigoris, & vis pondus, <lb/>aut virtus <var>.c.</var> ad angulum rectum cum <var>.o.a.</var> minimè trahens, amitttat. </s> - <s xml:space="preserve">Hinc quoque co <lb/>rollarium quoddam Å¿equetur, quò d quantò propinquius erit centrum <var>.o.</var> libræ cen-<lb/>tro regionis elementaris, tantò quo que minus erit graue.</s> - </p> - <div type="float" level="4" n="1"> - <handwritten xlink:label="hd-0155-01" xlink:href="hd-0155-01a"/> - <handwritten xlink:label="hd-0155-02" xlink:href="hd-0155-02a"/> - </div> - <figure position="here"> - <image file="0155-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0155-01"/> - </figure> - <pb o="144" rhead="IO. BAPT. BENED." n="156" file="0156" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0156"/> - </div> - <div type="section" level="3" n="4"> - <head style="it" xml:space="preserve">Quemadmodum exÅ¿upradictis cauÅ¿is omnes staterarum & <lb/>uectium cauſæ dependeant.</head> - <head xml:space="preserve">CAP. IIII.</head> - <p> - <s xml:space="preserve">VIs brachij longioris alicuius Å¿tateræ, aut vectis, maior breuioris, ab ijs, quæ in Å¿u <lb/>perioribus capitibus diximus, ideÅ¿t <reg norm="quod" type="simple">ê™</reg> nitatur <reg norm="pendeatuem" type="context">pendeatuẽ</reg> magis aut minus à <lb/>centro pondus in extremitate brachij maioris poÅ¿itum, oboritur. </s> - <s xml:space="preserve">Quamobrem illud <lb/>à nobis primò eÅ¿t cognoÅ¿cendum, Å¿tateras, aut vectes, puras mathematicas li-<lb/>neas non eſſe, Å¿ed naturales, hincque exiÅ¿tere corpora cum materia coniuncta. </s> - <s xml:space="preserve">Nunc <lb/>igitur imaginemur <var>.n.s.</var> eam Å¿uperficiem eſſe, quæ Å¿ecundum longitudinem axem Å¿ta <lb/>teræ Å¿cindit. </s> - <s xml:space="preserve">& Å¿upponamus ipÅ¿ius centrum eſſe primum in <var>.i.</var> & maius brachium eſſe <lb/> <var>.i.u</var>: minus autem <var>.i.n.</var> & lineam verticalem <var>.i.o.</var> quæ tanta Å¿it, quanta eÅ¿t Å¿piſſitu-<lb/>do, aut craſſities ipÅ¿ius Å¿tateræ à ſuperiori latere ad inferius, ad faciliorem intelligen-<lb/>tiam, Å¿upponendo <var>.n.s.</var> <reg norm="parallelogrammam" type="context">parallelogrãmam</reg>. </s> - <s xml:space="preserve">PoÅ¿itis igitur duobus ponderibus æquali-<lb/> - <anchor type="handwritten" xlink:label="hd-0156-01a" xlink:href="hd-0156-01"/> - bus in extremitatibus brachiorum, experientia innoteÅ¿cit, <reg norm="quod" type="simple">ê™</reg> pondus ad <var>.u.s.</var> appen-<lb/>Å¿um, viol entiam faciet ponderi appenÅ¿o ad <var>.n.x.</var> Å¿ed nos volumus inueÅ¿tigare <reg norm="causam" type="context">causã</reg> <lb/>huius effectus, quæ à nemine vnquam literarum monumentis, <reg norm="quod" type="simple">ê™</reg> Å¿ciam, conÅ¿ignata <lb/> - <anchor type="handwritten" xlink:label="hd-0156-02a" xlink:href="hd-0156-02"/> - fuit. </s> - <s xml:space="preserve">Iam diximus Å¿tateram, aut vectem materialem eſſe & <var>.n.s.</var> eius Å¿uperficiem me-<lb/>diam, Å¿upponendo <var>.i.</var> eſſe centrum quo nititur dicta Å¿tatera aut vectis; </s> - <s xml:space="preserve">Cum hocer-<lb/>go ita Å¿e habeat, Å¿int <var>.u.s.</var> et <var>.n.x.</var> lineæ inclinationum ponderum, & imaginemur, <reg norm="quod" type="simple">ê™</reg> <lb/>dicta pondera pendeant à punctis <var>.u.</var> et <var>.n.</var> vt reuera pendent, etiam Å¿i appenÅ¿a eſſent <lb/>Å¿ub <var>.s.</var> et <var>.x.</var> quia punctum <var>.u.</var> & punctum <var>.n.</var> ita coniuncta Å¿unt cum <var>.s.</var> et <var>.x.</var> ut qui <reg norm="vnum" type="context">vnÅ©</reg> <lb/>trahit alterum quoque trahat. </s> - <s xml:space="preserve">Imaginemur quoque duas lineas <var>.i.u</var>: <var>i.n.</var> et <var>.i.e.</var> quÄ™ <lb/><var>i.e.</var> faciat angulum <var>.o.i.e.</var> æqualem angulo <var>.o.i.n</var>. </s> - <s xml:space="preserve">Hinc clarè nobis patebit, Å¿i quis ipÅ¿i <lb/>e. pondus ipÅ¿ius <var>.u.</var> (<reg norm="quod" type="simple">ê™</reg> æquale eÅ¿t ponderi <var>.n.</var>) appenderet, id eandem planè vim habe <lb/>ret, quam pondus ipÅ¿ius <var>.n.</var> habet, & Å¿tateram neque Å¿urÅ¿um, neque deorÅ¿um moue-<lb/>ret, quia ambo pondera ad centrum <var>.i.</var> mediantibus lineis <var>.e.i.</var> et <var>.n.i.</var> exÄ™quo annite-<lb/>rentur, Å¿ed dicto pondere poÅ¿ito in .u: linea <var>.u.i.</var> per quam pondus centro annititur, <lb/>magis orizontalis quam <var>.e.i.</var> fit, & linea <var>.u.s.</var> inclinationis longius diÅ¿tans à centro <var>.i.</var> <lb/> - <anchor type="handwritten" xlink:label="hd-0156-03a" xlink:href="hd-0156-03"/> - quà m linea <var>.e.t.</var> vnde huiuÅ¿modi pondus magis quoque liberum à centro <var>.i.</var> reÅ¿ultat. <lb/></s> - <s xml:space="preserve">magisque ponderoÅ¿um, quam cum erat in <var>.e.</var> ratione eorum, quæ primo & Å¿ecundo <lb/>capitibus diximus, & ob hanc cauÅ¿am Å¿uperat pondus poÅ¿itum in <var>.n</var>. </s> - <s xml:space="preserve">Sed Å¿i centrum <lb/>fuerit .in <var>.o.</var> imaginabimur duas lineas <var>.o.s.</var> et <var>.o.x.</var> & Å¿upponemus quòd pondera po-<lb/>Å¿ita Å¿int in <var>.s.</var> et <var>.x.</var> vnde exiÅ¿tente magis orizontali linea <var>.o.s.</var> quam erit <var>.o.x.</var> & linea <lb/><var>u.s.</var> inclinationis longius diÅ¿tante à centro <var>.o.</var> quà m linea <var>.e.t.</var> eius pondus erit <reg norm="quoque" type="simple">quoq;</reg> <lb/> - <anchor type="figure" xlink:label="fig-0156-01a" xlink:href="fig-0156-01"/> - <anchor type="figure" xlink:label="fig-0156-02a" xlink:href="fig-0156-02"/> - <pb o="145" rhead="DE MECHAN." n="157" file="0157" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0157"/> - grauius, quia tantò minus pendebit à centro <var>.o.</var> & ratiocinando, vt Å¿uperius dixi-<lb/>mus, inueniemus eundem effectum verum eſſe. </s> - <s xml:space="preserve">In Å¿tateris, rectè & propriè appella <lb/>ri poteÅ¿t <var>.x.i.s.</var> aut <var>.n.o.u.</var> orizontalis, Å¿ed in omni vectium Å¿pecie, hoc <reg norm="tantum" type="context">tãtum</reg> per quan <lb/>dam Å¿imilitudinem dicetur. </s> - <s xml:space="preserve">Idem contemplari licet Å¿upponendo centrum in medio <lb/>inter <var>.o.</var> et <var>.i.</var> quod vnuÅ¿quiÅ¿que ex Å¿e abÅ¿que alterius auxilio facile præſtare poterit.</s> - </p> - <div type="float" level="4" n="1"> - <handwritten xlink:label="hd-0156-01" xlink:href="hd-0156-01a"/> - <handwritten xlink:label="hd-0156-02" xlink:href="hd-0156-02a"/> - <figure xlink:label="fig-0156-01" xlink:href="fig-0156-01a"> - <image file="0156-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0156-01"/> - </figure> - <handwritten xlink:label="hd-0156-03" xlink:href="hd-0156-03a"/> - <figure xlink:label="fig-0156-02" xlink:href="fig-0156-02a"> - <image file="0156-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0156-02"/> - </figure> - </div> - </div> - <div type="section" level="3" n="5"> - <head style="it" xml:space="preserve">De quibuÅ¿dam rebus animaduerÅ¿ione dignis.</head> - <head xml:space="preserve">CAP.V.</head> - <p> - <s xml:space="preserve">NOn omittenda mihi <reg norm="videntur" type="context">vidẽtur</reg> quædam, quæ ad <reg norm="tractationem" type="context">tractationẽ</reg> vectium admodum <lb/>Å¿unt neceſſaria. </s> - <s xml:space="preserve">Quod autem quærimus, in eo conÅ¿iÅ¿tit, quòd aliqui vectes <lb/>adhibeantur ad opus, quorum centrum, quod Græci <reg norm="hypomochlion" type="context">hypomochliõ</reg> appellant vnum <lb/>eÅ¿t ex extremis ipÅ¿ius vectis, & pondus, quod Å¿urÅ¿um eleuari debet, inter ipÅ¿a-<lb/>met extrema iacet, propinquum tamen hypomochlio, vt exempli gratia, Å¿i vectis <lb/>eſſet infraÅ¿cripta figura <var>.o.s.u.x.</var> cuius hypomochlion eſſet in puncto <var>.o.</var> & pondus in <lb/>puncto <var>.n.</var> clarum erit, <reg norm="quod" type="simple">ê™</reg> cum eleuari debeat <var>.n.</var> oportebit quoque opera manus ele-<lb/>uari <var>.u</var>. </s> - <s xml:space="preserve">Nunc conÅ¿iderandum eÅ¿t quomodo pondus <var>.n.</var> annitatur ad <var>.u</var>. </s> - <s xml:space="preserve">Hanc ob cau <lb/>Å¿am imaginabimur rectas lineas <var>.n.o</var>: <var>n.i</var>: <var>n.e</var>: <var>n.t.</var> et <var>.n.u.</var> quarum <var>.n.i.</var> verÅ¿us mundi cen <lb/>trum Å¿it poÅ¿ita, et <var>.n.t.</var> faciat angulum <var>.i.n.t.</var> æqualem angulo <var>.i.n.o</var>. </s> - <s xml:space="preserve">Nunc ponendo ali <lb/>quam virtutem in <var>.i.</var> æquali inclinatione ad Å¿uperius conÅ¿tante, vt <var>.n.</var> ad inferius (re-<lb/>mota tamen grauitate materiæ vectis) huiuÅ¿modi virtus, totum pondus ipÅ¿ius <var>.n.</var> com <lb/>muni quadam Å¿cientiæ notione Å¿uÅ¿tinebit. </s> - <s xml:space="preserve">& Å¿i <reg norm="pondus" type="context">põdus</reg> ipÅ¿ius <var>.n.</var> eſſet in <var>.x.</var> è directo Å¿u-<lb/>per <var>.o.</var> totum pondus Å¿uper hypomochlio Å¿e haberet, & tanta virtus ipÅ¿ius hypomo-<lb/>chlij Å¿ufficeret ad reÅ¿iÅ¿tendum pro Å¿uÅ¿tinendo, quanta eÅ¿t grauitas ipÅ¿ius ponderis, <lb/>Å¿ed ipÅ¿um iterum ponamus in <var>.n.</var> ibi clarum erit, quòd Å¿i alia virtus à parte inÅ¿eriori <lb/>ad Å¿uperiorem vectis non opponitur, excepto tamen hypomochlio, oportebit virtu <lb/>te cuiuÅ¿dam partis ponderis <var>.n.</var> (abÅ¿que conÅ¿ideratione tamen, vt iam dixi, ponderis <lb/>materiæ vectis) vt vectis à parte <var>.s.u.</var> deprimatur, & dixi vnius cuiuÅ¿dam partis pon-<lb/>deris <var>.n.</var> quia alia <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> ponderis pars annititur ipÅ¿i hypomochlio <var>.o.</var> <reg norm="mediante" type="context">mediãte</reg> linea <lb/><var>o.n.</var> quæ angulos rectos cum <var>.o.x.</var> non facit. </s> - <s xml:space="preserve">Si autem à puncto <var>.t.</var> opponet Å¿eÅ¿e huiuÅ¿-<lb/>modi reÅ¿iÅ¿tentia, vt vectis non deprimatur, clarum erit communi Å¿cientia, <reg norm="quod" type="simple">ê™</reg> virtus <lb/>ponderis <var>.n.</var> diuiÅ¿a erit per medium æqualiter, cuius vna medietas Å¿uper <var>.o.</var> quieÅ¿cet, <lb/>& alia Å¿uper <var>.t.</var> mediantibus duabus lineis <var>.n.o.</var> et <var>.n.t</var>. </s> - <s xml:space="preserve">Imaginemur nunc reÅ¿iÅ¿tentiam <lb/>t. ablatam eſſe, <reg norm="poÅ¿itamque" type="simple">poÅ¿itamq;</reg> in <var>.e.</var> clarum quoque erit, <reg norm="quod" type="simple">ê™</reg> maior pars ponderis <var>.n.</var> ipÅ¿i <var>.e.</var> <lb/>annitetur beneficio lineæ <var>.n.e.</var> quà m ipÅ¿i <var>.o.</var> cum linea <var>.n.i.</var> inclinationis ipÅ¿i <var>.e.</var> Å¿it pro <lb/>pinquior quam <var>.o.</var> quia omnis reÅ¿iÅ¿tentia aut in <var>.i.</var> aut in <var>.e.</var> aut in <var>.t.</var> aut in <var>.u.</var> eÅ¿t loco <lb/>centri, quemadmodum eÅ¿t <var>.o.</var> & alter alterius opera iuuatur. </s> - <s xml:space="preserve">Si verò eadem reÅ¿iÅ¿ten <lb/>tia poÅ¿ita erit in <var>.u.</var> clarum quoque erit, <reg norm="quod" type="simple">ê™</reg> minor pars ponderis <var>.n.</var> annitetur ipÅ¿i <var>.u.</var> <reg norm="quam" type="context">quã</reg> <lb/>ipÅ¿i <var>.o.</var> cum dicta <var>.n.i.</var> à centro <var>.u.</var> longius quam à centro <var>.o.</var> diÅ¿ter, & proportio partis <lb/>ponderis <var>.n.</var> in <var>.o.</var> ad propor-<lb/>tionem partis ponderis <var>.n.</var> in <lb/> - <anchor type="figure" xlink:label="fig-0157-01a" xlink:href="fig-0157-01"/> - u. non erit <reg norm="Å¿ecundum" type="context">Å¿ecÅ©dum</reg> propor <lb/>tionem angulorum <var>.u.n.i.</var> et <lb/><var>o.n.i.</var> Å¿ed Å¿ecundum propor <lb/>tionem <var>.u.i.</var> ad <var>.i.o.</var> quod cla <lb/>rè compræhendi poteÅ¿t ab - <pb o="146" rhead="IO. BAPT. BENED." n="158" file="0158" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0158"/> - huius effectus conuerÅ¿o, ideÅ¿t, vt quemadmodum nunc Å¿upponuntur <var>.o.</var> et <var>.u.</var> eſſe duo <lb/>centra quibus <reg norm="Å¿uÅ¿tinetur" type="simple">Å¿uÅ¿tinet̃</reg> pondus <var>.e.</var> ipÅ¿ius <var>.n.</var> imaginemur <var>.n.</var> eſſe quoddam centrum à <lb/>quo pendeant duo pondera <var>.o.</var> et <var>.u.</var> Å¿ic inuicem proportionata, ut Å¿unt <var>.u.i.</var> et <var>.i.o.</var> <lb/>certe horum ponderum cauÅ¿a Å¿tatera <var>.o.s.</var> quam vectem appellabamus à nulla parte <lb/>inclinabitur. </s> - <s xml:space="preserve">Redeuntes nunc ad propoÅ¿itum, dicemus <reg norm="quod" type="simple">ê™</reg> annitente pondere ipÅ¿ius <var>.<lb/>n.</var> minus ad <var>.u.</var> quam ad <var>.o.</var> ideÅ¿t ad <var>.t.</var> minori vi opus erit in <var>.u.</var> quà m in <var>.t.</var> ad attollen-<lb/>dum pondus ipÅ¿ius <var>.n.</var> & Å¿ic per conÅ¿equens quantò longius erit punctum <var>.u.</var> ab <var>.t.</var> tan <lb/>tò minori quoque vi egebit, & conÅ¿equenter quando vis, aut reÅ¿iÅ¿tentia in <var>.u.</var> ita pro <lb/>portionata erit illi, quæ eÅ¿t ipÅ¿ius <var>.o.</var> vt eÅ¿t <var>.o.i.</var> ad <var>.i.u.</var> vectis non mouebitur. </s> - <s xml:space="preserve">Sed quan <lb/>do erit proportio maior, reÅ¿iÅ¿tentiæ ipÅ¿ius <var>.u.</var> ad eam, quæ eÅ¿t ipÅ¿ius <var>.o.</var> ea, quæ eÅ¿t <var>.o.<lb/>i.</var> ad <var>.i.u.</var> </s> - <s xml:space="preserve">tunc vectis à par-<lb/>teipÅ¿ius <var>.u.s.</var> eleuabitur, Å¿i <lb/> - <anchor type="figure" xlink:label="fig-0158-01a" xlink:href="fig-0158-01"/> - vero proportio minor eſſet <lb/>quà m.o.i. ad <var>.i.u.</var> </s> - <s xml:space="preserve">tunc ve-<lb/>ctis ab eadem parte depri-<lb/>metur.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0157-01" xlink:href="fig-0157-01a"> - <image file="0157-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0157-01"/> - </figure> - <figure xlink:label="fig-0158-01" xlink:href="fig-0158-01a"> - <image file="0158-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0158-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="6"> - <head style="it" xml:space="preserve">De ratione cuiuÅ¿dam uis adauctæ.</head> - <head xml:space="preserve">CAP. VI.</head> - <p> - <s xml:space="preserve">QVibuÅ¿dam in locis vtuntur <reg norm="quidam" type="context">quidã</reg> <reg norm="quodam" type="context">quodã</reg> <reg norm="inÅ¿trumento" type="context">inÅ¿trumẽto</reg> piÅ¿torio ad <reg norm="Å¿ubigendam" type="context context">Å¿ubigẽdã</reg> pa-<lb/>Å¿tam, vnius tantum hominis ui adhibita, quæ quidem machina cum mihi di-<lb/>gna contemplatione eſſe videatur, eius aliquam rationem proponere volui, pro cu-<lb/>ius deÅ¿criptione imaginemur planum, in quo Å¿edet ille, qui voluit paÅ¿tam, & in quo <lb/>ipÅ¿a paÅ¿ta eÅ¿t repoÅ¿ita <var>.T.S.D.</var> & triangulum <var>.T.A.S.</var> immobile perpendiculare-<lb/>qÌue Å¿uperficiei dicti plani, angulo autem <var>.A.</var> coniunctum lignum <var>.A.E.</var> vt Å¿emidiame <lb/>trum mobilem, & æqualem perpendiculari ipÅ¿ius trianguli, </s> - <s xml:space="preserve">unde <var>.A.</var> loco centri erit <lb/>et <var>.D.O.</var> Å¿it Å¿emidiameter, qui paÅ¿tam contundit, & ab eius extremo <var>.O.</var> (quod <var>.O.</var> <lb/>quando <var>.D.O.</var> orizontalis eÅ¿t, in baÅ¿i dicti trianguli reperitur) veniat lignum <var>.O.V.</var> <lb/>quod cum <var>.A.V.</var> Å¿it æquale perpendiculari imaginatæ ab angulo <var>.A.</var> baÅ¿i <var>.T.S.</var> <reg norm="deno- datum" type="context">deno-<lb/>datÅ©</reg> <reg norm="tantum" type="wordlist/context">tñ</reg> utvulgo <reg norm="dicitur" type="simple">dicit̃</reg> Å¿eu flexile in <var>.O.</var> & in <var>.V.</var> vt elleuare <reg norm="atque" type="simple">atq;</reg> deprimere Å¿emidiame <lb/>trum <var>.D.O.</var> poſſit, et <var>.V.O.</var> Å¿it æqualis <var>.A.V.</var> et <var>.V.</var> medium Å¿it inter <var>.A.</var> et <var>.E.</var> vnde <var>.A.V.</var> <lb/>cum <var>.O.V.</var> æquales erunt <var>.A.E.</var> Å¿unt deinde duo ligna <reg norm="perpendicularia" type="context">perpẽdicularia</reg> ab <var>.A.</var> ad baÅ¿im <lb/>fixa, & immobilia inter Å¿e adeò diÅ¿tantia, vt inter ipÅ¿a <reg norm="pertranÅ¿eant" type="context context">pertrãſeãt</reg> <var>.O.V.</var> et <var>.D.O.</var> Å¿upra <lb/>& infra, ne deuiet Å¿emidiametrum <var>.D.O</var>. </s> - <s xml:space="preserve">In extremitate deinde ipÅ¿ius <var>.E.</var> Å¿it lignum <lb/>quoddam tenue, vt digitus polex, ad angulos rectos cum <var>.A.E.</var> quod ab aliquo, qui <lb/>antedictam machinam Å¿tet, manibus teneatur, qui quidem homo idipÅ¿um lignum, <lb/>ideÅ¿t Å¿emidiametrum <var>.A.E.</var> à ſuperficie trianguli dicti, ad Å¿e trahendo, & deinde ver <lb/>Å¿us eundem triangulum impellendo, vim quandam maximam mediante Å¿emidia <lb/>metro <var>.D.O.</var> Å¿uper paÅ¿tam excitat.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius rei contemplatione volo vt Å¿ecundam hanc Å¿ubÅ¿criptam figuram <var>.b.a.<lb/>u.x.</var> imaginemur, in qua <var>.u.</var> exprimat <var>.A.</var> primæ figuræ, & <var>.a.</var> denotet <var>.O.</var> & <var>.o.V.</var> & <var>.x.<lb/>E.</var> imaginemur etiam <var>.u.a.</var> baÅ¿em trianguli <var>.a.u.o.</var> cui <var>.o.t.</var> perpendicularis dictæ baÅ¿i <var>.<lb/>u.a.</var> addatur. </s> - <s xml:space="preserve"><reg norm="HucuÅ¿que" type="simple">HucuÅ¿q;</reg> igitur <var>.u.o.</var> æqualis erit <var>.o.x.</var> & ipÅ¿i <var>.o.a.</var> imaginemur etiam <var>.a.o.</var> <lb/>vÅ¿que ad <var>.b.</var> ita productam vt <var>.o.b.</var> æqualis Å¿it <var>.o.a.</var> ponamus etiam pondus in <var>.a.</var> impel- - <pb o="147" rhead="DE MECHAN." n="159" file="0159" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0159"/> - lere verÅ¿us <var>.u.</var> vnde linea eius inclinationis Å¿it Å¿emper <var>.a.u.</var> Å¿upponamus etiam <var>.a.o.b.</var> <lb/>eſſe <reg norm="libram" type="context">librã</reg>, aut Å¿tateram, aut vectem, & <var>.o.</var> eius centrum, vnde vis, aut virtus ipÅ¿ius <var>.a.</var> <lb/>proportionalis erit ipÅ¿i <var>.o.t.</var> reÅ¿pectu virtutis, aut vis imaginatæ in <var>.b.</var> inclinationis <lb/>perpendicularis ipÅ¿i <var>.b.a.</var> quæ quidem virtus, aut vis in <var>.b.</var> proportionalis erit ipÅ¿i <var>.b.<lb/>o.</var> ex tertio capite huius tractatus; </s> - <s xml:space="preserve">Si ergo fuiſſet poÅ¿ita in <var>.b.</var> virtus quædarn ad an-<lb/>gulum rectum, trahens lineam <var>.b.o.</var> tam proportionatam virtuti perpendiculari ip-<lb/>Å¿ius <var>.a.</var> quam eÅ¿t <var>.o.t.</var> proportionata ipÅ¿i <var>.o.b.</var> Å¿tatera <var>.b.o.a.</var> non moueretur, Å¿ed quæuis <lb/>portio maior in <var>.b.</var> Å¿uperaret <var>.a.</var> cum autem fuerit <var>.o.x.</var> æqualis ipÅ¿i <var>.o.b.</var> <reg norm="idem" type="context">idẽ</reg> planè eue-<lb/> - <anchor type="figure" xlink:label="fig-0159-01a" xlink:href="fig-0159-01"/> - <anchor type="figure" xlink:label="fig-0159-02a" xlink:href="fig-0159-02"/> - niet, communi quadam Å¿cientia, ponen-<lb/>do virtutem <var>.b.</var> in <var>.x</var>. </s> - <s xml:space="preserve">Quantitas ergo virtu <lb/>tis in <var>.x.</var> quæ Å¿uperare debet reÅ¿iÅ¿tentiam <lb/>in <var>.a.</var> quæ ipÅ¿i <var>.u.</var> contraponitur, debet ha-<lb/>bere aliquantulum maioris proportionis <lb/>ad reÅ¿iÅ¿tentiam, quæ in <var>.a.</var> angulum re-<lb/>ctum efficeret cum <var>.a.o.</var> ea, quæ eÅ¿t <var>.o.t.</var> <lb/>ad <var>.o.x</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0159-01" xlink:href="fig-0159-01a"> - <image file="0159-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0159-01"/> - </figure> - <figure xlink:label="fig-0159-02" xlink:href="fig-0159-02a"> - <image file="0159-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0159-02"/> - </figure> - </div> - <pb o="148" rhead="IO. BABPT. BENED." n="160" file="0160" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0160"/> - </div> - <div type="section" level="3" n="7"> - <head style="it" xml:space="preserve">De quibuÅ¿dam erroribus Nicolai Tartaleæ circa pondera <lb/>corporum & eorum motus, quorum aliqui deÅ¿umpti <lb/>fuerunt à fordano Å¿criptore quodam antiquo.</head> - <head xml:space="preserve">CAP. VII.</head> - <p> - <s xml:space="preserve">CVm magis amici veritatis eſſe debeamus quà m cuiuÅ¿quam hominis, quemad-<lb/>modum AriÅ¿to. Å¿cribit, detegam hoc loco quoÅ¿dam errores Nicolai TartaleÄ™ <lb/>de ponderibus corporum, & velocitatibus motuum localium. </s> - <s xml:space="preserve">Et primum decipitur <lb/>is in .8. lib. Å¿uarum diuerÅ¿arum inuentionum in Å¿ecunda propoÅ¿itione, cum non ani-<lb/>maduerterit quanti momenti Å¿int extrinÅ¿ecæ reÅ¿iÅ¿tentiæ.</s> - </p> - <p> - <s xml:space="preserve">Subiectum quoque tertiæ propoÅ¿itionis eÅ¿t malè demonÅ¿tratum, quia idem pla-<lb/>nè ex eius demonÅ¿tratione iam dicta corporibus hætereogeneis, aut figura diuerÅ¿is <lb/>contingeret, quod ad velocitates attinet.</s> - </p> - <p> - <s xml:space="preserve">In quarta propoÅ¿itione, quod ad <reg norm="diÅ¿putandum" type="context context">diÅ¿putãdÅ©</reg> proponit <reg norm="non" type="context">nõ</reg> concludit melius. </s> - <s xml:space="preserve"><reg norm="autem" type="context">autẽ</reg> id <lb/>ab eo <reg norm="Å¿equitur" type="simple">Å¿equit̃</reg>, quod Archimedes in .6. propoÅ¿itione lib. primi de <reg norm="ponderibus" type="context">põderibus</reg> <reg norm="probauit" type="simple">ê“bauit</reg>.</s> - </p> - <handwritten/> - <p> - <s xml:space="preserve">Sed in Å¿ecunda parte quintÄ™ propoÅ¿itionis non uidet <reg norm="quod" type="simple">ê™</reg> uigore Å¿itus eo modo, quo <lb/>ipÅ¿e diÅ¿putat, nulla elicitur ponderis differentia. </s> - <s xml:space="preserve">quia Å¿i corpus <var>.B.</var> deÅ¿cendere debet <lb/>per arcum <var>.i.l.</var> corpus <var>.A.</var> aÅ¿cendere debet per arcum <var>.u.s.</var> æqualem, & Å¿imilem. eadem <lb/>quoque ratione Å¿ituatum, vt eÅ¿t arcus <var>.i.l.</var> vnde vt eÅ¿t facilè corpori <var>.B.</var> deÅ¿cendere <lb/>per arcum <var>.i.l.</var> difficile ita erit corpori <var>.A.</var> aÅ¿cendere per arcum <var>.u.s</var>. </s> - <s xml:space="preserve">HÄ™c autem qnin <lb/>ta propoÅ¿itio Tartaleæ eÅ¿t Å¿ecuuda quæſtio à Iordano propoÅ¿ita.</s> - </p> - <p> - <s xml:space="preserve">Quòd autem ad primum corollarium dictæ propoÅ¿itionis attinet, verum ille qui <lb/>dem Å¿cribit, eius tamen effectus cauÅ¿a & à Iordano prius, & ab ipÅ¿o poÅ¿tea citata, na-<lb/>tura Å¿ua vera non eÅ¿t. </s> - <s xml:space="preserve">quia vera cauÅ¿a per Å¿e ab eo oritur, <reg norm="quod" type="simple">ê™</reg> à centro libræ dependeat <lb/>vt primo cap. huius tractatus oÅ¿tendi. </s> - <s xml:space="preserve">Secundum verò corollarium falÅ¿um eſſe, ijs ra <lb/>tionibus quas nunc Å¿ubiungam, patebit. </s> - <s xml:space="preserve">Imaginemur <var>.u.</var> pro centro regionis ele-<lb/>mentaris, & libram <var>.b.o.a.</var> obliquam reÅ¿pectu ad <var>.u.</var> & brachijs æqualibus <reg norm="conÅ¿tantem" type="context">conÅ¿tãtem</reg>, <lb/>& pondera in <var>.a.</var> et in <var>.b.</var> etiam æqualia. </s> - <s xml:space="preserve">lineæ autem inclinationum Å¿int <var>.a.u.</var> et <var>.b.u.</var> <lb/>imaginemur etiam lineam <var>.o.u.</var> & à centro <var>.o.</var> libræ duas <var>.o.t.</var> et <var>.o.e.</var> perpendiculares <lb/>inclinationum lineis; </s> - <s xml:space="preserve">vnde pondus ipÅ¿ius <var>.a.</var> in huiuÅ¿modi Å¿itu tam erit proportiona <lb/>tum ponderi <var>.b.</var> quam proportionata erit linea <var>.o.t.</var> lineæ <var>.o.e.</var> ex eo <reg norm="quod" type="simple">ê™</reg> tertio cap. hu-<lb/>iustractatus probaui, Å¿ed linea <var>.o.t.</var> maior eÅ¿t linea <var>.o.e.</var> quod Å¿ic probo. </s> - <s xml:space="preserve">Imaginemur <lb/>triangulum <var>.u.a.b.</var> circunÅ¿criptum eſſe à circulo <var>.u.a.n.b.</var> cuius <var>.c.</var> Å¿it centrum, <reg norm="quod" type="simple">ê™</reg> erit <lb/>extra lineam <var>.u.o.</var> cum Å¿upponatur <var>.a.o.b.</var> obliquam eſſe reÅ¿pectu ad <var>.u.o</var>. </s> - <s xml:space="preserve">Imagine-<lb/>mur deinde à centro <var>.c.</var> lineam <var>.c.o.s.</var> vÅ¿que ad circunferentiam, quæ perpendicula-<lb/>ris erit ipÅ¿i <var>.a.b.</var> ex tertia lib. 3. Eucli. </s> - <s xml:space="preserve">Å¿i poÅ¿teà imaginemur duas lineas <var>.c.a.</var> et <var>.c.b.</var> ha <lb/>bebimus ex .8. lib. primi, angulum <var>.a.c.o.</var> æqualem angulo <var>.b.c.o</var>. </s> - <s xml:space="preserve">Vnde ex .25. lib. 3. <lb/>arcus <var>.a.s.</var> æqualis erit arcui <var>.b.s.</var> Å¿ed Å¿i imaginabimur <var>.u.o.</var> ad circunferentiam vÅ¿que <lb/>productam, clarum erit <reg norm="quod" type="simple">ê™</reg> arcum <var>.s.b.</var> Å¿ecaret in puncto <var>.n.</var> vnde arcus <var>.n.b.</var> minor erit <lb/>arcu <var>.n.a.</var> & Å¿ic etiam angulus <var>.n.u.b.</var> minor erit angulo <var>.n.u.a.</var> ex <ref id="ref-0024">ultima lib. 6.</ref> </s> - <s xml:space="preserve">Imagi-<lb/>nemur nunc alium quendam circulum, cuius <var>.o.u.</var> Å¿it diameter, cuius circunferentia <lb/>per duo puncta <var>.e.</var> et <var>.t.</var> <reg norm="prætergradiatur" type="simple">prætergradiat̃</reg>, cum in ipÅ¿is Å¿int angulirecti, quod quilibet ex <lb/>Å¿eratio cinando colligere poteÅ¿t, Å¿i .30. lib. 3. in mentem reuocauerit. </s> - <s xml:space="preserve">Sed cum angu-<lb/>lus <var>.o.u.t.</var> Å¿it maior angulo <var>.o.u.e.</var> arcus <var>.o.t.</var> maior erit arcu <var>.o.e.</var> ex vltima .6. vnde cor <lb/>da <var>.o.t.</var> maior erit corda ipÅ¿ius <var>.o.e.</var> ex conuerÅ¿o .27. lib. 3. quod eÅ¿t propoÅ¿itum. </s> - <s xml:space="preserve">Pon-<lb/> - <anchor type="handwritten" xlink:label="hd-0160-01a" xlink:href="hd-0160-01"/> - dusigitur ipÅ¿ius <var>.a.</var> in huiuÅ¿modi Å¿itu, pondere ipÅ¿ius <var>.b.</var> grauius erit. </s> - <s xml:space="preserve">Quod è directo ijs <lb/>repugnat quæ Tartalea in 2. parte quinræ propoÅ¿itionis ediÅ¿erit, & per conÅ¿equens <lb/>2. corollarij falÅ¿itatem oÅ¿tendit, vt eam quoque, quæ in 6. propoÅ¿itione latet. </s> - <s xml:space="preserve">quia <reg norm="cum" type="context">cÅ©</reg> - <pb o="149" rhead="DE MECHAN." n="161" file="0161" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0161"/> - proportio <reg norm="ponderis" type="context">põderis</reg> <var>.a.</var> ad pon <lb/>dusipÅ¿ius <var>.b.</var> eadem Å¿it cum <lb/>ea quÄ™ eÅ¿t <var>.o.t.</var> ad <var>.o.e.</var> Å¿ub co <lb/><reg norm="gnitionem" type="context">gnitionẽ</reg> noÅ¿tram cadere po <lb/>teÅ¿t, primum cognoÅ¿cendo <lb/>angulos obliquitatis librÄ™, <lb/>ideÅ¿t angulos <var>.b.o.u.</var> et <var>.a.o.<lb/>u.</var> quia oportet Å¿emper Å¿up-<lb/> - <anchor type="figure" xlink:label="fig-0161-01a" xlink:href="fig-0161-01"/> - ponere Å¿itum aliquem no-<lb/>tum. </s> - <s xml:space="preserve">Si nobis deinde co-<lb/>gnita erit proportio ipÅ¿ius <var>.<lb/>o.u.</var> ad <var>.o.b.</var> et. ad <var>.o.a.</var> aſſe-<lb/>quemur cognitionem angu <lb/>li <var>.b.</var> et <var>.o.a.u.</var> & per conÅ¿e-<lb/>quens ipÅ¿ius <var>.o.a.t.</var> eius reÅ¿i-<lb/>dui, vnde poÅ¿tea beneficio <lb/>angulorum <var>.e.</var> et <var>.t.</var> rectorum <lb/>& laterum <var>.o.b.</var> et <var>.o.a.</var> cogni <lb/>torum in cognitionem <var>.o.t.</var> <lb/>et <var>.o.e.</var> facile deueniemus.</s> - </p> - <div type="float" level="4" n="1"> - <handwritten xlink:label="hd-0160-01" xlink:href="hd-0160-01a"/> - <figure xlink:label="fig-0161-01" xlink:href="fig-0161-01a"> - <image file="0161-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0161-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="8"> - <head xml:space="preserve">CAP. VIII.</head> - <p> - <s xml:space="preserve">QVod autem idem Tartalea in .6. propoÅ¿itione, & Iordanus in Å¿ecunda parte. <lb/></s> - <s xml:space="preserve">Å¿ecundæ propoÅ¿itionis Å¿cribunt, maximum quoque errorem inÅ¿e continet. <lb/></s> - <s xml:space="preserve">Dicunt enim <reg norm="angulum" type="context">angulÅ©</reg> <lb/><var>h.a.f.</var> differentem ab <lb/>angulo <var>.d.b.f.</var> alia ra-<lb/>tione non eſſe quà m <lb/>per angulum conta-<lb/>ctus <reg norm="duorum" type="context">duorÅ©</reg> <reg norm="circulorum" type="context">circulorÅ©</reg>, <lb/>vt in Å¿ua figura Å¿cribit <lb/>Tartalea; </s> - <s xml:space="preserve">id quod fal-<lb/>Å¿iſſimum eÅ¿t. </s> - <s xml:space="preserve"><reg norm="Quam" type="context">Quã</reg> ob <lb/>cauÅ¿am in Å¿ubÅ¿cripta <lb/>figura Å¿it libra <var>.B.A.</var> <lb/> - <anchor type="figure" xlink:label="fig-0161-02a" xlink:href="fig-0161-02"/> - & eius centrum. </s> - <s xml:space="preserve">C et <var>.<lb/>u.</var> <reg norm="centrum" type="context">centrÅ©</reg> regionis ele <lb/>mentaris, et <var>.A.u.</var> et <var>.B.<lb/>u.</var> lineæ <reg norm="inclinationum" type="context">inclinationÅ©</reg>. <lb/></s> - <s xml:space="preserve">Imaginemur deinde <lb/>lineam <var>.B.K.</var> <reg norm="parallelam" type="context">parallelã</reg> <lb/>ipÅ¿i <var>.A.u.</var> quæ gyrum <var>.<lb/>B.F.A.</var> in puncto <var>.K.</var> <lb/>communi Å¿cientiæ <reg norm="prae cepto" type="simple">prÄ™Â<lb/>cepto</reg> Å¿cindet, & habe <lb/>bimus angulum <var>.K.B.<lb/>Z.</var> æqualem angulo <var>.<lb/>H.A.F.</var> ideÅ¿t <var>.u.A.F.</var> <lb/>(quia <var>.H.u.</var> et <var>.D.</var> <reg norm="unum" type="context">unÅ©</reg> <lb/>Å¿unt) cum ex .29. libr. <lb/>primi Euclidis angu- - <pb o="150" rhead="IO. BAPT. BENED." n="162" file="0162" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0162"/> - lus <var>u.A.C.</var> æqualis Å¿it <lb/>angulo <var>.K.B.T.</var> & an-<lb/>gulus <var>.C.A.F.</var> æqua-<lb/>lis angulo <var>.T.B.Z.</var> <reg norm="nunc" type="context">nÅ©c</reg> <lb/>comparatio eÅ¿t inter <lb/>angulum <var>.D.B.F.</var> & an <lb/>gulum <var>.K.B.Z.</var> miÅ¿tili-<lb/>neos, qui quidem duo <lb/>anguli, <reg norm="communem" type="context">cõmunem</reg> ha-<lb/>bent angulum miÅ¿tili <lb/>neum <var>.K.B.F.</var> quapro-<lb/> - <anchor type="figure" xlink:label="fig-0162-01a" xlink:href="fig-0162-01"/> - pter Å¿i angulus <var>.K.B.Z.</var> <lb/>miÅ¿tilineus maior eÅ¿t <lb/>angulo <var>.D.B.F.</var> miÅ¿ti-<lb/>lineo per angulum <var>.<lb/>K.B.Z.</var> contingentiæ, <lb/>circulorum ergo angu <lb/>lus miÅ¿tilineus com-<lb/>munis <var>.K.B.F.</var> æqualis <lb/>erit miÅ¿tilineo, angu-<lb/>lo <var>.D.B.F.</var> pars vide-<lb/>licet Å¿ui toto. </s> - <s xml:space="preserve">Omnis <lb/>autem error in quem <lb/>Tartalea, <reg norm="Iordanusque" type="simple">Iordanusq;</reg> <lb/>lapÅ¿i fuerunt ab eo, <reg norm="quod" type="simple">ê™</reg> <lb/> - <anchor type="handwritten" xlink:label="hd-0162-0a" xlink:href="hd-0162-01"/> - lineas inclinationum <lb/>pro parallelis viciſſim <lb/>Å¿umpÅ¿erunt, emana-<lb/>uit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0161-02" xlink:href="fig-0161-02a"> - <image file="0161-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0161-02"/> - </figure> - <figure xlink:label="fig-0162-01" xlink:href="fig-0162-01a"> - <image file="0162-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0162-01"/> - </figure> - <handwritten xlink:label="hd-0162-01" xlink:href="hd-0162-01a"/> - </div> - <p> - <s xml:space="preserve">Septima propoÅ¿itio Tartaleæ, quæ eÅ¿t <reg norm="quinta" type="simple">ê—nta</reg> quæſtio Iordani mihi <reg norm="videtur" type="simple">videt̃</reg> excipien-<lb/>da riÅ¿u, cum pondus ipÅ¿ius <var>.A.</var> ponderi ipÅ¿ius <var>.B.</var> exiÅ¿tens æquale, grauius Å¿it pondere <lb/>eiuÅ¿dem <var>.B.</var> ratione minoris aperturæ anguli contingentiæ in <var>.A.</var> quam in <var>.B.</var> in quo <lb/>idem error committitur, qui in præcedenti committebatur, cum Å¿cilicet ipÅ¿e putet <lb/>lineas <var>.A.E.</var> et <var>.B.D.</var> figuræ ab eo confictæ Å¿ibi inuicem eſſe parallelas, quæ etiam Å¿i <lb/>æquidiÅ¿tantes eſſent (vnde angulus <var>.E.A.G.</var> minor eſſet angulo <var>.D.B.F.</var>) non eam ta <lb/>men ob cauÅ¿am huiuÅ¿modiangulorum differentia cauÅ¿a eſſet differentiæ <reg norm="grauitatum" type="context">grauitatÅ©</reg> <lb/>ipÅ¿orum <var>.A.</var> et <var>.B.</var> ob ea quæ cap .4. huius tractatus poÅ¿ui.</s> - </p> - <p> - <s xml:space="preserve">Octaua autem propoÅ¿itio, quæ eÅ¿t .6. quæſtio Iordani Iongè melius demonÅ¿tratur <lb/>ab Archi. in .6. lib. primi de ponderibus, cum nec à Iordano, nec à Tartalæa probata <lb/>fuerit, cum ijdem non probauerint præcedentes, quas in dicta .8. </s> - <s xml:space="preserve">TartalÄ™a citat, qui <lb/>neque etiam probat nonam .10. 11. 12. et .13. cum ad prÄ™cedentes probandas mini <lb/>mè acceſſerit.</s> - </p> - <p> - <s xml:space="preserve">Quartadecima verò, quæ eÅ¿t .10. quęſtio Iordani, duas ob cauÅ¿as eÅ¿t falÅ¿a, quarum <lb/>vna eÅ¿t, <reg norm="quod" type="simple">ê™</reg> (Å¿upponendo <var>.A.D.E.G.B.</var> eſſe vnum brachium librÄ™ et <var>.A.</var> punctum <reg norm="centri" type="context">cẽtri</reg> <lb/>eiuÅ¿dem, et <var>.D.</var> pondus Ä™quale ponderi <var>.E.</var> & lineas inclinationum <var>.D.K.</var> et <var>.E.M.</var>) an <lb/>guli <var>.K.D.E.</var> et <var>.M.E.G.</var> Å¿ibi <reg norm="inuicem" type="context">inuicẽ</reg> <reg norm="non" type="context">nõ</reg> Å¿unt Ä™quales; </s> - <s xml:space="preserve"><reg norm="cum" type="context">cÅ©</reg> ille angulus Å¿it intrinÅ¿ecus, hic <lb/>verò extrinÅ¿ecus & oppoÅ¿itus dicto intrinÅ¿eco <reg norm="vnius" type="simple">vniê°</reg> <reg norm="trianguli" type="context">triãguli</reg> terminati à . <var>D.E.</var> à <var>.D.K.</var> - <pb o="151" rhead="DE MECHAN." n="163" file="0163" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0163"/> - et <var>.E.M.</var> lineis productis vÅ¿que ad centrum regionis elementaris, vnde dictus angu-<lb/>lus <var>.M.E.G.</var> maior eÅ¿t alio, ex .16 lib. primi Eucli. </s> - <s xml:space="preserve">Qua ratione fit, vt hanc ob cauÅ¿am <lb/>E. grauius Å¿it ipÅ¿o <var>.D.</var> cum minus dependeat à centro <var>.A.</var> vt primo cap. huius tractatus <lb/>iam dixi. </s> - <s xml:space="preserve">Alia quoque eÅ¿tratio, qua dictum <var>.E.</var> grauius fit ipÅ¿o <var>.D.</var> quÄ™ quidem eÅ¿t <lb/>maior diÅ¿tantia à centro <var>.A.</var> libræ, per Å¿imiles rationes capit .4. huius tractatus ci-<lb/>tatas.</s> - </p> - <p> - <s xml:space="preserve">Decimaquinta <reg norm="quoque" type="simple">quoq;</reg> nil penitus valet, quÄ™ eÅ¿t .11. quęſtio Iordani, cuius Autho-<lb/>ris opuÅ¿culum opera Traiani BibliopolÄ™ Venetijs è tenebris in lucem emerÅ¿it.</s> - </p> - </div> - <div type="section" level="3" n="9"> - <head style="it" xml:space="preserve">QuòdÅ¿ummaratione Å¿tateræper æqualia interualla <lb/>Å¿int diuiſæ.</head> - <head xml:space="preserve">CAP. IX.</head> - <p> - <s xml:space="preserve">MAgna cum ratione <reg norm="diuiduntur" type="context">diuidÅ©tur</reg> Å¿tateræ per interualla Ä™qualia, in libras, aut in <lb/>vncias, aut quoquo alio modo. </s> - <s xml:space="preserve">Nam Å¿it Å¿tatera exempli gratia <var>.a.b.</var> <lb/>& punctum, <reg norm="quod" type="simple">ê™</reg> eam Å¿uÅ¿tinet Å¿it <var>.c.</var> & vas illud, <reg norm="quod" type="simple">ê™</reg> continetid, quod ponderari debet <lb/>f. </s> - <s xml:space="preserve">Imaginemur nunc quod pondus brachij <var>.c.b.</var> ab una parte, & pondus brachij <var>.c.a.</var> <reg norm="cum" type="context">cÅ©</reg> <lb/>eo, <reg norm="quod" type="simple">ê™</reg> eÅ¿t dicti vaÅ¿is <var>.f.</var> ab altera parte, Å¿int cauſę, quibus Å¿tatera <var>.a.b.c.</var> Å¿tet orizonta-<lb/>lis. cui Å¿ic orizontali manenti imaginemur ad punctum <var>.a.</var> adiunctum eſſe pondus, <lb/>veluti vnius librÄ™. & ad punctum <var>.d.</var> tam diÅ¿tanti à <var>.c.</var> ut eÅ¿t <var>.a.</var> ab ipÅ¿o <var>.c.</var> aliud quoque <lb/>pondus vnius libræ <reg norm="additum" type="context">additÅ©</reg> eſſe, vnde <reg norm="coni" type="context">cõi</reg> <reg norm="quadam" type="context">quadã</reg> Å¿cientia Å¿tatera, non mouebitur Å¿itu. <lb/></s> - <s xml:space="preserve"><reg norm="quia" type="simple">ê—a</reg> exiÅ¿tentibus duobus hiÅ¿ce ponderibus æqualibus, altero in <var>.d.</var> & altero in <var>.a.</var> remo <lb/>ta cum eſſent <var>.d.b.</var> et <var>.f.</var> abÅ¿que dubio <var>.a.d.</var> non mutaret Å¿itum, Å¿ed <var>.d.b.</var> et, f. in Å¿itu, in <lb/>quo reperiuntur, à centro paribus viribus prÄ™dita Å¿unt. </s> - <s xml:space="preserve">Addendo igitur <var>.d.b.</var> ipÅ¿i <var>.d.</var> <lb/>et <var>.f.</var> ipÅ¿i .a: Å¿umma earum, æqualibus quoque viribus conÅ¿tabunt. ex communi Å¿en-<lb/>tentia, quæ habet Å¿i Ä™qualibus addas Ä™qualia, tota quoque fient Ä™qualia. </s> - <s xml:space="preserve">Si verò <lb/>ponderi ipÅ¿ius <var>.a.</var> aliud adderetur eidem Ä™quale, haberemus in <var>.a.</var> duplum pon-<lb/>dus ei <reg norm="quod" type="simple">ê™</reg> eÅ¿t ipÅ¿ius <var>.d.</var> Å¿ed volentes vt Å¿olum cum pondere ipÅ¿ius <var>.d.</var> Å¿tatera Å¿tet orizon <lb/>talis, Å¿i dictum pondus ipÅ¿ius <var>.d.</var> longè diÅ¿tabit à centro <var>.c.</var> per duplum ipÅ¿ius <var>.c.a.</var> ideÅ¿t <lb/>ipÅ¿ius <var>.c.d.</var> id <reg norm="quod" type="simple">ê™</reg> volumus aſſeque-<lb/>mur, beneficio Å¿upradictarum ra <lb/> - <anchor type="figure" xlink:label="fig-0163-01a" xlink:href="fig-0163-01"/> - tionum, adiuti opera Å¿extÄ™ lib. priÂ<lb/>mi de <reg norm="ponderibus" type="context">põderibus</reg> Archimedis. </s> - <s xml:space="preserve">Et <lb/>Å¿i quis aliud <reg norm="quoque" type="simple">quoq;</reg> pondus adiun <lb/>geret ipÅ¿i <var>.a.</var> æquale illi priori, ad <lb/><reg norm="efficiendum" type="context">efficiẽdum</reg>, vt Å¿tatera Å¿emper ori <lb/>zontalis maneret, oporteret, vt <reg norm="pondus" type="context">põdus</reg> ipÅ¿ius <var>.d.</var> ab <var>.c.</var> longè diÅ¿taret, ita vt huiuÅ¿modi <lb/>diÅ¿tantia tripla eſſet primæ, & Å¿ic per quoÅ¿dam quaÅ¿i gradus interualla redderentur <lb/>æqualia.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0163-01" xlink:href="fig-0163-01a"> - <image file="0163-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0163-01"/> - </figure> - </div> - <pb o="152" rhead="IO. BAPT. BENED." n="164" file="0164" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0164"/> - </div> - <div type="section" level="3" n="10"> - <head style="it" xml:space="preserve">Quòd line a circularis non habe at concauum cum con-<lb/>uexo coniunctum, & quod Aristo. cir caproportio <lb/>nes motuum aberrauerit.</head> - <head xml:space="preserve">CAP.X.</head> - <p> - <s xml:space="preserve">ARiÅ¿toteles in principio quæſtionum Mechanicarum ait lineam, quæ terminat <lb/> - <anchor type="handwritten" xlink:label="hd-0164-01a" xlink:href="hd-0164-01"/> - circulum videtur conuexum habere coniunctum cum concauo, quod falÅ¿um <lb/>eÅ¿t: </s> - <s xml:space="preserve">quia huiuÅ¿modi linea partes nullas Å¿ecundum latitudinem habet, (vt ipÅ¿e etiam <lb/>confirmat) Å¿ed eÅ¿t idem conuexum circuli: </s> - <s xml:space="preserve">linea verò quæ terminus eÅ¿t Å¿uperficiei <lb/>ambientis, & amplectentis circulum eÅ¿t eadem concauitas dictæ Å¿uperficiei eun-<lb/>dem circulum ambientis, quæ nullam conuexitatem habet. </s> - <s xml:space="preserve">& hæ duæ Å¿unt lineæ, <lb/>quarum vna diuerÅ¿a eÅ¿t ab alia, neque altera alterius, quod ad conuexum, & ad con-<lb/>cauum attinet.</s> - </p> - <div type="float" level="4" n="1"> - <handwritten xlink:label="hd-0164-01" xlink:href="hd-0164-01a"/> - </div> - <p> - <s xml:space="preserve">Sed illud, quod AriÅ¿toteles Å¿cribit de duplici reÅ¿pectu motus vnius puncti Å¿ecun <lb/>dum vnam datam pro portionem, non Å¿ufficit, ille enim Å¿ic ait.</s> - </p> - <p> - <s xml:space="preserve">Sit proportio Å¿ecundum quam latum fertur, quam habet <var>.A.B.</var> ad <var>.A.C.</var> et <var>.A.</var> qui <lb/>dem feratur verÅ¿us .B: <var>A.B.</var> verò Å¿ubterferatur verÅ¿us <var>.M.C.</var> latum autem Å¿it <var>.A.</var> <reg norm="quidem" type="context">quidẽ</reg> <lb/>ad <var>.D.</var> vbi autem eÅ¿t <var>.A.B.</var> verÅ¿us <var>.E</var>. </s> - <s xml:space="preserve">Quoniam igitur lationis erat proportio, quam <var>.<lb/>A.B.</var> habet ad <var>.A.C.</var> neceſſe eÅ¿t & <var>.A.D.</var> ad <var>.A.E.</var> hanc habere rationem. </s> - <s xml:space="preserve">Simile igi <lb/> - <anchor type="handwritten" xlink:label="hd-0164-02a" xlink:href="hd-0164-02"/> - tur eÅ¿t pro portione paruum quadr ilaterum maiori. </s> - <s xml:space="preserve">Quamobrem etc.</s> - </p> - <div type="float" level="4" n="2"> - <handwritten xlink:label="hd-0164-02" xlink:href="hd-0164-02a"/> - </div> - <p> - <s xml:space="preserve">Cui reÅ¿pondeo, punctum <var>.A.</var> quod mouetur in linea <var>.A.M.</var> ab <var>.A.</var> verÅ¿us <var>.M.</var> vÅ¿que <lb/>ad <var>.F.</var> non moueriab aliqua proportione determinata magis quà m ab alia: </s> - <s xml:space="preserve">vnde <reg norm="non" type="context">nõ</reg> <lb/>Å¿olum poſſumus imaginari dictum punctum <var>.A.</var> moueri ab <var>.A.</var> vÅ¿que ad <var>.F.</var> eiuÅ¿dem <lb/>velocitatis Å¿ub alia quadam proportione, Å¿ed etiam Å¿ub alia, quæ iam datæ contraria <lb/>Å¿it, vt eÅ¿t proportio ipÅ¿ius <var>.A.C.</var> ad <var>.A.B.</var> <reg norm="imaginantes" type="context">imaginãtes</reg> moueri <var>.A.</var> verÅ¿us <var>.C.</var> et <var>.A.C.</var> ver <lb/>Å¿us <var>.B.M.</var> delatam. </s> - <s xml:space="preserve">Dico etiam idem <var>.A.</var> moueri vÅ¿que ad <var>.F.</var> Å¿ecundum proportio-<lb/>nem ipÅ¿ius <var>.A.O.</var> ad <var>.A.N</var>. </s> - <s xml:space="preserve">Quamobrem imaginemur à puncto <var>.F.</var> lineam <var>.F.H.</var> cum <lb/>linea <var>.F.A.</var> efficere angu-<lb/>lum æqualem angulo <var>.O.<lb/>P.A.</var> & à puncto <var>.A.</var> <reg norm="lineam" type="context">lineã</reg> <lb/> - <anchor type="figure" xlink:label="fig-0164-01a" xlink:href="fig-0164-01"/> - <var>A.H.</var> <reg norm="cum" type="context">cÅ©</reg> linea <var>.A.F.</var> face-<lb/>re <reg norm="angulum" type="context">angulÅ©</reg> <reg norm="æqualem" type="context">æqualẽ</reg> angulo <lb/><var>O.A.P.</var> unde angulus <var>.H.</var> <lb/>æqualis erit angulo <var>.O.</var> <lb/>ex .32. libr. primi Eucl. <lb/></s> - <s xml:space="preserve">& <reg norm="triangulum" type="context">triangulÅ©</reg> <var>.A.H.F.</var> Ä™qui <lb/>angulum erit triangulo <var>.<lb/>A.O.P</var>. </s> - <s xml:space="preserve">Quam ob <reg norm="causam" type="context">causã</reg> <lb/><reg norm="eadem" type="context">eadẽ</reg> proportio erit <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <lb/><var>A.H.</var> ad <var>.F.H.</var> quÄ™ <reg norm="enipÅ¿ius" type="context">ẽipÅ¿ius</reg> <lb/><var>A.O.</var> ad <var>.O.P.</var> punctum <lb/>igitur <var>.A.</var> vÅ¿que ad <var>.F.</var> mouetur Å¿ecundum proportionem etiam ipÅ¿ius <var>.A.O.</var> ad <var>.O.P.</var> <lb/>HuiuÅ¿modi igitur conÅ¿ideratio, ab AriÅ¿totele facta, nullius eÅ¿t momenti.</s> - </p> - <div type="float" level="4" n="3"> - <figure xlink:label="fig-0164-01" xlink:href="fig-0164-01a"> - <image file="0164-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0164-01"/> - </figure> - </div> - <pb o="153" rhead="DE MECHAN." n="165" file="0165" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0165"/> - </div> - <div type="section" level="3" n="11"> - <head style="it" xml:space="preserve">Quod Aristo. in prima mechanicarum quæstionum eius quod <lb/>inquir it, uer am cauÅ¿am non attulerit.</head> - <head xml:space="preserve">CAP. XI.</head> - <p> - <s xml:space="preserve">QVærens AriÅ¿toteles vnde fiat, vt eæ libræ, quæ brachia habent alijs longiora, <lb/>Å¿int exactiores cæteris, ait hoc euenire ratione maioris velocitatis extremo <lb/>rum earundem. </s> - <s xml:space="preserve">Quod verum non eÅ¿t; </s> - <s xml:space="preserve">quia hîc effectus nil aliud eÅ¿t, quam clarius pro <lb/>ponere ob omnium oculos obliquitatem brachiorum à linea orizontali, & oÅ¿tende-<lb/>re etiam facilius à dicto orizontali Å¿itu exire brachia iam dicta. </s> - <s xml:space="preserve">Quæ quidem per Å¿e <lb/>neque à velocitate, neque à tarditate motus, Å¿ed à ratione vectis, & à ma-<lb/>iori interuallo inter Å¿ecundum Å¿itum extremorum à primo proficiÅ¿cuntur. </s> - <s xml:space="preserve">Vt exem-<lb/>pli gratia, imaginemur magnam libram <var>.A.B.</var> orizontalem, cuius centrum Å¿it <var>.E.</var> et <lb/>pondus <var>.B.</var> maius Å¿it pondere ipÅ¿ius <var>.A.</var> vnde conceditur, quòd ob hanc rationem di-<lb/>cta libra Å¿itum mutabit, qui Å¿ecundus Å¿itus Å¿it in <var>.H.F</var>. </s> - <s xml:space="preserve">Imaginemur etiam <reg norm="paruam" type="context">paruã</reg> <reg norm="quan- dam" type="context">quã-<lb/>dam</reg> libram <var>.a.e.b.</var> orizontalem, quæ pondera habeat <var>.a.</var> et <var>.b.</var> æqualia duobus ponde <lb/>ribus alterius libræ & Å¿ecundus Å¿itus Å¿it in <var>.h.f.</var> ita tamen vt anguli circa <var>.e.</var> æquales <lb/>Å¿int ijs, qui Å¿unt circa <var>.E.</var> ideÅ¿t <var>.b.e.f.</var> Å¿it Ä™qualis <var>.B.E.F</var>. </s> - <s xml:space="preserve">Nunc dico Å¿itum <var>.H.F.</var> <reg norm="exa- ctiorem" type="context">exa-<lb/>ctiorẽ</reg> futurum & clariorem Å¿itu <var>.h.e.f.</var> ratione interualli <var>.B.F.</var> maioris, interuallo <var>.<lb/>b.f.</var> quod <var>.B.F.</var> in eadem proportione maior eÅ¿t ipÅ¿o <var>.b.f.</var> in qua <var>.B.E.</var> maius eÅ¿t <var>.b.e.</var> <lb/>quod autem interuallum <var>.B.F.</var> breuiori, aut longiori temporis Å¿pacio quam <var>.b.f.</var> Å¿it fa <lb/>ctum, nil planè refert. </s> - <s xml:space="preserve">Ratione vectis deinde, dico <reg norm="quod" type="simple">ê™</reg> Å¿i Å¿upponemus duas libras pa-<lb/>res <reg norm="æqualesque" type="simple">æqualesqÌ;</reg> in omni alio reÅ¿pectu, præter quà m in brachiorum longitudine, pon-<lb/>dus <var>.B.</var> maiorem vim habebit ad deprimendum brachium <var>.E.B.</var> quà m pondus <var>.b.</var> quia <lb/>libræ materiales, cum Å¿uÅ¿tineantur ab <var>.E.e.</var> & non à puncto mathematico, Å¿ed <lb/>à linea, aut Å¿uperficie naturali in materia exiÅ¿tente. </s> - <s xml:space="preserve">vnde aliqua reÅ¿iÅ¿tentia ipÅ¿i mo-<lb/>tui brachiorum oritur, & hanc ob cauÅ¿am, Å¿upponendo hanc reÅ¿iÅ¿tentiam æqualem <lb/>tam in <var>.E.</var> quà m in <var>.e.</var> clarum erit ob ea, quæ in cap .4. huius tractatus oÅ¿tendi <var>.B.</var> cum <lb/>minus dependeat ab <var>.E.</var> aut minus quoque eidem <var>.E.</var> annitatur, ponderoÅ¿um magis <lb/>futurum, quam <var>.b.</var> & hac de cauÅ¿a mouebit ad partem inferiorem, maiori cum agilita <lb/>te, brachium <var>.E.B.</var> multo magis etiam illud ipÅ¿um deprimet, ideÅ¿t maiorem etiam an <lb/>gulum <var>.B.E.F.</var> quà m erit angulus <var>.b.e.f.</var> faciet.</s> - </p> - <figure position="here"> - <image file="0165-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0165-01"/> - </figure> - <pb o="154" rhead="IO. BAPT. BENED." n="166" file="0166" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0166"/> - </div> - <div type="section" level="3" n="12"> - <head style="it" xml:space="preserve">De uer a cauÅ¿a Å¿ecundæ, & tertiæ quæstionis mechanicæ <lb/>ab AriÅ¿totele nonperÅ¿pecta.</head> - <head xml:space="preserve">CAP. XII.</head> - <p> - <s xml:space="preserve">ARiÅ¿toteles in Å¿ecunda quæſtionum mechanicarum quærens illius rationem Å¿ic <lb/>Å¿cribit.</s> - </p> - <p> - <s xml:space="preserve">Cur Å¿iquidem Å¿urÅ¿um fuerit Å¿partum quando deorÅ¿um lato pondere quiÅ¿piam id <lb/>admouet rurÅ¿us aÅ¿cendit libra: </s> - <s xml:space="preserve">Å¿i autem deorÅ¿um conÅ¿titutum fuerit non aÅ¿cendit, <lb/>Å¿ed manet? </s> - <s xml:space="preserve">an quia Å¿urÅ¿um quidem Å¿parto exiÅ¿tente plus libræ extra perpendiculum <lb/>Å¿i (Å¿partum enim eÅ¿t perpendiculum) quare neceſſe eÅ¿t deorÅ¿um ferriid, quod plus <lb/>eÅ¿t, quare & cætera.</s> - </p> - <p> - <s xml:space="preserve">Sed vera cauÅ¿a, vnde fiat, vt Å¿i Å¿partum fuerit Å¿urÅ¿um, & brachium vnum <lb/>ipfius libræ deprimendo, & idem liberum deinde permittendo, ad Å¿itum ori-<lb/>zontalem redeat, non Å¿olum eÅ¿t maior quantitas ponderis brachiorum quæ iam præ <lb/>tergreſſa eÅ¿t vltra verticalem lineam, Å¿ed etiam eÅ¿t longitudo brachij eleuati, quæ vl <lb/>tra verticalem lineam reperitur, vnde eius extremi pondus redditur grauius in pro-<lb/>portione, quam in hoc exemplo proponam, Å¿it <var>.A.B.</var> libra in Å¿itu orizontali, cuius <lb/>Å¿partum Å¿it <var>.E.</var> Å¿uper ipÅ¿am. </s> - <s xml:space="preserve">& deprimentes brachium ipÅ¿ius <var>.A.</var> vÅ¿que ad <var>.F.</var> eius Å¿itus <lb/>Å¿it in <var>.F.H.</var> vnde medium <reg norm="punctum" type="context">pÅ©ctum</reg> <var>.G.</var> prætergreſſum erit lineam verticalem <var>.V.Z.</var> ver <lb/>Å¿us <var>.B.</var> quæ <var>.V.Z.</var> Å¿ecabit brachium <var>.F.G.</var> in puncto <var>.D.</var> vnde <var>.D.H.</var> longius erit ipÅ¿o <var>.<lb/>F.D</var>. </s> - <s xml:space="preserve">Nunc nobis Å¿upponendum eÅ¿t id, <lb/>quod veriſſimum exiÅ¿tit, dictam Å¿cilicet li <lb/> - <anchor type="figure" xlink:label="fig-0166-01a" xlink:href="fig-0166-01"/> - bram in Å¿itu <var>.F.H.</var> <reg norm="etiam" type="context">etiã</reg> Å¿i Å¿uÅ¿tineatur à pun-<lb/>cto <var>.E.</var> idem tamen futurum ac Å¿i Å¿uÅ¿tenta-<lb/>retur in puncto <var>.D.</var> vnde Å¿equitur, quod <lb/>pondus appenÅ¿um ex ipÅ¿a <var>.H.</var> ita grauius <lb/>reddatur, ipÅ¿o <var>.F.</var> in eadem propor-<lb/>tione, quæ maioreÅ¿t <var>.D.H.</var> ipÅ¿o <var>.D.F.</var> ob <lb/>rationes quas in primis huius tractatus ca-<lb/>pitibus poÅ¿ui, vt etiam Å¿i <var>.D.H.</var> quodmate <lb/>riale eſſe Å¿upponitur, nullam planÅ¿e4; </s> - <s xml:space="preserve">graui-<lb/>tatem haberet, <reg norm="Å¿olustamem" type="context">Å¿olustamẽ</reg> exceſſus vis pon <lb/>deris in <var>.H.</var> poÅ¿iti, longè maior pondere in <lb/>F. collocato pro maiorilongitudine ipÅ¿ius <lb/><var>D.H.</var> Å¿ufficiat. ad præſtandum vt libra ad <lb/>Å¿itum orizontalem redeat.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0166-01" xlink:href="fig-0166-01a"> - <image file="0166-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0166-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">In Å¿ecunda deinde huius quęſtionis par <lb/> - <anchor type="handwritten" xlink:label="hd-0166-01a" xlink:href="hd-0166-01"/> - te, in qua Å¿cribit libram in Å¿itu, in quo poÅ¿i <lb/>ta eÅ¿t, firmam manere, toto cÄ™lo aberrat, quia <reg norm="neceſſarium" type="context">neceſſariÅ©</reg> eÅ¿t, vt omninò cadat, eò <reg norm="uÅ¿que" type="simple">uÅ¿q;</reg> <lb/>quò Å¿partum Å¿urÅ¿um remaneat: </s> - <s xml:space="preserve">ablato tamen omni impedimento, quod nulla eget <lb/>probatione, cum natura Å¿ua clariſſimè pateat.</s> - </p> - <div type="float" level="4" n="2"> - <handwritten xlink:label="hd-0166-01" xlink:href="hd-0166-01a"/> - </div> - <p> - <s xml:space="preserve">CauÅ¿a, deinde, vera tertiæ quæſtionis non eÅ¿t ea, quam AriÅ¿toteles ponit, Å¿ed hu-<lb/>iuÅ¿modi effectus ab eo, quod capitibus .4. et .5. huius tractatus propoÅ¿ui originem <lb/>habet.</s> - </p> - <pb o="155" rhead="DE MECHAN." n="167" file="0167" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0167"/> - </div> - <div type="section" level="3" n="13"> - <head style="it" xml:space="preserve">Quòd AriÅ¿totelisratio in 6. quæſtione poÅ¿it a non Å¿it admittenda.</head> - <head xml:space="preserve">CAP. XIII.</head> - <p> - <s xml:space="preserve">VOlens AriÅ¿toteles rationem proponere, vnde fiat, vt nauis velocius moueatur <lb/>cum antennam altiorem quà m cum depræſſiorem habet, id ad vectis ratio-<lb/>nem refert, quod verum <reg norm="non" type="context">nõ</reg> eÅ¿t. </s> - <s xml:space="preserve">HuiuÅ¿modi enim ratione nauis tardius potius, quà m <lb/>velocius ferri deberet, quia quantò altius eÅ¿t velum, vi venti impulÅ¿um, <reg norm="tantò" type="context">tãtò</reg> magis <lb/>proram ipÅ¿ius nauis in aquam demergit. </s> - <s xml:space="preserve">Sed huiuÅ¿modi effectus à maiori potius <lb/>quantitate venti quam recipit, quà m ab alia aliqua cauÅ¿a oritur, quia ventus liberius <lb/><reg norm="vehementiusque" type="simple">vehementiusqÌ;</reg> in altiore parte, quà m in depræſſione vagatur & perflat.</s> - </p> - </div> - <div type="section" level="3" n="14"> - <head style="it" xml:space="preserve">Quòdrationes ab AriÅ¿totele de octaua quæstione confictæ <lb/>Å¿ufficient es non Å¿int.</head> - <head xml:space="preserve">CAP. XIIII.</head> - <p> - <s xml:space="preserve">RAtiones etiam ab AriÅ¿totele propoÅ¿itæ pro indaganda octauæ quæſtionis ve-<lb/>ritate, in qua quærit vnde fiat, vt corpora rotundæ figuræ, ad <reg norm="voluendum" type="context">voluendÅ©</reg> Å¿int <lb/>faciliora reliquis, quarum reuolutionum corporum tres Å¿pecies aſſignat, <reg norm="quarum" type="context">quarÅ©</reg> vna <lb/>eÅ¿t, vt rotarum <reg norm="curruum" type="context">curruÅ©</reg>; </s> - <s xml:space="preserve">altera vt rotarum puteorum, aut trochlearum, quibus hauri-<lb/>tur aqua; </s> - <s xml:space="preserve">& tertia, vt paruorum vaÅ¿orum a figulis fabricatorum, <reg norm="Å¿ufficientes" type="context">Å¿ufficiẽtes</reg> <reg norm="non" type="context">nõ</reg> Å¿unt.</s> - </p> - <p> - <s xml:space="preserve">Incipiens autem à prima dico dubium non eſſe, quin tangente corpore aliquo ro <lb/>tundo aliquod planum mediante Å¿olo quodam puncto contingat, quemadmodum <lb/>probat TheodoÅ¿ius in .3. lib. primi & Vitellio in .71. lib. primi, & <reg norm="ducendo" type="context">ducẽdo</reg> per <reg norm="centrum" type="context">centrÅ©</reg> <lb/>Å¿phæræ lineam vÅ¿que ad punctum contactus, ipÅ¿a erit perpendicularis plano contin-<lb/>genti Å¿phÄ™ram dictam, vt probat <reg norm="idem" type="context">idẽ</reg> ThÄ™odoÅ¿ius in .4. lib. primi <reg norm="Alhazem" type="context">Alhazẽ</reg> in .25. quar-<lb/>ti, & Vitellio in .7. primi. </s> - <s xml:space="preserve">Verum etiam eÅ¿t omnem inclinationem ponderoÅ¿am huiuÅ¿ <lb/>modi corporis homogÄ™nei totam hanc lineam æqualiter omni ex parte circundare; <lb/></s> - <s xml:space="preserve">cuius quidem rei exemplum in carta deÅ¿cribere poſſumus mediante figura circulari <lb/>hîc Å¿ubÅ¿cripta <var>.a.n.e.u.</var> contigua lineæ rectæ <var>.b.d.</var> in puncto <var>.a.</var> vnde <var>.e.o.a.</var> perpendicu <lb/>laris erit ipÅ¿i <var>.b.d.</var> ex .17. lib. 3. Eucli. & <reg norm="tantum" type="context">tantÅ©</reg> ponderis habebimus à parte <var>.a.u.e.</var> quan <lb/>tum ab ipÅ¿a <var>.a.n.e</var>. </s> - <s xml:space="preserve">Nuncigitur Å¿i imaginabimur ductum eſſe centrum verÅ¿us <var>.u.</var> per <lb/>lineam <var>.o.u.</var> parallelam ipÅ¿i <var>.a.d.</var> clarum nobis <lb/>erit, <reg norm="quod" type="simple">ê™</reg> <reg norm="abÅ¿que" type="simple">abÅ¿q;</reg> vlla difficultate aut reÅ¿iÅ¿tentia <reg norm="idem" type="context">idẽ</reg> <lb/> - <anchor type="figure" xlink:label="fig-0167-01a" xlink:href="fig-0167-01"/> - ducemus, quia huiuÅ¿modi centrum ab inferiori <lb/>parte ad Å¿uperiorem, nunquam mutabit Å¿itum <lb/>reÅ¿pectu <reg norm="diÅ¿tantiæ" type="context">diÅ¿tãtiæ</reg> Å¿eu interualli, quÄ™ inter ipÅ¿um <lb/>lineamqÌue <var>.a.d.</var> intercedit, <reg norm="quod" type="simple">ê™</reg> quidem centrum <lb/>in Å¿e colligittotum pondus figurÄ™ <var>.a.n.e.u.</var> & be <lb/>neficio lineæ <var>.e.o.a.</var> illud ipÅ¿um puncto <var>.a.</var> in li-<lb/>nea <var>.b.a.d.</var> committit, productum <var>.a.</var> nil refert, <lb/>vt magis, aut minus verÅ¿us ipÅ¿um <var>.d.</var> aut verÅ¿us <lb/>b. <reg norm="dirigatur" type="simple">dirigat̃</reg>; </s> - <s xml:space="preserve">ita vt <reg norm="cum" type="context">cÅ©</reg> non oporteat vt huius figuræ <lb/><reg norm="pondus" type="context">põdus</reg>, vna vice, magis eleuetur, quà m alia, Å¿ed <lb/>Å¿emper Ä™qualiter Å¿uper lineam <var>.b.a.d.</var> quieÅ¿cat.</s> - <pb o="156" rhead="IO. BAPT. BENED." n="168" file="0168" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0168"/> - <s xml:space="preserve"><reg norm="Sitque" type="simple">Sitq;</reg> Å¿emper diuiÅ¿um à linea <var>.a.o.e.</var> per medium, Å¿equitur communi quodam con-<lb/>ceptu, nullam nobis difficultatem oborituram, dictum centrum ad quam volueri-<lb/>mus partem ducendo, quemadmodum à qualibet alia figura, quæ perfectè rotunda <lb/>non eſſet, emergeret; </s> - <s xml:space="preserve">Vt <reg norm="exempli" type="context">exẽpli</reg> gratia, Å¿i imaginabimur pentagonum <var>.K.i.h.f.l.</var> quie <lb/>Å¿cere <reg norm="Å¿uper" type="simple">Å¿uê‘</reg> <reg norm="eandem" type="context">eandẽ</reg> <reg norm="lineam" type="context">lineã</reg> <var>.a.b.K.</var> ita ut <reg norm="primum" type="context">primÅ©</reg> <reg norm="totum" type="context">totÅ©</reg> latus <var>.i.K.</var> in linea <var>.b.K.</var> <reg norm="extendatur" type="context simple">extẽdat̃</reg>, <reg norm="ducen- do" type="context">ducẽ-<lb/>do</reg> poÅ¿teà centrum <var>.o.</var> (ponamus.) verÅ¿us <var>.l.</var> dubium non eÅ¿t, quin oporteat, vt dictum <lb/>centrum <var>.o.</var> à linea <var>.b.d.</var> eleuetur, ab <reg norm="eademque" type="simple">eademq;</reg> magis diÅ¿tet, voluens Å¿e per <reg norm="arcum" type="context">arcÅ©</reg> vnum <lb/>circuli, <reg norm="qui" type="simple">ê—</reg> <reg norm="pro" type="simple">ê“</reg> Å¿uo Å¿emidiametro habeat <var>.o.K.</var> quÄ™ maior eÅ¿t ipÅ¿a <var>.o.a.</var> ex .18. li. primi Eu <lb/>cli. vnde Å¿i à puncto <var>.K.</var> imaginabimur lineam <var>.K.c.</var> reÅ¿picientem centrum regionis <lb/>elementaris, dubium non eÅ¿t, quin Å¿i velimus transferre <reg norm="centrum" type="context">cẽtrum</reg> hoc à priori Å¿itu <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> <lb/>ad dictam lineam, oporteat addere pondus parti ipÅ¿ius <var>.l.</var> quæ à linea <var>.K.c.</var> fuit Å¿ecta, <lb/>aut aliquid de ipÅ¿o pondere partis centri detrahere. </s> - <s xml:space="preserve">quod quibuÅ¿uis modis fiat, ar-<lb/>duum certè eÅ¿t ad efficiendum; </s> - <s xml:space="preserve">neque hoc etiam accidit figuræ perfectè rotundæ, <lb/>cum <reg norm="centrum" type="context">cẽtrum</reg> <reg norm="quod" type="simple">ê™</reg> perfectè in medio ipÅ¿ius ponderis eÅ¿t, reperiatur Å¿emper in linea per-<lb/>pendiculari ipÅ¿i plano, in quo animaduertendum eÅ¿t, <reg norm="quod" type="simple">ê™</reg> etiam Å¿i ipÅ¿um planum ap-<lb/>pellem; </s> - <s xml:space="preserve">pro plano tamen perfecto intelligi nolo, Å¿ed pro Å¿uperficie perfectè <reg norm="Å¿phaeri- ca" type="simple">Å¿phÄ™ri-<lb/>ca</reg> circa centrum à corporibus grauibus expetitum; </s> - <s xml:space="preserve">nam ratione magnæ amplitudi-<lb/>nis huiuÅ¿modi Å¿uperficiei, nullam differentiam notatu dignam à perfecto aliquo pla <lb/>no exigui interualli ad curuitatem eiuÅ¿dem Å¿uperficiei imaginari poterimus. </s> - <s xml:space="preserve">Sed ut <lb/>redeamus ad Å¿ermonem de reuolutione figuræ rotundæ Å¿uÅ¿ceptum, <reg norm="clarum" type="context">clarÅ©</reg> igitur erit <lb/>quamlibet minimam vim (vt ita dicam) quÄ™ trahat, aut impellat centrum <var>.o.</var> verÅ¿us <var>.u.</var> <lb/>huiuÅ¿modi figuram reuoluturam, cuius media pars ad trahendum, aut impellendum <lb/>punctum <var>.e.</var> Å¿ufficiere; </s> - <s xml:space="preserve">Imaginemur autem <reg norm="quod" type="simple">ê™</reg> li <lb/>nea <var>.n.o.u.</var> eſſet libra <reg norm="quÄ™dam" type="context">quÄ™dã</reg> in figura perfectè <lb/> - <anchor type="figure" xlink:label="fig-0168-01a" xlink:href="fig-0168-01"/> - rotunda <var>.a.n.e.u.</var> poÅ¿ita, & vis, quÄ™ trahere cen <lb/>trum deberet, diuiÅ¿a eſſet per medium, cuius <lb/>medietas appenÅ¿a eſſet extremitati <var>.u.</var> diame-<lb/>tri <var>.n.o.u.</var> <reg norm="clarum" type="context">clarÅ©</reg> erit, <reg norm="quod" type="simple">ê™</reg> abÅ¿que vlla difficultate <lb/>reuolueret figuram Å¿uper lineam <var>.b.a.d.</var> verÅ¿us <var>.<lb/>d.</var> quia huius vis, aut pondus <reg norm="nullum" type="context">nullÅ©</reg> contra pon <lb/>dus haberet vltra centrum <var>.o.</var> uerÅ¿us <var>.n.</var> <reg norm="quod" type="simple">ê™</reg> cen-<lb/>trum <var>.o.</var> perpetuo quieÅ¿cit <reg norm="Å¿uper" type="simple">Å¿uê‘</reg>. a. in linea <var>.e.o.<lb/>a.</var> per medium diuidente Å¿emper totum pon-<lb/>dus figurÄ™ Å¿uppoÅ¿itÄ™. </s> - <s xml:space="preserve">Tantò facilius ergo tota <lb/>dicta vis ap <lb/> - <anchor type="figure" xlink:label="fig-0168-02a" xlink:href="fig-0168-02"/> - <anchor type="figure" xlink:label="fig-0168-03a" xlink:href="fig-0168-03"/> - plicata cen <lb/>tro, <reg norm="ipsum" type="context">ipsÅ©</reg> ver <lb/>Å¿us <var>.u.</var> <reg norm="trahens" type="context">trahẽs</reg> <lb/>per lineam <lb/><reg norm="parallelam" type="context">parallelã</reg> ip <lb/>Å¿i <var>.a.d.</var> <reg norm="dictam" type="context">dictã</reg> <lb/>figuram re-<lb/>uolueret. </s> - <s xml:space="preserve">Et <lb/>Å¿i linea qua <lb/>dictum cen <lb/>trum trahi-<lb/>tur ab ipÅ¿o - <pb o="157" rhead="DE MECHAN." n="169" file="0169" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0169"/> - <var>b.a.d.</var> non æquediÅ¿taret, Å¿ed Å¿urÅ¿um traheret Å¿uper <var>.u.</var> aut Å¿ubter, aliquid de Å¿ua vi vir <lb/><reg norm="tuteque" type="simple">tuteqÌ;</reg> amitteret, & tantò plus, quantò inclinata magis eſſet verÅ¿us <var>.a.o.e.</var> & tandem <lb/>cum eſſet vnita cum <var>.a.o.e.</var> aut ad Å¿uperius, aut ad inferius quantalibet ui, etiam Å¿i in-<lb/>finita, figuram extra Å¿itum primæ lineæ <var>.a.o.e.</var> non moueret, Å¿ed Å¿i Å¿urÅ¿um traheret Å¿e <lb/>iungeret eam à linea <var>.b.a.d.</var> non ob id tamen efficeret, ut centrum <var>.o.</var> exiret extra pri <lb/>mam lineam <var>.a.o.e</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0167-01" xlink:href="fig-0167-01a"> - <image file="0167-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0167-01"/> - </figure> - <figure xlink:label="fig-0168-01" xlink:href="fig-0168-01a"> - <image file="0168-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0168-01"/> - </figure> - <figure xlink:label="fig-0168-02" xlink:href="fig-0168-02a"> - <image file="0168-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0168-02"/> - </figure> - <figure xlink:label="fig-0168-03" xlink:href="fig-0168-03a"> - <image file="0168-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0168-03"/> - </figure> - </div> - <p> - <s xml:space="preserve">Secunda verò Å¿pecies, tribus reuolutionum modis, abÅ¿que axis mutatione conÅ¿ta <lb/>re poteÅ¿t, ideÅ¿t modo, quo reuoluuntur trochleæ mediante fune, & quo reuoluuntur <lb/>aliquæ rotæ, in quibus aliquod animal incedit; </s> - <s xml:space="preserve">& quo reuoluuntur illæ, quæ in homi <lb/>nis manu circunuoluuntur medio alicuius manubrij inflexi. </s> - <s xml:space="preserve">Hi omnes modi cum <lb/>circulari figura magis, <reg norm="quam" type="context">quã</reg> cum alia quauis, faciliores euadunt. </s> - <s xml:space="preserve">Et primò Å¿i priorem <lb/>modum conÅ¿iderabimus, vt mediante fune quælibet figura, quæ circularis non Å¿it, <lb/>voluatur, Å¿upponamus exemplo debere reuolui pentagonum æquiangulum <var>.a.e.i.o.<lb/>u.</var> circa centrum <var>.c.</var> mediante fune <var>.q.u.a.e.i.p.</var> neceſſariò occurrent (in hac figura an-<lb/>gulorum, <reg norm="laterumque" type="simple">laterumqÌ;</reg> diÅ¿parium) plures inæqualitates, quæ reuolutionem eiuÅ¿dem fi-<lb/>guræ irregularem efficient; </s> - <s xml:space="preserve">quarum vna erit, quod duæ partes funis, ideÅ¿t <var>.u.q.</var> et <var>.i.p.</var> <lb/>non erunt in vna <reg norm="eademque" type="simple">eademqÌ;</reg> inter Å¿e diÅ¿tantia Å¿emper, quod facile intellectu erit, Å¿i ima <lb/>ginabimur ductas eſſe lineas <var>.a.i</var>: <var>u.i</var>: et <var>.i.c.t.</var> Å¿i funis duo pondera habebit alterum <lb/>altero maius, Å¿uis extremis appenÅ¿a, vnde debeat figura virtute ponderis maioris cir <lb/>cunuolui: </s> - <s xml:space="preserve">dictæ duæ partes <var>.u.q.</var> et <var>.i.p.</var> eiuÅ¿dem funis, <reg norm="mundi" type="context">mÅ©di</reg> centrum, dum firmæ ma <lb/>nebunt, reÅ¿picient; </s> - <s xml:space="preserve">Å¿ed permittentes pondera libera; </s> - <s xml:space="preserve">maius, efficiens vt circunuolua-<lb/>tur figura; </s> - <s xml:space="preserve">efficiet, vt aliquando vnum exlateribus, eiuÅ¿dem figuræ mundi <reg norm="quoque" type="simple">quoq;</reg> cen <lb/>trum reÅ¿piciet, vt in <lb/> - <anchor type="figure" xlink:label="fig-0169-01a" xlink:href="fig-0169-01"/> - figura <var>.A.</var> <reg norm="Å¿icque" type="simple">Å¿icqÌ;</reg> etiam <lb/>linea <var>.i.c.t.</var> (pro <reg norm="exem- plo" type="context">exẽ-<lb/>plo</reg>) erit menÅ¿ura di-<lb/>Å¿tantiæ funium inter <lb/>ipÅ¿as, & deinde <reg norm="circum" type="context">circÅ©</reg> <lb/>uoluendo etiam di-<lb/>Å¿tabuntinter Å¿e per li <lb/><reg norm="neam" type="context">neã</reg> <var>.i.a.</var> aut <var>.i.u.</var> vt in <lb/>figura <var>.B.</var> <reg norm="innotuit" type="context">Ä©notuit</reg> <reg norm="exem" type="context">exẽ</reg> <lb/>plo, & Å¿ic etiam ali-<lb/>quando erunt magis <lb/><reg norm="diÅ¿tantes" type="context">diÅ¿tãtes</reg>, quà m linea <lb/><var>t.i.</var> & minus quà m.i.<lb/>a: </s> - <s xml:space="preserve">nunquam tamen minus quam <var>.t.i.</var> neque magis <reg norm="quam" type="context">quã</reg> <lb/><var>i.a.</var> aut <var>.i.u.</var> quæ Å¿unt æquales; </s> - <s xml:space="preserve">Quæ quidem varietas, <lb/> - <anchor type="figure" xlink:label="fig-0169-02a" xlink:href="fig-0169-02"/> - in hanc, & in illam partem impellet partes penden-<lb/>tes funis, vnde æqualiter non trahent. </s> - <s xml:space="preserve">Idem dico, Å¿i <lb/>extrema <var>.q.</var> et <var>.p.</var> eſſent quoque Å¿emper in vna <reg norm="eademque" type="context simple">eadẽqÌ;</reg> <lb/>diÅ¿tantia; </s> - <s xml:space="preserve">neque à corpore <reg norm="ponderoÅ¿o" type="context">põderoÅ¿o</reg> eſſent attracta, <lb/>quia aliæ partes ipÅ¿ius <var>.u.q.</var> et <var>.i.p.</var> ex Å¿upradictis ratio-<lb/>nibus vnam <reg norm="eademque" type="simple">eademqÌ;</reg> diÅ¿tantiam <reg norm="non" type="context">nõ</reg> Å¿emper <reg norm="Å¿eruarent" type="context">Å¿eruarẽt</reg>. <lb/></s> - <s xml:space="preserve">vnde fieret vt cum diuerÅ¿is angulis tam <var>.i.p.</var> <reg norm="quam" type="context">quã</reg> <var>.u.q.</var> <lb/><reg norm="traherent" type="context">traherẽt</reg> Å¿emidiametros <var>.c.i</var>: <var>c.e</var>: <var>c.a</var>: <var>c.u.</var> et <var>.c.o.</var> quia <reg norm="non" type="context">nõ</reg> <lb/>Å¿emper traherent ope Å¿eu virtute anguli æqualis ipÅ¿i <var>.<lb/>c.i.p</var>. </s> - <s xml:space="preserve">Hæc autem inÄ™ qualitas communis eÅ¿t omnibus - <pb o="158" rhead="IO. BAPT. BENED." n="170" file="0170" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0170"/> - figuris rectilineis tam paris, quà m diÅ¿paris numeri. </s> - <s xml:space="preserve">Sed aliam quandam maiorem <lb/>inÄ™qualitatem habent hæ figuræ numeri diÅ¿paris, quæ eÅ¿t, quòd <reg norm="quando" type="context">quãdo</reg> linea <var>.t.i.</var> tam <lb/> <var>.u.q.</var> quà m ipÅ¿i <var>.i.p.</var> <lb/> - <anchor type="figure" xlink:label="fig-0170-01a" xlink:href="fig-0170-01"/> - <reg norm="perpendicularis" type="simple context">ê‘pẽdicularis</reg> fuerit, <lb/>ideÅ¿t <reg norm="quando" type="context">quãdo</reg> <var>.t.i.</var> cum <lb/>dictis partibus funis <lb/>angulos rectos con-<lb/>Å¿tituerit, <reg norm="tunc" type="context">tÅ©c</reg> ratione <lb/><reg norm="longitudinis" type="context">lõgitudinis</reg> ipÅ¿ius <var>.c.<lb/>i.</var> maioris quam <var>.t.<lb/>c.</var> (quia cum Å¿it <var>.c.i.</var> <reg norm="ae- qualis" type="simple">Ä™-<lb/>qualis</reg> ipÅ¿i <var>.c.a.</var> et <var>.c.a.</var> <lb/>maior ipÅ¿a <var>.c.t</var>: <var>c.i.</var> <lb/>etiam maior Å¿it ipÅ¿a <var>.<lb/>c.t.</var>) pondus aut vis <lb/>ipÅ¿ius <var>.p.</var> Å¿uperabit <reg norm="eam" type="context">eã</reg> <lb/>quæ eÅ¿t ipÅ¿ius <var>.q.</var> Å¿ed <lb/>quando <var>.t.</var> erit in oppoÅ¿ita parte, et <var>.i.</var> in ea, quæ eÅ¿t <lb/>ipÅ¿ius .t: q. <reg norm="eandem" type="context">eãdem</reg> ob cauÅ¿am Å¿uperabit <var>.p.</var> & Å¿ic mo <lb/> - <anchor type="figure" xlink:label="fig-0170-02a" xlink:href="fig-0170-02"/> - tum faciet irregularem, & <reg norm="non" type="context">nõ</reg> vniformem; </s> - <s xml:space="preserve">& obid <lb/>etiam perarduum, præter ictus, quos infligunt an-<lb/>guli in partem pendentem <reg norm="aÅ¿cendentem" type="context">aÅ¿cendẽtem</reg> funis, <reg norm="quan- do" type="context">quã-<lb/>do</reg> vnum exlateribus vnitur cum fune.</s> - </p> - <div type="float" level="4" n="2"> - <figure xlink:label="fig-0169-01" xlink:href="fig-0169-01a"> - <image file="0169-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0169-01"/> - </figure> - <figure xlink:label="fig-0169-02" xlink:href="fig-0169-02a"> - <image file="0169-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0169-02"/> - </figure> - <figure xlink:label="fig-0170-01" xlink:href="fig-0170-01a"> - <image file="0170-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0170-01"/> - </figure> - <figure xlink:label="fig-0170-02" xlink:href="fig-0170-02a"> - <image file="0170-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0170-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Aliam inÄ™qualitatem habent figuræ pares, quæ <lb/>etiam in imparibus cernitur, etÅ¿i aliquantulum di-<lb/>uerÅ¿a; </s> - <s xml:space="preserve">quæ ab eo oritur, quod funes Å¿it modò ma-<lb/>gis, modo minus propinquÄ™ centro; </s> - <s xml:space="preserve">quæ inæqualis <lb/>diÅ¿tantia, maiorem <reg norm="minoremque" type="simple">minoremqÌ;</reg> vim Å¿uper dictum <lb/>centrum ob rationes in Å¿ecunda parte cap. decimi <lb/>huius tractatus propoÅ¿itas, gignit. </s> - <s xml:space="preserve">Nulla autem <lb/>ex ijs inæqualitatibus circulari figuræ contingit. </s> - <s xml:space="preserve">Illud verò, quod de pentagonis fi-<lb/>guris dixi, omnibus aliis figuris diÅ¿paribus accommodari poteÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Secundus modus eÅ¿t earum rotarum, in quibus aliquod animal incedit, quæ Å¿i cir-<lb/>culares non eſſent, tantò difficilius voluerentur, quantò pauciores angulos haberent. <lb/></s> - <s xml:space="preserve">quod cum per Å¿e pateat, non demonÅ¿trabo. </s> - <s xml:space="preserve">Si ergo quantò plures angulos habebit <lb/>dicta figura, tantò ad circunuoluendum hoc modo agilior erit. </s> - <s xml:space="preserve">Circularis igitur fi-<lb/> - <anchor type="handwritten" xlink:label="hd-0170-01a" xlink:href="hd-0170-01"/> - gura, quæ ex infinitis angulis efficitur, omnium agillima erit.</s> - </p> - <div type="float" level="4" n="3"> - <handwritten xlink:label="hd-0170-01" xlink:href="hd-0170-01a"/> - </div> - <p> - <s xml:space="preserve">Tertius modus eÅ¿t earum rotarum, quæ manubrium habent, quæ etiam quantò <lb/>pauciores angulos habebunt, tanto <reg norm="quoque" type="simple">quoq;</reg> difficiliores reddentur, tam ratione inimi <lb/>citiæ: </s> - <s xml:space="preserve">quam exercet cum vacuo natura, quà m <reg norm="violentiÄ™" type="context">violẽtiÄ™</reg>, quam anguli aeri faciunt, eum <lb/>expellendo, vt ipÅ¿i occupent locum, quem ipÅ¿e <reg norm="aer" type="conjecture">aér</reg> implebat. </s> - <s xml:space="preserve">Quod nullo modo po <lb/>teÅ¿t euenire circulari figuræ.</s> - </p> - <p> - <s xml:space="preserve">Nunc nobis ad dicendum reÅ¿tat de Å¿pecie reuolutionis rotarum, quæ parallelæ <lb/>Å¿unt orizonri, quibus accidit poſſe volui primo <reg norm="tertioque" type="simple">tertioqÌ;</reg> modo Å¿ecundÄ™ Å¿peciei, & ob <lb/>id Å¿i circulares non erunt, eadem Å¿ubibunt incommoda, de quibus in Å¿ecunda illa Å¿pe <lb/>cie loquuti Å¿umus. </s> - <s xml:space="preserve">Å¿ed circulares rotæ huius tertiæ Å¿peciei ad reuoluendum erunt re-<lb/>liquis eò faciliores, <reg norm="quod" type="simple">ê™</reg> vno <reg norm="Å¿olum" type="context">Å¿olÅ©</reg> polo nituntur; </s> - <s xml:space="preserve">Quod alijs nequaquam conceditur.</s> - </p> - <pb o="159" rhead="DE MECHAN." n="171" file="0171" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0171"/> - <p> - <s xml:space="preserve">Super hac tertia Å¿pecie formari poteÅ¿t problema, vnde fiat, vt quieÅ¿cens huiuÅ¿-<lb/>modi rota parallela orizonti Å¿uper vnum punctum, & quantò fieri poteÅ¿t exiÅ¿tens <reg norm="ae- qualis" type="simple">Ä™-<lb/>qualis</reg>, Å¿i eam circunuoluamus maiore qua poterimus ui, & <reg norm="eandem" type="context">eãdem</reg> poÅ¿tea dimitten-<lb/>tes non perpetuò circunuoluatur.</s> - </p> - <p> - <s xml:space="preserve">Hoc quidem, quatuor fit ob cauÅ¿as. quarum prima eÅ¿t, quia huiuÅ¿modi motus, eius <lb/>rotæ non Å¿it naturalis. </s> - <s xml:space="preserve">Å¿ecunda eÅ¿t, quia etiamÅ¿i rota Å¿uper punctum mathematicum <lb/>quieÅ¿ceret, oporteret tamen vt Å¿uperius <reg norm="alterum" type="context">alterÅ©</reg> haberet polum, qui ipÅ¿am <reg norm="orizontalem" type="context">orizontalẽ</reg> <lb/>teneret, qui quidem munimento aliquo corporeo indigeret; </s> - <s xml:space="preserve">vnde fricatio quÄ™dam <lb/>conÅ¿equeretur, ex qua reÅ¿iÅ¿tentia prodiret.</s> - </p> - <p> - <s xml:space="preserve">Tertia eÅ¿t, quia aer contiguus eam perpetuò aÅ¿tringit, <reg norm="hocque" type="simple">hocqÌ;</reg> modo eius motui <lb/>reÅ¿iÅ¿tit.</s> - </p> - <p> - <s xml:space="preserve">Quarta eÅ¿t, quia quÄ™libet pars corporea, quÄ™ à ſe mouetur, impetu eidem à quali-<lb/>bet extrinÅ¿eca virtute mouente impręſſo, habet naturalem inclinationem ad rectum <lb/>iter, non autem curuum, vnde Å¿i à dicta rota particula aliqua Å¿uÄ™ circunferentiæ <reg norm="diÅ¿ium" type="context">diÅ¿iÅ©</reg> <lb/>geretur, abÅ¿que dubio per aliquod temporis Å¿patium pars Å¿eparata recto itinere fer <lb/>retur per aerem, vt exemplo à fundis, quibus iaciuntur lapides, Å¿umpto, cognoÅ¿ce <lb/>re poÅ¿sumus, in quibus, impetus motus impręſſus naturali quadam propenÅ¿ione <lb/>rectum iter peragit, cum euibratus lapis, per lineam rectam contiguam giro, quem <lb/>primo faciebat, in puncto, in quo dimiſſus fuit, rectum iter inÅ¿tituat, vt rationi con-<lb/>Å¿entaneum eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Eadem, quoque ratione fit, vt quantò maior eÅ¿t aliqua rota, tantò maiorem quo <lb/>que impetum, & impreſſionem motus eius circunferentiæ partesrecipiant, vnde <reg norm="Å¿ae pe" type="simple">ſęÂ<lb/>pe</reg> euenit, vt dum eam Å¿iÅ¿tere volumus, id <reg norm="cum" type="context">cÅ©</reg> labore & cum diſſicultate agamus ; </s> - <s xml:space="preserve">quia <lb/>quantò maior eÅ¿t diameter vnius circuli, tantò minus curua eÅ¿t eiuÅ¿dem circunferen <lb/>tia, & tantò propius accedit angulum eiuÅ¿dem circunferentiæ ad quantitatem duo-<lb/>rum angulorum rectorum rectilineorum, ideÅ¿t circunferentia ad rectitudinem linea <lb/>rem. </s> - <s xml:space="preserve">Vnde earundem partium dictæ circunferentiæ motus ad inclinationem Å¿ibi à <lb/>natura tributam, quæ eÅ¿t incedendi per lineam rectam, magis accedit.</s> - </p> - </div> - <div type="section" level="3" n="15"> - <head style="it" xml:space="preserve">Quod Aristotelis ratio none queÅ¿tionis <lb/>admittendanon Å¿it.</head> - <head xml:space="preserve">CAP. XV.</head> - <p> - <s xml:space="preserve">VEra ratio nonæ quęſtionis à ſecunda parte decimi cap. huius tractatus, & non <lb/>aliunde, accerÅ¿iri debet.</s> - </p> - </div> - <div type="section" level="3" n="16"> - <head style="it" xml:space="preserve">Quod Aristotelis rationes de decima queÅ¿tione <lb/>Å¿int reijciende.</head> - <head xml:space="preserve">CAP. XVI.</head> - <p> - <s xml:space="preserve">ARiÅ¿totelis rationes, vnde fiat, vt facilius moueantur libræ vacuæ, quà m plenè <lb/>ad propoÅ¿itam diÅ¿putationem non pertinent; </s> - <s xml:space="preserve">quia Å¿emper ineunda eÅ¿t ratio <lb/>proportionis virtutis mouentis Å¿uper mobile; </s> - <s xml:space="preserve">quod ipÅ¿e non fecit.</s> - </p> - <pb o="160" rhead="IO. BABPT. BENED." n="172" file="0172" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0172"/> - <p> - <s xml:space="preserve">Sit exempli gratia libra <var>.a.i.e.</var> quæ in vtraque extremitate vnciam vnam Å¿olum <lb/>ponderis obtineat, & Å¿it libra <var>.n.i.u.</var> æqualis priori, quæ pro Å¿ingula extremitate <reg norm="vnam" type="context">vnã</reg> <lb/> - <anchor type="handwritten" xlink:label="hd-0172-01a" xlink:href="hd-0172-01"/> - ponderis libram habeat. </s> - <s xml:space="preserve">AriÅ¿toteles admiratur, quòd addendo ipÅ¿i <var>.e.</var> mediam pon <lb/>deris vnciam, brachium <var>.i.e.</var> velocius cadat, quà m <reg norm="adijciendo" type="context">adijciẽdo</reg> <reg norm="ipsam" type="context">ipsã</reg> <reg norm="mediam" type="context">mediã</reg> <reg norm="vnciam" type="context">vnciã</reg> ipÅ¿i <var>.u.</var> <lb/>brachij <var>.i.u</var>. </s> - <s xml:space="preserve">Quod à duabus cauÅ¿is proficiÅ¿citur, quarum prior eÅ¿t, magna differentia <lb/>proportionis vnius libræ ad medietatem vnius vnciæ, ad proportionem vnius vnciæ <lb/>ad ipÅ¿am medietatem, quia Å¿i pondus adiectum extremo <var>.u.</var> dimidiæ eſſet libræ , & <lb/>cum eadem tarditate brachium moueret, optimo iure in admirationem poſſet Ari-<lb/>Å¿toteles duci. </s> - <s xml:space="preserve">Sed hoc fieri non poſſet, quia ipÅ¿um deprimeret cum eadem quaÅ¿i ve <lb/>locitate, qua media vncia brachium <var>.i.e</var>. </s> - <s xml:space="preserve">Dixi autem quaÅ¿i, quia nonnihil diÅ¿crimi-<lb/>nis intercederet, quod proficiÅ¿citur à ſecunda ratione. </s> - <s xml:space="preserve">Et hæc, reÅ¿iÅ¿tentia eÅ¿t , quæ <lb/>oritur à ſparto, quia quantò maius pondus continet libra, tantò magis præmit Å¿par <lb/>tum in loco, in quo Å¿uÅ¿tinetur; </s> - <s xml:space="preserve">vnde maior reÅ¿iÅ¿tentia in circunuolutione <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> Å¿par <lb/>ti, in loco, in quo quieÅ¿cit, exoritur, quia ipÅ¿um eÅ¿t corpus materiale. </s> - <s xml:space="preserve">Si quis autem <lb/>vellet, vt brachium <var>.i.u.</var> eadem agilitate, qua <var>.i.e.</var> deÅ¿cenderet, oporteret, vt propor-<lb/>tio dimidiæ librÄ™ adiectæ ponderi ipÅ¿ius <var>.u.</var> <lb/> - <anchor type="figure" xlink:label="fig-0172-01a" xlink:href="fig-0172-01"/> - quod eÅ¿t vnius libræ, vim Å¿uam haberet, <lb/>quæ excederet reÅ¿iÅ¿tentiam Å¿ui Å¿parti (me-<lb/>dio brachiorum maiorum ijs qui Å¿unt <var>.a.i.<lb/>e.</var>) ita proportionatam, vt proportionata <lb/>eÅ¿t vis dimidiæ vnciæ ipÅ¿i e. iunctæ, reÅ¿iÅ¿ten <lb/>tiæ Å¿ui Å¿parti. </s> - <s xml:space="preserve">HuiuÅ¿modi rationes cum ro-<lb/>tis grauioribus <reg norm="leuioribusque" type="simple">leuioribusq;</reg>, & ijs, quÄ™ à cor <lb/>poribus quibuÅ¿libet grauibus impelluntur, accommodatæ fuerint, titubantem intel <lb/>lectum confirmabunt.</s> - </p> - <div type="float" level="4" n="1"> - <handwritten xlink:label="hd-0172-01" xlink:href="hd-0172-01a"/> - <figure xlink:label="fig-0172-01" xlink:href="fig-0172-01a"> - <image file="0172-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0172-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="17"> - <head style="it" xml:space="preserve">De uer a cauÅ¿a .12. questionis mechanice.</head> - <head xml:space="preserve">CAP. XVII.</head> - <p> - <s xml:space="preserve">VEra ratio, cur multò longius corpus aliquod graue impellatur funda, quam <lb/>manu, inde oritur, quòd circunuoluendo fundam, maior impræſſio impetus <lb/>motus fit in corpore graui, quà m fieret manu, quod corpus liberatum deinde cum <lb/>fuerit à funda, natura duce, iter <reg norm="fuum" type="context">fuÅ©</reg> à puncto, à quo proÅ¿ilijt, per lineam contiguam <lb/>giro, quem poÅ¿tremò faciebat, Å¿uÅ¿cipit. </s> - <s xml:space="preserve"><reg norm="Dubitandumque" type="simple">DubitandumqÌ</reg> non eÅ¿t, quin dicta funda <lb/>maior impetus motus dicto corpori imprimi poſſit, <reg norm="cum" type="context">cÅ©</reg> ex multis circumactibus, ma-<lb/>ior Å¿emper impetus dicto corpori accedat. </s> - <s xml:space="preserve">Manus autem eiuÅ¿dem corporis motus, <lb/>dum illud ipÅ¿um circunuoluitur (pace AriÅ¿totelis dixerim) centrum non eÅ¿t, neque <lb/>funis eÅ¿t Å¿emidiameter. </s> - <s xml:space="preserve">Immo manus quam maximè fieri poteÅ¿t in orbem cietur; <lb/></s> - <s xml:space="preserve">qui quidem motus in orbem, vt circumagatur etiam ipÅ¿um corpus, cogit, quod qui-<lb/>dem corpus, naturali quadam inclinatione, exiguo quodam impetu iam incÄ™pto, <lb/>vellet recta iter peragere, vt in Å¿ubÅ¿cripta figura patet, in qua <var>.e.</var> Å¿ignificat manum <var>.a.</var> <lb/>corpus <var>.a.b.</var> lineam rectam tangentem girum <var>.a.a.a.a.</var> quando corpus liberum rema-<lb/>net. </s> - <s xml:space="preserve">Verum quidem eÅ¿t, impręſſum illum impetum, continuò paulatim decreÅ¿cere <lb/>vnde Å¿tatim inclinatio grauitatis eiuÅ¿dem corporis Å¿ubingreditur, quæ Å¿eÅ¿e miÅ¿cens <lb/>cum impręſſione facta per vim, non permittit, vt linea <var>.a.b.</var> longo tempore recta per - <pb o="161" rhead="DE MECHAN." n="173" file="0173" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0173"/> - maneat, Å¿ed citò fiat curua, cum dictum corpus <var>.a.</var> duabus virtutibus moueatur, qua-<lb/>rum vna eÅ¿t, violentia impræſſa, & alia natura, contra opinionem Tartaleæ, qui ne-<lb/>gat corpus aliquod motibus violen <lb/> - <anchor type="figure" xlink:label="fig-0173-01a" xlink:href="fig-0173-01"/> - to & naturali Å¿imul & Å¿emel moueri <lb/>poſſe. </s> - <s xml:space="preserve"><reg norm="Neque" type="simple">Neq;</reg> eÅ¿t <reg norm="Å¿ilentio" type="context">Å¿ilẽtio</reg> <reg norm="prætereundus" type="context">prætereÅ©dus</reg> <lb/>hac in re <reg norm="quidam" type="simple context">ê—dã</reg> notatu <reg norm="dignus" type="simple">dignê°</reg> effectus, <lb/>qui eiuÅ¿modi eÅ¿t, <reg norm="quod" type="simple">ê™</reg> quanto magis <lb/>creÅ¿cit impetus in corpore <var>.a.</var> cauÅ¿a <lb/>tus ab augumento velocitatis giri <lb/>ipÅ¿ius <var>.e.</var> <reg norm="tantò" type="context">tãtò</reg> magis oportet, vt Å¿en-<lb/>tiat Å¿e trahi manus à dicto corpore <lb/>a. mediante fune, quia quantò ma-<lb/>ior impetus motus ipÅ¿i <var>.a.</var> eÅ¿t impręſ <lb/>Å¿us, tantò magis dictum corpus <var>.a.</var> <lb/>ad rectum iter peragendum incli-<lb/>natur, vnde vt recta incedat, tantò <lb/>maiore quoque vi trahit.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0173-01" xlink:href="fig-0173-01a"> - <image file="0173-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0173-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="18"> - <head style="it" xml:space="preserve">De decimatertia questione.</head> - <head xml:space="preserve">CAP. XVIII.</head> - <p> - <s xml:space="preserve">DEcimatertia quæſtio ad vectem omnino eÅ¿t referenda. </s> - <s xml:space="preserve">Imaginari debemus <lb/>axem cylindrici iugi, hypomochlion eſſe. </s> - <s xml:space="preserve">Quod reÅ¿tat, illud ipÅ¿um totum de <lb/>pendet à .4. <reg norm="quintoque" type="simple">quintoqÌ;</reg> cap. huius tractatus. </s> - <s xml:space="preserve">Vna tamen differentia inter hanc machi-<lb/>nam, <reg norm="vectemque" type="simple">vectemqÌ;</reg> reperitur, quæ eÅ¿t, <reg norm="quod" type="simple">ê™</reg> iugum aliquam reÅ¿iÅ¿tentiam pro coniunctione <lb/>calcata in loco, in quo voluitur, magis quà m hypomochlion vecti efficiat.</s> - </p> - </div> - <div type="section" level="3" n="19"> - <head style="it" xml:space="preserve">De decimaquart a queÅ¿tione.</head> - <head xml:space="preserve">CAP. XIX.</head> - <p> - <s xml:space="preserve">RAtiones etiam decimæquartæ quæſtionis dependent ab ijs, quæ Å¿unt vectis, vt <lb/>exempli gratia Å¿it lignum <var>.a.b.c.d.</var> frang endum in medio, annitendo genibus <lb/>in punctum <var>.o.</var> clariſſimè tunc videbimus, <reg norm="quod" type="simple">ê™</reg> tenentes marlus longè à medio, in locis <lb/>a. et <var>.c.</var> facilius <reg norm="minorique" type="simple">minoriqÌ;</reg> cum labore illum frangemus, quà m Å¿i eaÅ¿dem vicinas me-<lb/>dio eiuÅ¿dem ligni in locis <var>.e.</var> et <var>.i.</var> poneremus. </s> - <s xml:space="preserve">Cuius rei rationes <reg norm="eædem" type="context">eædẽ</reg> Å¿unt <reg norm="cum" type="context">cÅ©</reg> ijs, quæ <lb/>primis huius tractatus capitibus propoÅ¿itæ fuerunt. </s> - <s xml:space="preserve">Imaginemur lineas rectas ductas <lb/>à puncto <var>.o.</var> ad loca <var>.a.e.i.</var> et <var>.c.</var> hinc manifeÅ¿tè perÅ¿piciemus eorum, quæ iam diximus <lb/>ratione, <reg norm="quod" type="simple">ê™</reg> loca <var>.e.</var> et <var>.i.</var> mediantibus duabus lineis <var>.e.o.</var> et <var>.i.o.</var> magis annitentur <var>.o.</var> cen <lb/>tro, quà m loco <var>.a.</var> et <var>.c.</var> <reg norm="duarum" type="context">duarÅ©</reg> <reg norm="linearum" type="context">linearÅ©</reg> <var>.a.o.</var> et <var>.c.o.</var> beneficio; </s> - <s xml:space="preserve">vnde vim <reg norm="quoque" type="simple">quoq;</reg> maiorem <lb/><reg norm="habebunt" type="context">habebÅ©t</reg> <lb/> - <anchor type="figure" xlink:label="fig-0173-02a" xlink:href="fig-0173-02"/> - in <var>.a.</var> et <var>.c.</var> <lb/>quà m in <lb/>e. et <var>.i</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0173-02" xlink:href="fig-0173-02a"> - <image file="0173-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0173-02"/> - </figure> - </div> - <pb o="162" rhead="IO. BAPT. BENED." n="174" file="0174" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0174"/> - </div> - <div type="section" level="3" n="20"> - <head style="it" xml:space="preserve">De uer a r atione .17. queÅ¿tionis.</head> - <head xml:space="preserve">CAP. XX.</head> - <p> - <s xml:space="preserve">DEcimaÅ¿eptima quæſtio ab AriÅ¿totele haud benè percepta fuit, quia is non ac-<lb/>commodat partes vectis Å¿uis locis. </s> - <s xml:space="preserve">Quamobrem imaginemur duos vectes <var>.<lb/>a.o.n.</var> et <var>.o.e.u.</var> quorum centra, quæ hypomochlia appellantur Å¿int <var>.o.</var> & pondera, <lb/>quæ Å¿unt attollenda Å¿int <var>.a.</var> et <var>.e.</var> inter Å¿e æqualia, & diÅ¿tantiÄ™ Å¿int <var>.a.o.</var> et <var>.e.o.</var> Å¿ibi <reg norm="inuicem" type="context">inuicẽ</reg> <lb/>æquales, Å¿ed <var>.o.n.</var> æqualis Å¿it ipÅ¿i <var>.o.u</var>: clarum erit, <reg norm="quod" type="simple">ê™</reg> ad eleuandum <var>.a.</var> oportebit depri <lb/>mere <var>.n.</var> & ad eleuandum <var>.e.</var> oportebit attollere <var>.u</var>. </s> - <s xml:space="preserve">Et quia omnia Å¿upponuntur æ qua <lb/> - <anchor type="handwritten" xlink:label="hd-0174-01a" xlink:href="hd-0174-01"/> - lia, clarum quoque erit, commu-<lb/>ni Å¿cientia, tantam virtutem in <lb/> - <anchor type="figure" xlink:label="fig-0174-01a" xlink:href="fig-0174-01"/> - n. quanta Å¿ufficiet ad <reg norm="attollendum" type="context">attollendÅ©</reg> <lb/>a. in <var>.u.</var> <reg norm="quoque" type="simple">quoq;</reg> Å¿uffecturam ad ele-<lb/>uandum <var>.e.</var> quia <reg norm="cum" type="context">cÅ©</reg> æqualibus an <lb/>gulis ijs, quibus duæ virtutes <var>.a.</var> <lb/>et <var>.n.</var> annituntur <var>.o.</var> centro, ita <var>.e.</var> <lb/>et <var>.u.</var> è contrario Å¿uo centro <var>.o.</var> an <lb/>nituntur. </s> - <s xml:space="preserve">& omnes rationes pro <lb/>vecte <var>.a.o.n.</var> quarto <reg norm="quintoque" type="simple">quintoqÌ;</reg> huius tracta-<lb/> - <anchor type="figure" xlink:label="fig-0174-02a" xlink:href="fig-0174-02"/> - tus capitibus citatæ, vecti <var>.o.e.u.</var> vt Å¿atis Å¿u <lb/><reg norm="perque" type="simple">perqÌ;</reg> dixi in dicto capit .5. conuenire poÅ¿-<lb/>Å¿unt.</s> - </p> - <div type="float" level="4" n="1"> - <handwritten xlink:label="hd-0174-01" xlink:href="hd-0174-01a"/> - <figure xlink:label="fig-0174-01" xlink:href="fig-0174-01a"> - <image file="0174-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0174-01"/> - </figure> - <figure xlink:label="fig-0174-02" xlink:href="fig-0174-02a"> - <image file="0174-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0174-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Nunc Å¿it aliqua pars ligni cindenda Å¿e-<lb/>cundum venulas Å¿uas <var>.d.e.f.g.</var> & Å¿it cuneus <lb/><var>a.b.c.</var> qui vi mallei <var>.P.</var> vÅ¿que ad <var>.t.x.</var> pene-<lb/>trarit. </s> - <s xml:space="preserve">Hinc clarum erit, quòd apertura <lb/><var>i.m.r.</var> ligni, poÅ¿t quam infigitur cuneus Å¿e <lb/>cundum venas, longÃor erit parte <var>.x.b.t.</var> cu <lb/>nei, quæ ingreſſa eÅ¿t. </s> - <s xml:space="preserve">Oportet nunc ima-<lb/>ginari duos vectes Å¿imiles Å¿upradictæ <var>.u.e.<lb/>o.</var> in hunc modum, vt puncta <var>i.r.</var> lìni Å¿int <lb/>loco <var>.u.</var> extremi <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> vectis, et <var>.t.x.</var> loco vir <lb/>tutis applicatæ ipÅ¿i <var>.u.</var> & reÅ¿iÅ¿tentia circa <lb/>punctum <var>.m.</var> loco ponderis <var>.e.</var> vectis <var>.o.e.u.</var> <lb/>dicti, & pars <var>.K.</var> quaÅ¿i immediata poÅ¿t <var>.m.</var> <lb/>verÅ¿us extremitatem <var>.f.e.</var> ligni, Å¿it loco hy-<lb/>pomochlij <var>.o</var>. </s> - <s xml:space="preserve">Hinc fiet vt quanto longio <lb/>res erunt lineæ <var>.i.m.K.</var> et <var>.r.m.K.</var> tantò quo <lb/>que facilius virtutes <var>.t.x.</var> impellent <var>.i.r</var>.</s> - </p> - <pb o="163" rhead="DE MECHAN." n="175" file="0175" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0175"/> - </div> - <div type="section" level="3" n="21"> - <head style="it" xml:space="preserve">De uera & intrinÅ¿eca cauÅ¿a trocble arum.</head> - <head xml:space="preserve">CAP. XXI.</head> - <p> - <s xml:space="preserve">PRo intelligenda vera, & intrinÅ¿eca ratione, vnde fiat ut multitudo rotularum in <lb/>trochleis cauÅ¿a Å¿it, ut exigua vis Å¿urÅ¿um moueat, aut attollat <reg norm="pondera" type="context">põdera</reg> magna. </s> - <s xml:space="preserve">Ima <lb/>ginemur duas hîc Å¿ubÅ¿criptas trochlæas explicatas tranÅ¿uerÅ¿aliter in hunc modum, <lb/>ideÅ¿t Å¿it <reg norm="paruum" type="context">paruÅ©</reg> <reg norm="tignum" type="context">tignÅ©</reg> <var>.a.b.</var> fixum & <reg norm="parallelum" type="context">parallelÅ©</reg> orizonti. cui Å¿int rotulæ appenÅ¿e ab infe <lb/>riori parte ad Å¿uperiorem <reg norm="huicque" type="simple">huicqÌ;</reg> è regione <reg norm="oppoÅ¿itus" type="simple">oppoÅ¿itê°</reg> Å¿it aliud <reg norm="tignum" type="context">tignÅ©</reg> <var>.c.d.</var> quod moueri <lb/>poſſit ab imo ad Å¿umum, Å¿uper quod totidem Å¿int rotulæ aut radij, <reg norm="cum" type="context">cÅ©</reg> annexa poÅ¿tea <lb/>fuerit funis puncto <var>.b.</var> fixo, eam faciendo pertranÅ¿ire per rotulas tam à parte Å¿upe-<lb/>riore, quam ab inferiore; </s> - <s xml:space="preserve">& appenÅ¿um deinde cum erit paruo illi tigno <var>.c.d.</var> mobili <lb/>pondus <var>.E.</var> ducendo poÅ¿tmodum extremum <var>.f.</var> funis tranÅ¿euntis per rotulas, idem pla <lb/>nè fiet quod à trochlÄ™is Å¿imul unitis fieri Å¿olet. </s> - <s xml:space="preserve">Cuius quidem effectus ratio Å¿ub no-<lb/>Å¿tram cognitionem cadet facilius in huiuÅ¿modi figura. </s> - <s xml:space="preserve">Imaginemur Å¿eparatim Å¿ta-<lb/>teram <var>.g.h.</var> cuius <reg norm="centrum" type="context">cẽtrum</reg> Å¿it <var>.K.</var> ita Å¿itum, ut brachium <var>.g.k.</var> Å¿it duplum ad brachium <var>.K.<lb/>h.</var> Å¿upponendo igitur in puncto <var>.g.</var> pondus, aut virtutem mouentem unius libræ, & in <lb/>h. duarum librarum, <reg norm="abÅ¿que" type="simple">abÅ¿q;</reg> dubio hæ duæ uirtutes in huiuÅ¿modi diÅ¿tantijs à centro <lb/> - <anchor type="handwritten" xlink:label="hd-0175-01a" xlink:href="hd-0175-01"/> - Ä™quales <reg norm="inuicem" type="context">inuicẽ</reg> <reg norm="erunt" type="context">erÅ©t</reg>, ob rationes prioribus capitibus iam allatas, & Å¿tatera orizontalis <lb/>manebit. </s> - <s xml:space="preserve">Vnde clarum erit, <reg norm="quod" type="simple">ê™</reg> quæuis etiam exigua virtus adiuncta ipÅ¿i <var>.g.</var> mouebit <lb/>Å¿tateram extra orizontalem Å¿itum. </s> - <s xml:space="preserve">Nunc Å¿i puncto <var>.i.</var> ex æquo medio inter <var>.g.</var> et <var>.K.</var> <lb/>applicata erit virtus ipÅ¿ius <var>.h.</var> non amplius conÅ¿iderato brachio <var>.K.h.</var> inclinante uirtu-<lb/>te ipÅ¿ius <var>.i.</var> eandem partem verÅ¿us, in quam inclinabat, quando erat in <var>.h.</var> Å¿ed uirtus ip <lb/>Å¿ius <var>.g.</var> inclinet contrario modo, <reg norm="diuerÅ¿oque" type="simple">diuerÅ¿oqÌ;</reg> ab eo, quo inclinabat prius; </s> - <s xml:space="preserve">clarum <reg norm="quoque" type="simple">quoq;</reg> <lb/>erit, communi conceptu, & ob ea, quæ cap .5. huius tractatus Å¿unt dicta <var>.g.h.</var> Å¿emper <lb/>in eodem Å¿itu abÅ¿que motu manÅ¿uram, <reg norm="hancque" type="simple">hancqÌ;</reg> Å¿tateram appellabimus mobilem, & <lb/>primam. </s> - <s xml:space="preserve">Imaginemur nunc à puncto <var>.e.</var> fixo deÅ¿cendere funem <var>.e.K.</var> quæ fulciat pun <lb/>ctum <var>.K.</var> extremum diametri <var>.g.K.</var> quam intelligo pro diametro vnius ex rotulis infe <lb/>rioribus trochleæ; </s> - <s xml:space="preserve">& Å¿it <var>.n.l.m.</var> diameter vnius ex rotulis Å¿uperioribus alterius parui <lb/>tigni defixi à parte inclinationis ipÅ¿ius <var>.g.</var> & parallela diametro <var>.g.K.</var> cuius diametri <lb/>centrum fixum Å¿it <var>.l.</var> & Å¿it coniunctum <var>.g.</var> punctum, à fune cum puncto <var>.m.</var> quæ <reg norm="tam" type="context">tã</reg> per-<lb/>pendicularis Å¿it primo diametro <var>.g.i.K.</var> quà m Å¿ecundo <var>.n.m.</var> ideÅ¿t ita vt anguli <var>.n.m.g.</var> <lb/> - <anchor type="figure" xlink:label="fig-0175-01a" xlink:href="fig-0175-01"/> - <pb o="164" rhead="IO. BAPT. BENED." n="176" file="0176" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0176"/> - et <var>.m.g.k.</var> Å¿int recti. </s> - <s xml:space="preserve">Imaginemur <reg norm="quoque" type="simple">quoq;</reg> virtutem ipÅ¿ius <var>.g.</var> applicatam eſſe extremo <var>.<lb/>n.</var> cum inclinatione tamen contraria, ideÅ¿t ad inferiorem partem, quæ quidem virtus <lb/>communi quodam conceptu eandem poſſidebit vim Å¿uÅ¿tentandi immobilem diame <lb/>trum <var>.g.i.k.</var> quam habebat, <reg norm="quando" type="wordlist">qñ</reg> erat in <var>.g.</var> cum inclinatione ad Å¿uperiorem partem, <lb/>& Å¿ic etiam diameter <var>.n.l.m.</var> non magis ab una, quà m ab alia parte declinabit, quia <lb/>cum quædam virtus in <var>.n.</var> reperiatur æqualis medietati uirtutis ipÅ¿ius <var>.i.</var> quæ uirtus ip <lb/>Å¿ius <var>.i.</var> uim habet deprimendi ipÅ¿um <var>.g.</var> ideÅ¿t <var>.m.</var> pro dimidia Å¿ui ipÅ¿ius parte, Å¿equitur <var>.<lb/>n.m.</var> debere immobilem permanere. </s> - <s xml:space="preserve">Nunc Å¿i alia diameter rotulæ mobilis erit de-<lb/>Å¿umpta, quæ Å¿it <var>.p.q.o.</var> cuius centrum Å¿it <var>.q.</var> in Å¿itu parallelo ipÅ¿i <var>.n.l.m.</var> & Å¿ic collocata, <lb/>vt coniungendo <var>.o.</var> cum <var>.n.</var> anguli <var>.m.n.o.</var> et <var>.n.o.p.</var> Å¿int recti: </s> - <s xml:space="preserve">Å¿i imaginati fuerimus <reg norm="tranÅ¿" type="context">trãſ</reg> <lb/>latum eſſe <reg norm="pondusipÅ¿ius" type="simple">pondusipÅ¿iê°</reg> <var>.n.</var> in <var>.o.</var> <reg norm="cum" type="context">cÅ©</reg> <reg norm="eadem" type="context">eadẽ</reg> inclinatione ad depræſſiorem partem, illud ip <lb/>Å¿um, ac Å¿i eſſet in <var>.n.</var> communi conceptu, Å¿ine alicuius diametri mutatione præſtabit. <lb/></s> - <s xml:space="preserve">Et Å¿i centrum <var>.q.</var> fixum eſſet, & extremo <var>.p.</var> appoÅ¿itum fuiſſet pondus ipÅ¿ius <var>.o.</var> cum in <lb/>clinatione ad Å¿uperiorem partem, idem etiam planè pręſtaret, etiam Å¿i nullum ullius <lb/>diametri Å¿itum, communi Å¿cientia, mutaret, cum extremum <var>.m.</var> deorÅ¿um Å¿it ductum <lb/>à . g. uirtute dimidiæ partis ipÅ¿ius <var>.i.</var> & ab alia huic Å¿imili <var>.m.</var> quoque deorÅ¿um Å¿it tra-<lb/>ctum ab .o: quod quidem <var>.o.</var> deorÅ¿um eÅ¿t alteratum, ob inclinationem ad Å¿uperius <lb/>à uirtute poÅ¿ita in <var>.p.</var> Å¿upponendo centrum <var>.q.</var> fixum. </s> - <s xml:space="preserve">Sed Å¿i loco centri fixi, imagina <lb/>bimur in <var>.q.</var> pondus aliquod æquale ipÅ¿i <var>.i.</var> quod duplum erit in uirtute ad eam, quæ <lb/>eÅ¿t ipÅ¿ius <var>.p.</var> & ipÅ¿ius quoque .g: Å¿equetur <reg norm="etiam" type="context">etiã</reg> eadem immobilitas horum trium dia-<lb/>metrorum. </s> - <s xml:space="preserve">Quia cum Å¿it huiuÅ¿modi pondus Å¿eu virtus in <var>.q.</var> cum inclinatione con-<lb/>traria virtuti in <var>.p.</var> quæ æquipollet dimidiæ parti ipÅ¿ius <var>.q.</var> & Å¿ic ei quæ eÅ¿t ipÅ¿ius <var>.o.</var> Å¿i-<lb/>militer quia <var>.o.</var> tractum eÅ¿t Å¿upra ab <var>.n.</var> virtute ipÅ¿ius <var>.g.</var> quod <var>.m.</var> deorÅ¿um trudit; </s> - <s xml:space="preserve">idcir <lb/>co quanta erit vis quam habebit virtus in <var>.q.</var> ferendi deorÅ¿um diametrum <var>.p.o.</var> tanta <lb/>quoque virtutes ipÅ¿orum <var>.p.</var> et <var>.o.</var> æquales, & æqualiter diÅ¿tantes à <var>.q.</var> ipÅ¿um ad Å¿upe-<lb/>riorem partem inclinabunt. </s> - <s xml:space="preserve">Quamobrem nec aÅ¿cender, nec deÅ¿cendet, nec locum <lb/>mutabit. </s> - <s xml:space="preserve">Supponamus nunc quartum diametrum rotulæ <var>.s.t.r.</var> quæ Å¿it Å¿ecunda rotu <lb/>larum fixarum, parallela ipÅ¿i <var>.p.o.</var> & in eo Å¿itu, quo coniungendo extrema <var>.r.p.</var> anguli <lb/><var>o.p.r.</var> et <var>.p.r.s.</var> Å¿int recti, & imaginemur virtutem ipÅ¿ius <var>.p.</var> reperiri in <var>.s.</var> cum inclinatio <lb/>ne tamen contraria, ideÅ¿t deorÅ¿um verÅ¿us, ex his <reg norm="idem" type="context">idẽ</reg> quoque planè Å¿equetur, ideÅ¿t <reg norm="quod" type="simple">ê™</reg> <lb/>nulla <reg norm="harum" type="context">harÅ©</reg> quatuor diametrorum mouebitur. </s> - <s xml:space="preserve">quia eundem <reg norm="effectum" type="context">effectÅ©</reg> <reg norm="cum" type="context">cÅ©</reg> inclinatione <lb/>deorÅ¿um verÅ¿us efficeret dicta virtus in <var>.s.</var> quem in <var>.p.</var> cum inclinatione Å¿urÅ¿um verÅ¿us. <lb/></s> - <s xml:space="preserve">et iam dictum eÅ¿t virtutem ipÅ¿ius <var>.g.</var> dimidium virtutis ipÅ¿ius <var>.i.</var> trahere <var>.m.</var> quæ <reg norm="mediam" type="context">mediã</reg> <lb/> - <anchor type="handwritten" xlink:label="hd-0176-01a" xlink:href="hd-0176-01"/> - te <var>.n.</var> attrahit <var>.o.</var> eodem robore, et <var>.s.</var> eadem vi trahit <var>.p.</var> medio ipÅ¿ius <var>.r</var>. </s> - <s xml:space="preserve">HucuÅ¿que <reg norm="Å¿cien- tificè" type="context">Å¿ciẽ-<lb/>tificè</reg> nouimus pondus, aut virtutem ipÅ¿ius <var>.s.</var> quæ eÅ¿t dimidium <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <var>.i.</var> Å¿uÅ¿tinere uim <lb/>ipÅ¿orum <var>.i.</var> et <var>.q.</var> nam quater tantum, quanta ipÅ¿amet virtus ipÅ¿ius <var>.s.</var> eſſe conÅ¿picitur. <lb/></s> - <s xml:space="preserve">Et Å¿i adiunctÄ™ nobis eſſent duæ aliæ diametri cum ijÅ¿dem planè conditionibus <reg norm="ijÅ¿dem" type="context">ijÅ¿dẽ</reg> <lb/>rationibus vtentes, cognoÅ¿ceremus quod eadem medietas ipÅ¿ius <var>.i.</var> Å¿exies tantum <reg norm="pon" type="context">põ</reg> <lb/>deris, quanta ipÅ¿a exiÅ¿teret, Å¿eÅ¿tineret. </s> - <s xml:space="preserve">Vnde <reg norm="manifeÅ¿tum" type="context">manifeÅ¿tÅ©</reg> euadit, <reg norm="quod" type="simple">ê™</reg> eidem medietati <lb/>ipÅ¿ius <var>.i.</var> in <var>.s.</var> nonnihil virtutis addendo, dictæ diametri, illicò <reg norm="mouerentur" type="context">mouerẽtur</reg> Å¿itu. </s> - <s xml:space="preserve">Et quia <lb/>rotulæ in quolibet puncto, aliquam diametrum habent, neceſſariò Å¿equitur <reg norm="quod" type="simple">ê™</reg> infe-<lb/>riores ad Å¿uperiores accedere debeant. </s> - <s xml:space="preserve">Attamen Å¿i forte extremum immobile ip-<lb/>Å¿ius funis non pendet à puncto <var>.e.</var> trochleæ Å¿uperioris, Å¿ed alligatum fuerit ad <reg norm="medium" type="context">mediÅ©</reg> <lb/>inferioris trochleæ ut ad punctum <var>.i.</var> ope unius trochleÄ™ Å¿uperioris immobilis vt in fi <lb/>gura <var>.A.</var> videre licet, clarè patebit <reg norm="quod" type="simple">ê™</reg> à tribus virtutibus æqualibus pondus in <var>.i.</var> <reg norm="poÅ¿itum" type="context">poÅ¿itÅ©</reg> <lb/>Å¿uÅ¿tinebitur: </s> - <s xml:space="preserve">hoc eÅ¿t à <var>.g.</var> ab <var>.i.</var> & ab <var>.k.</var> <reg norm="quarum" type="context">quarÅ©</reg> vnaquÄ™que tertia pars erit ipÅ¿ius <var>.i.</var> in con <lb/>c<unsure/>ontrariam <reg norm="partem" type="context">partẽ</reg>, hoc eÅ¿t tertia pars reÅ¿iÅ¿tentiæ. </s> - <s xml:space="preserve">propterea <reg norm="quod" type="simple">ê™</reg> ex æquo inter Å¿e <reg norm="diÅ¿tant" type="context">diÅ¿tãt</reg>. - <pb o="165" rhead="DE MECHAN." n="177" file="0177" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0177"/> - <var>g.i.</var> et .K: </s> - <s xml:space="preserve">Quà propter augebitur virtus per numeros impares, hoc modo; </s> - <s xml:space="preserve">Nam <var>.g.</var> <lb/>eſſet tertia pars reÅ¿iÅ¿tentiÄ™, quemadmodum prius media erat. </s> - <s xml:space="preserve">Idem infero de <var>.m.n.<lb/>o.p.r.</var> et <var>.s</var>. </s> - <s xml:space="preserve">Sed cum oporteat pondus <var>.q.</var> tantum eſſe vt <reg norm="Å¿uffieiant" type="context">Å¿uffieiãt</reg> reÅ¿iÅ¿tentiæ in <var>.o.</var> et <var>.p.</var> <lb/>ipÅ¿um Å¿uÅ¿tinere, idcirco ipÅ¿um pondus <var>.q.</var> Å¿ubÅ¿eÅ¿quialter erit <reg norm="ponderi" type="context">põderi</reg> in <var>.i.</var> poÅ¿iti. </s> - <s xml:space="preserve">Qua-<lb/>propter <var>.s.</var> quinta pars erit ponderum <var>.i.</var> et <var>.q</var>. </s> - <s xml:space="preserve">Deinde Å¿i adhuc. duo diametri vnus in-<lb/>ferior, alter verò Å¿uperior additi fuerint cum pondere æquali <var>.q.</var> ad medium diame-<lb/>tri inferioris, </s> - <s xml:space="preserve">tunc pondus <var>.s.</var> erit Å¿eptima pars trium ponderum <var>.i.q.</var> & tertij additi, ex <lb/> - <anchor type="figure" xlink:label="fig-0177-01a" xlink:href="fig-0177-01"/> - Å¿upradictis rationibus. </s> - <s xml:space="preserve">Et quia virtus Å¿uÅ¿ti <lb/> - <anchor type="figure" xlink:label="fig-0177-02a" xlink:href="fig-0177-02"/> - nens totale pondus trochleæ inferiori ap-<lb/>penÅ¿um in tot diuiditur partes æquales, <lb/>quot Å¿unt diametri orbiculorum trochleæ <lb/>inferioris, quando extremum immobile fu <lb/>nis alligatum fuerit trochleÄ™ Å¿uperiori, vt <lb/>puta in puncto <var>.e.</var> cum verò alligatum fue-<lb/>rit trochleæ inferiori, virtus primi diame-<lb/>tri <var>.g.i.K.</var> trochleæ inferioris Å¿emper Å¿eÅ¿qui <lb/>altera erit vnicuique aliorum <reg norm="diametrorum" type="context">diametrorÅ©</reg> <lb/>ideò virtus reÅ¿iÅ¿tentiÄ™ alterius extremi mo <lb/>bilis funis, puta <var>.s.</var> Å¿ubmultiplex erit totalis <lb/>ponderis, eo modo quo diximus, cuius vir <lb/>tus, Å¿eu grauitas diuiditur Å¿eu diÅ¿trubuitur <lb/>diametris inferioris trochleæ vt dictum eÅ¿t.</s> - </p> - <div type="float" level="4" n="1"> - <handwritten xlink:label="hd-0175-01" xlink:href="hd-0175-01a"/> - <handwritten xlink:label="hd-0176-01" xlink:href="hd-0176-01a"/> - <figure xlink:label="fig-0175-01" xlink:href="fig-0175-01a"> - <image file="0175-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0175-01"/> - </figure> - <figure xlink:label="fig-0177-01" xlink:href="fig-0177-01a"> - <image file="0177-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0177-01"/> - </figure> - <figure xlink:label="fig-0177-02" xlink:href="fig-0177-02a"> - <image file="0177-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0177-02"/> - </figure> - </div> - </div> - <div type="section" level="3" n="22"> - <head style="it" xml:space="preserve">Depropria cauÅ¿a .24. quæſtionis.</head> - <head xml:space="preserve">CAP. XXII.</head> - <p> - <s xml:space="preserve">VEra cauÅ¿a effectus, qui vigeÅ¿imaquarta quæſtione exprimitur, adhuc à nemine <lb/>(quod Å¿ciam) animaduerÅ¿a fuit, licet non Å¿it admodum ardua vel obÅ¿cura. </s> - <s xml:space="preserve">Ima <lb/>ginemur ergo duos circulos <var>.c.f.</var> et <var>.b.g.</var> concentricos, <reg norm="itaque" type="simple">itaq;</reg> Å¿imul coniunctos, vt Å¿i ip <lb/>Å¿orum vnus feratur in orbem, alius quoque circumagatur, eo modo, quo curruum ro <lb/>tæ voluuntur. </s> - <s xml:space="preserve">Et imaginemur primò Å¿uper lineam <var>.f.i.</var> reuolui maiorem, & quando <lb/>idem circulus erit in <var>.l.</var> dictam lineam <var>.f.i.</var> tangere circunferentiam eiuÅ¿dem in pun- - <pb o="166" rhead="IO. BAPT. BENED." n="178" file="0178" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0178"/> - cto <var>.c.</var> vnde linea <var>.g.m.</var> mediante <var>.K.</var> continget circunferentiam circuli minoris in pun <lb/>cto .b: et <var>.K.g.</var> ex .34. primi Eucli. æqualis erit ipÅ¿i <var>.f.l.</var> quia ex .17. tertii, anguli <var>.f.</var> et <var>.g.</var> <lb/>Å¿unt æquales, vnde ex .28. primi <var>.f.l.</var> et <var>.g.K.</var> Å¿unt parallelæ. </s> - <s xml:space="preserve">& Å¿ic erunt <var>.k.l.</var> cum <var>.f.g.</var> ex <lb/>eadem Å¿upradicta. </s> - <s xml:space="preserve">Ratio autem, qua arcus <var>.g.b.</var> tranÅ¿ierit lineam <var>.g.K.</var> maiorem ipÅ¿a, <lb/>eÅ¿t, quia dum mouetur, quodlibet punctum ipÅ¿ius <var>.g.b.</var> virtute reuolutionis ipÅ¿ius <var>.f.c.</var> <lb/>omne punctum eiuÅ¿dem arcus <var>.g.b.</var> vlterius verÅ¿us <var>.K.</var> quam Å¿i moueretur virtute re-<lb/>uolutionis ipÅ¿ius <var>.g.b.</var> Å¿uper lineam <var>.g.m.</var> defertur. </s> - <s xml:space="preserve">vt exempli gratia, quando virtute <lb/>reuolutionis maioris circuli, centrum <var>.a.</var> reperitur in Å¿itu lineæ <var>.l.K.</var> punctum <var>.g.</var> confe <lb/>cerit iter <var>.g.u.</var> & punctum <var>.b.</var> iter <var>.b.K.</var> etiam reliqua omnia puncta inter <var>.g.b.</var> magna <lb/>itinera egerint, cum à magno circulo Å¿int ante delata. </s> - <s xml:space="preserve">Imaginemur quoque hos cir <lb/>culos eſſe delatos virtute reuolutionis circuli minoris, & <reg norm="partem" type="context">partẽ</reg> <var>.g.t.</var> rectè <var>.g.m.</var> dimen-<lb/>Å¿am fuiſſe ab arcu <var>.g.b</var>. </s> - <s xml:space="preserve"><reg norm="Quando" type="context">Quãdo</reg> ergo <var>.b.</var> erit in <var>.t.</var> factum erit iter <var>.b.t.</var> ab ipÅ¿o <var>.b.</var> et <var>.g.</var> fa-<lb/>ciet iter <var>.g.n.</var> quÄ™ itinera alijs multò breuiora Å¿unt, quia breuioribus cruribus reuolu-<lb/>ta Å¿unt dicta puncta; </s> - <s xml:space="preserve">& Å¿ic dico de reliquis omnibus punctis inter <var>.g.</var> et <var>.b.</var> & in hoc ca <lb/>Å¿u punctum <var>.f.</var> erit in <var>.q.</var> & punctum <var>.c.</var> erit in <var>.e</var>. </s> - <s xml:space="preserve">Quamobrem omnia puncta <reg norm="contingen- tiæ" type="context">cõtingen-<lb/>tiæ</reg> inter <var>.f.</var> et <var>.c.</var> non Å¿olum non erunt delata anteà , Å¿ed potius à primo Å¿itu retrorÅ¿um <lb/>erunt repulÅ¿a. </s> - <s xml:space="preserve">Vnde non eÅ¿t, quòd in tantam admirationem ducamur Å¿i dum reuol <lb/>uitur circulus maior, arcus <var>.g.b.</var> circuli minoris, totam lineam <var>.g.K.</var> tranÅ¿ire videtur, <lb/>& dum reuoluitur minor, apparet arcum <var>.f.c</var>: maius iter quam ab <var>.f.</var> ad <var>.e.</var> non facere, <lb/>cum maiore Å¿eÅ¿e in orbem ferente, quodlibet punctum arcus <var>.g.b.</var> ad vnam <reg norm="eandemque" type="context simple">eandẽq;</reg> <lb/>partem duos motus obtineat. </s> - <s xml:space="preserve">vt exempli gratia punctum <var>.b.</var> non Å¿olum mouetur ver <lb/>Å¿us <var>.m.</var> quòd circa centrum <var>.a.</var> feratur, cum ipÅ¿um etiam centrum moueatur verÅ¿us <var>.m.</var> <lb/>Å¿ed quia prÄ™ter hoc deferantur quoque à circulo maiori verÅ¿us <var>.m.</var> vÅ¿que ad lineam <var>.<lb/>k.l</var>. </s> - <s xml:space="preserve">Dum verò minor circulus in girum ducitur, habet quodlibet punctum arcus <var>.f.c.</var> <lb/>duos motus contrarios, quorum alter verÅ¿us <var>.i.</var> virtute reuolutionis circuli minoris, <lb/>& alter ex eo, <reg norm="quod" type="simple">ê™</reg> dictus circulus maior circa centrum <var>.a.</var> voluatur, vnde omne <reg norm="punctum" type="context">punctÅ©</reg> <lb/>contactus circuli maioris cum recta <var>.f.i.</var> tetrorÅ¿um pellitur verÅ¿us <var>.x</var>.</s> - </p> - <figure position="here"> - <image file="0178-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0178-01"/> - </figure> - <pb o="167" rhead="DE MECHAN." n="179" file="0179" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0179"/> - </div> - <div type="section" level="3" n="23"> - <head style="it" xml:space="preserve">De uer a cauÅ¿a .30. quæstionis.</head> - <head xml:space="preserve">CAP. <sic comment="should be XXIII">XXIIII.</sic></head> - <p> - <s xml:space="preserve">VEra ratio, cur homo dum Å¿edet ( non tamen Turcarum more ) Å¿i velit <lb/>Å¿eÅ¿e in pedes erigere, calcaneos retrahit, vt efficiat angulum acutum, cum <reg norm="fae- moribus" type="simple">fÄ™-<lb/>moribus</reg> coxis à parte inferiori, & ventrem inclinat, ad conÅ¿tituendum etiam angu <lb/>lum acutum in Å¿uperiori parte, ea eÅ¿t; </s> - <s xml:space="preserve">vt totius corporis pondus, ex Ä™quo, ideÅ¿t ab <lb/>oppoÅ¿itis partibus circundet lineam rectam, quæ tranÅ¿it per locum, in quo conquie <lb/>Å¿cunt pedes verÅ¿us mundi centrum. </s> - <s xml:space="preserve">ideÅ¿t, ut edatur Ä™quilibrium ponderis ipÅ¿ius cor-<lb/>poris circum lineam illam, quÄ™ Å¿ub pedibus inÅ¿eruit pro Å¿parto. </s> - <s xml:space="preserve">Vnde aperiendo, <lb/>deinde dictos duos angulos circa dictam <reg norm="lineam" type="context">lineã</reg>, abÅ¿que vlla difficultate erigitur cor-<lb/>pus, & abÅ¿que periculo in alterutram partem cadendi.</s> - </p> - </div> - <div type="section" level="3" n="24"> - <head style="it" xml:space="preserve">Deratione .35. & ultimæ quæstionis.</head> - <head xml:space="preserve">CAP. <sic comment="should be XXIIII">XXV.</sic></head> - <p> - <s xml:space="preserve">VEra ratio, quare, quÄ™ reperiuntur in vorticibus aquarum, Å¿emper verÅ¿us <lb/>medium ipÅ¿arum vertiginum vniuntur, inde promanat, quod media <lb/>vertiginum Å¿emper depreſſiora Å¿unt. </s> - <s xml:space="preserve">vnde quòd dicta corpora ad medium acce-<lb/>dant, nihil aliud eÅ¿t, quà m ipÅ¿a corpora Å¿uo pondere grauitateqÌue deÅ¿cendere, figu <lb/>ra enim vorticibus eÅ¿t quaÅ¿i conica, & concaua cum angulo deorÅ¿um, & gyro baÅ¿is <lb/>Å¿urÅ¿um. </s> - <s xml:space="preserve">Atque hæc vera eÅ¿t huius effectus cauÅ¿a, & non ea quam AriÅ¿toteles ponit, <lb/>à <unsure/> quo aliarum omnium quæſtionum, quas ego omiÅ¿i rationes Å¿unt benè propoÅ¿itæ.</s> - </p> - <pb o="168" n="180" file="0180" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0180"/> - </div> - </div> - <div type="chapter" level="2" n="4"> - <head xml:space="preserve">DISPVTATIONES <lb/>DE QVIBVSDAM PLACITIS <lb/>ARISTOTELIS.</head> - <p style="it"> - <s xml:space="preserve">TANTA A eÅ¿t certè AriÅ¿totelis amplitudo at que authoritas, vt dif-<lb/>ficillimum ac periculoÅ¿um Å¿it quidpiam Å¿cribere contra quam <lb/>ipÅ¿e docuerit, & mihi præſertim, cui Å¿emper viÅ¿a est viri <lb/>illius Å¿apientia admirabilis. </s> - <s xml:space="preserve">Veruntamen studio veritatis im-<lb/>pulÅ¿us, cuius ipÅ¿e amore in Å¿eipÅ¿um Å¿iviueret excitaretur, in me <lb/>dium <reg norm="quædam" type="context">quædã</reg> proferre non dubitaui, in quibus me inconcußa mathematicæ philoÅ¿ophiæ <lb/>baÅ¿is, cui Å¿emper inÅ¿iÅ¿to, ab eo dißentire coegit.</s> - </p> - <div type="section" level="3" n="1"> - <head style="it" xml:space="preserve">Qualiter & ubi AriÅ¿toteles de uelocitate motuum natura-<lb/>lium localium aliter tractauerit quam nos <lb/>Å¿entiamus.</head> - <head xml:space="preserve">CAP.I.</head> - <p> - <s xml:space="preserve">VOlens AriÅ¿toreles probare vacuum non eſſe in rerum natura .8. cap. lib. 4. phyÂ<lb/>Å¿icorum ait, idem corpus per varia <reg norm="diuerÅ¿aque" type="simple">diuerÅ¿aqÌ;</reg> media, vt per <reg norm="aerem" type="context">aerẽ</reg>, & per <reg norm="aquam" type="context">aquã</reg> <lb/>Å¿i moueretur, proportionem velocitatis eiuÅ¿dem corporis per aerem, ei, quæ per <lb/>aquam fit, vnam <reg norm="eandemque" type="simple">eandemqÌ;</reg> futuram cum ea, quæ eÅ¿t Å¿ubtilitatis aereÄ™ ad Å¿ubtilitatem <lb/>aquæ. </s> - <s xml:space="preserve">In poÅ¿trema autem parte eiuÅ¿dem capitis Å¿ic Å¿cribit: </s> - <s xml:space="preserve">Nam cum ea quÄ™ ma-<lb/>iorem vel ponderis velleuitatis pręſtantiam habent, Å¿i Å¿imili Å¿igura Å¿int, <reg norm="Å¿pacium" type="context">Å¿paciÅ©</reg> par, <lb/>& æquale, maiore celeritate conficere cernamus, ea quam magnitudines inter Å¿e ha <lb/>bent, proportione: </s> - <s xml:space="preserve">profectò idem etiam perinane fieret. </s> - <s xml:space="preserve">Aliam quoque rationem <lb/>proponit phyloÅ¿ophus .2. cap. Å¿exti phyÅ¿icorum Å¿cribens eademmet proportione, <lb/>qua tempus diuiditur, magnitudinem etiam diuidi. </s> - <s xml:space="preserve">Sexto autem cap. primi de cÅ“-<lb/>lo Å¿cribit, tempora eandem proportionem habere, quam habentè conuerÅ¿o ponde-<lb/>ra; </s> - <s xml:space="preserve">vt Å¿i media pars vnius ponderis, vnius horæ Å¿patio moueretur, vniuerÅ¿um pondus <lb/>in media hora moueretur. </s> - <s xml:space="preserve">Secundo cap. lib. 3. de cÅ“lo duobus in locis apertè comÂ<lb/>monÅ¿trat velocitatem corporis minoris, maiori corpori comparatam, in eadem exi-<lb/>Å¿tere proportione, in qua dicta corpora adinuicem relata exiÅ¿tunt. </s> - <s xml:space="preserve">Quinto cap. eiuÅ¿Â<lb/>dem lib. idem affirmat, exemplo ab igne deÅ¿umpto. </s> - <s xml:space="preserve">Ex alijs etiam plurimis locis <lb/>cognoÅ¿ci poteÅ¿t, Å¿enÅ¿iſſe AriÅ¿totelem duo corpora eadem Å¿pecie, & figura prædita <lb/>eandem planè proportionem in Å¿uorum motuum velocitatibus, quam in Å¿uis ma-<lb/>gnitudinibus habent, retinere. </s> - <s xml:space="preserve">Alij quoque permulti eandem opinionem retinue <lb/>runt, & <reg norm="omnium" type="context">omniÅ©</reg> poÅ¿tremus Nicolaus Tartalea, Å¿ecunda propoÅ¿itione vigeÅ¿iminoni <lb/>quæſiti octaui libri, vbi profitetur Å¿e demonÅ¿tratiuè probare hanc propoÅ¿itio-<lb/>nem veram exiÅ¿tere; </s> - <s xml:space="preserve"><reg norm="neque" type="simple">neq;</reg> videt quà m magna reÅ¿iÅ¿tentiarum Å¿it differentia, quætam <lb/>ex diuerÅ¿itate figurarum, quà m ex magnitudinum varietate exoriri poteÅ¿t; </s> - <s xml:space="preserve">quas qui <lb/>dem diuerÅ¿itates ne conÅ¿iderat quidem.</s> - </p> - <pb o="169" rhead="DISPVTATIONES." n="181" file="0181" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0181"/> - </div> - <div type="section" level="3" n="2"> - <head style="it" xml:space="preserve">Quædam Å¿upponenda ut conÅ¿tet cur circa uelocit atem motuum <lb/>natur alium localium ab AriÅ¿totelis placitis <lb/>recedamus.</head> - <head xml:space="preserve">CAP. II.</head> - <p> - <s xml:space="preserve">CVM Å¿uÅ¿ceperimus prouinciam probandi quod AriÅ¿toteles circa motus <lb/>locales naturales deceptus fuerit, Å¿unt quædam primo veriſſima & obie-<lb/>cta intellectus perÅ¿e cognita pręſupponenda, ac primum quælibet duo corpora, <lb/>grauia, aut leuia, area æquali, <reg norm="Å¿imilique" type="simple">Å¿imiliqÌ</reg> figura, Å¿ed ex materia diuerÅ¿a conÅ¿tantia, <reg norm="eodem qÌue" type="context">eodẽÂ<lb/>qÌue</reg> modo Å¿itum habentia, eandem proportionem velocitatis inter Å¿uos motus loca <lb/>les naturales, ut inter Å¿uamet pondera aut leuitates in vno <reg norm="eodemque" type="simple">eodemqÌ;</reg> medio, Å¿eruatu-<lb/>ra. </s> - <s xml:space="preserve">Quod quidem natura Å¿ua notiſſimum eÅ¿t Å¿i conÅ¿iderabimus non aliunde maio-<lb/>rem tarditatem, aut velocitatem gigni, quà m à .4. cauÅ¿is (dummodo medium vnifor <lb/>mè Å¿it & quietum) ideÅ¿t à maiori aut minori pondere aut leuitate; </s> - <s xml:space="preserve">à diuerÅ¿a figura; <lb/></s> - <s xml:space="preserve">à ſitu ciuÅ¿dem figuræ diuerÅ¿o, reÅ¿pectu lineÄ™ directionis, quæ recta inter mundi cen-<lb/>trum, & circunferentiam extenditur; </s> - <s xml:space="preserve">& ab inæquali magnitudine. </s> - <s xml:space="preserve">Vnde patebit, <lb/>quòd figuram non variando, nec in qualitate nec in quantitate, neque eiuÅ¿dem figu-<lb/>ræ Å¿itum, motum fore proportionatum virtuti mouenti, quæ erit pondus aut leuitas. <lb/></s> - <s xml:space="preserve">Quod autem de qualitate, de quantitate & Å¿itu eiuÅ¿dem figuræ dico, reÅ¿pectu reÅ¿i-<lb/>ftentiæ ipÅ¿ius medii dico: </s> - <s xml:space="preserve">Quia diffimilitudo aut inÄ™qualitas figurarum, aut Å¿itus di-<lb/>uerÅ¿us non <reg norm="parum" type="context">parÅ©</reg> alterat dictorum corporum motus, cum figura parua facilius diui-<lb/>dat continuitatem medij, quam magna; </s> - <s xml:space="preserve">vt etiam cÄ™lerius idem facit acuta, quà m ob <lb/>tufa; </s> - <s xml:space="preserve">& illa quæ cum angulo, qui antecedat mouebitur velocius quà m illa, quæ Å¿ecus. <lb/></s> - <s xml:space="preserve">QuotieÅ¿cunque igitur duo corpora vnam <reg norm="eandemque" type="simple">eandemqÌ;</reg> reÅ¿iÅ¿tentiam ipÅ¿orum Å¿uperfi-<lb/>ciebus, aut habebunt aut recipient, eorum motus inter Å¿eipÅ¿os eodem planè modo <lb/>proportionati conÅ¿urgent, quo erunt ipÅ¿orum virtutes mouentes: </s> - <s xml:space="preserve">& è conuerÅ¿o, quo <lb/>tieÅ¿cunque duo corpora vnam <reg norm="eandemque" type="simple">eandemqÌ;</reg> grauitatem, aut leuitatem, & diuerÅ¿as reÅ¿i <lb/>Å¿tentias habebunt, eorum motus inter Å¿eipÅ¿os eandem <reg norm="proportionem" type="context">proportionẽ</reg> Å¿ortientur, <reg norm="quam" type="context">quã</reg> <lb/>habebunteorum reÅ¿iÅ¿tentiæ conuerÅ¿o modo; </s> - <s xml:space="preserve">quæ quidem reÅ¿iÅ¿tentiæ inter Å¿eipÅ¿as, <lb/>eandem proportionem quà m ipÅ¿arum Å¿uperficies habebunt, aut in qualitate Å¿ola fi <lb/>guræ, aut in quantitate Å¿ola, aut in Å¿itu, aut in aliquibus ex dictis rebus, eo tamen mo <lb/>do, quiÅ¿uperius poÅ¿itus fuit, vt Å¿cilicet corpus illud quod alteri comparatum, æqua-<lb/>lis erit ponderis, aut leuitatis, Å¿ed minoris reÅ¿iÅ¿tentiæ, exiÅ¿tet velocius altero, in <reg norm="eadem" type="context">eadẽ</reg> <lb/>proportione, cuius Å¿uperficies reÅ¿iÅ¿tentiam Å¿uÅ¿cipit minorem ea quæ alterius eÅ¿t cor-<lb/>poris, ratione facilioris diuiÅ¿ionis continuitatis aeris, aut aquæ; </s> - <s xml:space="preserve">Vt exempli gratia, <lb/>Å¿i proportio Å¿uperficiei corporis maioris Å¿uperficiei minoris Å¿eÅ¿quitertia eſſet, pro-<lb/>portio velocitatis dicti corporis maioris, velocitati corporis minoris, eſſet Å¿ubÅ¿eÅ¿qui <lb/>tertia; </s> - <s xml:space="preserve">vnde velocitas minoris corporis, maior eſſet velocitate corporis maioris, <reg norm="quem" type="context">quẽ</reg> <lb/>admodum quaternarius numerus ternario maior exiÅ¿tit.</s> - </p> - <p> - <s xml:space="preserve">Aliud quoque Å¿upponendum eÅ¿t, velocitatem Å¿cilicet motus naturalis alicuius <lb/>corporis grauis, in diuerÅ¿is medijs, propor-<lb/>tionatam eſſe ponderi eiuÅ¿dem corporis in <lb/>ijÅ¿dem medijs; </s> - <s xml:space="preserve">Vt exempli gratia, Å¿i pondus <lb/> - <anchor type="figure" xlink:label="fig-0181-01a" xlink:href="fig-0181-01"/> - totale alicuius corporis grauis Å¿ignificatum <lb/>crit ab <var>.a.i.</var> quo corpore poÅ¿ito in aliquo me- - <pb o="170" rhead="IO. BABPT. BENED." n="182" file="0182" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0182"/> - dio minus denÅ¿o, quà m ipÅ¿um Å¿it, (quia in medio Å¿e denÅ¿iore Å¿i poner etur, non graue <lb/>e<gap extent="2"/>et, Å¿ed leue, quemadmodum Archimedes oÅ¿tendit) illud medium Å¿ubtrahat par-<lb/>tem <var>.e.i.</var> vnde pars <var>.a.e.</var> eiuÅ¿dem ponderis libera manear; </s> - <s xml:space="preserve">& poÅ¿ito deinde eodem cor <lb/>pore in aliquo alio medio denÅ¿iore, minus tamen denÅ¿o quam ipÅ¿um Å¿it corpus, <lb/>hoc medium Å¿ubtrahat partem <var>.u.i.</var> dicti ponderis, vnde pars <var>.a.u.</var> ei uÅ¿dem <lb/>ponderis remanebit. </s> - <s xml:space="preserve">Dico proportionem velocitatis eiuÅ¿dem corporis per <reg norm="medium" type="context">mediÅ©</reg> <lb/>minus denÅ¿um, ad velocitatem eiuÅ¿dem per medium magis denÅ¿um futuram vt <var>.a.e.</var> <lb/>ad <var>.a.u.</var> vt eÅ¿t etiam rationi conÅ¿onum, magis quà m Å¿i dicamus huiuÅ¿modi velocitates <lb/>eſſe, vt <var>.u.i.</var> ad <var>.e.i.</var> cum velocitates à virtutibus mouentibus Å¿olum (cum figura vna, <lb/><reg norm="eademque" type="simple">eademqÌ;</reg> in qualitate, quantitate <reg norm="Å¿ituque" type="simple">Å¿ituqÌ</reg> erit) proportionentur. </s> - <s xml:space="preserve">Quæne<unsure/>nc diximus, <lb/>planè Å¿imilia Å¿untijs, quæ Å¿upra Å¿cripÅ¿imus, quia idem eÅ¿t dicere, proportionem velo <lb/>citatum, duorum corporum hetereogeneorum, Å¿ed Å¿imilium figura, & magnitudine <lb/>æ qualium, in vno Å¿olo medio, æqualem eſſe <lb/>proportioni ponderum ipÅ¿orum, vt Å¿i dicam?</s> - <s xml:space="preserve"><unsure/> <lb/>proportionem velocitatum vnius Å¿olum cor-<lb/> - <anchor type="figure" xlink:label="fig-0182-01a" xlink:href="fig-0182-01"/> - poris per diuerÅ¿a media eandem eſſe cum ea. <lb/>quæ eÅ¿t <reg norm="ponderum" type="context">ponderÅ©</reg> dicti corporis in iÅ¿idem medijs.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0181-01" xlink:href="fig-0181-01a"> - <image file="0181-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0181-01"/> - </figure> - <figure xlink:label="fig-0182-01" xlink:href="fig-0182-01a"> - <image file="0182-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0182-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="3"> - <head style="it" xml:space="preserve">Poſſe uelocitatem alicuius corporis proportionem contrariam <lb/>in diuerÅ¿is medijs habere cum denÅ¿itate eorum.</head> - <head xml:space="preserve">CAP. III.</head> - <p> - <s xml:space="preserve">POſſibile eÅ¿t in rerum natura corpus aliquod huiuÅ¿modi denÅ¿itate præditum re-<lb/>periri, vt velocitas eius motus naturalis per aerem, velocitati per aquamita pro <lb/>portionata exiÅ¿tat, vt eÅ¿t <reg norm="denÅ¿itas" type="context">dẽſitas</reg> aquæ denÅ¿itati aeris. </s> - <s xml:space="preserve">DenÅ¿itas aquæ notetur (exem-<lb/>pli gratia) per <var>.u.i.</var> & ea, quæaeris eÅ¿t per <var>.e.i.</var> & pondus alicuius corporis in aere per <lb/><var>e.a.</var> & pondus eiuÅ¿dem corporis in aqua per <var>.u.a.</var> ita tamen, quod eadem proportio <lb/>Å¿it <var>.e.a.</var> ad <var>.u.a.</var> vt <var>.u.i.</var> ad <var>.e.i.</var> vnde per vltimam Å¿uppoÅ¿itionem <reg norm="præcedentis" type="context">præcedẽtis</reg> capitis, pro <lb/>portio velocitatis prædicti corporis per aerem, <lb/>proportioni eiuÅ¿dem corporis per aquam erit, vt <lb/> - <anchor type="figure" xlink:label="fig-0182-02a" xlink:href="fig-0182-02"/> - <var>e.a.</var> ad <var>.u.a.</var> ergo per .11. quinti, vt <var>.u.i.</var> ad <var>.e.i</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0182-02" xlink:href="fig-0182-02a"> - <image file="0182-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0182-02"/> - </figure> - </div> - </div> - <div type="section" level="3" n="4"> - <head style="it" xml:space="preserve">OÅ¿citanter ab AriÅ¿totele nonnibil prolatum cap 8. <lb/>lib. 4 PhyÅ¿icorum.</head> - <head xml:space="preserve">CAP. IIII.</head> - <p> - <s xml:space="preserve">EX Å¿upradictis patet in vniuerÅ¿um non eſſe verum quod AriÅ¿to .8. cap .4. lib. phyÂ<lb/>Å¿icorum Å¿cribit, velocitates Å¿cilicet motuum alicuius corporis per diuerÅ¿a me-<lb/>dia, proportionatas eſſe denÅ¿itatibus eorundem mediorum. </s> - <s xml:space="preserve">Quocirca, Å¿it propor-<lb/>tio <var>.u.i.</var> ad <var>.e.i.</var> vt <reg norm="denÅ¿itatis" type="context">dẽſitatis</reg> aquÄ™ ad <reg norm="aeream" type="context">aereã</reg> <reg norm="denÅ¿itatem" type="context">dẽſitatem</reg> .et <var>.e.a.</var> ad <var>.u.a.</var> vt ponderis alicuius <lb/>corporis in aere ad pondus eiuÅ¿dem in aqua, ita tamen vt maior aut minor propor-<lb/>tio Å¿it <var>.e.a.</var> ad <var>.u.a.</var> quam <var>.u.i.</var> ad <var>.e.i.</var> vnde exiÅ¿tente proportione velocitatis per <reg norm="aerem" type="context">aerẽ</reg> - <pb o="171" rhead="DISPVTATIONES." n="183" file="0183" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0183"/> - ad velocitatem per aquam vt <var>.e.a.</var> ad <var>.a.u.</var> non erit ergo vt <var>.u.i.</var> ad <var>.e.i</var>. </s> - <s xml:space="preserve">Ob hanc igitur <lb/>cauÅ¿am nimis diſſentaneum eÅ¿t rationi, opi-<lb/>nari proportionem velocitatis omnium cor <lb/>porum grauium per aerem vnam <reg norm="eandemque" type="simple">eandemqÌ;</reg> <lb/> - <anchor type="figure" xlink:label="fig-0183-01a" xlink:href="fig-0183-01"/> - eſſe cum velocitate eorundem per aquam, <lb/>quemadmodum AriÅ¿toteles Å¿enÅ¿it.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0183-01" xlink:href="fig-0183-01a"> - <image file="0183-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0183-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="5"> - <head style="it" xml:space="preserve">Exempla dictorum.</head> - <head xml:space="preserve">CAP.V.</head> - <p> - <s xml:space="preserve">POnamus, exempli gratia, aquam eſſe in denÅ¿itate dupla ad aerem, & aliquod <lb/>graue corpus in aqua duplum ad denÅ¿itatem ipÅ¿ius aquæ, vnde dictum corpus in <lb/>denÅ¿itate ad aerem quadruplum erit; </s> - <s xml:space="preserve">quam ob cauÅ¿am, mediam Å¿ui ponderis tota-<lb/>lis partem in aqua, & in aere quartam partem, ex .7. lib. de inÅ¿identibus aquæ ab Ar-<lb/>chimede conÅ¿cripto, amitteret. </s> - <s xml:space="preserve">Moueretur igitur in aqua virtute illius mediæ partis <lb/><reg norm="ponderis" type="context">põderis</reg> Å¿ui, in aere <reg norm="autem" type="wordlist">aÅ©t</reg> uirtute <reg norm="trium" type="context">triÅ©</reg> <reg norm="quartarum" type="context">quartarÅ©</reg>; </s> - <s xml:space="preserve">vnde proportio facultatis <reg norm="mouentis" type="context">mouẽtis</reg> dicti <lb/>corporis in aere ad facultatem mouentem eiuÅ¿de m in aqua Å¿eÅ¿quialtera erit. </s> - <s xml:space="preserve"><reg norm="hocque" type="simple">hocqÌ;</reg> <lb/>corpus appelletur <var>.A</var>. </s> - <s xml:space="preserve">Sit aliud quoque corpus, quod <var>.B.</var> nominetur, Å¿imile figura, & <lb/>magnitudine corporea corpori <var>.A.</var> Å¿ed <reg norm="denÅ¿itate" type="context">dẽſitate</reg>, in proportione Å¿eÅ¿quialtera ad <reg norm="aquam" type="context">aquã</reg>, <lb/>& denÅ¿ius erit aere in proportione tripla. </s> - <s xml:space="preserve">quamobrem corpus <var>.A.</var> grauius erit cor-<lb/>pore <var>.B.</var> in aere in proportione Å¿eÅ¿quialtera, vnde etiam velocius erit ipÅ¿o <var>.B.</var> in aere <lb/>in eadem proportione, Å¿ed corpus <var>.B.</var> in aere, duplo maius pondus habebit, <reg norm="quam" type="context">quã</reg> in <lb/>aqua, cum in aere remaneant ei duæ ponderis tertiæ partes, & in aqua vna tantum, <lb/>ita vt AriÅ¿toteli concedam corpus <var>.B.</var> in aere, quam in aqua velocius futurum in ea-<lb/>dem proportione, in qua, aqua eÅ¿t <reg norm="denÅ¿ior" type="context">dẽſior</reg> aere, ex Euclidis vndecima propoÅ¿itione <lb/>lib. quinti. </s> - <s xml:space="preserve">Sed præter hæc omnia, Å¿i corpus <var>.A.</var> eſſet etiam velocius in aere, <reg norm="quam" type="context">quã</reg> in <lb/>aqua, in eadem proportione, Å¿equeretur ex .16. dicti lib. quinti proportionem velo-<lb/>citatis <var>.A.</var> in aqua ad <reg norm="velocitatem" type="context">velocitatẽ</reg> ipÅ¿ius <var>.B.</var> in aqua etiam Å¿eÅ¿quialteram eſſe. </s> - <s xml:space="preserve">Sed cum <lb/>corpus <var>.A.</var> in denÅ¿itate ad aquam <reg norm="duplum" type="context">duplÅ©</reg> Å¿it, & corpus <var>.B.</var> <reg norm="Å¿eÅ¿quialterum" type="context">Å¿eÅ¿quialterÅ©</reg> ad ipÅ¿am <reg norm="aquam" type="context">aquã</reg>, <lb/>Å¿equetur <reg norm="proportionem" type="context">proportionẽ</reg> ponderis ipÅ¿ius <var>.A.</var> ad <reg norm="pondus" type="context">põdus</reg> ipÅ¿ius <var>.B.</var> in aqua eſſe in propor-<lb/>tione dupla; </s> - <s xml:space="preserve">Vnde ex primo Å¿uppoÅ¿ito capitis Å¿ecundi proportio velocitatis <var>.A.</var> ad <lb/>velocitatem <var>.B.</var> in aqua dupla erit, non Å¿eÅ¿quialtera. </s> - <s xml:space="preserve">Si ergo proportio velocitatis <var>.<lb/>A.</var> ad eam quæ eÅ¿t <var>.B.</var> in aqua dupla eÅ¿t, & ea, quæeÅ¿t <var>.B.</var> in aere, ad eam, quæ eÅ¿t ipÅ¿ius <lb/>per aquam eÅ¿t etiam dupla (vnde ea quÄ™ eÅ¿t <var>.A.</var> per aquam Ä™qualis erit ei, quæ eÅ¿t <var>.B.</var> <lb/>peraerem, ex .9. lib. quinti) & cum ea, quæ eÅ¿t <var>.A.</var> Å¿it ei, quæ eÅ¿t <var>.B.</var> per aerem Å¿eÅ¿qui-<lb/>altera, erit ergo ea, quæ eÅ¿t <var>.A.</var> per aerem, ei, quæ eÅ¿t Å¿uimet ipÅ¿ius per aquam Å¿eÅ¿qui <lb/>altera, non autem dupla, ex .7. eiuÅ¿dem libr. quinti. </s> - <s xml:space="preserve">HiÅ¿ce rationibus accedimus ad <lb/>confirmandam veritatem vltimi Å¿uppoÅ¿iti cap .2. proportionem videlicet velocita <lb/>tis <reg norm="motus" type="simple">motê°</reg> naturalis in diuerÅ¿is medijs <reg norm="alicuius" type="simple">alicuiê°</reg> corporis <reg norm="ponderoÅ¿i" type="context">põderoÅ¿i</reg> in ipÅ¿is medijs eſſe ean <lb/>dem cum ea, quæeÅ¿t inter pondera <lb/> - <anchor type="figure" xlink:label="fig-0183-02a" xlink:href="fig-0183-02"/> - dicti corporis in dictis medijs. de ijs <lb/>tamen medijs intelligendo, quæ un-<lb/>ctuoÅ¿a, aut pinguia non Å¿unt, ut Å¿unt <lb/>oleum, lac, aut huiuÅ¿modialia, quæà <lb/>qualibet minima qualitate frigoris <lb/>aut caloris alterantur, & impermea-<lb/>biles fiunt.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0183-02" xlink:href="fig-0183-02a"> - <image file="0183-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0183-02"/> - </figure> - </div> - <pb o="172" rhead="IO. BAPT. BENED." n="184" file="0184" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0184"/> - </div> - <div type="section" level="3" n="6"> - <head style="it" xml:space="preserve">Quod proportiones ponderum eiuÅ¿dem corporis in diuerÅ¿is medijs pro <lb/>portiones eorum mediorum denÅ¿it atum non Å¿eruant. Unde ne-<lb/>ceßariò inæquales proportiones uelocitatum <lb/>producuntur.</head> - <head xml:space="preserve">CAP. VI.</head> - <p> - <s xml:space="preserve">OMne corpus graue variat proportionem ponderis per diuerÅ¿a media, vnde <lb/>proportiones velocitatum inæquales exiÅ¿tunt. </s> - <s xml:space="preserve">Vt exempli gratia, Å¿i fue-<lb/>rit corpus <var>.A.</var> cuius pondus totale Å¿it <var>.o.a.</var> quod in aqua diminutum Å¿it ratione partis <var>.<lb/>e.o.</var> ita vt ei Å¿olum relinquatur pondus <var>.a.e.</var> & in aeie adempta Å¿it ei pars <var>.i.o.</var> vnde Å¿o <lb/>lum remaneat pondus <var>.a.i</var>. </s> - <s xml:space="preserve">Supponamus aliud <reg norm="quoque" type="simple">quoq;</reg> medium in eadem proportio-<lb/>ne minus denÅ¿um, quà m aer, quemadmodum aer minus denÅ¿us eÅ¿t, aqua, in quo, cor <lb/>pus <var>.A.</var> ammittat partem <var>.t.o.</var> ponderis Å¿ui, vnde ex .7. lib. de inÅ¿identibus aquæ Ar-<lb/>chimedis, eadem proportio erit <var>.e.o.</var> ad <var>.i.o.</var> quæ eÅ¿t <var>.i.o.</var> ad <var>.t.o</var>. </s> - <s xml:space="preserve">Supponamus <reg norm="quoque" type="simple">quoq;</reg> <lb/>eandem proportionem eſſe <var>.a.i.</var> ad <var>.a.e.</var> eÅ¿t <var>.e.o.</var> ad <var>.i.o.</var> </s> - <s xml:space="preserve">tunc dico non futuram ean-<lb/>dem proportionem <var>.t.a.</var> ad <var>.a.i.</var> quæ eÅ¿t <var>.i.o.</var> ad <var>.t.o</var>. </s> - <s xml:space="preserve">Cum Å¿it ergo proportio <var>.a.i.</var> <lb/>ad <var>.a.e.</var> ut <var>.e.o.</var> ad <var>.i.o.</var> erit diÅ¿iunctim <var>.e.i.</var> ad <var>.e.a.</var> vt <var>.e.i.</var> ad <var>.i.o</var>. </s> - <s xml:space="preserve">Quare ex .9. libr. quinÂ<lb/>ti erit <var>.a.e.</var> æqualis <var>.i.o.</var> Å¿ed cum ita Å¿ehabeat <var>.e.o.</var> ad <var>.i.o.</var> vt <var>.i.o.</var> ad <var>.t.o.</var> ita quoque <lb/>Å¿e habebit, ex vndecima quinti <var>.a.i.</var> ad <var>.e.a.</var> ut <var>.i.o.</var> ad <var>.t.o</var>. </s> - <s xml:space="preserve">Cum autem (vt vidimus). <var>a.e.</var> <lb/>Ä™qualis Å¿it ipÅ¿i <var>.i.o.</var> non poterit eſſe proportio <var>.t.a.</var> ad <var>.i.a.</var> vt eÅ¿t <var>.o.i.</var> ad <var>.t.o.</var> quia Å¿i <lb/>hoc eſſet, eſſet etiam diÅ¿iunctim proportio <var>.i.t.</var> ad <var>.i.a.</var> vt eÅ¿t <var>.i.t.</var> ad <var>.t.o.</var> & ex Å¿upradicta <lb/>9. lib. quinti <var>.a.i.</var> æqualis eſſet <var>.t.o</var>. </s> - <s xml:space="preserve">Maximum autem inconueniens eſſet <var>.t.o.</var> minorem <lb/><var>o.i.</var> ideÅ¿t minorem <var>.a.e.</var> æqualem eſſe <var>.a.i.</var> quæ maior eÅ¿t <var>.a.e</var>. </s> - <s xml:space="preserve">OÅ¿tenÅ¿iuè tamen idem <lb/>hoc modo probari poteÅ¿t, vt exiÅ¿tente <var>.i.o.</var> Ä™quali ipÅ¿i <var>.a.e.</var> per conÅ¿equens <reg norm="quoque" type="simple">quoq;</reg> erit <lb/>minor ipÅ¿a <var>.a.i.</var> cum <var>.a.e.</var> pars Å¿it ipÅ¿ius <var>a.i</var>. </s> - <s xml:space="preserve"><reg norm="Pereandem" type="context">Pereãdem</reg> tamen rationem <var>.o.t.</var> minoreÅ¿t <var>.<lb/>o.i</var>. </s> - <s xml:space="preserve">Tanto magis igitur minor erit <var>.t.o.</var> ipÅ¿a <var>.i.a</var>. </s> - <s xml:space="preserve">Vnde ex .8. libri quinti maiorem pro <lb/>portionem habebit <var>.i.t.</var> <lb/>ad <var>.t.o.</var> quam ad <var>.i.a.</var> & <lb/>ex .28. <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> lib <var>.i.o.</var> ad <lb/><var>t.o.</var> <reg norm="maiorem" type="context">maiorẽ</reg> proportio-<lb/> - <anchor type="figure" xlink:label="fig-0184-01a" xlink:href="fig-0184-01"/> - <reg norm="nem" type="context">nẽ</reg> habebit, quà m.t.a. <lb/>ad <var>.i.a.</var> ex .12. igitur di-<lb/>cti quinti maiorem pro <lb/>portionem habebit <var>.i.a.</var> ad <var>.e.a.</var> quà m.t.a. ad <var>.i.a.</var> ita ergo Å¿e habebunt ipÅ¿orum velo-<lb/>citates.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0184-01" xlink:href="fig-0184-01a"> - <image file="0184-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0184-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="7"> - <head style="it" xml:space="preserve">Corpora grauia aut leuia eiuÅ¿dem figur æ et materiæ Å¿ed inæqualis <lb/>magnitudinis, in Å¿uis motibus natur alibus uelocit atis, in eo <lb/>dem medio, proportionem longè diuerÅ¿am Å¿eruatura <lb/>eße quam AristoteliuiÅ¿um fuerit.</head> - <head xml:space="preserve">CAP. VII.</head> - <p> - <s xml:space="preserve">ESt mihi nunc probandum <reg norm="quod" type="simple">ê™</reg> in uno <reg norm="eodemque" type="simple">eodemqÌ;</reg> mcdio duo corpora inæqualia, Å¿ed <lb/>Å¿imili figura & materia, mouebuntur naturali motu, diuerÅ¿a tamen ratione ab - <pb o="173" rhead="DISPVTATIONES." n="185" file="0185" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0185"/> - ea, quà m AriÅ¿toteles præſcripÅ¿it.</s> - </p> - <p> - <s xml:space="preserve">Sintigitur corpora <var>.a.</var> et <var>.o.</var> inæqualia, <reg norm="eadentamen" type="context">eadẽtamen</reg> figura & materia prædita, quo-<lb/>rum <var>.a.</var> maius Å¿it, & per conÅ¿equens in eadem quoque proportione grauius ipÅ¿o <var>.o.</var> in <lb/>qua eÅ¿t maius, communi omnium Å¿ententia.</s> - </p> - <p> - <s xml:space="preserve">Scribit ergo AriÅ¿toteles proportionem velocitatis corporis <var>.a.</var> ad eam, quæ eÅ¿t <lb/>corporis <var>.o.</var> (naturaliterÅ¿e vnoquoque mouente) eandem futuram, quæ eÅ¿t magnitu <lb/>dinis, aut grauitatis corporis <var>.a.</var> ad magnitudinem, aut grauitatem corporis <var>.o</var>. </s> - <s xml:space="preserve">Ima-<lb/>ginemur igitur corpus u. eadem magnitudine & figura, qua corpus <var>.a.</var> præditum eÅ¿t, <lb/>Å¿ed eandem grauitatem obtinere, quæ communicata eÅ¿t corpori <var>.o.</var> quod ex quauis <lb/>materia conÅ¿ter. </s> - <s xml:space="preserve">Hinc ex primo Å¿uppoÅ¿ito Å¿ecundi capitis certi erimus proportio-<lb/>nem velocitatis corporis <var>.a.</var> Å¿i comparetur cum velocitate corporis <var>.u.</var> futuram, vt <reg norm="eam" type="context">eã</reg>, <lb/>quæ eÅ¿t ponderis corporis <var>.a.</var> ad pondus ipÅ¿ius corporis <var>.u</var>. </s> - <s xml:space="preserve">Ex .9. igitur lib. quinti Eu-<lb/>cli. cogitur fateri AriÅ¿toteles velocitatem corporis <var>.o.</var> eſſe vnam <reg norm="eandemque" type="simple">eandemqÌ;</reg> in Å¿pe-<lb/>cie, quæ eÅ¿t corporis <var>.u</var>. </s> - <s xml:space="preserve">Quod primo Å¿uppoÅ¿ito cap. Å¿ecundi huius lib. planè repugnaÂ<lb/>ret. </s> - <s xml:space="preserve">Igitur hæc AriÅ¿totelis opinio falÅ¿a eÅ¿t. </s> - <s xml:space="preserve">Idem quoque probaretur mediante cor <lb/>pore <var>.i.</var> æquali magnitudine, <reg norm="Å¿imilique" type="simple">Å¿imiliqÌ;</reg> figura cum corpore <var>.o.</var> prædito, Å¿ed, quòd ad <lb/>quantitatem attinet, æquali corpori <var>.a.</var> vnde ex primo Å¿uppoÅ¿ito cap. Å¿ecundi huius li <lb/>bri in eadem pro <lb/>portione <reg norm="velocius" type="simple">velociê°</reg> <lb/>eſſet corpore <var>.o.</var> <lb/> - <anchor type="figure" xlink:label="fig-0185-01a" xlink:href="fig-0185-01"/> - in qua grauius eÅ¿t. <lb/>ex .9. igitur quin-<lb/>ti cogitur AriÅ¿to-<lb/>teles affirmare <reg norm="tam" type="context">tã</reg> <lb/>velox eſſe corpus <lb/>a. <reg norm="quam" type="context">quã</reg> eÅ¿t corpus <lb/>i. vnde idem pla-<lb/>nè inconueniens emergit ex Å¿ecundo Å¿uppoÅ¿ito cap. Å¿ecundi huius lib..</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0185-01" xlink:href="fig-0185-01a"> - <image file="0185-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0185-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="8"> - <head style="it" xml:space="preserve">Quod duo corpor a in æqualia eiuÅ¿dem materia in diuerÅ¿is <lb/>medijs eandem uelocitatis proportionem <lb/>retinebunt.</head> - <head xml:space="preserve">CAP. VIII.</head> - <p> - <s xml:space="preserve">QVælibet duo corpora inæqualia Å¿imili tamen figura & eadem materia con-<lb/>Å¿tantia, naturaliter Å¿e per diuerÅ¿a media mouentia, vnam <reg norm="eandemque" type="simple">eandemqÌ;</reg> Å¿em-<lb/>per proportionem velocitatum Å¿eruant.</s> - </p> - <p> - <s xml:space="preserve">Sint duo corpora <var>.A.</var> et <var>.B.</var> Å¿ibi inuicem inæqualia quorum <var>.A.</var> Å¿it maius, Å¿ed Å¿imile <lb/>figura & idem materia, <lb/>cuius pondus totaleÅ¿it <var>.<lb/> - <anchor type="figure" xlink:label="fig-0185-02a" xlink:href="fig-0185-02"/> - x.o.</var> & pondus totaleip <lb/>Å¿ius <var>.B.</var> Å¿it <var>.u.s</var>. </s> - <s xml:space="preserve">Imagine-<lb/>mur quoque corpus <var>.A.</var> <lb/>poÅ¿itum in aqua amitte <lb/>re <reg norm="partem" type="context">partẽ</reg> <var>.o.e.</var> ponderis. - <pb o="174" rhead="IO. BAPT. BENED." n="186" file="0186" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0186"/> - <var>o.x.</var> et <var>.B.</var> quoque in eodem loco amittere <var>.c.s.</var> et <var>.A.</var> in <reg norm="aem" type="context">aẽ</reg><unsure/>re partem <var>.i.o.</var> et <var>.B.</var> partem. <lb/> <var>.t.s</var>. </s> - <s xml:space="preserve">Nunc quia corpus aqueum, cui correÅ¿pondet <var>.e.o.</var> æquale eÅ¿t ipÅ¿i <var>.A.</var> & corpus <lb/>aqueum, cui correÅ¿pondet <var>.c.s.</var> æquale eÅ¿t i pÅ¿i <var>.B.</var> vt eÅ¿t ab Archimede <reg norm="probatum" type="context">probatÅ©</reg>: </s> - <s xml:space="preserve">com <lb/>muni quadam Å¿cientiæ ratione, Å¿equitur eandem proportionem futuram <var>.o.x.</var> ad <var>.e.o.</var> <lb/>quæ eÅ¿t <var>.u.s.</var> ad <var>.c.s.</var> ob <reg norm="eaÅ¿demque" type="simple">eaÅ¿demqÌ;</reg> rationes idem erit de <var>.x.o.</var> ad <var>.i.o.</var> ut <var>.u.s.</var> ad <var>.t.s.</var> & <reg norm="idem" type="context">idẽ</reg> <lb/>etiam erit de <var>.o.x.</var> ad <var>.s.u.</var> vt de <var>.e.o.</var> ad <var>.c.s.</var> vt etiam de <var>.o.i.</var> ad <var>.s.t</var>. </s> - <s xml:space="preserve">Vnde ex .19. lib. <lb/>quintà erit de <var>.x.i.</var> ad <var>.u.t.</var> quemadmodum de <var>.x.o.</var> ad <var>.u.s.</var> idem dico de <var>.x.e.</var> ad <var>.u.c</var>. </s> - <s xml:space="preserve">Ex <lb/>11. igitur dicti lib. erit. de <var>.x.i.</var> ad <var>.u.t.</var> quemadmodum de <var>.x.e.</var> ad <var>.u.c.</var> ex quibus <reg norm="quidem" type="context">quidẽ</reg> <lb/>proportionibus, Å¿i Å¿ubtra <lb/> - <anchor type="figure" xlink:label="fig-0186-01a" xlink:href="fig-0186-01"/> - hantur proportiones @reÅ¿i <lb/><reg norm="Å¿tentiarum" type="context">Å¿tẽtiarum</reg> extrinÅ¿ecus <reg norm="ad- uenentium" type="context">ad-<lb/>uenẽtium</reg>, proportiones <lb/>quæ remanebunt, exter-<lb/>tio communi axiomate <lb/>ab Eucli. in principio priÂ<lb/>mi lib. poÅ¿ito, ad inuicem <lb/>erunt æquales, Å¿ecundum quas eorundem corporum Å¿unt velocitates.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0185-02" xlink:href="fig-0185-02a"> - <image file="0185-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0185-02"/> - </figure> - <figure xlink:label="fig-0186-01" xlink:href="fig-0186-01a"> - <image file="0186-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0186-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="9"> - <head style="it" xml:space="preserve">Anrectè Aristoteles diÅ¿eruerit de proportionibus mo-<lb/>tuum in uacuo.</head> - <head xml:space="preserve">CAP. IX.</head> - <p> - <s xml:space="preserve">CVm verò AriÅ¿toteles circa finem cap .8. lib. 4. phyÅ¿icorum Å¿ubiungit quod ea-<lb/>dem proportione dicta corpora mouerentur in vacuo, vt in pleno, id pace <reg norm="eius" type="simple">eiê°</reg> <lb/><reg norm="dictum" type="context">dictÅ©</reg> Å¿it planè <reg norm="erroneum" type="context">erroneÅ©</reg> eÅ¿t. </s> - <s xml:space="preserve">quia in pleno dictis corporibus Å¿ubtrahitur proportio reÅ¿i <lb/>Å¿tentiarum extrinÅ¿ecarum à proportione ponderum, vt velocitatum proportio re-<lb/>maneat, quÄ™ nulla eſſet, Å¿i dictarum reÅ¿iÅ¿tentiarum proportio, ponderum propor-<lb/>tioni æqualis eſſet, & hanc ob cauÅ¿am diuerÅ¿am velocitatum proportionem in va-<lb/>cuo haberent ab ea, quæ eÅ¿t in pleno.</s> - </p> - </div> - <div type="section" level="3" n="10"> - <head style="it" xml:space="preserve">Quòd in uacuo corpor a eiuÅ¿dem materiæ æquali uelocita-<lb/>te mouerentur.</head> - <head xml:space="preserve">CAP.X.</head> - <p> - <s xml:space="preserve">QVòd Å¿upradicta corpora in vacuo naturaliter pari velocitate mouerentur, <lb/>hac ratione aſſero.</s> - </p> - <p> - <s xml:space="preserve">Sint enim duo corpora <var>.o.</var> et <var>.g.</var> omogenea, et <var>.g.</var> Å¿it dimidia pars ipÅ¿ius <var>.o.</var> Å¿int alia <lb/>quoque duo corpora <var>.a.</var> et <var>.e.</var> omogenea primis, quorum quodlibet æquale Å¿it ipÅ¿i <var>.g.</var> <lb/>& imaginatione comprÄ™hendamus ambo poÅ¿ita in extremitatibus alicuius lineæ, cu <lb/>ius medium Å¿it <var>.i.</var> clarum erit, tantum pondus habiturum, punctum <var>.i.</var> quantum <reg norm="centrum" type="context">centrÅ©</reg> <lb/>ipÅ¿ius <var>.o.</var> quod <var>.i.</var> virtute corporis <var>.a.</var> et <var>.e.</var> in vacuo, <lb/> - <anchor type="figure" xlink:label="fig-0186-02a" xlink:href="fig-0186-02"/> - eadem velocitate moueretur, quacentrum ipÅ¿ius .<lb/>o: </s> - <s xml:space="preserve">cum autem difiuncta eſſent dicta corpora <var>.a.</var> et <var>.e.</var> <lb/>à dicta linea, non ideo aliquo modo Å¿uam velocita - <pb o="175" rhead="DISPVTATIONES." n="187" file="0187" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0187"/> - tem mutarent, quorum quodlibet eſſet quoque tam velox, quam eÅ¿t .g: igitur <var>.g.</var> <lb/>tam velox eſſet quam <var>.o</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0186-02" xlink:href="fig-0186-02a"> - <image file="0186-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0186-02"/> - </figure> - </div> - </div> - <div type="section" level="3" n="11"> - <head style="it" xml:space="preserve">Corpora licet inæqualia eiuÅ¿dem materiæ & figuræ, Å¿ireÅ¿iÅ¿ten-<lb/>tias habuerint ponderibus proportionales <lb/>æqualiter mouebuntur.</head> - <head xml:space="preserve">CAP. XI.</head> - <p> - <s xml:space="preserve">EAdem ratione, quam cap. antecedente præſcripÅ¿imus, poſſet oÅ¿tendi, Å¿i duo cor-<lb/>pora <var>.o.</var> et <var>.g.</var> Å¿uas reÅ¿iÅ¿tentias, ita ad inuicem proportionatas haberent, utÅ¿unt <lb/>eorum pondera, in pleno pari velocitate prædita eſſe, quod in fine capitis noni leui <lb/>ter attigi, quia punctum <var>.i.</var> tam velox eſſet, ut centrum ipÅ¿ius <var>.o.</var> cum à tanto pondere <lb/>i. motum eſſet; </s> - <s xml:space="preserve">quanto centrum ipÅ¿ius <var>.o.</var> atquetan <lb/> - <anchor type="figure" xlink:label="fig-0187-01a" xlink:href="fig-0187-01"/> - tam reÅ¿iÅ¿tentiam duo corpora <var>.a.</var> et <var>.e.</var> <reg norm="quanta" type="context">quãta</reg> ipÅ¿um <lb/>o. Å¿olum haberet ex hypotheÅ¿i, dicta tamen corpo <lb/>ra <var>.a.</var> et <var>.e.</var> tam Å¿eparata, quam coniuncta, eandem <lb/>velocitatem retinerent <var>.g.</var> igitur tam velox eſſet, <lb/>quam <var>.o</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0187-01" xlink:href="fig-0187-01a"> - <image file="0187-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0187-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="12"> - <head style="it" xml:space="preserve">Maior hic demonÅ¿ir atur eſſe proportio ponder is corpor is den <lb/>Å¿ioris ad pondus minus denÅ¿i in <reg norm="medijs" type="lig">medijs</reg> <reg norm="denÅ¿ioribus" type="context">dẽſioribus</reg>, quam <lb/>Å¿it eorundem corporum in medio minus denÅ¿o, nec <lb/>corporum ponder a Å¿eruare proportionem <lb/>denÅ¿itatis mediorum.</head> - <head xml:space="preserve">CAP. XII.</head> - <p> - <s xml:space="preserve">PRopoÅ¿ita nobis cum fuerint duo corpora <var>.A.</var> et <var>.B.</var> area corporea æqualia, quo-<lb/>rum <var>.A.</var> denÅ¿ius Å¿it ipÅ¿o <var>.B.</var> probabo in medio magis denÅ¿o, maiorem proportio <lb/>nem futuram ponderis ipÅ¿ius <var>.A.</var> ad pondus <var>.B.</var> quà m in medio minus denÅ¿o.</s> - </p> - <p> - <s xml:space="preserve">Sit igitur <var>.p.g.</var> pondus totale ipÅ¿ius corporis <var>.A.</var> et <var>.q.k.</var> ipÅ¿ius corporis <var>.B.</var> vnde <var>.p.g.</var> <lb/>maius erit ipÅ¿o <var>.q.k</var>. </s> - <s xml:space="preserve">Sit quoque <var>.o.g.</var> pondus, quod medium magis denÅ¿um Å¿ubtra-<lb/>hit à pondere <var>.p.g.</var> et <var>.n.k.</var> Å¿it pondus, quod idem medium Å¿ubtrahit à pondere <var>.q.k.</var> et <lb/><var>f.g.</var> Å¿it pondus, quod medium minus denÅ¿um Å¿ubtrahit à <var>.p.g.</var> et <var>.i.k.</var> illud, quodid@m <lb/><reg norm="medium" type="context">mediÅ©</reg> Å¿ubtrahit ab <var>.q.k.</var> vnde <var>.o.g.</var> æquale erit <var>.n.k.</var> et <var>.f.g.</var> ipÅ¿i <var>.i.k.</var> quia quod ad <reg norm="aream" type="context">areã</reg> <lb/>attinet, corpora Å¿upponuntur æqualia, vnde proportio <var>.p.f.</var> ad <var>.q.i.</var> maior erit ea, quæ <lb/>eÅ¿t <var>.o.f.</var> ad <var>.n.i.</var> communi <lb/> - <anchor type="figure" xlink:label="fig-0187-02a" xlink:href="fig-0187-02"/> - Å¿cientiæ notione, quia Å¿i <lb/>Å¿cinderet <reg norm="aliquis" type="simple">aliê—s</reg>.p.f. in pun <lb/>cto <var>.c.</var> ita. vt <var>.c.f.</var> æquale eÅ¿-<lb/>Å¿et ipÅ¿i <var>.q.i.</var> proportio <var>.c.f.</var> <lb/>ad <var>.q.i.</var> eſſet vt ea, quæ eÅ¿t <var>.<lb/>o.f.</var> ad <var>.n.i.</var> (hoc eÅ¿t nulla) - <pb o="176" rhead="IO. BAPT. BENED." n="188" file="0188" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0188"/> - Å¿ed proportio <var>.p.f.</var> ad <var>.q.i.</var> maior eſſet ea, quæ eÅ¿t <var>.c.f.</var> ad <var>.q.i.</var> ex. octaua lib. quinti, vn-<lb/>de ex .12. eiuÅ¿dem lib. maior eſſet <var>.p.f.</var> ad <var>.q.i.</var> quà m.o.f. ad <var>.n.i.</var> ex .33. igitur eiuÅ¿dem, <lb/>maior erit proportio <var>.p.o.</var> ad <var>.q.n.</var> quà m.p.f. ad <var>.q.i</var>. </s> - <s xml:space="preserve">Sic quoque Å¿e habebunt ad inui <lb/>cem velocitates, quod eÅ¿t propoÅ¿itum. </s> - <s xml:space="preserve">Cum autem proportio <var>.p.o.</var> ad <var>.q.n.</var> maior Å¿it, <lb/>quà m.p.f. ad <var>.q.i.</var> permurando igitur maior erit proportio <var>.p.o.</var> ad <var>.p.f.</var> quam <var>.q.n.</var> ad <var>.<lb/>q.i.</var> aut euerÅ¿im maior erit proportio <var>.q.i.</var> ad <var>.q.n.</var> quà m.p.f. ad <var>.p.o.</var> vnde Å¿i proportio <lb/><var>p.f.</var> ad <var>.p.o.</var> eſſet ac ea, quæ eÅ¿t <var>.o.g.</var> ad <var>.f.g.</var> non eſſet <var>.q.i.</var> ad <var>.q.n.</var> ut eÅ¿t <var>.o.g.</var> ad <var>.f.g.</var> aut <lb/>vt <var>.n.k.</var> ad <var>.i.k.</var> quodidem <lb/>eÅ¿t, de quibus quidem re-<lb/> - <anchor type="figure" xlink:label="fig-0188-01a" xlink:href="fig-0188-01"/> - bus, exemplis propoÅ¿itis <lb/>quinto capite <reg norm="mentionem" type="context">mẽtionem</reg> <lb/>feci.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0187-02" xlink:href="fig-0187-02a"> - <image file="0187-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0187-02"/> - </figure> - <figure xlink:label="fig-0188-01" xlink:href="fig-0188-01a"> - <image file="0188-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0188-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Velocitatibus autem Å¿e-<lb/>quentibus pondera, Å¿equi <lb/>tur proportionem veloci-<lb/>citatum duorum corporum hetereogeneorum eandem non eſſe per diuerÅ¿a media, <lb/>contra id, quod Å¿equeretur Å¿i AriÅ¿totelis opinionem .8. cap. lib. 4. phyÅ¿icorum re-<lb/>ciperemus.</s> - </p> - </div> - <div type="section" level="3" n="13"> - <head style="it" xml:space="preserve">Longe aliter ueritatem Å¿e habere quam Aristoteles <lb/>doceat in fine libri Å¿eptimi phyÅ¿icorum.</head> - <head xml:space="preserve">CAP. XIII.</head> - <p> - <s xml:space="preserve">NOn tam facile eÅ¿t aſſignare proportionem velocitatum duorum corporum na <lb/>turalium, quam AriÅ¿toteles vltimo cap. lib. 7. phyÅ¿icorum putauit.</s> - </p> - <p> - <s xml:space="preserve">Quamobrem Å¿int duo corpora <var>.B.</var> et <var>.D.</var> materia <reg norm="magnitudineque" type="simple">magnitudineqÌ;</reg> diuerÅ¿a, pondere <lb/>tamen, & figura Å¿imilia, & proportio reÅ¿iÅ¿tentiarum, quas recipiunt à medio <reg norm="dum" type="context">dÅ©</reg> mo-<lb/>uentur, Å¿it. ut <var>.o.i.</var> ad <var>.a.e.</var> denotentur deinde velocitates totales abÅ¿que vlla reÅ¿iÅ¿ten-<lb/>tia ab <var>.a.u.</var> et <var>.o.c.</var> quæ æquales erunt ad inuicem per communem Å¿cientiam ex Å¿up-<lb/>poÅ¿ito, Å¿int alia deinde duo corpora <var>.V.</var> et <var>.M.</var> eodem modo Å¿e habentia ut prima <var>.B.</var> <lb/>et <var>.D.</var> in eodem medio, Å¿ed ex diuerÅ¿a materia ab ea, quæ eÅ¿t illorum duorum corpo <lb/>rum, magnitudine tamen & figura ijÅ¿dem Å¿imilia: </s> - <s xml:space="preserve">Å¿ignificentur quoque eo-<lb/>rundem reÅ¿iÅ¿tentiæ per <var>.t.s.</var> et <var>.n.r.</var> & eorundem velocitates à nulla ex reÅ¿iÅ¿tentijs di-<lb/>minutæ, per <var>.n.x.</var> et <var>.t.g.</var> vnde <var>.n.r.</var> æqualis erit <var>.a.e.</var> et <var>.t.s.</var> ipÅ¿i <var>.o.i.</var> et <var>.n.x.</var> ipÅ¿i <var>.t.g</var>: <var>n.x.</var> ta-<lb/>men et <var>.t.g.</var> non erunt Ä™qualia <var>.a.u.</var> et <var>.o.c</var>. </s> - <s xml:space="preserve">Sed exempli gratia, ponamus ea eſſe mi-<lb/>nora. </s> - <s xml:space="preserve">Supponamus nunc <var>.e.u.</var> velocitatem eſſe quæ remanet ipÅ¿i <var>.B.</var> cum applicata <lb/>erit reÅ¿iÅ¿tentia <var>.a.e.</var> dicto corpori <var>.B.</var> quæ diminutam facit totam <var>.a.u.</var> per <var>.a.e.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.i.c.</var> <lb/>ea, quÄ™ remanet ipÅ¿i <var>.o.c.</var> corporis <var>.D.</var> et <var>.r.x.</var> ea, quæ remanet <var>.n.x.</var> corporis <var>.V.</var> et <var>.s.g.</var> <lb/>ea, quæ eÅ¿t ex <var>.t.g.</var> corporis <var>.M</var>. </s> - <s xml:space="preserve">Vnde communi omnium <reg norm="conÅ¿enÅ¿u" type="context">cõſenÅ¿u</reg> aſſequemur <var>.e.u.</var> ma <lb/>iorem futuram <var>.r.x.</var> et <var>.i.c.</var> ipÅ¿a <var>.s.g</var>. </s> - <s xml:space="preserve">Scindatur deinde <var>.a.m.</var> ad Ä™qualitatem <var>.n.x.</var> et <var>.o.z.</var> <lb/>ipÅ¿ius <var>.t.g.</var> vnde <var>.a.m.</var> ad <var>.o.z.</var> et <var>.m.u.</var> ad <var>.z.c.</var> æquales habebimus, ut quoque <var>.e.m.</var> ad <var>.r.<lb/>x.</var> et <var>.i.z.</var> ad <var>.s.g.</var> quamobrem <var>.e.m.</var> maior erit ipÅ¿a <var>.z.i.</var> maior igitur erit proportio <var>.z.c.</var> <lb/>ad <var>.z.i.</var> quà m.m.u. ad <var>.m.e.</var> (quia <var>.z.c.</var> ad <var>.z.i.</var> ita Å¿e habet vt <var>.m.u.</var> ad <var>.i.z.</var> ex .7. lib. quin-<lb/>ti, Å¿ed <var>.m.u.</var> ad <var>.i.z.</var> maior eÅ¿t quam ad <var>.m.e.</var> ex .8. dicti lib. vnde ex .12. eiuÅ¿dem <var>.z.c.</var> ad <lb/>ad <var>.z.i.</var> maior erit, quà m.m.u. ad <var>.m.e</var>. </s> - <s xml:space="preserve">Ergo ex .28. maior proportio erit <var>.c.i.</var> ad <var>.z.i.</var> - <pb o="177" rhead="DISPVTATIONES." n="189" file="0189" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0189"/> - quam <var>.u.</var> ad <var>.m.e.</var> & ex .27. maior erit proportio <var>.c.i.</var> ad <var>.u.e.</var> quam <var>.z.i.</var> ad <var>.e.m.</var> ideÅ¿t <var>.s.</var> g <lb/>ad <var>.r.x.</var> quod AriÅ¿toteli in mentem non venerat. </s> - <s xml:space="preserve">Alijs quoque modis idem proba-<lb/>ri poteÅ¿t, vt Å¿i diceret aliquis, maiorem proportionem eſſe <var>.e.m.</var> ad <var>.m.u.</var> quam <var>.i.z.</var> ad <lb/><var>z.c.</var> (quia <var>.e.m.</var> ad <var>.m.u.</var> eadem eÅ¿t ratio vt ad <var>.z.c.</var> ex .7. quinti, Å¿ed proportio <var>.e.m.</var> ad <var>.<lb/>z.c.</var> maior eÅ¿t quam <var>.i.z.</var> ad <var>.z.c.</var> ex .8. eiuÅ¿dem, ergo ea, quæ eÅ¿t <var>.e.m.</var> ad <var>.m.u.</var> ex .12. ma <lb/>for erit, quam <var>.i.z.</var> ad <var>.z.c.</var>) vnde componendo, ea quæ eÅ¿t <var>.e.u.</var> ad <var>.m.u.</var> maior erit illa, <lb/>quæ eÅ¿t <var>.i.c.</var> ad <var>.z.c.</var> & <reg norm="permutando" type="context">permutãdo</reg>, quam ea, quæ eÅ¿t <var>.e.u.</var> ad <var>.i.c.</var> ea, quæ eÅ¿t <var>.m.u.</var> ad <var>.z.c.</var> <lb/>& ex .33. quinti, ea, quæ eÅ¿t <var>.e.m.</var> ad <var>.i.z.</var> maior erit ea, quæ eÅ¿t <var>.e.u.</var> ad <var>.i.c</var>.</s> - </p> - <figure position="here"> - <image file="0189-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0189-01"/> - </figure> - </div> - <div type="section" level="3" n="14"> - <head style="it" xml:space="preserve">Quid Å¿equatur ex Å¿upradistis.</head> - <head xml:space="preserve">CAP. XIIII.</head> - <p> - <s xml:space="preserve">EX præcedenti capite manifeÅ¿tè depræhenditur, in vniuerÅ¿um AriÅ¿totelis opi-<lb/>nionem veram non eſſe in prima parte vltimi capitis. lib. 7. phyÅ¿icorum; </s> - <s xml:space="preserve">quia <lb/>in eo loco Å¿upponens ipÅ¿e corpus <var>.B.</var> prÄ™cedentis capitis eſſe dimidiam partem ipÅ¿ius <lb/>D. quantum ad aream corpoream Å¿pectat (Å¿unt tamen pondere ad inuicem æqualia) <lb/>ait <var>.B.</var> futurum duplo velocius ipÅ¿o <var>.D</var>. </s> - <s xml:space="preserve">Ego verò præcedenti capite accepi <var>.e.u.</var> pro <lb/>velocitate reÅ¿idua corporis <var>.B.</var> (Å¿ubtracta ea tamen parte, quam ei reÅ¿iÅ¿tentia adimit, <lb/>quæ erat <var>.e.a.</var>) et <var>.i.c.</var> pro ea, quæ eÅ¿t corporis <var>.D.</var> et <var>.r.x.</var> pro ea, quæ eÅ¿t corporis <var>.V.</var> et <var>.<lb/>s.g.</var> pro ea, quæ eÅ¿t corporis <var>.M</var>. </s> - <s xml:space="preserve">Dicat nunc AriÅ¿toteles, quÄ™ nam harum duarum pro <lb/>portionum dupla erit? </s> - <s xml:space="preserve">quia Å¿i earum aliqua talis erit, alia nullo modo eſſe poterit, <lb/>vt iam oÅ¿tendi, etiamÅ¿i duo corpora <var>.V.</var> et <var>.M.</var> eaÅ¿dem conditiones habeant, quas <var>.B.</var> <lb/>et <var>.D</var>. </s> - <s xml:space="preserve">Ratio autem, quæ AriÅ¿totelem induxerit ad illud credendum, nulla alia eſſe <lb/>potuit, quà m quod putarit reÅ¿iÅ¿tentias proportionatas eſſe magnitudinibus corpo-<lb/>reis, ideÅ¿t quemadmodum <var>.B.</var> erat corporaliter dimidia pars ipÅ¿ius <var>.D.</var> Å¿ic etiam habe <lb/>ret medietatem eius reÅ¿iÅ¿tentiæ, quam habuiſſet corpus <var>.D</var>. </s> - <s xml:space="preserve">Quod etÅ¿i verum eſſet, <lb/>non tamen Å¿equeretur neceſſariò in quibuÅ¿libet corporibus futuram velocitatum <lb/>proportionem eandem, quæ reÅ¿iÅ¿tentiarum eÅ¿t, vt Å¿uperiore capite oÅ¿tendimus.</s> - </p> - </div> - <div type="section" level="3" n="15"> - <head style="it" xml:space="preserve">Numrestè Å¿enÅ¿erit PhiloÅ¿ophus reÅ¿istentias proportionales <lb/>eße cum corporibus mobilibus.</head> - <head xml:space="preserve">CAP. XV.</head> - <p> - <s xml:space="preserve">QVòd AriÅ¿toteles crediderit reÅ¿iÅ¿tentias proportionatas eſſe corporibus, erra-<lb/>uit. </s> - <s xml:space="preserve">Si Å¿uperficies ijÅ¿dem proportionatæ eſſent, dubium non eÅ¿t, quin <lb/>reÅ¿iÅ¿tentiæ quoque ipſæ, ijÅ¿dem proportionatæ exiÅ¿terent, Å¿upponendo eas Å¿imiles <lb/>Å¿itu, dum eadem corpora mouerentur. </s> - <s xml:space="preserve">Sed eadem proportio non eÅ¿t inter Å¿uperfi- - <pb o="178" rhead="IO. BAPT. BENED." n="190" file="0190" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0190"/> - cies, & quæ inter corpor a <reg norm="reperitur" type="simple">reperit̃</reg>: </s> - <s xml:space="preserve">AriÅ¿toteles igitur in eo defecit. </s> - <s xml:space="preserve">Quòd <reg norm="autem" type="context">autẽ</reg> inter <lb/>Å¿uperficies non eadem Å¿it proportio, quæ inter corpora extat, Å¿i primo ad Å¿phÄ™ricas <lb/>mentem verterimus, intelligemus proportionem eam, quæ inter duas Å¿phæras repe <lb/>ritur triplam Å¿emper exiÅ¿tere ei, quæ eÅ¿t inter ipÅ¿arum diametros ex vltima .12. libr. <lb/>Euclid. </s> - <s xml:space="preserve">EÅ¿t autem proportio, quæ eÅ¿t inter Å¿uperficies Å¿phÄ™ricas Ä™qualis ei, quæ eÅ¿t <lb/>ipÅ¿orum circulorum maiorum ex .16. lib. quinti, cum ex .31. primi de Å¿phæra & cy-<lb/>lindro Archimedis, omnis Å¿phærica Å¿uperficies quadrupla, Å¿it maiori circulo ipÅ¿ius <lb/>Å¿phÄ™ræ, Å¿ed proportio, quæ eÅ¿t inter dictos circulos, eÅ¿t dupla ei, quæ eÅ¿t inter <reg norm="eorun- dem" type="context context">eorÅ©-<lb/>dẽ</reg> diametros ex .2. lib. 12. Euc. </s> - <s xml:space="preserve">ergo <reg norm="proportio" type="simple">ê“portio</reg>, quæ eÅ¿t inter corpora, Å¿eÅ¿quialtera erit <lb/>ei, quæ eÅ¿t Å¿uperficierum, & non æqualis, ut AriÅ¿toteles putauit. </s> - <s xml:space="preserve">Idem de corporibus <lb/>Å¿imilibus à ſuperficiebus planis terminatis dico, ratiocinando mediante .36. lib. 11. <lb/>et .18. Å¿exti, vnde cognoÅ¿cemus proportionem corporum, proportioni laterum, tri-<lb/>plam futuram, & Å¿uperficierum proportionem, laterum proportioni duplam. </s> - <s xml:space="preserve">Quare <lb/>corporum proportio, ei, quæ Å¿uperficierum eÅ¿t, Å¿eÅ¿quialtera erit, ita ut Å¿i velocitates <lb/>extitiſſent ad inuicem proportionatæ, vt Å¿uperficies, proportio velocitatis corporis <var>.<lb/>B.</var> ei, quæ eÅ¿t corporis <var>.C.</var> fuiſſet Å¿ubÅ¿eÅ¿quialtera proportioni corporum, & non æqua <lb/>lis eidem.</s> - </p> - </div> - <div type="section" level="3" n="16"> - <head style="it" xml:space="preserve">FdipÅ¿um aliter demonÅ¿tr atur.</head> - <head xml:space="preserve">CAP. XVI.</head> - <p> - <s xml:space="preserve">ALio quoque modo probari poteÅ¿t non eſſe in vniuerÅ¿um verum id, quod Ari-<lb/>Å¿toteles in prima parte capitis vltimi lib. 7. phyÅ¿icorum ait, Å¿ic Å¿cribens.</s> - </p> - <p> - <s xml:space="preserve">Si <var>.A.</var> quidem Å¿it id quod mouet <var>.B.</var> verò id quod mouetur, et <var>.C.</var> Å¿it longitudo per <lb/>quam, et <var>.D.</var> tempus in quo eÅ¿t motum, in tempore nimirum Ä™quali, potentia æqua-<lb/>lis <var>.A.</var> dimidium ipÅ¿ius <var>.B.</var> per duplum mouebit ipÅ¿ius <var>.C.</var> per ipÅ¿um autem <var>.C.</var> in dimi <lb/>dio temporis <var>.D.</var> Å¿ic enim erit rationis Å¿imilitudo.</s> - </p> - <p> - <s xml:space="preserve">Sit ergo corpus <var>.o.</var> Å¿eptimi capitis pondere æquali corpori <var>.u.</var> eiuÅ¿dem capitis, Å¿ed <lb/>area corporea minusipÅ¿o <var>.u.</var> pro medietate. </s> - <s xml:space="preserve">Simile tamen figura. </s> - <s xml:space="preserve">Imaginemur <reg norm="nunc" type="context">nÅ©c</reg> <lb/>tertium aliud corpus omogeneum ipÅ¿i <var>.u.</var> quod Å¿it <var>.i.</var> magnitudine & figura Å¿imile ipÅ¿i <lb/>o. vnde minor erit ipÅ¿o <var>.u.</var> pro media parte, & hanc ob cauÅ¿am ipÅ¿um <var>.u.</var> erit duplo ma <lb/>gis graue, quà m ipÅ¿um <var>.i.</var> & per conÅ¿equens ipÅ¿um quoque <var>.o.</var> duplo grauius erit <reg norm="quam" type="context">quã</reg> <lb/>Å¿it ipÅ¿um <var>.i.</var> ex .7. libr. quinti Euclidis. </s> - <s xml:space="preserve">IpÅ¿um ergo corpus <var>.o.</var> duplo velocius erit, <lb/>quà m ipÅ¿um <var>.i.</var> ex primo Å¿uppoÅ¿ito cap .2. huius lib. </s> - <s xml:space="preserve">Vnde ex .9. quinti, velocitas ipÅ¿ius <lb/>i. æqualis eſſet ei, quæ eÅ¿t ipÅ¿ius u. cum AriÅ¿toteles Å¿cribat <var>.o.</var> quoque futurum duplo <lb/>velocius ipÅ¿o <var>.u.</var> <reg norm="quod" type="simple">ê™</reg> cap .7. huius lib. falÅ¿um eſſe demonÅ¿traui.</s> - </p> - </div> - <div type="section" level="3" n="17"> - <head style="it" xml:space="preserve">De alio Aristo. lapÅ¿u.</head> - <head xml:space="preserve">CAP. XVII.</head> - <p> - <s xml:space="preserve">SCribit AriÅ¿toteles in ultimo cap. lib. 7. phyÅ¿icorum in hunc modum. <lb/></s> - <s xml:space="preserve">Si duo quædam Å¿eorÅ¿um per tantum Å¿patium tanto tempore duo Å¿eorÅ¿um pon <lb/>dera mouent, & compoÅ¿ita per longitudinem æqualem, <reg norm="Ä™qualiuem" type="context">Ä™qualiuẽ</reg> in tempore, com-<lb/>poÅ¿itum ex ponderibus <reg norm="vtriÅ¿que" type="simple">vtriÅ¿q;</reg> mouebunt, eÅ¿t enim in eis eadem ratio.</s> - </p> - <pb o="179" rhead="DISPVTATIONES." n="191" file="0191" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0191"/> - <p> - <s xml:space="preserve">Quod in vniuerÅ¿um nec etiam poteÅ¿t eſſe verum in pleno, quia cap .14. iam pro-<lb/>baui, non eandem proportionem eſſe inter Å¿uperÅ¿icies corporum, & ipÅ¿a corpora.</s> - </p> - </div> - <div type="section" level="3" n="18"> - <head style="it" xml:space="preserve">Quomodo dignoÅ¿catur proportio uelocitatis duorum Å¿imilium <lb/>corporum omogeniorum inaqualium.</head> - <head xml:space="preserve">CAP. XVIII.</head> - <p> - <s xml:space="preserve">ETiam Å¿i reperire in qua proportione motus naturaliter moueantur duo corpo-<lb/>ra, figura & materia Å¿imilia, inÄ™qualia tamen ad inuicem, non facile Å¿it, oÅ¿ten-<lb/>dam tamen qua ratione id conÅ¿equi poſſimus.</s> - </p> - <p> - <s xml:space="preserve">Proponantur nobis, exempli gratia, duo corpora <var>.a.</var> et <var>.o.</var> Å¿phÄ™rica, inÄ™qualia inui-<lb/>cem, omogenea tamen materia, quorum <var>.a.</var> maius Å¿it; </s> - <s xml:space="preserve">Å¿i voluerimus inuenire in qua <lb/>nam velocitatis proportione naturaliter mouerentur. </s> - <s xml:space="preserve">Volo vt inquiratur corpus <var>.i.</var> <lb/>Å¿phÄ™ricum, alia tamen & diuerÅ¿a materia conÅ¿tans, Å¿ed pondere Ä™quale corpori <var>.o.</var> & <lb/>Å¿uperÅ¿icie tam proportionata Å¿uperficiei corp oris <var>.a.</var> quà m eÅ¿t ea, quæ eÅ¿t Å¿ui ponde-<lb/>ris ad pondus ipÅ¿ius <var>.a</var>. </s> - <s xml:space="preserve">Hoc facto, indagetur, quænam erit proportio inter Å¿u-<lb/>perficies corporum <var>.i.</var> et <var>.o.</var> quÄ™ Å¿emper dupla eÅ¿t, vel Å¿ubdupla ei quæ eÅ¿t diametro-<lb/>rum; </s> - <s xml:space="preserve">ut iam cap .15. dixi, & hÄ™c proportio Å¿uperficierum Å¿phÄ™ricarum <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <var>.o.</var> et <var>.i.</var> Å¿ub <lb/>trahatur ab æqualitate, quod igitur remanebit, erit proportio <reg norm="velocitatum" type="context">velocitatÅ©</reg> inter duo <lb/>corpora <var>.o.</var> et <var>.i.</var> ideÅ¿t inter <var>.o.</var> et <var>.a.</var> vt exempli gratia, Å¿i proportio Å¿uperficiei <var>.o.</var> Å¿uperfi <lb/>ciei ipÅ¿ius <var>.i.</var> Å¿eÅ¿quitertiα<unsure/> eſſet, Å¿ub <lb/>trahendo eam ab Ä™qualitate, rema-<lb/> - <anchor type="figure" xlink:label="fig-0191-01a" xlink:href="fig-0191-01"/> - neret <reg norm="proportio" type="simple">ê“portio</reg> Å¿ubÅ¿eÅ¿quitertia, vnde <lb/>velocitas corporis maioris ( quod in <lb/>pręſenti loco Å¿upponitur eſſe <var>.o.</var>) ei, <lb/>quÄ™ eÅ¿t corporis minoris, quale eÅ¿t <lb/>corpus <var>.i.</var> Å¿ubÅ¿eÅ¿quitertia eſſet; </s> - <s xml:space="preserve">aut <lb/>dicamus quòd <var>.i.</var> eſſet velocius ipÅ¿o <lb/>o. in proportione Å¿eÅ¿quitertia ex Å¿e <lb/>cundo Å¿uppoÅ¿ito Å¿ecundi capitis huius libri. </s> - <s xml:space="preserve">Sed <var>.i.</var> tam velox eÅ¿t quam ipÅ¿um <var>.a.</var> ex <num value="11">.<lb/>11.</num> cap. ergo proportio velocitatis ipÅ¿ius <var>.a.</var> Å¿eÅ¿quitertia erit ei. quæ eÅ¿t ipÅ¿ius <var>.o</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0191-01" xlink:href="fig-0191-01a"> - <image file="0191-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0191-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="19"> - <head style="it" xml:space="preserve">Quam Å¿it inanis ab AriÅ¿totele Å¿uÅ¿cepta demonÅ¿tratio quod <lb/>uacuum non detur.</head> - <head xml:space="preserve">CAP. XIX.</head> - <p> - <s xml:space="preserve">EX ijs, quæ Å¿uperius <reg norm="demonÅ¿trauimus" type="context">demõſtrauimus</reg> facilè cognoÅ¿ci poteÅ¿t irritam eſſc eam ratio <lb/>nem, quam AriÅ¿toteles .8. cap. lib. 4. phyÅ¿icorum ad deÅ¿truendum vacuum, <reg norm="con" type="context">cõ</reg> <lb/>finxit. </s> - <s xml:space="preserve">Vtigitur idem facilius oÅ¿tendamus, compræhendamus imaginatione infini-<lb/>ta media corporea, quorum vnum altero rarius Å¿it, in qua placuerit nobis ex propor <lb/>tionibus, incipiendo ab uno, imaginemur etiam corpus <var>.Q.</var> denÅ¿ius primo medio, cu-<lb/>ius corporis, totalis grauitas Å¿it <var>.a.b.</var> & poÅ¿itum in ipÅ¿o medio, amittat partem <var>.e.b.</var> ip-<lb/>Å¿ius grauitatis, & in Å¿ecundo medio amittat <var>.i.b.</var> & Å¿ic per gradus vnde nobis patebie<unsure/> - <pb o="180" rhead="IO. BAPT. BENED." n="192" file="0192" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0192"/> - dicto corpori <var>.Q</var>. </s> - <s xml:space="preserve">Nunquam remanÅ¿uram Å¿uam totalem grauitatem <var>.a.b.</var> in quolibet <lb/>ex-dictis medijs. </s> - <s xml:space="preserve">Nunc Å¿i quærat à me AriÅ¿toteles proportionem velocitatis corpo-<lb/>ris <var>.Q.</var> per vacuum ad velocitatem dicti corporis per plenum, ego ei proponam pro-<lb/>portionem ipÅ¿ius <var>.a.b.</var> ad <var>.a.e.</var> exempli gratia, dicens, <reg norm="quod" type="simple">ê™</reg> <reg norm="quemadmodum" type="wordlist">quẽadmodum</reg> <var>.a.b.</var> maius eÅ¿t <lb/>ip Å¿o <var>.a.e.</var> Å¿ic etiam corpus <var>.Q.</var> velocius erit in vacuo, quà m in pleno, dicti autem ple-<lb/>ni denÅ¿itatem appellabimus <var>.e.b</var>. </s> - <s xml:space="preserve">AriÅ¿toteles dicet nunc, <reg norm="quod" type="simple">ê™</reg> aliud quoddam medium <lb/>in eadem proportione Å¿ubtilius ipÅ¿o <var>.e.b.</var> deÅ¿umatur; </s> - <s xml:space="preserve">quemadmodum <var>.a.e.</var> minus eÅ¿t <lb/>ipÅ¿o <var>.a.b.</var> Å¿it ergo iÅ¿tud <var>.i.b.</var> in quo AriÅ¿toteles credit corpus Q. futurum tam velox ut <lb/>in vacuo, in quo aberrat, <reg norm="quia" type="simple">ê—a</reg> proportio velocitatis corporis <var>.Q.</var> in medio <var>.i.b.</var> ad velo <lb/>citatem eiuÅ¿dem in medio <lb/><var>e.b.</var> ita Å¿e hà bebit, ut <var>.i.a.</var> ad <lb/> - <anchor type="figure" xlink:label="fig-0192-01a" xlink:href="fig-0192-01"/> - <var>e.a.</var> ex ultimo Å¿uppoÅ¿ito ca <lb/>pit .2. huius libr. quæ minor <lb/>eſſet ea, quæ eÅ¿t <var>.a.b.</var> ad <var>.a.e.</var> ex .8. lib. quinti Eucli.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0192-01" xlink:href="fig-0192-01a"> - <image file="0192-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0192-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="20"> - <head style="it" xml:space="preserve">Non Å¿atis dilucidè AriÅ¿totelem de loco ratiocinatum fuiße.</head> - <head xml:space="preserve">CAP. XX.</head> - <p> - <s xml:space="preserve">QVæ AriÅ¿toteles de loco Å¿cribit multas in Å¿e continent difficultates. </s> - <s xml:space="preserve">Primum, <lb/>cap .4. lib. 4. phyÅ¿icorum ait, omne corpus eſſe in Å¿uo proprio loco, Å¿upponen <lb/>do vnum centrum pro loco grauium, et unam circunferentiam pro loco leuium cor <lb/>porum. </s> - <s xml:space="preserve">Sed quomodo punctum poteÅ¿t eſſe locus ipÅ¿ius corporis, cum omni dimen <lb/>Å¿ione <reg norm="capacitateque" type="simple">capacitateqÌ;</reg> Å¿it denudatum? </s> - <s xml:space="preserve">vnde Å¿i <reg norm="centrum" type="context">centrÅ©</reg> locus eſſet corporum grauium, om <lb/>nia dicta corpora grauia, extra proprium locum exiÅ¿terent, quia nullum ex iis eÅ¿t, <reg norm="quod" type="simple">ê™</reg> <lb/>Å¿it in centro. </s> - <s xml:space="preserve">Adde quod neque hoc cum loci definitione ab ipÅ¿o poÅ¿ita conÅ¿entiret <lb/>cum ipÅ¿e dicat in eodem cap. locum eſſe Å¿uperÅ¿iciem quandam, & non interuallum, <lb/>licet huiuÅ¿modi definitio falÅ¿a appareat primo ex <reg norm="inconuenienti" type="context">incõuenienti</reg> falÅ¿o, quod ipÅ¿e hinc <lb/>Å¿equuturum dicit, ideÅ¿t, quod Å¿i locus interuallum eſſet, infinita loca exiÅ¿terent, quod <lb/>reuera nec ob hanc cauÅ¿am inconueniens exiÅ¿tit, quia eodem planè modo quo ali-<lb/>quod corpus poteÅ¿t eſſe infinita corpora, (quod ipÅ¿e diceret in potentia) Å¿ic etiam in <lb/>teruallum aliquod poſſet eſſe infinita interualla. </s> - <s xml:space="preserve">Cum autem dicat Å¿uperficies cor-<lb/>poris ambientis eſſe locum eius corporis, quod continetur, cogitur dicere lineam, <lb/>quæ circundat Å¿uperficiem, Å¿uperficiei locum eſſe, & puncta ipÅ¿ius lineæ, quod reue <lb/>ra abÅ¿urdum eÅ¿t. </s> - <s xml:space="preserve">Locus corporis eÅ¿t interuallum illud eadem magnitudine & figu-<lb/>ra, qua corpus ipÅ¿um prÄ™ditum eÅ¿t, quod Å¿i non eſſet, Å¿ed eſſet Å¿uperficies, quemad-<lb/>modum AriÅ¿toteles voluit, maximum inconueniens Å¿equeretur, Å¿cilicet æquales lo-<lb/>cos capere inÄ™qualia corpora, aut corpora æqualia, locos inÄ™quales occupare, quod <lb/>Å¿citu facillimum eÅ¿t, cum Theon Å¿uper PtolomÄ™i AlmageÅ¿tum iam probarit Å¿phæ-<lb/>ricam Å¿uperficiem maius interuallum corporeum continere, quà m aliam <reg norm="quanuis" type="context">quãuis</reg> Å¿u-<lb/>perficiem dictæ Å¿phÄ™ricæ æqualem, vnde poſſent facilè reperiri duo loci, quorum al-<lb/>ter millies altero maior eſſet, capaces tamen corporum æqualium, aut reperiri duo <lb/>corpora, quorum alterum millies maius eſſet altero, quÄ™ tamen corpora apta eſſent <lb/>ad occupandos locos Ä™quales, quamuis AriÅ¿toteles dicat, locum, neque maiorem ne <lb/>que minorem eſſe debere locato. </s> - <s xml:space="preserve">Sed interualla corporea Ä™qualia à quauis figura <lb/>terminata, continebunt Å¿emper corpora Ä™qualia. </s> - <s xml:space="preserve">Corporeum igitur interuallum eÅ¿t - <pb o="181" rhead="DISPVTATIONES." n="193" file="0193" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0193"/> - reuera locus corpori adÄ™quatus, cum corpus in interuallum Å¿uperÅ¿iciale non intret, <lb/>quam @is interuallum corporeum ingrediatur. </s> - <s xml:space="preserve">Et hoc modo <reg norm="nullum" type="context">nullÅ©</reg> eÅ¿t corpus, quod <lb/>in m@ do aut extra mundum ( dicat autem AriÅ¿toteles quicquid voluerit ) locum <lb/>Å¿uum non habeat.</s> - </p> - </div> - <div type="section" level="3" n="21"> - <head style="it" xml:space="preserve">V<unsure/>trum bene Aristoteles Å¿enÅ¿erit de infinito.</head> - <head xml:space="preserve">CAP. XXI.</head> - <p> - <s xml:space="preserve">TRactans AriÅ¿toteles in fine quinti cap. lib. 3. phyÅ¿icorum de infinito ait, impoÅ¿Â<lb/>Å¿ibile cum Å¿it inuenire locum infinitum, & omne corpus in loco cum Å¿it, impoÅ¿ <lb/>Å¿ibile quoque eſſe in rerum natura aliquod: </s> - <s xml:space="preserve">infinitum corpus reperiri. </s> - <s xml:space="preserve">Omittamus <lb/>quòd cum AriÅ¿toteles debuerit beneficio loci deÅ¿truere infinitum, ordine peruerÅ¿o <lb/>de infinito prius, quà m de loco diÅ¿putationem inÅ¿tituat; </s> - <s xml:space="preserve">Å¿ed dicamus ipÅ¿um intelli-<lb/>gere de infinito corporeo, & cum probauerimus corporis locum eſſe corporeum in <lb/>teruallum, non autem Å¿uperficiem, neque opus Å¿it in definitione interualli mentio <lb/>nem aliquam facere terminorum, vnde ipÅ¿um infinitum eſſe poteÅ¿t, neque aliqua ra <lb/>tione de hac re dubitari poteÅ¿t; </s> - <s xml:space="preserve">hoc modo nullum inconueniens Å¿equeretur, quòd <lb/>extra cÄ™lum reperiri poſſit corpus aliquod infinitum, quamuis, id ipÅ¿e nulla euiden-<lb/>ti ratione inductus perneget. </s> - <s xml:space="preserve">SenÅ¿it quoque, abÅ¿que eo, <reg norm="quod" type="simple">ê™</reg> aliquam rationem propo <lb/>nat, aliquid extra cÅ“lum reperiri quemadmodum apparet ex fine cap .9. lib. primi <lb/>de cÅ“lo, cum etiam ait cap .8. lib. 8. phyÅ¿icorum, infinitas partes alicuius continui eÅ¿-<lb/>Å¿e Å¿olum in potentia, non item in actu, hoc non eÅ¿t illico concedendum, quia Å¿i omne <lb/>totum continuum, & re ipÅ¿a exiÅ¿tens, in actu eÅ¿t, omnis quoque eius pars erit in actu, <lb/>quia Å¿tultum eſſet credere, ea quæ actu Å¿unt, ex ijs, quæ potentia exiÅ¿tunt, componi. <lb/></s> - <s xml:space="preserve">Neque etiam dicendum eÅ¿t continuationem earundem partium efficere, vt poten-<lb/>tia Å¿int ipſæ partes, & omni actu priuatæ; </s> - <s xml:space="preserve">Sit exempli gratia linea recta <var>.a.u.</var> continua <lb/>quæ deinde diuidatur in puncto <var>.e.</var> per æqualia, dubium non eÅ¿t, quin ante <reg norm="diuiÅ¿ionem" type="context">diuiÅ¿ionẽ</reg>, <lb/>medietas <var>.a.e.</var> tam in actu (licet coniuncta cum alia <var>.e.u.</var>) reperiretur, quà m totum .2. <lb/>u. licet à ſenÅ¿u diÅ¿tincta non eſſet. </s> - <s xml:space="preserve">Idem affirmo de medietate <var>.a.e.</var> ideÅ¿t de quarta <lb/>parte totius <var>.a.u.</var> & pariter de octaua, de milleÅ¿ima, & de quauis, ita vt eſſentia actua <lb/>lis infiniti hoc modo tutò concedi poſſit, <reg norm="cum" type="context">cÅ©</reg> ita Å¿it in natura. </s> - <s xml:space="preserve">Sed peius etiam Å¿enÅ¿it <lb/>AriÅ¿toteles eodem loco capitis quinti lib. 3. phyÅ¿icorum, negando infinitum poſſe <lb/>connumerari inter quantitates, dicens vnam aliquam quantitatem intelligi vt cubi <lb/>tum, tricubitum, & cætera; </s> - <s xml:space="preserve">vbi non conÅ¿iderat eadem etiam ratione intelligi poſſe <lb/>aliquam quantitatem <reg norm="infinitorum" type="context">infinitorÅ©</reg> cubitorum, & in quantitatis definitione nullam eÅ¿-<lb/>Å¿e neceſſitatem terminorum, vt exempli gratia in definitione numeri, non eÅ¿t neceÅ¿ <lb/>Å¿itas alicuius determinati numeri, quia multitudo, non minus infinita, quà m finita, <lb/>intelligi poteÅ¿t. </s> - <s xml:space="preserve">Vbi poÅ¿teà cap .8. libr .4. phyÅ¿icorum ait nullam eſſe differentiam <lb/>inter infinitum, & vacuum, reuera nihil abÅ¿urdius hoc dicere fingereue poterat.</s> - </p> - <pb o="182" rhead="IO. BAPT. BENED." n="194" file="0194" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0194"/> - </div> - <div type="section" level="3" n="22"> - <head style="it" xml:space="preserve">Exagitatur ab AriÅ¿totele adductatemporis definitio.</head> - <head xml:space="preserve">CAP. XXII.</head> - <p> - <s xml:space="preserve">CVM Å¿enÅ¿erit AriÅ¿toteles <reg norm="tempus" type="context">tẽpus</reg> abÅ¿que motu eſſe <reg norm="non" type="context">nõ</reg> poſſe, ea tamen ab <reg norm="inui- cem" type="context">inui-<lb/>cẽ</reg> Å¿eparans, <reg norm="volens" type="context">volẽs</reg> definire <reg norm="tempus" type="context">tẽpus</reg> ait, <reg norm="ipsum" type="context">ipsÅ©</reg> eſſe <reg norm="motus" type="simple">motê°</reg> <reg norm="menÅ¿uram" type="context">menÅ¿urã</reg> <reg norm="numerumque" type="context simple">numerÅ©qÌ;</reg>. </s> - <s xml:space="preserve">Quæ <lb/>quidem definitio, natura Å¿ua non eÅ¿t bona, quia tempus, neque numerus eÅ¿t, neque <lb/><reg norm="etiam" type="context">etiã</reg> eÅ¿t <reg norm="menÅ¿ura" type="context">mẽſura</reg> motus <reg norm="per" type="simple">ê‘</reg> Å¿e, Å¿ed <reg norm="tantum" type="wordlist">tm̃</reg> <reg norm="per" type="simple">ê‘</reg> <reg norm="accidens" type="context">accidẽs</reg>, quia nihil eÅ¿t, <reg norm="quod" type="wordlist">qÄ‘</reg> numeret aut menÅ¿uret <lb/>aliud, quod non Å¿it <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> Å¿peciei <reg norm="cum" type="context">cÅ©</reg> illo quod <reg norm="menÅ¿uratur" type="context">mẽſuratur</reg>, aut numero <reg norm="circunÅ¿cribitur" type="simple">circunÅ¿cribit̃</reg>, <lb/>vt <reg norm="exempli" type="context">exẽpli</reg> gratia, nulla <reg norm="vnquam" type="context">vnquã</reg> Å¿uperficies <reg norm="per" type="simple">ê‘</reg> Å¿e numerabit aut <reg norm="menÅ¿urabit" type="context">mẽſurabit</reg> <reg norm="lineam" type="context">lineã</reg>, aut cor-<lb/>pus; </s> - <s xml:space="preserve">neclinea Å¿uperficiem <reg norm="aliquam" type="context">aliquã</reg>, aut corpus: </s> - <s xml:space="preserve">nec corpus <reg norm="lineam" type="context">lineã</reg> <reg norm="aliquam" type="context">aliquã</reg> aut Å¿uperfi-<lb/>ciem; </s> - <s xml:space="preserve">Sed linea lineam menÅ¿urabit; </s> - <s xml:space="preserve">Å¿uperficies Å¿uperficiem; </s> - <s xml:space="preserve">& corpus corpus; </s> - <s xml:space="preserve"><reg norm="etianÅ¿i" type="context">etiãſi</reg> <lb/>tam vna ex iis quantitatibus quà m altera Å¿it continua. </s> - <s xml:space="preserve">Cum verò motus non Å¿it tem <lb/>pus, neque tempus Å¿it motus, Å¿ed inter Å¿e maximè differant, Å¿equetur ex iis, alterum <lb/>nullo modo per Å¿e eſſe menÅ¿uram alterius, niÅ¿i per accidens. </s> - <s xml:space="preserve">Et Å¿i alicui videtur, <reg norm="quod" type="simple">ê™</reg> <lb/>ad Å¿ignificandam aliquam quantitatem motus, dicere huiuÅ¿modi operationem dua-<lb/>rum horarum, aut duorum dierum, aut duorum annorum Å¿patio completam eſſe, Å¿it <lb/>ponere tantum tempus: </s> - <s xml:space="preserve">animaduertere debet hoc Å¿impliciter non eſſe verum, quia <lb/>horarum, dierum. </s> - <s xml:space="preserve">& annorum interualla, imaginatione <reg norm="concipiuntur" type="context">concipiÅ©tur</reg> vt motus corpo-<lb/>rum cÄ™leÅ¿tium, Å¿ine quibus, neque anni, neque dies, neque horÄ™ exiÅ¿terent, <reg norm="etiam" type="context">etiã</reg> Å¿i om <lb/>nis motus Å¿it (vt ita dicam) locatus in tempore, ut corpus in loco, vnde motus motu, <lb/>& tempus tempore, non autem aliud ab alio menÅ¿uratur. </s> - <s xml:space="preserve">Tempus ex neceſſita-<lb/>te (phyloÅ¿ophicè tamen loquendo) res eÅ¿t æterna, motus non item, quia diuerÅ¿is mo <lb/>dis terminari poteÅ¿t & ceſſare, & interim dum ceſſabit quieÅ¿cet corpus, quod <lb/>primo mouebatur. </s> - <s xml:space="preserve">nihilominus tamen, tempus continuabit curÅ¿um Å¿uum. </s> - <s xml:space="preserve">Tempus <lb/>igitur potius locus motus erit dicendum, quà m numerus aut menÅ¿ura eius, & tale eÅ¿t, <lb/>vt conÅ¿umatum uideatur à continuò quodam fluxu vnius inÅ¿tantis, quemadmodum <lb/>iam dixi in .38. capite meÄ™ gnomonicæ, & cum dico ab vno inÅ¿tanti, vnum in Å¿pecie, <lb/>& non in numero intelligo, quod à ſenÅ¿ibus noÅ¿tris percipi non poteÅ¿t, <reg norm="neque" type="simple">neq;</reg> etiam <lb/>notari, quia nouum Å¿emper inÅ¿tans nobis occurrit. </s> - <s xml:space="preserve">& Å¿i aliquis aliquod <reg norm="exemplum" type="context">exemplÅ©</reg> (lar <lb/>go modo) incompræhenÅ¿ibilitatis ipÅ¿ius inÅ¿tantis deÅ¿ideraret, imaginetur rotam ali <lb/>quam albam, in qua Å¿it nigrum aliquod punctum Å¿enÅ¿ibile, aut è contra rotam <reg norm="nigram" type="context">nigrã</reg> <lb/>imaginetur, in qua Å¿it punctum album, quæ rota velociſſimè moueatur; </s> - <s xml:space="preserve">huiuÅ¿modi <lb/>punctum, nullo modo aſſignari poterit, magis ab una parte quà m ab altera; </s> - <s xml:space="preserve">immo Å¿e <lb/>Å¿e nobis offeret Å¿emper in forma lineæ circularis. </s> - <s xml:space="preserve">poſſumus aliquo modo etiam Å¿u-<lb/>mere exemplum à ſono, quia omnis chorda cuiuÅ¿libet inÅ¿trumenti muÅ¿ici, dum Å¿o-<lb/>nus editur, tremit, unde huiuÅ¿modi Å¿onus, appellari poteÅ¿t aggregatum aliquod ex <lb/>innumerabilibus Å¿onis. </s> - <s xml:space="preserve">eodem modo Å¿e habet Å¿onus, quem Ä™dunt campanÄ™, & omnia <lb/>inÅ¿trumenta tam naturalia, quà m artificialia, quæ quantò velocius <reg norm="tremunt" type="context">tremÅ©t</reg>, tanto acu <lb/><reg norm="tiorem" type="context">tiorẽ</reg> generant Å¿onum, & quantò tardius, tantò grauiorem. </s> - <s xml:space="preserve">Neque eÅ¿t quòd in ad-<lb/>mirationem ducamur, quòd Å¿enÅ¿ui unum aliquod continuum appareat id, quod di-<lb/>Å¿cretorum eÅ¿t multitudo ( non putet tamen aliquis me negare continuitatem Å¿ucceÅ¿ <lb/>Å¿iuam ipÅ¿ius temporis) quod clare cognoÅ¿ci poteÅ¿t à niue, aut à chryÅ¿tallo, aut à vi-<lb/>tro, aut à ſaccaro in minutiſſimas partes redacto, quæ continuam aliquam <reg norm="albedinem" type="context">albedinẽ</reg> <lb/>nobis ad inÅ¿piciendum offerunt, quod nihil aliud eÅ¿t, quà m innumerabilis quædam <lb/>multitudo minutorum reflexorum. </s> - <s xml:space="preserve"><reg norm="Idem" type="context">Idẽ</reg> dico de Å¿puto, & qualibet Å¿puma, & quan- - <pb o="183" rhead="DISPVTATIONES." n="195" file="0195" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0195"/> - to minutiora Å¿unt corpuÅ¿cula à quibus vt à ſpeculis reflectitur lumen, tantò magis ag <lb/><reg norm="gregatum" type="context">gregatÅ©</reg> illud <reg norm="album" type="context">albÅ©</reg> apparet. </s> - <s xml:space="preserve">Hæc <reg norm="autem" type="context">autẽ</reg> exempla <reg norm="cum" type="context">cÅ©</reg> Å¿int, nec non largo modo Å¿umpta, <lb/><reg norm="mirum" type="context">mirÅ©</reg> non erit Å¿i claudicare <reg norm="videbuntur" type="context simple">videbÅ©t̃</reg>. </s> - <s xml:space="preserve">Sed ut ad <reg norm="motum" type="context">motÅ©</reg>, & <reg norm="tempus" type="context">tẽpus</reg> reuertamur ( quæ Å¿unt <lb/><reg norm="continua" type="context">cõtinua</reg> Å¿ucceſſiua) AriÅ¿toteles in definiendo tempore, non reduxit in mentem, quod <lb/>Å¿cribit decimo metaphyÅ¿icÄ™ et .4. cap. Å¿ecundo. libr. de cÄ™lo, omnia videlicet, ab eo, <lb/>quod minimum eÅ¿t in Å¿uo genere, menÅ¿urari, & ex Å¿eipÅ¿o in phyÅ¿icorum libris, tem-<lb/>pus non eÅ¿t de genere motus; </s> - <s xml:space="preserve">ergo eius ipÅ¿ius rationum ui, tempus non erit menÅ¿ura <lb/>motus, Å¿ed motus quidem poteÅ¿t menÅ¿urare motum, videlicet velocior minus velo-<lb/>cem, & breuior longiorem; </s> - <s xml:space="preserve">& <reg norm="numerus" type="simple">numerê°</reg> <reg norm="menÅ¿uratur" type="simple">menÅ¿urat̃</reg> numero, & tempus tempore in quan <lb/>tum longum eÅ¿t, aut breue, non in quantum velox, aut tardum; </s> - <s xml:space="preserve">Nullum autem in-<lb/>conueniens Å¿equetur Å¿umendo tempus tam ptoportionale motui, quam locus cor-<lb/>pori, quia motus decem milliarium, quæ aliquis vnius horæ Å¿patio conficiat, erit pro <lb/>portionalis corpori denÅ¿o, & motus vnius milliaris eadem hora peracti, proportiona <lb/>lis erit corpori raro; </s> - <s xml:space="preserve">& quemadmodum corpus denÅ¿um occupat minus interuallum <lb/>loci, contra quam fiat in corpore raro: </s> - <s xml:space="preserve">Å¿ic etiam motus velox breuiori temporis Å¿pa-<lb/>tio peragetur, quam tardus.</s> - </p> - </div> - <div type="section" level="3" n="23"> - <head style="it" xml:space="preserve">Motum rectum eſſe continuum, uel dißentiente <lb/>AriÅ¿totele.</head> - <head xml:space="preserve">CAP. XXIII.</head> - <p> - <s xml:space="preserve">ARiÅ¿toteles .8. capi .8. phyÅ¿icorum ait impoſſibile eſſe aliquid per <reg norm="lineam" type="context">lineã</reg> rectam <lb/>nunc vno modo, nunc altero, ideÅ¿t eundo, & redeundo per dictam lineam in <lb/>extremis abÅ¿que quiete moueri. </s> - <s xml:space="preserve">Id quod contrà poſſibile eſſe dico. </s> - <s xml:space="preserve">Pro Å¿pecula-<lb/>tione cuius rei imaginemur circulum <var>.u.a.n.</var> motu continuo circa centrum <var>.o.</var> in <reg norm="quam" type="context">quã</reg> <lb/>libet partem, aut <reg norm="dextram" type="context">dextrã</reg>, aut <reg norm="Å¿iniÅ¿tram" type="context">Å¿iniÅ¿trã</reg> ferri; </s> - <s xml:space="preserve">& imaginemur <reg norm="punctum" type="context">pÅ©ctum</reg> <var>.b.</var> extra ipÅ¿um, ubi <lb/>magis nobis videbitur, à quo ducantur duæ lineæ recte <var>.b.u.</var> et <var>.b.n.</var> contiguæ ipÅ¿i cir-<lb/>culo in punctis <var>.u.</var> et <var>.n</var>. </s> - <s xml:space="preserve">Imaginatione quoque inter has duas lineas, alteram quæ Å¿it <var>.<lb/>u.n.</var> aut <var>.c.d.</var> aut <var>.e.f.</var> aut <var>.g.h.</var> conÅ¿tituamus in quali <lb/> - <anchor type="figure" xlink:label="fig-0195-01a" xlink:href="fig-0195-01"/> - bet parte, Å¿umemus etiam punctum <var>.a.</var> circun-<lb/>ferentiæ dicti circuli, à quo vÅ¿que ad <var>.b.</var> lineam <var>.<lb/>b.a.</var> imaginemur <reg norm="fixam" type="context">fixã</reg> in <var>.b.</var> Å¿ed quod remanat mo <lb/>bile, Å¿ecundum quod mouebitur punctum <var>.a.</var> vn-<lb/>de <reg norm="aliquando" type="context">aliquãdo</reg> hæc linea erit eadem cum <var>.b.u.</var> & ali <lb/>quando cum <var>.b.n.</var> & aliquando ab <var>.b.u.</var> verÅ¿us <var>.b.<lb/>n.</var> proficiÅ¿cetur, & aliquando ab <var>.b.n.</var> verÅ¿us <var>.b.u.</var> <lb/>vt accidit lineæ directionis, & retrogradationis <lb/>planetarum, vnde circulus <var>.u.a.n.</var> erit vt epiciclus <lb/>et <var>.b.</var> vt terræ centrum. </s> - <s xml:space="preserve">Clarum nunc erit, quòd <lb/>quando linea <var>.b.a.</var> eadem erit cum <var>.b.u.</var> aut cum <lb/><var>b.n.</var> non quieÅ¿cet, quia in inÅ¿tanti reuertetur, quia <lb/><var>b.u.</var> et <var>.b.n.</var> in puncto, <reg norm="dictum" type="context">dictÅ©</reg> circulum tangunt, & <lb/>dicta <var>.b.a.</var> interÅ¿ecabit Å¿emper aliquam ex dictis <lb/><var>u.n.</var> aut <var>.c.d.</var> aut <var>.e.f.</var> aut <var>.g.h.</var> quod interÅ¿ectionis <lb/>punctum Å¿it <var>.t</var>. </s> - <s xml:space="preserve">Imaginemur nunc quod <reg norm="Å¿ecundum" type="context context">Å¿ecÅ©dÅ©</reg> <lb/>punctum <var>.t.</var> aliquid per aliquam ex dictis lineis - <pb o="184" rhead="IO. BABPT. BENED." n="196" file="0196" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0196"/> - moueatur, clarum erit quod tale aliquid, nunquam quieÅ¿cet, etiam Å¿i Å¿it in quouis ex <lb/>tremo. </s> - <s xml:space="preserve">AriÅ¿totelis igitur opinio, tuta non eÅ¿t.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0195-01" xlink:href="fig-0195-01a"> - <image file="0195-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0195-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="24"> - <head style="it" xml:space="preserve">Idem uir grauisÅ¿imus an bene Å¿enÅ¿erit de motibus corporum <lb/>uiolentis & natur alibus.</head> - <head xml:space="preserve">CAP. XXIIII.</head> - <p> - <s xml:space="preserve">ARiÅ¿toteles in fine .8. phyÅ¿icorum Å¿entit corpus per vim motum, & Å¿eparatum à <lb/>primo mouente, moueri, aut motum eſſe per aliquod tempus ab aere, aut ab <lb/>aqua, quæ ipÅ¿um <reg norm="Å¿equuntur" type="context">Å¿equÅ©tur</reg>. quod fieri non poteÅ¿t; </s> - <s xml:space="preserve">quia imo aer, qui in locum defer-<lb/>tum à corpore Å¿ubintrat ad fugandum vacuum, non Å¿olum hoc corpus non impellit, <lb/>Å¿ed potius id cohibet à motu, quia aer per vim à corpore ducitur retrò, & diuiÅ¿us à <lb/>parte anteriori à dicto corpore, reÅ¿iÅ¿tit Å¿imiliter, & quantum dictus aer in dicta parte <lb/>condenÅ¿atur, tantum in poÅ¿teriori rarefit, vnde per vim Å¿eÅ¿e rarefaciens non permit-<lb/>tit, vt dictum corpus cum ea velocitate fugiat, cum qua aufugeret, quia omne agens <lb/>in agendo patitur. </s> - <s xml:space="preserve">Quamobrem cum aer à dicto corpore rapiatur, corpus quoque <lb/>ipÅ¿um ab aere rapitur. </s> - <s xml:space="preserve">HuiuÅ¿modi autem rarefactio aeris, naturalis non eÅ¿t, Å¿ed vio <lb/>lenta; </s> - <s xml:space="preserve">& hanc ob cauÅ¿am reÅ¿iÅ¿tit, & ad Å¿e trahit, Å¿ed non Å¿ufferente natura, vt inter <reg norm="vnum" type="context">vnÅ©</reg> <lb/>& aliud ex dictis corporibus reperiatur vacuum; </s> - <s xml:space="preserve">iccirco funt hæc Å¿emper contigua, <lb/>& mobile corpus aerem deÅ¿erere cum nequeat, eius velocitas impeditur. </s> - <s xml:space="preserve">HuiuÅ¿mo <lb/>di igitur corporis Å¿eparatim à primo mouente velo citas oritur à quadam naturali im <lb/>pręſſione, ex impetuofitate recepta à dicto mobili, quæ imprÄ™sÅ¿io & impetuoÅ¿itas, <lb/>in motibus rectis naturalibus continuò creÅ¿cit, cum perpetuò inÅ¿e cauÅ¿am <reg norm="mouentem" type="context context">mouẽtẽ</reg>, <lb/>ideÅ¿t propenÅ¿ionem eundi ad locum ei à natura asÅ¿ignatum habeat. </s> - <s xml:space="preserve">AriÅ¿to .8. cap. <lb/>primi lib. de cÅ“lo, dicere non deberet <reg norm="qui" type="simple">ê—</reg> quantò propius accedit corpus ad <reg norm="terminum" type="context">terminÅ©</reg> <lb/>ad quem, tantò magis Å¿it velox; </s> - <s xml:space="preserve">Å¿ed potius, <reg norm="qui" type="simple">ê—</reg> quantò longius diÅ¿tat à termino à quò <lb/>tantò velocius exiÅ¿tit. </s> - <s xml:space="preserve">quia tantò maior fit femper impræsfio, quantò magis moue-<lb/>tur naturaliter corpus, & continuò nouum impetum recipit, cum in fe motus caufam <lb/>contineat, quæ eÅ¿t inclinatio ad locum Å¿uum eundi, extra quem per vim confiftit. <lb/></s> - <s xml:space="preserve">Neque etiam rectè Å¿eripÅ¿it AriÅ¿to .9. cap. lib. 8. phyficorum et .2. lib. primi de cÅ“lo <lb/>eſſe aliquem motum ex recto & circulari mixtum, <reg norm="qui" type="simple">ê—</reg> omninò imposÅ¿ibile eÅ¿t.</s> - </p> - </div> - <div type="section" level="3" n="25"> - <head style="it" xml:space="preserve">Motum rectum & natur alem non eſſe primo & per Å¿e <lb/>quicquid AriÅ¿toteli uiÅ¿um Å¿it.</head> - <head xml:space="preserve">CAP. XXV.</head> - <p> - <s xml:space="preserve">MOtus rectus corporum naturalium Å¿urÅ¿um, aut dcorfum, non eft naturalis pri <lb/>mò & per Å¿e, quia motus naturalis perpetuus eÅ¿t, aut vt melius dicam, inceÅ¿-<lb/>fabilis, & alius eſſe non poteft quà m circularis, <reg norm="nullaque" type="simple">nullaqÌ;</reg> pars cum fuo toto coniun-<lb/>cta, alium motum naturalem habere poteft, quà m eum, qui eft totius. </s> - <s xml:space="preserve">fi autem à ſuo <lb/>toto diuulfa atque difiuncta fit, <reg norm="libereque" type="simple">libereqÌ;</reg> vagetur, Å¿pontè, & quà m breuisÅ¿ima poteft <lb/>via, ad locum, <reg norm="Å¿uitotius" type="simple">Å¿uitotiê°</reg> à natura Å¿tatutum proficiÅ¿citur. </s> - <s xml:space="preserve">hic motus primò, & per fe di-<lb/>cti corporis, naturalis non eft, cum à caufa naturæ fuæ contraria fit generatus, ideft, - <pb o="185" rhead="DISPVTATIONES." n="197" file="0197" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0197"/> - ab co quod fit extra Å¿uum locum, vbi contra naturam Å¿uam reperitur. </s> - <s xml:space="preserve">Vnde hu-<lb/>iuÅ¿modi motus, partim & non omninò, naturalis eft. </s> - <s xml:space="preserve">Is autem proprius eÅ¿t & natura <lb/>lis motus, qui dicti corporis eſſentiam conÅ¿eruat. </s> - <s xml:space="preserve">hoc autem non præſtat hic rectus, <lb/>cum deÅ¿truat, ergò hic motus primò & per Å¿e naturalis non eft.</s> - </p> - </div> - <div type="section" level="3" n="26"> - <head style="it" xml:space="preserve">Omne corpus eſſe in loco proprio graue, ut Aristoteli placuit, <lb/>non eft admittendum.</head> - <head xml:space="preserve">CAP. XXVI.</head> - <p> - <s xml:space="preserve">ARift .4. cap. lib. 4. de cÄ™lo fic Å¿cribit.</s> - </p> - <p> - <s xml:space="preserve">Suo enim in loco grauitatem habent omnia præter ignem, fignum cuius eft <lb/>vtrem inflatum plus ponderis, quam vacuum habere, & c.</s> - </p> - <p> - <s xml:space="preserve">Quo in loco, manifeftè indicat Å¿e caufam nec grauitatis, nec leuitatis corporum <lb/>naturalium nofce, quæ eft denfitas auto raritas corporis grauis, aut leuis, maior denÅ¿i-<lb/>tate, aut raritate medij permeabilis, in quo reperitur.</s> - </p> - <p> - <s xml:space="preserve">Exemplum <reg norm="qui" type="simple">ê—</reg> ipÅ¿e de vtre inflato proponit, debuiſſet Å¿altem ei oculos ad verita-<lb/>tem, quæ clarisÅ¿imè fulget, inÅ¿piciendum aperire. </s> - <s xml:space="preserve">VerisÅ¿imum eÅ¿t, vtrem inflatum <lb/>plus ponderis habere quà m vacuum, aut quando aer in eo non eft per vim inclufus.</s> - </p> - <p> - <s xml:space="preserve">Ratio autem huius rei eft, quia quando inflatus eft, ea quantitas aeris, in eum <lb/>per vim iniecti, minorem occupat locum, quà m Å¿i eidem liberè vagari permit-<lb/>teretur, vnde violenter, quodam modo, con denfata eft, & quia corpus denfum in <lb/>minus denfo, femper deÅ¿cendit, & minus denÅ¿um in magis denÅ¿o aÅ¿cendit. </s> - <s xml:space="preserve">Hanc ob <lb/>caufam vter inflatus plenus corpore magis denÅ¿o, quà m eft medium quod eum cir-<lb/>cundat, deÅ¿cendit, non quia aer inaere, aut aqua in aqua fit grauis.</s> - </p> - </div> - <div type="section" level="3" n="27"> - <head style="it" xml:space="preserve">Haud admittendam opinionem Principis Peripateticorum <lb/>de circulo, & Å¿pbæra.</head> - <head xml:space="preserve">CAP. XXVII.</head> - <p> - <s xml:space="preserve">CVm Ariftoteles fenÅ¿erit circulum eſſe figurarum Å¿uperficialium <reg norm="primam" type="context">primã</reg>, & <reg norm="Å¿phae ram" type="simple context">Å¿phÄ™Â<lb/>rã</reg> eſſe <reg norm="primam" type="context">primã</reg> <reg norm="corporearum" type="context">corporearÅ©</reg> <reg norm="prope" type="wordlist">ê‘ê‘</reg> <reg norm="earum" type="context">earÅ©</reg> periferias, decipitur. </s> - <s xml:space="preserve">Sunt enim vltimæ, <lb/>non primæ. </s> - <s xml:space="preserve">Sunt quidem (in quò rectè Å¿entit) perfectè, licet rationem huius rei non <lb/>nouerit. </s> - <s xml:space="preserve">Nam <reg norm="centrum" type="context">centrÅ©</reg> cuiuÅ¿libet rei, eiuÅ¿dem rei <reg norm="principium" type="context">principiÅ©</reg> eft, & eÄ™ figurÄ™, quæ ipÅ¿um <lb/>æqualiter circundant, poſſunt appellari perÅ¿ectæ, Å¿iue Å¿int Å¿uperficiales, Å¿iue corpo-<lb/>reæ, & ècontrà illæ, quæ contrario modo Å¿e habent, imperfectæ. </s> - <s xml:space="preserve">Quòd autem per-<lb/>Å¿ectum eÅ¿t, licet natura fit primum, eÅ¿t tamen vltimum generatione. </s> - <s xml:space="preserve">Sed quando <lb/>Ariftoteles duas dictas figuras pronuntiauit primas, vt perfectas, prioritate Å¿cilicet <lb/>ea, quæ oritur à perfectione, verum dixit; </s> - <s xml:space="preserve">fed quando de figuris Å¿uperficialibus lo-<lb/>quens, vult circulum effe primum, quia ab vna <reg norm="tantum" type="context">tãtum</reg> linea terminetur; </s> - <s xml:space="preserve"><reg norm="non" type="context">nõ</reg> minus pro <lb/>circulo, quam pro oxigonia Å¿eu elipÅ¿i, aut cucurbitali, aut aliis multis figuris ab vna <lb/>tantum linea terminatis concludit. </s> - <s xml:space="preserve">Neque etiam hæc ratio perfectionem circuli <reg norm="mon" type="context">mõ</reg> <lb/>Å¿trat, quia aliæ figuræ, à lineis curuis terminatÄ™, eandem conditionem fortiuntur. <lb/></s> - <s xml:space="preserve">Circulus Å¿phÄ™<reg norm="raque" type="simple">raqÌ;</reg>, non ex vno Å¿olo angulo recto conÅ¿tant, vt idem Ariftoteles putat - <pb o="186" rhead="IO. BAPT. BENED." n="198" file="0198" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0198"/> - cap .4. lib. 4. de cÄ™lo, etiam fi triangulus ex duobus angulis rectis conÅ¿urgat, Å¿ed Å¿unt <lb/>figurÄ™ infinitorum angulorum rectorum, & hanc ob cauÅ¿am à me dicuntur vltimæ & <lb/>perfectÄ™, quia infinito nihil addi poteÅ¿t. </s> - <s xml:space="preserve">Numerus angulorum rectorum circuli, eft <lb/>minor duplo infinito per duo infinita angulorum contingentiæ, quæ duo infinita mi <lb/>nora funt quouis angulo acuto rectilineo, & numerus angulorum rectorum <reg norm="folidorum" type="context">folidorÅ©</reg> <lb/>Å¿phÄ™ræ, minor eft quadruplo infinito per .4. infinita angulorum Å¿olidorum <reg norm="contingen- tiæ" type="context">cõtingen-<lb/>tiæ</reg>, quæ .4. infinita, minora Å¿unt quouis angulo Å¿olido acuto terminato à tribus pla-<lb/>nis. </s> - <s xml:space="preserve">Triangulus inter figuras planas Å¿uperÅ¿iciales eft primus, & circulus vltimus; </s> - <s xml:space="preserve">& <lb/>pyramis quadrilatera, inter corpora eft prima, & Å¿phÄ™ra vltima.</s> - </p> - </div> - <div type="section" level="3" n="28"> - <head style="it" xml:space="preserve">Occultam fuiße grauisÅ¿imo Stagirit & canÅ¿am Å¿cintilla-<lb/>tionis Å¿tellarum.</head> - <head xml:space="preserve">CAP. XXVIII.</head> - <p> - <s xml:space="preserve">VBi AriÅ¿toteles ait Å¿cintillationem Å¿tellarum Å¿ieriratione aÅ¿pectus @oÅ¿tri ob, ma <lb/>ximam diÅ¿tantiam, maximum errorem committit, vt etiam facid quum putat <lb/>vifionem fieri extramittendo, contra id, quod alio loco, immo contra veritatem ip <lb/>Å¿am afferuit. </s> - <s xml:space="preserve">Scintillatio ergo Å¿tellarum, neque aÅ¿pectus noÅ¿tri ratione, neque ali-<lb/>cuius mutationis earundem Å¿tellarum, Å¿ed ab inæqualitate motus corporum diapha <lb/>norum mediorum naÅ¿citur, <reg norm="quemadmodum" type="wordlist">quẽadmodum</reg> clarè cernitur, quòd fi inter aliquod obie <lb/>ctum, & nos, aliquis Å¿umus, qui aÅ¿cendat, intercefferit, videbimus obiectum illud qua <lb/>Å¿i tremere. </s> - <s xml:space="preserve">Hoc autem tantò magis fiet, quantò magis diÅ¿tabit obiectum ab ipÅ¿o fu <lb/>mo; </s> - <s xml:space="preserve">vnde admirationi locus non erit, fi ftellas fixas magis Å¿cintillare, quam errantes <lb/>cernamus. </s> - <s xml:space="preserve">Lumen Å¿tellæ ad oculum noÅ¿trum accedens, perpetuò per diuerfas dia-<lb/>phaneitates penetrat, medio continuorum motuum corporum mediorum, vnde <lb/>continuò eorum lumen variatur, & hoc in <reg norm="longinquis" type="context">lõginquis</reg> magis, quà m in propinquis Å¿tel <lb/>lis apparet, quemadmodum ab exemplo de fumo allato, & etiam ab aliquibus vi-<lb/>tris ex Å¿uperficie non plana, Å¿ed irregulari conÅ¿tantibus, quilibet cognoÅ¿cere poteft.</s> - </p> - </div> - <div type="section" level="3" n="29"> - <head style="it" xml:space="preserve">Daricontinuum infinitum motum Å¿uper rectam at que <lb/>finitam lineam.</head> - <head xml:space="preserve">CAP. XXIX.</head> - <p> - <s xml:space="preserve">OMnes hactenus Å¿enÅ¿erunt imposfibile eſſe dari per <reg norm="imaginationem" type="context">imaginationẽ</reg> motum con-<lb/>tinuum & <reg norm="perpetuum" type="context">perpetuÅ©</reg> <lb/> - <anchor type="figure" xlink:label="fig-0198-01a" xlink:href="fig-0198-01"/> - Å¿uper vnam lineam rectam <lb/>finit: </s> - <s xml:space="preserve">in quo <reg norm="tantum" type="wordlist/context">tñ</reg> decipiuntur. <lb/></s> - <s xml:space="preserve">Imaginemur <reg norm="ion" type="context">iõ</reg> duas lineas <lb/>parallelas <var>.a.b.</var> et <var>.t.x.</var> <reg norm="quarum" type="context">quarÅ©</reg> <lb/><var>b.a.</var> fit <reg norm="infinita" type="context">Ä©finita</reg> à qualibet par <lb/>te, & in ea imaginemur pun <lb/>ctum <var>.a.</var> moueri continuò ad <lb/>quam voluerimus partem, <lb/>& - <pb o="187" rhead="DISPVTATIONES." n="199" file="0199" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0199"/> - & in linea <var>.t.x.</var> imaginemur punctum fixu@, quod fit <var>.c.</var> imaginemur etiam inter <var>.c.</var> @. <lb/>a. vnam lineam rectam <var>.c.a.</var> & inter duas parallelas dictas <var>.r.x.</var> fixam, & motus punct@i <lb/>fit ab <var>.b.</var> verfus <var>.a.</var> ita ut <var>.c.a.</var> fecet <var>.r.x.</var> in puncto <var>.i.</var> quod interfectionis punctum mo-<lb/>uebitur ab <var>.r.</var> verfus <var>.x.</var> continuò, in tempore infinito, neque vnquam idem erit cum <lb/>puncto <var>.x</var>.</s> - </p> - <div type="float" level="4" n="1"> - <figure xlink:label="fig-0198-01" xlink:href="fig-0198-01a"> - <image file="0198-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0198-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="30"> - <head style="it" xml:space="preserve">Non eſſe Å¿olis calorem à motu localι ipÅ¿ius corporis Å¿olaris, <lb/>ut AriÅ¿toteli placuit.</head> - <head xml:space="preserve">CAP. XXX.</head> - <p> - <s xml:space="preserve">JD nullo planè modo eÅ¿t admittendum quod Ariftoteles credidit calorem folis à <lb/>motu locali ipÅ¿iuÅ¿met corporis folaris, & non à lumine, prouenire, quemadmo-<lb/>dum manifeftè aſſerit primo metheororum cap .3. circa finem fic fcribens.</s> - </p> - <p> - <s xml:space="preserve">Vtigitur repor gignatur atque calor, folis latio duntaxat, Å¿atis eÅ¿t eſſicere, & c. Å¿ed <lb/>cap .7. lib. 2. de cælo fic Å¿eribit, Caliditas autem ab ipÅ¿is, <reg norm="lumenque" type="simple">lumenqÌ;</reg> ideo fit, quia aer <lb/>ab illorum motione fricatur.</s> - </p> - <p> - <s xml:space="preserve">Vbi non folum oftendit fe opinari, quòd motus corporum cÅ“leÅ¿tium fit caufa ca <lb/>loris, Å¿ed eriam luminis, paulò autem poÅ¿t dicit, Å¿uperiorum autem corporum vnum <lb/>quodque fertur in Å¿phæra, vt ipÅ¿a quidem non igniantur. </s> - <s xml:space="preserve">Opinio profecto abfur-<lb/>da. </s> - <s xml:space="preserve">Nam cùm corpus Å¿olate fixum fit in Å¿pisfitudine Å¿ui orbis deferentis, fe-<lb/>cundum communem opinionem, non mouetur per fe, Å¿ed accidentaliter, cum Å¿ei-<lb/>licet fertur à dicto Å¿uo orbe, vnde fieri poteÅ¿t, vt in motu fui orbis, nullum ex <lb/>orbibus fuorum deferentium augis fricet, fed fi fricaret, id faceret mediante vno fo <lb/>lo puncto, vt cuilibet, aliquantulum in mathematicis verfato patet. </s> - <s xml:space="preserve">Quam ob cau <lb/>Å¿am, rationi <reg norm="confentaneum" type="context">cõfentaneum</reg> non eſſet credere, quòd tantùm caloris gigneretur. </s> - <s xml:space="preserve">Quod <lb/>tamen fi posÅ¿ibile eſſet, quid ergo fricatio Å¿uperficierum orbis Å¿ui, cum iis, quæ funt <lb/>deferentium augis efficeret? </s> - <s xml:space="preserve"><reg norm="Quando" type="context">Quãdo</reg> tamen hoc fieret, vt Å¿cilicet à fricatione fuper <lb/>ficierum procederet calor, nil planè diferiminis inter hyemen, & æftatem intercede <lb/>ret, nec inter calorem diei, & noctis, nec inter unam horam, aut alteram; </s> - <s xml:space="preserve">fed fecun-<lb/>dum Ariftotelis rationes, Venus, <reg norm="Mercuriusque" type="simple">MercuriusqÌ;</reg>, magis calefacere quam fol de <reg norm="berent" type="context">berẽt</reg>, <lb/>cum ita Å¿int veloces vt ipÅ¿e Sol, & eodem magis propinqua terræ. </s> - <s xml:space="preserve">Verum Ari-<lb/>Å¿totelis <reg norm="temporibus" type="context">tẽporibus</reg>, <reg norm="nullum" type="context">nullÅ©</reg> <reg norm="alium" type="context">aliÅ©</reg> <reg norm="planetam" type="context">planetã</reg> quam folem <reg norm="putabant" type="context">putabãt</reg> philofophi Å¿upra <reg norm="Lunam" type="context">Lunã</reg> eÅ¿-<lb/>Å¿e. </s> - <s xml:space="preserve">Atque etiam <reg norm="contigeret" type="context">cõtigeret</reg> menfe Decembri, quam Iunio, magis inualeÅ¿ceret calor, <lb/>cum huiuÅ¿modi menÅ¿e Å¿olad nos propius accedat, quam menfe Iunii. per differen-<lb/>tiam maiorem diametro regionis elementaris, (nam folaris eccentricitas maior eft <lb/>Å¿emidiametro <reg norm="elementaris" type="context">elemẽtaris</reg> regionis) non confiderans Ariftoteles differentiam ca-<lb/>loris, quæ naÅ¿citur ex eo, <reg norm="qui" type="simple">ê—</reg> Sol aut altius Å¿upra orizontem excurrat, aut infra <reg norm="eundem" type="context">eundẽ</reg> <lb/>feratur; </s> - <s xml:space="preserve">neque eam, quÄ™ à longitudine, aut breuitate diei proficiÅ¿citur. </s> - <s xml:space="preserve">Sed quia Ari <lb/>Å¿toteles eodem cap tertio Metheororum intelligit de motu rapto, ideÅ¿t diurno, Å¿iue <lb/>dicamus vniuerfali, hinc Å¿equi deberet, <reg norm="qui" type="simple">ê—</reg> Sol maiorem caloris uim menÅ¿e Martij & <lb/>Septembris, quà m aliis menfibus, profunderet, quia in iiÅ¿dem temporibus, Å¿ol virtu <lb/>te huiuÅ¿modi motus velocior exiftat, quà m alio quolibet tempore anni, cum tunc <lb/>per æquatorem circuÅ©oluatur. </s> - <s xml:space="preserve">Multa quoque alia incommoda Å¿equerentur Å¿i Ari <lb/>Å¿torelis rationes admitteremus. </s> - <s xml:space="preserve">Sed clarè uidemus, mediante refl exione aut refra-<lb/>ctione radiorum folarium, <reg norm="qui" type="simple">ê—</reg> vniente Å¿eÅ¿e lumine, unitur quoque, & augetur calor, <lb/>atque omnis res ad comburendum apta accenditur, & inflammatur. </s> - <s xml:space="preserve">In lumine igi- - <pb o="188" rhead="IO. BAPT BENED." n="200" file="0200" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0200"/> - tur continetur calor, & non in motu ipÅ¿ius Å¿olis, & ita in lumine Å¿edem habet, ut fi Å¿ol <lb/>quieÅ¿ceret, neque in orbe Å¿uo circumager etur, infÄ™liciſſima eſſet ea regio, in cuiu; <lb/></s> - <s xml:space="preserve">Zenith ipÅ¿e reperiretur.</s> - </p> - </div> - <div type="section" level="3" n="31"> - <head style="it" xml:space="preserve">Vnde caloris Å¿olis prode at incrementum & state, et byeme <lb/>decrementum.</head> - <head xml:space="preserve">CAP. XXXI.</head> - <p> - <s xml:space="preserve">CVm capite Å¿uperiore oÅ¿tenderim calorem Å¿olis non aliunde, quà m à lumine <lb/>prouenire, oÅ¿tendam nunc ex ordine, ex quot, <reg norm="quibusque" type="simple">quibusqÌ;</reg> cauÅ¿is oriatur magna <lb/>differentia eius caloris æſtatis ad hyemem, quarum nonnullæ ab antiquis obÅ¿erua-<lb/>tæ fuerunt, aliæ autem à nemine, quod Å¿ciam. </s> - <s xml:space="preserve">Sunt autem quinque ad minus eæ cau <lb/>ſæ, quarum vna eft diuturna folis mora, tempore æſtatis Å¿upra orizontem, quæ cau-<lb/>Å¿a ab antiquis pofita, & citata fuit. </s> - <s xml:space="preserve">Aliam quoque huius rei cauÅ¿am iidem antiqui <lb/>dicebant eſſe propinquitatem Å¿olis noftro Zenith, Å¿ed hæc cauÅ¿a immediata non <lb/>eÅ¿t, quia ab ea tres caufæ immediatæ naÅ¿cuntur; </s> - <s xml:space="preserve">quarum vna eft maior unio radij re <lb/>flexi cum radio incidenti; </s> - <s xml:space="preserve">Å¿ecunda maior quantitas luminis in Å¿uperficie terrÄ™; <lb/></s> - <s xml:space="preserve">tertia, minor <reg norm="reÅ¿iÅ¿tentia" type="context">reÅ¿iÅ¿tẽtia</reg> vaporum ipÅ¿i radio luminoÅ¿o facta; </s> - <s xml:space="preserve">quarta verò eft impresfio <lb/>caloris facta in terra, quæ cum aliis caufis coniuncta calorem adauget. quæ quidem <lb/>caufæ nemini adhuc, quod fciam, in <reg norm="mentem" type="context">mentẽ</reg> venerunt. </s> - <s xml:space="preserve">Quòd autem attinet ad ma-<lb/>iorem coniunctionem radii reÅ¿lexi cum incidente, quiÅ¿que, uel Å¿altem mediocriter <lb/>in cathoptricÄ™ cognitione verÅ¿atus hoc verum eſſe cognoÅ¿cet. </s> - <s xml:space="preserve">Vt hoc tamen in-<lb/>noteÅ¿cat facilius. </s> - <s xml:space="preserve">Imaginemur <var>.q.p.</var> et <var>.b.d.</var> eſſe duas particulas Ä™quales Å¿uperficiei <lb/>ipfius terræ, Å¿uper quas cadant duo radii luminofi Å¿olis <var>.e.q.</var> et <var>.n.d.</var> quorum <var>.e.q.</var> fit ad <lb/>modum obliquus, et <var>.n.d.</var> quaÅ¿i perpendicularis, vnde radii reÅ¿lexi <var>.p.a.</var> et <var>.b.u.</var> aÅ¿cen <lb/>dent cum angulis æqualibus eis, qui funt Å¿uorum cadentium, cum omnis angulus re-<lb/>flexionis femper æqualis Å¿it angulo Å¿uæ incidentiÄ™, vt cuilibet in cathoptrica, vel me <lb/>diocriter verfato pater. </s> - <s xml:space="preserve">Mixtio autem primorum obliquorum erit <var>.q.o.p.</var> & ea, quæ <lb/>eft minus obliquorum <var>.b.i.d.</var> quorum duorum triangulorum nullus unquam erit, qui <lb/>dubitare posfit <var>.q.o.p.</var> non eſſe minorem <var>.b.i.d.</var> cum anguli <var>.q.</var> et <var>.p.</var> trianguli <var>.q.o.p.</var> a-<lb/>cutiores Å¿int angulis <var>.b.</var> et <var>.d.</var> trianguli <var>.b.i.d.</var> ex Å¿uppoÅ¿ito. </s> - <s xml:space="preserve">Quòd uero attinet ad ma <lb/>iorem quantitatem luminis Å¿uper terræ Å¿uperÅ¿iciem; </s> - <s xml:space="preserve">Imaginemur radium <var>.a.q.</var> cuius <lb/>reÅ¿pectu etiam imagine mur duos Å¿uperficiei terræ Å¿itus, quorum vnus fit <var>.q.o</var>: cui di-<lb/>ctus radius fit perpendicularis, & alter <var>.q.p.</var> cui radius <var>.a.q.</var> ex obliquo incidat. </s> - <s xml:space="preserve">Ima-<lb/>ginemur ergo triangulum <var>.q.o.p.</var> cuius angulus <var>.o.</var> rectus eÅ¿t ex Å¿uppoÅ¿ito, unde <var>.q.o.</var> <lb/>minor erit <var>.q.p.</var> ex .18. primi Euclidis. </s> - <s xml:space="preserve">hinc fit, vt Å¿uper <var>.q.o.</var> cadat vniuerÅ¿um lumen, <lb/>quod Å¿uper <var>.q.p.</var> diffunditur. </s> - <s xml:space="preserve">Sit <var>.q.u.</var> æqualis <var>.q.o.</var> & fit imaginatione protracta <var>.u.n.</var> <lb/>æquidiftans <var>.p.o.a.</var> vnde <var>.q.u.</var> illuminata erit à radio <var>.n.q.</var> minore radio <var>.a.q.</var> ergo mi-<lb/>nus calida erit Å¿uperficies <var>.q.u.</var> ipÅ¿ius terræ, quà m.q.o. quia maius lumen in Å¿e maio-<lb/>rem calorem includit: </s> - <s xml:space="preserve">quod manifeÅ¿tè apparet in radiorum vnione mediante refle-<lb/>xione, aut refractione. </s> - <s xml:space="preserve">Sed quod attinet ad minorem refiÅ¿tentiam vaporum ad ip-<lb/>Å¿um radium luminoÅ¿um, etfi primo capite meæ Gnomonicæ leuiter id attigerim, ni <lb/>hilominus tamen, & idem ipÅ¿um hoc loco proponam. </s> - <s xml:space="preserve">Denotetur, exempli gratia, <lb/>Å¿uperficies terræ ab <var>.o.g.</var> et ea, quæ eft vaporum ab <var>.n.a.</var> Å¿upponatur etiam Å¿ol in fitu.</s> - <pb o="189" rhead="DISPVTATIONES." n="201" file="0201" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0201"/> - <s xml:space="preserve">q. qui Å¿it Zenith <reg norm="puncti" type="context">pÅ©cti</reg> <var>.o.</var> & <reg norm="etiam" type="context">etiã</reg> in <var>.p.</var> ipÅ¿i <reg norm="orizonti" type="context">orizõti</reg> propinquus, aut extra Zenith, cuius <lb/>duos radios <var>.q.o.</var> et <var>.p.o</var>. </s> - <s xml:space="preserve">Imaginemur, quorum duæ partes <var>.a.o.</var> et <var>.n.o.</var> erunt aliquo <lb/>modo ab ipÅ¿is vaporibus offuÅ¿catæ, Å¿ed <var>.o.n.</var> breuior eÅ¿t <var>.o.a.</var> ex .7. lib. 3. Eucli. </s> - <s xml:space="preserve">mino-<lb/>rem ergo reÅ¿iÅ¿tentiam habebit à vaporibus Å¿ol in Zenith, quà m extra cundem com -<lb/>morans, & quantò longius erit idem ab ipÅ¿o Zenith, tanto maiorem reÅ¿iÅ¿tentiam à <lb/>dictis vaporibus inferri ex eadem .7. lib. 3. Eucli. dicemus.</s> - </p> - <figure position="here"> - <image file="0201-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0201-01"/> - </figure> - <figure position="here"> - <image file="0201-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0201-02"/> - </figure> - </div> - <div type="section" level="3" n="32"> - <head style="it" xml:space="preserve">Nullum corpus Å¿enÅ¿us expers à ſono offendi, præterquam <lb/>Aristoteles crediderit.</head> - <head xml:space="preserve">CAP. XXXII.</head> - <p> - <s xml:space="preserve">POſſe Å¿onum corpus aliquod, quod Å¿enÅ¿u Å¿it deÅ¿titutum, vt AriÅ¿toteles .9. cap. li-<lb/>br .2. de cælo putauit, offendere, eÅ¿t falÅ¿um.</s> - </p> - <p> - <s xml:space="preserve">Corpus enim non niÅ¿i à corpore poteÅ¿t lædi, non ergo à ſono, cum Å¿onus corpus <lb/>non Å¿it. </s> - <s xml:space="preserve">Sed aer, & ignis, cum è contra Å¿int corpora, hoc facilè præſtare poſſunt im-<lb/>plendo aliquem locum velociter ad excludendum vacuum; </s> - <s xml:space="preserve">vnde generatur Å¿onus. <lb/></s> - <s xml:space="preserve">Quod hucuÅ¿que à nemine animaduerÅ¿um fuiſſe comperio.</s> - </p> - <pb o="190" rhead="IO. BAPT. BENED." n="202" file="0202" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0202"/> - </div> - <div type="section" level="3" n="33"> - <head style="it" xml:space="preserve">Pytagoreorum opinionem de Å¿onitu corporum cælestium non <lb/>fuiſſe ab Aristotele Å¿ublatam.</head> - <head xml:space="preserve">CAP. XXXIII.</head> - <p> - <s xml:space="preserve">SEnÅ¿erunt Pythagorici orbes cæleÅ¿tes dum circunuoluuntur, non autem corpora <lb/>Å¿tellarum Å¿olum, æd ere Å¿onu. </s> - <s xml:space="preserve">Quibus dum AriÅ¿toteles contradicere cogitat, <lb/>maximè fauet. </s> - <s xml:space="preserve">Eatamen opinio è phyloÅ¿ophorum Å¿cholis eÅ¿t explodenda, quia aut <lb/>orbes Å¿unt Å¿ibi ipÅ¿is contigui, aut inuicem diÅ¿tantes: </s> - <s xml:space="preserve">Å¿i ab inuicem diÅ¿tant (quod <lb/>nemo adhuc conceſſit, quia hac ratione vacuum introduceretur) clarum eÅ¿t, quod <lb/>cum Å¿e minime tangant, Å¿onum edere nequeunt: </s> - <s xml:space="preserve">Si verò eorum vnus alteri Å¿it conti <lb/>guus, <reg norm="neque" type="simple">neq;</reg> etiam ab ipÅ¿is Å¿onus reÅ¿ultare poterit, quia pro certo putandum eÅ¿t, <reg norm="ipÅ¿orum" type="context">ipÅ¿orÅ©</reg> <lb/>Å¿uperficies tam politas eſſe, ac lenas, vt nihil omnino aÅ¿peritatis, aut inæqualitatis <lb/>contineant. </s> - <s xml:space="preserve">Vt exempli gratia, Å¿i aliquis duo Å¿pecula plana inuicem confricaret, nul <lb/>lum planè Å¿onum audiret, Å¿ed Å¿i hoc faceret cum duabus Å¿uperficiebus a Å¿peris, <reg norm="Å¿onum" type="context">Å¿onÅ©</reg> <lb/>perÅ¿entiret, & tanto maiorem; </s> - <s xml:space="preserve">quantò aÅ¿periores eſſent dictæ Å¿uperficies, & qui vult <lb/>vtarcus lirÄ™, ex corda Å¿onum eliciat, colophonia dictum arcum illinet, vt aÅ¿periorem <lb/>reddat. </s> - <s xml:space="preserve">Neceſſarium quoque eÅ¿t vt tremat Å¿iue trepidet corpus, quod <reg norm="Å¿onum" type="context">Å¿onÅ©</reg> edere <lb/>debet; </s> - <s xml:space="preserve">Neque etiam abÅ¿que aere Å¿onus efficipotelt, quia aer Å¿onat ingrediendo <lb/>velociter ad implendum locum, vt non remaneat vacuus. </s> - <s xml:space="preserve">Sed Å¿upponendo in æche <lb/>rea regione neque aerem eſſe, neque corpus aliquod fluidum, clarè patebit orbes <lb/>cÅ“leÅ¿tes ex Å¿eÅ¿e nullum emittere Å¿onum. </s> - <s xml:space="preserve">Idem affirmo de fricatione Å¿uperficiei con <lb/>cauæ infimi orbis lunaris cum conuexa materiæ à dicto orbe contentæ, Å¿uperioribus <lb/>rationibus fultus, vt etiam experientia à corpore aliquo fluido, quod in alio velociÅ¿ <lb/>Å¿imè moueretur deÅ¿umpta fretus, cuius corporis Å¿uperficies tamen lenis eſſet, à quo <lb/>Å¿onus non gigneretur. </s> - <s xml:space="preserve">Et non minus dicere poſſum, corpus fluidum moueri in con-<lb/>tinente loco immobili, quam dictum corpus continens illud eſſe, quod moueretur, <lb/>& non fluidum corpus. </s> - <s xml:space="preserve">Cuius rei poſſumus etiam exemplum habere à quouis <lb/>corpore perfectè rotundo, quod circa Å¿uum axem velociſſimè moueatur, nullum Å¿o-<lb/>num efficiet, quia nullam aeris partem extra Å¿uum <reg norm="locum" type="context">locÅ©</reg> impellit dum mouetur non <lb/>Å¿ecundum totum, Å¿ed Å¿ecundum Å¿uas partes, quarum quælibet abÅ¿que reÅ¿iÅ¿tentia im-<lb/>mediatè Å¿ubintrat locum alterius, abÅ¿que temporis interpoÅ¿itione. </s> - <s xml:space="preserve">nec huiuÅ¿modi <lb/>locum aliquo modo eadem materia dicti corporis, quod circunuoluitur: </s> - <s xml:space="preserve">deÅ¿titutum <lb/>dimittat. </s> - <s xml:space="preserve">Sed Å¿i Pythagorici de alia quadam harmoniæ Å¿pecie ab ea, quæ eÅ¿t Å¿ono-<lb/>rum, vt à diuerÅ¿is velocitatibus motuum, aut à diuerÅ¿is magnitudinibus aut diÅ¿tantiis, <lb/>aut Å¿tellarum influxibus intellexiſſent, rectè Å¿enÅ¿iſſent exparte, non autem omnino, <lb/>quia ea harmoniam efficere nequeunt, quæ ad <reg norm="inuicem" type="context">inuicẽ</reg> Å¿ecundum interualla harmoni-<lb/>ca proportionata non Å¿unt, vt Å¿unt dupla, Å¿eÅ¿quial tera, Å¿eÅ¿quitertia, Å¿eÅ¿quiquarta, Å¿eÅ¿-<lb/>quiquinta, Å¿upertripartientia quintas, <reg norm="Å¿uperbipartientia" type="context">Å¿uperbipartiẽtia</reg> tertias, & quÄ™ ab ijs <reg norm="dependent" type="context">dependẽt</reg> <lb/>ideÅ¿t coniuncta Å¿unt cum duplis; </s> - <s xml:space="preserve">de conÅ¿onantijs loquendo. de diſſonantiis idem di <lb/>co, quæ harmonicis inÅ¿eruiunt modulationibus, vt <reg norm="Å¿eÅ¿quioctauum" type="context">Å¿eÅ¿quioctauÅ©</reg>, <reg norm="Å¿eÅ¿quinonum" type="context">Å¿eÅ¿quinonÅ©</reg>, Å¿eÅ¿qui <lb/><reg norm="quintundecimum" type="context context">quintÅ©decimÅ©</reg>, <reg norm="Å¿equiuigeÅ¿imunquartum" type="context context">Å¿equiuigeÅ¿imÅ©quartÅ©</reg>, <reg norm="Å¿eÅ¿quioctuogeÅ¿imum" type="context">Å¿eÅ¿quioctuogeÅ¿imÅ©</reg>, & Å¿uperbipartiens vigeÅ¿i <lb/>masquintas. </s> - <s xml:space="preserve"><reg norm="Verum" type="context">VerÅ©</reg> quidem eÅ¿t nonnulla harmonica interualla in aÅ¿pectibus <reg norm="comperta" type="context">cõperta</reg> <lb/>fuiſſe, vt Prolomeus oÅ¿tendit, & alii quoque aſſerunt. </s> - <s xml:space="preserve">ineÅ¿t tamen huic rei nonnihil <lb/>difficultatis. </s> - <s xml:space="preserve">vt exempli gratia, Å¿i Å¿ubtrahamus diateſſaron extra diapaÅ¿on, remanet <lb/>diapente, & Å¿i à diapente Å¿ubtrahamus Å¿emiditonum, remanet ditonum (quæ duæ - <pb o="191" rhead="DISPVTATIONES." n="203" file="0203" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0203"/> - conÅ¿onantiæ, eum habent reÅ¿pectum ad inuicem, quem habent diateſſaron, & dia <lb/>pente, quia quemadmodum Å¿emiditonum, & ditonum Å¿imul coniuncta, compo <lb/>nunt diapente, Å¿ic diateſſaron, & diapente Å¿imul vnita componunt diapaÅ¿on; </s> - <s xml:space="preserve">& <reg norm="quem" type="context">quẽ</reg> <lb/>admodum terminus, qui diuidit diapaÅ¿on in <reg norm="diateſſaron" type="context">diateſſarõ</reg>, & diapente, eÅ¿t mediator ha@ <lb/>monicus inter extrema diapaÅ¿on diuiÅ¿i, Å¿ic etiam terminus, qui diuidit <reg norm="diapente" type="context">diapẽte</reg> in Å¿e-<lb/>miditonum, & ditonum, mediator eÅ¿t harmonicus inter extrema ipÅ¿ius diapente diui <lb/>Å¿i) Å¿ubtrahendo deinde à diapaÅ¿on Å¿emiditonum remanet exachordum maius, & ab <lb/>eodem diapaÅ¿on Å¿ubtrahendo ditonum remanet exachordum minus, quÄ™ <reg norm="quidem" type="context">quidẽ</reg> <reg norm="non" type="context">nõ</reg> <lb/>accidunt aÅ¿pectuum circulo, quia Å¿ubtrahendo aÅ¿pectum quadratum ab oppoÅ¿ito, <lb/>remanet aliud quadratum, & Å¿ubtrahendo Å¿extilem à trino remanet quoque alius <lb/>Å¿extilis. </s> - <s xml:space="preserve">Quòd autem attinet ad motus, ad magnitudines, ad diÅ¿tantias, & ad influ-<lb/>xus, nihil eÅ¿t, quod hiÅ¿ce proportionibus conueniat, Å¿ed quia hæc omnia <reg norm="dependent" type="context">depẽdent</reg> <lb/>ab <reg norm="infinita" type="context">Ä©finita</reg>, & diuina <reg norm="prouidentia" type="simple context">ê“uidẽtia</reg> Dei, neceſſariò fit vt iÅ¿tæ velocitates, eæ magnitudines, <lb/>diÅ¿tantiæ, & influxus, talem ordinem, & reÅ¿pectum inter Å¿eipÅ¿a, & vniuerÅ¿um <reg norm="habeant" type="context">habeãt</reg>, <lb/>qualis perfectiſſimus Å¿it.</s> - </p> - </div> - <div type="section" level="3" n="34"> - <head style="it" xml:space="preserve">Deraro et denÅ¿o nonnulla, minus diligenter à Peripateticis <lb/>perpenÅ¿a.</head> - <head xml:space="preserve">CAP. XXXIIII.</head> - <p> - <s xml:space="preserve">ANtiqui Peripatetici de videndo in hyeme animalium halitu. </s> - <s xml:space="preserve">Id, quod in æſta <lb/>te non euenit, malè diÅ¿putauerunt, quia hoc naÅ¿citurà condenÅ¿atione hali <lb/>tus, quÄ™ ab ambiente frigore fit. </s> - <s xml:space="preserve">quia halitus is abore, aut naÅ¿o animalis <reg norm="exiens" type="context">exiẽs</reg> <lb/>non eÅ¿t purus aer attractus primò, Å¿ed mixtus eÅ¿t cum quodam vapore excrementi-<lb/>tio, & Å¿ubtili, quo Å¿emper ab ea parte <reg norm="euacuatur" type="simple">euacuat̃</reg> corpus, qui Å¿tatim ab aere frigido cir-<lb/>cundatur, & denÅ¿atur, quam ob cauÅ¿am ab ipÅ¿o ea luminis pars reflectitur, quæ eum <lb/>penetrare non poteÅ¿t, quod in hypocauÅ¿tis, <reg norm="huiuÅ¿modique" type="simple">huiuÅ¿modiqÌ;</reg> calidis locis non fit. </s> - <s xml:space="preserve">Idem <lb/>exemplo ab aqua Å¿tatim à ciÅ¿ternis, aut profundis puteis in hyeme extracta compro <lb/>bari poteÅ¿t, quia tunc temporis, huiuÅ¿modi aqua, cum magis calida Å¿it, quà m fri-<lb/>gida, emittit vaporem, qui facillimè videtur, ob rationem iam dictam, quod <lb/>in æſtate non cernitur in aqua, etÅ¿i ea magis calida eſſet, quam ea, quæ in hyeme <lb/>hauritur.</s> - </p> - <p> - <s xml:space="preserve">Ratio autem, quam ab antiperiÅ¿taÅ¿i deÅ¿umptam citarunt iidem ad inquirendum, <lb/>cur aqua Å¿ubterranea magis calida, aut minus frigida, hyberno tempore, quà m ea, <lb/>quæ eÅ¿t Å¿upra terram Å¿it, vana eÅ¿t, quia hoc non aliunde fit, quà m ab eo, <reg norm="quod" type="simple">ê™</reg> terræ por-<lb/>ri à frigoris Å¿iccitate Å¿int clauÅ¿i, vnde vapores & exalationes non tam facilè exire poÅ¿ <lb/>Å¿unt. </s> - <s xml:space="preserve">quamobrem calefiunt Å¿ubterraneæ partes. </s> - <s xml:space="preserve">Fimum, fÅ“num, frumentum hac in <lb/>re Å¿unt nobis exemplo, in quibus ſępiſſimè viÅ¿um eÅ¿t ignem accendi.</s> - </p> - <p> - <s xml:space="preserve">Priore illa quoque ratione de antiperiÅ¿taÅ¿i dicta, volunt philoÅ¿ophi maiorem ca-<lb/>liditatem hyÄ™me, quà m ęſtate in animalium Å¿tomacho contineri, non animaduerten <lb/>tes Å¿iccitatem, frigiditatis partes Å¿uperficiales corporis, <reg norm="reÅ¿tringentem" type="context">reÅ¿tringẽtem</reg>, Å¿anguinem ver <lb/>Å¿us originem Å¿uam impellere, qui in eo loco copioÅ¿ior cum Å¿it, eas partes tunc tem-<lb/>poris calefacit magis.</s> - </p> - <p> - <s xml:space="preserve">Neque etiam ijdem nouerunt cauÅ¿am, vnde fiat, ut in æſtate impleto vaÅ¿e vitreo, <lb/>aut argenteo, aut ex materia non poroÅ¿a conÅ¿tante, aqua frigida, vas Å¿udet, quod - <pb o="192" rhead="IO. BAPT. BENED." n="204" file="0204" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0204"/> - tempore hyemis, non niÅ¿i in calidis locis euenit, quem Å¿udorem, dicebantipÅ¿i, eſſe <lb/>eandem aquam, quæ per porros vaÅ¿is exiret, quod falÅ¿iſſimum eÅ¿t, quia Å¿i per porros <lb/>aqua frigida exiret multò magis exiret calida, cum Å¿ubtilior Å¿it, & ad penetrandum <lb/>aptior. </s> - <s xml:space="preserve">Sed hoc non aliunde oritur, quà m à condenÅ¿atione aeris vas circundantis, <lb/>cauÅ¿ata à frigiditate vaÅ¿is refrigerati ab aqua, quemadmodum tempore hyberno <lb/>clarè videmus mane Å¿uperficies interiores vitri feneÅ¿trarum Å¿udare, quia <reg norm="extrinÅ¿ecum" type="context">extrinÅ¿ecÅ©</reg> <lb/>frigus refrigerando vitrum, intrinÅ¿ecum aerem Å¿ibi contiguum congelat.</s> - </p> - <p> - <s xml:space="preserve">Neque Å¿ilentio inuoluendum eÅ¿t, nec AriÅ¿totelem, neque alium ex Å¿uis fautoribus <lb/>animaduertiſſe denÅ¿um, & rarum eſſe cauÅ¿am ventorum. </s> - <s xml:space="preserve"><reg norm="Rarum" type="context">RarÅ©</reg> autem & denÅ¿um, me <lb/>diante calore & frigore fit, & Å¿i à partibus, in omogeneis, licet <reg norm="argumentari" type="context">argumẽtari</reg>, de toto <lb/>deducat conÅ¿equentiam qui velit, obÅ¿eruans in calidis æſtatis diebus, dum aliqua nu <lb/>becula ad Solem cooperiendum incedit, ibi Å¿tatim agitationem aeris Å¿entiri; </s> - <s xml:space="preserve">ea verò <lb/>nubecula prætergreſſa cum fuerit, & in ea parte, aer ad priÅ¿tinam raritatem cauÅ¿a-<lb/>tam à calore Solis redierit, quieÅ¿cit; </s> - <s xml:space="preserve">huiuÅ¿modi autem aeris agitatio, à nulla certè ex <lb/>halatione proficiÅ¿citur, Å¿ed à motu Å¿olum locali, quem dum condenÅ¿atur, facit. </s> - <s xml:space="preserve">Om <lb/>ne denÅ¿um natura Å¿ua frigidum eÅ¿t; </s> - <s xml:space="preserve">omne rarum calidum, & è conuerÅ¿o. </s> - <s xml:space="preserve">Et frigida <lb/>aura, quæ à flabellis cauÅ¿atur, non Å¿olum à nouo aere qui nos tangit, Å¿ed etiam à den-<lb/>Å¿o, quod in agitatione eiuÅ¿dem aeris fit, naÅ¿citur.</s> - </p> - <p> - <s xml:space="preserve">Cum autem de raritate & denÅ¿itate diÅ¿putationem Å¿uÅ¿ceperim, non Å¿ine ratione <lb/>mihi <reg norm="videtur" type="simple">videt̃</reg> <reg norm="illorum" type="context">illorÅ©</reg> <reg norm="opinionem" type="context">opinionẽ</reg> <reg norm="explodendam" type="context context">explodẽdã</reg> eſſe, qui LunÄ™ maculas <reg norm="non" type="context">nõ</reg> aliud eſſe dixerunt, <lb/>quà m aliquas partes rariores aliis eiuÅ¿dem Lunæ partibus, non obÅ¿eruantes rarum, & <lb/>denÅ¿um, proportionabilia lumini, quod ab huiuÅ¿modi corporibus reflectitur, non eÅ¿Â<lb/>Å¿e. </s> - <s xml:space="preserve">quia corpus aliquod rarum aliquando aptum erit ad reflectendum maius lumen, <lb/>quà m corpus minus rarum ut manifeÅ¿tè apparet à nubibus reflecti lumen: </s> - <s xml:space="preserve">quod <lb/>ab aere non fit. </s> - <s xml:space="preserve">Non defuerunt qui contrarium dixerunt, ideÅ¿t, eas LunÄ™ partes, den <lb/>Å¿iores eſſe; </s> - <s xml:space="preserve">neque unquam aliquis fuit qui de diaphano, aut opaco mentionem fece <lb/>rit, quia melius eÅ¿t credere, eas partes diaphanas, Å¿iue perÅ¿picuas magis eſſe, quà m a-<lb/>lias, quÄ™ per aliquod Å¿patium, Å¿olis radio ingreſſum permittant, & aliÄ™ partes <reg norm="cum" type="context">cÅ©</reg> Å¿int <lb/>opacæ ipÅ¿um à ſuperficie reflectant. </s> - <s xml:space="preserve">diuerÅ¿a tamen ratione à ſpeculo, cum in pleni-<lb/>lunio tota ferè LunÄ™ pars illuminata cernatur, quamuis dictum lumen extenÅ¿iuè & in <lb/>tenÅ¿iuè Å¿it minus eo, quod ipÅ¿a in nouilunio recipit. </s> - <s xml:space="preserve">Indignum autem mihi videtur <lb/>ijs reÅ¿pondere, qui dixerunt huiuÅ¿modi maculas, terræ vmbras exiÅ¿tere, cum craſſiſſi-<lb/>mæ ignorantiæ tenebris Å¿int circunfuÅ¿i, vt <reg norm="etiam" type="context">etiã</reg> fuit Cornelius Agrippa, qui primo de <lb/>occulta philoÅ¿ophia dicens Å¿e noſſe modum <reg norm="quendam" type="context">quendã</reg> naturalem à Pythagora inuen-<lb/>tum, quo in Luna id totum, quod ipÅ¿e Å¿uper Å¿peculum Å¿cripÅ¿iſſet, videretur. </s> - <s xml:space="preserve">oÅ¿tendit <lb/>manifeÅ¿tè Å¿e ignorare luminum <reg norm="vmbrarumque" type="simple">vmbrarumqÌ;</reg> naturam. </s> - <s xml:space="preserve">quia nulla vnquam vmbra ge <lb/>nerari poteÅ¿t à corpore, quod aut opacum non Å¿it, aut officio opaci non fungatur, <lb/>vt nunc dicemus de diaphaneitate aquæ. </s> - <s xml:space="preserve">Neque corpus opacum illuminatum <reg norm="adum- brare" type="context">adÅ©-<lb/>brare</reg> poteÅ¿t, niÅ¿i opacum illud in linea recta Å¿itum obtineat, quæ inter lucidum & il <lb/>luminatum extenditur. </s> - <s xml:space="preserve">Neque etiam reÅ¿pondebimus ijs, qui Å¿entiunt quotieÅ¿cun-<lb/>que nulla eſſet terra, Å¿ed totus hic globus eſſet aqua, toties non futuram eclipÅ¿im lu-<lb/>narem, ratione diaphaneitatis aquæ. </s> - <s xml:space="preserve">Quod <reg norm="falÅ¿iſſimum" type="context">falÅ¿iſſimÅ©</reg> eÅ¿t, quia omne corpus <reg norm="Å¿phae- ricum" type="simple">Å¿phÄ™-<lb/>ricum</reg> quantumuis diaphanum Å¿it, dummodo Å¿it denÅ¿ius aere, luminoÅ¿os radios re-<lb/>frangit, & eos ad inuicem interÅ¿ecare facit, qui deinde vltra interÅ¿ectionis <reg norm="punctum" type="context">punctÅ©</reg> di-<lb/>Å¿gregantur, ita vt amittant illuminationis actum. </s> - <s xml:space="preserve">Adde <reg norm="quod" type="simple">ê™</reg> etÅ¿i huiuÅ¿modi corpus <lb/>aqueum, Å¿phÄ™ricum non eſſet, Å¿ed cubicum, illud Å¿uper <reg norm="aliquam" type="context">aliquã</reg> ex eius Å¿uperficiebus <lb/>ad angulos rectos radius Å¿olaris percuteret, non eum tamen penetraret, quia dictus <lb/>radius perpetuò debilitatur, & eò magis, quo maiorem profunditatem in diaphano - <pb o="193" rhead="DISPVTATIONES." n="205" file="0205" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0205"/> - eius corporis, quod Å¿it denÅ¿ius aere acquirit, nec totus radius vnquam dictum corpus <lb/>ingr editur, cum ab eius Å¿uperficie magna pars reflectatur. </s> - <s xml:space="preserve">ReÅ¿iÅ¿tit ergo huiuÅ¿mo-<lb/>di corpus lumini, & quantò magis Å¿piſſum aut profundum exiÅ¿tit, tantò, validius reÅ¿i <lb/>Å¿tit. </s> - <s xml:space="preserve">Habemus huius rei teÅ¿tes, piÅ¿catores vnionum, in ijs mundi partibus, quæ pau-<lb/>cis ab hinc annis HiÅ¿panorum opera nobis innotuerunt, qui aÅ¿firmanu ad maris <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> <lb/>fundum lumen Solis non peruenire.</s> - </p> - <p> - <s xml:space="preserve">Immediata ratio, cur nebulÄ™ in ijs locis in quibus <reg norm="conÅ¿piciuntur" type="context">cõſpiciuntur</reg> permaneant, & <reg norm="uunc" type="context">uÅ©c</reg> <lb/>altiores, nunc vero depręſſiores cernantur, non ea eÅ¿t, quam AriÅ¿toteles cap .3. lib. 1. <lb/>metheororum proponit, Å¿ed inde oritur, quòd Å¿int eædem denÅ¿iores ea parte aeris, <lb/>quæ ipÅ¿is Å¿upereminet, & rariores e a, quæ ijÅ¿dem Å¿ubiacet. </s> - <s xml:space="preserve">Quòd autem alicuius cor <lb/>poris denÅ¿itas maior ea, quæ eÅ¿t medij, in quo reperitur, cauÅ¿a Å¿it, vt ipÅ¿um corpus de <lb/>Å¿cendat, & maior raritas eiuÅ¿dem corporis, ea, quæ eÅ¿t quoque me dij, efficiat, vt di-<lb/>ctum corpus aÅ¿cendat, iam Archimedes in lib. de inÅ¿identibus aquæ docuit.</s> - </p> - <p> - <s xml:space="preserve"><reg norm="Rectiſſimem" type="context">Rectiſſimẽ</reg> inÅ¿tituit natura, vt corpora denÅ¿iora verÅ¿us loca anguÅ¿tiora, & minora <lb/>(intelligendo ea loca orbicularis figuræ) quæ ad centrum propius accedunt, & rario <lb/>ra ad ampliora loca, & maius Å¿patium occupantia, Å¿eÅ¿e reciperent. </s> - <s xml:space="preserve">tum quia eadem <lb/>quantitas materiæ condenÅ¿atæ, eget minori loco quam rarefacta, <reg norm="cum" type="context">cÅ©</reg> etiani, quia <reg norm="cum" type="context">cÅ©</reg> <lb/>corpus denÅ¿um non ita ad velocitatem motus localis, vt rarum, idoneum Å¿it, ad eas <lb/>partes accedat, quæ motibus tardioribus magis Å¿unt aptæ, corpora <reg norm="autem" type="context">autẽ</reg> rara ad eas, <lb/>quæ velocioribus motibus Å¿unt aptiores Å¿eÅ¿e transferant. </s> - <s xml:space="preserve">præterquam <reg norm="quod" type="simple">ê™</reg> reuera ap-<lb/>pareat pro maiori parte, corpus magis denÅ¿um, minus diaphanum; </s> - <s xml:space="preserve">aut magis <reg norm="opacum" type="context">opacÅ©</reg> <lb/>futurum, quà m rarum, licet ſæpiſſimè videamus contrarium, vt Å¿uperius innuimus. <lb/></s> - <s xml:space="preserve">eÅ¿t tamen naturale <reg norm="proportionatumque" type="simple">proportionatumqÌ;</reg> magis opacum denÅ¿o, & diaphanum raro, <lb/>quà m è contra. </s> - <s xml:space="preserve">Quamobrem Å¿umma ratione inducta natura voluit, vt corpora ma <lb/>gis opaca, aut minus diaphana, magis vicina centro colligantur, vt Å¿patium, quòd re <lb/>manet, abÅ¿que vllo impedimento à radijs Å¿olaribus penetrari poſſit. </s> - <s xml:space="preserve">Tres autem eæ <lb/>cauſæ, quas hoc loco poÅ¿ui, propriæ Å¿unt, immediatæ, & per Å¿e, ex quibus fit, vt corpo <lb/>ra denÅ¿iora deÅ¿cendant, & rariora aÅ¿cendant in mediis minus denÅ¿is, aut minus raris <lb/>dictorum corporum, quæ à nemine, <reg norm="quod" type="simple">ê™</reg> Å¿ciam, hucuÅ¿que propoÅ¿itæ fuerunt.</s> - </p> - <p> - <s xml:space="preserve">Qui autem aſſerunt cucurbitæ, quam apponunt chirurgi, effectum ex eo naÅ¿ci, <reg norm="quod" type="simple">ê™</reg> <lb/>calidi Å¿it attrahere, valdè aberrant à vero quia hoc, non niÅ¿i à raro, & à denÅ¿o imme-<lb/>diatè, à calido & frigido cauÅ¿atis efficitur, quia aer in cucurbita rarefactus à calore <lb/>& per conÅ¿equens dilatatus, Å¿tatim vt à dicto calore deÅ¿eritur, iterum condenÅ¿atur & <lb/>tantò citius, quantò aer ambiens frigidior exiÅ¿tet, & quia eadem materia cum con-<lb/>denÅ¿ata fuerit minorem Å¿emper occupat locum, reÅ¿tringens igitur Å¿eÅ¿e in cucurbi-<lb/>ta aer dum condenÅ¿atur, neceſſariò fit, ne ulla, Å¿cilicet pars vacua remaneat, <reg norm="quod" type="simple">ê™</reg> cum <lb/>alius aer ingredi cucurbitam nequeat aliud corpus ingrediatur. </s> - <s xml:space="preserve">Idem cum amphora <lb/>in qua nullum aliud, quam aèreum Å¿it corpus experiri poſſumus, Å¿i <reg norm="eam" type="context">eã</reg> ad ignem pri-<lb/>mò calefactam, deinde <reg norm="cum" type="context">cÅ©</reg> ore in amplo aliquo cyatho, aut alio vaÅ¿e. </s> - <s xml:space="preserve">vino, aut aqua <lb/>pleno vbi videbimus huiuÅ¿modi liquorem Å¿tatim Å¿urÅ¿um ferri, quia dum calefit am-<lb/>phora, rarefit quoque aer qui in ea continetur, & quia rateÅ¿cit dilaratur, & quia <lb/>dilatatur, eget maiore loco; </s> - <s xml:space="preserve">& ideo magna pars eius foras exit; </s> - <s xml:space="preserve">Cum verò ea aeris <lb/>portio, quæ intus remanÅ¿erit, iterum condenÅ¿atur ob defectum caloris, reÅ¿tringitur, <lb/><reg norm="minorique" type="simple">minoriqÌ;</reg> indiget loco; </s> - <s xml:space="preserve">Quod cum ita Å¿e habeat, neceſſarium eÅ¿t, ne aliquis locus va <lb/>cuus remaneat, vt aliud quoddam corpus ingrediatur, cum ad <reg norm="ingrediendum" type="context">ingrediẽdum</reg> aeri non <lb/>patuerit aditus. </s> - <s xml:space="preserve">quod Å¿i corpus admodum non erit fluxile, aut humidum, ita vt ingre <lb/>di amphoram poſſit ita amphorÄ™ hærebit, vt non cito diuelli poſſit, & eo modo ſępe - <pb o="194" rhead="IO. BAPT. BENED." n="206" file="0206" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0206"/> - <reg norm="cum" type="context">cÅ©</reg> admiratione <reg norm="videmus" type="simple">videmê°</reg> fragile vas <reg norm="vitreum" type="context">vitreÅ©</reg> <reg norm="magnum" type="context">magnÅ©</reg>, & graue <reg norm="lapideum" type="context">lapideÅ©</reg> corpus eleuare. <lb/></s> - <s xml:space="preserve">Sed vt ad denÅ¿um & ad rarum redeamus, mihi videtur frigidum eſſe conÅ¿equentem <lb/>qualitatem denÅ¿i, & calidum rari, quia quæuis res dum calefit, rarefit, & quælibet <lb/>materia dum refrigeratur, Å¿imul condenÅ¿atur. </s> - <s xml:space="preserve">Qua ratione fit, vt terra frigidior <lb/>Å¿it aqua, & ignis calidior Å¿it aere.</s> - </p> - <p> - <s xml:space="preserve">Nec propriè locutus eÅ¿t AriÅ¿toteles .9. & .10. capite primi lib. & Å¿ecundo Å¿ecundi <lb/>metheororum cum dixerit <reg norm="calorem" type="context">calorẽ</reg> Solis eum eſſe, qui Å¿urÅ¿um humores, <reg norm="vaporesque" type="simple">vaporesqÌ;</reg> eue <lb/>hat, quia Sol nil aliud facit, quam calefacere, cuius caloris ratione, ea materia rarefit, <lb/>& ob rarefactionem leuior facta aÅ¿cendit, non quia Å¿urÅ¿um à Sole feratur.</s> - </p> - <p> - <s xml:space="preserve">QuÄ™ Å¿ubÅ¿equuntur, cum raro ac denÅ¿o Å¿imbolum habere videntur. </s> - <s xml:space="preserve">cum raro, Å¿cili-<lb/>cet calidum, humidum, leue, Å¿ublime, diaphanum, lumen, clarum, lux, <reg norm="album" type="context">albÅ©</reg>, dies, mo-<lb/>tus, velox, Å¿implex, diÅ¿gregatum, molle, lene, acutum, Å¿ubtile, coctum, <reg norm="Å¿paciosum" type="context">Å¿paciosÅ©</reg>, <lb/>dulce, voluptas, audacia, lætitia, liberalitas, veritas, induÅ¿tria, amor, miÅ¿ericordia, hu-<lb/>manitas, Å¿anitas, vita, & iis Å¿imilia. </s> - <s xml:space="preserve">Cum denÅ¿o verò frigidum, Å¿iccum, graue, imum, <lb/>opacum, vmbra, obÅ¿curum, tenebræ, nigrum, nox, quies, tardum, mixtum, congrega <lb/>tum, durum, aÅ¿perum, ob tuÅ¿um, craſſum, crudum, anguÅ¿tum, amarum, dolor, cimor, <lb/><reg norm="melancholia" type="context">melãcholia</reg>, auaritia, mendacium, inertia, odium, crudelitas, feritas, infirmitas, mors, <lb/>& ijs Å¿imilia.</s> - </p> - <p> - <s xml:space="preserve">Verum eÅ¿t quod ea ratio, qua AriÅ¿toteles ait aerem humidum eſſe, parui eÅ¿t mo-<lb/>menti, quia Å¿imiliter deigne inferri poſſet, qui facilius à termino alieno, <reg norm="quam" type="context">quã</reg> aer, aut <lb/>aqua terminari poteÅ¿t.</s> - </p> - </div> - <div type="section" level="3" n="35"> - <head style="it" xml:space="preserve">Motum rectum curuo poſſe comparari etiam diÅ¿entiente <lb/>AriÅ¿totele.</head> - <head xml:space="preserve">CAP. XXXV.</head> - <p> - <s xml:space="preserve">SEd vt ad <reg norm="AriÅ¿totelem" type="context">AriÅ¿totelẽ</reg> redeamus, rectè dicere non poteÅ¿t motum rectum ad <reg norm="curuum" type="context">curuÅ©</reg> <lb/>comparabilem non eſſe .4. cap. lib. 7. phyÅ¿icorum, vbi errat quoque dicens repe <lb/>riri non poſſe lineam aliquam rectam alicuius circuli circunferentiæ æqualem. </s> - <s xml:space="preserve">quia <lb/>Archimedes iam probauit in lib. de quadratura circuli, triangulum illum orthogo-<lb/>nium, cuius vnum ex lateribus circundantibus angulum rectum æquale eſſet Å¿emi-<lb/>piametro alicuius circuli, & aliud circunferentiæ, æqualem futurum dicto circulo. </s> - <s xml:space="preserve">Il <lb/>lud igitur triangulum orthogonium, quod æquale erit alicui circulo, & habebit ali-<lb/>quod ex Å¿uis lateribus circundantibus angulum rectum æquale Å¿emidiametro dicti <lb/>circuli, aliud quoque latus ipÅ¿um angulum rectum circundans, ex neceſſitate, <reg norm="circun- ferentiæ" type="context">circÅ©-<lb/>ferentiæ</reg> dicti circuli habebit æquale. </s> - <s xml:space="preserve">PoteÅ¿t igitur dari vna quædam recta linea <reg norm="ae- qualis" type="simple">Ä™-<lb/>qualis</reg> circulari contra AriÅ¿totelis opinionem, qui non benè reuocauit in mentem, <lb/>quod Å¿cripÅ¿it de relatiuis, cum dixit quadraturam circuli poſſe quidem dari, etÅ¿i <reg norm="tunc" type="context">tÅ©c</reg> <lb/><reg norm="temporis" type="wordlist">tê‘is</reg> de ea <reg norm="non" type="context">nõ</reg> <reg norm="haberetur" type="simple">haberet̃</reg> Å¿cientia. </s> - <s xml:space="preserve">Si <reg norm="igitur" type="simple">igit̃</reg> dicta quadratura dari <reg norm="pont" type="context">põt</reg>, poteÅ¿t <reg norm="etiam" type="context">etiã</reg> dari vna <lb/>recta linea Ä™qualis circunferentiæ <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> circuli, ob rationes <reg norm="iam" type="context">iã</reg> dictas. </s> - <s xml:space="preserve">Sed Å¿i AriÅ¿t. <lb/>dixiſſet, circularem corporum cÄ™leÅ¿tium motum, comparabilem non eſſerecto cor-<lb/>porum elementarium, verum dixiſſet, non quia eorum alter circularis, alter ve-<lb/>rò Å¿it rectus, Å¿ed quia cÅ“leÅ¿tis regularis Å¿it, neque modò tardus, modò velox, <lb/>Å¿ed vnam Å¿emper & eandem velocitatem retinens, <reg norm="motus" type="simple">motê°</reg> <reg norm="autem" type="wordlist">aÅ©t</reg>, qui eÅ¿t <reg norm="corporum" type="context">corporÅ©</reg> elemen - <pb o="195" rhead="DISPVTATIO NES." n="207" file="0207" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0207"/> - tarium è contrà ſe habeat, præter id, <reg norm="quod" type="simple">ê™</reg> nunquam fuit neque Å¿it futurus aliquis <reg norm="horum" type="context">horÅ©</reg> <lb/>rectorum, qui naturales dicuntur, qui tam velociter moueatur, ut motus cÅ“li, quia Å¿i <lb/>voluerimus conÅ¿iderare motum diurnum .24. horarum, Å¿ecundum opinionem com-<lb/>munem, reperiemus calculando, Lunam in quadraturis cum Sole, dum inuenitur in <lb/>æquatore, Å¿ingulis horarum minutis moueri per .500. milliaria Italica vel circa, & in <lb/>coniunctionibus, & oppoÅ¿itionibus ipÅ¿ius Solis .1000. vel circa, & Solem tempore <reg norm="ae- quinoctiorum" type="simple context">Ä™-<lb/>quinoctiorÅ©</reg> .18000. & <reg norm="Saturnum" type="context">SaturnÅ©</reg> circa æquatoris <reg norm="Å¿itum" type="context">Å¿itÅ©</reg> .260000. & <reg norm="amplius" type="simple">ampliê°</reg> de Å¿tellis <reg norm="autem" type="wordlist">aÅ©t</reg> <lb/>fixis circa æquatorem poÅ¿itis quiuis cogitet; </s> - <s xml:space="preserve">quod reuera diffi cillimum quibuÅ¿dam <lb/>videbitur, quod quidem non occurrit <reg norm="Å¿ecundum" type="context">Å¿ecÅ©dum</reg> pulcherrimam AriÅ¿tarchi Å¿amij opi-<lb/>nionem, diuinitus à Nicolao Copernico expreſſam, contra quam nil planè valent <lb/>rationes ab AriÅ¿totele; </s> - <s xml:space="preserve">neque etiam à Ptolomeo propoÅ¿itÄ™. </s> - <s xml:space="preserve">Motu verò proprio, quo <lb/>libet horÄ™ minuto, Sol <reg norm="mouetur" type="simple">mouet̃</reg> per milliaria circa .48. </s> - <s xml:space="preserve">Luna <reg norm="quando" type="context">quãdo</reg> <reg norm="coniuncta" type="context">cõiuncta</reg> eÅ¿t, aut op <lb/>poÅ¿ita reperitur Soli .36. milliaria, & in quadraturis .18. Saturnus .24. Iupiter .40: <lb/>Mars .100: Venus .26: Mercur .5. </s> - <s xml:space="preserve">Sed Saturnus motu rapido, vno horæ minuto <reg norm="mo- uetur" type="simple">mo-<lb/>uet̃</reg> circa .260000. milliaria, vt diximus Iupiter circa .170000. Mars .75000. Venus. <lb/>10000, Mercurius .2000. corpus <reg norm="autem" type="context">autẽ</reg> elementare, & Å¿i <reg norm="moueretur" type="simple">moueret̃</reg> motu recto hoc <reg norm="mon" type="context">mõ</reg>, <lb/>& velocius etiam corpore cÄ™leÅ¿ti, non obÅ¿eruans <reg norm="tamen" type="wordlist">tamẽ</reg> uniformitatem, ut dictum cÅ“ <lb/>leÅ¿te facit, cum eodem nullo modo comparari poſſet, quia rectus dictus naturalis, <lb/>Å¿uam Å¿emper velocitatem adauget, ob continuam impreſſionem, quam recipit à cau <lb/>Å¿a perpetuò coniuncta cum ipÅ¿o corpore, quÄ™ eÅ¿t propenÅ¿io illa naturalis eundi bre-<lb/>uiori quadam via ad locum Å¿uum, ita vt etiam Å¿i dictum corpus elementare à motu <lb/>tardiore ad velociorem, Å¿uperare poſſet <reg norm="motum" type="context">motÅ©</reg> alicuius corporis cÄ™leÅ¿tis, ij duo motus <lb/>interÅ¿ecarent Å¿eÅ¿e in vno Å¿olo <reg norm="puncto" type="context">pÅ©cto</reg>, quod diuidi <reg norm="diÅ¿tribuique" type="simple">diÅ¿tribuiqÌ;</reg> in partes nequiret, ideÅ¿t <lb/>non niÅ¿i in vno Å¿olo temporis inÅ¿tanti redderentur æquales, vt ita dicam. </s> - <s xml:space="preserve"><reg norm="Neque" type="simple">Neq;</reg> <reg norm="Å¿olum" type="context">Å¿olÅ©</reg> <lb/>loquor de circulari cÅ“leÅ¿ti cum recto elementari, Å¿ed de qualibet alia motuum Å¿pe-<lb/>cie, Å¿iue Å¿int ambo recti, Å¿iue ambo curui, quando aliquis eorum irregularis erit.</s> - </p> - </div> - <div type="section" level="3" n="36"> - <head style="it" xml:space="preserve">Minus Å¿ufficienter exploÅ¿am fuiſſe ab AriÅ¿totele opinionem cre-<lb/>dentium plures mundos exiÅ¿tere.</head> - <head xml:space="preserve">CAP. XXXVI.</head> - <p> - <s xml:space="preserve">MAior ratio, qua AriÅ¿toteles eorum opinionem, qui plures eſſe mundos dixe <lb/>runt, refutare nititur, in eo conÅ¿iÅ¿tit, quod is credat partes terræ, quæ alijs <lb/>mundis aſſignarentur, ad huius mundi centrum inclinationem habere, & Å¿ic ignem <lb/>illorum, propenÅ¿ionem habiturum ad circunferentiam huius.</s> - </p> - <p> - <s xml:space="preserve">Quæ certè ratio tam debilis eÅ¿t, vt per Å¿e cadat, non conÅ¿ideransipÅ¿e, quòd Å¿i <lb/>eſſent dicti mundi, eorum quilibet Å¿uum proprium centrum, <reg norm="Å¿uamque" type="simple">Å¿uamqÌ;</reg> propriam cir-<lb/>cunferentiam haberet, <reg norm="terrasque" type="simple">terrasqÌ;</reg> & ignes haberent inclinationem ad centra circunfe-<lb/><reg norm="rentiasque" type="simple">rentiasqÌ;</reg> Å¿uorum mundorum, abÅ¿que eo, <reg norm="quod" type="simple">ê™</reg> vna terra, alterius centrum appeteret; </s> - <s xml:space="preserve">vt <lb/>exempli gratia, Å¿i doctiſſimi AriÅ¿tarchi opinio eÅ¿t vera, rationi <reg norm="quoque" type="simple">quoq;</reg> conÅ¿entaneum <lb/>erit maximè, vt quod Lunæ contingit, cuilibet <reg norm="etiam" type="context">etiã</reg> ex aliis quinque planetis eue-<lb/>niat, ideÅ¿t, vt quemadmodum Luna Å¿uorum epicyclorum ope <reg norm="circum" type="context">circÅ©</reg> terram voluitur, <lb/>quaÅ¿i per circunferentiam alterius cuiuÅ¿dam epicycli, in quo terra Å¿it inÅ¿tar centri <lb/>naturalis (ideÅ¿t Å¿it in medio) delati ab orbe annuo circa Solem; </s> - <s xml:space="preserve">Sic etiam Saturnus, <lb/>Iupiter, Mars, Venus, atque Mercurius, cir cum aliquod corpus in medio Å¿ui epici- - <pb o="196" rhead="IO. BABPT. BENED." n="208" file="0208" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0208"/> - cli maioris, Å¿itum habens, voluantur; </s> - <s xml:space="preserve">quod quidem corpus, & aliquem quoque ha-<lb/>beat motum circa Å¿uum axem, Å¿it opacum, ijs conditionibus, quæ terræ Å¿unt Å¿imi-<lb/>les, præditum exiÅ¿tat, & in dicto epyciclo Å¿int res Å¿imiles iÅ¿tis lunaribus.</s> - </p> - </div> - <div type="section" level="3" n="37"> - <head style="it" xml:space="preserve">Anrectè loquutus Å¿it PhyloÅ¿opbus de extenÅ¿ione luminis <lb/>per uacuum.</head> - <head xml:space="preserve">CAP. XXXVII.</head> - <p> - <s xml:space="preserve">ARriÅ¿toteles Å¿ecundo lib. de anima Å¿entit <reg norm="quod" type="simple">ê™</reg> per vacuum non extenderetur <reg norm="lu- mem" type="context">lu-<lb/>mẽ</reg>, quod procederet à corpore lucido. </s> - <s xml:space="preserve">Quod veriÅ¿imile <reg norm="non" type="context">nõ</reg> eÅ¿t; </s> - <s xml:space="preserve"><reg norm="quia" type="simple">ê—a</reg> <reg norm="quemadmo dum" type="wordlist">quẽadmoÂ<lb/>dum</reg> quantò rarius eÅ¿t aliquod corpus, tanto aptius eÅ¿t vt diaphanum exiÅ¿tat; </s> - <s xml:space="preserve">& <reg norm="quan- tò" type="context">quã-<lb/>tò</reg> rarius eÅ¿t dictum corpus, tantò minorem quantitatem materiæ contineat; </s> - <s xml:space="preserve">Å¿ic <reg norm="quam" type="context">quã</reg> <lb/>tò magis diaphanum eÅ¿t, cum ex perexigua materia conÅ¿tet, tantò magis liber tran-<lb/>Å¿itus luminis patet; </s> - <s xml:space="preserve">Vnde quantò minor quantitas materiæ erit in dicto Å¿patio, tan <lb/>tò nitidius pertranÅ¿ibit lumen. </s> - <s xml:space="preserve">Sequitur ergo, quòd vbi nulla eſſet materia, totum <lb/>lumen libere tranÅ¿iret. </s> - <s xml:space="preserve">Color cÄ™ruleus quem videmus in profunditate aquæ, & ae-<lb/>ris, color eÅ¿t a quæ & aeris, qui denotat reÅ¿iÅ¿tentiam factam ab aere & ab aqua ipÅ¿i lu <lb/>mini; </s> - <s xml:space="preserve">Quod quidem lumen ubi corpus aliquod non eſſet, minime reflecteretur, Å¿ed <lb/>abÅ¿que vllo impedimento rectà tranÅ¿iret.</s> - </p> - </div> - <div type="section" level="3" n="38"> - <head style="it" xml:space="preserve">An rectè phyloÅ¿ophiÅ“<unsure/> penus AriÅ¿toteles Å¿enÅ¿erit de loco im-<lb/>pellendo à pyramide.</head> - <head xml:space="preserve">CAP. XXXVIII.</head> - <p> - <s xml:space="preserve">ARiÅ¿toteles .8. cap. lib. 3. de cÅ“lo, diÅ¿putans contra antiquos de elementorum <lb/>figuris, ait pyramidem implere poſſe locum corporeum. </s> - <s xml:space="preserve">quod verum non <lb/>eÅ¿t. </s> - <s xml:space="preserve">Cubus quidem id facit ab .8. enim cubis perfectè impletur locus, Å¿ed non <lb/>item .12. pyramides, ut AriÅ¿toteles Å¿enÅ¿it (ideÅ¿t Å¿ex Å¿uper aliquam exagonam figu-<lb/>ram Å¿uperficialem & Å¿exÅ¿ub eadem) id præſtant, cum potius maius vacuum rema-<lb/>neatad quamlibet partium Å¿upra, & infra, quam plenum. </s> - <s xml:space="preserve">Rectius AriÅ¿toteles <lb/>egiſſet, Å¿i probaſſet ratione immobilitatis conuenire pyramidem terræ, quam cu-<lb/>bum. </s> - <s xml:space="preserve">quamuis, de horum corporum altero, Å¿it Å¿tultum hoc credere. </s> - <s xml:space="preserve">decepti tamen <lb/>fuerunt antiqui, credentes cubum ad motum minus idoneum eſſe, quam reliqua <lb/>quatuor corpora regularia (loquor autem habita volubilitatis ratione) quia pyra-<lb/>midale eÅ¿t illud, quod ita Å¿e habet, vt multis rationibus probari poteÅ¿t, quarum vna <lb/>hæc nobis Å¿ufficiet. </s> - <s xml:space="preserve">Scimus iam ex communi conceptu corpus Å¿phÄ™ricum eſſe ma-<lb/>gis volubile, <reg norm="inÅ¿tabileque" type="simple">inÅ¿tabileqÌ;</reg>, quà m alia Å¿int. </s> - <s xml:space="preserve">Illud ergo corpus, cuius figura ad Å¿phæri-<lb/>cam magis accedet, ad uoluendum, & ad mouendum facilius erit quouis alio, quod <lb/>æqualis Å¿it quantitatis, & Å¿ibi omogeneum materia, vt exempli gratia corpus .20. ba <lb/>Å¿ium ad voluendum, & ad mouendum promptius erit eo, quod ex .12. conÅ¿tat, & id, <lb/>quod eÅ¿t .12. eo, quod eÅ¿t .8. & id, quod eÅ¿t .8. eo, quod eÅ¿t .6. & id, quod eÅ¿t .6. vt <lb/>cubus eÅ¿t, eo, quod eÅ¿t .4. cuiuÅ¿modi eÅ¿t pyramidale. </s> - <s xml:space="preserve">Huc accedit, quòd pyrami-<lb/>dale corpus aliam conditionem habet, quà m cubicum, cum in quauis facie inalte- - <pb o="197" rhead="DISPVTATIONES." n="209" file="0209" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0209"/> - rabile Å¿it, cubicum autem econtrà ſit alterabile vndequaque, <reg norm="Å¿uaque" type="simple">Å¿uaqÌ;</reg> quadrata in <reg norm="rhum- bos" type="context">rhÅ©-<lb/>bos</reg> mutare poſſit, iiÅ¿dem exiÅ¿tentibus lateribus.</s> - </p> - </div> - <div type="section" level="3" n="39"> - <head style="it" xml:space="preserve">Examinatur quam ualida Å¿it ratio Aristotelis de <lb/>inalterabilitate CÅ“li.</head> - <head xml:space="preserve">CAP. XXXIX.</head> - <p> - <s xml:space="preserve">ARriÅ¿toteles textu .22. primi lib. de CÅ“lo ita inquit.</s> - </p> - <p> - <s xml:space="preserve">Accidit autem, & hoc per Å¿enÅ¿um Å¿ufficienter, quo ad humanam dixiſſe fi-<lb/>dem, & omni prÄ™terito tempore Å¿ecundum traditam inuicem memoriam, nihil vi-<lb/>detur tranÅ¿mutatum neque Å¿ecundum totum vltimum cÄ™lum, neque Å¿ecundum par-<lb/>tem ipÅ¿ius propriam vllam.</s> - </p> - <p> - <s xml:space="preserve">Hoc autem in loco AriÅ¿to. non conÅ¿iderauit, <reg norm="quod" type="simple">ê™</reg> Å¿imiliter de terra dici poſſet, quan <lb/>do ipÅ¿a ita eminus proÅ¿piceretur, imo abÅ¿que dubio putandum eÅ¿t, <reg norm="quod" type="simple">ê™</reg> Å¿i terra luce So <lb/>lis prædita eſſet, & aliquis ipÅ¿am ab octauo orbe vellet videre, nullo pacto cerne-<lb/>ret, cum Å¿idera illa quæ primæ magnitudinis vocantur, & quæ pluÅ¿quam centies ma <lb/>iora ip<unsure/>Å¿a terra putantur non niÅ¿i vt puncta videantur.</s> - </p> - <pb o="198" n="210" file="0210" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0210"/> - </div> - </div> - <div type="chapter" level="2" n="5"> - <head xml:space="preserve">IN QVINTVM <lb/>EVCLIDIS LIBRVM</head> - <p type="title" style="it"> - <s xml:space="preserve">QVamuis omnia libri quinti Euclid. uerißima Å¿int. <lb/></s> - <s xml:space="preserve">Animaduertimus tamen permultos Å¿umma <reg norm="cum" type="context">cÅ©</reg> <lb/>difficultate <reg norm="eorum" type="context">eorÅ©</reg> demonstr ationes percipere. </s> - <s xml:space="preserve">PrÅ“<unsure/>-<lb/>cipuè ubi quint a, aut Å¿eptima deffinitiones eiuÅ¿-<lb/>dem libri neceſſariÅ“<unsure/> Å¿unt. </s> - <s xml:space="preserve">IllÅ“ enim adeo obÅ¿curÅ“ <lb/>uidentur, ut longè facilius admißuri Å¿int hÅ“c no-<lb/>Å¿tra poÅ¿tulat at anquam clarior a. </s> - <s xml:space="preserve">At que etiam tanquam intellectui <lb/>commodiora, quam Å¿it illud quintum <reg norm="idemque" type="simple">idemqÌê«</reg> ultimum postulatum <lb/>eiuÅ¿dem in primo libro poÅ¿itum, de line a duas alias Å¿ecante. </s> - <s xml:space="preserve">Quan-<lb/>doquidem <reg norm="ijs" type="lig">ijs</reg> noÅ¿tris postulatis admißis, Å¿equentia Theoremata per <lb/>facillima reddentur.</s> - </p> - <div type="section" level="3" n="1"> - <div type="section" level="4" n="1"> - <head style="it" xml:space="preserve">Horum autem primum est.</head> - <p> - <s xml:space="preserve"><emph style="sc">Qvod</emph> tota compoÅ¿ita ex æquali numero partium æqualium, Å¿unt inuicem <lb/>æqualia.</s> - </p> - <p> - <s xml:space="preserve">VtÅ¿i quis diceret omnes proportiones quæ <reg norm="compoÅ¿itæ" type="context">cõpoÅ¿itæ</reg> Å¿unt ex æquali numero alia-<lb/>rum proportionum inuicem æqualium, Å¿unt etiam inuicem æquales, quod Eucli-<lb/>des conatur demonÅ¿trare in .22. et .23. quinti libri.</s> - </p> - </div> - <div type="section" level="4" n="2"> - <head xml:space="preserve">SECVNDVM.</head> - <p> - <s xml:space="preserve"><emph style="sc">Qvod</emph> Å¿i à totis æqualibus detractæ fuerint æquales partes, quæ remanent erunt <lb/>partes inuicem æquales.</s> - </p> - <p> - <s xml:space="preserve">Et è conuerÅ¿o Å¿i æqualibus æqualia addas compoÅ¿ita erunt inuicem æqualia.</s> - </p> - <p> - <s xml:space="preserve">Quod in ipÅ¿is proportionibus hoc loco Å¿emper intelligendum eÅ¿t.</s> - </p> - </div> - <div type="section" level="4" n="3"> - <head xml:space="preserve">TERTIVM.</head> - <head style="it" xml:space="preserve">QuÄ™ est εuclidis Å¿eptima propoÅ¿itio.</head> - <p> - <s xml:space="preserve"><emph style="sc">Qvod</emph> Å¿i fuerint plures termini æquales inuicem, ratio Å¿eu proportio vnius ip-<lb/>Å¿orum ad alium tertium terminum maiorem, minoremúe, Å¿ed eiuÅ¿dem generis, erit <lb/>cadem quæ cuiuÅ¿uis alterius termini ad eundem tertium. </s> - <s xml:space="preserve">Et è conuerÅ¿o, quæ fuerit <lb/>proportio tertij termini ad vnum prædictorum æqualium, eadem erit, Å¿pecie, cum <lb/>alio eorundem terminorum.</s> - </p> - <pb o="199" rhead="IN QVINT. LIB. EVCLI." n="211" file="0211" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0211"/> - </div> - <div type="section" level="4" n="4"> - <head xml:space="preserve">QVARTVM.</head> - <head style="it" xml:space="preserve">εuclidis uerò nona propoÅ¿itio.</head> - <p> - <s xml:space="preserve"><emph style="sc">Qvotiescvnqve</emph> proportio vnius plurium terminorum collatorum cum ali <lb/>quo tertio eiuÅ¿dem generis, eadem fuerit cum ea quæ eÅ¿t cuiuÅ¿uis alterius dictorum <lb/>terminorum cum eodem tertio, aut proportio dicti tertij, cum aliquo dictorum, ea-<lb/>dem fuerit cum ea quæ ipÅ¿ius eÅ¿t ad aliquem alium eorundem <reg norm="terminorum" type="context">terminorÅ©</reg>, tunc eiuÅ¿-<lb/>modi termini, æquales erunt inter Å¿e.</s> - </p> - </div> - <div type="section" level="4" n="5"> - <head xml:space="preserve">QVINTVM.</head> - <head style="it" xml:space="preserve">Euclidis uerò octaua propoÅ¿itio.</head> - <p> - <s xml:space="preserve"><emph style="sc">Qvoties</emph> plures erunt termini, quorum vnus fuerit maior altero, Å¿i compa-<lb/>rentur alicui tertio eiuÅ¿dem generis, proportio maioris adtertium illum, maior erit <lb/>ea, quæ eÅ¿t minoris ad prædictum tertium, & proportio illius tertij ad maiorem, mi-<lb/>nor erit ea quæ eiuÅ¿dem tertij ad minorem terminum comparati.</s> - </p> - </div> - <div type="section" level="4" n="6"> - <head xml:space="preserve">SEXTVM.</head> - <head style="it" xml:space="preserve">εuclidis uerò decima propoÅ¿itio.</head> - <p> - <s xml:space="preserve"><emph style="sc">Qvoties</emph> proportio vnius, ex pluribus terminis comparatis ad aliquem ter-<lb/>tium, maior fuerit proportione alicuius alterius dictorum cum eodem tertio, primus <lb/>ille terminus, altero maior erit. </s> - <s xml:space="preserve">Et quoties proportio tertij termini ad vnum <reg norm="quam" type="context">quã</reg> <lb/>ad alterum terminum maior fuerit, eiuÅ¿modi terminus altero minor erit.</s> - </p> - </div> - <div type="section" level="4" n="7"> - <head xml:space="preserve">SEPTIMVM.</head> - <head style="it" xml:space="preserve">Euclidis uerò undecima propoÅ¿itio.</head> - <p> - <s xml:space="preserve"><emph style="sc">Proportiones</emph>, quarum vnaquÄ™que cum aliqua tertia æqualis eÅ¿t, ipſæ quo-<lb/>que inter Å¿e Å¿unt æquales. </s> - <s xml:space="preserve">Vtillud, Quæ vni & eidem Å¿unt æqualia, Å¿ibi inuicem <lb/>Å¿unt æqualia.</s> - </p> - </div> - <div type="section" level="4" n="8"> - <head xml:space="preserve">OCTAVVM.</head> - <head style="it" xml:space="preserve">εuclidis uerò duodecima propoÅ¿itio.</head> - <p> - <s xml:space="preserve"><emph style="sc">Qvotiescvnqve</emph> proportio vnius ex pluribus antecedentibus cum Å¿uo ex <lb/>pluribus conÅ¿equentibus, æqualis fuerit ei cuiuÅ¿uis alterius dictorum <reg norm="antecedentium" type="context">antecedentiÅ©</reg>, <lb/>cum Å¿uo plurium <reg norm="conÅ¿equentium" type="context">cõſequentium</reg>, proportio totius aggregati antecedentium cum to-<lb/>to aggregato conÅ¿equentium, dictæ primÄ™ proportioni Ä™qualis erit, nempe illius an <lb/>tecedentis ad Å¿uum conÅ¿equens.</s> - </p> - <pb o="200" rhead="IO. BAPT. BENED." n="212" file="0212" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0212"/> - </div> - <div type="section" level="4" n="9"> - <head xml:space="preserve">NONVM.</head> - <head style="it" xml:space="preserve">Euclidis uero tertiadecima propoÅ¿itio.</head> - <p> - <s xml:space="preserve"><emph style="sc">Qvotiescvnqve</emph> aliqua proportio plurium proportionum inuicem æqua-<lb/>lium, tertia aliqua proportione, maior aut minor fuerit, quælibet prædictarum æqua <lb/>lium inter Å¿e, tertia illa proportione maior aut minor pariter erit.</s> - </p> - </div> - <div type="section" level="4" n="10"> - <head xml:space="preserve">DECIMVM.</head> - <p> - <s xml:space="preserve"><emph style="sc">Qvotiescvnqve</emph> fuerint ex vna parte plurestermini (Å¿iue coniuncti Å¿iue di-<lb/>Å¿iuncti Å¿int) æquales Å¿inguli vni tertio termino; </s> - <s xml:space="preserve">ex altera verò parte totidem fuerint <lb/>alteri tertio termino æquales, proportio aggregati priorum terminorum ad <reg norm="Å¿uum" type="context">Å¿uÅ©</reg> ter-<lb/>tium, æqualis erit proportioni aggregati reliquorum terminorum ad Å¿uum tertium, <lb/>& è conuerÅ¿o, ita Å¿e habebit primus tertius terminus ad Å¿uos multos terminos, Å¿icut <lb/>Å¿e habet Å¿ecundus tertius terminus ad Å¿uos Å¿imul Å¿umptos.</s> - </p> - </div> - <div type="section" level="4" n="11"> - <head xml:space="preserve">VNDECIMVM.</head> - <p> - <s xml:space="preserve">Aggregatum ex partibus proportiona litatis continuæ, quod inter maximum, & <lb/>minimum terminum omnium terminorum proportionalium compræhenditur, Å¿em <lb/>per multiplex eÅ¿t ad Å¿ingulas partiales proportiones, ex quibus ipÅ¿um componitur.</s> - </p> - </div> - <div type="section" level="4" n="12"> - <head xml:space="preserve">DVODECIMVM.</head> - <p> - <s xml:space="preserve">Quæuis proportio quocunque modo diuiÅ¿a fuerit, ex iis partibus componitur, in <lb/>quas diuiditur.</s> - </p> - <p style="it"> - <s xml:space="preserve">Cum enim bæ præpoÅ¿itiones Å¿int ita conÅ¿picuæ ipÅ¿i intellectui, ut <reg norm="abÅ¿que" type="simple">abÅ¿q;</reg> dubio inter obie <lb/>ct a ipÅ¿ius intellectus connumerari poſſint, nullus Å¿anæ mentis eas negabit.</s> - </p> - </div> - </div> - <div type="section" level="3" n="2"> - <div type="section" level="4" n="1"> - <head xml:space="preserve">THEOR.I. II. ET III.</head> - <p> - <s xml:space="preserve">PRimum, Å¿ecundum, & tertium theorema quinti Euclidis ab ipÅ¿o Å¿atis exactè de <lb/>monÅ¿tratur, Å¿tudioÅ¿us itaque autorem conÅ¿ulat.</s> - </p> - </div> - <div type="section" level="4" n="2"> - <head xml:space="preserve">THEOREM. IIII.</head> - <p> - <s xml:space="preserve">QVartum vero Theorema Eu-<lb/> - <anchor type="figure" xlink:label="fig-0212-01a" xlink:href="fig-0212-01"/> - clidis ego Å¿ic <reg norm="demonÅ¿trarem" type="context">demonÅ¿trarẽ</reg>. <lb/></s> - <s xml:space="preserve">Å¿it, verbi gratia, proportio <var>.a.</var> ad <var>.b.</var> <lb/>quæ eÅ¿t <var>.c.</var> ad <var>.d.</var> Å¿umptis multiplici-<lb/>bus <var>.e.</var> et <var>.f.</var> ad <var>.a.</var> et <var>.c.</var> æqualiter, item <lb/>multiplicibus <var>.g.</var> et <var>.h.</var> ad <var>.b.</var> et <var>.d.</var> dico <lb/>proportionem <var>.e.</var> ad <var>.g.</var> eſſe eandem <lb/>quæ eÅ¿t <var>.f.</var> ad <var>.h</var>. </s> - <s xml:space="preserve">Habemus enim ex .10 <lb/>poÅ¿tulato præmiſſo, eandem futuram <lb/>proportionem <var>.e.</var> ad <var>.a.</var> quæ eÅ¿t <var>.f.</var> ad <var>.c.</var> <lb/>& ita <var>.b.</var> ad <var>.g.</var> quæ eÅ¿t <var>.d.</var> ad <var>.h.</var> ex præ-<lb/>Å¿uppoÅ¿ito verò <reg norm="cum" type="context">cÅ©</reg> Å¿ic Å¿e habeat <var>.a.</var> ad <lb/>b. Å¿icut <var>.c.</var> ad <var>.d.</var> erit ex primo poÅ¿tula-<lb/>to <reg norm="eadem" type="context">eadẽ</reg> proportio <var>.e.</var> ad <var>.g.</var> quæ eÅ¿t <var>.f.</var> <lb/>ad <var>.h</var>. </s> - <s xml:space="preserve">Nam proportio <var>.e.</var> ad <var>.g.</var> compo <lb/>nitur ex eis quæ Å¿unt <var>.e.</var> ad .a: et <var>.a.</var> ad <var>. - <pb o="201" rhead="IN QVINT. LIB. EVCLI." n="213" file="0213" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0213"/> - b.</var> et <var>.b.</var> ad <var>.g.</var> & Å¿imiliter proportio <var>.f.</var> ad <var>.h.</var> <reg norm="componitur" type="context">cõponitur</reg> ex eis quæſunt <var>.f.</var> ad <var>.c.</var> et <var>.c.</var> ad <var>.d.</var> <lb/>et <var>.d.</var> ad <var>.h</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0212-01" xlink:href="fig-0212-01a"> - <image file="0212-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0212-01"/> - </figure> - </div> - </div> - <div type="section" level="4" n="3"> - <head xml:space="preserve">THEOR.V. ET VI.</head> - <p> - <s xml:space="preserve"><emph style="sc">Circa</emph> 5. et .6. theorema nihil notandum occurrit.</s> - </p> - </div> - <div type="section" level="4" n="4"> - <head xml:space="preserve">THEOR. VII. VIII. IX.X. XI. XII. XIII.</head> - <p> - <s xml:space="preserve">THeoremata à .6. in .13. cum Å¿int de obiectis intelligibilibus, Å¿ine vllo medio, <lb/>ab intellectu cognitis, inter axiomata à me relata fuerunt .7. inquam quinti <lb/>Euclid. fecimus tertium PoÅ¿tulatum, .8. quintum, .9. quartum, .10. Å¿extum, .11. Å¿eptiÂ<lb/>mum, .12. octauum, .13. nonum.</s> - </p> - </div> - <div type="section" level="4" n="5"> - <head xml:space="preserve">THEOREM. XIIII.</head> - <p> - <s xml:space="preserve">QVartumdecimum Theorema ex Euclide demonÅ¿trabitur, mutatis tantum <lb/>theorematibus ab interprete notatis, ita vt loco .7. 8. noni, & decimi citetur <lb/>tertium .5. 4. et .6. poÅ¿tulatum à me propoÅ¿itum.</s> - </p> - </div> - <div type="section" level="4" n="6"> - <head xml:space="preserve">THEOR. XV.</head> - <p> - <s xml:space="preserve">QVintumdecimum Theorema Å¿ic demonÅ¿trabo; </s> - <s xml:space="preserve">Sit, exempli gratia, a. termi-<lb/>nus antecedens. et <var>.b.</var> conÅ¿equens, qui-<lb/>bus duo multiplices Å¿umantur <var>.c.</var> et <var>.d</var>. </s> - <s xml:space="preserve">Dico <lb/> - <anchor type="figure" xlink:label="fig-0213-01a" xlink:href="fig-0213-01"/> - eandem proportionem habiturum <var>.c.</var> ad <var>.d.</var> <lb/>quam <var>.a.</var> ad <var>.b.</var> habet. </s> - <s xml:space="preserve">In primis enim manife-<lb/>Å¿tè patet quamlibet partem ipÅ¿ius <var>.c.</var> habitu-<lb/>ram eandem proportionem cum qualibet par <lb/>te <var>.d.</var> quam habet <var>.a.</var> ad <var>.b.</var> quare ex .7. et .8. po <lb/>Å¿tulato propoÅ¿itum eluceÅ¿cet.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0213-01" xlink:href="fig-0213-01a"> - <image file="0213-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0213-01"/> - </figure> - </div> - </div> - <div type="section" level="4" n="7"> - <head xml:space="preserve">THEOREM. XVI.</head> - <p> - <s xml:space="preserve">SExtumdecimum theorema Å¿ic demonÅ¿trabitur. </s> - <s xml:space="preserve">Sit, exempli cauÅ¿a, eadem pro <lb/>portio <var>.a.</var> ad <var>.b.</var> quæ eÅ¿t <var>.c.</var> ad <var>.d</var>. </s> - <s xml:space="preserve">Dico <reg norm="quod" type="simple">ê™</reg> ita Å¿e habebit <var>.a.</var> ad <var>.c.</var> Å¿icut <var>.b.</var> ad <var>.d</var>. </s> - <s xml:space="preserve">Cogi-<lb/>temus itaque alterum iÅ¿torum terminorum <var>.c.</var> aut <var>.b.</var> medium inter <var>.a.</var> et <var>.d</var>. </s> - <s xml:space="preserve">quare <lb/>primum intelligamus <var>.b.</var> inter <var>.a.</var> et. d proportio ipÅ¿ius <var>.a.</var> ad <var>.d.</var> componetur ex ea quÄ™ <lb/>eÅ¿t <var>.a.</var> ad <var>.b.</var> & ea quæ eÅ¿t <var>.b.</var> ad <var>.d.</var> ex .12. poÅ¿tulato. </s> - <s xml:space="preserve">Et ex eodem, illa ipÅ¿a proportio <var>.<lb/>a.</var> ad <var>.d.</var> pariter componetur ex ea quæ eÅ¿t <var>.a.</var> ad <var>.c.</var> & ea quæ eÅ¿t <var>.c.</var> ad <var>.d.</var> Å¿umpto <var>.c.</var> pro <lb/>medio termino. </s> - <s xml:space="preserve">Ex quo Å¿equitur, aggregatum duarum proportionum, videlicet <var>.a.</var> <lb/>ad <var>.b.</var> et <var>.b.</var> ad <var>.d.</var> æquale eſſe aggregato <var>.a.</var> ad <var>.c.</var> et <var>.c.</var> ad <var>.d.</var> ex quibus aggregatis æqua-<lb/>libus Å¿i duas proportiones æquales Å¿ubtraxerimus, eam videlicet quæ eÅ¿t <var>.a.</var> ad <var>.b.</var> & il <lb/>lam quæ eÅ¿t <var>.c.</var> ad <var>.d.</var> Å¿upererunt duæ proportiones <lb/>inter Å¿e æquales. </s> - <s xml:space="preserve">erit enim proportio <var>.a.</var> ad <var>.c.</var> æqua <lb/> - <anchor type="figure" xlink:label="fig-0213-02a" xlink:href="fig-0213-02"/> - lis proportioni <var>.b.</var> ad <var>.d.</var> ex prima parte Å¿ecundi po <lb/>Å¿tulati diuiÅ¿im.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0213-02" xlink:href="fig-0213-02a"> - <image file="0213-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0213-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Alia etiam ratione idipÅ¿um <reg norm="demonÅ¿trari" type="context">demõſtrari</reg> poteÅ¿t, <lb/>Å¿umpto <var>.b.</var> pro medio termino inter <var>.a.</var> et .c: et <var>.c.</var> <lb/>pro termino medio inter <var>.b.</var> et <var>.d</var>. </s> - <s xml:space="preserve">quare propor-<lb/>tio <var>.a.</var> ad <var>.c.</var> componetur ex <var>.a.</var> ad <var>.b.</var> et <var>.b.</var> ad <var>.c.</var> illa <lb/>verò quæ eÅ¿t <var>.b.</var> ad <var>.d.</var> ex <var>.b.</var> ad <var>.c.</var> et <var>.c.</var> ad <var>.d.</var> ex .12. - <pb o="202" rhead="IO. BAPT. BENED." n="214" file="0214" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0214"/> - poÅ¿tulato. </s> - <s xml:space="preserve">Sed cum proportio <var>.a.</var> ad <var>.b.</var> Ä™qualis Å¿it <lb/> - <anchor type="figure" xlink:label="fig-0214-01a" xlink:href="fig-0214-01"/> - proportioni <var>.c.</var> ad <var>.d.</var> communis autem <var>.b.c</var>: propor <lb/>tio. </s> - <s xml:space="preserve">itaque <var>.a.</var> ad <var>.c.</var> æqualis erit <var>.b.</var> ad <var>.d.</var> ex Å¿ecunda <lb/>parte .2. poÅ¿tulati compoÅ¿itè, & Å¿ic habebimus pro <lb/>poÅ¿itum, ita quòd quotieÅ¿cunque <reg norm="dabuntur" type="context">dabÅ©tur</reg> .4. <reg norm="quam" type="context">quã</reg> <lb/>titates ex una parte proportionales, illæ ipſæ ex <lb/>altera proportionales erunt.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0214-01" xlink:href="fig-0214-01a"> - <image file="0214-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0214-01"/> - </figure> - </div> - </div> - <div type="section" level="4" n="8"> - <head xml:space="preserve">THEOR. XVII.</head> - <p> - <s xml:space="preserve">DEcimiÅ¿eptimi theorematis hæc eÅ¿t demonÅ¿tratio. </s> - <s xml:space="preserve">Ita Å¿e ha beat <var>a.c.b.</var> ad <var>.c.<lb/>b.</var> Å¿icut Å¿e habet <var>.d.f.e.</var> ad <var>.f.e</var>. </s> - <s xml:space="preserve">Probo ita Å¿e habere <var>.a.c.</var> ad <var>.c.b.</var> Å¿icut Å¿e habet <var>.d.<lb/>f.</var> ad <var>.f.e</var>. </s> - <s xml:space="preserve">Cogitemus itaque alterum terminum Å¿cilicet <var>.n.f.</var> qui Å¿ic Å¿e habeat. ad <var>.f.e.</var> <lb/>Å¿icut Å¿e habet <var>.a.c.</var> ad <var>.c.b</var>. </s> - <s xml:space="preserve">Quare ex præcedenti theoremate ita Å¿e habebit <var>.a.c.</var> ad <var>.n.<lb/>f.</var> Å¿icut Å¿e habet <var>.c.b.</var> ad <var>.f.e.</var> & ex .8 poÅ¿tulato ita Å¿e habebit <var>.a.c.b.</var> ad <var>.n.f.e.</var> Å¿icut Å¿e ha-<lb/>bet <var>.c.b.</var> ad <var>.f.e</var>. </s> - <s xml:space="preserve">Sed cum ex præſuppoÅ¿ito ita Å¿e habeat <var>.a.c.b.</var> ad <var>.c.b.</var> Å¿icut Å¿e habet <var>.<lb/>d.f.e.</var> ad <var>.f.e.</var> ideo ex præcedenti theoremate ita Å¿e habebit <var>.a.c.b.</var> ad <var>.d.f.e.</var> Å¿icut Å¿e ha <lb/>bet <var>.c.b.</var> ad <var>.f.e.</var> demonÅ¿tratum autem eÅ¿t ita Å¿e habere <var>.c.b.</var> ad <var>.f.e.</var> Å¿icut Å¿e habet <var>.a.c.b.</var> <lb/>ad <var>.n.f.e</var>. </s> - <s xml:space="preserve">Quare ex .7. poÅ¿tulato proportio <var>.a.c.b.</var> ad <var>.d.f.</var> e, æqualis erit proportioni <var>.a.<lb/>c.b.</var> ad <var>.n.f.e.</var> & ex .4. poÅ¿tulato <var>.d.f.e.</var> æqualis erit <var>.n.f.e</var>. </s> - <s xml:space="preserve">Itaque ex 3. poÅ¿tulato primi <lb/>Euclidis <var>.f.d.</var> æqualis erit <var>.n.f</var>. </s> - <s xml:space="preserve">Quamob <lb/>rem proportio <var>.a.c.</var> ad <var>.d.f.</var> Ä™qualis erit <lb/> - <anchor type="figure" xlink:label="fig-0214-02a" xlink:href="fig-0214-02"/> - proportioni <var>.a.c.</var> ad <var>.n.f.</var> ex Å¿ecunda par-<lb/>te tertij axiomatis præmiſſi. </s> - <s xml:space="preserve">Igitur ita <lb/>Å¿e habebit <var>.a.c.</var> ad <var>.d.f.</var> Å¿icut <var>.c.b.</var> ad <var>.f.e.</var> ex <lb/>7. poÅ¿tulato. </s> - <s xml:space="preserve">& Å¿ic ex præcedenti theo-<lb/>remate ita Å¿e habebit <var>.a.c.</var> ad <var>.c.b.</var> Å¿icut <var>.d.f.</var> ad <var>.f.e.</var> quod erat propoÅ¿itum: </s> - <s xml:space="preserve">QuotieÅ¿-<lb/>cunque igitur dabuntur .4. quantitates coniunctim proportionales, diuiÅ¿im quoque <lb/>proportionales erunt.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0214-02" xlink:href="fig-0214-02a"> - <image file="0214-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0214-02"/> - </figure> - </div> - </div> - <div type="section" level="4" n="9"> - <head xml:space="preserve">THEOREM. XVIII.</head> - <p> - <s xml:space="preserve">THeorema .18. hac ratione demonÅ¿trari poteÅ¿t. </s> - <s xml:space="preserve">Detur proportio <var>.a.c.</var> ad <var>.c.b.</var> Å¿i-<lb/>milis ei quæ eÅ¿t <var>.d.f.</var> ad <var>.f.e.</var> probo ita Å¿e habere <var>.a.c.b.</var> ad <var>.c.b.</var> Å¿icut Å¿e habet <var>.d.f.<lb/>e.</var> ad <var>.f.e</var>. </s> - <s xml:space="preserve">In primis notum eÅ¿t ex .16. theoremate ita Å¿e habiturum, <var>a.c.</var> ad <var>.d.f.</var> Å¿i <lb/>cut <var>.c.b.</var> ad <var>.f.e</var>. </s> - <s xml:space="preserve">Quare ex .8. poÅ¿tulato ita <lb/>Å¿e habebit <var>.a.c.b.</var> ad <var>.d.f.e.</var> Å¿icut <var>.c.b.</var> ad <var>.f.e.</var> <lb/> - <anchor type="figure" xlink:label="fig-0214-03a" xlink:href="fig-0214-03"/> - </s> - <s xml:space="preserve">Itaque ex .16. theoremate ita Å¿e habebit <var>.<lb/>a.c.b.</var> ad <var>.c.b.</var> Å¿icut <var>.d.f.e.</var> ad <var>.f.e</var>. </s> - <s xml:space="preserve">Quod erat <lb/>propoÅ¿itum. </s> - <s xml:space="preserve">QuotieÅ¿cunque igitur .4. <lb/>quantitates dabuntur vnius <reg norm="eiuÅ¿demque" type="simple">eiuÅ¿demqÌ;</reg> generis diÅ¿iunctim proportionales, coniun-<lb/>ctim quoque proportionales erunt.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0214-03" xlink:href="fig-0214-03a"> - <image file="0214-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0214-03"/> - </figure> - </div> - </div> - <div type="section" level="4" n="10"> - <head xml:space="preserve">THEOREM. XIX.</head> - <p> - <s xml:space="preserve">THeorema .19. Å¿atis quidem apud Euclidem demonÅ¿tratur: </s> - <s xml:space="preserve">eius tamentertia <lb/>pars commodius hac ratione demonÅ¿trari poterit (nempe) quod cum Å¿it pro- - <pb o="203" rhead="IN QVINT. LIB. EVCLI." n="215" file="0215" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0215"/> - portio <var>.a.</var> ad <var>.b.</var> quæ eÅ¿t <var>.c.</var> ad <var>.d.</var> probabo ita Å¿e habituram proportionem <var>.b.</var> ad <var>.a.</var> Å¿icut <lb/>Å¿e habet <var>.d.</var> ad <var>.c.</var> hoc argumento: </s> - <s xml:space="preserve">Å¿i <var>.a.</var> ad <var>.b.</var> ita Å¿e <lb/>habet Å¿icut <var>.c.</var> ad <var>.d.</var> ex .16. theoremate ita Å¿e ha <lb/> - <anchor type="figure" xlink:label="fig-0215-01a" xlink:href="fig-0215-01"/> - bebit <var>.a.</var> ad <var>.c</var>, Å¿icut <var>.b.</var> ad <var>.d</var>. </s> - <s xml:space="preserve">Quare Å¿ic Å¿e habebit <lb/>b. ad <var>.d.</var> Å¿icut <var>.a.</var> ad <var>.c</var>. </s> - <s xml:space="preserve">Itaque ex eodem .16. ita Å¿e <lb/>Å¿e habebit <var>.b.</var> ad <var>.a.</var> Å¿icut <var>.d.</var> ad <var>.c</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0215-01" xlink:href="fig-0215-01a"> - <image file="0215-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0215-01"/> - </figure> - </div> - </div> - <div type="section" level="4" n="11"> - <head xml:space="preserve">THEOREM. XX.</head> - <p> - <s xml:space="preserve">QVamuis .20. theorema apud Eucli. perfectè demonÅ¿tratum fuerit, poteÅ¿t ni-<lb/>hilominus & hac via demonÅ¿trari. </s> - <s xml:space="preserve">Sic Å¿e habeat proportio <var>.a.</var> ad <var>.b.</var> Å¿icut Å¿e <lb/>habet <var>.c.</var> ad <var>.d.</var> & proportio <var>.b.</var> ad <var>.e.</var> Å¿icut <var>.d.</var> ad <var>.<lb/>f</var>. </s> - <s xml:space="preserve">Dico <reg norm="quod" type="simple">ê™</reg> Å¿i <var>.a.</var> maius fuerit <var>.e.</var> pariter <var>.c.</var> maius <lb/> - <anchor type="figure" xlink:label="fig-0215-02a" xlink:href="fig-0215-02"/> - erit <var>.f.</var> & Å¿i <var>.a.</var> minus fuerit .e: c. <reg norm="quoque" type="simple">quoq;</reg> minus erit <lb/>f. Å¿in verò Ä™quale, <reg norm="ent" type="context">ẽt</reg> æquale erit. </s> - <s xml:space="preserve">Nam ex pri <lb/>mo poÅ¿tulato certi Å¿umus ita Å¿e habere pro <lb/><reg norm="portionem" type="context">portionẽ</reg> <var>.a.</var> ad <var>.e.</var> Å¿icut Å¿e habet proportio <var>.c.</var> ad <lb/>p. </s> - <s xml:space="preserve">Quare ex .12. theor <reg norm="propoÅ¿itum" type="simple context">ê“poÅ¿itÅ©</reg> <reg norm="manifeÅ¿tum" type="context">manifeÅ¿tÅ©</reg> erit.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0215-02" xlink:href="fig-0215-02a"> - <image file="0215-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0215-02"/> - </figure> - </div> - </div> - <div type="section" level="4" n="12"> - <head xml:space="preserve">THEOREM. XXI.</head> - <p> - <s xml:space="preserve">VIgeÅ¿imum primum theorema, Å¿atis apud Eucli. probatum, nihilominus præ-<lb/>Å¿cripto nunc modo demonÅ¿trari poterit.</s> - </p> - </div> - <div type="section" level="4" n="13"> - <head xml:space="preserve">THEOREM. XXII. XXIII.</head> - <p> - <s xml:space="preserve">DVO hæc theoremata in primum poÅ¿tulatum collegimus. <lb/></s> - <s xml:space="preserve">Sequentia verò cum exactè apud Eucli. demonÅ¿trentur non eÅ¿t cur nos in <lb/>ijs immoremur.</s> - </p> - <pb o="204" n="216" file="0216" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0216"/> - </div> - </div> - </div> - <div type="chapter" level="2" n="6"> - <head xml:space="preserve">PHYSICA, <lb/>ET MATHEMATICA <lb/>RESPONSA.</head> - <head style="it" xml:space="preserve">FO. BAPTISTAE BεNεDICTI PATRITII <lb/>Veneti, PhiloÅ¿ophi Mathematici.</head> - <div type="preface" level="3" n="1"> - <head xml:space="preserve">Ad Lectorem.</head> - <p> - <s xml:space="preserve">VT Nilmagis virtutis eÅ¿t proprium, quà m <lb/>agitari, & inceßabili motu prodeße. </s> - <s xml:space="preserve">Ac velu <lb/>ti fulgidum Å¿ydus ante oculos <reg norm="Å¿pectantium" type="context">Å¿pectantiÅ©</reg> com <lb/>micare. </s> - <s xml:space="preserve">Ita mihi mathematicis <reg norm="ijsque" type="simple">ijsq;</reg> maxi <lb/>mè philoÅ¿ophicis Å¿peculationibus dedito, Å¿apiÅ¿-<lb/>Å¿imè, ut in principium Å¿ummorum aulis, & <lb/>amplißimis ciuitatibus degenti, ubi multa Å¿em <lb/>per Nobilium mir a curioÅ¿itate, Å¿ciendi deÅ¿iderio, & conferendicu <lb/>piditate referta, uerÅ¿antur ingenia, contigit, modo ab his, modo ab <lb/>illis, aut uerbis tentari, aut literis prouocari ad diſſerendum, de <lb/>his, in quorum Å¿tudijs uerÅ¿amur. </s> - <s xml:space="preserve">Quarum concertationum & re <lb/>Å¿ponÅ¿ionum, quoniam non omnino indigna exiÅ¿timaui, quæmemoriæ<unsure/> <lb/>comendarentur, partem aliquam apud me conÅ¿eruaui. </s> - <s xml:space="preserve">Vbi uerò <lb/>per ocium licuit, collegi, relegi, ac tandem de manu mittere decreui. <lb/></s> - <s xml:space="preserve">Tum ut Å¿cientia ipÅ¿a quo magis diffundetur, creÅ¿cat; </s> - <s xml:space="preserve">& quicquid <lb/>ualeo, Å¿ine inuidia in communem utilitatem conferam. </s> - <s xml:space="preserve">Tum ut ui-<lb/>rorum præctantiβimorum, qui me Å¿uis interrogationibus excitaue <lb/>runt, quantum in me erit, gratitudinis ergo, nomina reddam im-<lb/>mortalia, & eorum exemplo alios, ocio Å¿ordidiore abiecto, quod Å¿olet <lb/>ourialium præcipuè excelÅ¿a ingenia corrumpere, ad Å¿ciÅ¿cit andum <lb/>conferendum, & diſſerendum, derebus <reg norm="Å¿erijs" type="lig">Å¿erijs</reg>, & quæuÅ¿ui aliquan-<lb/>do eße poßint, & <reg norm="quandoque" type="simple">quandoq;</reg> euulgari mereantur, alliciam. </s> - <s xml:space="preserve">Tuinte-<lb/>rim nostris laboribus fruere, & nostram diligentiam boni, & æqui <lb/>conÅ¿ule, & Vale.</s> - </p> - <pb o="205" n="217" file="0217" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0217"/> - </div> - <div type="section" level="3" n="1"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DETEMPORVM <lb/>EMENDATIONE <lb/>IO. BAPTIST AE BENEDICTI <lb/>Patritij Veneti, PhiloÅ¿ophi <lb/>Mathematici.</head> - <head style="it" xml:space="preserve">AD SERENISS. CMANVELEM PHILIB. <lb/>Allobrogum & Å¿ubalpinarum gentium Ducem <lb/>Inuictiβimum.</head> - <head xml:space="preserve">EPISTOLA.</head> - <p> - <s xml:space="preserve"><emph style="sc">MIrvm</emph>, Quà m lectione epiÅ¿tolæſeu (vt vocant) Breuis <var>.S.D.<lb/>N</var>. </s> - <s xml:space="preserve">Gregorij XIII. Pont. Max. quod ad me nuper tua CelÅ¿itu <lb/>do miÅ¿it ex Nicea, vt meam de ea re Å¿ententiam proferrem, <lb/>delectatus Å¿im; </s> - <s xml:space="preserve">ex quo, non tantum recta illius mens ac verè <lb/>Å¿ancta cogitatio, Å¿ed etiam aperta <reg norm="maximaque" type="simple">maximaqÌ;</reg>, Å¿i ad exitum per <lb/><reg norm="ducatur" type="simple">ducat̃</reg>, imo Å¿ummè neceſſaria vniuerÅ¿o orbi vtilitas percipi <lb/>poteÅ¿t; </s> - <s xml:space="preserve">qua de re memini cum CelÅ¿itudine tua aliquando Å¿er-<lb/>monem habuiſſe. </s> - <s xml:space="preserve">Vidi præterea cum ipÅ¿o breui tranÅ¿-<lb/>miſſum compendium Domini AloiÅ¿ij Lilij: </s> - <s xml:space="preserve">cuius mihi Å¿ententia perplacet, de corre <lb/>ctione eius diei, qui 134. quoque anno præter, neceſſitatem, gignitur. </s> - <s xml:space="preserve">qui Å¿anè dies <lb/>perpetuæ retrogradationis ingreſſus Solis in Zodiaci Å¿igna, cauÅ¿a fuit. </s> - <s xml:space="preserve">quod ita per-<lb/>Å¿picuè patebit. </s> - <s xml:space="preserve">Cum Numa Pompilius anni curÅ¿um correxit <reg norm="emendauitque" type="simple">emendauitqÌ;</reg>, ea Å¿anè <lb/>mente id videtur præſtitiſſe, vt principium Ianuarij primi menÅ¿is anni, præcisè in ip <lb/>Å¿o hyemalis Å¿oltitij puncto collocaretur. </s> - <s xml:space="preserve">quod hac tempeÅ¿tate, dictam ob cauÅ¿am <lb/>adeò retroceſſit, vt circa vndecimam diem Decembris eſſe reperiatur. </s> - <s xml:space="preserve">quod Å¿i cen-<lb/>teÅ¿imo trigeÅ¿imo quarto quoque anno detractus dies vnus fuiſſet, nihil erroris pror-<lb/>Å¿us accidiſſet. </s> - <s xml:space="preserve"><reg norm="Atque" type="simple">Atq;</reg> dies hic (vt alias CelÅ¿it. tuæ Å¿ignificaui) inde generatur, quod quar <lb/>to quoque anno addentes nos ad quarti anni dies .365. diem horarum .24. ob erro-<lb/>rem annuum horarum quinque minutorum .49. Å¿ecundorum ferè .16. (anni æqualis <lb/>Å¿iue medij) fallimur quarto quoque anno in minutis .42. <reg norm="Å¿ecundis" type="context">Å¿ecÅ©dis</reg> propè .56. amplius <lb/>quà m par Å¿it minutis Å¿cilicet .10. Å¿ecundis ferè .44. Å¿ingulis annis; </s> - <s xml:space="preserve">qui numerus .134. <lb/>multiplicatus, diem penè horarum .24. conÅ¿tituit; </s> - <s xml:space="preserve">penè inquam, quia minutum <reg norm="vnum" type="context">vnÅ©</reg> <lb/>deeſſet <reg norm="tantummodo" type="wordlist">tm̃modo</reg>, & Å¿ecunda .44. Å¿i <reg norm="decem" type="context">decẽ</reg> illa minuta, & .44. Å¿e cunda annua, exquiÅ¿ita eÅ¿ <lb/>Å¿ent atque perfecta; </s> - <s xml:space="preserve">quæ tamen differentia nullius adeo eſſet momenti, aut certè pe-<lb/>rexigui, vt vix exactis .111086. annis, diem vnum afferret. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> planè neceſſaria <lb/>eiuÅ¿modi eſſet emendatio, <reg norm="aptaque" type="simple">aptaqÌ;</reg> eius ratio à D. </s> - <s xml:space="preserve">Lilio oÅ¿tenditur, prout etiam Pe-<lb/>trus Pitatus VeronenÅ¿is tradidit, in eo, quem de vera anni quantitate tractatu con-<lb/>Å¿cripÅ¿it, nempe vt tribus primis centeÅ¿imis annis, centeÅ¿imus quiÅ¿que annus commu-<lb/>nis Å¿it, quartus Å¿ubÅ¿equens centeÅ¿imus intercalaris: </s> - <s xml:space="preserve">quod Å¿anè fierineceſſe eÅ¿t. </s> - <s xml:space="preserve">Nam - <pb o="206" n="218" file="0218" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0218"/> - cùm tribus centeÅ¿imis <reg norm="communibus" type="context">cõmunibus</reg>, tres quartas diei partes plus æquo detraxerimus, <lb/>non enim centeÅ¿imo quoque anno, Å¿ed centeÅ¿imo trigeÅ¿imo quarto, dictus dies de-<lb/>trahi debet, poÅ¿tquam tres integros dies, qui quadringentis detrahendi erant, tre-<lb/>centorum annorum Å¿pacio detraxerimus; </s> - <s xml:space="preserve">Å¿itqÌue 134. penè tertia pars .400. quarto <lb/>annorum centenario, tres quartæ diei partes recuperabuntur; </s> - <s xml:space="preserve">atque ita in fine qua-<lb/>dringentorum annorum omnia exactè Å¿uo loco reÅ¿tituta erunt. </s> - <s xml:space="preserve">Idcirco dictus iam <lb/>quadringenteÅ¿imus annus intercalaris & non communis conÅ¿tituendus erit, non alia <lb/>de cauÅ¿a, quam vt biſſexti ordinem Å¿equamur.</s> - </p> - <p> - <s xml:space="preserve">Is verò modus, qui à D. </s> - <s xml:space="preserve">Lilio traditus eÅ¿t, de ratione inueniendi Å¿ingulis menÅ¿ibus <lb/>Nouilunij diem, interdum fallere nos poſſet vno die; </s> - <s xml:space="preserve">prout Ianuario proximè <lb/>lapÅ¿o accidit; </s> - <s xml:space="preserve">quo ex præſcripto modo nouilunij, dies nonus illius menÅ¿is eſſe debuiÅ¿ <lb/>Å¿et, qui fuit tamen dies Å¿eptimus, Å¿exta decima hora cum dimidia poÅ¿t meridiem. </s> - <s xml:space="preserve">Ne <lb/>que etiam tutum eÅ¿t, via integrorum dierum, nulla habita horarum aut minutorum <lb/>ratione, nec minus ea, quæ à Pitato tradita eÅ¿t, mediorum Å¿eu æqualium <reg norm="motuum" type="context">motuÅ©</reg> pro <lb/>gredi: </s> - <s xml:space="preserve">At cenÅ¿erem potius veros motus Å¿equendos eſſe ex calculis exactarum tabu-<lb/>larum, quales Prutenicas eſſe iudico; </s> - <s xml:space="preserve">Et cum Å¿olius PaÅ¿chæ cauÅ¿a laboremus hac in <lb/>re, pleniluniorum verorum, in multos annos tabulas formarem, quæ æquinoctia ver <lb/>nalia Å¿equuntur, cum aſſignatione diei PaÅ¿chatis præcisè, prout fecit Pitatus; </s> - <s xml:space="preserve">non <lb/>via tamen æqualium pleniluniorum Å¿ed verorum. </s> - <s xml:space="preserve">Porrò quod ad PaÅ¿chatis cele-<lb/>brationem attinet, rationi conÅ¿entaneum eÅ¿t, concilij Niceni decretum ea de re Å¿er <lb/>uari, prima Å¿cilicet dominica die poÅ¿t primum plenilunium, quod æquinoctium ver-<lb/>nale Å¿equitur; </s> - <s xml:space="preserve">hoc tamen animaduerÅ¿o, Å¿i dictum plenilunium primum poÅ¿t æquino-<lb/>ctium contingens, <reg norm="diem" type="context">diẽ</reg> dominicum Å¿ortiretur; </s> - <s xml:space="preserve">nulla ratione tali die PaÅ¿cha celebran-<lb/>dum eſſe; </s> - <s xml:space="preserve">verum Å¿ubÅ¿equenti, ne cum HÄ™breis conÅ¿entiat EccleÅ¿ia ChriÅ¿ti: </s> - <s xml:space="preserve">quæ fuit <lb/>cauÅ¿a, vt in decreto concilij Niceni Å¿tatutum Å¿it, à quartadecima, in vigeÅ¿imam pri-<lb/>mam celebrari debere: </s> - <s xml:space="preserve">Quod mihi Petrus Pitatus non animaduertiſſe videtur, cum <lb/>ex <reg norm="eius" type="simple">eiê°</reg> <reg norm="Å¿ententia" type="context">Å¿entẽtia</reg> in Å¿uis tabulis die PaÅ¿chate declarata, huiuÅ¿ce anni PaÅ¿ca <reg norm="celebrandum" type="context">celebrandÅ©</reg> <lb/>fuerit .23. </s> - <s xml:space="preserve">Martij, ipÅ¿omet de plenilunij non tantum æqualis, Å¿ed veri.</s> - </p> - <p> - <s xml:space="preserve">Dies autem PaÅ¿chatum elapÅ¿orum, quos hactenus examinaui, reperi omnes con <lb/>cordare cum ea regula, quam nonnulli de die carnis priuij tradiderunt. </s> - <s xml:space="preserve">nempe pri-<lb/>mum diem martis poÅ¿t nouilunium Februarij, carnis priuij diem eſſe; </s> - <s xml:space="preserve">non autem <reg norm="cum" type="context">cÅ©</reg> <lb/><reg norm="sanctione" type="context">sãctione</reg> Patrum concilij Niceni, qua Å¿tatuerunt à vigeÅ¿ima prima Martii dirigen-<lb/>dum eſſe PaÅ¿chatis diem, vt potè qui Å¿ibi perÅ¿uaÅ¿erunt, circa eum diem <reg norm="æquinoctium" type="context">æquinoctiÅ©</reg> <lb/>perpetuò eſſe debere; </s> - <s xml:space="preserve">prout tunc temporis erat. </s> - <s xml:space="preserve">Non <reg norm="itaque" type="simple">itaq;</reg> error accidit, quod Pa <lb/>Å¿cha ex huiuÅ¿modi Å¿uppoÅ¿itione concilij, poÅ¿t vigeÅ¿imam primam lunÄ™ celebretur, <lb/>cum Å¿eruata regula concilij non fuerit. </s> - <s xml:space="preserve">Prout <reg norm="manifeÅ¿tum" type="context">manifeÅ¿tÅ©</reg> eÅ¿t de PaÅ¿chate anni .1566. <lb/>celebrato .14. </s> - <s xml:space="preserve">Aprilis (quÄ™ fuit .24. lunÄ™) quod .7. dicti menÅ¿is <reg norm="celebrandum" type="context">celebrãdum</reg> erat. </s> - <s xml:space="preserve">Tum <lb/>anno .1569. 10. </s> - <s xml:space="preserve">Aprilis Å¿olenne fuit PaÅ¿ca, quod tertia eiuÅ¿dem eſſe debuerat. </s> - <s xml:space="preserve">Anno <lb/>deinde 1572. 6. Aprilis, dies fuit PaÅ¿chatis, quæ .30. Martij futura erat, anno vero <lb/>1575. in tertiam Aprilis PaÅ¿cha incidit, caÅ¿urum in .27. Martii.</s> - </p> - <p> - <s xml:space="preserve">Cum igirur (vt ex diplomate ad CelÅ¿it. tuam miſſo patet) <var>S.D.N.</var> mens Å¿it <reg norm="atque" type="simple">atq;</reg> vo <lb/>luntas, ut quiÅ¿que liberè in medium proferat quid hac dere Å¿entiat: </s> - <s xml:space="preserve">quædam mihi <lb/>non omnino præmittenda occurrunt, quæ tantis cÅ“ptis non nihil adiumenti for-<lb/>taſſe adferre queant.</s> - </p> - <p> - <s xml:space="preserve">Atque illud in primis non tantum ut corrigatur Calendarium ob PaÅ¿cha cætera-<lb/>qÌue feÅ¿ta mobilia ab illo manantia, vt decreto concilij Niceni Å¿ancitum eÅ¿t, Å¿cilicet <lb/>vt ipÅ¿um PaÅ¿cha celebretur prima dominica poÅ¿t primum plenilunium, quod æqui- - <pb o="207" n="219" file="0219" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0219"/> - noctium vernale proximè Å¿equitur; </s> - <s xml:space="preserve">verum etiam quò anni principium emendetur, <lb/>Å¿cilicet vt ad Å¿uum verum principium reuocetur annus. </s> - <s xml:space="preserve">Nempè ad diem hyemalis <lb/>Å¿olÅ¿titij, quæ prima Ianuarij dies eſſe debet.</s> - </p> - <p> - <s xml:space="preserve">Deinde, tot dierum menÅ¿es conÅ¿tituantur, quot hac noÅ¿tra tempeÅ¿tate, Å¿ol in ipÅ¿is <lb/>Zodiaci Å¿ignis verÅ¿atur. </s> - <s xml:space="preserve">PoÅ¿tremò, quædam feÅ¿ta immobilia in alios dies <reg norm="transferam" type="context">transferã</reg> <lb/>tur, <reg norm="celebrenturque" type="simple">celebrenturqÌ;</reg> aptis temporibus: </s> - <s xml:space="preserve">quod à <var>.S.D.N.</var> mente diſſentire minimè vide-<lb/>tur. </s> - <s xml:space="preserve">cum non magis de feÅ¿tis mobilibus quam immobilibus agat, imo etiam planè <lb/>æquum Å¿it, vt habeatur vtrorunque ratio, quò Å¿tatutis temporibus celebrentur.</s> - </p> - <p> - <s xml:space="preserve">Vt autem ad primam Ianuarij <reg norm="diem" type="context">diẽ</reg> verum principium anni reuocetur; </s> - <s xml:space="preserve">cenÅ¿erem <lb/>ex eo anno, quem corrigere voluerimus, non modò dies .10. eſſe detrahendos, <reg norm="verum" type="context">verÅ©</reg> <lb/>etiam vnum & uiginti, illo ipÅ¿o anno; </s> - <s xml:space="preserve"><reg norm="idque" type="simple">idqÌ;</reg> duplici via; </s> - <s xml:space="preserve">aut partiendo menÅ¿es, atque <lb/>ex illis demendo eos dies, qui minus ad rem hanc facere videbuntur, ac tum rema-<lb/>neat annus trecentorum quadraginta quatuor dierum ita vt decem menÅ¿es Å¿int die. <lb/></s> - <s xml:space="preserve">rum duorum Å¿patio Å¿olito breuiores, alter menÅ¿is vno deficiat: </s> - <s xml:space="preserve">aut conÅ¿tituto <reg norm="Decem- bri" type="context">Decẽ-<lb/>bri</reg> dicti anni dierum decem, dies autem ille, qui decimum <reg norm="proximem" type="context">proximẽ</reg> Å¿equitur, Å¿it & <lb/>primus Ianuarij, & dies Å¿olÅ¿titij ob quam cauÅ¿am exiÅ¿timarem conÅ¿ultiſſimum eiuÅ¿ <lb/>modi annum eſſe mileÅ¿imum quingenteÅ¿imum Å¿eptuageÅ¿imum nonum. </s> - <s xml:space="preserve">Quo quam <lb/>primum <var>.S.D.N</var>. </s> - <s xml:space="preserve">Pontifex max. Å¿uis temporibus huius correctionis manifeÅ¿tos effe-<lb/>ctus experiri & perpendere, atque diÅ¿poÅ¿itionem anni non Å¿olum principio, Å¿ed <reg norm="cae teris" type="simple">cÄ™Â<lb/>teris</reg> partibus Å¿uis in vniuerÅ¿um tam concinnè <reg norm="apteque" type="simple">apteqÌ;</reg> reÅ¿pondere, & aÅ¿trorum moti-<lb/>bus, & EccleÅ¿iæ Å¿acroÅ¿anctæ Å¿anctionibus, Å¿e authore lætari poſſit.</s> - </p> - <p> - <s xml:space="preserve">Omnino <reg norm="itaque" type="simple">itaq;</reg> iudico detrahendos eſſe vnum & viginti dies elapÅ¿i erroris: </s> - <s xml:space="preserve">non de <lb/>cem tantum, quo hyemmalis conuerÅ¿io ad initium Ianuarij reuocetur; </s> - <s xml:space="preserve">idqÌue ne à <lb/>communi opinione de ipÅ¿o anni principio veritas diÅ¿crepet, quæ principium Ianua-<lb/>rij, anni principium arbitratur. </s> - <s xml:space="preserve">etenim cum credant omnes <reg norm="annum" type="context">annÅ©</reg> à Ianuario inchoa-<lb/>ri, veritas autem ipÅ¿a Å¿ic Å¿e habeat, vt nobis Å¿eptentrionalibus tunc inchoet annus, <lb/>cum ad nos Sol accedere incipit, aut dies augetur; </s> - <s xml:space="preserve">non conuenit principia eiuÅ¿mo-<lb/>di Å¿eparata & diÅ¿crepantia eſſe. </s> - <s xml:space="preserve">Et hanc fuiſſe Numæ Pompilio mentem credibile <lb/>eÅ¿t, qui ad annum Romuli decem menÅ¿ium, Ianuarium & Februarium addidit, vt <lb/>principium Ianuarij principium eſſet anni: </s> - <s xml:space="preserve">cuius rei argumentum eſſe poteÅ¿t, quod <lb/>C. </s> - <s xml:space="preserve">Iulij Cæſaris temporibus (qui multis annis poÅ¿t Numam fuit) <reg norm="atque" type="simple">atq;</reg> vti Pont. </s> - <s xml:space="preserve">Max. <lb/></s> - <s xml:space="preserve">corrigendorum feÅ¿torum curam Å¿uÅ¿cepit hyemale Å¿olÅ¿titium per aliquot dies retro-<lb/>ceſſerat; </s> - <s xml:space="preserve">nec mirum tamen eſſet, Å¿i Numæ temporibus, exactè prima Ianuarij die non <lb/>fuiſſet hyemale Å¿olÅ¿titium, adhuc pubeÅ¿cente in Italia AÅ¿tronomia.</s> - </p> - <p> - <s xml:space="preserve">HuiuÅ¿modi autem correctio dierum .21. poÅ¿t .2300. annos à Numa, quæ Å¿it per-<lb/>petuo Å¿eruitura, media emendatione ea, quæ de tribus centeÅ¿imis annis communi-<lb/>bus, & quarto intercalari, Å¿uperius propoÅ¿ita fuit, non repudianda ei videatur, qui <lb/>Å¿ciet, qua ratione Numæ Pompilij annus corrigeretur, octauo quoque anno, inter-<lb/>calando annum vltimum medijs diebus .90. quo prima dies Ianuarij ad verum prin <lb/>cipium anni, hoc eÅ¿t hyemale Å¿olÅ¿titium, reduceretur.</s> - </p> - <p> - <s xml:space="preserve">Alio item argumento cuique patere poteÅ¿t, priÅ¿cos Romanos Å¿tatuiſſe annum ab <lb/>hyemali Å¿olÅ¿titio initium Å¿umere, vt inquit Ouidius primo FaÅ¿torum.</s> - </p> - <p> - <s xml:space="preserve">Bruma noui prima eÅ¿t, <reg norm="veterisque" type="simple">veterisqÌ;</reg> nouiſſima Solis.</s> - </p> - <p> - <s xml:space="preserve">Principium capiunt PhÄ™bus, & annus idem.</s> - </p> - <p> - <s xml:space="preserve">co quod diem naturalem à medio noctis inchoarent, ab eo puncto Å¿cilicet, quo Sol <lb/>ad noÅ¿trum hemiÅ¿pherium accedere incipit.</s> - </p> - <p> - <s xml:space="preserve">Tribuebant igitur veteres diei, atque anno principium ab eo puncto, quo Sol - <pb o="208" n="220" file="0220" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0220"/> - ad nos accedit: </s> - <s xml:space="preserve">cum punctum Zodiaci, quod tropicum hyemalem Capricorni nobis <lb/>producit, reÅ¿pondeat puncto meridiani Å¿ub terra, in quo Sol Å¿emel in die reperitur: <lb/></s> - <s xml:space="preserve">Quòd apertè norunt hi, qui Å¿ub polo boreali conÅ¿tituti Å¿unt. </s> - <s xml:space="preserve"><reg norm="Atque" type="simple">Atq;</reg> facilè diÅ¿cerne <lb/>re poſſumus, diem Å¿cilicet & annum, quaÅ¿i Å¿ibi ad inuicem medio Å¿uarum partium <lb/>reÅ¿pondere; </s> - <s xml:space="preserve">Å¿olÅ¿titium inquam hyemale, mediæ nocti, æſtiuum meridiei, æquino-<lb/>ctium vernale ortui Solis, autumnale occaÅ¿ui. </s> - <s xml:space="preserve">Quam tamen Å¿imilitudinem, multò <lb/>quam nos manifeÅ¿tius deprehendunt, hi qui (ut diximus) Å¿ub polo borcali verÅ¿antur.</s> - </p> - <p> - <s xml:space="preserve">Quod Å¿i quis dubitet hac ratione correcto anno, quo nam pacto ad calculos coe-<lb/>leÅ¿tes motus medijs tabulis aÅ¿tronomicis hactenus in lucem æditis redigi poſſint, id <lb/>facilimum Å¿anè erit, exempli gratia; </s> - <s xml:space="preserve">aliquis planetÄ™ Å¿itum, aut alicuius Å¿tellÄ™ fixæ, quo <lb/>cunque die menÅ¿is anni correcti inuenire cupit, detrahat ex huiuÅ¿modi <reg norm="tempore" type="context">tẽpore</reg> dies <num value="21">.<lb/>21.</num> ab Aera ChriÅ¿ti, cum reÅ¿iduo Å¿upputet Å¿tellam, cuius Å¿itum Å¿cire deÅ¿iderat; </s> - <s xml:space="preserve">Å¿um-<lb/>pta quacunque tabula, Å¿upputatio erit exacta: </s> - <s xml:space="preserve">Cuius ratio cuilibet manifeÅ¿ta erit, <lb/>qui Å¿ciet annum vt potè .1579. dierum .344. tantummodo conÅ¿titutum fuiſſe. </s> - <s xml:space="preserve">Nam <lb/>in ijÅ¿dem locis cÅ“li prima die Ianuarij correcti, erunt Å¿tellæ quibus eſſe Å¿olebant .11 <lb/>Decembris præcedentis anni ex Å¿upputatione tabularum: </s> - <s xml:space="preserve">atque ita deinceps. </s> - <s xml:space="preserve">Alia <lb/>præterea via idem perfici poſſet inuentione omnium motuum cÄ™leÅ¿tium ipÅ¿o princi <lb/>pio anni .1580. correcti: </s> - <s xml:space="preserve">hoc Å¿tatuto, vt hi motus radices eſſent Aeræ <var>S.D.N.</var> Grego <lb/>rij XIII. quod Å¿i alio <reg norm="tempore" type="context">tẽpore</reg> quiÅ¿piam motus cÄ™leÅ¿tes ad calculos redigere voluerit, <lb/>Å¿upputabit ab Aera huiuÅ¿modi, quæ anno .1580. principium habuerit: </s> - <s xml:space="preserve">Quæ vt nobi <lb/>lius nomen Å¿ortiatur, <reg norm="idque" type="simple">idqÌ;</reg> merito ex nomine Gregorij. XIII. Pont. Max. appelletur; <lb/></s> - <s xml:space="preserve">exemplo antiquarum, quæ ex Principum nominibus Å¿unt appellate: </s> - <s xml:space="preserve">vt tanto Pontifi <lb/>ci, cùm ex alijs multi, tum etiam ex hac non infima re, inter mortales immortale no <lb/>men comparetur. </s> - <s xml:space="preserve">Ei verò Å¿ummæ, quæ ex huiuÅ¿modi Aera Gregoriana ex tabu-<lb/>lis colligetur, ipÅ¿iuÅ¿met Aeræ radices addantur, vt exactus calculus habeatur. </s> - <s xml:space="preserve">Et <lb/>hæc Å¿it primæ Å¿ententiæ noÅ¿træ explicatio.</s> - </p> - <p> - <s xml:space="preserve">Altera erit numerum dierum menÅ¿ium anni alia ratione quam nunc Å¿e habeat, or <lb/>dinandum eſſe: </s> - <s xml:space="preserve">nempe vt Ianuarius, Nouember <reg norm="atque" type="simple">atq;</reg> December dies .29. Å¿inguli con <lb/>tineant, Februarius, Martius, & October .30. </s> - <s xml:space="preserve">Aprilis, Maius, AuguÅ¿tus, & September <lb/>dies .31. Iunius, ac Iulius .32. atque id hac potiſſimum de cauÅ¿a, vt Sol unum <reg norm="quodque" type="simple">quodqÌ;</reg> <lb/>Å¿ignum calendis menÅ¿rum ingredi poſſit. </s> - <s xml:space="preserve">Nam detractis (ut dictum eÅ¿t) diebus .21. <lb/>& reuocato ingreſſu Solis in principium Capricorni ad principium Ianuarij, in quo <lb/>Å¿igno hac noÅ¿tra tempeÅ¿tate, Sol, dies propè .29. & quartam vnam verÅ¿atur: </s> - <s xml:space="preserve">Å¿i Ianua <lb/>rius .29. dies continebit, exactis hiÅ¿ce diebus, ingredietur Aquarium circa princi-<lb/>pium Februarii; </s> - <s xml:space="preserve">hæret autem hoc noÅ¿tro ſæculo in Aquario Sol dies propè .29. cum <lb/>dimidio; </s> - <s xml:space="preserve">quare Å¿i Februarius erit .30. dierum, elapÅ¿is ipÅ¿is diebus, Sol ingredietur pi-<lb/>Å¿ces circa principium Martii: </s> - <s xml:space="preserve">& Å¿ic de cæteris.</s> - </p> - <p> - <s xml:space="preserve">Quamobrem Å¿i generali correctione annus emendandus erit, pulcherrimè acci-<lb/>det, Å¿i menÅ¿es anni cum duodecim partibus cÅ“leÅ¿tibus, itineris annui Solis, concor-<lb/>dauerint; </s> - <s xml:space="preserve"><reg norm="eiÅ¿que" type="simple">eiÅ¿qÌ;</reg> aptè reÅ¿ponderint. </s> - <s xml:space="preserve">Qua ex re, variÄ™ vtilitates promanabunt, pręſertim <lb/>Nautis, Agricolis, Medicis, & alijs qui vera principia, & interualla temporum per <lb/>Å¿pecta habebunt: </s> - <s xml:space="preserve">terminos item & interualla incrementi & diminutionis dierum & <lb/>noctium, & eorundem æqualitatis. </s> - <s xml:space="preserve">Exempli cauÅ¿a, Å¿cient omnes principium Ianua-<lb/>rij, eſſe non modo anni principium, verum etiam hyemis, eſſe minimam anni diem, <lb/>& eius noctem maximam; </s> - <s xml:space="preserve">principium incrementi diei, & diminutionis noctis; </s> - <s xml:space="preserve">atque <lb/>etiam omnia illa, quæ ex huiuÅ¿modi conuerÅ¿ione Solis ad nos dependent. </s> - <s xml:space="preserve">Pariter <lb/>Å¿cient omnes primam diem Iulij, non tantum æqualiter annum diuidere, Å¿ed prin - <pb o="209" rhead="EPISTOLA." n="221" file="0221" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0221"/> - cipìum quoque eſſe ęſtatis, maximam diem, noctem minimam totius anni; </s> - <s xml:space="preserve">princi-<lb/>pium diminutionis diei & incrementi noctis, vnà <reg norm="etiam" type="context">etiã</reg> ea, quÄ™ Solis conuerÅ¿ionem <lb/>ad auÅ¿trales Å¿equuntur.</s> - </p> - <p> - <s xml:space="preserve">Neconon intelliget vnuÅ¿quiÅ¿que primam diem Aprilis, <reg norm="primamque" type="simple">primamqÌ;</reg> Octobr. æqui-<lb/>noctiorum dies eſſe; </s> - <s xml:space="preserve">primam autem diem Aprilis, initium veris; </s> - <s xml:space="preserve">Octobris Autumni; <lb/></s> - <s xml:space="preserve">Item Aprilis diem eſſe eum, quo dies noctis prolixitatem vincere incipit: </s> - <s xml:space="preserve">Octobris, <lb/>quo nox diei longitudinem Å¿uperat, & alia huiuÅ¿modi, quæ ab æquinoctijs <reg norm="dependent" type="context context">depẽdẽt</reg>.</s> - </p> - <p> - <s xml:space="preserve">Si vero quiÅ¿piam obijciat, modum hunc noÅ¿trum & ordinem perpetuum eſſe non <lb/>poſſe, ob motum augis Solis; </s> - <s xml:space="preserve">quod punctum cum fuerit in principio Capricorni, <reg norm="tunc" type="context">tÅ©c</reg> <lb/>Sol hærebit in Å¿igno Sagittarij .32. diebus, totidem in Capricorno, in Geminis vero <lb/>29. totidem in Cancro; </s> - <s xml:space="preserve">ex quo Å¿equetur prioribus <reg norm="contrarius" type="context">cõtrarius</reg> effectus; </s> - <s xml:space="preserve">huic ego reÅ¿pon <lb/>debo, tale <reg norm="quidpiam" type="context">quidpiã</reg> non euenturum, niÅ¿i exactis ab hoc anno annis .240@0. quod Å¿i <lb/>mundus poÅ¿thac totidem annis, quot fuit antehac, perdurauerit, punctus augis non <lb/>amplius à ſitu præſenti, quà m .45. gradibus diÅ¿tabit. </s> - <s xml:space="preserve">Verum demus <reg norm="modum" type="context">modÅ©</reg> noÅ¿trum <lb/>& regulam in annos ter, aut quater mille Å¿ubÅ¿eruire poſſe, nec amplius, certè hoc <lb/>toto tempore nullius momenti penè erit, quæ accidere poterit mutatio, tametÅ¿i ela-<lb/>pÅ¿is quatuor millibus annorum Februarius eſſe debebit .29. dierum. Aprilis & No-<lb/>uember .30. Iunius & October .31. AuguÅ¿tus .32. in aliis verò menÅ¿ibus nihil mutan-<lb/>dum erit. </s> - <s xml:space="preserve">Ecce quam Å¿it nullius momenti mutatio.</s> - </p> - <p> - <s xml:space="preserve">Quæ Å¿i Iulij Cæſaris temporibus fuiſſent animaduerÅ¿a nunquam omiſſa fuiſſent, <lb/>Å¿ed Å¿cientiæ AÅ¿tronomicæ nondum (vt ita dicam) confirmata ætas, cum alibi, <reg norm="tum" type="context">tÅ©</reg> maxi <lb/>mè in Italia, quo minus hæc aut Å¿cirentur aut Å¿tatuerentur impediebat.</s> - </p> - <p> - <s xml:space="preserve">Tertia ratio eÅ¿t, vt non <reg norm="Å¿olum" type="context">Å¿olÅ©</reg> feÅ¿ta mobilia, verum <reg norm="ent" type="context">ẽt</reg> immobilia ad meliorem regu <lb/>lam (ut dictum eÅ¿t) reuocentur, Å¿i Å¿uis temporibus celebranda erunt. </s> - <s xml:space="preserve">Quorum <reg norm="primum" type="context">primÅ©</reg> <lb/>eÅ¿t Natiuitas Domini, & quæ ab ea pendent; </s> - <s xml:space="preserve">nempe Circuncifio, Epiphania, Purifi-<lb/>catio, Annunciatio, & Natiuitas Io. BaptiÅ¿tæ. </s> - <s xml:space="preserve">ita vt dies Natalis Domini celebretur <lb/>prima die anni, cum Dei filius naÅ¿ci voluerit circa verum principium anni, quod à <lb/>Å¿olÅ¿titio hyemali initium ducit, & in ipÅ¿o principio diei naturalis ex <reg norm="Romanorum" type="context">RomanorÅ©</reg> Å¿en <lb/>tentia, media Å¿cilicet nocte, tanquam qui Å¿ummæ lætitiæ principium, poÅ¿t longos & <lb/>graues filiorum Adæ mærores, eſſet allaturus. </s> - <s xml:space="preserve">Nec forÅ¿an Ianuarij nomini, à vete-<lb/>ribus Iano bifronti dicati hæc mutatio non conueniret, cum in ip Å¿o Å¿eruatore, duæ <lb/>veluti frontes & formæ vnitæ Å¿int, duæ Å¿cilicet naturæ diuina & humana. </s> - <s xml:space="preserve">Hac ratio <lb/>ne abuÅ¿us tolletur, natus ex diuerÅ¿is moribus Tabulariorum, quorum alij monumen-<lb/>ta, Å¿eu quæ uocant InÅ¿trumenta, à die Natiuitatis Domini incohant, alij à Circunci <lb/>Å¿ione, alij à Calendis Martij, nonnulli à PaÅ¿chate; </s> - <s xml:space="preserve">quæ varietas innumerabiles lites <lb/>affert & abuÅ¿us propè infinitos, ob dubiam & ancipitem Å¿cripturam. </s> - <s xml:space="preserve">Indictionum <lb/>præterea ordini, hic noÅ¿ter modus nihil officiet; </s> - <s xml:space="preserve">celebrato Natali celebrabitur Cir-<lb/>cunciÅ¿io octaua Ianuarij. </s> - <s xml:space="preserve">Epiphania .13. eiuÅ¿dem. </s> - <s xml:space="preserve">Purificatio .11. Februarij quæ erit <lb/>40: </s> - <s xml:space="preserve">dies à Natiuitate Å¿eruatoris. </s> - <s xml:space="preserve">Prima Aprilis Annunciatio Virginis Å¿olennis erit, <lb/>ipÅ¿o nempè die æquinoctij, natiuitas Diui Io: </s> - <s xml:space="preserve">BaptiÅ¿tÄ™ celebrabitur Prima Iulij die <lb/>quæ erit Å¿olÅ¿titij æſtiui, cum illa diminutionem capit. vtrectè Diuus AuguÅ¿tinus il-<lb/>la verba Io: </s> - <s xml:space="preserve">BaptiÅ¿tÄ™ interpretatus fuerit. </s> - <s xml:space="preserve">Illum opportet creÅ¿cere, me autem minui: <lb/></s> - <s xml:space="preserve">in quibus Å¿ic tantus Doctor philoÅ¿ophatur, vt tempus etiam natiuitatis Å¿erui & do-<lb/>mini præclare notet dicens, natus eÅ¿t Å¿eruus cum decreÅ¿cunt dies, natus eÅ¿t Dominus <lb/>cùm creÅ¿cere incipiunt.</s> - </p> - <p> - <s xml:space="preserve">InÅ¿ignes etiam Theologi admonuerunt habendam rationem eſſe nonnullorum <lb/>feÅ¿torum, vt Diui Antonij, diuorum Fabiani & SebaÅ¿tiani, & aliorum Å¿anctorum, - <pb o="210" rhead="IO. BAPT. BENED." n="222" file="0222" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0222"/> - fi forte in octauam Epiphaniæ inciderint: </s> - <s xml:space="preserve">Verum hÄ™c <var>.S.D.N.</var> curæ erunt, ut in aptiÅ¿ <lb/>Å¿ima tempora transferantur.</s> - </p> - <p> - <s xml:space="preserve">Admonuerunt præterea transferendos eſſe dies feÅ¿tos Beati Stephani, Ioannis, <lb/>& Innocentium, vt quemadmodum factum eÅ¿t hactenus, diem natalis proximè Å¿e <lb/>quantur, ob multorum Doctorum, non recentium modo, Å¿ed etiam antiquorum ob <lb/>Å¿eruantiam; </s> - <s xml:space="preserve">qui Å¿uis omelijs & concionibus multa piè, de myÅ¿teriis Å¿ucceſſionis FeÅ¿to <lb/>rum huiuÅ¿imodi tradiderunt.</s> - </p> - <p> - <s xml:space="preserve">Cuperent etiam præclari Theologi diem Aſſumptionis Beatæ virginis incidere <lb/>in primam Septembris, Natiuitatem autem in .25. vt quemadmodum toto illo men <lb/>Å¿e in Å¿igno Virginis Å¿ol verÅ¿abitur, ita EccleÅ¿ia Der in cÄ™lebrandi tantæ Virginis ma <lb/>tris Deilaudibus occupetur.</s> - </p> - <p> - <s xml:space="preserve">Atque hÄ™c Å¿unt SerenisÅ¿ime Princeps, quÄ™ longa & attenta cogitatione à me exa <lb/>minata, atque perpenÅ¿a fuerunt; </s> - <s xml:space="preserve">quæſitam diligenter & accuratè expendentur ab <lb/>his, quorum intereÅ¿t, quam mihi apta & rationi conÅ¿entanea, ac vera penitus, imo <lb/>(quod me magis afficit) etiam tibi viÅ¿a fuerunt; </s> - <s xml:space="preserve">non dubito quin placitura Å¿int; </s> - <s xml:space="preserve">& vo <lb/>tis Å¿ummi Pont. aliqua ex parte Å¿atisfactura. </s> - <s xml:space="preserve">eò magis quòd te iubente, & cogitata à <lb/>me, & Å¿cripta fuerint. </s> - <s xml:space="preserve">Vale Princeps Sereniſſime, & qua Å¿oles hylaritate cÄ™tera no <lb/>Å¿tra, etiam has breues vigilias Å¿uÅ¿cipe & foue. </s> - <s xml:space="preserve">Dat. AuguÅ¿tæ Taurinorum Kal. <lb/></s> - <s xml:space="preserve">Aprilis. <num value="1578">MDLXXVIII</num>. </s> - </p> - <p> - <s xml:space="preserve">T. CelÅ¿itudinis.</s> - </p> - <p> - <s xml:space="preserve">Deditiſſimus Mathematicus.</s> - </p> - <p> - <s xml:space="preserve">Io. Bap. Benedictus.</s> - </p> - <pb o="211" n="223" file="0223" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0223"/> - </div> - </div> - <div type="section" level="3" n="2"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE CIRCVLO <lb/>AMBIENTE QVADRILATERVM.</head> - <head xml:space="preserve">AD SERENISS. CAROLVM EMANVELEM <lb/>Pedemontis Principem.</head> - <p> - <s xml:space="preserve">PRoblema quod à celÅ¿itudine tua nobis proponitur non Å¿olum poſſibile eÅ¿t, Å¿ed <lb/>facile etiam ad Å¿oluendum, hoc eÅ¿t quod circulus talis inueniatur, qui poſſit cir <lb/>cunÅ¿cribere, Å¿eu capere quadrilaterum ex quatuor datis rectis lineis terminatum, vel <lb/>Å¿ic, datis quatuor rectis lineis ex quibus quadrilaterum poſſit eftici, tale efficiatur vt <lb/>circa ipÅ¿um, circulus poſſit circunÅ¿cribi.</s> - </p> - <p> - <s xml:space="preserve">Sint igitur .4. lineæ propoÅ¿itæ <var>.b.d</var>: <var>q.b</var>: <var>a.q</var>: et <var>.a.d.</var> ex <reg norm="quibus" type="simple">quibê°</reg> poſſibile Å¿it quadrilate <lb/>rum conÅ¿titui, tale vero conÅ¿tituatur, vt aliquis circulus poſſit ipÅ¿um circunÅ¿cribere. <lb/></s> - <s xml:space="preserve">imaginemur autem hoc factum eſſe, quod quidem quadrilaterum Å¿it <var>.a.d.q.b.</var> cuius <lb/> - <anchor type="figure" xlink:label="fig-0223-01a" xlink:href="fig-0223-01"/> - <anchor type="figure" xlink:label="fig-0223-02a" xlink:href="fig-0223-02"/> - <anchor type="figure" xlink:label="fig-0223-03a" xlink:href="fig-0223-03"/> - <anchor type="figure" xlink:label="fig-0223-04a" xlink:href="fig-0223-04"/> - <anchor type="figure" xlink:label="fig-0223-05a" xlink:href="fig-0223-05"/> - <pb o="212" rhead="IO. BAPT. BENED." n="224" file="0224" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0224"/> - diametri Å¿int <var>.q.d.</var> et <var>.a.b.</var> quæ Å¿e inuicem interÅ¿ecent in puncto <var>.o.</var> vnde cum anguli <lb/>contra Å¿e poÅ¿iti circa <var>.o.</var> æquales inuicem Å¿int ex .15. primi Eucli. </s> - <s xml:space="preserve">& angulus <var>.a.q.d.</var> æ-<lb/>qualis angulo <var>.a.b.d.</var> & angulus <var>.q.b.a.</var> æqualis angulo <var>.q.d.a.</var> et <var>.b.q.d.</var> angulo <var>.b.a.d.</var> <lb/>ex .20. tertij tunc triangulus <var>.a.o.q.</var> Å¿imilis erit triangulo <var>.d.o.b.</var> et <var>.q.o.b.</var> Å¿imilis trian-<lb/>gulo <var>.a.o.d.</var> ex definitione. </s> - <s xml:space="preserve">Vnde eadem proportio erit ipÅ¿ius <var>.q.o.</var> ad <var>.b.o.</var> quæ ipÅ¿ius <lb/><var>q.a.</var> ad <var>.b.d.</var> & ipÅ¿ius <var>.b.o.</var> ad <var>.o.d.</var> eadem quæ <var>.q.b.</var> ad <var>.a.d.</var> & ipÅ¿ius <var>.q.o.</var> ad <var>.o.a.</var> eadem <lb/>quæ <var>.q.b.</var> ad <var>.a.d.</var> proportio igitur <var>.q.o.</var> ad <var>.o.d.</var> cognita nobis erit, vt compoÅ¿ita ex <lb/>ea quæ eÅ¿t <var>.q.o.</var> ad <var>.o.b.</var> ex <var>.o.b.</var> ad <var>.o.d.</var> quæ nobis cognitÄ™ Å¿unt, mediante <lb/>proportione ipÅ¿ius <var>.q.a.</var> ad <var>.b.d.</var> & ipÅ¿ius <var>.q.b.</var> ad <var>.a.d.</var> proportio Å¿imiliter ipÅ¿ius <var>.b.o.</var> <lb/>ad <var>.o.a.</var> nobis cognita erit, vt compoÅ¿ita ex proportione ipÅ¿ius <var>.b.o.</var> ad <var>.o.q.</var> & <lb/>ipÅ¿ius <var>.o.q.</var> ad <var>.o.a.</var> cognitis, mediante proportione ipÅ¿ius <var>.b.d.</var> ad <var>.q.a.</var> & ipÅ¿ius <var>.q.b.</var> ad <lb/><var>a.d.</var> cum <reg norm="autem" type="context">autẽ</reg> proportio ipÅ¿ius <var>.q.o.</var> ad <var>.o.b.</var> nobis cognita Å¿it, </s> - <s xml:space="preserve">tunc nobis cognita erit <lb/>proportio ipÅ¿ius <var>.q.d.</var> ad <var>.a.b</var>. </s> - <s xml:space="preserve">Nam ut <var>.q.o.</var> ad <var>.o.b.</var> eÅ¿t vt <var>.a.o.</var> ad <var>.o.d.</var> ex Å¿imilitudine, <lb/></s> - <s xml:space="preserve">quare proportio compoÅ¿iti ex primo, & quarto terminorum ad compoÅ¿itum ex .2. & <lb/>tertio, cognita erit. </s> - <s xml:space="preserve">Å¿ed quod fit ex <var>.q.d.</var> in <var>.a.b.</var> cognitum nobis eÅ¿t, vt æquale duobus <lb/>productis, hoc eÅ¿t ex <var>.q.a.</var> in <var>.d.b.</var> & ex <var>.q.b.</var> in <var>.d.a.</var> ex Å¿ecunda primi AlmageÅ¿ti. </s> - <s xml:space="preserve">quæ <lb/>producta nobis cognita Å¿unt, cum nobis data Å¿int eorum latera. </s> - <s xml:space="preserve">Quapropter facta <lb/>cum fuerit figura quadrilatera rectangula Å¿imilis alicui alterirectangulæ figuræ pro <lb/>ductæ à duobus lateribus inuicem ita proportionatis, vt Å¿e habet <var>.q.d.</var> ad <var>.a.b.</var> æqua-<lb/>lis tamen duobus productis, hoc eÅ¿t producto ex <var>.q.a.</var> in <var>.d.b.</var> & ex <var>.q.b.</var> in <var>.d.a.</var> ex <lb/>doctrina, 25. Å¿exti Eucli quæ quidem figura, exempli gratia, Å¿it <var>.u.t.</var> eius verò latera <lb/>Å¿int <var>.u.n.</var> et <var>.n.t.</var> Hæc enim dico æqualia eſſe <var>.q.d.</var> et <var>.b.a.</var> hoc eÅ¿t <var>.n.t.</var> maius maio-<lb/>ri <var>.b.a.</var> et <var>.u.n.</var> minus minori <var>.q.d</var>. </s> - <s xml:space="preserve">Quod ita probabo. </s> - <s xml:space="preserve">cogitemus rectangulum <var>.s.r.</var> <lb/>productum eſſe ex duobus lateribus <var>.q.d.</var> et <var>.a.b.</var> Å¿ed, <var>s.n.</var> æqualis Å¿it <var>.q.d.</var> et <var>.n.r.</var> æqua-<lb/>lis <var>.a.b.</var> <reg norm="Å¿intque" type="simple">Å¿intqÌ;</reg> duæ lineæ <var>.s.n.</var> et <var>.n.t.</var> inuicem directè coniunctæ, vnde <var>.u.n.</var> directè <lb/>coniuncta etiam erit cum <var>.n.r.</var> ex quo rectangulum <var>.u.t.</var> æquale erit rectangulo <var>.s.r.</var> ex <lb/>communi conceptu, <reg norm="eademque" type="simple">eademqÌ</reg> proportio erit <var>.u.n.</var> ad <var>.n.t.</var> quÄ™ <var>.s.n.</var> ad <var>.n.r.</var> eo <reg norm="quod" type="simple">ê™</reg> ita fa-<lb/>ctum fuit, cum autem ita Å¿it <var>.u.n.</var> ad <var>.n.t.</var> vt <var>.s.n.</var> ad <var>.n.r.</var> </s> - <s xml:space="preserve">tunc permutando ita erit <var>.n.t.</var> ad <lb/><var>n.r.</var> vt <var>.u.n.</var> ad <var>.n.s.</var> Å¿ed quia ita eÅ¿t <var>.u.n.</var> ad <var>.n.r.</var> vt <var>.s.n.</var> ad <var>.n.t.</var> ex 15. Å¿exti, </s> - <s xml:space="preserve">tunc permutan <lb/>do ita erit <var>.n.r.</var> ad <var>.n.t.</var> vt <var>.n.u.</var> ad <var>.n.s.</var> </s> - <s xml:space="preserve">quare ex 11. quinti ita erit <var>.n.t.</var> ad <var>.n.r.</var> vt <var>.n.r.</var> ad <var>.n.<lb/>t.</var> quapropter ex neceſſitate Å¿equitur <var>.n.t.</var> et <var>.n.r.</var> inuicem æquales eſſe, et <var>.u.n.</var> Å¿imiliter <lb/>cum <var>.n.s</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0223-01" xlink:href="fig-0223-01a"> - <image file="0223-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0223-01"/> - </figure> - <figure xlink:label="fig-0223-02" xlink:href="fig-0223-02a"> - <image file="0223-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0223-02"/> - </figure> - <figure xlink:label="fig-0223-03" xlink:href="fig-0223-03a"> - <image file="0223-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0223-03"/> - </figure> - <figure xlink:label="fig-0223-04" xlink:href="fig-0223-04a"> - <image file="0223-04" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0223-04"/> - </figure> - <figure xlink:label="fig-0223-05" xlink:href="fig-0223-05a"> - <image file="0223-05" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0223-05"/> - </figure> - </div> - <p> - <s xml:space="preserve">Inuentæ nunc cum fuerint duæ diametri <var>.q.d.</var> et <var>.a.b.</var> ipÅ¿ius quadrilateri, difficile <lb/>non erit eius angulos inuenire, eo <reg norm="quod" type="simple">ê™</reg> mediante <var>.a.b.</var> cognita, Å¿imul cum <var>.b.d.</var> et <var>.a.d.</var> da <lb/>tis, faciemus triangulum <var>.a.b.d.</var> vel <reg norm="mediante" type="context">mediãte</reg> <var>.q.d.</var> et <var>.q.a.</var> et <var>.a.d.</var> cognitis faciemus <reg norm="triam" type="context">triã</reg> <lb/>gulum <var>.a.q.d.</var> ex .22. primi. </s> - <s xml:space="preserve">Vnde cum centrum circuli circunÅ¿criptibilis cuiuÅ¿uis di-<lb/>ctorum triangulorum ex quinta quarti inuentum fuerit, triangulum reliquum, ab eo <lb/>dem circulo circunÅ¿criptum erit, ex communi Å¿cientia.</s> - </p> - <p> - <s xml:space="preserve">SEd vt ipÅ¿a operatio facilior fiat, Sint eÄ™dem lineæ <var>.b.d</var>: <var>b.q</var>: <var>a.q.</var> et <var>.a.d.</var> ex quibus <lb/>poſſit <reg norm="quadrilaterum" type="context">quadrilaterÅ©</reg> effici. </s> - <s xml:space="preserve">Videatur deinde primò quas volumus oppoÅ¿itas Å¿ibi <lb/>inuicem eſſe, ponatur ergò ut <var>.q.a.</var> et <var>.b.d.</var> velimus oppoÅ¿itas inuicem facere, et <var>.q.b.</var> <lb/>cum <var>.a.d.</var> Å¿imiliter, accipiemus nunc <var>.K.</var> cuiuÅ¿uis magnitudinis, cui comparetur <var>.e.</var> <lb/>ita proportionata, vt <var>.q.b.</var> eÅ¿t ipÅ¿i <var>.a.d.</var> ex doctrina .10. Å¿exti Eucli. vel accipiatur <var>.a.d.</var> <lb/>vice <var>.K.</var> et <var>.q.b.</var> vice <var>.e.</var> quod idem erit, & expeditius, inuenietur Å¿imiliter <var>.h.</var> ita pro-<lb/>portionata ad <var>.e.</var> et <var>.g.</var> ad <var>.k.</var> vt <var>.b.d.</var> eÅ¿t ad <var>.q.a.</var> vel <var>.g.</var> ad <var>.h.</var> vt <var>.a.d.</var> ipÅ¿i <var>.q.b.</var> quod <reg norm="idem" type="context">idẽ</reg> erit.</s> - </p> - <p> - <s xml:space="preserve">Hoc facto coniungantur inuicem directè <var>.g.</var> et <var>.e.</var> quarum compoÅ¿itum Å¿it <var>.g.e.</var> & <lb/>ita duæ <var>.K.</var> et <var>.h.</var> ex quibus Å¿it <var>.K.h</var>. </s> - <s xml:space="preserve">Nunc ex iÅ¿tis duabus lineis <var>.e.g.</var> et <var>K.h.</var> fiat paral- - <pb o="213" rhead="EPISTOLAE" n="225" file="0225" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0225"/> - lelogrammum <var>.Z.</var> deinde fiant alia duo parallelogramma rectangula quorum vnum <lb/>Å¿it ex <var>.q.a.</var> in <var>.b.d.</var> reliquum verò Å¿it ex <var>.q.b.</var> in <var>.a.d.</var> quæ quidem Å¿int <var>.f.m</var>.</s> - </p> - <p> - <s xml:space="preserve">Quo facto deÅ¿ignetur rectangulum <var>.u.t.</var> ex .25. Å¿exti, quod æquale Å¿it duobus re-<lb/>ctangulis <var>.f.</var> et <var>.m.</var> Å¿imile tamen <var>.Z.</var> cuius rectanguli vnum latus correÅ¿pondet <var>.e.g.</var> reli-<lb/>quum verò <var>.K.h.</var> in proportione, Å¿ed in æqualitate, vnum correÅ¿pondet <var>.q.d.</var> <reg norm="reliquum" type="context">reliquÅ©</reg> <lb/>vero <var>.a.b.</var> diametris ipÅ¿ius quadrilateri.</s> - </p> - <p> - <s xml:space="preserve">Accipiatur nunc latus illud quod correÅ¿pondet <var>.K.h.</var> hoc eÅ¿t ipÅ¿i <var>.a.b.</var> maius Å¿cili-<lb/>cet, & Å¿imul cum <var>.b.d.</var> et <var>.a.d.</var> formetur <reg norm="triangulum" type="context context">triãgulÅ©</reg> <var>.a.b.d.</var> ex .22. primi Eucli. circa quod <lb/>circunÅ¿cribatur circulus ex .5. quarti. </s> - <s xml:space="preserve">& inuentum erit quod quÄ™rebamus.</s> - </p> - <figure position="here"> - <image file="0225-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0225-01"/> - </figure> - <figure position="here"> - <image file="0225-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0225-02"/> - </figure> - <figure position="here"> - <image file="0225-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0225-03"/> - </figure> - <figure position="here"> - <image file="0225-04" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0225-04"/> - </figure> - <figure position="here"> - <image file="0225-05" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0225-05"/> - </figure> - <pb o="214" n="226" file="0226" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0226"/> - </div> - </div> - <div type="section" level="3" n="3"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">PER EVNDEM PARALLELVM <lb/>abÅ¿que correctione Å¿emper nauigari <lb/>non poſſe.</head> - <head style="it" xml:space="preserve">Vbi not antur Petri <reg norm="Nonij" type="lig">Nonij</reg> lapÅ¿us in correctione erroris nauis. <lb/>Et alij Petri Medinæ errores.</head> - <head xml:space="preserve">ILLVSTRISSIMO ANDREAE PROVANAE <lb/>Leinici Domino, FruzaÅ¿ci comiti, Aequiti Torquato, inthimo Sere-<lb/>niÅ¿simi Sabaudiæ Ducis ConÅ¿iliario, <reg norm="eiuÅ¿que" type="simple">eiuÅ¿qÌ;</reg>, & Å¿acræ religio-<lb/>nis Å¿anctorum Mauritij, & Lazari ClaÅ¿si Præfecto.</head> - <p> - <s xml:space="preserve"><emph style="sc">INter</emph> Eximiastuas virtutes, reinauticæ peritia IlluÅ¿tris emicat <lb/>merito ad te Å¿cribendum duxi, quod ad eam facultatem perti-<lb/>nens excogitaui, Å¿imul cum quibuÅ¿dam alijs inÅ¿trumentis, vt non-<lb/>nihil commodi attuliſſe videar maritimis negotijs, & aliqua ex <lb/>parte animi mei erga te propenÅ¿ionem indicauiſſe.</s> - </p> - <p> - <s xml:space="preserve">PEr vnum <reg norm="eum" type="context">eÅ©</reg><reg norm="demque" type="simple">demqÌ;</reg> parallelum in primis <reg norm="abÅ¿que" type="simple">abÅ¿q;</reg> aliqua corre <lb/>ctione Å¿emper nauigari poſſe, omninò nego. </s> - <s xml:space="preserve"><reg norm="Nam" type="context">Nã</reg>, <reg norm="verum" type="context">verÅ©</reg> eÅ¿tid quod Petrus Nonius in <lb/>initio Å¿ui operis oÅ¿tendit, ideÅ¿t nauim verÅ¿us æquatorem Å¿emper declinare: </s> - <s xml:space="preserve">qui <reg norm="cum" type="context">cÅ©</reg> <lb/>corrigit errorem, fallitur, cum ipÅ¿e, eandem nauim, parallelam æquatori in vno ver <lb/>ticali ipÅ¿i æquatori propinquiori, & non in primo parallelo dirigit, itaque exiÅ¿timat <lb/>in fine itineris, vbi deÅ¿cribit punctum <var>.o.</var> eam in eodem parallelo priori repe-<lb/>riri debere, quod <reg norm="verum" type="context">verÅ©</reg> <reg norm="non" type="context">nõ</reg> eÅ¿t, quia ea correctio efficit, vt motus nauis <reg norm="effectum" type="context">effectÅ©</reg> <reg norm="cuiuÅ¿dam" type="context">cuiuÅ¿dã</reg> <lb/><reg norm="deÅ¿cenÅ¿us" type="context">deÅ¿cẽſus</reg> <reg norm="per" type="simple">ê‘</reg> Å¿caligradum pręſtet, in quo à gradu in <reg norm="gradum" type="context">gradÅ©</reg> fiat deÅ¿cenÅ¿us, Å¿ed Å¿i per gra <lb/>dus <reg norm="tantum" type="wordlist">tm̃</reg> aÅ¿cenderet <reg norm="quantum" type="context context">quãtÅ©</reg> <reg norm="deÅ¿cendit" type="context">deÅ¿cẽdit</reg>, <reg norm="dubium" type="context">dubiÅ©</reg> <reg norm="non" type="context">nõ</reg> eÅ¿t quin in fine ita eſſet Å¿e habitura <reg norm="quem" type="context">quẽ</reg> <lb/>ad modum in medio & in principio, cum verò Å¿emper deÅ¿cendat, abÅ¿que vlla aÅ¿cen-<lb/>Å¿ione, neceſſariò Å¿ic Å¿emper procedens, remota cum eſſent impedimenta terræ, Å¿ub <lb/>æquatore reperiretur, Å¿ub quo perpetuò circuiret globum.</s> - </p> - <p> - <s xml:space="preserve">Idem Å¿ub quolibet meridiano præſtare poteÅ¿t, ideÅ¿t vno <reg norm="eodemque" type="simple">eodemqÌ;</reg> vento circun-<lb/>uerti: </s> - <s xml:space="preserve">Å¿ed per alios circulos quam per hos duos (Å¿iue circulus magnus Å¿iue paruus) id <lb/>nunquam perfectè efficere poteÅ¿t, de parallelis iam manifeÅ¿tum eÅ¿t, cum impetus na <lb/>turalis corporum, quæ mota Å¿unt Å¿int Å¿emper in Å¿uperficiebus circulorum maiorum, <lb/>quorum circunferentiÄ™ cum circunferentijs minorum, præter quam per vnum quod <lb/>dam punctum quando adinuicem contiguæ Å¿unt, aut per duo ideÅ¿t cum Å¿e Å¿e interÅ¿e-<lb/>cant non communicant, ita quod ad efficiendum, vt triremis aliqua, aut nauis, per <lb/>aliquem ex parallelis ad æquatorem moueatur, neceſſario Å¿it futurum, vt ratione <reg norm="con- tiguitatis" type="context">cõ-<lb/>tiguitatis</reg> & non continuitatis eam moueri curemus. </s> - <s xml:space="preserve">quia ratione continuitatis om-<lb/>ninò fieri non <reg norm="pont" type="context">põt</reg>, aut conÅ¿tet virtus mouens remis, aut velis. </s> - <s xml:space="preserve">Sed per quemlibet <reg norm="alium" type="context">aliÅ©</reg> <lb/>circulum maiorem, qui non Å¿it aut æquator aut aliquis ex meridianis, eÅ¿t penitus im <lb/>poſſibile. </s> - <s xml:space="preserve">ideÅ¿t vt vnius venti vi nauis impellatur. </s> - <s xml:space="preserve">Quod vt clarè pateat, Å¿it orizon <var>.<lb/>a.c.b.d.</var> & æquator <var>.c.q.t.d.</var> & vnus meridianorum Å¿it <var>.a.r.n.t.b.</var> in quo <var>.n.</var> Å¿it Zenit Å¿ub <lb/>quo primum nauis reperiatur et <var>.r.</var> Å¿it polus Å¿eptentrionalis. </s> - <s xml:space="preserve">Ponamus etiam quod - <pb o="215" rhead="EPISTOLA." n="227" file="0227" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0227"/> - azimut <var>.f.q.u.n.p.</var> conÅ¿tituat angulum <var>.a.n.p.</var> Å¿eu <var>.f.n.b.</var> cum meridiano graduum .45. <lb/>vnde tot graduum eruntarcus <var>.a.p.</var> et <var>.b.f.</var> orizontis, quapropter punctum <var>.f.</var> commu-<lb/>ne ipÅ¿ius orizontis cum azimut, erit medio in loco inter <var>.b.</var> et <var>.c.</var> & ideo quarta <var>.n.f.</var> ip <lb/>Å¿ius azimut Å¿ecabit quartam <var>.c.t.</var> ipÅ¿ius æquatoris in puncto <var>.q.</var> & habebimus triangu-<lb/>Ium <var>.q.t.n.</var> cuius angulus <var>.t.</var> rectus erit, & angulus <var>.n.</var> cognitus Å¿imul cum latere <var>.n.t.</var> la-<lb/>titudinis loci, quibus rebus mediantibus deueniemus in cognitionem lateris <var>.q.n.</var> la-<lb/>teris <var>.q.t.</var> & anguli <var>.q.</var> ex .4. primi Copernici Å¿i voluerimus.</s> - </p> - <p> - <s xml:space="preserve">Ponamus nunc nauem à puncto <var>.n.</var> diÅ¿cedere Å¿eu iter facere verÅ¿us <var>.u.</var> punctum, & <lb/> - <anchor type="figure" xlink:label="fig-0227-01a" xlink:href="fig-0227-01"/> - in ipÅ¿o <var>.u.</var> reperiri, iam in hoc Å¿itu ha-<lb/>bebimus angulum huius Å¿ecundi me-<lb/>ridiani <var>.r.u.p.</var> qui quidem in hoc caÅ¿u <lb/>minor eſſet angulo <var>.r.n.p.</var> extrinÅ¿eco <lb/>trianguli <var>.r.u.n.</var> ex conuerÅ¿o Å¿ecundæ <lb/>partis .48. <reg norm="propoÅ¿itionis" type="simple">ê“poÅ¿itionis</reg> tertij lib. de <reg norm="trian gulis" type="context">triãÂ<lb/>gulis</reg> Monteregij, Å¿eu ex .13. primi Me <lb/>nelai, cuius anguli <var>.u.</var> arcus orizontalis <lb/>Å¿it <var>.x.e.</var> qui quidem minor erit arcu <var>.a.<lb/>p.</var> vt patet ratione anguli <var>.r.u.e.</var> mino-<lb/>ris, ergo alius ventus nauem impellet à <lb/>puncto .11. verÅ¿us <var>.q.</var> diuerÅ¿us ab illo qui <lb/>prius ab <var>.n.</var> verÅ¿us <var>.u.</var> eam impellebat.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0227-01" xlink:href="fig-0227-01a"> - <image file="0227-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0227-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Vnde clarè patet verum eſſe quod <lb/>dico, hoc eÅ¿t quod aliquo modo fieri <lb/>non poteÅ¿t, vt nauis ab aliquo loco ad <lb/>alium, breuiſſimo interuallo ire poſſit <lb/>ideÅ¿t per gyrum circuli maioris Å¿phæræ vno tantummodo <reg norm="vento" type="context">vẽto</reg> eam impellente, præ <lb/>ter quam in Ä™quatore, Å¿eu in aliquo quouis meridianorum, nos autem ire per gyrum <lb/>alicuius paralleli dementia eſſet, niÅ¿i neceſſitas cogeret.</s> - </p> - <p> - <s xml:space="preserve">HuiuÅ¿modi demonÅ¿trationis ope, quantum decipiatur Petrus Medina cap .6. lib. <lb/>3. cognoÅ¿citur, vbi Å¿ic Å¿cribit; </s> - <s xml:space="preserve">Vbicunque locorum reperiatur homo, aliquem circu-<lb/>lum qui vniuerÅ¿um ambiat imaginatione Å¿ibi confingens, per totum eum circulum <lb/>vno <reg norm="eodemque" type="simple">eodemqÌ;</reg> vento nauigatio Å¿uÅ¿cipitur. </s> - <s xml:space="preserve">Ex hac etiam demonÅ¿tratione, quà m fal <lb/>Å¿a Å¿it charta maritima patet, cuius beneficio exiÅ¿timant nautÄ™ Å¿e per breuiſſimum iter <lb/>a loco ad locum vehi etiamÅ¿i dicti loci non Å¿int ambo in æquatore, aut in aliquo me <lb/>ridiano, Å¿ed extra dictos circulos vnico tantum vento impellente & Å¿i in paruis æquo <lb/>fibus hic error parum depræhenditur, forte tamen in magno Oceano clarè pateret. <lb/></s> - <s xml:space="preserve">In Å¿uperius igitur dicta demonÅ¿tratione iam oÅ¿tendi, quod Å¿i velimus vehi ab vno lo <lb/>co ad alium beneficio alicuius circuli maioris, præter duos iam dictos, hoc fieri non <lb/>poteÅ¿t vno <reg norm="eodemque" type="simple">eodemqÌ;</reg> <reg norm="vento" type="context">vẽto</reg> impellente. </s> - <s xml:space="preserve">Vnde Å¿equitur, omnia ea interualla quæ vno <lb/><reg norm="eodemque" type="simple">eodemqÌ</reg> vento tranÅ¿ibimus futura longiora, præterquam in duobus dictis circulis <lb/>æquinoctiali & meridiano.</s> - </p> - <p> - <s xml:space="preserve">Cum verò Petrus Medina cap .7. volens probare chartam maritimam bonam eÅ¿-<lb/>Å¿e, planiÅ¿phÄ™rium Prolomei & Iordani citat, non animaduertit quam diuerÅ¿o mo-<lb/>do a charta maritima huiuÅ¿modi inÅ¿trumentum Å¿it fabricatum, cum exceptis orizon <lb/>te recto, & meridiano in dicto inÅ¿trumento quilibet alius circulus Å¿it circulus, Å¿iue Å¿it <lb/>almicantarat, Å¿iue azimur, Å¿iue æquator, Å¿iue tropicus, Å¿iue zodiacus, Å¿iue alius quiuis <lb/>circulus, eum in charta maritima ne vna quidem Å¿it linea, quÄ™ non Å¿it recta, quolibet <lb/>nomine vocetur.</s> - </p> - <pb o="216" rhead="IO. BABPT. BENED." n="228" file="0228" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0228"/> - <p> - <s xml:space="preserve">Superius poÅ¿itæ meæ demonÅ¿trationis ope, deuenimus in cognitionem magnitu <lb/>dinis arcus <var>.n.q.</var> cognoÅ¿cimus etiam angulum <var>.n.q.t.</var> vnde nobis <reg norm="manifeÅ¿tum" type="context">manifeÅ¿tÅ©</reg> eſſet quo <lb/>vento oporteret iter facere. cum à puncto <var>.q.</var> nauis aliqua diÅ¿ceſſura eſſet, in eodem <lb/>azimut propoÅ¿ito. </s> - <s xml:space="preserve">Idem etiam dico de puncto <var>.u.</var> cum cogniti eſſent arcus <var>.n.u.</var> et <var>.n.<lb/>r.</var> vt Å¿upponitur, Å¿imul cum angulo <var>.r.n.u.</var> vnde cognitus eſſet nobis angulus <var>.n.u.r.</var> ex <lb/>11. primi lib. Copernici, ex quo ventus nobis cognitus foret.</s> - </p> - <p> - <s xml:space="preserve">Modus autem quem idem Medina cap .9. lib. tertij ad cognoÅ¿cendam diÅ¿tantiam <lb/>vnius meridiani ab alio præſcribit, in genere eÅ¿t falÅ¿us, etiam Å¿i is ab antiquis eum de <lb/>Å¿umat, qui, hic non viderunt quam magna inter meridianos differentia Å¿it interuallo <lb/>rum eorum quæ Å¿unt vicina polis & eorum quæ Å¿unt circa æquatorem.</s> - </p> - <p> - <s xml:space="preserve">FalÅ¿us eÅ¿t etiam modus ab eo traditus ad cognoÅ¿cendos gradus longitudinis per <lb/>medium itineris cogniti in quouis parallelo extra æquatorem facti, & hoc cap .14. li <lb/>bri tertij eiuÅ¿dem, & primo cap. lib. 4. <reg norm="continetur" type="context">cõtinetur</reg>, vbi .17. leucas cum dimidia cuilibet <lb/>gradui tam paralleli quà m meridiani aſſignat.</s> - </p> - <p> - <s xml:space="preserve">FalÅ¿um eÅ¿t etiam quod ab eo aſſeritur, Solem, cum reperiretur in æquatore, circa <lb/>eos qui Å¿ub ipÅ¿o æquatore habitant, vnius diei <reg norm="noctisque" type="simple">noctisqÌ;</reg> Å¿patio per omnes uentos cir-<lb/>cunuolui. </s> - <s xml:space="preserve">quia illis æquator idem eÅ¿t cum verticali, qui duos tantum rhumbos pro-<lb/>ducit, ideÅ¿t orientis, & occidentis: </s> - <s xml:space="preserve">hic verò error, in Å¿ecundo cap. lib. 6. habetur.</s> - </p> - <p> - <s xml:space="preserve">FalÅ¿um eÅ¿t etiam quod profert Solem ijs qui habitant Å¿phæram obliquam, qua-<lb/>libet hora tertia, regulariter ab vno rhumbo ad alium ex præcipuis ideÅ¿t ab vno azi <lb/>mut ad alium progredi, quemadmodum eadem cap .2. lib. 6. et .7. cap. Å¿eptimi libr. <lb/>Å¿cribit. </s> - <s xml:space="preserve">Huius autem rei falÅ¿itas ita facile depræhendetur, ponamus hemiÅ¿phÄ™rium <lb/>orientale, verbi gratia, cuius meridianus Å¿it <var>.p.z.b.</var> æquator <reg norm="autem" type="wordlist">aÅ©t</reg> <var>.e.m.</var> vnus verò paral <lb/>lelorum Å¿eptentrionalium Å¿it <var>.c.a.</var> in quo Solem exiÅ¿tere ponamus, orizon autem Å¿it <var>.<lb/>b.m.</var> zenit vero <var>.z.</var> polus arcticus <var>.p.</var> Å¿it poÅ¿tea azimut <var>.z.q.</var> à meridiano diÅ¿tans per gra <lb/>dus .45. qui quidem azimut in hoc hemiÅ¿phÄ™rio erit rhumbus illius venti, quem uul-<lb/>go Itali Sirocum dicunt, et <var>.z.m.</var> Å¿it azimut verticalis qui in hoc hemiÅ¿phærio erit <reg norm="ron" type="context">rõ</reg> <lb/>bus venti orientalis, ita <reg norm="quod" type="simple">ê™</reg> Å¿ecundum Medinam à rhumbo <var>.z.m.</var> ad <var>.z.q</var>. </s> - <s xml:space="preserve">SolabÅ¿oluet <lb/>Å¿patium <reg norm="temporis" type="context">tẽporis</reg> trium horarum, & aliud <lb/>æquale temporis Å¿patium abÅ¿oluet à <reg norm="rhum" type="context">rhÅ©</reg> <lb/> - <anchor type="figure" xlink:label="fig-0228-01a" xlink:href="fig-0228-01"/> - bo <var>.z.q.</var> ad <var>.z.b.</var> ex ipÅ¿o Medina, vnde ar-<lb/>cus <var>.a.o.</var> paralleli eſſet graduum .45. & <lb/>item arcus <var>.o.c</var>.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0228-01" xlink:href="fig-0228-01a"> - <image file="0228-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0228-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Ponamus <reg norm="nunc" type="context">nÅ©c</reg> Solem reperiri in Ä™qua <lb/>tore, vbi per ipÅ¿um Medinam arcus <var>.u.m.</var> <lb/>Å¿imiliter eſſet graduum .45. & Å¿ic <var>.u.e</var>: pro <lb/>tracto ergo arcu <var>.p.o.f.</var> palam erit ar-<lb/>cum <var>.f.e.</var> fore graduum .45. Å¿ed cum <lb/>arcus <var>.e.u.</var> Å¿it graduum .45. ex Å¿uppoÅ¿ito <lb/>ipÅ¿ius Medinæ, Å¿equeretur arcum <var>.e.f.</var> <lb/>æqualem eſſe arcui <var>.e.u.</var> pars igitur æqua <lb/>lis erit Å¿uo toto.</s> - </p> - <p> - <s xml:space="preserve">Id etiam quod Petrus Nonius pagina <lb/>124. et .125. lib. de arte nauigandi con-<lb/>tra nautas de diÅ¿tantijs Solis à meridiano Å¿cribit, hanc opinionem Petrià Medina & <lb/>corum qui idem ei perÅ¿uaÅ¿erunt falÅ¿am eſſe demonÅ¿trat.</s> - </p> - <pb o="217" rhead="EPISTOLAE." n="229" file="0229" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0229"/> - <p> - <s xml:space="preserve">FalÅ¿um eÅ¿t etiamid quod cap .3. lib. 6. pronuntiat, ita dicens.</s> - </p> - <p> - <s xml:space="preserve">Quod cum verum eſſet à parte <reg norm="orientali" type="context">oriẽtali</reg> inÅ¿ularum quæ azore dicuntur, pyxidem <lb/>verÅ¿us eum ventum qui vulgò Græcus dicitur, & ab occidentali verÅ¿us eum qui Ma <lb/>giÅ¿ter dicitur, vergere, huius rei nulla eÅ¿t ratio.</s> - </p> - <p> - <s xml:space="preserve">Ego enim huiuÅ¿modi rationem reperiri poſſe contendo, quæ talis eÅ¿t, quia pars <lb/>roſæ (ut vocant) à magnete tacta, ad aliquod punctum, aut Å¿itum globi terrÄ™, in eo-<lb/>dem meridiano inÅ¿ularum, quæ Azore dicuntur, vltra Å¿itum poli arctici in terra diri-<lb/>geretur, ita vt Å¿itus dicti poli in terra eſſet in dicto meridiano, inter locum qui ab in <lb/>dice roſæ aut pyxidis reÅ¿piceretur, & dictas inÅ¿ulas, id quod Å¿uperius Å¿cripto meridia <lb/>no facile cognoÅ¿ci poteÅ¿t, Å¿umendo pro inÅ¿ulis Å¿itum <var>.e.</var> in meridiano et <var>.z.</var> pro polo, <lb/>et <var>.p.</var> pro loco qui à pyxide Å¿it viÅ¿us, <reg norm="imaginando" type="context">imaginãdo</reg> deinde pyxidem in <var>.f.</var> magis orienta <lb/>li quam eÅ¿t <var>.e.</var> clarum eÅ¿t lineam quæ reÅ¿picit (ponamus) <var>f.p.</var> verÅ¿us Græcum & ab alia <lb/>parte verÅ¿us MagiÅ¿trum declinare.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">De Armilla Nautica.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">CVm ſæpe viderim quam in magnis æquoribus nos fallant, <reg norm="atque" type="simple">atq;</reg> decipiant mari <lb/>timæ, Å¿eu nauigatoriæ chartÄ™, quemadmodum aliquoties inter nos Å¿ermonem <lb/>habuimus: </s> - <s xml:space="preserve">in id totus incubui vt aliquam machinam excogitarem, quæ difficilis non <lb/>eſſet, <reg norm="efficeretque" type="simple">efficeretqÌ;</reg> vt nauis Å¿uper aquÄ™ globum, beneficio circulorum maiorum, quam <lb/>optimè poſſet, ideÅ¿t breuiſſimo itinere ab uno loco <reg norm="and" type="context">ãd</reg> alium ferretur. </s> - <s xml:space="preserve">Id <reg norm="quod" type="simple">ê™</reg> mihi ex <lb/>animi voto Å¿ucceſſurum putaui, beneficio <reg norm="quinque" type="simple">quinq;</reg> circulorum circundantium <reg norm="aliquem" type="context">aliquẽ</reg> <lb/>globum <reg norm="terreÅ¿trem" type="context">terreÅ¿trẽ</reg> & maritimum, quales ij Å¿unt qui in inferiori Germania à Gerardo <lb/>Mercatore Å¿truuntur, qui vno pede cum dimidio diametri conÅ¿tet, ideÅ¿tÅ¿eÅ¿quipede.</s> - </p> - <p> - <s xml:space="preserve">Sit ergo, exempli gratia, huiuÅ¿modi globus <var>.a.b.d.</var> circa quem duo circuli, aut cirÂ<lb/> - <anchor type="figure" xlink:label="fig-0229-01a" xlink:href="fig-0229-01"/> - <pb o="218" rhead="IO. BAPT. BENED." n="230" file="0230" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0230"/> - culares lineæ ex aurichalco applicentur inuicem coniuncti per medium ad angulos <lb/>rectos, quorum prior <var>.f.e.g.</var> in Å¿e globi polos mediantibus extremitatibus axis mun-<lb/>di contineat, qui quidem poli à punctis Å¿uarum interÅ¿ectionum per <reg norm="quartam" type="context">quartã</reg> ex æquo <lb/>in punctis <var>.f.</var> et <var>.g.</var> ita diÅ¿tent, vt globus circa eoÅ¿dem, in Å¿itu longitudinis mundi vol-<lb/>ui poſſit. </s> - <s xml:space="preserve">HuiuÅ¿modi autem circulus, æquatoris deferens appelletur.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0229-01" xlink:href="fig-0229-01a"> - <image file="0229-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0229-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Secundus autem circulus Å¿it <var>.h.e.K.</var> cum primo ad angulos rectos in puncto <var>.e.</var> & <lb/>in Å¿uo oppoÅ¿ito connexus, & is appellabitur æquator, & poli <var>.f.g.</var> primi poli dicentur.</s> - </p> - <p> - <s xml:space="preserve">Circa huiuÅ¿modi duos circulos, alios <reg norm="etiam" type="context">etiã</reg> duos exiÅ¿tere <reg norm="vellem" type="context">vellẽ</reg> Å¿imul <reg norm="coniunctos" type="context context">cõiÅ©ctos</reg> medio <lb/>ad angulos rectos. </s> - <s xml:space="preserve">In quibus <reg norm="quidem" type="context">quidẽ</reg> interÅ¿ectionis punctis Å¿int duo poli, qui hos duos <lb/>circulos cum Å¿ecundo priorum ideÅ¿t cum æquatore in duobus punctis inuicem op-<lb/>poÅ¿itis connectant; </s> - <s xml:space="preserve">quæ æquatoris puncta à punctis interÅ¿ectionis eiuÅ¿dem cum Å¿uo <lb/>deferente, ratione vna quarta diÅ¿tent, quorum duorum circulorum primus Å¿it <var>.n.i.<lb/>m.</var> quem deferentem azimut appellabimus; </s> - <s xml:space="preserve">Å¿ecundus <var>.r.n.s.m.</var> azimut locorum noÂ<lb/>minabimus. </s> - <s xml:space="preserve">eorundem interÅ¿ectionis rectæ, puncta Å¿int <var>.n.</var> et <var>.m.</var> à quibus duo poli ex <lb/>aurichalco confecti Å¿imiles primis <var>.n.h.</var> et <var>.m.K.</var> vÅ¿que ad puncta <var>.h.</var> et <var>.K.</var> æquatoris <lb/>perueniant, qui Å¿pisÅ¿itudinem æquatoris diÅ¿tantem à puncto <var>.e.</var> vna quarta <reg norm="penetrent" type="context">penetrẽt</reg>, <lb/>ita vt æquator circum circa <var>.n.h.</var> et <var>.m.K.</var> in Å¿itu latitudinis mundi verti queat. </s> - <s xml:space="preserve">Et <lb/>hos, Å¿ecundos polos nominabimus.</s> - </p> - <p> - <s xml:space="preserve">Alius deinde circulus <var>.q.i.p.</var> duos poÅ¿teriores circulos ambiat, cum deferente ta-<lb/>men azimut mediantibus duobus polis in puncto <var>.i.</var> & in Å¿uo oppoÅ¿ito exÄ™quo diÅ¿tan <lb/>tibus à ſecundis polis vnius quartæ Å¿patio iungatur. </s> - <s xml:space="preserve">Ita vt dictum deferens azimut <lb/>circa hos tertios polos volui poſſit, atque hunc circulum <var>.q.i.p.</var> orizontem vniuerÅ¿a <lb/>lem vocabimus. </s> - <s xml:space="preserve">Hic vero orizon Å¿uper quatuor quartas circuli, aut Å¿uper quatuor <lb/>paruis columnis, ut fieri Å¿olet innixis Å¿uæ baÅ¿i, ita ponatur, vt moueri non poſſit.</s> - </p> - <p> - <s xml:space="preserve">Primus autem circulus <var>.f.e.g.</var> deferens æquatoris in .4. partes æquales diuidatur, <lb/>quarum quælibet .90. gradibus conÅ¿tet, incipiendo ab interÅ¿ectionibus <var>.e.</var> & eius op <lb/>poÅ¿ito æquatoris, & numeri in polis <var>.f.</var> et <var>.g.</var> globi finem Å¿ortiantur. </s> - <s xml:space="preserve">Diuidatur etiam <lb/>æquator <var>.h.e.K.</var> in .360. partes incipientes à puncto <var>.e.</var> verÅ¿us <var>.K.</var> deferens autem azi <lb/>mut <var>.n.i.m.</var> ab omni diuiÅ¿ione liber maneat, Å¿ed azimut <var>.n.s.m.r.</var> in .360. gradus inci <lb/>piendo à puncto <var>.n.</var> verÅ¿us <var>.r.</var> diuidatur.</s> - </p> - <p> - <s xml:space="preserve">Orizon autem <var>.q.i.p.</var> diuidatur in quartas, quarum quælibet Å¿it nonaginta <reg norm="graduum" type="context">graduÅ©</reg> <lb/>incipiendo à puncto <var>.i.</var> & eius oppoÅ¿ito ideÅ¿t à polis poÅ¿tremis & terminando in pun-<lb/>ctis <var>.q.</var> et <var>.p.</var> in medio ipÅ¿orum polorum, & quarta <var>.i.p.</var> orientalis Å¿eptentrionalis, et <var>.i.<lb/>q.</var> orientalis meridiana appellentur. </s> - <s xml:space="preserve">& Å¿ic ordine Å¿eruato occidentales.</s> - </p> - <p> - <s xml:space="preserve">Præterea prÄ™parata Å¿it quædam quarta, ex aurichalco, circuli æqualis ipÅ¿i orizon-<lb/>ti, & in .90. gradus diÅ¿tincta quæ cum quauis Å¿uarum extremitatum ipÅ¿i zenit, in azi-<lb/>mut applicari poſſit, quemadmodum circa globos cÄ™leÅ¿tes fieri Å¿olet; </s> - <s xml:space="preserve">quÄ™ quidem ad <lb/>cognoÅ¿cendam altitudinem poli ipÅ¿ius globi ab orizonte nobis inÅ¿eruiet.</s> - </p> - <p> - <s xml:space="preserve">Atque hac ratione hanc noÅ¿tram machinam perfectè abÅ¿oluemus <reg norm="quam" type="context">quã</reg> appellan-<lb/>dam eſſe Armillam nauticam Å¿entio. </s> - <s xml:space="preserve">Hic autem illud non omittam, concauum <lb/>duorum priorum circulorum à ſuperficie globi non nimis diÅ¿tare debere & con-<lb/>cauum aliorum à ſuperficie conuexa priorum longe poÅ¿itos eſſe <reg norm="non" type="context">nõ</reg> debere, & con <lb/>cauam orizontis à conuexa Å¿ecundorum procul abeſſe non debere.</s> - </p> - <p> - <s xml:space="preserve">Neque illud etiam prætermittendum eÅ¿t, opere pretium fore Å¿i in interÅ¿ectione <lb/>e. priorum, erit foramen elicum, vt clauo elico ex aurichalco confecto, poſſimus <lb/>Å¿iÅ¿tere globum, quando oportuerit, ne amplius circa primos Å¿uos polos <var>.f.g.</var> circun-<lb/>uoluatur, cum <reg norm="Å¿itus" type="simple">Å¿itê°</reg> fuerit. </s> - <s xml:space="preserve">Inde etiam laudo vt in azimut <var>.r.n.s.m.</var> è regione deferen <lb/>tis æquatoris, ideÅ¿t <var>.f.e.g.</var> aliud quoddam foramen huiuÅ¿modi Å¿it poÅ¿itum, in quo - <pb o="219" rhead="EPISTOL AE." n="231" file="0231" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0231"/> - clauus elicus vÅ¿que ad circulum <var>.f.e.g.</var> perueniens, æquatorem Å¿iÅ¿tere poſſit, ne circa <lb/>Å¿ecundos polos <var>.h.</var> et <var>.K.</var> amplius moueatur quum noluerimus eum mutare Å¿itum.</s> - </p> - <figure position="here"> - <image file="0231-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0231-01"/> - </figure> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">DeuÅ¿u Armillæ nauticæ.</head> - <p> - <s xml:space="preserve">V Tautem noÅ¿tra Armilla nautica vti poſſimus pyxidem nos prius oportebit <lb/>habere, diuerÅ¿am tamen ab ijs, quibus nautæ hactenus vÅ¿i fuere: </s> - <s xml:space="preserve">nolo <lb/>enim vt <reg norm="tam" type="context">tã</reg> craſſa minerua beneficio <reg norm="ventorum" type="context">vẽtorum</reg> communium circa hanc rem nos gera <lb/>mus, Å¿ed ratione graduum orizontis in .360. partes diÅ¿tincti, atque ob hanc cauÅ¿am <lb/>Å¿entio, vt ima pars pyxidis penitus detecta videatur, & in .360. partes dinidatur, <reg norm="nilque" type="simple">nilqÌ</reg> <lb/>aliud quam quandam lanceo lam Å¿upra eius acum eſſe volo, quæ dum mouebitur na <lb/>uis, per gradus quamlibet orizontis partem oÅ¿tendet; </s> - <s xml:space="preserve">hos autem .360. gradus, ita Å¿e <lb/>habere volo, vt quÄ™libet quarta .90. contineat, <reg norm="Å¿upputatioque" type="simple">Å¿upputatioqÌ;</reg> à linea meridiana inci-<lb/>piat, & in verticali deÅ¿inat, vt huiuÅ¿modi diuiÅ¿io cum ea, quæ eÅ¿t orizontis Armillæ <lb/>eadem Å¿it.</s> - </p> - <p> - <s xml:space="preserve"><reg norm="Pręſupponantur" type="context">Pręſupponãtur</reg> nunc in globo duo loci extra æquatorem, & in diuerÅ¿is meridia-<lb/>nis quomodolibet a dinuicem diÅ¿tantes, à quorum vno ad alium Å¿it nauigandum iti-<lb/>nere quo ad fieri poterit breuiori, ideÅ¿t per gyrum circuli maioris, dixi autem extra <lb/>æquatorem, ideÅ¿t vt ambo, nec in æquatore, nec in uno <reg norm="eodemque" type="simple">eodemqÌ;</reg> meridiano <reg norm="exiÅ¿tant" type="context">exiÅ¿tãt</reg>, <lb/>quia vt alià s dixi in huiuÅ¿modi locis, vnico tantum vento comite, iter conficere <lb/>poſſumus.</s> - </p> - <pb o="220" rhead="IO. BAPT. BENED." n="232" file="0232" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0232"/> - <p> - <s xml:space="preserve">Volo primum vt mediante <reg norm="circunuolutione" type="context">circÅ©uolutione</reg> globi circa primos polos <var>.f.g.</var> & æqua <lb/>toris circa Å¿ecundos <var>.h.K.</var> hoc eÅ¿t per longitudinem, & latitudinem, hi duo loci in <lb/>globo propoÅ¿iti Å¿ub azimut <var>.r.n.s.m.</var> Å¿ecundorum circulorum <reg norm="Å¿itum" type="context">Å¿itÅ©</reg> Å¿ortiantur, qui azi-<lb/>mut orizontem in punctis <var>.q.</var> et <var>.p.</var> Å¿emper ad angulos rectos diÅ¿peÅ¿cit <reg norm="ibique" type="simple">ibiq;</reg> globum <lb/>ita quieÅ¿cere vt circa polos <var>.f.g.</var> non voluatur, & æquatorem etiam Å¿ic Å¿irmare, vt cir-<lb/>ca Å¿ecundos polos <var>.h.K.</var> non vertatur faciamus.</s> - </p> - <p> - <s xml:space="preserve">Quod cum factum fuerit, Å¿ecundorum circulorum primus, qui eÅ¿t <var>.n.i.m.</var> deferens <lb/>azimut, circa tertios polos <var>.i.</var> & eius oppoÅ¿itum, eo uÅ¿que voluatur quouÅ¿que prior <lb/>globi locus, ideÅ¿t is a quo iter eÅ¿t incohandum per .90. gradus azimut diÅ¿tet ab ori-<lb/>zonte, ideÅ¿t Å¿ub zenit orizontis <var>.q.i.p.</var> Å¿it poÅ¿itus, quemadmodum, exempli gratia, Å¿i <lb/>punctum <var>.a.</var> dicti primi loci globi rationem indueret, & borealius eſſet, mediante <lb/>circunuolutione circuli <var>.n.i.m.</var> circa dictos tertios polos æqualiter diÅ¿tans ab <var>.q.</var> et <var>.p.</var> <lb/>ideÅ¿t per .90. gradus poneretur Å¿ub <var>.r</var>.</s> - </p> - <p> - <s xml:space="preserve">ConÅ¿ideretur deinde vbi æquator <var>.h.e.K.</var> Å¿ecundus circulus duorum primorum, <lb/>ab orizonte <var>.q.i.p.</var> Å¿ecabitur, exempli gratia, in puncto <var>.c.</var> quartÄ™ orientalis Å¿epten-<lb/>trionalis eiuÅ¿dem orizontis. </s> - <s xml:space="preserve">Videatur deinde quot nam gradibus conÅ¿tabit ar-<lb/>cus <var>.i.c.</var> & per totidem gradus conÅ¿tituatur extremitas Å¿eptentrionalis lineæ meridia-<lb/>nÄ™ pyxidis nauticÄ™, diÅ¿tantis à cuÅ¿pide Å¿eptentrionali ipÅ¿ius lanceolæ orientem ver-<lb/>Å¿us, mediante nauis circunuolutione. </s> - <s xml:space="preserve">vnde ipÅ¿amet nauis in huiuÅ¿modi Å¿itu azimut, <lb/>qui per duos hos loc<unsure/>os tranÅ¿it, dirigetur, eſſiciendo vt eius prora verÅ¿us <reg norm="locum" type="context">locÅ©</reg> ad <reg norm="quem" type="context">quẽ</reg> <lb/>voluerimus tendere dirigatur. </s> - <s xml:space="preserve">Cum verò vela ventis dabimus, tot milliarium <lb/>Å¿eu leucarum iter conficiemus, quot quarta pars vnius gradus requirit. </s> - <s xml:space="preserve">& dum <lb/>hociter abÅ¿oluitur, ille qui prÄ™eÅ¿t naui, defferentem azimut <var>.n.i.m.</var> circa Å¿uos polos <var>.i.</var> <lb/>& eius oppoÅ¿itum, Å¿ic circunuoluat, vt interÅ¿e ctio azimut <var>.r.n.s.m.</var> cum orizonte <var>.q.i.<lb/>p.</var> diÅ¿tet à prima ratione dictæ quartÄ™ partis vnius gradus, conÅ¿tituendo Å¿ecundum lo <lb/>cum, proximiorem zenit, ratione dictæ quartæ partis gradus azimut. </s> - <s xml:space="preserve">HiÅ¿ce ita pera <lb/>ctis, obÅ¿eruetur deinde vbiæquator <var>.h.e.K.</var> hac Å¿ecunda vice interÅ¿ecabit orizontem <lb/><var>q.i.p.</var> quod quidem interÅ¿ectionis punctum Å¿emper appelletur <var>.c.</var> quod dico non am <lb/>plius in eadem diÅ¿tantia manÅ¿urum, ut prius à puncto <var>.i.</var> Å¿ed aut longius diÅ¿tabit, aut <lb/>propius accedet, vt in præſenti exemplo. </s> - <s xml:space="preserve">quemadmodum ex Å¿e manifeÅ¿tum eÅ¿t, cú <lb/>poli globi, ideÅ¿t Ä™quatoris Å¿int extra azimut, vt præſupponitur, quia loci Å¿unt in diuer <lb/>Å¿is meridianis.</s> - </p> - <p> - <s xml:space="preserve">Pro huiuÅ¿modi autem diÅ¿tantiæ ratione denuo dirigatur nauis prout æquator <var>.h.<lb/>e.K.</var> in orizonte <var>.q.i.p.</var> nobis oÅ¿tendet, atque hoc modo omnium iter quaÅ¿i breuiſſi-<lb/>mum fiet. </s> - <s xml:space="preserve">dico autem, quaÅ¿i, quia omnibus modis neceſſariò conficitur iter contor-<lb/>tum & in formam Å¿erpigineæ lineæ. </s> - <s xml:space="preserve">Applicantes deinde per vices extremitatem <lb/>quartæ appoÅ¿itæ (de qua Å¿uperius mentionem fecimus) ipÅ¿i zenit <var>.r.</var> efficientes ut per <lb/>Å¿itum poli globi pertranÅ¿eat, deueniemus in cognitionem altitudinis eiuÅ¿dem ab <lb/>orizonte, & per conÅ¿equens quantum itineris per latitudinem eiuÅ¿dem globi pere-<lb/>gerit. </s> - <s xml:space="preserve">mediante deinde interÅ¿ectione orizontis <var>.q.i.p.</var> cum æquatore, cognoÅ¿cemus <lb/><reg norm="quantum" type="context">quãtum</reg> itineris per longitudinem eiuÅ¿dem globi, in ipÅ¿o Ä™quatore fuerit peractum,.</s> - </p> - <pb o="221" rhead="EPISTOLAE." n="233" file="0233" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0233"/> - </div> - <div type="letter" level="4" n="4"> - <head style="it" xml:space="preserve">Inctrumentum adortum, & occaÅ¿um LunacognoÅ¿cendum <lb/>qualibet anni die.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">ECce tibi vir IlluÅ¿triÅ¿s. <reg norm="modum" type="context">modÅ©</reg> <reg norm="conficiendi" type="context">conficiẽdi</reg> inÅ¿trumenti nuper à me <reg norm="inuenti" type="context">inuẽti</reg>, vt tibi Å¿i <lb/>gnificaui, quo Å¿cire poſſis fermè in dies, qua hora (de aÅ¿tronomicis loquor) ad <lb/>determinatum parallelum & abÅ¿que multa Å¿upputatione, etiam abÅ¿que AÅ¿trolabio <lb/>Luna oriatur <reg norm="occidatque" type="simple">occidatqÌ;</reg>. </s> - <s xml:space="preserve">In quo inÅ¿trumento poteris etiam videre quo in Å¿igno Sol, <lb/>& ſæpius itidem Luna permeat, & huiuÅ¿ce aÅ¿pectus cum Sole, atque longitudinem <lb/>diei <reg norm="noctisque" type="simple">noctisqÌ;</reg> toto anni tempore exactè diÅ¿cernere.</s> - </p> - <p> - <s xml:space="preserve">Circularis lamina ex argento, aut ære, aliaúe materia paranda eÅ¿t, in cu-<lb/>ius Å¿uperficie ambarum facierum Zodiacus delineabitur, modo inferius depicto, <lb/>deinde pro anno quinque circuli Å¿ibi inuicem <reg norm="concentrici" type="context">cõcentrici</reg>, at reÅ¿pectu Zodiaci excen <lb/>trici cÄ™labuntur in ea, adeo vt vtriuÅ¿que centri diÅ¿tantia Å¿it pro .32. parte Å¿emidia-<lb/>metri concauitatis Zodiaci è regione locis augis, temporis qui noÅ¿tra ætate circa Å¿i-<lb/>nem Å¿ecundi gradus cancri inuenitur, eandem viam, in hoc, Å¿equuti, quam Stofle-<lb/>rus in dorÅ¿o Aftrolabij docet. </s> - <s xml:space="preserve">At nomina menÅ¿ium media ponantur inter duos <lb/>maiores circulos, poÅ¿tea inter Å¿ecundum, & tertium ab vna facierum laminæ, ar-<lb/>cus Å¿emidiurni, ab altera vero arcus Å¿eminocturni, per quinos quoÅ¿que dies collo-<lb/>centur, ita exactè, vt hic Å¿ubtus videbis. </s> - <s xml:space="preserve">adeo vt numeri dierum & ipÅ¿orum dierum <lb/>Å¿igna Å¿int in interuallis vicinioribus centro communi dictorum quinque circulorum.</s> - </p> - <p> - <s xml:space="preserve">PoÅ¿teaquam ab vna & altera <reg norm="facierum" type="context">facierÅ©</reg> laminæ hæc inÅ¿culpra fuerint, aliæ duæ circu <lb/>lares laminæ, magnitudinis Å¿emidiametri minimi quinque circulorum accipiantur: <lb/></s> - <s xml:space="preserve">quarum vna pro ortu, & altera pro occaÅ¿u Lunæ deÅ¿eruiet. </s> - <s xml:space="preserve">In qualibet ipÅ¿arum <lb/><reg norm="conÅ¿tituentur" type="context">conÅ¿tituẽtur</reg> circuli quatuor, eo modo qui paulo inferius cernitur, quos omnes diui-<lb/>demus in triginta Å¿pacia æqualia: </s> - <s xml:space="preserve">& in interuallo <reg norm="quod" type="wordlist">qÄ‘</reg> inter duos primos circulos poÅ¿i <lb/>tum eÅ¿t, triginta dies <reg norm="annotabimus" type="simple">annotabimê°</reg> qui ipÅ¿os Lunætriginta dies <reg norm="pręſcribent" type="context">pręſcribẽt</reg>, vt in figara.</s> - </p> - <p> - <s xml:space="preserve">PoÅ¿tmodum in lamina quæ ortus Lunæ indicabit, ac duorum maiorum <reg norm="circulo- rum" type="context">circulo-<lb/>rÅ©</reg> interuallo è regione numeri .1. videlicet primi diei, ponemus horas .12. & minuta <num value="48">.<lb/>48.</num> ex aduerÅ¿o diei Å¿ecundi ho .13. et min .36. ex oppoÅ¿ito tertij ho 14. min .24. & Å¿ic <lb/>Å¿ucceſſiuè augendo per min .48. & indicem è diuerÅ¿o diei .30. <reg norm="Å¿tatuendo" type="context">Å¿tatuẽdo</reg>, qui coitus <lb/>Lunæ cum Sole Å¿igniÅ¿icabit: </s> - <s xml:space="preserve">atque lineas aÅ¿pectuum, vt inferius videre eÅ¿t facilè in <lb/>ueniemus.</s> - </p> - <p> - <s xml:space="preserve">Altera in lamina quæ occaÅ¿um Lunæ indicabit, poÅ¿tquam diÅ¿tincta fuerit, vt alte-<lb/>ra .30. dies ac cÄ™teræ lineæ, eo modo quo in Å¿uperiori collocabuntur, at numeri in-<lb/>terualli maioris, aliter diÅ¿ponentur, vt potè ex aduerÅ¿o diei primi Å¿olum .48. minu-<lb/>ta deÅ¿cribi debent, è directo Å¿ecundi diei ponenda erit hora vna cum minutis .36. & <lb/>è regione tertij inÅ¿cribentur .2. horæ, & min .24. & Å¿ic ex ordine per .48. minuta au-<lb/>gendo.</s> - </p> - <p> - <s xml:space="preserve">Nunc lamina ortus LunÄ™, cum anno arcuum Å¿eminocturnorum, & illa occaÅ¿us cum <lb/>anno arcuum Å¿emidiurnorum <reg norm="concentrari" type="context">concẽtrari</reg> debet, & ita noÅ¿trum inÅ¿trumentum perfe-<lb/>ctum erit & abÅ¿olutum.</s> - </p> - <p> - <s xml:space="preserve">Quoties igitur voluerimus medio inÅ¿trumento dignoÅ¿cere fermè in tali orizonte <lb/>qua hora Luna oriatur, ita neceſſe erit volubilem rotam ortus flectere, ut index ve <lb/>niat è regione diei menÅ¿is in quo talis operatio fit & talirota Å¿irma manente perÅ¿pi- - <pb o="222" rhead="IO. BAPT. BENED." n="234" file="0234" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0234"/> - cere ex aduerÅ¿o diei Lunæ, numerum horarum & minutorum in maiori interuallo <lb/>ipÅ¿ius rotæ notatorum, qui cum arcu Å¿eminocturno anni, quo cum in ipÅ¿a rectitudi-<lb/>ne centri conueniet colligetur, & Å¿umma quæ ex tali Å¿upputatione proueniet aper-<lb/>tas faciet horas aÅ¿tronomicas, quibus ferè etÅ¿i non exactè in die propoÅ¿ito Luna <lb/>orietur. </s> - <s xml:space="preserve">IdipÅ¿um fiet pro occaÅ¿u Lunæ.</s> - </p> - <p> - <s xml:space="preserve"><emph style="sc">DIem</emph> ætatis Lunæ iam totus orbis Å¿cit inuenire, media Å¿upputatione nu-<lb/>meri EpactÄ™ currentis cum numero menÅ¿ium, Å¿umpto principio à Martio, <lb/>adiunctis diebus menÅ¿is currentis, & detracto numero .30. à ſumma prÄ™dicta, Å¿i ab ip <lb/>Å¿a dictus numerns .30. Å¿uperatur.</s> - </p> - <p> - <s xml:space="preserve">Sed ne aliquis putet Å¿ufficere tantummodo additionem quatuor quintarum ho-<lb/>rÄ™ qualibet die. à nouilunio inchoando, Å¿ciendum eÅ¿t huiuÅ¿modi receſſum Lunæ <lb/>(quamuis non ita exactæ fiat) non <reg norm="computandum" type="context">computandÅ©</reg> eſſe ab orizonte aliquo, Å¿ed à recto, <lb/>Å¿eu à meridiano quod idem eÅ¿t, quemadmodum <reg norm="vnicuique" type="simple">vnicuiq;</reg> mediocriter erudito pa-<lb/>tere poteÅ¿t. </s> - <s xml:space="preserve">At propoÅ¿itum nobis non eÅ¿t Å¿cire qua hora Luna in meridiano repe-<lb/>riatur, Å¿ed in noÅ¿tro obliquo orizonte, in parte orientali Å¿eu occidentali, </s> - <s xml:space="preserve">propterea <lb/>igitur addendus eÅ¿t, ei Å¿ummæ temporis, qua Luna diÅ¿tat à meridiano, arcus Å¿emi-<lb/>diurnus, vel Å¿eminocturnus illius loci Zodiaci, in quo Luna reperitur illa die in pro <lb/>poÅ¿ito parallelo, vt Å¿ciatur proximÄ™, qua hora (ex aÅ¿tronomicis) Luna erit in ori-<lb/>zonte <reg norm="orientali" type="context">oriẽtali</reg>, vel occidentali dicti paralleli. </s> - <s xml:space="preserve">Å¿upra dicta enim additio quatuor quin <lb/>tarum horæ tantummodo, Å¿ufficiens erit temporibus æquinoctij, Å¿ed aliis anni tem-<lb/>poribus falli ratione iam dicta.</s> - </p> - <pb o="223" n="235" file="0235" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0235"/> - <figure position="here"> - <image file="0235-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0235-01"/> - <caption style="it" xml:space="preserve">Pro Lunæ ortu. <lb/>Ad lati .45.</caption> - </figure> - <figure position="here"> - <image file="0235-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0235-02"/> - </figure> - <pb o="224" n="236" file="0236" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0236"/> - <figure position="here"> - <image file="0236-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0236-01"/> - <caption style="it" xml:space="preserve">Pro Lunæ occaÅ¿u. <lb/>Ad lati .45.</caption> - </figure> - <figure position="here"> - <image file="0236-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0236-02"/> - </figure> - <pb o="225" n="237" file="0237" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0237"/> - </div> - </div> - <div type="section" level="3" n="4"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE LVCERNA SPIRITALI QVAM SERENISS. <lb/>Sabaudiæ Duce <var>.D.</var> meo collendiÅ¿s. anno .1570. conÅ¿truxi.</head> - <head style="it" xml:space="preserve">CLARISS. FRANCISCO BARBARO VENETORVM <lb/>apud Sereniſſimum Sabaudiæ Ducem Oratori IlluÅ¿triſſimo.</head> - <p> - <s xml:space="preserve"><emph style="sc">HEron</emph> varias ac diuerÅ¿as hydraulicas, & Å¿piritales machinas propoÅ¿uit, in <lb/>ter quastamen nullam Å¿imilem, ei <reg norm="quam" type="context">quã</reg> ego Sereniſſimo Sabaudiæ Duci <lb/>conÅ¿truxi, deÅ¿cribit, quæ quidem fuit Lucerna, & erat huiuÅ¿modi, vt à <lb/>magno aliquo vaÅ¿e oleo pleno Å¿upra alicuius triclinij tabulatum poÅ¿ito, <lb/>Å¿ubtilis quidam tubus perpendi-<lb/> - <anchor type="figure" xlink:label="fig-0237-01a" xlink:href="fig-0237-01"/> - culariter per tabulatum exiret, <lb/>& in dictum triclinium vÅ¿que ad <lb/>medium deÅ¿cenderet, ita tamcn <lb/>vt hic Å¿olus tubus, non item <lb/>vas oleo plenum cerneretur, cu-<lb/>ius <reg norm="quidem" type="context">quidẽ</reg> tubi inferiori extremi-<lb/>tati iunctum eſſet quoddam par-<lb/>uum receptaculum olei, Å¿imile co <lb/>operculo alicuius pyxidis, è cuius <lb/>ambitu prope baÅ¿im multi diuer-<lb/>Å¿i quæ tubi æquales & orizonta-<lb/>les, cuiuÅ¿uis longitudinis proÅ¿ili-<lb/>rent, quorum quilibet in extremi <lb/>tate Å¿ua, exiguam quandam pyra <lb/><reg norm="midem" type="context">midẽ</reg>, <reg norm="appensam" type="context context">appẽsã</reg> haberet, in qua elli <lb/><reg norm="chnum" type="context">chnÅ©</reg> eſſet <reg norm="cum" type="context">cÅ©</reg> mixo. </s> - <s xml:space="preserve"><reg norm="oleum" type="context">oleÅ©</reg> deinde <lb/>medio <reg norm="perpendicularis" type="simple context">ê‘pẽdicularis</reg> tubi ad rece <lb/><reg norm="ptaculum" type="context">ptaculÅ©</reg> extrinÅ¿ece deÅ¿cendebat, <lb/>& per alios tubos ad nutriendas <lb/>flammas dum <reg norm="arderent" type="context">arderẽt</reg> ferebatur: <lb/></s> - <s xml:space="preserve">at vero <reg norm="cum" type="context">cÅ©</reg> eædem erant extinctæ <lb/>ne minima quidem olei gutta de <lb/>Å¿cendebat: </s> - <s xml:space="preserve">id quod eos qui aÅ¿ta-<lb/>bant in admirationem trahebat.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0237-01" xlink:href="fig-0237-01a"> - <image file="0237-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0237-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Hæcautem lucerna Å¿ic erat <reg norm="con" type="context">cõ</reg> <lb/>Å¿tructa. </s> - <s xml:space="preserve">Vas <reg norm="olearium" type="context">oleariÅ©</reg> cylindricum <lb/>vt in Å¿ubÅ¿cripta figura patet, cuiuÅ¿ <lb/>uis magnitudinis, omni ex parte <lb/>clauÅ¿um faciendum curaui, ita ta <lb/>men vt eius coopereulum <reg norm="aliquam" type="context">aliquã</reg> <lb/><reg norm="tulum" type="context">tulÅ©</reg> concauum eſſet, in cuius me <lb/>dio erat foramen <var>.e.</var> quod erat os <lb/>tubi <var>.e.g.</var> qui Å¿ub <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> vaÅ¿is fun-<lb/>do vÅ¿que ad <var>.g.</var> tranÅ¿ibat, Å¿ed po-<lb/>Å¿tea Å¿urÅ¿um, quaÅ¿i <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> ad cooper <lb/>culum in Å¿itu <var>.c.</var> ab inferiori parte <lb/>reflectebatur, & ibi <reg norm="terminabatur" type="simple">terminabat̃</reg>.</s> - <pb o="226" rhead="IO. BABPT. BENED." n="238" file="0238" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0238"/> - <s xml:space="preserve">Vnde oleum quod in vas infundebatur per foramen <var>.e.</var> dictum vas poÅ¿tea ingredie-<lb/>batur per foramen <var>.c</var>. </s> - <s xml:space="preserve">Habebat deinde tubum <var>.n.u.</var> rectum, qui à ſitu <var>.n.</var> propinquo co <lb/>operculo ad libellam extremi <var>.c.</var> incipiebat, & per fundum <reg norm="contignationemque" type="simple">contignationemqÌ;</reg> vÅ¿que <lb/>ad centrum Å¿upradictireceptaculi (circa quod <reg norm="tuborum" type="context">tuborÅ©</reg> ope appenfæ erant ellychnio-<lb/>rum pyramides) tranÅ¿ibat, atque huiuiuÅ¿modi tubi <var>.n.u.</var> extremitates tam Å¿uperius <lb/>quam inferius erant apertæ, & hic tubus aeris erat. </s> - <s xml:space="preserve">Præterea aliud quoddam fora-<lb/> - <anchor type="figure" xlink:label="fig-0238-01a" xlink:href="fig-0238-01"/> - men in vaÅ¿is fundo feceram, cui <lb/>paruum <reg norm="tubum" type="context">tubÅ©</reg> <var>.a.o.t.</var> reflexum, ita <lb/>tamen, vt <var>.o.</var> altius eſſet quam <var>.g.</var> <lb/><reg norm="aptaueram" type="context">aptauerã</reg>, <reg norm="atque" type="simple">atq;</reg> <reg norm="per" type="simple">ê‘</reg> hunc reflexum tu <lb/>bum <var>.a.o.t.</var> oleum vaÅ¿is exibat, <reg norm="quod" type="simple">ê™</reg> <lb/>per oÅ¿culum <var>.t.</var> in quendam cana-<lb/>lem tubo <var>.n.u.</var> inÅ¿ertum, ab extra <lb/>oleum effundebat, & ab exteriori <lb/>parte arundinis <var>.n.u.</var> ingredieba-<lb/>tur receptaculum appenÅ¿um ex-<lb/>tremo dicti tubi <var>.n.u.</var> quod extre-<lb/>mum <reg norm="apertum" type="context">apertÅ©</reg> erat, ut dixi, & à fun-<lb/>do receptaculi tantum <reg norm="diÅ¿tans" type="context">diÅ¿tãs</reg>, <reg norm="quam" type="context">quã</reg> <lb/>tum volebamus oleum ab ipÅ¿ius <lb/>receptaculi fundo altum exiÅ¿tere <lb/>quod quidem oleum Å¿tatim vt ad <lb/>oÅ¿culum dicti tubi <var>.n.u.</var> accedebat <lb/>id claudebat. </s> - <s xml:space="preserve">Vnde aeri <reg norm="ingrediem" type="context">ingrediẽ</reg> <lb/>di vas <var>.q.b.</var> non amplius patebat <lb/>aditus, & per conÅ¿equens, neque <lb/>amplius <reg norm="oleum" type="context">oleÅ©</reg> <reg norm="per" type="simple">ê‘</reg> <reg norm="tubum" type="context">tubÅ©</reg> <var>.a.o.t.</var> eflue-<lb/>bat, nec etiam per <reg norm="tubum" type="context">tubÅ©</reg> <var>.e.g.c.</var> aer <lb/>ingredi poterat, cum <var>.c.g.</var> Å¿emper <lb/>oleo exiÅ¿teret <reg norm="plenum" type="context">plenÅ©</reg> <reg norm="quum" type="context">quÅ©</reg> <var>.g.</var> magis <lb/><reg norm="quam" type="context">quã</reg> <var>.t.</var> <reg norm="adimum" type="context">adimÅ©</reg> deuergeret.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0238-01" xlink:href="fig-0238-01a"> - <image file="0238-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0238-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Quoties deinde oleum in vas <lb/>infundere volebamus, oportebat <lb/><reg norm="cum" type="context">cÅ©</reg> fumitate digiti claudere oÅ¿cu-<lb/>lum <var>.t.</var> exiguitubi <var>.a.o.t.</var> vnde aer <lb/><reg norm="impulsus" type="simple">impulsê°</reg> ab oleo tubi <var>.e.g.c.</var> extra <lb/>per tubum <var>n.u.</var> quouÅ¿que oleum <lb/>vaÅ¿is ad Ä™quilibrium ipÅ¿ius <var>.n.</var> per-<lb/>ueniebat, <reg norm="pertubum" type="context">pertubÅ©</reg> <var>.n.u.</var> ingredie-<lb/>batu. </s> - <s xml:space="preserve">& quando dictum oleum <reg norm="qui" type="simple">ê—</reg> <lb/>dictum tubum <var>.n.u.</var> extrinÅ¿ecè in-<lb/>trabat in receptaculum <var>.d.p.</var> nil <lb/>amplius olei in vas infundendum <lb/>erat, & oportebat alicuius digito <lb/>foramen <var>.u.</var> inferius arundinis <var>.n.<lb/>u.</var> claudi, & foramen <var>.t.</var> aperiri, vt <lb/><reg norm="per" type="simple">ê‘</reg> ipÅ¿um <var>.t.</var> aliqua portio olei exi-<lb/>ret, quia tunc quædam pars tubi <var>.e.g.</var> vacua reddebatur, & cum per <var>.t.</var> nil amplius - <pb o="227" rhead="EPISTOLAE." n="239" file="0239" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0239"/> - olei egrediebatur, aperiebatur <var>.u.</var> & peripÅ¿um <var>.t.</var> denuo tantum olei exire permitte-<lb/>bamus, quantum in receptaculo ad claudendum foramen <var>.u.</var> idoneum exiÅ¿teret. </s> - <s xml:space="preserve">Ra-<lb/>tio vero, quÄ™ me mouit, ut punctum <var>.g.</var> inferius ipÅ¿ius <var>.t.</var> conÅ¿tituam, eÅ¿t, quia <reg norm="cum" type="context">cÅ©</reg> clau-<lb/>Å¿um erit <var>.u.</var> per dictum <var>.t.</var> oleum non amplius egredietur, quia pondus olei in tubo <var>.c.<lb/>g.</var> maius euadet oleo quod vÅ¿que ad <var>.t.</var> progrederetur, tubum autem <var>.e.g.c.</var> reflexum <lb/>facio, ne cogamur claudere foramen <var>.e.</var> quia hoc difficile præſtaretur, tubum etiam <lb/><var>a.o.t.</var> Å¿urÅ¿um verÅ¿usreflexum conÅ¿titui, vt aerem ab ingreſſu per foramen <var>.t.</var> <reg norm="arcerem" type="context">arcerẽ</reg>, <lb/>quia huiuÅ¿modi aer nunquam deÅ¿cendit Å¿i corpus magis denÅ¿um non deÅ¿cendat.</s> - </p> - <p> - <s xml:space="preserve">Verum eÅ¿t, <reg norm="quod" type="simple">ê™</reg> melius erit, vt maiores difficultates euitemus, Å¿tatuere dictum <reg norm="tubum" type="context">tubÅ©</reg> <lb/><var>a.o.t.</var> ita curuum vt eÅ¿t .ω qui cum Å¿uo extremo inferiori ipÅ¿i <var>.n.u.</var> Å¿it contiguus ita ta-<lb/>men ut dictum extremum inferius <reg norm="non" type="context">nõ</reg> Å¿it inferius quam <var>.o.</var> quia totum oleum exiret.</s> - </p> - <p> - <s xml:space="preserve">Volui etiam vt Å¿uperior extremitas <var>.n.</var> tubi <var>.n.u.</var> Å¿it in aere vaÅ¿is & non in oleo, ne <lb/>per eam oleum exeat, quia cum extremitas <var>.u.</var> inferior Å¿it <var>.g.</var> totum oleum quod Å¿u-<lb/>peraret oÅ¿culum <var>.n.</var> per dictum tubum <var>.n.u.</var> ratione maioris ponderis egrederetur, <lb/><reg norm="quemadmodum" type="wordlist">quẽadmodum</reg> cuilibet, vel mediocriter in philoÅ¿ophicis rebus verÅ¿ato innoteÅ¿cet.</s> - </p> - <pb o="228" n="240" file="0240" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0240"/> - </div> - </div> - <div type="section" level="3" n="5"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DEFENSIO EPHEMERIDVM.</head> - <head style="it" xml:space="preserve">AD JLLVST.D. BERNARDVM <lb/>Trottum.</head> - <p> - <s xml:space="preserve"><emph style="sc">EDita</emph> erant Å¿cripta quædam, quorum titulus <emph style="sc">animadver-<lb/>siones in ephemeridas</emph>. </s> - <s xml:space="preserve">& breuis alia diÅ¿putatio de er <lb/>roribus calculorum AÅ¿tronomicorum. ac <reg norm="demum" type="context">demÅ©</reg> TheÅ¿es quæ-<lb/>dam typis datæ .11. AuguÅ¿ti .1581. quÄ™ omnia cum ad manus <lb/>meas perueniſſent, non potui, non eis animum admouere, <reg norm="cum" type="context">cÅ©</reg> <lb/>ibi de his Å¿tudijs <reg norm="ageretur" type="simple">ageret̃</reg>, in quibus partem non exiguam an-<lb/>norum meorum conÅ¿umpÅ¿i. </s> - <s xml:space="preserve">nec tamen Å¿cribere aliquid Å¿ta-<lb/>tueram; </s> - <s xml:space="preserve">tum quod exiÅ¿timarem viros AÅ¿tronomiæ peritos <lb/>facile, quanti facienda eſſent ea quæ edita erant iudicaturos, alijs verò haud gratam <lb/>futuram harum rerum tractationem. </s> - <s xml:space="preserve">Tum quod Å¿i ingenue meam Å¿ententiam profer <lb/>re voluiſſem<unsure/>, non poteram, Å¿ine maxima authoris moleÅ¿tia ferè omnia reprobare. <lb/></s> - <s xml:space="preserve">Quandoquidem Å¿olet vnuÅ¿quiÅ¿que indignationem concipere ex his, quæ Å¿uæ opi-<lb/>nioni repugnant, id omne maleuolentiæ potius quam veritatis Å¿tudio tribuens. </s> - <s xml:space="preserve">Qui <lb/>nimo cum nec deeſſent qui dicerent in meipÅ¿um directa ea tela fuiſſe, nullam fidem <lb/>eis adhibendam duxi, nec enim qui in ephemeri das inuehitur, me arguere poteÅ¿t, <lb/>qui nullas ephemeridas Å¿cripÅ¿i, nec tabulas compoÅ¿ui. </s> - <s xml:space="preserve">Nec Å¿i author quidpiam ex <lb/>his quæ à nobis edita fuiſſent impugnare voluiſſet, ægrè ferre debuiſſem, modo à ve <lb/>ritate nuÅ¿quam deuiaſſet. </s> - <s xml:space="preserve">Liberum enim eÅ¿t cuique Å¿cribere quodlibet. </s> - <s xml:space="preserve">nec Ari-<lb/>Å¿totelem afficit iniuria, quicunque illi fidem Å¿uam non accommodat, & Å¿i valdè ini-<lb/>quus Å¿it, quiÅ¿quis maiorum opiniones veras, & ab omnibus merito comprobatas <reg norm="non" type="context">nõ</reg> <lb/>admittit. </s> - <s xml:space="preserve">Hinc mihi Å¿atis <reg norm="oimbus" type="context">oÄ©bus</reg> feciſſe videbar, cum his qui de Å¿criptis illis me <reg norm="inter- rogauerant" type="context">inter-<lb/>rogauerãt</reg> reÅ¿pondiſſem, ea non Å¿atis firmis eſſe innixa <reg norm="fundamentis" type="context">fundamẽtis</reg>, & quod ad talia <lb/>tractanda opus fuiſſet exercitatiore iudicio. </s> - <s xml:space="preserve">Verumtamen cum tu vnus maxime om <lb/>nium deÅ¿ideres tibi clarius, quæ nam de his mea Å¿it Å¿ententia explicari, non tam tuis <lb/>precibus deuictus quam mea ipÅ¿ius cupiditate de te benemerendi impulÅ¿us, non Å¿u-<lb/>Å¿tineo diutius animum tuum hęſitantem relinquere. </s> - <s xml:space="preserve">Atque vt tibi adeo honeÅ¿ta cu <lb/>pienti <reg norm="morem" type="context">morẽ</reg> geram paucis hiÅ¿ce Å¿criptis incultis quidem, vt ab homine omni pror-<lb/>Å¿us facundia deÅ¿tituto exaratis, Å¿ed ex quibus nihilominus facile, atque perÅ¿picue, vt <lb/>Å¿pero, conceptum animi noÅ¿tri percipere poſſis, Å¿i tamen eam præſtantis ingenij tui <lb/>aciem adhibueris, qua Å¿oles intima quæque Å¿cientiarum penetrare, noÅ¿træ opinio-<lb/>nis Å¿ummam perÅ¿trinxi, quÄ™ ad te mittere decreui. </s> - <s xml:space="preserve">& quamuis ipÅ¿a res de qua agitur, <lb/>quæ exactiorem deÅ¿iderat expoÅ¿itionem, prolixiorem me eſſe cogerit quam voluiÅ¿-<lb/>Å¿em. </s> - <s xml:space="preserve">multa tamen me obmiÅ¿iſſe intelliges, non admodum neceſſaria his quibus <lb/>AÅ¿trologiæ noti Å¿unt termini, vt tuarum occupationum rationem me etiam habere <lb/>intelligeres, at que vt Å¿ummam oblectationem <reg norm="concipiam" type="context">cõcipiam</reg> animo Å¿i me tibi aliqua ex <lb/>parte Å¿atisfeciſſe intellexero ita humanitati tuæ gratiam habebo, quÄ™ mihi occaÅ¿io <lb/><reg norm="nem" type="context">nẽ</reg> <reg norm="praebuit" type="simple">p̃buit</reg>, imò verò me impulit, ad ea <reg norm="proferenda" type="context">proferẽda</reg>: </s> - <s xml:space="preserve">quæ grata eſſe poſſint tui Å¿imilibus, <lb/>ideÅ¿t <reg norm="praeclaro" type="simple">p̃claro</reg> & <reg norm="candido" type="context">cãdido</reg> ingenio præditis, <reg norm="atque" type="simple">atq;</reg> ad <reg norm="euellendam" type="context context">euellẽdã</reg> ex <reg norm="eorum" type="context">eorÅ©</reg> animis falÅ¿am opi- - <pb o="229" rhead="EPISTOL AE." n="241" file="0241" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0241"/> - nionem, Å¿i quam fortaſſe ex <reg norm="illorum" type="context">illorÅ©</reg> Å¿criptorum lectione <reg norm="conceperunt" type="context">cõceperunt</reg> circa ea, de qui-<lb/>bus nunc Å¿um acturus.</s> - </p> - <p> - <s xml:space="preserve"><reg norm="Quemadmodum" type="context">QuemadmodÅ©</reg> igitur ab hoc authore ter Å¿criptum fuit de <reg norm="contradictionibus" type="context">cõtradictionibus</reg>, Å¿iue <lb/>erroribus Ephemeridum, & earum calculos Å¿equentium, & de ratione qua cognoÅ¿ci <lb/>poteÅ¿t Å¿itus & locus alicuius Å¿uperioris planetæ, diuerÅ¿us ab eo, qui ab ipÅ¿is Epheme <lb/>ridibus aſſignatus eÅ¿t, ita diÅ¿putationem hanc meam diuidam in tres partes, quo Å¿ci <lb/>licet minus confusè, & magis diÅ¿tinctè à me Å¿cribatur, <reg norm="praeÅ¿upponendo" type="simple">p̃ſupponendo</reg>, vt animaduer-<lb/>tere potes, huius Å¿criptoris intentionem, aliam non fuiſſe, <reg norm="quam" type="context">quã</reg> <reg norm="oÅ¿tendere" type="context">oÅ¿tẽdere</reg>, quod Å¿cripto <lb/>res Ephemeridum diuerÅ¿imode eiuÅ¿dem temporis locum planetæ aſſignauere, & <lb/>quod eum faciant modo nimium velociter currere, modo nimium in vno Å¿igno mo-<lb/>rari, vt (exempli gratia) Martem interdum faciunt morari Å¿ex, aut Å¿eptem menÅ¿ibus <lb/>in vno Å¿igno. </s> - <s xml:space="preserve"><reg norm="Idque" type="simple">Idq;</reg> poÅ¿tea in cauÅ¿a eſſe ait, vt AÅ¿trologi indiciarij <reg norm="fallantur" type="context">fallãtur</reg>, & Å¿imul <lb/>careant certis fundamentis rationum quibus futura indicent, & prædicant. </s> - <s xml:space="preserve">Primum <lb/>ergo videndum eÅ¿t, quam rectè hic vÅ¿us Å¿it arte, & Å¿cientia, vt aliorum opiniones, & <lb/>Å¿cripta redarguere poſſet. </s> - <s xml:space="preserve">Deinde videbimus quomodo verum Å¿it, & poſſibile id <lb/>quod ab AÅ¿trologis hactenus creditum, atque traditum eÅ¿t, & qua ratione poſſint Å¿ie <lb/>ri veri calculi à peritis regularum Å¿cientiæ.</s> - </p> - <p> - <s xml:space="preserve">In primo igitur tractatu inÅ¿cripto AnimadnerÅ¿iones, præſupponit Author pro-<lb/>feſſores huius Å¿cientiæ neÅ¿cire inuenire vera loca planetarum, quia vtuntur Ephe-<lb/>meridibus, in quibus eorum loca non rectè Å¿unt notata. </s> - <s xml:space="preserve">Quod Å¿ecundum ipÅ¿um ori <lb/>tur, ex errore calculatorum, Å¿eu computiÅ¿tarum, potius quam ex varietate tabula-<lb/>rum, à quibus Ephemerides Å¿umptæ Å¿unt, hoc tamen verum non eÅ¿t, Ephemeridas, <lb/>Å¿cilicet, ita inter Å¿e differre, ratione errorum computiÅ¿tarum tantummodo, Å¿ed po-<lb/>tius ratione ipÅ¿arum tabularum, & Å¿i interdum contingere poſſit error aliquorum mi <lb/>nutorum, nec non graduum, non propterea Ephemerides ita Å¿pernendæ Å¿unt. </s> - <s xml:space="preserve">In <lb/>multis enim calculis, tales errores excuÅ¿abiles Å¿unt, cum ab innumerabilibus propè <lb/>accidentibus oriri poſſint, præſertim in calculis prutenicis.</s> - </p> - <p> - <s xml:space="preserve">Videatur deinde vbi is profert quinquageÅ¿imum enuntiatum centiloquij Ptole-<lb/>mÄ™i, Å¿atis mendoÅ¿e. </s> - <s xml:space="preserve">PtolemÄ™us enim ibi Å¿ic ait.</s> - </p> - <quote> - <s xml:space="preserve">Non obliuiÅ¿caris eſſe centum viginti coniunctiones, quæ Å¿unt in Å¿tellis erraticis, <lb/>in illis enim eÅ¿t maior Å¿cientia eorum quæfiunt in hunc <reg norm="mundum" type="context">mÅ©dum</reg> Å¿uÅ¿cipiendi incre-<lb/>mentum, & decrementum.</s> - </quote> - <p> - <s xml:space="preserve">Nam, neque eo in loco, neque alibi, PtolemÄ™us quidquam eius dicere voluit <lb/>quod ab hoc profertur.</s> - </p> - <p> - <s xml:space="preserve">Pergatur poÅ¿tea in pag .2. & videbitur hunc exiÅ¿timare abÅ¿urdum quod Saturni, <lb/>& Iouis coniunctio vera anni .1563. potuerit eſſe in Leone Å¿igno igneæ triplicitatis <lb/>cum eorum coniunctio vera anni .1544. fuerit in Scorpione, Å¿igno triplicitatis <lb/>aqueæ, & cum coitus eorum anni .1583. futurus Å¿it in PiÅ¿cibus, Å¿igno pariter tripli-<lb/>citatis aqueæ. </s> - <s xml:space="preserve">Ita enim ait.</s> - </p> - <quote> - <s xml:space="preserve">Nam poÅ¿tquam duæ Å¿tellæ coiuerint, non prius Å¿ub alio alterius triplicitatis Å¿igno <lb/>inter Å¿e Å¿unt conuenturæ, quam per omnia Å¿igna quæ eiuÅ¿dem ternarij cum primo ex <lb/>titerint prius <reg norm="coniungantur" type="context">coniungãtur</reg>. </s> - <s xml:space="preserve">Ita Å¿entit PtolemÄ™us, <reg norm="cæterique" type="simple">cæteriqÌ;</reg> non aÅ¿pernendi nominis <lb/>Aftronomi.</s> - </quote> - <p> - <s xml:space="preserve">Et <reg norm="tamen" type="wordlist">tamẽ</reg> PtolemÄ™us nunquam <reg norm="quidquam" type="context">quidquã</reg> huius rei attigit, & quamuis AlbumaÅ¿ar <lb/>& Alchibitius de eo <reg norm="loquantur" type="context">loquãtur</reg>, is tamen eosnon intellexit, cum illi ibi <reg norm="non" type="context">nõ</reg> agant de - <pb o="230" rhead="IO. BAPT. BENED." n="242" file="0242" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0242"/> - periodis apparentibus, aut veris, Å¿ed de mediocribus aut æqualibus, & quidem re-<lb/>ctè dicunt, quia lineæ eorum mediorum motuum non coeunt in aliquo Å¿igno alte-<lb/>rius triplicitatis, prius quam pertranÅ¿iuerint omnia Å¿igna illius, in qua incÄ™perunt. <lb/></s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> nullum inconueniens Å¿equitur, Å¿i in veris <reg norm="coniunctionibus" type="context">cõiunctionibus</reg> non reperitur hæc re-<lb/>gula. </s> - <s xml:space="preserve">Fieri enim poteÅ¿t, vt lineæ mediorum motuum coniungantur in vno Å¿igno, cor <lb/>pora verò eorum planetarum coeant in alio, cum rarò eueniat, vt linea medij mo-<lb/>tus, eadem Å¿it cum linea veri.</s> - </p> - <p> - <s xml:space="preserve">Nunc quidem <reg norm="tamen" type="wordlist">tamẽ</reg> non affirmauerim, nec ne gauerim eorum coniunctionem an <lb/>ni .1563. fuiſſe potius in Cancro, quam in Leone. </s> - <s xml:space="preserve">Sed tantum dicam <reg norm="vanum" type="context">vanÅ©</reg> eſſe cre <lb/>dere id eueniſſe propter Å¿imilem naturam, aut qualitatem Å¿ignorum. </s> - <s xml:space="preserve">Hunc enim <lb/>reÅ¿pectum non habent illi planetæ in verisÅ¿uis coniunctionibus. </s> - <s xml:space="preserve">Exempli autem <lb/>cauÅ¿a ponamus, quod rectè Å¿upputatæ fuerint coniunctiones annorum .1484. 1504. <lb/>& 1524. quod attinet ad differentiam duodecatemorij, Å¿cilicet prima in .24. gradu <lb/>Scorpij, Å¿ecunda in .20. Cancri, tertia in .10. PiÅ¿cium. </s> - <s xml:space="preserve">Cum Å¿ecunda anticipauerit <lb/>trigonum perfectum cum prima, gradibus .4. & tertia anticipauerit trigonum perfe-<lb/>ctum <reg norm="cum" type="context">cÅ©</reg> Å¿ecunda gradibus .10. Å¿i forte prima vt facta fuit in .24. gradu Scorpij facta <lb/>fuiſſet in .2. gradu eiuÅ¿dem, <reg norm="planum" type="context">planÅ©</reg> eÅ¿t <reg norm="quod" type="simple">ê™</reg> <reg norm="Å¿ecunda" type="context">Å¿ecÅ©da</reg> facta fuiſſet in .28. gradu geminorum <lb/>& tertia in .18. Aquarij, quæ Å¿igna Å¿unt diuerſæ triplicitatis ab illa Cancri. </s> - <s xml:space="preserve">InÅ¿uper <lb/>Å¿i coniunctio anni .1544. quæ fuit in .28. gradu Scorpij fuerit recta <reg norm="correÅ¿pondens" type="context">correÅ¿pondẽs</reg> <reg norm="prae cedenti" type="simple">prÄ™Â<lb/>cedenti</reg>, anni .1524. per gra .18. Å¿ine dubio Å¿i coniunctio anni .1524. facta fuiſſet in <lb/>18. gradu Aquarij, illa anni .1544. fuiſſet in .6. Scorpij Å¿igni alterius triplicitatis <reg norm="quam" type="context">quã</reg> <lb/>Å¿int Gemini. </s> - <s xml:space="preserve">Præterea, vt anno .1544. <reg norm="coniunctio" type="context">cõiunctio</reg> facta eÅ¿t in .28. gra. Scorpij, & 1563 <lb/>in .29. Cancri, ponendo eas eſſe rectas, quod attinet ad Å¿uperandum trigonum vno <lb/>gradu, Å¿i anno .1544. facta fuiſſet in .30. Scorpij, anno .1563. proculdubio facta fuiÅ¿-<lb/>Å¿et in primo gradu Leonis. </s> - <s xml:space="preserve">Et Å¿uppoÅ¿itis ijs interuallis, quæ Å¿uperÅ¿unt, aut deÅ¿unt per <lb/>fectis trigonis, Å¿i coniunctio anni .1524. fuiſſet in .20. gradu PiÅ¿cium, anno .1544. <lb/>fuiſſet in .8. Sagittarij. </s> - <s xml:space="preserve">Quæ quidem omnia aduerÅ¿antur opinioni huius Å¿criptoris.</s> - </p> - <p> - <s xml:space="preserve">Quodautem opinatur coniunctionem anni .1583. fore in Ariete, Å¿ic dicens pagi <lb/>na Å¿ecunda.</s> - </p> - <quote> - <s xml:space="preserve">Non erit ab re Å¿i & eandem Saturni, & Iouis coniunctionem in primo igneæ tri-<lb/>plicitatis Å¿igno, quod eÅ¿t Aries futuram afferamus anno .1583. Å¿i ab accidentibus no <lb/>bis licet, vt ab omnibus paſſim conceditur, planetarum loca diÅ¿cernere.</s> - </quote> - <p> - <s xml:space="preserve">In eo fallitur, <reg norm="nam" type="context">nã</reg> <reg norm="neque" type="simple">neq;</reg> Saturnus, <reg norm="neque" type="simple">neq;</reg> Iupiter, <reg norm="errant" type="context">errãt</reg> à vero per .9: nec .8. gra. ac ne <reg norm="per" type="simple">ê‘</reg> <lb/>4. <reg norm="quidem" type="context">quidẽ</reg> in <reg norm="qui" type="simple">ê—</reg> buÅ¿uis Ephemeridibus aut tabulis. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> videbit eiuÅ¿modi <reg norm="coniunctionem" type="context context">cõiunctionẽ</reg> <lb/>contra <reg norm="Å¿ententiam" type="context">Å¿ententiã</reg> <reg norm="Å¿uam" type="context">Å¿uã</reg> fieri in PiÅ¿cibus, <reg norm="non" type="context">nõ</reg> <reg norm="autem" type="wordlist">aÅ©t</reg> in Ariete. (vt poÅ¿tea res ipÅ¿a nos docuit <lb/>Å¿ub <reg norm="menÅ¿e" type="context">mẽſe</reg> Aprili. poÅ¿t <reg norm="quidem" type="simple context">ê—dẽ</reg> <reg norm="Å¿criptam" type="context">Å¿criptã</reg> <reg norm="hanc" type="context">hãc</reg> <reg norm="epiÅ¿tolam" type="context">epiÅ¿tolã</reg>, <reg norm="vulgarique" type="simple">vulgariqÌ;</reg> Å¿ermone <reg norm="tranÅ¿miÅ¿sam" type="context context">trãſmiÅ¿sã</reg>, Å¿ed <reg norm="an- tequam" type="context">an-<lb/>tequã</reg> in <reg norm="latinum" type="context">latinÅ©</reg> <reg norm="tranÅ¿lata" type="context">trãſlata</reg>, & huic volumini inÅ¿erta <reg norm="cum" type="context">cÅ©</reg> alijs Typographo <reg norm="committeretur" type="context">cõmitteretur</reg>.)</s> - </p> - <p> - <s xml:space="preserve">Vbi poÅ¿tea meminit magnæ periodi annorum .960. non tantum ei cogitandum <lb/>erat <reg norm="hanc" type="context">hãc</reg> fuiſſe opinionem <reg norm="antiquorum" type="context">antiquorÅ©</reg>, vt videri <reg norm="pont" type="context">põt</reg> apud <reg norm="AlbumaÅ¿arem" type="context">AlbumaÅ¿arẽ</reg> & <reg norm="Alchibicium" type="context">AlchibiciÅ©</reg>, <lb/>Å¿ed etiam perpendendum an eſſet vera, <reg norm="priuÅ¿quam" type="context">priuÅ¿quã</reg> ei adhæreret. </s> - <s xml:space="preserve">Hic enim fuit vnus <lb/>ex erroribus illius ætatis, quæ nondum penetrauerat intima huius Å¿cientiæ. </s> - <s xml:space="preserve">Sunt <lb/>tamen illi antiqui excuÅ¿atione aliqua digni. </s> - <s xml:space="preserve">Ponebant enim vigeÅ¿imo quoque an-<lb/>no præcisè fieri mediam coniunctionem Saturni <reg norm="cum" type="context">cÅ©</reg> Ioue, & in quolibet Å¿igno eiuÅ¿-<lb/>dem triplicitatis <reg norm="coniungi" type="context">cõiungi</reg> quater. </s> - <s xml:space="preserve">Itaque in qualibet triplicitate dicebant eos coire <lb/>duodecies.</s> - </p> - <pb o="231" rhead="EPISTOL AE." n="243" file="0243" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0243"/> - <p> - <s xml:space="preserve">Quod Å¿ecundum primum Å¿uppoÅ¿itum finiebatur Å¿pacio annorum .240. qui nume <lb/>rus fit. ex .20. duodecies multiplicatis. </s> - <s xml:space="preserve">Et quia triplicitates Å¿unt .4. ideo credebant <lb/>in Å¿pacio annorum .960. qui numerus fit ex .240. quater multiplicato, perfici .48. <reg norm="con- iunctiones" type="context">cõ-<lb/>iunctiones</reg>, priuÅ¿quam redirent ad Å¿e coniungendos in eodem loco, ubi prius iun-<lb/>cti fuiſſent. </s> - <s xml:space="preserve">Primum autem Å¿uppoÅ¿itum, quod vigeÅ¿imo quoque anno <reg norm="iungerentur" type="context">iÅ©gerentur</reg>, <lb/>colligebant Å¿ic ratiocinantes. </s> - <s xml:space="preserve">Si Saturnus annis .30. peragit Å¿uum curÅ¿um per om-<lb/>nia Å¿igna Zodiaci, Iupiter autem peragit eum annis .12. Saturnus ambulauerit .4. Å¿i <lb/>gna, et .4. quintas partes Å¿igni, Å¿iue gra .24. dum Iupiter peragit integrum ambitum <lb/>ideÅ¿t annis .12. </s> - <s xml:space="preserve">Itaque deÅ¿unt ei anni .8. ad perueniendum ad .20. quibus .8. annis Sa-<lb/>turnus <reg norm="perambulat" type="simple">ê‘ambulat</reg> Å¿igna tria & <reg norm="quintam" type="context">quintã</reg> <reg norm="partem" type="context">partẽ</reg> unius Å¿igni <var>.i.</var> gradus .6. qui iuncti dictis Å¿i <lb/>gnis .4. & gra .24. faciunt Å¿igna .8. quæ Iupiter item percurrit in annis .8. atque ita in <lb/>annis .20. </s> - <s xml:space="preserve">Iupiter percurrit .20. Å¿igna <reg norm="antequam" type="context">antequã</reg> perueniat ad <reg norm="Saturnum" type="context">SaturnÅ©</reg>, cum Saturnus eo <lb/>dem tempore perfecerit curÅ¿um Å¿ignorum .8. </s> - <s xml:space="preserve">Eandem concluÅ¿ionem etiam fortaÅ¿ <lb/>Å¿e collegerant ex dictis Å¿uppoÅ¿itis, dicentes, Å¿i Saturnus annis .30. ambulat .12. Å¿igna <lb/>proculdubio annis .20. ambulat .8. Å¿igna, quo tempore Iupiter perambulat .20. ad ra <lb/>tionem .12. Å¿ignorum in annis .12.</s> - </p> - <p> - <s xml:space="preserve">Verum hoc Å¿uppoÅ¿itum non eÅ¿t bonum, quoniam, Å¿i ita eſſet, coniunctiones <reg norm="horum" type="context">horÅ©</reg> <lb/>duorum planetarum nunquam exirent ex vna triplicitate, & non modo .960. <reg norm="quoque" type="simple">quoq;</reg> <lb/>anno, Å¿ed etiam Å¿exageÅ¿imo rurÅ¿us coniungerentur in eodem puncto. </s> - <s xml:space="preserve">nec coniunctio <lb/>nes eorum (Å¿emper autem intelligo de medijs) unquam egrederentur ex illis tribus <lb/>Å¿ignis Zodiaci.</s> - </p> - <p> - <s xml:space="preserve">Sed periodus æqualis Saturni, eÅ¿t dierum circiter .10740. atque ita minor an .30. <lb/>atque etiam .29. cum dimidio periodus autem æqualis Iouis, eÅ¿t circiter .4328. vt <lb/>ego eam comperio, quidquid alij <reg norm="dicant" type="context">dicãt</reg>, <reg norm="vtque" type="simple">vtqÌ;</reg> planius alias oÅ¿tendam. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> hæc per <lb/>iodus Iouis, etiam minor eÅ¿t ann .12. prætermittendo in Å¿upputatione <reg norm="tam" type="context">tã</reg> Saturni <reg norm="quam" type="context">quã</reg> <lb/>Iouis quaÅ¿dam minutias horarum & earum partium, quæ hac in re pto nihilo habe-<lb/>ri poſſunt. </s> - <s xml:space="preserve">Atque his duabus periodis eccentricorum duorum planetarum poſſu-<lb/>mus cognoÅ¿cere interuallum quod erit inter vtramque mediam coniunctionem, hoc <lb/>modo agendo, & ratiocinando.</s> - </p> - <p> - <s xml:space="preserve">Si Saturnus diebus .10740. circuit gradus .360. diebus .4328. qui Å¿unt periodus <lb/>Iouis, conficiet gradus .145. & min .4. ideÅ¿t min .8704. & eadem regula inueniemus <lb/><reg norm="quod" type="simple">ê™</reg> Saturnus .30. <reg norm="quibuÅ¿que" type="simple">quibuÅ¿qÌ;</reg> diebus, conÅ¿iciet min .60. & Å¿ecunda .20. </s> - <s xml:space="preserve">Iupiter autem Å¿in-<lb/>gulis .30. diebus, conficiet min .149. & Å¿ecunda .43. vnde Å¿ubtrahendo minuta Satur-<lb/>ni à minutis Iouis, Å¿upererunt min .89. cum Å¿ecun .23. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> Iupiter .30. <reg norm="quibuÅ¿que" type="simple">quibuÅ¿qÌ;</reg> die-<lb/>bus velocitate curÅ¿us, Å¿uperabit Saturnum minutis .89. cum Å¿ecundis .23. </s> - <s xml:space="preserve"><reg norm="Atque" type="simple">Atq;</reg> dicen-<lb/>do, Å¿i minuta .89. cum Å¿ecundis .23 dant nobis dies .30. Å¿upradicta, minuta .8704. da-<lb/>bunt nobis dies .2921. quibus iunctis cum diebus .4328. periodi Iouis, efficientur dies <lb/>7249. ideÅ¿t anni Aegiptij .19. cum diebus .314. & hæc erit æqua periodus temporis <lb/>inter vtranque coniunctionem horum duorum Altiorum planetarum. </s> - <s xml:space="preserve">Vt autem pla <lb/>nius oÅ¿tendatur hanc operationem rectam eſſe (nam demonÅ¿trationem <reg norm="Å¿peculatiuam" type="context">Å¿peculatiuã</reg> <lb/>huius operationis in .113. Theoremate noÅ¿træ ArithmeticÄ™ <reg norm="cuique" type="simple">cuiq;</reg> videre licet) fieri <lb/>poteÅ¿t his alijs calculis.</s> - </p> - <p> - <s xml:space="preserve">Si Saturnus diebus .10740. tranÅ¿it per gra .360. in Å¿pacio dierum .2921. tranÅ¿ibit <lb/>per gradus .97. min .54. quibus iunctis cum gra .145. min .4. Å¿upra notatis, efticientur <lb/>gra .242. min .58. </s> - <s xml:space="preserve">Deinde, Å¿i Iupiter Å¿patio dierum .4328. tranÅ¿it per gra .360. igitur <lb/>Å¿patio .2921. per eandem regulam inueniemus eum tranÅ¿ire gradus .242. mi .58. qui <lb/>numerus par eÅ¿t illi Saturni. </s> - <s xml:space="preserve">Cum ergo Iupiter confecerit vnum ambitum poÅ¿t con - <pb o="232" rhead="IO. BAPT. BENED." n="244" file="0244" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0244"/> - iunctionem cum Saturno, vt rurÅ¿us aſſequatur Saturnum, tranÅ¿eundum ei erit <reg norm="per" type="simple">ê‘</reg> gra <num value="242">.<lb/>242.</num> min .58. iter confectum à Saturno toto tempore annorum .19. & dierum .314. <lb/>ad rationem graduum .360. diebus .10740. (poſſumus <reg norm="etiam" type="context">etiã</reg> dicere gra .243. quia præ <lb/>termiÅ¿imus quaÅ¿dam exiguas particulas periodorum perfectorum cuiuÅ¿que planetÄ™ <lb/>in Å¿uperioribus Å¿upputationibus.) </s> - <s xml:space="preserve">Illos verò gradus .43. Iupiter conÅ¿iciet diebus <num value="2921">.<lb/>2921.</num> ad rationem graduum .360. diebus .4328. </s> - <s xml:space="preserve"><reg norm="Atque" type="simple">Atq;</reg> ita vt diximus, ab vna coniun-<lb/>ctione ad aliam intererunt anni .19. </s> - <s xml:space="preserve">Aegiptij cum diebus .314. vel circa.</s> - </p> - <p> - <s xml:space="preserve">Nunc autem vt videatur an tabulæ AlfonÅ¿i conueniant cum hoc <reg norm="numero" type="wordlist">nr̃o</reg> calculo, <reg norm="con- Å¿iderabimus" type="context">cõ-<lb/>Å¿iderabimus</reg>, <reg norm="quod" type="simple">ê™</reg> Era (vt vocant) dicti temporis annorum .19. cum diebus .314. eÅ¿t dua-<lb/>rum tertiarum Å¿exagenarum, Å¿ecundæ nullius, & 53. <reg norm="primarum" type="context">primarÅ©</reg> Å¿iue dierum. </s> - <s xml:space="preserve">Et per <reg norm="hanc" type="context">hãc</reg> <lb/>Eram colligendo motum mediocrem, tum Saturni, tum Iouis, omiſſis radicibus, & in <lb/>cipiendo ab Ariete, comperiemus <reg norm="quod" type="simple">ê™</reg> <reg norm="vtriuÅ¿que" type="simple">vtriuÅ¿q;</reg> planetæ lineæ eiuÅ¿modi motus tranÅ¿i <lb/>bunt per min .56. tertij gradus Sagittarij, ideÅ¿t coniunctæ erunt.</s> - </p> - <p> - <s xml:space="preserve">In fine poÅ¿tea Å¿ecundæ periodi, cuius era erit .4. tertiarum, Å¿ecundæ .1. et .47. pri-<lb/>marum Å¿exagenarum, locus mediocris <reg norm="vtriuÅ¿que" type="simple">vtriuÅ¿q;</reg> erit in min .56. gra. Å¿exti Leonis. </s> - <s xml:space="preserve">In <lb/>fine verò tertiæ periodi, cuius era erit .6. tertiarum .2. Å¿ecundarum, et .41. primæ, lo-<lb/>cus eorum mediocris inuenietur in .56. minuto gradus .9. Arietis. </s> - <s xml:space="preserve"><reg norm="Atque" type="simple">Atq;</reg> ita deinceps <lb/>in fine <reg norm="cuiusque" type="simple">cuiusqÌ;</reg> periodi, locus <reg norm="eorum" type="context">eorÅ©</reg> mediocris coniunctim Å¿emper diÅ¿tabit à loco me <lb/>diocri præcedentis coniunctionis gradibus .117. ideÅ¿t in trigono antecedenti, minus <lb/>gra .3. </s> - <s xml:space="preserve">Vnde apparet has coniunctiones procedere in contrariam partem reÅ¿pectu or <lb/>dinis <reg norm="Å¿ignorum" type="context">Å¿ignorÅ©</reg> Zodiaci, Å¿ed reÅ¿pectu ordinis graduum <reg norm="Å¿ignorum" type="context">Å¿ignorÅ©</reg>, Å¿emper <reg norm="progrediuntur" type="simple">progrediunt̃</reg> or <lb/>dine per ternos gradus nunquam retrogradientes. </s> - <s xml:space="preserve">Hinc Å¿e quitur, vt non duodecies <lb/>in omni triplicitate coniungantur hi duo planetÄ™, vt antiqui putauerunt, Å¿ed decies <lb/>tantum. </s> - <s xml:space="preserve">& ad Å¿ummum ter in Å¿ingulo Å¿igno, Å¿patio annorum .198. & dierum .220. aut <lb/>circiter, non autem .240. nec .242. </s> - <s xml:space="preserve">Atque decem vices comprehendunt gra .27. & <lb/>vltima vice inueniuntur in Å¿igno Å¿equenti alterius triplicitatis. </s> - <s xml:space="preserve">Exempli gratia, po-<lb/>namus <reg norm="quod" type="simple">ê™</reg> prima vice <reg norm="coniungantur" type="context simple">cõiungant̃</reg> in gra .2. Arietis, Å¿ecunda coniunctio erit in .5. </s> - <s xml:space="preserve">Sagit-<lb/>tarij, tertia. in .8. Leonis. quarta in .11. Arietis, quinta in .14. Sagit .6. in .17. Leonis. <lb/>Å¿eptima in .20. Arietis, octaua in .23. Sagittarij, nona in .26. Leonis, decima in .29. <lb/>Arietis, et vndecima erit in gra .2. Capricorni Å¿igni <reg norm="Å¿equentis" type="context">Å¿equẽtis</reg> triplicitatis. </s> - <s xml:space="preserve">Decem igiÂ<lb/>tur interualla Å¿ingula annorum .19. & dierum .314. faciunt annos .198. & dies .220. <lb/>Immo pertabulas AlfonÅ¿i, eiuÅ¿modi periodus non modo non reperitur <reg norm="annorum" type="context">annorÅ©</reg> .242 <lb/>nec .240. vt antiqui credidere, Å¿ed tribus diebus minor annis .198. & diebus .220. id-<lb/>eÅ¿t per dictas tabulas inuenitur eſſe annorum .198. & dierum .217. tantum, qui nume <lb/>rus multiplicatus per .4. triplicitates, efficiet periodum maiorem, quæ erit annorum <lb/>794. & dierum .138. quo tempore dicti planetæ redeunt ad eundem locum vbi pri-<lb/>mum Å¿e coniunxere.</s> - </p> - <p> - <s xml:space="preserve">Vt exempli gratia, locus mediocris Saturni & Iouis in fine annorum .198. dierum <lb/>217. reperitur in gradu .30. Sagittarij. </s> - <s xml:space="preserve">Si quæſiuerimus hunc locum per <reg norm="aggregatum" type="context">aggregatÅ©</reg> <lb/>annorum .794. & dierum .138. cum annis .198. & diebus .217. quorum Å¿umma eÅ¿t <num value="992">.<lb/>992.</num> & dies .355. inuenietur locus mediocris ipÅ¿orum <reg norm="planetarum" type="context">planetarÅ©</reg> in dicto vltimo gra <lb/>du Sagittarij. </s> - <s xml:space="preserve">Sed Å¿i quęſiuerimus eorum locum mediocrem per aggregatum anno <lb/>rum .198. & dierum .217. cum annis .960. quod erit Å¿umma annorum .1158. & <reg norm="dierum" type="context">dierÅ©</reg> <lb/>217. reperiemus Iouem in gradu .18. </s> - <s xml:space="preserve">Sagittarij & Saturnum in .16. Leonis diÅ¿tanti-<lb/>bus inter Å¿e duabus eorum lineis motuum mediocrium gra. circiter .122. </s> - <s xml:space="preserve"><reg norm="Atque" type="simple">Atq;</reg> Iupi-<lb/>ter præcedet, & oportebit <reg norm="quod" type="simple">ê™</reg> coniunctio eorum mediocris fuerit multis annis ante <lb/>omittendo (vt dixi) radices, quia Å¿atis eÅ¿t inuenire interuallum inter lineas eorum me <lb/>diorum motuum.</s> - </p> - <pb o="233" rhead="EPISTOL AE." n="245" file="0245" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0245"/> - <p> - <s xml:space="preserve">Debebat igitur author animaduerÅ¿ionum non quaÅ¿i cæcus eæcos Å¿equi, Å¿ed prius <lb/>laborare, vt certior fieret, an interuallum annorum .960. </s> - <s xml:space="preserve">Verum eſſet.</s> - </p> - <p> - <s xml:space="preserve">Sed peius eÅ¿t, <reg norm="quod" type="simple">ê™</reg> idem author paulo inferius citat coniunctiones horum duorum <lb/>planetarum anni .1493. et .1512. quas neÅ¿cio vnde Å¿umpÅ¿erit.</s> - </p> - <p> - <s xml:space="preserve">Nam, etÅ¿i inter hos annos eÅ¿t interuallum annorum .19. tamen tantum abeÅ¿t, vt <lb/>coiuerint dictis annis, vt Saturnus anno .1493. ante finem AuguÅ¿ti fuerit in .28. gra-<lb/>du Aquarij, Iupiter verò in .28. Leonis ex diametro oppoÅ¿iti. </s> - <s xml:space="preserve">Et anno .1512. per to <lb/>tum menÅ¿em Iunium & AuguÅ¿tum, Saturnus fuerit in Libra, Iupiter verò in Ariete, <lb/>itaque inter Å¿e Å¿imiliter oppoÅ¿iti, & Å¿i perfecta oppoÅ¿itio non fuit poÅ¿tea niÅ¿i ad <reg norm="finem" type="context">finẽ</reg> <lb/>Iunij ann .1513. & locus Monteregij ab eo citatus, vbi ait eum ponere coniunctio-<lb/>nem anni .1484. in gra .23. min .4. </s> - <s xml:space="preserve">Scorpij, eÅ¿t mendoÅ¿us. </s> - <s xml:space="preserve">Nam ipÅ¿e Montere gius po <lb/>nit dictam coniunctionem in mi .42. gra .24. non autem in min .4. ipÅ¿ius gradus. </s> - <s xml:space="preserve">Sed <lb/>hic error nullius eÅ¿t momenti, fortaſſe qui impræſſorum incuria irrepÅ¿it.</s> - </p> - <p> - <s xml:space="preserve">Pergatur poÅ¿tea obÅ¿ecro ad paginam .3. ipÅ¿arum AnimaduerÅ¿ionum, vbi hic co-<lb/>natur oÅ¿tendere calculatores non obÅ¿eruaſſe verum modum, Å¿ic dicens.</s> - </p> - <quote> - <s xml:space="preserve">Anno .1484. Nouembris .25. Saturno locum conÅ¿tituit Monteregius in grad .23. <lb/>min .4. Scorpij. </s> - <s xml:space="preserve">Anno poÅ¿tmodum Å¿ubÅ¿equenti qui eÅ¿t .1485. eundem in min .7. Sa-<lb/>gitarij collocat .21. Februarij die. </s> - <s xml:space="preserve"><reg norm="Interque" type="simple">InterqÌ;</reg> tempora duo interÅ¿unt menÅ¿es dies .26. <lb/></s> - <s xml:space="preserve">At cum ex motus Å¿ui natura Saturnus hoctemporis Å¿pacio gradus .4. non debeat <reg norm="tran- Å¿cendere" type="context">trã-<lb/>Å¿cendere</reg>, Å¿it tamen inter <reg norm="vtrunque" type="simple">vtrunq;</reg> tempus differentia graduum .7. minutorum .3. quæ <lb/>ratione Å¿ui motus requirunt menÅ¿es .6. vt eos perficiat, conÅ¿tat pluÅ¿quam tribus men <lb/>Å¿ibus fallere nos Saturnum.</s> - </quote> - <p> - <s xml:space="preserve">Hic videre licet quam veram viam hic Å¿ecutus Å¿it ad aperiendos errores Epheme <lb/>ridum, & miÅ¿eri Monteregij, qui Saturnum claudum facit tantum itineris conficere <lb/>tribus <reg norm="menÅ¿ibus" type="context">mẽſibus</reg>, <reg norm="quantum" type="context context">quãtÅ©</reg> vix confeciſſet <reg norm="menÅ¿ib" type="context">mẽſib</reg>. Å¿ex. </s> - <s xml:space="preserve">Sed fortaſſe <reg norm="ratiocinatur" type="simple">ratiocinat̃</reg> hoc modo.</s> - </p> - <p> - <s xml:space="preserve">Si motus naturalis Saturni facit vt circumeat totum cÄ™lum annis .30. igitur menÅ¿i-<lb/>bus .30. conficiet duodecimam partem circuitus, cum menÅ¿es .30. Å¿int duodecima <lb/>pars annorum .30. & quia duodecima pars circuitus cÄ™li intelligitur conÅ¿tare ex .30. <lb/>gradibus, igitur quilibet menÅ¿is poÅ¿tulabit gradum vnum. </s> - <s xml:space="preserve">Ideo illi .6. aut .7. gradus <lb/>poÅ¿tulant tempus, amplius menÅ¿ium Å¿ex.</s> - </p> - <p> - <s xml:space="preserve">Atque eiuÅ¿modi mira ratiocinatio poteÅ¿t in .2. exemplo eius, inÅ¿cripto.</s> - </p> - <quote> - <s xml:space="preserve">Deeodem ex eodem</s> - </quote> - <p> - <s xml:space="preserve">Vbi miratur, <reg norm="quod" type="simple">ê™</reg> Monteregius faciat Saturnum ambulare gra .9. min .10. in menÅ¿i-<lb/>bus .7. & diebus .6. </s> - <s xml:space="preserve">Ad quod iter Saturnc Å¿eni opus eſſet Å¿altem menÅ¿ibus .9. eius <lb/>iudicio.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i hoc miratur, quid dicturus fuiſſet, Å¿i animaduertiſſet, quod idem calculator <lb/>Monteregius facit Saturnum ambulare immo volare gra .9. min .48. non in 7. Å¿ed in <lb/>2. menÅ¿ibus cum dimidio, videlicet à .10. die Iunij vÅ¿que ad .26. AuguÅ¿ti <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> an-<lb/>ni .1504.</s> - </p> - <p> - <s xml:space="preserve">Quid Å¿i etiam animaduertiſſet <reg norm="quod" type="simple">ê™</reg> à .10. die Iunij Å¿upradicti vÅ¿que ad .16. Ianuarij <lb/>anni Å¿equentis, faciunt Saturnum, Å¿urÅ¿um, deorÅ¿um curÅ¿itare amplius gra .17. mi .54. <lb/>Imino Å¿i animaduertiſſet, quod anno .1524. </s> - <s xml:space="preserve">Stoflerinus ab initio anni, <reg norm="vÅ¿quem" type="context">vÅ¿quẽ</reg> ad <lb/>medium Maium, ideÅ¿t menÅ¿ib .4. cum dimidio, facit Saturnum ambulare gra .15. </s> - <s xml:space="preserve">Pro <lb/>fectò ob has velocitates, eius iudicio, tam abÅ¿urdas, obÅ¿tupuiſſet.</s> - </p> - <p> - <s xml:space="preserve">Vbi autem in tergo eiuÅ¿dem paginÄ™ ait, quod gradibus .13. min .42. reÅ¿pondent <lb/>menÅ¿es .19. errauit in calculo, nam ex eiuÅ¿modi tempore Å¿ecundum eius regulam ef- - <pb o="234" rhead="IO. BAPT. BENED." n="246" file="0246" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0246"/> - ficerentnr Å¿inguli ambitus Saturni ad rationem annorum amplius .40.</s> - </p> - <p> - <s xml:space="preserve">Videamus nunc vbi agit de Ioue, & reperiemus <reg norm="quod" type="simple">ê™</reg> in primo exemplo circa <reg norm="annum" type="context">annÅ©</reg>. <lb/>1484. reprÄ™hendit Monteregium, quia facit Iouem ambulare gradus .14. cum min <num value="6">.<lb/>6.</num> in menÅ¿ibus .2. diebus .4. ad quod iter, vt ipÅ¿e ait, opus eſſet Å¿altem <reg norm="menÅ¿ibus" type="context">mẽſibus</reg> .11. <reg norm="atque" type="simple">atq;</reg> <lb/>ita Å¿ecundum ipÅ¿um, Ioui opus eſſet anno vno pro Å¿ingulo medio Å¿igno. </s> - <s xml:space="preserve">Vbi bonus <lb/>hic vir pariter cæcutit.</s> - </p> - <p> - <s xml:space="preserve">Idem in Å¿ecundo exemplo Å¿umpto à Stoflerino ait, <reg norm="quod" type="simple">ê™</reg> Ioui ad curÅ¿um vnius gra-<lb/>dus, & min .5. opus eÅ¿t diebus .30. non autem menÅ¿ibus .7. & diebus .28. vbi oÅ¿tendit, <lb/>Å¿e paruum diÅ¿crimen facere inter Iouem, & Saturnum.</s> - </p> - <p> - <s xml:space="preserve">Miratur poÅ¿tea <reg norm="quod" type="simple">ê™</reg> Stoflerinus faciat laborare generoÅ¿um Iouem ferè menÅ¿ibus <lb/>Å¿ex in vno gradu. </s> - <s xml:space="preserve">Sed multo magis, vt puto, miratus eſſet, Å¿i vidiſſet, quod idem <lb/>Stoflerus in eodem anno facit, quod Iupiter die .4. </s> - <s xml:space="preserve">Ianuarij Å¿it in eodem puncto, in <lb/>quo poÅ¿tea reperitur die vltima AuguÅ¿ti. </s> - <s xml:space="preserve">At fortaſſe dici poſſet, quod Iupiter pro-<lb/>pter prudentiam, & bonitatem Å¿uam factus eÅ¿t R ex omnium Deorum, vt ait Home-<lb/>rus, & ideo expulit è Å¿ede Saturnum, & aÅ¿cendit in altiori cÅ“lo. </s> - <s xml:space="preserve">Vnde euenit vt fa-<lb/>ctus fuerit lentior in curÅ¿u, Saturnus autem velocior. </s> - <s xml:space="preserve">Aut iam tot annos eſſe na-<lb/>tum Iouem, vt iure credi poſſit eum iam factum eſſe Å¿enem, & pariter tardio-<lb/>rem in Å¿e <reg norm="mouendo" type="context">mouẽdo</reg>. </s> - <s xml:space="preserve">aut <reg norm="tunc" type="context">tÅ©c</reg> temporis illum detentum fuiſſe in Å¿ibi dilecta Arcadia <reg norm="cum" type="context">cÅ©</reg> <lb/>CaliÅ¿to. </s> - <s xml:space="preserve">Aut fortaſſe erat in alta Å¿pecula intentus audiendo ingenti certami-<lb/>ni Timoclis & Damidis, vnde pendebat exitium aut gloria familiæ Å¿uæ, nam alio-<lb/>quin Stoflerus non depræhendiſſet eum tam otioÅ¿um & morantem. </s> - <s xml:space="preserve">Sediam relin-<lb/>quamus Saturnum & Iouem, & ad Martem veniamus.</s> - </p> - <p> - <s xml:space="preserve">Ferox & inquietus Mars, qui Å¿emper bella & ignes Å¿pirare Å¿olet, etiam, & ipÅ¿e ab <lb/>AÅ¿trologis factus eÅ¿t piger, & languidus, vt velint eum nonnunquam commorari <lb/>in vno Å¿igno Å¿ex aut Å¿eptem menÅ¿ibus; </s> - <s xml:space="preserve">quod nullo pacto placet authori Animaduer-<lb/>Å¿ionum, cum pag .4. ita Å¿cribat.</s> - </p> - <quote> - <s xml:space="preserve">Quod citra notam, ab omnibus creditur poſſe obÅ¿eruari, quamuis à nobis non ac <lb/>cipiatur.</s> - </quote> - <p> - <s xml:space="preserve">Itaque ei videtur impoſſibile. </s> - <s xml:space="preserve">Quia Mars peragit Å¿uum eircuitum minus .2. an-<lb/>nis. </s> - <s xml:space="preserve">Sed audacior fuiſſe videtur, qui voluerit arguere tot egregios viros antiquos, <lb/>& recentiores, qui vti diligentes rerum cÅ“leÅ¿tium obÅ¿eruatores, ipÅ¿is oculis certi fa <lb/>cti Å¿unt tam de his effectibus Martis, quam aliorum, vnde coacti Å¿unt fingere tantam <lb/>magnitudinem eius epicycli, cum ipÅ¿e nunquam obÅ¿eruauerit motum, nec huius nec <lb/>alterius planetæ, Å¿ed tantum viderit eius moram in Ephemeride Å¿criptam. </s> - <s xml:space="preserve">Si enim <lb/>Å¿altem diceret, Å¿e aliquo tempore obÅ¿eruaſſe iter Martis, & comperuiſſe aliorum opi <lb/>nionem falÅ¿am, attuliſſet aliquem colorem Å¿ententiæ Å¿uæ. </s> - <s xml:space="preserve">Sed Å¿i obÅ¿eruaſſet, non <lb/>Å¿cripÅ¿iſſet poÅ¿tea contra, vt puto. </s> - <s xml:space="preserve">Res enim ita Å¿e habet, quod Mars in omni circui <lb/>tu Å¿ui epicycli tranÅ¿iens per inferiorem partem ipÅ¿ius epicycli, Å¿emper commoratur <lb/>multis menÅ¿ibus in vno duodecatemorio Zodiaci, Å¿cilicet .6. et .7. menÅ¿ibus, atque <lb/>etiam amplius, quod quidem ego ſæpe obÅ¿eruaui, præſertim anno .1565. et .1566. <lb/>hoc ordine. </s> - <s xml:space="preserve">Primum inÅ¿piciens Ephemeridas Å¿tadij, reperi <reg norm="quod" type="simple">ê™</reg> Mars Å¿ecundum eum <lb/>egrediebatur retrogradationem circa diem .12. Ianuarij anni 1566. in .16. grad. <lb/>Geminorum. </s> - <s xml:space="preserve">Et Å¿imiliter quod anno .1565. die vltima AuguÅ¿ti Mars futurus erat in <lb/>eodem Å¿upradicto loco, priuÅ¿quam retrogradi inciperet. </s> - <s xml:space="preserve">PoÅ¿tea inueni, quod poÅ¿t <lb/>retrogradationem die .11. Aprilis .1566. </s> - <s xml:space="preserve">Idem Mars futurus erat in gra .16. Cancri, <lb/><reg norm="itaque" type="simple">itaq;</reg> in his .30. gradibus à .16. Geminorum ad .16. Cancri conÅ¿umebatur Å¿patium <lb/>menÅ¿ium .7. & dierum .11.</s> - </p> - <pb o="235" rhead="EPISTOLAE." n="247" file="0247" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0247"/> - <p> - <s xml:space="preserve">Quo Å¿upputato, Å¿umpÅ¿i inÅ¿trumenta, & ad experimentum me paraui, & vltima <lb/>nocte menÅ¿is AuguÅ¿ti anni .1565. reperi Martem eſſe in dicto gradu geminorum vt <lb/>Å¿cribebat Stadius. </s> - <s xml:space="preserve">Deinde Å¿ingulis ebdomadibus obÅ¿eruans retrogradationem; </s> - <s xml:space="preserve">vidi <lb/>circa finem Octobris quod retrogradi incipiebat, & ea retrogradatio perÅ¿euerauit <lb/>vÅ¿que ad medium menÅ¿em Ianuarium, aut circiter, anni .1566. obÅ¿eruaui poÅ¿tea <lb/>etiam Å¿itum eiuÅ¿dem planetæ die .11. </s> - <s xml:space="preserve">Aprilis Å¿equentis <reg norm="eumque" type="simple">eumqÌ;</reg><unsure/> inueni in gradu .16. <lb/>Cancri, vti eum poÅ¿uerat Stadius. </s> - <s xml:space="preserve">Atque ita experimentum meum conuenit cum <lb/>calculo Stadij, <reg norm="comperique" type="simple">comperiqÌ;</reg> eum non erraſſe: </s> - <s xml:space="preserve">Et Å¿ic quiÅ¿que binis <reg norm="quibusque" type="simple">quibusqÌ;</reg> annis pote-<lb/>rit certior fieri de veritate. </s> - <s xml:space="preserve">Si autem delectationis cauÅ¿a id experiri volueris, expe-<lb/>ctato primam retrogradationem Martis, cuius initium Å¿ecundum Stadium futurum <lb/>eÅ¿t circa diem .20. </s> - <s xml:space="preserve">Nouembris anni .1582. & finis circa diem .10. Februar .1583. <lb/>circa grad .9. Cancri, & animaduerte quando Mars erit circa dictum gra .9. Can-<lb/>cri prius quam retrogradi incipiat, quod erit circa diem .19. Septem .1582. </s> - <s xml:space="preserve">Dein-<lb/>de aÅ¿pice quum erit in grad .9. Leonis, quod erit circa diem .7. Mai .1583. & vide-<lb/>bis <reg norm="quod" type="simple">ê™</reg> ipÅ¿e Mars in his gra .30. <reg norm="morabitur" type="simple">morabit̃</reg> <reg norm="per" type="simple">ê‘</reg> menÅ¿es .7. & dies .18. <reg norm="atque" type="simple">atq;</reg> vt eius rei pericu-<lb/>lum facias, obÅ¿erua noctem præcedentem diei .19. </s> - <s xml:space="preserve">Septem .1582. locum <reg norm="longitudinis" type="context">lõgitudinis</reg> <lb/>eius Å¿tellæ, & idem poÅ¿tea obÅ¿erua nocte præcedente diei .7. Mai, aut nocte Å¿equen-<lb/>ti .1583. & inter duos hoÅ¿ce terminos obÅ¿erua aliqua alia nocte Å¿tatum eius. </s> - <s xml:space="preserve">Mani-<lb/><reg norm="feÅ¿toque" type="simple">feÅ¿toqÌ;</reg> videbis Martem conÅ¿umere totum dictum tempus in hoc duodecatemorio. <lb/></s> - <s xml:space="preserve">Et quicunque aliquid intelligit in hac facultate quamuis non viderit PtolomÄ™i <lb/>AlmageÅ¿tum, minori labore poſſet per calculos Å¿cientificos colligere verita-<lb/>tem, Å¿uppoÅ¿itis tamen terminis Å¿criptis in theoricis planetarum. </s> - <s xml:space="preserve">Qui enim vidit <lb/>AlmageÅ¿tum vel reuolutiones orbium cÅ“leÅ¿tium Nicolai Copernici, non poteÅ¿t de <lb/>hoc vllo pacto dubitare. </s> - <s xml:space="preserve">Sed qui nondum tantopere progreſſus eÅ¿t, <reg norm="Å¿altem" type="context">Å¿altẽ</reg> capiat <reg norm="huius" type="simple">huiê°</reg> <lb/>rei notitiam vniuerÅ¿alem, hoc modo. </s> - <s xml:space="preserve">Supponat primum eccentricitatem deferen-<lb/>tis epicycli Martis, eſſe .6. partium <reg norm="talium" type="context">taliÅ©</reg>, quales Å¿unt Å¿exageÅ¿imæ Å¿emidiametri ipÅ¿ius <lb/>deferentis, & Å¿emidiametrum epicycli eſſe, partium <reg norm="Å¿upradictarum" type="context">Å¿upradictarÅ©</reg> .39. cum dimidia, <lb/>& quod argumenta vera, in temporibus primarum Å¿tationum ( cum epicyclus eÅ¿t in <lb/>auge, aut in eius oppoÅ¿ito, aut in <reg norm="lungitudinib" type="context">lÅ©gitudinib</reg>. me dio cribus ) <reg norm="iam" type="context">iã</reg> ab antiquis rectè Å¿uppu <lb/>tata Å¿int, Å¿icuti Å¿unt. </s> - <s xml:space="preserve">Et præſupponat motum diurnum centri epicycli. min .31. cum di <lb/>midio, quamuis reuera Å¿it min .31. & Å¿ecundorum .27. aut circiter, nunc <reg norm="quidem" type="context">quidẽ</reg> præter <lb/>mittens, quod vnus habeat reÅ¿pectum ad augem mediam epicycli, & alter ad cen-<lb/>trum æquantis. </s> - <s xml:space="preserve">Atque his præſuppoÅ¿itis fingat ( exempli gratia ) quod centrum <lb/>epicycli Å¿it in quauis longitudinum mediarum, & Mars in prima maxima æqua-<lb/>tione argumenti, Å¿cilicet in prima linea, quæ attingens epicyclum, à centro mundi <lb/>pergat ad circunferentiam Zodiaci, quæ erit illa linea <reg norm="contingentiæ" type="context">cõtingentiæ</reg> a qua <reg norm="proficiÅ¿cens" type="context">proficiÅ¿cẽs</reg> <lb/>Mars perget ad lineam primæ Å¿tationis, vt poÅ¿tea retrogradiatur, veluti Å¿i in infrapo <lb/>Å¿ita figura maiori, <reg norm="centrum" type="context context">cẽtrÅ©</reg> <reg norm="mundi" type="context">mÅ©di</reg> eſſet <var>.o.</var> & vnus arcus <reg norm="eccentrici" type="context">eccẽtrici</reg> eſſet <var>.a.b.c.d.</var> & vna ex li <lb/>neis medio cribus longitudinum eſſet <var>.o.c.f.</var> & centrum epicycli <var>.c.</var> qui notabitur per <lb/><var>a.f.e.g.</var> & lineæ contingentes epicyclum in punctis <var>.i.</var> et <var>.t.</var> Å¿int notatæ <var>.o.i.</var> et <var>.o.t.</var> & li-<lb/>nea primæ Å¿tationis <var>.o.n.b.</var> & linea Å¿ecundæ <var>.o.u.d.</var> Å¿i igitur Mars eſſet in puncto <var>.i.</var> an-<lb/>gulus <var>.i.o.e.</var> maximæ æquationis argumenti eſſet gra .40. minut .55. <reg norm="quanuis" type="context">quãuis</reg> talis maxi <lb/>ma æquatio argumenti in longitudinibus mediocribus AlfonÅ¿i ponatur eſſe gra .41. <lb/>minut .10. quod euenit quia calculatores ipÅ¿arum tabularum interuallum <var>.o.c.</var> quod <lb/>in eo Å¿itu epicycli interponitur inter centrum mundi, & centrum dicti epicycli, ac-<lb/>ceperunt partium Å¿exaginta præcisè, nihili facientes minuta illa .18. aut circiter, quÄ™ <lb/>verè Å¿unt præter dictas partes .60. quandoquid em euenit vt dictum interuallum in - <pb o="236" rhead="IO. BAPT. BENED." n="248" file="0248" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0248"/> - tali Å¿itu epicycli Å¿it baÅ¿is vnius trianguli orthogonij, cuius vnum ex illis duobus late-<lb/>ribus eÅ¿t Å¿emidiameter eccentrici partium .60. prÄ™cisè, aliud eÅ¿t interuallum eccen-<lb/>tricitatis partium .6. eiuÅ¿modi. </s> - <s xml:space="preserve">Angulus ergo <var>.i.o.c.</var> vt dixi, erit partium .40. minu <num value="55">.<lb/>55.</num> qui angulus continuò variatur Å¿ecundum Å¿itum epicycli. </s> - <s xml:space="preserve">& cum centrum <lb/>eius eÅ¿t in auge eccentrici. eÅ¿t minimus <reg norm="quam" type="context">quã</reg> eſſe poſſit. </s> - <s xml:space="preserve"><reg norm="eÅ¿tque" type="simple">eÅ¿tqÌ;</reg> tantum grad .36. min <num value="46">.<lb/>46.</num> & in oppoÅ¿ito ipÅ¿ius augis eÅ¿t grad .47. min .1. maximus quam alibi vnquam Å¿it, <lb/>& Å¿ic continuò variatur, Å¿ecundum Å¿itum, quem habet epicyclus in eccentrico. </s> - <s xml:space="preserve">Qui <lb/>quidem angulus inuenitur per doctrinam .27. et .28. libri primi Monteregij de trian <lb/>gulis. </s> - <s xml:space="preserve">Nam triangulus <var>.c.i.o.</var> eÅ¿t Å¿emper rectangulus in puncto <var>.i.</var> & latus <var>.c.i.</var> reÅ¿pectu <lb/>Å¿emidiametri eÅ¿t datum. </s> - <s xml:space="preserve">Quod <var>.c.i.</var> erit veluti partium .39. cum dimidia, et dictum <lb/>interuallum <var>.o.c.</var> veluti pat<unsure/> cium .60. min .18. & quia datur nobis etiam eccentricitas <lb/>veluti partium .60. talium, & cum <var>.c.o.</var> Å¿it linea veri motus epicycli, & latus Å¿imiliter <lb/>vnius trian guli, cuius duo latera Å¿unt Å¿upradicta, Å¿cilicet Å¿emidiameter eccentrici, & <lb/>eccentricitas, inter Å¿e compræhendentes angulum datum. </s> - <s xml:space="preserve">Nam Å¿emper præſuppo <lb/>nitur datus locus centri ipÅ¿ius epicycli, cum ipÅ¿e eÅ¿t extra augem aut oppoÅ¿itum eius <lb/>quia in auge linea <var>.o.c.</var> conÅ¿tat ex Å¿emidiametro eccentrici & interualli eccentricita-<lb/>tis. </s> - <s xml:space="preserve">& in eius oppoÅ¿ito, ipÅ¿a linea <var>.o.c.</var> eÅ¿t minor dicto Å¿emidiametro eccentrici per in <lb/>teruallum dictæ eccentricitatis. </s> - <s xml:space="preserve">Vnde etiam poſſumus extra augem, vel oppoÅ¿itum <lb/>eius cognoÅ¿cere <var>.o.c.</var> tanquam latus dicti trianguli duorum laterum <reg norm="cum" type="context">cÅ©</reg> angulo cogni <lb/>torum. </s> - <s xml:space="preserve"><reg norm="Idque" type="simple">IdqÌ;</reg> per .49. propoÅ¿itionem libri primi <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> Monteregij cum Å¿cilicet dictus <lb/>angulus <reg norm="non" type="context">nõ</reg> fuerit rectus. </s> - <s xml:space="preserve">Nam Å¿i fuerit rectus videbitur per .27. et .28. Å¿upra citatas.</s> - </p> - <p> - <s xml:space="preserve">Cum igitur hab eamus angulum <var>.c.o.i.</var> gra .40. mi .55. angulus <var>.o.c.i.</var> tanquam reli-<lb/>quus exrecto, erit grad .49. mi .5. cui reÅ¿pondet arcus <var>.i.g.</var> epicycli confectus à Marte <lb/>in diebus circiter .105. ad rationem min .28. aut circiter in Å¿ingulos dies, prætermiÅ¿-<lb/>Å¿is nunc quidem minutijs cum exigui momenti Å¿it error .15. aut .20. dierum ad verifi <lb/>cationem longæ morÄ™ Martis in vno duodecatemorio, atque per hoc tempus cen-<lb/>trum epicycli conficit gradus .55. min .7. aut circiter, ad rationem minutorum .31. <reg norm="cum" type="context">cÅ©</reg> <lb/>dimidio in Å¿ingulos dies. qui numerus graduum .55. min .7: differt à numero <reg norm="graduum" type="context">graduÅ©</reg>. <lb/>40. min .55. maximæ æquationis argumenti gradibus .14. mi .12. nec refert quod gra <num value="55">.<lb/>55.</num> min .7. habeant reÅ¿pectum ad centrum æquantis, magis quam ad centrum <reg norm="mundi" type="context">mÅ©di</reg>, <lb/>quia differentia non eÅ¿t tanta, vt poſſit inducere errorem menÅ¿ium. </s> - <s xml:space="preserve">Hinc Å¿equitur <lb/>quod in fine dictorum dierum .105. </s> - <s xml:space="preserve">Mars erit in linea <var>.o.c.</var> veri motus epicycli, Å¿ed <lb/>gradibus .14. min .12. vlterius quam in primo loco, in quo erat in Zodiaco, & erit in <lb/>medio Å¿uæ retrogradationis. </s> - <s xml:space="preserve">Sed quoniam Mars manifeÅ¿tè retrogradi non incipit <lb/>in puncto <var>.i.</var> conting entiæ, imo ab illo puncto vÅ¿que ad terminum primæ Å¿tationis li <lb/>neæ <var>.o.n.</var> interponitur arcus <var>.i.n.</var> epicycli, qui eÅ¿t graduum .32. minu .14. </s> - <s xml:space="preserve"><reg norm="Idque" type="simple">IdqÌ;</reg> cogno-<lb/>Å¿citur Å¿ubtrahendo arcum <var>.f.i.n.</var> graduum .163. mi .9. qui eÅ¿t inter augem, & primam <lb/>Å¿tationem, à gradibus .180. ( qui arcus <var>.f.i.n.</var> erit verum argumentum, quod Å¿i-<lb/>militer variatur Å¿ecundum Å¿itum epicycli, etÅ¿i eiuÅ¿modi varietas, nobis <reg norm="non" type="context">nõ</reg> eÅ¿t magni <lb/>momenti, vnde poſſumus præſupponere, quod <var>.c.</var> centrum epicycli non alteret <reg norm="in- teruallum" type="context">in-<lb/>teruallÅ©</reg> <var>.c.o.</var> à centro <reg norm="mundi" type="context">mÅ©di</reg>, <reg norm="cum" type="context">cÅ©</reg> non posÅ¿it intercedere, error <reg norm="menÅ¿ium" type="context context">mẽſiÅ©</reg> reliquum verò <var>.g.<lb/>n.</var> graduum .16. min .51. Å¿ubtrahendo ex arcu <var>.g.i.</var> graduum .49. minuti .5. vnde reli-<lb/>quus nobis erit arcus <var>.n.i.</var> graduum .32. min .14. in eiuÅ¿modi tamen Å¿itu mediocrium <lb/>longitudinum. </s> - <s xml:space="preserve">Nunc hic arcus epicycli graduum .32. mi .14. fit à ſtella Martis die-<lb/>bus .69. ad rationem Å¿upradictam, omittendo quod ipÅ¿a Å¿tella habeat reÅ¿pectum ad <lb/>augem mediocrem epicycli, & quod dicta aux mediocris mutet diÅ¿tantiam à vera <lb/>propter motum epicycli, quod nunc quidem parui refert, in quibus diebus .69. cen- - <pb o="237" rhead="EPISTOLAE." n="249" file="0249" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0249"/> - trum epicycli conficit gra .36. min .13. ad rationem Å¿upradictam. </s> - <s xml:space="preserve">ReÅ¿tat nunc no-<lb/>bis inuenire angulum <var>.b.o.c.</var> in centro <reg norm="mundi" type="context">mÅ©di</reg> inter duas lineas, <var>b.o.</var> et <var>.c.o.</var> <reg norm="quarum" type="context">quarÅ©</reg> prior <lb/>eÅ¿t primæ Å¿tationis, altera eÅ¿t veri motus epicycli, quod facilè intelligemus per di-<lb/>ctam .49. lib. 1. Monteregij, cum duo latera <var>.n.c.</var> et <var>.c.o.</var> & angulus <var>.n.c.o.</var> Å¿int nobis no <lb/>ta. </s> - <s xml:space="preserve">Hoc autem fiet fingendo lineam <var>.n.h.</var> perpendiculerem ad <var>.o.c.</var> quæ tanquam Å¿i-<lb/>nus anguli <var>.n.c.h.</var> erit partium .28986. talium qualium <var>.n.c.</var> eſſet partium .100000. & <lb/><var>c.h.</var> tanquam Å¿inus anguli <var>.c.n.h.</var> reÅ¿tantis ex vno recto, erit partium .95706. dicendo <lb/>poÅ¿tea Å¿i <var>.n.e.</var> tanquam Å¿inus totalis partium .100000. dat nobis <var>.n.h.</var> partium .28986 <lb/>quid dabit nobis diameter <var>.n.c.</var> tanquam partium .39. mi .30. inueniemus <var>.n.h.</var> venire <lb/>nobis ex partibus 11. mi .27. & idem faciendo de <var>.c.h.</var> inueniemus quod veniet no-<lb/>bis partium .37. mi .48. quibus Å¿ubtractis extota <var>.c.o.</var> quÄ™ eÅ¿t <reg norm="partium" type="context">partiÅ©</reg> .60. mi .18. reliqua <lb/>erit nobis <var>.h.o.</var> partium .22. min .30. capiendo poÅ¿tea radicem <reg norm="quadratam" type="context">quadratã</reg> Å¿ummæ qua <lb/>drati <var>.n.h.</var> cum quadrato <var>.h.o.</var> veniet nobis <var>.n.o.</var> partium .25. min .12. talium qualis <var>.n.<lb/>h.</var> eÅ¿t partium .11. min .27. Å¿i igitur ad <var>.o.n.</var> tanquam partium .25. min .12. reÅ¿pondet <var>.<lb/>n.h.</var> partium .11. minuti .27. linea <var>.n.h.</var> ad <var>.o.n.</var> tanquam partium 100000. reÅ¿pon <lb/>debit part .45436. tanquam Å¿inus anguli <var>.n.o.h.</var> qui angulus erit gra .27. minut .1. Å¿ub <lb/>tracto poÅ¿tea hoc angulo ab angulo <var>.c.o.i.</var> graduum .40. minut .55. remanebit an <lb/>gulus <var>.n.o.i.</var> graduum .13. minut .54. inter lineam contingentiæ, & lineam primæ <lb/>Å¿tationis in eiuÅ¿modi Å¿itu. </s> - <s xml:space="preserve">Et ideo Mars acceſſerit ad lineam <var>.o.c.</var> veri motus epi-<lb/>cycli. </s> - <s xml:space="preserve">Sed quia linea <var>.o.i.</var> contingentiæ, propter motum centri epicycli, in dictis die-<lb/>bus .69. confecerit gradus .36. minut .13. ( præſuppoÅ¿ita Å¿emper eadem di-<lb/>Å¿tantia <var>.o.c.</var> quamuis nonnulla Å¿it differentia, quam nunc prætermittemus ) & Mars in <lb/>dicto tempore retrogreſſus fuerit per dictum angulum gra .13. mi .54. quibus dedu-<lb/>ctis, ex .36. & min .13. reÅ¿tabunt gra .22. min .19. itaque in diebus .69. </s> - <s xml:space="preserve">Mars promo-<lb/>tus fuerit a primo Å¿itu gra .22. min .19. aut circiter, prius quam retrogradatio eius in-<lb/>cipiat eſſe appa-<lb/>rens.</s> - </p> - <figure position="here"> - <image file="0249-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0249-01"/> - </figure> - <p> - <s xml:space="preserve">Nunc à prima <lb/>Å¿tatione vÅ¿que ad <lb/><reg norm="lineam" type="context">lineã</reg> veri motus <lb/>epicycli Å¿unt gra <num value="16">.<lb/>16.</num> min .51. ipÅ¿ius <lb/>epicycli, vt Å¿upra <lb/><reg norm="vidimus" type="simple">vidimê°</reg> quos Mars <lb/>tranÅ¿it in diebus <lb/>36. aut circiter ad <lb/>rationem min .28. <lb/>in Å¿ingulos dies, <lb/>quo tempore cen <lb/>trum epicycli, in <lb/>tali diÅ¿tantia à <reg norm="cen- tro" type="context">cẽ-<lb/>tro</reg> mundi confice <lb/>ret gra .18. mi .54. <lb/>ad rationem min <num value="31">.<lb/>31.</num> cum dimidio <lb/>in Å¿ingulos dies, <lb/><reg norm="quibus" type="simple">quibê°</reg> deductis ex <lb/>gra .27. min .1. an- - <pb o="238" rhead="IO. BAPT. BENED." n="250" file="0250" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0250"/> - guli <var>.c.o.n.</var> remanebunt gra .8. min .7. pro numero dimidiæ retrogradationis quum <lb/>Mars erit in linea <var>.o.c.</var> veri motus epicycli. </s> - <s xml:space="preserve">Quibus gradibus .8 min .7. Å¿ubtractis à <lb/>grad .22. minu .19. per quos Mars progreſſus erat, Å¿upererunt grad .14. minut .12. <lb/>quibus ipÅ¿e in media retrogradatione exiÅ¿tens in linea <var>.o.c.</var> veri motus epicycli pro-<lb/>motus erit à principio primi Å¿itus. </s> - <s xml:space="preserve">Quod cum eo concordat quod Å¿upra diximus cir <lb/>ca hos gradus .14. min .12. vltra primum Å¿itum in Å¿patio dierum .105. vt Å¿upra, ad <reg norm="tam" type="context">tã</reg> <lb/>tum enim aÅ¿cendunt .69. et .36. nunc fingendo <reg norm="quod" type="simple">ê™</reg> Mars pergat in Å¿uo motu compo-<lb/>Å¿ito qui conÅ¿tat ex his duobus circulis, vi eccentrici, & epicycli ( quanquam <lb/>vt dixi omittimus illam Å¿ummam Å¿ubtilitatem Å¿eu Å¿crupuloÅ¿i<unsure/> atem <reg norm="continuæ" type="context">cõtinuæ</reg> inæqua-<lb/>litatis diÅ¿tantiæ centri epicycli à centro mundi, & præterimus etiam <reg norm="irregularitatem" type="context">irregularitatẽ</reg> <lb/>eius circa centrum mundi, propter regularitatem eius circa centrum æquantis, <lb/>atque etiam miſſum facimus motum epicycli recti à ſua media auge ) fingendo <lb/>inquam, quod Mars dicto motu Å¿uo pergat, vÅ¿que ad punctum Å¿ecundæ Å¿ta-<lb/>tionis, præteribunt alij dies .36. vt prius, quibus iunctis cum .105. fiet Å¿um-<lb/>ma dierum .141. & Mars retrogreſſus erit per alios grad .8. minut .7. quibus Å¿ub-<lb/>tractis a gradibus .14. m in .12. per quos progreſſus erat Å¿upererunt. gra .6. min .5. qui <lb/>bus ipÅ¿e Mars in fine Å¿uæ retrogradationis promotus erit à primo loco vnde moueri <lb/>cÅ“pit. </s> - <s xml:space="preserve">Inter hanc igitur Å¿ecundam Å¿tationem lineæ <var>.o.u.d.</var> & lineam <var>.o.t.</var> Å¿ecundæ con <lb/>tingentiæ, Mars diebus .69. vt prius, confecerit gradus .32. min .14. Å¿ui epicycli, & eo-<lb/>dem tempore linea contingens <var>.o.t.</var> ambulauerit gra .36. min .13. vt prius, à quo itine <lb/>re Å¿ubtracto angulo Å¿ecundo <var>.d.o.t.</var> graduum .13. min .54. Å¿upererunt gra .22. min .19 <lb/>vti prius, quos Mars ambulauerit directè & apparenter, quibus additis ad gra .6. mi <lb/>nut .5. quibus Mars progreſſus erat à principio motus, fient gra .28. min. circiter .24. <lb/>quibus proceſſerit à priori loco in diebus .177. ideÅ¿t in .141. et .36. qui Å¿unt ferè m. n <lb/>Å¿es .6. </s> - <s xml:space="preserve">Itaque Mars partim Å¿urÅ¿um partim, deorÅ¿um ambulans detentus erit <reg norm="menÅ¿ibus" type="context">mẽſibus</reg> <lb/>6. in grad .28. </s> - <s xml:space="preserve">Zo-<lb/>diaci, <reg norm="atque" type="simple">atq;</reg> Å¿i finxe <lb/> - <anchor type="figure" xlink:label="fig-0250-01a" xlink:href="fig-0250-01"/> - rimas quod epicy <lb/>clus <reg norm="moueatur" type="simple">moueat̃</reg> ver-<lb/>Å¿us oppoÅ¿itum au-<lb/>gis, <reg norm="longior" type="context">lõgior</reg> erit mo <lb/>ra planetæ in eiuÅ¿ <lb/>modi duodecate-<lb/>morio, propter au <lb/>gumentum æqua-<lb/>tionis argumenti. <lb/></s> - <s xml:space="preserve">Itaque probata à <lb/>nobis eÅ¿t poſſibi-<lb/>litas huius moræ <lb/>Mattis. </s> - <s xml:space="preserve">Quod qui <lb/>dem mihi Å¿uffice-<lb/>re videtur <reg norm="non" type="context">nõ</reg> mo-<lb/>do cibi, Å¿ed etiam <lb/>cuiuis, qui harum <lb/>Å¿cientiarum prin-<lb/>cipia teneat. </s> - <s xml:space="preserve">Ne-<lb/>que enim <reg norm="nunc" type="context">nÅ©c</reg> do-<lb/>cere volo eos qui - <pb o="239" rhead="EPISTOLAE." n="251" file="0251" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0251"/> - in ijs Å¿unt conÅ¿umati, nec curam mihi Å¿uÅ¿cipere erudiendi imperitos. </s> - <s xml:space="preserve">Satis igitur Å¿it <lb/>oÅ¿tendiſſe, quod qui Å¿cripÅ¿it Martem commorari poſſe tam multos menÅ¿es in vno Å¿i <lb/>gno, non impoſſibilem rem tradidit. </s> - <s xml:space="preserve">Immo per obÅ¿eruationes huius veritatis mil-<lb/>lies factas, AÅ¿trologi fecere Å¿upradictas Å¿uppoÅ¿itiones neceſſarias ad <reg norm="reducendum" type="context">reducẽdum</reg> in <lb/>Å¿uas cauÅ¿as, & ad regulam, eiuÅ¿modi veriſſimos effectus.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0250-01" xlink:href="fig-0250-01a"> - <image file="0250-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0250-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Non oportebat autem Å¿criptorem harum animaduerÅ¿ionum tantopere eiuÅ¿modi <lb/>mora commoueri, Å¿ed cogitare <reg norm="quod" type="simple">ê™</reg> fortaſſe calculi facti fuerunt eo tempore quo mi-<lb/>Å¿er Mars à Vulcano rete vinctus erat. </s> - <s xml:space="preserve">Vnde cum nonita celeriter Å¿e expedire poſſet <lb/>iter eius Å¿egnius peractum fuit. </s> - <s xml:space="preserve">Aut <reg norm="quod" type="simple">ê™</reg> <reg norm="quum" type="context">quÅ©</reg> vulneratus fuit in bello Troiano, vis eius <lb/>& agilitas per aliquantulum temporis imminuta fuit. </s> - <s xml:space="preserve">Atque Å¿i hic etiam intellexiÅ¿ <lb/>Å¿et eum <reg norm="aliquando" type="context">aliquãdo</reg> fuiſſe in poteÅ¿tate Othi, & Ephialtis <reg norm="vinctum" type="context">vinctÅ©</reg> & carceri incluÅ¿um <reg norm="men- Å¿es" type="context">mẽ-<lb/>Å¿es</reg> tredecim, dum ab Eribea Å¿olutus fuit, vt tu, antiquos Å¿equens, eleganter Å¿cribis <lb/>in illis tuis pulcherimis dialogis. non exiÅ¿timaſſet, credo, tam abÅ¿urdum quod alius <lb/><reg norm="eum" type="context">eÅ©</reg> detinuiſſet Å¿ex aut Å¿eptem menÅ¿ibus, Å¿ed operam dediſſet vt a te intelligeret quid <lb/>Å¿ibi vellet tam longa captiuitas.</s> - </p> - <p> - <s xml:space="preserve">Sed vt ad rem redeamus. </s> - <s xml:space="preserve">Idem pag .4. ait, quod verus motus Martis diÅ¿tat à me-<lb/>dio circiter dies .8. Å¿upponens medium motum eſſe dierum .683. <reg norm="quod" type="simple">ê™</reg> etiam falÅ¿um eÅ¿t. <lb/></s> - <s xml:space="preserve">Sed vtcunque Å¿it, fallitur. </s> - <s xml:space="preserve">Solet enim periodus veri motus Martis eſſe die-<lb/>rum circiter .708. modo paulo plus, modo paulo minus, & interdum poteÅ¿t etiam <lb/>eſſe multo breuior, Å¿icuti erit à die 3. </s> - <s xml:space="preserve">Decembris anni .1593. <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> ad initium Iunij <num value="1595">.<lb/>1595.</num> </s> - <s xml:space="preserve">Tunc enim erit tantum dierum .545. & non quidem Å¿ine ratione, nam dicto <lb/>initio Decembris Mars paulo ante cæperiteſſe directus, cum centrum epicycli erit <lb/>circa medium Tauri, & eius Å¿tella in principio Arietis & initio Iunij .1595. </s> - <s xml:space="preserve">Mars pa <lb/>rum diÅ¿tabit ab initio retrogradationis, regreſſus tamen ad initium ipÅ¿ius Arietis, & <lb/>centrum epicycli erit circa medium Aquarij, in cuius Å¿igni medio, hac ætate repe-<lb/>ritur oppoÅ¿itum augis, & in quo Å¿itu, æquationes <reg norm="argumenti" type="context">argumẽti</reg> Å¿unt, quam maximÄ™ eſſe <lb/>poſſint, quum centrum epicycli circuiuerit Å¿olum circiter tres quartas totius ambi-<lb/>tus, & Mars circuiuerit per partem Å¿uperiorem epicycli circiter gradus .252. </s> - <s xml:space="preserve">Hoc au <lb/>tem dico, vt oÅ¿tendam poſſibilitatem huius eius extraordinariæ velocitatis. </s> - <s xml:space="preserve">Nam <lb/><reg norm="quicunque" type="simple">quicunq;</reg> voluerit poterit certior fieri, per calculum partium motus Martis.</s> - </p> - <p> - <s xml:space="preserve">Vbiautem poÅ¿tea idem author miratur interualla, quæ <reg norm="ponuntur" type="context">ponÅ©tur</reg> inter coniunctio <lb/>nes Iouis, & Martis in eodem Å¿igno, <reg norm="eaque" type="simple">eaqÌ;</reg> vocat errores maximos, oÅ¿tendit Å¿e non re <lb/>ctè conÅ¿ideraſſe motus eorum. </s> - <s xml:space="preserve">Et præcipuè primum miratur <reg norm="quod" type="simple">ê™</reg> inter annum .1528. <lb/>et .1553. </s> - <s xml:space="preserve">Iupiter & Mars nunquam coeant in Leone, cum hæ duæ coniunctiones in <lb/>ter Å¿e diÅ¿tent ann .25. afferens pro ratione, quod hæc duo Å¿ydera, altero quoque anno <lb/>coniunguntur, Å¿ic dicens.</s> - </p> - <quote> - <s xml:space="preserve">Qui Å¿ciet has duas Å¿tellas Å¿ecundo quoque anno inter Å¿e coniungendas, mirabitur <lb/>quomodo non poterunt numeratores, huiuÅ¿modi animaduertere errores.</s> - </quote> - <p> - <s xml:space="preserve">Et præter hanc <reg norm="rationem" type="context">rationẽ</reg> fortaſſe <reg norm="ent" type="context">ẽt</reg> conÅ¿iderauit, <reg norm="quod" type="simple">ê™</reg> in dicto temporis interuallo Iu <lb/>piter <reg norm="semper" type="context">sẽper</reg> fuit in Leone, vt ann .1540. et .1541. </s> - <s xml:space="preserve">Mars <reg norm="autem" type="wordlist">aÅ©t</reg> in eo ſæpe fuit. </s> - <s xml:space="preserve">Vnde im <lb/>poſſibile eilvidetur eos non conueniſſe in dicto Å¿igno. </s> - <s xml:space="preserve"><reg norm="Idemque" type="simple">IdemqÌ;</reg> dici poteÅ¿t de alijs <lb/>coniunctionibus eorundem planetarum, atque has differentias temporum inter di-<lb/>ctas coniunctiones ipÅ¿e tribuit erroribus calculorum Ephemeridum, non autem ta-<lb/>bularum, vt Å¿upra dixit. </s> - <s xml:space="preserve">Å¿ed neÅ¿cio quare vellet dictos planetas coire in Leone, Å¿i <lb/>quum Iupiter in eo erat anno .1540. et .1541. & in eo <reg norm="deambulabatur" type="simple">deambulabat̃</reg>, Mars interea <lb/>erat <reg norm="mon" type="context">mõ</reg> in Libra, modo in Scorpione, Sagittario, Capricorno, & alijs Å¿ignis <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> ad <lb/>Cancrum, in quo cum repertus fuit anno .1541. cogitans congredi cum loue in Leo - <pb o="240" rhead="IO. BAPT. BENED." n="252" file="0252" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0252"/> - ne, comperit eum inde aufugiſſe. </s> - <s xml:space="preserve"><reg norm="Idque" type="simple">IdqÌ;</reg> fortaſſe, Iupiter data opera fecit, vt huiuÅ¿-<lb/>modi AÅ¿trologos in admirationem induceret.</s> - </p> - <p> - <s xml:space="preserve">Idem dico de alijs coniunctionibus horum duorum.</s> - </p> - <p> - <s xml:space="preserve">Quod poÅ¿tea ait, eos Å¿ecundo quoque anno coniungi, animaduertendum eÅ¿t, <reg norm="quia" type="simple">ê—a</reg> <lb/>(vt <reg norm="iam" type="context">iã</reg> dixi) duæ Å¿unt Å¿pecies coniunctionum, quarum vna eÅ¿t linearum eorum me-<lb/>diorum motuum, altera corporum eorum, Å¿altem in longitudine, cum ambo inue-<lb/>niuntur in eodem circulo, qui tranÅ¿it per polos ecclipticæ, nam eos inueniri in <reg norm="eadem" type="context">eadẽ</reg> <lb/>linea recta <reg norm="tranÅ¿eunte" type="context context">trãſeÅ©te</reg> per centrum mundi, rarisÅ¿imum eÅ¿t. </s> - <s xml:space="preserve">Atque coniunctio Å¿upra di-<lb/>ctarum linearum vocatur media, & inter Iouem & Martem fieri Å¿olet Å¿patio <reg norm="dierum" type="context">dierÅ©</reg>. <lb/>816. cum dimidio, aut circiter. </s> - <s xml:space="preserve">Altera <reg norm="dicitur" type="simple">dicit̃</reg> vera, Å¿iue apparens, & irregulatiſſima, <lb/>quæ quidem non Å¿eruat tempus determinatum. </s> - <s xml:space="preserve">Quare quamuis altero <reg norm="quoque" type="simple">quoq;</reg> an-<lb/>no coniungantur; </s> - <s xml:space="preserve">& Iupiter duodenis annis tranÅ¿eat per totum Zodiacum, non ideo <lb/>neceſſe eÅ¿t, vt in Å¿patio .24. annorum coniungantur in Å¿ingulis Å¿ignis, nunquam in eo <lb/>deÅ¿icientes, vtipÅ¿e credit loquens de veris coniunctionibus apparentibus, eo quod <lb/>Å¿int irregulatiſſimæ, vt dixi.</s> - </p> - <p> - <s xml:space="preserve">Atque Å¿i quis velit inuenire periodum coniunctionum mediocrium horum duo-<lb/>rum planetarum, ita faciendum erit. </s> - <s xml:space="preserve">Sumat periodum motus mediocris Iouis, quÄ™ <lb/>eÅ¿t dierum .4328. & Martis, quæ eÅ¿t dierum .687. in quo tempore Martis, Iupiter am <lb/>bulat gra .57. min .8. & diebus .30. conficit. grad .2. minut .29. & Å¿ecun .23. ad ratio-<lb/>nem gra .360. in diebus .4328. </s> - <s xml:space="preserve">Mars verò ad rationem <reg norm="graduum" type="context">graduÅ©</reg> .360. in diebus .687. <lb/>Å¿ingulis .30. diebus conficit. gra .15. mi .43. <reg norm="Å¿ecum" type="context">Å¿ecÅ©</reg> .14. vnde differentia inter eos eÅ¿t gra <lb/>duum .13. mi .15. <reg norm="Å¿ecum" type="context">Å¿ecÅ©</reg> .51. per quam diuidendo productum graduum .57. min .8. in <lb/>dies .30. <reg norm="obuenient" type="context">obueniẽt</reg> dies .129. & duæ tertiæ. </s> - <s xml:space="preserve">quibus addendo periodum Martis fient <num value="816">.<lb/>816.</num> cum dimidio, aut circiter. </s> - <s xml:space="preserve">Atque hÄ™c eÅ¿t periodus infallibilis mediarum con <lb/>iunctionum Iouis cum Marte.</s> - </p> - <p> - <s xml:space="preserve">Nunc venientes ad tabulas AnimaduerÅ¿ionum, videbimus hæc mirabilia eius, in <lb/>quo conÅ¿iÅ¿tant & vbi Å¿int tam multi inÅ¿ignes errores.</s> - </p> - <p> - <s xml:space="preserve">Primum igitur neminem later quod calculus Saturni, à Leouitio editus, difert à <lb/>calculo Stadij circiter gra .2. aut .3. cum Leouitius faciat eum progredi per <reg norm="tantum" type="context">tãtum</reg> in <lb/>teruallum, modo plus, modo minus, & Å¿imiliter Iouem. </s> - <s xml:space="preserve">Å¿ed longe minori diffe-<lb/>rentia, & ſępe gra .1. minus, atque in alijs planetis differunt, modo plus, modo minus. <lb/></s> - <s xml:space="preserve">Huic igitur <reg norm="mirum" type="context">mirÅ©</reg> videtur, quod vnus ex his calculatoribus detineat Saturnum plu-<lb/>ribus menÅ¿ibus in vno Å¿igno, & alter in alio, non animaduertens dictam <reg norm="differentiam" type="context">differentiã</reg> <lb/>eſſe eius rei cauÅ¿am. </s> - <s xml:space="preserve">Miratur item, quod vnus ex is faciat Saturnum morari paucis <lb/>menÅ¿ibus in vno Å¿igno, alter vero eum ibi detineat integris annis. </s> - <s xml:space="preserve">Vt exempli gra-<lb/>tia, verÅ¿us finem Å¿uæ tabulæ Saturni, dicit quod Leouitius eum carceri includit in <lb/>geminis annis .2. menÅ¿e vno, & diebus .9. </s> - <s xml:space="preserve">Stadius vero clementior eum liberat intra <lb/>menÅ¿es .3. & dies .14. </s> - <s xml:space="preserve">Sed hic non cogitat, quod Stadius facit eum ingredi in gemi-<lb/>nos anno .1559. die .10. Iunii, & ambulare directum vÅ¿que ad diem .6. Septembris, <lb/>eiuÅ¿dem anni gra .6. min .34. <reg norm="eumque" type="simple">eumqÌ;</reg> poÅ¿tea retrogradum inde exire die .22. Decem. <lb/>eiuÅ¿dem anni, cum ingreditur in Taurum, vbi partim retrogradus, & partim dire-<lb/>ctus manet vÅ¿que ad diem .20. Februa .1560. rediens poÅ¿tea in geminos, in quibus <lb/>manet vÅ¿que ad diem Iunii .1561. & inde ingreditur in Cancrum, <reg norm="ambulatque" type="simple">ambulatqÌ;</reg> dire-<lb/>ctus. gra .4. min .59. vÅ¿que ad diem .4. Octob. </s> - <s xml:space="preserve">Vnde retrogradiens rurÅ¿um intratin Ge <lb/>minos die .28. Decemb. eiuÅ¿dem anni, at que ibi partim retrogradus partim directus <lb/>manet vÅ¿que ad diem .12. Apr .1562. itaque in pluribus vicibus facit eum morari in <lb/>Geminis dies circiter .816. ideÅ¿t circiter menÅ¿es .27. Å¿umpÅ¿it autem hic Å¿criptor bre - <pb o="241" rhead="EPISTOL AE." n="253" file="0253" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0253"/> - uiſſimam moram cauÅ¿a comparationis cum calculo Leouitij, vt faceret differentiam <lb/>apparere maiorem. </s> - <s xml:space="preserve">Tamen in quouis dictorum temporum nunquam inuenietur <lb/>Leouitius differre à Stadio plus gradibus tribus integris. </s> - <s xml:space="preserve">Idem fecit in multis alijs lo <lb/>cis dictorum virorum eos conferens tum in Saturno tum in Ioue, & Marte, <reg norm="putans" type="context">putãs</reg> ma <lb/>gnum eſſe errorem, <reg norm="quod" type="simple">ê™</reg> planeta non perambulet totum Å¿ignum, in quod eÅ¿t ingreſſus <lb/>vel directus vel totum retrogradus. </s> - <s xml:space="preserve">Atque hæc opinio Å¿imilis eÅ¿t Å¿uperiori de con <lb/>iunctionibus veris Saturni, & Iouis, vbi dicit quod nunquam coniunguntur in vno Å¿i <lb/>gno alterius triplicitatis, niÅ¿i perfecerit coniunctionem in omnibus Å¿ignis primæ tri <lb/>plicitatis. </s> - <s xml:space="preserve">Verum vt Å¿uperÅ¿edeam vlterius diÅ¿putare, mihi videtur, quod hactenus <lb/>dixi, poſſe tibi Å¿atisfacere, quod attinet ad Å¿ciendam Å¿ententiam mean Å¿uper dictis <lb/>AnimaduerÅ¿ionibus latinè Å¿criptis. </s> - <s xml:space="preserve">Hoc tamen non prætermittam, <reg norm="quod" type="simple">ê™</reg> hic non ani-<lb/>maduertit, <reg norm="nempe" type="context">nẽpe</reg> <reg norm="quod" type="simple">ê™</reg> <reg norm="differentiæ" type="context">differẽtiæ</reg> locorum <reg norm="planetarum" type="context">planetarÅ©</reg> quæ <reg norm="suntinter" type="context">sÅ©tinter</reg> ephemeridas Leouitij <lb/>& Stadij, euenere, quia vnus Å¿upputat <reg norm="cum" type="context">cÅ©</reg> radicibus, & <reg norm="fundamentis" type="context">fundamẽtis</reg> AlfonÅ¿i alter verò <lb/>Reinoldi ex Copernico recentius obÅ¿eruatis, ita idem euenire poterit futuris tem-<lb/>poribus, Å¿i Å¿upputati fuerint dicti motus, & loci cum recentioribus obÅ¿eruationibus <lb/>cum impoſſibile Å¿it tam Å¿ubtiliter, tanqÌue perfectè Å¿upputare loca & <reg norm="motus" type="simple">motê°</reg> eorum, <lb/>vtlungo interuallo temporis non comperiantur in eis aliquæ differentiÄ™, cuius rei re <lb/>medium eÅ¿t Å¿emper Å¿equi recentiores obÅ¿eruationes & tabulas.</s> - </p> - <p> - <s xml:space="preserve">Atque vt tibi Å¿atisfaciam etiam circa alia Å¿cripta vulgari lingua edita menÅ¿ibus .4 <lb/>poÅ¿t latina, etÅ¿i intelligere potes, qualia poſſint eſſe alia eius Å¿cripta, ex ijs quæ Å¿upra <lb/>dicta Å¿unt, atque etiam ex eo, quod dicit Å¿emiÅ¿iſſe multa exempla Å¿uarum Animad <lb/>uerÅ¿ionum in varias terras, illis qui profitentur has Å¿cientias, aut earum Å¿tudioÅ¿i Å¿unt, <lb/>nec <reg norm="quenquam" type="context">quenquã</reg> inueniſſe qui ad <reg norm="tam" type="context">tã</reg> laudabilem prouinciam motus Å¿it, nec vidiſſe, <reg norm="quod" type="simple">ê™</reg> ali <lb/>quis reÅ¿ponderit eius rationibus; </s> - <s xml:space="preserve">laudabilem prouinciam, autem puto, <reg norm="quod" type="simple">ê™</reg> intelligat <lb/>correctionem ephemeridum, verens, ne culpa calculatorum, qui eas Å¿umpÅ¿ere e ta-<lb/>bulis, tam differentes Å¿int, vt <reg norm="quibuÅ¿dam" type="context">quibuÅ¿dã</reg> locis cap .1. </s> - <s xml:space="preserve">Videtur, & præcipuè vbi Å¿ic ait.</s> - </p> - <quote xml:lang="it"> - <s xml:space="preserve">Perche eſſendo impoſſibile alli Å¿tudioÅ¿i di dette Å¿cientiæ di non Å¿eruirÅ¿i delle <lb/>ephemeridi, maggiormente a quelli che non Å¿anno Å¿eruirÅ¿i delle tauole, e cono-<lb/>Å¿cendo d'incorrere in errori Å¿enza hauerui altro rimedio, Å¿arebbono forzati di ab <lb/>bandonare i Å¿tudij loro.</s> - </quote> - <p> - <s xml:space="preserve">Quanquam circa finem dicti capitis redeat in meliorem viam & aduerÅ¿etur Å¿i-<lb/>bijpÅ¿i vbi Å¿ic ait.</s> - </p> - <quote xml:lang="it"> - <s xml:space="preserve">Che poi eſſi poſſeſſori della Å¿cienza, & c.</s> - </quote> - <p> - <s xml:space="preserve">Etiam aperiam tibi, quæ mea Å¿it de ijs Å¿ententia.</s> - </p> - <p> - <s xml:space="preserve">Hicigitur in Å¿criptis Italicis, vt morderet aliquem ex ijs, qui eius Å¿uperiora Å¿cripta <lb/>non laudauerant, occaÅ¿ionem capit aperiendi aliquos illius errores, per editionem <lb/>collationis quorundam calculorum a Å¿e collectorum illius, atque etiam aliorum, cu <lb/>ius calculi Å¿unt in Å¿ecunda, & Å¿eptima figura. </s> - <s xml:space="preserve">Sed prius quam veniamus ad defenÅ¿io <lb/>nem harum duarum figurarum vide obÅ¿ecro quam alienum ei videatur, quod alij <lb/>dixerint differentiam ephemeridum non eſſe magni momenti, non afferens reÅ¿pe-<lb/>ctum vllum, qui enim dixerunt eiuÅ¿modi differentiam non eſſe magni momen-<lb/>ti id dixerunt habito reÅ¿pectu ad Å¿ignum in quo eÅ¿t planeta, vt (exempli gra-<lb/>tia) quamuis in ponendo loco Saturni Leouitius interdum differat à Stadio gra <lb/>dibus .3. quum vterque eum ponat in eodem Å¿igno, tuncid nullius momenti eÅ¿t, <lb/>& Å¿ic in coniunctionibus aut alijs aÅ¿pectibus duo, aut .3. gradus non faciunt alteratio <lb/>nem Å¿enÅ¿ibilem, cum virtus coniunctionum, & aÅ¿pectuum inÅ¿it, & duret per mul-<lb/>tos gradus ante aut poÅ¿t ipÅ¿um punctum. </s> - <s xml:space="preserve">Nec quicquam <reg norm="tamen" type="wordlist">tamẽ</reg> eÅ¿t qui dubitet, quin <lb/>præſtaret Å¿cire Å¿ubtiliter ipÅ¿um punctum. </s> - <s xml:space="preserve">Nec vnquam fuit aliquis qui negauerit <reg norm="re ferre" type="conjecture" resp="paul">re - <pb o="242" rhead="IO. BAPT. BENED." n="254" file="0254" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0254"/> - ferre</reg> vt anni directionum correÅ¿pondeant gradibus æquatoris. </s> - <s xml:space="preserve">Et præterea in ephe <lb/>meridibus videntur certè motus & aÅ¿pectus luminarium, quamuis inÅ¿it <reg norm="differentia" type="context">differẽtia</reg> mi <lb/>nutorum. </s> - <s xml:space="preserve">Nam non differunt gradibus, præter Å¿itum parum diÅ¿tantem à vero om-<lb/>nium planetarum, quorum cognitio in cÅ“lo, quamuis circa eorum locum error eÅ¿-<lb/>Å¿et gra .10. tamen in hoc prodeſſet, & tempus aÅ¿pectus eorum, etÅ¿inon diei præcisè, <lb/>quia influentia eiuÅ¿modi a Å¿pectuum, præterquam Lunæ durat multis diebus, & non <lb/>vno tantum. </s> - <s xml:space="preserve">præterquam quod ipſæ ephemerides <reg norm="oÅ¿tendunt" type="context">oÅ¿tendÅ©t</reg> nobis tempus <reg norm="ecclipÅ¿ium" type="context">ecclipÅ¿iÅ©</reg>, <lb/>in quo certènon differunt nec diebus nec multis horis, & itidem multa alia.</s> - </p> - <p> - <s xml:space="preserve">Non Å¿unt igitur contemnendæ ephemerides, nec habendæ pro re nullius pretij, <lb/>vt hic ait.</s> - </p> - <p> - <s xml:space="preserve">Quod attinet ad illa alia, quæ hic vocat errores ephemeridum, tam de apparenti <lb/>coniunctione Saturni cum Ioue in Å¿ignis alterius triplicitatis prius quam perÄ™gerit <lb/>præcedentem, quam de faciendo currere <reg norm="Saturnum" type="context">SaturnÅ©</reg>, & de retinendo Ioue, de <reg norm="detinem" type="context">detinẽ</reg> <lb/>do Marte .6. aut .7. menÅ¿ibus in vno Å¿igno, de Marte, & Ioue non coeuntibus Å¿ingu-<lb/>lis .24. annis in quolibet Å¿igno, & cius generis alia, minime verum eÅ¿t quod Å¿int er-<lb/>rores, quamuis huic præbuerint occaÅ¿ionem toties errandi.</s> - </p> - <p> - <s xml:space="preserve">Comparatio poÅ¿tea inter eius calculos Å¿umptos partim ex tabulis Iunctini, & par <lb/>tim ex ephemeridibus Stadij tan quam calculis Copernici, & calculos figurarum Å¿u-<lb/>per eis poÅ¿itarum Å¿upputatarum à diuerÅ¿is per ephemeridas AlfonÅ¿inas, etiam pro-<lb/>poÅ¿ita ab eo eÅ¿t ad oÅ¿tendendum magnam & monÅ¿truoÅ¿am differentiam, vt ait cap <num value="2">.<lb/>2.</num> vbi miratur, quod cum ex communi Å¿ententia calculi Copernici meliores Å¿int, cal <lb/>culatores dictarum figurarum potius eos Å¿umpÅ¿erint à tabulis AlfonÅ¿i, quam Coper <lb/>nici. </s> - <s xml:space="preserve">Quæ admiratio quam aliena Å¿it, conÅ¿iderandum permittam cuiuis intelligen-<lb/>ti harum Å¿acultatum, cum ſæpe accidere poſſit. </s> - <s xml:space="preserve">vt cum aliquis velit Å¿cire Å¿olum vni <lb/>uerÅ¿alia alicuius geneÅ¿is, Å¿iue natiuitatis, cum non inueniantur ephemerides Coper-<lb/>nici, Å¿ed tantum AlfonÅ¿i, calculator vtatur tantum ephemeridibus, quas inuenit, <reg norm="tum" type="context">tÅ©</reg> <lb/>cauÅ¿a vitandi tædij calculi tabularum, qui magni laboris eÅ¿t, prÄ™cipuè in tabulis Pru <lb/>tenicis Reinoldi. </s> - <s xml:space="preserve">tum quia Å¿uperflua ei eÅ¿t Å¿umma Å¿ubtilitas, cum non curet laborare <lb/>circa directiones vt <reg norm="factum" type="context">factÅ©</reg> eÅ¿t pro <reg norm="Å¿ecunda" type="context">Å¿ecÅ©da</reg> figura ab hoc propoÅ¿ita, quæ erat anni .1551 <lb/>quo non <reg norm="inueniebantur" type="context">inueniebãtur</reg> ephemerides Copernicæ, quæ non editæ Å¿unt ante annum <lb/>1554. præter quam quod ille nobilis vir pro quo Å¿upputata fuit dicta Å¿ecunda natiui <lb/>tas dubitabat de anno, vt hic Å¿imiliter Å¿cit. </s> - <s xml:space="preserve">quare potuiſſet perdi tempus, & labor, Å¿i <lb/>Å¿upputata fuiſſet per tabulas Reinoldi, nam Iunctini tabulæ nondum editæ fuerant. <lb/></s> - <s xml:space="preserve">Calculus poÅ¿tea Å¿eptimæ figuræ, qui erat reuolutio dictæ Å¿ecundæ natiuitatis, <reg norm="duabus" type="simple">duabê°</reg> <lb/>de cauÅ¿is non factus eÅ¿t per tabulas prutenicas, primum, quia eius anni .1580. non <lb/>inueniebantur amplius ephemerides Copernicæ. </s> - <s xml:space="preserve">Ephemerides enim Stadij <reg norm="in- cipientes" type="context">in-<lb/>cipiẽtes</reg> ab ann .1554. <reg norm="deÅ¿inunt" type="context">deÅ¿inÅ©t</reg> ann .1576. & <reg norm="continuatæ" type="context">cõtinuatæ</reg> poÅ¿tea quæ perueniunt vÅ¿que <lb/>ad annum .1600. non peruenere ad manus calculatoris ante hunc annum .1581. </s> - <s xml:space="preserve">Al-<lb/>tera ratio eÅ¿t, quia in reuolutionibus, quoniam in eis non fiunt directiones, non po-<lb/>nuntur à doctis, ne minuta quidem. </s> - <s xml:space="preserve">quare non Å¿olum non curant eas Å¿upputare per <lb/>tabulas, Å¿ed nec exquiÅ¿itè quidem per ephemeridas. </s> - <s xml:space="preserve">Calculi poÅ¿tea ab hoc Å¿umpti <lb/>ex tabulis Iunctini, & poÅ¿iti Å¿ub dicta Å¿ecunda figura, adeò rectè facti Å¿unt, vt cum Å¿e <lb/>cundum ip Å¿as tabulas oporteat Saturnum eſſe circa .32. minutum gradus .23. Aqua-<lb/>rij, ipÅ¿e eum Å¿cribat in gra .11. mi .3. dicti Å¿igni. Iupiter Å¿imiliter qui Å¿ecundum dictas <lb/>tabulas inuenitur circa finem gradus .5. Cancri, ab eo ponitur in min .28. gra .19. eiuÅ¿ <lb/>dem. ex quibus planetis Saturnus in figura poÅ¿itus eÅ¿t in min .27. grad .23. Arietis, Iu <lb/>piter autem in min .3. gra .6. Cancri, </s> - <s xml:space="preserve">Vnde Å¿ecun dum verum, inter calculum Alfon - <pb o="243" rhead="EPISTOLAE." n="255" file="0255" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0255"/> - Å¿i & Iunctini in Saturno non erat differentia plus quam minu .5. & in Ioue min .4. Å¿ed <lb/>Å¿ecundum calculum huius in Saturno fuiſſet differentia gra .11. minu .54. & in Ioue <lb/>grad .13. min .35. </s> - <s xml:space="preserve">Atque hæ Å¿unt quidem differentiæ magnæ, & monÅ¿truoſæ vtipÅ¿e <lb/>eas vocat, vt etiam eÅ¿tilla Veneris, & Mercurij inter tertiam figuram, & eius calcu-<lb/>lum Å¿umptum, non quidem à tabulis laborioÅ¿is, Å¿ed à ſimplicibus ephemeridibus Sta <lb/>dij, quæ differentia <reg norm="non" type="context">nõ</reg> eÅ¿t quidem paucorum graduum, cum Å¿it tertiæ partis cÅ“li in <lb/>quolibet dictotum planetarum. </s> - <s xml:space="preserve"><reg norm="HuiuÅ¿modique" type="simple">HuiuÅ¿modiqÌ;</reg> monÅ¿tra certè non Å¿unt orta à ta-<lb/>bulis Å¿iue ephemeridibus diuerÅ¿is, Å¿ed Å¿unt partus huius authoris.</s> - </p> - <p> - <s xml:space="preserve">Pergens poÅ¿tea aſſiduè bonus hic vir hominibus dare Å¿pecimen doctrinæ Å¿uæ ape <lb/>riendo (vt conatur) aliorum errores, proponit duas differentias inter primam fi-<lb/>guram, & Å¿uum calculum Å¿uppoÅ¿itum Saturni, & Iouis. </s> - <s xml:space="preserve">Primum de Saturno ait, <reg norm="quod" type="simple">ê™</reg> <lb/>cum differentia Å¿it gra .1. min .30. oÅ¿tendit in directione, <reg norm="quod" type="simple">ê™</reg> accidens Å¿it euenturum <lb/>anno vno, & menÅ¿ibus Å¿ex ante, aut poÅ¿t, quaÅ¿i eiuÅ¿modi differentia eſſet partium æ-<lb/>quatoris, Å¿icuti eÅ¿t partium Zodiaci. </s> - <s xml:space="preserve">Idem dico de differentia Iouis. </s> - <s xml:space="preserve">Quod quidem, <lb/>manifeÅ¿tum eÅ¿t inditium Å¿cientiæ Å¿uæ, & quantum ea intelligat de quibus loquitur.</s> - </p> - <p> - <s xml:space="preserve">Quod poÅ¿tea attinet ad differentiam inter Copernicum & AlfonÅ¿um, circa <reg norm="Solem" type="context">Solẽ</reg>, <lb/>nullus eÅ¿t harum Å¿cientiarum peritus, qui id neÅ¿ciat, & Å¿imiliter de differentia Å¿itus <reg norm="cae li" type="simple">cÄ™Â<lb/>li</reg> in reuolutionibus annuis.</s> - </p> - <p> - <s xml:space="preserve">Quod vero ait Å¿eptimam Å¿iguram malè <reg norm="Å¿upputatam" type="context">Å¿upputatã</reg> fuiſſe, Å¿i non eÅ¿t maximus cer-<lb/>tè non eÅ¿t minimus monÅ¿truoÅ¿orum eius errorum. </s> - <s xml:space="preserve">Vbi itidem videri poteÅ¿t, quam <lb/>alienus hic Å¿it ab hac Å¿cientia. </s> - <s xml:space="preserve">Nam Å¿i Å¿altem curaſſet Å¿ibi ab aliquo Å¿upputandum <lb/>locum Solis per tabulas AlfonÅ¿i in inÅ¿tanti minutorum .36. pomeridianorum, certior <lb/>factus eſſet quod in illo puncto Sol inueniebatur in minu 54. grad .11. </s> - <s xml:space="preserve">Geminorum, <lb/>ideÅ¿t præterierat gra .10. cum min .54. vel Å¿i curaſſet Å¿ibi inueniendum tempus, per <lb/>dictas tabulas cum grad .10. min .54. </s> - <s xml:space="preserve">Geminorum vt faciendum eÅ¿t, Å¿equendo <reg norm="tamen" type="wordlist">tamẽ</reg> <lb/>AlfonÅ¿um, & non per calculum Solis poÅ¿itum in ephemeridibus, vt parum periti fa <lb/>cere Å¿olent, vidiſſet <reg norm="quod" type="simple">ê™</reg> inuenta eſſent min .36. pomeridiana. </s> - <s xml:space="preserve">Leuis tamen occaÅ¿io hu <lb/>ic fuit Å¿uÅ¿picandi eiuÅ¿modi tempus eſſe falÅ¿um, quod viderit in illa figura <reg norm="Solem" type="context">Solẽ</reg> po <lb/>Å¿itum eſſe cum gra .11. & non cum gra .10. min .54. non animaduertens ita notatum <lb/>fuiſſe Solem vt omnes alios planetas, Å¿cilicet Å¿ine minutis, quum, vt dixi, in reuolu <lb/>tionibus non adhibeatur tanta Å¿cruploÅ¿itas.</s> - </p> - <p> - <s xml:space="preserve">Quod deinde ait, in illa figura Solem poÅ¿itum eſſe in decima domo, & non in .9. <lb/>id relinquam iudicio eorum qui Å¿ciunt numerare domos, Å¿altem poÅ¿uiſſet authori-<lb/>tate Å¿ua Solem in dicta decima diuersè ab exemplo ei dato ab amico, vt oÅ¿tenderet <lb/>Å¿e dicere verum, vt in Å¿ecunda figura diÅ¿crepat ab ipÅ¿o exemplo in collocando Leo-<lb/>ne, Virgine, & Libra, & Scorpio, quos malè locauit, & Å¿i alii bene Å¿e habent.</s> - </p> - <p> - <s xml:space="preserve">Atque quod hactenus à me dictum eÅ¿t, Å¿atis Å¿it ad <reg norm="intelligendum" type="context">intelligẽdum</reg> quale Å¿it <reg norm="reliquum" type="context">reliquÅ©</reg> <lb/>dictæ eius diÅ¿putationis. </s> - <s xml:space="preserve">Sienim velim pergere notare omnia eius errorum loca, eÅ¿ <lb/>Å¿et mihi inanis labor, & tibi nimia moleÅ¿tia. </s> - <s xml:space="preserve">Et quamuis non defuerint præſtantiſſi-<lb/>mi viri, qui viÅ¿is eius Å¿criptis familiariter eum monuere, & tu ipÅ¿e, vt audiui, cum <reg norm="in- Å¿trumento" type="context">in-<lb/>Å¿trumẽto</reg> theoricÄ™ in <reg norm="manibus" type="simple">manibê°</reg> ei <reg norm="oÅ¿tenderis" type="context">oÅ¿tẽderis</reg> quo <reg norm="mon" type="context">mõ</reg> Mars poſſit morari amplius Å¿ex <reg norm="mem" type="context">mẽ</reg> <lb/>Å¿ibus in vno Å¿igno. </s> - <s xml:space="preserve">& præterea cum iam ab initio Taurinum aduenit, mecum com-<lb/>municauerit illa Å¿ua prima Å¿cripta, <reg norm="egoque" type="simple">egoqÌ;</reg> eum monuerim, quod in varijs <reg norm="rebus" type="simple">rebê°</reg> falleba <lb/>tur, <reg norm="diſſuaÅ¿erimque" type="simple">diſſuaÅ¿erimqÌ;</reg> ne ea imprimenda curaret, quia nullum honorem inde referret, <lb/>eum hortans, vt potius alijs rebus operam daret, atque ei dixerim quod ad animad-<lb/>uerÅ¿iones differentiarum ephemeridum attinet, quod id iam <reg norm="oens" type="context">oẽs</reg> animaduerterant. <lb/></s> - <s xml:space="preserve">Mihi reÅ¿pondit Å¿e decreuiſſe illa edere, vt poÅ¿tea fecit, & tot admonitionibus non - <pb o="244" rhead="IO. BABPT. BENED." n="256" file="0256" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0256"/> - acqu<unsure/>ieÅ¿cens, die .11. </s> - <s xml:space="preserve">AuguÅ¿ti edidit chartam illam impreſſam inuitans ad diÅ¿puta-<lb/>tionem quotquot adhÄ™rerent contrariæ Å¿ententiæ, volens Å¿uÅ¿tinere <reg norm="Martem" type="context">Martẽ</reg> non poÅ¿ <lb/>Å¿e commorari in vno Å¿igno amplius duobus menÅ¿ibus, Å¿upponens partem princi-<lb/>piorum ab omnibus admiſſorum, & in fine paginæ exponens modum, quo vtitur ad <lb/>probationem Å¿uæ intentionis. </s> - <s xml:space="preserve">Puto autem quodÅ¿ecum ratiocinabatur de Marte, <lb/>vt fecit de Saturno in Å¿cripto latino, hoc modo. </s> - <s xml:space="preserve">Si Mars in duobus annis ambulat <lb/>per omnia 12. Å¿igna, neceſſe eÅ¿t igitur, vtin menÅ¿ibus duobus ambulet per vnum Å¿i-<lb/>gum, cum menÅ¿es .2. Å¿int duo decima pars annorum duorum. </s> - <s xml:space="preserve">Sedibi Å¿tatim in ipÅ¿o <lb/>initio commitcit errorem graduum ferè .7. dicens, quod medius motus Martis inue-<lb/>niebatur Å¿ignorum .4. & gra .17. cum eo tempore dictus medius motus non eſſet reue <lb/>ra plus quam Å¿ign .4. grad .10. mi .36. verum hoc ad ea, quæ Å¿equuntur exigui eÅ¿t mo <lb/>menti. </s> - <s xml:space="preserve">Is poÅ¿tea particulatim colligit medium motum Martis ad diem .29. </s> - <s xml:space="preserve">Mai an-<lb/>ni .15 14. quem ait eſſe Å¿ignorum .9. gra .27. min .53. & tamen reuera erat <reg norm="tantum" type="context">tantÅ©</reg> Å¿igno <lb/>rum .9. gra .21. mi .29. Å¿ed miſſum faciamus etiam hunc errorem <reg norm="tanquam" type="context">tanquã</reg> à primo pen <lb/>dentem. </s> - <s xml:space="preserve">Cum deinde ibidem ponit centrum epicycli, Å¿imiliter errat, nam <lb/>centrum epicycli nunquam poni debet vbi eÅ¿t linea medij motus, niÅ¿i Å¿it in auge, aut <lb/>in oppoÅ¿ito aug is eccentrici, quia debebat collocare ipÅ¿um centrum <reg norm="tanto" type="context">tãto</reg> poÅ¿t linec<unsure/><reg norm="am" type="context">ã</reg> <lb/>medij motus, quanta erat æquatio centri, quia medium centrum Martistunc erat mi <lb/>nus Å¿ignis Å¿ex, & aux eccentrici eius erat in Å¿exto minuto grad .16. Leonis. </s> - <s xml:space="preserve">Tamen <lb/>hoc etiam leue eÅ¿t. </s> - <s xml:space="preserve">Præſupponamus igitur quod centrum epicycli cſſet in grad .28 <lb/>Capricorni, vt ipÅ¿e credidit, ideÅ¿t gradibus .7. vlterius quam erat reuera. </s> - <s xml:space="preserve">Ait poÅ¿tea <lb/>Å¿e comperiſſe Martem ambulaſſe Å¿igna .4. & grad .22. eius epicycli, Å¿ed non explicat <lb/>an intelligat de argumento medio, an de vero, quod vocatur æquatum, nam Å¿i intel <lb/>ligatur de medio, hoc eſſe non poteÅ¿t, cum <reg norm="medium" type="context">mediÅ©</reg> eſſet Å¿ignorum .4. gra .24. mi .35. <lb/>Å¿ed Å¿i intelligatur de vero, vt iure credendum eÅ¿t (alioquin etiam erraſſet) cer-<lb/>tè falÅ¿um eÅ¿t. </s> - <s xml:space="preserve"><reg norm="Nam" type="context">Nã</reg>, verum, erat Å¿ignorum .4. grad .29. minu .39. </s> - <s xml:space="preserve">Itaque Mars non di-<lb/>Å¿tabat à linea veri motus epicycli amplius gradibus .30. & minu .21. ipÅ¿ius epicycli, <lb/>& æquatio argumentiÅ¿ecundo correcta erat gra .44. minu .2. à quo Å¿ubtracta æquatio <lb/>ne centri, quæ erat gr .5. minu .4. (cum centrum epicycli deberet tanto Å¿pacio eſſe <lb/>poÅ¿t lineam medij motus quantum Å¿upra dixi) Å¿upererant gra .38. minu .58. adden-<lb/>di gradibus, & minu. medij motus, qui cum reuera eſſent grad .21. & minu .29. Ca-<lb/>pricorni, perueniebant ad minu .27. grad .1. PiÅ¿cium. </s> - <s xml:space="preserve">Sed præſuppoÅ¿ito <reg norm="Å¿ecundum" type="context">Å¿ecundÅ©</reg> <lb/>ipÅ¿um, quod medius motus eſſet grad .28. </s> - <s xml:space="preserve">Capricorni, & quod Mars eſſet non <reg norm="Å¿olum" type="context">Å¿olÅ©</reg> <lb/>vbi hic ait, Å¿ed etiam in prima linea contingentiæ epicycli, ideÅ¿t in prima linea ma <lb/>ximæ æquationis argumenti, & præſuppoÅ¿ito etiam quod dicta æquatio eſſet æqua-<lb/>lis illi, quam haberet ad medium Aquarij Å¿cilicet grad .47. quum centrum epicycli <lb/>eÅ¿t in oppoÅ¿ito augis, manifeÅ¿tum eÅ¿t, quod eiuÅ¿modi linea contingentiæ non tranÅ¿i <lb/>ret vltra grad .15. PiÅ¿cium, & tamen hic ait, quodlinea veri motus Martis vadit ad <lb/>grad .16. Arietis. </s> - <s xml:space="preserve">vnde oporteret, quod æquatio argumenti eſſet plus quam grad <num value="78">.<lb/>78.</num> </s> - <s xml:space="preserve">Quod Å¿i verum eſſet, & <var>.o.c.</var> etiam eſſet partium .54. Å¿ecundum diÅ¿tantiam pro-<lb/>ximiorem centro mundi, Å¿emidiameter epicycli eſſet eiuÅ¿modi partium .52. minut <num value="49">.<lb/>49.</num> & quum Mars eſſet in <var>.g.</var> ideÅ¿t in oppoÅ¿ito veræ augis epicycli, dum centrum epi <lb/>cycli eſſet in eiuÅ¿modi diÅ¿tantia à terra, diÅ¿tantia <var>.o.g.</var> ideÅ¿t à terra ad Martem non <lb/>eſſet plus, quam vna Å¿ola pars ex dictis, cum minut .11. cum partes .52. minu .49. <lb/>ad .54. Å¿int vt Å¿inus anguli gra .78. qui eÅ¿t partium .97814. ad Å¿inum totalem partium <lb/>100000. </s> - <s xml:space="preserve">Nam iam Å¿upra dixi, quod triangulus <var>.o.c.i.</var> eÅ¿t rectangulus. </s> - <s xml:space="preserve">Hinc Å¿eque- - <pb o="245" rhead="EPISTOLAE." n="257" file="0257" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0257"/> - retur, quod in in-<lb/>teruallo <var>.o.g.</var> <reg norm="vnius" type="simple">vniê°</reg> <lb/> - <anchor type="figure" xlink:label="fig-0257-01a" xlink:href="fig-0257-01"/> - partis, & min .11. <lb/>reÅ¿pectu <var>.o.c.</var> par-<lb/>tium .54. </s> - <s xml:space="preserve">Colloca <lb/>retur Å¿emidiame-<lb/>ter terrÄ™ cum Å¿piÅ¿-<lb/>Å¿itudine aeris, i-<lb/>gnis, cÄ™lorum Lu-<lb/>næ, Mercurij, Ve-<lb/>neris, & Solis, <reg norm="prae terquam" type="simple">prÄ™Â<lb/>terquam</reg> quod vt <lb/>inter Solem, & <lb/>terram Å¿unt circa <lb/>605. diametri ip <lb/>Å¿ius terræ, inter <lb/>terram, & Mar-<lb/>tem cum eſſet in <lb/>auge Å¿ui epicycli, <lb/>& epicyclus in au <lb/>ge eccentrici, in-<lb/>uenirentur cir--<lb/>ca .60000. dia--<lb/>metri eiuÅ¿dem ter <lb/>ræ, & tamen ea diÅ¿tantia Å¿iue interuallum non poteÅ¿t continere .5000. diametri ter-<lb/>ræ. </s> - <s xml:space="preserve">Et quod plus eÅ¿t, hic tam vaÅ¿tum facit hunc Å¿uum epicyclum, vt ambiente Mar-<lb/>te per inferiorem eius partem, neceſſe ei eſſet manere in vno duodecatemorio mul-<lb/>to plus quam .7. aut .8. menÅ¿. </s> - <s xml:space="preserve">vnde hic multo magis miraretur quam prius. </s> - <s xml:space="preserve">Hinc cer-<lb/>nere licet quam rectè facti Å¿int hi eius calculi.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0257-01" xlink:href="fig-0257-01a"> - <image file="0257-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0257-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Vt autem etiam hinc aliqua vtilitas capiatur (prætermiſſis inconuenientibus vna <lb/>cum falÅ¿is Å¿uppoÅ¿itis huius) Videamus ordine Å¿cientifico vbi poterat eſſe verus lo-<lb/>cus Martis, aut vero proximus, die .29. Mai anni .1514. quem hic exempli cauÅ¿a Å¿u-<lb/>mit. </s> - <s xml:space="preserve"><reg norm="Idque" type="simple">IdqÌ</reg> tam ad defenÅ¿ionem tabularum AlfonÅ¿i, quam ephemeridum ex eis col <lb/>lectarum. </s> - <s xml:space="preserve">quæ quidem exactæ Å¿unt, vt quiÅ¿que peritus Å¿acile videre poterit, non au-<lb/>tem calculatæ à tam Å¿tupidis hominibus, vt à vero aberrent etiam gradibus .46. vt <lb/>hic ait Å¿e depræhendiſſe.</s> - </p> - <p> - <s xml:space="preserve">Primum igitur Å¿upponemus eoÅ¿dem illos terminos, quos ipÅ¿e nec d@bet, nec po <lb/>teÅ¿t negare, præter ea quæ Å¿upra Å¿uppoÅ¿ita Å¿unt, nempe quod Å¿emidiameter epicycli <lb/>Å¿it partium .39. minu .30. & eccentricitas partium .6. talium qualium eÅ¿t Å¿emidiame-<lb/>ter deferentis diuiÅ¿us in .60. & quod dicto tempore aux eccentrici Martis eſſet cir-<lb/>ca minutum .5. grad .16. Leonis, Å¿cilicet graduum .135. min .5. & quod linea motus <lb/>me<unsure/>diocris eſſer circa minu .30. gradus .22. </s> - <s xml:space="preserve">Capricorni, & quod verum <reg norm="centrum" type="context">centrÅ©</reg> Mar <lb/>tis eſſet grad .151 minut .20. & quod argumentum verum eſſet grad .149. minu .39. <lb/><reg norm="atque" type="simple">atq;</reg> ita oÅ¿tendam, neque tabulas, neque ephemerides errare, ne quidem vno gra-<lb/>du, ac ne quidem multis minutis, non modò tam monÅ¿truoÅ¿a differentia, vt ipÅ¿e <lb/>ait.</s> - </p> - <p> - <s xml:space="preserve">Quare primum nobis Å¿cientificè inueniendum eÅ¿t, quanta eſſet diÅ¿tantia <var>.o.c.</var> - <pb o="246" rhead="IO. BAPT. BENED." n="258" file="0258" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0258"/> - præciÅ¿e ideÅ¿t interuallum inter centrum mundi, & centrum epicycli Martis in huiuÅ¿-<lb/>modi Å¿itu.</s> - </p> - <p> - <s xml:space="preserve">Fingemus igitur eccenticum Martis Å¿ignificatum per <var>.p.c.m.</var> cuius centrum Å¿it <var>.r.</var> <lb/>& lineam augis <var>.p.r.o.m.</var> in qua <reg norm="centrum" type="context">centrÅ©</reg> mundi Å¿it <var>.o.</var> centrum autem verum epicycli, <lb/>comprehendatur ab angulo <var>.p.o.c.</var> qui Å¿it graduum .151. min .30. Å¿ecundum Å¿uppoÅ¿i-<lb/>tum. </s> - <s xml:space="preserve">Quare in puncto <var>.c.</var> erit centrum epicycli. </s> - <s xml:space="preserve">Imaginemur ergo <var>.c.o.</var> productam à <lb/>parte <var>.o.</var> quouÅ¿que ab <var>.r.</var> centro deferentis veniat linea <var>.r.k.</var> perpendiculariter, faciens <lb/>angulum rectum in puncto. k & quoniam angulus <var>.r.o.c.</var> datur nobis graduum .151. <lb/>min .30. ideo cognoÅ¿cemus angulum <var>.r.o.k.</var> tanquam reliquum ex duobus rectis, qui <lb/>erit gra .28. min .30. & Å¿imiliter angu-<lb/> - <anchor type="figure" xlink:label="fig-0258-01a" xlink:href="fig-0258-01"/> - lum <var>.o.r.k.</var> tanquam reÅ¿iduum vnius <lb/>recti, qui erit gra .61. min .30. cuius Å¿i-<lb/>nus ideÅ¿t <var>.o.k.</var> erit partium .8788 1. et <var>.k.<lb/>r.</var> vt Å¿inus anguli <var>.r.o.k.</var> partium .47715 <lb/>talium qualium <var>.o.r.</var> eſſet 100000. Å¿ed <lb/>vt <var>.o.r.</var> eÅ¿t .6. latus <var>.o.k.</var> erit .5. & min .16 <lb/>et <var>.r.k.</var> partium .2. min .52. & quia <var>.r.c.</var> <lb/>cÅ¿t <reg norm="partium" type="context">partiÅ©</reg> 60. eiuÅ¿modi, Å¿i ab eius qua-<lb/>drato Å¿ubtractum fuerit quadratum ip <lb/>Å¿ius <var>.r.k.</var> reliquum erit nobis <reg norm="quadratum" type="context">quadratÅ©</reg> <lb/>ipÅ¿ius <var>.k.c.</var> cuius radix, ideÅ¿t <var>.k.</var> erit par-<lb/>tium .59. min .56. à qua <var>.c.k.</var> Å¿ubtrahen-<lb/>do poÅ¿tea <var>.k.o.</var> partium .5. minu .16. re-<lb/>manebit <var>.o.c.</var> partium .54. min .40. pro <lb/>diÅ¿tantia quæſita.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0258-01" xlink:href="fig-0258-01a"> - <image file="0258-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0258-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Fingamus poÅ¿tea epicyclum <var>.f.n.g.</var> <lb/>in quo argumentum verum graduum <num value="149">.<lb/>149.</num> minu .39. Å¿it arcus <var>.f.n.</var> vbi Mars inueniatur in <var>.n.</var> per quem punctum tranÅ¿eat li-<lb/>nea <var>.o.n.</var> veri motus Martis. </s> - <s xml:space="preserve">Deinde inueniamus angulum <var>.c.o.n.</var> æquationis <reg norm="argumem" type="context">argumẽ</reg> <lb/>ti, modo iam dicto, ideÅ¿t ducendo Å¿inum <var>.n.h.</var> arcus <var>.n.g.</var> qui arcus tanquam reliquus <lb/>argumenti veri, iam præſuppoÅ¿iti, ex dimidio circulo, erit graduum 30. minu .21. & <lb/><var>n.h.</var> eius Å¿inus partium .50528. Å¿inus Å¿imiliter anguli <var>.n.c.h.</var> et <var>.c.h.</var> tanquam Å¿inus an-<lb/>guli <var>.c.n.h.</var> reÅ¿tantis ex uno recto grad .59. minu .39. erit partium .86295. <reg norm="talium" type="context">taliÅ©</reg> qua-<lb/>lium <var>.c.n.</var> Å¿inus totus eſſet partium .100000. Å¿ed vt partium .39. & min .30. Å¿inus <var>.c.h.</var> <lb/>erit partium .34. min .5. et <var>.n.h.</var> partium .19. mi .57. reliquum poÅ¿tea <var>.h.o.</var> ex <var>.o.c.</var> par-<lb/>tium .20. min .35. quia iam Å¿upra inuenimus <var>.o.c.</var> eſſe partium eiuÅ¿modi .54. minu .40. <lb/>vnde <var>.o.n.</var> vt radix quadrata Å¿ummæ duorum <var>.n.h.</var> et <var>.h.o.</var> erit partium .28. minu .41. <lb/>talium qualium <var>.n.h.</var> inuenta fuit partium .19. min .57. quæ <var>.n.h.</var> erit poÅ¿tea partium, <lb/>69552. talium qualium <var>.n.o.</var> partium .100000. & Å¿umpta dicta <var>.n.h.</var> vt Å¿inus dictarum <lb/>partium, dabit nobis angulum <var>.n.o.h.</var> quæſitum gra .44. min .4. qui per tabulas Alfon <lb/>Å¿i inuentus eÅ¿t gra .44. min .2. par huic, vt dici poteÅ¿t. </s> - <s xml:space="preserve">Quiangulus gra .44. minu .4. <lb/>collectus cum angulo veri centri iam Å¿uppoÅ¿ito graduum .151. minu .20. & cum an-<lb/>gulo augis eccentrici Martis, Å¿imiliter Å¿uppoÅ¿itæ grad .135. min .5. dabit nobis Å¿um-<lb/>mam veræ diÅ¿tantiæ Martis à principio Arietis grad .330. min .29. quod aliud non <lb/>Å¿ignificat, niÅ¿i quod Mars inuenietur in minu .29. primi gradus PiÅ¿cium. </s> - <s xml:space="preserve">Et Stofle-<lb/>rus in Å¿uis ephemeridibus ponit eum in .22. minuto dicti primi gradus, cuius diffe- - <pb o="247" rhead="EPISTOL AE." n="259" file="0259" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0259"/> - rentia à tabulis <lb/>eÅ¿t minut .5. tan <lb/> - <anchor type="figure" xlink:label="fig-0259-01a" xlink:href="fig-0259-01"/> - <reg norm="tum" type="context">tÅ©</reg>, & à me<unsure/>o cal-<lb/>culo min .7. vide <lb/>licet minima.</s> - </p> - <div type="float" level="5" n="4"> - <figure xlink:label="fig-0259-01" xlink:href="fig-0259-01a"> - <image file="0259-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0259-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Nunc autem <lb/>nolui Å¿umere ip-<lb/>Å¿um angulum æ-<lb/>quationis à tabu <lb/>lis propter duas <lb/>rationes, <reg norm="primum" type="context">primÅ©</reg> <lb/>quia ne hic qui-<lb/>dem repræhen-<lb/>for in hoc voluit <lb/>credere dictis ta <lb/>bulis. </s> - <s xml:space="preserve">Sed id vo-<lb/>luit videre pro-<lb/>prijs oculis <reg norm="in" type="wordlist">Ä©</reg> Å¿ua <lb/>theorica Martis. <lb/></s> - <s xml:space="preserve">Vbi <reg norm="inuenit" type="wordlist">Ä©uenit</reg> quod <lb/>linea <var>.o.n.</var> tranÅ¿it <lb/>per gra .16. </s> - <s xml:space="preserve">Arie <lb/>tis. </s> - <s xml:space="preserve">Secunda ra-<lb/>tio eÅ¿t, vt videa-<lb/>tur quod dictæ tabulæ rectè Å¿upputatæ Å¿unt, Å¿uper dictis Å¿uppoÅ¿itis.</s> - </p> - <p> - <s xml:space="preserve">Sed vt videat quantus Å¿it medius motus Martis die .29. </s> - <s xml:space="preserve">Mai colligit fruÅ¿tatim, <reg norm="quod" type="simple">ê™</reg> <lb/>eleganter colligere poterat, vna opera in columellis ipÅ¿ius medij motus eiuÅ¿modi <lb/>Å¿tellæ per eram eiuÅ¿dem temporis, quæ erat .2. primarum Å¿exagenarum .33. Å¿ecunda <lb/>rum .32. tertiarum, et .52. quartarum.</s> - </p> - <p> - <s xml:space="preserve">Primum deinde Å¿uppoÅ¿itum quod Å¿cribit, Å¿cilicet, quod diameter epicycli Å¿um-<lb/>ptus in longitudine media Å¿it Å¿ignorum .2. & grad .19. vti Å¿uperfluum eÅ¿t, ita etiam fal <lb/>Å¿um, nam eiuÅ¿modi diameter in dicto loco, non occupat ad centrum mundi plus <reg norm="quam" type="context">quã</reg> <lb/>gra .66. min .28. ideÅ¿t Å¿igna .2. gra .6. min .28. quia proportio <var>.o.c.</var> ad Å¿emidiametrum <lb/>epicycli in eiuÅ¿modi loco eÅ¿t ut partium .60. minu .18. ad partes .39. min .30. quÄ™ duÄ™ <lb/>lineæ intellectÄ™, vt latera vnius trianguli rectanguli, habebunt pro baÅ¿i aliam lineam <lb/>partium Å¿imilium .72. mi .5. </s> - <s xml:space="preserve">Quæ intellecta vt Å¿inus totus dabit Å¿emidiametrum epi-<lb/>cycli partium .54798. tanquam Å¿inum Å¿ubiectum angulo gra .33. min .14. pro medie <lb/>tate illius, quod quæritur.</s> - </p> - <p> - <s xml:space="preserve">Nec prætermittenda mihi videtur ratio, qua credere poſſumus, hunc cogi-<lb/>taſſe, quod diameter epicycli compleat Å¿patium duorum Å¿ignorum cum gradibus. <lb/>19 quæ quidem ratio alia eſſe non poteÅ¿t, niÅ¿i quod cum iÅ¿te inuenerit, in <reg norm="comentarijs" type="context">comẽtarijs</reg> <lb/>Theoricarum, Å¿emidiametr um huiuÅ¿modi epicycli eſſe partium .39 min .30. talium <lb/>qualium Å¿unt .60. illæ quæ Å¿unt Å¿emidiametri huius eccentrici, dictas igitur partes .39 <lb/>mi .30. hic putauit eſſe gradus Zodiaci, & propterea dixit <reg norm="diametrum" type="context">diametrÅ©</reg> huiuÅ¿modi epicy <lb/>cli eſſe <reg norm="Å¿ignorum" type="context">Å¿ignorÅ©</reg> <reg norm="duorum" type="context">duorÅ©</reg>, & gra .19. qui numerus .79. duplus eÅ¿t numero .39. <reg norm="cum" type="context">cÅ©</reg> dimidio <lb/>hoc autem dixit accidere in longitudinibus medijs, quia Å¿i hic intellexiſſet de pro-<lb/>portione horum duorum diametrorum, quæ eÅ¿t ut .120. ad .79. non Å¿pecificaſſet lo- - <pb o="248" rhead="IO. BAPT. BENED." n="260" file="0260" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0260"/> - cum epicycli, cum ipÅ¿a proportio nullo modo alteratur exiÅ¿tente epicyclo vbi volue <lb/>ris ipÅ¿ius circunferentiæ eccentrici, Å¿ed angulus in centro mundi, cui Å¿ubiacet dictus <lb/>diameter epicycli, bene alteratur, propter inæqualem diÅ¿tantiam centri epicycli <lb/>ab ipÅ¿o centro mundi. </s> - <s xml:space="preserve">At Å¿i de tali angulo inferre voluiſſet, iam probaui ipÅ¿um <reg norm="con- tinere" type="context">cõ-<lb/>tinere</reg> Å¿olum gra .66. minu .28. exiÅ¿tente centro epicycli in longitudinibus medijs & <lb/>non gra .79. vt ipÅ¿e dicit.</s> - </p> - <p> - <s xml:space="preserve">Omitto poÅ¿tea, quod vbi mentionem facit coniunctionum Solis cum Marte au-<lb/>gium & earum oppoÅ¿itorum, non explicat an intelligat de veris an de medijs. </s> - <s xml:space="preserve">Nam <lb/>Å¿i ex eius modo loquendi accipiatur eum loqui de veris multum erraret.</s> - </p> - <p> - <s xml:space="preserve">Sed quia iam tibi moleÅ¿tum eſſe inciperet Å¿i diutius te detinerem in his conten-<lb/>tionibus aÅ¿tronomicis, vlterius non diÅ¿putabo. </s> - <s xml:space="preserve">Satis enim hactenus explicaui <reg norm="Å¿enten- tiam" type="context">Å¿entẽ-<lb/>tiam</reg> meam, vt oÅ¿tendiſſe videor quam mihi iucundum Å¿it tibi <reg norm="morem" type="context">morẽ</reg> gerere. </s> - <s xml:space="preserve">In quo <lb/>etiam hnmanitati tuæ gratiam habebo, <reg norm="quum" type="context">quÅ©</reg> petitione tua occaÅ¿ionem mihi de deris <lb/>efficiendi, vt tum amici tui (amant enim te omnia Å¿ublimia ingenia) tum alij, Å¿i <reg norm="quam" type="context">quã</reg> <lb/>falÅ¿am opinionem ex huius Benedicti Altæuillæ Å¿criptis Å¿umpÅ¿iſſent, <reg norm="eam" type="context">eã</reg> relinquant, <lb/>& per te hoc beneficium à me conÅ¿equantur, & huiuÅ¿modi occaÅ¿ionem, & iuuandi <lb/>hominum Å¿tudia & tibi gratum faciendi, honorificum, & per gratum mihi fuiſſe in-<lb/>telligant. </s> - <s xml:space="preserve">Vale & me vt Å¿oles ama.</s> - </p> - <p> - <s xml:space="preserve">Taurini pridie Kal. </s> - <s xml:space="preserve">Octobris .1581.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">De probatione diuiÅ¿ionis numerorum.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">INter alia quæ à me Å¿cire cupis, vir doctiſſime, hoc vnum eÅ¿t, vt ex literis tuis ac-<lb/>cepi, vnde Å¿it vt priÅ¿ci noÅ¿tri probatione numeri nouenarij potius quam Å¿epte-<lb/>narij vÅ¿i fuerint, & qua ratione non idem proueniat ex probatione numerorum <lb/>octonarij, Å¿enarij, vel quinarij, aut cuiuslibet alterius: </s> - <s xml:space="preserve">Vnde pariter oriatur quod in <lb/>partitionis probatione neceſſum Å¿it probationum euentus multiplicare cum proba-<lb/>tione diuiÅ¿oris, ac eam quæ eÅ¿t producti poÅ¿tea cum probatione fractionis in Å¿um-<lb/>mam colligere, & c. </s> - <s xml:space="preserve">Ad hæc in primis reÅ¿pondeo, cum aliquoties accidere poſſit ta <lb/>les probationes nos fallere poſſe, <reg norm="idque" type="simple">idqÌ;</reg> fi in tali Å¿umma Å¿imilis numerus, ut puta Å¿e-<lb/>ptem, aut nouem, plus vel minus æquo iuÅ¿touè poÅ¿itus fuerit, attamen per raro eueni <lb/>re poteÅ¿t, vt quis per nouenarium potius quam per Å¿eptenarium decipiatur. </s> - <s xml:space="preserve">Exem-<lb/>pli gratia, ponamus Å¿ummam eſſe .100. quam numerus nouenarius vndecies Å¿olum <lb/>ingreditur, at Å¿eptenarius quatuordecies, vnde quis <reg norm="ſępius" type="simple">ſępiê°</reg> ex Å¿eptenario, hacratione, <lb/>quam ex nouenario numero Å¿e poſſe errare facile depræhendet, efÅ¿i ex probatione <lb/>nouenarij magis quam Å¿eptenarij, vt practici Å¿cribunt, duabus de cauÅ¿is errare poſſi-<lb/>mus. </s> - <s xml:space="preserve">Alia tamen ratio mihi Å¿uppetit, ob quam credibile eÅ¿t ipÅ¿os potius nouena-<lb/>rio adiutos fuiſſe, quam Å¿eptenario, quæ eÅ¿t ob Å¿ui cum velocitatem tum <reg norm="facilitatem" type="context">facilitatẽ</reg>, <lb/><reg norm="neque" type="simple">neq;</reg> enim in Å¿eptenario eÅ¿t adeo facilis. </s> - <s xml:space="preserve"><reg norm="Nam" type="context">Nã</reg> quamuis, tam vna quam altera aliud <lb/>non Å¿it, quam numerorum ordines diuidere (Å¿i de Å¿ummis primo loquamur) aut è <lb/>Å¿umma Å¿uperÅ¿luum ordinum colligere, & videre an idemmet Å¿uperfluum ex eadem <lb/>Å¿umma emanet, attamen cum modus, qui in hoc adhiberi poteÅ¿t in nouenario <reg norm="quam" type="context">quã</reg> <lb/>in Å¿eptenario velocior Å¿it, & ob id probationem nouenarij Å¿eligunt potius quam <lb/>Å¿eptenarij.</s> - </p> - <pb o="249" rhead="EPISTOLAE." n="261" file="0261" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0261"/> - <p> - <s xml:space="preserve">Verum nolo te in ea, quæfalÅ¿a eÅ¿t, opinione conÅ¿iÅ¿tere, nonidem, & cum octona-<lb/>rio, Å¿enario, vel quinario, aut quouis alio numero poſſe efficere, cum eademmet ra <lb/>tio, quæ in Å¿eptenario, aut nouenario, <reg norm="ent" type="context">ẽt</reg> in cæteris perhibeatur. </s> - <s xml:space="preserve">Ponamus <reg norm="exemplum" type="context">exemplÅ©</reg> <lb/>hos tres or dinum numeros velle Å¿upputare, quorum primus Å¿it .679. Å¿ecundus .846. <lb/>& tertius .935. & illorum <reg norm="Å¿ummam" type="context">Å¿ummã</reg> .2460. nunc maiorem numerum primi ordinis ab <lb/>octonario menÅ¿i, proijciendo, remanebit .7. deinde maiorem numerum demendo à <lb/>Å¿ecundo or dine, reÅ¿iduum erit .6. ac Å¿i idem in tertio ordine fecerimus, erit nobis re-<lb/>liquum .7. </s> - <s xml:space="preserve">Demum tria hæc reÅ¿idua in vnum collecta .20. efficient, à quibus Å¿i nume <lb/>rum maiorem ab octonario menÅ¿um dempÅ¿eris, Å¿upererunt .4. & totidem à ſumma <num value="2460">.<lb/>2460.</num> remanebunt, reiecto maiori numero ab octonario menÅ¿o. </s> - <s xml:space="preserve">Atque idem me-<lb/>dio quouis alio numero, euenire poteÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Cuius ratio tam perÅ¿e clara atque euidens eÅ¿t, quod Å¿i Å¿ummam trium <reg norm="reliquorum" type="context">reliquorÅ©</reg>, <lb/>quæ eÅ¿t .20. à ſumma .2460. Å¿ubduxeris, remanebunt .2440. pro Å¿umma trium nume <lb/>rorum dictorum trium ordinum ab octonario menÅ¿orum, cui numero addito .16. pro <lb/>maiori numero Å¿ummÄ™ <reg norm="reliquorum" type="context">reliquorÅ©</reg>, qui ab octonario menÅ¿us Å¿it, Å¿upererunt .4. </s> - <s xml:space="preserve">At Å¿i per <lb/><reg norm="Å¿enarium" type="context">Å¿enariÅ©</reg> <reg norm="experimentum" type="context context">experimẽtÅ©</reg> feceris, remanebit <var>.o.</var> & Å¿ic de reliquis per ordinem <reg norm="procedendo" type="context">procedẽdo</reg>.</s> - </p> - <p> - <s xml:space="preserve">Verum poſſes Å¿ciÅ¿citari, quare velocius, exceſſus ordinum, potius per <reg norm="noue- narium" type="context">noue-<lb/>nariÅ©</reg>, quam per cæteros numeros, prout <reg norm="docent" type="context">docẽt</reg> practici, inueniri queat, videlicet ag <lb/>gregando prius duas figuras numerorum primæ Å¿ummæ, deinde alias duas. </s> - <s xml:space="preserve">Exem-<lb/>plum Å¿it primus ordo .679. colligendo .6. et .7. faciunt 13. & cum hæc Å¿umma Å¿it dua <lb/>rum figurarum, Å¿upputantur & ipſæ, è quibus prodeunt .4. & conÅ¿imilis erit proba-<lb/>tio numeri .67. facta per .9. quod idem eÅ¿t, ac Å¿i quis diuidat .67. per .9. ex quo reli-<lb/>qui erunt Å¿emper .4.</s> - </p> - <p> - <s xml:space="preserve">At quo ratio huiuÅ¿ce perÅ¿picuè dignoÅ¿ci poſſit, in primis Å¿ciendum eÅ¿t, cuique <lb/>ex Å¿e cognitum, atque exploratum eſſe, denarium numerum vnitate nouenarium Å¿u <lb/>perare, & ex hoc Å¿equitur, Å¿ex denarios continere in Å¿e Å¿ex nouenarios, & Å¿ex vni-<lb/>tates.</s> - </p> - <p> - <s xml:space="preserve">At Å¿ex vnitates, vna cum .7. faciunt .13. & quia in .13. eÅ¿t denarius, igitur in illo erit <lb/>vnitas Å¿upra .9. </s> - <s xml:space="preserve">Quæ vnitas addita ternario, præbet nobis Å¿uperfluum, per quod .67. <lb/>Å¿uperat .54. iunctum cum .9. Å¿cilicet Å¿ummam .63.</s> - </p> - <p> - <s xml:space="preserve">Idem dicinon poteÅ¿t de octonario, Å¿eptenario, vel Å¿enario, & de reliquis, quo-<lb/>niam numerus denariorum, in cæteris minoribus nouenario non præbet illico nu-<lb/>merum exceſſus maioris numeri, qui à numero probationis menÅ¿us eÅ¿t. </s> - <s xml:space="preserve">Et quod di <lb/>co de probatione aggregationis, idem intelligo de alijs operationibus, vt puta Å¿ub-<lb/>tractionis, multiplicationis, & partitionis Å¿eu diuiÅ¿ionis.</s> - </p> - <p> - <s xml:space="preserve">Vnde autem oriatur, vt in partitionis probatione opus Å¿it probationem euentus <lb/>cum diuiÅ¿ionis probatione multiplicare, & productum cum fractionis probatione <lb/>Å¿upputare, Å¿eu aggregare, tibi non erit ignotum, quoties animaduerteris, quod <lb/>productum ipÅ¿ius euentus cum diuiÅ¿ore, adiunctum fractioni, perpetuo Å¿e æquat nu <lb/>mero diuiÅ¿ibili. </s> - <s xml:space="preserve">Et quoniam numeri probationum Å¿unt partes, quæ remanent ex <lb/>ipÅ¿is totis, detractis maioribus numeris ab eo dimenÅ¿is, quo pro communi men-<lb/>Å¿ura vtimur (prout .7. vel .9. aut alium numerum, quem voluerimus) par eÅ¿t vt ex ip-<lb/>Å¿arum remanentibus partibus, velut ex ipÅ¿is totis idem fiat.</s> - </p> - <pb o="250" rhead="IO. BAPT. BENED." n="262" file="0262" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0262"/> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">De falacia operationis triangulorum Å¿phericorum.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">QVod diebus præteris tibi Å¿ignificaui, idem nunc confirmo, Å¿cilicet Å¿phærico-<lb/>rum triangulorum operationem ſæpe nos fallere, vt exempli gratia, Å¿i pro <lb/>poÅ¿itus nobis fuiſſet triangulus <var>.A.B.C.</var> cuius angulus <var>.A.</var> nobis datus eſſet graduum <num value="114">.<lb/>114.</num> mi <var>.o.</var> & eius latus <var>.A.</var> B, graduum .67. min .5. & latus <var>.A.C.</var> graduum .45. mi .10. <lb/>Å¿i reliquos angulos cum tertio latere etiam cognoÅ¿cere voluerimus, ex methodo .11 <lb/>primi Copernici propoÅ¿itum obtinebimus. </s> - <s xml:space="preserve">vnde latus <var>.B.C.</var> eſſet graduum .89. min <num value="30">.<lb/>30.</num> angulus vero <var>.C.</var> graduum .57. min .14. angulus autem <var>.B.</var> grad .48. min .38. </s> - <s xml:space="preserve">Qua-<lb/>re vltimus hic angulus <var>.B.</var> falÅ¿us eſſet, eo quod operatio paruorum triangulorum in <lb/>cauÅ¿a eÅ¿t, quotieÅ¿cunque eorum latera tam breuia Å¿int, ut non eccedant vnum gra-<lb/>dum, quare ipÅ¿orum angulorum veram quantitatem non tribuunt. </s> - <s xml:space="preserve">propterea igitur <lb/>cum voluerimus veram <reg norm="quantitatem" type="context">quãtitatem</reg> ipÅ¿ius anguli <var>.B.</var> oportet poÅ¿t quam inuenerimus <lb/>angulum <var>.C.</var> mediante arcu <var>.D.E.</var> Å¿upponere alium polum in <var>.B.</var> deinde producere. </s> - <s xml:space="preserve">B <lb/>A. vÅ¿que ad <var>.d.</var> et <var>.B.C.</var> vÅ¿que ad <var>.e.</var> imaginando <var>.B.d.</var> et <var>.B.e.</var> duas quartas eſſe magno-<lb/>rum circulorum, extendendo poÅ¿tea <var>.d.e.</var> vÅ¿que ad interÅ¿ectionem cum <var>.A.C.</var> & <reg norm="eum" type="context">eÅ©</reg> <lb/>dem ordinem proÅ¿equendo, </s> - <s xml:space="preserve">tunc <var>.e.d.</var> nobis oÅ¿tendet angulum <var>.B.</var> eſſe gra .40. mi .22 <lb/>quæ erit eius vera quantitas. </s> - <s xml:space="preserve">Cuius quidem rei experientiam poſſumus etiam fa-<lb/>cere, hoc modo, eÅ¿to, exempli gratia, quod nobis datus Å¿it angulus <var>.C.</var> graduum .57. <lb/>min .14. cum latere <var>.A.C.</var> gra .45. min .10. & latus <var>.B.C.</var> gra .89. min .30. </s> - <s xml:space="preserve">Tunc Å¿i ordi-<lb/>nem .11. dicti lib. Å¿e quemur, obtinebimus intentum, hoc modo Å¿cilicet Å¿upponendo <lb/>in <var>.A.</var> polum, & non in <var>.B.</var> ducendo etiam <var>.A.B.</var> et <var>.A.C.</var> Å¿ed <var>.A.B.</var> <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> ad gra .90. du-<lb/>cendo poÅ¿tea <var>.D.E.</var> ita quod ab omni parte concurrat cum latere <var>.B.C.</var> producto, vn <lb/>de tam <var>.f.C.B.F.</var> quam <var>.f.D.E.F.</var> erunt Å¿emicirculi magnorum circulorum. </s> - <s xml:space="preserve">quare <var>.C.<lb/>D.</var> nobis cognitus erit gra .44. min .50. & Å¿ic etiam angulus <var>.D.C.f.</var> gra .57. min .14. ex <lb/>4. dicti lib. poÅ¿tea habebimus <var>.F.l.</var> gra .60. min .54. & angulum <var>.f.</var> gra .53. mi .24. aggre <lb/>gatum poÅ¿tea <var>.f.C.</var> cum <var>.C.B.</var> habebimus <var>.f.B.</var> gra .150. min .24. qui Å¿i a Å¿emicirculo <reg norm="dem" type="context">dẽ</reg> <lb/>ptus fuerit, nobis remanebit <var>.B.F.</var> gra .29. mi .36. cum angulo <var>.F.</var> cognito <reg norm="cum" type="context">cÅ©</reg> Å¿it æqua-<lb/> - <anchor type="figure" xlink:label="fig-0262-01a" xlink:href="fig-0262-01"/> - <anchor type="figure" xlink:label="fig-0262-02a" xlink:href="fig-0262-02"/> - lis <var>.f.</var> eius oppoÅ¿ito. </s> - <s xml:space="preserve">Vnde ex dicta .4. co-<lb/>gnoÅ¿cemus angulum <var>.B.</var> gra .40. min .31. <lb/>qui ferè æ qualis eÅ¿t Å¿uperiori iam inuen-<lb/>to, nec ab ipÅ¿o differt niÅ¿i per min .9. quæ <lb/>quidem differentia parua eÅ¿t reÅ¿pectu al - <pb o="251" rhead="EPISTOLAE." n="263" file="0263" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0263"/> - terius differentiæ quam Å¿upra inuenerimus.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0262-01" xlink:href="fig-0262-01a"> - <image file="0262-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0262-01"/> - </figure> - <figure xlink:label="fig-0262-02" xlink:href="fig-0262-02a"> - <image file="0262-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0262-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Superius enim dixinon eſſe ponendum polum in <var>.B.</var> eo quod <var>.B.C.</var> Å¿it gra .89. mi <num value="30">.<lb/>30.</num> vnde nobis prodijſſet triangulus <var>.f.C.D.</var> trium valde paruorum laterum, quorum <lb/>latus <var>.C.D.</var> eſſet gra <var>.o.</var> mi .30. & latus <var>.f.l.</var> gra <var>.o.</var> mi .55. & latus <var>.F.D.</var> gra <var>.o.</var> mi .47. vn-<lb/>de angulus <var>.f.</var> gra .32. min .40. falÅ¿us eſſet, qui <reg norm="quidem" type="context">quidẽ</reg> poÅ¿tea nobis daret <var>.D.E.</var> gra .45 <lb/>minu .16. falÅ¿um Å¿imiliter.</s> - </p> - </div> - <div type="letter" level="4" n="4"> - <head style="it" xml:space="preserve">De paßione circuli bactenus incognita.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">DVbitandum quidem <reg norm="non" type="context">nõ</reg> eÅ¿t quin paſſiones circuli innumerabiles penè Å¿int, quÄ™ <lb/>quidem omnes ferè caÅ¿u inueniuntur, vt mihi nunc accidit, quam tibi mitto, <lb/>hæc autem eÅ¿t, quòd quadratum lineæ <var>.a.g.</var> in figura hic Å¿ubÅ¿cripta Å¿emper æquale <lb/>eÅ¿t ei producto, quod fit ex <var>.a.e.</var> in diametro circuli <var>.g.c.b.</var> Å¿imul Å¿umpto cum quadra <lb/>to inÅ¿criptibili in dicto circulo, & Å¿imul cum quadrato lineæ <var>.a.b.</var> <reg norm="contingentis" type="context">contingẽtis</reg> ipÅ¿um <lb/>circulum, Å¿upponendo <var>.a.g.</var> per centrum ipÅ¿ius circuli tranÅ¿ire.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius demonÅ¿tratione à centro <var>.e.</var> duco Å¿emidiametrum <var>.e.c.</var> <reg norm="perpendicularem" type="context">perpendicularẽ</reg> <lb/>ipÅ¿i <var>.g.a.</var> & à puncto <var>.c.</var> ad <var>.a.</var> duco <var>.c.a.</var> quæ Å¿ecabit circunferentiam ipÅ¿ius circuli in <reg norm="pum" type="context">pÅ©</reg> <lb/>cto <var>.d.</var> eo, quod angulus <var>.c.</var> acutus eÅ¿t. </s> - <s xml:space="preserve">Nunc ex .35. tertij, productum <var>.c.a.</var> in <var>.a.d.</var> æqua <lb/>le eÅ¿t quadrato <var>.a.b.</var> productum autem <var>.a.c.</var> in <var>.d.c.</var> æquale eÅ¿t quadrato inÅ¿criptibili in <lb/>circulo <var>.g.c.b.</var> ex .130. primi Vitellionis, <reg norm="in" type="wordlist">Ä©</reg> qua propoÅ¿itione ipÅ¿e Vitellio Å¿upplet pro <lb/>eo, quod in quinta propoÅ¿itione libri de lineis Å¿pirabilibus Archimedis deÅ¿ideratur, <lb/>Å¿ed quadratum <var>.a.c.</var> æquale eÅ¿t ijs duobus productis. per .2. Å¿ecundi Eucli. ergo qua-<lb/>dratum <var>.a.c.</var> æquale erit quadrato inÅ¿criptibili in circulo <var>.d.c.g.</var> & quadrato <var>.a.b.</var> Å¿ed <lb/>quadratum lineæ <var>.a.c.</var> æquale eÅ¿t duobus quadratis, hoc eÅ¿t lineæ <var>.a.e.</var> & lineæ <var>.e.c.</var> ex <lb/>pitagorica, </s> - <s xml:space="preserve">quare ex communi conceptu duo quadrata lineæ <var>.a.e.</var> & lineÄ™ <var>.e.c.</var> hoc eÅ¿t <lb/>lineæ <var>.e.g.</var> quod idem eÅ¿t, æqualia erunt duobus iam dictis, hoc eÅ¿t inÅ¿criptibili, <lb/>& ei, quod fit ex <var>.a.b.</var> Å¿ed quadratum lineæ <var>.a.g.</var> æquale eÅ¿t quadrato lineæ <var>.a.e.</var> & qua <lb/>drato quod fit ex <var>.e.g.</var> & duplo illius quod fit ex <var>.a.e.</var> in <var>.e.g.</var> hoc eÅ¿t producto <var>.a.e.</var> in <lb/>diametrum. </s> - <s xml:space="preserve">Quare quadratum lineæ <var>.a.g.</var> æquale eÅ¿t quadrato circunÅ¿criptibili, & <lb/>quadrato lineæ <var>.a.b.</var> & producto lineæ <var>.a.e.</var> in diametrum circuli <var>.d.c.g</var>.</s> - </p> - <p> - <s xml:space="preserve">Breuiori etiam methodo demonÅ¿trare poſſu <lb/> - <anchor type="figure" xlink:label="fig-0263-01a" xlink:href="fig-0263-01"/> - mus quadrata lineæ <var>.a.e.</var> et <var>.e.g.</var> æqualia eÅ¿-<lb/>Å¿e quadrato circunÅ¿criptibili, & quadrato lineæ <var>.<lb/>a.b.</var> ducendo lineam <var>.e.b.</var> quæ æqualis eÅ¿t lineæ <var>.<lb/>e.g.</var> tali methodo, hoc eÅ¿t, conÅ¿iderando, quod <lb/>quadratum inÅ¿criptibile Å¿emper duplum eÅ¿t qua <lb/>drato Å¿emidiametri, vel medietati circumÅ¿cri-<lb/>ptibili, quod quidem nihil aliud eÅ¿t, niÅ¿i æquale <lb/>eſſe ijs duobus quadratis, hoc eÅ¿t lineæ <var>.e.b.</var> & li-<lb/>neæ <var>.e.g.</var> Å¿ed quadratum lineæ <var>.a.e.</var> æquale eÅ¿t iis <lb/>duobus quadratis, hoc eÅ¿t lineæ <var>.a.b.</var> & lineæ <var>.b.e.</var> vnde quadrat um lineæ <var>.a.e.</var> cum <lb/>quadrato lineæ <var>.e.g.</var> æquale eÅ¿t quadrato circunÅ¿criptibili, Å¿imul collecto cum qua-<lb/>drato lineæ <var>.a.b</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0263-01" xlink:href="fig-0263-01a"> - <image file="0263-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0263-01"/> - </figure> - </div> - <pb o="252" rhead="IO. BAPT. BENED." n="264" file="0264" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0264"/> - </div> - <div type="letter" level="4" n="5"> - <head style="it" xml:space="preserve">Demonstrationes quarundam propoÅ¿itionum de quibus agit <lb/>Cardanus capite primo libro .16. de <lb/>Å¿ubtilitate.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">EA quæ Cardanus in primo cap. lib. 16. de Å¿ubtilitate ita Å¿cribit, quod Å¿i diame-<lb/>tros producatur extra quantumlibet, alia verò diametro in centro Å¿ecetur ad <lb/>rectos, ex huius fine <reg norm="&c." type="unresolved">&c.</reg> quæ quidem Å¿ecundum illum eÅ¿t vndecima proprietas cir <lb/>culi, quoniam te id non intelligere Å¿cribis, <reg norm="idemque" type="simple">idemqÌ;</reg> dicis etiam de duodecima, & Å¿i-<lb/>militer de tribus illis paſſionibus, quas ipſæ communes facit circulo, defectioni, Å¿eu <lb/>ellipÅ¿i, & hyperboli, tibi breuiter reÅ¿pondebo.</s> - </p> - <p> - <s xml:space="preserve">Circa vndecimam proprietatem circuli verum dicit. </s> - <s xml:space="preserve">Imaginemur circulum <var>.p.<lb/>d.q.</var> à duabus diametris, inuicem ad angulos rectos coniunctis, diuiÅ¿um <var>.p.d.</var> et <var>.d.g.</var> di <lb/>uidatur enim quarta <var>.q.d.</var> per quot partes æquales volueris, mediantibus punctis <var>.b.a.<lb/>o.</var> <reg norm="ducanturque" type="simple">ducanturqÌ;</reg> ab ijÅ¿dem punctis tot perpendiculares diametro <var>.d.g.</var> quæ Å¿int <var>.b.m.a.n.</var> <lb/>et <var>.o.s.</var> quæ quidem erunt parallelæ diametro <var>.q.p.</var> coniungatur deinde extremitas <var>.d.</var> <lb/>diametri <var>.d.g.</var> cum primo puncto <var>.b.</var> & protrahatur <var>.d.b.</var> vÅ¿que ad concurÅ¿um cum diz <lb/>metro <var>.p.q.</var> protracto in puncto, h. </s> - <s xml:space="preserve">Nunc dico <var>.q.h.</var> quæ adiacet diametro <var>.q.p.</var> æqua-<lb/>lem eſſe omnibus dictis perpendicularibus, quapropter coniungantur puncta <var>.m.a</var>: <lb/><var>n.o.</var> et <var>.s.q.</var> & producantur vÅ¿que ad adiacentem diametro <var>.q.p.</var> in punctis <var>.c.</var> et <var>.e.</var> vn <lb/>de habebimus angulos <var>.b.a.o.q.</var> inuicem æquales ex .26. tertij, cum verò <var>.o.s.a.n.</var> et <lb/><var>b.m.</var> parallelæ Å¿int ipÅ¿i <var>.p.h</var>. </s> - <s xml:space="preserve">tunc anguli <var>.b.h.c</var>: <var>a.c.e</var>: et <var>.o.e.q.</var> æquales erunt angulis <var>.d.<lb/>b.m</var>: <var>m.a.n.</var> et <var>.n.o.s.</var> ex .29. primi: </s> - <s xml:space="preserve">quare anguli <var>.h.c.e.q.</var> erunt inuicem æquales, vnde <lb/>ex .28. eiuÅ¿dem <var>.b.h</var>: <var>m.c</var>: <var>n.e.</var> et <var>.s.q.</var> erunt <reg norm="inuicem" type="context">inuicẽ</reg> parallelÄ™, & ex .34. <var>e.q.</var> æqualis erit <var>.<lb/>o.s.</var> et <var>.e.c.</var> æqualis <var>.n.a.</var> et <var>.m.b.</var> æqualis <var>.c.h.</var> verum eÅ¿t igitur propoÅ¿itum.</s> - </p> - <p> - <s xml:space="preserve">Duodecima vero <reg norm="proprietas" type="simple">ê“prietas</reg> eÅ¿t, ut Å¿i fuerit circulus <var>.a.b.e.q.</var> cuius duo diametriad <lb/>rectos coniuncti Å¿int <var>.a.e.</var> et <var>.q.b.</var> & diameter <var>.a.e.</var> protractus indeterminatè ad partem <lb/>e. </s> - <s xml:space="preserve">tunc Å¿i ab extremo <var>.b.</var> diametri <var>.q.b.</var> ducta fuerit <var>.b.n.u.</var> extra circulum, Å¿eu <var>.b.u.n.</var> in <lb/>tra circulum, vt in Å¿ubiecta figura patet, ita vt Å¿ecta Å¿it à circunferentia circuli in <reg norm="pum" type="context">pÅ©</reg> <lb/>cto <var>.n.</var> vel à diametro in puncto <var>.u.</var> Å¿emper id quod fit ex <var>.u.b.</var> in <var>.b.n.</var> æquale erit qua-<lb/>drato inÅ¿criptibili in dicto circulo, hoc autem diuerÅ¿imodè cognoÅ¿ci poteÅ¿t, tribus <lb/>enim modis ego inueni, quorum primus ita Å¿e habet. </s> - <s xml:space="preserve">Nam Å¿i punctus <var>.u.</var> fuerit ex-<lb/>tra circulum, ducantur <var>.b.e.</var> et <var>.e.n.</var> & habebimus duos triangulos <var>.b.n.e.</var> et <var>.b.e.u.</var> Å¿imi <lb/>les inuicem, eo, quod angulus <var>.b.</var> communis ambobus exiÅ¿tit, & angulus <var>.b.n.e.</var> æqua <lb/>lis eÅ¿t angulo <var>.b.e.u.</var> quod ita probatur, nam angulus <var>.b.n.e.</var> cum angulo <var>.b.a.e.</var> (ducta <lb/>cum fuerit <var>.b.a.</var>) æquatur duobus rectis ex .21. tertij, Å¿ed ex quinta primi angulus <var>.b.<lb/>e.a.</var> Ä™qualis eÅ¿t angulo <var>.b.a.e</var>: </s> - <s xml:space="preserve">quare angulus <var>.b.n.e.</var> cum angulo <var>.b.e.a.</var> Ä™quatur duobus <lb/>rectis, Å¿ed ex .13. eiuÅ¿dem angulus <var>.b.n.e.</var> cum angulo etiam <var>.e.n.u.</var> æquatur duobus re <lb/>ctis, ergo angulus <var>.e.n.u.</var> æquatur angulo <var>.b.e.a</var>. </s> - <s xml:space="preserve">quare angulus <var>.b.n.e.</var> æquatur <reg norm="etiam" type="context">etiã</reg> an-<lb/>gulo <var>.b.e.u.</var> vnde ex .32. eiuÅ¿dem reliquus angulus <var>.b.u.e.</var> æqualis erit reliquo angulo <lb/><var>b.e.n.</var> latera igitur erunt proportionalia ex .4. Å¿exti, vnde ita Å¿e habebit <var>.u.b.</var> ad <var>.b.<lb/>e.</var> vt <var>.b.e.</var> ad <var>.b.n.</var> ex .16. Å¿exti igitur <reg norm="verum" type="context">verÅ©</reg> erit propoÅ¿itum.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i punctus <var>.u.</var> intra circulum fuerit, triangulus <var>.b.e.n.</var> Å¿imilis erit triangulo <var>.b.u.<lb/>e.</var> nam angulus <var>.b.</var> ambobus communis erit. </s> - <s xml:space="preserve">Angulus vero <var>.b.n.e.</var> Ä™qualis eÅ¿t angulo <var>.<lb/>b.e.u.</var> ex .26. tertij, </s> - <s xml:space="preserve">quare ex .32. primi reliquus angulus <var>.b.e.n.</var> æqualis erit reliquo - <pb o="253" rhead="EPISTOLAE." n="265" file="0265" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0265"/> - angulo <var>.b.u.e.</var> vnde ex .4. Å¿exti eadem proportio erit ipÅ¿ius <var>.b.n.</var> ad <var>.b.e.</var> quæ <var>.b.e.</var> ad <lb/><var>b.u</var>. </s> - <s xml:space="preserve">quare ex .16. eiuÅ¿dem patebit propoÅ¿itum.</s> - </p> - <p> - <s xml:space="preserve">Secundus autem modus ita Å¿e habet, ducta <var>.q.n.</var> habebimus duo triangula ortho-<lb/>gonia Å¿imilia inuicem <var>.b.q.n.</var> et <var>.b.u.o.</var> eo quod angulus <var>.b.</var> communis ambobus exi-<lb/>Å¿tit, </s> - <s xml:space="preserve">quare ex .4. Å¿exti ita Å¿e habebit <var>.u.b.</var> ad <var>.b.o.</var> vt <var>.q.b.</var> ad <var>.b.n.</var> vnde ex .15. eiuÅ¿dem <lb/>quod fit ex <var>.u.b.</var> in <var>.b.n.</var> æquale erit ei, quod fit ex <var>.q.b.</var> in <var>.b.o</var>. </s> - <s xml:space="preserve">Sed ex .16. eiuÅ¿dem, <reg norm="quod" type="simple">ê™</reg> <lb/>fit ex <var>.q.b.</var> in <var>.b.o.</var> Ä™quatur quadrato <var>.b.e.</var> quia <var>.b.e.</var> media proportionalis eÅ¿t inter dia <lb/>metrum & Å¿emidiametrum eiuÅ¿dem circuli. ex .4. eiuÅ¿dem, </s> - <s xml:space="preserve">quare quod fit ex <var>.u.b.</var> in <lb/><var>b.n.</var> æquale erit quadrato ipÅ¿ius <var>.b.e</var>.</s> - </p> - <p> - <s xml:space="preserve">Tertius modus adiungitur, & eÅ¿t quod cum quadratum <var>.u.b.</var> exiÅ¿tente <var>.u.</var> extra cir-<lb/>culum æquale Å¿it ei, quod Å¿it ex <var>.u.b.</var> in <var>.b.n.</var> Å¿imul Å¿umpto cum eo, <reg norm="quod" type="simple">ê™</reg> fit ex <var>.u.b.</var> in <var>.u.n.</var> <lb/>ex Å¿ecunda Å¿ecundi, & idem quadratum <var>.u.b.</var> æquale duobus quadratis <var>.u.o.</var> et <var>.o.b.</var> ex <lb/>penultima primi, ideo duo dicta producta æqualia erunt dictis duobus quadratis <var>.o.</var> <lb/> - <anchor type="figure" xlink:label="fig-0265-01a" xlink:href="fig-0265-01"/> - u. Å¿cilicet et <var>.o.b.</var> Å¿ed quadratum <lb/>o u. æquatur ei, quod fit ex <var>.a.u.</var> <lb/>in <var>.e.u.</var> & ei quod fit. ex <var>.o.e.</var> in Å¿e <lb/>ipÅ¿am ex .6. Å¿ecundi, </s> - <s xml:space="preserve">quare duo <lb/><reg norm="iam" type="context">iã</reg> dicta producta æqualia erunt <lb/>duobus dictis quadratis, <var>o.b.</var> Å¿ci <lb/>licet. et <var>.o.e.</var> & ei quod fit ex <var>.a.<lb/>u.</var> in <var>.u.e.</var> Å¿ed quod fit ex <var>b.u.</var> in <var>.u <lb/>n.</var> æquale eÅ¿t ei quod fit ex <var>.a.u.</var> <lb/>in <var>.u.e.</var> ex .35. 3. <reg norm="relinquitur" type="simple">relinquit̃</reg> ergo <lb/>vt id <reg norm="quod" type="wordlist">qÄ‘</reg> fit ex <var>.u.b.</var> in <var>.b.n.</var> æqua-<lb/>le Å¿it <reg norm="duobus" type="simple">duobê°</reg> quadratis <var>.o.b.</var> et <var>.o.<lb/>e</var>. </s> - <s xml:space="preserve">quare & quadrato ipÅ¿ius <var>.b.e.</var> <lb/>ex Pitagorica.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0265-01" xlink:href="fig-0265-01a"> - <image file="0265-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0265-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Siautem <reg norm="punctum" type="context context">pÅ©ctÅ©</reg> <var>.u.</var> fuiſſet intra <lb/>circulum idem eueniret. </s> - <s xml:space="preserve">Nam <lb/>quadrato <var>.b.e.</var> <reg norm="æquantur" type="context">æquãtur</reg> duo qua <lb/>drata <var>.o.b.</var> et <var>.o.e.</var> Å¿ed vice qua-<lb/>drati <var>.o.e.</var> dicemus <reg norm="quadratum" type="context">quadratÅ©</reg> <var>.o.<lb/>u.</var> cum eo quod fit ex <var>.a.u.</var> in <var>.u.e.</var> <lb/>ex .5. Å¿ecundi, id eÅ¿t quadratum <var>.<lb/>o.u.</var> <reg norm="cum" type="context">cÅ©</reg> eo quod fit ex <var>.b.u.</var> in <var>.u.<lb/>n.</var> ex .34. tertij, vnde quadratum <lb/><var>b.e.</var> æquale erit quadrato <var>.o.b.</var> <lb/>& quadrato <var>.o.u.</var> ideÅ¿t quadrato <lb/><var>b.u.</var> ex Pitagorica Å¿imul <reg norm="cum" type="context">cÅ©</reg> pro-<lb/>ducto <var>.b.u.</var> in <var>.u.n.</var> ideÅ¿t producto <lb/><var>n.b.</var> in <var>.b.u.</var> quod æquale eÅ¿t qua <lb/>drat <var>o.b.u.</var> cum producto <var>.b.u.</var> in <lb/><var>u.n.</var> ex .3. Å¿ecundi.</s> - </p> - <p> - <s xml:space="preserve">Circa tres paſſiones commu-<lb/>nes poÅ¿tea circulo hyperboli, & <lb/>defectioni notandum eÅ¿t <reg norm="primam" type="context">primã</reg> <lb/>patere ex .36: primi Pergei, Å¿e- - <pb o="254" rhead="IO. BABPT. BENED." n="266" file="0266" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0266"/> - cundam verò ex .37. et .38. eiuÅ¿dem, </s> - <s xml:space="preserve">propterea quod in .37. probat mediante maiori <lb/>diametro ipÅ¿ius hyperbolis & defectionis, In .38. autem mediante minori diametro <lb/>ordinatè ad maiorem.</s> - </p> - <p> - <s xml:space="preserve">Tertia autem paſſio, non niÅ¿i circulo conuenit; </s> - <s xml:space="preserve">pace ipÅ¿ius Cardani dictum Å¿it.</s> - </p> - <p> - <s xml:space="preserve">Quapropter Å¿it circulus <var>.q.o.b.</var> cuius diameter Å¿it <var>.q.b.</var> contingentes vero ab extre <lb/>mitate diametri Å¿int <var>.d.b.</var> et <var>.q.g.</var> per punctum autem <var>.o.</var> quoduis, ipÅ¿ius <reg norm="circunferentiæ" type="context">circÅ©ferentiæ</reg>, <lb/>tranÅ¿eant <var>.b.o.g.</var> et <var>.q.o.d</var>. </s> - <s xml:space="preserve">tunc dico productum <var>.q.o.</var> in <var>.q.d.</var> vel <var>.b.o.</var> in <var>.b.g.</var> Ä™quale eÅ¿-<lb/>Å¿e quadrato <var>.q.b.</var> quod ita probo.</s> - </p> - <p> - <s xml:space="preserve">Nam angulus <var>.q.b.d.</var> Å¿eu <var>.b.q.g.</var> rectus eÅ¿t ex .17. tertij Eucli. et <var>.b.o.q.</var> Å¿imiliter re-<lb/>ctus ex .30. ipÅ¿ius lib. angulus verò <var>.b.q.d.</var> Å¿eu <var>.q.b.g.</var> communis eÅ¿t. </s> - <s xml:space="preserve">quare <var>.b.q.</var> media <lb/>proportionalis erit inter dictas lineas <var>.q.d.</var> et <var>.q.o.</var> & inter <var>.b.g.</var> et <var>.b.o</var>. </s> - <s xml:space="preserve">Vnde Å¿equetur <lb/>propoÅ¿itum ex .16.6. Eucli.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i circa diametrum <var>.q.b.</var> mente fingamus aliquam elipÅ¿im, quætangat ipÅ¿um <lb/> - <anchor type="figure" xlink:label="fig-0266-01a" xlink:href="fig-0266-01"/> - circulum duobus punctis me-<lb/>diantibus <var>.q.</var> et <var>.b.</var> (nam pluribus <lb/>eſſet impoſſibile, ex .27. quarti <lb/>Pergei) clarè patebit, quod <reg norm="pum" type="context">pÅ©</reg> <lb/>ctus <var>.o.</var> erit extra <reg norm="circunferentiam" type="context">circunferentiã</reg> <lb/>ipÅ¿ius defectionis, </s> - <s xml:space="preserve">quare ipÅ¿a cir <lb/>cunferentia Å¿ecabit <var>.b.g.</var> vel <var>.q.<lb/>d.</var> in alio puncto, vnde ipÅ¿i non <lb/>occurret id quod probauimus <lb/>de circulo.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0266-01" xlink:href="fig-0266-01a"> - <image file="0266-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0266-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Admiratus etiam Å¿um, ipÅ¿um <lb/>Cardanum dicere hyperbolem <lb/>ita vocari, eo quod angulus con <lb/>tentus ab axe ipÅ¿ius figuræ, & à <lb/>latere trigoni in hyperbole ma-<lb/>ior Å¿it quam in parabole, quod <lb/>eriam confirmat paulo inferius, <lb/>nam hoc verum non eÅ¿t, imo fal <lb/>Å¿iſſimum. </s> - <s xml:space="preserve">Talis enim Å¿ectio ita <lb/>nominata fuit, hoc eÅ¿t hyperbo <lb/>les, Å¿imili ratione, qua elipÅ¿is Å¿eu <lb/>defectio etiam vocata fuit, nam <lb/>Å¿icut in ipÅ¿a defectione quadra-<lb/>tum ordinatÄ™ <var>.l.m.</var> minor eÅ¿t pro <lb/>ducto lineæ <var>.e.m.</var> in <var>.e.t.</var> per figu <lb/>ram Å¿imilcm producto <var>.d.e.</var> in <var>.e.<lb/>t.</var> quæ eandem obtineat <reg norm="altitu- dinem" type="context">altitu-<lb/>dinẽ</reg> ipÅ¿ius <var>.e.m.</var> vt ipÅ¿e Pergeus <lb/>monÅ¿trat in .13. primi lib. ita in <lb/>hyperbole <reg norm="dictum" type="context">dictÅ©</reg> quadratum ex <lb/>cedit quantitatem illius figuræ, <lb/>per Å¿imilem dictæ vt in .12. <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <lb/>Pergei facilè videre eÅ¿t. </s> - <s xml:space="preserve">Å¿ed <reg norm="prae ter" type="simple">prÄ™Â<lb/>ter</reg> illas paſſiones, quas notat - <pb o="255" rhead="EPISTOL AE." n="267" file="0267" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0267"/> - Cardanus in Å¿upradicto capite, multæ aliæ Å¿unt, cum corollarium primæ tertij Eu-<lb/>cli. Å¿it paſſio propria ipÅ¿ius circuli, & idem dico de propoÅ¿itione .3. 4. 7. 8. 9. 11. 12. <lb/>13. 14. 15. 17. 18. 19. 20. 30. 31. ipÅ¿ius tertij lib. nec non de .8. 9. ct .10. tertijdecimi, & <lb/>de prima .3. 4. 5. 6. et .7. quartidecimi eiuÅ¿dem. </s> - <s xml:space="preserve">Idem infero de ea quod Å¿crip Å¿i Ma <lb/>rio Nizzolio, FranciÅ¿co Vimercato, FranciÅ¿co Contareno, Angelo AgrimenÅ¿ori, & <lb/>de alijs nonnullis à me excogitatis.</s> - </p> - </div> - </div> - <div type="section" level="3" n="6"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE FINE CORPORVM COELESTIVM, <lb/>& eorum motu.</head> - <head style="it" xml:space="preserve">IlluÅ¿tri vire, Philiberto Pingonio Sabaudo CuÅ¿iacenÅ¿ium <lb/>Baroni.</head> - <p> - <s xml:space="preserve">CVm antea meo nomine SebaÅ¿tianus noÅ¿ter omnia ferè tibi retuliſſet, inter alia, <lb/>quæ relin quebantur tibi <reg norm="dicenda" type="context">dicẽda</reg>, hoc vnum erat, quod Å¿i abÅ¿que lumine Å¿upe-<lb/>riori, in quem finem facta fuerint corpora cÅ“leÅ¿tia Å¿cire deÅ¿ideras, & humanam ra-<lb/>tionem Å¿equi volueris, putandum tibi non erit ea Å¿olum effecta eſſe, vt tam vile cor <lb/>pus, vt eÅ¿t terra aquis irrigata, animalia, & plantas regant, cum ea corpora Å¿int diuina, <lb/>in numero incomprÄ™henÅ¿ibilia, maximis ma gnitudinibus, & motibus velocisfimis, <lb/>prædita, id etiam minus putabunt hij, qui opinionem AriÅ¿tarchi Samij, & Nicolai <lb/>Copernici Å¿equuntur, quorum ratione fieri non poteÅ¿t, vt credant, eius, quod ex vni <lb/>uerÅ¿o reliquum eÅ¿t, alium finem non habere, quam regimen huius centri epicycli Lu <lb/>naris, vt illorum more loquar. </s> - <s xml:space="preserve"><reg norm="Quam" type="context">Quã</reg> enim turpe eſſet Å¿i centra aliorum epicyclorum <lb/>planetarum tali regimine priuarentur, id quod nullo modo cum ratione conÅ¿entit, <lb/>Å¿i tam vera eÅ¿t ea opinio, quemadmodum <reg norm="rationabiliorem" type="context">rationabiliorẽ</reg> eam <reg norm="exiÅ¿timant" type="context">exiÅ¿timãt</reg>. </s> - <s xml:space="preserve"><reg norm="Neque" type="simple">Neq;</reg> quid <lb/>quam valet opinio AriÅ¿totelis, qui corpora cÅ“leÅ¿tia, ab ortu, & interitu libera eſſe <lb/>fentit. </s> - <s xml:space="preserve">dicens Å¿uperioribus fÄ™culis, à noÅ¿tris antiquis nullam vnquam animaduerÅ¿am <lb/>fuiſſe alterationem in cÅ“lo, cum non videat Å¿i quis eſſet in cÅ“lo, <reg norm="neque" type="simple">neq;</reg> etiam obÅ¿erua <lb/>re poſſet alterationes quæ in terra, & circa terram fiunt, quæ in partibus, & non in to <lb/>to Å¿pectantur: </s> - <s xml:space="preserve">vnde etiam fieri poteÅ¿t, vt in cÅ“lo Å¿int particulares alterationes, <lb/>quæ à nobis tamen, qui ab illis longè diÅ¿tamus, non comprÄ™hendantur, terra, <reg norm="mareque" type="simple">mareqÌ;</reg> <lb/>(quamuis minimum reÅ¿pectu ipÅ¿ius terræ) ratione totius ita Å¿e Å¿emper <reg norm="habuerunt" type="context">habuerÅ©t</reg> <reg norm="quem" type="context">quẽ</reg> <lb/>admodum Å¿eÅ¿e habere corpora cÅ“leÅ¿tia videmus, Å¿ed alteratio, ratione tantum ali-<lb/>quarum minimarum partium quaÅ¿i inÅ¿enÅ¿ibilium, Å¿i cum toto comparentur fit. </s> - <s xml:space="preserve">Quis <lb/>enim Å¿cit, vt iam tibi dixi, quin, quemadmodum Luna circa terram voluitur, <reg norm="ipÅ¿aque" type="simple">ipÅ¿aqÌ;</reg> <lb/>terra Å¿it veluti centrum epicycli maioris eiuÅ¿dem, vt AriÅ¿tarchus Samius, & Nico-<lb/>laus Copernicus cenÅ¿uerunt; </s> - <s xml:space="preserve">Å¿ic etiam Saturnus, Iupiter, Mars, Venus, <reg norm="atque" type="simple">atq;</reg> Mercu <lb/>rius circa alia huiuÅ¿modi corpora, huic terræ Å¿imilia, in orbem agantur, quaÅ¿i Å¿pecu-<lb/>la, lumen Solis Å¿uo centro ex reflexione, deferentia (fuppoÅ¿ita dico vera illorum opi <lb/>nione) <reg norm="Nollem" type="context">Nollẽ</reg> tamen tibi è mente excidere, vt alià s te monui, <reg norm="quod" type="simple">ê™</reg> Å¿i communis opinio <lb/>vera eÅ¿t, neceſſario <reg norm="fatendum" type="context">fatendÅ©</reg> Å¿it corpus Å¿olare, <reg norm="dum" type="context">dÅ©</reg> in æquatore reperitur moru diurno <lb/>quolibet horÄ™ minuto, magis <reg norm="quam" type="context">quã</reg> <reg norm="decem" type="context">decẽ</reg> & <reg norm="Å¿eptem" type="context">Å¿eptẽ</reg> mille milliaria <reg norm="peragere" type="simple">ê‘agere</reg>, ideÅ¿t paulo mi <lb/>nus quam .18000. milliaria, Saturnum verò cum Å¿imiliter eÅ¿t in æquatore, eodem <reg norm="tem- poris" type="context">tẽ-<lb/>poris</reg> Å¿patio, quaÅ¿i tercentamille milliaria Italica conficere, & Å¿ic per gradus alia cor <lb/>pora velociora alijs moueri; </s> - <s xml:space="preserve">quæ quidem omnia, <reg norm="cum" type="context">cÅ©</reg> Å¿implici gyro terræ circa Å¿uum - <pb o="256" rhead="IO. BABPT. BENED." n="268" file="0268" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0268"/> - axem (vt dicunt) tolluntur, quod Å¿ufficit ad recipiendum lumen, & influentias illo-<lb/>rum corporum. </s> - <s xml:space="preserve">Et ita, veluti princeps corporum vniuerÅ¿i, intra vnum an-<lb/>num circa eam vertitur. </s> - <s xml:space="preserve">Ita etiam Å¿uſſiceret, vt ipÅ¿a terra circa dictum diuinum <lb/>corpus Å¿olare, interÅ¿ecando axem diurnum cum axe annuali (cum ab eo lumen, ca-<lb/>lorem, & influentiam Å¿uÅ¿cipere debeat) <reg norm="circunuolueretur" type="simple">circunuolueret̃</reg>. </s> - <s xml:space="preserve">Rationes autem a Ptolo-<lb/>meo in contrarium adductæ apud ipÅ¿os, nullÄ™ Å¿unt, quia quÄ™libet pars (vt inquiunt) <lb/>retinet naturam totius, præterquam <reg norm="quod" type="simple">ê™</reg> aer, & aqua, quæ ipÅ¿am terram <reg norm="circundant" type="context">circundãt</reg>, pla <lb/>nè eundem naturalem impetum motus obtineant, quitanto lentior eÅ¿t, quanto lon <lb/>gius diÅ¿tat aer ab ipÅ¿a terra, Å¿ecundum etiam talem opinionem, nulla neceſſitas fo-<lb/>ret, vt locus fixarum terminaretur aliquibus Å¿uperficiebus, conuexa Å¿cilicet, & de-<lb/>uexa.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">De Luce, Lumine, & Colore, De obiectuoculi, De lumine <lb/>Luna, & Rubedine nubium.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">QVod proximè quærebas, an Å¿it lux aliqua, quæ à corpore lucido non proue-<lb/>niat, mihi facilè ad conÅ¿iderandum videtur. </s> - <s xml:space="preserve">hic enim oportet, vt nos ad id <lb/>quod perpetuò videmus referamus, exiÅ¿timo autem te velle dicere lumen, non lu-<lb/>cem, quia propriè lux, qualitas ea viÅ¿ibilis appellatur, quæ eÅ¿t in corpore lucido, à <lb/>quo quidem corpore lumen effunditur; </s> - <s xml:space="preserve">lumen verò, ea qualitas eſſe dicitur, quæ ex <lb/>tra ipÅ¿um corpus reperitur, à luce, quæ in dicto corpore manet emanans. </s> - <s xml:space="preserve">vnde pa-<lb/>tet, nullam lucem abÅ¿que corpore Å¿ubiecto eſſe poſſe, id quod cum fieri quîret, <reg norm="idem" type="context">idẽ</reg> <lb/>de quolibet alio accidente dici poſſet, id eÅ¿t quod ex Å¿e, & abÅ¿que aliquo Å¿ubiecto <lb/>Å¿ubÅ¿iÅ¿teret.</s> - </p> - <p> - <s xml:space="preserve">Lumen deinde à luce proficiÅ¿ci patet, <reg norm="quod" type="simple">ê™</reg> penetrat diaphanum, neque aliquo mo-<lb/>do Å¿uum actum oÅ¿tendit, niÅ¿i, aut per incidentiam, aut ratione opaci, ex reflexione, <lb/>cuius Å¿uperficiei colorem induit. </s> - <s xml:space="preserve">Atque hæc eÅ¿t cauÅ¿a, vt inter crepuÅ¿culum matu-<lb/>tinum, aut veÅ¿pertinum, nox etiam Å¿i Å¿it Å¿erena, adeo obÅ¿cura nobis appareat, quam-<lb/>uis totum vniuerÅ¿um diaphanum, extra conum vmbræ, quæ ex terra prouenit Å¿it vn <lb/><reg norm="dique" type="simple">diq;</reg> radijs luminoÅ¿is Solis colluÅ¿tratum; </s> - <s xml:space="preserve">qui quidem radij, non niÅ¿i à ſuamet reflexio <lb/>ne à Luna, & ab alijs Å¿tellis (vt corporibus opacis, quæ reÅ¿iÅ¿tunt lumini, ne vlterius <lb/>penetrare poſſit, vnde retrò redit) comprehenduntur.</s> - </p> - <p> - <s xml:space="preserve">Ais etiam propria viÅ¿us obiecta plura eſſe, nominans pro vno, colorem, & lucem <lb/>pro alio. </s> - <s xml:space="preserve">Ego autem reÅ¿pondeo, obiectum oculi eſſe vnicum tantum, ideÅ¿t lumen. <lb/></s> - <s xml:space="preserve">Quod ad lucem Å¿pectat, iam tibi dixi, eam eſſe quandam qualitatem in corpore luci <lb/>do, & non extra ipÅ¿um poÅ¿itam, à quo quidem corpore, cum non exeat, oculi obie-<lb/>ctum eſſe nequit, Å¿ed lumen quidem ab ipÅ¿a luce productum. </s> - <s xml:space="preserve">Color etiam, qui eÅ¿t in <lb/>corpore colorato, obiectum oculi eſſe non poteÅ¿t, cum dictum corpus non deÅ¿erat, <lb/>Å¿ed lumen quidem ab eodem corpore reflexum, & huiuÅ¿modi corporis colore tin-<lb/>ctum: </s> - <s xml:space="preserve">vnde tam lumen incidens, quam reflexum colore eÅ¿t Å¿emper imbutum.</s> - </p> - <p> - <s xml:space="preserve">Illud quidem coloratum eÅ¿t qualitate lucis corporis lucidi, a ut medij, per quod <lb/>tranÅ¿it, Å¿ed hoc colore corporis, à quo reflectitur.</s> - </p> - <p> - <s xml:space="preserve">Neque etiam te ignorare volo, lumina reflexa colorata, non reflecti à puris pro-<lb/><reg norm="priisque" type="simple">priisqÌ;</reg> Å¿uperficiebus corporum coloratorum, eo <reg norm="quod" type="wordlist">qÄ‘</reg> pauca corpora tam opaca repe-<lb/>riuntur, ut immediatè lumen à ſuperficie propriè <reg norm="reflectant" type="context">reflectãt</reg>, Å¿ed lumen penetrat alim - <pb o="259" rhead="EPISTOL AE." n="271" file="0271" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0271"/> - quantulum dicta corpora, & ita illorum colore afficitur, vbi verò non penetrat, non <lb/>coloratur colore corporis illius.</s> - </p> - <p> - <s xml:space="preserve">Sed vt ad propoÅ¿itum redeamus, dico lum en tantum eſſe viÅ¿us obiectum, quod Å¿i <lb/>colore eÅ¿t imbutum, aut tale eÅ¿t ratione color ris lucis, quæ eum mittit, aut ratione me <lb/>dij per quem tranÅ¿it, aut ratione corporis, vnde reflectitur, etÅ¿i Å¿uperficies corporis <lb/>vnde lumen reflectitur eſſet omnino priuata colore, Å¿ub aÅ¿pectum non caderet, vt <lb/>etiam cum huiuÅ¿modi Å¿uperficies læuigata, & polita eÅ¿t Å¿ecundum <reg norm="continuitatem" type="context">continuitatẽ</reg> Å¿ua-<lb/>rum partium, videlicet, vt Å¿peculi radio tamen non profundante, & ideo perfectiffi <lb/>morum quorundam Å¿peculorum Å¿uperficies non cernuntur, Å¿ed lumen tantum re-<lb/>flexum, colore alicuius alterius Å¿uperficiei, aut à luce, corporis lucidi, aut à me-<lb/>dio per quem tranÅ¿it, conÅ¿picitur. </s> - <s xml:space="preserve">Ego verò non aſſero colorem non eſſe quid di-<lb/>uerÅ¿um à lumine, Å¿ed imagineris lumen eſſe veluti animam, aut Å¿ubÅ¿tantiam & colo <lb/>rem corporis formam accidentalem, cum nullum lumen à ſenÅ¿u viÅ¿us percipi poÅ¿-<lb/>Å¿it, quod aliquo modo colore non Å¿it imbutum: </s> - <s xml:space="preserve">& eundem reÅ¿pectum quem Å¿onus <lb/>ad auditum, lumen ad oculum habet, quia vt Å¿onus Å¿ecundum eam velocitatem, quæ <lb/>à motione aeris, aut aquæ, ex colliÅ¿ione <reg norm="aliorum" type="context">aliorÅ©</reg> corporum producitur ad euitan dum <lb/>vacuum, a cutus, vel grauis Å¿entitur, ita lumen originem ducens à corpore lucido per <lb/>medium diaphanum aeris, aut aquæ, aut alterius huiuÅ¿modi corporis ad oculum tran <lb/>Å¿it colorem lucis, aut medij per quod tranÅ¿it, aut vnde reflectitur induit.</s> - </p> - <p> - <s xml:space="preserve">Quod verò Luna nullum ex Å¿e habeat lumen, Å¿ufficiens inditium eÅ¿t nos ipÅ¿am <lb/>tantò magis obÅ¿curam videre, quantò magis in cono vmbræ terræ immergitur, & <lb/>Å¿i eo tempore ipÅ¿am videmus rubeo colore affectam, hoc enim accidit, quia radij Å¿o <lb/>lares vndequaque refranguntur à vaporibus ipÅ¿am terram circundantibus, quæ qui-<lb/>dem refractio fit verÅ¿us axem coni vmbræ terræ, </s> - <s xml:space="preserve">& propterea vmbra dicti coni non <lb/>eÅ¿t æqualiter obÅ¿cura, Å¿eu tenebroÅ¿a, circa vero <reg norm="axem" type="context">axẽ</reg> ipÅ¿ius coni, magis quam circa eius <lb/><reg norm="circunferentiam" type="context context">circÅ©ferentiã</reg>, obÅ¿cura <reg norm="verum" type="wordlist">vr̃</reg>, & quia corpus lunare tale eÅ¿t, vt facillimè recipiat <reg norm="qualecum" type="context">qualecÅ©</reg> <lb/>que lumen, quod etiam manifeÅ¿tè videtur dum ipÅ¿a Luna reperitur Å¿ecundum lon-<lb/>gitudinem inter Solem, & Venerem, quod pars Lunæ lumine Solis deÅ¿tituta, à lumi <lb/>ne Veneris aliquantulum illuÅ¿tratur, quod ego ſæpè vidi, & multis oÅ¿tendi. </s> - <s xml:space="preserve">Propte-<lb/>rea dum ipÅ¿a Luna in cono vmbræ terræ reperitur adhuc videtur. </s> - <s xml:space="preserve">Rubedo etiam il-<lb/>la nubium poÅ¿t Solis occaÅ¿um, vel ante ortum, aliunde non prouenit, niÅ¿i à qualitate <lb/><reg norm="vaporum" type="context">vaporÅ©</reg>, per quos Å¿olares radij tranÅ¿eunt, à quibus vaporibus, tali colore ipÅ¿i radij <lb/>afficiuntur, eomet modo quo radius, cuiuÅ¿uis corporis lucidit, <reg norm="tranÅ¿iens" type="context">trãſiens</reg> per vitrum, Å¿eu <lb/>aliud diaphanum coloratum.</s> - </p> - <pb o="258" rhead="IO. BABPT. BENED." n="270" file="0270" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0270"/> - </div> - </div> - <div type="section" level="3" n="7"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE ICTV BOMBARDAE SECVNDVM <lb/>diuerÅ¿as eleuationes. Et de quibuÅ¿dam erroribus Nico-<lb/>lai Tartaleæ, circa idem.</head> - <head style="it" xml:space="preserve">Fllustri D. IoÅ¿epho Cambiano ex Ruffia Dominis, aquiti <lb/>Å¿trenuo, & tormentis bellicis Serenißimi Ducis <lb/>Sabaudia Prafecto.</head> - <p> - <s xml:space="preserve">EXcogitaui quÄ™dam dum ocio frui licuit per abÅ¿entiam Ducis Sereniſſimi, <lb/>quæ ad te Å¿cribere placuit, vt Å¿i probaueris in lucem quandoque profer-<lb/>re non dubitem, Å¿i deÅ¿pexeris, ocius Å¿upprimam, Å¿unt autem huiuÅ¿modi.</s> - </p> - <p> - <s xml:space="preserve">Vnde fiat vt tormentum bellicum vehementi<unsure/> feriat ictu Å¿uperius delato <lb/>quam orizontali, vt Tartalea Å¿cribit, quæſito Å¿ecundo libr. primi quæſitorum, à ne-<lb/>mine adhuc (quod Å¿ciam) traditum eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Rationes verò Tartaleæ nullius Å¿unt momenti, quia Å¿i validæ eſſent, Å¿equeretur <lb/>vt inclinata bombarda, adeo vt angulus Å¿ub orizonte factus æqualis eſſet ei, qui Å¿u <lb/>pra orizontem eÅ¿t, ictum bombardÄ™ in vtroque huiuÅ¿modi Å¿itu eundem eſſe <reg norm="futurum" type="context">futurÅ©</reg>. <lb/></s> - <s xml:space="preserve">& Å¿i aliqua differentia oriretur ratione granitatis pilæ ab ipÅ¿a bombarda emiſſæ, hoc <lb/>fieret, vt Å¿cilicet velocior eſſet in motu inclinato quam in eleuato cum pondus, mo-<lb/>tui adeo non opponatur. </s> - <s xml:space="preserve">Id quod non ita se habet, vera enim cauÅ¿a vnde fiat, vt bom <lb/>barda eleuata vehementius feriat, quà m ea quæ eÅ¿t minus alta, eadem eÅ¿t ferè, in ge-<lb/>nere, cum ea, qua aliquod corpus materia magis denÅ¿a, Å¿ed Å¿imile & Ä™quale alteri cor <lb/>pori materiæ minus denſæ velocius mouetur ab vna eademqÌue, aut æquali vi <lb/>compulÅ¿um. </s> - <s xml:space="preserve">EÅ¿t eadem etiam in Å¿pecie ei, qua maiorem effectum producit <lb/>puluis, qui in locis Å¿ubterraneis ponitur quum vaÅ¿is optimè colligatis ferro in-<lb/>cluditur. </s> - <s xml:space="preserve">EÅ¿t etiam Å¿imilis ei, qua longius impellitur pila, qua ludimus, ab ali-<lb/>quo inÅ¿trumento ligneo, quando percutitur contra, quam cum Å¿ecundum Å¿uum mo-<lb/>tum proijcitur. </s> - <s xml:space="preserve">Id quod inde fit, quia virtus mouens maiori vi, & intenÅ¿iori huiuÅ¿-<lb/>modi corpus percutit, quia corpus quod moueri debet, quanto magis reÅ¿iÅ¿tit virtu-<lb/>ti mouenti (certum tamen terminum præſcribendo) in exiguo eo temporis Å¿patio, <lb/>tanto maiorem virtutem colligit, quæ ipÅ¿um deinde tanto cum impetu mo-<lb/>uet, & tanto magis impellens concomitatur, vt maiorem effectum efficiat, quam Å¿i <lb/>ad mouendum Å¿eÅ¿e facilè reddidiſſet. </s> - <s xml:space="preserve">Atque hoc Å¿upradictis ictibus eleuatis acci-<lb/>dit, quia grauitas pilæ, ea eÅ¿t quæ reÅ¿iÅ¿tens virtuti mouenti, dat ei commoditatem <lb/>colligendi dictam virtutem, multo magis quam eſſet ea, quæ ad depreſſiorem eleua <lb/>uationem eam <reg norm="immpelleret" type="context">Ä©mpelleret</reg>. </s> - <s xml:space="preserve">Et quia huiuÅ¿modi multiplicatio virtutis, nullam propor <lb/>tionem cum pondere pilæ gerit, volo inferre quod dum colligitur tanta virtus, col-<lb/>ligitur multo plus eo, quod ad impellendam dictam pilam Å¿ufficeret, ratione magnæ <lb/>velocitatis augumenti, quia quanto plus temporis ei conceditur ad commutandam <lb/>puluerem in ignem, tanto maior quantitasignis progignitur, vnde fit, vt tanto ma-<lb/>iori loco indigeat, quamobrem tanto magis impellit, Å¿ed vt dixi, tanta cum veloci-<lb/>tate <reg norm="adaugetur" type="simple">adauget̃</reg>, vt huiuÅ¿modi virtus longè Å¿uperet <reg norm="reÅ¿iÅ¿tentiam" type="context context">reÅ¿iÅ¿tẽtiã</reg> <reg norm="ponderis" type="context">põderis</reg> pilæ, & Å¿ic eÅ¿t cau <lb/>fa, ut effectus, quod <reg norm="experientia" type="context">experiẽtia</reg> innot eÅ¿cit <reg norm="producatur" type="simple">producat̃</reg>. </s> - <s xml:space="preserve">Sed ea ratio, qua Å¿eÅ¿e <reg norm="idem" type="context">idẽ</reg> author <lb/>in tertio quæſito ad aliquod impoſſibile, circa iter ipÅ¿ius pilæ Legatum HiÅ¿panum - <pb o="259" rhead="EPISTOL AE." n="271" file="0271" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0271"/> - reducere putat, nullo fundamento nititur, quia non eÅ¿t Å¿emper dicendum, quod <reg norm="quam" type="context">quã</reg> <lb/>to velocior Å¿it quædam pila, tanto rectius moueatur, quia ei dici poſſet, vÅ¿que ad cer <lb/>tum quendam terminum velocitatis, per tantum Å¿patij eam aptam eſſe, vt recta per <lb/>fectè moueatur, Å¿ed Å¿i velocius iret, non tamen futurum, vt per idem Å¿patium re-<lb/>ctius moueretur, Å¿ed quod per longius Å¿patium recta motum perageret, & Å¿ic nihil <lb/>haberet quod replicaret, præter quam quod ipÅ¿e Å¿upponit id quod in 18. quæſito <lb/>negat, in quo ait pilam uicinam orificio, non adeo uelocem eſſe, quam cum aliquan-<lb/>tulum ab eodem eÅ¿t rem ota, ratione reÅ¿iÅ¿tentiæ Å¿ui cyllindriaerei. </s> - <s xml:space="preserve">Sed quod pila, <lb/>recta eat quanto altior, aut depreſſior <reg norm="bombarda" type="context">bõbarda</reg> erit, <reg norm="ion" type="context">iõ</reg> fit, quia linea inclinationis na <lb/>turalis cum linea inclinationis uiolentæ angulum rectum non facit, </s> - <s xml:space="preserve">unde quanto lon <lb/>gius diÅ¿tat à recto huiuÅ¿modi <reg norm="angulus" type="simple">angulê°</reg>, Å¿iue Å¿it acutus Å¿iue obtuÅ¿us, tanto minorem uim <lb/>habet, eodem planè ferè modo quem tertio capite mei tractatus de rebus mechani <lb/>cis deÅ¿cripÅ¿i. </s> - <s xml:space="preserve">Quia in ictibus eleuatis, iter inclinationis violentæ ipÅ¿ius pilæ verÅ¿us <lb/>terminum ad quem, incipiendo à loco ipÅ¿ius pilæ cum itinere inclinationis natura-<lb/>lis, angulum obtuÅ¿um, & in ictibus inclinatis acutum conÅ¿tituit. </s> - <s xml:space="preserve">Neque etiam hic <lb/>prætermittam notatu dignum errorem, quem Tartalea eodem loco committit, <reg norm="cum" type="context">cÅ©</reg> <lb/>putet indifferenter aliquod corpus impellere, aut percutere maiori <reg norm="cum" type="context">cÅ©</reg> impetu cum <lb/>eÅ¿t in itinere recto. </s> - <s xml:space="preserve">Quia Å¿equeretur quod aliquod corpus graue perpendiculari-<lb/>ter Å¿urÅ¿um verÅ¿us proiectum, in qualibet parte Å¿ui itineris, Å¿emper fortius percute-<lb/>ret, quam in qualibet parte itineris alterius cuiuÅ¿uis eleuationis obliquæ, quod <reg norm="quam" type="context">quã</reg> <lb/>Å¿it falÅ¿um, tibi conÅ¿iderandum relinquo.</s> - </p> - <p> - <s xml:space="preserve">EÅ¿t etiam falÅ¿a ea ratio, quam in quarto quæſito idem adducit, quia aer in motu <lb/>non tantum durat, quantum ipÅ¿e putat, imò huiuÅ¿modi violenta agitatio, citò ceſſat <lb/>& citius etiam, quam Å¿i extra aliquam bombardam cum tanta violentia impulliſſet <lb/>Å¿aceum plumis plenum.</s> - </p> - <p> - <s xml:space="preserve">Ratio etiam quam in .18. quæſito de eo, quod pila pertran Å¿eat illud corpus cyllin <lb/>dricum aereum adducit, eÅ¿t planè vana, quia Å¿tatim aer, qui prius in <reg norm="bombarda" type="context">bõbarda</reg> erat <lb/>incluÅ¿us, extra ipÅ¿am <reg norm="erumpit" type="context">erÅ©pit</reg>, cedit, à <reg norm="pilaque" type="simple">pilaqÌ;</reg> diuiditur, vt Å¿i nunquam eam figuram in-<lb/>duiſſet, neque aer ambiens ei reÅ¿iÅ¿tit. </s> - <s xml:space="preserve">Sed quod velocior Å¿it in certa quadam diÅ¿tan <lb/>tia, quam in principio erat, Å¿i hoc <reg norm="verum" type="context">verÅ©</reg> eſſet, ab alia cauÅ¿a dependeret, quæ partim Å¿i <lb/>milis eſſet ei, quæ efficit, vt corpora in motibus naturalibus, cum longius diÅ¿tant à ter <lb/>mino vnde naturaliter Å¿eÅ¿e mouerunt, Å¿int velociora, quia per aliquod Å¿patium hu-<lb/>iuÅ¿modi corpus moueretur quemadmodum motu naturali cietur.</s> - </p> - <p> - <s xml:space="preserve">Ratio autem eius quare pila, aut globus bombardæ Å¿ibiletab eodem in Å¿eptimo <lb/>quæſito nil valet, quia hoc fit cum pila aliquam paruam concauitatem habet.</s> - </p> - <p> - <s xml:space="preserve">In .27. autem quæſito ait, quod retrotrahendo Å¿ignum, ictus altius tenderet, quod <lb/>poteÅ¿t etiam eſſe falÅ¿um, cum hocnon Å¿it neceſſarium, quia pila dum deÅ¿cendit, for-<lb/>taſſe tangeret Å¿copum.</s> - </p> - <pb o="260" rhead="IO. BAPT. BENED." n="272" file="0272" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0272"/> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">Deerroribus Ioannis Stadij.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">FIguram quam ponit Ioannes Stadius pag .147. in lib. Å¿uarum tabularum Prute-<lb/>nicarum, à Nicolao Copernico Å¿umpÅ¿it pag .64. à tergo in libr. reuolutionum <lb/>cÅ“leÅ¿tium, Å¿ed ipÅ¿e Stadius eam non intellexit, omitto, quod mutauerit characte-<lb/>res ipÅ¿ius figurÄ™, vt illa Å¿ua videatur, quod nihil refert, alterat etiam <reg norm="demonÅ¿trationem" type="context">demonÅ¿trationẽ</reg>, <lb/>Å¿ed ipÅ¿um putare <var>.i.K.</var> perpendicularem à centro circuli Å¿emper dependere, eÅ¿t intol <lb/>lerabilis error; </s> - <s xml:space="preserve">nec vnquam verificatur hoc, niÅ¿i quando punctum <var>.K.</var> interÅ¿ectionis <lb/>diametrorum parallelorum, forte reperitur in axe mundi. </s> - <s xml:space="preserve">Reliqua verò Å¿uæ demon <lb/>Å¿trationis, Å¿i non intelligis, minimè miror, eo quod ipÅ¿emet Stadius Å¿eipÅ¿um confun <lb/>dit. </s> - <s xml:space="preserve">Veram autem demonÅ¿tratio nem huiuÅ¿modi figuræ in dicto libr. Copernici cla-<lb/>rè videbis. </s> - <s xml:space="preserve">Quod verò diuersè cogitaui nunc acciptito.</s> - </p> - <p> - <s xml:space="preserve">Cum nobis cognita Å¿it maxima ecclipticæ declinatio, vt puta <var>.a.c.</var> Å¿i latitudo <reg norm="etiam" type="context">etiã</reg> <lb/>Å¿tellæ nobis data fuerit, vt puta <var>.c.e.</var> cognitus nobis erit totalis arcus <var>.a.e.</var> & eius Å¿inus <var>.<lb/>e.m.</var> & quia notus etiam nobis eÅ¿t Å¿inus arcus <var>.a.c.</var> hoc eÅ¿t <var>.c.n.</var> & corda <var>.e.f.</var> medio eius <lb/>arcus <var>.e.p.f.</var> minoris media circunferentia, per duplum latitudinis datæ, vnde <var>.e.l.</var> eius <lb/>dimidium nobis cognitum erit, vel vt Å¿inus arcus <var>.e.p.</var> cognitus etiam nobis eÅ¿t Å¿inus <var>.<lb/>q.g.</var> declinationis <var>.a.g.</var> datæ, cui æqualis eÅ¿t <var>.m.t.</var> ex .34. primi Euclid. </s> - <s xml:space="preserve">vnde <var>.e.t.</var> nobis <lb/>cognita remanet, cum verò duo trianguli <var>.i.c.n.</var> et <var>.t.e.K.</var> <reg norm="æquianguli" type="context">æquiãguli</reg> Å¿int, propter duas <lb/>parallelas <var>.e.m.</var> et <var>.n.c.</var> ex .28. primi, & propter duas <var>.a.b.</var> et <var>.g.h.</var> & propter duas <var>.c.d.</var> <lb/>et <var>.e.f.</var> eo quod ex communi Å¿cientia anguli <var>.c.</var> et <var>.e.</var> Å¿unt æquales, cum ex .29. dicti lib. <lb/><reg norm="vnuÅ¿quiÅ¿que" type="simple">vnuÅ¿quiÅ¿q;</reg> æqualis Å¿it angulo <var>.m.ω.i.</var> ita etiam infero de angulis <var>.e.K.t.</var> et <var>.c.i.n.</var> <reg norm="quorum" type="context">quorÅ©</reg> <lb/>vnuÅ¿quiÅ¿que æqualis eÅ¿t angulo <var>.ω.x.t.</var> & Å¿ic de alijs dico, co quod vnuÅ¿quiÅ¿que <reg norm="eorum" type="context">eorÅ©</reg> <lb/>æqualis eÅ¿t angulo <var>.m.</var> vnde cum cognitum nobis Å¿it latus <var>.n.c.</var> et <var>.c.i.</var> et <var>.t.e.</var> <reg norm="notum" type="context">notÅ©</reg> etiam <lb/>nobis erit <var>.e.K.</var> ex .19. Å¿eptimi, eo <reg norm="quod" type="simple">ê™</reg> <lb/>ex .4. Å¿exti Å¿unt inuicem proportio-<lb/> - <anchor type="figure" xlink:label="fig-0272-01a" xlink:href="fig-0272-01"/> - nalia, detrahendo poÅ¿tea <var>.e.K.</var> ab <var>.<lb/>e.l.</var> cognito, vel ècontra, hoc ab il-<lb/>lo, nobis innoteÅ¿cet <var>.K.l.</var> Å¿inus longi <lb/>tudinis Å¿tellæ.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0272-01" xlink:href="fig-0272-01a"> - <image file="0272-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0272-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Valde etiam miror id, quod di-<lb/>ctus Stadius pag .9. illius libr. Å¿cri-<lb/>bit, hoc eÅ¿t, Solem maiorem <lb/>eſſe Luna, Å¿olum .1644. vici-<lb/>bus, </s> - <s xml:space="preserve">propterea <reg norm="quod" type="simple">ê™</reg> cum affirmet So-<lb/>lem maiorem eſſe terra (vt etiam <lb/>in AlmageÅ¿to videre eÅ¿t) 166. vici <lb/>bus cum tribus quartis, terram ve-<lb/>ro maiorem Luna .39. vicibus cum <lb/>quarta parte, </s> - <s xml:space="preserve">tunc Solem oporte-<lb/>ret maiorem eſſe Luna .6545. vici-<lb/>bus, & non .1644.</s> - </p> - <pb o="261" rhead="EPISTOL AE." n="273" file="0273" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0273"/> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">Decognitione latitudinum stellarum.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">AD cognoÅ¿cendam latitudinem Å¿tellæ, <reg norm="eiusque" type="simple">eiusqÌ</reg> declinationem, Monteregius in <lb/>10. propoÅ¿itione .8. li. </s> - <s xml:space="preserve">AlmageÅ¿ti <reg norm="methodum" type="context">methodÅ©</reg> Å¿atis docuit, Å¿ed Å¿i alia aliqua metho <lb/>do hoc idem cognoÅ¿cere voluerimus, oportebit nos prius altitudinem poli cogno-<lb/>Å¿cere, deinde altitudinem meridianam ipÅ¿ius Å¿tellæ, nec non horam, <reg norm="quando" type="context">quãdo</reg> ipÅ¿a Å¿tel <lb/>la in meridiano Å¿upra terram reperitur, qua hora mediante, illicò cognoÅ¿cemus pun <lb/>ctum ecclipticæà meridiano interÅ¿ecto, eo tempore, quo Å¿tella cÅ“lum mediat Å¿u-<lb/>pra terram. </s> - <s xml:space="preserve">Et quia ex cognita altitudine poli, illico cognoÅ¿citur altitudo æqua-<lb/>toris, cuius altitudinis differentia ab altitudine Å¿tellæ eÅ¿t declinatio ipÅ¿ius Å¿tellæ, ha-<lb/>bebimus ideo eius declinationem cognitam; </s> - <s xml:space="preserve">qua mediante ad <reg norm="cognoÅ¿cendum" type="context">cognoÅ¿cendÅ©</reg> etiam <lb/>latitudinem ita faciemus.</s> - </p> - <p> - <s xml:space="preserve">Sit exempli gratia <var>.p.o.u.</var> meridianus <var>.u.a.</var> verò æquator <var>.e.a.</var> autem eccliptica, & <lb/>o. centrum aÅ¿tri <var>.u.o.</var> verò eius declinatio ab æquatore, et <var>.e.a.</var> arcus Ä™cclipticæ inter <lb/>æquatorem, & meridianum, hoc eÅ¿t minor quarta, et <var>.a.u.</var> aÅ¿cenÅ¿io recta ipÅ¿ius arcus, <lb/>et <var>.u.e.</var> Å¿it declinatio puncti <var>.e.</var> Ä™cclipticÄ™ ab æquatore, <reg norm="reÅ¿iduum" type="context">reÅ¿iduÅ©</reg> vero declinationis Å¿tel-<lb/>lÄ™ Å¿it <var>.o.e.</var> quæ <reg norm="omnia" type="wordlist">oÄ©a</reg> nobis cognita erunt, <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.t.</var> polus Ä™cclipticus, à quo per <var>.o.</var> vÅ¿que ad <lb/>Ä™cclipticam tranÅ¿eat quarta <var>.t.i.</var> in qua quÄ™rendus erit arcus <var>.o.i.</var> hoc modo.</s> - </p> - <p> - <s xml:space="preserve">Primum arcus <var>.o.u</var>: <var>e.u</var>: <var>e.o</var>: <var>a.e</var>: et <var>.a.u.</var> nobis cogniti Å¿unt, cum angulo <var>.a.</var> declinatio <lb/>nis Ä™clipticÄ™, & cum angulo <var>.u.</var> recto, vnde ex .4. primi Copernici, cognoÅ¿cemus angu <lb/>lum <var>.a.e.u.</var> collateralem, & eius <var>.o.e.i</var>. </s> - <s xml:space="preserve">quare in triangulo <var>.o.e.i.</var> cognoÅ¿cemus <reg norm="angulum" type="context">angulÅ©</reg> <lb/>e. et deinde <var>.i.</var> vt <reg norm="rectum" type="context">rectÅ©</reg>, & latus <var>.o.e.</var> ergò ex <reg norm="eadem" type="context">eadẽ</reg> .4. cognoÅ¿cemus <reg norm="arcum" type="context">arcÅ©</reg> <var>.o.i.</var> quæſitum, <lb/>& Å¿imiliter arcum <var>.e.i.</var> qui coniunctus vel <reg norm="demptus" type="context">dẽptus</reg> ab <var>.a.e.</var> tribuet nobis longitudinem <lb/>Å¿tellÄ™, Å¿ed quia huiuÅ¿modi operatio in paruis triangulis valde fallit. </s> - <s xml:space="preserve">Ideo tibi Å¿ua-<lb/>deo alia methodo, hoc facere, hoc eÅ¿t inuenire angulum <var>.o.</var> trianguli <var>.t.p.o.</var> cuius duo <lb/>latera <var>.t.p.</var> et <var>.p.o.</var> cognita nobis Å¿unt, cum angulo <var>.p</var>. </s> - <s xml:space="preserve">Nam <var>.o.p.</var> eÅ¿t complementum de <lb/>clinationis Å¿tellæ, et <var>.p.t.</var> eÅ¿t arcus coluri Å¿olÅ¿titiorum inter duos polos, & angulus <var>.p.</var> <lb/>reÅ¿iduum ex recto <var>.t.p.a.</var> duorum colurum dempto angulo. a, <var>p.u.</var> cognito aÅ¿cenÅ¿ionis <lb/>recte, vnde angulus <var>.u.o.s.</var> vt contrapoÅ¿itus cognitus remanet. </s> - <s xml:space="preserve">angulus verò <var>.u.</var> rectus <lb/>eÅ¿t, & arcus <var>.o.u.</var> cognitus, </s> - <s xml:space="preserve">quare cognitus <lb/>nobis erit arcus <var>.u.s.</var> & angulus <var>.u.s.o.</var> vnde <lb/> - <anchor type="figure" xlink:label="fig-0273-01a" xlink:href="fig-0273-01"/> - arcus <var>.a.s.</var> nobis cognitus remanebit <reg norm="cum" type="context">cÅ©</reg> an-<lb/>gulo <var>.a.s.i.</var> reÅ¿iduo ex duobus rectis. </s> - <s xml:space="preserve">Et quia <lb/>etiam angulus <var>.s.a.i.</var> cognitus eÅ¿t, cum Å¿it an <lb/>gulus maximÄ™ declinationis Zodiaci ab <lb/>æquatore. </s> - <s xml:space="preserve">Ideo in triangulo <var>.a.s.i.</var> cuius <lb/>duo anguli <var>.a.</var> et <var>.s.</var> cum latere <var>.a.s.</var> dantur, fa <lb/>cilè inueniemus arcum <var>.s.i.</var> <reg norm="cum" type="context">cÅ©</reg> arcu <var>.a.i.</var> Å¿ed <lb/><var>a.i.</var> erit longitudinis Å¿tellæ dempto poÅ¿tea <var>.<lb/>s.i.</var> ex <var>.s.o.</var> iam inuento habebimus arcum <var>.i.<lb/>o.</var> latitudinis ipÅ¿ius Å¿tellæ.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0273-01" xlink:href="fig-0273-01a"> - <image file="0273-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0273-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Hæc autem tibi Å¿cribo non vt ipÅ¿is vta-<lb/>ris, Å¿ed potius vt tibi morem <reg norm="geram" type="context">gerã</reg>, cum bre <lb/>uiſſima methodus Å¿it illa, <reg norm="quam" type="context">quã</reg> Monteregius <lb/>Å¿cripÅ¿it <reg norm="in" type="wordlist">Ä©</reg> .10. <reg norm="propoÅ¿itione" type="simple">ê“poÅ¿itione</reg> .8. li. in AlmageÅ¿t.</s> - </p> - <pb o="222" rhead="IO. BAPT. BENED." n="274" file="0274" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0274"/> - </div> - </div> - <div type="section" level="3" n="8"> - <div type="letter" level="4" n="1"> - <head style="it" xml:space="preserve">Qualiter circulus deÅ¿ignari poßit alios duos circulos <lb/>propoÅ¿itos includens.</head> - <head xml:space="preserve">CLARISS. PETRO PIZZAMANO.</head> - <p> - <s xml:space="preserve">SVperioribus diebus per tuas literas à me quæſiuiÅ¿ti, vt modum tibi Å¿cribere vel-<lb/>lem, quo circulus deÅ¿ignari poſſit circunÅ¿cribens alios duos propoÅ¿itos circulos. <lb/></s> - <s xml:space="preserve">Qua in re vt tibi Å¿atisfaciam quod maximè cupio ita rem accipe.</s> - </p> - <p> - <s xml:space="preserve">PropoÅ¿iti circuli Å¿int, aut inter Å¿e contigui, aut interÅ¿ecantes vel Å¿eparati. </s> - <s xml:space="preserve">EÅ¿to <reg norm="pri- mum" type="context">pri-<lb/>mÅ©</reg> contiguos eſſe, qui Å¿int <var>.d.b.</var> et <var>.f.q.</var> <reg norm="quorum" type="context">quorÅ©</reg> <var>.d.b.</var> maior Å¿it et <var>.f.q.</var> minor, <reg norm="eorum" type="context">eorÅ©</reg> vero <lb/>centra Å¿int .a et <var>.o.</var> <reg norm="punctum" type="context">punctÅ©</reg> autem <reg norm="contingentiÄ™" type="context">cõtingentiÄ™</reg> Å¿it <var>.i</var>. </s> - <s xml:space="preserve"><reg norm="Nunc" type="context">NÅ©c</reg> <reg norm="protrahatur" type="simple simple">ê“trahat̃</reg>. <var>b.a.o.q.</var> per <reg norm="centra" type="context">cẽtra</reg> eo <lb/>rum ab vna circunferentia ad aliam, quÄ™ quidem linea tranÅ¿ibit per punctum <var>.i.</var> ex <lb/>11, tertij Eucli. </s> - <s xml:space="preserve">deinde à diametro maiori abÅ¿cindatur <var>.i.e.</var> ad æqualitatem minoris <lb/>Å¿emidiametri, quo facto Å¿umatur diÅ¿tantia inter <var>.e.</var> et <var>.b.</var> circino mediante <reg norm="factoque" type="simple">factoqÌ</reg> cen <lb/>tro <var>.o.</var> Å¿cindatur, alio circini pede, circunferentia maioris circuli in puncto <var>.u.</var> à quo Å¿i <lb/>mente concipiemus duas lineas <var>.u.a.d.</var> et <var>.u.o.f.</var> tranÅ¿euntes per eorum centra <var>.a.</var> et <var>.o.</var> <lb/>vÅ¿que ad circunferentias in punctis <var>.d.</var> et <var>.f.</var> ipſę <reg norm="erunt" type="context">erÅ©t</reg> inuicem Ä™quales, eo quod <var>.e.i.</var> <reg norm="sum- pta" type="context">sÅ©-<lb/>pta</reg> fuit æqualis <var>.o.f.</var> et <var>.o.u.</var> æqualis <var>.e.b.</var> </s> - <s xml:space="preserve">quare <var>.u.f.</var> æqualis erit <var>.b.i.</var> Å¿ed <var>u.d.</var> <reg norm="etiam" type="context">etiã</reg> æqua <lb/>lis <var>.b.i.</var> ergo <var>.u.d.</var> æqualis erit <var>u.f.</var> & circulus, cuius <var>u.d.</var> vel <var>.u.f.</var> erit Å¿emidiameter, con-<lb/>tiguus erit ipÅ¿is propoÅ¿itis circulis ex conuerÅ¿o .11. iam dictæ. </s> - <s xml:space="preserve">Idem dico pro circu-<lb/>lis Å¿e inuicem Å¿ecantibus.</s> - </p> - <figure position="here"> - <image file="0274-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0274-01"/> - </figure> - <figure position="here"> - <image file="0274-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0274-02"/> - </figure> - <pb o="263" rhead="EPISTOL AE." n="275" file="0275" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0275"/> - <p> - <s xml:space="preserve">Sed Å¿i circuli propoÅ¿iti Å¿eiuncti fuerint, Å¿umatur <var>.b.i.</var> diameter maioris, qui fiat Å¿e-<lb/>midiameter vnius circuli circa centrum <var>.o.</var> & hic circulus vocetur <var>.h.x.</var> coniunga-<lb/>tur deinde Å¿emidiameter <var>.o.i.</var> minoris circuli cum Å¿emidiametro <var>.a.i.</var> circuli maio-<lb/>ris, & ex huiuÅ¿modi compoÅ¿ita linea, fiat vnus Å¿emidiameter <var>.a.x.</var> circuli <var>.x.n.</var> concen <lb/>trici cum maiori, & à puncto <var>.x.</var> interÅ¿ectionis horum circulorum (poÅ¿ito quod Å¿e in-<lb/>uicem interÅ¿ecent) ducantur per eorum centra <var>.x.a.</var> et <var>.x.o.</var> vÅ¿que ad ipÅ¿orum circun-<lb/>ferentias in punctis <var>.d.</var> et <var>.f.</var> duÄ™ <lb/>lineæ, vnde habebimus <var>.x.d.</var> <lb/>æqualem <var>.x.f.</var> eo quod tam in <lb/> - <anchor type="figure" xlink:label="fig-0275-01a" xlink:href="fig-0275-01"/> - <var>x.d.</var> quam in <var>.x.f.</var> reperiuntur <lb/>diametri, & Å¿emidiametri am-<lb/>borum circulorum, facto deni <lb/>que centro <var>.x.</var> vnius circuli, cu <lb/>ius Å¿emidiameter Ä™qualis Å¿it <lb/>vni earum <var>.x.d.</var> vel <var>.x.f.</var> folu-<lb/>tum erit problema, dicta ra-<lb/>tione.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0275-01" xlink:href="fig-0275-01a"> - <image file="0275-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0275-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Si verò diÅ¿tantia duorum <lb/>propoÅ¿itorum circulorum tanta fuerit, quod Å¿ecundi circuli nequeant Å¿e inuicem <lb/>tangere, vel Å¿ecare, tunc alia via incedendum erit, quÄ™ talis eÅ¿t & generalis. </s> - <s xml:space="preserve">Diuida-<lb/>tur tota <var>.q.b.</var> per æqualia in puncto <var>.z.</var> circa quod <reg norm="Å¿ignentur" type="context">Å¿ignẽtur</reg> duo puncta ab ipÅ¿o Ä™quidi <lb/>Å¿tantia <var>.K.</var> et <var>.p.</var> diÅ¿tantia vero <var>.a.K.</var> facta Å¿it Å¿emidiameter eſſe vnius circuli <var>.K.x.</var> circa <lb/>centrum <var>.a.</var> diÅ¿tantia autem <var>.o.p.</var> Å¿emidiameter alterius circuli <var>.p.x.</var> circa cen-<lb/>trum <var>.o.</var> qui quidem circuli Å¿e inuicem Å¿ecent in puncto <var>.x.</var> à quo cum ductÄ™ fue-<lb/>rinc <var>.x.a.d.</var> et <var>.x.o.f.</var> per centra dictorum circulorum, ipÅ¿e erunt <reg norm="inuicem" type="context">inuicẽ</reg> Ä™quales, eo <reg norm="quod" type="wordlist">qÄ‘</reg> <lb/>cum <var>.b.K.</var> æqualis Å¿it <var>.q.p.</var> igitur <var>.x.d.</var> et <var>.q.p.</var> erunt inuicem Ä™quales, Å¿ed <var>.f.x.</var> æqualis eÅ¿t <lb/><var>q.p</var>. </s> - <s xml:space="preserve">quare <var>.x.f.</var> æqualis erit <var>.x.d.</var> tunc Å¿i <var>.x.</var> centrum fuerit vnius circuli, cuius Å¿emidia-<lb/>mer Å¿it vna dictarum, problema Å¿olutum erit.</s> - </p> - <p> - <s xml:space="preserve">Talis etiam Å¿oiutio commo-<lb/>da erit ad inueniendum dictum <lb/> - <anchor type="figure" xlink:label="fig-0275-02a" xlink:href="fig-0275-02"/> - circulum cuiuÅ¿uis magnitudinis, <lb/>dato tamen <reg norm="quod" type="simple">ê™</reg> eius diameter, ma <lb/>ior Å¿it <var>.b.z.</var> cum in noÅ¿tra poteÅ¿ta <lb/>te Å¿it accipere puncta <var>.K.</var> et <var>.p.</var> pro <lb/>xima vel remota ab ipÅ¿o <var>.z.</var> ad li-<lb/>bitum. </s> - <s xml:space="preserve">Vnde abÅ¿que vlla diuiÅ¿io <lb/>neipÅ¿ius <var>.q.b.</var> per medium, Å¿atis <lb/>erit Å¿ignare puncta <var>.K.</var> et <var>.p.</var> dua-<lb/>bus diÅ¿tantijs mediantibus <var>.b.K.</var> <lb/>et <var>.q.p.</var> inuicem æqualibus, & <lb/>etiam propoÅ¿itis.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0275-02" xlink:href="fig-0275-02a"> - <image file="0275-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0275-02"/> - </figure> - </div> - <handwritten/> - <pb o="264" rhead="IO. BAPT. BENED." n="276" file="0276" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0276"/> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">Figuram Å¿uperficialem ellipÅ¿i Å¿imilem, ex datis axibus cir-<lb/>cino mediante delineari poſſe.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">FIguram Å¿uperficialem ellipÅ¿i Å¿imilem, ex datis axibus, circino mediante delinea <lb/>re cum volueris, ita facito.</s> - </p> - <p> - <s xml:space="preserve">Sit <var>.e.c.</var> Å¿emiaxis maior <var>.a.e.</var> verò minor, ad angulum rectum inuicem coniuncti, <lb/>tunc <var>.a.e.</var> producatur vÅ¿que ad <var>.o</var>. </s> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> <var>.a.o.</var> maior Å¿it quam diÅ¿tantia inter <var>.o.</var> et <var>.c.</var> quÄ™ <lb/>quidem <var>.a.o.</var> poſſet etiam dari, deÅ¿cribatur poÅ¿tea circulus <var>.a.d.b.</var> circa centrum <var>.o.</var> à <lb/>quo puncto protrahatur Å¿emidiameter <var>.o.b.</var> quæ cum <var>.a.o.</var> angulum rectum conÅ¿ti-<lb/>tuat, quÄ™ <var>.o.b.</var> erit æquidiÅ¿tans <var>.e.c.</var> ex .28. primi, ducatur poÅ¿tea <var>.b.c.d.</var> et <var>.o.t.d.</var> vnde <lb/>angulus <var>.t.c.d.</var> Ä™qualis erit angulo <var>.o.b.d.</var> ex .29. eiuÅ¿dem. </s> - <s xml:space="preserve">ex quinta autem anguli <var>.b.</var> <lb/>et <var>.d.</var> Å¿unt inuicem æquales, </s> - <s xml:space="preserve">quare etiam <lb/>& anguli <var>.d.</var> et <var>.c.</var> inuicem Ä™quales erunt, <lb/> - <anchor type="figure" xlink:label="fig-0276-01a" xlink:href="fig-0276-01"/> - & ex .6. eiuÅ¿dem <var>.t.c.</var> Ä™qualis erit <var>.t.d.</var> duca <lb/>tur poÅ¿tea <var>.d.x.h.</var> perpendicularis lineæ <var>.c.<lb/>e.</var> ita diÅ¿tans Å¿ub ipÅ¿a <var>.c.e.</var> vt arcus circula-<lb/>ris circa <var>.t.</var> delineatus ex Å¿emidiametro <var>.t.<lb/>d.</var> aptus Å¿it eam Å¿ecare, Å¿umpto poÅ¿tea <var>.r.</var> <lb/>tam diÅ¿tante ab <var>.e.</var> vt <var>.t.</var> reperitur ab ipÅ¿o <lb/>e. et <var>.z.</var> ab <var>.e.</var> vt <var>.o.</var> ab eodem, ducendo po-<lb/>Å¿tea duos alios arcus magnitudinis <reg norm="priorum" type="context">priorÅ©</reg> <lb/>circa centra <var>.r.</var> et <var>.z.</var> habebimus propoÅ¿i-<lb/>tum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0276-01" xlink:href="fig-0276-01a"> - <image file="0276-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0276-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Sed cum quis voluerit prius arcus mi-<lb/>norum circulorum delineare circa maio-<lb/>rem axem, fiant cuiuÅ¿uis magnitudinis, vt <lb/>in Å¿ecunda figura videre eÅ¿t, poÅ¿ito tamen quod eorum diameter, minor Å¿it minore <lb/>axe ipÅ¿ius figurÄ™, quorum circulorum vnus Å¿it <var>.c.d.</var> circa <var>.t.</var> eius centrum, deinde in axe <lb/>minori Å¿umatur <var>.a.x.</var> æqualis <var>.c.t.</var> & protrahatur <var>.t.x.</var> quÄ™ per Ä™qualia diuidatur in pun-<lb/>cto <var>.n.</var> à quo poÅ¿tea ducatur <var>.n.o.</var> ad angulos rectos <lb/> - <anchor type="figure" xlink:label="fig-0276-02a" xlink:href="fig-0276-02"/> - cum <var>.t.x.</var> vÅ¿que ad interÅ¿ectionem cum <var>.a.e.</var> in pun-<lb/>cto <var>.o.</var> minori axi producta cum oportuerit, quod <lb/>quidem punctum <var>.o.</var> centrum erit arcus <var>.d.a.</var> maio-<lb/>ris, eo quod <var>.o.t.</var> æqualis eſſet <var>.o.x.</var> ex .4. primi Eu-<lb/>cli. </s> - <s xml:space="preserve">vnde <var>.o.d.</var> æqualis eſſet <var>.o.a.</var> & circuli etiam in-<lb/>uicem contingentes in puncto <var>.d.</var> ex .11. tertij tam <lb/>in prima, quam in Å¿ecunda figura, Å¿umpto <reg norm="denique" type="simple">deniq;</reg> <lb/>puncto <var>.s.</var> tam remoto ab <var>.e.</var> quam <var>.o.</var> reperitur ab <lb/>eodem, ipÅ¿um, centrum erit alterius arcus oppoÅ¿i-<lb/>ti, poſſemus etiam <reg norm="abÅ¿que" type="simple">abÅ¿q;</reg> diuiÅ¿ione ipÅ¿ius, <var>t.x.</var> conÅ¿ti <lb/>tuere angulum <var>.x.t.o.</var> <reg norm="æqualem" type="context">æqualẽ</reg> angulo <var>.t.x.o.</var> vnde ex <lb/>6. primi haberemus <var>.o.t.</var> æqualem <var>.o.x</var>.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0276-02" xlink:href="fig-0276-02a"> - <image file="0276-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0276-02"/> - </figure> - </div> - <pb o="265" rhead="EPISTOLAE." n="277" file="0277" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0277"/> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">De inuentione axis propoÅ¿ite portionis datæ Å¿phæræ.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">VTaxem propoÅ¿itæ alicuius datæ Å¿phæræ inuenire poſſis ita tibi operandum eÅ¿t <lb/>vt gratia exempli. </s> - <s xml:space="preserve">PropoÅ¿ita nobis eÅ¿t Å¿phæra <var>.c.i.e.t.</var> diametri cognitæ. </s> - <s xml:space="preserve">pro <lb/>poÅ¿ita etiam eÅ¿t nobis eius portio <var>.n.e.u.</var> axis <var>.e.a.</var> cognitæ minoris Å¿emidiametro, da-<lb/>ta etiam nobis eÅ¿t proportio alterius portionis minoris hemiÅ¿phærio <var>.i.e.t.</var> ad por-<lb/>tionem <var>.n.e.u.</var> quæritur nunc quantus Å¿it axis <var>.e.x.</var> Å¿ecundæ portionis hoc eÅ¿t deÅ¿idera-<lb/>mus cognoÅ¿cere proportionem <var>.e.x.</var> ad <var>.e.a.</var> vel ad diametrum ipÅ¿ius Å¿pheræ.</s> - </p> - <p> - <s xml:space="preserve">Cuius gratia reperiatur primò proportio <reg norm="circunferentiæ" type="context">circÅ©ferentiæ</reg> maioris circuli ipÅ¿ius <reg norm="Å¿phae ræ" type="simple">Å¿phÄ™Â<lb/>ræ</reg> adeius diametrum, quæ ferè eÅ¿t vt .22. ad .7. ex Archimede.</s> - </p> - <p> - <s xml:space="preserve">Quo facto, inueniatur quantitas Å¿uperficialis huiuÅ¿modi maioris circuli, quæ Å¿em-<lb/>per æqualis eÅ¿t producto quod fit ex Å¿emidiametro in dimidium circunferentiæ ip-<lb/>fius circuli, ex eodem Archimede. </s> - <s xml:space="preserve">Et Å¿ic cognoÅ¿cemus quartam partem Å¿uperficiei <lb/>Å¿phæricæ Å¿phærÄ™ propoÅ¿ite ex .31. primi lib. de Å¿phæra, & cyllindro Archimedis.</s> - </p> - <p> - <s xml:space="preserve">Deinde Å¿umatur tertia pars producti, quod fit ex Å¿emidiametro in Å¿uperficiem <lb/>maioris circuli, & habebimus conum, cuius baÅ¿is erit circulus maior, altitudo verò <lb/>Å¿emidiameter propoÅ¿itæ Å¿phæræ ex .9. duodecimi Eucli.</s> - </p> - <p> - <s xml:space="preserve">Quadruplum poÅ¿tea huiuÅ¿modi coni, erit quantitas Å¿oliditatis, Å¿eu corporeitas to <lb/>tius Å¿phærÄ™ ex .32. dicti lib. Archimedis.</s> - </p> - <p> - <s xml:space="preserve">Imaginemur poÅ¿tea <reg norm="in" type="wordlist">Ä©</reg> Å¿phærica portione <var>.n.e.u.</var> <reg norm="lineam" type="context">lineã</reg> <var>.e.u.</var> à <reg norm="summitate" type="context">sÅ©mitate</reg> ad <reg norm="extremitatem" type="context">extremitatẽ</reg> <lb/>baÅ¿is, cuius <var>.e.u.</var> quantitatem cognoÅ¿cemus, hoc modo Å¿cilicet, fumendo <reg norm="radicem" type="context">radicẽ</reg> qua-<lb/>dratam producti <var>.c.e.</var> in <var>.e.a.</var> eo quod <lb/>quadratum <var>.e.u.</var> æquale eÅ¿t quadrato <lb/> - <anchor type="figure" xlink:label="fig-0277-01a" xlink:href="fig-0277-01"/> - <var>a.u.</var> & quadrato <var>.a.e.</var> ex penultima <lb/>primi Eucli. </s> - <s xml:space="preserve">hoc eÅ¿t producto quod <lb/>fit ex <var>.c.a.</var> in <var>.a.e.</var> ex .34. tertij <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg>, <lb/>& quadrato <var>.a.e.</var> hoc eÅ¿t producto, <lb/>quod fit ex <var>.c.e.</var> in <var>.e.a.</var> ex .3. Å¿ecundi <lb/>eiuÅ¿dem.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0277-01" xlink:href="fig-0277-01a"> - <image file="0277-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0277-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Inuenta poÅ¿tea <var>.e.u.</var> ponamus eam <lb/>vnius circuli Å¿emidiametrum eſſe, cu <lb/>ius Å¿uperficialis quantitas etiam inue <lb/>niatur, vt Å¿upra dictum eÅ¿t, quæ qui <lb/><reg norm="dem" type="context">dẽ</reg> æqualis erit Å¿uperficiei portionis <lb/><var>n.e.u.</var> ex .40. primi li. </s> - <s xml:space="preserve">Archimedis de <lb/>Å¿phæra, & cyllindro.</s> - </p> - <p> - <s xml:space="preserve">Hæc autem quantitas vltimo <reg norm="inuem" type="context">inuẽ</reg> <lb/>ta multiplicetur cum tertia parte Å¿e-<lb/>midiametri datæ Å¿phæræ, & habebi-<lb/>mus Å¿oliditatem vnius coni æqualis <lb/>aggregato Å¿oliditatis portionis <var>.n.e.<lb/>u.</var> Å¿imul Å¿umptÄ™<unsure/>, <reg norm="cum" type="context">cÅ©</reg> Å¿oliditate vnius co <lb/>ni, cuius axis Å¿it <var>.a.o.</var> <reg norm="reÅ¿iduum" type="context">reÅ¿iduÅ©</reg> Å¿emidia-<lb/>metri noÅ¿træ Å¿phæræ dempta <var>.a.e.</var> ba - <pb o="266" rhead="IO. BABPT. BENED." n="278" file="0278" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0278"/> - Å¿is verò eadem quæ eÅ¿t portionis, cuius diameter eÅ¿t <var>.n.u.</var> ex .9. 12. Eucli. & ex .42. id-<lb/>eÅ¿t vltima primi Archimedis de Å¿phæra, & cyllindro.</s> - </p> - <p> - <s xml:space="preserve">Nunc autem ex hoc aggregato iam vltimo dicto detrahatur conus, cuius <var>.o.a.</var> eÅ¿t <lb/>axis et <var>.n.u.</var> diameter baÅ¿is, qui quidem conus nobis cognitus eÅ¿t, cum <var>.a.n.</var> Å¿emidia-<lb/>meter eius baÅ¿is, nobis cognita Å¿it ex .34. 3. Eucli. </s> - <s xml:space="preserve">& Å¿ic quantitas eius baÅ¿is, & ita ter-<lb/>tia pars <var>.a.o.</var> eius axis, quÄ™ multiplicata cum dicta baÅ¿i, cuius <var>.n.u.</var> eÅ¿t diameter, produ <lb/>cit dictum conum, qui quidem conus, vt diximus, demptus cum fuerit ex dicto ag-<lb/>gre gato, relinquet nobis Å¿oliditatem portionis <var>.n.e.u.</var> vnde cognoÅ¿cemus proportio <lb/>nem iÅ¿tius portionis ad totam Å¿phæram propoÅ¿itam.</s> - </p> - <p> - <s xml:space="preserve">Sed cum nobis propoÅ¿ita Å¿it proportio portionis <var>.n.e.u.</var> ad portionem <var>.i.e.t.</var> cogno <lb/>Å¿cemus etiam Å¿oliditatem huius Å¿ecundÄ™ portionis <var>.i.e.t.</var> & Å¿imiliter <reg norm="proportionem" type="context">proportionẽ</reg> hu-<lb/>ius ad totam Å¿phÄ™ram, & ad <reg norm="reÅ¿iduum" type="context">reÅ¿iduÅ©</reg> <reg norm="etiam" type="context">etiã</reg> ipÅ¿ius Å¿phÄ™rÄ™ hoc eÅ¿t portioni <var>.i.c.t</var>.</s> - </p> - <p> - <s xml:space="preserve">Protrahatur nunc diameter <var>.c.e.</var> à parte <var>.e.</var> <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> quo <var>.e.f.</var> æqualis Å¿it <var>.e.o.</var> Å¿emidiame <lb/>tro Å¿phÄ™rÄ™, quæ quidem <var>.f.e.</var> diuidatur in puncto <var>.h.</var> ita vt proportio <var>.f.h.</var> ad <var>.h.e.</var> æqua-<lb/>lis Å¿it proportioni portionis <var>.i.c.t.</var> ad portionem <var>.i.e.t.</var> quod quidem hoc modo efficieÂ<lb/>tur. </s> - <s xml:space="preserve">applicabimus lineam <var>.f.q.</var> (indeterminatam) cum <var>.f.e.</var> ad quemuis angulum in <reg norm="pun- cto" type="context">pÅ©-<lb/>cto</reg> <var>.f.</var> in qua accipiemus duas lineas <var>.f.p.</var> et <var>p.q.</var> inuicem ita relatas, vt Å¿e habent in pro <lb/>portione duæ iam dictæ portiones, hoc eÅ¿t, vt <var>.i.c.t.</var> portio ad portionem <var>.i.e.t.</var> ducen <lb/>do poÅ¿tea <var>.q.e.</var> et <var>.p.h.</var> parallelam ad ipÅ¿am <var>.q.e.</var> diuiÅ¿am habebimus <var>.f.e.</var> in eadem pro <lb/>portione vt dictum eÅ¿t ex .2. Å¿exti, & .11 quinti Euclidis, vnde <var>.c.e</var>: <var>e.f.</var> et <var>.f.h.</var> nobis co <lb/>gnitÄ™ erunt.</s> - </p> - <p> - <s xml:space="preserve">Oportebit nos nunc cognoÅ¿cere quantitatem <var>.c.x.</var> hoc modo, videlicet, quÄ™ramus <lb/>quadratum, cuius <var>.c.x.</var> eius Å¿it radix, cui quadratum lineÄ™ <var>.c.e.</var> cognitum, ita Å¿it propor-<lb/>tionatum, vt eÅ¿t linea <var>.x.f.</var> ad lineam <var>.f.h.</var> quÄ™ nobis cognita eÅ¿t, quod rectè factum erit <lb/>ex eo, quod Å¿cripÅ¿it Archimedes in .4. Å¿ecundi de Å¿phÄ™ra, & cyllindro.</s> - </p> - <p> - <s xml:space="preserve">Sed quia Archimedes eo in loco Å¿upponit id, quod necipÅ¿e, nec alius adhuc inue <lb/>nit, niÅ¿i via naturali, hoc eÅ¿t tres partes Ä™quales ex proportione data effici, non erit in <lb/>conueniens etiam nobis hac via, circa hoc aliquid dicere.</s> - </p> - <p> - <s xml:space="preserve">Accipiemus igitur diametrum <var>.c.e.</var> cum addita <var>.e.f.</var> eius Å¿emidiametro, diuidemusÂ<lb/>qÌue <var>.f.e.</var> in puncto <var>.h.</var> vt Å¿upra factum fuit, applicabimus poÅ¿tea <var>.c.m.</var> indeterminatam <lb/>angulariter ad <var>.c.e.</var> à qua <var>.c.m.</var> accipiemus <var>.c.g.</var> æqualem <var>.f.h.</var> quÄ™remus deinde natu-<lb/>rali via punctum <var>.b.</var> ita ut protrahendo à puncto <var>.e.</var> (altero extremo diametri) <var>e.m.</var> pa <lb/>rallelam ad <var>.b.g.</var> ductam, erigendo <var>.b.d.</var> perpendicularem ad <var>.c.e.</var> in puncto <var>.b.</var> protra <lb/><reg norm="ctaque" type="simple">ctaqÌ;</reg> <var>.d.c.</var> quæ à diametro <var>.e.c.</var> deducta ab <var>.c.</var> incohando vÅ¿que ad <var>.x.</var> relinquat nobis <var>.<lb/>x.f.</var> Ä™qualem <var>.c.m</var>.</s> - </p> - <p> - <s xml:space="preserve">Cuius rei ratio eÅ¿t, quia quadratum <var>.c.e.</var> Å¿e habet ad quadratum <var>.c.d.</var> vt <var>.c.e.</var> ad <var>.c.<lb/>b.</var> ex .4. et .18. Å¿exti Eucl. </s> - <s xml:space="preserve">Å¿ed ex .4. ita Å¿e habet <var>.m.c.</var> ad <var>.c.g.</var> vt <var>.e.c.</var> ad <var>.b.c.</var> & cum Å¿it <var>.c.<lb/>g.</var> Ä™q alis <var>.f.h.</var> Å¿i <var>.c.m.</var> Ä™qualis fuerit <var>.f.x.</var> habebimus propoÅ¿itum. </s> - <s xml:space="preserve">Quod Å¿i quis per di-<lb/>Å¿cretum vel et hoc facere, ita ei agendum erit.</s> - </p> - <p> - <s xml:space="preserve">Ponamus exempli gratia totum diametrum <var>.c.e.</var> propoÅ¿itæ Å¿phæræ eſſe ut decem, <lb/><reg norm="proportionemque" type="simple">proportionemqÌ;</reg> reÅ¿iduÄ™ portionis <var>.i.c.t.</var> ad Å¿ecundam <var>.i.e.t.</var> hoc eÅ¿t <var>.f.h.</var> ad <var>.h.e.</var> Å¿eÅ¿qui-<lb/>alteram eſſe, vnde <var>.e.h.</var> bis tertia erit ìpÅ¿ius <var>.f.h.</var> <reg norm="totaque" type="simple">totaqÌ;</reg> linea <var>.c.f.</var> erit .15. et <var>.f.h.</var> erit .3. <lb/>& quadratum lineæ <var>.c.e.</var> erit .100.</s> - </p> - <p> - <s xml:space="preserve">Quærendo poÅ¿tea quadratum lineæ <var>.c.x.</var> cui quadratum <var>.c.e.</var> hoc eÅ¿t .100. ita pro-<lb/>portionatum Å¿it vt <var>.f.x.</var> ad <var>.f.h.</var> hoc eÅ¿t ad .3. Å¿i autem cogitauerimus <var>.c.x.</var> eſſe nouem <lb/>partium talium qualium <var>.c.e.</var> eÅ¿t decem, eius quadratum erit .81. et <var>.x.f.</var> erit .6. par-<lb/>tium talium qualium <var>.c.f.</var> eÅ¿t .15. dicendo poÅ¿tea Å¿i .100. dat .81. (ex regula de tribus) - <pb o="267" rhead="EPISTOL AE." n="279" file="0279" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0279"/> - <var>x.f.</var> hoc eÅ¿t .6. dabit .4. integra cum <num value="86">.<lb/> - <anchor type="figure" xlink:label="fig-0279-01a" xlink:href="fig-0279-01"/> - 86.</num> centeÅ¿imis, Å¿ed nos vellemus no-<lb/>bis prouenire tria, eo <reg norm="quod" type="simple">ê™</reg> ita eÅ¿t <var>.f.h.</var> <lb/>qua propter deÅ¿cendere nos oporte-<lb/>bit à nouem ad .8. & ab .8. ad .7. & à . <lb/>7. ad .6. </s> - <s xml:space="preserve">tunc inueniemus <var>.c.x.</var> oporte-<lb/>re eſſe circiter quinque cum duabus <lb/>tertijs, <reg norm="operando" type="context">operãdo</reg> poÅ¿tea ex regula de <lb/>tribus, Å¿i dixerimus quando .100. no-<lb/>bis dat .32. cum nona parte integri, <lb/></s> - <s xml:space="preserve">tunc nouem cum tertia parte integri <lb/>dabit .2. <reg norm="cum" type="context">cÅ©</reg> 296. de .300. hoc eÅ¿t .2. <lb/>cum circa .49. quinquageÅ¿imis, quæ <lb/>quidem quantitas, cum propinquiſſi <lb/>ma Å¿it lineæ <var>.f.h.</var> trium integrorum di <lb/>cemus <var>.c.x.</var> eſſe quinque integrorum <lb/>cum duabus tertijs partibus vnius in <lb/>tegri, et <var>.e.x.</var> reÅ¿iduum, hoc eÅ¿t axem <lb/>quæſitum portionis <var>.i.e.t.</var> eſſe circa .4 <lb/>integra cum tertia parte vnius inte-<lb/>gri.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0279-01" xlink:href="fig-0279-01a"> - <image file="0279-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0279-01"/> - </figure> - </div> - </div> - </div> - <div type="section" level="3" n="9"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE ERRORIBVS THOMAE PORCACHII <lb/>& Benedicti Bordonij in eorum inÅ¿ularijs.</head> - <head style="it" xml:space="preserve">Excellentißimo D. lo. BaptiÅ¿tæ Fæmello Ciui <reg norm="Decurionique" type="simple">DecurioniqÌê«</reg> Tau-<lb/>rinenÅ¿i PhiloÅ¿opho, Medico, & in Accademia eius <lb/>Ciuitatis Medicinæ Practicæ Ordinario, <reg norm="Pri- marioque" type="simple">Pri-<lb/>marioqÌê«</reg> profeßori celeberrimo.</head> - <p> - <s xml:space="preserve">DIj perdant tuas adeo moleÅ¿tas, & aſſiduas curas, quæ te nimis à ſuauiori-<lb/>bus Å¿tudijs diÅ¿trahunt, & à nobis longius abducunt. </s> - <s xml:space="preserve">Nam, ut tibi <reg norm="quietem" type="context">quietẽ</reg>, <lb/>ita mihi ingentem adimunt voluptatem. </s> - <s xml:space="preserve">Sed ne in aliquo erga te defi-<lb/>cere videar, quæ tibi olim promiÅ¿i, nunc mitto.</s> - </p> - <p> - <s xml:space="preserve">Negari quidem non poteÅ¿t, quin fuerit laborioÅ¿um opus Porcachij, & Benedicti <lb/>Bordonij, hoc eÅ¿t inÅ¿ularium, qui rectè etiam feciſſent, cum loqui eos oportebat de <lb/>terminis Å¿phæræ ratione Å¿itus locorum, Å¿i Å¿eipÅ¿os alicuius excellentis CoÅ¿mographi <lb/>conÅ¿ilio Å¿ubmiÅ¿iſſent. </s> - <s xml:space="preserve">ConÅ¿idera quæſo, quomodo admitti poſſit, id quod ait Por-<lb/>cachius initio Å¿ui operis, ideÅ¿t IÅ¿landiam Å¿ub Polo arctico iacere, inter auÅ¿trum, & <lb/>boream: </s> - <s xml:space="preserve">omittamus etiam quod idem in ProÄ™mio lib. Å¿ecundi, vbi ait Biarmiam, <lb/>(& non IÅ¿landiam) eſſe Å¿ub dicto polo arctico: </s> - <s xml:space="preserve">in <reg norm="eodemque" type="simple">eodemqÌ;</reg> principio repetit ipÅ¿am <lb/>IÅ¿landiam inter auÅ¿trum, & boream per centum leucas Germanicas extendi, dein-<lb/>de verÅ¿us occidentem, ea duo Å¿tupenda miracula conÅ¿pici. </s> - <s xml:space="preserve">Vide quæſo, quomodo <lb/>incolÄ™ Å¿ub aliquo ex polis, habere poſſint occidentem, orientem, magiÅ¿trum, <reg norm="auÅ¿trum" type="context">auÅ¿trÅ©</reg>, - <pb o="268" rhead="IO. BAPT. BENDE." n="280" file="0280" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0280"/> - & boream, & vt melius dicam aliquem rhombum. </s> - <s xml:space="preserve">Sed quomodo fieri poteÅ¿t, vt in-<lb/>Å¿ula IÅ¿landiæ Å¿it Å¿ub polo, eius tamen dies, & nox maior non Å¿it longior Å¿patio <reg norm="trium" type="context">triÅ©</reg> <lb/>menÅ¿ium? </s> - <s xml:space="preserve">vt ipÅ¿e pagina .62. in proÄ™mio Å¿ecundi lib. affirmat, quamuis hoc à Bordo <lb/>ne deÅ¿umat. </s> - <s xml:space="preserve">In quo <reg norm="vterque" type="simple">vterq;</reg> fallitur, <reg norm="Å¿entientes" type="context">Å¿entiẽtes</reg> huiuÅ¿modi diem ab ingreſſu Solis, in <lb/>principium geminorum incipere, & in egreſſu à Leone terminari, ideÅ¿t à .12. Maij ad <lb/>14. AuguÅ¿ti, quaÅ¿i Å¿i ab æquatore finis Leonis ita declinaret, vt principium gemino-<lb/>rum, & finis Aquarij, vt initium Sagittarij, nam ratio poÅ¿tulat, tantum de-<lb/>clinari ab æquatore finem quantum initium diei, vbi maximus dies .24. horas ex <lb/>cedit, & Å¿ic dico de noctibus: </s> - <s xml:space="preserve">vnde in huiuÅ¿modi regione, vbi per tres menÅ¿es conti <lb/>nuos Sol radios emittit, huiuÅ¿modi dies à medietate Tauri incipit, & in medietate <lb/>Leonis terminatur, quæ quidem loca æqualem declinationem habent, & Å¿ic nox <lb/>trium menÅ¿ium incipit à medietate Scorpionis, & in medietate Aquarij, eadem ra-<lb/>tione finitur.</s> - </p> - <p> - <s xml:space="preserve">Septima verò pag. <reg norm="idem" type="context">idẽ</reg> ait, dies Å¿olÅ¿titiales eſſe circa .24. Iunij. </s> - <s xml:space="preserve"><reg norm="quod" type="wordlist">QÄ‘</reg>, an <reg norm="tunc" type="context">tÅ©c</reg> eſſet <reg norm="verum" type="context">verÅ©</reg>, <lb/>tu ipÅ¿e videto. </s> - <s xml:space="preserve">Is præterca modus <reg norm="quem" type="context">quẽ</reg> ad <reg norm="inueniendum" type="context context">inueniẽdÅ©</reg> <reg norm="orientem" type="context">orientẽ</reg>, & <reg norm="occidentem" type="context">occidẽtem</reg> præſcribit <lb/>in eodem proÄ™mio pag .63. eÅ¿t tædioÅ¿us, cum Å¿emper expectare nos cogat æquino-<lb/>ctij tempus, cum alij modi reperiantur breuiores, qui in qualibet reuolutione primi <lb/>mobilis obÅ¿eruari poſſunt, quorum vnus erit mediante inuentione lineæ meridiane <lb/>orizontalis, eo modo, quo Å¿criptum eÅ¿t ab antiquis mediante Sole, aut Luna, quæ <lb/>luminaria in quolibet alio loco, <reg norm="præterquam" type="context">præterquã</reg> Å¿ub polo efficiunt, vt extremitas vmbræ <lb/>rectæ <reg norm="gnomonum" type="context">gnomonÅ©</reg> <reg norm="gyrum" type="context">gyrÅ©</reg> <reg norm="oxigonium" type="context">oxigoniÅ©</reg>, Å¿eu <reg norm="eclipticum" type="context">eclipticÅ©</reg> ducat, ideÅ¿t in ijs locis, <reg norm="quorum" type="context">quorÅ©</reg> zenit eÅ¿t <lb/>inter polum, & circulum arcticum, quemadmodum facit, vt alijs, exiÅ¿tentibus ipÅ¿is <lb/>luminaribus extra æquatorem, & circulos arcticos gyrum hyperbolicum reddant. <lb/></s> - <s xml:space="preserve">Sed id quod eidem Porcachio impoſſibile eſſe apud eos, qui habitant Å¿ub polo vi-<lb/>detur, ideÅ¿t vt multis rationibus, vt ipÅ¿e dicit, fieri non poſſit, ut fiat immediata quÄ™ <lb/>dam, & Å¿ubita mutatio à continuo die ad continuam noctem abÅ¿que eo quod ijs, <lb/>Å¿altem Å¿emel conceſſa Å¿int dies, & nox terminata duodecim horarum, eÅ¿t magis ad <lb/>mirandum impoſſibile, quod imaginari poſſimus, nam neceſſarium eſſet, ut orizon-<lb/>habitatorum Å¿ub polo Å¿ecaret æquatorem contra id, quod Å¿uperius admiÅ¿erat, id-<lb/>eÅ¿t <reg norm="orizontem" type="context context">orizõtẽ</reg> Biarmiæ, eſſe <reg norm="eundem" type="context context">eÅ©dẽ</reg> <reg norm="cum" type="context">cÅ©</reg> circulo æquinoctiali. </s> - <s xml:space="preserve">Vide etiam quid is ab anti-<lb/>quis colligat, loquens de iis, quæ in inÅ¿ula Taprobana ad finem pag .186. admirabi <lb/>lia Å¿unt, Å¿cribens eiuÅ¿dem inÅ¿ulÄ™ habitatoribus, Lunam Å¿uper terram non apparere <lb/>ab octauo uÅ¿que ad decimumÅ¿extum diem: </s> - <s xml:space="preserve">prÄ™ter quam, quod etiam Å¿cribit, in <lb/>eadem inÅ¿ula, tramuntanam non uideri, quod falÅ¿um eÅ¿t, quia hæc à polo arctico <lb/>circiter quatuor gradibus diÅ¿tat noÅ¿tris temporibus. </s> - <s xml:space="preserve">unde ab ijs qui Å¿unt Å¿ub æqua-<lb/>tore, cum ea Å¿upra orizontem eÅ¿t, conÅ¿pici poteÅ¿t, cum ijÅ¿dem Å¿ingulis diebus oria-<lb/>tur, & occidat. </s> - <s xml:space="preserve">Idem etiam pro re admirabili Å¿cribit, uideri Canopum, qui à po-<lb/>lo antarctico plus quam quadraginta gradibus diÅ¿tat.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">De erroribus Lucilli Philalthæi.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">QVod Lucillus PhilalthÄ™us tam eximius Mathematicus Å¿it, ut ipÅ¿um Anto-<lb/>nius Berga facit, ego quidem non uideo. </s> - <s xml:space="preserve">In Å¿uis enim commentariis de <lb/>CÅ“lo, dicit primum, Pyramidem, quÄ™ inter corpora regularia primum locum tenet - <pb o="269" rhead="EPISTOL AE." n="281" file="0281" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0281"/> - Å¿ex baÅ¿ibus conÅ¿tare, pag.15. 583. 632. et .647. </s> - <s xml:space="preserve">Omitto errorem ab eodem com-<lb/>miſſum in fine pag .39. ubi oleum grauius eſſe quam aquam fatetur, cum id ad res <lb/>mathematicas non Å¿pectet: </s> - <s xml:space="preserve">Omitto etiam quod idem neget aÅ¿trologiam pag .74. <lb/>79. & quod etiam dicat pag .89. </s> - <s xml:space="preserve">Deum eſſe ad orientem, non conÅ¿iderans aliqui-<lb/>bus populis noÅ¿trum orientem eſſe occidentem.</s> - </p> - <p> - <s xml:space="preserve">Quod idem ait pag .241. </s> - <s xml:space="preserve">AÅ¿trologiam eſſe antiquiorem AÅ¿tronomia eÅ¿t falÅ¿iſſi-<lb/>mum, quia iudiciaria Å¿emper præſupponit cognitionem Å¿itus Å¿tellarum, quæ ab A-<lb/>Å¿tronomia petitur. </s> - <s xml:space="preserve">Mouebit tibi riÅ¿um quod ait pag .307. his verbis.</s> - </p> - <quote> - <s xml:space="preserve">Verum propriè media dicitur illa, quæ rectam Å¿phæram omninò habet, quæ eun <lb/>dem polum orizontis & mundi obtinet, quæ orizontem habet diuidentem <reg norm="Å¿phæram" type="context">Å¿phærã</reg> <lb/>æquè Å¿ecundum angulos rectè.</s> - </quote> - <p> - <s xml:space="preserve">Paulo inferius continuans Å¿ermonem de Å¿phæra recta, ait.</s> - </p> - <quote> - <s xml:space="preserve">Et niÅ¿i tumor terræ, & gibum eſſet, ijs perpetuus eſſet dies Å¿ine nocte.</s> - </quote> - <p> - <s xml:space="preserve">Linea verò .56. ait habitatores Å¿phÄ™rÄ™ rectè habere .4. Å¿olÅ¿titia, Å¿eÅ¿e ipÅ¿um huius <lb/>rei planè ignarum prodens .310. autem pag. Å¿ic Å¿cribit.</s> - </p> - <quote> - <s xml:space="preserve">Quoniam repercutiuntur radij, & peridem centrum tranÅ¿eunt, ob id Å¿tupam ap <lb/>poÅ¿itam centro radius accendit.</s> - </quote> - <p> - <s xml:space="preserve">Quem quidem errorem ab Euclide deÅ¿umit, et .15. linea pag .636. repetit.</s> - </p> - <p> - <s xml:space="preserve">Si vis ridere, legito .16. primas lineas .357. pag. </s> - <s xml:space="preserve">Quod idem deinde dicat circa fi-<lb/>nem 396. pag. lucem eſſe Å¿ubÅ¿tantiam corporis lucidi & corpoream, Å¿ubijciam tuo <lb/>iudicio, vt etiam quod ait .397. pag. his vetbis vtens.</s> - </p> - <quote> - <s xml:space="preserve">Idcirco animalia illa, quæ nocte vagantur perpolita, dum volant, aerem terunt <lb/>nocturnum, & fulgent.</s> - </quote> - <p> - <s xml:space="preserve">Et pag .398.</s> - </p> - <quote> - <s xml:space="preserve">Multitudo radiorum non admodum facit ad excitandum calorem Å¿i Å¿olum inci-<lb/>dat Å¿ine repercuſſu, neorecta incidere iuuerit.</s> - </quote> - <p> - <s xml:space="preserve">Quod falÅ¿um eÅ¿t cum radius incidens longè magis quam reflexus calefaciat. </s> - <s xml:space="preserve">In fi <lb/>ne autem .405. Å¿ic Å¿cribit.</s> - </p> - <quote> - <s xml:space="preserve">Sol in ortu & in occaÅ¿u longius apparet, iccircò reuolui creditur. </s> - <s xml:space="preserve">Hinc etiam in <lb/>abÅ¿ide Å¿tare putatur, & in oppoÅ¿ito abÅ¿idis, vnde Å¿olÅ¿titia vocant, Å¿ed nobis in Can-<lb/>cro, antipodibus verò in Capricorno tum Sol abeſſe longius apparet vtriÅ¿que.</s> - </quote> - <p> - <s xml:space="preserve">An hoc quid peius dici poteÅ¿t? </s> - <s xml:space="preserve">Circa vero .40. lineam pag .459. Å¿ic Å¿cribit.</s> - </p> - <quote> - <s xml:space="preserve">Si enim alij planetæ, & Å¿tellæ fixÄ™ reciperent à Sole lumen, dum accederent ad So <lb/>lem, vel recederent, aut contra, Sol ad eas appropinquaret, & abÅ¿cederet, eaÅ¿dem-<lb/>lucis viciſſitudinis Å¿ubiret, quas Luna.</s> - </quote> - <quote> - <s xml:space="preserve">Hoc autem nondum deprÄ™henÅ¿um eÅ¿t, quin etiam Mercurius, Venus, Å¿uo interpo <lb/>Å¿itu, Solem occultarent nobis, vt Luna.</s> - </quote> - <quote> - <s xml:space="preserve">Paulo inferius Å¿ic ait. </s> - <s xml:space="preserve">RurÅ¿us æquè Saturnus, Jupiter, Mars, Å¿ubire deliquium, <lb/>more Lunæ, aut Å¿altem obiectu terræ inter Solem & ipÅ¿os, quia tum ob interpoÅ¿itam <lb/>terram non poſſent haurire lumen à Sole.</s> - </quote> - <p> - <s xml:space="preserve">Hæc verò omnia, talia Å¿unt, qualia ab ijs qui incipiunt intelligere Å¿phæram non <lb/>proferrentur. </s> - <s xml:space="preserve">Omittamus, quod ait deinde.</s> - </p> - <quote> - <s xml:space="preserve">Accedit quod Å¿i aÅ¿tra lumen à Sole acciperent eiuÅ¿dem caloris eſſent. </s> - <s xml:space="preserve">Itaque om <lb/>nia Å¿iccarent, & nulla eſſent frigidæ conÅ¿titutionis contra AÅ¿trologos.</s> - </quote> - <p> - <s xml:space="preserve">Quia hac ratione, Luna, quæ negari non poteÅ¿t, quin ab ipÅ¿o Sole <reg norm="lumem" type="context">lumẽ</reg> accipiat, <lb/>eiuÅ¿dem caloris eſſet cum eodem Sole. </s> - <s xml:space="preserve">Sunt ea etiam ridenda, quæ idem ait pag.<lb/>460. lineis .18. 19. 23. 26. 27. 29. quaÅ¿i ea lux infinita (vt ita dicam) magni Solis, non - <pb o="270" rhead="IO. BAPT. BENED." n="282" file="0282" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0282"/> - in alium finem Å¿it effecta quam ad illuminandam Å¿uperficiem huius excrementi ip-<lb/>Å¿ius vniuerÅ¿i ad vtilitatem hominum, imò, vt rectius dicam, <reg norm="animalium" type="context">animaliÅ©</reg>. </s> - <s xml:space="preserve">vide etiam <lb/>pag .632. et .633. vbi AriÅ¿totelem de implendo loco non intellexit, cum citet Å¿phæ-<lb/>ram, loco pyramidis, & inter .46. et .47. lineas dicat <reg norm="quadratum" type="context">quadratÅ©</reg> eſſe quid multiplex, <lb/>cum Å¿it vnicum tantum in Å¿pecie, quia Å¿pecies eÅ¿t quadrilateri, & quadranguli, Å¿ed <lb/>vbi in .6. linea pag .633. ait.</s> - </p> - <p> - <s xml:space="preserve">Item hexagonus.</s> - </p> - <p> - <s xml:space="preserve">Magnum errorem committit, vt etiam cum .12. linea .636. pag. Å¿cribens.</s> - </p> - <p> - <s xml:space="preserve">Pyramis, Å¿iue planum, Å¿iue Å¿olidum, habet acutiſſimum, & in .2. libr. de anima <lb/>pag .215. dicat de die poſſe videri Å¿tellas in Å¿peculo poÅ¿ito in vaÅ¿e aqua pleno, quod <lb/>reuera eÅ¿t valde abÅ¿urdum. </s> - <s xml:space="preserve">Alios eiuÅ¿dem errores tibi non patefacio, quia iam ni-<lb/>hil amplius otij mihi eÅ¿t, Å¿ed eos tu ipÅ¿e perÅ¿picere, & cognoÅ¿cere facilè poteris, & <lb/>multò plures quidem, quam putas.</s> - </p> - </div> - </div> - <div type="section" level="3" n="10"> - <div type="letter" level="4" n="1"> - <head style="it" xml:space="preserve">Cur maius lumen extenuet minus.</head> - <head xml:space="preserve">PIRRO DE ARZONIS.</head> - <p> - <s xml:space="preserve">EX tuis literis intellexi id, quod etiam Å¿ine ijs exploratum mihi erat. </s> - <s xml:space="preserve">Sed conce <lb/>do tantum eſſe dicere vbi eÅ¿t maius lumen, minus non diÅ¿cerni, quantum inter <lb/>diu Å¿tellas non videri: </s> - <s xml:space="preserve">immo eÅ¿t etiam magis vniuerÅ¿ale, quia idem multis aliis lumi-<lb/>nibus, præter ea quæ Å¿unt Å¿tellarum, ea ratione contingit, quia ingrediente per pupil-<lb/>lam, tam lumine maiori, quam minori, reflexum ipÅ¿ius maioris in oculo, in Å¿itu mino <lb/>ris, efficit, vt ipÅ¿um minus confundatur, & diÅ¿tingui nequeat, quemadmodum aper-<lb/>te cognoÅ¿ci poteÅ¿t in aliquo cubiculo, cuius parietes dealbati Å¿int, in quo, vnicum <lb/>tantum Å¿it exiguum foramen, per quod aliqua lumina reflexa ab obiectis extrinÅ¿ecis <lb/>intra ipÅ¿um cubiculum ingredi poſſint, vnde imagines <reg norm="obiectorum" type="context">obiectorÅ©</reg> in parietibus con-<lb/>Å¿piciuntur, Å¿ed Å¿i per idem foramen ingrederetur etiam primarius radius Solis, re-<lb/>flexus huiuÅ¿modi radij efficeret, vt dictæ imagines, magis aut minus euaneÅ¿cerent, <lb/>prout dictus reflexus radij Å¿olaris, maiori, minorÃue vi polleret.</s> - </p> - <p> - <s xml:space="preserve">Ad hoc tamen propoÅ¿itum, nolo tibi Å¿ilentio inuolui mirabilem quendam effe-<lb/>ctum eiuÅ¿modi rei. </s> - <s xml:space="preserve">Hoc eÅ¿t vt fiat foramen illud rotundum, magnitudinis tamen <lb/>vnius Å¿pecilli, quod foramen obturetur mediante vno illorum Å¿pecillorum, quæ pro <lb/>Å¿enibus (non breuis viÅ¿ionis) conficiuntur, hoc eÅ¿t quorum ambæ Å¿uperficies con <lb/>uexæ Å¿unt, non autem concauæ. </s> - <s xml:space="preserve">Deinde opponatur folium album papiri, adeo di <lb/>Å¿tans à foramine, vt extrinÅ¿eca obiecta in eo appareant. </s> - <s xml:space="preserve">Quæ quidem obiecta Å¿i à <lb/>Sole illuÅ¿trata fuerint, tam clara, & diÅ¿tincta videbuntur, vt nihil pulchrius dele-<lb/><reg norm="ctabiliusque" type="simple">ctabiliusqÌ;</reg> videri poterit, inuerÅ¿a tamen. </s> - <s xml:space="preserve">Sed Å¿i ea directa videre voluerimus. </s> - <s xml:space="preserve">hoc <lb/>optimè faciemus, mediante reflexione alicuius Å¿peculi plani.</s> - </p> - <pb o="271" rhead="EPISTOL AE." n="283" file="0283" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0283"/> - </div> - </div> - <div type="section" level="3" n="11"> - <div type="letter" level="4" n="1"> - <head style="it" xml:space="preserve">Cur byems valde frigida Å¿equatur ac<unsure/>tatem in qua <lb/>calor viguerit.</head> - <head xml:space="preserve">NOBILISSIMO, NECNON INGENIOSISSIMO <lb/>Gabrieli BuÅ¿chæ, MediolanenÅ¿i.</head> - <p> - <s xml:space="preserve">QVod dixi hyemem valde frigidam Å¿equi <reg norm="æſtatem" type="context">æſtatẽ</reg>, in qua calor viguerit, inde na <lb/>Å¿citur, quia calor terrÄ™, aquæ, & aeris, non eÅ¿t naturalis horum corporum, vt <lb/>eÅ¿t frigus, cum calor à Sole procedat, qui ea calefacit Å¿uo lumine, vnde quod æſtate <lb/>Sol præter modum calefaciat <reg norm="terram" type="context">terrã</reg>, ideo <reg norm="contingit" type="context">cõtingit</reg>, quod minora <reg norm="impedimenta" type="context">impedimẽta</reg> contra <lb/>ria Å¿ortiatur, & cum eandem poÅ¿tea deÅ¿erit, ad aliam partem æquatoris <reg norm="tranÅ¿migrans" type="context">tranÅ¿migrãs</reg> <lb/>terra ad Å¿uam qualitatem reddit, maiori cum impetu, eo modo, quo res in mo-<lb/>tibus localibus naturalibus, qui etiam terminos Å¿ibi prÄ™fixos, & conÅ¿titutos exce-<lb/>dunt, hinc etiam hyeme fit glacies, ex calefacta prius aqua, quæ durior poÅ¿tea eÅ¿t <lb/>atque frigidior alia. </s> - <s xml:space="preserve">AeÅ¿tas etià m quæ Å¿equitur hyemem valde frigidam, non <lb/>erit admodum calida, quia Sol inueniens contrarium naturale valde potens, non <lb/>tam facile illud pellere poteÅ¿t, vnde etiam Å¿i in Geminis, Cancro, & Leone, moram <lb/>trahat, non Å¿ufficit tamen ut magnum calorem imprimere poſſit. </s> - <s xml:space="preserve">Vnde Å¿equitur duas <lb/>æſtates quarum una Å¿equatur aliam, in eodem loco, uehementi calore præditas eÅ¿-<lb/>Å¿e non poſſe, quemadmodum nec duas hyemes exceſſiuo frigore, remotis tamen <lb/>accidentibus uentorum, pluuiarum, & niuium.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head xml:space="preserve">QVOD MALE SENSERIT NICOLAVS TARTA-<lb/>lea circa attractionem machinæ tormentalis.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">EFfectus, quem Å¿cribit Tattalea quęſito quinto primi lib. necnon quæſito <lb/>21. et .24. maxima cum ratione eſſe uidetur, non tamen ea quam ipÅ¿e in <lb/>quinto profert, quia uerum non eÅ¿t, vt quanto aliquid fit calidius, <reg norm="tanto" type="context">tãto</reg> ue-<lb/>hementius attrahat, eo quod Å¿i etiam huiuÅ¿modi res, in eodem calore, in <lb/>quo Å¿emel reperitur, firma maneret; </s> - <s xml:space="preserve">neque attraheret, neque aliquid impelleret. <lb/></s> - <s xml:space="preserve">Nam dum aliquod corpus calefit, dilatatur, & per conÅ¿equens circumcirca <reg norm="undique" type="simple">undiq;</reg> <lb/>trudit, & partes uaÅ¿is debiliores cedunt. </s> - <s xml:space="preserve">dum uerò dictum corpus re frigeratur, re-<lb/>Å¿tringitur, & dum in unum cogitur, Å¿i reperiatur in uaÅ¿e, quod aer, aqua, aut aliud <lb/>aliquod corpus ingredi nequeat, dictum uas à quo circundatur frangit, ne aliqua <lb/>pars loci uacua remaneat, Å¿ed Å¿i aliquod corpus ingredi poteÅ¿t, illud ipÅ¿um ad Å¿e at-<lb/>trahit, quemadmodum uidere licet in cucurbitulis. </s> - <s xml:space="preserve">Vnde Å¿equitur eam propoÅ¿i-<lb/>tionem, qua dicitur, calidi eÅ¿t attrahere, ueram non eſſe, quia Å¿i hoc fieret, quanto <lb/>aliquid calidus efficeretur, tanto magis attraheret, & ècontra, cum tamen planè <lb/>contrarium appareat, cum quanto magis aliquid calefit, tanto uehementius impel-<lb/>lat, & quanto magis frigefit, tanto plus attrahat. </s> - <s xml:space="preserve">Quapropter uerius dicemus, fri-<lb/>gidi eſſe attrahere, calidi uerò expellere, quamuis per accidens. </s> - <s xml:space="preserve">Ex quo Å¿equitur, ut <lb/>quanto calidior facta fuerit materia aliqua, aliquo loco determinata, redeundo po-<lb/>Å¿tea ad Å¿uam priorem frigiditatem, tanto minori loco indigeat, Å¿imiliter etiam <lb/>è conuerÅ¿o accidit, ut quanto frigidior <reg norm="reperitur" type="simple">reê‘itur</reg> talis materia, tanto maioriloco, po- - <pb o="272" rhead="IO. BAPT. BENED." n="284" file="0284" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0284"/> - Å¿tea egeat ipÅ¿a ualde calefacta. </s> - <s xml:space="preserve">Quod Tartalea in quinto quęſito non animaduer-<lb/>terat.</s> - </p> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">Solutiones aliqua, circa altimetriam.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">TVas literas accepi, <reg norm="tuasque" type="simple">tuasqÌ;</reg> dubitationes conÅ¿ideraui, quas quidem non inutiles <lb/>inueni, quo uerò ad primam, dico te oportere illud Theorema Å¿peculari or <lb/>dine huiuÅ¿modi methodi, uidelicet quod <reg norm="quotieÅ¿cunque" type="simple">quotieÅ¿cunq;</reg> habuerimus <reg norm="angulum" type="context">angulÅ©</reg> <reg norm="aliquem" type="context">aliquẽ</reg> <lb/>cuiufuis amplitudinis, puta <var>.A.R.V.</var> cuius duo latera <var>.R.A.</var> et <var>.R.V.</var> indeterminata <lb/>intelligantur, Å¿i ab aliquo puncto inter ipÅ¿as poÅ¿ito, puta <var>.u.</var> quod etiam uocetur <var>.i.</var> du <lb/>ctÄ™ fuerint .4. lineÄ™ ipÅ¿is dictis lateribus, hac Å¿cilicet <reg norm="conditione" type="context">cõditione</reg>, <reg norm="quod" type="wordlist">qÄ‘</reg> duÄ™ ex dictis .4. Å¿int <lb/>parallelÄ™ ipfis <reg norm="lateribus" type="simple">lateribê°</reg>, puta <lb/> - <anchor type="figure" xlink:label="fig-0284-01a" xlink:href="fig-0284-01"/> - <var>u.e.</var> et <var>.u.E.</var> reliquÄ™ uero duÄ™ <lb/>Å¿eccent ipÅ¿a latera, ut <var>V.u.<lb/>a.</var> et <var>.I.u.A</var>. </s> - <s xml:space="preserve">Dico nunc pro-<lb/>portionem <var>.e.A.</var> ad <var>.e.a.</var> ean <lb/>dem eſſe, quÄ™ <var>.E.V.</var> ad <var>.E.I.</var> <lb/>Nam Å¿cimus proportionem <lb/><var>E.i.</var> ad <var>.E.i.</var> eandem eſſe quÄ™ <lb/><var>e.i.</var> ad <var>.e.A.</var> ex fimilitudine <lb/><reg norm="triangulorum" type="context">triangulorÅ©</reg>, Å¿imiliter <reg norm="propor" type="simple">ê“por</reg> <lb/><reg norm="tionem" type="context">tionẽ</reg> <var>.E.u.</var> ad <var>.E.V.</var> <reg norm="eandem" type="context context">eãdẽ</reg> quÄ™ <lb/><var>e.a.</var> ad <var>.e.u.</var> </s> - <s xml:space="preserve">quare aggregata <lb/>ex iÅ¿tis erunt inuicem <reg norm="aequa- lia" type="simple">Ä™qua-<lb/>lia</reg>, uel Å¿i mauis ex Ä™qua pro <lb/>portionalitate, quod idem <lb/>eÅ¿t, ita Å¿e habebit <var>.E.I.</var> ad <var>.<lb/>E.V.</var> ut <var>.e.a.</var> ad <var>.e.A</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0284-01" xlink:href="fig-0284-01a"> - <image file="0284-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0284-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">SuppoÅ¿ito nunc plano orizontali <var>.V.E</var>. </s> - <s xml:space="preserve"><reg norm="Altitudineque" type="simple">AltitudineqÌ;</reg> inacceſſibili <var>.A.E</var>. </s> - <s xml:space="preserve">DuÄ™ ue-<lb/>rò Å¿tationes oculorum Å¿int <var>.V.</var> et <var>.I.</var> lineÄ™ autem uiÅ¿uales Å¿int <var>.V.A.</var> et <var>.I.A</var>. </s> - <s xml:space="preserve">Et quadra-<lb/>tum geometricum Å¿it <var>.b.e</var>. </s> - <s xml:space="preserve">Supponatur nunc pro prima dubitatione, quod in am-<lb/>babus Å¿tationibus filum perpendiculare Å¿eccet latus <var>.e.c.</var> non autem <var>.b.c.</var> (nam quan-<lb/>do in ambabus Å¿tationibus filum Å¿ecat latus <var>.b.c.</var> nullum tibi dubium oritur, imo ma <lb/>nifeÅ¿tè patent partes lateris <var>.b.c.</var> terminatas à <var>.b.</var> & à filo proportionales eſſe <var>.V.E.</var> & <lb/><var>I.E.</var> Å¿umpto <var>.E.</var> pro <var>.b.</var> et <var>.I.V.</var> pro punctis Å¿ecatis à filo, ex <reg norm="euidenti" type="context">euidẽti</reg> Å¿imilitudine trian-<lb/>gulorum quadrati cum triangulis <var>.A.E.V.</var> et <var>.A.E.I.</var>) Sed cum in pręſenti caÅ¿u repe-<lb/>riatur triangulum <var>.u.e.a.</var> minus, in Å¿tatione remotiori, Å¿imile triangulo maiori <var>.V.E.<lb/>A.</var> & triangulum maius <var>.i.e.a.</var> proximioris Å¿tationis, Å¿imile triangulo minori <var>.I.E.A.</var> <lb/>(quod in alio iam dicto, caÅ¿u non accidit, ut unum triangulorum, minus Å¿cilicet, Å¿i-<lb/>mile Å¿it uno triangulorum, maiori Å¿cilicet & è conuerÅ¿o) Non omnino abÅ¿que ratio <lb/>ne dubitas quo pacto fieri poſſit ut <var>.a.e.</var> remotioris Å¿tationis ad <var>.a.e.</var> propinquioris ita <lb/>Å¿e habeat quema dmodum <var>.I.E.</var> ad <var>.E.V</var>. </s> - <s xml:space="preserve">Quapropter Å¿i prÄ™cedentem figuram dili- - <pb o="273" rhead="EPISTOL AE." n="285" file="0285" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0285"/> - genter inÅ¿pexeris, omnis tua dubitatio euaneÅ¿cet. in qua figura apertè vi debis cor-<lb/>reÅ¿pondentiam talium triangulorum inter Å¿e, nec magis, nec minus quam in infra-<lb/>Å¿cripta hic figura cernere licet, quamuis in hac, triangula quadrati, Å¿eparata Å¿int ab <lb/>imaginarijs <var>.A.E.V.</var> et <var>.A.E.I.</var> in Å¿upradicta vero coniuncta, & inuicem communican <lb/>tia in puncto <var>.u.i.</var> quod quidem nihil refert. </s> - <s xml:space="preserve">Dempta igitur <var>.a.e.</var> minori ex <var>.a.e.</var> ma-<lb/>iori, reliquum ita Å¿e habebit ad <var>.a.e.</var> minorem, vt, <var>V.I.</var> ad <var>.I.E.</var> quod nunc tibi <lb/>clarè patebit. </s> - <s xml:space="preserve">Vnde ex te poteris ordinem operationis proÅ¿equi, vt in cognitionem <lb/>peruenias ipÅ¿ius <var>.I.E.</var> ipÅ¿ius <var>.A.E.</var> & ipÅ¿ius <var>.I.A.</var> vel <var>.V.A</var>.</s> - </p> - <figure position="here"> - <image file="0285-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0285-01"/> - </figure> - <p> - <s xml:space="preserve">Sed <reg norm="quando" type="context">quãdo</reg> in proximiori Å¿tatione latus <var>.b.c.</var> in remotiori vero latus <var>.c.e.</var> Å¿ecatur à fi <lb/>lo (pro Å¿ecunda dubitatione) </s> - <s xml:space="preserve">Tunc oportet imaginatione conÅ¿iderare latus <var>.b.c.</var> in <lb/>re motiori Å¿tatione diÅ¿tentum eſſe vÅ¿que ad filum in puncto <var>.n.</var> vbi videbis <reg norm="triangulum" type="context">triangulÅ©</reg> <var>.<lb/>u.b.n.</var> Å¿imile triangulo <var>.A.E.V.</var> ita vt <var>.i.b.a.</var> Å¿imile Å¿uo <var>.A.E.I.</var> reperitur, vbi tam in vno <lb/> - <anchor type="figure" xlink:label="fig-0285-02a" xlink:href="fig-0285-02"/> - <pb o="274" rhead="IO. BAPT. BENED." n="286" file="0286" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0286"/> - quam in altero <var>.i.b.</var> et <var>u.b.</var> correÅ¿pondebit ipÅ¿i <var>.A.E.</var> et <var>.b.n.</var> ipÅ¿i <var>.E.V.</var> et <var>.b.a.</var> ipÅ¿i <var>.E.I.</var> <lb/>quapropter iubeo, vt quæras quantum Å¿it latus <var>.b.n.</var> ex regula de tribus, dicens Å¿i <var>.a.e.</var> <lb/>tribuit mihi <var>.e.u.</var> quid mihi dabit <var>.u.b</var>? </s> - <s xml:space="preserve">eo quod <var>.a.e.u.</var> Å¿imile eÅ¿t <var>.u.b.n.</var> reperto autem <lb/>latere <var>.b.n.</var> ex quo dempto <var>.b.a.</var> breuioris diÅ¿tantiÄ™, reÅ¿iduum reÅ¿pondebit ipÅ¿i <var>.I.V.</var> vt <lb/>Å¿cis, vnde proÅ¿equendo operationem tibi cognitam, obtinebis intentum, hoc eÅ¿t co <lb/>gnoÅ¿ces reliqua interualla. </s> - <s xml:space="preserve">Nihil enim miror demonÅ¿trationem Tartaleæ circa hu <lb/>iuÅ¿modi operationem te minime Å¿atisfeciſſe.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0285-02" xlink:href="fig-0285-02a"> - <image file="0285-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0285-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Quod autem quarta propoÅ¿itio illius Å¿criptoris, de quo nuper <reg norm="mecum" type="context">mecÅ©</reg> locutus es, vt <lb/>mihi dixiÅ¿ti, tua Å¿it, hoc enimego, nec affirmare, nec negare audeo, quamuis in mul <lb/>tis cum tua manuÅ¿cripta concordet. </s> - <s xml:space="preserve">Nam ſępæ cogitationes hominum in idem co-<lb/>incidunt, vt pluries cenÅ¿uit AriÅ¿to.</s> - </p> - <figure position="here"> - <image file="0286-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0286-01"/> - </figure> - </div> - <div type="letter" level="4" n="4"> - <head style="it" xml:space="preserve">Demonstrationes quorundam problematum Nicolai Tartalea <lb/>cum <reg norm="alijs" type="lig">alijs</reg> operationibus circa eadem Å¿ubiecta.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">AMor erga te meus Å¿anè Å¿ingularis, nullo modo <reg norm="permittit" type="simple">ê‘mittit</reg>, vt ea quæ TartaleÄ™ Å¿cri <lb/>pta <reg norm="examinando" type="context">examinãdo</reg> inuenerim, non tibi <reg norm="communicem" type="context">cõmunicem</reg>. </s> - <s xml:space="preserve">Hæc autem Å¿unt circa quæ <lb/>dam illius Authoris problemata, quorum primum ab ipÅ¿o Tartalea <lb/>Å¿criptum in .3. quæſito libr .4. tale eÅ¿t, is vult locare .3500. homines, eodem modo, <lb/>quo præſupponit locatos eſſe .1000. ita vt quilibet hominum ordo Å¿iue vt vulgo di-<lb/>citur filtia Å¿it .49. quapropter multiplicat quadratum ipÅ¿ius .49. quod eÅ¿t .2401. <reg norm="cum" type="context">cÅ©</reg> <lb/>numero .3500. propoÅ¿ito, productum verò .8403500. diuidit per .1000. vt proue-<lb/>niat .8403. cuius radix quadrata eÅ¿t .91. pro numero hominum vniuÅ¿cuiuÅ¿que ordinis <lb/>propoÅ¿iti numeri .3500.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius operationis ratione, cogitemus rectangulum <var>.a.b.</var> 1000. hominum, et <var>.d.<lb/>b.</var> Å¿it vna filtia Å¿iue ordo .49. <reg norm="hominum" type="wordlist">hoÄ©um</reg>, cuius <reg norm="quadratum" type="context">quadratÅ©</reg> Å¿it <var>.b.c.</var> 2401. imaginemur <reg norm="etiam" type="context">etiã</reg> re <lb/>ctangulum <var>.A.B.</var> 3500. hominum, quod Å¿upponemus Å¿imile rectangulo <var>.a.b.</var> et <var>.B.C.</var> - <pb o="275" rhead="EPISTOL AE." n="287" file="0287" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0287"/> - Å¿it quadratum ipÅ¿ius <var>.D.B</var>. </s> - <s xml:space="preserve">Nunc Å¿upponendo <var>.A.B.</var> Å¿imile <var>.a.b.</var> clarum erit ex diffini-<lb/>tione Å¿imilium figurarum, quod eadem proportio erit <var>.A.D.</var> ad <var>.D.B.</var> quÄ™ <var>.a.d.</var> ad <var>.d.<lb/>b.</var> hoc eÅ¿t <var>.A.D.</var> ad <var>.D.C.</var> vt <var>.a.d.</var> ad <var>.d.c.</var> hoc eÅ¿t <var>.A.B.</var> ad <var>.B.c.</var> vt <var>.a.b.</var> ad <var>.b.c.</var> ex prima <lb/>Å¿exti, vel .18. Å¿eu .19. Å¿eptimi, </s> - <s xml:space="preserve">tunc cum dixerimus Å¿i <var>.a.b.</var> ita reÅ¿pondet ad <var>.b.c.</var> ergo <var>.A.<lb/>B.</var> correÅ¿pondet etiam ita ad <var>.B.C.</var> </s> - <s xml:space="preserve">quare ex regula de tribus rectè fit multiplicando <var>.<lb/>A.B.</var> per <var>.b.c.</var> productum verò diuidendo per <var>.a.b.</var> ex .15. Å¿exti vel .20. Å¿eptimi, cuius <lb/>prouentus radix quadrata erit quod quærebatur.</s> - </p> - <p> - <s xml:space="preserve">Sed aliter idem poſſe fieri Å¿peculatus Å¿um, hoc eÅ¿t multiplicando numerum .49. <lb/>ordinis .1000. hominum <reg norm="cum" type="context">cÅ©</reg> radice quadrata numeri .3500. propoÅ¿iti, productum ve-<lb/>rò diuidere per radicem quadratam ipÅ¿ius .1000. vnde prouentus .91. erit numerus <lb/>vnius ordinis .3500. numeri <reg norm="propoÅ¿iti" type="simple">ê“poÅ¿iti</reg>.</s> - </p> - <p> - <s xml:space="preserve">Cuius <reg norm="operationis" type="simple">oê‘ationis</reg> Å¿peculatio eÅ¿t iÅ¿ta. <lb/> - <anchor type="figure" xlink:label="fig-0287-01a" xlink:href="fig-0287-01"/> - </s> - <s xml:space="preserve">Sit <var>.a.b.</var> quadratum .1000. et <var>.a.c.</var> Å¿ua <lb/>radix et <var>.a.d.</var> rectangulum propoÅ¿i-<lb/>tum ipÅ¿ius .1000. et <var>.a.e.</var> vnus ordo. <lb/></s> - <s xml:space="preserve">Sit etiam <var>.A.B.</var> quadratum .3500. & <lb/><var>A.C.</var> eius radix et <var>.A.D.</var> <reg norm="rectangulum" type="context">rectangulÅ©</reg> <lb/>ipÅ¿ius numeri .3500. propoÅ¿iti, Å¿imile <lb/>tamen rectangulo <var>.a.d.</var> et <var>.A.E.</var> eius <lb/>vnus ordo. </s> - <s xml:space="preserve"><reg norm="Cum" type="context">CÅ©</reg> enim <var>.a.b.</var> æquale Å¿it <lb/><var>a.d.</var> et <var>.A.B</var>: <var>A.D.</var> <reg norm="tunc" type="context">tÅ©c</reg> <var>.a.c.</var> erit media <lb/>proportionalis inter <var>.a.e.</var> et <var>.e.d.</var> & Å¿ic <lb/><var>A.C.</var> erit etiam media proportiona <lb/>lis inter <var>.A.E.</var> et <var>.E.D.</var> per .16. Å¿exti, <lb/>Å¿eu .20. Å¿eptimi, & quia proporrio. A <lb/>E. ad <var>.E.D.</var> æqualis eÅ¿t proportioni <var>.<lb/>a.e.</var> ad <var>.e.d.</var> cum <var>.A.D.</var> Å¿upponatur Å¿i-<lb/>mile <var>.a.d.</var> ergo proportio <var>.A.E.</var> ad <var>.A<lb/>C.</var> Ä™qualis erit proportioni <var>.a.e.</var> ad <var>.a.<lb/>c.</var> quÄ™ medietates Å¿unt <reg norm="totorum" type="context">totorÅ©</reg> æqua-<lb/>lium, rectè igitur fiet Å¿i procedamus <lb/>ex regula de tribus, dicendo Å¿i <var>.a.c.</var> <lb/><reg norm="correÅ¿pondet" type="context">correÅ¿põdet</reg> <var>.a.e.</var> tùc <var>.A.C.</var> <reg norm="correÅ¿pon" type="context">correÅ¿põ</reg> <lb/>det <var>.A.E.</var> ex Å¿upradictis .15. Å¿exti. vel <lb/>20. Å¿eptimi.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0287-01" xlink:href="fig-0287-01a"> - <image file="0287-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0287-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Ratio verò quarti quæſiti per Å¿e <lb/>patet, quod eÅ¿t inuenire <reg norm="pauimentum" type="context">pauimentÅ©</reg> <lb/>Å¿eu aream quadratam, in qua poſſint <lb/>locari quot homines volueris, ita in <lb/>ter Å¿e Å¿iti, ut vnuÅ¿quiÅ¿que occupet <num value="7">.<lb/>7.</num> pedes ipÅ¿ius areÄ™ in longitudinem <lb/>et .3. per latitudinem à lateribus.</s> - </p> - <p> - <s xml:space="preserve">Seu ex propoÅ¿ito hominum nume <lb/>ro inuenire numerum ipÅ¿orum loca-<lb/>bilem in aliqua area quadrata, ita, <lb/>vt vnuÅ¿quiÅ¿que occupet .21. pedes <lb/>quadratos ipÅ¿ius areæ.</s> - </p> - <pb o="276" rhead="IO. BAPT. BENED." n="288" file="0288" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0288"/> - <p> - <s xml:space="preserve">Sed aliter idem fieri poſſe inueni, hoc eÅ¿t <reg norm="multiplicando" type="context">multiplicãdo</reg> radicem quadratam pro-<lb/>poÅ¿iti numeri hominum per .21. & productum item multiplicando per eandem radi <lb/>cem, & huiuÅ¿modi producti radicem diuiden do per .3. vnde prouentus eſſet nume-<lb/>rus hominum vnius ordinis. </s> - <s xml:space="preserve"><reg norm="Exempli" type="context">Exẽpli</reg> gratia proponu<unsure/>ntur .3600. homines, multiplica <lb/>bimus huiuÅ¿modi numeri radicem <lb/>quadratam hoc eÅ¿t .60. per .21. hoc <lb/> - <anchor type="figure" xlink:label="fig-0288-01a" xlink:href="fig-0288-01"/> - eÅ¿t per productum quod fit ex .7. <reg norm="cum" type="context">cÅ©</reg> <lb/>3. & reÅ¿ultabit nobis .1260. quod Å¿i <lb/>multiplicabitur, per .60. hoc eÅ¿t per <lb/>eandem radicem, reÅ¿ultabit nobis <num value="75600">.<lb/>75600.</num> cuius producti radix qua-<lb/>drata eÅ¿t ferè .275. qua diuiÅ¿a per .3 <lb/>proueniet nobis .91. pro hominum <lb/>numero vnius ordinis.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0288-01" xlink:href="fig-0288-01a"> - <image file="0288-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0288-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Cuiusratio eÅ¿t iÅ¿ta, cogitemus nu <lb/>merum .3600. propoÅ¿itum eſſe qua <lb/>dratum <var>.a.b.</var> (Å¿ed non areæ) cuius ra <lb/>dix .60. Å¿it <var>.a.c.</var> & quia hic numerus <num value="60">.<lb/>60.</num> intelligitur eſſe hominum, quo-<lb/>rum <reg norm="vnuÅ¿quiÅ¿que" type="simple">vnuÅ¿quiÅ¿q;</reg> occupat .21. pedes <lb/>quadratos Å¿uperficiales ex Å¿uppoÅ¿i-<lb/>to, </s> - <s xml:space="preserve">& propterea multiplicatur, 60. <lb/>cum .21. vnde nobis veniat .1260. <lb/>quadrati Å¿uperficiales pro vnoquo-<lb/>que ordine, & <reg norm="quia" type="simple">ê—a</reg>.b.c. vt. latus qua-<lb/>drati <var>.a.b.</var> habet tot ordines homi-<lb/>num Å¿imiliter, hoc eÅ¿t .60. igi-<lb/>tur multiplicando .60. cum .1260. <lb/>habebimus totalem Å¿uperficiem <var>.a.<lb/>b.</var> ex .75600. quadratis Å¿uperficiali-<lb/>bus, quæ quadrata imaginemur lo-<lb/>cata eſſe in quodam totali quadra-<lb/>to, quod Å¿it <var>.e.f.</var> cuius radix Å¿it <var>.e.</var> g <num value="275">.<lb/>275.</num> pedum qui diuidantur per .3. <lb/>hoc eÅ¿t per numerum pedum latitu-<lb/>dinis & prouenient nobis .91. pro <lb/>numero <reg norm="hominum" type="context">hominÅ©</reg> <reg norm="vniuÅ¿cuiuÅ¿que" type="simple">vniuÅ¿cuiuÅ¿q;</reg> ordi-<lb/>nis, diuidendo poÅ¿tea latus <var>.f.g.</var> per <lb/>numerum Å¿patij inter vnum, & <reg norm="alium" type="context">aliÅ©</reg> <lb/>ordinem, quod eÅ¿t .7. proueniet <lb/>nobis .39. pro numero ordinum.</s> - </p> - <p> - <s xml:space="preserve">Aliter, & breuius etiam poſſumus idem inuenire, hoc eÅ¿t multiplicando <reg norm="nume- rum" type="context">nume-<lb/>rÅ©</reg> <reg norm="propoÅ¿itum" type="context">propoÅ¿itÅ©</reg> <reg norm="hominum" type="context">hominÅ©</reg> <reg norm="cum" type="context">cÅ©</reg> rectangulo .21. vnde venietnobis <reg norm="productum" type="simple context">ê“ductÅ©</reg> .75600 quod pro <lb/><reg norm="ductum" type="context">ductÅ©</reg> Å¿i accipiemus vt <reg norm="quadratum" type="context">quadratÅ©</reg>, cuius radix erit .275. quæ diuidatur <reg norm="per" type="simple">ê‘</reg> .3. habebi-<lb/>mus <reg norm="propofitum" type="simple context">ê“pofitÅ©</reg>. </s> - <s xml:space="preserve">Cuius ratio <reg norm="pendet" type="context">pẽdet</reg> à ſupradicta, eo <reg norm="quod" type="wordlist">qÄ‘</reg> loco <reg norm="multiplicandi" type="context">multiplicãdi</reg> <var>.a.c.</var> (hoc eÅ¿t <num value="60">. - <pb o="277" rhead="EPISTOL AE." n="289" file="0289" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0289"/> - 60.</num>) per .21. deinde <reg norm="productum" type="context">productÅ©</reg> etiam multiplicare per <var>.b.c.</var> (hoc eÅ¿t .60.) breuius erit <lb/>multiplicare totum numerum .3600. per .21. cÄ™tera verò facere, vt diximus.</s> - </p> - <p> - <s xml:space="preserve">Sed <reg norm="vnaquæque" type="simple">vnaquæq;</reg> iÅ¿tarum operationum, aliquid imperfectionis patitur, eo quod <reg norm="cum" type="context">cÅ©</reg> <lb/>aliquis cuperet quadratum perfectum Å¿uperficiale habere, <reg norm="abÅ¿que" type="simple">abÅ¿q;</reg> aliquo defectu, vel <lb/>exceÅ¿fu, aliquid aliud adhuc facere oporteret, hoc eÅ¿t, inuentum cum fuerit quadra <lb/>tum <var>.e.f.</var> cum Å¿uis radicibus <var>.e.g.</var> et <var>.g.f.</var> pedum .275. vnaquaque, vt in dicto exemplo <lb/>factum eÅ¿t, oportebit <reg norm="numerum" type="context">numerÅ©</reg> quærere minorem ipÅ¿o .275. Å¿ed proximiorem men-<lb/>furabilem ab .3. & ab .7. quod facilè fiet Å¿i diuiÅ¿erimus .275. per .21. detrahendo fra-<lb/>cta diuiÅ¿ionis ab ipÅ¿o .275. quæ quidem fracta in hoc exemplo Å¿unt .2. vnde remane-<lb/>bit .273. pro numero laterum quadrati Å¿uperficialis, in quo poſſent locari .3549. ho-<lb/>mines, eo ordine quo Å¿upra dictum eÅ¿t, quorum Å¿cilicet vnuÅ¿quiÅ¿que obtineat .21. <lb/>pedes Å¿uperÅ¿iciales.</s> - </p> - </div> - </div> - <div type="section" level="3" n="12"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE INTERVALLIS MVSICIS.</head> - <head style="it" xml:space="preserve">Cypriano Rorè MuÅ¿ico celeberrimo.</head> - <p> - <s xml:space="preserve"><emph style="sc">OPinio</emph> Hectoris EuÅ¿onij Cypriane mi dilectiſſime, vera non eÅ¿t, quod ali <lb/>quisrectè poſſit intelligere rationes conÅ¿onantiarum muÅ¿icæ, ablque co <lb/>gnitione illarum mediante ipÅ¿o Å¿enÅ¿u, imo nemo <reg norm="pont" type="context">põt</reg> calere <reg norm="theoriam" type="context">theoriã</reg> mu <lb/>Å¿ices, niÅ¿i aliquo <reg norm="mon" type="context">mõ</reg> verÅ¿atus Å¿it in praxi. </s> - <s xml:space="preserve"><reg norm="Quando" type="wordlist">QÅ©o</reg> enim cognoÅ¿ci <reg norm="poterunt" type="context">poterÅ©t</reg> <lb/>quid nam Å¿int diapaÅ¿on, diapente, diateſſeron, ditonus, Å¿emiditonus, hexacordum <lb/>maius, aut minus, & conÅ¿onantiæ ex ijs cum diapaÅ¿on compoÅ¿itæ, abÅ¿que earum <lb/>praxi? </s> - <s xml:space="preserve">vnde Å¿equetur <reg norm="neque" type="simple">neq;</reg> etiam cognoÅ¿ci poſſe interualla diſſonantia. </s> - <s xml:space="preserve">Et purus <lb/>practicus non intelliget quid Å¿it octaua, quinta, quarta, tertia maior, tertia minor, <lb/>Å¿exta maior, Å¿exta minor, decima maior, decima minor, vndecima, duode-<lb/>cima, decimatertia maior, aut minor, aut decimaquinta, & aliæ, ita vt ad <lb/>comparandam perfectionem muÅ¿icæ neceſſarium Å¿it, & thÄ™oriam & praxim ad-<lb/>diÅ¿cere. </s> - <s xml:space="preserve">Cum prÄ™terea Ludouicus Folianus apertè monÅ¿trarit (etiam Å¿i id à diato-<lb/>nico Å¿intono Ptolomei deÅ¿umpÅ¿erit) reperiri duos tonos, maiorem, & minorem, id-<lb/>eÅ¿t Å¿eÅ¿quioctauum, & Å¿eÅ¿quinonum, & tria Å¿emitonia, maius, minus, & mini-<lb/>mum, ideÅ¿t Å¿eÅ¿quiquintumdecimum, qui eÅ¿t maius, Å¿eÅ¿quiuigeÅ¿imum quartum id-<lb/>eÅ¿t minimum, & mediocre, vt .27. ad .25. quæ proportio Å¿uperbipartiens vigeÅ¿i-<lb/>maÅ¿quintas appellatur, & cum cognouerit Å¿emiditonum conÅ¿onantem eſſe Å¿eÅ¿qui-<lb/>quintum, ditonum Å¿eÅ¿quiquartum, & hexachordum minus, vt .8. ad .5. quæ propor-<lb/>tio dicitur Å¿upertripartiens quintas, & hexachordum maius, vt .5. ad .3. hÄ™c autem vo <lb/>catur Å¿uperbipartienstertias; </s> - <s xml:space="preserve">omnium Å¿implicium conÅ¿onantiarum cognitioni, ex-<lb/>tremam impoÅ¿uit manum. </s> - <s xml:space="preserve">Et quia tibi etiam oÅ¿tendere promiÅ¿i in modulationibus - <pb o="278" rhead="IO. BAPT. BENED." n="290" file="0290" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0290"/> - hæc omnia interualla Å¿eruari, ideo ad te mitto Å¿eptem hic Å¿ubÅ¿cripta exempla, in <lb/>quorum primo, & Å¿ecundo, inter dieÅ¿im, et <var>.b.</var> in Å¿uperiori, agnoÅ¿ces interuallum mi <lb/>nimi Å¿emitonij, & Å¿i ibi Å¿it dieÅ¿is, tanquam terminus ad quem, et <var>.b.</var> tanquam termi-<lb/>nus à quo: </s> - <s xml:space="preserve">quod autem inter dieÅ¿im et <var>.b.</var> Å¿it Å¿emitonium minimum, facilè agnoÅ¿ces <lb/>Å¿i Å¿ubtraxeris <reg norm="decimam" type="context">decimã</reg> <reg norm="minorem" type="context">minorẽ</reg> à maiori, <reg norm="quam" type="context">quã</reg> facit Å¿uperius <reg norm="cum" type="context">cÅ©</reg> inferiori, ideÅ¿t <reg norm="cum" type="context">cÅ©</reg> baſſu.</s> - </p> - <p> - <s xml:space="preserve">Qua quidem modulatione tu etiam vÅ¿us es in cantilena illa, quæ Galica lingua <lb/>incipit. </s> - <s xml:space="preserve">Hellas comment. </s> - <s xml:space="preserve">Eadem, ego quoque in meis cantilenis latino Å¿ermo-<lb/>ne compoÅ¿itis, quæ Moreta vocantur aliquando vÅ¿us Å¿um.</s> - </p> - <p> - <s xml:space="preserve">Sed in tertio exemplo inuenies Å¿emitonium maius, neceſſariò genitum in Å¿upe-<lb/>riori, Å¿i Å¿extam maiorem cum baſſu eſſicere volueris, quia tenor, à ditono cum <lb/>Å¿uperiori ad diapentem, & ad vniÅ¿onum cum baſſu procedit, vbi quieÅ¿cit, progre-<lb/>diendo poÅ¿tea baſſus ad Å¿emiditonum cum tenore, </s> - <s xml:space="preserve">tunc Å¿i à proportione huius Å¿ep-<lb/>timæ, quæ eÅ¿t vt .9. ad .5. hoc eÅ¿t Å¿uperquadripartiensquintas demptum fuerit hexa-<lb/>chordum maius, Å¿eu Å¿exta maior, quæ eÅ¿t vt .5. ad .3. remanebit proportio .27. ad .25. <lb/>quæ maior eÅ¿t quam .32. ad .30.</s> - </p> - <p> - <s xml:space="preserve">In quarto <reg norm="exemplo" type="context">exẽplo</reg> habebis Å¿emitonium minus in Å¿uperiori, quod quidem remanet <lb/>ex Å¿ubtractione ditoni <reg norm="conÅ¿onantis" type="context context">cõſonãtis</reg> ab diateſſaron <reg norm="comprÄ™henÅ¿a" type="context">cõprÄ™henÅ¿a</reg> à ſuperiori cum tenore.</s> - </p> - <p> - <s xml:space="preserve">In quinto exemplo videbis tonum minorem, & tonum maiorem Å¿ucceſſiuè vnum <lb/>poÅ¿t alium in tenore, detrahendo primo <reg norm="Å¿emiditonum" type="context">Å¿emiditonÅ©</reg> à diateſſaron, quod Å¿uperius fa-<lb/>cit cum tenore, vel detrahendo diapente ab hexachordo maiori, quod facit tenor <lb/>cum baſſu, vnde remanet tonus minor Å¿eÅ¿quinonus, detrahendo poÅ¿tea diateſſaron <lb/>à diapente, quod Å¿uperius facit cum tenore, remanebit tonus maior Å¿eÅ¿quioctauus.</s> - </p> - <p> - <s xml:space="preserve">In Å¿exto exemplo deinde videbis tenorem aÅ¿cendere per duos tonos minores Å¿uc <lb/>ceſſiuè vnum poÅ¿t alium in tenore, Å¿i <reg norm="demp" type="context">dẽp</reg> Å¿eris <reg norm="Å¿emiditonum" type="context">Å¿emiditonÅ©</reg> à diateſſaron <reg norm="cum" type="context">cÅ©</reg> Å¿uperiori.</s> - </p> - <p> - <s xml:space="preserve">In .7. <reg norm="exemplo" type="context">exẽplo</reg> demum videbis <reg norm="Å¿uperiorem" type="context">Å¿uperiorẽ</reg> aÅ¿cendere per duos tonos maiores Å¿ucceÅ¿-<lb/>Å¿iuè <reg norm="vnum" type="context">vnÅ©</reg> poÅ¿t <reg norm="alium" type="context">aliÅ©</reg>, Å¿i dempÅ¿eris diateſſaron à diapente, quod facittenor <reg norm="cum" type="context">cÅ©</reg> Å¿uperiori.</s> - </p> - <figure position="here"> - <image file="0290-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0290-01"/> - </figure> - <pb o="279" rhead="EPISTOL AE." n="291" file="0291" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0291"/> - </div> - </div> - <div type="section" level="3" n="13"> - <div type="letter" level="4" n="1"> - <head style="it" xml:space="preserve">De eodem Å¿ubiecto.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">QVod alià s tibi dixi, verum eÅ¿t, quod neceſſarium nullo modo Å¿it, vt modulan-<lb/>do, deÅ¿inat cantilena in eodem tono (quod Græci phthongum appel-<lb/>lant) à quo incÄ™pit. </s> - <s xml:space="preserve">immo neceſſariò Å¿emper ferè, altius, aut depræſ-<lb/>Å¿ius terminatur, per differentiam alicuius interualli æqualis, vel multiplicis ipÅ¿i com <lb/>mati Å¿eÅ¿quioctuageÅ¿imæ, quod quidem comma, quamuis cantabile non Å¿it, inÅ¿enÅ¿i-<lb/>biliter tamen generatur, & toties ab aliqua parte ipÅ¿ius cantilenæ poſſet <reg norm="dictum" type="context">dictÅ©</reg> com-<lb/>ma gcnerari, verÅ¿us acutum, vel graue, quod in fine ipÅ¿ius cantilenÄ™, vocis phtongus <lb/>reperiatur diÅ¿tans à primo per interuallum alicuius toni Å¿eÅ¿quinoni, Å¿eu Å¿eÅ¿quioctaui <lb/>plus, minúsue, vt in Å¿ubÅ¿cripto exemplo clarè videre potes in prima figura, vbi Å¿u-<lb/>perius à <var>.g.</var> primÄ™ cellulæ ad <var>.g.</var> Å¿ecundæ, intereÅ¿t vnum <reg norm="comma" type="context">cõma</reg>, eo quod progrediens <lb/>Å¿uperius in prima cellula ipÅ¿ius cantilenæ à quarta ad quintam cum tenore, aÅ¿cendit <lb/>per tonum Å¿eÅ¿quioctauum, à prima cellula deinde ad Å¿ecundam, tenor aÅ¿cendit Å¿imi-<lb/>liter per tonum Å¿eÅ¿quioctauum cum tranÅ¿eat à quinta ad quartam, quod facit cum <lb/>Å¿uperiori, in Å¿ecunda cellula poÅ¿tea, cum Å¿uperius deÅ¿cendat à maiori Å¿exta ad quin <lb/>tam, quod facit cum baſſu, Å¿eu à quarta ad tertiam minorem, quod facit cum teno-<lb/>re, </s> - <s xml:space="preserve">tunc deÅ¿cendit per tonum Å¿eÅ¿quinonum, ita quod non reuertitur ad <reg norm="eundem" type="context">eundẽ</reg> <reg norm="phthon" type="context">phthõ</reg> <lb/>gum, vbi prius erat in prima cellula, Å¿ed reperitur per <reg norm="vnum" type="context">vnÅ©</reg> <reg norm="coomma" type="context">coõma</reg> altius, <reg norm="quod" type="wordlist">qÄ‘</reg> <reg norm="quidem" type="context">quidẽ</reg> <lb/><reg norm="comma" type="context">cõma</reg> eÅ¿t differentia inter <reg norm="tonum" type="context">tonÅ©</reg> <reg norm="Å¿eÅ¿quioctauum" type="context">Å¿eÅ¿quioctauÅ©</reg> & <reg norm="Å¿eÅ¿quinonum" type="context">Å¿eÅ¿quinonÅ©</reg>, vt alias tibi <reg norm="demonÅ¿traui" type="context">demõſtraui</reg>.</s> - </p> - <p> - <s xml:space="preserve">Progrediendo igitur hoc modo, videbis quod cum tenor à ſecunda cellula ad ter <lb/>tiam tranÅ¿eat à tertia minori ad quartam, quod facit cum Å¿uperiori, deÅ¿cendit per <lb/>tonum Å¿eÅ¿quinonum, vnde in tertia cellula altius remanet quam in prima per <reg norm="vnum" type="context">vnÅ©</reg> <lb/>comma, in qua tertia cellula, cum iterum tranÅ¿eat Å¿uperius à quarta ad quintam, <reg norm="quod" type="simple">ê™</reg> <lb/>facit cum tenore, eleuatur per tonum Å¿eÅ¿quioctauum, proÅ¿equendo deinde tali ordi <lb/>ne, vidc<unsure/>bis in quarta cellula cantilenam auctam per duo commata, in Å¿exta, <reg norm="autem" type="wordlist">aÅ©t</reg> cel-<lb/>lula per tria commata, in octaua verò per .4. commata, vnde hac merhodo, Å¿i can-<lb/>tilena prolixior debito eſſet, vel Å¿i talia interualla frequentiora reperirentur, poſſet <lb/>cantilena à principio ad finem differre per .9. commata, & plus etiam, quæ quidem <lb/> - <anchor type="figure" xlink:label="fig-0291-01a" xlink:href="fig-0291-01"/> - <pb o="280" rhead="IO. BABPT. BENED." n="292" file="0292" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0292"/> - interualla Å¿uperant tonum Å¿eÅ¿quinonum, & Å¿i eſſent .10. commata<unsure/> Å¿uperarent tonum <lb/>Å¿eÅ¿quioctauum, eo quod aggregatum ex .9. commatibus continetur Å¿ub iÅ¿tis duobus <lb/>terminis hoc eÅ¿t .150094635296999121. et .134217728000000000. quæ qui-<lb/>dem proportio maior eÅ¿t proportione Å¿eÅ¿quinona, Å¿umma verò .10. commatum con <lb/>tinetur Å¿ub .12157665459056928801. et .10737418240000000000. quæ pro <lb/>portio maior eÅ¿t tono Å¿eÅ¿quioctauo, quod autem dico de aÅ¿cenÅ¿u cantilenæ, idem aÅ¿-<lb/>Å¿ero de eiuÅ¿dem deÅ¿cenÅ¿u, & hoc non tantum per interuallum illius commatis, quod <lb/>eÅ¿t differentia toni maioris à minori, Å¿ed etiam per illud quod eÅ¿t differentia Å¿emito <lb/>nij maioris à minori, vt in Å¿ecundo exemplo hic Å¿ubÅ¿cripto videre eÅ¿t in deÅ¿cenÅ¿u <lb/>cantilenæ per comma & comma, vt differentia inter Å¿emitonia maiora<unsure/> & minora, <lb/>vbi in prima cellula diÅ¿cedens baſſus à quinta cum Å¿uperiori, & ab vniÅ¿ono cum te-<lb/>nore deÅ¿cendens ad tertiam minorem cum ipÅ¿o tenore, facit cum Å¿uperiori <reg norm="Å¿eptimam" type="context">Å¿eptimã</reg> <lb/>maiorem, quæ eÅ¿t vt .9. ad .5. Å¿uperquadripartiensquintas Å¿cilicet, à qua diÅ¿cedens <lb/>poÅ¿tea Å¿uperius, vt faciat cum baſſu Å¿extam maiorem, deÅ¿cendit per Å¿emitonium ma <lb/>ius, à qua Å¿exta maiori deÅ¿cendens baſſus, & aÅ¿cendens per quartam, efficit cum di-<lb/>cto Å¿uperiori <reg norm="tertiam" type="context">tertiã</reg> maiorem, à qua diÅ¿cedens Å¿uperius, vt efficiat quartam cum ipÅ¿o <lb/>baſſu (qui quidem baſſus tranÅ¿it in tenorem) aÅ¿cendit per Å¿emitonium minus, diffe-<lb/>rens à ſemitonio maiori per vnum comma, vnde cantilena remanet depreſſa per <lb/>vnum comma. </s> - <s xml:space="preserve">cum deinde idem faciat inter tertiam, & quartam cellulam, per a-<lb/>liud comma deÅ¿cendit, & Å¿ic toties facere poſſet, vt poÅ¿tremo valde deprimatur <lb/>cantilena à primo phthongo.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0291-01" xlink:href="fig-0291-01a"> - <image file="0291-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0291-01"/> - </figure> - </div> - <figure position="here"> - <image file="0292-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0292-01"/> - </figure> - <p> - <s xml:space="preserve">Quod autem hic Å¿upradictum eÅ¿t, clrca inÅ¿trumenta artificialia non accidit, qua <lb/>propter organa, & clauicimbula concordantur certo quodam ordine, ita vt omnes <lb/>conÅ¿onantiæ, excepta diapaÅ¿on, Å¿eu octaua, Å¿int imperfectæ, hoc eÅ¿t, aut diminutÄ™, <lb/>aut Å¿uperantes à inÅ¿to, vt exempli gratia, omnes quintæ Å¿unt diminutæ, quartæ verò <lb/>Å¿unt exceſſiuÄ™, quod quidem fit, vt tertiæ, & Å¿extæ, non multum auribus diſſonent, <lb/>eo quod Å¿i quintæ omnes, & quartæ, perfectæ eſſent, </s> - <s xml:space="preserve">tunc omnes Å¿extÄ™, & tertiæ in-<lb/>tollerabiles eſſent, & à perfectis differrent per vnum comma, quod manifeÅ¿tum no-<lb/>bis erit hoc modo, accipiamus tres diapentes, Å¿eu quintas, conÅ¿equenter Å¿ucceſſiuas <lb/>vnam poÅ¿t aliam, hoc eÅ¿t tres proportiones Å¿eÅ¿quialteras, quarum aggregatum erit <lb/>vt .27. ad .8. quæ proportio, dicitur tripla Å¿upertripartiensoctauas, & quæ à practicis - <pb o="281" rhead="EPISTOL AE." n="293" file="0293" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0293"/> - appellaretur tertia decima maior, vt exempli gratia, eſſet Gamaut cum Å¿ecundo ela <lb/>mi, </s> - <s xml:space="preserve">tunc talis tertiadecima valde odioÅ¿a eſſet Å¿enÅ¿ui auditus, à qua, Å¿i dempta Å¿uerit <lb/>diapaÅ¿on, Å¿eu octaua, remaneret quoddam hexachordum maius, Å¿eu Å¿exta maior, au-<lb/>ribus valde inimica, Å¿ub proportione .13. ad 8. Å¿ed hæc proportio differret à propor <lb/>tione Å¿uperbipartientetertias perfecti hexachordi maioris, hoc eÅ¿t Å¿extæ maioris <lb/>conÅ¿onantis, per proportionem Å¿eÅ¿quioctuageÅ¿imam, hoc eÅ¿t per vnum comma, <lb/>quod quidem eÅ¿t etiam differentia aggregati trium Å¿eÅ¿quialterarum, à tertiadeci-<lb/>ma maiori conÅ¿onanti, hoc eÅ¿t exceſſus proportionis triplæ Å¿upertripartientis octa-<lb/>uas, Å¿upra triplam Å¿eÅ¿quitertiam, quæ eÅ¿t Å¿umma ipÅ¿ius duplæ cum Å¿uperbi partien-<lb/>tetertias.</s> - </p> - <p> - <s xml:space="preserve">A tali Å¿umma igitur trium Å¿eÅ¿quialterarum efficitur tertiadecima maior diſſonans <lb/>excedens conÅ¿onantem per vnum comma (cuius proportio eÅ¿t .81. ad .80.) quæ con-<lb/>fonans continetur in proportione .10. ad .3. vt Å¿upra dixi.</s> - </p> - <p> - <s xml:space="preserve">Hæcigitur eÅ¿t vera ratio, propter quam debemus comma diÅ¿tribuere in organis <lb/>& clauicymbalis, cum ab aggregato trium quintarum producatur talis exceſſus Å¿u-<lb/>pra perfectam, Å¿eu conÅ¿onantem tertiamdecimam maiorem, quod quidem aggre-<lb/>gatum, cum demptum fuerit à quintadecima, relinquet nobis tertiam minorem <lb/>diſſonantem, & mancam, per eundem exceſſum à conÅ¿onanti. </s> - <s xml:space="preserve">quæ quidem tertia <lb/>minor diſſonans Å¿ubtracta à diapente Å¿eu quinta perfecta, relinquet nobis tertiam <lb/>maiorem diſſonantem, qu@ conÅ¿onantem excedit per eundem exceſſum comma-<lb/>tis, & hæc demum tertia maior diſſonans, dempta ex diapaÅ¿on, Å¿eu octaua, relin-<lb/>quet nobis hexachordum minus, hoc eÅ¿t Å¿extam minorem diſſonantem, & muti-<lb/>lam à conÅ¿onanti per eundem exceſſum commatis. </s> - <s xml:space="preserve">De huiuÅ¿modi verò commatis <lb/>diÅ¿tributione doctiſſimè Å¿cripÅ¿it Excellentiſſimus Zarlinus in Å¿ecunda parte InÅ¿titu-<lb/>tionum Harmonicarum.</s> - </p> - <p> - <s xml:space="preserve">Sed quia Å¿enÅ¿us auditus non poteÅ¿t exactè cognoÅ¿cere debitam quantitatem ex-<lb/>ceſſus, vel defectus, intendendo vel remittendo chordas inÅ¿trumentorum, ideo hanc <lb/>viam Å¿equutus Å¿um.</s> - </p> - <p> - <s xml:space="preserve">Sit exempli gratia, hic Å¿ubÅ¿criptus ordo lignorum tangentium Å¿eu pinarum inci-<lb/>piens ab <var>.G.</var> deÅ¿inens ad <var>.g.</var> ita quod inter ipÅ¿os terminos Å¿it ea conÅ¿onantia quæ vo-<lb/>catur vigeÅ¿imaÅ¿ecunda, quæro primum <var>.b.</var> inter <var>.D.E.</var> quod eÅ¿t nigrum ipÅ¿ius Ela-<lb/>mi grauiſſimum, quod groſſo modo facio conÅ¿onans cum <var>.G.</var> grauiſſimo per Å¿ex-<lb/>tam <reg norm="minorem" type="context">minorẽ</reg>, deinde <reg norm="cum" type="context">cÅ©</reg> ipÅ¿o pt<unsure/>imo <var>.b.</var> ipÅ¿ius elami concordo Å¿uum octauum & quin-<lb/>tumdecimum, quo perfectius poſſum, deinde accipio <var>.b.</var> molle Å¿ecundum ipÅ¿ius. b <lb/>fabmi quod concordo cum <var>.b.</var> primo ipÅ¿ius Elami per quintam imperfectam, dein-<lb/>de cum hoc <var>.b.</var> Å¿ecundo ipÅ¿ius bÅ¿abmi concordo Å¿ecundum <var>.f.</var> per quintam Å¿imiliter <lb/>imperfectam, cum quo <var>.f.</var> poÅ¿tea concordo tertium <var>.c.</var> per Å¿imilem quintam, quem <lb/>tertium <var>.c.</var> poÅ¿tea confero <reg norm="cum" type="context">cÅ©</reg> Å¿ecundo <var>.b.</var> ipÅ¿ius elami, ita quod inter Å¿e conÅ¿onent per <lb/>Å¿extam maiorem tolerabilem, & Å¿i Å¿ic inuenio, tunc nihil muto has treschordas hoc. <lb/> - <anchor type="table" xlink:label="table-0293-01a" xlink:href="table-0293-01"/> - <pb o="288" rhead="IO. BAPT. BENED." n="294" file="0294" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0294"/> - eÅ¿t <var>.b.</var> Å¿ecundum ipÅ¿ius bfabmi, f. Å¿ecundum, et <var>.c.</var> tertium, Å¿ed Å¿i dictum tertium <var>.c.</var> <lb/>valde diſſonans eſſet cum <var>.b.</var> Å¿ecundo ipÅ¿ius elami, </s> - <s xml:space="preserve">tunc ipÅ¿um <var>.c.</var> intendo, aut re-<lb/>mitto, quouÅ¿que aliquo modo Å¿it conÅ¿onans per Å¿extam maiorem aliquantulum ex <lb/>ceſſiuam cum <var>.b.</var> Å¿ecundo ipÅ¿ius elami, cum quo poÅ¿tea <var>.c.</var> conÅ¿onare aliquantulum fa <lb/>cio <var>.f.</var> Å¿ecundum per quintam defectiuam, & cum hoc<unsure/> demum <var>.b.</var> Å¿ecundum ipÅ¿ius <lb/>bfabmi, quo facto concordo Å¿ecundum <var>.c.</var> cum tertio per octauam, cum quo Å¿ecun-<lb/>do <var>.c.</var> poÅ¿tea concordo tertium <var>.g.</var> per talem quintam, quod ipÅ¿um tertium <var>.g.</var> cum Å¿e-<lb/>cundo <var>.b.</var> ipÅ¿ius bfabmi conÅ¿onet tolerabiliter per Å¿extam maiorem <reg norm="aliquantulum" type="context">aliquãtulum</reg> ex-<lb/>ceſſiuam, </s> - <s xml:space="preserve">deinde cum iÅ¿to tertio <var>.g.</var> concordo tertium <var>.d.</var> per talem quintam, ita quod <lb/>ipÅ¿um <var>.3.d.</var> concordet tolerabiliter cum <var>.2.f.</var> per Å¿extam maiorem exceſſiuam, poÅ¿tea <lb/>cum hoc <var>.3.d.</var> concordo <var>.2.d.</var> per octauam perfecte, cum quo <var>.2.d.</var> poÅ¿tea concordo <var>.<lb/>3.a.</var> per quintam, vt in alijs <reg norm="factum" type="context">factÅ©</reg> eÅ¿t, ita vt <reg norm="cum" type="context">cÅ©</reg> <var>.2.c.</var> conÅ¿onet talis Å¿exta maior, vt Å¿upra <lb/>dictum eÅ¿t, cum quo <var>.3.a.</var> poÅ¿tea concordo <var>.3.e.</var> per quintam, vt dictum eÅ¿t, ita quod <lb/>cum <var>.3.g.</var> faciat Å¿extam maiorem vt Å¿upra, poÅ¿tea cum hoc <var>.e.</var> concordo <var>.2.e.</var> per octa <lb/>uam, cum quo concordo <var>.b.</var> quadrum tertium per quintam, vt dictum eÅ¿t, ita quod <reg norm="cum" type="context">cÅ©</reg> <lb/><var>2.d.</var> faciat Å¿extam maiorem Å¿imilem alijs Å¿uperius dictis, cum quo <var>.b.</var> quadrato tertio <lb/>concordo tertium nigrum ipÅ¿ius <var>.f.</var> per quintam, ita quod cum <var>.3 a.</var> faciat Å¿extam ma-<lb/>iorem, vt Å¿upra, deinde cum hoc concordo <var>.2.f.</var> nigrum per octauam, cum quo, per <lb/>quintam concordo <var>3.c.</var> nigrum ita quod cum <var>.2.e.</var> faciat Å¿extam dictam, demum <reg norm="cum" type="context">cÅ©</reg> <lb/>hoc concordo <var>.4.g.</var> nigrum per quintam, ita quod faciat cum <var>.3.b.</var> quadrato Å¿extam <lb/>dictam, & Å¿ic ad vltimam quintam peruenio, Å¿upra quod <var>.g.</var> nigrum nulla quinta am-<lb/>plius reperitur, poÅ¿tea cum iÅ¿tis chordis concordo per octauas omnes alias ab acutis <lb/>ad graues.</s> - </p> - <div level="5" n="2" type="float"> - <xhtml:table xlink:label="table-0293-01" xlink:href="table-0293-01a"> - <xhtml:tr> - <xhtml:td xml:space="preserve">G A B b C * D b E F * g * a b b c * d b e f * g * a b b c * d b e f * g *</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 4.</xhtml:td> - </xhtml:tr> - </xhtml:table> - </div> - <xhtml:table> - <xhtml:tr> - <xhtml:td xml:space="preserve">G A B b C * D b E F * g * a b b c * d b e f * g * a b b c * d b e f * g *</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 4.</xhtml:td> - </xhtml:tr> - </xhtml:table> - <p> - <s xml:space="preserve">Valde etiam admiratione dignum eÅ¿t, quod perÅ¿ectiores quæque conÅ¿onan <lb/>tiæ, ita in harmonica diuiÅ¿ione Å¿ibi inuicem conueniant, vt dia paÅ¿on cum diapente, <lb/>cum diapaÅ¿ondiapente, cum ditono, cum hexachordo maiori cum bisdiapaÅ¿on, <reg norm="cum" type="context">cÅ©</reg> <lb/>decimaleptima maiori. </s> - <s xml:space="preserve">Nam in ipÅ¿a diapaÅ¿on, harmonicè <reg norm="locantur" type="context">locãtur</reg> diapente in par <lb/>te grauiori, & diateÅ¿laron in acutiori. </s> - <s xml:space="preserve">In diapente verò harmonicè locantur ditonus <lb/>in parte grauiori, & Å¿emiditonus in acutiori. </s> - <s xml:space="preserve">In ditono harmonicè locantur tonus <lb/>maior in parte grauiori, & tonus minor in acutiori. </s> - <s xml:space="preserve">In hexachordo maiori, harmo-<lb/>nicè locantur diateſſaron in parte grauiori, & ditonus in acutiori. </s> - <s xml:space="preserve">In diapaÅ¿ondia-<lb/>pente, harmonic<unsure/>è locantur diapaÅ¿on in parte grauiori, & diapente in acutiori. <lb/></s> - <s xml:space="preserve">In bisdiapaÅ¿on, ha@monicè locantur decima maior in parte grauiori & hexachor-<lb/>dum minus in acutiori. </s> - <s xml:space="preserve">In decimaÅ¿eptima maiori, harmonicè locantur diapaÅ¿ondia-<lb/>pente in parte grauiori, & hexachordum maius in parte acutiori. </s> - <s xml:space="preserve">Ita quod ronus <lb/>Å¿eÅ¿quioctauus in ditono, proportionalis eÅ¿t ipÅ¿i ditono in diapente. </s> - <s xml:space="preserve">Tonus verò Å¿eÅ¿-<lb/>quinonus in ipÅ¿o ditono, proportionalis eÅ¿t triemitonio, vel Å¿eÅ¿quitonio Å¿eu Å¿emidi-<lb/>tono (quod idem eÅ¿t) in diapente. </s> - <s xml:space="preserve">Ditonus autem in diapente, proportionalis eÅ¿t <lb/>ipÅ¿i diapente in diapaÅ¿on. </s> - <s xml:space="preserve">SeÅ¿quitonus verò in diapente, proportionalis eÅ¿t diateÅ¿-<lb/>Å¿aron in diapaÅ¿on. </s> - <s xml:space="preserve">Et Å¿ic de Å¿ingulis. </s> - <s xml:space="preserve">Ita quod tonus Å¿eÅ¿quioctauus in ditono, dito-<lb/>mus in diapente, diateſſaron in hexachordo maiori, diapente in diapaÅ¿on, diapaÅ¿on <lb/>in diapaÅ¿ondiapente, decimamaior in bisdiapaÅ¿on, diapaÅ¿ondiapente in decima- - <pb o="283" rhead="EPISTOL AE." n="295" file="0295" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0295"/> - Å¿eptima maiori, omnia Å¿ibi inuicem Å¿unt proportionalia, idem etiam dico de reli-<lb/>quis partibus, cum relatæ fuerint ad Å¿ua tota.</s> - </p> - <p> - <s xml:space="preserve">Nec alienum mihi videtur à propoÅ¿ito inÅ¿tituto, Å¿peculari modum generationis <lb/>ipÅ¿arum Å¿implicium <reg norm="conÅ¿onantiarum" type="context">conÅ¿onantiarÅ©</reg>; </s> - <s xml:space="preserve">qui quidem modus fit ex quadam æquatione per <lb/>@uſſionum, Å¿eu æquali concurÅ¿u vndarum aeris, vel conterminatione earum.</s> - </p> - <p> - <s xml:space="preserve">Nam, nulli dubium eÅ¿t, quin vniÅ¿onus Å¿it prima principalis <reg norm="audituque" type="simple">audituqÌ</reg> amiciſſima, <lb/>nec non magis propria conÅ¿onantia; </s> - <s xml:space="preserve">& Å¿i intelligatur, vt punctus in linea, vel vnitas <lb/>in numero, quam immediate Å¿equitur diapaÅ¿on, ei Å¿imillima, poÅ¿t hanc verò diapen <lb/>te, <reg norm="cæteræque" type="simple">cæteræqÌ;</reg>. </s> - <s xml:space="preserve">Videamus igitur ordinem concurÅ¿us percuſſionum terminorum, Å¿eu <lb/>vndarum aeris, vnde Å¿onus generatur.</s> - </p> - <p> - <s xml:space="preserve">Concipiatur igitur mente monochordus, hoc eÅ¿t chorda diÅ¿tenta, quæ cum diuiÅ¿a <lb/>fuerit in duas æquales partes à ponticulo, </s> - <s xml:space="preserve">tunc <reg norm="vnaquæque" type="simple">vnaquæq;</reg> pars eundem Å¿onum pro-<lb/>feret, & ambæ formabunt vniÅ¿onum, quia eodem tempore, tot percuſſiones in aere <lb/>faciet vna partium illius chordæ, quot & altera: </s> - <s xml:space="preserve">ita vt vndæ aeris Å¿imul eant, & æqua <lb/>liter concurrant, abÅ¿que ulla interÅ¿ectione, vel fractione illarum inuicem.</s> - </p> - <p> - <s xml:space="preserve">Sed cum ponticulus ita diuiÅ¿erit chordam, vt relicta Å¿it eius tertia pars ab vno la-<lb/>tere, ab alio vero, duÄ™ tertiÄ™, </s> - <s xml:space="preserve">tunc maior pars, dupla erit minori, & <reg norm="Å¿onabunt" type="context">Å¿onabÅ©t</reg> ipÅ¿am dia <lb/>paÅ¿on conÅ¿onantiam, percuſſiones vero terminorum ipÅ¿ius, tali proportione Å¿e inui-<lb/>cem habebunt, ut in qualibet Å¿ecunda percuſſione minoris portionis ipÅ¿ius chordæ, <lb/>maior percutiet, Å¿eu concurret cum minori, eodem temporis inÅ¿tanti, cum ne-<lb/>mo Å¿it qui neÅ¿ciat, quod quo longior eÅ¿t chorda, etiam tardius moueatur, </s> - <s xml:space="preserve">quare <lb/>cum longior dupla Å¿it breuiori, & eiuÅ¿dem intenÅ¿ionis tam vna quam altera, tunc eo <lb/>tempore, quo longior vnum interuallum tremoris perfecerit, breuior duo interual-<lb/>la conficiet.</s> - </p> - <p> - <s xml:space="preserve">Cum autem ponticulus ita diuiÅ¿erit chordam, ut ab uno latere relinquantur duæ <lb/>quintæ partes, ab alio verò tres quintæ, ex quibus partibus generatur conÅ¿onantia <lb/>diapente; </s> - <s xml:space="preserve">tunc clarè patet, quod eadem proportione tardius erit vnum interuallu<unsure/>m <lb/>tremoris maioris portionis, vno interuallo tremoris minoris portionis, quam ma-<lb/>i<unsure/>or portio habet ad minorem; </s> - <s xml:space="preserve">hoc eÅ¿t tempus maioris interualli ad tempus minoris <lb/>erit <reg norm="Å¿eÅ¿quialterum" type="context">Å¿eÅ¿quialterÅ©</reg> quare non <reg norm="conuenient" type="context">cõuenient</reg> Å¿imul, niÅ¿i perfectis tribus interuallis mino-<lb/>ris portionis, & duobus maioris; </s> - <s xml:space="preserve">ita quod eadem proportio erit numeri interuallo-<lb/>rum minoris portionis ad interualla maioris, quæ longitudinis maioris portionis ad <lb/>longitudinem minoris; </s> - <s xml:space="preserve">vnde productum numeri portionis minoris ipÅ¿ius chordæ <lb/>in numerum interuallorum motus ipÅ¿ius portionis, æquale erit producto numeri <lb/>portionis maioris in numerum interuallorum ipÅ¿ius maioris portionis; </s> - <s xml:space="preserve">quæ quidem <lb/>producta ita Å¿e habebunt, vt in diapaÅ¿on, Å¿it binarius numerus; </s> - <s xml:space="preserve">in diapente verò <lb/>Å¿enarius; </s> - <s xml:space="preserve">in diateſſaron duodenarius, in hexachordo maiori quindenarius; </s> - <s xml:space="preserve">in di-<lb/>tono vicenarius, in Å¿emiditono tricenarius, demum in hexachordo minori quadra <lb/>genarius: </s> - <s xml:space="preserve">qui quidem numeri non abÅ¿que mirabili analogia conueniunt inuicem.</s> - </p> - <p> - <s xml:space="preserve">Voluptas autem, quam auditui afferunt conÅ¿onantiæ fit, quia leniuntur Å¿enÅ¿us, <lb/>quemadmodum <reg norm="contra" type="context">cõtra</reg>, dolor qui à diſſonantijs oritur, ab aÅ¿peritate naÅ¿citur, id quod <lb/>facilè vide<unsure/>re poteris cum conchordantur organorum fiÅ¿tulæ.</s> - </p> - <pb o="284" rhead="IO. BAPT. BENED." n="296" file="0296" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0296"/> - </div> - </div> - <div type="section" level="3" n="14"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE IVSTITIA COMMVTATIVA.</head> - <head style="it" xml:space="preserve">FranciÅ¿co Ferrario AnciÅ¿a IuriÅ¿conÅ¿ulto <reg norm="Å¿enatorique" type="simple">Å¿enatoriqÌê«</reg> apud <lb/>Å¿ubalpinos grauißimo.</head> - <p> - <s xml:space="preserve"><emph style="sc">SAepivs</emph> inter nos dum oportunitas vicinarum ædium, & amoris mutui <lb/>vis, ad familiaria trahunt colloquia ego de meis mathematicis, tu de tuis <lb/>legibus, in quibus tractandis magnum tibi nomen comparaÅ¿ti loquuti Å¿u <lb/>mus. </s> - <s xml:space="preserve">Cum vero nonnunquam de mirabili iuÅ¿titiæ commutatÃuæ inÅ¿titu <lb/>to non ingratus incidiſſet Å¿ermo, dixi modum, quo formam Å¿uam à proportionali-<lb/>tate arithmetica diÅ¿iuncta, & non a coniuncta deÅ¿umat, à nemine literis proditum <lb/>eſſe, libet autem nunc per otium latius explicare. </s> - <s xml:space="preserve">dixi enim à diÅ¿iuncta, & non con-<lb/>iuncta proportionalitate, quia in coniuncta, Å¿eu continua nullo pacto fieri poteÅ¿t <lb/>talis commutatio, cum Å¿emper quatuor terminos ad minus tranÅ¿eat, vt nunc vide-<lb/>bimus.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Petrus ex Å¿uis bonis tribuat Ioanni aliquid valoris quinquagin <lb/>ta aureorum.</s> - </p> - <p> - <s xml:space="preserve">Vnde priuÅ¿quam Ioannes aliquid ex Å¿uis bonis retribuat Petro, bona ipÅ¿ius Pe-<lb/>tri diminuta erunt per quinquaginta aureos, bona verò ipÅ¿ius Ioannis, aucta toti-<lb/>dem aureis.</s> - </p> - <p> - <s xml:space="preserve">Ecce nunc quo pacto conftituti Å¿unt .4. termini in proportionalitate aritmetica, <lb/>per quos Å¿it talis permutatio, Å¿ed nondum æquata, niÅ¿i fiat æqualis retributio à Ioan-<lb/>ne ad Petrum, vt videbimus.</s> - </p> - <p> - <s xml:space="preserve">Cogitentur itaque .4. termini aritmeticè proportionales <var>.C.A.B.D</var>. </s> - <s xml:space="preserve">Ita quod <var>.A.</var> <lb/>mediante Å¿ignificentur bona Ioannis <var>.B.</var> vero Petri, prius quam Petrus aliquid ex bo <lb/>nis Å¿uis tribuat Ioanni. </s> - <s xml:space="preserve">Tunc Petrus Å¿ecat partem vnam ex <var>.B.</var> <reg norm="eamque" type="simple">eamqÌ;</reg> dat ipÅ¿i Ioan-<lb/>ni, vnde ipÅ¿i Petro remanet <var>.D</var>. </s> - <s xml:space="preserve">Ioanni autem <var>.C.</var> quatuor igitur termini conÅ¿tituti <lb/>Å¿unt <var>.B.D.C.A.</var> quorum <var>.B.</var> primus <var>.A.</var> quartus <var>.C.</var> uero tertius <var>.D.</var> <reg norm="autem" type="wordlist">aÅ©t</reg> Å¿ecundus, Å¿ed <lb/>B. et <var>.A.</var> Å¿unt in Å¿ua naturali mediocritate abÅ¿que defectu vel exceſſu Å¿ui ipÅ¿ius. </s> - <s xml:space="preserve">Non <lb/>ita tamen Å¿e habet <var>.C.</var> et <var>.D.</var> quia <var>.D.</var> deficit <var>.C.</var> autem excedit à ſua priori quantitate. <lb/></s> - <s xml:space="preserve">Nihilominus iÅ¿ti .4. termini conÅ¿tituti Å¿unt in ipÅ¿a aritmetica proportionalitate, nam <lb/>eadem quantitate qua <var>.D.</var> diminuta eÅ¿t à <var>.B.</var> eadem <var>.C.</var> aucta eÅ¿t Å¿upra <var>.A</var>.</s> - </p> - <p> - <s xml:space="preserve">Sed quia <var>.B.</var> et <var>.A.</var> tantummodo iuÅ¿ti Å¿unt termini <var>.C.</var> uerò et <var>.D.</var> iniuÅ¿ti, vt ad Å¿uam <lb/>priorem æqualitatem reuertantur, oportebit ex <var>.C.</var> Å¿ecare aliquam partem æqualis <lb/>valoris ei, qua <var>.C.</var> Å¿uperat <var>.A.</var> vel qua <var>.D.</var> minor eÅ¿t <var>.B.</var> & ipÅ¿am partem addere ipÅ¿i <var>.D.</var> <lb/>vt bona Petri reuertantur ad priorem Å¿uam quantitatem ipÅ¿ius <var>.B.</var> & bona Ioannis <lb/>remaneant æqualia <var>.A.</var> vt prius.</s> - </p> - <p> - <s xml:space="preserve">Quare neceſſarium non eÅ¿t, vt talis proportionalitas Å¿it coniuncta (vt inquit Eu <lb/>f<gap/>atius Å¿eu Michael EpheÅ¿ius, <lb/>Å¿uper quinto capite libr. quin-<lb/> - <anchor type="figure" xlink:label="fig-0296-01a" xlink:href="fig-0296-01"/> - ti Ethicorum) tribus terminis <lb/>contenta, imò oportet ut ipÅ¿a <lb/>diÅ¿iuncta Å¿it, ut diximus, vbi <lb/>non eÅ¿t neceſſe quod <var>.A.</var> æqualis Å¿it <var>.B.</var> aliquo modo.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0296-01" xlink:href="fig-0296-01a"> - <image file="0296-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0296-01"/> - </figure> - </div> - <pb o="285" rhead="EPISTOL AE." n="297" file="0297" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0297"/> - </div> - </div> - <div type="section" level="3" n="15"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE MOTV MOLAE, ET TROCHI, DE AMPVL-<lb/>lis aquæ, de claritate aeris, & Lunæ noctu fulgentis, de æter-<lb/>nitate temporis, & infinito Å¿pacio extra <lb/>CÅ“lum, CÅ“<reg norm="lique" type="simple">liqÌ;</reg> figura.</head> - <head style="it" xml:space="preserve">Illust. Ioanni Paulo Capra NouarienÅ¿i Sabaudia Ducis boÅ¿picij <lb/>Magistro, viro ingeny praÅ¿tantia, & morum cando-<lb/>re, non minus quam familia nobili-<lb/>tate conÅ¿picuo.</head> - <p> - <s xml:space="preserve">SI vera eſſet animorum illa tranÅ¿migratio quam Å¿ibi Italicæ Å¿apientiæ Pa-<lb/>ter Pythagoras effinxerat, tuam, <reg norm="meamque" type="simple">meamqÌ;</reg> exiÅ¿timarem animam canis, <lb/>quandoque venatici fuiſſe.</s> - </p> - <p> - <s xml:space="preserve">Quæris à me literis tuis, an motus circularis alicuius molæ molendina <lb/>rie, Å¿i Å¿uper aliquod punctum, quaÅ¿i <reg norm="mathematicum" type="context">mathematicÅ©</reg>, quieÅ¿ceret, poſſet eſſe perpetuus, <lb/>cum aliquando eſſet mota, Å¿upponendo etiam eandem eſſe perfectè rotundam, & <lb/>lÄ™uigatam. </s> - <s xml:space="preserve">ReÅ¿pondeo huiuÅ¿modi motum nullo modo futurum perpetuum, nec <lb/>etiam multum duraturum, quia præterquam quod ab aere qui ei circumcirca <reg norm="aliquam" type="context">aliquã</reg> <lb/>reÅ¿iÅ¿tentiam facit Å¿tringitur, eÅ¿t etiam reÅ¿iÅ¿tentia partium illius corporis moti, quæ <lb/>cum motæ Å¿unt, natura, impetum habent efficiendi iter directum, vnde cum Å¿imul <lb/>iunctæ Å¿int, & earum vna continuata cum alia. </s> - <s xml:space="preserve">dum circulariter mouentur patiuntur <lb/>violentiam, & in huiuÅ¿modi motu per vim vnitæ manent, quia quanto magis mo-<lb/>uentur, tanto magis in ijs creÅ¿cit naturalis inclinatio recta eundi, vnde tanto magis <lb/>contra Å¿uammet naturam voluuntur, ita vt Å¿ecundum naturam quieÅ¿cant, quia cum <lb/>eis proprium Å¿it, quando Å¿unt motæ, eundi recta, quanto violentius voluuntur, tan-<lb/>to magis vna reÅ¿iÅ¿tit alteri, & quaÅ¿i retrò reuocat eam, quam antea reperitur habere.</s> - </p> - <p> - <s xml:space="preserve">Ab eiuÅ¿modi inclinatione rectitudinis motus partium alicuius corporis rotundi <lb/>fit, vt per aliquod temporis Å¿pacium, trochus cum magna violentia Å¿eipÅ¿um circun-<lb/>agens, omninò rectus quieÅ¿cat Å¿uper illam cuÅ¿pidem ferri quam habet, non incli-<lb/>nans Å¿e verÅ¿us mundi centrum, magis ad vnam <reg norm="partem" type="context">partẽ</reg>, quam ad aliam, cum quælibet <lb/>Å¿uarum partium in huiuÅ¿modi motu non inclinet omnino verÅ¿us <reg norm="mundi" type="context">mÅ©di</reg> centrum, Å¿ed <lb/>multo magis per tranÅ¿uerÅ¿um ad angulos rectos cum linea directionis, aut verticali, <lb/>aut orizontis axe, ita vt neceſſariò huiuÅ¿modi corpus rectum Å¿tare debeat. </s> - <s xml:space="preserve">Et quod <lb/>dico ipÅ¿as partes non omninò inclinare verÅ¿us mundi centrum, id ea ratione dico, <lb/>quia non abÅ¿olutè Å¿unt unquam priuatæ huiuÅ¿modi inclinatione, quÄ™ efficit vt ipÅ¿um <lb/>corpus eo puncto nitatur. </s> - <s xml:space="preserve">Verum tamen eÅ¿t, quod quanto magis eÅ¿t velox, tan-<lb/>to minus premit ipÅ¿um punctum, imò ipÅ¿um corpus <reg norm="tanto" type="context">tãto</reg> magis leue remanet. </s> - <s xml:space="preserve">Id <reg norm="quod" type="wordlist">qÄ‘</reg> <lb/>apertè patet <reg norm="Å¿umendo" type="context">Å¿umẽdo</reg> <reg norm="exemplum" type="context context">exẽplÅ©</reg> pilÄ™ alicuius arcus, aut <reg norm="alicuius" type="simple">alicuiê°</reg> alterius <reg norm="inÅ¿trumenti" type="context">inÅ¿trumẽti</reg>, Å¿eu ma <lb/>chinæ miſſilis, quæ pila quanto eÅ¿t velocior, in motu violento, tanto maiorem pro-<lb/>penÅ¿ionem habet rectius eundi, vnde verÅ¿us mundi centrum tanto minus inclinat, <lb/>& hanc ob cauÅ¿am leuior redditur. </s> - <s xml:space="preserve">Sed Å¿i clarius, hanc veritatem videre cupis, <lb/>cogita illud corpus, Trochum Å¿cilicet, dum velociſſime circunducitur Å¿ecari, Å¿eu <lb/>diuidi in multas partes, vnde uidebis illas omnes, non illico uerÅ¿us mundi centrum - <pb o="286" rhead="IO. BAPT. BENED." n="298" file="0298" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0298"/> - deÅ¿cendere, Å¿ed recta orizontaliter, vt ita dicam, moueri. </s> - <s xml:space="preserve">Id quod à nemine ad-<lb/>huc (quod Å¿ciam) in trocho eÅ¿t obÅ¿eruatum. </s> - <s xml:space="preserve">Ab huiuÅ¿modi motu trochi, aut hu-<lb/>ius generis corporis, clarè perÅ¿picitur, quà m errent peripatetici circa motum uio-<lb/>lentum alicuius corporis, qui exiÅ¿timant aerem qui Å¿ubintrat ad occupandum locum <lb/>à corpore relictum, ipÅ¿um corpus impellere, cum ab hoc, magis effectus contrarius <lb/>naÅ¿catur.</s> - </p> - <p> - <s xml:space="preserve">Quod deinde ampullæ iungantur in aqua, non fit ratione Å¿impathiæ, de qua lo-<lb/>quitur FracaÅ¿torus, nam per accidens iunguntur, quia cum alia ad aliam accedit, quÄ™ <lb/>libet earum tentat aÅ¿cendere ab ea parte, à qua inuicem hærent, quemadmodum <lb/>efficiunt iuxta labrum vaÅ¿is, ea enim Å¿uperficies a quæ vicina circunferentiæ vaÅ¿is ali <lb/>quantulum aÅ¿cendit in vaÅ¿e, qui non eÅ¿t omnino plenus.</s> - </p> - <p> - <s xml:space="preserve">Ad id deinde quod de claritate noctium Å¿cribis, miror cur non videas, quod <reg norm="quam" type="context">quã</reg> <lb/>to magis obÅ¿cura nox apparet, non dico ratione nubium, Å¿ed diÅ¿tantiæ Solis Å¿ub <lb/>orizonte ab eodem orizonte, tanto magis claram, & luminoÅ¿am Å¿eÅ¿e nobis oÅ¿ten-<lb/>dit Luna in quintadecima, quia cum Sol eÅ¿t in Sagittario, & Capricorno, Luna eÅ¿t <lb/>in Geminis, & in Cancro, vnde in media nocte, eius radius per valde exiguam <reg norm="quam" type="context">quã</reg> <lb/>titatem vaporum tranÅ¿it, quia tunc ipÅ¿a eÅ¿t valde propinqua axi orizontis, & præ-<lb/>terea in huiuÅ¿modi tempore anni & noctis, aer eÅ¿t magis purgatus, quà m in qualibet <lb/>alia temporis parte, quia hieme Sol non poteÅ¿t excitare multos vapores, & ij, qui <lb/>at tolluntur, nocte à frigore Å¿tatim congellati ratione grauitatis <reg norm="decidunt" type="context">decidÅ©t</reg>, </s> - <s xml:space="preserve">unde rema-<lb/>net aer multo clarior, qua ratione apparent Å¿tellæ minutæ, & CÅ“lum ijÅ¿dem ma-<lb/>gis ornatum, quà m in quolibet alio anni tempore.</s> - </p> - <p> - <s xml:space="preserve">Dicere deinde, quemadmodum hic mundus eÅ¿t ætatis Å¿eptem, aut octomillium <lb/>annorum, ita nunc potuiſſet eſſe (Å¿i Deus voluiſſet) ætatis quinquagintamillium; </s> - <s xml:space="preserve">er <lb/>go erat tempus; </s> - <s xml:space="preserve">ita Å¿e habet, ac Å¿i diceremus, quemadmodum hic mundus eÅ¿t tan-<lb/>tæ magnitudinis, ita etiam quinquagies maior eſſe potuiſſet, ergo eÅ¿t Å¿patium, aut <lb/>interuallum corporeum, quod eum capere potuiſſet.</s> - </p> - <p> - <s xml:space="preserve">Illud, nihil, AriÅ¿totelis extra CÄ™lum, nullo modo nobis inÅ¿eruit pro eiuÅ¿dem CÅ“ <lb/>li Å¿phÄ™rica rotunditate, cum cuiuÅ¿que alterius ex infinitis figuris CÄ™lum ipÅ¿um eſſe <lb/>poſſit, Å¿ecundum Å¿uam Å¿uperficiem conuexam. </s> - <s xml:space="preserve">Nam CÅ“lum ea ratione Å¿phÄ™ricum <lb/>non eÅ¿t, quod magis Å¿it capax, quia ei innumerahiles alias figuras adeo magnas po <lb/>terat concedere cauÅ¿a diuina: </s> - <s xml:space="preserve">Å¿ed Å¿phæricum eÅ¿t effectum, ne partem aliquam habe <lb/>ret Å¿ui termini Å¿uperfluam, quia nullum corpus à breuiori termino quam à ſphærico <lb/>terminari poteÅ¿t.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">Derèuolutione rota putealis & <reg norm="alijs" type="lig">alijs</reg> <lb/>problematibus.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">FVnis cui appenÅ¿a eÅ¿t Å¿itula, longè facilius axi inuoluitur, Å¿i ipÅ¿i axi affixa Å¿it rota. <lb/></s> - <s xml:space="preserve">atque item commodius eò fiet, quo amplior rota erit, & axis exilior. <lb/></s> - <s xml:space="preserve">Commodiſſimè autem, Å¿i ipÅ¿a rotæ extrema circunferentia, ex materia minori, <lb/>& denÅ¿iori, ac proinde grauiori conÅ¿tabit. </s> - <s xml:space="preserve">Cuius rei ratio multiplex eÅ¿t. </s> - <s xml:space="preserve">Nem-<lb/>pe quia omne corpus graue, aut Å¿ui natura, aut vi motum, in Å¿e recipit impreſſio- - <pb o="287" rhead="EPISTOL AE." n="299" file="0299" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0299"/> - nem & impetum motus, ita vt Å¿eparatum à virtute mouente per aliquod temporis <lb/>Å¿patium ex Å¿eipÅ¿o moueatur. </s> - <s xml:space="preserve">nam Å¿i Å¿ecundum naturam motu cieatur, Å¿uam veloci-<lb/>tatem Å¿emper augebit, cum in eo, impetus & impreſſio Å¿emper augeantur, quia <lb/>coniunctam habet per petuò virtutem mouentem. </s> - <s xml:space="preserve">Vnde manu mouendo rotam, ab <lb/><reg norm="eaque" type="simple">eaqÌ;</reg> eam remouendo rota Å¿tatim non quieÅ¿cet, Å¿ed per aliquod temporis Å¿patium <lb/>circunuertetur.</s> - </p> - <p> - <s xml:space="preserve">Secunda cauÅ¿a eÅ¿t, quia quoduis gr aue corpus, aut per naturam, aut per vim mo-<lb/>tum, rectitudinem itineris naturaliter appetat, quod clarè cognoÅ¿cere poſſumus, <lb/>proijciendo lapides funda, & circunducentes brachium, nam funes tanto maius <lb/>pondus acquirunt, & manum tanto magis onerant, quanto velocius voluitur funda, <lb/>& incitatur motus, quod ab appetitu naturali inÅ¿ito ei corpori per <reg norm="lineam" type="context">lineã</reg> rectam pro-<lb/>grediendi procedit. </s> - <s xml:space="preserve">Vnde fit, vt pondus circunferentiæ ipÅ¿ius rotæ, tanto facilius cir-<lb/>cunuoluatur, & ex Å¿eipÅ¿o tanto longiori tempore moueatur, quanto longius diÅ¿tat à <lb/>centro, cum eius iter tanto minus Å¿it curuum. </s> - <s xml:space="preserve">Hanc igitur ob cauÅ¿am, rota, quanto <lb/>maior erit, <reg norm="eiuÅ¿que" type="simple">eiuÅ¿qÌ;</reg> pondus tanto magis vicinum circunferentiæ, tanto magis durabit <lb/>impetus motus aſſumptus.</s> - </p> - <p> - <s xml:space="preserve">Tertia cauÅ¿a eÅ¿t, quod funis dum circunuoluitur, vicinius axi mathematico reuo-<lb/>lutionis, quam corpus graue circunferentiæ rotæ, ratione vectis, cum rota eÅ¿t in mo <lb/>tu, eius impetus non obtinet reÅ¿iÅ¿tentiam æqualem à contrario pondere aquæ in Å¿itu <lb/>la poÅ¿itæ.</s> - </p> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">De machina, qua aquam impellit & Å¿ubleuat.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">VNde Å¿it vt in fonte mandauerim, <lb/> - <anchor type="figure" xlink:label="fig-0299-01a" xlink:href="fig-0299-01"/> - vas Å¿eu mortarium in quod in-<lb/>greditur inÅ¿trumentum, quod aquam <lb/>impellit, diametrum Å¿uæ concauitatis, <lb/>habere non oportere maiorem dia-<lb/>metro fiÅ¿tulæ, per quam debet aÅ¿cende <lb/>re aqua, ratio eÅ¿t, quia Å¿i maius eſſet, <lb/>neceſſarium eſſet aliquod inÅ¿trumen-<lb/>tum quo aqua impelleretur multo gra <lb/>uius toto corpore aqueo, quod aptum <lb/>eſſet implere aliquam fiÅ¿tulam adeo <lb/>altam, vt eÅ¿t fons, quæ tamen eſſet <lb/>adeo lata vt eÅ¿t mortarium.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0299-01" xlink:href="fig-0299-01a"> - <image file="0299-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0299-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Sit exempli gratia, tota fiÅ¿tula, Å¿eu <lb/>hirundo, per quam aÅ¿cendit aqua <var>.f.</var> <lb/>mortarium verò Å¿it <var>.a.u.</var> quod tam <reg norm="altum" type="context">altÅ©</reg> <lb/>Å¿it vt <var>.f.</var> Å¿ed <var>.f.</var> anguÅ¿tior ipÅ¿o <var>.a.u</var>. </s> - <s xml:space="preserve">Nunc <lb/> - <anchor type="figure" xlink:label="fig-0299-02a" xlink:href="fig-0299-02"/> - cum repleta fuerint hæc duo vaÅ¿a, ma-<lb/>nifeÅ¿tum erit, quod aqua ipÅ¿ius <var>.f.</var> Å¿uffi-<lb/>ciens erit ad <reg norm="reÅ¿iÅ¿tendum" type="context">reÅ¿iÅ¿tẽdum</reg> toti aquæ <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <lb/><var>a.u.</var> & aqua <var>.a.u.</var> reÅ¿iÅ¿tet aquæ <var>.f.</var> quam-<lb/>uis aqua <var>.a.u.</var> maioris quantitatis Å¿it, & <lb/>ponderis ipÅ¿a <var>.f.</var> hoc autem euenit ex <lb/>eo quod aqua <var>.a.u.</var> <reg norm="non" type="context">nõ</reg> impellit aquam - <pb o="288" rhead="IO. BAPT. BENED." n="300" file="0300" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0300"/> - f. toto Å¿uo pondere, </s> - <s xml:space="preserve">propterea quod pondus diuiditur proportionaliter Å¿upra ba-<lb/>Å¿im vaÅ¿is.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0299-02" xlink:href="fig-0299-02a"> - <image file="0299-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0299-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Sit exempli <reg norm="gratia" type="wordlist">gr̃a</reg> vas aliquod <var>.b.d.n.m.</var> conicæ figuræ, Å¿eu <reg norm="truncus" type="context">trÅ©cus</reg> coni concaui aqua <lb/>plenus, cuius orificij diameter Å¿it <var>.b.d.</var> & multiplex diametro <var>.m.n.</var> infimæ baÅ¿is. </s> - <s xml:space="preserve">co-<lb/>gitemus etiam <var>.b.d.</var> diuiÅ¿um in tot partes, quarum <reg norm="vnaquæque" type="simple">vnaquæq;</reg> æqualis Å¿it <var>.m.n.</var> imagi-<lb/><reg norm="nemurque" type="simple">nemurqÌ;</reg> tot lineas perpendiculares deÅ¿cendere verÅ¿us mundi centrum ad puncta <var>r.<lb/>c.m.</var> et <var>.t.x.m.</var> vt in Å¿ubÅ¿cripta hic figura videre eÅ¿t, per quas cogitemus tot Å¿uperfi-<lb/>cies curuas <reg norm="conicasque" type="simple">conicasqÌ;</reg>, inter quas, mente concipienda eÅ¿t aqua, quÄ™ pondere Å¿uo quie <lb/>Å¿cet Å¿upra maiorem Å¿uperficiem illa, quæ æque diÅ¿tans eſſet mundi centro, Å¿eu quam <lb/>Å¿upra baÅ¿im <var>.m.n.</var> vt exempli gratia conÅ¿ideretur aqua inter <var>.g.m.</var> et <var>.s.r.</var> cuius pondus <lb/>diÅ¿tribuitur fecundum latitudinem <var>.m.r.</var> quæ maior eÅ¿t <var>.g.s.</var> cogitemus igitur <var>.m.c.</var> æ-<lb/>qualem eſſe <var>.g.s.</var> manifeÅ¿tum erit, quod <var>.m.c.</var> non Å¿uÅ¿tinebit totum pondus a quæ, quæ <lb/>inter <var>.g.m.</var> et <var>.s.r.</var> reperitur, eo quod omnis pars aquæ ad perpendiculum inclinat ver-<lb/>Å¿us mundi centrum, quapropter fundus Å¿eu baÅ¿is <var>.m.n.</var> non Å¿uÅ¿tinet aliud pondus <reg norm="quam" type="context">quã</reg> <lb/>aquæ <var>.f.m.</var> Å¿ed Å¿i quis hoc in dubium reuocaret dicens, quod aqua circunÅ¿cribens Å¿i-<lb/>tum corporis aquei <var>.f.m.</var> impellit lateraliter dictum corpus aqueum, reÅ¿pondendum <lb/>eÅ¿t, quod ex æquo huius corporis <var>.f.m.</var> aqua impellit etiam aquam circunÅ¿tantem, <lb/>eo, quod Å¿unt corpora homogenea, cum in corporibus homogeneis æquales partes <lb/>habeant æquales vires.</s> - </p> - <p> - <s xml:space="preserve">Sed redeundo ad vaÅ¿a <var>.a.u.</var> et <var>.f.</var> dico quod Å¿icut aqua <var>.f.</var> Å¿ufficit ad <reg norm="reÅ¿iÅ¿tendum" type="context">reÅ¿iÅ¿tendÅ©</reg> aquæ <lb/><var>a.u.</var> ita quodlibet aliud pondus Ä™quale <var>.f.</var> cuiuÅ¿uis materiæ, in fiÅ¿tula <var>.f.</var> poÅ¿itum, Å¿uffi-<lb/>ciens erit, dummodo illud corpus ita Å¿it adæquatum concauitati fiÅ¿tulæ <var>.f.</var> quod non <lb/>permittat tranÅ¿itum aliquem aquæ vel <lb/> - <anchor type="figure" xlink:label="fig-0300-01a" xlink:href="fig-0300-01"/> - aeris inter conuexum ipÅ¿ius corporis, <lb/>& deuexum fiÅ¿tulæ <var>.f.</var> & hoc ex Å¿e Å¿atis <lb/>patet, Å¿ed in vaÅ¿e <var>.a.u.</var> cum ex hypothe <lb/>Å¿i latius Å¿it ipÅ¿o <var>.f.</var> nullum aliud corpus <lb/>Å¿ufficiens erit ad reÅ¿iÅ¿tendum aquæ ip-<lb/>Å¿ius <var>.f.</var> quin tam graue Å¿it, quam tota <lb/>aqua <var>.a.u.</var> exiÅ¿tente <var>.a.u.</var> tam alto quam <lb/>f. </s> - <s xml:space="preserve">Vnde Å¿i aqua ipÅ¿ius <var>.f.</var> nil plus eſſet <lb/>quam vna tantummodo libra, & vas <var>.a.<lb/>u.</var> exiÅ¿teret latius ipÅ¿o <var>.f.</var> in decupla pro <lb/>portione, </s> - <s xml:space="preserve">tunc in ipÅ¿o <var>.a.u.</var> oporteret <lb/>corpus adæquatum ipÅ¿i concauitati po <lb/>nere, cuius pondus eſſet decem libra-<lb/>rum, vt Å¿ufficeret ad Å¿uÅ¿tinendum <reg norm="aquam" type="context">aquã</reg> <lb/>ipÅ¿ius <var>.f.</var> & ad im<unsure/><reg norm="pellendum" type="context">pellendÅ©</reg> ipÅ¿am <reg norm="aquam" type="context">aquã</reg> <var>.<lb/>f.</var> deberet eſſe plus quam decem libra-<lb/>rum. </s> - <s xml:space="preserve">Ponamus nunc illud corpus, ita <lb/> - <anchor type="figure" xlink:label="fig-0300-02a" xlink:href="fig-0300-02"/> - denÅ¿ius eſſe aqua, vt maius <reg norm="interuallum" type="context">interuallÅ©</reg> <lb/>non occupet, quam <var>.o.e.</var> corpus igitur <lb/><var>o.e.</var> Å¿ufficiens erit ad impellendum <lb/>aquam <var>.f.</var> & non eo minus.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0300-01" xlink:href="fig-0300-01a"> - <image file="0300-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0300-01"/> - </figure> - <figure xlink:label="fig-0300-02" xlink:href="fig-0300-02a"> - <image file="0300-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0300-02"/> - </figure> - </div> - <pb o="289" rhead="EPISTOL AE." n="301" file="0301" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0301"/> - </div> - </div> - <div type="section" level="3" n="16"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">NOVA SOLVTIO PROBLEMATIS DE VASE <lb/>pleno liquoris.</head> - <head style="it" xml:space="preserve">Nicolao Caluxio Serenißimi Ducis Sabaudia à ſecretis.</head> - <p> - <s xml:space="preserve"><emph style="sc">QVod</emph> à me poÅ¿tulas eÅ¿t problema ab alijs iam Å¿criptum, Å¿ed illud tibialio <lb/>medio Å¿oluam.</s> - </p> - <p> - <s xml:space="preserve">Proponitur vas <reg norm="plenum" type="context">plenÅ©</reg> liquore aliquo, puta aqua, <reg norm="quod" type="simple">ê™</reg> tres habeat fiÅ¿tulas <lb/>ad baÅ¿im, quarum vnaquæque poſſit euacuare ipÅ¿um vas, inæquales ta-<lb/>men, ita quod prima tam lata Å¿it, vt Å¿patio vnius horæ poſſit ipÅ¿um euacuare to-<lb/>tum; </s> - <s xml:space="preserve">Å¿ecunda vero Å¿patio duarum horarum, tertia autem Å¿patio trium hora-<unsure/>-<lb/>rum. </s> - <s xml:space="preserve">Tunc quæritur quanto tempore omnes tres fiÅ¿tulæ Å¿imul apertæ euacua-<lb/>bunt ipÅ¿um vas. </s> - <s xml:space="preserve">Ad hoc volo vt quæratur primo quanta pars aquæ vnaquÄ™quÄ™ fi-<lb/>Å¿tula euacuabit in aliquo dato tempore, quod facilè eÅ¿t, vt puta, prima fiÅ¿tu-<lb/>la, Å¿patio dimidiæ horæ euacuabit dimidium vas, eo quod Å¿patio integræ horæ po-<lb/>teÅ¿t totum euacuare, Å¿ecunda fiÅ¿tula, eodem temporis Å¿patio, euacuabit quartam <lb/>partem ipÅ¿ius vaÅ¿is, tertia verò fiÅ¿tula, eodemmet Å¿patio temporis dimidiæ horæ, <lb/>euacuabit Å¿extam partem ipÅ¿ius vaÅ¿is, quæ omnia fracta Å¿imul collecta faciunt vnde-<lb/>cim duodecimas partes totius vaÅ¿is, vnde manifeÅ¿tum erit, quod omnes fiÅ¿tulæ pari-<lb/>ter apertæ, Å¿patio dimidiÄ™ horæ euacuabunt vndecim duodecimas partes totius a-<lb/>quæ, Å¿ed nos cupimus Å¿cire, quanto tempore, totum vas euacuabitur, apertis omni <lb/>bus fiÅ¿tulis, quapropter dicemus ita; </s> - <s xml:space="preserve">Si vndecim duodecimæ partes conÅ¿umunt mi-<lb/>nuta .30. temporis, quantum conÅ¿ument omnes partes aquæ? </s> - <s xml:space="preserve">quæ Å¿unt .12. quare ex <lb/>regula de tribus prouenient nobis minuta .32. cum .8. vndecimis vnius minuti, hoc <lb/>eÅ¿t cum .43. Å¿ecundis horæ ferè, vel Å¿iaccipiemus tres quartas vnius horæ, </s> - <s xml:space="preserve">tunc pri-<lb/>ma fiÅ¿tula emittet tres quartas partes totius aquæ, Å¿ecunda, tres octauas <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> aquÄ™, <lb/>tertia verò, quarta pars, tunc omnia, hæc collecta, faciunt vnum integrum cum tri <lb/>bus octauis. </s> - <s xml:space="preserve">Si dixerimus igitur quando vnum integrum cum tribus octauis abÅ¿u-<lb/>mit .45. minuta temporis, ergo illud Å¿olum integrum abÅ¿umet idem vt Å¿upra hoc eÅ¿t <lb/>min .32. cum .8. vndecimis vnius minuti vel .43. Å¿ecundis. </s> - <s xml:space="preserve">Cuius rei Å¿peculatio <reg norm="tam" type="context">tã</reg> con <lb/>iuncta eÅ¿t operationi, quòd vna cognita, reliqua Å¿tatim cognoÅ¿citur.</s> - </p> - <p> - <s xml:space="preserve">Idem eueniet de implendo vaÅ¿e tribus Å¿imilibus fiÅ¿tulis mediantibus.</s> - </p> - <p> - <s xml:space="preserve">Secundum quæſitum ab alijs traditum, tuum etiam, aliter quoque poteÅ¿t Å¿olui, <lb/>propterea non prÄ™termittam tibi Å¿atisfacere.</s> - </p> - <p> - <s xml:space="preserve">Problema itaque tale eÅ¿t, vt Å¿it vas aliquod in <reg norm="quod" type="wordlist">qÄ‘</reg> infunditur aqua per tres fiÅ¿tu-<lb/>las, Å¿ed dum infunditur aqua, eadem egreditur per duas alias fiÅ¿tulas in fundo <lb/>vaÅ¿is poÅ¿itas, Å¿ed tres Å¿uperiores Å¿int inuicem proportionatæ, vt Å¿upradictum <lb/>eÅ¿t, primaqÌue inferiorum talis Å¿it, vt Å¿patio .4. horarum poſſit totum vas euacua-<lb/>re, Å¿ecunda autem poſſit Å¿patio .6. horarum idem facere, vnde ex Å¿upradictis, vas im <lb/>plebitur à tribus fiÅ¿tulis Å¿uperioribus, clauÅ¿is exiÅ¿tentibus inferioribus, Å¿patio tempo <lb/>ris <reg norm="minutorum" type="context">minutorÅ©</reg> .32. <reg norm="cum" type="context">cÅ©</reg> .8. vndecimis hoc eÅ¿t min .32. cum .43. Å¿ecundis, deinde per duas <lb/>fiÅ¿tulas inferiores poſſet euacuari Å¿patio <reg norm="temporis" type="context">tẽporis</reg> horarum .2. et. mi .24. exÅ¿upradictis.</s> - </p> - <p> - <s xml:space="preserve">Supponamus igitur omnes fiÅ¿tulas operari Å¿patio temporis minutorum .32. cum <lb/>Å¿ecundis .43. </s> - <s xml:space="preserve">tunc manifeÅ¿tum eÅ¿t quod vas non implebitur, eo Å¿patio min .32. cum <lb/>Å¿ecundis .43. Å¿ed tanta aqua deficiet, quanta ab inferioribus fiÅ¿tulis eo Å¿patio tempo <lb/>ris min .32. Å¿ecun .43. poteÅ¿t euacuari, </s> - <s xml:space="preserve">quare proportio partis vaÅ¿is vacuæ, ad totum <lb/>vas, erit vt min .33. ferè ad horas .2. min .24. quod per Å¿e patet, </s> - <s xml:space="preserve">tunc Å¿i demptum fue- - <pb o="290" rhead="IO. BAPT. BENED." n="302" file="0302" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0302"/> - rit tempus .33. minutorum ex h oris .2. min .24. reliquum erit hora .1. min .51. vnde <lb/>proportio aquæ, quæ in vaÅ¿e reperitur, ad eam, quæ totum vas implet, erit vt .111. <lb/>ad .144. </s> - <s xml:space="preserve">Quare nunc poſſumus rectè dicere ex regula de tribus Å¿i .111. indigent mi-<lb/>nuta .33. temporis, ergo .144. indigent min .43. horæ, in quo tempore implebitur to-<lb/>tum vas omnibus fiÅ¿tulis operantibus.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">Aliæ circuli noua paßiones.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">VTad aÅ¿cendendum ignis, & ad <reg norm="deÅ¿cendendum" type="context">deÅ¿cendendÅ©</reg> quicquid graue natum eÅ¿t, ita ad <lb/>Å¿peculandum humanus intellectus. </s> - <s xml:space="preserve">nec quieÅ¿cit, dum poteÅ¿t, eÅ¿t enim ver-<lb/>Å¿atile, <reg norm="agitandoque" type="simple">agitandoqÌ;</reg> Å¿eÅ¿e cauÅ¿is rerum immiÅ¿cere, & abditum aliquid rimari, <lb/>conatur, & eÅ¿t in nobis, quaÅ¿i Diogenes quidam in Dolio.</s> - </p> - <p> - <s xml:space="preserve">Tibi igitur mitto quod vltimò inueni, alias Å¿cilicet nouas circuli paſſiones, <lb/>quæ ita Å¿e <reg norm="habent" type="context">habẽt</reg>. </s> - <s xml:space="preserve">Sit circulus <var>.a.b.c.</var> in quo Å¿it <var>.a.d.</var> latus quadrati inÅ¿criptibilis in ipÅ¿o <lb/>circulo, ct <var>.b.c.</var> Å¿it diameter ad rectos cum <var>.a.d.</var> in puncto <var>.e.</var> quod medium erit inter <lb/>a. et <var>.d.</var> ex .3. tertij Eucli. Å¿it Å¿imiliter <var>.a.f.</var> contingens ipÅ¿um circulum in puncto <var>.a.</var> quæ <lb/>protracta Å¿it vÅ¿que ad punctum <var>.f.</var> interÅ¿ectionis cum diametro protracto, quod ita <lb/>eueniet cum anguli <var>.a.e.f.</var> et <var>.f.a.e.</var> minores Å¿int duobus rectis, eo quod angulus <var>.f.a.e.</var> <lb/>acutus Å¿it, cum <var>.a.d.</var> tranÅ¿eat inter centrum et <var>.f</var>.</s> - </p> - <p> - <s xml:space="preserve">Dico nunc quod productum diametri <var>.b.c.</var> in parte <var>.c.e.</var> ipÅ¿ius, æqualis erit produ-<lb/>cto ipÅ¿ius <var>.c.f.</var> in <var>.a.d</var>. </s> - <s xml:space="preserve">Protrahatur imaginatione <var>.b.a.</var> et <var>.a.c.</var> </s> - <s xml:space="preserve">vnde ex .26. tertij Euclid. <lb/>habebimus angulum <var>.d.a.c.</var> æqualem angulo <var>.a.b.c</var>. </s> - <s xml:space="preserve">Å¿ed ex .31. eiuÅ¿dem angulus <var>.f.a.<lb/>c.</var> æqualis eÅ¿t angulo <var>.b</var>. </s> - <s xml:space="preserve">quare æqualis erit angulo <var>.d.a.c.</var> & ita habebimus per .3. Å¿exti <lb/>eandem proportionem <var>.f.c.</var> ad <var>.c.e.</var> quæ <var>.f.a.</var> ad <var>.a.e.</var> Å¿ed <var>.a.f.</var> eÅ¿t æqualis Å¿emidiametro <lb/>circuli propoÅ¿iti, </s> - <s xml:space="preserve">propterea quod Å¿i producta fuerit à puncto <var>.a.</var> ad centrum <var>.o.</var> Å¿emi <lb/>diameter <var>.a.o.</var> hæc cum <var>.o.e.</var> faciet dimidium angulirecti, cum ex Å¿uppoÅ¿ito <var>.a.d.</var> la-<lb/>tus Å¿it quadrati inÅ¿criptibilis in ipÅ¿o circulo. </s> - <s xml:space="preserve">& cum <var>.a.f.</var> rectum ex .17. tertij, vnde an <lb/>gulus <var>.f.</var> erit Å¿imiliter medietas recti ex .32. primi, </s> - <s xml:space="preserve">quare ex .6. eiuÅ¿dem <var>.a.f.</var> æqualis <lb/>erit <var>.a.o</var>. </s> - <s xml:space="preserve">Ergo cum proportio <var>.f.c.</var> ad <var>.c.e.</var> Å¿it. vt <var>.f.a.</var> ad <var>.a.e.</var> erit Å¿imiliter vt <var>.b.c.</var> ad <var>.a.d.</var> <lb/>hoc eÅ¿t ut dupli ad duplum, vnde ex .15. Å¿exti <lb/>manifeÅ¿tum erit propoÅ¿itum, ex quo alia paÅ¿-<lb/> - <anchor type="figure" xlink:label="fig-0302-01a" xlink:href="fig-0302-01"/> - Å¿io oritur, hoc eÅ¿t, quod productum <var>.f.c.</var> in <var>.a.<lb/>d.</var> æ quale Å¿it qua drato ipÅ¿ius <var>.a.c.</var> ratio eÅ¿t, quia <lb/>quadratum <var>.a.c.</var> æ quale eÅ¿t producto <var>.b.c.</var> in <var>.c.<lb/>e.</var> eo quod <var>.a.c.</var> media proportionalis eÅ¿t inter <var>.<lb/>b.c.</var> et <var>.c.e.</var> ex Å¿imilitudine triangulorum <var>.a.b.c.</var> <lb/>et <var>.e.a.c.</var> nam anguli <var>.b.a.c.</var> et <var>.a.e.c.</var> recti Å¿unt <lb/>et <var>.c.</var> <reg norm="communis" type="context">cõmunis</reg>, vnde <var>.b.</var> erit æqualis <var>.e.a.c.</var> ex .32 <lb/>primi, Å¿equitur etiam, quod <var>.a.c.</var> Å¿it media pro <lb/>portionalis inter <var>.a.d.</var> et <var>.f.c.</var> & hæc etiam erit <lb/>alia circuli paſſio, & quia <var>.a.c.</var> eÅ¿t latus octago-<lb/>ni igitur tale latus <reg norm="medium" type="context">mediÅ©</reg> proportionale erit <lb/>inter latus quadrati. et <var>.f.c.</var> <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> circuli, quÄ™ <lb/>quidem <var>.f.c.</var> eÅ¿t una portio diametri quadrati circunÅ¿criptibilis ipÅ¿um circulum inter <lb/>circulum & angulum ipÅ¿ius quadrati.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0302-01" xlink:href="fig-0302-01a"> - <image file="0302-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0302-01"/> - </figure> - </div> - <pb o="291" rhead="EPISTOLAE." n="303" file="0303" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0303"/> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">Quod incendium, ex reflexione radiorum Å¿olarium, non fiat in cen <lb/>tro Å¿peculi Å¿pharici, & aliquid contra Cardanum, & <lb/>de moturadiorum Å¿olarium.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">ITerum tibi dico, quod radij illi Å¿olares, quià diuerÅ¿is punctis ipÅ¿ius Å¿olaris corpo-<lb/>ris veniunt, tranÅ¿euntes per centrum Å¿peculi Å¿phærici concaui, quamuis à ſuper-<lb/>ficie Å¿peculi ad centrum ipÅ¿um reflectantur, vt alÃâs tibi dixi, nihilominus nullo mo <lb/>do poſſunt aliquod obiectum incendere duabus ex cauÅ¿is, quarum vna eÅ¿t, quia cum <lb/>Sol valde remotus Å¿it à nobis, valde etiam acutus generatur angulus coni radiorum <lb/>in centro Å¿peculi, vnde à parua Å¿uperficie ipÅ¿ius Å¿peculi reflectuntur, </s> - <s xml:space="preserve">quare pauciſſi-<lb/>mi radij Å¿unt qui reflectantur in ipÅ¿o centro, & propterea non Å¿ufficiunt ad combu <lb/>Å¿tionem alicuius obiecti. </s> - <s xml:space="preserve">Alia verò cauÅ¿a eÅ¿t, quod quamuis multi, & Å¿ufficientes <lb/>radij fuiſſentad <reg norm="comburendum" type="context context">cõburendÅ©</reg> velociter quoduis obiectum. </s> - <s xml:space="preserve">impoſſibile tamen omnino <lb/>eÅ¿set, vt aliquod obiectum comburerent, propterea quod cum radij incidentes de-<lb/>beant per centrum tranÅ¿ire, obiectum combuÅ¿tibile, vt opacum, obÅ¿taret ipÅ¿is radijs, <lb/>ne vlterius tranÅ¿irent, vnde nulla fieret reflexio, Å¿ed etiam Å¿i dicti radij in centro re <lb/>flexi, Å¿ufficerent ad combuÅ¿tionem, incidentes hoc magis eſſicerent. </s> - <s xml:space="preserve">& ita abÅ¿que <lb/>vllo Å¿peculo, omnia & in quolibet loco comburerentur, quod manifeÅ¿tè falÅ¿um eÅ¿t. <lb/></s> - <s xml:space="preserve">DeÅ¿ine igitur mihi citate Lucillum Philalteum, qui in philoÅ¿ophia mathematica <lb/>fuit omnium imperitiſſimus. </s> - <s xml:space="preserve">Verum Å¿peculum vÅ¿torium illud eÅ¿t quod ab Alhazem <lb/></s> - <s xml:space="preserve">Deinde à Vitellìone deÅ¿cribitur.</s> - </p> - <p> - <s xml:space="preserve">Quod deinde verum Å¿it, <reg norm="vmbram" type="context">vmbrã</reg> vniuÅ¿cuiuÅ¿que corporis opaci à Sole productam <lb/>Å¿emper eſſe centum <reg norm="nouemque" type="simple">nouemqÌ;</reg> vicibus maiorem diametro eiuÅ¿dem corporis, nego.</s> - </p> - <p> - <s xml:space="preserve">Imaginemur <var>.s.</var> 1. diametrum eſſe illius circuli, quo vltimi radij Å¿olares veniunt tan <lb/>gentes corpus cuius diameter Å¿it <var>.c.e.</var> et <var>.a.i.</var> Å¿it diameter alterius circuli eiuÅ¿dem cor-<lb/>poris Å¿olaris à quo vltimi radij veniunt tangentes corpus, cuius diameter Å¿it <var>.f.g.</var> in <lb/>eadem diÅ¿tantia, & eodem Å¿itu prioris corporis. </s> - <s xml:space="preserve">Tunc conus vmbræ ipÅ¿ius <var>.f.g.</var> Å¿it <var>.f.<lb/>g.q.</var> & ipÅ¿ius <var>.c.e.</var> Å¿it <var>.c.n.e.</var> centrum autem Å¿olare Å¿it <var>.o.</var> conorum verò axes Å¿int <var>.t.n.q</var>. <lb/></s> - <s xml:space="preserve">tunc ex Å¿uppoÅ¿ito <var>.q.f.a</var>: <var>n.c.s</var>: <var>n.e.l</var>: et <var>.q.g.i.</var> erunt omnes contigui corpori Å¿olari, vn-<lb/>de ex .17. tertij Eucli. anguli <var>.o.a.q.</var> et <var>.o.s.n.</var> erunt recti. </s> - <s xml:space="preserve">protracta deinde cum fu erit <lb/><var>a.s.</var> habebimus angulos <var>.u.a.s.</var> et <var>.u.s.a.</var> minores duobus rectis. </s> - <s xml:space="preserve">Quare <var>.n.s.</var> concurret <lb/> - <anchor type="figure" xlink:label="fig-0303-01a" xlink:href="fig-0303-01"/> - <pb o="292" rhead="IO. BAPT. BENED." n="304" file="0304" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0304"/> - cum <var>.a.q.</var> in puncto <var>.u</var>. </s> - <s xml:space="preserve">Nunc verò Å¿i vmbra <var>.t.q.</var> tanto maior eÅ¿t <var>.f.g.</var> quanto .109. eÅ¿t <lb/>vno et <var>.t.n.</var> etiam <reg norm="tanto" type="context">tãto</reg> maior <var>.c.e.</var> ergò eadem proportio erit <var>.q.t.</var> ad <var>.t.f.</var> quÄ™ <var>.n.t.</var> ad <var>.t.<lb/>c.</var> Å¿ed cum angulus <var>.t.</var> communis Å¿it ambobus triangulis <var>.q.t.f.</var> et <var>.n.t.c.</var> Å¿equitur ex .6. <lb/>Å¿exti dictos triangulos æ quiangulos eſſe. </s> - <s xml:space="preserve">Vnde Å¿i anguli <var>.t.n.c.</var> et <var>.t.q.f.</var> æ quales inui <lb/>cem Å¿unt, ergo <var>.q.f.</var> æquidiÅ¿tans erit <var>.n.c.</var> quod eÅ¿t impoſſibile, quia nunc demonÅ¿tra-<lb/>uimus ipÅ¿as concurrere in puncto <var>.u</var>. </s> - <s xml:space="preserve">Quare non eÅ¿t eadem proportio <var>.q.t.</var> ad <var>.t.f.</var> quæ <lb/><var>n.t.</var> ad <var>.t.c.</var> decipitur ergo Cardanus in .4. lib. de Å¿ubtilitate.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0303-01" xlink:href="fig-0303-01a"> - <image file="0303-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0303-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Circa illud deinde quod à me quæris, hoc eÅ¿t, quæ Å¿it cauÅ¿a, quod nos videmus <lb/>radium Å¿olarem tardiſſimè moueri, cum alias tibi dixerim ipÅ¿um qualibet hora cir-<lb/>ca terram quindecim gradus perficere, reÅ¿pondeo, quod radius ille quem videmus, <lb/>exempli gratia, in aliquo cubiculo, nunquam eÅ¿t idem numero, Å¿ed quia ipÅ¿i radij <lb/>nullo modo differunt inter Å¿e, niÅ¿i in numero, proptera putamus eundem Å¿emper eſſe, <lb/>cum Å¿emper alius, atque alius Å¿it, quorum vnuÅ¿quiÅ¿que (de illis loquor, qui ad hunc <lb/>terræ globum perueniunt) circa terram reuoluitur Å¿patio .24. horarum, & cum quili <lb/>bet circulus diuidatur in .360. gradus, quorum vigeÅ¿imaquarta pars eÅ¿t .15. verum <lb/>eÅ¿t igitur, quod tibi iam dixeram.</s> - </p> - <figure position="here"> - <image file="0304-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0304-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="17"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">OPERATIONES DIVERSAE AB ALIIS <lb/>Michaelis Stifelij.</head> - <head style="it" xml:space="preserve">Conrado Terl.</head> - <p> - <s xml:space="preserve"><emph style="sc">QVod</emph> in .2. exemplo. II. cap. Stifelius Å¿cribit in .3. lib. pag .282. non nego <lb/>quin pulchrum Å¿it, Å¿ed alijs pulchrioribus modis poſſumus illud idem de-<lb/>monÅ¿trare; </s> - <s xml:space="preserve">cogita igitur Å¿uperficiem rectangulam, cuius medietas Å¿it <reg norm="triam" type="context">triã</reg> <lb/>gulus rectangulus <var>.a.b.g.</var> vnde ex Å¿uppoÅ¿ito nobis cognita erit Å¿uperficies <lb/>ipÅ¿ius trianguli, tanquam dimidium totius parallelogrammi rectanguli cogniti. <lb/></s> - <s xml:space="preserve">Quare ex .25. Å¿ecundi triangulorum <reg norm="Monteregij" type="context">Mõteregij</reg>, cognita nobis <reg norm="erunt" type="context">erÅ©t</reg> latera <var>.a.b.</var> et <var>.b.g</var>.</s> - </p> - <p> - <s xml:space="preserve">Alia etiam breuiori methodo idem poſſumus eſſicere, mediante angulo <var>.b.</var> recto, <lb/>eo quod cum nobis cognita Å¿it Å¿uperficies trianguli Å¿imul <reg norm="cum" type="context">cÅ©</reg> baÅ¿i <var>.a.g.</var> cognita etiam <lb/>nobis fit perpendicularis <var>.b.d.</var> à puncto <var>.b.</var> ad baÅ¿im, & conÅ¿equenter cognitum no-<lb/>bis erit productum ipÅ¿ius <var>.a.d.</var> in <var>.d.g.</var> & quia nobis cognita eÅ¿t <var>.a.g.</var> & eius medietas, - <pb o="293" rhead="EPISTOL AE." n="305" file="0305" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0305"/> - ideo vnaquæque eius pars <var>.a.d.</var> et <var>.d.g.</var> Å¿imiliter nobis cognita erit ex quinta Å¿ecundi <lb/>Eucl. </s> - <s xml:space="preserve">vnde ex penultima primi habebimus propoÅ¿itum.</s> - </p> - <p> - <s xml:space="preserve">Poſſumus item circulum mente concipere cuius <var>.a.g.</var> Å¿it diameter, & ab eius cen-<lb/>tro <var>.e.</var> protracta cum fuerit <var>.e.b.</var> quæ nobis cognita erit, vt medietas ipÅ¿ius <var>.a.g.</var> de cu <lb/>ius potentia, dempta <reg norm="cum" type="context">cÅ©</reg> fuerit potentia <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <var>b.o.</var> remanebit nobis potentia ipÅ¿ius <var>.d.<lb/>e.</var> & ita eius longitudo, quæ addita medietati <var>.e.g.</var> & detracta à dimidio <var>.e.d.</var> erunt <lb/>nobis cognitæ <var>.a.d.</var> et <var>.d.g.</var> vnde <var>.b.g.</var> et <var>.b.d.</var> remanebunt nobis cognitæ ex dicta pe-<lb/>nultima primi Eucli. </s> - <s xml:space="preserve">huiuÅ¿modi figuram videbis in dicto .25. problemate .2. li. Mon-<lb/>tisregij.</s> - </p> - <p> - <s xml:space="preserve">Aliter etiam poſſumus hoc idem efficere.</s> - </p> - <p> - <s xml:space="preserve">Sit rectangulus hic Å¿ubÅ¿criptus <var>.a.b.c.u.</var> Å¿uperficiei cognitÄ™ Å¿imul cum diametro <var>.a.<lb/>c.</var> extendatur imaginatione <var>.b.c.</var> vÅ¿que ad, f. ita quod <var>.c.f.</var> æqualis Å¿it <var>.c.u.</var> intelligan-<lb/><reg norm="turque" type="simple">turqÌ;</reg> quadrata <var>.g.f</var>: <var>g.u.</var> ct <var>.u.f.</var> vnde <reg norm="summa" type="context">sÅ©ma</reg> <reg norm="quadratorum" type="context">quadratorÅ©</reg> <var>.g.u</var>:u.f. cognita nobis erit ex <lb/>penultima primi. </s> - <s xml:space="preserve">nam <var>.a.c.</var> data nobis fuit, quare <reg norm="Å¿ummam" type="context">Å¿ummã</reg> <var>.g.u</var>:u.b: et <var>.u.f.</var> cognoÅ¿ce-<lb/>mus, cui <reg norm="summæ" type="context">sÅ©mæ</reg> addito Å¿uplemento <var>.d.e.</var> æ quali <var>.u.<lb/>b.</var> dabit nobis <reg norm="cognitum" type="context">cognitÅ©</reg> quadrarum <var>.g.f.</var> totale, qua <lb/> - <anchor type="figure" xlink:label="fig-0305-01a" xlink:href="fig-0305-01"/> - re cognoÅ¿cetur eius radix <var>.b.f.</var> cognita igitur <var>.b.f.</var> <lb/>cum pro ducto <var>.b.u.</var> illico ex .5. Å¿ecundi cognoÅ¿ce-<lb/>tur <var>.b.c.</var> et <var>.c.f.</var> forte cognita <var>.b.f.</var> diuiÅ¿a <reg norm="per" type="simple">ê‘</reg> æqualia <lb/>in puncto <var>.t.</var> & per inæqualiz in <reg norm="puncto" type="context">pÅ©cto</reg> <var>.c</var>. </s> - <s xml:space="preserve">Nam qua <lb/><reg norm="dratum" type="context">dratÅ©</reg> ipÅ¿ius <var>.t.f.</var> cognitum, Ä™quatur <reg norm="rectangulo" type="context">rectãgulo</reg> <var>.b.u.</var> <lb/><reg norm="cum" type="context">cÅ©</reg> quadrato ipÅ¿ius <var>.t.c.</var> <reg norm="dempto" type="context">dẽpto</reg> igitur rectangulo, <var>b.<lb/>u.</var> ex quadrato ipÅ¿ius <var>.t.f.</var> relinquetur quadratum <lb/><reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <var>.t.c.</var> cognitum & eius radix <var>.t.c.</var> qua addita ipÅ¿i <lb/>medietati <var>.b.t.</var> & <reg norm="dempta" type="context">dẽpta</reg> ex medietate <var>.f.t.</var> relinque-<lb/>tur propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0305-01" xlink:href="fig-0305-01a"> - <image file="0305-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0305-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Similiter de tertio exemplo eiuÅ¿dem Stifelij <lb/>infero.</s> - </p> - <p> - <s xml:space="preserve">Sit rectangulus <var>.a.b.c.u.</var> cuius diametri <var>.a.c.</var> quantitas, Å¿imul cum proportione late <lb/>rum <var>.b.c.</var> et <var>.b.a.</var> nobis data Å¿it. </s> - <s xml:space="preserve">cum autem Å¿cire voluerimus eius Å¿uperficiem <var>.b.u.</var> cla-<lb/>rum eÅ¿t, quod cum nobis data Å¿it proportio <var>.b.c.</var> ad <var>.b.a.</var> illico cognoÅ¿cemus <reg norm="etiam" type="context">etiã</reg> pro-<lb/>portionem quadrati ipÅ¿ius <var>.b.c.</var> ad quadratum ip-<lb/> - <anchor type="figure" xlink:label="fig-0305-02a" xlink:href="fig-0305-02"/> - Å¿ius <var>.b.a.</var> cum dupla Å¿it ei quæ <var>.b.c.</var> ad <var>.b.a.</var> ita etiam <lb/>& aggregati dictorum quadratorum ad quadra-<lb/>tum ipÅ¿ius <var>.b.a.</var> hoc eÅ¿t nota erit nobis proportio <lb/>quadrati ipÅ¿ius <var>.a.c.</var> diagonalis ad quadratum ip-<lb/>Å¿ius <var>.a.b.</var> idem dico de quadrato <var>.b.c.</var> ideÅ¿t quod <lb/>proportio quadrati ipÅ¿ius <var>.a.c.</var> ad quadratum <var>.b.c.</var> <lb/>cognita nobis erit, Å¿ed <var>.a.c.</var> data nobis fuit, qua-<lb/>re cognoÅ¿cemus etiam omnia dicta quadrata eo-<lb/><reg norm="rumque" type="simple">rumqÌ;</reg> radices <var>.a.b.</var> et <var>.b.c.</var> </s> - <s xml:space="preserve">quare & Å¿uperficiem re-<lb/>ctanguli quæſitam.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0305-02" xlink:href="fig-0305-02a"> - <image file="0305-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0305-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Quartum exemplum etiam faciliori via poteÅ¿t <lb/>Å¿olui, propterea, quod cum nobis cognita Å¿it ba-<lb/>Å¿is trianguli cum Å¿umma reliquorum laterum, & <lb/><reg norm="cum" type="context">cÅ©</reg> angulo oppoÅ¿ito baÅ¿i ipÅ¿ius reliqua cognita no <lb/>bis emergunt ex .15. problemate Å¿ecundi lib. de Triangulis ipÅ¿ius Monteregii.</s> - </p> - <pb o="294" rhead="IO. BAPT. BENED." n="306" file="0306" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0306"/> - <p> - <s xml:space="preserve">Vel Å¿i tibi placet, accipe hanc aliam methodum à me excogitatum.</s> - </p> - <p> - <s xml:space="preserve">Duplicetur <reg norm="triangulum" type="context">triangulÅ©</reg> <var>.a.b.c.</var> <reg norm="orthogonium" type="context">orthogoniÅ©</reg>, & fiat <reg norm="rectangulum" type="context">rectangulÅ©</reg> <var>.b.u.</var> vt in mea figura <lb/>Å¿ecundi exempli hic vides. </s> - <s xml:space="preserve"><reg norm="producaturque" type="simple">producaturqÌ;</reg> <var>.b.c.</var> quouÅ¿que <var>.c.f.</var> æqualis Å¿it <var>.c.u.</var> vnde <var>.b.f.</var> <lb/>cognita nobis erit ex hypotheÅ¿i, </s> - <s xml:space="preserve">quare cognoÅ¿cemus etiam quadratum <var>.g.f.</var> à quo <lb/><reg norm="demptum" type="context">demptÅ©</reg> cum fuerit <reg norm="aggregatum" type="context">aggregatÅ©</reg> quadratorum <var>.g.u.</var> et <var>.u.f.</var> nobis <reg norm="cognitum" type="context">cognitÅ©</reg> (nam quadra <lb/>ta <var>.g.u.</var> et <var>.u.f.</var> æqualia Å¿unt quadrato ipÅ¿ius <var>.a.c.</var> diagonalis datÄ™) remanebit aggrega-<lb/>tum <reg norm="Å¿upplementorum" type="context context">Å¿upplemẽtorÅ©</reg> cognitum, </s> - <s xml:space="preserve">quare eius medietas cognoÅ¿cetur ideÅ¿t <var>.b.u.</var> vndæ ex <num value="5">.<lb/>5.</num> Å¿ecundi Eucli. vt Å¿uperius diximus cognoÅ¿cetur etiam <var>.b.c.</var> et <var>.c.f.</var> diÅ¿tinctæ.</s> - </p> - <p> - <s xml:space="preserve">Idem aſſero de <reg norm="exemplo" type="context">exẽplo</reg> Gemmæ FriÅ¿ij à Stifelio citato in Appendice regulæ falÅ¿i.</s> - </p> - <p> - <s xml:space="preserve">Sit gratia exempli rectangulum hicÅ¿ubÅ¿criptum <var>.a.b.</var> datæ Å¿uperficiei data etiam <lb/>nobis Å¿it proportio <var>.a.e.</var> ad <var>.e.b.</var> laterum producentium, <reg norm="cogitemusque" type="simple">cogitemusqÌ;</reg> <var>.a.e.</var> producta <lb/>vÅ¿que ad <var>.o.</var> ita vt <var>.e.o.</var> æqualis Å¿it ipÅ¿i <var>.e.b.</var> imagine <lb/> - <anchor type="figure" xlink:label="fig-0306-01a" xlink:href="fig-0306-01"/> - mus <reg norm="etiam" type="context">etiã</reg> perfectum eſſe quadratum <var>.b.o.</var> vnde ex <lb/>prima Å¿exti Å¿eu .18. vel .19. Å¿eptimi vel .15. quinti <lb/>eadem proportio erit ipÅ¿ius <var>.a.b.</var> ad <var>.b.o.</var> vt <var>.a.e.</var> ad <lb/><var>e.o.</var> vel ad <var>.e.b.</var> </s> - <s xml:space="preserve">quare ex regula de tribus, cogno-<lb/>Å¿cemus quadratum <var>.b.o.</var> & eius <reg norm="radicem" type="context">radicẽ</reg> <var>.e.o.</var> & ex ea <lb/>demregula cognoÅ¿cemus <var>.a.e.</var> cum cognita nobis Å¿it <var>.e.o.</var> Å¿imul cum proportione <var>.e.o.</var> <lb/>ad <var>.e.a</var>.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0306-01" xlink:href="fig-0306-01a"> - <image file="0306-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0306-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">Quod circulus Å¿it figura infinitorum angulorum hoc eÅ¿t <lb/>ultima poligoniarum.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">SEd quod idem Stifelius in Appendice Å¿ecundi libri dicat circulum eſſe figuram <lb/>poligoniam, non eÅ¿t ita mirandum, nam & alij multi doctiſſimi viri hanc <lb/>veritatem cognouerunt, de Leone BaptiÅ¿ta Alberto nihil dicam, cum ipÅ¿e fateatur <lb/>hoc accepiſſe à philoÅ¿ophis, vt etiam refert AriÅ¿t. de Å¿phæratertio de cÅ“lo. </s> - <s xml:space="preserve">conÅ¿i-<lb/>dera quæſo in circulo, quod cum angulus contingentiæ Å¿it angulus, quamuis <reg norm="omnium" type="context">omniÅ©</reg> <lb/>acutorum rectilineorum anguÅ¿tiſſimus, vnde ex communi ratione Å¿equitur reliquum <lb/>ex duobus rectis rectilineis eſſe angulum, & Å¿i omnium obtuÅ¿orum rectilineorum Å¿it <lb/>ampliſſimum, tanto magis igitur erit angulus, id quod remanet ex duobus rectis re <lb/>ctilineis, detractis <reg norm="cum" type="context">cÅ©</reg> fuerint duobus angulis contingentiæ, qui quidem angulus erit <lb/>in quouis puncto circunferentiæ ipÅ¿ius circuli, idem intelligendum eÅ¿t de Å¿phæra, <lb/>cuius angulus eÅ¿t reÅ¿iduum ex quatuor rectis Å¿olidis, detractis cum fuerint quatuor <lb/>angulis contingentiæ <reg norm="Å¿olidisque" type="simple">Å¿olidisqÌ;</reg>.</s> - </p> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">Explanatio .25. Problematis lib. 2. Monteregij.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">QVod in .25. problemate .2. lib. de triangulis Monteregium non intelligas, mi-<lb/>rum non eÅ¿t, eo quod quandoque bonus dormitat Homerus. </s> - <s xml:space="preserve">Puto enim il-<lb/>lud problema ab ipÅ¿o Monteregio non fuiſſe viÅ¿itatum. </s> - <s xml:space="preserve">Sed ne me aliquo modo <lb/>culpes, accipe hanc <reg norm="aliam" type="context">aliã</reg> <reg norm="methodum" type="context">methodÅ©</reg> a me aliter <reg norm="etiam" type="context">etiã</reg> <reg norm="excogitatam" type="context">excogitatã</reg> in eadem ipÅ¿ius figura.</s> - </p> - <pb o="295" rhead="EPISTOL AE." n="307" file="0307" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0307"/> - <p> - <s xml:space="preserve"><reg norm="PropoÅ¿itum" type="context">PropoÅ¿itÅ©</reg> Å¿it nobis triangulum <var>.a.b.g.</var> cuius baÅ¿is data Å¿it cum area, Å¿eu perpendi-<lb/>culari <var>.a.d.</var> cum angulo etiam <var>.a.</var> ad cognoÅ¿cendum autem <var>.a.b.</var> et <var>.b.g.</var> cogitemus circu <lb/>lum <var>.a.b.q.g.</var> circunÅ¿cribere ipÅ¿um triangulum cuius diameter <var>.p.q.</var> ad rectos Å¿e-<lb/>cet baÅ¿im <var>.b.g.</var> in puncto <var>.m.</var> cogitemus etiam <var>.b.p.</var> et <var>.p.g.</var> vnde ex .20. ter-<lb/>tij Euclid. angulus <var>.b.p.g.</var> æqualis erit <lb/> - <anchor type="figure" xlink:label="fig-0307-01a" xlink:href="fig-0307-01"/> - angulo <var>.a.</var> & angulus <var>.m.p.b.</var> erit eius di <lb/>midium, quod ex te ipÅ¿o cognoÅ¿ces, & <lb/><reg norm="angulus" type="simple">angulê°</reg> <var>.p.b.m.</var> Å¿imiliter cognoÅ¿cetur, <lb/></s> - <s xml:space="preserve">quare ex .29. primi eiuÅ¿dem Montere <lb/>gij cognoÅ¿cemus <var>.p.m.</var> et <var>.p.b.</var> (nam <var>.b.<lb/>m.</var> datum fuit, vt dimidium totius ba-<lb/>Å¿is <var>.b.g.</var>) ducta poÅ¿tea <var>.b.q.</var> ex <reg norm="eadem" type="context">eadẽ</reg> .29. <lb/>cognoÅ¿cemus <var>.p.q.</var> cum <var>.p.b.</var> iam cogni <lb/>ta fuerit, à qua <var>.p.q.</var> (diametro) <reg norm="dempta" type="context">dẽpta</reg> <lb/><var>p.m.</var> remanebit <var>.q.m.</var> cognita, <reg norm="cum" type="context">cÅ©</reg> qua <lb/>iuncta cum fuerit <var>.m.t.</var> æquali <var>.a.d.</var> per <lb/>pendiculari, dabitur <var>.q.t.</var> et <var>.t.p.</var> inter <lb/>quas <var>.a.t.</var> media proportionalis loca-<lb/>tur, </s> - <s xml:space="preserve">quare cognoÅ¿cemus <var>.a.t.</var> quæ Å¿inus <lb/>eÅ¿t arcus <var>.a.p.</var> vnde cognitus erit arcus <lb/><var>a.p.</var> Å¿ed arcus <var>.p.g.</var> cognitus eÅ¿t median <lb/>te angulo <var>.p.b.g.</var> cognito, qui quidem <lb/>arcus <var>.p.g.</var> Å¿i coniunctus fuerit cum arcu <var>.p.a.</var> cognoÅ¿cemus compoÅ¿itum <var>.a.g.</var> & eius <lb/>chorda Å¿imiliter (hoc eÅ¿t <reg norm="Å¿ecundum" type="context">Å¿ecundÅ©</reg> latus) qua cognita, illico cognoÅ¿cemus chordam <lb/><var>a.b.</var> hoc eÅ¿t tertium latus trianguli propoÅ¿iti.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0307-01" xlink:href="fig-0307-01a"> - <image file="0307-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0307-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="4"> - <head style="it" xml:space="preserve">Quæ<unsure/>dam not and a in Federicum Comandinum.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">PVtabas enim me ioco dixiſſe Federicum Comandinum non omnino irrepræ-<lb/>henÅ¿ibilem eſſe, vide igitur, quod Å¿cribit in quinto lemmate in decimam <lb/>propoÅ¿itionem libr .2. de inÅ¿identibus aquæ Archimedis, volens demonÅ¿tra-<lb/>re eandem eſſe proportionem <var>.l.b.</var> ad <var>.b.m.</var> quæ <var>.c.e.</var> ad <var>.e.a.</var> vbi eÅ¿t aliquo modo pro-<lb/>lixum, mediante linea <var>.c.p.</var> cum Å¿uis partibus, citans etiam antecedens lemma extra <lb/>propoÅ¿itum, eo quod nec in antecedente lemmate, nec in alio, ipÅ¿e vnquam proba <lb/>uerit proportionem <var>.c.d.</var> ad <var>.d.q.</var> eſſe, vt <var>.l.b.</var> @d.b.m. Å¿ed ne putes me falli, tibi demon <lb/>Å¿trabo non eſſe neceſſarium ducere lineam <var>.c.m.p.</var> vel <var>.q.p.</var> eo quod <reg norm="cum" type="context">cÅ©</reg> per quintam <lb/>lib. de quadratura parabolÄ™ Archimedis, ita Å¿it <var>.c.d.</var> ad <var>.d.e.</var> vt <var>.l.b.</var> ad <var>.b.m.</var> exiÅ¿tente <lb/><var>a.c.</var> dupla ipÅ¿i <var>.d.c.</var> et <var>.e.c.</var> dupla ipÅ¿i <var>.g.c.</var> et <var>.l.d.</var> dupla ipÅ¿i <var>.l.b</var>: erit, primo componen-<lb/>do <var>.c.e.</var> ad <var>.e.d.</var> vt <var>.l.d.</var> ad <var>.d.m.</var> & per æqualitatem proportionum, ita erit <var>.e.g.</var> ad <var>.e.d.</var> <lb/>vt <var>.b.d.</var> 2d.d.m. & per .19. quinti Eucli. ita erit <var>.e.g.</var> ideÅ¿t <var>.g.c.</var> ad <var>.g.d.</var> vt <var>.b.d.</var> ideÅ¿t <var>.l.b.</var> <lb/>ad <var>.b.m.</var> Å¿ed <var>.c.g.</var> ad <var>.g.d.</var> eft vt <var>.c.e.</var> ad <var>.e.a.</var> ratio eÅ¿t, quia componendo ita eÅ¿t <var>.c.d.</var> ad <var>.d.<lb/>g.</var> vt <var>.c.a.</var> ad <var>.a.e.</var> & hoc eÅ¿t, quia permutando, ita eÅ¿t <var>.a.c.</var> ad <var>.d.c.</var> vt <var>.a.e.</var> ad <var>.d.g.</var> & hoc <lb/>verum eÅ¿t ex .19. quinti eo quod totius <var>.a.c.</var> ad totum <var>.d.c.</var> eft vt abÅ¿ciſſi <var>.e.c.</var> ad abÅ¿ciÅ¿ <lb/>Å¿um <var>.g.c.</var> vt Å¿upradixi.</s> - </p> - <pb o="296" rhead="IO. BAPT. BENED." n="308" file="0308" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0308"/> - <p> - <s xml:space="preserve">Sed etiam alio vniuerÅ¿aliori modo potes probare, quod ita Å¿it <var>.u.x.</var> ad <var>.x.y.</var> vt <var>.c.e.</var> <lb/>ad <var>.e.a.</var> cogitando in linea <var>.c.a.</var> punctum quoddam quod vocabimus Å¿imiliter <var>.y.</var> in <lb/>tali Å¿itu locatum, quod diuidat <var>.c.a.</var> eadem proportione qua <var>.y.</var> diuidit <var>.u.s.</var> vnde cum <lb/><var>e.s.</var> diuiÅ¿a eodem modo etiam Å¿it à puncto <var>.s.</var> ex Å¿upradicta quinta lib. de quadratura <lb/>parabolæ, erit igitur proportio <var>.a.y.</var> ad <var>.y.c.</var> vt <var>.e.s.</var> ad <var>.s.c.</var> per .11. quinti Eucli. </s> - <s xml:space="preserve">& com <lb/>ponendo ita erit <reg norm="totius" type="simple">totiê°</reg> <var>.a.c.</var> ad totum <var>.y.c.</var> vt abÅ¿cisÅ¿i <var>.s.c.</var> ad abÅ¿ciÅ¿sum <var>.s.c.</var> </s> - <s xml:space="preserve">quare reÅ¿idui <lb/><var>a.e.</var> ad reÅ¿iduum <var>.y.s.</var> erit vt totius <var>.a.c.</var> ad totum <var>.y.c.</var> & permutando, ita erit <var>.a.c.</var> ad <var>.a.<lb/>e.</var> vt <var>.y.c.</var> ad <var>.y.s.</var> & diuidendo, ita erit <var>.<lb/>c.e.</var> ad <var>.e.a.</var> ut <var>.c.s.</var> ad <var>.s.y.</var> & quia pun-<lb/> - <anchor type="figure" xlink:label="fig-0308-01a" xlink:href="fig-0308-01"/> - ctum <var>.s.</var> diuidit <var>.c.a.</var> eodem modo, quo <lb/>x. diuidit <var>.u.s.</var> per Å¿upradictam <reg norm="quintam" type="context">quintã</reg>, <lb/>ergo ita erit <var>.c.s.</var> ad <var>.s.y.</var> in linea <var>.c.a.</var> vt <lb/><var>u.x.</var> ad <var>.x.y.</var> </s> - <s xml:space="preserve">vnde ex .11. quinti <var>.c.e.</var> ad <lb/><var>e.a.</var> erit, vt <var>.u.x.</var> ad <var>.x,y</var>. </s> - <s xml:space="preserve">quare Å¿equitur, <lb/>primum, Å¿ecundum, tertium, & quartum lemma Å¿uperflua eſſe.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0308-01" xlink:href="fig-0308-01a"> - <image file="0308-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0308-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Quod deinde ponit pro corellario in fine .6. lemmatis, aliter quam per .6. lemma <lb/>poteÅ¿t demonÅ¿trari, hoc mode. </s> - <s xml:space="preserve">Nam Å¿uperius demonÅ¿trauimus eandem propor-<lb/>tionem eſſe <var>.l.b.</var> ad <var>.b.m.</var> quæ <var>.c.e.</var> ad <var>.e.a.</var> <reg norm="idem" type="context">idẽ</reg> dico de proportione <var>.u.x.</var> ad <var>.x.y.</var> & om-<lb/>nium æquidiÅ¿tantium ad <var>.h.e.</var> quibus rationibus mediantibus codem modo Å¿cies, <reg norm="quod" type="simple">ê™</reg> <lb/><var>u.y.</var> ad <var>.y.r.</var> erit, vt <var>.c.d.</var> ad <var>.d.c.</var> & ita dico de omnibus <reg norm="æquidiÅ¿tantibus" type="context">æquidiÅ¿tãtibus</reg>. ad <var>.h.e.</var> </s> - <s xml:space="preserve">vnde <var>.l.b.</var> <lb/>ad <var>.b.m.</var> erit vt <var>.u.x.</var> ad <var>.x.y.</var> et <var>.l.m.</var> ad <var>.m.d.</var> vt <var>.u.y.</var> ad <var>.y.r.</var> per .11. quinti, Å¿ed cum Å¿it <var>.l.<lb/>b.</var> ad <var>.b.m.</var> vt <var>.u.x.</var> ad <var>.x.y.</var> componendo erit <var>.l.m.</var> ad <var>.b.m.</var> vt <var>.u.x.</var> ad <var>.x.y.</var> & euerÅ¿im <var>.b.<lb/>m.</var> ad <var>.m.b.</var> erit, vt <var>.x.y.</var> ad <var>.y.u.</var> & per æquam proportionalitatem erit <var>.b.m.</var> ad <var>.m.d.</var> vt<unsure/> <lb/><var>x.y.</var> ad <var>.y.r.</var> quod eÅ¿t propoÅ¿itum.</s> - </p> - <p> - <s xml:space="preserve">Non video etiam, quare ipÅ¿e ducat lineam <var>.s.r.</var> cum in ipÅ¿o contextu nihil Å¿aciac <lb/>de dicta <var>.s.r</var>.</s> - </p> - <p> - <s xml:space="preserve">Comentum poÅ¿tea contextus <var>.P.</var> pulchrius eſſet, Å¿i diceret, quod cum ita Å¿it totius, <lb/><var>l.a.</var> ad totum <var>.a.d.</var> Å¿ic Å¿e habebit abÅ¿ciſſum <var>.a.i.</var> ad abÅ¿ciſſum <var>.a.z.</var> eo quod ita eÅ¿t, vt Å¿cis, <lb/>hoc eÅ¿t in proportione dupla, ergo reÅ¿idui <var>.i.l.</var> ad reÅ¿iduum <var>.d.z.</var> erit vt totius <var>.a.l.</var> ad <lb/>totum <var>.a.d.</var> hoc eÅ¿t in proportione dupla.</s> - </p> - </div> - <div type="letter" level="4" n="5"> - <head style="it" xml:space="preserve">De ViÅ¿u.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">RAtio vnde Å¿iat, vt videamus diÅ¿tinctè omnes eolores, cum in qualibet aeris par <lb/>te, quo lumina reÅ¿lexa poſſunt peruenire mixta Å¿int, & non diÅ¿tincta, oritur à <lb/>paruitate ipÅ¿ius pupillæ oculorum, & à magna expanÅ¿ione virtutis viÅ¿iuæ in Å¿uperÅ¿i-<lb/>cie concaua orbis continentis humores diaphanos oculorum per ramuÅ¿culos nerui <lb/>optici remotè ab ipÅ¿a pupilla. </s> - <s xml:space="preserve">& quamuis radii luminoÅ¿i frangantur ab vnoquoque <lb/>humore diuerÅ¿imodè, hoc nihilominus maximè iuuat ad diÅ¿tinctionem radiorum, <lb/>Å¿ed & Å¿i directè procederent, idem ferè eueniret, non tamen Å¿uis locis, cogita exem-<lb/>pli gratia lineam <var>.a.u.e.</var> vt communis Å¿ectio cuiuÅ¿dam plani Å¿ecantis Å¿phæram oculi, <lb/>per centrum ipÅ¿ius, & pupillæ, et <var>.o.</var> punctum Å¿it proximum centro ipÅ¿ius pupillæ, <lb/>Å¿ed interius aliquantulum, extra <reg norm="autem" type="context">autẽ</reg> <reg norm="oculum" type="context">oculÅ©</reg>, Å¿int varij colores, vt <var>.c.n.t.</var> in dicto plano.</s> - </p> - <p> - <s xml:space="preserve">Iam nulli dubium eÅ¿t quod lumina quæ producuntur ab <var>.c.n.t.</var> ad <var>.o.</var> in ipÅ¿o <var>.o.</var> mi- - <pb o="297" rhead="EPISTOL AE." n="309" file="0309" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0309"/> - xta, & non diÅ¿tincta, procedendo igitur vlteriusipÅ¿i radij citra <var>.o</var>. </s> - <s xml:space="preserve">tunc <reg norm="diÅ¿gregantur" type="context">diÅ¿gregãtur</reg>, <lb/>& Å¿eparantur abinuicem, & <reg norm="cum" type="context">cÅ©</reg> perueniunt ad lineam <var>.a.u.e.</var> Å¿entiuntur diÅ¿tincti alij <lb/>ab alijs. </s> - <s xml:space="preserve">Cuius quidem rei, exemplum manifeÅ¿tum accipere poſſumus à quouis <lb/>cubiculo ex omni parte clauÅ¿o, quod tranÅ¿itum <reg norm="nullum" type="context">nullÅ©</reg> permittat radijs luminoÅ¿is, ni <lb/>Å¿i per aliquod paruum foramen, in quo foramine, & extra ipÅ¿um cubiculum, omnes <lb/>radij mixti erunt, Å¿ed in obiecto pariete ipÅ¿ius cu-<lb/> - <anchor type="figure" xlink:label="fig-0309-01a" xlink:href="fig-0309-01"/> - biculi videb untur diÅ¿tincti, vnde Å¿equitur, quòd <lb/>quo remotius erit obiectum <var>.c.n.t.</var> ab <var>.o.</var> tanto acu-<lb/>tior erit angulus <var>.c.o.t.</var> & Å¿uus contrapoÅ¿itus Å¿imili-<lb/>ter, & per conÅ¿equens linea <var>.e.u.a.</var> breuior erit, & <lb/><reg norm="punctum" type="context">punctÅ©</reg> <var>.o.</var> propinquius etiam erit ipÅ¿i lineæ <var>.a.u.e.</var> <lb/>quæ omnia efficiunt, vt nobis obiectum <var>.c.t.</var> <reg norm="paruum" type="context">paruÅ©</reg>, <lb/>& minus diÅ¿tinctum, Å¿eu magis confuÅ¿um appareat.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0309-01" xlink:href="fig-0309-01a"> - <image file="0309-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0309-01"/> - </figure> - </div> - </div> - </div> - <div type="section" level="3" n="18"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE APPARENTI DISTANTIA PARTIV M <lb/>hæmiÅ¿phærij.</head> - <head style="it" xml:space="preserve">AnÅ¿elmo Fucaro.</head> - <p> - <s xml:space="preserve">GRatæ mihi tuæ literæ <reg norm="fuerunt" type="context">fuerÅ©t</reg>, quibus <reg norm="oÅ¿tendis" type="context">oÅ¿tẽdis</reg> non <reg norm="paruum" type="context">paruÅ©</reg> <reg norm="deÅ¿iderium" type="context">deÅ¿ideriÅ©</reg> <reg norm="Å¿ciendi" type="context">Å¿ciẽdi</reg> vnde <lb/>fiat, quod cum dies illuceÅ¿cit, & eÅ¿t Å¿erena pars CÅ“li, circa axem orizontis <lb/>demiſſior appareat, quam aliæ partes, <reg norm="cum" type="context">cÅ©</reg> ab alijs (quod <reg norm="Å¿ciam" type="context">Å¿ciã</reg>) Å¿atis expreſſum <reg norm="non" type="context">nõ</reg> fue <lb/>rit, Å¿ed quia de eo à me aliquid Å¿cire deÅ¿ideras dicam quod mihi <reg norm="verum" type="wordlist">vr̃</reg>. </s> - <s xml:space="preserve">Scias non <reg norm="Å¿olum" type="context">Å¿olÅ©</reg> <lb/><reg norm="multitudinem" type="context">multitudinẽ</reg> <reg norm="obiectorum" type="context">obiectorÅ©</reg> <reg norm="oppoÅ¿itorum" type="context">oppoÅ¿itorÅ©</reg> efficere, vt aliqua res alia longius diÅ¿tare videat, <lb/>vt alij <reg norm="putarunt" type="context">putarÅ©t</reg>, Å¿ed etiam diuerÅ¿itates colorum, quamobrem cum decipiamur, cre-<lb/>dentes CÅ“lum eſſe præditum colore cÄ™ruleo, cum is color, aeri, non CÅ“lo <lb/>conueniat, & videntes huiuÅ¿modi colorem circa axem <reg norm="orizontis" type="context">orizõtis</reg> magis denÅ¿um, <reg norm="quam" type="context">quã</reg> <lb/>verÅ¿us ipÅ¿um orizontem, ratione exiguæ reÅ¿lexionis, à pauca quantitate vaporum <lb/>inter noÅ¿trum Å¿itum, & reÅ¿lexionis locum, iudicamus CÅ“lum proximiorem eſſe cir-<lb/>ca dictum axem, quam Å¿int aliæ partes; </s> - <s xml:space="preserve">præterquam, quod is color, qui videtur <lb/>terminare, aut impedire radium viÅ¿ualem (aduertas tamen me hac in re platonicum <lb/>non eſſe) eo Å¿emper propinquior eſſe videtur, qui ei locum dat, & hanc ob cauÅ¿am <lb/>videntes nos <reg norm="denÅ¿itatem" type="context context">dẽſitatẽ</reg> cÄ™rulei circa axem orizontis, & cernentes amplitudinem gy <lb/>ri aliarum partium, adducimur, vt putemus <reg norm="eam" type="context">eã</reg> <reg norm="partem" type="context">partẽ</reg> <lb/> - <anchor type="figure" xlink:label="fig-0309-02a" xlink:href="fig-0309-02"/> - viciniorem eſſe. </s> - <s xml:space="preserve"><reg norm="Neque" type="simple">Neq;</reg> illud <reg norm="etiam" type="context">etiã</reg> <reg norm="omittam" type="context">omittã</reg> hoc <reg norm="etiam" type="context">etiã</reg> fie <lb/>ri ratione imaginationis, vnde <reg norm="etiam" type="context">etiã</reg> multis <reg norm="contrarium" type="context">contrariÅ©</reg> <lb/>euenire poteÅ¿t, ideÅ¿t vt eis magis profundum videa <lb/>tur <reg norm="CÅ“lum" type="context">CÅ“lÅ©</reg>, circa axem orizontis, quam vicinum gy <lb/>ro <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> <reg norm="orizontis" type="context">orizõtis</reg>, iudicantibus <reg norm="eam" type="context">eã</reg> <reg norm="partem" type="context">partẽ</reg> <reg norm="longinquio" type="context">lõginquio</reg> <lb/><reg norm="rem" type="context">rẽ</reg> eſſe, quæ Å¿eÅ¿e magis <reg norm="obÅ¿curam" type="context">obÅ¿curã</reg> oculo <reg norm="demonÅ¿trat" type="context">demõſtrat</reg>, & <lb/><reg norm="eam" type="context">eã</reg> <reg norm="propinquiorem" type="context">propinquiorẽ</reg> quÄ™ Å¿eÅ¿e <reg norm="clariorem" type="context">clariorẽ</reg> oÅ¿tendit, vt ei <reg norm="ent" type="context">ẽt</reg> <lb/>cont<unsure/>ingere poteÅ¿t, qui <reg norm="Å¿ubÅ¿criptam" type="context">Å¿ubÅ¿criptã</reg> <reg norm="Å¿iguram" type="context">Å¿igurã</reg> <reg norm="cubicam" type="context">cubicã</reg> non <lb/><reg norm="quidem" type="context">quidẽ</reg> <reg norm="ductam" type="context">ductã</reg> Å¿ecundú <reg norm="ordinem" type="context">ordinẽ</reg> opticè, Å¿ed ita, vt om-<lb/>nia latera oppoÅ¿ita <reg norm="inuicem" type="context">inuicẽ</reg> Å¿int parallela, proÅ¿piciet, <lb/>ideÅ¿t <var>.a.i.</var> ad <var>.e.t.</var> et <var>.c.u.</var> ad <var>.o.n.</var> et <var>.a.i.</var> ad <var>.c.u.</var> et <var>.e.t.</var> ad <lb/><var>o.n.</var> vnde Å¿equitur, vt aliquando quadratum <var>.a.o.</var> <lb/>videbitur citra, et <var>.i.n.</var> vltra <reg norm="dictum" type="context">dictÅ©</reg> cubum aliquando <lb/>verò èconuerÅ¿o.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0309-02" xlink:href="fig-0309-02a"> - <image file="0309-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0309-02"/> - </figure> - </div> - <pb o="298" rhead="IO. BAPT. BENED." n="310" file="0310" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0310"/> - </div> - </div> - <div type="section" level="3" n="19"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE PHILOSOPHIA MATHEMA TICA.</head> - <head style="it" xml:space="preserve">Dominico PiÅ¿ano.</head> - <p> - <s xml:space="preserve">SI omnia vno colore conÅ¿tarent, & corporum vmbræ à luminibus non di-<lb/>Å¿tinguerentur, neque diuerÅ¿itas Å¿itus, lumina, quæ veniunt ad oculum non al-<lb/>teraret; </s> - <s xml:space="preserve">perinde eſſet, ac Å¿i eſſemus cÅ“ci. </s> - <s xml:space="preserve">Miror quod cum in AriÅ¿totele Å¿is <lb/>verÅ¿atus, in tuis tamen Å¿criptis philoÅ¿ophum à Mathematico Å¿epares, quaÅ¿i mathe-<lb/>maticus non Å¿it adeò philoÅ¿ophus, vt eÅ¿t naturalis, & metaphyÅ¿icus, cum multo ma <lb/>gis quam ij philoÅ¿ophus Å¿it appellandus, Å¿i ad veritatem Å¿uarum concluÅ¿ionum reÅ¿pi <lb/>ciamus. </s> - <s xml:space="preserve">Verum <reg norm="quidem" type="context">quidẽ</reg> eÅ¿t, te in huiuÅ¿modi errore <reg norm="Å¿olum" type="context">Å¿olÅ©</reg> non verÅ¿ari; </s> - <s xml:space="preserve">Å¿ed grauius eÅ¿t, <lb/>quod cum vos videatis etiam res morales Å¿ub philoÅ¿ophiÄ™ <reg norm="appellationem" type="context">appellationẽ</reg> cadere, non <lb/>animaduertatis diuinas Å¿cientias mathematicas etiam philoÅ¿ophiæ nomine ornan-<lb/>das eſſe. </s> - <s xml:space="preserve">Quod Å¿i eiuÅ¿dem nomen penitius conÅ¿iderare velimus, inueniemus aper-<lb/>tè, mathematico magisillud ipÅ¿um quà m cuilibet alio conuenire, cum nullus ex alijs <lb/>tam certo Å¿ciat id quod affirmat quam mathematicus, neque aliquis Å¿it, qui in co-<lb/>gnitionis, & Å¿cientiæ cupiditatem magis ducatur, vt apertè patet, cum nec etiam ipÅ¿i <lb/>Å¿enÅ¿ui det locum, neque aliquid præſupponat, quod non Å¿it ita verum & intellectui <lb/>notum, vt nulla quæuis porentia, illud eſſe falÅ¿um oÅ¿tendere queat. </s> - <s xml:space="preserve">Sed quia <lb/>Græci, qui ad placitum nomina rebus impoÅ¿uerunt, voluerunt etiam, non Å¿olum <lb/>mathematica, Å¿ed etiam naturalia, metaphyÅ¿ica, & moralia, Å¿ub communi philoÅ¿o-<lb/>phiæ nomine contineri. </s> - <s xml:space="preserve"><reg norm="Vtaunt" type="context">VtaÅ©t</reg> tibi Å¿atisfaciam authoritate AriÅ¿totelis, quem tanto-<lb/>pere colis, primum conÅ¿idera, nunquam eum de philoÅ¿opho <reg norm="mentionem" type="context">mẽtionem</reg> facere quin <lb/>prius aperiat de quo philoÅ¿opho loquatur, atque hoc Å¿emper præſtat, exceptis qui-<lb/>buÅ¿dam locis, vt cap .2. lib. 4. <reg norm="MetaphyÅ¿icorum" type="context">MetaphyÅ¿icorÅ©</reg>, vbi de philoÅ¿opho in genere <reg norm="loquens" type="context">loquẽs</reg>, <lb/>ait, proprium eſſe philoÅ¿ophi. vt res omnes Å¿peculetur atque hoc in principio quin <lb/>ti textus aſſerit, cum in quarto iam oÅ¿tenderit mathematicum eſſe philoÅ¿ophum: <lb/></s> - <s xml:space="preserve">omitto quod in .2. textu Å¿ecundi phyÅ¿icorum idem affirmet, æquum eſſe appellare <lb/>philoÅ¿ophiam Å¿cientiam veritatis, & finem Å¿peculatiuæ exiÅ¿tere veritatem. </s> - <s xml:space="preserve">An non <lb/>idem in primo cap .6. metaphiÅ¿icæ philoÅ¿ophiam Å¿peculatiuam, mathematicis phy <lb/>Å¿icis & Å¿upernaturalibus rebus contineri? </s> - <s xml:space="preserve">An non idem paulo inferius Å¿cribit phyÅ¿i-<lb/>cam primam futuram, Å¿i aliæ Å¿ubÅ¿tantiæ quam naturales non reperirentur? </s> - <s xml:space="preserve">conÅ¿idera <lb/>deinde quid dicat in fine tertij cap. lib. 11. quo loco nil clarius eſſe poteÅ¿t, lege etiam <lb/>quæ .6. cap. eiuÅ¿dem libri ab eodem adducuntur, & quæ in .8. cap .12. libri textu .44. <lb/>apertè ponuntur. </s> - <s xml:space="preserve">Quod Å¿i hæc tibi non Å¿ufficiunt, vereor ne tuus morbus deÅ¿pe-<lb/>ratus euadat.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">De imaginatione Å¿pecierum.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">QVod dixi domino Tadeo eÅ¿t, quod aliquas particularium Å¿pecies, perfectè & <lb/>integrè imaginari poſſumus, alias non item, id tibi melius exemplo innote-<lb/>Å¿c<unsure/>et. </s> - <s xml:space="preserve">Proponatur tibi triangulus æquilaterus datæ magnitudinis, <reg norm="datique" type="simple">datiqÌ;</reg> coloris, hu <lb/>iuÅ¿modi enim particularis, potes imaginatione tibi fingere integram Å¿peciem, tota <lb/><reg norm="lemque" type="simple">lemqÌ;</reg> ei adæquatam, Å¿ed Å¿i aliquam Å¿peciem aliquando vniuerÅ¿aliorem imaginatio - <pb o="299" rhead="EPISTOL AE." n="311" file="0311" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0311"/> - ne concipere velles, quemadmodum vnius trianguli Ä™quilateri, tali magnitudine, <lb/>Å¿ed non præfinito colore conÅ¿tantis, hoc minime præſtare poſſes. </s> - <s xml:space="preserve">quia nullam rem <lb/>viÅ¿ibilem priuatam colore imaginari poſſumus. </s> - <s xml:space="preserve">nec etiam potes imaginari <reg norm="Å¿peciem" type="context">Å¿peciẽ</reg> ali <lb/>cuius trianguli æquilateri, indeterminatæ magnitudinis, & indefiniti coloris, quæ <lb/>cuilibet particulari cuiuÅ¿uis magnitudinis, & coloris poÅ¿tea applicari queat. </s> - <s xml:space="preserve">Species <lb/>deinde alicuius trianguli Ä™quicruri, aut vnius trianguli laterum inÄ™qualium, aut <reg norm="triam" type="context">triã</reg> <lb/>guli in genere, aut tandem figuræ, conÅ¿iderato tu ipÅ¿e, an poſſit Å¿ub imaginationem <lb/>cadere. </s> - <s xml:space="preserve">Poſſumus quidem huiuÅ¿modi Å¿peciem (ratione mediante) intelligere, vn <lb/>de quamlibet Å¿peciem rei particularis viÅ¿ibilis, compoÅ¿itæ, ex figura, magnitudine, <lb/>& colore, perfectè imaginari poſſimus, & huiuÅ¿modi conceptus erit Å¿pecialiſſima <lb/>Å¿pecies, quia in infinito Å¿uorum indiuiduorum, nunquam fiet, vt aliquod eorum, ali <lb/>quo modo ab alijs differre poſſit; </s> - <s xml:space="preserve">admonens te, nil reperiri, quod differat, aut in Å¿e <lb/>partem aliquam habeat, quod aliquid aliud non obtineat, quin dicta differentia Å¿it <lb/>Å¿pecifica, eius tamen Å¿olum partis quæ differt ab alia duorum indiuiduorum, vnius, <lb/><reg norm="eiuÅ¿demque" type="simple">eiuÅ¿demqÌ;</reg> Å¿peciei. </s> - <s xml:space="preserve">quia Å¿i eÅ¿t in magnitudine nulla planè magnitudo reperitur, quÄ™ <lb/>Å¿ua Å¿pecie non Å¿it dotata, quod Å¿i non eſſet, inter res omnes nulla æqualitas <lb/>eluceret: </s> - <s xml:space="preserve">& Å¿i in figura, & colore, idem affirmo, aliter nulla res fimilis eſſet <lb/>alteri, neque aliqua Å¿imilitu lo reperiretur. </s> - <s xml:space="preserve">Idem de quolibet alio obiecto <lb/>Å¿enÅ¿ibili dico. </s> - <s xml:space="preserve">Ratio autem eorum omnium quæ dixi eÅ¿t, quia imaginatiua nihil <lb/>aliud intellectui oÅ¿tendere poteÅ¿t, quam id quod recipit à ſenÅ¿u, & cum Å¿enÅ¿us, <lb/>alio modo moueri non poſſit quam Å¿upradicto, hanc ob cau Å¿am verum eÅ¿t, <reg norm="quicquid" type="simple">ê—cquid</reg> <lb/>Å¿cripÅ¿i. </s> - <s xml:space="preserve">Vnde triangulum Ä™quilaterum datÄ™ magnitudinis, erit genus triangulorum <lb/><reg norm="Ä™quilaterum" type="context">Ä™quilaterÅ©</reg> <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> datæ magnitudinis, Å¿ed <reg norm="diuerÅ¿orum" type="context">diuerÅ¿orÅ©</reg> <reg norm="colorum" type="context">colorÅ©</reg>, erit <reg norm="etiam" type="context">etiã</reg> Å¿pecies trianguli <lb/>æquilateri indeterminatæ magnitudinis, & hic deinde erit Å¿pecies trianguli, & hic <lb/>poÅ¿tea Å¿pecies figuræ. </s> - <s xml:space="preserve">Idem de alijs omnibus rebus per gradus dico, quÄ™ Å¿icut à ſen <lb/>Å¿u, ita etiam ab imaginatione longè recedunt, adeo vt has Å¿pecies Å¿pecialiſſimas tan-<lb/>tum, ideÅ¿t eas Å¿olùm, quas hic Å¿uperius deÅ¿cripÅ¿i, integrè capere poſſit: </s> - <s xml:space="preserve">at verò gene <lb/>ra, quanto vniuerÅ¿aliora Å¿unt, ab eadem imaginatione, tanto longius diÅ¿tant.</s> - </p> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">De maculis Lunæ, & eius lumine.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">MAculæ Lunæ, nihil aliud Å¿unt, quà m partes ipÅ¿ius LunÄ™ magis perÅ¿picuæ, à qui <lb/>bus, lumen non refleſſum, Å¿ed penetrans, nobis occultatur; </s> - <s xml:space="preserve"><reg norm="quemadmodum" type="context">quemadmodÅ©</reg> <lb/>via lactea, nihil aliud eÅ¿t, quam pars octaui orbis magis opaca, à qua lumen Solis re-<lb/>fleſſum, Å¿eÅ¿e nobis oÅ¿tendit. </s> - <s xml:space="preserve">Quod autem Maurolicus Å¿cribit folio .64. cap. de aÅ¿tro <lb/>rum fulÅ¿ionibus, circa <reg norm="Lunam" type="context">Lunã</reg>, eÅ¿t falÅ¿um; </s> - <s xml:space="preserve">primo, quia non conÅ¿iderat <reg norm="differentiam" type="context">differentiã</reg> inten-<lb/>Å¿ionis luminum inter Venerem, & Lunam, cum <reg norm="lumem" type="context">lumẽ</reg> illius Å¿it magis intenÅ¿um, quam <lb/>Lunæ, quia quilibet qui Å¿ano Å¿it <reg norm="onculo" type="context">õculo</reg>, facile poteÅ¿t compræhendere, Å¿i Lu-<lb/>na eſſet, vbi eÅ¿t Venus, aut Venus vbireperitur Luna (quibus in locis eiuÅ¿dem ma <lb/>gnitudinis nobis apparerent) ipÅ¿a Luna à Venere longè Å¿uperaretur, & excedere-<lb/>tur Å¿plendore, & lumine, ita vt Å¿i etiam verum eſſet, quod per tres gradus inter-<lb/>ualli Å¿eÅ¿e nobis proderet Å¿exageÅ¿ima pars luminis (quod in quadraturis nec in vllo <lb/>alio Å¿itu verum e<unsure/>uadit, reÅ¿pectu ad Solem, ideÅ¿t vt tres gradus differentiæ Å¿itus, con <lb/>Å¿tituant Å¿exag eÅ¿imam partem differentiæ luminis reÅ¿pectu noÅ¿tri) non ideo tamen - <pb o="300" rhead="IO. BAPT. BENED." n="312" file="0312" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0312"/> - dictum lumen conÅ¿piceretur, quia non Å¿ufficit extenÅ¿io luminis, cum eiuÅ¿dem inten <lb/>Å¿io Å¿it etiam neceſſaria. </s> - <s xml:space="preserve">Sed id quoque tibi dico, quod etiam Å¿i dicta Å¿exageÅ¿ima <lb/>pars totius luminis lunaris, eadem intenÅ¿ione Å¿plendoris, & luminis Veneris, in tali <lb/>diÅ¿tantia trium graduum à Sole prædita eſſet, non eam <reg norm="tamen" type="wordlist">tamẽ</reg> videremus, ratione ob <lb/>liquitatis curuÄ™, & Å¿phæricÄ™ Å¿uperficiei Lunæ, reÅ¿pectu noÅ¿tri, in huiuÅ¿modi Å¿itu: </s> - <s xml:space="preserve">id <reg norm="qui" type="simple">ê—</reg> <lb/>tibi ita demonÅ¿tratum volo.</s> - </p> - <p> - <s xml:space="preserve">Pars Å¿uperficialis lunaris globi, quæ nos reÅ¿picit Å¿it <var>.a.p.u.</var> quam accipere poſſu-<lb/>mus pro medietate ipÅ¿ius Å¿uperficiei totalis, eo quod reÅ¿pectu noÅ¿tri viÅ¿us, inÅ¿enÅ¿ibi <lb/>liter, ab ipÅ¿a medietate differat, pars autem à Sole viÅ¿a Å¿it <var>.u.q.a.</var> cogitemus etiam cir <lb/>culum <var>.a.p.u.q.</var> vnum eſſe ex maioribus ipÅ¿ius globi, cuius Å¿uperficies <reg norm="tranÅ¿eat" type="context">trãſeat</reg> per ocu <lb/>lum vidontis, vnde pars eius <var>.a.p.u.</var> diuidet vmbram per æqualia, reliqua verò pars <var>.<lb/>a.q.u.</var> diuidet per æqualia lumen ipÅ¿ius Lunæ à Sole receptum, ita quod pars illumi <lb/>nata, erit medietas <var>.u.q.a.</var> exceſſus verò, cum noÅ¿tro viÅ¿ui incompræhenÅ¿ibilis Å¿it, pro <lb/>nihilo reputetur, cuius cauÅ¿a eÅ¿t, maxima illa diÅ¿tantia, quæ inter Solem, & Lunam <lb/>reperitur, quamuis Sol maior Å¿it Luna multis millibus vicium, eo quod tunc inter So <lb/>lem, & Lunam reperiantur plus quam .570. diametri terræ.</s> - </p> - <p> - <s xml:space="preserve">Supponamus nunc Lunam remotam eſſe à loco ipÅ¿ius <reg norm="coniunctionis" type="context">cõiunctionis</reg> cum Sole per <lb/>3. gradus. </s> - <s xml:space="preserve">vnde <reg norm="quemadmodum" type="wordlist">quẽadmodum</reg> prius <lb/> - <anchor type="figure" xlink:label="fig-0312-01a" xlink:href="fig-0312-01"/> - lumen erat in gyro <var>.a.q.u.</var> nunc re-<lb/>periatur in gyro <var>.x.q.t.</var> ita quod <var>.t.u.</var> <lb/>erit Å¿exageÅ¿ima pars ipÅ¿ius <var>.a.p.u.</var> <reg norm="quod" type="wordlist">qÄ‘</reg> <lb/>à vero Å¿enÅ¿ibiliter non diÅ¿cedit. <lb/></s> - <s xml:space="preserve">Imaginentur nunc duæ rectæ lineæ <lb/>ductæ ab oculo <var>.d.</var> ad puncta <var>.t.</var> et <var>.u.</var> <lb/>verum tamen eÅ¿t quod linea <var>.d.u.</var> Å¿e-<lb/>cabit <reg norm="arcum" type="context">arcÅ©</reg> <var>.t.u.</var> Å¿ed ita propinqua <reg norm="pum" type="context">pÅ©</reg> <lb/>cto <var>.u.</var> quod erit ei ferè contingens, <lb/>vnde abÅ¿que Å¿enÅ¿ibili errore poſſu-<lb/>mus arcum <var>.t.u.</var> intelligere inter duas <lb/>lineas <var>.d.t.</var> et <var>.d.u.</var> quapropter tale lu-<lb/>men compræhendetur, ferè, Å¿ub an-<lb/>gulo <var>.t.d.u.</var> quem quidem angulum <lb/>oportet nos videre, cuius magnitu-<lb/>dinis exiÅ¿tat, reÅ¿pectu totalis anguli <lb/><var>a.d.u.</var> protracta cum fuerit <var>.d.a</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0312-01" xlink:href="fig-0312-01a"> - <image file="0312-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0312-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Producatur primo <var>.d.t.</var> vÅ¿que ad <lb/>diametrum in puncto <var>.i.</var> deinde per <lb/>puncta <var>.a.</var> et <var>.u.</var> ducatur arcus <var>.a.e.u.</var> cir <lb/>ca <var>.d.</var> <reg norm="centrum" type="context">cẽtrum</reg>, ad quem ducatur linea <var>.<lb/>d.t.i.</var> in puncto <var>.e.</var> Å¿ed quia, cum dia-<lb/>meter <var>.a.u.</var> tam breuis Å¿it reÅ¿pectu di <lb/>Å¿tantiæ à terra, tempore interlunij, <lb/>vnde minor <reg norm="centeÅ¿ima" type="context">cẽteÅ¿ima</reg> parte ipÅ¿ius di-<lb/>Å¿tantiæ exiÅ¿tit, <reg norm="Å¿equitur" type="simple">Å¿equit̃</reg> nos poſſe <reg norm="abÅ¿que" type="simple">abÅ¿q;</reg> <lb/>Å¿enÅ¿ibili errore cogitare, à puncto <var>.d.</var> <lb/>ad quoduis punctum ipÅ¿ius diametri <lb/>omnes lineas ad angulos rectos cum <lb/>ipÅ¿o diametro, & inÅ¿enÅ¿ibilis inæqua - <pb o="301" rhead="EPISTOL AE" n="313" file="0313" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0313"/> - litatis à linea <var>.d.o</var>. </s> - <s xml:space="preserve">Accipiemus igitur <var>.t.i.</var> pro Å¿inu arcus <var>.t.u.</var> qui eÅ¿t graduum .3. hoc <lb/>eÅ¿t Å¿exageÅ¿ima pars Å¿emicirculi graduum .180. quapropter <var>.t.i.</var> erit partium .5233. ta-<lb/>lium qualium <var>.o.u.</var> eÅ¿t .100000. cuius <var>.t.i.</var> quadratum demptum cum fuerit à quadra-<lb/>to Å¿emidiametri <var>.o.t.</var> relinquet nobis quadratum ipÅ¿ius <var>.o.i.</var> quæ quidem <var>.o.i.</var> vt radix <lb/>quadrata, erit partium .99862. talium qualium Å¿emidiameter eÅ¿t .100000. vnde <var>.i.u.</var> <lb/>reÅ¿iduum diametri, remanebit partium .138. </s> - <s xml:space="preserve">Vel Å¿ic, cum cognitus Å¿it nobis arcus <var>.<lb/>t.u.</var> illicò cognoÅ¿cemus Å¿inum arcus <var>.p.t.</var> <reg norm="complementum" type="context">complementÅ©</reg> vnius quartæ, qui Å¿inus æ qua-<lb/>lis erit ferè arcui <var>.o.i.</var> partium .99862. vnde <var>.i.u.</var> erit, vt dictum eÅ¿t, partium .138. quÄ™ <lb/>quidem <var>.i.u.</var> æqualis eÅ¿t ferè Å¿inui arcus <var>.u.e.</var> & ita etiam <var>.u.e.</var> </s> - <s xml:space="preserve">quare Å¿i diuiÅ¿a fuerit to-<lb/>ta <var>.a.u.</var> <reg norm="partium" type="context">partiÅ©</reg> .200000. per .138. proueniet nobis .1449. & Å¿ic angulus <var>.t.d.u.</var> erit vna <lb/>partium .1449. anguli <var>.a.d.u</var>. </s> - <s xml:space="preserve">Confideremus igitur quomodo fieri poteÅ¿t, vt oculo <lb/>compræhendatur hæc tam parua particula luminis lunaris.</s> - </p> - </div> - </div> - <div type="section" level="3" n="20"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">SOLVTIONES ALIQVAE.</head> - <head style="it" xml:space="preserve">Paulo Aemilio Raifestaim.</head> - <p> - <s xml:space="preserve">PoÅ¿t eas literas quas proximè ad te dedi, FranciÅ¿cus Monardus mihi retulit tuas <lb/>nonnullas dubitationes circa noÅ¿trum Theorema Arithmeticum .116. quarum <lb/>prima eÅ¿t, quod Å¿i numerus <var>.a.</var> <reg norm="cogitatus" type="simple">cogitatê°</reg>, eſſet æqualis .4. </s> - <s xml:space="preserve">tunc ipÅ¿e non eſſet multiplex <lb/>ipÅ¿i .4. de quo tamen nullam mentionem feci. </s> - <s xml:space="preserve">Idem etiam inquis, Å¿i <var>.a.</var> fuiſſet .5. 6. 7. <lb/>nec non .1. 2. et .3. </s> - <s xml:space="preserve">Cui reÅ¿pondi, quod <reg norm="quanuis" type="context">quãuis</reg> nullam fecerim mentionem de æqua <lb/>litate ipÅ¿ius <var>.a.</var> cum .4. nihiltamen refert, </s> - <s xml:space="preserve">propterea quod quando ita fuiſſet, nihi-<lb/>lominus eaÅ¿dem conditiones Å¿ubiret, <reg norm="quemadodum" type="context">quemadodÅ©</reg> Å¿i fuiſſet duplus, triplus, aut qua <lb/>druplus. eo quod à genere multiplici, æqualitas, formam diuerÅ¿am non induat. </s> - <s xml:space="preserve">Qua <lb/>re idem eueniet Å¿i <var>.a.</var> fuerit .4. 5. 6. 7. vt Å¿i eſſet .8. 9. 10. et .11. & Å¿ic de cæteris, excepto <lb/>quod in proprijs multiplicibus, vel in Å¿uperantibus ipÅ¿is multiplicibus <var>.a.</var> menÅ¿urare <lb/>tur ab ipÅ¿o .4. plus quam Å¿emel. </s> - <s xml:space="preserve">Quod autem dicis. de .1. 2. et .3. nihil eÅ¿t, quia, <lb/>vt in Å¿ecunda Å¿umma, hoc eÅ¿t in tertio termino maximo, reliquus tertius terminus, <lb/>ideÅ¿t .9. non compræhendetur, ita nobis indicabit primum numerum Å¿umptum mi <lb/>norem eſſe quaternario. </s> - <s xml:space="preserve">Quæ omnia, exipÅ¿a noÅ¿tra thÄ™oria ibidem expreſſa ma-<lb/>nifeÅ¿tantur. </s> - <s xml:space="preserve">Quid autem circa hoc Frater Lucas dicat, neÅ¿cio, quia ipÅ¿ius opus <lb/>ad manus meas nunquam peruenit, Å¿atis enim mihi fuit, in Tartalea hanc praxim <lb/>vidiſſe, ratio vero nullibi à me reperta fuit. </s> - <s xml:space="preserve">Tartalea enim multos citat authores, <lb/>quorum Å¿cripta ego nunquam vidi, vt Leonardi PiÅ¿ani, ProÅ¿docimi, Petri Borghi, <lb/>Fratris Lucæ, Ioannis Sfortunati, <reg norm="cæterorumque" type="simple">cæterorumqÌ;</reg> Å¿imilium.</s> - </p> - <pb o="302" rhead="IO. BABPT. BENED." n="314" file="0314" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0314"/> - </div> - </div> - <div type="section" level="3" n="21"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">ELIPSIM PROPOSITAM QVALITER</head> - <head xml:space="preserve">quadrare valeamus.</head> - <head style="it" xml:space="preserve">IlluÅ¿tri Uiro FranciÅ¿co Mendo Zzæ</head> - <p> - <s xml:space="preserve">QVod antea tuo nomine fecerat Marcus Antonius amicus noÅ¿ter Å¿ufficie-<lb/>bat. </s> - <s xml:space="preserve">Sed quia, quæ nunc à me petis, talia Å¿unt, vt Å¿ine tripartita <reg norm="aequa- liter" type="simple">Ä™qua-<lb/>liter</reg> aliqua data proportione non poſſit aliquis exactè intentum perfice-<lb/>re, nihilominus, Å¿uppoÅ¿ita di <lb/> - <anchor type="figure" xlink:label="fig-0314-01a" xlink:href="fig-0314-01"/> - cta diuiÅ¿ione, reliqua facilia <reg norm="erunt" type="context">erÅ©t</reg>. </s> - <s xml:space="preserve"><reg norm="Primum" type="context">PrimÅ©</reg> <lb/>enim eÅ¿t. </s> - <s xml:space="preserve">PropoÅ¿itam EllipÅ¿im qua-<lb/>drare.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0314-01" xlink:href="fig-0314-01a"> - <image file="0314-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0314-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Sit <reg norm="igitur" type="simple">igit̃</reg> EllipÅ¿is propoÅ¿ita <var>.a.b.d.c.</var> cu-<lb/>ius axes Å¿int <var>.a.b.</var> et <var>.d.c.</var> dati, Å¿eu <reg norm="reperti" type="simple">reê‘ti</reg> ex <lb/>47. <reg norm="Å¿ecundi" type="context">Å¿ecÅ©di</reg> Pergei, <reg norm="Å¿intque" type="simple">Å¿intqÌ;</reg> duo circuli <var>.a.e.<lb/>b.f.</var> et <var>.g.d.h.c.</var> circa eaÅ¿dem diametros, <lb/><reg norm="tunc" type="context">tÅ©c</reg> proportio <var>.a.b.</var> ad <var>.d.c.</var> <reg norm="dimidium" type="context">dimidiÅ©</reg> erit <lb/>proportionis circulorum ex .2. 12. Eu-<lb/>clid. </s> - <s xml:space="preserve">Å¿ed proportio <var>.a.b.</var> ad <var>.d.c.</var> æqualis <lb/>eÅ¿t proportioni maioris circuli ad Elli <lb/>pÅ¿im .ex .5. Archimedis in lib. de conoÂ<lb/>idalibus, quapropter proportio Elli-<lb/>pÅ¿is ad minorem circulum altera me-<lb/>dietas erit totius proportionis circulo-<lb/>rum, hoc eÅ¿t maioris ad minorem, qua <lb/>re EllipÅ¿is media proportionalis erit <lb/>inter eos circulos. </s> - <s xml:space="preserve">Nunc verò cum <lb/>ex Archimede repertÄ™ fuerint duæ fi-<lb/>guræ rectilineæ æquales duobus circu <lb/>lis iam dictis, & inter has, reperta fue <lb/>rit alia media proportionalis propoÅ¿i-<lb/>tum obtinebimus.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">Spheroidem propoÅ¿itam cubare.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">PRopoÅ¿ita Å¿phæroides erit, aut prolata, aut oblonga, Å¿it prius prolata, <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.a.b.</var> <lb/>diameter circuli, qui eam per æqualia Å¿ecat, circa quam <var>.a.b.</var> vt circa axem in-<lb/>telligatur Å¿phæroides oblonga, cuius Å¿piſſitudo Å¿it <var>.d.c.</var> axis prolatæ, cogitemus <reg norm="nunc" type="context">nÅ©c</reg> <lb/>duas Å¿phæras <var>.a.e.b.f.</var> et <var>.g.d.h.c.</var> circa dictos axes. </s> - <s xml:space="preserve">Vnde quatuor corpora habebi-<lb/>mus, hoc eÅ¿t duas Å¿phæras, & duas Å¿phæroides, quas probabo continuas proportio-<lb/>nales inuicem eſſe.</s> - </p> - <p> - <s xml:space="preserve">ConÅ¿ideremus igitur duos conos rectos, quorum <var>.a.b.</var> diameter Å¿it eorum baÅ¿ium, <lb/>altitudo autem maioris, æqualis Å¿it Å¿emidiametro majori, hoc eÅ¿t medietati <var>.a.b.</var> al- - <pb o="303" rhead="EPISTOLAE." n="315" file="0315" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0315"/> - titudo verò minoris, æqualis Å¿it Å¿emidiametro minori, hoc eÅ¿t medietati <var>.d.c.</var> vnde <lb/>habebimus proportionem coni maioris ad conum minorem, <reg norm="eandem" type="context">eãdem</reg> quæ eÅ¿t diame <lb/>tri maioris ad diametrum minorem, quod ex .2. parte .11. duodecimi Eucli. </s> - <s xml:space="preserve">nec non <lb/>ex .9. eiuÅ¿dem manifeÅ¿tum eÅ¿t, Å¿ed conus minor, eÅ¿t quarta pars Å¿phæroidis prolatæ <lb/>ex .29. </s> - <s xml:space="preserve">Archimedis in lib. de conoidalibus, & conus maior, eÅ¿t etiam quarta pars <lb/>Å¿phæræ, ex .32. primi lib. de Å¿phæra, & cyllindro, </s> - <s xml:space="preserve">quare ex communi Å¿cientia, <reg norm="eadem" type="context">eadẽ</reg> <lb/>proportio erit Å¿phæræ maioris ad Å¿phæroidem prolatam, quæ <var>.a.b.</var> ad <var>.d.c.</var> Å¿ed pro-<lb/>portio <var>.a.b.</var> ad <var>.d.c.</var> eÅ¿t tertia pars proportionis maioris Å¿phæræ ad <reg norm="minorem" type="context">minorẽ</reg>. </s> - <s xml:space="preserve">ConÅ¿idere <lb/>mus <reg norm="nunc" type="context">nÅ©c</reg> alios duos conos rectos, vnius & <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> baÅ¿is, <reg norm="cuius" type="simple">cuiê°</reg> diameter Å¿it <var>.d.c.</var> Å¿ed altitu <lb/>do maioris, æqualis Å¿it Å¿emidiametroÅ¿phÄ™rÄ™ maioris, altitudo verò minoris, Å¿it æqua <lb/>lis Å¿emidiametro minoris Å¿phæræ, vnde ex dictis rationibus habebimus <reg norm="proportio- nem" type="context">proportio-<lb/>nẽ</reg> maioris coni ad <reg norm="minorem" type="context">minorẽ</reg>, vt quæ eÅ¿t <var>.o.b.</var> ad <var>.o.d.</var> hoc eÅ¿t vt <var>.a.b.</var> ad <var>.d.c.</var> & ex dictis <reg norm="pro poÅ¿itionibus" type="simple">ê“Â<lb/>poÅ¿itionibus</reg> ita Å¿e habebit Å¿phæroides oblonga ad Å¿phÄ™ram minorem vt <var>.a.b.</var> ad <var>.d.<lb/>c.</var> hoc eÅ¿t tertia pars proportionis Å¿phæræ maioris ad minorem. </s> - <s xml:space="preserve">Quare proportio <lb/>Å¿phæroidis prolatæ ad oblongam, erit reliqua tertia pars proportionis maioris <reg norm="Å¿phae ræ" type="simple">Å¿phÄ™Â<lb/>ræ</reg> ad minorem. </s> - <s xml:space="preserve">Quapropter hæc quatuor corpora continua proportionalia inui-<lb/>cem erunt.</s> - </p> - <p> - <s xml:space="preserve">Nunc verò quærenda eÅ¿t inter <var>.a.b.</var> & Å¿uas duas tertias partes vna media pro por-<lb/>tionalis, quæ Å¿it <var>.K.</var> & ex Archimede, inuentum Å¿it quadratum Ä™quale circulo, cuius <lb/>Å¿it <var>.K.</var> diameter. </s> - <s xml:space="preserve">Vnde proportio circuli (cuius <var>.a.b.</var> eÅ¿t diameter) ad circulum cu-<lb/>ius <var>.K.</var> eÅ¿t diameter, Å¿eÅ¿quialtera erit ex .2. 12. Eucli.</s> - </p> - <p> - <s xml:space="preserve">Ducatur deinde quadratum lineæ <var>.K.</var> in lineam <var>.a.b.</var> & proueniet nobis cor-<lb/>pus quoddam, quod æquale erit Å¿phærÄ™ maiori, ex corellario .32. primi de Å¿phÄ™ra & <lb/>cyllindro, cuius corporis, latus cubus Å¿it <var>.m</var>.</s> - </p> - <p> - <s xml:space="preserve">Idem facere oportebit mediante <var>.d.c.</var> minoris Å¿phærÄ™, cuius corporis cubica ra-<lb/>dix Å¿it <var>.n</var>.</s> - </p> - <p> - <s xml:space="preserve">Nunc verò inter <var>.m.</var> et <var>.n.</var> inueniantur duÄ™ mediÄ™ proportionales <var>.s.t.</var> & ex <var>.s.</var> pro-<lb/>ducatur cubus, qui Ä™qualis erit Å¿phÄ™roidi prolatæ propoÅ¿iti, cubus vero <var>.t.</var> æqualis <lb/>erit Å¿phÄ™roidi oblongÄ™, cuius axis eſſet <var>.a.b</var>.</s> - </p> - <p> - <s xml:space="preserve">Si autem Å¿phÄ™roides oblonga nobis propoÅ¿ita fuiſſet, eodem methodo Å¿oluere-<lb/>tur problema.</s> - </p> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">Quadratum circulis mediantibus deÅ¿ignare.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">MOdus autem conficiendi quadratum ex circulis Å¿upra datam lineam, vt Do-<lb/>minum GaÅ¿parem docui, facillimus eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Sit enim linea <var>.b.a.</var> 46. propoÅ¿itionis primi Euclidis, <reg norm="poÅ¿itoque" type="simple">poÅ¿itoqÌ;</reg> pede immobli circi-<lb/>ni in puncto <var>.a.</var> Å¿ecundum quantitatem lineæ <var>.a.b.</var> propoÅ¿itÄ™ fiat circulus, Å¿imiliter cir-<lb/>ca punctum <var>.b.</var> alius circulus eiuÅ¿dem magnitudinis, </s> - <s xml:space="preserve">erecta deinde Å¿ola <var>.a.c.</var> perpendi <lb/>culari ipÅ¿i <var>.a.b.</var> ex puncto <var>.a.</var> ipÅ¿a Å¿ecabitur à circunferentia circuli. cuius centrum eÅ¿t <var>.<lb/>a.</var> in puncto <var>.c.</var> vnde <var>.a.c.</var> æqualis erit <var>.a.b.</var> poÅ¿ito demum pede immobili ipÅ¿ius circi <lb/>ni in puncto <var>.c.</var> Å¿ecundum longitudinem ipÅ¿ius <var>.c.a.</var> fiat alius circulus, qui æqualis erit <lb/>reliquis duobus circulis cum eorum Å¿emidiametri æquales Å¿int, & hic vltimo factus <lb/>Å¿ecabit circulum, cuius <reg norm="centrum" type="context">centrÅ©</reg> eÅ¿t <var>.b.</var> in <reg norm="puncto" type="context">pÅ©cto</reg> <var>.d.</var> à quo cum ductæ fuerint <var>.d.c.</var> et <var>.d.b.</var> - <pb o="304" rhead="IO. BAPT. BENED." n="316" file="0316" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0316"/> - rectè habebimus quod volumus. </s> - <s xml:space="preserve">nam omnia latera Å¿unt inuicem Ä™qualia ex condi-<lb/>ti onibus circuli, angulus autem <var>.a.</var> rectus effectus fuit, </s> - <s xml:space="preserve">tunc Å¿i imaginatione cogita-<lb/>ta fuerit diameter <var>.b.c.</var> ex .8. primi, concludemus angulum <var>.d.</var> eſſe rectum </s> - <s xml:space="preserve">deinde ex .5 <lb/>et .32. eiuÅ¿dem concludemus etiam reliquos angulos rectos eſſe.</s> - </p> - <p> - <s xml:space="preserve">Circa verò id quod mihi Å¿cripÅ¿iÅ¿ti de igne perpetuo putans nugas eſſe, quod Ro-<lb/>mæ inuentæ fuerint lucernÄ™ ardentes in Å¿epulchris antiquorum. </s> - <s xml:space="preserve">Ego quid em mi-<lb/>nimè puto eas nugas eſſe, propterea quod tales lucernas non vnus tantum aut duo <lb/>viderint, Å¿ed multi homines fide digniſſimi. </s> - <s xml:space="preserve">PrÄ™tera cum aisid nulla ratione poſſe <lb/>fieri. </s> - <s xml:space="preserve">ReÅ¿pondeo quod maxima ratione poſſibile eſſe puto, quam quidem ra-<lb/>tionem ita eſſe oportet, quod primum lucerna Å¿it perfectè circuncluÅ¿a, vt <lb/>materia in ea conÅ¿tituta nullo modo exire poſſit, </s> - <s xml:space="preserve">deinde quod materia inflamabilis <lb/>talis Å¿it, vt excrementum fuliginoÅ¿um ex flamma tranÅ¿miſſum, tangendo Å¿uperfi-<lb/>ciem deuexam ipÅ¿ius lucernæ, aptum Å¿it in <reg norm="priÅ¿tinum" type="context">priÅ¿tinÅ©</reg> <reg norm="humorem" type="context">humorẽ</reg> conge<unsure/>lari, Å¿iue transfor-<lb/>mari, vnde materia prima per tres formas perpetuò tranÅ¿ibit, hoc eÅ¿t per humorem, <lb/>Å¿iue oleum tale, vt diximus, per ignem, Å¿eu flammam, & per vaporem, Å¿eu exhala-<lb/>tionem fuliginoÅ¿am aptam condenÅ¿ari, atque in priorem humorem illicò reuerti.</s> - </p> - </div> - </div> - <div type="section" level="3" n="22"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE DIVISIONE TRIANGVLI SECVNDVM <lb/>propoÅ¿itam proportionem.</head> - <head style="it" xml:space="preserve">Michaeli Angelo MuciaÅ¿co.</head> - <p> - <s xml:space="preserve"><emph style="sc">QVod</emph> mihi proponis, tale eÅ¿t, vt Å¿cilicet tibi modum Å¿cribam diuidendi <lb/>triangulum propoÅ¿itum Å¿ecundum datam proportionem à linea tranÅ¿eun <lb/>te per punctum notatum extra triangulum.</s> - </p> - <p> - <s xml:space="preserve">Tr<unsure/><reg norm="iangulum" type="context">iangulÅ©</reg> <reg norm="igitur" type="simple">igit̃</reg> à te mihi propoÅ¿itum Å¿it <var>.n.o.u.</var> conÅ¿idero <reg norm="primum" type="context">primÅ©</reg> quod Å¿i <lb/>quis ipÅ¿um diuiÅ¿erit in duas partes mediante <var>.e.s.</var> parallela ad <var>.n.u.</var> ea proportione, <lb/>quam mihi proponis. </s> - <s xml:space="preserve">deinde inuenerit in dicta <var>.e.s.</var> punctum <var>.r.</var> per quod tranÅ¿iens <lb/>alia linea à puncto <var>.p.</var> propoÅ¿ito, ita quod efficiat duo triangula <var>.m.r.e.</var> et <var>.r.s.x.</var> inui-<lb/>cem æqualia, problema Å¿olutum erit. <lb/> - <anchor type="figure" xlink:label="fig-0316-01a" xlink:href="fig-0316-01"/> - eo quod triangulum <var>.m.o.x.</var> æquale <lb/>eſſet triangulo <var>.e.o.s.</var> & quadrilate-<lb/>rum reÅ¿iduum <var>.m.n.u.x.</var> etiam Ä™quale <lb/>eſſet quadrilatero <var>.e.n.u.s</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0316-01" xlink:href="fig-0316-01a"> - <image file="0316-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0316-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Sed dum punctum <var>.r.</var> uenarer, alia <lb/> - <anchor type="figure" xlink:label="fig-0316-02a" xlink:href="fig-0316-02"/> - via mihi in mentem venit, cognoui <lb/>igitur quod quum propoÅ¿itum expe-<lb/>ditum fuiſſer, hoc eÅ¿t, <reg norm="quod" type="simple">ê™</reg> Å¿i à puncto p. <lb/>protracta eſſet linea <var>.p.m.</var> quÄ™ trian-<lb/>gulum <var>.n.o.u.</var> in duas partes inuicem <lb/>ita proportionatas diuiÅ¿iſſet, vt Å¿e ha <lb/>bet <var>.A.</var> et <var>.B.</var> ita Å¿e haberet <reg norm="productum" type="context">productÅ©</reg> <lb/> - <anchor type="figure" xlink:label="fig-0316-03a" xlink:href="fig-0316-03"/> - <var>n.o.</var> in <var>.o.u.</var> ad productum <var>.m.o.</var> in <var>.o.<lb/>x.</var> vt trianguli <var>.n.o.u.</var> ad triangulum <lb/><var>m.o.x.</var> quod quidem non eÅ¿t diffi-<lb/>cile Å¿peculari, ex methodo .24. Å¿exti, - <pb o="305" rhead="EPISTOLAE." n="317" file="0317" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0317"/> - eo quod tam proportio producti <var>.n.o.</var> in <var>.o.u.</var> ad productum <var>.m.o.</var> in <var>.o.x.</var> quam pro-<lb/>portio trianguli <var>.n.o.u.</var> ad triangulum <var>.m.o.x.</var> componitur ex proportione <var>.u.o.</var> ad <var>.o.<lb/>x.</var> & ex proportion <var>e.n.o.</var> ad <var>.m.o.</var> vnde proportio dictorum productorum nobis co-<lb/>gnita erit, eo quod cum nobis cognita Å¿it proportio <var>.A.</var> ad <var>.B.</var> vt data, cognita etiam <lb/>nobis erit coniuncta, hoceÅ¿t <var>.A.B.</var> ad <var>.B</var>. </s> - <s xml:space="preserve">& propterea ea quæ trianguli <var>.n.o.u.</var> ad <reg norm="trian- gulum" type="context">triã-<lb/>gulum</reg> <var>.m.o.x.</var> & Å¿imiliter productorum. </s> - <s xml:space="preserve">Quæſiui poÅ¿tea modum inueniendi duas <lb/>dictas lineas <var>.m.o.</var> et <var>.o.x.</var> & cognoui quod Å¿i producta fuerit <var>.p.i.</var> æquidiÅ¿tans li-<lb/>neæ <var>.o.x.</var> <reg norm="producendoque" type="simple">producendoqÌ</reg> <var>.o.n.</var> quouÅ¿que cum <var>.p.i.</var> Å¿e interÅ¿ecarent in puncto <var>.i.</var> inuenien <lb/>do poÅ¿tea lineam quandam, quæ ducta cum <var>.p.i.</var> efficeret rectangulum æquale rectan <lb/>gulo cognito quod ex <var>.m.o.</var> in <var>.o.x.</var> poteÅ¿t fieri, quod cognitum dico, eo quod nobis <lb/>cognita eÅ¿t proportio data, & rectangulum etiam <var>.n.o.</var> in <var>.o.u.</var> deinde Å¿ecando ab <var>.o.<lb/>n.</var> partem æqualem lineæ iam inuentæ, quæ Å¿it <var>.o.t</var>. </s> - <s xml:space="preserve">Inueniendo poÅ¿tea, ex .28. Å¿exti <lb/>lineam <var>.o.m.</var> cuius productum in <var>.m.t.</var> æquale Å¿it producto <var>.t.o.</var> in <var>.o.i.</var> vnde ex .15. eiuÅ¿ <lb/>dem proportio <var>.o.i.</var> ad <var>.m.o.</var> eadem eſſet, quæ <var>.m.t.</var> ad <var>.o.t.</var> & componendo, ita Å¿e ha-<lb/>beret <var>.m.i.</var> ad <var>.m.o.</var> vt <var>.m.o.</var> ad <var>.o.t.</var> Å¿ed ex .4. Å¿exti, ita eſſet <var>.p.i.</var> ad <var>.o.x.</var> vt <var>.m.i.</var> ad <var>.m.o</var>. <lb/></s> - <s xml:space="preserve">quare ex .11. quinti, ita eſſet <var>.p.i.</var> ad <var>.o.x.</var> vt <var>.m.o.</var> ad <var>.o.t.</var> vnde ex .15. Å¿exti productum <var>.<lb/>o.x.</var> in <var>.m.o.</var> æquale eſſet producto. p, i. in <var>.o.t.</var> & Å¿ic haberemus intentum.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0316-02" xlink:href="fig-0316-02a"> - <image file="0316-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0316-02"/> - </figure> - <figure xlink:label="fig-0316-03" xlink:href="fig-0316-03a"> - <image file="0316-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0316-03"/> - </figure> - </div> - <p> - <s xml:space="preserve">Sed Å¿i punctum <var>.m.</var> caderet in punctum <var>.n.</var> idem eſſet, Å¿i vorò punctum <var>.m.</var> tranÅ¿iret <lb/>n. oporteret nos facere hoc in latere <var>.n.u.</var> ipÅ¿um quærendo in linea <var>.n.u.</var> ducendo pri <lb/>mum lineam <var>.p.i.</var> <reg norm="æquidiÅ¿tantem" type="context">æquidiÅ¿tantẽ</reg> <var>.u.x.</var> & producendo <var>.u.n.</var> ad partem <var>.u.</var> proÅ¿equendo, <reg norm="quod" type="simple">ê™</reg> <lb/>Å¿uperius iam dictum eÅ¿t.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">Idem facere de parallelogr ammo.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">DAtum parallelogrammum in duas partes diuidere, Å¿ecundum aliquam datam <lb/>proportionem à linea tranÅ¿eunte per punctum propoÅ¿itum.</s> - </p> - <p> - <s xml:space="preserve">Sit exempli gratia, datum parallelogrammum <var>.b.u.</var> datum verò punctum <var>.o.</var> extra <lb/>figuram, proportio autem ea Å¿it, quæ <var>.A.</var> ad <var>.B.</var> vt Å¿upra. </s> - <s xml:space="preserve">Nunc diuidatur primò re-<lb/>ctangulum datum per æqualia, mediante linea <var>.r.c.</var> parallela ambobus lateribus <var>.b.x.</var> <lb/>et <var>.s.u.</var> quæ quidem linea diuidatur in puncto <var>.i.</var> ita quod eadem proportio Å¿it <var>.r.i.</var> ad <var>.<lb/>i.c.</var> vt <var>.A.</var> ad <var>.B.</var> protrahatur deinde à puncto <var>.o.</var> linea <var>.o.i.q.</var> quæ Å¿ecabit ambo duo la-<lb/>tera <var>.b.x.</var> vel <var>.s.u.</var> intra terminos eorum, vel tantum <var>.b.x.</var> reliquum verò extra termi-<lb/>nos <var>.s.u</var>.</s> - </p> - <p> - <s xml:space="preserve">Nunc autem Å¿i intra dictos terminos tranÅ¿ibit, vt in prima figura videre potes, <lb/>problema Å¿olutum erit, eo quod <lb/> - <anchor type="figure" xlink:label="fig-0317-01a" xlink:href="fig-0317-01"/> - Å¿i à puncto <var>.i.</var> protracta fuerit <var>.p.<lb/>d.</var> pa rallela ad <var>.u.x.</var> habebimus <lb/>ex prima Å¿exti eandem propor-<lb/>tionem <var>.s.d.</var> ad <var>.p.x.</var> ut <var>.r.i.</var> ad <var>.i.c.</var> <lb/>hoc eÅ¿t vt <var>.A.</var> ad <var>.B.</var> Å¿ed <reg norm="triangulus" type="context">triãgulus</reg> <lb/><var>i.e.d.</var> æqualis eÅ¿t triangulo <var>.i.q.p.</var> <lb/>vt tibi facilè patebit, vnde qua-<lb/>drilaterum <var>.e.q.u.x.</var> æquale erit <lb/>quadrilatero <var>.d.u.</var> ex communi - <pb o="306" rhead="IO. BAPT. BENED." n="318" file="0318" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0318"/> - Å¿cientia. </s> - <s xml:space="preserve">Quare ex .9. quinti, ita erit <var>.s.d.</var> ad dictum <var>.d.u.</var> vt ad quadrilaterum <var>.e.q.u.<lb/>x.</var> hoc eÅ¿t vt <var>.A.</var> ad <var>.B.</var> ex .11. eiuÅ¿dem.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0317-01" xlink:href="fig-0317-01a"> - <image file="0317-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0317-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Sed Å¿i punctum <var>.q.</var> fuerit extra ut in .2. figura videre eÅ¿t. </s> - <s xml:space="preserve">tunc manifeÅ¿tum erit, <reg norm="quod" type="simple">ê™</reg> <lb/>triangulus <var>.e.x.t.</var> maior erit pa-<lb/>rallelogrammo <var>.d.u.</var> per triangu <lb/> - <anchor type="figure" xlink:label="fig-0318-01a" xlink:href="fig-0318-01"/> - lum <var>.q.t.u.</var> cum triangulus <var>.q.i.p.</var> <lb/>æqualis triangulo <var>.d.i.e.</var> excedat <lb/>quadrilaterum <var>.i.t.u.p.</var> per trian <lb/>gulum <reg norm="dictum" type="context">dictÅ©</reg> <var>.q.t.u.</var> quapropter <lb/>cum diuiÅ¿us fuerit triangulus <var>.e.<lb/>x.t.</var> mediante linea <var>.o.n.K.</var> ita <reg norm="quod" type="simple">ê™</reg> <lb/><reg norm="quadrilaterum" type="context">quadrilaterÅ©</reg> <var>.e.n.K.t.</var> Å¿it æquale <lb/>triangulo <var>.q.t.u.</var> ex doctrina præ <lb/>cedenti, habebimus propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0318-01" xlink:href="fig-0318-01a"> - <image file="0318-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0318-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">Idem de frusto trianguli.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">SEd Å¿i quadrilaterum dictum eſſet fruÅ¿tum alicuius <reg norm="trianguli" type="context">triãguli</reg> ut in figura <var>.A.</var> hic Å¿ub <lb/>Å¿cripta videre eÅ¿t, Å¿uppoÅ¿ita, <var>b.d.</var> parallela ad <var>.u.p.</var> ita faciendum eſſet, ducendo <lb/>Å¿cilicet parallelam <var>.u.x.</var> ad <var>.b.p.</var> quæ producatur vÅ¿que ad concurÅ¿um cum <var>.b.d.</var> <lb/>in puncto <var>.x.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> proportio data inter <var>.t.a.</var> et <var>.a.e.</var> quas duas lineas cogitemus inuicem <lb/>directè coniunctas, </s> - <s xml:space="preserve">tunc diuidatur tota <var>.t.e.</var> <lb/> - <anchor type="figure" xlink:label="fig-0318-02a" xlink:href="fig-0318-02"/> - in puncto <var>.i.</var> ita vt <var>.t.i.</var> ad <var>.i.e.</var> Å¿it vt quadrilate <lb/>ri <var>.p.d.</var> ad trigonum <var>.u.d.x</var>. </s> - <s xml:space="preserve">deinde diuidatur <lb/><var>t.i.</var> in puncto r. tali modo vt <var>.t.r.</var> ad <var>.r.i.</var> Å¿e ha-<lb/>beat vt <var>.t.a.</var> ad <var>.a.e.</var> quo facto ex doctrina <reg norm="prae cedenti" type="simple">prÄ™Â<lb/>cedenti</reg> diuidatur totum parallelogram--<lb/>mum <var>.p.x.</var> mediante linea <var>.o.q.</var> Å¿ecundum <lb/>quod Å¿e habet <var>.t.r.</var> ad <var>.r.e</var>. </s> - <s xml:space="preserve">Atque ita Å¿olu-<lb/>tum erit problema, vt exte ipÅ¿o ratiotina-<lb/>ri facile potes.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0318-02" xlink:href="fig-0318-02a"> - <image file="0318-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0318-02"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="4"> - <head style="it" xml:space="preserve">Fdem de quadrilatero in genere.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">SEd Å¿i nullum latus parallelum reliquo erit, ita faciendum erit. </s> - <s xml:space="preserve">Å¿i Å¿it tale quadrila <lb/>terum <var>.b.d.u.p.</var> oportet vt ipÅ¿um conuertamus in triangulum, producendo duo <lb/>quÄ™uis eius latera oppoÅ¿ita uÅ¿que ad interÅ¿ectionem ut pote <var>.u.p.</var> et <var>.d.b.</var> in puncto <var>.x.</var> <lb/>quo facto, Å¿upponemus <var>.o.</var> eſſe punctum datum, proportio verò data Å¿it <var>.t.r.</var> ad <var>.r.i.</var> ad <lb/>iungatur deinde <var>.i.e.</var> ad <var>.t.i.</var> ad quam <var>.e.i.</var> ipÅ¿a <var>.t.i.</var> Å¿e habeat vt quadrilaterum <var>.b.d.u.p.</var> - <pb o="307" rhead="EPISTOLAE." n="319" file="0319" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0319"/> - Å¿e habet ad triangulum <var>b.p.x.</var> ducatur poÅ¿tea <var>.o.q.</var> quæ diuidat totale triangulum <var>.d.<lb/>u.x.</var> in duas partes inuicem ita proportionatas, ut Å¿e habent <var>t.r.</var> et <var>.r.e.</var> quæ quidem <lb/>partes Å¿int <var>.c.d.u.q.</var> et <var>.c.q.x.</var> ut in primo problemate tibi monÅ¿traui, & habebis pro-<lb/>poÅ¿itum, dato quod punctum <var>.c.</var> Å¿it inter <lb/>b. et <var>.d</var>.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i forte linea <var>.o.q.</var> Å¿ecabit <var>.b.x.</var> hoc <lb/> - <anchor type="figure" xlink:label="fig-0319-01a" xlink:href="fig-0319-01"/> - eÅ¿t Å¿i punctum <var>.c.</var> eſſet inter <var>.b.</var> et <var>.x.</var> mani-<lb/>feÅ¿tum eÅ¿t, quod <var>.c.q.</var> Å¿ecaret <var>.b.p.</var> in pun-<lb/>cto <var>.y.</var> vnde in tali caÅ¿u, alio modo ope-<lb/>randum eſſet, hoc eÅ¿t ducendo <var>.b.u.</var> quæ <lb/>diuideret quadrilaterum in duo triangu-<lb/>la, & ut Å¿e haberet triangulum <var>.b.d.u.</var> ad <lb/>triangulum <var>.b.p.u.</var> vellem vt ita Å¿ecaretur <lb/><var>t.i.</var> in puncto <var>.n.</var> vt ita Å¿e haberet <var>.t.n.</var> ad <var>.n.<lb/>i.</var> ut dictum eÅ¿t de iÅ¿tis duobus triangulis, <lb/></s> - <s xml:space="preserve">deinde prout Å¿e habet <var>.n.r.</var> ad <var>.r.i.</var> ita Å¿eca-<lb/>res triangulum <var>.b.p.u.</var> mediante linea <var>.o.<lb/>K.</var> ex doctrina primi problematis, & ita haberes propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0319-01" xlink:href="fig-0319-01a"> - <image file="0319-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0319-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="5"> - <head style="it" xml:space="preserve">Idem de Pentagono, Exagono, & de reliquis.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">PEntagonum, Å¿eu hexagonum, vel alias quaÅ¿uis multilateras figuras propoÅ¿itas its <lb/>diuidere, vt dictum eÅ¿t de trilateris, & quadrilateris.</s> - </p> - <p> - <s xml:space="preserve">Sit exempli gratia pentagonus <var>.a.d.u.p.b.</var> quem Å¿ecare volumus <reg norm="mediante" type="context">mediãte</reg> linea <var>.o.<lb/>q.</var> in duas partes inuicem Å¿e habentes, vt Å¿e habent <var>.t.r.</var> et <var>.r.i.</var> oportet igitur ut ipÅ¿um <lb/>pentagonum reducas ad quadrilaterum <var>.x.a.d.u.</var> quod diuidatur Å¿ecundum præce-<lb/>dentem doctrinam, vt Å¿e habet <var>.t.r.</var> ad <var>.r.e.</var> <lb/>vnde Å¿i punctum <var>.q.</var> incidit inter <var>.p.</var> et <var>.u</var>. </s> - <s xml:space="preserve">tunc <lb/>habebis propoÅ¿itum, Å¿i verò incidet inter <var>.<lb/> - <anchor type="figure" xlink:label="fig-0319-02a" xlink:href="fig-0319-02"/> - p.</var> et <var>.x.</var> clarum erit quod linea <var>.o.q.</var> Å¿ecabit <lb/>latus <var>.p.b.</var> trianguli <var>.b.x.p.</var> in puncto <var>.y.</var> qua-<lb/>propter duces lineam <var>.a.p.</var> vt claudat trian-<lb/>gulum <var>.a.b.p.</var> <reg norm="diuidaturque" type="simple">diuidaturqÌ;</reg> <var>.t.i.</var> in puncto <var>.n.</var> ita <lb/>vt <var>.t.n.</var> ad <var>.n.i.</var> Å¿e habeat, vt quadrilaterum. a<unsure/> <var>.<lb/>d.u.p.</var> ad <reg norm="triangulum" type="context">triãgulum</reg> <var>.a.b.p</var>. </s> - <s xml:space="preserve">deinde <reg norm="hunc" type="context">hÅ©c</reg> trian <lb/>gulum <var>.a.b.p.</var> diuidas mediante linea <var>.o.K.</var> <lb/>vt <var>.n.r.</var> ad <var>.r.i.</var> ex doctrina primi problematis <lb/>& habebis propoÅ¿itum. </s> - <s xml:space="preserve">Idem dico de hexa <lb/>gono, reducendo ipÅ¿um ad pentagonum, & <lb/>item de eptagono, ipÅ¿um reducendo ad exa <lb/>gonum, & idem infero de infinito ipÅ¿arum <lb/>Å¿uperficialium figurarum rectilinearum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0319-02" xlink:href="fig-0319-02a"> - <image file="0319-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0319-02"/> - </figure> - </div> - <pb o="308" rhead="IO. BAPT. BENED." n="320" file="0320" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0320"/> - </div> - <div type="letter" level="4" n="6"> - <head style="it" xml:space="preserve">De duobus triangulis equalibus inter lineas <lb/>inuicem inclinatas.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">TV mihi vltimò proponis duas lineas rectas <var>.b.f.</var> et <var>.q.s.</var> in eadem Å¿uperficie pla-<lb/>na, non tamen inuicem æqu idiÅ¿tantes, proponis etiam <var>.n.t.</var> in eadem Å¿uperfi-<lb/>cie, quæ vnamquamque priorum Å¿ecat, proponis etiam lineam <var>.h.</var> tali conditione, <lb/>quod nulli dictarum Å¿it parallela, </s> - <s xml:space="preserve">deinde Å¿cire cupis qua arte aliquis poſſet ducere <var>.<lb/>c.u.</var> parallelam ad <var>.h.</var> ita quod Å¿ecando <var>.n.t.</var> conÅ¿tituat duos triangulos <var>.n.o.u.</var> et <var>.t.o.e.</var> <lb/>inuicem æquales.</s> - </p> - <p> - <s xml:space="preserve">Facita, producas primò duas primas lineas à parte, in qua inuicem inclinantur, <lb/>vÅ¿que ad concurÅ¿um in puncto <var>.i</var>. </s> - <s xml:space="preserve">deinde à puncto <var>.n.</var> duces <var>.n.c.</var> <reg norm="parallelam" type="context">parallelã</reg> ad <var>.h.</var> poÅ¿tea <lb/>ex .25. Å¿exti Eucli. conÅ¿titues <reg norm="triangulum" type="context">triãgulum</reg> <var>.i.u.e.</var> Å¿imile triangulo <var>.i.c.n.</var> æquale tamen <lb/>triangulo <var>.i.t.n.</var> & Å¿olutum erit problema. <lb/></s> - <s xml:space="preserve">VelÅ¿ic, inuenies <var>.i.e.</var> mediam <lb/> - <anchor type="figure" xlink:label="fig-0320-01a" xlink:href="fig-0320-01"/> - proportionalem inter <var>.i.c.</var> & <lb/><var>i.t.</var> duces poÅ¿tea <var>.e.u.</var> paralle-<lb/>lam lineÄ™ <var>.h.</var> vel <var>.c.n.</var> quod <reg norm="idem" type="context">idẽ</reg> <lb/>erit ex .30. primi Eucli. </s> - <s xml:space="preserve">& Å¿o-<lb/>lutum erit problema.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0320-01" xlink:href="fig-0320-01a"> - <image file="0320-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0320-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Nam ex .17. Å¿exti eadem <lb/>proportio erittrianguli <var>.i.c.<lb/>n.</var> ad triangulum <var>.i.e.u.</var> ut <var>.i.c.</var> <lb/>ad <var>.i.t</var>. </s> - <s xml:space="preserve">Quare ut trianguli <var>.i.<lb/>c.n.</var> ad <reg norm="triangulum" type="context">triangulÅ©</reg> <var>.i.t.n.</var> ex pri-<lb/>ma Å¿exti, et .11. quinti. </s> - <s xml:space="preserve">Vnde <lb/>ex .9. eiuÅ¿dem <var>.i.e.u.</var> æqualis <lb/>erit <var>.i.t.n</var>. </s> - <s xml:space="preserve">Quapropter <var>.o.n.u.</var> <lb/>æqualis etiam erit <var>.o.e.t</var>.</s> - </p> - </div> - </div> - <div type="section" level="3" n="23"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">SOLVTIONES NONNVLLAE QVOR VNDAM <lb/>problematum.</head> - <head style="it" xml:space="preserve">ThaodoÅ¿io à Raifestaim.</head> - <p> - <s xml:space="preserve"><emph style="sc">DVritandvm</emph> profecto non eÅ¿t, quin quotidie hominibus Å¿tudioÅ¿is ali-<lb/>quid noui deÅ¿it, quemadmodum, quod tibi nunc occurrit, mihi non-<lb/>nunquam accidit, hoc eÅ¿t inuenire orizontem, cui aliqua propoÅ¿ita Å¿tel <lb/>la oriatur cum gradu ipÅ¿ius longitudinis. </s> - <s xml:space="preserve">pro <reg norm="cuius" type="simple">cuiê°</reg> rei operatione te prius <lb/>Å¿cire oportebit vtrum Å¿tella in Å¿ignis aÅ¿cendentibus, vel deÅ¿cendentibus reperiatur, <lb/>hoc eÅ¿t in Å¿ignis, quÄ™ à Capricorno ad Cancrum procedunt, vel in illis, quÄ™ à Can-<lb/>cro ad Capricornum numerantur, </s> - <s xml:space="preserve">propterea quod Å¿i in Å¿ignis aÅ¿cendentibus inue-<lb/>nitur, Å¿ciendum eÅ¿t, quod Å¿upra talem orizontem polus mundi auÅ¿tralis attollitur, <lb/>Å¿ed Å¿i in Å¿ignis deÅ¿cendentibus reperitur, </s> - <s xml:space="preserve">tunc polus borealis eleuatur Å¿upra dictum - <pb o="309" rhead="EPISTOL AE." n="321" file="0321" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0321"/> - orizontem, vt exempli gratia, canicula quæ à Græcis <reg norm="Prochyon" type="context">Prochyõ</reg> vocatur, reperitur in <lb/>24. minuto vigeÅ¿imi gradus Cancri, quapropter polus borealis eleuatur Å¿upra ori-<lb/>zontem, cui ipÅ¿a oritur cum eodem gradu, & minuto eclipticÄ™ illius Å¿igni. </s> - <s xml:space="preserve">Å¿ed <lb/>quia volumus etiam Å¿cire veram quantitatem arcus eleuationis huiuÅ¿modi poli, pro <lb/>pterea accipiemus in tabula generali Monteregij numerum qui vocatur radix aÅ¿cen <lb/>Å¿ionum, èregione numeri longitudinis ipÅ¿ius Å¿tellæ, qui quidem numerus in præſen <lb/>ti exemplo erit gra .107. cum minutis .53. qui eÅ¿t <reg norm="cuiuÅ¿dam" type="context">cuiuÅ¿dã</reg> arcus æquatoris, qui inci-<lb/>pit in principio Arietis, & in circulo latitudinis deÅ¿init, hoc eÅ¿t ab orizonte quæſi-<lb/>to, ita quod talis numerus erit aÅ¿cenÅ¿io obliqua huiuÅ¿modi puncti eclipticæ illi ori-<lb/>zonti, qua aÅ¿cenÅ¿ione mediante, Å¿imul cum gradu, & minuto longitudinis in tabulis <lb/>aÅ¿cenÅ¿ionum obliquarum, inueniemus gradum, & minutum altitudinis pollaris, <reg norm="quod" type="wordlist">qÄ‘</reg> <lb/>quærebatur, eodem ordine ac methodo, quo vtimur ad inueniendum in tabulis po-<lb/>Å¿itionum, polum circuli poÅ¿itionis alicuius aÅ¿tri, mediante declinatione & diÅ¿tantia <lb/>à meridiano ciuÅ¿dem aÅ¿tri, vt Å¿cis. </s> - <s xml:space="preserve">Vnde in præſenti exemplo eleuatio poli borea <lb/>lis Å¿upra talem orizontem erit gra .7. cum minutis .45.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i Å¿tella fuerit in medietate aÅ¿cendente, tunc certi erimus polum auÅ¿tralem Å¿u <lb/>per dictum orizontem attolli, nam idem eÅ¿t quærere altitudinem vnius <reg norm="polorum" type="context">polorÅ©</reg> mun <lb/>di à tali orizonte, quod diÅ¿tantiam dicti poli à circulo Å¿ecundum quem longitudo <lb/>terminatur, qui etiam latitudinis dicitur, eo quod tunc temporis talis circulus vnus <lb/>& idem eÅ¿t cum orizonte. </s> - <s xml:space="preserve">Sumatur ergo exempli gratia Å¿tella, quæ in ore piÅ¿cis au <lb/>Å¿tralis eÅ¿t, quÄ™, pro nunc, Å¿it in gradu .20. cum minutis .14. </s> - <s xml:space="preserve">Aquarij longitudinis, & <lb/>in gradu .23. cum nullo minuto meridianæ latitudinis. </s> - <s xml:space="preserve">Tunc certi erimus orizon-<lb/>tem, cui dicta Å¿tella oritur cum eiuÅ¿modi puncto eclipticæ, depreſſum eſſe à parte <lb/>auÅ¿trali Å¿ub <reg norm="illoque" type="simple">illoqÌ</reg> polo, Å¿ed quia propoÅ¿itum eÅ¿t Å¿cire etiam quantitatem huiuÅ¿mo-<lb/>di depræſſionis, reperiemus in tabula generali gradum, & minutum æquatoris, cor-<lb/>reÅ¿pondentem tali puncto longitudinis à circulo latitudinis terminato, qui quidem <lb/>numerus in præſenti exemplo erit gra .317. cum minutis .46. & hic numerus, vt dixi <lb/>mus eÅ¿t aÅ¿cen. obli. ad dictum orizontem, vbi polus auÅ¿tralis attollitur, & deÅ¿cenÅ¿io <lb/>obliqua, vbi polus borealis eleuatur. </s> - <s xml:space="preserve">Quapropter Å¿i à .317. gradibus cum minutis <lb/>46. demptus fuerit dimidius circulus gra .180. remanebunt gra .137. cum minutis .46 <lb/>& punctus oppoÅ¿itus gradibus .20. cum .14. minutis Aquarij eÅ¿t in eodem numero <lb/>Leonis, & mediantibus iÅ¿tis gradibus .137. min .46. aÅ¿cenÅ¿ionis, cum grad .20. min <num value="14">.<lb/>14.</num> Leonis inueniemus eleuationem poli borealis ab orizonte in tabulis aÅ¿cenÅ¿io-<lb/>num obliquarum Monteregij, hoc eÅ¿t gra .17. min .53. & eadem altitudo erit poli <lb/>auÅ¿tralis Å¿upra orizontem à quo Fomahant cum dicto puncto eclipticæ oritur, in qua <lb/>longitudine dicta Å¿tella reperitur.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i propoÅ¿itus nobis fuerit punctus eclipticæ, cum quo aliqua Å¿tella oritura Å¿it, <lb/>& oporteat inuenire vbi, hoc eÅ¿t orizontem huiuÅ¿modi ortus, eleuatione poli arti <lb/>ci, Å¿eu antarctici Å¿upra talem orizontem, ita operandum eſſet.</s> - </p> - <pb o="310" rhead="IO. BAPT. BENED." n="322" file="0322" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0322"/> - <p> - <s xml:space="preserve">Sit exempli gratia Å¿tella <var>.o.</var> ecli <lb/> - <anchor type="figure" xlink:label="fig-0322-01a" xlink:href="fig-0322-01"/> - ptica verò <var>.d.q.</var> æquator autem <var>.<lb/>p.q.</var> punctus verò eclipticæ, cum <lb/>quo Å¿tella oritura Å¿it <var>.e.</var> orizon de <lb/><reg norm="mum" type="context">mÅ©</reg> <var>.o.e.</var> vbi Å¿tella oriri poſſit <reg norm="cum" type="context">cÅ©</reg> <lb/>puncto <var>.e</var>. </s> - <s xml:space="preserve">Nam cum Å¿tella pro-<lb/>ponitur, datur etiam eius longi-<lb/>tudo, nec non latitudo, </s> - <s xml:space="preserve">quare ar-<lb/>cus <var>.a.q.</var> & arcus <var>.a.o.</var> nobis cogni <lb/>tus erit, cum Å¿upponatur arcus <var>.a.<lb/>o.</var> eſſe circuli latitudinis, et <var>.a.o.</var> <lb/>Iatitudo ipÅ¿ius Å¿tellæ, & angulus <lb/>a. rectus erit, & quia punctum <var>.e.</var> <lb/>datur, ergo arcus <var>.a.o.</var> & arcus <var>.a.<lb/>e.</var> Å¿imul <reg norm="cum" type="context">cÅ©</reg> angulo <var>.a.</var> recto cogni-<lb/>ti Å¿unt, vnde ex .11. primi lib. coÂ<lb/>pernici, angulus <var>.a.e.o.</var> cognoÅ¿ce-<lb/>tur, & angulus <var>.q.e.o.</var> Å¿imiliter, vt <lb/>reÅ¿iduum ex duobus rectis quo. e <lb/>mediante cum angulo <var>.q.</var> declina <lb/>tionis ab æquatore, <reg norm="medianteque" type="simple">medianteqÌ;</reg> <lb/>latere <var>.q.e.</var> cognito, cognitus quo <lb/>que nobis erit angnlus <var>.e.t.q.</var> ex <lb/>12. eiuÅ¿dem. </s> - <s xml:space="preserve">qui quidem angulus <lb/>erit altitudinis æquatoris ab ori-<lb/>zonte quæſito, qui demptus à <lb/>90. gradibus, dabit altitudinem <lb/>poli ab orizonte quæſito.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0322-01" xlink:href="fig-0322-01a"> - <image file="0322-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0322-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Inuenire poÅ¿tea gradum eclipticÄ™, cum quo Å¿tella data oriatur ad orizontem pro <lb/>poÅ¿itum, nullius eÅ¿t difficul@atis.</s> - </p> - <p> - <s xml:space="preserve">Ponamus exempli gratia, aliquem Å¿cire velle gradum eclipticæ, cum quo canicu-<lb/>la oritur ad orizontem, cui polus boreus eleuatur per gradus .44. quæ canicula Å¿up-<lb/>ponatur habere gradus .19. cum min .24. </s> - <s xml:space="preserve">Cancri longitudinis, & gra .16. min .10. lati-<lb/>tudinis meridianæ, quærere primum oportet eius declinationem ex doctrina .2. pro <lb/>blematis tabularum directionum Monteregij, quæ erit graduum .6. cum minutis .5. <lb/>Å¿eptentrionalis, </s> - <s xml:space="preserve">deinde inuenire eius aÅ¿cenÅ¿ionem rectam ex doctrina .4. problema <lb/>tis eiuÅ¿dem Monteregij, quæ erit gra .108. mi .42. </s> - <s xml:space="preserve">deinde <reg norm="mediante" type="context">mediãte</reg> declinatione iam <lb/>inuenta in tabulis differentiarum aÅ¿cenÅ¿ionalium Å¿ub polo .44. accipiemus differen-<lb/>tiam aÅ¿cenÅ¿ionum, qua differt recta ab obliqua, quæ in præſenti exemplo erit gra <num value="5">.<lb/>5.</num> min .55. quæ dempta ab aÅ¿cenÅ¿ione recta Å¿tellæ, vt præſens exemplum exigit, relin <lb/>quet nobis aÅ¿cenÅ¿ionem obliquam Å¿tellæ propoÅ¿itæ ad polum. gra .44. quæ erit gra <num value="102">.<lb/>102.</num> minu .47. qua mediante, in tabulis aÅ¿cenÅ¿ionum obliquarum poli .44. habebi-<lb/>mus gradum & minutum eclipticÄ™ cum quo Å¿tella oritur. </s> - <s xml:space="preserve">quod in caÅ¿u noÅ¿tro erit <lb/>gra .1. min .8. Leonis, Å¿ed Å¿i tecum non fuerint tabulæ dictæ, potes eleganter omnia <lb/>hæc perficere via triangulorum Å¿phæricorum.</s> - </p> - <pb o="311" rhead="EPISTOL AE." n="323" file="0323" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0323"/> - <p> - <s xml:space="preserve">Via triangul@rum<unsure/> idem facere.</s> - </p> - <p> - <s xml:space="preserve">Sit <reg norm="exempli" type="context">exẽpli</reg> gr@tia <var>.q.b.</var> æquator, ecliptica verò <var>.q.a.</var> propoÅ¿itus <reg norm="autem" type="wordlist">aÅ©t</reg> orizon Å¿it <var>.o.c.d.</var> <lb/>& Å¿tella data Å¿it <var>.o.</var> in orientali parte orizontis, circulus verò <var>.a.o.</var> ille Å¿it, qui <reg norm="tranÅ¿iens" type="context">tranÅ¿iẽs</reg> <lb/>per polos eclipti<unsure/>cæ & per centrum Å¿tellæ terminat longitudinem ipÅ¿ius Å¿tellæ, & in <lb/>ipÅ¿o Å¿it eius latitudo. </s> - <s xml:space="preserve">Nunc propoÅ¿itum Å¿it inuenire arcum <var>.d.q.</var> eo quod illicò Å¿cie <lb/>mus punctum <var>.d.</var> qua propter oportet nos prius cognoÅ¿cere arcum <var>.d.a.</var> qui demptus, <lb/>vel additus arcui <var>.a.q.</var> prius cognito ex Å¿uppoÅ¿ito (nam data nobis eÅ¿t longitudo, & <lb/>latitudo Å¿tellæ) dabit nobis <var>.d.q</var>.</s> - </p> - <p> - <s xml:space="preserve">Cum igitur voluerimus arcum <var>.d.a.</var> cognoÅ¿cere, ita faciemus. </s> - <s xml:space="preserve">nam <var>.q.a.</var> cognitus <lb/>nobis eÅ¿t ex Å¿uppoÅ¿ito vt dictum eÅ¿t. </s> - <s xml:space="preserve">angulus quoque <var>.a.q.b.</var> qui declin tionis eclipti <lb/>cæ ab æquatore eÅ¿t, angulus deinde <var>.a.</var> (trianguli <var>.a.b.q.</var>) rectus eÅ¿t, ergo ex .4. primi <lb/>copernici cogn@tus nobis erit arcus <var>.<lb/>a.b.</var> nec non angulus <var>.a.b.q.</var> vnde an-<lb/> - <anchor type="figure" xlink:label="fig-0323-01a" xlink:href="fig-0323-01"/> - gulus <var>.o.b.e.</var> reÅ¿iduus ex duobus re-<lb/>ctis in duobus primis hic Å¿ubÅ¿criptis <lb/>figuris nobis itidem cognitus erit, <lb/>etiam & arcus <var>.b.o.</var> reÅ¿iduus Å¿iue com <lb/>poÅ¿itus ex ar cu <var>.a.o.</var> cognito ex Å¿up-<lb/>poÅ¿ito <reg norm="cum" type="context">cÅ©</reg> Å¿it arcus latitud nis ab ecli-<lb/>ptica. </s> - <s xml:space="preserve">Tunc<unsure/> in triangulo <var>.o.b.e.</var> co-<lb/>gnoÅ¿cimus latus <var>.o.b.</var> & angulum <var>.o.<lb/>b.e.</var> nec non angulum <var>.b.e.o.</var> qui eÅ¿t <lb/>altitudinis æquatoris ab orizonte , <lb/></s> - <s xml:space="preserve">quare ex .12. dicti lib. cognitus nobis <lb/>erit angulus <var>.b.o.e</var>. </s> - <s xml:space="preserve">ConÅ¿ideremus <lb/>deinde triangulum <var>.a.o.d.</var> cuius angu <lb/>lus <var>.a.</var> rectus eÅ¿t, & angulus <var>.a.o.d.</var> <reg norm="cum" type="context">cÅ©</reg> <lb/>latere <var>.a.o.</var> etiam cognitus, vnde ex <lb/>Å¿upradicta .4. nobis cognitus erit ar-<lb/>cus <var>.a.d.</var> & conÅ¿equenter cognoÅ¿ce-<lb/>mus at cum <var>.d.q.</var> eius reÅ¿iduum, Å¿eu <lb/>compoÅ¿itum, quem quærebamus.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0323-01" xlink:href="fig-0323-01a"> - <image file="0323-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0323-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Sed Å¿i hac via inuenire deÅ¿ideras, <lb/>cui orizonti propoÅ¿ita Å¿tella oriatur <lb/>cum eodem eclipticÄ™ puncto <var>.a.</var> lon-<lb/>gitudinis, hoc aliud nihil eſſet, quam <lb/>cognoÅ¿cere amplitudinem anguli <var>.a.<lb/>b.q.</var> eo quod talis orizon, idem cir-<lb/>culus eſſet <var>.a.b.o.</var> vnde cum quis Å¿ci-<lb/>ret vnum illorum angulorum quem <lb/>æquator efficit cum orizonte, reli-<lb/>qua illicò ei innoteÅ¿cent, Å¿ed dictus <lb/>angulus <var>.b.</var> iam diximus quomodo <lb/>cognoÅ¿catur.</s> - </p> - <pb o="312" rhead="IO. BAPT. BENED." n="324" file="0324" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0324"/> - <p> - <s xml:space="preserve">Ponamus nos Å¿cire velle <reg norm="punctum" type="context">pÅ©ctum</reg> <lb/> - <anchor type="figure" xlink:label="fig-0324-01a" xlink:href="fig-0324-01"/> - eclipticÄ™, cum quo Procyon oritur <lb/>polo .44. o. dato, quod Å¿tella in gra <num value="19">.<lb/>19.</num> cum min .24. </s> - <s xml:space="preserve">Cancri, reperiatur <lb/>diÅ¿tans ab ecliptica per gra .16. min <num value="10">.<lb/>10.</num> meridiem verÅ¿us. </s> - <s xml:space="preserve">vnde <reg norm="arcus" type="simple">arcê°</reg> <var>.a.q.</var> <lb/>erit gra .70. min .36. <reg norm="eiusque" type="simple">eiusqÌ;</reg> Å¿inus par-<lb/>tium 94321. talium qualium totalis <lb/>eÅ¿t .100000. arcus verò <var>.a.o.</var> gra .16. <lb/>minut .10. Å¿inus erit 27845. angu-<lb/>lus autem <var>.a.q.e.</var> declinationis zodia <lb/>ci ab Ä™quatore grad .23. min .30. cu-<lb/>ius Å¿inus eÅ¿t .39875. </s> - <s xml:space="preserve">Quare ex Å¿upra-<lb/>dictis rationibus angulus <var>.a.b.q.</var> erit <lb/>gra .82. mi .24. cuius <reg norm="Å¿inus" type="simple">Å¿inê°</reg> erit .99122. <lb/>arcus vero <var>.a.b.</var> gra .22. minu .17. cu-<lb/>ius Å¿inus erit .37945. </s> - <s xml:space="preserve">angulus deinde <lb/><var>o.e.b.</var> trianguli <var>.o.e.b.</var> eÅ¿t gra .46. mi <var>.<lb/>o.</var> altitudinis æquatoris ab orizonte, <lb/>cuius Å¿inus eÅ¿t .71934. angulus Å¿imili <lb/>ter <var>.o.b.e.</var> medio coniuncti, quibus <lb/>rectus perÅ¿icitur, arcus etiam <var>.o.b.</var> no <lb/>tus eÅ¿t grad .6. min .7. cuius Å¿inus eÅ¿t <num value="10655">.<lb/>10655.</num> cum Å¿it differentia inter ar-<lb/>cus <var>.a.b.</var> et <var>.a.o.</var> cognitos.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0324-01" xlink:href="fig-0324-01a"> - <image file="0324-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0324-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Quare ex .12. iam Å¿upradicta an-<lb/>gulus <var>.e.o.b.</var> hoc eÅ¿t <var>.a.o.d.</var> erit. grad <num value="36">.<lb/>36.</num> min .39. cuius Å¿inus erit .59693. <lb/></s> - <s xml:space="preserve">deinde per .4. cognitus erit nobis an <lb/>gulus <var>.a.d.o.</var> gra .55. min .5. cuius <reg norm="Å¿inus" type="simple">Å¿inê°</reg> <lb/>erit .81998. arcus verò <var>.d.o.</var> gra .19. <lb/>min .51. cuius Å¿inus erit .33957 ar-<lb/>cus autem gra .11. min .42. cuius <reg norm="Å¿inus" type="simple">Å¿inê°</reg> <lb/>erit .20270. vnde <reg norm="arcus" type="simple">arcê°</reg> <var>.d.q.</var> reÅ¿iduus <lb/>ex <var>.a.q.</var> erit gra .58. min .54. complementum <reg norm="autem" type="wordlist">aÅ©t</reg> quartæ erit gra .31. mi .6. hoc eÅ¿t gra <num value="1">.<lb/>1.</num> Å¿igni Leonis. cum min .6.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">De Å¿phæroide duplæ Å¿pbær æ propoÅ¿it æ.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">MOdus autem inueniendi Å¿phæroidem ex dato axe, quod duplum Å¿it Å¿phæ-<lb/>ra propoÅ¿ita, talis eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Sit exempli gratia <var>.a.b.c.</var> Å¿phæra propoÅ¿ita. cuius Å¿emidia meter Å¿it <var>.o.c.</var> Å¿emiaxis <lb/>vero Å¿phæroidis Å¿it <var>.d.x.</var> cuius dimidium Å¿it <var>.u.x.</var> </s> - <s xml:space="preserve">tunc ex doctrina .9. Å¿exti Euclid. inue <lb/>niatur <var>.g.h.</var> media proportionalis inter <var>.u.x.</var> et <var>.c.o.</var> </s> - <s xml:space="preserve">deinde Å¿icut Å¿e habet <var>.u.x.</var> ad <var>.g.h.</var> - <pb o="313" rhead="EPISTOL AE." n="325" file="0325" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0325"/> - faciemus, quod diameter <var>.a.b.</var> dictæ Å¿phæræ ita Å¿e habcat ad <var>.e.f.</var> ex .10. Å¿exti, quæ <lb/><var>e.f.</var> erit reliqua axis quæſita. </s> - <s xml:space="preserve">Vnde conÅ¿tituta cum fuerit ellipÅ¿is <var>.d.f.t.e.</var> ex dictis axi-<lb/>bus, </s> - <s xml:space="preserve">deinde circumuertendo ellipÅ¿im circa maiorem axem, conÅ¿tituemus Å¿phæroi-<lb/>dem oblongam, Å¿i autem circumuertemus ipÅ¿am circa minorem axim conÅ¿tituemus <lb/>Å¿phæroidem prolatam.</s> - </p> - <p> - <s xml:space="preserve">Quod autem talis operatio rationalis Å¿it, nulli dubium erit, que<unsure/>tieÅ¿cunque co-<lb/>gnoÅ¿cet conum rectum <var>.e.u.f.</var> æqualem eſſe cono recto <var>.a.c.b.</var> ex .2. parte .12. duodeci <lb/>mi Euclid. </s> - <s xml:space="preserve">& quod cum conus <var>.e.d.f.</var> duplus Å¿it cono <var>.e.u.f.</var> ex lemmate collecto ab <lb/>11. duodecimi, conus <var>.e.d.f.</var> duplus exiÅ¿tit etiam cono <var>.a.c.b.</var> ex .7. quinti. </s> - <s xml:space="preserve">Cum de-<lb/>inde ex .32. primi lib. de Å¿phæra, & cyllindro Å¿phæra <var>.a.c.b.q.</var> quadrupla Å¿it cono <var>.a.<lb/>c.b.</var> ipÅ¿a conÅ¿equenter dupla erit cono <var>.e.d.f.</var> Å¿ed ex .29. primi de conoidalibus, dimi <lb/>dium Å¿phæroidis <var>.e.d.f.t.</var> hoc eÅ¿t <var>.e.d.f.</var> dupla eÅ¿t cono <var>.e.d.f</var>. </s> - <s xml:space="preserve">Quare talis medietas <lb/>æqualis eÅ¿t Å¿phæræ propoÅ¿itæ, totaqÌue Å¿phæroides dupla erit Å¿phærÄ™ datæ. </s> - <s xml:space="preserve">Quod <lb/>autem dico de proportione dupla, idem infero de qualibet alia, Å¿umendo <var>.u.x.</var> ita pro <lb/>portionatam ad <var>.d.x.</var> vt proponitur.</s> - </p> - <p> - <s xml:space="preserve">SphÄ™ram autem inuenire quæ dimidia Å¿it Å¿phæroidis propoÅ¿itæ nullius erit nego-<lb/>tij, quotieÅ¿cunque inuentus fuerit modus diuidendi vnam datam proportionem in <lb/>tres æquales partes.</s> - </p> - <p> - <s xml:space="preserve">Sit propoÅ¿ita Å¿phæroides <var>.e.f.d.t.</var> cuius axes ex conÅ¿equentia dantur <var>.e.f.</var> et <var>.d.t.</var> quÄ™ <lb/>quidem Å¿phæroides Å¿it primo oblonga, et <var>.u.x.</var> Å¿it dimidium axis maioris. </s> - <s xml:space="preserve">imagine-<lb/>tur etiam conus <var>.e.u.f.</var> vt Å¿upra. </s> - <s xml:space="preserve">Imaginetur etiam factum eſſe, quod proponitur, hoc <lb/>eÅ¿t, vt Å¿phæra <var>.a.b.c.q.</var> Å¿it dimidium ipÅ¿ius Å¿phæroidis, vnde conus <var>.a.c.b.</var> æqualis erit <lb/>cono <var>.e.u.x.</var> vt Å¿upra demonÅ¿tratum eÅ¿t, & Å¿it <var>.g.h.</var> media proportionalis inter <var>.u.x.</var> et <lb/><var>o.c</var>. </s> - <s xml:space="preserve">Iam viÅ¿um Å¿uperius fuit, quod eadem proportio erat ipÅ¿ius <var>.u.x.</var> ad <var>.g.h.</var> quæ <var>.a.b.</var> <lb/>ad <var>.e.f.</var> </s> - <s xml:space="preserve">quare eadem quæ <var>.o.b.</var> ad <var>.e.x.</var> Å¿ed <var>.u.x.</var> et <var>.e.x.</var> dantur. </s> - <s xml:space="preserve">inter quas <var>.g.h.</var> et <var>.o.b.</var> vel <lb/><var>o.c.</var> (nam <var>.o.c.</var> æqualis eÅ¿t <var>.o.b.</var>) mediÄ™ proportionales Å¿unt, eo quod cum <var>.g.h.</var> media <lb/>proportionalis Å¿it inter <var>.u.x.</var> et <var>.o.c.</var> & proportio <var>.o.b.</var> ad <var>.e.x.</var> æqualis Å¿it ei, quæ <var>.u.x.</var> <lb/>ad <var>.g.h.</var> hoc eÅ¿t ei quæ <var>.g.h.</var> ad <var>.o.c.</var> vel. ad <var>.o.b.</var> </s> - <s xml:space="preserve">quare quotieÅ¿cunque inuentæ fuerint <var>.<lb/>g.h.</var> et <var>.o.c.</var> vel <var>.o.b.</var> mediæ proportionales inter <var>.d.x.</var> et <var>.x.e.</var> ipÅ¿a <var>.o.c.</var> vel <var>.o.b.</var> erit Å¿emi <lb/>diameter Å¿phæræ quæſitÄ™. </s> - <s xml:space="preserve">eodem modo faciendum erit Å¿i Å¿phÄ™roides fuerit prolata.</s> - </p> - <figure position="here"> - <image file="0325-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0325-01"/> - </figure> - <pb o="314" rhead="IO. BABPT. BENED." n="326" file="0326" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0326"/> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">Modus inueniendi duo triangula <reg norm="varijs" type="lig">varijs</reg> conditionibus <lb/>affecta.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">QVod etiam quæris ita Å¿e habet, duo Å¿cilicet triangula inuenire, æqualia dua-<lb/>bus Å¿uperficiebus rectilineis propoÅ¿itis, quÄ™ quidem triangula Å¿int eiuÅ¿dem <lb/>alritudinis, & quod <reg norm="vnunquodque" type="context">vnÅ©quodque</reg> habeat angulum æqualem angulo pro<unsure/>poÅ¿ito, & <reg norm="quod" type="simple">ê™</reg> <lb/>alius angulus vnius, cum alio alterius, æquetur duobus rectis.</s> - </p> - <p> - <s xml:space="preserve">Sint exempli gratia duæ propoÅ¿itæ Å¿uperflcies <var>.c.y.</var> duo verò anguli dati Å¿int <var>.r.s.</var> <lb/>cum voluerimus inuenire duo triangula (quæ Å¿int <var>.a.i.u.</var> et <var>.n.t.x.</var> ) tali conditio-<lb/>ne prædita, quod angulus, a. æqualis Å¿it angulo <var>.s.</var> & angulus <var>.t.</var> angulo <var>.r.</var> & quod <lb/>angulus <var>.x.</var> Å¿imul cum angulo <var>.u.</var> <reg norm="æ- quentur" type="context">æ-<lb/>quẽtur</reg> duobus rectis, & quod <reg norm="triam" type="context">triã</reg> <lb/> - <anchor type="figure" xlink:label="fig-0326-01a" xlink:href="fig-0326-01"/> - <reg norm="gulum" type="context">gulÅ©</reg> <var>.a.i.u.</var> æquale Å¿it Å¿uperficiei <var>.<lb/>c.</var> reliquum verò Å¿uperficiei <var>.y.</var> <lb/>Ex duabus Å¿uperficiebus <var>.c.</var> et <var>.y.</var> <lb/>conÅ¿tituemus duo quadrata, per vl <lb/>timam Å¿ecundi Eucli. accipiemus, <lb/></s> - <s xml:space="preserve">deinde duo latera tetragonica ip-<lb/>Å¿orum quadratorum, & inuenie-<lb/>mus tertiam lineam in continua <lb/>proportionalitate cum illis lateri-<lb/>bus ex .10. Å¿exti, Å¿eruabimus po-<lb/>Å¿tea extremas illarum, quæ Å¿int <var>.z.</var> <lb/>et <var>.l.</var> quarum proportio, <reg norm="eadem" type="context">eadẽ</reg> erit, <lb/>quæ inter duas propoÅ¿itas Å¿uperfi-<lb/>cies reperitur ex .18. Å¿exti, accipie <lb/>mus, deinde lineam aliquam cu-<lb/>inÅ¿uis longitudinis, quæ Å¿it <var>.q.g.</var> Å¿u-<lb/>pra quam conÅ¿tituemus in puncto <lb/>q. angulum <var>.m.q.g.</var> Ä™qualem angu-<lb/>lo <var>.s.</var> & angulum <var>.m.q.K.</var> æqualem <lb/>angulo <var>.r.</var> ex .23. primi, poÅ¿tea ve-<lb/>rò à quouis puncto ipÅ¿ius lineæ <var>.q.<lb/>m.</var> puta <var>.o.</var> ducetur <var>.o.f.</var> vÅ¿que ad <var>.q.<lb/>g.</var> quorÅ¿um volueris, producendo <lb/>ipÅ¿am <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg>. ad <var>.d.</var> ita quod propor-<lb/>tio <var>f.o.</var> ad <var>.o.d.</var> Å¿it vt <var>.z.</var> ad <var>.l.</var> ex .10. <lb/>Å¿exti, ducendo poÅ¿tea à puncto <var>.d.</var> <lb/>lineam <var>.d.h.E.</var> parallelam lineæ <var>.q.<lb/>g.</var> & quia ex .2. primi Vitellionis <var>.<lb/>h.E.</var> Å¿ecatur ab <var>.q.K.</var> in puncto <var>.b.</var> <lb/>protrahemus <var>.b.o.p.</var> vnde ex Å¿imi-<lb/>litudine triangulorum habebimus <lb/>proportionem <var>.p.o.</var> ad <var>.o.b.</var> vt <var>.f.o.</var> - <pb o="315" rhead="EPISTOL AE." n="327" file="0327" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0327"/> - ad <var>.o.d.</var> hoc eÅ¿t vt <var>.z.</var> ad <var>.l.</var> hoc eÅ¿t vt <var>.c.</var> ad <var>.y</var>. </s> - <s xml:space="preserve">quare <reg norm="triangulum" type="context">triangulÅ©</reg> <var>.p.q.o.</var> ita erit proportio <lb/><reg norm="natum" type="context">natÅ©</reg> triangulo <var>.o.q.b.</var> vt <var>.c.</var> ad <var>.y.</var> conÅ¿tituo deinde ex .25. fexti duo triangula Å¿imi-<lb/>lia duobus <var>.p.q.o.</var> et <var>.o.q.b.</var> <reg norm="æqualiaque" type="simple">æqualiaqÌ</reg> <var>.c.</var> et <var>.y.</var> quÄ™ Å¿int <var>.a.i.u.</var> et <var>.n.t.x.</var> Å¿ecetur poÅ¿tea <var>.q.<lb/>g.</var> in puncto <var>.æ.</var> ita, quod <var>.q.æ.</var> æqualis Å¿it <var>.i.a.</var> duco poÅ¿tea <var>.æ.e.</var> æquidiÅ¿tantem. ad <var>.p.b.</var> <lb/>& Å¿ic habebimus duo triangula <var>.q.x.æ.</var> ct <var>.q.x.</var> e, vt quærebantur, quamuis duo trian <lb/>gula <var>.a.i.u.</var> et <var>.t.n.x.</var> eaÅ¿dem habeant conditiones.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0326-01" xlink:href="fig-0326-01a"> - <image file="0326-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0326-01"/> - </figure> - </div> - </div> - </div> - <div type="section" level="3" n="24"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE IMPERFECTA SOLVTIONE PROBLE-<lb/>matis Nicolai Tartaleæ ad Cardanum. De animad-<lb/>uerÅ¿ione in Ptolomeum. Deincendio carbo-<lb/>num à vento.</head> - <head style="it" xml:space="preserve">Clariβimo Dominico MoreÅ¿ino.</head> - <p> - <s xml:space="preserve"><emph style="sc">SCio</emph> propoÅ¿itam tibi quæſtionem te diu agitauiſſe, nectamen Å¿olutio-<lb/>nem aſſequi potuiſſe, aduerte igitur ipÅ¿am falÅ¿am, ideÅ¿t impoſſibilem <lb/>eſſe, quemadmodum etiam decimumoctauum quæſitum propoÅ¿itum à <lb/>Cardano Tartaleæ, ab ipÅ¿o Tarralea Å¿olutum minimè fuit. </s> - <s xml:space="preserve">Quiquidem <lb/>Tartalea vult circulum deÅ¿cribi circa triangulum per quintam libri quarti Euclidis, <lb/>vt in fine ferè quintæ partis Å¿uarum menÅ¿urarum affirmat, neque videt in quinta <lb/>quarti Euclidem vti vndecima primi, & in vndecima primi, quarta aut octaua eiuÅ¿-<lb/>dem, quas, ipÅ¿e Euclides oÅ¿tenÅ¿iuè non demonÅ¿trauit. </s> - <s xml:space="preserve">Quapropter oportebat Tar-<lb/>taleam demonÅ¿traſſe omnes propoÅ¿itiones ad hoc neceſſarias oÅ¿tenÅ¿iuè <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> ad pri-<lb/>mas indemonÅ¿trabiles, quia ad demonÅ¿trandam Å¿cientificè <reg norm="aliquam" type="context">aliquã</reg> propoÅ¿itionem, <lb/>aut à propoÅ¿itione in propoÅ¿itionem vÅ¿que ad prima principia vniuerÅ¿alia ( vt ali-<lb/>quando ego feci) eÅ¿t retrogradandum, aut ab ipÅ¿is principijs incipiendum Å¿ucceſſi-<lb/>uè eouÅ¿que progrediendo donec ad propoÅ¿itionem quam demonÅ¿trare volumus <lb/>perueniamus.</s> - </p> - <p> - <s xml:space="preserve">Quod ad Ptolomeum in geographia attinet, dico eum mihi non Å¿atisfacere, cum <lb/>Å¿umit portionem arcus circuli maioris inter vnam ciuitatem, & aliam, ea ratione <lb/>quam deÅ¿cribit. </s> - <s xml:space="preserve">Quod Å¿i vÅ¿us fuiſſet modo Menelai, ab ipÅ¿omet deinde in <reg norm="Å¿uum" type="context">Å¿uÅ©</reg> Al-<lb/>mageÅ¿tum vÅ¿urpato, aut Monteregij triangulorum Å¿phÄ™ricorum, quem Copernicus <lb/>adhibuit (qui tamen modus, tempore Ptolomei, nondum fortaſſe in lucem vene-<lb/>rat) bene egiſſet.</s> - </p> - <p> - <s xml:space="preserve">Quod deinde ad Å¿uum illud inÅ¿trumentum geometricum attinet, eÅ¿t <reg norm="imperfectum" type="context">imperfectÅ©</reg>, <lb/>vt oÅ¿tendi domino Pandulfo.</s> - </p> - <p> - <s xml:space="preserve">Motum autem aeris, aut mauis ventum, accendere ignem, non Å¿olum ratione an <lb/>tiperiÅ¿taÅ¿is, quam affers euenit, Å¿ed etiam quia à carbonibus accenÅ¿is totam excre <lb/>mentitiam materiam, quæ eos circundat, auferat.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">Alia dilucidatio propoÅ¿itionis .25. lib. 2. Monteregij.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">SCribiste non intelligere .25. propoÅ¿itionem lib. 2. Monteregij. cum necÅ¿cias <lb/>reperire diametrum circuli circunÅ¿criptibilis circa propoÅ¿itum triangu- - <pb o="316" rhead="IO. BAPT. BENED." n="328" file="0328" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0328"/> - lum, cuius data Å¿it b aÅ¿is tantummodo Å¿imul cum angulo, qui ipÅ¿i baÅ¿i opponitur.</s> - </p> - <p> - <s xml:space="preserve">Imagineris igitur triangulum datum eſſe obtuÅ¿iangulum <var>.a.b.g.</var> cuius baÅ¿i <var>.b.<lb/>g.</var> Å¿it nobis data Å¿imul cum angulo <var>.a.</var> ei oppoÅ¿ito, obtuÅ¿oqÌue; </s> - <s xml:space="preserve">ConÅ¿idera etiam cir-<lb/>culum <var>.a.b.g.q.</var> ipÅ¿um trian gulum circunÅ¿cribentem, cuius diameter <var>.q.e.p.</var> tranÅ¿eat <lb/>per <var>.m.</var> punctum medium ipÅ¿ius <var>.b.g.</var> <reg norm="tunc" type="context">tÅ©c</reg> protractis imaginatione <var>.e.g.</var> et <var>.g.p.</var> certi eri-<lb/>mus angulos. circa <var>.m.</var> rectos eſſe ex .3 tertij Eucli. <reg norm="angulumque" type="simple">angulumqÌ</reg> <var>.q.e.g.</var> duplum eſſe an <lb/>gulo <var>.q.p.g.</var> ex .19. eiuÅ¿dem, vnde æqualem angulo <var>.a.</var> qui etiam duplus eÅ¿t angulo <var>.q.<lb/>p.g.</var> quapropter proportio arcus <var>.q.g.</var> ad arcum <lb/> - <anchor type="figure" xlink:label="fig-0328-01a" xlink:href="fig-0328-01"/> - <var>g.p.</var> tibi cognita erit, & proportio etiam chor-<lb/>de <var>.p.g.</var> ad Å¿inum <var>.m.g.</var> arcus <var>.g.p.</var> & quia <var>.m.g.</var> vt <lb/>dimidium ipÅ¿ius <var>.b.g.</var> tibi data eÅ¿t, cognoÅ¿ces <lb/>etiam <var>.p.g.</var> vt <var>.m.g.</var> & Å¿ic tertium latus <var>.m.p.</var> trian-<lb/>guli orthogonij <var>.p.m.g.</var> & q<unsure/>a ex .34. tertij quod <lb/>fit ex <var>.p.m.</var> in <var>.m.q.</var> eÅ¿t æquale ei quod fit ex <var>.b.m.</var> <lb/>in <var>.m.g.</var> ideo cum diuiÅ¿um fuerit productum <var>.b.</var> <lb/>m in <var>.m.g.</var> per <var>.p.m.</var> proueniet <var>.m.q.</var> quapropter <lb/>habebis totum <var>.q.p</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0328-01" xlink:href="fig-0328-01a"> - <image file="0328-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0328-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Idem efficies, Å¿i <reg norm="cum" type="context">cÅ©</reg> angulus <var>.a.</var> acutus fuiſſet.</s> - </p> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">Modus inueniendi puncta elliptica via Pergei.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">MOdus inueniendi puncta elliptica, via .21. primi lib. Pergei ex datis axibus, <lb/>vt vbi alias Å¿ignificati, talis eÅ¿t. <lb/> - <anchor type="figure" xlink:label="fig-0328-02a" xlink:href="fig-0328-02"/> - </s> - <s xml:space="preserve">Sit exempli gratia maior axis propo-<lb/>Å¿itus <var>.a.c.</var> minor autem <var>.b.d.</var> cum ergo <lb/>volueris inuenire punctum circunfe-<lb/>rentiæ correÅ¿pondentem puncto <var>.e.</var> <lb/>maioris axis, inueniemus primò la-<lb/>tus tetragonicum producti <var>.a.g.</var> in <var>.g.<lb/>c.</var> quod Å¿it <var>.h.</var> <reg norm="latusque" type="simple">latusqÌ</reg> <reg norm="tetragonicum" type="context">tetragonicÅ©</reg> pro-<lb/>ducti <var>.a.e.</var> in <var>.e.c.</var> quod Å¿it <var>.i</var>. </s> - <s xml:space="preserve">deinde in-<lb/>ueniemus lineam <var>.K.</var> tertiam in con-<lb/>tinua proportionalitate cum <var>.h.</var> et <var>.i.</var> <lb/>vnde <var>.i.</var> erit media proportionalis in-<lb/>ter <var>.h.</var> et <var>.K.</var> & vt <var>.h.</var> proportionalis erit <lb/>ad <var>.K.</var> inueniemus <var>.e.f.</var> cui <var>.g.d.</var> medie-<lb/>tas Å¿ecundi axis ita Å¿e habeat, quæ po <lb/>Å¿tea iuncta axi maiori, ad angulosrectos in puncto <var>.e.</var> dabit Å¿itum puncti <var>.f.</var> quæſiti ex <lb/>dicta .21. primi lib. Pergei, Å¿ed talis modus prolixus eÅ¿t.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0328-02" xlink:href="fig-0328-02a"> - <image file="0328-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0328-02"/> - </figure> - </div> - <pb o="317" rhead="EPISTOL AE." n="329" file="0329" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0329"/> - <p> - <s xml:space="preserve">Accipeigitur huncalium.</s> - </p> - <p> - <s xml:space="preserve">Sit propoÅ¿itus maior axis <var>.q.p.</var> minor verò <var>.e.c.</var> ad angulos rectos Å¿e inuicem <lb/>Å¿ecantes in puncto <var>.o.</var> deÅ¿cribatur circulus <var>.q.n.p.a.</var> cuius diameter Å¿it axis maior, in <lb/>quo accipiatur punctum, quod volueris, vt puta <var>.u.</var> à quo protrahatur <var>.u.b.</var> paralle-<lb/>la ad <var>.o.c.n.</var> deÅ¿ignetur poÅ¿tea Å¿eparatim circulus <var>.u.b.n.</var> cuius diameter æqualis Å¿it Å¿e <lb/>midiametro prioris circuli, ita etiam fiat circulus <var>.u.i.c.</var> contingens circulum <var>.u.b.n.</var> <lb/>in puncto <var>.u.</var> cuius diameter Å¿it <var>.u.c.</var> æqualis dimidio axi minori. </s> - <s xml:space="preserve">accipiatur deinde in <lb/>circulo maximo longitudo <var>.u.b.</var> quæ collocetur in circulo mediocri à puncto <var>.u.</var> quæ <lb/>Å¿ecabitur à minimo circulo in puncto <var>.i.</var> cum itaque longitudo <var>.u.i.</var> menÅ¿urata fue-<lb/>rit in <var>.u.b.</var> maximi circuli à puncto <var>.u.</var> habebimus propoÅ¿itum.</s> - </p> - <p> - <s xml:space="preserve">Cuius reiratio eÅ¿t, quia <var>.u.b.</var> mediocris circuli diuiditur à gyro minimi in puncto <lb/>i. eadem proportione, qua diuiÅ¿a eÅ¿t <var>.u.n.</var> in puncto <var>.c.</var> quod manifeÅ¿tum eÅ¿t exÅ¿imi-<lb/>litudine triangulorum <var>.u.b.n.</var> et <var>.u.i.c.</var> imaginatæ cum fuerint duæ <var>.b.n.</var> et <var>.i.c.</var> Å¿ed ita <lb/>eſſe oportet parallelas maximi circuli, quotieÅ¿cunque circunferentia ipÅ¿ius ellipÅ¿is <lb/>tranÅ¿itura Å¿it per <var>.c.</var> vt in .51. cap. meæ gnomonicæ oÅ¿tenÅ¿um fuit.</s> - </p> - <figure position="here"> - <image file="0329-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0329-01"/> - </figure> - </div> - <div type="letter" level="4" n="4"> - <head style="it" xml:space="preserve">Modus deÅ¿ignandi angulum, certo modo conditionatum.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">NVllius reuera difficultatis mihi videtur eſſe, quotieÅ¿cunque nobis propoÅ¿ita <lb/>fuerint duo puncta <var>.a.</var> et <var>.b.</var> Å¿imul cum <lb/>angulo <var>.d.</var> <reg norm="necnon" type="context">necnõ</reg> linea <var>.g.</var> ducere duas lineas <lb/> - <anchor type="figure" xlink:label="fig-0329-02a" xlink:href="fig-0329-02"/> - à dictis punctis terminatas, quæ <reg norm="conÅ¿tituant" type="context">conÅ¿tituãt</reg> <lb/>angulum æqualem dato, & ipſæ directè <reg norm="con" type="context">cõ</reg> <lb/>iunctæ conÅ¿tituant lineam æqualem da-<lb/>tæ. </s> - <s xml:space="preserve">Nam ducatur linea indefinita per <lb/>puncta propoÅ¿ita, cuius lineæ, pars illa, quÄ™ <lb/>intercepta fuerit inter dicta puncta, diui-<lb/>datur per æqualia in puncto <var>.o.</var> etiam & li-<lb/>nea data, quarum medietates accipio in <lb/>linea indefinitè protracta à puncto <var>.o.</var> me- - <pb o="318" rhead="IO. BAPT. BENED." n="330" file="0330" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0330"/> - dio, vt vna earum Å¿it <var>.o.c.</var> reliqua verò Å¿it <var>.o.e.</var> </s> - <s xml:space="preserve">deinde aperiatur circinus quantum <var>.<lb/>o.c.</var> <reg norm="poÅ¿itoque" type="simple">poÅ¿itoqÌ;</reg> vno pede in <var>.b.</var> <reg norm="deÅ¿ignentur" type="context">deÅ¿ignẽtur</reg> cum altero duo arcus <var>.n.K.</var> poÅ¿ito iterum vno <lb/>pede in <var>.a.</var> deÅ¿ignentur alij duo arcus inter-<lb/>Å¿ecantes primos in punctis <var>.n.K</var>. </s> - <s xml:space="preserve">Deinde à <lb/> - <anchor type="figure" xlink:label="fig-0330-01a" xlink:href="fig-0330-01"/> - puncto <var>.n.</var> ad <var>.K.</var> ducetur linea <var>.n.K.</var> quæ per <lb/>punctum <var>.o.</var> tranÅ¿ibit, quam <var>.n.K.</var> mente <reg norm="con- cipio" type="context">cõ-<lb/>cipio</reg>, vt axis minor <reg norm="vnius" type="simple">vniê°</reg> ellipÅ¿is, cuius <var>.e.c.</var> <lb/>Å¿it axis maior, quibus axibus mediantibus <lb/>deÅ¿ignetur ellipÅ¿is <var>.n.c.K.e.</var> conÅ¿idero dein <lb/>de <var>.a.b.</var> vt <reg norm="chordanvnius" type="context">chordãvnius</reg> circuli, Å¿eu portio-<lb/>nis circularis, quæ capax Å¿it vnius anguli <lb/>æqualis angulo <var>.d.</var> propoÅ¿ito, ex .32. tertij <lb/>Euclid. cuius circunferentia, circunferen-<lb/>tiam ipÅ¿ius ellipÅ¿is Å¿ecabit in <reg norm="duobus" type="simple">duobê°</reg> punctis <lb/>quorum <reg norm="vnum" type="context">vnÅ©</reg> Å¿it <var>.i.</var> à quo protractæ cum fue <lb/>rint duæ lineæ <var>.a.i.</var> et <var>.i.b.</var> habebis propoÅ¿itum, cum <var>.a.i.</var> iuncta cum <var>.i.b.</var> æquetur <var>.e.</var> c, <lb/>ex .52. tertij Pergei.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0329-02" xlink:href="fig-0329-02a"> - <image file="0329-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0329-02"/> - </figure> - <figure xlink:label="fig-0330-01" xlink:href="fig-0330-01a"> - <image file="0330-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0330-01"/> - </figure> - </div> - </div> - </div> - <div type="section" level="3" n="25"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">EMENDATIO CVIVSDAM FALSI MODI <lb/>delineandi horologia Italica orizontalia.</head> - <head style="it" xml:space="preserve">Foanni Paulo Dardano.</head> - <p> - <s xml:space="preserve"><emph style="sc">MOdvs</emph> delineandi horologia Italica orizontalia, quem tibi monÅ¿trauit <lb/>neÅ¿cio quis, ni fallor, talis eÅ¿t. </s> - <s xml:space="preserve">DeÅ¿ignato meridiano <var>.l.b.m.q.</var> <reg norm="ductisque" type="simple">ductisqÌ;</reg> duo <lb/>bus diametris <var>.l.m.</var> et <var>.b.</var> q, inuicem ad rectos in centro <var>.g.</var> quorum <var>.l.m.</var> Å¿it <lb/>verticalis <var>.b.q.</var> vero <reg norm="orizontalis" type="context">orizõtalis</reg>, <reg norm="ductoque" type="simple">ductoqÌ;</reg> diametro <var>.f.h.</var> tropici Cancri Å¿e-<lb/>cundum altitudinem poli datam, <reg norm="deÅ¿criptoque" type="simple">deÅ¿criptoqÌ;</reg> dimidio circulo <var>.f.z.h.</var> ipÅ¿ius paralleli, <lb/><reg norm="inuentoque" type="simple">inuentoqÌ;</reg> puncto <var>.z.</var> horæ propoÅ¿itÄ™, & ab eo ducta per pendiculari <var>.z.r.</var> ad <var>.f.h.</var> & à <lb/>puncto <var>.r.</var> ducta <var>.r.o.y.</var> parallela ad diametrum <var>.q.b.</var> orizontalem, ducis poÅ¿tea <var>.f.ω.</var> et <var>.<lb/>r.t.</var> vÅ¿que ad orizontalem <var>.q.b.</var> parallelas ad diametrum <var>.l.m.</var> verticalem. </s> - <s xml:space="preserve">Determi-<lb/>nato poÅ¿tea gnomone <var>.g.s.</var> in orizontis axe, <reg norm="ductaque" type="simple">ductaqÌ;</reg> vmbrarum linea <var>.s.K.</var> parallela <lb/>orizontali, <reg norm="ductaque" type="simple">ductaqÌ;</reg> <var>.y.g.K.</var> ad terminandam <var>.s.K.</var> delineas deinde Å¿eparatim circulum <lb/><var>q.x.b.n.</var> magnitudinis prioris, qui quidem circulus Å¿ignificet orizontem ipÅ¿um, in <lb/>quo ductis diametris <var>.q.g.b.</var> et <var>.l.g.m.</var> accipis in diametro <var>.q.g.b.</var> puncta <var>.a.</var> et <var>.ω.</var> ita à <reg norm="cen tro" type="context">cẽÂ<lb/>tro</reg> <var>.g.</var> diÅ¿tantia, vt Å¿unt in diametro orizontali prioris circuli, ducis poÅ¿tea per pun-<lb/>ctum <var>.a.</var> lineam <var>.x.a.n.</var> ad rectos cum dicto diametro, </s> - <s xml:space="preserve">deinde per tria puncta <var>.n.ω.x.</var> <lb/>tranÅ¿ire facis circunferentiam circuli per <reg norm="quintam" type="context">quintã</reg> quarti Euclidis, poÅ¿tea in dicto dia-<lb/>metro accipis punctum <var>.t.</var> ita diÅ¿tans à centro, & ex eadem parte, vt in priori circulo, <lb/>à quo puncto ducis <var>.t.u.</var> parallelam <var>.x.n.</var> vÅ¿que ad circunferentiam <var>.x.ω.n.</var> in puncto <var>.u.</var> <lb/>quo facto, ducis à centro <var>.g.</var> per punctum <var>.u.</var> ipÅ¿ius circularis circunferentiæ <var>.g.u.</var> inde-<lb/>terminatam, quam poÅ¿tea terminas in puncto <var>.K.</var> ita quod <var>.g.K.</var> æqualis Å¿it <var>.s.K</var>. </s> - <s xml:space="preserve">Dicis <lb/>poÅ¿tea punctum <var>.K.</var> in eodem Å¿itu reperiri, reÅ¿pectu duorum diametrorum <var>.q.b.</var> me-<lb/>ridiani. et <var>.l.m.</var> verticalis, vt decet, & oportet punctum horæ propoÅ¿itÄ™ exiÅ¿tere.</s> - </p> - <p> - <s xml:space="preserve">Quod quidem dico eſſe falÅ¿um, propterea quod perpendiculares quas cogita-<lb/>mus cadere à punctis circunferentiæ cuiuÅ¿uis paralleli Å¿upra quemuis orizontem ob - <pb o="319" rhead="EPISTOL AE." n="331" file="0331" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0331"/> - liquum Å¿ecantem æquatorem, omnes caduntin gyro elliptico, oxygonio, Å¿eu de-<lb/>fectionali, & non circulari. </s> - <s xml:space="preserve">Vnde per Å¿upradicta tria puncta <var>.n.ω.x.</var> oporteret tranÅ¿i <lb/>re talem circunferentiam, & non <reg norm="circularem" type="context">circularẽ</reg>, quæ circunferentia eſſet vnius ellipÅ¿is, <lb/>cuius minor axis in diametro <var>.b.q.</var> eſſet .ab <var>.ω.</var> vÅ¿que ad <var>.i.</var> terminum Å¿ini <var>h.i.</var> arcus <var>.h.b.</var> <lb/>in analemate, maior verò axis eſſet magnitudinis <var>.f.h.</var> diametri paralleli, quæ <reg norm="tranÅ¿iÅ¿- Å¿et" type="context">trãſiÅ¿-<lb/>Å¿et</reg> per punctum <var>.c.</var> medium inter <var>.ω.</var> et <var>.i.</var> quÄ™ quidem circunfere ntia tota eſſet intra cir <lb/>culum <var>.q.n.b.x.</var> <reg norm="con" type="context">cõ</reg><reg norm="tiguaque" type="simple">tiguaqÌ;</reg> gyro <var>.q.n.b.x.</var> in punctis <var>.n.x</var>.</s> - </p> - <p> - <s xml:space="preserve">Si ergò circunferentia <var>.n.ω.x.</var> eſſet elliptica tunc punctum <var>.u.</var> in orizonte illud eſſet <lb/>vbi caderet Å¿inus altitudinis horÄ™, et <var>.t.u.</var> æqualis eſſet <var>.r.z.</var> communi Å¿ectioni paralle <lb/>li cum almicantarat ex .34. primi Euclid. et <var>.u.g.</var> æqualis eſſet <var>.o.y.</var> communi Å¿ectioni <lb/>almicantarat cum meridiano, vel cum azimut illius horæ ex .4. primi, cum <var>.g.t.</var> æqua <lb/>lis Å¿it ipÅ¿i <var>.o.r.</var> et <var>.t.u.</var> ipÅ¿i <var>.r.z.</var> & angu <lb/> - <anchor type="figure" xlink:label="fig-0331-01a" xlink:href="fig-0331-01"/> - lus. t trianguli <var>.g.t.u.</var> rectus, quem-<lb/>admodum <var>.r.</var> qui compræhenditur <lb/>ab <var>.z.r.</var> et <var>.r.o.</var> vnde anguli <var>.K.g.m.</var> <lb/>et <var>.K.g.b.</var> rectè Å¿e haberent, diÅ¿tan-<lb/>tia verò inter <var>.K.</var> et <var>.g.</var> <reg norm="iam" type="context">iã</reg> rectè Å¿um-<lb/>pta fuit.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0331-01" xlink:href="fig-0331-01a"> - <image file="0331-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0331-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Sed quia punctum <var>.u.</var> vt <reg norm="plurimum" type="context">plurimÅ©</reg> <lb/>(in gyro circulari Å¿umptum) extra <lb/>puncta interÅ¿ectionum ipÅ¿ius circu <lb/>laris gyri cum elliptico reperìtur, <lb/></s> - <s xml:space="preserve">propterea efficit angulos <var>.K.g.m.</var> <lb/>et <var>.K.g.b.</var> falÅ¿os, & non æquales il-<lb/>lis, qui fiunt ab azimut horæ cum <lb/>verticali, & cum meridiano, quæ <lb/>omnia ex cap .52. meæ gnomonicÄ™ <lb/>facilè videre potes.</s> - </p> - <p> - <s xml:space="preserve">Nectacere volo quod punctum <lb/>u. verum, hoc eÅ¿t ellipticum, inue-<lb/>niri poſſet ea via quam Å¿cripÅ¿i in <lb/>eodem .52. cap. qua mediante do-<lb/>cui demum inuenire punctum <var>.Ï€.</var> <lb/>orizontis, quamuis in præſenti caÂ<lb/>Å¿u <var>.ω.λ.</var> perpendicularis eſſet Å¿upra <lb/>minorem axem ipÅ¿ius ellipÅ¿is, <reg norm="quan- uis" type="context">quã-<lb/>uis</reg> Å¿upra maiorem axem, quod ta-<lb/>men minimè mutat ordinem, imò <lb/>rationes eÄ™dem Å¿unt, tam in vna, <lb/>quam in alia operatione, Å¿ed vt il-<lb/>licò <reg norm="idipsum" type="context">idipsÅ©</reg> habeas, fac vt <var>.t.u.</var> æqua <lb/>lis. Å¿it <var>.r.z</var>. </s> - <s xml:space="preserve">& tunc punctum <var>.K.</var> erit <lb/><reg norm="quæſitum" type="context">quæſitÅ©</reg>, quod ego in .52. cap. meÄ™ <lb/>gnomonicÄ™, ijs verbis Å¿ignificaui.</s> - </p> - <quote> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> mediis binis triangulis ijs, <lb/><reg norm="medioque" type="simple">medioqÌ;</reg> azimut Solis pariter ho-<lb/>rologia fabricari poterunt.</s> - </quote> - <pb o="320" rhead="IO. BAPT. BENED." n="332" file="0332" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0332"/> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">De Horologio perpendiculari ad oriz ontem rectum.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">MOdus quem tibi Å¿cribere promiÅ¿i delineandi lineas horarias communes in <lb/>pariete perpendiculariter ad orizontem rectum, declinantem à meridiano, <lb/>Å¿umendus eÅ¿t ex .46. cap. meæ gnomonicæ, hocÅ¿cilicet ordine.</s> - </p> - <p> - <s xml:space="preserve">Sit exempli gratia, orizon hic Å¿ubÅ¿criptus. or. oc <var>.M.S.</var> diuiÅ¿us à meridiana <var>.M.<lb/>S.</var> et verticali Å¿eu æquinoctiali. or. oc. </s> - <s xml:space="preserve">SitqÌue <var>.e.t.</var> communis Å¿ectio muri cum ori-<lb/>zonte, et <var>.g.n.</var> Å¿it gnomon perpendicularis ipÅ¿i muro, vnde ex dictis in mea gnomo-<lb/>nica, cognoÅ¿cemus in ipÅ¿a murali orizontali totam <var>.e.t.</var> inter meridianam orizonta-<lb/>lem, & æquinoctialem orizontalem, cognoÅ¿cemus etiam partem <var>.g.t.</var> ipÅ¿ius æ-<lb/>quinoctialis orizontalis, quam quidem accipiamus in rectitudine ipÅ¿ius muralis ori <lb/>zontalis, quæ quidem Å¿it <var>.t.G.</var> quo facto erigatur <var>.G.A.</var> ad rectos cum <var>.G.t.e.</var> & cir-<lb/>cum <var>.G.e.</var> deÅ¿ignetur vna medietas circuli verÅ¿us <var>.e.</var> cuiuÅ¿uis magnitudinis, quæ di-<lb/>uiÅ¿a in .12. partes Ä™quales, Å¿ignificabit <reg norm="medietatem" type="context">medietatẽ</reg> æquatoris, <reg norm="protrahanturque" type="simple">protrahanturqÌ;</reg> lineæ oc <lb/>cultæ à centro <var>.G.</var> per Å¿ectiones circunferentiÄ™ dimidij circuli, quæ fignificabunt <reg norm="con" type="context">cõ</reg> <lb/>munes Å¿ectiones æquatoris cum circulis horarijs communibus, quo facto oportet, vt <lb/>à puncto <var>.t.</var> protrahatur <var>.t.s.</var> ad rectos cum murali orizontali, quæ quidem <var>.t.s.</var> Å¿ignifi-<lb/>cabit communem Å¿ectionem æquatoris cum muro propoÅ¿ito, & erit Ä™quÄ™diÅ¿tans me <lb/>ridianæ murali ex .6. vndecimi Eucli. eo quod ex .19. eiuÅ¿dem <reg norm="vnaquæque" type="simple">vnaquæq;</reg> illarum, per <lb/>pendicularis eÅ¿t tali orizonti. </s> - <s xml:space="preserve">Videantur nunc puncta communia iÅ¿ti <var>.t.s.</var> & occultis <lb/>protractis à centro <var>.G.</var> medietatis circularis, per quæ puncta protrahantur à puncto <var>.<lb/>e.</var> tot lineæ, punctum enim <var>.e.</var> Å¿ignificat punctum axis mundi, & meridianæ in mu-<lb/>ro propoÅ¿ito, eo quod in tali Å¿itu Å¿phæræ rectæ, dictum punctum reperitur in orizon <lb/>re, cum <var>.M.s.</var> non Å¿olum Å¿it meridiana orizontalis, Å¿ed etiam axis mundi, </s> - <s xml:space="preserve">deinde nul <lb/>li dubium eÅ¿t quin meridiana muralis Å¿it perpendicularis orizontali murali <var>.e.t.</var> à <lb/>puncto <var>.e</var>. </s> - <s xml:space="preserve">Sed quia dimidium <lb/> - <anchor type="figure" xlink:label="fig-0332-01a" xlink:href="fig-0332-01"/> - harum <reg norm="linearum" type="context">linearÅ©</reg> horariarum erit <lb/>Å¿ub orizontali <var>.e.t.G.</var> alterum <lb/>vero <reg norm="dimidium" type="context">dimidiÅ©</reg> Å¿upra ipÅ¿am, opor <lb/>tet quod quÄ™ Å¿upra Å¿unt produ-<lb/>cantur à parte .oe. Å¿ub orizonta-<lb/>li, ab alia parte meridianæ, & <lb/>talis erit effigies horologij mura <lb/>lis in hoc Å¿phÄ™rÄ™ Å¿itu, hoc eÅ¿t ver <lb/>Å¿us quartam orientalem auÅ¿tra-<lb/><reg norm="lemque" type="simple">lemqÌ;</reg>, vnde orizontalis <var>.e.t.</var> <lb/>erit Å¿emper horæ .6. matutinæ, <lb/>Å¿ecunda verò ab ipÅ¿a erit horæ. <lb/>7. tertia <reg norm="autem" type="context">autẽ</reg> horæ .8. & Å¿ic dein-<lb/>ceps.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0332-01" xlink:href="fig-0332-01a"> - <image file="0332-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0332-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">QuotieÅ¿cunque verò angu-<lb/>lus <var>.n.g.e.</var> minor erit maxima <lb/>Solis declinatione, & Sol fuerit <lb/>in parte auÅ¿trali ab æquatore <reg norm="cum" type="context">cÅ©</reg> - <pb o="321" rhead="EPISTOL AE." n="333" file="0333" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0333"/> - maiori numero declinationis quam fuerit angulus <var>.n.g.e</var>. </s> - <s xml:space="preserve">tunc talis paries illumina-<lb/>bitur ab ipÅ¿o Sole à mane vÅ¿que ad veÅ¿peram.</s> - </p> - <p> - <s xml:space="preserve">Huius quidem rei Å¿peculatio, vnicuique manifeÅ¿ta erit, qui rationes .46. cap. no-<lb/>Å¿træ gnomonicæ prius intellexerit, vbi manifeÅ¿tè apparet proportionem Å¿emidiame <lb/>tri horologij (Å¿i ita eam appellare licet) ad Å¿emidiametrum æquatoris horarij Å¿em-<lb/>per eſſe, vt <var>.e.t.</var> ad <var>.t.g.</var> hoc eÅ¿t proportio maioris inæqualitatis. </s> - <s xml:space="preserve">nolo etiam prætermic <lb/>tere. </s> - <s xml:space="preserve">quin te admoneam, vt nullo pacto confidas in longioribus vmbris, eo quod val <lb/>de nos decipiant, cum Å¿emper iuÅ¿to breuiores Å¿int.</s> - </p> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">Declar atio quorundam verborum noÅ¿træ<unsure/> Gnomonicæ. <lb/><reg norm="DefenÅ¿ioque" type="simple">DefenÅ¿ioqÌê«</reg> nostra contra Christophorum Clauium.</head> - <head xml:space="preserve">AD EVNDEM DARDANVM.</head> - <p> - <s xml:space="preserve">TVas demum accepi literas, qui <lb/> - <anchor type="figure" xlink:label="fig-0333-01a" xlink:href="fig-0333-01"/> - bus mihi Å¿ignificas te totum <num value="52">.<lb/>52.</num> caput meæ gnomonicæ intelle-<lb/>xiſſe, præter illa verba, quæ etiam <lb/>Å¿uperioribus diebus ad te Å¿cripÅ¿i, <lb/>hoc eÅ¿t.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0333-01" xlink:href="fig-0333-01a"> - <image file="0333-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0333-01"/> - </figure> - </div> - <quote> - <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> medijs binis triangulis ijs, <lb/><reg norm="medioque" type="simple">medioqÌ;</reg> azimut Solis, pariter ho-<lb/>rologia fabricari poterunt.</s> - </quote> - <p> - <s xml:space="preserve">Quapropter nealiquid tibi de-<lb/>Å¿it, Å¿cire debes, menihil aliud, eo <lb/>in loco inferre voluiſſe, quà m <reg norm="quod" type="wordlist">qÄ‘</reg> <lb/>punctum horæ propoÅ¿itæ in plano <lb/>horologij orizontali reperiri po-<lb/>teÅ¿t, ope longitudinis vmbræ gno-<lb/>monis, & eius declinationis à ver-<lb/>ticali linea, Å¿eu à meridiana orizon <lb/>tali, iam in ipÅ¿o horologij plano <lb/>ductis.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿it analemma <var>.<lb/>l.q.m.b.</var> in quo <var>.l.m.</var> Å¿it verticalis <var>.q.<lb/>b.</var> verò orizontalis <var>.f.n.h.</var> autem Å¿it <lb/>Å¿emicirculus, cuiuÅ¿uis paralleli æqui <lb/>noctiali, cuius diameter fit <var>.f.h.</var> et <var>.<lb/>n.</var> Å¿it Solis locus in ipÅ¿o parallelo: </s> - <s xml:space="preserve">n.<lb/>r. autem Å¿it rectus Å¿inus arcus <var>.f.n</var>: et <var>.<lb/>r.o.z.</var> Å¿ectio communis ipÅ¿ius almi-<lb/>cantarat cum meridiano, et <var>.s.a.</var> <reg norm="con" type="context">cõ</reg> <lb/>munis Å¿ectio azimut Solis cum pla-<lb/>no horologij, et <var>.s.g.</var> gnomon, et <var>.x.<lb/>g.a.</var> radius Solis <var>.z.u.</var> verò Å¿inus alti-<lb/>tudinis ipÅ¿ius Solis, vbi videre po-<lb/>tes duo triangula dicta eſſe <var>.z.u.g.</var> - <pb o="322" rhead="IO. BAPT. BENED." n="334" file="0334" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0334"/> - et <var>.g.s.a.</var> quibus mediantibus cognoÅ¿citur longitudo vmbræ gnomonis hoc eÅ¿t <var>.s.a</var>.</s> - </p> - <p> - <s xml:space="preserve">Cum autem dico, <reg norm="medioque" type="simple">medioqÌ;</reg> azimut Solis, nihil aliud Å¿igniÅ¿icare volo, niÅ¿i angu-<lb/>lum, quem terminat linea azimutalis horologij, hoc eÅ¿t vmbra gnomonis cum li-<lb/>nea meridiana, Å¿eu cum verticali in ipÅ¿o plano horologij. </s> - <s xml:space="preserve">qui quidem anguli, æqua <lb/>les Å¿unt ijs, qui in triangulo conÅ¿tituto ex <var>.n.r.</var> ex <var>.r.o.</var> & ex <var>.o.z.</var> reperiuntur, cuius qui <lb/>dem trianguli, angulus puncti <var>.r.</var> rectus eÅ¿t, angulus verò terminatus ab <var>.n.r.</var> et <var>.o.z.</var> il <lb/>le eÅ¿t quem conÅ¿tituit azimut cum verticali, vel ipÅ¿i æqualis, vt coalternus, reliquus <lb/>verò in <reg norm="puncto" type="context">pÅ©cto</reg> <var>.o.</var> ille eÅ¿t <reg norm="quem" type="context">quẽ</reg> azimut facit <reg norm="cum" type="context">cÅ©</reg> meridiano, vel ipÅ¿i Ä™qualis vt coalternus.</s> - </p> - <p> - <s xml:space="preserve">Vnde quotieÅ¿cunque volueris in aliquo plano, orizonti parallelo, lineas hora-<lb/>rias ducere, iudico optimum fore Å¿i Å¿eparatim deÅ¿ignatæ fuerint hæ tres figuræ, hoc <lb/>eÅ¿t analemma meridianum, vel azimutale, vt ita dicam, </s> - <s xml:space="preserve">deinde parallelus <reg norm="inÅ¿eruiens" type="context">inÅ¿eruiẽs</reg> <lb/>pro tropicis, vt ego feci cap .51. meæ gnomonicæ, quæ duæ figuræ, <reg norm="Å¿ufficientes" type="context">Å¿ufficiẽtes</reg> <reg norm="erunt" type="context">erÅ©t</reg> <lb/>pro omnibus horologijs, tam ori-<lb/> - <anchor type="figure" xlink:label="fig-0334-01a" xlink:href="fig-0334-01"/> - zontalibus quam muralibus, non <lb/>tamen omninò, ideo pro orizon-<lb/>talibus, tertiam figuram Å¿eparatam <lb/>deÅ¿ignaui, quÄ™ erit circulus <var>.H.I.<lb/>K.</var> eiuÅ¿dem magnitudinis cum ana-<lb/>lemmate, in quo, ductis duobus <lb/>dia metris inuicom ad rectos, quo-<lb/>rum vnus <var>.H.I.</var> Å¿ignificet orizon-<lb/>talem lineam, reliqua verò <reg norm="ver- ticalem" type="context">ver-<lb/>ticalẽ</reg>, ducatur poÅ¿tea <var>.s.a.</var> tam di-<lb/>Å¿tans ab <var>.H.I.</var> quanta eÅ¿t longitudo <lb/>gnomonis horologij orizontalis, <lb/>cogitemus, </s> - <s xml:space="preserve">deinde hunc circulum <lb/>communem eſſe omnibus azimut <lb/>necnon plano horologij, in cuius <lb/>circunferentia à puncto <var>.k.</var> nadir ip-<lb/>Å¿ius zenit, accipiantur arcus æqua-<lb/>les ijs ipÅ¿orum azimut, quos termi-<lb/>nat zenit, & ipÅ¿i almicantarat, vt <lb/>exempli gratia, accipiemus arcum <lb/><var>k.L.</var> æqualem arcui <var>.L.z.</var> ipÅ¿ius ana-<lb/>lemmatis, ducta poÅ¿tea linea occul <lb/>ta <var>.o.L.</var> Å¿ignabimus azimutalem <var>.s.a.</var> <lb/>in puncto <var>.a.</var> vbi hæ duæ lineæ Å¿ein <lb/>uicem Å¿ecant, & Å¿ic habebimus iu-<lb/>Å¿tam <reg norm="quantitatem" type="context">quãtitatem</reg> ipÅ¿ius vmbræ gno <lb/>monis <var>.s.o.</var> tali hora, </s> - <s xml:space="preserve">deinde in ori-<lb/>zontali <var>.H.I.</var> Å¿umatur <var>.o.r.</var> à centro <var>.<lb/>o.</var> Ä™qualis ei quÄ™ in analemate repe <lb/>ritur, quæ vna portio eÅ¿t commu-<lb/>nis Å¿ectionis meridiani cum almi-<lb/>cantarat, terminata ab axe orizon <lb/>tis, & à diametro paralleli. </s> - <s xml:space="preserve">Deinde - <pb o="323" rhead="EPISTOL AE." n="335" file="0335" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0335"/> - <reg norm="ducatur" type="simple">ducat̃</reg>. <var>r.V.</var> ad rectos cum <var>.H.I.</var> vÅ¿que ad circunferentiam, in qua accipiatur <var>.r.n.</var> æqua <lb/>lis ei quæ eÅ¿t in parallelo, ducatur poÅ¿tea <var>.o.n.M.</var> & habe bimus triangulum <var>.o.r.n.</var> Å¿i-<lb/>milem <reg norm="æqualemque" type="simple">æqualemqÌ;</reg> triangulo iam Å¿upradicto. </s> - <s xml:space="preserve">Vnde angulus <var>.H.o.M.</var> ei æqualis erit, <lb/>quem azimut facit cum meridiano, & angulus <var>.M.o.k.</var> ei Ä™qualis, quem azimut con-<lb/>Å¿tituit cum verticali, ita quod Å¿i talis circulus <var>.H.k.I.</var> eſſet planum horologij orizon-<lb/>talis, Å¿uppoÅ¿ito <var>.o.</var> pro pede gnomonis, Å¿ecando poÅ¿tea <var>.o.M.</var> in puncto <var>.i.</var> ita vt <var>.o.i.</var> <lb/>æqualis eſſet <var>.s.a.</var> dato quod <var>.o.M.</var> ducta Å¿it ad partem Å¿ibi conuenientem, reÅ¿pectu <var>.o.<lb/>k.</var> ipÅ¿a pro verticali Å¿uppoÅ¿ita, quod tibirelinquo, cum hoc facillimum Å¿it, tunc pun <lb/>ctum <var>.i.</var> eſſet quod quærebamus. </s> - <s xml:space="preserve">Quod verò de vno puncto dico, idem de omni-<lb/>bus infero.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0334-01" xlink:href="fig-0334-01a"> - <image file="0334-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0334-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Vbi verò mihi Å¿ignificas ChriÅ¿tophorum Clauium, me duobus in locis meæ gno <lb/>monicæ redarguere, iam vidi. </s> - <s xml:space="preserve">Circa primum locum igitur, qui eÅ¿t in pagin .161. <lb/>ita inquit.</s> - </p> - <quote> - <s xml:space="preserve">Non enim deÅ¿unt, qui vel omninò negent, inter quos eÅ¿t Ioannes BaptiÅ¿ta Be-<lb/>nedictus in Å¿ua gnomonica cap .70. et .71. vbialia, & multo longiore ratione cona-<lb/>tur arcus Å¿ignorum deÅ¿cribere, vel certe dubitent, hoc modo rectè poſſe deÅ¿cribi ar-<lb/>cus Å¿ignorum, cum rationem non videant, qua hæc noÅ¿tra deÅ¿criptio quam quidem <lb/>omnes Å¿criptores Å¿ine vlla demonÅ¿tratione tradunt nitatur.</s> - </quote> - <p> - <s xml:space="preserve">AbÅ¿que dubio raptim tranÅ¿currit illa capita .70. 71. </s> - <s xml:space="preserve">Reuerendus Clauius alio quin <lb/>non Å¿cripÅ¿iſſet, quòd ego alia & multo longiore ratione conatus Å¿im arcus Å¿ignorum <lb/>deÅ¿cribere & c. præſertim cum eadem prorÅ¿us ratio, quæ ibi à me tradita eÅ¿t, illa Å¿it, <lb/>quam ip Å¿e Å¿uis Å¿criptis inÅ¿eruit.</s> - </p> - <p> - <s xml:space="preserve">Meus igitur modus in dictis capitibus traditus, minime diÅ¿crepat ab eo, Å¿ed ab il-<lb/>lorum modo, quorum opinio eÅ¿t interualla <var>.e.h</var>: <var>h.u</var>: u. n; </s> - <s xml:space="preserve">n.m. et <var>.m.d.</var> meæ figuræ in <lb/>pagi .75. poÅ¿itæ, æqualia eſſe interuallis <var>.e.h</var>:h.u:u.n:n.m. et <var>.m.d.</var> præcedentis figuræ, <lb/>qui etiam Å¿upponunt <var>.t.e.</var> meæ figuræ .75. eſſe directè coniuncta cum linea <var>.e.h.u.n.<lb/>m.d.</var> & propterea verÅ¿us finem .73. pag. dixi.</s> - </p> - <quote> - <s xml:space="preserve">Aduertat autem quam diligentiſſime quiÅ¿que ne Å¿e decipi patiatur à ſubÅ¿cripta fi <lb/>gura Å¿emicirculi <var>.Q.æ.m.</var> cum reliquis lineis ductis, ex antiquorum more, & c.</s> - </quote> - <p> - <s xml:space="preserve">Eo quod non defuerunt aliqui, ex vetuÅ¿tioribus (quorum Å¿cripta ad meas manus <lb/>peruenerunt) qui Å¿umentes interualla <var>e.h</var>:h.u. & c. figuræ. pag .75. æqualia illis figu-<lb/>ræ pag .74. <reg norm="putauerunt" type="context">putauerÅ©t</reg> lineam <var>.t.e.</var> directè coniunctam eſſe cum <var>.e.h.</var> & c. quod quidem <lb/>maximi erroris cauÅ¿a erat, </s> - <s xml:space="preserve">& propterea cap .71. verum modum oÅ¿tendi, Å¿eruando il <lb/>lam eandem Å¿uppoÅ¿itionem, hoc eÅ¿t quod interÅ¿titia <var>.e.h</var>: <var>h.u</var>: & c. figurÄ™ pag .75. æqua <lb/>lia Å¿int interÅ¿titijs <var>.e.h</var>: <var>h.u.</var> & c. præcedentis figuræ, & ideò in dicto cap .71. dixi.</s> - </p> - <quote> - <s xml:space="preserve">SuppoÅ¿ito deinde <var>.f.e.b.</var> lineam meridianam eſſe in plano orizontali, cÄ™terÄ™ lineÄ™ <lb/>horariÄ™ erunt prÄ™dictÄ™.</s> - </quote> - <p> - <s xml:space="preserve">Stantibus igitur his Å¿uppoÅ¿itis, vt habeantur omnia Å¿cientificè, volui, vt intellige-<lb/>retur pyra mis qua drilatera, eo modo quo dixi, cap .71. vbi clarè patet eandem pyra <lb/>midem eſſe, quam Pater Clauius (tacitè) poÅ¿uit in figura horologij, vt ipÅ¿e docuit <lb/>propoÅ¿itione Å¿ecunda, lib. Å¿ecundi, cuius baÅ¿is eÅ¿t triangulum <var>.H.I.F.</var> Å¿uæ figurÄ™ (exem <lb/>pli gratia pro quinta hora poÅ¿t meridiana) Alterum verò triangulum à me cogita-<lb/>tum, terminatum ab <var>.t.e</var>: <var>e.d</var>: et. ab <var>.t.d.</var> eleuata in mea figura, eÅ¿t in Å¿ua <reg norm="triangulum" type="context">triãgulum</reg> <var>.D.<lb/>I.F.</var> & propterea dixi.</s> - </p> - <quote> - <s xml:space="preserve">Nam <var>.t.e.</var> et <var>.e.d.</var> <reg norm="vtræque" type="simple">vtræq;</reg> in plano horologii non Å¿unt, quamuis in plano æquatoris <lb/>tres Å¿int, & c.</s> - </quote> - <p> - <s xml:space="preserve">Angulus verò <var>.e.</var> quem dico rectum eſſe, in Å¿ua figura eÅ¿t angulus <var>.D.I.F.</var> & mea - <pb o="324" rhead="IO. BAPT. BENED." n="336" file="0336" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0336"/> - <var>t.d.</var> imaginata, eÅ¿t Å¿ua <var>.D.F</var>. </s> - <s xml:space="preserve">Tertium deinde triangulum, quod in mea figura ter-<lb/>minatur ab <var>.t.d.</var> ab <var>.f.d.</var> & ab <var>f.t.</var> in Å¿ua eÅ¿t triangulum <var>.D.F.H.</var> vnde mea <var>.f.t.</var> reÅ¿pon-<lb/>det Å¿uÄ™ <var>.H.D.</var> & mea <var>.f.d.</var> Å¿uæ <var>.H.F.</var> & mea <var>.t.d.</var> Å¿uÄ™ <var>.D.F</var>. </s> - <s xml:space="preserve">Quartum autem triangulum <lb/><var>f.t.e.</var> in mea figura, reÅ¿pondet Å¿uo <var>.H.D.I.</var> & meum punctum <var>.t.</var> Å¿uo. </s> - <s xml:space="preserve">D, Nunc triangu <lb/>lum rectangulum, quod dico Å¿eparatim conÅ¿tituere, eÅ¿t illud tertium dictum corre-<lb/>Å¿pondens Å¿uo <var>.D.F.H.</var> vt ipÅ¿e facit in <reg norm="Å¿equenti" type="context">Å¿equẽti</reg> figura, quod ipÅ¿e vocat <var>.D.C.H.</var> & <reg norm="meus" type="simple">meê°</reg><unsure/> <lb/>radius <var>.t.x.</var> in Å¿ua figura, ille eÅ¿t qui terminatur ab <var>.D.</var> & ab initio Tauri, & Virginis.</s> - </p> - <p> - <s xml:space="preserve">Et quamuis ego non Å¿cripÅ¿erim talem Å¿iguram, vt ipÅ¿e fecit, nihilominus ipÅ¿am <lb/>verbis deÅ¿cripÅ¿i eomet modo, & propterea dixi.</s> - </p> - <quote> - <s xml:space="preserve">Quam <reg norm="diuiÅ¿ionem" type="context">diuiÅ¿ionẽ</reg>, Å¿i in triangulo Å¿eorÅ¿um deÅ¿cripto inuenire voluerimus, res erit <lb/>inuentu facillima, cum rectum angulum <var>.f.t.d.</var> (reÅ¿pondentem Å¿uo <var>.H.D.C.</var>) prÄ™dicti <lb/>trianguli tertij ea ratione diuiÅ¿erimus, & c.</s> - </quote> - <p> - <s xml:space="preserve">Quapropter Reuerendus Clauius non animaduertit meam rationem aliam non <lb/>eſſe, nec puncto longiorem Å¿ua, cum eademmet ipÅ¿a Å¿it.</s> - </p> - <p> - <s xml:space="preserve">Citaui etiam MunÅ¿terum cap .30. eo quod in ea impreſſione, quam tunc prÄ™ mani <lb/>bus habui, vidi in ea figura, quam ipÅ¿e vocat fundamentum horologiorum, literam <lb/>c. poÅ¿itam eſſe loco <var>.f.</var> et <var>.f.</var> loco <var>.c.</var> quod cauſæ fuit, vt omnia mendoÅ¿a viderentur, re <lb/>centiores autem impreſſiones correctæ Å¿unt.</s> - </p> - <p> - <s xml:space="preserve">RurÅ¿us alio in loco mihi accidit vt reprÄ™henderim Alexandrum Piccolomineum <lb/>in libris de Å¿phÄ™ra, qui quidem dicebat eas figuras Å¿uperficiales, quæ paucioribus an <lb/>gulis circunÅ¿criberentur, capaciores eſſe alijs, dummodo earum periphæriæ eſſent <lb/>æquales.</s> - </p> - <p> - <s xml:space="preserve">Nunc autem correctÄ™ Å¿unt eo in loco impreſſiones, & qui non viderit primas, pu-<lb/>tabit me immeritò ipÅ¿um repræhendere.</s> - </p> - <p> - <s xml:space="preserve">Idem etiam dico de eo capite ipÅ¿ius Piccolominei, in ijÅ¿dem libris, vbi tractat de <lb/>modo, quo vÅ¿i Å¿unt antiqui ad diuidendum zodiacum in .12. Å¿igna, quod erat circa <lb/>finem quarti libri.</s> - </p> - <p> - <s xml:space="preserve">Nunc verò, in recentioribus impreſſionibus, illud caput poÅ¿itum non eÅ¿t. </s> - <s xml:space="preserve">ImpreÅ¿ <lb/>Å¿iones autem illæ, vbi talia dixit, duæ fuerunt, quarum prima erat anni .1540. Å¿ecun <lb/>da verò .1552 Venetijs apud Andream Puteum.</s> - </p> - <p> - <s xml:space="preserve">Alius verò locus ipÅ¿ius Reuerendi Clauij, contra meas reprÄ™henÅ¿iones, eÅ¿t circa <lb/>finem pag .298. & circa .299. vbi ita Å¿cribit.</s> - </p> - <quote> - <s xml:space="preserve">Ex his liquido conÅ¿tat, non rectè à Ioan. BaptiÅ¿ta Benedicto in Å¿ua gnomonica ca <lb/>pit .49. reprÄ™hendi hancrationem deÅ¿cribendi horologij declinantis, qua omnes fe-<lb/>rè alij Å¿criptores vtuntur, quoniam, vt ex demonÅ¿tratione à nobis allata conÅ¿tat, re-<lb/>ctè per eam lineæ ho rariÄ™ in plano, quod à verticali declinat ducuntur. </s> - <s xml:space="preserve">Modus au <lb/>tem quem eo loco pręſcribit differentem ab eo, quem nos tradidimus certus etiam <lb/>eÅ¿t, Å¿ed nulla ratione noÅ¿tro contrarius, quia nos conÅ¿tituimus <var>.D.E.F.</var> angulum de-<lb/>clinationis plani à verticali circulo propriè dicto, ipÅ¿e autem loco huius anguli aſſu-<lb/>mit angulum declinationis eiuÅ¿dem plani à Meridiano circulo, vnde mirum non eÅ¿t <lb/>modum ipÅ¿ius à noÅ¿tro diÅ¿crepare. </s> - <s xml:space="preserve">Quod Å¿i <reg norm="conÅ¿titueremus" type="context">cõſtitueremus</reg> <var>.D.E.F.</var> angulum decli-<lb/>natio nis plani à Meridiano, ut ipÅ¿e (quemadmodum forÅ¿itan ab alijs putauit fieri) <lb/>& in reliqua deÅ¿criptione progrederemur, vt tradidimus, proculdubio <reg norm="horologium" type="context">horologiÅ©</reg> <lb/>declinans perperam deÅ¿criberetur, vt rectè docet.</s> - </quote> - <p> - <s xml:space="preserve">Optimè Å¿cripÅ¿iſſet Reuerendus Clauius, Å¿i verum fuiſſet, quod antiqui Å¿umerent <lb/>declinationem Å¿uperius <reg norm="dictam" type="context">dictã</reg> à verticali propriè dicto, & non à meridiano. </s> - <s xml:space="preserve">Sed ego <lb/>dico, authores à me citatos. capit .49. meÄ™ gnomonicÄ™ Å¿umere dictam declinatio- - <pb o="325" rhead="EPISTOLAE." n="337" file="0337" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0337"/> - nem planià meridiano, & non à dicto verticali.</s> - </p> - <p> - <s xml:space="preserve">Con Å¿idera primum in MunÅ¿tero cap .16. Å¿uæ horologiographiæ, vbi clarè docet <lb/>accipere angulum compræhenÅ¿um inter meridianum, & planum propoÅ¿itum, vbi <lb/>etiam ponit quandam figuram ædificij cum pariete Å¿uper quo deÅ¿ignatum eÅ¿t quod <lb/>dam horologium, & vbi Å¿e manifeÅ¿tè declarar, ita dicens.</s> - </p> - <quote> - <s xml:space="preserve">Nam ipÅ¿arum partium complementum. propoÅ¿itum indicabit angulum, quan-<lb/>tus videlicet fuerit arcus eiuÅ¿dem circuli <var>.d.e.f.g.</var> à puncto <var>.g.</var> vÅ¿que ad productam li-<lb/>neam meridianam interceptus, qui vnà cum ipÅ¿o <var>.f.g.</var> quadrantem integrare videtur, <lb/>vt in Å¿equenti figura: </s> - <s xml:space="preserve">quoniam arcus <var>.f.g.</var> eÅ¿t Å¿exaginta partium, qualium <var>.e.</var> f<unsure/>. <reg norm="quadrans" type="context">quadrãs</reg> <lb/>nonaginta, vnde concluditur reliquam partem hoc eÅ¿t, datum inclinationis <reg norm="angulum" type="context">angulÅ©</reg>, <lb/>fore partium triginta Å¿imilium.</s> - </quote> - <p> - <s xml:space="preserve">Orontius verò cap .13. <reg norm="ijÅ¿dem" type="context">ijÅ¿dẽ</reg> vtitur verbis, cum figura Å¿imili ad reliqua autem ipÅ¿ius <lb/>R. Clauij, videnda nondum mihi otium fuit. </s> - <s xml:space="preserve">quod Å¿i dabitur, tibi libenter<unsure/> dicam <lb/>quid Å¿entiam.</s> - </p> - </div> - </div> - <div type="section" level="3" n="26"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE MODO DVCENDI LINEAS HORARIAS <lb/>Å¿uper cyllindro immobili.</head> - <head style="it" xml:space="preserve">Hieronymo Ferrerio artium & Medicina Doctori peritißimo.</head> - <p> - <s xml:space="preserve"><emph style="sc">DEsignare</emph> horarias lineas Å¿uper cyllindro immobili, ad orizontemqÌue <lb/>perpendiculariter erecto difficile tibi non erit, (quod à me poÅ¿tulaÅ¿ti) Å¿i <lb/>modum .53. cap. meæ gnomonicæ obÅ¿eruaueris, accipiendo tamen pro <lb/>linea orizontali in tabula non aliquam rectam lineam, Å¿ed circularem, <lb/>Å¿imilemque circunferentiæ ipÅ¿ius cyllindri, dico autem Å¿imilem, eo quod Å¿i gn o-<lb/>mon <var>.o.x.</var> Å¿upra tabulam Å¿ignatus, & perpendicularis ipÅ¿i orizontali circulari <var>.b.i.x.</var> <lb/>eſſet dimidia, vel tertia vel quarta pars gnomonis cyllindro infixi, oporteret, vt <lb/>Å¿emidiameter circuli <var>.b.i.x.</var> etiam <lb/>eſſet medietas, vel tertia, aut quar <lb/> - <anchor type="figure" xlink:label="fig-0337-01a" xlink:href="fig-0337-01"/> - ta pars Å¿emidiametri cyllindri, vt <lb/>omnes arcus huiuÅ¿modi circuli in <lb/>ter ipÅ¿os azimut intercepti Å¿imiles <lb/>Å¿int arcubus cyllindri, quod à <lb/>te ipÅ¿o facilè videre Å¿cientificè po <lb/>teris. </s> - <s xml:space="preserve">reliqua nihil mutanda erunt <lb/>ab eo, quod Å¿cripÅ¿i circa figuram <num value="53">.<lb/>53.</num> cap. vt dixi. </s> - <s xml:space="preserve">Vnde inuenta <lb/>cum fuerit diÅ¿tantia orizontalis <lb/>puncti <var>.b.</var> à pede gnomonis <var>.x.</var> nec <lb/>non quantitas azi mutalis muralis <lb/><var>b.t.</var> quæ Å¿emper ab orizontali per <lb/>pendiculariter deÅ¿cendit, illicò <lb/>punctum <var>.t.</var> horæ propoÅ¿itæ in cy-<lb/>lindro inuenietur.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0337-01" xlink:href="fig-0337-01a"> - <image file="0337-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0337-01"/> - </figure> - </div> - <pb o="326" rhead="IO. BAPT. BENED." n="338" file="0338" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0338"/> - <p> - <s xml:space="preserve">Nunc verò cum duo puncta alicuius horariÄ™ lineæ inuenta fuerint, quæ à Solis Å¿i-<lb/>tu in diuerÅ¿is parallelis efficiuntur, Å¿i voluerimus ipÅ¿am lineam <reg norm="horariam" type="context">horariã</reg> ducere, Å¿cien <lb/>dum primò eÅ¿t ipÅ¿am lineam horariam eſſe communem Å¿ectionem circuli horarij, <lb/>illius horæ cum Å¿uperficie cyllindrica, </s> - <s xml:space="preserve">& propterea ellipticam, vt oÅ¿tendit Serenus <lb/>in .19. primi lib. quod etiam ellicere poſſumus ab eo, quod Archimedes in .10. pro-<lb/>poÅ¿itione libr. de conoidalibus, Å¿cribit. </s> - <s xml:space="preserve">Quapropter oporter nos inÅ¿trumen-<lb/>tum prius componere, modo circini, Å¿ed trium crurum, quæ omnia in eadem <lb/>plana Å¿uperficie Å¿int, ea tamen arte factum, vt quodlibet illorum poſſimus pro-<lb/>longare, necnon contrahere, ut cum duo extrema firmata fuerint, media poÅ¿-<lb/>Å¿it circunduci circa centrum, Å¿eu punctum commune illarum interÅ¿ectionum <reg norm="Å¿imulque" type="simple">Å¿imulqÌ;</reg> <lb/>poſſit produci, necnon abbreuiari vel augeri, & diminui, vt mediante Å¿ua extremi-<lb/>tate inÅ¿eriori poſſimus delineare gyrum ellipticum horarium, dum <reg norm="centrum" type="context">cẽtrum</reg> ipÅ¿orum <lb/>crurum adhæreat extremitati gnomonis, reliquæ vero extremitates ipÅ¿orum <reg norm="crurum" type="context">crurÅ©</reg> <lb/>Å¿int Å¿upra puncta inuenta ipÅ¿ius horæ. </s> - <s xml:space="preserve">oportet etiam vt hoc inÅ¿trumentum à tergo <lb/>ipÅ¿orum crurum habeat in Å¿uperiori parte Å¿uperficiem quandam <reg norm="Å¿emicircularem" type="context">Å¿emicircularẽ</reg>, quÄ™ <lb/>Å¿it vice vnius partis illius Å¿uperficiei, in qua Å¿upponuntur omnia crura inÅ¿trumenti, <lb/>& hoc quantum fieri poteÅ¿t, quod quidem fieri debet, ne crus medium, hoc eÅ¿t mo <lb/>bile, exeat à tali Å¿uperficie, Å¿eu declinet ab ea, quæ Å¿emper Å¿upponitur in Å¿itu circuli <lb/>horarij talis horæ. </s> - <s xml:space="preserve">oportet etiam, vt iuxta circunferentiam dimidij circuli Å¿int duo <lb/>gyri eiuÅ¿dem materiæ inter Å¿e parum diÅ¿tantes, ita ut crura poſſint moueri, intra hos <lb/>gyros, & dimidium circulum, & quod inter hos gyros locatæ Å¿int duæ cochleæ, Å¿eu <lb/>duo helices, vt quando voluerimus, poſſimus fir-<lb/>mare ipÅ¿a crura extrema, dum eorum extremitates <lb/> - <anchor type="figure" xlink:label="fig-0338-01a" xlink:href="fig-0338-01"/> - fuerint Å¿upra puncta inuenta illius horæ, </s> - <s xml:space="preserve">deinde in <lb/>dorÅ¿o iÅ¿tius inÅ¿trumenti, circa centrum coniunctio <lb/>nis, rectè factum erit Å¿i aliqua concauitas fuerit, in <lb/>qua, extremitas gnomonis poſſit locari, dum duce-<lb/>re voluerimus aliquam horariam lineam.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0338-01" xlink:href="fig-0338-01a"> - <image file="0338-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0338-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Tale inÅ¿trumentum excogitaui ad fugiendum <lb/>tædium inueniendi dictam ellipticam ex punctis.</s> - </p> - <p> - <s xml:space="preserve">Nunc autem Å¿ciendum eÅ¿t, quod vnus tantum-<lb/>modo gnomon Å¿ufficiens non erit pro tota die æſti-<lb/>ua, neque duo, niÅ¿i valde breues fuerint reÅ¿pectu <lb/>Å¿emidiametri cyllindri, & in Å¿itu medio quartarum <lb/>meridionalium noÅ¿tro orizonti, quorum autem <lb/>longitudo ita inuenienda eſſet.</s> - </p> - <p> - <s xml:space="preserve">Sit exempli gratia circulus <var>.a.b.e.u.</var> cyllindri ori <lb/>zontis vice, <reg norm="diuiÅ¿usque" type="simple">diuiÅ¿usqÌ;</reg> à duobus diametris <var>.d.e.</var> et <var>.c.<lb/>f.</var> quarum <var>.c.f.</var> Å¿it pro meridiana: </s> - <s xml:space="preserve">d.e. autem pro verticali, <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> e. punctus orientalis: </s> - <s xml:space="preserve">d. <lb/>verò <reg norm="occidentalis" type="context">occidẽtalis</reg> <var>.f.</var> autem meridionalis. et <var>.c.</var> Å¿eptentrionalis, <reg norm="computeturque" type="simple">computeturqÌ;</reg> maxima. <lb/></s> - <s xml:space="preserve">Solis amplitudo ab <var>.f.</var> verÅ¿us <var>.e.</var> quæ terminetur ab <var>.q.</var> ita <reg norm="quodarcus" type="simple">quodarcê°</reg> <var>.f.q.</var> minor Å¿it <reg norm="quam" type="context">quã</reg> <lb/>graduum .45. aliter impoſſibile eſſet duobus <reg norm="tantummodo" type="context">tantÅ©modo</reg> gnomonibus mediantibus <lb/>tota die æſtiua horas videre.</s> - </p> - <p> - <s xml:space="preserve">Quo facto ducatur ab .q: <var>q.p.</var> contingens circulum & à centro circuli <var>.o.</var> per pun-<lb/>ctum <var>.u.</var> medium quartæ ducatur <var>.o.u.i.</var> vÅ¿que ad contingentem <var>.q.p.</var> vnde <var>.u.i.</var> longitu <lb/>do erit vniuÅ¿cuiuÅ¿que gnomonis, qui gnomones infixi erunt in medio dictarum <lb/>quartarum.</s> - </p> - <pb o="327" rhead="EPISTOL AE." n="339" file="0339" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0339"/> - <p> - <s xml:space="preserve">HuiuÅ¿modi rei ratio per Å¿e nota erit quotieÅ¿cunque cogitauerimus verum arcum <lb/><var>e.b.</var> amplitudinis æſtiuæ, <reg norm="protractaque" type="simple">protractaqÌ;</reg> <var>.o.b.</var> quÄ™ parallela erit <var>.q.p.</var> vnde cum Sol tem-<lb/>pore æſtiuo orietur, tunc radios Å¿uos emittet via iÅ¿tarum æquidiÅ¿tantium linearum.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i longiores gnomones cuperes, oportebit eos tres eſſe, quorum vnus erit <lb/>orientalis in puncto <var>.e.</var> alter occi-<lb/>dentalis in puncto <var>.d.</var> reliquus ve-<lb/> - <anchor type="figure" xlink:label="fig-0339-01a" xlink:href="fig-0339-01"/> - rò meridionalis in puncto <var>.f.</var> quo-<lb/>rum <reg norm="vnuÅ¿quiÅ¿que" type="simple">vnuÅ¿quiÅ¿q;</reg> poteÅ¿t eſſe maior <lb/>tertia parte Å¿emidiametri cyllin-<lb/>dri, Å¿ed Å¿i voluerimus Å¿cire <reg norm="quan- tum" type="context">quan-<lb/>tÅ©</reg> ad plus poſſit eſſe longus vnuÅ¿-<lb/>quiÅ¿que illorum, ita faciendum <lb/>erit.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0339-01" xlink:href="fig-0339-01a"> - <image file="0339-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0339-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Faciemus <reg norm="quadratum" type="context">quadratÅ©</reg> <var>.o.a.h.u.</var> ex <lb/>Å¿emidiametro dicti circuli, a dia-<lb/>metro poÅ¿tea <var>.o.h.</var> huiuÅ¿modi qua <lb/>drati Å¿ubtrahatur Å¿emidiameter <var>.<lb/>o.e.</var> circuli, reÅ¿iduum verò <var>.e.h.</var> ip-<lb/>Å¿ius diametri <var>.o.h.</var> quadrati, erit <reg norm="lon" type="context">lõ</reg> <lb/>gitudo gnomonis, vbi Å¿imul appa <lb/>ret huiuÅ¿modi rei ratio, eo quod <lb/>cum gnomon <var>.e.h.</var> orientalis deÅ¿i-<lb/>net operari, illico meridionalis <var>.<lb/>f.g.</var> Å¿ubintrabit, poÅ¿t hunc verò occidentalis <var>.d.K.</var> monÅ¿trabit reliquum diei.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">Earundem line arum deÅ¿criptio Å¿aper conum rectum.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">CVm Å¿uper datum conum re-<lb/> - <anchor type="figure" xlink:label="fig-0339-02a" xlink:href="fig-0339-02"/> - ctum idem facere volueris <lb/>eÅ¿to conus <var>.A.</var> & R. qui diuiÅ¿us ima <lb/>ginatione Å¿it à quodam plano per <lb/>axem, & communis Å¿ectio Å¿it trian <lb/>gulus <var>.A.</var> & R. in quo plano cogite <lb/>mus gnomonem infixum ad rectos <lb/>vbi volueris, qui Å¿it <var>.p.t.o.</var> cogitem <lb/>etiam <var>.l.t.m.</var> aliud eſſe planum (in <lb/>quo Å¿it gnomon) quod conum Å¿e <lb/>cet, quæ quidem Å¿ectio, circularis <lb/>erit, ex .4. primi Pergei. </s> - <s xml:space="preserve">imagine-<lb/>mur etiam Å¿uperficiem <var>.p.s.</var> eſſe azi <lb/>mut in quo gnomonreperirur, Å¿u-<lb/><reg norm="perficiemque" type="simple">perficiemqÌ;</reg> <var>.e.s.</var> azimut propoÅ¿itæ <lb/>horÄ™, <reg norm="angulumque" type="simple">angulumqÌ;</reg> <var>.e.o.a.</var> contrapoÅ¿i <lb/><reg norm="tum" type="context">tÅ©</reg> angulo altitudinis Solis ab ori- - <pb o="328" rhead="IO. BAPT. BENED." n="340" file="0340" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0340"/> - zonte; </s> - <s xml:space="preserve">cogitemus etiam lineam <var>.A.t.i.x.</var> illud coni latus eſſe, qu od à ſummitate verÂ<lb/>Å¿us baÅ¿im tranÅ¿it per medium latitudinis ipÅ¿ius gnomonis, concipiamus etiam mente <lb/><var>e.a.</var> communem Å¿ectionem eſſe trianguli Å¿upra dicti cum azimut horæ, necnon pun-<lb/>ctum <var>.K.</var> eſſe commune radio Solis <var>.o.a.</var> & Å¿uperficiei conicæ, quod quidem eÅ¿t illud <lb/>quod quæritur, hoc Å¿cilicet modo. </s> - <s xml:space="preserve">Primum cognoÅ¿cimus angulum <var>.p.A.t.</var> vt medie <lb/>tas anguli totius coni, & angulum <var>.p.</var> rectum, vnde <var>.t.</var> tam intrinÅ¿ecus, quam extrinÅ¿e-<lb/>custrianguli <var>.A.p.t.</var> nobis cognitus erit. </s> - <s xml:space="preserve">Nunc cum angulus <var>.A.t.o.</var> cognoÅ¿catur, Å¿i <lb/>gnomon <var>t.o.</var> fixus fuerit in Å¿uperficie conica, ita qd cum latere <var>.A.t.</var> eſſiciat <reg norm="angulum" type="context">angulÅ©</reg> <lb/><var>A.t.o.</var> & lateraliter faciat angulosrectos cum Å¿uperficie conica, ad quod efficiendum <lb/>nulla eÅ¿t difficultas, cognoÅ¿cendo deinde <var>.A.t.</var> Å¿imul cum angulis <var>.A.</var> et <var>.t.</var> intrinÅ¿ecis <lb/>trianguli ortogonij <var>.A.p.t.</var> cognoÅ¿cemus <var>.p.t.</var> et <var>.A.p.</var> vnde etiam tota <var>.o.p.</var> Å¿ed cogno <lb/>Å¿cendo <var>.o.p.</var> cum angulo <var>.p.o.e.</var> (angulus enim <var>.p.o.e.</var> cognoÅ¿citur ex hypotheÅ¿i cum <lb/>Å¿it inter azimut Solis & azimut gnomonis) cum angulo <var>.o.p.e.</var> recto cognoÅ¿cemus <var>.p.<lb/>e.</var> et <var>.o.e.</var> </s> - <s xml:space="preserve">deinde cum nobis nota Å¿it <var>.o.e.</var> cum angulo altitudinis Solis <var>.e.o.a.</var> & angu-<lb/>lo <var>.o.e.a.</var> recto cognoÅ¿c emus longitudinem azimutalis <var>.e.a.</var> necnon quantitatem <var>.a.o.</var> <lb/>Imaginata poÅ¿tea <var>.a.q.</var> æquidiÅ¿tante <var>.e.p.</var> habebimus <var>.p.q.</var> æqualem <var>.a.e.</var> ex .34. primi <lb/>Eucli. </s> - <s xml:space="preserve">Vnde duabus <var>.o.p.</var> et <var>.p.q.</var> mediantibus, <reg norm="cognitiÅ¿que" type="simple">cognitiÅ¿qÌ;</reg> cum angulo recto <var>.p.</var> cogno <lb/>Å¿cemus <var>.o.q.</var> nec non angulum <var>.o.q.<lb/>p.</var> quo mediante, necnon median-<lb/>te angulo <var>.q.A.t.</var> et <var>.A.q.</var> cognita, co <lb/> - <anchor type="figure" xlink:label="fig-0340-01a" xlink:href="fig-0340-01"/> - gnoÅ¿cemus <var>.A.i.</var> et <var>.q.i.</var> quÄ™ <var>.q.i.</var> dem <lb/>pta à <var>.q.o.</var> relinquet nobis <reg norm="cognitam" type="context">cognitã</reg> <lb/><var>i.o</var>. </s> - <s xml:space="preserve">Et quia <var>.o.i.q.</var> et <var>.o.K.a.</var> Å¿emper <lb/>Å¿unt in eadem Å¿uperficie Å¿ecante co <lb/>num, quæ etiam Å¿ecat Å¿uperficiem <lb/>trianguli <var>.A.q.x.</var> ad rectos ex .18. vn <lb/>decimi, cum linea <var>.u.n.</var> perpendicu <lb/>laris Å¿it Å¿uperficiei trianguli <var>.A.q.i.</var> <lb/>ex .8. dicti, quia parallela eÅ¿t <var>.l.p.</var> quÄ™ <lb/>perpendicularis eÅ¿t Å¿uperficiei <reg norm="trian- guli" type="context">triã-<lb/>guli</reg> <var>.o.p.q.</var> ex .4. eiuÅ¿dem, Å¿equitur, <lb/>quod talis Å¿ectio ( quæ intelligatur <lb/>per <var>.u.K.i.n.</var>) Å¿emper erit elliptica, <lb/>vel parabole, Å¿eu hyperbole, <reg norm="prout" type="simple">ê“ut</reg> <lb/>linea <var>.o.i.q.</var> Å¿ecabit latus coni, oppo <lb/>Å¿itum lateri <var>.A.i.</var> diÅ¿tento in ipÅ¿a Å¿uperficie conica, Å¿eu ad Å¿uperiorem partem produ <lb/>ctum, velipÅ¿i parallelum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0339-02" xlink:href="fig-0339-02a"> - <image file="0339-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0339-02"/> - </figure> - <figure xlink:label="fig-0340-01" xlink:href="fig-0340-01a"> - <image file="0340-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0340-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Supponamus nunc dictam lineam <var>.o.q.</var> Å¿ecare dictum oppoÅ¿itum latus lateri <var>.A.i.</var> <lb/>verÅ¿us baÅ¿im, vnde Å¿ectio <var>.u.K.i.n.</var> erit elliptica. </s> - <s xml:space="preserve">quod facile cognitu eÅ¿t <reg norm="mediante" type="context">mediãte</reg> com <lb/>paratione angulorum <var>.A.q.i.</var> et <var>.q.A.i.</var> interÅ¿e, eo quod Å¿i eſſent Ä™quales, dicta Å¿ect o <lb/>barabola eſſet ex .27. primi Eucli. et .11. primi Pergei, Å¿ed Å¿i angulus <var>.A.q.i.</var> maior eÅ¿-<lb/>Å¿et angulo <var>.q.A.i.</var> Å¿ectio eſſet ellipÅ¿is, ex ultimo poÅ¿tulato primi Euclid. </s> - <s xml:space="preserve">& ex .13. pri-<lb/>mi Pergei, Å¿ed Å¿i dictus angulus <var>.A.q.i.</var> minor eſſet angulo <var>.A.</var> tunc Å¿ectio eſſet hyper-<lb/>bole ex dicto poÅ¿tulato & ex .12. primi Pergei. </s> - <s xml:space="preserve">Sit ergo primum vt <reg norm="dictum" type="context">dictÅ©</reg> eÅ¿t, hoc <lb/>eÅ¿t, quod Å¿ectio eſſet oxygonia, ideÅ¿t elliptica, Å¿eu defectio (quod idem eÅ¿t,) Å¿epa-<lb/>ratim oportebit nos ellipÅ¿im deÅ¿ignare <reg norm="Å¿imilem" type="context">Å¿imilẽ</reg> <reg norm="Ä™qualemque" type="context simple">Ä™qualẽqÌ;</reg> ei, quæ eÅ¿t <var>.u.K.i.n.</var> <reg norm="quod" type="wordlist">qÄ‘</reg> <reg norm="quidem" type="context">quidẽ</reg> <lb/>difficile non erit, quotieÅ¿cunque Å¿uos axes inuenerimus, maiorem Å¿cilicet, & mino- - <pb o="329" rhead="EPISTOL AE." n="341" file="0341" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0341"/> - rem, quæ ita reperientur, efficiemus primo anguium coni, qui Å¿it <var>.i.A.b.</var> quem diui-<lb/>demus per æqualia mediante <var>.A.q.</var> conÅ¿tituendo <var>.A.i.</var> huius anguli æqualem <var>.A.i.</var> Å¿u-<lb/>perficiei conicæ et <var>.A.q.</var> diuidentem, æqualem parti <var>.A.q.</var> axis coni, ducendo poÅ¿tea <lb/>ab <var>.i.</var> per <var>.q.</var> lineam vnam quouÅ¿que concurrat <var>.A.b.</var> in puncto <var>.b.</var> habebimus <var>.i.b.</var> pro <lb/>maiori axi ipÅ¿i ellipÅ¿is, quod per Å¿e clarum eÅ¿t, cuius medietas Å¿it <var>.i.c.</var> Å¿ed <var>.i.q.</var> ipÅ¿ius <var>.i.<lb/>b.</var> æqualis eÅ¿t ipÅ¿i <var>.q.i.</var> ipÅ¿ius coni, ex quarta primi Eucli. et <var>.q.b.</var> ipÅ¿ius <var>.i.b.</var> æqualis alte <lb/>ri parti inuiÅ¿ibili. </s> - <s xml:space="preserve">Reliquum eÅ¿t, vt reperiamus minorem axem, quem vocabimus <var>.<lb/>f.r.</var> ducatur ergo primum <var>.q.a.u.n.</var> ad rectos cum <var>.i.b.</var> <reg norm="æqualisque" type="simple">æqualisqÌ;</reg> ei quæ eÅ¿t coni, & diui <lb/>Å¿a Å¿imiliter in <var>.a.</var> quæ <var>.u.n.</var> ipÅ¿ius coni nobis cognita eÅ¿t ex lateribus <var>.A.u.</var> et <var>.A.n.</var> & ex <lb/>angulo coni, et <var>.a.q.</var> æqualis eÅ¿t <var>.e.p.</var> ex .34. primi. </s> - <s xml:space="preserve">Nunc certi erimus ex .21. primi <lb/>Pergei, quod eadem proportio erit quadrati <var>.u.q.</var> ad quadratum ipÅ¿ius <var>.f.c.</var> quæ pro-<lb/>ducti ipÅ¿ius <var>.i.q.</var> in <var>.q.b.</var> ad productum ipÅ¿ius <var>.i.c.</var> in <var>.c.b.</var> & cum cognita nobis Å¿int <lb/>hæc tria producta hoc eÅ¿t <var>.i.q.</var> in <var>.q.b.</var> et <var>.i.c.</var> in <var>.c.b.</var> et <var>.u.q.</var> in Å¿eipÅ¿a, cognoÅ¿cemus <reg norm="etiam" type="context">etiã</reg> <lb/>quartum ipÅ¿ius <var>.f.c.</var> & fic <var>.f.c.</var> <reg norm="eiuÅ¿que" type="simple">eiuÅ¿qÌ;</reg> duplum <var>.f.r.</var> cogniti nobis itaque cum Å¿int hi duo <lb/>axes <var>.i.b.</var> et <var>.f.r.</var> formabimus ellipÅ¿im. </s> - <s xml:space="preserve">Deinde producemus axim <var>.b.i.</var> à part <var>e.i.</var> quo-<lb/>uÅ¿que <var>.i.o.</var> æqualis Å¿it ei quæ extra conum eÅ¿t, dein-<lb/> - <anchor type="figure" xlink:label="fig-0341-01a" xlink:href="fig-0341-01"/> - de ducemus <var>.o.a.</var> quæ circunferentiam ellipticam <lb/>Å¿ecabit in puncto <var>.K.</var> vnde habebimus quantita-<lb/>tem ipÅ¿ius <var>.o.K.</var> et <var>.K.i.</var> rectam. </s> - <s xml:space="preserve">inde mediante cir-<lb/>cino Å¿i acceperimus rectam diÅ¿tantiam ab <var>.i.</var> ad <var>.K.</var> <lb/>in ellipÅ¿i, </s> - <s xml:space="preserve">deinde firmando pedem circini in pun-<lb/>cto <var>.i.</var> in Å¿uperficie conica, & cum alio Å¿ignando <lb/>lineam vnam curuam ad partem <var>.K.</var> in Å¿uperficie <lb/>conica, Å¿umendo poÅ¿tea interuallum <var>.o.K.</var> extra el <lb/>lipÅ¿im, </s> - <s xml:space="preserve">deinde firmando vnum pedem circini in <lb/>extre mitate gnomonis, cum alio poÅ¿tea Å¿ignan-<lb/>do aliam lineam curuam in Å¿uperficie ipÅ¿ius coni, <lb/>quæ primam Å¿e cet in puncto <var>.K.</var> hoc erit punctum <lb/>quæſitum horÄ™ propoÅ¿itæ in Å¿uperficie conica <lb/>propoÅ¿ita.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0341-01" xlink:href="fig-0341-01a"> - <image file="0341-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0341-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Sed Å¿i talis Å¿ectio fuerit parabole, vel hyperbo <lb/>le, tunc mediante Å¿uo diametro <var>.i.q.</var> cum baÅ¿i <var>.u.<lb/>q.n.</var> cognita, deÅ¿ignabimus ipÅ¿am Å¿ectionem <var>.u.i.</var> n <lb/>ope mei <reg norm="inÅ¿trumenti" type="context">inÅ¿trumẽti</reg> in calce meÄ™ gnomonicæ de <lb/>Å¿cripti, </s> - <s xml:space="preserve">deinde diuiÅ¿a <var>.u.q.</var> in <var>.a.</var> <reg norm="pro" type="simple">ê“</reg><reg norm="ductaque" type="simple">ductaqÌ;</reg> <var>q.i.</var> <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> <lb/> - <anchor type="figure" xlink:label="fig-0341-02a" xlink:href="fig-0341-02"/> - ad <var>.o.</var> <reg norm="ductaque" type="simple">ductaqÌ;</reg> <var>.o.a.</var> habebimus punctum <var>.K</var>. </s> - <s xml:space="preserve">Reli-<lb/>qua facienda Å¿unt, vt dictum eÅ¿t de ellipÅ¿i.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0341-02" xlink:href="fig-0341-02a"> - <image file="0341-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0341-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Inuenta modo cum fuerint duo puncta eiuÅ¿-<lb/>dem horæ propoÅ¿itÄ™, ducemus ab vno ad a-<lb/>liud, lineam horariam mediante circino trium <lb/>crurum, quem tibi Å¿cripÅ¿i nudius tertius pro cyl <lb/>lindro, quæ <reg norm="quidem" type="context">quidẽ</reg> linea crit portio gyri ellipÅ¿is, <lb/>Å¿eu hyperbolÄ™, vel parabolÄ™, vt à te ipÅ¿o cogi-<lb/>tare potes.</s> - </p> - <pb o="330" rhead="IO. BAPT. BENED." n="342" file="0342" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0342"/> - </div> - </div> - <div type="section" level="3" n="27"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">QVAEDAM NOTATV DIGNA IN <lb/>Ptolomeum.</head> - <head style="it" xml:space="preserve">Bartolomeo Christino Serenißimi SabaudiÅ“ Ducis apparitore.</head> - <p> - <s xml:space="preserve">EX tuis literis cognoui quo erga me animo eſſes, <reg norm="qualique" type="simple">qualiqÌ;</reg> voluntate, Å¿ed ne <lb/>tua pulcherrima Å¿tudia aliquo modo imperfecta <reg norm="relinquantur" type="simple">relinquant̃</reg>, vel ego tibi <lb/>deeſſe videar, dum Problemata geographica Magni Ptolomei conÅ¿i-<lb/>deras, aduerte, quod Å¿i putares in figura .6. cap. libr .7. geographiÄ™ eiuÅ¿-<lb/>dem (vt multi credunt) lineam <var>.V.*.</var> Å¿ecare circunferentiam <var>.A.D.</var> in puncto <var>.G.</var> ita <lb/>vt punctus <var>.G.</var> Å¿it tropici æſtiui, ideÅ¿t arcum <var>.D.G.</var> eſſe graduum .24. cum illis inci-<lb/>deres in maximum errorem. </s> - <s xml:space="preserve">Quapropter conÅ¿idera quæ nunc tibi Å¿cribo.</s> - </p> - <p> - <s xml:space="preserve">Sit circulus <var>.A.B.C.D.</var> huius centrum <var>.E.</var> <reg norm="Å¿upponaturque" type="simple">Å¿upponaturqÌ;</reg> Å¿emidiameter <var>.E.D.</var> eſſe <lb/>partium 120. quarum <var>.E.Z.</var> in alio Å¿emidiametro <var>.C.E.</var> ei orthogonaliter coniuncto, <lb/>talium Å¿it .17. in Å¿emidiametro vero <var>.E.A.</var> accipiatur <var>.E.S.</var> talium .24. et <var>.E.V.</var> 64. vn <lb/>de <var>.S.V.</var> erit partium 40. Å¿imilium.</s> - </p> - <p> - <s xml:space="preserve">Erigatur deinde <var>.S.*.</var> ad rectos cum <var>.E.A.</var> in puncto <var>.S.</var> quæ terminetur ab inter-<lb/>Å¿ectione lineÄ™ ductæ per puncta <var>.Z.D.</var> in puncto <var>.*.</var> ducatur demum <var>.V.*.</var> quæ Å¿eca-<lb/>bit circunf rentiam <var>.A.D.</var> in puncto <var>.G</var>. </s> - <s xml:space="preserve">Quæratur nunc quantitas ipÅ¿ius <var>.G.D.</var> <lb/>Ad quod efficiendum quærenda primum eÅ¿t quantitas ipÅ¿ius <var>.S.*.</var> quam illico co <lb/>gnoÅ¿cemus ex regula de tribus, cum dixerimus, Å¿i@ 17. dat nobis .120. quid dabit .41. <lb/>(nam duo triangula <var>.Z.E.D.</var> et <var>.Z.S.*.</var> Å¿unt inuicem Å¿imilia, cum <var>.S.*.</var> parallela Å¿it <lb/>ipÅ¿i <var>.E.D.</var>) vnde <var>.S.*.</var> proueniet nobis ex Å¿imilibus partibus .289. cum fracto, quod <lb/>r<unsure/>eijciamus ob minorem laborem.</s> - </p> - <p> - <s xml:space="preserve">Producantur poÅ¿tea <var>.V.*.</var> et <var>.E.D.</var> vÅ¿que ad eorum concurÅ¿um in puncto <var>.ω.</var> quæ-<lb/><reg norm="remusque" type="simple">remusqÌ;</reg> quanta Å¿it <var>.E.ω.</var> ex eadem regula, cum dixerimus, Å¿i .40. dat nobis .289. quid <lb/> - <anchor type="figure" xlink:label="fig-0342-01a" xlink:href="fig-0342-01"/> - <pb o="331" rhead="EPISTOL AE." n="343" file="0343" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0343"/> - dabit .64. (nam duo triangula <var>.V.S.*.</var> et <var>.V.E.ω.</var> Å¿unt inuicem Å¿imilia eadem ratio-<lb/>ne) vnde <var>.E.ω.</var> veniet nobis extalibus partibus .462.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0342-01" xlink:href="fig-0342-01a"> - <image file="0342-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0342-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Coniungatur nunc quadratum ipÅ¿ius <var>.E.V.</var> quod eÅ¿t .4096. cum quadrato ipÅ¿ius <var>.<lb/>E.ω.</var> quod eÅ¿t .213444. & habebimus quadratum ipfius <var>.V.ω.</var> talium <reg norm="partium" type="context">partiÅ©</reg> .217540.</s> - </p> - <p> - <s xml:space="preserve">Dicemus poÅ¿tea Å¿i .217540. dat nobis .4096. quid dabit quadratum ipÅ¿ius <var>.V.ω.</var> vt <lb/>Å¿inus totus quod eÅ¿t .10000000000. vnde veniet pro quadrato ipÅ¿ius <var>.V.E.</var> talium <lb/>partium, Å¿uperficialium Å¿cilicet .18827211. cuius radix erit .13721. & erit Å¿inus an-<lb/>guli <var>.V.ω.E.</var> qui erit grad .7. min .53. vnde angulus <var>.ω.V.E.</var> erit grad .82. min .7. eius <lb/>vero Å¿inus erit partium .99054.</s> - </p> - <p> - <s xml:space="preserve">Nunc autem quia angulus <var>.E.V.ω.</var> eÅ¿t acutus, imaginemur <var>.E.Å”.</var> ductam eſſe ad reÂ<lb/>ctos ipÅ¿i <var>.V.ω.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> etiam ducta ipÅ¿a <var>.E.G</var>. </s> - <s xml:space="preserve">Vnde habebimus angulum <var>.Å”.E.V.</var> gra-<lb/>duum .7. min .53. eius vero Å¿inus <var>.Å”.V.</var> partium .13721. (propter Å¿imilitudinem trianÂ<lb/>gulorum <var>.E.Å”.V.</var> et <var>.ω.E.V.</var>) talium Å¿cilicet, qualium <var>.E.V.</var> fuerit .100000. </s> - <s xml:space="preserve">Sed qua-<lb/>lium <var>.E.V.</var> eÅ¿t .64. talium erit .8. cum tribus quartis, cuius <var>.Å”.V.</var> quadratum erit parÂ<lb/>tium .76. cum dimidio Å¿imilium Å¿ed Å¿uperÅ¿icialium, quo quidem quadrato dempto <lb/>ex quadrato ipÅ¿ius .64. quod eÅ¿t .4096. remanebit quadratum ipÅ¿ius <var>.E.Å”.</var> partium <num value="2871">.<lb/>2871.</num> quo etiam quadrato <var>.E.Å”.</var> dempto ex quadrato <var>.E.G.</var> partium .14400. remaneÂ<lb/>bit quadratum ipÅ¿ius <var>.Å”.G.</var> partium .11529. cuius radix <var>.Å”.G.</var> erit partium .107. <reg norm="talium" type="context">taliÅ©</reg> <lb/>qualium <var>.E.G.</var> eÅ¿t .120. Å¿ed qualium <var>.E.G.</var> erit .100000. talium <var>.Å”.G.</var> erit partium <num value="89166">.<lb/>89166.</num> quæ vt Å¿inus anguli <var>.Å”.E.G.</var> habebit pro ipÅ¿o angulo, gra .63. min .5. qui colleÂ<lb/>cti cum gra .7. min .53. anguli <var>.V.E.Å”.</var> dabunt totum angulum <var>.A.E.G.</var> grad .70. <lb/>min .58. cuius complementum ex grad .90. erit <var>.G.D.</var> graduum .19. min .2. & non .24. <lb/>vt omnes ferè putant.</s> - </p> - </div> - </div> - <div type="section" level="3" n="28"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE REFLEXIONIBVS RADIORVM.</head> - <head style="it" xml:space="preserve">Excellentißimo PhiloÅ¿opbo FranciÅ¿co Vimercato.</head> - <p> - <s xml:space="preserve"><emph style="sc">QVoniam</emph> non videbatur quieÅ¿cere animus tuus, cum paucis ab hinc die-<lb/>bus tibi Å¿iÅ¿citanti reÅ¿pondiſſem, nec tamen rationem omnium, quæ dixe-<lb/>ram exactè explicare per tem poris anguÅ¿tiam potuiſſem, cogitaui ad te <lb/>per hanc occaÅ¿ionem Å¿cribens, & iam dicta repetere, & omnium tibi ra-<lb/>tionem Å¿ubiungere, & vt mihi plenius Å¿atisfaciam, & tibi commodè perlegenti faci <lb/>lius Å¿it veritatem intueri. </s> - <s xml:space="preserve">ScripÅ¿iÅ¿ti enim in tuis diÅ¿putationibus, vir doctiſſime, quod <lb/>omnis res viÅ¿a per <reg norm="Å¿peculum" type="context">Å¿peculÅ©</reg> <reg norm="quodcunque" type="context">quodcÅ©que</reg>, Å¿ub breuiſſimis lineis <reg norm="compræhendatur" type="context">cõpræhendatur</reg> à vifu.</s> - </p> - <p> - <s xml:space="preserve">PropoÅ¿itio hæc non eÅ¿t vniuerÅ¿aliter vera (quamuis etiam ab alijs omnibus pro ta <lb/>li poÅ¿ita Å¿it) cum in Å¿peculis concauis non Å¿emper verificetur, vt nunc tibi demon-<lb/>Å¿trabo.</s> - </p> - <p> - <s xml:space="preserve">EÅ¿to quod linea recta <var>.b.d.</var> tangat circulum <lb/> - <anchor type="figure" xlink:label="fig-0343-01a" xlink:href="fig-0343-01"/> - <var>b.o.q.n.</var> qui Å¿it communis Å¿ectionis Å¿up crficiei re <lb/>flexionis, & Å¿phæricÄ™ alicuius Å¿peculi Å¿phærici <lb/>concaui, & punctum contingentiæ Å¿it <var>.b.</var> à quo <lb/>exeant duæ lineæ <var>.b.q.</var> et <var>.b.n.</var> efficientes duos an <lb/>gulos inuicem æquales circa perpendicularem <var>.<lb/>b.c.</var> res autem viÅ¿a primò Å¿it in ipÅ¿a circunferen-<lb/>tia huiuÅ¿modi circuli in puncto <var>.n.</var> oculus vero in <lb/>puncto <var>.q.</var> ipÅ¿ius circunferentiÄ™. </s> - <s xml:space="preserve">Dico nunc duas - <pb o="332" rhead="IO. BAPT. BENED." n="344" file="0344" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0344"/> - lineas <var>.b.q.</var> et <var>.b.n.</var> Å¿imul Å¿umptas longiores eſſe omnibus alijs lineis exeuntibus ab ip <lb/>Å¿is punctis <var>.q.n.</var> quæ in aliquo puncto dictæ circunferentiæ Å¿imul concurrant.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0343-01" xlink:href="fig-0343-01a"> - <image file="0343-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0343-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Sint igitur aliæ duæ <var>.q.o.</var> et <var>.n.o.</var> quas probare volo Å¿imul Å¿umptas, eſſe minores dua <lb/>bus Å¿imul Å¿umptis <var>.q.b.</var> et <var>.n.b</var>. </s> - <s xml:space="preserve">Nam ex .20. tertij Eucli. cognoÅ¿cimus angulos <var>.q.b.n.</var> <lb/>et <var>.q.o.n.</var> inuicem æquales eſſe, & Å¿imiliter angulos <var>.b.n.o.</var> et <var>.b.q.o</var>. </s> - <s xml:space="preserve">deinde ex .15. pri <lb/>mi eiuÅ¿dem habemus angulos contra Å¿e poÅ¿itos, <lb/>circa <var>.a.</var> eſſe etiam inuicem Ä™quales. </s> - <s xml:space="preserve">Vnde ex .4 <lb/> - <anchor type="figure" xlink:label="fig-0344-01a" xlink:href="fig-0344-01"/> - Å¿exti, habebimus proportionem <var>.a.b.</var> ad .a<lb/>o. eandem eſſe, quæ <var>.a.n.</var> ad <var>.a.q.</var> & Å¿ic .b<lb/>n. ad <var>.o.q</var>. </s> - <s xml:space="preserve">Quare ita erit <var>.a.b.n.</var> ad <var>.a.o.q.</var> vt <var>.a.n</var> <lb/>ad <var>.a.q.</var> Å¿ed cum <var>.a.n.</var> maior Å¿it <var>.q.a.</var> ex .18. primi, <lb/>eo quod angulus <var>.b.q.n.</var> (qui æqualis eÅ¿t angulo <var>.<lb/>b.n.q.</var> ex .5. eiuÅ¿dem) maior eÅ¿t angulo <var>.a.n.q.</var> <lb/>qui pars eÅ¿t ipÅ¿ius <var>.b.n.q.</var> ergo latera Å¿imul Å¿um-<lb/>pta <var>.a.b.n.</var> maiora erunt lateribus <var>.a.o.q.</var> Å¿ed ex <num value="20">.<lb/>20.</num> primi <var>.a.b.n.</var> <reg norm="etiam" type="context">etiã</reg> maior erit <var>.a.n.</var> vnde ex .25. <lb/>quinti <var>.q.a.b.n.</var> maior erit <var>.n.a.o.q</var>. </s> - <s xml:space="preserve">quare Å¿equi-<lb/>tur verum eſſe propofitum.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0344-01" xlink:href="fig-0344-01a"> - <image file="0344-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0344-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Sed Å¿i oculus eſſet in <var>.u.</var> quemadmodum in Å¿ubÅ¿cripta hic <reg norm="Å¿ecunda" type="context">Å¿ecÅ©da</reg> figura videre eÅ¿t, <lb/>res autem viÅ¿ibilis in <var>.n.</var> ambo extra dictum circulum, eÅ¿to etiam primum <var>.b.u.</var> æqua-<lb/>lis <var>.b.n.</var> probabo Å¿imiliter <var>.u.b.n.</var> maiores eſſe <var>.u.o.n</var>. </s> - <s xml:space="preserve">Nam angulus <var>.o.</var> maior eÅ¿t angu-<lb/>lo <var>.b.</var> eo quod Å¿i circulum <var>.u.b.n.</var> cogitemus circunÅ¿cribere triangulum <var>.u.b.n.</var> ducen-<lb/>do vÅ¿que ad Å¿uam circunferentiam <var>.o.n.</var> in puncto <var>.s.</var> deinde ducendo <var>.u.s.</var> habebimus <lb/>ex .20. tertij angulum <var>.u.s.n.</var> <reg norm="æqualem" type="context">æqualẽ</reg> angulo <var>.u.b.n.</var> Å¿ed <reg norm="cum" type="context">cÅ©</reg> angulus <var>.u.o.n.</var> exterior trian <lb/>guli <var>.u.o.s.</var> exiÅ¿tat, ipÅ¿e maior erit angulo <var>.s.</var> ex .16. primi. </s> - <s xml:space="preserve">duco poÅ¿tea <var>.o.q.</var> parallelam <lb/>ad <var>.u.s.</var> quæ Å¿ecabit <var>.a.u.</var> in puncto <var>.q.</var> & habebimus angulum <var>.a.o.q.</var> Ä™qualem angulo <var>.<lb/>n.s.u.</var> ex .29. eiuÅ¿dem, hoc eÅ¿t angulo <var>.n.b.u.</var> fed ex Å¿u-<lb/> - <anchor type="figure" xlink:label="fig-0344-02a" xlink:href="fig-0344-02"/> - pradictis rationibus, lineæ <var>.q.b.n.</var> Å¿imul Å¿umptæ maio-<lb/>rem efficient longitudinem, quam <var>.q.o.n</var>. </s> - <s xml:space="preserve">Nunc cum <lb/>ipÅ¿i <var>.q.b.</var> addita fuerit <var>.u.q.</var> & vice <var>.q.o.</var> Å¿umpta fuerit ali-<lb/>qua linea minor ipÅ¿a <var>.u.q.o.</var> eo amplius <var>.u.q.b.n.</var> maior <lb/>erit, quod quidem hoc modo faciendum. </s> - <s xml:space="preserve">Acci-<lb/>piatur <var>.o.u.</var> vt comes <var>.o.n.</var> quæ minor eÅ¿t ambabus <var>.o.<lb/>q.</var> et <var>.q.u.</var> ex .20. primi, ita enim habebimus <reg norm="propoÅ¿itum" type="context">propoÅ¿itÅ©</reg>. <lb/></s> - <s xml:space="preserve">Å¿ed breuiori modo hoc ipÅ¿um videbis ex prÄ™cedenti, <lb/>& ex .21. primi Euclid. </s> - <s xml:space="preserve">Nam ex præcedenti <var>.u.b.n.</var> lon-<lb/>gior eÅ¿t ipÅ¿a <var>.u.s.n.</var> ex .21. autem primi <var>.u.s.n.</var> longior eÅ¿t <lb/>ipÅ¿a <var>.u.o.n.</var> ergo verum eÅ¿t propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0344-02" xlink:href="fig-0344-02a"> - <image file="0344-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0344-02"/> - </figure> - </div> - <figure position="here"> - <image file="0344-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0344-03"/> - </figure> - <p> - <s xml:space="preserve">Si verò radius incidentiæ <reg norm="non" type="context">nõ</reg> fuerit æqualis radio <lb/>reflexionis, Å¿it vt in hac Å¿ubÅ¿cripta tertia figura vide <lb/>re eÅ¿t <var>.u.b.p</var>.</s> - </p> - <p> - <s xml:space="preserve">Cum autem probauerim longitudinem <var>.u.b.n.</var> ma <lb/>iorem eſſe longitudine <var>.u.o.n.</var> coniungatur <var>.n.p.</var> cum <lb/><var>u.b.n</var>. </s> - <s xml:space="preserve">deinde. ab <var>.o.</var> ad <var>.p.</var> ducatur <var>.o.p.</var> quæ minor <lb/>erit longitudine <var>.o.n.p.</var> ex .20. primi, & illicò <lb/>manifeÅ¿tabitur verum eſſe propoÅ¿itum, etiam hoc <lb/>tertio modo.</s> - </p> - <pb o="333" rhead="EPISTOL AE." n="345" file="0345" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0345"/> - <p> - <s xml:space="preserve">Si <reg norm="autem" type="context">autẽ</reg> res viÅ¿ibilis <reg norm="oculusque" type="simple">oculusqÌ;</reg> ambo fuerint intra circulum, <reg norm="tunc" type="context">tÅ©c</reg> poſſibile eſſet quod <lb/><reg norm="longitudo" type="context">lõgitudo</reg> <var>.u.b.n.</var> modo maior, modo minor, modo verò æqualis eſſet ipÅ¿a <var>.u.o.n.</var> <reg norm="nunc" type="context">nÅ©c</reg>. <lb/></s> - <s xml:space="preserve">Quod etiam affirmo de <var>.u.b.p.</var> Å¿imiliter etiam eueniet Å¿i vnus terminorum <var>.u.</var> vel <var>.n.</var> <lb/>fuerit intra circunferentiam, reliquus verò extra ipÅ¿am.</s> - </p> - <p> - <s xml:space="preserve">ConÅ¿ideremus nunc hic inÅ¿raÅ¿criptam .4. figuram vbi <var>.d.b.p.</var> Å¿it circunferentia oxy <lb/>gonia Å¿eu elliptica (quod idem eÅ¿t) cuius maior axis Å¿it <var>.d.p.</var> in quo, duo termini <var>.u.n.</var> <lb/>Å¿int centra eius generationis: </s> - <s xml:space="preserve">b.x. verò Å¿it minor axis. </s> - <s xml:space="preserve">Imaginemur etiam circulum <var>.<lb/>b.o.x.</var> cuius Å¿emidiameter Å¿it <var>.c.b.</var> non maior medietate minoris axis, ne circunferen-<lb/>tia huiuÅ¿modi circuli Å¿ecet circunferentiam oxygoniam. </s> - <s xml:space="preserve">Cogitemus etiam circu-<lb/>lum <var>.b.e.</var> cuius Å¿emidiameter, minor non Å¿it minori axe <var>.b.x.</var> ipÅ¿ius oxygoniæ, ne Å¿e <lb/>inuicem Å¿ecent huiuÅ¿modi circunferentiæ, Å¿int etiam ambo eorum centra in linea <var>.b.<lb/>x.</var> minoris axis, & punctum <var>.b.</var> Å¿it commune vnicuique earum periphæriarum, vnde <lb/>minor circulus, totus intra, maior autem, totus extra ipÅ¿am <reg norm="figuram" type="context">figurã</reg> oxygoniam erit. <lb/></s> - <s xml:space="preserve">Nunc ad partem <var>.o.r.e.</var> vbi non communicant inuicem ipſæ circunferentiæ ducan-<lb/>tur <var>.n.o.r.e</var>: <var>u.o</var>: <var>u.r</var>: et <var>.u.e.</var> & per <var>.b.</var> et <var>.r.</var> cogitetur tranÅ¿ire alium circulum, cuius cen-<lb/>trum in axe <var>.b.x.</var> Å¿it <var>.t.</var> <reg norm="omnesque" type="simple">omnesqÌ;</reg> iÅ¿ti circuli imaginentur trium diuerÅ¿orum Å¿phærico-<lb/>rum Å¿peculorum, vnde pro genera <lb/>tione <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> oxygoniÄ™, Å¿eu ex .52. ter <lb/>tij Pergei, habebis longitudinem <var>.<lb/> - <anchor type="figure" xlink:label="fig-0345-01a" xlink:href="fig-0345-01"/> - u.r.n.</var> Ä™qualem eſſe longitudini <var>.u.b.<lb/>n.</var> & ei, quæ eÅ¿t <var>.u.o.n.</var> (vt minor ip <lb/>Å¿a <var>.u.r.n.</var> ex .21. primi Euclidis) mi-<lb/>nor ipÅ¿a <var>.u.b.n.</var> & longitudinem <var>.u.<lb/>e.n.</var> (vt maior ipÅ¿a <var>.u.r.n.</var> ex eadem <num value="21">.<lb/>21.</num> primi Eucli.) maior ipÅ¿a <var>.u.b.n</var>. <lb/></s> - <s xml:space="preserve">Sed Å¿i quis vellet hoc demonÅ¿trare <lb/>ope circuli, <reg norm="vnius" type="simple">vniê°</reg> <reg norm="tantummodo" type="context context">tãtÅ©modo</reg> Å¿peculi, <lb/><reg norm="multiplicando" type="context">multiplicãdo</reg> ipÅ¿as oxygonias <reg norm="quem- admodum" type="wordlist">quẽ-<lb/>admodum</reg> de ipÅ¿is circulis fecimus, obtineret Å¿imiliter propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="4"> - <figure xlink:label="fig-0345-01" xlink:href="fig-0345-01a"> - <image file="0345-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0345-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">Solutio dubitationis.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">RAtionalis eÅ¿t dubitatio tua, <lb/> - <anchor type="figure" xlink:label="fig-0345-02a" xlink:href="fig-0345-02"/> - vtrum (<reg norm="cum" type="context">cÅ©</reg> circulus minor hoc <lb/>eÅ¿t <var>.b.o.</var> habeat Å¿uum centrum in mi <lb/>nori axe inter centrum oxygoniæ, <lb/>et .b: exiÅ¿tente <var>.b.</var> extremo axis mi-<lb/>noris, <reg norm="communeque" type="simple">communeqÌ;</reg> ambobus circun-<lb/>ferentijs circuli Å¿cilicet & oxigoniÄ™) <lb/>dictus circulus minor, plura puncta <lb/>communia habeat cum ipÅ¿is circun-<lb/>ferentijs.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0345-02" xlink:href="fig-0345-02a"> - <image file="0345-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0345-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Cui dubitationi <reg norm="reÅ¿pondeo" type="context">reÅ¿põdeo</reg> quod <lb/>quotieÅ¿cunque centrum alicuius cir <lb/>culi fuerit idem cum <var>.c.</var> centro oxy-<lb/>goniæ, vel inter <var>.c.</var> et <var>.b.</var> in interual-<lb/>lo Å¿cilicet minoris axis, exiÅ¿tente <var>.b.</var> <lb/>Å¿ua extremitate communi ambabus - <pb o="334" rhead="IO. BABPT. BENED." n="346" file="0346" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0346"/> - circunferentijs, ipÅ¿as circunferentias inuicem contiguas eſſe oportebit in puncto <var>.b.</var> <lb/>tantummodo.</s> - </p> - <p> - <s xml:space="preserve">EÅ¿to primum quod centrum <var>.c.</var> commune exiÅ¿tat, vt dictum eÅ¿t. </s> - <s xml:space="preserve">Å¿it etiam centrum <lb/>vnius circuli, cuius diameter Å¿it <reg norm="idem" type="context">idẽ</reg> <reg norm="cum" type="context">cÅ©</reg> maiori axe <var>.d.p.</var> & in gyro oxygoniæ accipia-<lb/>tur punctum <var>.f.</var> proximum <var>.b.</var> quantum fieri poterit, </s> - <s xml:space="preserve">tunc protrahatur <var>.f.a.e.</var> parallela <lb/>ipÅ¿i <var>.g.c.</var> vÅ¿que ad gyrum maioris circuli in puncto <var>.e.</var> quæ cum <var>.d.p.</var> rectos efficiec <lb/>angulos. ex .29. primi Eucli. </s> - <s xml:space="preserve"><reg norm="Å¿ecabitque" type="simple">Å¿ecabitqÌ;</reg> gyrum circuli <var>.b.o.</var> minoris in puncto <var>.t.</var> quod di <lb/>co eſſe intra oxygoniam, <reg norm="Å¿eparatumque" type="simple">Å¿eparatumqÌ;</reg> ab <var>.f</var>. </s> - <s xml:space="preserve">Quapropter duco <var>.c.e.</var> quæ Å¿ecabit cir-<lb/>cunferentiam circuli minoris in <reg norm="puncto" type="context">pÅ©cto</reg> <var>.o.</var> à quo puncto duco etiam <var>.o.i.</var> parallelam ad <lb/><var>e.a</var>. </s> - <s xml:space="preserve">Deinde conÅ¿idero, quod ex ra-<lb/>tionibus ab Archimede adductis in <lb/> - <anchor type="figure" xlink:label="fig-0346-01a" xlink:href="fig-0346-01"/> - quinta propoÅ¿itione libri de conoi-<lb/>dalibus, & Å¿phæroidibus, eadem <lb/>proportio erit <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <var>.g.c.</var> ad <var>.b.c.</var> quÄ™ <lb/>ipÅ¿ius <var>.e.a.</var> ad <var>.f.a.</var> vnde permutando <lb/>ita erit ipÅ¿ius <var>.g.c.</var> ad <var>.e.a.</var> vel <var>.b.c.</var> ad <lb/><var>f.a.</var> hoc eÅ¿t ipÅ¿ius <var>.e.c.</var> ad <var>.e.a.</var> vt <var>.o.c.</var> <lb/>ad <var>.f.a.</var> Å¿ed ex Å¿imilitudine triangu-<lb/>lorum, & ex .11. quinti, ita <reg norm="etiam" type="context">etiã</reg> erit <lb/>ipÅ¿ius <var>.o.c.</var> ad <var>.o.i.</var> vt <var>.o.c.</var> ad <var>.f.a</var>. </s> - <s xml:space="preserve">Vn-<lb/>de Å¿equitur <var>.o.i.</var> æqualem eſſe <var>.f.a.</var> <lb/>Å¿ed ex .14. tertij Eucli <var>.t.a.</var> minor eÅ¿t <var>.<lb/>o.i</var>. </s> - <s xml:space="preserve">Quare minor etiam erit ipÅ¿a <var>.f.<lb/>a</var>. </s> - <s xml:space="preserve">Vnde punctum <var>.t.</var> intra oxygo-<lb/>niam erit, & conÅ¿equenter Å¿epara-<lb/>tum .ab <var>.f</var>.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0346-01" xlink:href="fig-0346-01a"> - <image file="0346-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0346-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Sed Å¿i centrum circuli minoris <lb/>fuerit inter <var>.c.</var> et <var>.b.</var> hoc eÅ¿t eccentri-<lb/>cum ipÅ¿ius oxygoniæ, ipÅ¿e tanget concentricum in puncto <var>.b.</var> tantummodò, vt in .3. <lb/>Euclidis libro probatur. </s> - <s xml:space="preserve">Vnde tanto magis diÅ¿tans erit punctum <var>.t.</var> à puncto <var>.f.</var> quod <lb/>erit propoÅ¿itum.</s> - </p> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">Alterius dubitationis Å¿olutio.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">VNde autem fiat, quod à ſpeculis planis, obiectorum imagines, ita diÅ¿tantes <lb/>vltra Å¿uperficiem ipÅ¿ius Å¿peculi videantur, vt obiecta citra ipÅ¿am Å¿uperficiem <lb/>reperiuntur.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius rei Å¿cientia, tres cognitiones nos primum habere oportet, quarum pri-<lb/>ma eÅ¿t. </s> - <s xml:space="preserve">Vnde fiat, quod obiecti imago in catheto incidentiæ videatur. </s> - <s xml:space="preserve"><reg norm="Secunda" type="context">SecÅ©da</reg>. </s> - <s xml:space="preserve">vn-<lb/>de efficiatur, quod angulus reflexionis, Å¿emper æqualis Å¿it angulo incidentiæ.</s> - </p> - <p> - <s xml:space="preserve">Terria demum. </s> - <s xml:space="preserve">Vnde naÅ¿catur quod radius incidentiæ Å¿imul cum radio reflexio-<lb/>nis Å¿it in quodam plano Å¿ecante Å¿uperficiem Å¿peculi Å¿emper ad rectos, quod qui-<lb/>dem planum vocatur Å¿uperficies reflexionis. </s> - <s xml:space="preserve">HuiuÅ¿modi tres paſſiones, ab omnibus <lb/>Å¿pecularijs conÅ¿ideratæ Å¿unt, Å¿ed rationes ab illis traditæ, mihi non Å¿atisfaciunt.</s> - </p> - <pb o="335" rhead="EPISTOLAE." n="347" file="0347" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0347"/> - <p> - <s xml:space="preserve">Nam circa æqualitatem angulorum reflexionis & incidentiæ, iam tibi probaui <lb/>illud non vniuerÅ¿aliter euenire à breuitate aggregati radiorum incidentiæ reflexio-<lb/><reg norm="nisque" type="simple">nisqÌ;</reg>. </s> - <s xml:space="preserve">Sed hoc naÅ¿citur potius ab eo, quod cum radius incidentiæ non poſſit Å¿uper <lb/>ficiem corporis opaci penetrare, reflectit, vt citra ipÅ¿am <reg norm="cum" type="context">cÅ©</reg> angulo æquali ei, quem <lb/>faceret cum eadem Å¿uperficie vltra ipÅ¿am Å¿i tranÅ¿iuiſſet.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia Å¿it <var>.a.</var> obiectum <var>.b.</var> <reg norm="autem" type="context">autẽ</reg> oculus in figura <var>.A.</var> et <var>.c.e.</var> Å¿uperficies ipÅ¿ius <lb/>Å¿peculi <var>.d.</var> verò Å¿it punctum ipÅ¿ius Å¿uperficiei, à quo ad oculum reflectitur imago ip-<lb/> - <anchor type="figure" xlink:label="fig-0347-01a" xlink:href="fig-0347-01"/> - Å¿ius <var>.a</var>. </s> - <s xml:space="preserve">Nunc Å¿i radius <var>.a.d.</var> incidentiæ, recta <lb/>incederet Å¿ub <var>.c.e.</var> efficeret angulum <var>.e.d.h.</var> <lb/>æqualem angulo <var>.c.d.a.</var> eius contrapoÅ¿ito, <lb/>Å¿ed quia impeditur ipſæ radius ab opacitate <lb/>ipÅ¿ius Å¿peculi <var>.c.e.</var> ne vlterius incedat, propte <lb/>rea reflectitur ab ipÅ¿a Å¿uperficie Å¿peculi, con-<lb/>Å¿tituens cum ipÅ¿a angulum <var>.e.d.b.</var> æqualem <lb/>angulo <var>.e.d.h.</var> Å¿ed quia angulus <var>.c.d.a.</var> eÅ¿t <reg norm="etiam" type="context">etiã</reg> <lb/>Ä™qualis ipÅ¿i angulo <var>.e.d.h.</var> propterea angulus <var>.e.d.b.</var> Ä™qualis exiÅ¿tit angulo <var>.c.d.</var> a; </s> - <s xml:space="preserve">per <lb/>accidens igitur Å¿equitur <var>.a.d.</var> et <var>.d.b.</var> Å¿imul Å¿umptas, breuiorem facere longiludinem <lb/>omni alia, quæ ab ipÅ¿a Å¿uperficie <var>.c.e.</var> ad eadem puncta <var>.a.b.</var> ducta eſſet, </s> - <s xml:space="preserve">quare natu-<lb/>ræintentio eÅ¿t efficere angulum <var>.e.d.b.</var> æqualem angulo <var>.e.d.h.</var> vnde ex accidenti po <lb/>Å¿tea Å¿equitur, ipÅ¿um æqualem eſſe angulo <var>.c.d.a.</var> & deinde <reg norm="quod" type="wordlist">qÄ‘</reg> lineæ <var>.a.d.</var> et <var>.d.b.</var> con-<lb/>Å¿tituant longitudinem breuiorem. </s> - <s xml:space="preserve">Quare illud quod omnes putabant eſſe primum <lb/>& perÅ¿e, vltimum eÅ¿t, & exaccidenti.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0347-01" xlink:href="fig-0347-01a"> - <image file="0347-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0347-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Quare vero Å¿uperficies, quæ vocatur reflexionis, in qua Å¿unt duæ lineÄ™, hoc eÅ¿t <lb/>incidentiÄ™, <reg norm="reflexionisque" type="simple">reflexionisqÌ;</reg>, Å¿emper Å¿it perpendicularis Å¿uperficiei ipÅ¿ius Å¿peculi: </s> - <s xml:space="preserve">Hæc <lb/>eÅ¿t ratio, quia cum quilibet radius incidentiæ, perpendicularis ipÅ¿i Å¿uperficiei Å¿pe-<lb/>culi, in Å¿eipÅ¿o reflectit, ex ijÅ¿dem dictis rationibus, hoc eÅ¿t, quia cum tali angulo vult <lb/>reflecti, cum quali tranÅ¿iret, ita etiam purandum eÅ¿t, quodradius incidens obliquus, <lb/>cum in Å¿eipÅ¿um non poſſit redire, quia non eÅ¿t perpendicularis Å¿uperficiei Å¿peculi, <lb/>reflectitur tamen per planum erectum ipÅ¿i Å¿uperficiei Å¿peculi, vt in eo, cui magis re-<lb/>Å¿iÅ¿tit Å¿uperficies corporis opaci, quà m alicui alij plano ipÅ¿ius infiniti inclinatorum <lb/>planorum, ab vtraque parte ipÅ¿ius plani perpendicularis, quod vnum etiam tan-<lb/>tummodo eÅ¿t, & in quo, radius maiorem vim obtinet reflectendi, Å¿eu in eo, in quo <lb/>radius ipÅ¿e cum maiori reÅ¿iÅ¿tentia repercutitur à ſuperficie corporis opaci.</s> - </p> - <p> - <s xml:space="preserve">PoÅ¿tremo <reg norm="Å¿ciendum" type="context context">Å¿ciẽdÅ©</reg> vnde oriatur, <reg norm="quod" type="simple">ê™</reg> rei viÅ¿ibilis imago, à ſpeculo plano reflexa, Å¿em <lb/>per in catheto incidentiæ videatur.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius rei ratione cognoÅ¿cendum primò eÅ¿t, quo modo fit perfecta <reg norm="Å¿implexque" type="simple">Å¿implexqÌ;</reg> <lb/>viÅ¿io, & non reflexa, deinde proÅ¿equemur ad reliqua huius tertiæ propoÅ¿itionis.</s> - </p> - <p> - <s xml:space="preserve">Animaduertendum igitur eÅ¿t, quod <reg norm="quotieÅ¿cunque" type="simple">quotieÅ¿cunq;</reg> obiectum aliquod viÅ¿ibile aÅ¿pi <lb/>cimus, nos nunquam perfectè illud comprehendere poſſumus, niÅ¿i in puncto con-<lb/>curÅ¿us, Å¿eu interÅ¿ectionis axium viÅ¿ualium, Å¿eu radialium ( vt ita loquar ) <reg norm="quam" type="context">quã</reg> <reg norm="inter- Å¿ectionem" type="context">inter-<lb/>Å¿ectionẽ</reg>, nos efficimus ope reuolutionis oculorum <reg norm="adinuicem" type="context">adinuicẽ</reg>, hoc eÅ¿t voluendo vnum <lb/>verÅ¿us alium, ita vt in Å¿itu ipÅ¿ius obiecti, Å¿einuicem Å¿ecent axes iam dicti, </s> - <s xml:space="preserve">tunc enim <lb/>vtroque oculo mediante, exacte rem perÅ¿picimus, cÄ™teris .8. circunÅ¿tantijs non ob-<lb/>Å¿tantibus.</s> - </p> - <p> - <s xml:space="preserve">Vnde Å¿tantibus oculis in tali Å¿itu, altero reÅ¿pectu alterius, Å¿i eorum alter tectus; <lb/></s> - <s xml:space="preserve">Å¿eu velatus fuerit, tune alio tantummodo oculo mediante, videbimus obiectum, <lb/>in ea diÅ¿tantia, exactius, quam in quauis alia propinquiori, & remotiori.</s> - </p> - <pb o="336" rhead="IO. BAPT. BENED." n="348" file="0348" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0348"/> - <p> - <s xml:space="preserve">Animal igitur, Å¿ecundum diÅ¿tantiam obiecti, oculum accommodat ad recipien-<lb/>dum quam exactiſſimè Å¿peciem ipÅ¿ius obiecti, & hoc voluendo ambos oculos, vnum <lb/>verÅ¿us alium, ita quod interÅ¿ectio axium Å¿it in Å¿itu Å¿eu loco dicti obiecti, nam tunc vi <lb/>dent ambo vel aliquis eorum Å¿olus, in tali diÅ¿tantia exactè obiectum videbit.</s> - </p> - <p> - <s xml:space="preserve">Vnde Å¿equitur obiectum viÅ¿ibile, compræhenÅ¿ibile non eſſe ab vno tantummodo <lb/>oculo in quolibet Å¿itu axis ipÅ¿ius oculi, Å¿ed in eo, vbi alius axis interÅ¿ecatur à dicto. <lb/></s> - <s xml:space="preserve">Quæ quidem interÅ¿ectio poteÅ¿t fieri propinqua, vel remota à viÅ¿u, ad certos tamen <lb/>terminos vÅ¿que.</s> - </p> - <p> - <s xml:space="preserve">De huiuÅ¿modi axium viÅ¿ualium interÅ¿ectione Å¿cribit Alhazem in .2. et .15. propo <lb/>Å¿itione tertij lib. Vitellio verò in .32. et .45. eiuÅ¿dem.</s> - </p> - <p> - <s xml:space="preserve">Quod igitur dico, verum eÅ¿t, ideÅ¿t, quod Å¿i vno tantummodo oculo aÅ¿piciemus <lb/>obiectum aliquod, ipÅ¿um nunquam perfectè proÅ¿picietur, niÅ¿i cum oculus ita Å¿itus <lb/>fuerit, vt eius axis cum axe alterius in loco obiecti Å¿e inuicem Å¿ecent, quamuis alter <lb/>oculus nihil videat, <reg norm="cum" type="context">cÅ©</reg> <reg norm="autem" type="wordlist">aÅ©t</reg> duobus oculis in tali Å¿itu <reg norm="conÅ¿titutis" type="context">cõſtitutis</reg> <reg norm="obiectum" type="context">obiectÅ©</reg> videmus, vnum <lb/>tantummodo nobis cernere videbimur, & Å¿i extra talem punctum interÅ¿ectionis ip-<lb/>Å¿um obiectum poÅ¿itum fuerit, tunc duo talia, obiecta nobis apparebunt, Å¿ed huiuÅ¿ <lb/>modi rei cauÅ¿am alias tibi manifeÅ¿tabo.</s> - </p> - <p> - <s xml:space="preserve">His igitur cognitis, ponamus aliquam <lb/> - <anchor type="figure" xlink:label="fig-0348-01a" xlink:href="fig-0348-01"/> - Å¿peculi Å¿uperficiem eſſe <var>.g.h.</var> in figura <var>.B.</var> <lb/>obiectum autem viÅ¿ibile <var>.b.</var> oculos vero <var>.a.</var> <lb/>et <var>.u.</var> punctum autem <var>.n.</var> in Å¿uperficie Å¿pecu <lb/>li, à quo imago ipÅ¿ius <var>.b.</var> reflectit ad <var>.a.</var> & <lb/>punctum <var>.t.</var> à quo reflectitur ad <var>.u.</var> et <var>.c.e.</var> <lb/>Å¿it <reg norm="communis" type="context">cõmunis</reg> Å¿ectio Å¿uperficiei reflexionis <lb/>radiorum <var>.b.n.a.</var> et <var>.c.f.</var> Å¿it communis Å¿ectio <lb/>Å¿uperficiei reflexionis radiorum <var>.b.t.u.</var> qua <lb/>rum <reg norm="vnaquæque" type="simple">vnaquæq;</reg> Å¿uperficies reflexionis, ere-<lb/>cta eÅ¿t ad Å¿uperficiem Å¿peculi <var>.g.h.</var> vt Å¿upra <lb/>diximus. </s> - <s xml:space="preserve">Nunc ex .19. vndecimi Eucl. Å¿equitur communem Å¿ectionem harum dua-<lb/>rum Å¿uperficierum. (b.c.d. Å¿cilicet) ad rectos etiam eſſe Å¿upra Å¿uperficiem Å¿peculi <var>.g.<lb/>h.</var> cum qua <var>.b.c.</var> quælibet linearum <var>.a.n.</var> vel <var>.u.t.</var> reflexarum ( productÄ™ cum fuerint ) <lb/>Å¿einuicem interÅ¿ecabunt eo quod duo anguli <var>.d.c.n.</var> et <var>.d.n.c.</var> Å¿imul collecti minores <lb/>Å¿unt duobus rectis, & ita <var>.d.c.t.</var> cum <var>.d.t.c.</var> cum anguli <var>.a.n.e.</var> et <var>.u.t.f.</var> reflexi, ipÅ¿is con-<lb/>trapoÅ¿iti, æquales Å¿int angulis <var>.b.n.c.</var> et <var>.b.t.c.</var> incidentiæ, quorum <reg norm="vnuÅ¿quiÅ¿que" type="simple">vnuÅ¿quiÅ¿q;</reg> ex .32. <lb/>primi, minor eÅ¿t recto.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0348-01" xlink:href="fig-0348-01a"> - <image file="0348-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0348-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Dico etiam quod in eodem puncto huiuÅ¿modi catheti <var>.b.c.d.</var> in quo interÅ¿ecabi-<lb/>tur à linea <var>.a.n.</var> in eodem Å¿ecabitur à linea <var>.u.t.</var> & quod punctum dicti concurÅ¿us, tan-<lb/>tum depreſſum erit Å¿ub Å¿uperficie Å¿peculi <var>.g.h.</var> quantum <var>.b.</var> Å¿upra ipÅ¿am reperietur. <lb/></s> - <s xml:space="preserve">Nam anguli <var>.b.n.c.</var> et <var>.d.n.c.</var> Å¿unt inuicem æquales, <reg norm="angulique" type="simple">anguliqÌ;</reg> <var>.b.c.n.</var> et <var>.d.c.n.</var> recti <var>.c.n.</var> <lb/>verò communis ambobus triangulis <var>.b.c.n.</var> et <var>.d.c.n.</var> vnde ex .26. primi Eucli. latus <var>.d.<lb/>c.</var> commune, vt trianguli <var>.d.c.n.</var>æquale erit lateri communi <var>.b.c.</var> vt trianguli <var>.b.c.n.</var> <lb/>Idem etiam dico de latere <var>.d.c.</var> vt ipÅ¿ius trianguli <var>.d.c.t.</var> quod æquatur lateri <var>.b.c.</var> vt <lb/>trianguli <var>.b.c.t</var>. </s> - <s xml:space="preserve">Vnde cum <var>.b.c.</var> vnum, & idem Å¿it: </s> - <s xml:space="preserve">d.c. igitur etiam erit, & ipÅ¿um <reg norm="vnum" type="context">vnÅ©</reg> <lb/>& idem, quod erit propoÅ¿itum.</s> - </p> - <p> - <s xml:space="preserve">Nunc autem cum hi duo radij Å¿einuicem Å¿ecent in puncto <var>.d.</var> ergo in ipÅ¿o puncto <var>.<lb/>d.</var> videbimur nobis videre <reg norm="imaginem" type="context">imaginẽ</reg> obiecti .b: <reg norm="cum" type="context">cÅ©</reg> ope <reg norm="duorum" type="context">duorÅ©</reg> <reg norm="iÅ¿torum" type="context">iÅ¿torÅ©</reg> <reg norm="radiorum" type="context">radiorÅ©</reg> <var>.n.a.</var> et <var>.t.<lb/>u.</var> ita inuicem <reg norm="Å¿itorum" type="context">Å¿itorÅ©</reg>, videamur nobis <reg norm="imaginem" type="context">imaginẽ</reg> proÅ¿picere. </s> - <s xml:space="preserve">Vnde Å¿i in tali caÅ¿u, vnus - <pb o="337" rhead="EPISTOLAE." n="349" file="0349" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0349"/> - oculorum clauderetur, nihilominus cum reliquo obiectum vidiſſemus in <reg norm="eodem" type="context">eodẽ</reg> ipÅ¿o <lb/>loco <var>.d.</var> & non in alio ex Å¿uperius dictis rationibus.</s> - </p> - <p> - <s xml:space="preserve">Et Å¿i Å¿tantibus ijs terminis volueremus pupillam oculi <var>.u.</var> verÅ¿us aliam <var>.a.</var> ad aÅ¿pi-<lb/>ciendum punctum <var>.n.</var> in Å¿uperficie <var>.g.h.</var> ipÅ¿ius Å¿peculi, hoc eÅ¿t Å¿i fecerimus quod axes <lb/>viÅ¿uales Å¿einuicem Å¿ecarent in ipÅ¿o puncto <var>.n</var>. </s> - <s xml:space="preserve">tunc videremur nobis videre duas <lb/>imagines ipÅ¿ius obiecti <var>.b.</var> intra Å¿peculum, eo quod obiectum, propter hoc non <lb/>ceſſaret reflectere ad oculos ab ipÅ¿is punctis <var>.n.</var> et <var>.t.</var> quapropter recipiendo ra-<lb/>dium <var>.t.u.</var> in Å¿itu axis oculi <var>.u.</var> & radium <var>.n.a.</var> in Å¿itu axis oculi <var>.a.</var> hi axes ex neceſſitate <lb/>(vt probauimus ) Å¿einuicem Å¿ecant in puncto <var>.d.</var> vnde vnam tantummodo imaginem <lb/>ipÅ¿ius obiecti nobis apparebit.</s> - </p> - <p> - <s xml:space="preserve">Ex his igitur omnibus potes facilè videre omnem imaginem, cuiuÅ¿uis obiecti, re-<lb/>flexam à ſpeculo, reperiri in ipÅ¿o catheto incidentiæ, cum ipÅ¿e Å¿emper Å¿it communis <lb/>Å¿ectio duarum Å¿uperficierum reflexionis, in quo catheto concurrunt ipſæ axes vi-<lb/>Å¿uales.</s> - </p> - <p> - <s xml:space="preserve">ExijÅ¿dem etiam dictis rationibus facile compræhendere poteris, vnde fiat, vt vi-<lb/>deamus imaginem reflexam à ſpeculis Å¿phÄ™ricis concauis citra <reg norm="ipÅ¿orum" type="context">ipÅ¿orÅ©</reg> Å¿uperficiem, & <lb/>non vltra. </s> - <s xml:space="preserve">Quod <reg norm="nunquam" type="context">nunquã</reg> euenit, niÅ¿i quando <reg norm="punctum" type="context">punctÅ©</reg> <var>.d.</var> interÅ¿ectionis <reg norm="ipÅ¿orum" type="context">ipÅ¿orÅ©</reg> <reg norm="radiorum" type="context">radiorÅ©</reg> <lb/>viÅ¿ualium (quod alio in loco non fit, niÅ¿i in catheto incidentiæ hoc eÅ¿t in communi <lb/>Å¿ectione duarum Å¿uperficierum reflexionis. </s> - <s xml:space="preserve">Dato quod obiectum non Å¿it in vna ea-<lb/>demqÌue Å¿uperficie, in qua reperti fuerint axes viÅ¿uales, hoc eÅ¿t dato, <reg norm="quod" type="simple">ê™</reg> ambo axes <lb/>viÅ¿uales non Å¿int in vna <reg norm="eademque" type="simple">eademqÌ;</reg> Å¿uperficie reflexionis) reperitur citra & non vltra Å¿u <lb/>perficiem ipÅ¿ius Å¿peculi.</s> - </p> - <p> - <s xml:space="preserve">Ad cuius rei euidentiam non <reg norm="prætermittam" type="context">prætermittã</reg> dicere, quod cum debeant Å¿emper Å¿u-<lb/>perficies reflexionum perpendiculares eſſe, velad rectos Å¿ecare Å¿uperficiem ipÅ¿ius <lb/>Å¿peculi, ipÅ¿arum communes Å¿ectiones cum Å¿uperficie Å¿peculi Å¿phÄ™rici, Å¿emper erunt <lb/>circunferentiæ magnorum circulorum illius Å¿phæræ, cuius portio eÅ¿t Å¿peculum <lb/>propoÅ¿itum, vt etiam Vitellio affirmat in prima Å¿exti libri. </s> - <s xml:space="preserve">Vnde vnuÅ¿quiÅ¿que ca-<lb/>thetus incidentiæ tranÅ¿ibit per centrum Å¿peculi, cum ipÅ¿e Å¿it communis Å¿ectio dua-<lb/>rum Å¿uperficierum reflexionis, </s> - <s xml:space="preserve">quare in ipÅ¿o catheto erit punctum interÅ¿ectionis ip <lb/>Å¿orum axium viÅ¿ualium ex neceſſitate, vt videbimus, Å¿i vnam tantummodo <reg norm="imaginem" type="context">imaginẽ</reg> <lb/>obiecti nobis videremur videre.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, Å¿int duæ Å¿uperficies reflexionis Å¿peculi Å¿phærici concaui <var>.b.n.c.a.</var> <lb/>et <var>.b.t.c.u.</var> <reg norm="obiectumque" type="simple">obiectumqÌ;</reg> Å¿it <var>.b.</var> oculi autem Å¿int <var>.a.u.</var> punctum verò Å¿uperficiei Å¿peculi, à <lb/>quo obiectum emittit reflexionem Å¿uÄ™ <lb/>imaginis ad oculum <var>.a.</var> Å¿it <var>.n.</var> <reg norm="punctum" type="context">pÅ©ctum</reg> au-<lb/> - <anchor type="figure" xlink:label="fig-0349-01a" xlink:href="fig-0349-01"/> - <anchor type="figure" xlink:label="fig-0349-02a" xlink:href="fig-0349-02"/> - tem à quo eandem reflectit oculo <var>.u.</var> Å¿it <lb/>t. communis autem Å¿ectio harum dua-<lb/>rum Å¿uperficierum Å¿it <var>.b.c.</var> Å¿ed <var>.x.</var> <reg norm="centrum" type="context">centrÅ©</reg> <lb/>Å¿it Å¿peculi, radius verò incidentiÄ™ Å¿uper <lb/>ficiei <var>.b.n.c.</var> erit <var>.b.n.</var> cuius reflexus Å¿it <var>.n.<lb/>a.</var> radij autem alterius Å¿uperficiei erunt <lb/><var>b.t.</var> et <var>.t.u</var>. </s> - <s xml:space="preserve">Imaginemur nunc duos Å¿emi <lb/>diametros <var>.x.n.</var> et <var>.x.t.</var> quæ angulos <var>.b.n.<lb/>a.</var> et <var>.b.t.u.</var> per æqualia diuidant ex Å¿up-<lb/>poÅ¿ito.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0349-01" xlink:href="fig-0349-01a"> - <image file="0349-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0349-01"/> - </figure> - <figure xlink:label="fig-0349-02" xlink:href="fig-0349-02a"> - <image file="0349-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0349-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Nunc ijs Å¿uppoÅ¿itis, Å¿i vnam tantum-<lb/>modo obiecti imaginem videbimus, - <pb o="338" rhead="IO. BAPT. BENED." n="350" file="0350" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0350"/> - clarum erit ex rationibus Å¿upradictis nos ipÅ¿am videre in <reg norm="communi" type="context">cõmuni</reg> concurÅ¿u ipÅ¿orum <lb/>axium viÅ¿ualium, qui axes cum reperiantur vnà cum ipÅ¿is radijs reflexis <var>.n.a.</var> et <var>.t.u.</var> <lb/>ex neceſſitate Å¿einuicem <reg norm="Å¿ecabunt" type="context">Å¿ecabÅ©t</reg> in catheto <var>.b.c.</var> cum extendantur in ipÅ¿is Å¿uperficie-<lb/>bus reflexionum, quæ Å¿uperficies nihil aliud commune inuicem habent, quam cathe <lb/>tum dictum <var>.b.c.</var> Å¿it igitur in puncto <var>.d</var>.</s> - </p> - <p> - <s xml:space="preserve">Ex his dictis alia oritur neceſſitas, hoc eÅ¿t, quod quotieÅ¿cunque vnam tantummo <lb/>do imaginem obiecti <var>.b.</var> videmus, dato quod duæ Å¿uperficies reflexionis Å¿int, & non <lb/>vna tantum, tunc angulos <var>.n.</var> et <var>.t.</var> Å¿emper inuicem æquales eſſe oportebit. </s> - <s xml:space="preserve">Vnde ar-<lb/>cus <var>.n.c.</var> et <var>.t.c.</var> ex neceſſitate inuicem æquales erunt.</s> - </p> - <p> - <s xml:space="preserve">Scimas enim ex .3. Å¿exti Euclid. quod eadem proportio erit ipÅ¿ius <var>.b.n.</var> ad <var>.n.<lb/>d.</var> quÄ™ ipÅ¿ius <var>.b.x.</var> ad <var>.x.d.</var> & ipÅ¿ius <var>.b.t.</var> ad <var>.t.d.</var> Å¿imiliter, </s> - <s xml:space="preserve">quare ipÅ¿iusb <var>.n.</var> ad <var>.n.<lb/>d.</var> erit vt ipÅ¿ius <var>.b.t.</var> ad <var>.t.d</var>. </s> - <s xml:space="preserve">Vnde Å¿equitur <var>.b.n.</var> æqualem eſſe ipÅ¿i <var>.b.t.</var> et <var>.n.d.</var> <lb/>ipÅ¿i <var>.t.d.</var> vt à medio circulo <var>.E.</var> potes videre, quamuis etiam <var>.b.</var> non eſſet extremum <lb/>diametri, Å¿ed vbicunque volueris in ipÅ¿o diametro, vel <reg norm="etiam" type="context">etiã</reg> protracta, eo quod pun-<lb/>ctum <var>.n.</var> & punctum <var>.t.</var> in eodem Å¿emicirculo, vel in æqualibus Å¿emicirculis, non <reg norm="poÅ¿sent" type="context">poÅ¿sẽt</reg> <lb/>aliter in ipÅ¿a circunferentia locari, <reg norm="eandem" type="context">eãdem</reg> Å¿eruando proportionem <var>.b.n.</var> ad <var>.n.d.</var> vt <var>.b.<lb/>t.</var> ad <var>.t.d.</var> </s> - <s xml:space="preserve">propterea quod in omni alio Å¿itu exiÅ¿tente puncto <var>.t.</var> ipÅ¿a <var>.b.t.</var> eſſet aut maior <lb/>aut minor ipÅ¿a <var>.b.n.</var> et <var>.t.d.</var> aut minor, aut maior ipÅ¿a <var>.t.d.</var> ex .7. & 14. tertij Eucli. </s> - <s xml:space="preserve">vnde <lb/>aut maior, aut minor proportio eſſet ipÅ¿ius <var>.b.t.</var> ad <var>.t.d.</var> quam ipÅ¿ius <var>.b.n.</var> ad <var>.n.d.</var> & non <lb/>eadem.</s> - </p> - <p> - <s xml:space="preserve">Nunc è conuerÅ¿o Å¿i <var>.b.n.</var> et <var>.b.t.</var> Å¿unt Å¿ibi inuicem æquales, & Å¿ic <var>.n.d.</var> cum <var>.t.d.</var> Å¿equi-<lb/>tur ex .8. primi Eucli. angulos <var>.n.</var> et <var>.t.</var> inuicem æquales eſſe.</s> - </p> - <p> - <s xml:space="preserve">Ab ijÅ¿dem Å¿peculationibus potes etiam videre vnde accidat quod partes Å¿uperio <lb/>res alicuius obiecti reflexæ à tali Å¿peculo concauo videntur nobis inferiores eſſe, & <lb/>inferiores appareant Å¿uperiores, & dextræ Å¿iniÅ¿træ, & Å¿iniÅ¿træ dextræ. </s> - <s xml:space="preserve">quod autem <lb/>hucuÅ¿que demonÅ¿traui de Å¿peculis planis, & Å¿phæricis concauis, ratiocinare tu ijÅ¿dem <lb/>medijs circa Å¿phærica conuexa, vbi clarè videbis puncta huiuÅ¿modi Å¿peculi conuexi, <lb/>à quibus reflectitur imago obiecti ad ambos oculos, Å¿emper oportere æquidiÅ¿tantia <lb/>eſſe à <reg norm="puncto" type="context">pÅ©cto</reg> communi ipÅ¿ius Å¿uperficiei Å¿peculi, & catheto incidentiæ, dum unam tan <lb/>tummodo imaginem ipÅ¿ius obiecti videmus, & à diuerÅ¿is Å¿uperficiebus reflexionum.</s> - </p> - <p> - <s xml:space="preserve">Nolo etiam prætermittere, quod nunc mihi Å¿uccurrit, hoc eÅ¿t quod poſſet ali-<lb/>quis duos Å¿itus inuenire, vnum pro oculo, alterum verò pro obiecto, reÅ¿pectu alicu-<lb/>ius Å¿peculi concaui, Å¿phÄ™roidis prolatæ, vt reflexio ipÅ¿ius obiecti videretur, vt linea <lb/>diuidens per æqualia ipÅ¿um Å¿peculum. </s> - <s xml:space="preserve">ReÅ¿pectu verò alicuius Å¿peculi concaui Å¿phæ-<lb/>roidis oblongæ, vt reflexio obiecti ad oculum veniret à tota Å¿uperficie ipÅ¿ius Å¿pecu-<lb/>li, vnde tota Å¿uperficies ipÅ¿ius Å¿peculi videretur colorata illo colore cuius eſſet <lb/>obiectum, quæ quidem paſſiones <reg norm="pendent" type="context">pendẽt</reg> à .48. tertij lib. ipÅ¿ius Pergei, vt ex te ipÅ¿o fa <lb/>cile videre potes, propter æqualitatem angulorum reflexionis, & incidentiæ.</s> - </p> - <p> - <s xml:space="preserve">Opinio autem mea, quam Å¿cire cupis de imagine obiecti reflexa, quam putas eÅ¿-<lb/>Å¿e in Å¿uperficie Å¿peculi, hæc eÅ¿t, quod nec in Å¿uperficie, nec ultra, nec citra eam eÅ¿t ip <lb/>Å¿a imago, quod autem vltra non Å¿it, hoc puto nulli dubium eſſe. </s> - <s xml:space="preserve">eadem etiam ra-<lb/>tione non erit citra Å¿uperficiem Å¿peculi concaui, quamuis ipÅ¿am nos compræhenda-<lb/>mus in concurÅ¿u radiorum viÅ¿ualium, tam ab vno Å¿peculo quam ab alio reflexione <lb/>facta. </s> - <s xml:space="preserve">Sed quòd ipÅ¿a neque Å¿it in ipÅ¿a Å¿peculi Å¿uperficie, manifeÅ¿tum erit ex hoc, <reg norm="quod" type="simple">ê™</reg> <lb/>duo Å¿pectantes in eodem Å¿peculo, duas diuerÅ¿as imagines vident, tres, <reg norm="autem" type="wordlist">aÅ©t</reg> tres, qua-<lb/>tuor, quatuor, & Å¿ic deinceps, vnde tot eſſent imagines Å¿upra Å¿uperficiem Å¿peculi, <lb/>quot obiecta, <reg norm="quod" type="simple">ê™</reg> tamen ita non eÅ¿t, nec plus eÅ¿t in vno loco ipÅ¿a imago, quam in alio, - <pb o="339" rhead="EPISTOLAE." n="351" file="0351" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0351"/> - niÅ¿i in obiecto ipÅ¿o, lumen enim abipÅ¿o obiecto reflexum, Å¿eipÅ¿um diffundit vndi-<lb/>que, & radijipÅ¿ius lu<unsure/>minis reflexi, vt plurimum Å¿einuicem Å¿ecant. </s> - <s xml:space="preserve">Vnde in ipÅ¿o ae-<lb/>re funt omnes miÅ¿ti. </s> - <s xml:space="preserve">Quapropter natura Å¿agacifÅ¿ima pupillam oculi animalibus tam <lb/>paruam conÅ¿truxit ad Å¿uperficiem tam amplæ Å¿phæræ ipÅ¿ius oculi, vt diÅ¿tinctæ vide-<lb/>rentur omnia obiecta.</s> - </p> - <p> - <s xml:space="preserve">Nolo etiam tibi tacere, quod <reg norm="quotieÅ¿cunque" type="simple">quotieÅ¿cunq;</reg> oculorum pupillæ poÅ¿itæ fuerint inter <lb/>cathetum incidentiæ, & Å¿uperficiem <lb/> - <anchor type="figure" xlink:label="fig-0351-01a" xlink:href="fig-0351-01"/> - <anchor type="figure" xlink:label="fig-0351-02a" xlink:href="fig-0351-02"/> - Å¿peculi Å¿phærici concaui, vt puta in li-<lb/>neis <var>.d.t.</var> et <var>.t.n.</var> in figura <var>.D</var>. </s> - <s xml:space="preserve">tunc nullo <lb/>pacto poſſemus videre vnam imagi-<lb/>nem obiecti, Å¿ed duas nec non confu-<lb/>sè, </s> - <s xml:space="preserve">propterea <reg norm="quod" type="simple">ê™</reg> nullo pacto radij <var>.t.d.</var> <lb/>et <var>.n.t.</var> reflexi poterint. </s> - <s xml:space="preserve">ambo vniri <reg norm="cum" type="context">cÅ©</reg> <lb/>ambobus axibus viÅ¿ualibus, eo quod <lb/>axes vifuales nunquam poſſunt inui-<lb/>cem interÅ¿ecari poÅ¿t viÅ¿um, Å¿ed Å¿em-<lb/>per ante ipÅ¿um, vnde nec inuicem pa-<lb/>ralleli poſſunt eſſe.</s> - </p> - <div type="float" level="5" n="4"> - <figure xlink:label="fig-0351-01" xlink:href="fig-0351-01a"> - <image file="0351-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0351-01"/> - </figure> - <figure xlink:label="fig-0351-02" xlink:href="fig-0351-02a"> - <image file="0351-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0351-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Dico etiam, quod Å¿i obiectum inci <lb/>derit in eadem Å¿uperficie, in qua duo <lb/>axes viÅ¿uales, vel radij reflexi <reg norm="reperiuntur" type="context">reperiÅ©tur</reg>, hoc eÅ¿t in vna <reg norm="eademque" type="simple">eademqÌ;</reg> Å¿uperficie reflexio-<lb/>nis, </s> - <s xml:space="preserve">tunc locus imaginis non erit in catheto incidentiæ, eo quod interfectio axium <lb/>uifualium non erit in ipÅ¿o catheto Å¿ed extra, in qua interÅ¿ectione fit viÅ¿io vnius tan-<lb/>tummodo imaginis, quod antiqui non animaduerterunt. </s> - <s xml:space="preserve">Hoc autem dico deÅ¿pe-<lb/>culo Å¿phærico concauo.</s> - </p> - </div> - <div type="letter" level="4" n="4"> - <head style="it" xml:space="preserve">Speculatio cuiuÅ¿dam propoÅ¿itionis aritbmetica.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">SPeculatio vltimæ propoÅ¿itionis quam numerorum via inueni, hæc eÅ¿t. </s> - <s xml:space="preserve">Imagi-<lb/>nemur triangulum <var>.r.e.o.</var> abÅ¿ciÅ¿um à circulo, in cuius circunferentia Å¿it punctum <lb/>r. Å¿uperioris anguli ipÅ¿ius trianguli, vel etiam non Å¿it abÅ¿ciÅ¿um dummodo protrahan <lb/>tur lineæ <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> ad circunferentiam, à quo ad oppoÅ¿itum latus <reg norm="defcendant" type="context">defcẽdant</reg> duæ <var>.r.K.</var> et <var>.r.<lb/>f.</var> ita <reg norm="quod" type="simple">ê™</reg>. <var>K.o.</var> æqualis Å¿it <var>.f.e.</var> vnde hæc .4. lineæ Å¿ecabuntur à circulo dicto in punctis <var>.n.<lb/>c.b.u</var>. </s> - <s xml:space="preserve">Dico nunc <reg norm="quod" type="simple">ê™</reg> producta <var>.o.r.n.</var> et <var>.e.r.u.</var> æqualia erunt productis <var>.K.r.c.</var> et <var>.f.r.b.</var> <lb/> - <anchor type="figure" xlink:label="fig-0351-03a" xlink:href="fig-0351-03"/> - <anchor type="figure" xlink:label="fig-0351-04a" xlink:href="fig-0351-04"/> - <pb o="340" rhead="IO. BAPT. BENED." n="352" file="0352" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0352"/> - quapropter cogitemus <var>.r.a.</var> indeterminatam tranÅ¿ire per centrum <var>.s.</var> ipÅ¿ius circuli, Å¿i-<lb/>militer etiam <var>.r.i.</var> ad punctum medium lateris <var>.e.o.</var> deinde à tribus punctis, <var>e.i.o.</var> ima-<lb/>ginemur tres perpendiculares ad <var>.r.a.</var> hoc eÅ¿t <var>.e.a</var>: <var>i.d.</var> et <var>.o.q.</var> & vbi circulus Å¿ecat <var>.r.a.</var> <lb/>fit punctum <var>.g.</var> protractis deinde <var>.g.n</var>: <var>g.x</var>: et <var>.g.u.</var> habebimus triangulum <var>.a.e.r.</var> Å¿imi-<lb/>lem triangulo <var>.g.u.r.</var> vnde clarum erit productum <var>.g.r.a.</var> æquale eſſe producto <var>.e.r.u.</var> <lb/><reg norm="productumque" type="simple">productumqÌ</reg> <var>.g.r.q.</var> æquale eſſe producto <var>.o.r.n.</var> nam trianguli <var>.g.r.n.</var> et <var>.o.r.q.</var> Å¿unt in-<lb/>u<unsure/>icem Å¿imiles, Å¿ed productum <var>.g.r.a.</var> Å¿imul cum producto <var>.g.r.q.</var> duplum eÅ¿t producto <var>.<lb/>g.r.d.</var> ex prima Å¿exti, eo quod <var>.a.r.q.</var> dupla eÅ¿t <var>.d.r.</var> & ideo productum <var>.e.r.u.</var> Å¿imul <reg norm="cum" type="context">cÅ©</reg> <lb/>producto <var>.o.r.n.</var> duplum erit producto <var>.i.r.x.</var> quod quidem æquale eÅ¿t producto <var>.g.r.<lb/>d.</var> ex Å¿imilibus rationibus iam Å¿upradictis. </s> - <s xml:space="preserve">Nunc ex Å¿imilibus rationibus producta <var>.f.<lb/>r.b.</var> et <var>.K.r.c.</var> dupla erunt producto <var>.i.r.x</var>. </s> - <s xml:space="preserve">quare prima producta æqualia erunt Å¿ecun-<lb/>dis. </s> - <s xml:space="preserve">Quod eÅ¿t propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0351-03" xlink:href="fig-0351-03a"> - <image file="0351-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0351-03"/> - </figure> - <figure xlink:label="fig-0351-04" xlink:href="fig-0351-04a"> - <image file="0351-04" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0351-04"/> - </figure> - </div> - <p> - <s xml:space="preserve">Ab huiuÅ¿modi demonÅ¿trat ione facilè videre poteris non eſſe generaliter verum, <lb/>id quod Nicolaus Tartalea inquit .43. quæſito vltimæ partis Å¿uorum tractatuum, hoc <lb/>eÅ¿t centrum circuli <var>.r.n.g.</var> Å¿emper eſſe in perpendiculari, quæ à puncto <var>.r.</var> ad lineam <var>.e.<lb/>o.</var> tranÅ¿it, protracta ipÅ¿a <var>.e.o.</var> quantum volueris, imò in quacunque alia linea ipÅ¿um eÅ¿ <lb/>Å¿e poteÅ¿t, nec non in aliqua parallela ipÅ¿i <var>.e.o.</var> quemadmodum ex te ipÅ¿o, medianti-<lb/>bus, hic Å¿upradictis rationibus videre poteris, vnde ex neceſſitate Å¿equitur illud pro <lb/>blema Å¿emper ferè falÅ¿um eſſe.</s> - </p> - <figure position="here"> - <image file="0352-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0352-01"/> - </figure> - <figure position="here"> - <image file="0352-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0352-02"/> - </figure> - </div> - <div type="letter" level="4" n="5"> - <head style="it" xml:space="preserve">Alia Å¿peculatio circa breuitatem radiorum incidentium <lb/>& reflexorum.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">ALius modus quem exercitationis gratia vltimò cogitaui, ad demonÅ¿trandum <lb/>breuitatem radiorum incidentium, & reflexorum in Å¿peculo plano, nunc ad <lb/>te Å¿cribo, quamuis prolixior ali quantulum Å¿it eo, quod ab antiquis traditus eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Imaginemur itaque <reg norm="lineam" type="context">lineã</reg> <var>.p.h.</var> pro <reg norm="communi" type="context">cõmuni</reg> Å¿ectione Å¿uperficiei reflexionis <reg norm="cum" type="context">cÅ©</reg> Å¿pe-<lb/>culo <var>.r.a.</var> verò et <var>.a.b.</var> pro radijs dictis, qui Å¿emper <reg norm="faciunt" type="context">faciÅ©t</reg> angulos <var>.b.a.h.</var> et <var>.r.a.p.</var> <reg norm="inuicem" type="context">inuicẽ</reg> - <pb o="341" rhead="EPISTOL AE." n="353" file="0353" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0353"/> - æquales. </s> - <s xml:space="preserve">Nunc protrahantur duæ <var>.r.o.</var> et <var>.b.o.</var> ab iiÅ¿dem punctis <var>.b.r.</var> ad aliud punctum, <lb/>quod volueris ipÅ¿ius lineæ <var>.p.h.</var> quas probabo <reg norm="longiores" type="conjecture">longiores'</reg> (Å¿imul Å¿umptas) eſſe priori-<lb/>bus. </s> - <s xml:space="preserve">Imaginemur igitur duas perpendiculares, Å¿eu cathetos <var>.b.i.</var> et <var>.q.r.a.</var> punctis <var>.b.<lb/>r.</var> ad <var>.p.h.</var> <reg norm="abÅ¿ciſſaque" type="simple">abÅ¿ciſſaqÌ</reg> Å¿it linea <var>.o.b.</var> in puncto <var>.x.</var> ita quod <var>.b.x.</var> æqualis Å¿it ipÅ¿i <var>.b.a.</var> quod <lb/>nulli dubium erit poſſe effici, cum <var>.o.b.</var> <reg norm="longiot" type="context">lõgiot</reg> Å¿it <var>.b.a.</var> co quod opponatur angulo ob-<lb/>tuÅ¿o ipÅ¿ius trianguli <var>.b.a.o.</var> quÄ™ <var>.o.b.</var> Å¿imiliter protrahatur vÅ¿que ad <var>.d.</var> ita quod <var>.b.d.</var> <lb/>æqualis Å¿it <var>.x.b.</var> </s> - <s xml:space="preserve">deinde protrahatur <var>.o.i.</var> quouÅ¿que <var>.i.h.</var> æqualis Å¿it <var>.a.i</var>. </s> - <s xml:space="preserve">In alia parte po-<lb/>Å¿tea idem faciendum eÅ¿t Å¿ecando <var>.a.r.</var> in puncto <var>.u.</var> ita quod <var>.u.r.</var> æqualis Å¿it <var>.r.o.</var> efficien <lb/>do <var>.r.s.</var> æqualem <var>.r.u.</var> et <var>.q.p.</var> æquale <var>.q.o.</var> vnde habebimus <reg norm="productum" type="context">productÅ©</reg> <var>.o.d.</var> in <var>.o.x.</var> æqua <lb/>le producto <var>.o.h.</var> in <var>.o.a.</var> & productum <var>.a.s.</var> in <var>.a.u.</var> æquale producto <var>.a.p.</var> in <var>.a.o.</var> exiÅ¿tis <lb/>rationibus. </s> - <s xml:space="preserve">Nam cum quadratum ipÅ¿ius <var>.o.b.</var> æquale Å¿it duobus quadratis <var>.o.i.</var> et <var>.i.<lb/>b.</var> ex penultima primi Eucli. ipÅ¿a quadrata <var>.o.i.</var> et <var>.i.b.</var> æqualia erunt producto <var>.o.d.</var> in <lb/><var>o.x.</var> Å¿imul Å¿umpto cum quadrato <var>.b.x.</var> ex .6. Å¿ecundi, hoc eÅ¿t ipÅ¿i producto Å¿imul Å¿um-<lb/>pto cum quadrato <var>.b.a.</var> hoc eÅ¿t ipÅ¿i producto Å¿imul Å¿umpto cum duobus quadratis <var>.a.<lb/>i.</var> et <var>.i.b.</var> Å¿ed quia productum <var>.o.h.</var> in <var>.o.a.</var> Å¿imul Å¿umpto cum quadrato <var>.a.i.</var> Ä™quatur qua <lb/>drato <var>.o.i.</var> ideo productum <var>.o.h.</var> in <var>.o.a.</var> Å¿imul Å¿umptum cum quadrato <var>.a.i.</var> & cum qua-<lb/>drato <var>.i.b.</var> æquale erit producto <var>.o.d.</var> in <var>.o.x.</var> Å¿imul Å¿umpto <reg norm="cum" type="context">cÅ©</reg> duobus quadratis dictis <lb/>hoc eÅ¿t ipÅ¿ius <var>.a.i.</var> et <var>.i.b.</var> quÄ™ quadrata dempta cum fuerint ab vtraque parte, tunc cer <lb/>ti erimus producta eſſe inuicem æqualia. </s> - <s xml:space="preserve">Idem dico de alijs ex altera parte. </s> - <s xml:space="preserve">Nunc <lb/>imaginemur protractam eſſc <var>.a.e.</var> parallelam ipÅ¿i <var>.o.b.</var> & habebimus proportionem <lb/>ipÅ¿ius <var>.a.b.</var> ad <var>.a.i.</var> maiorem eſſe ea quæ eÅ¿t ipÅ¿ius <var>.a.e.</var> ad eandem <var>.a.i.</var> cum <var>.a.b.</var> maior <lb/>Å¿it ipÅ¿a <var>.a.e.</var> vt oppoÅ¿ita angulo obtuÅ¿o, quapropter proportio <var>.x.b.</var> ad <var>.a.i.</var> maior erit <lb/>ea quæ eÅ¿t <var>.o.b.</var> ad <var>.o.i</var>. </s> - <s xml:space="preserve">Iam enim Å¿cis proportionem <var>.o.b.</var> ad <var>.o.i.</var> eſſe, vt <var>.a.e.</var> ad <var>.a.i.</var> ex <lb/>Å¿imilitudine triangulorum. </s> - <s xml:space="preserve">quare proportio <var>.b.d.</var> ad <var>.i.h.</var> maior erit proportione <var>.o.b.</var> <lb/>ad <var>.o.i.</var> <reg norm="tunc" type="context">tÅ©c</reg> ex .27. quinti <reg norm="permutando" type="simple context">ê‘mutãdo</reg> <reg norm="proportio" type="simple">ê“portio</reg> <var>.b.d.</var> ad <var>.b.o.</var> maior erit proportione <var>.i.h.</var> <lb/>ad <var>.i.o.</var> & ex .26. <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> <reg norm="componendo" type="context context">cõponẽdo</reg> maior <reg norm="proportio" type="simple">ê“portio</reg> erit <var>.o.d.</var> ad <var>.o.b.</var> ea quÄ™ eÅ¿t <var>.o.h.</var> ad. o <lb/>i. & <reg norm="permutando" type="context">permutãdo</reg> maior ipÅ¿ius <var>.o.d.</var> ad <var>.o.h.</var> ea quæ <var>.o.b.</var> ad <var>.o.i.</var> & ex .33. maior ipÅ¿ius <var>.b.<lb/>d.</var> ad <var>.i.h.</var> ea quæ <var>.o.d.</var> ad <var>.o.h</var>. </s> - <s xml:space="preserve">Sed vt <var>.b.a.</var> ad <var>.a.i.</var> ita eÅ¿t <var>.a.r.</var> ad <var>.a.q.</var> ex Å¿imilitudine <reg norm="triam" type="context">triã</reg> <lb/>gulorum. </s> - <s xml:space="preserve">Erit igitur <var>.a.r.</var> ad <var>.a.q.</var> maior proportio, ea quæ eÅ¿t <var>.o.b.</var> ad <var>.o.i.</var> & exijÅ¿dem <lb/>Å¿upradictis rationibus maior erit proportio ipÅ¿ius <var>.s.a.</var> ad <var>.p.a.</var> ea quæ eÅ¿t <var>.a.r.</var> ad <var>.a.q.</var> <lb/>Å¿ed cum iam probatum fuit proportio <lb/> - <anchor type="figure" xlink:label="fig-0353-01a" xlink:href="fig-0353-01"/> - nem <var>.b.d.</var> ad <var>.i.h.</var> hoc eÅ¿t <var>.a.b.</var> ad <var>.a.i.</var> ma <lb/>iorem eſſe <var>.o.d.</var> ad <var>.o.h.</var> ergo eo ma-<lb/>gis maior erit proportio ipÅ¿ius <var>.a.s.</var> ad <lb/><var>a.p.</var> ca quæ <var>.o.d.</var> ad <var>.o.h.</var> Å¿ed cum ex .15 <lb/>Å¿exti, eadem Å¿it proportio <var>.o.d.</var> ad <var>.o.a.</var> <lb/>quæ <var>.o.h.</var> ad <var>.o.x.</var> et <var>.s.a.</var> ad <var>.o.a.</var> quÄ™ <var>a.p.</var> <lb/>ad <var>.a.u.</var> </s> - <s xml:space="preserve">tunc erit <reg norm="permutando" type="context">permutãdo</reg> eadem <lb/>proportio ipÅ¿ius <var>.o.d.</var> ad <var>.o.h.</var> quæ <var>.o.a.</var> <lb/>ad <var>.o.x.</var> & ipÅ¿ius <var>.a.o.</var> ad <var>.a.u.</var> quemad-<lb/>modum ipÅ¿ius <var>.a.s.</var> ad <var>.a.p</var>. </s> - <s xml:space="preserve">Quare maior proportio erit ipÅ¿ius <var>.a.o.</var> ad <var>.a.u.</var> quam <var>.a.</var> o<unsure/>. <lb/>ad <var>.o.x</var>. </s> - <s xml:space="preserve">Vnde Å¿equitur <var>.o.x.</var> maiorem eſſe <var>.a.u.</var> ex .8. quinti, ergo <var>.b.x.o.r.</var> longior erit <lb/>ipÅ¿a <var>.b.a.u.r</var>. </s> - <s xml:space="preserve">Quod eÅ¿t propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0353-01" xlink:href="fig-0353-01a"> - <image file="0353-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0353-01"/> - </figure> - </div> - <pb o="342" rhead="IO. BAPT. BENED." n="354" file="0354" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0354"/> - <p> - <s xml:space="preserve">Alia etiam via poſſumus idem concludere. </s> - <s xml:space="preserve">Imaginemur maiorem axem alicu-<lb/>ius ellipÅ¿is tranÅ¿ire per duo puncta <var>.r.</var> et <var>.b.</var> Å¿upponendo ipÅ¿a puncta, ea eÅ¿le, quæ ita <lb/>axem diuidunt, vt Å¿ingula produ-<lb/> - <anchor type="figure" xlink:label="fig-0354-01a" xlink:href="fig-0354-01"/> - cta fectionum Å¿int, vt inquit Per-<lb/>geus. </s> - <s xml:space="preserve">imaginemur, etiam <var>.p.h.</var> con <lb/>tiguam eſſe ipÅ¿i ellipÅ¿i in <reg norm="puncto" type="context">pÅ©cto</reg> <var>.a.</var> <lb/>vnde Å¿i protractæ fuerint duæ <var>.r.a.</var> <lb/>et <var>.b.a.</var> habebimus ex .48. tertijip-<lb/>Å¿ius Pergei angulos <var>.b.a.h.</var> et <var>.r.a.<lb/>p.</var> inuicem æquales. </s> - <s xml:space="preserve">Ducendo <lb/>poÅ¿tea ad quoduis punctum ipÅ¿ius <lb/><var>p.h.</var> duas <var>.b.o.</var> et <var>.r.o.</var> certi erimus, <lb/>quod Å¿ecabuntur à gyro oxygo-<lb/>nio, quarum vna Å¿ecta Å¿it in pun-<lb/>cto <var>.i.</var> ducta poÅ¿tea <var>.i.r.</var> clarum erit ex .52. dicti, quod longitudo <var>.b.i.r.</var> æqualis erit lon <lb/>gitudini <var>.b.a.r.</var> & minor ipÅ¿a <var>.b.o.r.</var> ex .21. primi Euclid.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0354-01" xlink:href="fig-0354-01a"> - <image file="0354-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0354-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="6"> - <head style="it" xml:space="preserve">Deerrore Euclidis circa Å¿peculum vstorium.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">VErum Å¿peculum vÅ¿torium, illud non eÅ¿t, quod ab Euclide traditum fuit, & <reg norm="quod" type="simple">ê™</reg> <lb/>tu etiam putas, Nam Euclides errat, cum credat radios reflexos à ſuperficie <lb/>Å¿phærica concaua Å¿einuicem in centro Å¿peculi interÅ¿ecare. </s> - <s xml:space="preserve">Nam cum omnes lineÄ™ <lb/>recte à centro, & cir cunferentia alicuius Å¿phæræ terminatæ, Å¿int eidem circunferen-<lb/>tiæ perpendiculares, Å¿equeretur ex neceſſitate radios incidentiæ etiam perpendicu <lb/>lares eidem Å¿uperficiei eſſe, cum anguli incidentiæ Å¿emper æquales Å¿int angulis re-<lb/>flexionis, vnde etiam ex neceſſitate Å¿equeretur punctum corporis lucidi, à quo radij <lb/>luminoÅ¿i excunt, in centro Å¿peculi reperiri. </s> - <s xml:space="preserve">quod quidem falÅ¿iſſimum eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Alia etiam via poſſum hanc oÅ¿tendere impoſſibilitatem, & tibi probabo, quod <lb/>in nullo aliquo puncto poſſunt inuicem conuenire ipÅ¿i radijrefle xi omnes.</s> - </p> - <p> - <s xml:space="preserve">Sit igitur <var>.l.a.c.</var> <reg norm="conis" type="context">cõis</reg> Å¿ectio Å¿uperficiei reflexionis cum Å¿peculo, cuius centrum Å¿it <var>.o.</var> <lb/>punctum verò lucidum Å¿it <var>.g.</var> <reg norm="protrahaturque" type="simple">protrahaturqÌ</reg> <var>.g.o.a</var>. </s> - <s xml:space="preserve">Nunc autem primum dico, quod <lb/>radij reflexi à punctis diuerÅ¿arum <reg norm="diÅ¿tantiarum" type="context">diÅ¿tantiarÅ©</reg> ab <var>.a.</var> non <reg norm="coincident" type="context">coincidẽt</reg> inuicem in aliquo <lb/>puncto lineÄ™ <var>.g.o.a</var>: Å¿int ergo duo puncta <var>.u.</var> et <var>.r.</var> diuerÅ¿arum <reg norm="diÅ¿tantiarum" type="context">diÅ¿tantiarÅ©</reg> ab <var>.a.</var> à quibus <lb/>veniant duo radij incidentiæ <var>.g.r.</var> et <var>.g.u.</var> radius verò reflexus ab <var>.r.</var> Å¿it <var>.r.e.</var> protrahatur <lb/><var>u.e.</var> quam dico effe non poſſe radium reflexum ab <var>.u.</var> quotieÅ¿cunque eius incidens <lb/>deÅ¿cendat ab <var>.g</var>. </s> - <s xml:space="preserve">Protrahantur ergo duæ lineæ <var>.o.r.</var> et <var>.o.u.</var> vnde cum dixerit aliquis <lb/><var>u.e.</var> <reg norm="reflexum" type="context">reflexÅ©</reg> eſſe ipÅ¿ius <var>.g.u.</var> igitur anguli <var>.g.u.o.</var> et <var>.o.u.e.</var> erunt inuicem æquales, & Å¿ic <lb/>etiam erunt duo <var>.g.r.o.</var> et <var>.o.r.e.</var> vnde ex tertia Å¿exti & .11. quinti Eucli. proportio <var>.g.<lb/>u.</var> ad <var>.u.e.</var> æqualis eſſet ei, quæ <var>.g.r.</var> ad <var>.r.e.</var> quod quidem impoſſibile eſſe demonÅ¿tra-<lb/>bo, eo quod cum <var>.g.u.</var> maior Å¿it <var>.g.r.</var> ex .8. tertij, erit ex .8. quinti proportio ipÅ¿ius <var>.g.u.</var> <lb/>ad <var>.r.e.</var> maior proportione ipÅ¿ius <var>.g.r.</var> ad <var>.r.e.</var> Å¿ed ex .7. tertij <var>.u.e.</var> minor eÅ¿t <var>.r.e.</var> erit igi-<lb/>tur ex dicta .8. quinti maior proportio <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <var>.g.u.</var> ad <var>.u.e.</var> quam <var>.g.u.</var> ad <var>.r.e.</var> vnde eo ma - <pb o="343" rhead="EPISTOL AE." n="355" file="0355" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0355"/> - gis erit maior proportio ipÅ¿ius <var>.g.u.</var> ad <var>.u.e.</var> <lb/> - <anchor type="figure" xlink:label="fig-0355-01a" xlink:href="fig-0355-01"/> - quam ipÅ¿ius <var>.g.r.</var> ad <var>.r.e.</var> ergo non æqualis, <lb/>quapropter impoſſibile eÅ¿t <var>.u.e.</var> eſſe radium <lb/>reflexum incidentis radij <var>.g.u</var>. </s> - <s xml:space="preserve">Vnde Å¿equi <lb/>tur concurÅ¿um radiorum reflexorum à ſpe-<lb/>culo Å¿phærico concauo non eſſe in vno, & <lb/>e<unsure/>odem puncto ipÅ¿ius catheti incidentiæ, <lb/>quando à ſitu non æquidiÅ¿tanti ab ipÅ¿o ca-<lb/>theto <reg norm="reflectuntur" type="context">reflectÅ©tur</reg>, ex hac Å¿peculatione <reg norm="etiam" type="context">etiã</reg> <lb/>videre licet, verum eſſe id quod in .3. EpiÅ¿to <lb/>la tibi Å¿cripÅ¿i nempe, quod quotieÅ¿cunque <lb/>axes viÅ¿uales, vel radij reflexi, in vna ea-<lb/><reg norm="demque" type="simple">demqÌ</reg> Å¿uperficie reflexionis fuerint, </s> - <s xml:space="preserve">tunc <lb/>imago obiecti nullo modo videbitur in ca-<lb/>theto incidentiæ, in Å¿peculo Å¿phærico con-<lb/>cauo.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0355-01" xlink:href="fig-0355-01a"> - <image file="0355-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0355-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="7"> - <head style="it" xml:space="preserve">Alterius dubit ationis Å¿olutio.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">NOn abÅ¿que ratione dubitas, vtrum etiam in Å¿phæricis Å¿peculis conuexis idem <lb/>accidat, hoc eÅ¿t, an radij reflexi à punctis inÄ™qualis diÅ¿tantiæ à catheto inciden <lb/>tiæ conueniant inuicem in eodem catheto.</s> - </p> - <p> - <s xml:space="preserve">Ad quod reÅ¿pondeo, non concurrere in dicto catheto, Å¿ed extra ipÅ¿um, & Å¿imi-<lb/>liter extra ipÅ¿um vide bitur imago.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius rei ratione, imaginemur Å¿uperficiem reflexionis alicuius Å¿peculi Å¿phæ-<lb/>rici conuexi <var>.b.d.h.g.</var> cuius communis Å¿ectio cum Å¿uperficie Å¿phærica Å¿it linea <lb/>circularis <var>.d.e.h.</var> et <var>.o.</var> eius <reg norm="centrum" type="context">cẽtrum</reg>, à quo protrahatur <var>.g.b.</var> indeterminata, et <var>.o.g.</var> Å¿it Å¿e <lb/>midiameter circuli <var>.d.g.h.</var> et <var>.o.c.</var> Å¿it plus medietate ipÅ¿ius <var>.o.g.</var> <reg norm="accipiaturque" type="simple">accipiaturqÌ;</reg> linea <var>.e.c.</var> <lb/>minor ipÅ¿a <var>.o.c.</var> Å¿ed maior ipÅ¿a <var>.c.g.</var> <lb/> - <anchor type="figure" xlink:label="fig-0355-02a" xlink:href="fig-0355-02"/> - quod difficile non erit, locando im <lb/>mobilem pedem circini in puncto <var>.<lb/>c.</var> aperiendo ipÅ¿um aliquantulum <lb/>plus quam <var>.c.g.</var> Å¿ed minus quam <var>.c.<lb/>o.</var> Å¿ignando circunferentiam <var>.d.e.h.</var> <lb/>in puncto <var>.e.</var> quod ex .7. tertij poſſi-<lb/>bile eÅ¿t, protrahatur poÅ¿tea <var>.o.e.f.</var> <lb/>indeterminatè. </s> - <s xml:space="preserve">Facicmus deinde <lb/>angulum <var>.f.e.b.</var> æqualem angulo <var>.o.<lb/>e.c.</var> protracta poÅ¿tea cum fuerit <var>.c.<lb/>e.K.</var> indeterminatè, <reg norm="habebimus" type="simple">habebimê°</reg> duos <lb/>angulos <var>.b.e.f.</var> et <var>.f.e.K.</var> æquales in-<lb/>uicem mediante .15. primi, ita <reg norm="quod" type="wordlist">qÄ‘</reg> Å¿i <lb/>radius incidens veniet à puncto <var>.b.</var> <lb/>ad <var>.e.</var> reflexus erit <var>.e.K.</var> qui quidem - <pb o="344" rhead="IO. BAPT. BENED." n="356" file="0356" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0356"/> - refleyus Å¿ecabit cathetum <var>.b.o.</var> in puncto <var>.c.</var> intra Å¿peculum, nec dubitandum eÅ¿t quin <lb/>linea <var>.e.b.</var> Å¿ectura Å¿it <var>.b.o.</var> eo quod cum angulus <var>.o.e.c.</var> Å¿it maior angulo <var>.e.o.c.</var> ex .19. <lb/>primi, & Å¿imiliter angulus <var>.b.e.f.</var> Å¿equitur ex .13. dicti, angulos <var>.b.e.o.</var> et <var>.e.o.b.</var> eſſe mi <lb/>nores duobus rectis, vnde ex penultima petitione primi, duæ lineæ <var>.b.e.</var> et <var>.o.b.</var> <reg norm="inuicem" type="context">inuicẽ</reg> <lb/>concurrent. </s> - <s xml:space="preserve">Quare poſſumus ex hoc, quoddam corollarium extrahere, hoc eÅ¿t <lb/><reg norm="neceſſarium" type="context">neceſſariÅ©</reg> <reg norm="semper" type="context">sẽper</reg> exiÅ¿tat, vt linea <var>.c.e.</var> minor eſſe linea <var>.c.o</var>. </s> - <s xml:space="preserve">Sed vnde eueniat quod ip <lb/>Å¿a neceſſariò debeat Å¿emper maior eſſe ipÅ¿a <var>.c.g.</var> clarum eÅ¿t ex .7. tertij Eucli. </s> - <s xml:space="preserve">Nunc <lb/>imaginemur ductas eſſe duas <reg norm="tangentes" type="context">tãgentes</reg> <var>.b.d.</var> et <var>.b.h.</var> & ab <var>.e.</var> <reg norm="ipsam" type="context">ipsã</reg> <var>.e.i.</var> vnde certi erimus, <lb/>quod ab interuallo inter <var>.h.</var> et <var>.d.</var> punctum <var>.b.</var> <reg norm="ponſſibile" type="context">põſſibile</reg> Å¿it vt reflectatur. </s> - <s xml:space="preserve">Accipiamus <lb/>nunc <var>.p.c.</var> minorem medietate ipÅ¿ius <var>.b.c.</var> & à puncto <var>.p.</var> imaginemur tangentem <var>.p.q.</var> <lb/>in puncto <var>.q.</var> prorractaqÌue Å¿it <var>.b.q.</var> vt radius incidentiæ, </s> - <s xml:space="preserve">tunc dico, radium reflexum <lb/>ipÅ¿ius <var>.b.q.</var> <reg norm="non" type="context">nõ</reg> concurrere in eodem puncto <var>.c.</var> ipÅ¿ius catheti, Å¿i vero dixeris <reg norm="quod" type="simple">ê™</reg> Å¿ic. </s> - <s xml:space="preserve">EÅ¿to <lb/><reg norm="igitur" type="simple">igit̃</reg> radius dictus <var>.c.q.s</var>. </s> - <s xml:space="preserve">Imaginemur <reg norm="tangentem" type="context context">tãgentẽ</reg> <var>.e.i.</var> in puncto <var>.e.</var> vnde ex .18. quinti Alha <lb/>zem, vel .12. Å¿exti Vitellionis proportio <var>.b.i.</var> ad <var>.i.c.</var> erit, vt <var>.b.o.</var> ad <var>.o.c.</var> & Å¿imiliter erit <lb/>ipÅ¿ius <var>.b.p.</var> ad <var>.p.c.</var> vt <var>.b.o.</var> ad <var>.o.c.</var> ex eadem. </s> - <s xml:space="preserve">Quare ex .11. quinti Eucli. proportio ip <lb/>Å¿ius <var>.b.p.</var> ad <var>.p.c.</var> erit vt ipÅ¿ius <var>.b.i.</var> ad <var>.i.c.</var> Å¿ed quia <var>.p.b.</var> vt pars ipÅ¿ius <var>.b.i.</var> minor eÅ¿t ip-<lb/>Å¿a, ergo ex .14. dicti <var>.p.c.</var> minor erit ipÅ¿a <var>.c.i.</var> hoc eÅ¿t totum minus Å¿ua parte, quod eÅ¿t <lb/>impoſſibile, </s> - <s xml:space="preserve">quare non in ipÅ¿o catheto videbitur imago ipÅ¿ius obiecti.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0355-02" xlink:href="fig-0355-02a"> - <image file="0355-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0355-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Aliud notandum etiam cernere potes ex ipÅ¿is Å¿peculis Å¿phæricis conuexis, hoc eÅ¿t <lb/>quod poſſibile Å¿it aliquoties, radium reflexum concurrere cum catheto incidentiæ <lb/>extra Å¿peculum inter puncta <var>.g.</var> et <var>.p.</var> vt exempli gratia .Å¿i punctus <var>.p.</var> eſſet exactè <lb/>in medio inter <var>.b.</var> et g. </s> - <s xml:space="preserve">tunc punctum <var>.c.</var> ipÅ¿ius concurÅ¿us cum catheto incidentiæ eſſet <lb/>inter <var>.g.</var> et <var>.p.</var> eo quod <reg norm="cum" type="context">cÅ©</reg> linea <var>.p.q.</var> debeat @iui lere <reg norm="angulum" type="context">angulÅ©</reg> <var>.b.</var> q, c. <reg norm="per" type="simple">ê‘</reg> Ä™qualia, oportebit <lb/>c. poÅ¿itum eſſe inter <var>.g.</var> et <var>.p.</var> quia angulus <var>.g.q.p.</var> maior eÅ¿t angulo <var>.p.q.b.</var> vt per te faci <lb/>le potes ratiotinari, imaginando cir <lb/> - <anchor type="figure" xlink:label="fig-0356-01a" xlink:href="fig-0356-01"/> - culum circa <reg norm="triangulum" type="context">triãgulum</reg> <var>.g.q.b.</var> & dia <lb/>merrum perpendicularem .ad <var>.g.b.</var> <lb/>in puncto <var>.p.</var> producendo poÅ¿tea <var>.q.<lb/>p.</var> <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> ad <reg norm="alteram" type="context">alterã</reg> <reg norm="partem" type="context">partẽ</reg> circunferen-<lb/>tiæ ipÅ¿ius circuli. </s> - <s xml:space="preserve"><reg norm="argumentando" type="context context">argumẽtãdo</reg> dein-<lb/>de mediante vltima Å¿exti, illud <reg norm="idem" type="context">idẽ</reg> <lb/>po@es etiam Å¿cire ex .22. quinti Alha <lb/>zeni. & ex .26. Å¿exti Vitellionis. </s> - <s xml:space="preserve">vn-<lb/>de Å¿i ad ambas pupillas venerint ra <lb/>dij reflexi ipÅ¿ius obiecti <var>.b.a.</var> duobus <lb/>punctis huiuÅ¿modi Å¿peculi, ita di-<lb/>Å¿tantibus à puncto <var>.g.</var> vt <var>.q</var>. </s> - <s xml:space="preserve">tunc com <lb/>mune punctum concurÅ¿us axium vi <lb/>Å¿ualium erit in catheto inter <var>.g.p.</var> <lb/>vbi apparebit imago ex Å¿uperius di <lb/>ctis rationibus, ita vt <reg norm="non" type="context">nõ</reg> Å¿olum con <lb/>cauis, Å¿ed etiam conuexis hoc accidere poſſit.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0356-01" xlink:href="fig-0356-01a"> - <image file="0356-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0356-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">In planis autem <reg norm="nunquam" type="context">nunquã</reg> hoc poteÅ¿t euenire, vt tibi alias dixi, eo quod Å¿i <reg norm="acceperi- mus" type="conjecture">accéperi-<lb/>mus</reg> <reg norm="rectam" type="context">rectã</reg> <var>.m.r.</var> pro <reg norm="coni" type="context">cõi</reg> Å¿ectione <reg norm="Å¿uperficiei" type="simple">Å¿uê‘ficiei</reg> <var>.l.t.x.</var> reflexionis & <reg norm="Å¿uperficiei" type="simple">Å¿uê‘ficiei</reg> Å¿peculi, <reg norm="punctumque" type="context context simple">pÅ©ctÅ©qÌ;</reg> <lb/>lucidum <var>.l.</var> <reg norm="protractoque" type="simple">protractoqÌ;</reg> catheto <var>.l.r.t.</var> <reg norm="lineisque" type="simple">lineisqÌ;</reg> incidentiæ <var>.l.x.</var> et <var>.l.m.</var> reflexionis etiam <lb/><var>x.y.</var> et <var>.m.z.</var> cum anguli <var>.l.x.r.</var> et <var>.y.x.h.</var> et <var>.r.x.t.</var> æquales inuicem Å¿int, & Å¿ic anguli <var>.l.m.<lb/>r.</var> et <var>.z.m.h.</var> et <var>.r.m.t.</var> erit <var>.r.t.</var> tam pro triangulo <var>.r.x.t.</var> quam pro triangulo <var>.r.m.t.</var> æqua <lb/>lis <var>.r.l.</var> ex .26. primi, ita quod Å¿emper in puncto <var>.t.</var> <reg norm="conuenient" type="context">conueniẽt</reg> omnes radij reflexi ipÅ¿ius - <pb o="345" rhead="EPISTOL AE." n="357" file="0357" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0357"/> - puncti <var>.l.</var> clarum igitur nunc habes, quod in Å¿phærico concauo, Å¿eu conuexo, non <lb/>omnes radij reflexi conueniunt in vno, <reg norm="eodemque" type="simple">eodemqÌ;</reg> puncto catheti incidentiæ, quemad <lb/>modum in planis accidit, in quibus Å¿emper vnum, & idem punctum eÅ¿t ipÅ¿is commu <lb/>ne in ipÅ¿o incidentiæ catheto.</s> - </p> - <p> - <s xml:space="preserve">Non prætermittam etiam hunc alium breuiorem modum Å¿peculandi <reg norm="æqualita- tem" type="context">æqualita-<lb/>tẽ</reg> depreſſionis imaginis Å¿ub Å¿peculo plano, ei quæ Å¿upra reperitur ipÅ¿ius obiecti, in ca <lb/>theto incidentiæ, quemadmodum nu nc <lb/>vltimò diximus, hoc eÅ¿t quod cum <lb/> - <anchor type="figure" xlink:label="fig-0357-01a" xlink:href="fig-0357-01"/> - imago obiecti <var>.l.</var> reflexa à puncto <var>.<lb/>x.</var> reperiatur in linea <var>.y.x.t.</var> & ima-<lb/>go eiuÅ¿dem obiecti reflexa à pun-<lb/>cto <var>.m.</var> reperiatur in linea <var>.z.m.t.</var> & <lb/>iÅ¿tæ duæ lineÄ™ Å¿einuicem Å¿ecent in <lb/>puncto <var>.t.</var> ipÅ¿ius catheti, exiÅ¿tente <var>.<lb/>r.t.</var> æquali <var>.r.l.</var> vt nunc vidimus, er-<lb/>go Å¿emper imago reflexa à ſpecu-<lb/>lo plano, nobis apparebit <reg norm="in" type="wordlist">Ä©</reg> ipÅ¿o ca <lb/>theto, tam vltra Å¿peculum, quam ci <lb/>tra ipÅ¿um, <reg norm="repertum" type="simple context">reê‘tÅ©</reg> fuerit <reg norm="ipsum" type="context">ipsÅ©</reg> <reg norm="obiectum" type="context">obiectÅ©</reg> <lb/>quod nec Alhazem, nec Vitellio, <lb/>nec alius aliquis (quod Å¿ciam) ad huc Å¿cientificè demonÅ¿trauit. </s> - <s xml:space="preserve">exempla enim vel ex <lb/>perientia non faciunt Å¿cire. </s> - <s xml:space="preserve">Credo etiam te non dubitare quin duæ lineæ <var>.y.x.</var> et <var>.z.<lb/>m.</var> inuicem concurrant, cum anguli <var>.t.x.m.</var> et <var>.t.m.x.</var> minores Å¿int duobus rectis cum <lb/>æquales Å¿int angulis <var>.l.x.m.</var> et <var>.l.m.x</var>.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0357-01" xlink:href="fig-0357-01a"> - <image file="0357-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0357-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="8"> - <head style="it" xml:space="preserve">De rotunditate vmbræterræ in ecclipÅ¿ibus Lunaribus.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">ROtunditas vmbræ in ecclipÅ¿i-<lb/> - <anchor type="figure" xlink:label="fig-0357-02a" xlink:href="fig-0357-02"/> - bus lunaribus oritur <reg norm="tam" type="context">tã</reg> à rotun <lb/>ditate maris, <reg norm="quam" type="context">quã</reg> terræ, & Å¿i terra eÅ¿-<lb/>Å¿et <reg norm="etiam" type="context">etiã</reg> cuiuÅ¿uis alterius figurÄ™, <reg norm="quam" type="context">quã</reg> <lb/>Å¿phæricÄ™, dummodo aqua impleret <lb/><reg norm="locum" type="context">locÅ©</reg> Å¿phÄ™riceitatis à terra <reg norm="derelictum" type="context">derelictÅ©</reg>, <lb/>nihilominus vmbra eſſet rotunda, <lb/>quÄ™ quidem ab aqua produceretur, <lb/><reg norm="quanuis" type="context">quãuis</reg> Alexander Piccolhomineus <lb/> - <anchor type="figure" xlink:label="fig-0357-03a" xlink:href="fig-0357-03"/> - aliter Å¿entiat in libro de magnitudi-<lb/>ne terrÄ™, & aquæ.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0357-02" xlink:href="fig-0357-02a"> - <image file="0357-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0357-02"/> - </figure> - <figure xlink:label="fig-0357-03" xlink:href="fig-0357-03a"> - <image file="0357-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0357-03"/> - </figure> - </div> - <p> - <s xml:space="preserve"><reg norm="Sciendum" type="context context">SciẽdÅ©</reg> enim eÅ¿t, quod omne cor <lb/>pus in Å¿e habens <reg norm="aliquantulum" type="context">aliquantulÅ©</reg> opaci-<lb/>tatis, Å¿emper debilitat <reg norm="radium" type="context">radiÅ©</reg> lumino <lb/>Å¿um, & <reg norm="tanto" type="context">tãto</reg> magis, <reg norm="quanto" type="context">quãto</reg> magis in ip <lb/>Å¿o corpore radius penetrat, <reg norm="etiam" type="context">etiã</reg> & Å¿i <lb/>ad rectos incideret ipÅ¿e radius Å¿upra <lb/><reg norm="Å¿uperficiem" type="context">Å¿uperficiẽ</reg> <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> corporis. </s> - <s xml:space="preserve"><reg norm="Exempli" type="context">Exẽpli</reg> gra <lb/>tia, eÅ¿to <var>.q.p.</var> corpus a <reg norm="queum" type="context">queÅ©</reg>, cuius pro <lb/>funditas diuidatur in partibus <var>.d.K</var>: <lb/><var>K.s</var>: et <var>.s.f.</var> à puncto verò lucido <var>.b.</var> - <pb o="346" rhead="IO. BABPT. BENED." n="358" file="0358" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0358"/> - deÅ¿cendat radius <var>.b.d.K.s.f.</var> ad <reg norm="libitum" type="context">libitÅ©</reg> hoc eÅ¿t rectè vel obliquè, cuius pars <var>.b.d.</var> in ipÅ¿o <lb/>aere exiÅ¿tat. </s> - <s xml:space="preserve">Nunc manifeÅ¿tum erit partem <var>.b.d.</var> ipÅ¿ius radij clariorem Å¿eu minus im <lb/><reg norm="peditam" type="context">peditã</reg> eſſe quam <var>.d.K.</var> quod ex eo etiam cognoÅ¿cere poſſumus quia <var>.b.d.</var> reflectitur à <lb/>puncto <var>.d.</var> Å¿uperficiei corporis a quei, quapropter minus luminoÅ¿a remanebit pars <var>.d.<lb/>K.</var> cum non tota claritas <var>.b.d.</var> deÅ¿cendat in corpore aqueo, Å¿ed vna eius pars reflecta-<lb/>tur, reliqua verò tantummodò deÅ¿cendat, </s> - <s xml:space="preserve">deinde pars <var>.K.s.</var> ex neceſſitate debilior <lb/>erit ipÅ¿a <var>.d.K.</var> eo quod Å¿uccedit poÅ¿t ipÅ¿am <var>.d.K.</var> propter hoc etiam, quia cum corpus <lb/>aqueum habeat aliquantulum opacitatis, radius <var>.d.K.</var> ab omni puncto ipÅ¿ius Å¿piſſitu-<lb/>dinis a quæ continuo reflectitur, quÄ™ quidem reflexio eÅ¿t illud lumen cÄ™ruleum, quod <lb/>in profunditate ipÅ¿ius aquÄ™ nobis apparet. </s> - <s xml:space="preserve">Cum igitur reflexio ipÅ¿a Å¿emper detra-<lb/>hat ab ipÅ¿o radio luminoÅ¿o, reÅ¿iduum verò Å¿it id quod penetrat, ideo <var>.K.s.</var> erit vna <lb/>pars tantummodò luminis ipÅ¿ius <var>.d.K</var>: in <var>.s.f.</var> verò aliqua pars luminis ipÅ¿ius <var>.K.s.</var> & Å¿ic <lb/>continuò debilitatur radius, ita quod ad nihilum vÅ¿que deuenit, & vltra tale cor-<lb/>pus remanebit vmbra, quaÅ¿i Å¿i ipÅ¿um corpus eſſet perfectè opacum, cuius rei cauÅ¿a, <lb/>eÅ¿t illa continua reflexio, vt diximus, quæ continuò adimit aliquid ex ipÅ¿o radio, <lb/>nec permittit eum totum tranÅ¿ire.</s> - </p> - <p> - <s xml:space="preserve">Quapropter mirandum non eÅ¿t eos, qui margaritas quærunt in fundo maris nul-<lb/>lum ibi videre lumen. </s> - <s xml:space="preserve">Nihilominus vmbra maris, quam dico nos poſſe videre in <lb/>Å¿uperficie corporis lunaris, ab alia etiam ratione prouenire poſſet. </s> - <s xml:space="preserve">Imaginemur enim <lb/>aggregatum terrÄ™, <reg norm="marisque" type="simple">marisqÌ;</reg> eſſe tantummodò aqueum, quod quidem eſſet perfectè <lb/>Å¿phæricum ratione centri grauitatis, <reg norm="Å¿upponamusque" type="simple">Å¿upponamusqÌ;</reg> <reg norm="ipsum" type="context">ipsÅ©</reg> eſſe valde diaphanum, ita <lb/>quod radij Å¿olares ipÅ¿um penetraſſent. </s> - <s xml:space="preserve">Tunc dico quod in Å¿uperficie corporis luna-<lb/>ris produceret vmbram. </s> - <s xml:space="preserve">Pro cuius intelligentia cogitemus Å¿ubÅ¿criptam hic figuram <lb/><var>b.h.q.a.e.</var> eſſe Å¿phÄ™ram aliquam cryÅ¿tallinam, & ad partem <var>.b.h.q.</var> Å¿it radius lumino-<lb/>Å¿us Å¿olaris qui ipÅ¿am illuminet, cuius radij extremitates Å¿int <var>.d.b.l.</var> et <var>.p.q.r.</var> Å¿upponen-<lb/>do <var>.d.l.</var> et <var>.p.r.</var> terminos eſſe vnius plani Å¿ecantis ipÅ¿um radium per axem, </s> - <s xml:space="preserve">tunc vide-<lb/>bis ipÅ¿um radium <var>.b.p.q.d.</var> <reg norm="tranÅ¿eun- tem" type="context">tranÅ¿eÅ©-<lb/> - <anchor type="figure" xlink:label="fig-0358-01a" xlink:href="fig-0358-01"/> - tem</reg> ipÅ¿am Å¿phæram, congregari Å¿eu <lb/><reg norm="condenÅ¿ari" type="context">condẽſari</reg>, ob vniformem refractio-<lb/>nem, vÅ¿que ad punctum <var>.o.</var> deinde; <lb/>propter rectitudinem ipÅ¿ius diffu-<lb/>Å¿ionis, vltra punctum <var>.o.</var> ipÅ¿um dila-<lb/>tari, diÅ¿gregari, Å¿eu rarefieri, <reg norm="quouÅ¿que" type="simple">quouÅ¿q;</reg> <lb/>nullius illuminationis actum habeat .<lb/>vt: exempli gratia <var>.o.t.</var> et <var>.o.s.</var> eius par <lb/>tes, ita quod interualla <var>.c.o.b.</var> et <var>.u.</var> <lb/> - <anchor type="figure" xlink:label="fig-0358-02a" xlink:href="fig-0358-02"/> - <var>o.q.</var> relinquerentur priuata lumini-<lb/>bus, vnde vmbroÅ¿a remanerent. di-<lb/><reg norm="Å¿tantiaque" type="simple">Å¿tantiaqÌ;</reg> ab <var>.o.</var> ad Å¿uperficiem Å¿phÄ™ri <lb/>cam corporis <var>.b.e.d.q.</var> non Å¿olum <reg norm="non" type="context">nõ</reg> <lb/>maior eÅ¿t diametro ipÅ¿ius Å¿phæræ; <lb/></s> - <s xml:space="preserve">imo minor, vt à te ipÅ¿o experiri po-<lb/>tes. </s> - <s xml:space="preserve">PoÅ¿ito igitur aliquo obiecto <lb/>opaco in loco <var>.K.o.g.</var> eius Å¿uperficies <lb/>intercepta inter <var>.K.</var> et <var>.g.</var> adumbrata <lb/>erit, excepto puncto <var>.o</var>. </s> - <s xml:space="preserve">PoÅ¿ito dein <lb/>de ipÅ¿o obiecto in loco <var>.n.y.x.m.</var> <reg norm="eius" type="simple">eiê°</reg> <lb/>partes <var>.y.n.</var> et <var>.x.m.</var> remanebunt lu- - <pb o="347" rhead="EPISTOL AE." n="359" file="0359" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0359"/> - mine deÅ¿titutæ <reg norm="interuallumque" type="simple">interuallumqÌ;</reg> tantummodò inter <var>.y.x.</var> illuminatum erit, Å¿ed Å¿i in <lb/>loco <var>.c.u.</var> poÅ¿itum fuerit, </s> - <s xml:space="preserve">tunc totum <var>.c.u.</var> illuminatum erit, Å¿ed debili modo propter <lb/>detractionem factam à reflexione in Å¿uperficie corporis Å¿phærici, vt Å¿upra diximus.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0358-01" xlink:href="fig-0358-01a"> - <image file="0358-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0358-01"/> - </figure> - <figure xlink:label="fig-0358-02" xlink:href="fig-0358-02a"> - <image file="0358-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0358-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">PoÅ¿ito deinde obiecto in loco <var>.i.z.H.f.</var> tunc partes <var>.z.i.</var> et <var>.H.f.</var> rectos Solis radios <lb/>habebunt cum aliquibus refractis, Å¿ed <var>.z.H.</var> pauciſſimum habebit lumen, pro-<lb/>pter diÅ¿gregationem radiorum. </s> - <s xml:space="preserve">PoÅ¿ito poÅ¿tea ipÅ¿o obiecto in loco <var>.t.l.r.s.</var> tanto <lb/>minus lumen habebit pars <var>.l.r.</var> propter dictam <reg norm="diÅ¿gregationem" type="context">diÅ¿gregationẽ</reg>, Å¿eu <reg norm="diſſipationem" type="context">diſſipationẽ</reg> radio <lb/>rum, & Å¿ic Å¿ucceſſiuè quanto remotius poÅ¿itum fuerit ipÅ¿um obiectum, tanto minus <lb/>illuminabitur. </s> - <s xml:space="preserve">vnde ita remotum poterit locari, ut nullus actus luminis in eo <lb/>videatur, de radijs Å¿cilicet, qui per Å¿phæram chryÅ¿tallinam tranÅ¿ibunt, Å¿ed videbi-<lb/>tur vmbra ipÅ¿ius Å¿phÄ™rÄ™ in obiecto propoÅ¿ito, cum nullum actum illuminationis in <lb/>eo loco obiecti habeant radij tranÅ¿euntes per dictam Å¿phÄ™ram. </s> - <s xml:space="preserve">quapropter partes <var>.<lb/>t.l.</var> et <var>.r.s.</var> illuminatæ erunt à Sole, et <var>.l.r.</var> omnino lumine deÅ¿tituta.</s> - </p> - <p> - <s xml:space="preserve">Quòd vero tolerabilior Å¿it oculis radius reflexus Solis à ſuperÅ¿icie aquæ, quà m <lb/>à ſuperficie alicuius Å¿peculi, oritur ab eo, quod Å¿upra diximus, hoc eÅ¿t, quod ma-<lb/>gna parsipÅ¿ius luminis penetrat in aquam, & non totum reflectit, quod quidem non <lb/>accidit Å¿peculis opacis.</s> - </p> - </div> - </div> - <div type="section" level="3" n="29"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE LONGITVDINE DVORVM LATERVM <lb/>cuiuÅ¿uis trianguli Å¿upra tertium.</head> - <head style="it" xml:space="preserve">Hieronymo Fenarolo.</head> - <p> - <s xml:space="preserve"><emph style="sc"><reg norm="QVod" type="conjecture">QVo'd</reg></emph> quælibet duo latera continentia rectum angulum cuiuÅ¿uis triangu-<lb/>li orthogonij, longiora Å¿int tertio latere, per diametrum circuli in eo in-<lb/>Å¿cripti, ab alijs iam demonÅ¿tratum fuit. </s> - <s xml:space="preserve">Sed quòd quælibet duo latera <lb/>cuiuÅ¿uis trianguli longiora Å¿int tertio per latus tetragonicum, quadrupli <lb/>producti cuiuÅ¿uis lineæ deÅ¿cendentis ab angulo contento à dictis duobus lateribus <lb/>ad oppoÅ¿itam partem circuli inÅ¿cripti, in partem extrinÅ¿ecam ipÅ¿ius lineæ, nullus <lb/>(quod Å¿ciam) vnquam Å¿cripÅ¿it, vel animaduertit.</s> - </p> - <p> - <s xml:space="preserve">Sit exempli gratia triangulus <var>.a.b.c.</var> quem volueris, in quo deÅ¿cribatur circulus <var>.<lb/>u.s.n.</var> & puncta contingentiæ Å¿int eadem <var>.u.s.n.</var> à puncto vero <var>.a.</var> deÅ¿cendat linea <var>.a.<lb/>i.e.</var> quæ terminetur à circunferentia in puncto <var>.e.</var> ipÅ¿ius circunferentiæ, vbi volue-<lb/>ris. </s> - <s xml:space="preserve">Dico nunc latera <var>.a.b.</var> et <var>.a.c.</var> longiora eſſe latere <var>.b.c.</var> per latus <reg norm="tetragonicum" type="context">tetragonicÅ©</reg> qua-<lb/>drupli producti ipÅ¿ius <var>.a.e.</var> in <var>.a.i</var>. </s> - <s xml:space="preserve">Nam certi Å¿amus ex vltima parte penultimæ ter-<lb/>tij Eucli <var>.n.c.</var> et <var>.s.c.</var> æquales inuicem eſſe, & Å¿imiliter <var>.b.s.</var> et <var>.b.u.</var> vnde ex communi <lb/>conceptu dicta latera maiora erunt <lb/> - <anchor type="figure" xlink:label="fig-0359-01a" xlink:href="fig-0359-01"/> - ipÅ¿o <var>.b.c.</var> per <var>.a.u.</var> et <var>.a.n.</var> quæ duæ <lb/>partes Å¿unt inuicem æquales di-<lb/>cta ratione, & quadratum lineæ <lb/>æqualis aggregato earum, eſſet qua <lb/>druplum quadrato cuiuÅ¿uis earum <lb/>ex .4. Å¿ecundi, Å¿ed ex penultima ter <lb/>tij, productum <var>.a.e.</var> in <var>.a.i.</var> æquale eÅ¿t <lb/>quadrato ipÅ¿ius <var>.a.u.</var> vel ipÅ¿ius <var>.a.n</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0359-01" xlink:href="fig-0359-01a"> - <image file="0359-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0359-01"/> - </figure> - </div> - <pb o="348" rhead="IO. BAPT. BENED." n="360" file="0360" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0360"/> - <p> - <s xml:space="preserve">Verum eÅ¿t igitur quod <var>.a.b.</var> cum <var>.a.c.</var> longiores Å¿int ipÅ¿a <var>.b.c.</var> per latus terrago. <lb/>nicum quadrupli eius quod fit. ex <var>.a.e.</var> in <var>.a.i.</var> quod fuit propoÅ¿itum.</s> - </p> - <p> - <s xml:space="preserve">Illud etiam non eÅ¿t Å¿pernendum, quod quotieÅ¿cunque data fuerint omnia latera <lb/>alicuius trianguli, illicò poſſumus cognoÅ¿cere puncta <var>.u.n.s.</var> contingentiæ circuli in <lb/>Å¿cripti, ope vltimæ partis penultimæ tertij, eo quod ex illa iam Å¿cimus, quod de-<lb/>trahendo <var>.b.c.</var> ex aggregato aliorum duorum laterum, remanebit <var>.u.a.</var> et <var>.a.n.</var> qua-<lb/>rum vnaquÄ™que nota erit, cum illarum quælibet, medietas Å¿it reÅ¿idui cogniti, detra <lb/>hendo poÅ¿tea vnam <reg norm="illarum" type="context">illarÅ©</reg> ab altero <lb/> - <anchor type="figure" xlink:label="fig-0360-01a" xlink:href="fig-0360-01"/> - duorum laterum <var>.a.b.</var> vel <var>.a.c.</var> rema <lb/>nebit <var>.u.b.</var> vel <var>.c.n.</var> Ä™qualis <var>.b.s.</var> vel <var>.c.<lb/>s.</var> vnde Å¿imiliter nobis innoteÅ¿cet <lb/>punctum <var>.s.</var> cum duobus punctis <var>.u.</var> <lb/>ct <var>.n.</var> à quibus duobus punctis, Å¿i <lb/>duÄ™ perpendiculares ad talia latera <lb/>ductæ fuerint, vbi hæe perpendicu <lb/>lares Å¿einuicem Å¿ecabunt, ibi <reg norm="cen- trum" type="context">cen-<lb/>trÅ©</reg> circuli inÅ¿criptibilis erit in trian <lb/>gulo propoÅ¿ito.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0360-01" xlink:href="fig-0360-01a"> - <image file="0360-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0360-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Inter alia, quæ tibi dixi de Iride, quod memoria non tenes, nihil aliud eÅ¿t niÅ¿i <lb/>quod cum Iris videtur, non eodem loco ab omnibus videtur, quia reflexio eÅ¿t, & <lb/>vt reflexio luminis à ſpeculo non omnibus ab eodem puncto fit, ita etiam tibi dixi <lb/>de Iride.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">De InÅ¿trumento oxygonio, Å¿eu elliptico.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">QVod aliquando à me audiuiÅ¿ti falÅ¿um non eÅ¿t, Å¿cilicet poſſibile eſſe (vt <lb/>Å¿peculatus Å¿um) particulare inÅ¿trumentum fabricari ad deÅ¿ignandum oxy-<lb/>goniam, Å¿eu ellipticam Å¿ectionem, quæ à Pergeo defectio appellatur, quod quidem <lb/>inÅ¿trumentum valde diuerÅ¿um eÅ¿t ab alijs, quÄ™ alià s inueni, pro ipÅ¿is conicis Å¿ectio <lb/>nibus delineandis. </s> - <s xml:space="preserve">OccaÅ¿ionem <reg norm="autem" type="wordlist">aÅ©t</reg> huiuÅ¿imodi inÅ¿trumenti inueniendi mihi præ <lb/>buit <reg norm="Å¿ecunda" type="context">Å¿ecÅ©da</reg> dubij Å¿olutio <reg norm="quam" type="context">quã</reg> feci ann .1568. grauiſſ. philoÅ¿opho FranciÅ¿co Vimer <lb/>cato, <reg norm="nam" type="context">nã</reg> <reg norm="cum" type="context">cÅ©</reg> viderim in ea figura <var>.f.a.</var> Å¿emper <reg norm="æqualem" type="context">æqualẽ</reg> eſſe <var>.o.i.</var> Å¿uæ parallelæ Å¿cilicet, <lb/>vnde cum recta linea fuerit protracta per <var>.o.</var> et <var>.f.</var> ipÅ¿a foret Å¿emper <reg norm="Ä™quidiÅ¿tans" type="context">Ä™quidiÅ¿tãs</reg> <var>.d.p.</var> ex <lb/>33. primi Eucli. </s> - <s xml:space="preserve">Venit mihi in mentem modus conÅ¿truendi hoc Å¿ubÅ¿criptum inÅ¿tru-<lb/>mentum, tali ordine, videlicet, <reg norm="coniungendo" type="context">coniungẽdo</reg> Å¿eptem hic Å¿ubnotatas lineas materia-<lb/>les <var>.z.r</var>: <var>u.n</var>: <var>e.h</var>: <var>e.c</var>: <var>c.l</var>: <var>l.s.</var> et <var>.s.e.</var> Å¿imul, hoc modo, Å¿cilicet <reg norm="Å¿abricando" type="context">Å¿abricãdo</reg> quadrila-<lb/>terum æquilaterum <var>.c.e.s.l.</var> hac conditione, quod immobili exiÅ¿tente puncto <var>.c.</var> in li <lb/>nea <var>.z.r.</var> reliqua omnia mobilia exiÅ¿tant, hoc eÅ¿t quod <reg norm="punctum" type="context">punctÅ©</reg> <var>.s.</var> moueatur per di-<lb/>ctam lineam <var>.z.r.</var> & immobili exiÅ¿tente puncto <var>.e.</var> vt extremum lineæ <var>.e.h.</var> hoc eÅ¿t <lb/>coniuncto extremo <var>.e.</var> lineæ <var>.e.h.</var> cum angulo <var>.c.e.s.</var> reliqua puncta lineæ ipÅ¿ius <var>.e.h.</var> <lb/>moueantur per <var>.l.</var> & per duas parallelas <var>.u.n.</var> et <var>.z.r.</var> longitudo vero <var>.e.h.</var> Å¿it compo-<lb/>Å¿ita ex duplo vnius lateris ipÅ¿ius quadrilateris. </s> - <s xml:space="preserve">Oportet deinde quod punctum <var>.f.</var> <lb/>Å¿emper vnum, & idem Å¿it ipÅ¿ius parallelæ <var>.u.n.</var> moueatur tamen per <var>.e.h.</var> quod qui-<lb/>dem punctum illud erit, quod vnam <reg norm="portionem" type="context">portionẽ</reg> circunferentiæ oxygoniæ Å¿ectonis - <pb o="349" rhead="EPISTOL AE." n="361" file="0361" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0361"/> - deÅ¿ignabit, puncta verò <var>.o.</var> et <var>.K.</var> vt puncta laterum <var>.c.e.</var> et <var>.s.e.</var> æquædiÅ¿tantia à <lb/>punctis <var>.c.</var> et <var>.s.</var> <reg norm="eadem" type="context">eadẽ</reg> Å¿emper Å¿int, ita tamen vt puncta lineæ <var>.u.n.</var> Å¿emper diuerÅ¿a exi <lb/><reg norm="Å¿tant" type="context">Å¿tãt</reg>, & quodlibet ipÅ¿ius quadrilateri latus, æquale Å¿it medietati maioris axis ipÅ¿ius <lb/>oxygoniæ Å¿ectionis delineandæ, et <var>.c.o.</var> Å¿eu <var>.s.K.</var> (quod idem eÅ¿t) Å¿it æqualis medie <lb/>tati axis minoris dictæ Å¿ectionis, et <var>.z.r.</var> æqualis duplo <var>.e.h.</var> vnde, quando puncta <var>.e.</var> et <var>.<lb/>l.</var> coniuncta Å¿imul erunt, Å¿imiliter coniunctæ Å¿imul erunt <var>.c.e.</var> et <var>.e.s.</var> cum <var>.c.l.</var> et <var>.l.s.</var> - <anchor type="figure" xlink:label="fig-0361-01a" xlink:href="fig-0361-01"/> - <anchor type="figure" xlink:label="fig-0361-02a" xlink:href="fig-0361-02"/> - </s> - <pb o="350" rhead="IO. BAPT. BENED." n="362" file="0362" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0362"/> - <s xml:space="preserve">Quapropter puncta <var>.e.l.f.</var> et <var>.p.</var> extremum axis maioris, in eodem met loco erunt, <lb/>hoc eÅ¿t in aliquo extremorum maioris axis, & cum punct is <var>.s.</var> coniunct is fuerit cum <lb/>centro <var>.c.</var> punctus <var>.f.</var> parallelÄ™ <var>.u.p.</var> in extremo axis minoris erit, & in eodem loco erit <lb/>cum <var>.o.</var> & cum <var>.K</var>. </s> - <s xml:space="preserve">In extremitatibus verò lineæ <var>.z.r.</var> neceſſe eÅ¿t, vt Å¿int duo <reg norm="puncta" type="context">pÅ©cta</reg> fer <lb/>rea, ad firmandum ipÅ¿am <var>.z.r.</var> Å¿uper Å¿ubiectam lineam Å¿ignificantem maiorem axem <lb/>propoÅ¿itÄ™ Å¿ectionis.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0361-01" xlink:href="fig-0361-01a"> - <image file="0361-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0361-01"/> - </figure> - <figure xlink:label="fig-0361-02" xlink:href="fig-0361-02a"> - <image file="0361-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0361-02"/> - <caption style="it" xml:space="preserve">Instrumentum <lb/>oxigonium</caption> - </figure> - </div> - <figure position="here"> - <image file="0362-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0362-01"/> - </figure> - <figure position="here"> - <image file="0362-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0362-02"/> - </figure> - <pb o="351" rhead="EPISTOL AE." n="363" file="0363" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0363"/> - <p> - <s xml:space="preserve">Volo etiam quod ad partem <var>.c.l.s.</var> quadrilateri conÅ¿tituta Å¿it alia parallela ad <var>.z.<lb/>r.</var> & in æquali diÅ¿tantia ab ipÅ¿a quemadmodum <var>.u.n.</var> diÅ¿tat ad eademmet <var>.z.r.</var> ad ean <lb/>dem operationem faciendam. </s> - <s xml:space="preserve">Vnde in vno tantummodo itinere puncti <var>.s.</var> ab <var>.r.</var> <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> <lb/>ad <var>.c.</var> deÅ¿ignabimus quartam partem Å¿ectionis, conuerÅ¿o poÅ¿tea inÅ¿trumento, hoc eÅ¿t <lb/>poÅ¿ito puncto <var>.r.</var> vbi prius erat <var>.z.</var> et <var>.z.</var> vbi erat <var>.r.</var> aliam delineabimus quartam, & <lb/>Å¿ic ad oppoÅ¿itam partem ipÅ¿ius <var>.z.r.</var> faciendum erit. </s> - <s xml:space="preserve">Hoc inÅ¿trumentum poſſumus <lb/>etiam ita conÅ¿truere, vt puncta <var>.o.</var> et <var>.K.</var> poſſint collocari in laterihus <var>.c.e.</var> et <var>.e.s.</var> vbi no <lb/>bis magis libuerit, ita vt licebit in qualibet proportione <reg norm="axium" type="context">axiÅ©</reg> propoÅ¿ita, oxygoniam <lb/>deÅ¿ignare. </s> - <s xml:space="preserve">Nam <var>.c.o.</var> erit longitudo dimidij axis minoris, et <var>.c.e.</var> dimidij maioris.</s> - </p> - </div> - </div> - <div type="section" level="3" n="30"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE CONSTITVTIONE TRIANGVLI <lb/>orthogonij conditionati.</head> - <head style="it" xml:space="preserve">Domino Ludouico de Rocchaforte.</head> - <p> - <s xml:space="preserve"><emph style="sc">QVod</emph> à me poÅ¿tulas, non eÅ¿t admodum difficile, cupis enim triangulum <lb/>orthogonium, exempli gratia <var>.o.i.e.</var> in figura <var>.A.</var> ita conÅ¿tituere, vt di-<lb/>uiÅ¿um Å¿it à perpendiculari <var>.a.i.</var> & quod proportio <var>.o.e.</var> ad <var>.o.i.</var> Å¿it vt <var>.o.i.</var> ad <lb/><var>i.e.</var> & quod quadrati <var>.o.i.</var> ad quadratum <var>.o.a.</var> Å¿it vt <var>.e.i.</var> ad <var>.e.a.</var> & quadra <lb/>tum <var>.o.i.</var> ad quadratum <var>.e.i.</var> Å¿it .ut <var>.o.a.</var> ad <var>.e.a</var>. </s> - <s xml:space="preserve">Quæ omnia in promptu veniunt, quo <lb/>tieÅ¿cunque <var>.o.e.</var> fuerit diameter alicuius circuli, <reg norm="diuiÅ¿aque" type="simple">diuiÅ¿aqÌ;</reg> in puncto <var>.a.</var> Å¿ecundum pro <lb/>portionem habentem medium <reg norm="duoque" type="simple">duoqÌ;</reg> extrema, protracta deinde perpendiculari <var>.a.<lb/>i.</var> ad <var>o.e.</var> uÅ¿que ad circunferentiam, <reg norm="coniunctæque" type="simple">coniunctæqÌ;</reg> <var>.o.i.</var> et <var>.i.e</var>: tale triangulum, omnia <lb/>Å¿upradicta in Å¿e continebit.</s> - </p> - <p> - <s xml:space="preserve">Nam ex .30. tertij angulus <var>.i.</var> rectus erit, & ex .8. Å¿exti <var>.o.i.</var> erit media proportio-<lb/>nalis inter <var>.o.e.</var> et <var>.o.a.</var> et <var>.e.i.</var> inter <var>.o.e.</var> <lb/> - <anchor type="figure" xlink:label="fig-0363-01a" xlink:href="fig-0363-01"/> - et <var>.a.e.</var> Å¿ed quia ex diuiÅ¿ione facta in <reg norm="pum" type="context">pÅ©</reg> <lb/>cto <var>.a.</var> etiam <var>.o.a.</var> erit media proportio-<lb/>nalis inter totum & reÅ¿iduum, ideo ex <num value="11">.<lb/>11.</num> quinti ita erit <var>.o.e.</var> ad <var>.e.i.</var> vt <var>.o.e.</var> ad <var>.<lb/>o.a.</var> vnde ex .9. eiuÅ¿dem <var>.a.o.</var> erit æqua-<lb/>lis <var>.e.i.</var> & ideo <var>.o.i.</var> erit media proportio <lb/>nalis inter <var>.o.e.</var> et <var>.e.i</var>. </s> - <s xml:space="preserve">Sed quia propor-<lb/>tio <var>.e.i.</var> ad <var>.a.e.</var> <reg norm="eadem" type="context">eadẽ</reg> eÅ¿t, quÄ™ ipÅ¿ius <var>.o.e.</var> ad <lb/><var>o.a</var>. </s> - <s xml:space="preserve">tunc videbis ex .18. Å¿exti, quod pro <lb/>portio quadrati <var>.o.i.</var> ad quadratum <var>.o.a.</var> <lb/>erit vt <var>.e.i.</var> ad <var>.e.a.</var> cum vero duo trian-<lb/>guli <var>.o.i.a.</var> et <var>.a.i.e.</var> Å¿int inuicem Å¿imiles <lb/>ex Å¿upradicta .8. Å¿exti, </s> - <s xml:space="preserve">tunc videbis ex <lb/>18. et .17. eiuÅ¿dem dictos <reg norm="triangulos" type="context">triãgulos</reg> ean <lb/>dem habere inter Å¿e proportionem, quÄ™ <lb/>eÅ¿t inrer quadrata ipÅ¿ius <var>.o.i.</var> et <var>.i.e.</var> vnde <lb/>ex prima Å¿exti ita Å¿e inuicem habebunt <var>.<lb/>a.o.</var> et <var>.a.e</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0363-01" xlink:href="fig-0363-01a"> - <image file="0363-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0363-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Circa eam verò difficultatem quam - <pb o="352" rhead="I O. BAPT. BENED." n="364" file="0364" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0364"/> - habes in circulo .ω vbi fateris te non videre qua ratione eadem proportio Å¿it qua-<lb/>drati <var>.u.o.</var> ad quadratum <var>.o.n.</var> vt lineæ <var>.o.a.</var> ad lineam <var>.o.e.</var> partes diametri <var>.o.i.</var> ipÅ¿ius <lb/>circuli, terminatæ à perpendicularibus <var>.u.a.</var> et <var>.n.e</var>.</s> - </p> - <p> - <s xml:space="preserve">Hoc neceſſario contingit, propterea quod cum fuerint protractæ <var>.u.i.</var> et <var>.n.i.</var> <reg norm="tunc" type="context">tÅ©c</reg> <lb/>habebimus ad partem <var>.o.u.i.</var> triangulum <var>.o.u.i.</var> diuiÅ¿um in duo triangula Å¿imilia ipÅ¿i <lb/>totali triangulo. </s> - <s xml:space="preserve">Idem etiam dico ad partem <var>.o.n.i.</var> vnde ex tali Å¿imilitudine habe-<lb/>bimus <var>.o.u.</var> mediam proportionalem inter <var>.o.i.</var> et <var>.o.a.</var> et Å¿ic <var>.o.n.</var> erit media proportio <lb/>nalis inter <var>.o.i.</var> et <var>.o.e.</var> </s> - <s xml:space="preserve">quare ex .16. Å¿exti, quadratum <var>.o.u.</var> æquale erit producto ipÅ¿ius <lb/><var>o.i.</var> in <var>.o.a.</var> & quadratum <var>.o.n.</var> æquale producto <var>.o.i.</var> in <var>.o.e.</var> Å¿ed ex prima eiuÅ¿dem, ea <lb/>dem proportio eÅ¿t ipÅ¿ius <var>.o.a.</var> ad <var>.o.e.</var> quæ producti ipÅ¿ius <var>.o.i.</var> in <var>o.a.</var> ad productum <var>.o.<lb/>i.</var> in <var>.o.e.</var> </s> - <s xml:space="preserve">quare, ex <reg norm="communi" type="context">cõmuni</reg> conceptu, ita erit quadrati <var>.o.u.</var> ad quadratúm.o.n. </s> - <s xml:space="preserve">Et hÄ™c <lb/>eÅ¿t alia circuli paſſio.</s> - </p> - <p> - <s xml:space="preserve">Reliqua verò difficultas quam te habere Å¿cribis, eÅ¿t, quare cum duæ lineÄ™ <lb/><var>a.u.</var> et <var>.b.s.i.</var> Å¿int inuicem Ä™quales, diuiſæ verò non æquali modo, Å¿ed tali, quod <var>.a.</var> <lb/>maior Å¿it quam <var>.u.</var> et <var>.b.s.</var> maior quam <var>.i.</var> quomodo poteÅ¿t fieri, quod Å¿i <var>.u.</var> maior fue-<lb/>rit <var>.i.</var> proportio <var>.a.</var> ad <var>.i.</var> maior Å¿it quam ipÅ¿ius <var>.b.s.</var> ad <var>.u</var>.</s> - </p> - <p> - <s xml:space="preserve">Hoc etiam ex neceſſitate cuenit, eo <lb/>quod Å¿i accepta fuerit <var>.t.n.</var> æqualis <var>.u.</var> ab <lb/> - <anchor type="figure" xlink:label="fig-0364-01a" xlink:href="fig-0364-01"/> - <reg norm="ipÅ¿aque" type="simple">ipÅ¿aqÌ;</reg> abÅ¿ciÅ¿a fuerit <var>.t.</var> æqualis <var>.i.</var> & ab <var>.<lb/>b.s.</var> abÅ¿ciÅ¿a <var>.s.</var> æqualis <var>.n.</var> habebimus <var>.a.</var> et <lb/>b. inuicem æ quales, vnde habebis ma-<lb/>iorem propor tionem ipÅ¿ius <var>.b.</var> ad <var>.t.</var> <reg norm="quam" type="context">quã</reg> <lb/>s. ad <var>.n.</var> quod cum clarum per Å¿e Å¿it, tibi <lb/>relinquo. </s> - <s xml:space="preserve">Å¿ed ex .27. quinti, proportio <lb/>b. ad. s, maior erit quam <var>.t.</var> ad <var>.n.</var> & ex <lb/>28. <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> <reg norm="proportio" type="simple">ê“portio</reg> <var>.b.s.</var> ad <var>.s.</var> maior erit, <lb/>quam <var>.t.n.</var> ad <var>.n.</var> & ex .27. maior propor <lb/>tio erit ipÅ¿ius <var>.b.s.</var> ad <var>.n.t.</var> quam <var>.s.</var> ad <var>.n.</var> <lb/>ergo ex .33. maior erit ipÅ¿ius <var>.b.</var> ad <var>.t.</var> <reg norm="quam" type="context">quã</reg> <lb/><var>b.s.</var> ad <var>.n.t.</var> hoc eÅ¿t maior ipÅ¿ius <var>.a.</var> ad <lb/>i. quam <var>.b.s.</var> ad <var>.u.</var> quod eÅ¿t propo-<lb/>Å¿itum.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0364-01" xlink:href="fig-0364-01a"> - <image file="0364-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0364-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Id verò de quo me interrogas <reg norm="nempe" type="context">nẽpe</reg> de <lb/>diÅ¿tinctione orbium cÄ™leÅ¿tium, ortum <lb/>habet à communi opinione motuum <lb/>fixarum. </s> - <s xml:space="preserve">Nam cum putauerint philo-<lb/>Å¿ophi ipÅ¿as moueri, Å¿emper eandem <reg norm="Å¿eruando" type="context">Å¿eruãdo</reg> inuicem diÅ¿tantiam, non Å¿ine ratione <lb/>crediderunt eas fixas eſſe eodem in orbe, idem etiam poÅ¿tea de planetis opinaue-<lb/>runt. </s> - <s xml:space="preserve">Hoc eÅ¿t, vnumquemque, aliquo in orbe, fixo exiÅ¿tere.</s> - </p> - <pb o="353" rhead="EPISTOL AE." n="365" file="0365" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0365"/> - </div> - </div> - <div type="section" level="3" n="31"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE MODO DIVIDENDI PARABOLAM <lb/>propoÅ¿itam Å¿ecundum datam proportionem.</head> - <head style="it" xml:space="preserve">Pamphilo Gothfrid.</head> - <p> - <s xml:space="preserve"><emph style="sc">QVod</emph> à me quæris, eÅ¿t quidem poſſibile, non tamen adhuc inuentum, quo <lb/>niam nemo ad <reg norm="hunc" type="context">hÅ©c</reg> vÅ¿que diem diuiÅ¿it vnam datam proportionem in tres <lb/>æquales partes, Å¿ed Å¿i hoc pro facto conceſſeris, nunc tibi morem geram. <lb/></s> - <s xml:space="preserve">Nam proponis n. ihi parabolem <var>.x.b.e.</var> cum proportione <var>.p.</var> ad <var>.q.</var> <reg norm="cupiÅ¿que" type="simple">cupiÅ¿qÌ;</reg> <lb/>Å¿cire modum diuidendi ipÅ¿am parabolem vna mediante linea parallela ipÅ¿i baÅ¿i, ita <lb/>vt eandem habeat proportionem tota parabola ad partem abÅ¿ciſſam, quæ eÅ¿t inter <var>.<lb/>p.</var> et <var>.q</var>. </s> - <s xml:space="preserve">Ad quod faciendum, Å¿upponendum primò datam proportionem inter <var>.<lb/>p.</var> et <var>.q.</var> diuiÅ¿am eſſe in tres partes æquales, duabus lineis mediantibus <var>.n.</var> et <var>.u.</var> quæ me <lb/>diæ proportionales vocabuntur inter <var>.p.</var> et <var>.q</var>. </s> - <s xml:space="preserve">deinde à quouis puncto circunferentiÄ™ <lb/>ipÅ¿ius figuræ ducatur parallela baÅ¿i <var>.x.e.</var> poÅ¿tea verò per puncta media harum dua-<lb/>rum <reg norm="æquidiÅ¿tantium" type="context">æquidiÅ¿tantiÅ©</reg> protrahatur <var>.g.b.</var> quæ diameter erit Å¿ectionis, ex 28. Å¿ecundi Per-<lb/>gei, </s> - <s xml:space="preserve">diuidatur deinde hæc diameter in puncto <var>.a.</var> ita quod eadem proportio Å¿it ipÅ¿ius <lb/><var>b.g.</var> ad <var>.b.a.</var> quæ ipÅ¿ius <var>.p.</var> ad <var>.u.</var> quod tibi facile erit, Å¿ecando à linea <var>.p.</var> partem <var>.i.</var> æqua <lb/>lem ipÅ¿i <var>.u.</var> tali modo poÅ¿tea diuidendo <var>.b.g.</var> ex .12. Å¿exti, ducatur a puncto <var>.a.</var> ipÅ¿a <var>.d.<lb/>h.</var> parallclam ipÅ¿i <var>.x.e.</var> & habebitur propoÅ¿itum.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius reiratione, Å¿cies primum quod <var>.h.d.</var> diuiÅ¿a erit à diametro <var>.b.g.</var> per æqua <lb/>lia ex .7. primi Pergei, vel Å¿i cogitabimus aliquam lineam tangentem ipÅ¿am parabo <lb/>lam in puncto <var>.b</var>. </s> - <s xml:space="preserve">tunc ex quinta Å¿ecundi ipÅ¿ius Pergei habebimus ipÅ¿am eſſe paralle-<lb/>lam <var>.e.x.</var> & ex .30. primi Eucli. erit Å¿imiliter æquidiÅ¿tans <var>.d.h.</var> vnde ex .46. primi eiuÅ¿-<lb/>dem Pergei <var>.h.a.</var> æqualis erit <var>.d.a</var>. </s> - <s xml:space="preserve">Protrahatur deinde <var>.e.b</var>: d b: <var>x.b.</var> et <var>.h.b.</var> vnde ex .17 <lb/>lib. de quadratura parabolæ Archimedis, habebimus eandem proportionem Å¿uper <lb/>ficiei totalis parabolæ <var>.x.b.e.</var> ad trigonum <var>.x.b.e.</var> quæ portionis <var>.h.b.d.</var> ad Å¿uum <reg norm="tri- gonum" type="context">tri-<lb/>gonÅ©</reg>, eo quod <reg norm="tam" type="context">tã</reg> vna quà m alia erit Å¿eÅ¿quitertia, <reg norm="eius" type="simple">eiê°</reg> <reg norm="etiam" type="context">etiã</reg> medietates Å¿ic Å¿e <reg norm="habebunt" type="context">habebÅ©t</reg>.</s> - </p> - <p> - <s xml:space="preserve">Vnde permutando, proportio medietatis totalis parabolÄ™ ad medietatem partia <lb/>lem ipÅ¿ius, æqualis erit proportioni trianguli <lb/><var>g.b.e.</var> ad triangulum <var>.a.b.d.</var> Å¿ed ex .20. primi <lb/>Pergei, eadem eÅ¿t proportio quadrati ipÅ¿ius <var>.<lb/> - <anchor type="figure" xlink:label="fig-0365-01a" xlink:href="fig-0365-01"/> - g.e.</var> ad quadratum ipÅ¿ius <var>.a.d.</var> quæ <var>.b.g.</var> ad <var>.b.a.</var> <lb/>hoc eÅ¿t, vt <var>.g.e.</var> ad <var>.a.o.</var> ex Å¿imilitudine triangu-<lb/>lorum, & quia <var>.b.g.</var> ad <var>.b.a.</var> eÅ¿t Å¿icut <var>.p.</var> ad <var>.u.</var> ita <lb/>igitur erit quadrati ipÅ¿ius <var>.g.e.</var> ad quadratum <lb/>ipÅ¿eus <var>.a.d.</var> </s> - <s xml:space="preserve">quare <var>.g.e.</var> ad <var>.a.d.</var> erit ut p. ad <var>.n.</var> <lb/>ex .18. Å¿exti Euclid. </s> - <s xml:space="preserve">Å¿ed cum ex .24. eiuÅ¿dem <lb/>proportio trianguli <var>.b.g.e.</var> ad triangulum <var>.b.<lb/>a.d.</var> compoÅ¿ita Å¿it ex proportione <var>.g.e.</var> ad <var>.a.<lb/>d.</var> er. ex <var>.g.b.</var> ad <var>.b.a.</var> hoc eÅ¿t <var>.g.e.</var> ad <var>.a.o.</var> & <lb/>quia <reg norm="proportio" type="simple">ê“portio</reg> <var>.g.e.</var> ad <var>.a.o.</var> æqualis eÅ¿t ei quæ <var>.p.</var> <lb/>ad. u ex .11. quinti Euclid. </s> - <s xml:space="preserve">& proportio <var>.g.e.</var> <lb/>ad <var>.a.d.</var> æqualis eÅ¿t ei quæ <var>.p.</var> ad <var>.n.</var> hoc eÅ¿t vt <var>.u.</var> <lb/>ad <var>.q.</var> ergo proportio trianguli <var>.b.g.e.</var> ad trian-<lb/>gulum <var>.b.a.d.</var> compoÅ¿ita erit ex ca quÄ™ <var>.p.</var> ad <var>.u.</var> <lb/>& ex ea quæ <var>.u.</var> ad <var>.q.</var> æqualis ergo erit ei, quæ <lb/>p. ad <var>.q.</var> & ita medietates parabolarum, & eorum dupla.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0365-01" xlink:href="fig-0365-01a"> - <image file="0365-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0365-01"/> - </figure> - </div> - <pb o="354" rhead="IO. BABPT. BENED." n="366" file="0366" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0366"/> - <head style="sc" xml:space="preserve">COROLLARIVM.</head> - <p> - <s xml:space="preserve">Proportio maioris portionis ad minorem Å¿emper erit Å¿eÅ¿quialtera proportioni <lb/>ipÅ¿ius <var>.b.g.</var> ad <var>.a.b.</var> eo quod cum Å¿it proportio totalis portionis ad partialem vt trian-<lb/>guli <var>.b.g.e.</var> ad <var>.b.a.d.</var> & hæc Å¿eÅ¿quialtera proportioni ipÅ¿ius <var>.g.e.</var> ad <var>.a.o.</var> hoc eÅ¿t vt ip-<lb/>Å¿ius <var>.b.g.</var> ad <var>.b.a.</var> ideo proportio ipÅ¿arum portionum erit Å¿imiliter Å¿eÅ¿quialtera pro-<lb/>portioni diametrorum.</s> - </p> - <p> - <s xml:space="preserve">Deinde Å¿i protractæ fuerint <var>.b.d.</var> et <var>.g.e.</var> quouÅ¿que conueniant in puncto <var>.z.</var> habe <lb/>bis inter <var>.g.z.</var> et <var>.a.o.</var> duas <var>.g.e.</var> et <var>.a.d.</var> medias proportionales in proportionalitate con <lb/>tinua, eo quod cum (ex ijs quæ Å¿upra diximus.). <var>a.d.</var> media proportionalis Å¿it inter <var>.<lb/>g.e.</var> et <var>.a.o.</var> & proportio <var>.g.z.</var> ad <var>.g.e.</var> vt ipÅ¿ius <var>.a.d.</var> ad <var>.a.o.</var> eo quodipÅ¿ius <var>.g.z.</var> ad <var>.a.d.</var> <lb/>& ipÅ¿ius <var>.g.e.</var> ad <var>.a.o.</var> eÅ¿t vt ipÅ¿ius <var>.b.g.</var> ad <var>.b.a.</var> ex Å¿imilitudine triangulorum, ideo di-<lb/>ctæ <reg norm="proportiones" type="simple">ê“portiones</reg> erunt <reg norm="inuicem" type="context">inuicẽ</reg> æquales. </s> - <s xml:space="preserve">Vnde permutatim ita erit ipÅ¿ius <var>.g.z.</var> ad <var>.g.e.</var> <lb/>vt ipÅ¿ius <var>.a.d.</var> ad <var>.a.o.</var> & ut ipÅ¿ius <var>.g.e.</var> ad <var>.a.d</var>.</s> - </p> - <p> - <s xml:space="preserve">Amplius etiam dico, quod proportio pa <lb/> - <anchor type="figure" xlink:label="fig-0366-01a" xlink:href="fig-0366-01"/> - rabolæ totalis ad partialem, eadem eÅ¿t, quÄ™ <lb/>cubi ipÅ¿ius <var>.g.e.</var> ad cubum ipÅ¿ius <var>.a.d.</var> & ex <reg norm="con" type="context">cõ</reg> <lb/>Å¿equenti, vt cuborum earundem baÅ¿ium, eo <lb/>quod cum Å¿it, ex .36. vndecimi Euclid. pro-<lb/>portio cubi ipÅ¿ius <var>.g.e.</var> ad cubum ipÅ¿ius <var>.a.d.</var> <lb/>tripla ei quæ ipÅ¿ius <var>.g.e.</var> ad <var>.a.d.</var> ideo æqualis <lb/>erit ei quÄ™ trianguli <var>.b.g.e.</var> ad triangulum <var>.b.<lb/>a.d.</var> cum proportio horum duorum triangu <lb/>lorum compoÅ¿ita Å¿it (vt Å¿upra vidimus) ex <lb/>ea quæ <var>.g.e.</var> ad <var>.a.o.</var> & ex ea quæ <var>.g.e.</var> ad <var>.a.d.</var> <lb/>& hæc medietas illius, Å¿ed trianguli ita Å¿e in <lb/>uicem habenr, vt parabolÄ™, </s> - <s xml:space="preserve">quare ipſæ para-<lb/>bolæ Å¿einuicem habebunt, vt cubi ipÅ¿arum <lb/>baÅ¿ium.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0366-01" xlink:href="fig-0366-01a"> - <image file="0366-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0366-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">Cubum fabricare æqualem pyramidi propoÅ¿itæ.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">CVbum fabricare æqualem propoÅ¿itæ pyramidi quadrilateræ, nullius erit diffi-<lb/>cultatis, Å¿uppoÅ¿ita tamen pro reperta diuiÅ¿ione cuiuÅ¿uis datæ proportionis in <lb/>tres partes æquales. </s> - <s xml:space="preserve">Nam ex .6. duodecimi Eucli. patet omne corpus Å¿erratile d-ui <lb/>Å¿ibile eſſe in tres pyramides quadrilateras æquales, Å¿cimus etiam quod cuilibet py-<lb/>ramidi quadrilateræ poteÅ¿t reperiri Å¿uum Å¿erratile. </s> - <s xml:space="preserve">Sit igitur propoÅ¿ita pyramis qua <lb/>drilatera <var>.m.g.f.h.</var> cuius Å¿erratile ita inueniemus, ducendo primum <var>.h.i.</var> parallelam <lb/>ipÅ¿i <var>.g.f.</var> et <var>.f.i.</var> ipÅ¿i <var>.g.h.</var> in Å¿uperficie trianguli <var>.f.g.h.</var> et <var>.m.K.</var> ipÅ¿i <var>.g.h.</var> in Å¿uperficie <lb/>trianguli <var>.m.g.h.</var> & æqualem dictæ <var>.g.h.</var> ducetur poÅ¿tea <var>.K.h.</var> et <var>.K.i.</var> & habebimus cor <lb/>pus <var>.f.K.g.</var> Å¿erratile, & triplum pyramidi propoÅ¿itæ. </s> - <s xml:space="preserve">Nunc duplicemus ipÅ¿um, du-<lb/>cendo <var>.K.x.</var> in Å¿uperficie trianguli <var>.i.k.h.</var> parallelam, <reg norm="æqualemque" type="simple">æqualemqÌ;</reg> ipÅ¿i <var>.i.h.</var> et <var>.m.y.</var> <lb/>in Å¿uperficie trianguli <var>.f.m.g.</var> parallelam, Ä™<reg norm="qualemque" type="simple">qualemqÌ;</reg> ipÅ¿i <var>.f.g.</var> ducatur poÅ¿tea <var>.g.y.</var> et <var>.h.<lb/>x.</var> quarum <reg norm="vnaquæque" type="simple">vnaquæq;</reg> æqualis erit ipÅ¿i <var>.f.m.</var> vnde habebimus corpus <var>.f.x.</var> parallelepe-<lb/>pidum, & Å¿excuplum ipÅ¿i pyramidi propoÅ¿itæ.</s> - </p> - <pb o="355" rhead="EPISTOL AE." n="367" file="0367" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0367"/> - <p> - <s xml:space="preserve">Inueniatur nunc quadratum <var>.u.n.</var> æquale Å¿extæ parti Å¿uperficiei <var>.f.i.g.h.</var> quod per <lb/>Å¿e facile erit, </s> - <s xml:space="preserve">deinde accipiatur altitudo corporis <var>.f.x.</var> ducendo vnam perpendicula <lb/>rem à puncto <var>.m.</var> ad baÅ¿im <var>.f.g.h.</var> quÄ™ Å¿it <var>.n.e.</var> qua mediante, cum quadrato <var>.u.n.</var> fabri <lb/>cetur Å¿olidum parallelepepidum <var>.u.e.</var> quod erit æquale dictæ pyramidi ex .33. vnde-<lb/>cimi Euclid.</s> - </p> - <p> - <s xml:space="preserve">Repertæ nunc Å¿int duæ mediæ proportionales <var>.r.s.</var> inter <var>.n.e.</var> et <var>.n.p.</var> quarum <var>.s.</var> Å¿it <lb/>proximior ipÅ¿i <var>.u.p.</var> ex qua <var>.s.</var> Å¿i conÅ¿titutus fuerit cubus, habebimus propoÅ¿itum.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius rei ratione, cogitemus corpus <var>.u.e.</var> productum eſſe vÅ¿que ad <var>.a.o.</var> per lon-<lb/>gitudem <var>.s.</var> latus dicti cubi, qui quidem cubus Å¿it <var>.d.b.</var> vnde proportio corporis <var>.u.e.</var> <lb/>ad corpus <var>.e.o.</var> erit, vt Å¿uperficiei <var>.p.e.</var> ad Å¿uperficiem <var>.t.e.</var> ex .33. undecimi, ipſæ verò <lb/>Å¿uperficies Å¿ibi inuicem erunt vt <var>.n.e.</var> ad <var>.e.a.</var> ex prima Å¿exti, </s> - <s xml:space="preserve">quare proportio corpo <lb/>ris <var>.u.e.</var> ad corpus <var>.e.o.</var> dupla erit proportioni ipÅ¿ius <var>.s.</var> ad <var>.n.p.</var> Å¿ed cum ex .33 vndeci-<lb/>mi, proportio cubi <var>.d.b.</var> ad corpus <var>.e.o.</var> Å¿it vt <reg norm="quadratum" type="context">quadratÅ©</reg> <var>.q.b.</var> ad quadratum <var>.o.a.</var> & cum <lb/>proportio <var>.q.b.</var> ad <var>.o.a.</var> dupla Å¿it ei quæ <var>.q.o.</var> ad <var>.o.t.</var> ex .18. Å¿exti, erit igitur proportio <lb/>cubi <var>.d.b.</var> ad corpus <var>.e.o.</var> dupla ei quæ <var>.q.o.</var> ad <var>.o.t.</var> hoc eÅ¿t ei quæ <var>.s.</var> ad <var>.n.p.</var> Å¿ed ita erat <lb/>corporis <var>.u.e.</var> ad corpus <var>.e.o.</var> </s> - <s xml:space="preserve">quare ex .9. quinti, cubus <var>.d.b.</var> æqualis erit corpor<unsure/>i.u.e. <lb/>hoc eÅ¿t pyramidi propoÅ¿itæ.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i oportebit cubum maiorem vel minorem ipÅ¿a pyramide reperire, in qua <lb/>proportione tibi placuerit, </s> - <s xml:space="preserve">tunc opus erit aliud quadratum inuenire, quod in ea <lb/>proportione Å¿e habeat ad quadratum <var>.u.n.</var> quam volueris, quo mediante Å¿imul cum <lb/>altitudine pyramidis conÅ¿equemur propoÅ¿itum.</s> - </p> - <p> - <s xml:space="preserve">Aduertendum tamen quod fabri-<lb/>care ipÅ¿um corpus Å¿erratile <var>.k.f.h.</var> & Å¿o <lb/> - <anchor type="figure" xlink:label="fig-0367-01a" xlink:href="fig-0367-01"/> - lidum <var>.f.x.</var> neceſſarium non eÅ¿t, niÅ¿i pro <lb/>demonÅ¿tratione. </s> - <s xml:space="preserve"><reg norm="idemque" type="simple">idemqÌ;</reg> dico de alijs <lb/>Å¿olidis, nam pro Å¿implici operatione <lb/>huiuÅ¿modi problematis, abÅ¿que ali-<lb/>qua re neceſſaria ad Å¿peculandum, ita <lb/>faciendum erit.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0367-01" xlink:href="fig-0367-01a"> - <image file="0367-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0367-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Data pyramide <var>.m.f.g.h.</var> accipe <reg norm="eius" type="simple">eiê°</reg> <lb/>alitudinem à <reg norm="puncto" type="context">pÅ©cto</reg> <var>.m.</var> vÅ¿que ad Å¿uper <lb/>ficiem baÅ¿is <var>.f.g.h.</var> quæ Å¿it <var>.n.e.</var> accipe <lb/>deinde latus letragonicum quadrati <var>.<lb/> - <anchor type="figure" xlink:label="fig-0367-02a" xlink:href="fig-0367-02"/> - u.n.</var> æqualis tertiæ partis ipÅ¿ius baÅ¿is <var>.f.<lb/>g.h.</var> quod latus Å¿it <var>.n.p.</var> inter quod, et <var>.<lb/>n.e.</var> inuentæ cum fuerint duæ lineæ <lb/>mediæ proportiona es <var>.s.</var> et <var>.r.</var> <reg norm="quarum" type="context">quarÅ©</reg> <var>.<lb/>s.</var> proximior Å¿it <var>.n.p.</var> quæ <reg norm="quidem" type="context">quidẽ</reg> <var>.s.</var> erit <lb/>latus cubi quæſiti.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0367-02" xlink:href="fig-0367-02a"> - <image file="0367-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0367-02"/> - </figure> - </div> - <pb o="356" rhead="IO. BAPT. BENED." n="368" file="0368" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0368"/> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">Duplex modus par allelam orizontalem alicui muro propoÅ¿ito <lb/>una tantummodo statione ducendi.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">DVcere parallelam orizontalem alicui muro recto propoÅ¿ito vna tantummodò <lb/>Å¿tatione, non Å¿olum poſſibile eÅ¿t Å¿ed etiam facile.</s> - </p> - <p> - <s xml:space="preserve">Sit exempli gratia murus rectus <var>.a.d.</var> Å¿itus verò <var>.o.n</var>. </s> - <s xml:space="preserve">Si cupimus ducere <var>.n.u.</var> <lb/>parallelam dicto muro, accipiatur quadratum geometricum, Å¿eu Å¿cala altimetra <lb/>vel aliquod Å¿imile inÅ¿trumentum, quo mediante à ſitu <var>.o.</var> videbimus punctum <var>.q.</var> <lb/>quod volueris ipÅ¿ius muri, <reg norm="dexteram" type="context">dexterã</reg> <lb/>verÅ¿us, inferius tamen. ipÅ¿o <var>.o.</var> vnde <lb/> - <anchor type="figure" xlink:label="fig-0368-01a" xlink:href="fig-0368-01"/> - formatum habebimus triangulum <var>.<lb/>n.o.q</var>. </s> - <s xml:space="preserve">Quo facto ad partem <reg norm="Å¿iniÅ¿tram" type="context">Å¿iniÅ¿trã</reg> <lb/>cum eodem angulo <var>.n.o.q.</var> oporte-<lb/>bit nos inuenire punctum aliquod <var>.<lb/>p.</var> in dicta Å¿uperficie muri, </s> - <s xml:space="preserve">& tunc <lb/>habebimus angulum <var>.n.o.p.</var> æqua-<lb/>lem angulo <var>.n.o.q.</var> vnde angulus <var>.q.<lb/>n.p.</var> nobis cognitus erit, <reg norm="duoque" type="simple">duoqÌ;</reg> late <lb/>ra <var>.n.q.</var> et <var>.n.p.</var> erunt inuicem æqua-<lb/>lia, ex .26. primi Euclid. cum angu-<lb/>li <var>.q.o.n.</var> et <var>.q.n.o.</var> Å¿int æquales angu <lb/>lis <var>.p.o.n.</var> et <var>.p.n.o.</var> & latus <var>.o.n.</var> com <lb/>mune, vnde angulus <var>.q.n.g.</var> extrinÅ¿e <lb/>cus trianguli <var>.p.q.n.</var> <reg norm="reÅ¿iduusque" type="simple">reÅ¿iduusqÌ;</reg> ex <lb/>duobus rectis nobis cognitus erit, <lb/>etiam & eius medictas <var>.q.n.u.</var> æqua <lb/>lis angulo <var>.p.q.n.</var> eo quod ex .5. pri-<lb/>mi, anguli <var>.q.p.</var> Å¿unt inuicem æquales, & ex .32. eiuÅ¿dem, æquales Å¿unt extrinÅ¿eco <var>.q.n.<lb/>g.</var> & ex 27. <var>n.u.</var> erit parallela ipÅ¿i <var>.q.p</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0368-01" xlink:href="fig-0368-01a"> - <image file="0368-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0368-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Aliter etiam poſſumus idem efficere, Å¿umendo duo illa puncta in Å¿uprem a linea <lb/>orizontali ipÅ¿ius muri ad Å¿uperiorem partem aÅ¿piciendo, quemadmodum ad infe-<lb/>riorem, quod vnum & idem erit, dummodò non aÅ¿piciamus orizontaliter, eo quod <lb/>nos oportet Å¿uperficiem conicam producere, linea viÅ¿uali mediante. </s> - <s xml:space="preserve">cognoÅ¿cere auÂ<lb/>tem angulum <var>.q.n.p.</var> facile erit, conÅ¿tituendo primò inÅ¿trumentum in Å¿itu trianguli <var>.<lb/>o.n.q.</var> <reg norm="aÅ¿piciendoque" type="simple">aÅ¿piciendoqÌ;</reg> punctum <var>.c.</var> in Å¿uperficie <var>.n.q.o.</var> & Å¿ic in alia parte, exiÅ¿tente in-<lb/>Å¿trumento in Å¿itu trianguli <var>.o.p.n.</var> aÅ¿picere oportet punctum <var>.e.</var> proximum puncto <var>.n.</var> <lb/>vbi poſſit metiri angulum <var>.c.n.e</var>.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i Å¿itus puncti <var>.n.</var> talis eſſet, vt ab eo non poſſet aliquis murum videre ad re-<lb/>ctos angulos, aÅ¿piceremus punctum <var>.q.</var> Å¿ub orizontali ab oculis noÅ¿tris, in orizontali <lb/>tamen puncti <var>.n.</var> ita quod angulus <var>.o.n.q.</var> rectus exiÅ¿tat, quo facto obÅ¿eruando angu-<lb/>lum <var>.n.o.q.</var> eo mediante, medianteqÌue <var>.n.o.</var> cum angulo <var>.o.n.q.</var> cognoÅ¿cemus <lb/>quantitatem diÅ¿tantiæ <var>.n.q.</var> idem etiam faciendum eÅ¿t cum alio puncto <var>.p.</var> quod <lb/>volueris, & mediantibus duobus punctis inuicem proximis <var>.c.e.</var> cognoÅ¿catur an- - <pb o="357" rhead="EPISTOL AE." n="369" file="0369" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0369"/> - gulus <var>.p.n.q.</var> vnde ex methodo .56. <lb/> - <anchor type="figure" xlink:label="fig-0369-01a" xlink:href="fig-0369-01"/> - primi triangulorum Monteregij, <lb/>cognoÅ¿cemus reliqua trianguli <var>.<lb/>q.p.n</var>. </s> - <s xml:space="preserve">ConÅ¿tituendo poÅ¿tea angu-<lb/>lum <var>.q.n.u.</var> æqualem angulo <var>.n.q.p.</var> <lb/>propoÅ¿itum habebimus.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0369-01" xlink:href="fig-0369-01a"> - <image file="0369-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0369-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Si etiam puncta <var>.q.p.</var> lineæ <var>.q.p.</var> <lb/>orizontali in eodem plano non exi <lb/>Å¿terent cum puncto <var>.n.</var> nihil refer-<lb/>ret, dummodo in pauimento <reg norm="notem" type="context">notẽ</reg> <lb/>tur <reg norm="puncta" type="context">pÅ©cta</reg> <var>.c.e.</var> proxima <var>.n.</var> in ijÅ¿dem <lb/>Å¿uperficiebus triangulorum <var>.n.o.p.</var> <lb/>et <var>.n.o.q.</var> vnde <var>.n.c.</var> et <var>.n.e.</var> erunt <reg norm="com- munes" type="context">cõ-<lb/>munes</reg> Å¿ectiones dictarum Å¿uperficierum cum Å¿uperficie pauimenti Å¿upra quam fit <lb/>Å¿tatio.</s> - </p> - </div> - </div> - <div type="section" level="3" n="32"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">CONI RECTI DIVISIO A PLANO <lb/>parallelo baÅ¿i Å¿ecundum datam proportionem.</head> - <head style="it" xml:space="preserve">Rapbaeli de Auria.</head> - <p> - <s xml:space="preserve"><emph style="sc">QVotiescvnqve</emph> volueris conum rectum diuidere à plano parallelo ba-<lb/>Å¿i Å¿ecundum vnam datam proportionem, nullius tibi erit difficultatis, con <lb/>ceſſa <reg norm="tamen" type="wordlist">tamẽ</reg> pro inuenta diuiÅ¿ione cuiuÅ¿uis propoÅ¿itÄ™ proportionis per tres <lb/>æquales partes.</s> - </p> - <p> - <s xml:space="preserve">Sit exempli gratia conus rectus <var>.a.b.c.</var> Å¿ecandus vt dictum eÅ¿t, accipiatur latus <lb/>ipÅ¿ius, quod Å¿it <var>.a.c.</var> <reg norm="ipÅ¿umque" type="simple">ipÅ¿umqÌ;</reg> diuidatur in puncto <var>.d.</var> Å¿ecundum illam proportionem <lb/>quam deÅ¿ideras, hoc eÅ¿t ipÅ¿ius <var>.a.c.</var> ad <var>.a.d.</var> quo facto, inter totum <var>.a.c.</var> et <var>.a.d.</var> inuenian <lb/>tur duæ lineæ proportionales, quarum maior Å¿it <var>.a.i.</var> </s> - <s xml:space="preserve">tunc Å¿i conus <var>.a.b.c.</var> Å¿ectus fue-<lb/>rit à plano per punctum <var>.i.</var> parallelo baÅ¿i, habebimus quod quærebamus.</s> - </p> - <p> - <s xml:space="preserve">Cuius rei ratio, primò eÅ¿t, quia quotieÅ¿cunque conus aliquis Å¿ectus fuerit ab ali-<lb/>quo plano parallelo baÅ¿i ipÅ¿ius, pars Å¿uperior Å¿imilis Å¿emper erit totali cono, quod <lb/>ita probo, cogitemus conum Å¿ectum eſſe <lb/>à plano per axem <var>.a.l.</var> vnde ex .3. primi <lb/> - <anchor type="figure" xlink:label="fig-0369-02a" xlink:href="fig-0369-02"/> - Pergei, talis Å¿ectio triangularis erit, quæ <lb/>Å¿it <var>.a.b.c.</var> et <var>.b.c.</var> diameter erit baÅ¿is.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0369-02" xlink:href="fig-0369-02a"> - <image file="0369-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0369-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Imaginemur deinde <var>.K.i.</var> communem <lb/>eſſe Å¿ectionem huiuÅ¿modi trianguli cum <lb/>plano parallelo ipÅ¿i baÅ¿i, </s> - <s xml:space="preserve">tunc tale <reg norm="planum" type="context">planÅ©</reg>, <lb/>circulare erit ex .4. primi ipÅ¿ius Pergei <var>.K.<lb/>i.</var> verò, eius diameter erit, et <var>.a.m.</var> <reg norm="Å¿uus" type="simple">Å¿uê°</reg> axis.</s> - </p> - <p> - <s xml:space="preserve">Cum verò <var>.a.l.</var> Å¿it perpendicularis ipÅ¿i <lb/>baÅ¿i conitotalis, eo quod rectus Å¿upponi-<lb/>tur, ideo eadem <var>.a.m.l.</var> erit perpendicula <lb/>ris eriam ipÅ¿i Å¿ecundo plano circulari, ex <lb/>conuerÅ¿a .14. vndecimi Euclid. </s> - <s xml:space="preserve">vnde ex - <pb o="358" rhead="IO. BAPT. BENED." n="370" file="0370" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0370"/> - Å¿ecunda definitione eiuÅ¿dem libr <var>.a.m.l.</var> efficiet angulos rectos cum duabus <var>.b.c.</var> et <var>.K.<lb/>i.</var> in punctis <var>.m.</var> et <var>.l.</var> et <var>.k.i.</var> parallela erit ipÅ¿i <var>.b.c.</var> ex .28. primi, quod etiam poteÅ¿t con <lb/>cludi mediante .16. vndecimi, cum <var>.k.i.</var> et <var>.b.c.</var> Å¿int communes Å¿ectiones duorum pla <lb/>norum cum triangulari. </s> - <s xml:space="preserve">Deinde ex .29. primi anguli <var>.a.i.m.</var> et <var>.a.c.l.</var> erunt inuicem <lb/>æquales, idem etiam dico de angulis <var>.a.k.i.</var> et <var>.a.b.c.</var> anguli poÅ¿tea ad <var>.a.</var> communes <lb/>Å¿unt triangulis <var>.l.a.c.</var> et <var>.m.a.i.</var> vt triangulis <var>.l.a.b.</var> et <var>.m.a.k</var>. </s> - <s xml:space="preserve">Vnde ex .4. Å¿exti, eadem <lb/>proportio erit ipÅ¿ius <var>.m.i.</var> ad <var>.l.c.</var> & ipÅ¿ius <var>.m.k.</var> ad <var>.l.b.</var> vt ipÅ¿ius <var>.a.m.</var> ad <var>.a.l</var>. </s> - <s xml:space="preserve">Quare ex <lb/>vndecima quinti, ita erit ipÅ¿ius <var>.m.k.</var> ad <var>.l.b.</var> vt ipÅ¿ius <var>.m.i.</var> ad <var>.l.c.</var> & ex .13. eiuÅ¿dem, ita <lb/>erit ipÅ¿ius <var>.k.i.</var> ad <var>.b.c.</var> vt <var>.m.i.</var> ad <var>.l.c.</var> Å¿ed ipÅ¿ius <var>.m.i.</var> ad <var>.l.c.</var> eÅ¿t vt ipÅ¿ius <var>.a.m.</var> ad <var>.a.l.</var> quod <lb/>iam dictum eÅ¿t, vnde ex .11. dicta, ita erit ipÅ¿ius <var>.k.i.</var> ad <var>.b.c.</var> vt ipÅ¿ius <var>.a.m.</var> ad <var>.a.l.</var> & ex <lb/>16. dicti ita erit ipÅ¿ius <var>.a.m.</var> ad <var>.k.i.</var> vt ipÅ¿ius <var>.a.l.</var> ad <var>.b.c</var>. </s> - <s xml:space="preserve">Quare ex definitione ab Eu-<lb/>cli. poÅ¿ita in .11, lib. pars coni Å¿uperior Å¿imilis erit cono totali.</s> - </p> - <p> - <s xml:space="preserve">Deinde Å¿ciendum eÅ¿t illud quod Euclid. Å¿cribit in .10. duodecimi lib. hoc eÅ¿t, <reg norm="quod" type="simple">ê™</reg> <lb/>proportio duarum pyramidum inuicem <lb/>Å¿imilium, triplicata eÅ¿t ei diametrorum <lb/> - <anchor type="figure" xlink:label="fig-0370-01a" xlink:href="fig-0370-01"/> - Å¿uarum baÅ¿ium, hoc eÅ¿t, quod proportio <var>.<lb/>b.c.</var> ad <var>.k.i.</var> tertia pars erit proportionis to <lb/>tius pyramidis <var>.a.b.c.</var> partiali pyramidi <var>.a.<lb/>k.i.</var> Å¿ed ita eÅ¿t ipÅ¿ius <var>.a.c.</var> ad <var>.a.i.</var> vt ipÅ¿ius <var>.b.<lb/>c.</var> ad <var>.k.i.</var> ex .4. Å¿exti cum trianguli <var>.a.b.c.</var> <lb/>et <var>.a.k.i.</var> Å¿int æquianguli, quod ex ijs, quÄ™ <lb/>Å¿uperius diximus facile comprÄ™henditur. <lb/></s> - <s xml:space="preserve">Quare <reg norm="proportio" type="simple">ê“portio</reg> <var>.a.c.</var> ad <var>.a.i.</var> tertia pars erit <lb/>proportionis totius coni <var>.a.b.c.</var> ad eius par <lb/>tem abÅ¿ciſſam <var>.a.k.i.</var> Å¿ed eadem proportio <lb/>ipÅ¿ius <var>.a.c.</var> ad <var>.a.i.</var> erat etiam tertia pars pro <lb/>portionis ipÅ¿ius <var>.a.c.</var> ad <var>.a.d</var>. </s> - <s xml:space="preserve">Quare ex com <lb/>muni conceptu, proportio totius pyramidis, ad partem abÅ¿ciſſam, æqualis erit pro-<lb/>portioni ipÅ¿ius <var>.a.c.</var> ad <var>.a.d</var>.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0370-01" xlink:href="fig-0370-01a"> - <image file="0370-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0370-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">De differentia caloris Solis propter vaporum<unsure/> <lb/>altitudinem.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">NOlo, mihi credas, Å¿ed ex rationibus, quas tibi Å¿cribo conÅ¿idera, quod quo <lb/><reg norm="tieÅ¿cunque" type="simple">tieÅ¿cunq;</reg> craſſities vel <reg norm="denÅ¿itas" type="context">dẽſitas</reg> <reg norm="vaporum" type="context">vaporÅ©</reg>, Å¿eu altitudo, maior eſſet ea, quÄ™ nunc re-<lb/>peritur, </s> - <s xml:space="preserve">tunc minor differentia eſſet inter maiorem <reg norm="minoremque" type="simple">minoremqÌ;</reg> calorem Solis, quam <lb/>nunc Å¿entiamus. </s> - <s xml:space="preserve">Pro cuius rei euidentia, imaginemur in hac Å¿ubÅ¿cripta figura, li-<lb/>neam <var>.o.a.</var> pro Å¿emidiametro terræ, et <var>.a.c.</var> pro craſſitie vaporum, vt nunc Å¿e <lb/>habet, et <var>.a.d.</var> pro maiori craſſitie, imaginemurqÌue lineam <var>.a.b.</var> quaÅ¿i perpen-<lb/>dicularem ad <var>.o.a.</var> quæ abÅ¿ciſſa Å¿it in puncto u. à circunferentia <var>.c.u.</var> inferiori prio-<lb/>rum vaporum.</s> - </p> - <p> - <s xml:space="preserve">Tunc dico minorem eſſe proportionem ipÅ¿ius <var>.a.b.</var> ad <var>.a.d.</var> quam ipÅ¿ius <var>.a.u.</var> ad <var>.a.<lb/>c.</var> cogitemus ergo protractas eſſe lineas <var>.o.b</var>: <var>d.b</var>: <var>c.u.</var> et <var>.c.n.</var> quæ <var>.c.n.</var> Å¿ecabit <var>.a.u.</var> in - <pb o="359" rhead="EPISTOL AE." n="371" file="0371" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0371"/> - puncto <var>.i.</var> ex communi conceptu, & <lb/> - <anchor type="figure" xlink:label="fig-0371-01a" xlink:href="fig-0371-01"/> - parallcla erit ipÅ¿i <var>.d.b.</var> ex. Å¿ecunda par-<lb/>te Å¿ecundæ Å¿exti, vnde ex prima parte <lb/>ciuÅ¿dem, ita eritipÅ¿ius <var>.b.i.</var> ad <var>.i.a.</var> vt <var>.d.<lb/>c.</var> ad <var>.c.a.</var> & coniunctim ita erit ipÅ¿ius <var>.b.<lb/>a.</var> ad <var>.a.i.</var> vt ipÅ¿ius <var>.d.a.</var> ad <var>.a.c.</var> & permu <lb/>tatim ipÅ¿ius <var>.a.b.</var> ad <var>.a.d.</var> erit, vt <var>.a.i.</var> <lb/>ad <var>.a.c.</var> Å¿ed cum <var>.a.u.</var> maior Å¿it ipÅ¿a <var>.a.i.</var> <lb/>vt omne totum maius eÅ¿t Å¿ua parte. <lb/></s> - <s xml:space="preserve">maior proportio erit ipÅ¿ius <var>.a.u.</var> ad <var>.a.<lb/>c.</var> quam ipÅ¿ius <var>.a.i.</var> ad <var>.a.c.</var> hoc eÅ¿t quam <lb/>ipÅ¿ius <var>.a.b.</var> ad <var>.a.d</var>. </s> - <s xml:space="preserve">Verum igitur eÅ¿t <lb/>propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0371-01" xlink:href="fig-0371-01a"> - <image file="0371-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0371-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">De differentia caloris Solis reÅ¿pectu altitudinis ipÅ¿ius.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">QVodà me poÅ¿tulas deinde, ita Å¿e habet. </s> - <s xml:space="preserve">Inquis enim, quod cum differentia <lb/>inter maiorem, <reg norm="minoremque" type="simple">minoremqÌ;</reg> calorem, oriatur etiam ex differentia maioris <lb/>quantitatis vaporum ad minorem, per quam quantitatem vaporum rranÅ¿it lumen <lb/>Solis (vt alias etiam tibi dixi) velles nunc Å¿cire quantitatem ipÅ¿ius differentiÄ™, quæ <lb/>inter duas Solis datas altitudines Å¿upra orizontem reperitur.</s> - </p> - <p> - <s xml:space="preserve">Quapropter imaginemur circulum <var>.a.e.</var> pro magno terræ, et <var>.z.b.d.</var> pro magno <lb/>vaporum, Å¿upponatur etiam quod angulus <var>.z.o.d.</var> vel <var>.z.a.b.</var> qui Å¿unt inuicem fe-<lb/>rè æquales, Å¿it angulus diÅ¿tantiæ Solis à zenit, <var>z.a.</var> verò Å¿it Å¿piſſitudo vaporum, et <var>.a.<lb/>b.</var> radius tranÅ¿iens per vapores dictos. </s> - <s xml:space="preserve">nunc <lb/> - <anchor type="figure" xlink:label="fig-0371-02a" xlink:href="fig-0371-02"/> - quæratur proportio, quæ eÅ¿t inter <var>.a.b.</var> et <var>.a.<lb/>z.</var> qua inuenta, angulo <var>.z.a.b.</var> mediante, <lb/>quæremus eandem mediante angulo <var>.z.a.b.</var> <lb/>maiore priori, velipÅ¿o minore, vnde cogno <lb/>Å¿cemus differentiam duarum <var>.a.b.</var> quæ qui-<lb/>dem inæquales inuicem erunt, eo quod Å¿up <lb/>ponatur <var>.a.z.</var> immutabilis, & hoc ita facie-<lb/>mus. </s> - <s xml:space="preserve">Imaginabimur <var>.o.b.</var> quæ claudat trian <lb/>gulum <var>.a.b.o.</var> & quia <var>.a.z.</var> cognita eÅ¿t quam <lb/>Alhazem docetinuenire, cognoÅ¿cimus <reg norm="etiam" type="context">etiã</reg> <lb/><var>o.a.</var> vt Å¿emidiametrum terræ, vnde <var>.o.b.</var> et <var>.<lb/>o.a.</var> duo latera trianguli <var>.a.o.b.</var> cognita <reg norm="erunt" type="context">erÅ©t</reg> <lb/>Å¿imul cum angulo <var>.o.a.b.</var> reÅ¿iduo duorum re <lb/>ctorum, eo quod reliquus <var>.z.a.b.</var> datus eÅ¿t. <lb/></s> - <s xml:space="preserve">Quare <var>.a.b.</var> cognita erit reÅ¿pectu <var>.o.a.</var> et <var>.o.<lb/>b.</var> et <var>.a.z.</var> quæ eÅ¿t eorum differentia. </s> - <s xml:space="preserve">Nunc <lb/>Å¿i idem faciemus cum alia <var>.a.b.</var> Å¿ub diuerÅ¿o <lb/>angulo, habebimus propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0371-02" xlink:href="fig-0371-02a"> - <image file="0371-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0371-02"/> - </figure> - </div> - <pb o="360" rhead="IO. BAPT. BENED." n="372" file="0372" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0372"/> - </div> - </div> - <div type="section" level="3" n="33"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">NOTABILES ERRORES ORONTII <lb/>& Tartaleæ.</head> - <head style="it" xml:space="preserve">Cornelio Bitonto.</head> - <p> - <s xml:space="preserve"><emph style="sc">PArvvs</emph> error non fuit, vt putabat Orontius, quodanguli triangulorum <lb/>æquicrurium inuicem æqualium, baÅ¿ibus oppoÅ¿iti, ijÅ¿dem baÅ¿ibus propor <lb/>tionales eſſent, cuius opinionis cauÅ¿a fuit quod nunquam viderit vel me <lb/>minerit eius quod Ptolomeus Å¿cripÅ¿it lib. primo AlmageÅ¿ti, vbi de diÅ¿pro <lb/>portionalitate chordarum <reg norm="arcuumque" type="simple">arcuumqÌ;</reg> tractat, vel quod Å¿cribit Vitellio lib. primo pro <lb/>poÅ¿itione .35. Å¿eu lib. quarto, propoÅ¿itione .21. quod idem eÅ¿t. </s> - <s xml:space="preserve">Sed nec ego tibi pro <lb/>ponam id quod Å¿cribit Nicolaus Tartalea diuiÅ¿ioni .28. quinti capitis quartæ partis <lb/>Å¿uorum tractatuum, eo quod non exactè Å¿cientificè Å¿cripÅ¿erit, nec vniuerÅ¿aliter, <reg norm="quan- uis" type="context">quã-<lb/>uis</reg> talis propoÅ¿itio poſſit Å¿cientificè Å¿cribi, accipiendo <var>.b.c.</var> in eius figura, pro latere <lb/>octagoni, vnde angulus <var>.a.e.b.</var> duplum foret angulo <var>.b.e.c.</var> collocato poÅ¿tea <var>.b.c.</var> in <lb/>arcu <var>.a.b.</var> punctum <var>.c.</var> medium fuiſſet dicti arcus, et <var>.e.c.</var> diuideret <var>.a.b.</var> per æqualia, <lb/>ex quinta primi, nec non ad rectos ex .3. tertij, vnde ex .18. primi, clare vidiſſemus <lb/>non eſſe proportionem <var>.a.b.</var> ad <var>.b.c.</var> vt anguli ad angulum. </s> - <s xml:space="preserve">Sed vniuerÅ¿aliori modo <lb/>poſſumus hoc Å¿peculari. </s> - <s xml:space="preserve">Nam manifeÅ¿tè Å¿cimus, eandem eſſe proportionem circun <lb/>ferentiæ ad diametrum in omnibus circulis tam maioribus, quam minoribus. <lb/></s> - <s xml:space="preserve">Sint igitur duo anguli <var>.a.e.b.</var> et <var>.c.e.b.</var> cuiuÅ¿uis amplitudinis, quorum latera <var>.e.a</var>: <var>e.b</var>: <lb/>et <var>.e.c.</var> Å¿int inuicem æqualia, protrahatur <var>.b.a.</var> et <var>.b.c</var>. </s> - <s xml:space="preserve">Tunc dico maiorem proportio <lb/>nem eſſe anguli <var>.a.e.b.</var> ad angulum <var>.b.e.c.</var> quam <var>.a.b.</var> ad <var>.c.b.</var> ducatur enim <var>.b.g.</var> ita <reg norm="quod" type="simple">ê™</reg> <lb/>faciat angulum <var>.g.b.c.</var> æqualem angulo <var>.e.b.a.</var> protracta poÅ¿tea <var>.c.g.</var> quæ idem faciat <lb/>in puncto <var>.c.</var> vnde <var>.g.b.</var> et <var>.g.c.</var> æquales inuicem erunt ex .6. primi, & quia angulus <var>.a.</var> <lb/>æqualis eÅ¿t angulo <var>e.b.a.</var> ex quinta eiuÅ¿dem, ideo ex .32. dicti, et .4. Å¿exti, horum <lb/>duorum triangulorum latera, erunt inuicem proportionalia. </s> - <s xml:space="preserve">ConÅ¿tituto deinde <var>.g.</var> <lb/>centro, & Å¿ecundum Å¿emidiametrum <var>.g.b.</var> vel <var>.g.c.</var> quod idem eÅ¿t, deÅ¿cripto circu-<lb/>lo <var>.b.i.c.</var> necnon circulo <var>.b.c.a.</var> circa centrum <var>.e.</var> ope Å¿emidiametri <var>.e.b.</var> et <var>.e.a.</var> vn <lb/>de iÅ¿te circulus eritillo maior, cum <var>.e.b.</var> maior Å¿it <var>.g.b.</var> ex .14. quinti. cum ex .14. tertij <lb/><var>a.b.</var> longior Å¿it <var>.c.b.</var> Å¿ed ex vltima definitione tertij, arcus <var>.b.i.c.</var> et <var>.b.c.a.</var> erunt in-<lb/>uicem Å¿imiles, hoc eÅ¿t proportio totius cir-<lb/>cunferentiæ circuli <var>.b.i.c.</var> ad arcus <var>.b.i.c.</var> ea-<lb/> - <anchor type="figure" xlink:label="fig-0372-01a" xlink:href="fig-0372-01"/> - dem erit, quæ totius circunferentiæ circuli <lb/><var>b.c.a.</var> ad arcus <var>.b.c.a.</var> Å¿ed proportio diame-<lb/>tri ad circunferentiam eÅ¿t vt diametri ad cir <lb/>cunferentiam, vt Å¿upra diximus; </s> - <s xml:space="preserve">Quare ex <lb/>proportionum æqualitate, vt Å¿emidiametri <lb/>ad circunferentiam erit, vt Å¿emidiametri <lb/>ad circunferentiam, & per eandem propor <lb/>tionum Ä™qualitatem, proportio <var>.e.b.</var> ad <reg norm="arcum" type="context">arcÅ©</reg> <lb/><var>b.c.a.</var> erit, vt <var>.g.b.</var> ad arcum <var>.b.i.c.</var> & per ean <lb/>dem æqualitatem, ita erit <var>.a.b.</var> chordæ ad ar <lb/>cum <var>.b.c.a.</var> vt <var>.c.b.</var> chordæ ad arcum <var>.b.i.c.</var> <lb/>& permutando, ita erit chordæ <var>.a.b.</var> ad chor <lb/>dam <var>.c.b.</var> vt arcus <var>.b.c.a.</var> ad arcum <var>.b.i.c.</var> Å¿ed <lb/>arcus <var>.b.i.c.</var> maior eÅ¿t arcu <var>.b.d.c.</var> ex commu - <pb o="361" rhead="EPISTOL AE." n="373" file="0373" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0373"/> - ni Å¿cicntia. </s> - <s xml:space="preserve">Quare maior proportio erit acus <var>.b.c.a.</var> ad arcum <var>.b.d.c.</var> quam ad arcum <lb/><var>b.i.c.</var> ex .8. quinti. </s> - <s xml:space="preserve">Vnde ex vltima Å¿exti et .12. quinti, proportio anguli <var>.a.e.b.</var> ad an-<lb/>gulum <var>.c.e.b.</var> maior erit quam chordæ, Å¿iue baÅ¿is <var>.a.b.</var> ad chordam Å¿iue baÅ¿im <var>.c.b</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0372-01" xlink:href="fig-0372-01a"> - <image file="0372-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0372-01"/> - </figure> - </div> - </div> - </div> - <div type="section" level="3" n="34"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE CAVSA SVSPENSIONIS NVBIVM <lb/>in aere contra Antonium Bergam.</head> - <head style="it" xml:space="preserve">Clarißimo FranciÅ¿co Venerio.</head> - <p> - <s xml:space="preserve">EGo enim non tantum miror ea quæ mihi Å¿cripÅ¿iÅ¿ti de opinione OrtenÅ¿ij <lb/>quantum quod Antonius Berga putat nubes à Sole Å¿upenÅ¿as teneri, id pla <lb/>nè falÅ¿um eÅ¿t, vera cauÅ¿a huiuÅ¿modi effectus, alia nulla eÅ¿t, niÅ¿i earundem <lb/>raritas hoc eÅ¿t, cum rariores Å¿int ipÅ¿o aere Å¿ubiecto, </s> - <s xml:space="preserve">propterea Å¿upra <reg norm="ipsum" type="context">ipsÅ©</reg> <lb/>natant & Å¿tant Å¿ub eo qui rarior ipÅ¿is eÅ¿t, eo quod corpora rariora poÅ¿ita in medio <lb/>non tam raro, aÅ¿cendunt, & denÅ¿iora in medio minus denÅ¿o deÅ¿cendunt. </s> - <s xml:space="preserve">Nam Å¿i <lb/>Sol ipÅ¿as nubes Å¿uÅ¿penÅ¿as in aere teneret, hoc interdiu tantummodo fieret, Å¿ed no <lb/>ctu, cur non deÅ¿cendunt vÅ¿que ad terram, & in eodem loco Å¿emper manent? </s> - <s xml:space="preserve">Scien-<lb/>dum igitur eÅ¿t nubes aÅ¿cendere in altum quouÅ¿que inueniant aerem eiuÅ¿dem ra-<lb/>ritatis cuius ipſæ Å¿unt. </s> - <s xml:space="preserve">Raritas enim & denÅ¿itas non Å¿unt res viÅ¿ibiles niÅ¿i per acci-<lb/>dens, quemadmodum etiam leuitas, & grauitas, opacitas verò & diaphaneitas ma <lb/>gis <reg norm="compræhenduntur" type="context">compræhendÅ©tur</reg>, opacitas enim ex reflexione radiorum luminoÅ¿orum, diapha <lb/>neitas verò compræhenditur ex penetratione ipÅ¿orum radiorum, opacitas autem nu <lb/>bis non eÅ¿t denÅ¿itas, cum valde diuerÅ¿a Å¿it denÅ¿itas ab opacitate, Å¿icut raritas ab dia-<lb/>phaneitate, vt alià s dixi. </s> - <s xml:space="preserve">Et quando dicit, quod Sol calefaciendo aerem ipÅ¿am nu <lb/>bem ambientem, rarefaciat eum magis quam ipÅ¿am nubem reÅ¿pondeo, hoc verum <lb/>non eſſe, proptere<unsure/>a quodradius Solis non multum calefacit ea corpora, quæ ip Å¿i per <lb/>mittunt liberum tranÅ¿itum. </s> - <s xml:space="preserve">vnde corpora quanto magis diaphana Å¿unt tanto minus <lb/>ab ipÅ¿o radio luminoÅ¿o calefiunt, Å¿ed ea quæ magis opaca Å¿unt, magis etiam calefiunt <lb/>& per conÅ¿equens magis rarefiunt, cum calidi Å¿it per Å¿e rarefacere, & non attrahere, <lb/>vt ipÅ¿e & ferè omnes alij putant.</s> - </p> - </div> - </div> - <div type="section" level="3" n="35"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE RATIONE EXTENSIONIS FVNIS <lb/>cuiuÅ¿dam libramenti, & de quadam Å¿imboleita-<lb/>te circuli cum ellipÅ¿i.</head> - <head style="it" xml:space="preserve">Angelo Ferrario Serenißimi Ducis Sabaudia <lb/>AgrimenÅ¿ori expertißimo.</head> - <p> - <s xml:space="preserve"><emph style="sc">TIbi</emph> in mentem veniet, quod cum Å¿uperioribus diebus in villa lucenti, in <lb/>qua degebat Sereniſſimus Dux noÅ¿ter, dum viridarium ad æquilibrium <lb/>reducebas, eſſemus, à te quæſiui an Å¿cires vnde fieret, vt Å¿tante libramen-<lb/>to ad angulos rectos Å¿upra Å¿uum pedem, funis quæ extrema eiuÅ¿dem li-<lb/>bramenti cum pede in formam trianguli æquicruris coniungit, magis diÅ¿tentus exi-<lb/>Å¿teret, quam cum dictum libramentum cum pede obliquum remanet, ita vt huiuÅ¿- - <pb o="362" rhead="IO. BAPT. BENED." n="374" file="0374" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0374"/> - modifunis cum libramento triangulum Å¿calenum conÅ¿titueret.</s> - </p> - <p> - <s xml:space="preserve">Exempli gratia, ponamus lineam <var>.d.b.c.</var> eſſe libramentum .et <var>.b.e.u.</var> eius pedem, <lb/>funem autem, qui aliquando cum libramento facit triangulum iÅ¿ocellum, & aliquan <lb/>do Å¿calenum, eſſe <var>.d.e.c.</var> eÅ¿to etiam quod in figura <var>.A.</var> dictus triangulus <var>.d.e.c.</var> Å¿it iÅ¿o-<lb/>cellus, & in figura <var>.B.</var> Å¿calenus. </s> - <s xml:space="preserve">Tunc quæſiui à te an Å¿cires rationem, quare <lb/>funis <var>.d.e.c.</var> in figura <var>.A.</var> eſſet diÅ¿tenÅ¿us, & in figura <var>.B.</var> laxus quemadmodum vide-<lb/>bamus. </s> - <s xml:space="preserve">cum mihireÅ¿ponderis, neÅ¿cio quid, quod nunc memoria <reg norm="non" type="context">nõ</reg> teneo, Å¿ed quia <lb/>pollicitus Å¿um metibi eam afferre, propterea nunc ad te mitto. </s> - <s xml:space="preserve">Scias ergo huiuÅ¿-<lb/>modirationem nihil aliud eſſe niÅ¿i quod in figura <var>.A.</var> duæ lineæ <var>.c.e.</var> et <var>.d.e.</var> Å¿imul è <lb/>directo iunctæ longiores Å¿int illis, quÄ™ reperiuntur in figura <var>.B.</var> Å¿ed quia funis tam in <lb/>figura <var>.B.</var> quam in figura <var>.A.</var> vnus, & idem eÅ¿t, ideo in figura <var>.B.</var> laxatus eÅ¿t, & non in <lb/>tenÅ¿us, ut in figura <var>.A</var>. </s> - <s xml:space="preserve">Sed vt huiuÅ¿modi veritatis certam notitiam habeas, infraÅ¿cri <lb/>ptum circulum mente concipe <var>.f.e.i.</var> cuius Å¿emidiameter, æqualis Å¿it <var>.b.e.</var> & diame-<lb/>ter Å¿it <var>.f.i.</var> in quo imaginare eſſe tuum <lb/>libramentum <var>.d.b.c.</var> & figuras <var>.A.</var> et <var>.B.</var> <lb/> - <anchor type="figure" xlink:label="fig-0374-01a" xlink:href="fig-0374-01"/> - <anchor type="figure" xlink:label="fig-0374-02a" xlink:href="fig-0374-02"/> - & pr obabo lineas <var>.d.e.c.</var> figurÄ™ <var>.A.</var> lon <lb/>giores eſſe lineis <var>.d.e.c.</var> figuræ <var>.B</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0374-01" xlink:href="fig-0374-01a"> - <image file="0374-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0374-01"/> - </figure> - <figure xlink:label="fig-0374-02" xlink:href="fig-0374-02a"> - <image file="0374-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0374-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Imaginemur igitur lineam <var>.b.e.</var> eſſe <lb/>dimidium minoris axis <reg norm="alicuius" type="simple">alicuiê°</reg> ellipÅ¿is <lb/>cuius quidem figuræ ponamus <var>.d.</var> et <var>.c.</var> <lb/>centra ipÅ¿ius circunÅ¿criptionis eſſe, cu <lb/>ius <reg norm="circunferentia" type="context">circunferẽtia</reg>, nullidubium eÅ¿t, quin <lb/>extra propoÅ¿itum circulum tranÅ¿itura, <lb/>& in vno tantummodo puncto ipÅ¿um <lb/>circulum tactura Å¿it, qui exiÅ¿tat <var>.e.</var> <lb/> figuræ <var>.A.</var> Å¿eparatum tamen à puncto <lb/>e. figuræ <var>.B</var>. </s> - <s xml:space="preserve">Tunc Å¿i protracta fue-<lb/>rit linea <var>.d.e.</var> figuræ <var>.B.</var> vÅ¿que ad gi <lb/> - <anchor type="figure" xlink:label="fig-0374-03a" xlink:href="fig-0374-03"/> - rum ellipticum in puncto <var>.g.</var> à quo <lb/>ad punctum <var>.c.</var> ducta etiam Å¿it linea <lb/><var>g.c</var>. </s> - <s xml:space="preserve">tunc <reg norm="manifeÅ¿tum" type="context">manifeÅ¿tÅ©</reg> erit duas lineas <lb/><var>d.e.</var> et <var>.e.c.</var> figuræ <var>.A.</var> Å¿imul iunctas, <lb/>æquales eſſe duabus <var>.d.g.</var> et <var>.g.c.</var> Å¿i-<lb/>mul poÅ¿itis, vt etiam ex .52. tertij <lb/>Pergei facilè videre eÅ¿t, Å¿ed ex .21. <lb/>primi Euclid. iam certò Å¿cimus <var>.d.g.c.</var> longiores eſſe <var>.d.e.c.</var> Å¿iguræ <var>.B.</var> ergo <var>.d.e.c.</var> figu-<lb/>ræ <var>.A.</var> longiores Å¿unt <var>.d.e.c.</var> figuræ <var>.B.</var> quod eÅ¿t propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0374-03" xlink:href="fig-0374-03a"> - <image file="0374-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0374-03"/> - </figure> - </div> - <p> - <s xml:space="preserve">Quod etiam mihinunc circa hoc Å¿uccurrit, tibi libenter Å¿ignifico, hoc eÅ¿t, quod <lb/>Å¿icut in ellipÅ¿i duæ lineæ <var>.d.e.e.c.</var> figuræ <var>.A.</var> Å¿imul iunctæ, Å¿unt Å¿emper æquales duabus <lb/>lineis <var>.d.g.g.c.</var> in longitudine, ita in circulo duæ <var>.d.e.e.c.</var> figuræ <var>.A.</var> æquales Å¿unt in <lb/>potentia duabus <var>.d.e.e.c.</var> figurÄ™ <var>.B</var>.</s> - </p> - <p> - <s xml:space="preserve">ManifeÅ¿tum enim primum eÅ¿t ex penultima primi in figura <var>.A.</var> quadratum <var>.e.c.</var> <lb/>æquale eſſe duobus quadratis Å¿cilicet <var>.e.b.</var> et <var>.b.c.</var> & quadratum <var>.e.d.</var> æquale duobus <var>.<lb/>e.b.</var> et <var>.b.d</var>. </s> - <s xml:space="preserve">Quare quadrata <var>.e.c.</var> et <var>.e.d.</var> æqualia Å¿unt quadratis <var>.e.b.</var> figuræ <var>.A.</var> et <var>.e.<lb/>b.</var> figurÄ™. B et <var>.b.c.</var> et <var>.b.d.</var> hoc eÅ¿t duplo quadrati <var>.e.a.</var> (ducta cum fuerit <var>.e.a.</var> perpen-<lb/>dicularis ad <var>.c.b.d.a.</var>) duplo quadrati <var>.a.b.</var> ex penultima primi, & duplo quadrati <var>.b.<lb/>c</var>. </s> - <s xml:space="preserve">Sed quadrata <var>.d.e.</var> et <var>.e.c.</var> figurÄ™ <var>.B.</var> æqualia Å¿unt duplo quadrati <var>.a.e.</var> & quadrato <var>a.d.</var> - <pb o="363" rhead="EPISTOL AE." n="375" file="0375" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0375"/> - & qua drato <var>.a.c.</var> ex <reg norm="eadem" type="context">eadẽ</reg>. </s> - <s xml:space="preserve">Nunc videndum eÅ¿t <reg norm="vtrum" type="context">vtrÅ©</reg> <reg norm="duplum" type="context">duplÅ©</reg> quadrati <var>.a.e.</var> <reg norm="cum" type="context">cÅ©</reg> duplo qua <lb/>drati <var>.b.a.</var> <reg norm="cum" type="context">cÅ©</reg> duplo quadrati <var>.b.c.</var> Å¿it æquale duplo quadrati <var>.a.e.</var> <reg norm="cum" type="context">cÅ©</reg> quadrato <var>.a.d.</var> & <lb/>cum quadrato <var>.a.c</var>. </s> - <s xml:space="preserve">Sed quia tam ex vna parte quà m ex alia habemus duplum qua-<lb/>drati <var>.a.e</var>. </s> - <s xml:space="preserve">Videndum igitur erit vtrum duplum quadrati <var>.a.b.</var> Å¿imul cum duplo qua-<lb/>drati <var>.b.c.</var> Ä™quale Å¿it quadrato <var>.a.c.</var> cum quadrato <var>.a.d.</var> Å¿ed hoc manifeÅ¿tum eÅ¿t .ex .10. <lb/>Å¿ecundi Euclidis, dato quod <reg norm="punctum" type="context">punctÅ©</reg> <var>.a.</var> Å¿it inter <var>.f.</var> et <var>.d.</var> Å¿ed Å¿i fuerit inter <var>.d.</var> et <var>.b.</var> hoc <lb/>manifeÅ¿tum erit ex .9. Å¿ecundi dicti, nihilominus accipe hunc alium modum.</s> - </p> - <p> - <s xml:space="preserve">Sit hic Å¿ubÅ¿criptum quadratum <var>.D.</var> ex <var>.a.c.</var> in Å¿eipÅ¿a producta, cuius diameter Å¿it <lb/><var>a.n.</var> <reg norm="protrahanturque" type="simple">protrahanturqÌ</reg> parallelÄ™ <var>.d.h</var>: <var>b.K</var>: <var>l.m.o.</var> et <var>.r.q.s.</var> <reg norm="eique" type="simple">eiqÌ;</reg> addatur <var>.c.p.</var> ad <var>.a.c.</var> æqua-<lb/>lis tamen <var>.d.a.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> protracta <var>.p.u.</var> vÅ¿que ad <var>.m.o.u.</var> vnde habebimus <var>.a.n.</var> pro totali <lb/>quadrato, et <var>.p.s.</var> pro partiali, & æquali quadrato lineæ <var>.a.d</var>. </s> - <s xml:space="preserve">Videndum nunc eÅ¿t, <reg norm="vtrum" type="context">vtrÅ©</reg> <lb/>hÄ™c duo quadrata æqualia Å¿int duobus quadratis lineæ <var>.a.b.</var> & duobus lineæ <var>.b.c.</var> <reg norm="Nam" type="context">Nã</reg> <lb/>duo quadrata lineæ <var>.b.c.</var> Å¿int <var>.K.o.</var> et <var>.h.l.</var> videndum nunc eÅ¿t utrum reÅ¿iduum Ä™quale <lb/>Å¿it duobus quadratis lineÄ™ <var>.a.b.</var> quorum vnum Å¿it <var>.m.b.</var> alterum verò <var>.l.p.</var> quod Å¿upe-<lb/>rat <var>.l.c.</var> et <var>.s.p.</var> figuræ <var>.D.</var> per Å¿upplementum <var>.o.t.</var> cui æquale eÅ¿t parallelogrammum <var>.h.<lb/>m.</var> figuræ <var>.D.</var> Å¿ed Å¿i punctus <var>.a.</var> poÅ¿itus fuerit inter <var>.d.</var> et <var>.b.</var> conÅ¿tituto quadrato <var>.d.u.</var> <reg norm="cum" type="context">cÅ©</reg> <lb/>omnibus parallelis, vtin figura <var>.C.</var> viderelicet, in qua figura videbimus quadrata <var>.r.<lb/>n.</var> et <var>.d.r.</var> Ä™quari duplo quadratorum <var>.l.n.</var> et <var>.r.l.</var> nam in quadrato <var>.r.n.</var> ipÅ¿a duo quadra-<lb/>ta <var>.l.n.</var> et <var>.r.l.</var> capiuntur, reliquum eÅ¿t igitur vt videamus an duo Å¿upplementa <var>.l.t.</var> et <var>.l.<lb/>s.</var> cum quadrato <var>.d.r.</var> Å¿int æqualia dictis <reg norm="quadratis" type="typo">qÌuadratis</reg> <var>.l.n.</var> et <var>.r.l.</var> Å¿ed quadratum <var>.d.l.</var> æ qua-<lb/>tur quadrato <var>.l.n.</var> videndum igitur eÅ¿t, <lb/>an duo Å¿upplementa <var>.l.t.</var> et <var>.l.s.</var> cum qua <lb/> - <anchor type="figure" xlink:label="fig-0375-01a" xlink:href="fig-0375-01"/> - drato <var>.d.r.</var> Å¿int æqualia duobus quadra <lb/>tis <var>.d.l.</var> et <var>.r.l.</var> Å¿ed quadratum <var>.d.l.</var> æqua-<lb/>tur quadrato <var>.d.r.</var> & Å¿upplemento <var>.l.t.</var> <lb/>mediante <var>.q.l.</var> & Å¿upplemento <var>.r.b.</var> Å¿up-<lb/>plementum verò <var>.l.s.</var> Å¿uperat <reg norm="Å¿upplemem" type="context">Å¿upplemẽ</reg> <lb/>tum <var>.r.b.</var> per quantitatem <reg norm="æqualem" type="context">æqualẽ</reg> qua-<lb/>drato <var>.r.l.</var> </s> - <s xml:space="preserve">quare duo Å¿upplementa <var>.l.t.</var> <lb/>et <var>.l.s.</var> cum quadrato <var>.d.r.</var> æquantur qua <lb/>drato <var>.d.l.</var> <reg norm="cum" type="context">cÅ©</reg> quadrato <var>.l.r.</var> verum igitur eÅ¿t duas <var>.d.e.e.c.</var> figuræ <var>.A.</var> æquales eſſe in <lb/>potentia duabus <var>d.e.e.c.</var> figurÄ™ <var>.D.</var> quæ quidem affectio circuli, à nemine fuit adhuc <lb/>(quod Å¿ciam) detecta.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0375-01" xlink:href="fig-0375-01a"> - <image file="0375-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0375-01"/> - </figure> - </div> - <pb o="364" rhead="IO. BAPT. BENED." n="376" file="0376" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0376"/> - </div> - </div> - <div type="section" level="3" n="36"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE AVGMENTO PONDERIS CORPORIS <lb/>ad Å¿tateram appenÅ¿i, & quadam alia demonÅ¿tratione, <lb/>& quibuÅ¿dam erroribus Tartaleæ.</head> - <head style="it" xml:space="preserve">Mutio Groto.</head> - <p> - <s xml:space="preserve">SI ea quæ à me audiuiÅ¿ti non credis, conÅ¿idera quæſo libram Å¿eu Å¿tateram <lb/><var>o.a.</var> cuius centrum non longitudinis Å¿ed ponderum Å¿it <var>.i.</var> quÄ™ Å¿tatera, vt ori <lb/>zontaliter conÅ¿iÅ¿tat, oportebit pondus extremitatis <var>.o.</var> ita Å¿e habere <lb/>ad pondus extremitatis <var>.a.</var> ut <var>.a.i.</var> Å¿e habet ad <var>.o.i.</var> quod te Å¿cire puto, ima <lb/>ginemur nunc d uas lineas <var>.a.e.</var> et <var>.o.n.</var> paralle las <reg norm="infinitasque" type="simple">infinitasqÌ;</reg> & à puncto <var>.n.</var> immobili, <lb/>& fixo extra Å¿tateram, tranÅ¿eat per <var>.i.</var> linea <var>.n.i.e</var>. </s> - <s xml:space="preserve">Cogitemus etiam punctum <var>.e.</var> inter <lb/>Å¿ectionis ipÅ¿ius <var>.n.i.e.</var> cum <var>.a.e.</var> progredi vniformiter <reg norm="continuòque" type="simple">continuòqÌ;</reg> ab <var>.a.</var> per lineam <var>.a.e.</var> <lb/>vnde punctum <var>.i.</var> interÅ¿ectionis ipÅ¿ius <var>.n.i.e.</var> cum <var>.a.i.o.</var> Å¿emper vicinius fiet puncto <var>.o.</var> <lb/>nec unquam cum illo vnum erit, quamuis moueatur tempore infinito. </s> - <s xml:space="preserve">Nunc autem <lb/>dico, quod cum Å¿tateram <var>.o.i.a.</var> oporteat Å¿emper orizontalem eſſe virtute ponderis, <lb/>o. oportebit pundus <var>.o.</var> in infinitum etiam augeri, <reg norm="quotieÅ¿cunque" type="simple">quotieÅ¿cunq;</reg> pondus <var>.a.</var> nunquam <lb/>diminui voluerimus vel econtra hoc in infinitum diminui, Å¿i illud nunquam augeri <lb/>voluerimus.</s> - </p> - <p> - <s xml:space="preserve">Sedre vera non putabam te indigere aliqua demonÅ¿tratione, quod linea <var>.b.h.</var> di-<lb/>uiÅ¿a Å¿it per æqualia à <unsure/> linea <var>.c.a.</var> cum hæc perpendicularis Å¿it ab <var>.a.</var> ad baÅ¿im <var>.g.d.</var> in <reg norm="triam" type="context">triã</reg> <lb/>gulo orthogonio <var>.g.a.d.</var> & cum Å¿it <var>.b.h.</var> perpendicularis ad <var>.a.o.</var> ex Å¿uppoÅ¿ito quæ <var>.a.<lb/>o.</var> in Å¿e habet punctum medium baÅ¿is <var>.g.d.</var> nec <reg norm="non" type="context">nõ</reg> illud anguli recti <var>.a.</var> quod per Å¿e cla <lb/>riſſimum eÅ¿t, cum iam Å¿cis <var>.o.</var> eſſe centrum circuli circundantis triangulum <var>.g.a.d.</var> or-<lb/>thogonium, et <var>.g.d.</var> eius diameter, vnde <var>.o.a.</var> æquabitur ipÅ¿i <var>.o.g.</var> quapropter angulus <lb/>o. <reg norm="am" type="context">ã</reg>. g. æquabitur angulo <var>.g.</var> ex quinta primi, </s> - <s xml:space="preserve">deinde ex .32. eiuÅ¿dem, angulus <var>.h.</var> æqua <lb/>bitur angulo <var>.d.</var> eo quod an gulus <var>.e.</var> rectus eÅ¿t, quemadmodum et <var>.a.</var> Å¿ed angulus <var>.d.</var> <lb/>æqualis eÅ¿t angulo <var>.g.a.c</var>. </s> - <s xml:space="preserve">& propterea angulus <var>.h.</var> erit etiam æqualis angulo <var>.h.a.u.</var> <lb/>vnde <var>.h.u.</var> æqualis erit ipÅ¿i <var>.u.<lb/>a.</var> ex .6. primi, cum poÅ¿tea angulus <var>.<lb/> - <anchor type="figure" xlink:label="fig-0376-01a" xlink:href="fig-0376-01"/> - o.a.d.</var> æqualis Å¿itangulo <var>.d.</var> ex quinÂ<lb/>ta primi erit angulus <var>.a.b.e.</var> æqua-<lb/>lis angulo <var>.g.</var> ex .32. dicta, eo quod <lb/>e. rectus eÅ¿t, & ex eadem æqualis <lb/>erit angulo <var>.d.a.c.</var> vnde <var>.u.b.</var> erit <lb/>æqualis ipÅ¿i <var>.u.a.</var> ex .6. dicti, & ideo <lb/>æqualis eric ipÅ¿i <var>.u.h</var>. </s> - <s xml:space="preserve">Reliqua ve-<lb/>rò illius propoÅ¿itionis credo ex te <lb/>omnia poſſe <reg norm="intelligere" type="context">Ä©telligere</reg>, excepto, <reg norm="quod" type="simple">ê™</reg> <lb/>vt tibi Å¿ignificaui Å¿i à <reg norm="puncto" type="context">pÅ©cto</reg> <var>.i.</var> com-<lb/>muni ipÅ¿i <var>.a.c.u.</var> & circunferentiæ, <lb/>ducta fuerit <var>.i.x.</var> ad <reg norm="punctum" type="context">pÅ©ctum</reg> <var>.x.</var> com <lb/>mune vni parallelæ à <reg norm="puncto" type="context">pÅ©cto</reg> <var>.g.</var> ipÅ¿i <lb/><var>h.b.</var> & circunferentiæ, quod di-<lb/>cta <var>.i.x.</var> ad rectos erit ipÅ¿i <var>.a.b.d.</var> eo <lb/>quod cum angulus <var>.a.g.x.</var> æqualis - <pb o="365" rhead="EPISTOL AE." n="377" file="0377" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0377"/> - Å¿it angulo <var>.a.h.b.</var> propter æquidiÅ¿tantiam dictam, æqualis etiam erit angulo <var>.d.</var> & ar-<lb/>cus <var>.a.x.</var> æqualis arcui <var>.a.g.</var> vnde angulus <var>.a.i.x.</var> æqualis erit <var>.d.</var> Å¿ed angulus <var>.i.a.d.</var> com-<lb/>munis eÅ¿t triangulis <var>.c.a.d.</var> et <var>.i.a.t</var>. </s> - <s xml:space="preserve">quare angulus <var>.a.t.i.</var> rectus erit, vt <var>.c.</var> hoc eÅ¿t <var>.i.x.</var> per <lb/>pendicularis erit ipÅ¿i <var>.a.d</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0376-01" xlink:href="fig-0376-01a"> - <image file="0376-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0376-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Sed vbitibi Å¿cripÅ¿i circa finem illius epiÅ¿tolæ, Tartaleam erraſſe in quinta propo-<lb/>Å¿itione primi lib. Å¿uæ nouæ Å¿cientiæ, non Å¿ine ratione illud Å¿cripÅ¿i. </s> - <s xml:space="preserve">Nam, inquit ipÅ¿e, <lb/>nullum corpus æquè graue poteÅ¿t in aliquo temporis Å¿patio moueri motu naturali, <lb/><reg norm="violentoque" type="simple">violentoqÌ;</reg> Å¿imul miÅ¿tis. </s> - <s xml:space="preserve">Vbi decipitur, eo quod non animaduertit incrementum ve <lb/>locitatis vnius motus, Å¿imul eſſe cum decremento velocitatis alterius, <reg norm="eodemque" type="simple">eodemqÌ;</reg> tem <lb/>pore, vt manifeÅ¿tè patet in itinere corporis, ab ipÅ¿o pro exemplo aſſumpto, hoc eÅ¿t <lb/>quod velocitas motus in Å¿patio <var>.c.d.</var> creÅ¿cit vt naturalis, & decreÅ¿cit vt violenta. </s> - <s xml:space="preserve"><reg norm="nam" type="context">nã</reg> <lb/>creÅ¿cit orizontem verÅ¿us & decreÅ¿cit in remotione à linea <var>.a.b.</var> Å¿ed Å¿i à puncto <var>.c.</var> ad <lb/>punctum <var>.d.</var> motus eſſet purè violentus, vt putat Tartalea, corpus illud minimè de-<lb/>Å¿cenderet, eo quod uirtus mouens, in <var>.a.</var> poÅ¿ita, nullo pacto poteÅ¿t talem effectum ef-<lb/>ficere, vnde ab ipÅ¿a natura prouenit deÅ¿cenÅ¿io illius corporis propter <reg norm="grauitatem" type="context">grauitatẽ</reg>, <reg norm="quam" type="context">quã</reg> <lb/>dictum corpus habet in tali medio, aeris Å¿cilicet, & non ex violentia aliqua. </s> - <s xml:space="preserve">Sed Å¿i <lb/>dixiſſet ipÅ¿e, illum motum eſſe purum naturalem, hoc eſſet falÅ¿um, eo quod purus <lb/>naturalis motus alicuius corporis non impediti, extra locum Å¿uum, Å¿it per lineam re <lb/>ctam, & non per curuam, vt videre eÅ¿t inter <var>.c.</var> et <var>.d</var>.</s> - </p> - <p> - <s xml:space="preserve">In vltima propoÅ¿itione deinde eiuÅ¿dem lib. quæ .6. eÅ¿t decipitur Å¿imiliter, & hæc <lb/>deceptio oritur ab ignoratione quintæ, & à putando motum naturalem non eſſe cau <lb/>Å¿am ipÅ¿ius deÅ¿cenÅ¿us per Å¿patium <var>.c.d</var>. </s> - <s xml:space="preserve">Sed quia tibi Å¿ignificaui expeditiorem viam <lb/>repeririad <reg norm="cognoÅ¿cendam" type="context">cognoÅ¿cendã</reg> proportionem inter <var>.a.h.</var> et <var>.a.e.</var> in vltima propoÅ¿itione Å¿e-<lb/>cundi lib. ipÅ¿ius Tartaleæ, ipÅ¿am nunc tibi Å¿cribo. </s> - <s xml:space="preserve"><reg norm="Nam" type="context">Nã</reg> iam Å¿cis angulum <var>.h.l.i.</var> diui-<lb/>Å¿um eſſe per æqualia ab <var>.P.l.</var> & quod <var>.a.h.</var> et <var>.h.p.</var> Ä™quales inuicem Å¿unt ex .6. primi Eu-<lb/>cli. </s> - <s xml:space="preserve">vnde <var>.p.i.</var> et <var>.a.h.</var> æquales erunt inuicem Å¿imiliter, Å¿ed ex .3. Å¿exti ita eÅ¿t ipÅ¿ius <var>.a.l.</var> <lb/>ad <var>.l.i.</var> vt ipÅ¿ius <var>.a.p.</var> ad <var>.p.i.</var> & coniunctim ita erit <var>.a.l.i.</var> ad <var>.l.i.</var> vt <var>.a.i.</var> ad <var>.p.i.</var> Å¿ed <var>.a.l.</var> cogni <lb/>ta eÅ¿t ex eius quadrato, et <var>.l.i.</var> etiam, cum æqualis Å¿it ipÅ¿i <var>.a.i.</var> vnde ex regula de tribus <lb/>notam habebimus <var>.p.i.</var> reÅ¿pectu <var>.a.i.</var> & ita reÅ¿pectu <var>.a.e.</var> Å¿i hypotheÅ¿es ipÅ¿ius Tartaleæ <lb/>veræ Å¿unt.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">Alia demonstratio impoßibilitatis diuidendi per æqualia <lb/>proportionem Å¿uperparticularem in <lb/>diÅ¿cretis.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">QVod à me poÅ¿tulas, hoc eÅ¿t Å¿cientiam impoſſibilitatis diuidendi per æqualia <lb/>proportionem Å¿uperparticularem in numeris Å¿atis à Campano in .8. octaui <lb/>potes habere, Iacobus Faber StapulenÅ¿is etiam idem tractat<unsure/> in libello Å¿uÄ™ muÅ¿icæ <lb/>demonÅ¿tratæ. </s> - <s xml:space="preserve">Sed Å¿i etiam alia via idem deÅ¿ideras, quamuis longiori, nih<unsure/>ilomi-<lb/>nus vniuerÅ¿aliori, conÅ¿idera duos numeros <var>.g.</var> et <var>.h.</var> inuicem relatos Å¿ecundum pro-<lb/>portionem Å¿uperparcicularem, quam volueris. </s> - <s xml:space="preserve">Tunc dico impoſſibile eÅ¿le, vt per <lb/>æqualia diuidatur, quod Å¿i dixeris poſſibile eſſe, Å¿it per te <var>.K.</var> medius numerus - <pb o="366" rhead="IO. BAPT. BENED." n="378" file="0378" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0378"/> - proportionalis inter <var>.g.</var> et <var>.h</var>. </s> - <s xml:space="preserve">quare <var>.g.</var> et <var>.h.</var> non erunt minimi in ea proportione, quia <lb/>vnitas diuiÅ¿ibilis eſſet Å¿i <var>.g.h.</var> minimi fuiſſent, quod non conceditur, Å¿int igitur mini <lb/>mi in dicta proportione <var>.a.</var> et <var>.b.</var> quorum differentia erit vnitas, vt Å¿cis, <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.c.</var> quadra <lb/>tum ipÅ¿ius <var>.g.</var> et <var>.d.</var> quadratum ipÅ¿ius <var>.K</var>. </s> - <s xml:space="preserve">tunc clarum erit ex .11. octaui, quod propor-<lb/>tio ipÅ¿ius c. ad <var>.d.</var> eadem erit quæ <var>.g.</var> ad <var>.h.</var> hoc eÅ¿t vt ipÅ¿ius <var>.a.</var> ad <var>.b.</var> vnde Å¿i vnus termi. <lb/>norum <var>.a.</var> vel <var>.b.</var> eſſet quadratus, reliquus etiam quadratus eſſet ex .22. octaui, & ex <lb/>16. eiuÅ¿dem, inter <var>.a.</var> et <var>.b.</var> reperiretur aliquis medius numerus proportionalis, quod <lb/>fieri non poteÅ¿t ex hypotheÅ¿i, cum inter <var>.a.</var> et <var>.b.</var> nullus Å¿it numerus, quia differunt in <lb/>ter Å¿e per vnitatem tantummodo. </s> - <s xml:space="preserve">Nunc autem cum nullus numerorum <var>.a.</var> vel <var>.b.</var> qua <lb/>dratus Å¿it, ponatur quod <var>.f.</var> quadratus Å¿it ipÅ¿ius <var>.b.</var> et <var>.e.</var> Å¿it productum ipÅ¿ius <var>.a.</var> in <var>.b.</var> vn <lb/>de ex .18. Å¿eptimi, proportio ipÅ¿ius <var>.e.</var> ad <var>.f.</var> erit vt. ipÅ¿ius <var>.a.</var> ad <var>.b.</var> hoc eÅ¿t vt ipÅ¿ius <var>.c.</var> ad <lb/>d. quapropter <var>.e.</var> erit quadratus ex .22. octaui, cuius latus tetragonicum eſſet <reg norm="medium" type="context">mediÅ©</reg> <lb/>proportionale inter <var>.a.</var> et <var>.b.</var> ex .20. Å¿eptimi, quod eÅ¿t impoſſibile, vt iam dixi, cum <var>.a.</var> <lb/>et <var>.b.</var> Å¿int inui cem conÅ¿equentes, vnus poÅ¿t alium immediatè.</s> - </p> - <p> - <s xml:space="preserve">Superius enim dixi hunc modum eſſe vniuerÅ¿alem, <lb/>hoc eÅ¿t quod hac methodo poſſumus in cognitionem <lb/>vcnire, quod non Å¿olum in duas æquales partes diui-<lb/> - <anchor type="figure" xlink:label="fig-0378-01a" xlink:href="fig-0378-01"/> - di non poſſit, Å¿ed nec in tres, nec quatuor nec quot vo <lb/>lueris. </s> - <s xml:space="preserve">Primum enim quod non in tres diuidatur à te <lb/>ipÅ¿o cognoÅ¿ces ope <reg norm="cuborum" type="context">cuborÅ©</reg> vice <reg norm="quadratorum" type="context">quadratorÅ©</reg>, opevero <lb/><reg norm="cenÅ¿uum" type="context">cenÅ¿uÅ©</reg> <reg norm="cenÅ¿uum" type="context context">cẽſuÅ©</reg>, ve<unsure/>l qui cognouerit eam <reg norm="proportionem" type="context">proportionẽ</reg> <lb/>eſſe indiuiÅ¿ibilem per æqualia, illicò etiam cognoÅ¿cet <lb/>indiuiÅ¿ibilem eſſe per quatuor partes, ope verò pri-<lb/>morum relatorum, cognoÅ¿cet non eſſe diuiÅ¿ibilem per <lb/><reg norm="quinque" type="simple">quinq;</reg> partes, & Å¿ic de cÄ™teris, Å¿ed mediantibus ijs <lb/>quas Å¿cripÅ¿i de iÅ¿tis dignitatibus in libro <reg norm="ThÄ™orematum" type="context">ThÄ™orematÅ©</reg> <lb/>arithmeticorum.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0378-01" xlink:href="fig-0378-01a"> - <image file="0378-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0378-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Id autem quod IlluÅ¿triſſimus Daniel Barbarus Å¿cri <lb/>bit in quinta parte Å¿uæ perÅ¿pectiuæ, Å¿i Å¿upra aliquo im <lb/>mobili, atque magno pariete facere volueris, te opor <lb/>tebit hoc ex reflexione radij Å¿olaris à ſpeculo plano <lb/>perficere.</s> - </p> - </div> - </div> - <div type="section" level="3" n="37"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE INVENTIONE DIAMETRI <lb/>circuli circunÅ¿cribentis triangulum.</head> - <head style="it" xml:space="preserve">Francbino Triuultio.</head> - <p> - <s xml:space="preserve"><emph style="sc">QVod</emph> mihi nunc proponis eÅ¿t triangulum, cuius baÅ¿is cum angulo Å¿ibi op <lb/>poÅ¿ito dantur. </s> - <s xml:space="preserve"><reg norm="Vellesque" type="simple">VellesqÌ;</reg> diametrum circuli apti eum triangulum circnn-<lb/>Å¿cribere inuenire in diÅ¿creto.</s> - </p> - <p> - <s xml:space="preserve">Sit igitur triangulum <var>.a.b.g.</var> cuius baÅ¿is <var>.b.g.</var> Å¿imul cum angulo <var>.a.</var> ei op-<lb/>poÅ¿ito data Å¿it in numeris. </s> - <s xml:space="preserve">Imaginetur ergo circulas circunÅ¿eribens ipÅ¿um triangu-<lb/>lum <var>.b.p.g.q.</var> cuius diameter Å¿it <var>.q.p.</var> perpendicularis eius baÅ¿i <var>.b.g.</var> vnde <var>.b.g.</var> diuiÅ¿a <lb/>erit per æqualia ab ipÅ¿o diametro in puncto <var>.m.</var> per tertiam tertij, protrahatur etiam - <pb o="367" rhead="EPISTOL AE." n="379" file="0379" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0379"/> - e<unsure/> <var>.g.</var> vnde angulus <var>.g.e.q.</var> æqualis erit angulo <var>.b.a.g.</var> portionis, cum duplus Å¿it angulo <lb/><var>q.p.g.</var> medietati anguli ipÅ¿ius portionis ex .19. tertij, ita quod angulus <var>.q.e.g.</var> nobis <lb/>cognitus erit, & Å¿imiliter arcus <var>.g.q.</var> & conÅ¿equenter ar-<lb/>cus <var>.p.g.</var> reÅ¿iduum medij circuli, & Å¿ic <var>.m.g.</var> eius Å¿inus re <lb/> - <anchor type="figure" xlink:label="fig-0379-01a" xlink:href="fig-0379-01"/> - ctus, & etiam chorda <var>.p.g.</var> vt dupla Å¿inus dimidij arcus <var>.<lb/>p.g.</var> & Å¿ic <var>.p.m.</var> eius Å¿inus verÅ¿us, vel vt tertium latus trian <lb/>guli orthogonij <var>.p.g.m.</var> vnde nobis cognita erit propor <lb/>tio ipÅ¿ius <var>.b.g.</var> (quæ dupla eÅ¿t ipÅ¿i <var>.m.g.</var>) ad <var>.m.p.</var> & quia <lb/>productum <var>.p.m.</var> in <var>.m.q.</var> æquale eÅ¿t ei, quod fit ex <var>.b.m.</var> <lb/>in <var>m.g.</var> ex .34. tertij, quapropter nobis cognita erit pars <lb/><var>q.m.</var> quæ cum <var>.p.m.</var> complet totum diametrum <var>.q.p.</var> vn <lb/>de nobis cognita erit proportio ipÅ¿ius <var>.b.g.</var> ad <var>.q.p.</var> qua <lb/>mediante cognoÅ¿cemus diametrum Å¿ecundum partes il <lb/>las quibus propoÅ¿ita Å¿uerit <var>.b.g</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0379-01" xlink:href="fig-0379-01a"> - <image file="0379-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0379-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Hoc autem problema non in numeris Å¿ed in continuo ab Euclid. ponitur in .32<unsure/>. <lb/>tertij.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">De inuentione alterius trianguli conditionati.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">QVotieÅ¿cunque etiam inuenire voluerimus triangulum aliquem, puta <var>.n.q.o.</var> <lb/>æqualem triangulo <var>.t.</var> (exempli gratia) propoÅ¿ito, qui habeat angulum <var>.n.</var> æ-<lb/>qualem angalo <var>.a.</var> dato, latera vero continentia ipÅ¿um angulum <var>.n.</var> Å¿int inuicem pro-<lb/>portionata vt <var>.x.</var> et <var>.y.</var> ita faciemus, accipiemus lineam <var>.n.m.</var> cuius volueris magnitu-<lb/>dinis, Å¿upra quam conÅ¿tituemus triangulum <var>.m.n.p.</var> æqualem triangulo <var>.t.</var> hac metho-<lb/>do, hoc eÅ¿t prolungando latus <var>.r.z.</var> trianguli <var>.t.</var> quod Å¿it <var>.r.e.</var> ita vt duplum Å¿it ipÅ¿i <var>.r.z.</var> <lb/>ducendo poÅ¿tea <var>.c.e.</var> habebimus ex .38. primi triangulum <var>.t.</var> eſſe dimidium totius <lb/>trianguli <var>.r.c.e.</var> deÅ¿ignabimus deinde ex .44. dicti Å¿uperficiem <var>.p.n.m.b.</var> parallelo <lb/>grammam <reg norm="æqualemque" type="simple">æqualemqÌ;</reg> triangu <lb/>lo <var>.r.c.e.</var> habentem angulum <var>.<lb/> - <anchor type="figure" xlink:label="fig-0379-02a" xlink:href="fig-0379-02"/> - n.</var> æqualem angulo <var>.a.</var> ducatur <lb/>poÅ¿tea <var>.p.m.</var> & habebimus <reg norm="triam" type="context">triã</reg> <lb/>gulum <var>.m.n.p.</var> æqualem <var>.t.</var> cum <lb/>angulo <var>.n.</var> æquali angulo <var>.a.</var> pro <lb/>ducatur poÅ¿tea <var>.n.p.</var> ita vt <var>.n.K.</var> <lb/>Å¿e habeat .ad <var>.n.m.</var> quemadmo <lb/>dum <var>.x.</var> ad <var>.y.</var> quod erit facilli-<lb/>mum producendo <var>.n.m.</var> et <var>.n.<lb/>K.</var> indeterminatè Å¿i oportuerit, <lb/></s> - <s xml:space="preserve">deinde eas ad æqualitatem Å¿e-<lb/>can<unsure/>do ipÅ¿is <var>.x.</var> et <var>.y.</var> efficiendo <lb/>exempli gratia quod <var>.n.i.</var> Å¿it <lb/>æqualis ipÅ¿i <var>.x.</var> et <var>.n.u.</var> ipÅ¿i <var>.y.</var> du <lb/>cendo poÅ¿tea <var>.u.i.</var> deinde à puncto <var>.m.</var> ducendo <var>.m.K.</var> æquidiÅ¿tanter <var>.u.i.</var> ex .31. <lb/>primi. </s> - <s xml:space="preserve">& Å¿ic habebimus ex .4. Å¿exti proportionem <var>.x.</var> ad <var>.y.</var> eſſe inter <var>.n.K.</var> et <var>.n.</var> - <pb o="368" rhead="IO. BAPT. BENED." n="380" file="0380" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0380"/> - m. inuenies poÅ¿tea ex .9. eiuÅ¿-<lb/> - <anchor type="figure" xlink:label="fig-0380-01a" xlink:href="fig-0380-01"/> - dem lineam aliquam mediam <lb/>proportionalem inter <var>.n.K.</var> et <var>.<lb/>n.p.</var> quæ Å¿it <var>.n.o.</var> duces poÅ¿tea <lb/><var>o.q.</var> parallelam ipÅ¿i <var>.m.K.</var> & ha <lb/>bebis propoÅ¿itum, eo quod <reg norm="cum" type="context">cÅ©</reg> <lb/>Å¿it proportio trianguli <var>.n.m.K.</var> <lb/>ad triangulum <var>.n.m.p.</var> vt <var>.n.K.</var> <lb/>ad <var>.n.p.</var> ex prima Å¿exti, duo <reg norm="triam" type="context">triã</reg> <lb/>guli <var>.m.p.n.</var> et <var>.n.q.o.</var> æquales <lb/>erunt inuicem, ex .17. eiuÅ¿dem <lb/>& ex .9. quinti, & proportio <var>.<lb/>o.n.</var> ad <var>.n.q.</var> erit, vt <var>.x.</var><unsure/> ad <var>.y.</var> ex <num value="11">.<lb/>11.</num> dicti, cum ex .4. Å¿exti Å¿it vt <var>.<lb/>n.k.</var> ad <var>.n.m</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0379-02" xlink:href="fig-0379-02a"> - <image file="0379-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0379-02"/> - </figure> - <figure xlink:label="fig-0380-01" xlink:href="fig-0380-01a"> - <image file="0380-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0380-01"/> - </figure> - </div> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">De producto conditionato.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">PRoponis deinde mihi duas rectas lineas, vni quarum, vis vt aliam quandam di-<lb/>rectè coniungam, ita quod productum huius aggregati in lineam adiunctam <lb/>æquale Å¿it quadrato alterius.</s> - </p> - <p> - <s xml:space="preserve">Vt exempli gratia Å¿i fuerint duæ lineæ <var>.e.d.</var> et <var>.e.f.</var> opor-<lb/><reg norm="teretque" type="simple">teretqÌ;</reg> nos ad lineam <var>.e.f.</var> aliam lineam puta <var>.f.c.</var> vel <var>.e.b.</var> <reg norm="iun gere" type="context">iÅ©Â<lb/> - <anchor type="figure" xlink:label="fig-0380-02a" xlink:href="fig-0380-02"/> - gere</reg>, ita longam, vt productum totius compoÅ¿iti <var>.e.c.</var> vel <var>.<lb/>f.b.</var> in <var>.f.c.</var> vel <var>.e.b.</var> eſſet æquale quadrato ipÅ¿ius <var>.e.d</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0380-02" xlink:href="fig-0380-02a"> - <image file="0380-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0380-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Hoc enim nu llius eſſet difficultatis, eo quod <reg norm="quotieÅ¿cun- que" type="context">quotieÅ¿cÅ©-<lb/>que</reg> <var>.e.d.</var> coniuncta erit cum <var>.e.f.</var> ad rectos, <reg norm="diuiÅ¿aque" type="simple">diuiÅ¿aqÌ;</reg> per me <lb/>dium à puncto <var>.a.</var> à quo ducta <var>.a.d.</var> deinde Å¿ecundum Å¿emi-<lb/>diametrum <var>.a.d.</var> deÅ¿ignato circulo <var>.b.d.c.</var> & protracta <var>.e.f.</var> <lb/>à qua volueris parte vÅ¿que ad circunferentiam in <reg norm="puncto" type="context">pÅ©cto</reg> <var>.c.</var> <lb/>Å¿eu in puncto <var>.b.</var> habebimus intentum, eò quod Å¿i produ-<lb/>cta fuerit <var>.e.f.</var> etiam ab alia parte, vÅ¿que ad circunferentiam, habebimus <var>.b.e.</var> æqua-<lb/>lem ipÅ¿i <var>.f.c.</var> ex communi conceptu, & productum <var>.e.c.</var> in <var>.e.b.</var> æqualem quadra-<lb/>to ipÅ¿ius <var>.e.d.</var> ex .34. tertij, cum ex .3. eiuÅ¿dem <var>.e.d.</var> medietas Å¿it chordæ arcus dupli <lb/><var>b.d</var>.</s> - </p> - <p> - <s xml:space="preserve">De lapÅ¿u verò lapidis verÅ¿us mundi centrum, dum ipÅ¿um attingere, ac præterire <lb/>poſſet, de quo me interrogas. </s> - <s xml:space="preserve">Dico Nicolaum Tartaleam, nec non FranciÅ¿cum <lb/>Maurolicum rectè Å¿enÅ¿iſſe, malè verò Alexandrum Piccolhomineum, & exemplum <lb/>Maurolici optimum eſſe, quod tamen Å¿i capere non potes, crede Å¿altem authoritati <lb/>bus talium virorum, qui tantum in ijs Å¿cientijs Å¿uperant ipÅ¿um Alexandrum Piccol-<lb/>homineum, quantum à Sole cætera Å¿uperantur aÅ¿tra.</s> - </p> - <p> - <s xml:space="preserve">Lapis igitur ille tranÅ¿iret centrum, <reg norm="reddiretque" type="simple">reddiretqÌ;</reg>, cum diminutione tamen motus im <lb/>preſſi, eo fermè modo vt Å¿cribunt iudicioÅ¿iſſimi illi viri, donec poÅ¿t multas reddi-<lb/>tiones Å¿urÅ¿um, <reg norm="deorÅ¿umque" type="simple">deorÅ¿umqÌ;</reg> quieÅ¿ceret circa centrum mundi. </s> - <s xml:space="preserve">Lucidioris tamen intelli - <pb o="369" rhead="EPISTOL AE." n="381" file="0381" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0381"/> - gentiæ gratia cogita Å¿ilum illum (exempli adducti ab illis doctiſſimis viris) cui pon <lb/>dus appenÅ¿um eÅ¿t, æqualem eſſe axi orizontis, hoc eÅ¿t eius extremitatem immobi-<lb/>lem eſſe in primo mobili, & in ipÅ¿o zenit tui orizontis, </s> - <s xml:space="preserve">tunc arcus motionis ipÅ¿ius la <lb/>pidis per tantum interuallum, quantum eÅ¿t diameter terræ, inÅ¿enÅ¿ibiliter differret à <lb/>linea recta, & cum lapis diÅ¿tans à centro mundi per Å¿emidiametrum terræ, iret re-<lb/><reg norm="diretque" type="simple">diretqÌ;</reg>, vt Å¿cis, ergo idem faceret Å¿i Å¿ilum longius eſſet per dictum terræ Å¿emidia-<lb/>metrum, ita vt poſſetipÅ¿um centrum attingere, nam differentia illa Å¿emidiametri <lb/>terræ, ferè nulla eÅ¿t reÅ¿pectu Å¿emidiametri ipÅ¿ius primi mobilis.</s> - </p> - </div> - </div> - <div type="section" level="3" n="38"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">AN PENTAGONVS AB ALBERTO DVRERO <lb/>deÅ¿criptus æquiangulus Å¿it.</head> - <head style="it" xml:space="preserve">Conrado Neubart.</head> - <p> - <s xml:space="preserve">SI non credis Pentagonum ab Alberto Durero Å¿uper datam lineam deÅ¿i-<lb/>gnatum, æquiangulum non eſſe. </s> - <s xml:space="preserve">Fingamus hic Å¿ubiectam Å¿iguram Å¿imi-<lb/>lem ei quæ à Durero ponitur, in qua primò, ducta Å¿it linea <var>.o.a.</var> & habe <lb/>bimus angulum <var>.a.o.b.</var> graduum .60. talium qualium duo recti fuerint gra <num value="360">.<lb/>360.</num> vel .30. talium qualium duo recti fuerint .180. nam ex Å¿uppoÅ¿ito, arcus <var>.a.b.</var> eÅ¿t <lb/>Å¿exta pars totius circunferentiæ, angulus vero <var>.b.o.d.</var> rectus eÅ¿t, eo quod <var>.b.o.q.</var> rectus <lb/>etiam Å¿it, </s> - <s xml:space="preserve">quare angulus <var>.d.o.a.</var> reÅ¿iduus ex recto erit graduum .60. talium, ut rectus <lb/>eÅ¿t .90. angulus verò <var>.o.a.c.</var> erit gra .15. eorundem.</s> - </p> - <p> - <s xml:space="preserve">Ducatur deinde perpendicularis <var>.a.e.</var> ad <var>.o.d.</var> quæ vt Å¿inus anguli <var>.a.o.e.</var> erit par-<lb/>tium .86602. talium qualium <var>.a.o.</var> erit .100000. quæ quidem <var>.o.a.</var> vt chorda arcus <var>.a.<lb/>o.</var> eÅ¿t partium .51762. talium qualium <var>.a.d.</var> vel <var>.a.c.</var> Å¿emidiameter eÅ¿t .100000.</s> - </p> - <p> - <s xml:space="preserve">Nam Å¿inus dimidij arcus <var>.a.o.</var> (exi <lb/>Å¿tente <var>.a.o.</var> graduum .30.) eÅ¿t <reg norm="partium" type="context">partiÅ©</reg> <num value="25881">.<lb/> - <anchor type="figure" xlink:label="fig-0381-01a" xlink:href="fig-0381-01"/> - 25881.</num> ex quo <var>.a.e.</var> erit partium <num value="44827">.<lb/>44827.</num> talium qualium <var>.a.d.</var> erit <lb/>100000. vnde angulus <var>.a.d.o.</var> cuius Å¿i <lb/>nus eÅ¿t <var>.a.e.</var> erit graduum .26. min .38 <lb/>qui quidem angulus, Å¿umptus cum an <lb/>gulo <var>.a.o.d.</var> erit gra .86. min .38. </s> - <s xml:space="preserve">Dem <lb/>pta denique hac Å¿umma ex duobus <lb/>rectis gra .180. reliquum erit gra .93. <lb/>min .22. ideÅ¿t angulus <var>.o.a.d.</var> cui addi <lb/>tus cum fuerit angulus <var>.o.a.c.</var> gra .15. <lb/>talium, habebimus angulum <var>.c.a.d.</var> <lb/>graduum .108. min .22. exuperantem <lb/>verum angulum pentagoni per min <num value="22">.<lb/>22.</num> vel Å¿ic, cum inuentus fuerit angu-<lb/>lus <var>.a.d.o.</var> gra .26. min .38. Å¿i ex vno re <lb/>cto demptus fuerit, relinquetur an-<lb/>gulus <var>.d.a.e.</var> gra .63. min .22. qui qui-<lb/>dem collectus cum fuer it cum angu-<lb/>lo <var>.e.a.o.</var> reÅ¿iduo ex re cto dempto angulo <var>.a.o.e.</var> grad .60. qui <var>.e.a.o.</var> eÅ¿t grad .30. & - <pb o="370" rhead="IO. BAPT. BENED." n="382" file="0382" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0382"/> - e<unsure/>tiam collectus cum angulo <var>.o.a.c.</var> grad .15. hi tres anguli efficient angulum <var>.d.a.c.<lb/>d.</var> ctum grad .108. min .22.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0381-01" xlink:href="fig-0381-01a"> - <image file="0381-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0381-01"/> - </figure> - </div> - <head style="it" xml:space="preserve">Examinatio anguli <var>.u.</var></head> - <p> - <s xml:space="preserve">Ducatur <var>.d.n.</var> quam quidem <var>.d.n.</var> cognoÅ¿cemus vt Å¿inus anguli <var>.d.o.n.</var> gra .45. nam <lb/>angulus ei contrapoÅ¿itus <var>.q.o.p.</var> eÅ¿t dimidium recti, </s> - <s xml:space="preserve">quare <var>.d.n.</var> erit partium .70710. <lb/>talium qualium <var>.d.o.</var> fuerit .100000. Å¿ed <var>.d.o.</var> eÅ¿t partium .115270. qualium <var>.a.d.</var> eÅ¿t <num value="100000">.<lb/>100000.</num> nam <var>.e.d.</var> vt Å¿inus anguli <var>.e.a.d.</var> gra .63. min .22. eÅ¿t partium .89389. <var>o.e.</var> ve-<lb/>ro eÅ¿t partium .50000. talium qualium <var>.a.o.</var> eÅ¿t .100000. vt Å¿inus anguli <var>.e.a.o.</var> gra .30. <lb/>Å¿ed vt <var>.a.o.</var> eÅ¿t <reg norm="partium" type="context">partiÅ©</reg> .51762. </s> - <s xml:space="preserve">hoc eÅ¿t vt <var>.a.d.</var> eÅ¿t .100000. ipÅ¿a <var>.o.e.</var> erit <reg norm="partium" type="context">partiÅ©</reg> .25881 <lb/>quæ iuncta cum fuerit cum <var>.e.d.</var> efficiet <var>.d.o.</var> partium .115270. vt dictum eÅ¿t, quapro-<lb/>pter cum <var>.d.n.</var> Å¿it partium .70710. talium qualium <var>.d.o.</var> fuerit .100000. ipÅ¿a <var>.d.n.</var> erit <lb/>partium .81507. talium qualium <var>.d.o.</var> erit .115270. ideÅ¿t qualium <var>.d.a.</var> vel <var>.d.u.</var> erit <lb/>100000. quæ quidem <var>.d.n.</var> eÅ¿t Å¿inus anguli <var>.d.u.n.</var> graduum Å¿cilicet .54. 36. cuius du-<lb/>plum erit gra .109. mi .12. debebattamen eſſe .108. <var>m.o</var>.</s> - </p> - <head style="it" xml:space="preserve">Examinatio anguli <var>.d.</var></head> - <p> - <s xml:space="preserve">Accipe angulum <var>.a.d.o.</var> gra .26. <lb/> - <anchor type="figure" xlink:label="fig-0382-01a" xlink:href="fig-0382-01"/> - min .38. vt Å¿upra, cui applica angu-<lb/>lum <var>.o.d.n.</var> gra .45. min <var>.o.</var> Å¿imul cum <lb/>angulo <var>.u.d.n.</var> reÅ¿iduo ex recto gra <lb/>duum .35. minu .24. & conficies an-<lb/>gulum <var>.a.d.u.</var> grad .107. minu .2. & <lb/>habebis propoÅ¿itum, quem tamen <lb/>oportebat eſſe gra .108. min <var>.o</var>.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0382-01" xlink:href="fig-0382-01a"> - <image file="0382-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0382-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Quod autem omnia rectè Å¿uppu-<lb/>tata Å¿int, ex Å¿umma omnium angulo <lb/>rum patere poteÅ¿t. </s> - <s xml:space="preserve">nam collectis om <lb/>nibus quinque angulis <var>.a.c.d.f.u.</var> Å¿i-<lb/>mul, hoc eÅ¿t grad .108. minu .22. cum <lb/>gra .107. min .2. cum grad .12. effi-<lb/>cient grad .540. min <var>.o.</var> Å¿umma æqua <lb/>lis Å¿ex angulis rectis.</s> - </p> - <pb o="371" rhead="EPISTOL AE." n="383" file="0383" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0383"/> - </div> - </div> - <div type="section" level="3" n="39"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">ALIA DEMONSTRATIO NONÆ, ET DECIMÆ <lb/>Å¿ecundi Euclidis.</head> - <head style="it" xml:space="preserve">Petro Catenæ.</head> - <p> - <s xml:space="preserve"><emph style="sc">QVamvis</emph> nona ac decima Å¿ecundi Euclid. aliter à Comandino & Mau-<lb/>rolico demonÅ¿tratæ fuerint, nihilominus mihi etiam viÅ¿um eÅ¿t non nihil <lb/>meo moræ in eas tibi Å¿cribere, vt Å¿enÅ¿ibiliter quoque cognoÅ¿cas il-<lb/>las veras eſſe.</s> - </p> - <p> - <s xml:space="preserve">EÅ¿to linc<unsure/>a.a.b. pro nona propoÅ¿itione, diuiÅ¿a per æqualia in <var>.c.</var> per inæqualia verò <lb/>in <var>.d.</var> quadratum autem <var>.a.d.</var> Å¿it <var>.d.e.</var> quadratum verò <var>.d.b.</var> Å¿it <var>.d.i.</var> quadratum <var>.a.c.</var> <lb/>Å¿it <var>.c.f.</var> & quadratum <var>.c.d.</var> Å¿it <var>.c.K.</var> clarum enim erit <var>.K.h.</var> æqualem exiÅ¿tere ipÅ¿i <var>.a.c.</var> <lb/>Å¿eccetur igitur <var>.e.h.</var> in <var>.g.</var> ita vt <var>.h.g.</var> Ä™qualis exiÅ¿tat ipÅ¿i <var>.K.h.</var> vnde <var>.g.e.</var> æqualls erit <lb/><var>c.d.</var> perficiatur etiam quadratum <var>.h.n.</var> vnde in totali quadrato <var>.a.h.</var> habe bis <reg norm="duplum" type="context">duplÅ©</reg> <lb/>quadrati partis <var>.c.d.</var> nempe <var>.c.K.</var> et <var>.f.g.</var> & quadratum <var>.a.u.</var> cum gnomone <var>.u.g.h.k.</var> <lb/>cui deficit quadratum æquale <var>.d.i.</var> quadrato, vt Å¿int etiam duo quadrata partis <var>.a.c</var>.</s> - </p> - <p> - <s xml:space="preserve">In decima <reg norm="autem" type="wordlist">aÅ©t</reg> propoÅ¿itione, <reg norm="quadratum" type="context">quadratÅ©</reg> totalis lineæ <var>.a.d.</var> Å¿it <var>.d.e.</var> & lineæ <var>.b.d.</var> Å¿it <var>.b.i.</var> et <lb/><var>c.d.</var> Å¿it <var>.d.n.</var> et <var>.a.c.</var> Å¿it <var>.c.f.</var> et <var>.f.e.</var> Å¿it <var>.e.u.</var> vnde <var>.n.u.</var> æquale erit quadrato <var>.b.i.</var> vnde in qua <lb/>drato totali <var>.a.h.</var> videbis duo quadrata æqualia <var>.f.c.</var> et <var>.g.k.</var> partis <var>.a.c.</var> & quadratum <var>.c.<lb/>K.</var> cum gnomone <var>.n.f.e.g.</var> cui addito quadrato <var>.b.i.</var> habebis duplum quadrati partis <var>.<lb/>c.d</var>.</s> - </p> - <figure position="here"> - <image file="0383-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0383-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="40"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE STELLA CASSIOPEIÆ.</head> - <head style="it" xml:space="preserve">Annibali Raymundo AÅ¿trologo Peritißimo.</head> - <p> - <s xml:space="preserve"><emph style="sc">POstqvam</emph> tua doctiſſima Å¿cripta perlegi, conÅ¿ideraui, quod Å¿i à mul-<lb/>titudine exhalationum in regione elementari acciderit anno .1572. & <lb/>1573. vt totos Å¿ex menÅ¿es ab omnibus per vniuerÅ¿um terrarum orbem <lb/>viÅ¿a fuerit Å¿tella illa, quæ eÅ¿t in angulo Å¿eptentrionali quadrilateri Caſſio - <pb o="372" rhead="IO. BAPT. BENED." n="384" file="0384" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0384"/> - peiæ tam lucida, vt ipÅ¿o lucifero videretur rutilantior <reg norm="atque" type="simple">atq;</reg> cæterarum (abÅ¿que vlla <lb/>aÅ¿pectus diuerÅ¿itate) magis Å¿cintillans. </s> - <s xml:space="preserve">Quî fieri poterat, vt Å¿tellæ quæ ab illa pa-<lb/>rum diÅ¿tant, alioqui multo maiores, non etiam illa clariores apparuerint? </s> - <s xml:space="preserve">Å¿ed Å¿i ali-<lb/>quis diceret eam exhalationem non ita fortaſſe dilatari, vt inter nos, & aliam ali-<lb/>quam Å¿tellam interponeretur. </s> - <s xml:space="preserve">Tunc ego reÅ¿ponderem neceſſariò Å¿equi debere ta-<lb/>lem exhalationem, tantam latitudinem occupare, quod aliquibus populis aliam <lb/><reg norm="aliquam" type="context">aliquã</reg> <reg norm="Å¿tellam" type="context">Å¿tellã</reg> circunuicinãhac ipÅ¿a de qua mentionem Å¿ecimus redderet lucidiorem. <lb/></s> - <s xml:space="preserve">Sed cum hoc perÅ¿pectum fuerit nulli, Å¿equebatur lucem illam ab ipÅ¿is exhalatio-<lb/>nibus elementaribus haud poſſe oriri: </s> - <s xml:space="preserve">quod nobis Å¿cintillatio illa maxima perma-<lb/>gno fuit inditio, Å¿i phas eÅ¿t credere, <reg norm="nam" type="context">nã</reg> quo magis aliquod coeleÅ¿te corpus Å¿cintillat, <lb/>eo longius à nobis diÅ¿tare.</s> - </p> - <p> - <s xml:space="preserve">Verum quoniam efflagitaÅ¿ti à me vt aliquid circa huiuÅ¿ce rei Å¿peculationem tibi <lb/>Å¿cribam, idcirco tibi morem gerere volens paucis Å¿ubiungam.</s> - </p> - <p> - <s xml:space="preserve">ConÅ¿idera primo hanc Å¿ubÅ¿criptam primam figuram, in qua <var>.c.a.e.</var> Å¿ignatur pro <lb/>Globo terreÅ¿tri cuius <var>.i.</var> centrum Å¿it et <var>.u.o.n.</var> pro conuexo ignis, Å¿ed <var>.K.x.s.</var> pro orbe <lb/>octauo <var>.x.</var> autem pro Å¿tella iam Å¿uperius dicta, quæ Å¿emper fuit, eÅ¿t, & erit, quamuis <lb/>cæteris tribus nunc obÅ¿curior Å¿it. </s> - <s xml:space="preserve">Accipiantur deinde duo loca in Å¿uperÅ¿icie terræ, <lb/>quÄ™ Å¿int <var>.c.</var> et <var>.e.</var> diametraliter inuicem oppoÅ¿ita, ita quod circa eorum orizontes poÅ¿ <lb/>Å¿ibile Å¿it Å¿tellam <var>.x.</var> videre, radijs ipÅ¿ius Å¿tellæ mediantibus <var>.x.n.e.</var> et <var>.x.u.c.</var> <reg norm="quorum" type="context">quorÅ©</reg> par-<lb/>tes <var>.n.c.</var> et <var>.u.e.</var> ita breues Å¿int, reÅ¿pectu eorum <reg norm="totorum" type="context">totorÅ©</reg>, vt vix Å¿excenteÅ¿ima pars Å¿it vna <lb/><reg norm="quæque" type="simple">quæq;</reg> illarum, nec non <var>.c.e.</var> ita breuis reÅ¿pectu Å¿emidiametri octauæ Å¿phæræ, quod <lb/>vix Å¿it vna ex partibus decemmillibus, vt Å¿cis, Å¿equitur quod recta terminata ab <var>.u.</var> <lb/>et <var>.n.</var> minor Å¿enÅ¿ibiliter non Å¿it ipÅ¿o terræ diametro <var>.c.e.</var> cum duo hæc interualla ex <lb/>triangulorum Å¿imilitudine Å¿e habeant vt <var>.x.i.</var> ad <var>.x.o.</var> hoc eÅ¿t ferè vt .602. ad .601. vn-<lb/>de anguli <var>.n.e.c.</var> et <var>.u.c.e.</var> à rectis minime differre videbuntur, cum eorum differen-<lb/>tia certo modo minima Å¿it. </s> - <s xml:space="preserve">ductÄ™ poÅ¿tea <reg norm="cum" type="context">cÅ©</reg> fuerint duæ diagonales <var>.e.u.</var> et <var>.n.c.</var> <reg norm="termi- nabunt" type="context">termi-<lb/>nabÅ©t</reg> angulos <var>.n.e.u.</var> et <var>.e.n.c.</var> <reg norm="inuicem" type="context">inuicẽ</reg> ferè Ä™quales, <reg norm="idem" type="context">idẽ</reg> aſſero de angulis <var>.u.c.n.</var> et <var>.e.u.c</var>.</s> - </p> - <p> - <s xml:space="preserve">Supponatur nunc primò tuam exhalationem Å¿ublimatam eſſe ad Å¿upremas par-<lb/>tes elementaris regionis circum circa lineam <var>.o.i</var>: </s> - <s xml:space="preserve">tunc clarum eſſet quod Å¿i ratione hu <lb/>iuÅ¿modi exhalationis Å¿tella <var>.x.</var> ita lucida viÅ¿a fuerit tam aÅ¿picientibus ab <var>.e.</var> quam ab <lb/>c. exhalatio minoris latitudinis quam <var>.u.n.</var> eſſe non poterat, hoc eÅ¿t, quam terræ dia-<lb/>meter, cum idem in longitudine ferè Å¿it, Å¿ed punctum <var>.u.</var> Å¿atis videri poteÅ¿t ab oculo <lb/>in <var>.e.</var> & punctum <var>.n.</var> ab oculo in <var>.c.</var> vt alias tibi probaui, ratione refractionis radiorum <lb/>per diuerÅ¿a diafana tranÅ¿euntium. </s> - <s xml:space="preserve">Nunc producti cum fuerint ij duo radij <var>.e.u.</var> et <var>.c.<lb/>n.</var> vÅ¿que ad octauum orbem ad puncta <var>.s.</var> et <var>.K.</var> reliquum erit nos videre quantitates <lb/>graduum arcus <var>.s.x.</var> et <var>.k.x.</var> Å¿ed <var>.s.x.</var> Å¿ubiacet a ngulo <var>.s.e.x.</var> et <var>.k.x.</var> angulo <var>.k.c.x.</var> qui qui <lb/>quidem anguli nihil differunt Å¿enÅ¿ibiliter ac Å¿i eſſent in centro <var>.i</var>. </s> - <s xml:space="preserve">Et cum Å¿uperius di-<lb/>xerimus angulos <var>.s.e.x.</var> et <var>.k.c.x.</var> Å¿enÅ¿ibiliter mi nime differre ab angulis <var>.c.n.e.</var> et <var>.e.<lb/>u.c.</var> Å¿i cognouerimus quantitatem iÅ¿torum, cognita etiam nobis erit quantitas <lb/>ill orum.</s> - </p> - <p> - <s xml:space="preserve">Cum igitur Å¿emidiameter elementaris regionis maior Å¿it Å¿emidiametro terræ, vt <lb/>33. ad vnum, & cogitata <var>.c.n.</var> vt dicta Å¿emidiameter, quia Å¿enÅ¿ibiliter ab ea minime <lb/>diffe<unsure/>rt, nunc Å¿i Å¿upponatur dicta <var>.n.c.</var> vt <reg norm="baÅ¿istrianguli" type="context">baÅ¿istriãguli</reg> orthogonij eſſe <reg norm="partium" type="context">partiÅ©</reg> .100000 <lb/>& dixerimus Å¿i <var>.c.n.</var> vt partium .33. præbet nobis <var>.c.e.</var> duarum partium, quid nobis <lb/>pręſtabit eadem <var>.c.n.</var> vt partium .100000. vnde proueniet nobis <var>.c.e.</var> vt <reg norm="partium" type="context">partiÅ©</reg> .6060. <lb/>cuius angulus <var>.c.n.e.</var> erit graduum .3. & min .29. ita etiam erit angulus <var>.k.</var> e<unsure/> <var>.x.</var> cuius ar- - <pb o="373" rhead="EPISTOL AE." n="385" file="0385" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0385"/> - c<unsure/>us <var>.s.x.</var> eorundem graduum erit, <lb/> - <anchor type="figure" xlink:label="fig-0385-01a" xlink:href="fig-0385-01"/> - & minutorum. </s> - <s xml:space="preserve">idem dico de arcu <var>.<lb/>x.k</var>. </s> - <s xml:space="preserve">Sed circa dictam Å¿tellam om-<lb/>nes aliæ non diÅ¿tant huiuÅ¿modi in <lb/>teruallo.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0385-01" xlink:href="fig-0385-01a"> - <image file="0385-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0385-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Nihilominus nec tu nec alij pe <lb/>ripatetici qui hanc Å¿equuti Å¿unt <lb/>opinionem exhalationum, ad Å¿er <lb/>uandam nullitatem diucrÅ¿itatis <lb/>aÅ¿pectus, affirmant poſſe tam lon <lb/>ge à terra <reg norm="aÅ¿cendere" type="context">aÅ¿cẽdere</reg> exhalationes, <lb/>imo nec attingere Å¿upremas tertiÄ™ <lb/>regionis a eris partes, ita ut non <reg norm="in- grediantur" type="context">in-<lb/>grediãtur</reg> Å¿uum <reg norm="igneum" type="context">igneÅ©</reg> orbem, qui <lb/>quidem orbis Å¿ecundum illorum <lb/>opinionem incipit non valde lon-<lb/>gè à ſuperÅ¿icie terræ, vt <reg norm="iam" type="context">iã</reg> in mea <lb/>conÅ¿ideratione contra Antonium <lb/>Bergam probaui, Å¿ed demus, <reg norm="quod" type="simple">ê™</reg> <lb/>dictæ exhalationes aÅ¿cenderint <lb/>per decem Å¿emidiametros terræ, <lb/>diÅ¿currendo poÅ¿tea Å¿ic, cum <var>.c.n.</var> ut <lb/>decem, nobis dat <var>.c.e.</var> vt duo, <reg norm="quid" type="simple">ê—d</reg> <lb/>dabit nobisipÅ¿a <var>.c.n.</var> vt .100000. <lb/>& proueniet nobis <var>.c.e.</var> vt .200000. <lb/>cuius Å¿inus angulus erit gra .11. mi <num value="32">.<lb/>32.</num> & ita erunt anguli <var>.s.e.x.</var> et <var>.k.<lb/>c.x.</var> & Å¿ic eorum arcus <var>.s.x.</var> et <var>.k.x.</var> <lb/>Å¿ed quis vnquam dubitabit <reg norm="quod" type="simple">ê™</reg> in <lb/>tanto interuallo à dicta Å¿tella non <lb/>fint aliæ multæ ipÅ¿a maiores? </s> - <s xml:space="preserve">li-<lb/>neas vero <var>.e.o.r.</var> et <var>.c.o.t.</var> duxi, vt <lb/>videres effectum maioris aÅ¿pectus <lb/>diuerÅ¿itatis ab oculis <var>.e.</var> et <var>.c.</var> in cir-<lb/>culo altitudinis quando <var>.o.</var> fuiſſet <lb/>punctum illud lucidiſſimum, & <lb/>non <var>.x</var>.</s> - </p> - <pb o="374" rhead="IO. BAPT. BENED." n="386" file="0386" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0386"/> - <p> - <s xml:space="preserve">At poterit aliquis mihi obijcere quod cum <var>.i.o.</var> fuiſſet longior <var>.i.e.</var> per decem vi-<lb/>ces tantummodo, exiÅ¿tente oculo in <var>.e.</var> uel <var>.c.</var> per gradus .90. ab <var>.a</var>. </s> - <s xml:space="preserve">tunc punctus <var>.u.</var> vel <lb/>n. ab ipÅ¿o oculo non videretur ob terræ globoÅ¿itatem. </s> - <s xml:space="preserve">Imaginemur igitur à puncto <lb/>u. recta <var>.u.b.</var> tangens quartam <var>.a.e.</var> in puncto <var>.b.</var> vt in Å¿ecunda figura videre eÅ¿t, in qua <lb/>ducantur <var>.c.b</var>: <var>i.b</var>: et <var>.i.u.</var> quæ <var>.i.u.</var> Å¿ecabit arcum <var>.c.</var> <lb/> - <anchor type="figure" xlink:label="fig-0386-01a" xlink:href="fig-0386-01"/> - b. in puncto <var>.p.</var> per æqualia et <var>.c.b.</var> Å¿imiliter in pun-<lb/>cto <var>.y.</var> quod nulli dubium eÅ¿t, cum <var>.c.u.</var> æqualis Å¿it <var>.<lb/>u.b.</var> ex .35. tertij Euclidis, </s> - <s xml:space="preserve">unde ex octaua primi an-<lb/>gulus <var>.c.i.u.</var> æqualis erit angulo <var>.u.i.b.</var> & ideo arcus <var>.<lb/>c.p.</var> æquabitur arcui <var>.p.b.</var> Å¿ed ex .4. primi <var>.c.y.</var> æqua <lb/>lis erit ipÅ¿i <var>.y.b</var>.</s> - </p> - <div type="float" level="5" n="2"> - <figure xlink:label="fig-0386-01" xlink:href="fig-0386-01a"> - <image file="0386-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0386-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Nunc Å¿uppoÅ¿ita <var>.c.i.</var> decima parte ipÅ¿ius <var>.c.u.</var> nemi <lb/>ni dubium erit quod cum <var>.u.i.</var> Å¿ubtendatur angulo <lb/>recto <var>.u.c.i.</var> (iam Å¿upra diximus angulum <var>.c.</var> Å¿enÅ¿ibi-<lb/>liter minime differre à recto) ipÅ¿a vt Å¿inus totus erit <lb/>partium .100000. cuius quadratum cum diuiÅ¿um <lb/>fuerit in partes æquales centum & <reg norm="vnam" type="context">vnã</reg>, illarum vna <lb/>æqualis erit quadrto <var>.c.i.</var> reliquÄ™ vero quadrato ip-<lb/>Å¿ius <var>.u.c.</var> ex proportione duplicata quadratorum ad <lb/>eam quam continent eorum latera. </s> - <s xml:space="preserve">Sed quadra-<lb/>tum ipÅ¿ius <var>.u.i.</var> eÅ¿t partium .10000000000. </s> - <s xml:space="preserve">quare <lb/>quadratum <var>.c.i.</var> erit .99009900. cuius radix <var>.c.i.</var> <lb/>erit partium .9950. vnde quadratum ipÅ¿ius <var>.c.u.</var> erit <lb/>partium .9900990100. cuius radix <var>.u.c.</var> erit <reg norm="partium" type="context">partiÅ©</reg> <lb/>99500. vnde angulus <var>.c.i.u.</var> erit graduum .84. & mi <lb/>nu .17. & angulus <var>.c.u.i.</var> qui reÅ¿pondet Å¿inui <var>.c.i.</var> erit <lb/>gra .5. & min .43. cuius duplum, hoc eÅ¿t angulus <var>.c.<lb/>u.b.</var> erit grad .11. min .26. æqualis ferè angulo iam <lb/>Å¿upradicto. </s> - <s xml:space="preserve">Å¿ed <var>.c.y.</var> Å¿inus anguli <var>.c.i.y.</var> erit Å¿imiliter <lb/>partium .99500. talium vt <var>.c.i.</var> Å¿unt .100000. Å¿ed vt <lb/><var>c.i.</var> eÅ¿t partium .9950. </s> - <s xml:space="preserve">tunc <var>.c.y.</var> erit partium .9900 <lb/>hoc eÅ¿t quaÅ¿i decima pars ipÅ¿ius <var>.c.u.</var> </s> - <s xml:space="preserve">quare Å¿i ocu-<lb/>lus in <var>.e.</var> non videbit punctum <var>.u.</var> hoc punctum be-<lb/>ne videbitur ab oculo in <var>.b.</var> abÅ¿que Å¿enÅ¿ibili dimi-<lb/>nutione anguli in puncto <var>.u.</var> vt probauimus.</s> - </p> - </div> - </div> - <div type="section" level="3" n="41"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE MAGNITVDINIBVS FIGVRARVM <lb/>iÅ¿operimetrarum.</head> - <head style="it" xml:space="preserve">Domino Ioanni Mariæ Agatio.</head> - <p> - <s xml:space="preserve"><emph style="sc">QVamvis</emph> à Theone Å¿upra Ptolomei AlmageÅ¿tum Å¿ufficienter traditum <lb/>Å¿it de magnitudinibus figurarum iÅ¿opetimetrarum, nihilominus vt tibi <lb/>morem geram, ea nunc Å¿cribo, quæ mihi in mentem venerunt contra <lb/><reg norm="Alexandrum" type="context">Alexãdrum</reg> Piccolhomineum, <reg norm="antequam" type="context">antequã</reg> aliquid ipÅ¿ius Theonis vidiſſem - <pb o="375" rhead="EPISTOLAE." n="387" file="0387" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0387"/> - Alexander Piccolhomineus in libro primo de mundi Å¿phæra vbi tractat de <reg norm="caeliro- tunditate" type="simple">cÄ™liro-<lb/>tunditate</reg>, ita inquit.</s> - </p> - <quote xml:lang="it"> - <s xml:space="preserve">Oltre di queÅ¿to, douendo il decimo cielo contenere & in Å¿e chiudere tutte le co-<lb/>Å¿e, è conueneuol coÅ¿a il penÅ¿are, che foſſe fatto di quella più capace figura che eÅ¿-<lb/>Å¿er poſſa, la qual è la figura rotunda, però che Å¿i può trar da molti luoghi d'Euclide <lb/>che Å¿i come Å¿e noi ciimmagineremo più figure Å¿uperficiali talmente che tutte le li-<lb/>nee de l'vna congionte inÅ¿ieme, Å¿ieno vguali à tutte le linee pur inÅ¿iememente com <lb/>poÅ¿te di qual Å¿i voglia de l'altre figure, ne Å¿eguirà , che quella figura Å¿arà più capa-<lb/>ce la qual haurà manco angoli, & quella capaciſſima che Å¿arà ſenza alcuno come è <lb/>la figura circolare, & c.</s> - </quote> - <p> - <s xml:space="preserve">Cogitemus igitur primò de triangulo æquilate-<lb/>ro & quadrato iÅ¿operimetris, Å¿it enim triangulus æ-<lb/>quilaterus <var>.o.b.g.</var> quadratum verò <var>.b.l.</var> quorum pe-<lb/>riferiæ inuicem æquales Å¿int. </s> - <s xml:space="preserve">Dico quadratum ma-<lb/> - <anchor type="figure" xlink:label="fig-0387-01a" xlink:href="fig-0387-01"/> - ioris Å¿uperficiei eſſe ipÅ¿o triangulo. </s> - <s xml:space="preserve">Accipio pri-<lb/>mum lineam <var>.f.h.</var> eiuÅ¿dem longitudinis quæ vnius <lb/>periferiæ dictarum figurarum, quam punctis <var>.r.K.</var> <lb/>mediantibus diuido in tres Ä™quas partes, in quatuor <lb/>verò mediantibus punctis <var>.l.x.i.</var> vnde proportio to-<lb/>tius <var>.f.h.</var> ad <var>.K.h.</var> erit vt <var>.l.h.</var> ad <var>.i.h.</var> ideÅ¿t tripla, & per <lb/>16. quinti erit <var>.f.h.</var> ad <var>.l.h.</var> vt <var>.k.h.</var> ad <var>.i.h.</var> per .19. verò <lb/><var>f.h.</var> ad <var>.f.l.</var> vt <var>.K.h.</var> ad <var>.K.i.</var> Å¿ed <var>.f.l.</var> eÅ¿t quarta pars ip-<lb/>Å¿ius <var>.f.h.</var> ergo <var>.k.i.</var> erit quarta pars ipÅ¿ius <var>.k.h</var>. </s> - <s xml:space="preserve"><reg norm="Conium" type="context">ConiÅ©</reg> <lb/>gantur enim ambo iÅ¿tæ figuræ vt hic inferius vides, <lb/>vnde <var>.a.g.</var> erit quarta pars ipÅ¿ius <var>.b.g.</var> diuiÅ¿a poÅ¿tea <var>.<lb/>b.g.</var> per æqualia in <var>.c.</var> erit <var>.a.c.</var> æqualis <var>.a.g</var>. </s> - <s xml:space="preserve">Ducatur <lb/>deinde <var>.o.c.</var> quÄ™ per .8. primi, nec <reg norm="non" type="context">nõ</reg> ex definitione, <lb/>perpendicularis erit ipÅ¿i <var>.b.g.</var> ergo etiam <reg norm="quadratum" type="context">quadratÅ©</reg> <lb/>b q. Å¿upra <var>.b.g.</var> <reg norm="producoque" type="simple">producoqÌ;</reg> <var>.o.c.</var> vÅ¿que ad <var>.m.</var> nam nul <lb/>li dubium eÅ¿t quin <var>.o.c.</var> breuior Å¿it <var>.o.g.</var> ex .18. vel .48 <lb/>primi cui æquatur <var>.q.g.</var> diuido etiam <var>.c.m.</var> per æqua <lb/>lia in puncto <var>.e.</var> <reg norm="ducoque" type="simple">ducoqÌ;</reg> <var>t.e.p.</var> æquidiÅ¿tantem <var>.b.g.</var> <lb/>vnde habebimus duo quadrata <var>.e.g.</var> et <var>.e.b.</var> Å¿ed <lb/>quadratum <var>.b.l.</var> æquatur quadrato ipÅ¿ius <var>.c.a.</var> <lb/>cum duplo illius quod fit ex <var>.b.c.</var> in <var>.c.g.</var> vt patet <lb/>ex .9. Å¿ecundi, hoc eÅ¿t æquatur quadrato <var>.c.a.</var> & re-<lb/>ctangulo <var>.t.g</var>. </s> - <s xml:space="preserve">Deinde vt Å¿e habet <var>.p.g.</var> ad <var>.o.e.</var> ita Å¿e habet <var>.u.p.</var> ad <var>.u.e.</var> ex Å¿imilitudine <lb/>triangulorum. </s> - <s xml:space="preserve">Sed <var>.p.g.</var> maior eÅ¿t ipÅ¿a <var>.o.e.</var> cum <var>.p.g.</var> æqualis Å¿it <var>.e.m.</var> </s> - <s xml:space="preserve">quare triangu-<lb/>lus <var>.u.g.p.</var> maior erit triangulo <var>.o.e.u.</var> ex .17. Å¿exti. </s> - <s xml:space="preserve">Similiter dico maiorem eſſe trian <lb/>gulum <var>.b.d.t.</var> triangulo <var>.e.o.d.</var> vnde Å¿equitur rectangulum <var>.t.g.</var> maiorem eſſe triangu-<lb/>lo <var>.b.o.g.</var> Å¿ed quadratum <var>.b.l.</var> eÅ¿t etiam maior ipÅ¿o rectangulo <var>.t.g.</var> ex quadrato ipÅ¿ius <lb/><var>c.a.</var> vt diximus, tanto igitur maior erit triangulo <var>.b.o.g</var>.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0387-01" xlink:href="fig-0387-01a"> - <image file="0387-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0387-01"/> - </figure> - </div> - <pb o="376" rhead="IO. BABPT. BENED." n="388" file="0388" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0388"/> - <p> - <s xml:space="preserve">Poſſumus etiam probare quod periferia quadrati æqualis triangulo æquilatero <lb/>minor Å¿it periferia ipÅ¿ius trianguli æquilateri. </s> - <s xml:space="preserve">Cogita triangulum æquilaterum hic <lb/>Å¿ubÅ¿criptum <var>.d.l.q.</var> cuius baÅ¿is <var>.l.q.</var> diuiÅ¿a Å¿it per æqualia à perpendiculari <var>.d.o.</var> <reg norm="deÅ¿cri- ptumque" type="context simple">deÅ¿criÂ<lb/>ptÅ©qÌ;</reg> Å¿it rectangulum <var>.o.g.</var> quod æquale erit triangulo <var>.d.l.q.</var> Å¿ed periferia trianguli <lb/>maior eÅ¿t periferia rectanguli, nam <var>.l.q.</var> æqualis eÅ¿t <var>.o.q.</var> cum <var>.d.g.</var> Å¿ed <var>.q.d.</var> maior eÅ¿t <var>.o.<lb/>d.</var> ex .18. primi, vnde <var>.l.d.</var> maior etiam <var>.q.g.</var> cum ex .34. dicti latera oppoÅ¿ita ipÅ¿ius re <lb/>ctanguli Å¿int inuicem æqualia, accipiamus poÅ¿tea <var>.e.c.</var> æqualem <var>.o.d.</var> et <var>.c.h.</var> indire-<lb/>ctum æqualem <var>.o.q.</var> circa quem diametrum <var>.e.h.</var> intelligatur circulus <var>.e.i.h.k.</var> et. à punÂ<lb/>cto <var>.c.</var> dirigatur perpendicularis <var>.k.i.</var> ad <var>.e.h.</var> vnde ex .3. tertij <var>.c.i.</var> æqualis erit <var>.c.k.</var> & ex <lb/>34. quod fit ex <var>.c.i.</var> in <var>.c.k.</var> hoc eÅ¿t quadratum ipÅ¿ius <var>.c.i.</var> æquale erit ei quod fit .ex <var>.e.c.</var> <lb/>in <var>.c.h.</var> hoc eÅ¿t rectangulo <var>.g.o.</var> hoc eÅ¿t triangulo <var>.d.l.q.</var> Å¿ed <var>.e.h.</var> eÅ¿t dimidium perife-<lb/>riÄ™ ipÅ¿ius rectanguli <var>.g.o.</var> quæ minor eÅ¿t di midio periferiæ trianguli <var>.d.l.q.</var> vt vidimus <lb/>et <var>.i.k.</var> eÅ¿t dimidium periferiÄ™ quadrati ipÅ¿ius <var>.i.c.</var> & minor etiam ipÅ¿a <var>.e.h.</var> ex .14. tertij <lb/></s> - <s xml:space="preserve">quare verum eÅ¿t propoÅ¿itum.</s> - </p> - <figure position="here"> - <image file="0388-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0388-01"/> - </figure> - <p> - <s xml:space="preserve">Sed quando periferiæ Å¿unt inuicem æquales, poſſumus etiam breuiter videre id <lb/>quod Å¿upradiximus, hoc eÅ¿t, quod quadratum, maius Å¿it triangulo æquilatero. </s> - <s xml:space="preserve">Nam <lb/>cum <var>.b.g.</var> Å¿eÅ¿quitertia Å¿it ad <var>.b.a.</var> ergo <var>.b.g.</var> erit vt .4. et <var>.b.a.</var> ut .3. vnde <var>.b.q.</var> erit vt .16 <lb/>et <var>.b.l.</var> vt .9. et <var>.c.q.</var> vt .8. </s> - <s xml:space="preserve">quare <var>.b.l.</var> maius erit ipÅ¿o <reg norm="rectangulo" type="context">rectãgulo</reg> <var>.c.q.</var> Å¿ed <var>.c.q.</var> maius eÅ¿t <reg norm="triam" type="context">triã</reg> <lb/>gulo <var>.b.o.g.</var> cum <var>.q.g.</var> quæ æqualis eÅ¿t <var>.o.g.</var> maior Å¿it <var>.o.c.</var> ex .18. vel penultima primi, <lb/>nam Å¿i <var>.q.g.</var> æqualis eſſet <var>.o.c.</var> </s> - <s xml:space="preserve">tunc <var>.c.q.</var> æqualis eſſet triangulo <var>.b.o.g.</var> ex .41. primi.</s> - </p> - <p> - <s xml:space="preserve">Alia etiam via maiores noÅ¿tri vÅ¿i Å¿unt quæ generalis eÅ¿t vt in Theone Å¿upra Al-<lb/>mageÅ¿tum videre eÅ¿t, medijs perpendicularibus à centris ad latera figurarum, Å¿ed <lb/>quia <reg norm="differentia" type="context">differẽtia</reg> longitudinum ipÅ¿arum perpendicularium alio medio inueniri poteÅ¿t, <lb/>eo quo ipÅ¿i vÅ¿i Å¿unt, prætermittere nolo quin tibi Å¿cribam.</s> - </p> - <p> - <s xml:space="preserve">Ego enim ita diÅ¿curro.</s> - </p> - <p> - <s xml:space="preserve">Sint duæ figuræ iÅ¿operimetrÄ™ æquilaterÄ™ & æquiangulæ, puta primò trian-<lb/>gulum & quadratum quorum centra Å¿int <var>.e.</var> et <var>.o.</var> à quibus centris ad latera Å¿int per-<lb/>pendiculares <var>.e.n.</var> et <var>.o.u.</var> vnde <var>.n.</var> et <var>.u.</var> diuident latera per æqualia vt Å¿cis, ducantur <lb/>poÅ¿tea <var>.e.t.</var> et <var>.o.a.</var> ad angulos dictorum laterum, vnde habebimus angulum <var>.o.a.u.</var> <reg norm="di- midium" type="context">di-<lb/>midiÅ©</reg> recti, et <var>.e.t.n.</var> tertia pars vnius recti, vt ex te ipÅ¿o videre potes, </s> - <s xml:space="preserve">quare angulus - <pb o="377" rhead="EPISTOLAE." n="389" file="0389" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0389"/> - a. Å¿eÅ¿quialter erit angulo <var>.t.</var> quod vt clarius videas cogita lineam <var>.b.d.</var> cuius medietas <lb/>Å¿it <var>.c.d.</var> tertia verò pars illius Å¿it <var>.g.d.</var> </s> - <s xml:space="preserve">tunc dico <var>.c.d.</var> Å¿eÅ¿quialteram eſſe ipÅ¿i <var>.g.d.</var> Å¿it enim <lb/><var>f.d.</var> duplum ipÅ¿ius <var>.g.d.</var> </s> - <s xml:space="preserve">quare <var>.f.d.</var> erunt duæ tertiæ totius lineÄ™ <var>.b.d.</var> & quia eadem pro <lb/>portio eÅ¿t totius <var>.b.d.</var> ad <var>.c.d.</var> quæ <var>.f.d.</var> ad <var>.g.d.</var> ergo permutando eadem erit totius <var>.b.<lb/>d.</var> ad <var>.f.d.</var> quæ <var>.c.d.</var> ad <var>.g.d</var>. </s> - <s xml:space="preserve">Sed <var>.b.d.</var> ad <var>.f.d.</var> Å¿eÅ¿quialtera eÅ¿t, verum igitur erit quod an-<lb/>gulus <var>.a.</var> Å¿eÅ¿quialter Å¿it ipÅ¿i <var>.t.</var> deinde <var>.t.n.</var> eÅ¿t Å¿eÅ¿quitertia ipÅ¿i <var>.a.u.</var> vt Å¿uperius vidimus .<lb/>in eorum duplis. </s> - <s xml:space="preserve">Å¿cimus etiam <var>.n.e.</var> eſſe dimidium ipÅ¿ius <var>.t.e.</var> co quod cum <var>.e.t.n.</var> Å¿it <lb/>tertia pars vnius recti, angulus, <var>t.e.n.</var> erit duo tertia vnius recti, vnde <var>.e.n.</var> erit latus. <lb/>exagoni æquilateris inÅ¿criptibilis circulo cuius diameter Å¿it <var>.e.t.</var> </s> - <s xml:space="preserve">quare <var>.e.t.</var> dupla erit <lb/>ipÅ¿i <var>.e.n.</var> in longitudine, Å¿ed quadrupla in potentia: </s> - <s xml:space="preserve">t.n. vero tripla in potentia ipÅ¿i <var>.n.<lb/>e.</var> ex penultima primi, quæ omnia etiam ex .8. tertijdecimi. Eucli. elicere potes, Å¿ed <lb/><var>c.n.</var> erat Å¿exquitertia ipÅ¿i <var>.a.u.</var> in longitudine, hoc eÅ¿t ipÅ¿i <var>.o.u.</var> nam <var>.o.u.</var> æqualis eÅ¿t ipÅ¿i <lb/><var>a.u</var>. </s> - <s xml:space="preserve">quare <var>.n.t.</var> erit minus quam dupla in potentia ipÅ¿i <var>.o.u.</var> hoc eÅ¿t, vt .16. ad .9. ergo <lb/>maior proportio erit ipÅ¿ius <var>.t.n.</var> in potentia ad <var>.n.e.</var> quam ad <var>.o.u.</var> </s> - <s xml:space="preserve">quare etiam in lon <lb/>gitudine, maior proportio erit ipÅ¿ius <var>.t.n.</var> ad <var>.n.e.</var> quam ad <var>.o.u.</var> vnde <var>.o.u.</var> longior erit <lb/>ipÅ¿a <var>.n.e.</var> quod eÅ¿t propoÅ¿itum.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i <var>.o.a.u.</var> eſſet pentagonus æquilaterus & æquiangulus, Å¿imiliter probabo per-<lb/>pendicularem <var>.o.u.</var> longiorem eſſe <var>.n.e.</var> ipÅ¿ius trianguli æquilateri, dummodo Å¿int iÅ¿o-<lb/>perimetrÄ™. </s> - <s xml:space="preserve">Sit enim <var>.a.u.</var> dimidium lateris pentagoni ex Å¿uppoÅ¿ito, cuius centrum Å¿it <lb/>o. </s> - <s xml:space="preserve">tunc proportio <var>.t.n.</var> ad <var>.a.u.</var> erit Å¿uperbipartienstertias, vt ex ordine iam hic Å¿upradi <lb/>cto à te facillimè elicere potes, hoc eÅ¿t, vt .5. ad .3. et <var>.a.u.</var> minor erit <var>.o.u.</var> eo quod <lb/>angulus <var>.o.</var> minor erit angulo <var>.a.</var> nam angulus <var>.o.</var> erit quinta pars <reg norm="duorum" type="context">duorÅ©</reg> rectorum, hoc <lb/>eÅ¿t duæ quintæ vnius recti, vnde angulus <var>.a.</var> reÅ¿iduum vnius recti erit tres quin-<lb/>tæ vnius recti, </s> - <s xml:space="preserve">quare angulus <var>.a.</var> maior ericangulo <var>.o.</var> & conÅ¿equenter latus <var>.o.u.</var> ma-<lb/>ius latere <var>.a.u.</var> Å¿ed <var>.t.n.</var> minor eÅ¿t quam tripla in potentia ad <var>.a.u.</var> eo quod erit vt .25. <lb/>ad .9. cum in longitudine Å¿it vt .5. ad .3. Å¿ed dicta <var>.t.n.</var> tripla eÅ¿t in potentia ad <var>.e.n.</var> qua-<lb/>re <var>.a.u.</var> maior erit ipÅ¿a <var>.e.n.</var> Å¿ed <var>.o.u.</var> maior eÅ¿t ipÅ¿a <var>.a.u.</var> vt diximus, igitur multo magis <var>.<lb/>o.u.</var> maior eÅ¿t ipÅ¿a <var>.a.u.</var> vt <reg norm="diximus" type="simple">diximê°</reg> & <reg norm="conÅ¿equenter" type="context context">cõſequẽter</reg> multo magis <var>.o.u.</var> maior erit ipÅ¿a <var>.n.e</var>.</s> - </p> - <p> - <s xml:space="preserve">QuotieÅ¿cunque enim cognoÅ¿cimus proportionem anguli <var>.o.</var> ad angulum <var>.a.</var> quod <lb/>quidem facillimum eÅ¿t, nec non proportionem <var>.t.n.</var> ad <var>.a.u.</var> quod, etiam illico cogno-<lb/>Å¿citur, </s> - <s xml:space="preserve">tunc exÅ¿cientia cordarum & arcuum omnia etiam facillimè innueniuntur. <lb/></s> - <s xml:space="preserve">Verum circa <reg norm="triangulum" type="context">triangulÅ©</reg> æquilaterum, & pentagonum, alium <reg norm="modum" type="context">modÅ©</reg> inueni, Å¿ed aliquan <lb/>tulum prolixiorem.</s> - </p> - <figure position="here"> - <image file="0389-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0389-01"/> - </figure> - <pb o="378" rhead="IO. BAPT. BENED." n="390" file="0390" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0390"/> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">De incommenÅ¿ur abilitate, in longitudine perpendicu-<lb/>laris trianguli æquilateri cum eiuÅ¿dem latere.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">ID quod à me poÅ¿tulas eÅ¿t omnino impoſſibile, velles enim duos numeros inueni <lb/>re inter Å¿e ita Å¿e habentes, vt Å¿e habent perpendicularis in triangulo æquilatero <lb/>cum vno eius laterum, quod vero hoc fieri non poſſit, conÅ¿idera in figura præcedenti <lb/>triangulum æquilaterum <var>.d.l.q.</var> cuius perpendicularis Å¿it <var>.d.o.</var> quæ diuidit <var>.l.q.</var> per <lb/>æqualia in <var>.o.</var> vnde ex .4. Å¿ecundi Euclidis, quadratum <var>.l.q.</var> (ideÅ¿t <var>.d.q.</var>) quadruplum <lb/>erit quadrato <var>.o.q.</var> & ex penultima primi Ä™quale quadratis <var>.d.o.</var> et <var>.o.q.</var> </s> - <s xml:space="preserve">quare erit Å¿eÅ¿-<lb/>quitertium quadrato ipÅ¿ius <var>.d.o.</var> & ita quadratum <var>.d.o.</var> erit triplum quadrato ipÅ¿ius <var>.<lb/>o.q.</var> hæe autem proportiones non Å¿unt vt numeri quadrati ad numerum quadratum <lb/>quod Å¿i ita fuiſſent, Å¿equeretur ternarium numerum eſſe quadratum ex .22. octaui. <lb/></s> - <s xml:space="preserve">Cum igitur non Å¿int vt numeri quadrati ad numerum quadratum, Å¿equitur ex Å¿epti-<lb/>ma decimi <var>.d.o.</var> eſſe incommenÅ¿urabilem ipÅ¿i <var>.l.q.</var> Å¿eu <var>.d.q.</var> in longitudine.</s> - </p> - <p> - <s xml:space="preserve">Vel dicamus ita, proportio quadrati ipÅ¿ius <var>.l.q.</var> ad quadratum ipÅ¿ius <var>.o.d.</var> eÅ¿t in ge <lb/>nere Å¿uperparticulari, cum Å¿it Å¿eÅ¿quitertia, vnde quadratum ipÅ¿ius <var>.d.o.</var> numeris da-<lb/>ri non poteÅ¿t, eo quod Å¿i dabilis fuiſſet, Å¿equeretur, quod inter quadratum ipÅ¿ius. l<unsure/> <var>.<lb/>q.</var> & ipÅ¿ius <var>.d.o.</var> eſſet aliquis numerus medius proportionalis ex .16. octaui, vnde ex <lb/>octaua eiuÅ¿dem vnitas diuiÅ¿ibilis eſſet, quod fieri non poteÅ¿t.</s> - </p> - <figure position="here"> - <image file="0390-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0390-01"/> - </figure> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">De triangulo & Pentagono æquilatero</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">MOdum quem conÅ¿ideraui circa triangulum æquilaterum & pentagonum, vt <lb/>tibi Å¿ignificaui ita Å¿e habet.</s> - </p> - <p> - <s xml:space="preserve">Probandum primò eÅ¿t pentagonum altiorem eſſe triangulo Å¿ibi iÅ¿operimetro. <lb/></s> - <s xml:space="preserve">Iam tibi notam puto proportionem lateris trianguli ad latus pentagoni eſſe vt .5. <lb/>ad .3.</s> - </p> - <p> - <s xml:space="preserve">Sit igitur pentagonus <var>.b.d.m.g.v.</var> cuius periferia diÅ¿tenta Å¿it <var>.K.z.</var> baÅ¿is autem <var>.m.<lb/>g.</var> bifariam diuiÅ¿a Å¿it in puncto <var>.a.</var> <reg norm="ductaque" type="simple">ductaqÌ;</reg> <var>.a.b</var>: <var>b.g.</var> et <var>.b.m.</var> clarum erit <var>.a.b.</var> perdicu-<lb/>larem eſſe ad <var>.m.g.</var> ex .8. primi Eucli. cum <var>.b.m.</var> et <var>.b.g.</var> (baÅ¿es triangulorum <var>.b.d.m.</var> - <pb o="379" rhead="EPISTOLAE." n="391" file="0391" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0391"/> - et <var>.b.u.g.</var>) Å¿int inuicem æquales ex .4. eiuÅ¿dem.</s> - </p> - <p> - <s xml:space="preserve">Accipiatur deinde vel intelligatur <var>.g.p.</var> æqualis duabus te<unsure/>rtijs ipÅ¿ius <var>.a.g.</var> ducaturÂ<lb/>qÌue <var>.b.p.</var> quam probabo maiorem eſſe duplo ipÅ¿ius <var>.a.p.</var> vnde maior erit latere ipÅ¿ius <lb/>trigoni æquilateris, cuius dimidium eÅ¿t <var>.a.p.</var> Å¿cimus enim ipÅ¿um latus Å¿e habere ad <var>.m.<lb/>g.</var> vt quinque ad .3. ita etiam <var>.a.p.</var> ad <var>.a.g.</var> vt diximus.</s> - </p> - <p> - <s xml:space="preserve">Cum <reg norm="autem" type="context">autẽ</reg> angulus <var>.a.b.g.</var> Å¿it quarta pars anguli <var>.b.g.a.</var> ex .10. quarti & quinta pars <lb/>vnius recti ex .32. primi, dictus angulus erit graduum .18. et <var>.a.g.</var> erit partium .30902. <lb/>et <var>.a.b.</var> partium .95015 et <var>.a.p.</var> 51503. vnde ex penultima primi latus <var>.b.p.</var> erit par-<lb/>tium .108075. duplum vero ipÅ¿ius <var>.a.p.</var> erit .103006. latus igitur dicti trigoni, quod <lb/>ab <var>.p.</var> erigitur, Å¿ecabit perpendicularem <var>.a.b.</var> Å¿ub <var>.b.</var> hoc eÅ¿t inter <var>.b.</var> et <var>.a.</var> ex penultima <lb/>primi. </s> - <s xml:space="preserve">Finiatur enim triangulus æquicrurus <var>.b.q.p.</var> quem probaui maiorem eſſe æ-<lb/>quilatero iÅ¿operimetro pentagono propoÅ¿ito, <reg norm="ducaturque" type="simple">ducaturqÌ;</reg> <var>.u.p.</var> ducatur etiam <var>.u.n.</var> pa-<lb/>rallela ipÅ¿i <var>.b.g.</var> quæ concludet triangulum <var>.g.u.n.</var> Å¿imilem triangulo <var>.m.b.g.</var> eo quod <lb/>cum angulus <var>.m.b.g.</var> æqualis Å¿it angulo <var>.b.g.u.</var> ex .16. tertij, per .27. primi <var>.m.b.</var> et <var>.g.u.</var> <lb/>erunt inuicem <reg norm="æquidiÅ¿tantes" type="context">æquidiÅ¿tãtes</reg>, vnde angulus <var>.b.m.g.</var> æqualis erit angulo <var>.u.g.n.</var> et. ex .29. <lb/>angulus <var>.g.u.n.</var> æqualis erit angulo <var>.u.g.b</var>. </s> - <s xml:space="preserve">quare etiam angulo <var>.g.b.m.</var> & angulus <var>.u.n.<lb/>g.</var> angulo <var>.b.g.m.</var> ex .32. eiuÅ¿dem, </s> - <s xml:space="preserve">vnde ex .4. Å¿exti proportio <var>.g.n.</var> ad <var>.g.m.</var> erit .vt <var>.g.u.</var> <lb/>ad <var>.m.b.</var> Å¿ed cum <var>.g.u.</var> maior Å¿it dimidio ipÅ¿ius <var>.b.g.</var> ex .20. primi, hoc eÅ¿t maior dimi-<lb/>dio ipÅ¿ius <var>.b.m.</var> ergo <var>.g.n.</var> etiam maior erit ipÅ¿a <var>.g.a.</var> quapropter maior erit ipÅ¿a <var>.g.p.</var> <lb/>cum <var>.g.p.</var> minor Å¿it ipÅ¿a <var>.g.a.</var> ex hypotheÅ¿i, ducta deinde cum fuerit <var>.b.n.</var> habebimus <lb/>triangulum <var>.b.n.g.</var> <reg norm="æqualem" type="context">æqualẽ</reg> triangulo <var>.b.u.g.</var> & <reg norm="maiorem" type="context">maiorẽ</reg> <reg norm="triangulo" type="context">triãgulo</reg> <var>.b.p.g.</var> ex prima Å¿exti <lb/>vel quia totum maius eÅ¿t Å¿ua parte. </s> - <s xml:space="preserve">Triangulus igitur <var>.b.u.g.</var> maior eÅ¿t triangu-<lb/>lo <var>.b.p.g</var>. </s> - <s xml:space="preserve">quare triangulus <var>.b.u.o.</var> maior erit triangulo <var>.g.o.p.</var> ex communi conceptu, <lb/>idem infero ab alia parte dictarum figurarum. </s> - <s xml:space="preserve">Quare pentagonus <var>.b.d.m.g.u.</var> maior <lb/>erit triangulo <var>.b.q.p.</var> quem probauimus maiorem eſſe triangulo æquilatero Å¿ibi iÅ¿o-<lb/>perimetro.</s> - </p> - <figure position="here"> - <image file="0391-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0391-01"/> - </figure> - </div> - <div type="letter" level="4" n="4"> - <head style="it" xml:space="preserve">Comparatio periferiarum quadrati & trianguli aquilateri circunÅ¿criptorum ab eodem circulo.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">QVod autem periferia quadrati in eodem circulo inÅ¿cripti, in quo Å¿it triangu-<lb/>lus æquilaterus, longior Å¿it periferia ipÅ¿ius trianguli æquilateri, abÅ¿que vllo - <pb o="380" rhead="IO. BAPT. BENED." n="392" file="0392" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0392"/> - negotio cordarum & arcuum poſſumus geometricè demonÅ¿trare quod valde de-<lb/>Å¿ideras.</s> - </p> - <p> - <s xml:space="preserve">Quapropter Å¿it circulus <var>.b.a.e.q.</var> in quo Å¿it <reg norm="triangulum" type="context">triangulÅ©</reg> æquilaterum <var>.b.e.n.</var> & quadra <lb/>tum <var>.b.a.q.u.</var> cuius periferiam probabo longiorem eſſe periferia trianguli. </s> - <s xml:space="preserve">Sit enim <lb/>diameter circuli <var>.b.q.</var> qui etiam erit diameter quadrati, vt à te Å¿cire potes. </s> - <s xml:space="preserve">Sit etiam <lb/><reg norm="punctum" type="context">punctÅ©</reg> <var>.b.</var> commune tam anguli quadrati quam trianguli. </s> - <s xml:space="preserve">vnde Å¿equitur quod dictus <lb/>diameter Å¿ecabit latus <var>.n.e.</var> trianguli ad rectos & per æqualia in <var>.t</var>. </s> - <s xml:space="preserve">Nam cum arcus <var>.b.<lb/>e.</var> æqualis Å¿it arcui <var>.b.n.</var> ex .27. tertij, remanet vt arcus <var>.q.e.</var> equalis Å¿it arcui <var>.q.n.</var> vnde <lb/>angulus <var>.q.b.e.</var> æqualis erit angulo <var>.q.b.n.</var> ex .26. eiuÅ¿dem. </s> - <s xml:space="preserve">quare ex .4. primi anguli <lb/>ad <var>.t.</var> erunt recti, et <var>.n.t.</var> æqualis erit ipÅ¿i <var>.t.e.</var> vt diximus.</s> - </p> - <p> - <s xml:space="preserve">Deinde <var>.b.e.</var> et <var>.q.a.</var> Å¿einuicem <reg norm="Å¿ecant" type="context">Å¿ecãt</reg> in puncto <var>.o.</var> vt ex Å¿e clarum patet, ducatur po <lb/>Å¿tea <var>.q.e.</var> vnde habebimus angulum <var>.b.e.q.</var> rectum ex .30. tertij, </s> - <s xml:space="preserve">quare ex .18. primi <var>.q.<lb/>o.</var> longior erit ipÅ¿a <var>.q.e.</var> et <var>.q.e.</var> longior erit ipÅ¿a <var>.e.t</var>. </s> - <s xml:space="preserve">quare <var>.q.o.</var> longior erit ipÅ¿a <var>.t.e</var>.</s> - </p> - <p> - <s xml:space="preserve">Vt probemus poÅ¿tea <var>.b.a.o.</var> longiorem eſſe ipÅ¿a <var>.b.e.</var> producatur <var>.b.a.</var> ita quod <var>.a.<lb/>p.</var> æqualis Å¿it ipÅ¿i <var>.a.o.</var> <reg norm="ducaturque" type="simple">ducaturqÌ;</reg> <var>o.p.</var> et <var>.a.e.</var> cum autem ex iam dicta .30. tertij angulus <lb/><var>b.a.o.</var> <reg norm="rectus" type="simple">rectê°</reg> Å¿it, erit angulus <var>.o.a.p.</var> Å¿imiliter <reg norm="rectus" type="simple">rectê°</reg> ex .13. primi, vnde ex .5. et .32. <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> <lb/>angulus <var>.a.p.o.</var> erit dimidium recti, & Å¿imiliter, exijÅ¿dem, angulus <var>.b.q.a.</var> eÅ¿t dimidium <lb/>recti </s> - <s xml:space="preserve">quare angulus <var>.a.p.o.</var> æqualis erit angulo <var>.a.q.b.</var> Å¿ed angulus <var>.a.e.b.</var> æqualis eÅ¿t an <lb/>gulo <var>.a.q.b.</var> ex .20. tertij, ergo angulus <var>.b.p.o.</var> æqualis erit angulo .b, <var>e.a.</var> angulus vero <lb/><var>a.b.e.</var> communis eÅ¿t ambobus triangulis <var>.a.b.e.</var> et <var>.o.b.p</var>. </s> - <s xml:space="preserve">quare ex .32. primi anguli <var>.<lb/>b.a.e.</var> et <var>.b.o.p.</var> reliqui ex duobus rectis æqua <lb/> - <anchor type="figure" xlink:label="fig-0392-01a" xlink:href="fig-0392-01"/> - les inuicem erunt. </s> - <s xml:space="preserve">Quare ex quarta Å¿exti, <lb/>et .18. quinti proportio <var>.b.o.</var> ad <var>.b.p.</var> erit, vt <lb/><var>b.a.</var> ad <var>.b.e.</var> Å¿ed ex .18. primi <var>.b.o.</var> maior eÅ¿t <lb/>ipÅ¿a <var>.b.a</var>. </s> - <s xml:space="preserve">quare ex .14. quinti <var>.b.p.</var> maior erit <lb/>ipÅ¿a <var>.b.e.</var> Å¿ed <var>.b.p.</var> æquatur ipÅ¿is <var>.b.a.</var> cum <var>.a.</var> o <lb/>ex hypoteÅ¿i, ergo <var>.b.a.</var> cum <var>.a.o.</var> maior erit <lb/>ipÅ¿a <var>.b.e.</var> Å¿ed <var>.q.o.</var> maior erat ipÅ¿a <var>.t.e.</var> vt Å¿upe <lb/>rius vidimus, </s> - <s xml:space="preserve">quare <var>.b.a.</var> cum <var>.a.o.</var> et <var>.o.q.</var> ma <lb/>ior eÅ¿t ipÅ¿a <var>.b.e.</var> cum <var>.e.t.</var> hoc eÅ¿t dimidium <lb/>periferiÄ™ ipÅ¿ius quadrati, <reg norm="maius" type="simple">maiê°</reg> erit dimidio <lb/>periferiÄ™ <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <reg norm="trianguli" type="context">triãguli</reg> propoÅ¿iti, </s> - <s xml:space="preserve">quare ex 14. <lb/>dicta tota periferia dicti trianguli, Å¿imiliter <lb/>probarem de omnibus alijs figuris regulari <lb/>bus eodem circulo inÅ¿criptis.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0392-01" xlink:href="fig-0392-01a"> - <image file="0392-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0392-01"/> - </figure> - </div> - </div> - </div> - <div type="section" level="3" n="42"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">CONSIDERATIONES NONNVLLÆ IN <lb/>Archimedem.</head> - <head style="it" xml:space="preserve">Doct ßimo atque Reuerendo Domino Vincentio <lb/>Mercato.</head> - <p> - <s xml:space="preserve"><emph style="sc">QVod</emph> tibi alià s dixi verum eÅ¿t, intellectum Å¿cilicet non omninò quieÅ¿cere cir <lb/>ca illas duas Archimedis propoÅ¿itiones, quæ in translatione Tartaleæ Å¿unt <lb/>Å¿ub numeris .4. et .5. & in impreſſione BaÅ¿ileæ Å¿ub numeris .6. et .7. vbi <lb/>tractat - <pb o="381" rhead="EPISTOLAE." n="393" file="0393" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0393"/> - tractat de centris libræ, Å¿eu Å¿tateræ: </s> - <s xml:space="preserve">A Å¿pice igitur in .4. Å¿upradicta, quod cum appen-<lb/>ſæ fuerint omnes illæ partes ponderum, partibus longitudinis ipÅ¿ius <var>.l.K.</var> in qua volo <lb/>vt à punctis <var>.e.</var> et <var>.d.</var> imagineris duas lineas <var>.e.o.</var> et <var>.d.u.</var> inuicem æquales, & ferè per-<lb/>pendiculares ipÅ¿i <var>.l.K.</var> hoc eÅ¿t reÅ¿picientes mundi centrum; </s> - <s xml:space="preserve">imagineris etiam <var>.o.u.</var> <lb/> - <anchor type="handwritten" xlink:label="hd-0393-01a" xlink:href="hd-0393-01"/> - quæ Å¿it paralle la ipÅ¿i <var>.l.k.</var> quæ diuiÅ¿a Å¿it in puncto <var>.i.</var> Å¿upra <var>.g</var>. </s> - <s xml:space="preserve">Hinc nulli dubium erit, <lb/>cum <var>.g.</var> fuerit centrum totius ponderis appenÅ¿i ipÅ¿i <var>.l.K.</var> quod <var>.i.</var> Å¿imiliter erit centrum <lb/>cum directe locatum Å¿it Å¿upra <var>.g.</var> hoc eÅ¿t in eadem directionis linea, quod quidem <lb/>non indiget aliqua demonÅ¿tratione, cum per Å¿e Å¿atis pateat. </s> - <s xml:space="preserve">Vnde ex communi <lb/>conceptu <var>.o.</var> erit centrum ponderis appenÅ¿i ipÅ¿i <var>.l.h.</var> et <var>.u.</var> erit centrum ponderis ap-<lb/>penÅ¿i. ipÅ¿i <var>h.K</var>. </s> - <s xml:space="preserve">Scimus <reg norm="igitur" type="simple">igit̃</reg> <var>.i.</var> eſſe <reg norm="centrum" type="context">cẽtrum</reg> duorum, hoc eÅ¿t ipÅ¿ius <var>.l.h.</var> & ipÅ¿ius <var>.h.k.</var> con <lb/>tinuatorum per totam <var>.l.k</var>. </s> - <s xml:space="preserve">Nunc ergo Å¿i conÅ¿ideremus <var>.l.k.</var> diuiÅ¿am eſſe, hoc eÅ¿t di-<lb/>Å¿iunctam in puncto <var>.h.</var> inueniemus nihilominus <var>.i.</var> centrum eſſe dictorum ponderum, <lb/>& quod tantum eÅ¿t, ipÅ¿am eſſe <reg norm="continuam" type="context">continuã</reg>, quantum diuiÅ¿am in dicto puncto <var>.h.</var> neque <lb/>ex hoc, punctum <var>.i.</var> erit magis vel minus centrum duorum ponderum <var>.l.h.</var> et <var>.h.k.</var> quo <lb/>rum vnum pendet totum ab <var>.o.</var> aliud verò totum ab <var>.u.</var> & hoc modo in longitudine <var>.<lb/>o.u.</var> diuiÅ¿a vt dictum eÅ¿t, habebimus propoÅ¿itum.</s> - </p> - <div type="float" level="5" n="1"> - <handwritten xlink:label="hd-0393-01" xlink:href="hd-0393-01a"/> - </div> - <p> - <s xml:space="preserve">Reliquam propoÅ¿itionem tibi relinquo.</s> - </p> - <p> - <s xml:space="preserve">Illa verò propoÅ¿itio, quam tibi dixi Archimedem tacuiſſe in huiuÅ¿modi materia <lb/>eÅ¿t, quod Å¿i duo pondera æquilibrant ab extremis alicuius Å¿tateræ, in certis præfixis <lb/>diÅ¿tantijs à centro. </s> - <s xml:space="preserve">Tunc dico Å¿i eorum vno manente alterum moueatur remotius <lb/>ab ipÅ¿o centro quod illud deÅ¿cendet, & Å¿i vicinius ipÅ¿i centro appenÅ¿um fuerit aÅ¿cen-<lb/>det. </s> - <s xml:space="preserve">Hæc enim propoÅ¿itio quotidie omnibus in locis videtur, ipÅ¿am verÅ¿o4; </s> - <s xml:space="preserve">puto Ar <lb/>chimedem prætermiÅ¿iſſe ob facilitatem, cum ab antedicta ferè dependeat.</s> - </p> - <p> - <s xml:space="preserve">Sit exempli gratia Å¿tatera <var>.a.u.</var> cuius centr um Å¿it <var>.i.</var> & pondera <var>.u.a.</var> appenÅ¿a, Å¿ein-<lb/> - <anchor type="handwritten" xlink:label="hd-0393-02a" xlink:href="hd-0393-02"/> - uicem habeant vt <var>.i.u.</var> et <var>.i.a.</var> Å¿e inuicem habent. </s> - <s xml:space="preserve">Nunc dico quod Å¿i pondus ipÅ¿ius <var>.u.</var> <lb/>poÅ¿itum fuerit vicinius centro vt puta in <var>.o.</var> inmoto exiÅ¿tente pondere, a. quod bra-<lb/>chium <var>.i.o.u.</var> aÅ¿cendet, & è conuerÅ¿o, Å¿i remotius poÅ¿itum fuerit, deÅ¿cendet.</s> - </p> - <div type="float" level="5" n="2"> - <handwritten xlink:label="hd-0393-02" xlink:href="hd-0393-02a"/> - </div> - <p> - <s xml:space="preserve"><reg norm="Ponatur" type="simple">Ponat̃</reg> ergo vt <reg norm="dictum" type="context">dictÅ©</reg> eÅ¿t in <var>.o.</var> vicinius <reg norm="centro" type="context">cẽtro</reg>, quapropter brachium <var>.i.o.</var> <reg norm="breuius" type="simple">breuiê°</reg> erit <lb/>brachio <var>.i.u.</var> vnde minor proportio erit ipÅ¿ius <var>.i.o.</var> ad <var>.i.a.</var> quà m.i.u. ad eundem <var>.a.i.</var> & <lb/>conÅ¿equenter quam ponderis ipÅ¿ius <var>.a.</var> (quod Å¿it <var>.n.e.</var>) ad pondus ipÅ¿ius <var>.u</var>. </s> - <s xml:space="preserve">Quare Å¿i cx <lb/>pondere <var>.n.e.</var> dempta fuerit <var>.e.</var> pars eius, ita quod reliqua pars <var>.n.</var> Å¿e habeat ad pondus <lb/>o. vt Å¿e habet. i<unsure/> <var>.o.</var> ad <var>.i.a.</var> tunc Å¿tatera non mouebitur; </s> - <s xml:space="preserve">addita verò parte <var>.e.</var> ex com-<lb/>muni conceptu, a. deÅ¿cendet vnde <var>.o.</var> aÅ¿cenderet conuerÅ¿um verò ex Å¿imilibus ratio-<lb/>nibus per te concludes.</s> - </p> - <figure position="here"> - <image file="0393-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0393-01"/> - </figure> - <pb o="382" rhead="IO. BAPT. BENED." n="394" file="0394" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0394"/> - <p> - <s xml:space="preserve">In eo quod à me petis, mittendo te ad Eutotium, tibi non Å¿atisfacerem, cum Eu-<lb/>totius citet Å¿extum librum Pergei, quem nunquam vidimus, <reg norm="Å¿upponatque" type="simple">Å¿upponatqÌ;</reg> ea, quæ nec <lb/>ipÅ¿e nec alius vnquam quod Å¿cimus probauit.</s> - </p> - <p> - <s xml:space="preserve">DeÅ¿ideras enim demonÅ¿trationem illius quod Archimedes dicit inter primam, <lb/>& Å¿ecundam propoÅ¿itionem Å¿ecundi libri, vbi tractat de centris grauium, propte-<lb/>rea quod illud Å¿upponit pro manifeÅ¿to.</s> - </p> - <p> - <s xml:space="preserve">Sit enim figura hic Å¿ubÅ¿cripta, ferè Å¿imilis parabolæ poÅ¿itæ in .2. propoÅ¿itione di <lb/>cti libri, vt in impreſſione BaÅ¿ileenÅ¿i habetur, <reg norm="Å¿intque" type="simple">Å¿intqÌ;</reg> diuiſæ duæ <var>.a.b.</var> et <var>.b.c.</var> per æqua <lb/>lia à punctis <var>.x.</var> et <var>.u.</var> <reg norm="protractisque" type="simple">protractisqÌ;</reg> <var>.f.x.</var> et <var>.u.i.</var> ad <var>.b.d.</var> quæ inuicem etiam erunt parallelÄ™ <lb/>ex .30. primi Eucli. </s> - <s xml:space="preserve">vnde ipſæ etiam, diametri erunt ipÅ¿arum portionum: </s> - <s xml:space="preserve">vt ex eo col <lb/>ligere eÅ¿t, quod in .49. primi lib. Pergei probatur. </s> - <s xml:space="preserve">Imaginando poÅ¿tea ad puncta <var>.b.<lb/>f.</var> er<unsure/> <var>.i.</var> tres contingentes, manifeÅ¿tum erit punctum <var>.b.</var> illud eſſe quod terminat alti-<lb/>tudinem huiuÅ¿modi portionis, et <var>.f.</var> et <var>.i.</var> terminantia altitudines partialium, ex .5. Å¿eÂ<lb/>cundi ipÅ¿ius Pergei, eo quod dictæ contingentes paralellæ erunt ipÅ¿is baÅ¿ibus, vnde <lb/>trianguli inÅ¿cripti, eaÅ¿dem habebunt altitudines, quas portiones ipſæ, quod erit ex <lb/>mente Archimedis. </s> - <s xml:space="preserve">Et Å¿ic deinceps poteris multiplicare angulos Å¿iguræ rectilineæ <lb/>in parabola, quæ deÅ¿ignata erit vt deÅ¿iderat Archimedes, qui quidem dicit, quod <lb/>protractæ cum fuerint aliæ deinceps poÅ¿t <var>.f.i.</var> ipſæ inuicem Ä™quidiÅ¿tantes <reg norm="erunt" type="context">erÅ©t</reg>, diuiſę-<lb/>qÌue peræqualia ab <var>.d.b.</var> quod <reg norm="quanuis" type="context">quãuis</reg> <reg norm="verum" type="context">verÅ©</reg> Å¿it, <reg norm="tantum" type="wordlist/context">tñ</reg> ab Eutotio non Å¿atis <reg norm="demonÅ¿tratum" type="context context">demõſtratÅ©</reg> <lb/>eÅ¿t, cum Å¿upponat <var>.a.f.b.</var> æqualem eſſe ipÅ¿i <var>.b.i.c.</var> probare volens eius diametros æqua <lb/>les eſſe abÅ¿que aliqua citata ratione, quæ quidem ratio eſſet conuerÅ¿um .4. propoÅ¿i-<lb/>tionis libri de conoidalibus. </s> - <s xml:space="preserve">Sed oporteret nos <reg norm="etiam" type="context">etiã</reg> videre .6. librum ipÅ¿ius Pergei, <lb/>& propterea tibi non Å¿atisfacerem.</s> - </p> - <p> - <s xml:space="preserve">EÅ¿to igitur, ut inuenta Å¿it linea <var>.K.</var> cuius productum in <var>.u.i.</var> æquale Å¿it qua drato ip <lb/>Å¿ius <var>.u.c.</var> inuenta etiam Å¿it linea <var>.h.</var> cuius productum cum <var>.f.x.</var> æquale Å¿it quadrato ip-<lb/>Å¿ius <var>.a.x.</var> vnde ex conuerÅ¿o .49. primi ipÅ¿ius Pergei, proportio ipÅ¿ius <var>.K.</var> ad <var>.b.c.</var> erit ut <lb/>ipÅ¿ius <var>.b.c.</var> ad <var>.b.d.</var> & ipÅ¿ius <var>.h.</var> ad <var>.a.b.</var> vt ipÅ¿ius <var>.a.b.</var> ad <var>.b.d</var>. </s> - <s xml:space="preserve">Erit igitur ex .16. Å¿exti Eucl. <lb/>quadratum <var>.b.c.</var> æquale producto ipÅ¿ius <var>.K.</var> in <var>.b.d.</var> & quadratum <var>.a.b.</var> æquale produ-<lb/>cto ipÅ¿ius <var>.h.</var> in <var>.b.d.</var> & ex prima Å¿exti, ita erit ipÅ¿ius <var>.K.</var> ad <var>.h.</var> vt producti quod fit ex <var>.K.</var> <lb/>in <var>.b.d.</var> ad productum ipÅ¿ius <var>.h.</var> in <var>.b.d.</var> hoc eÅ¿t vt quadrati ipÅ¿ius <var>.b.c.</var> ad quadratum ip <lb/>Å¿ius <var>.b.a.</var> ex .16. et .11. quinti, hoc eÅ¿t vt quadrati ipÅ¿ius <var>.u.c.</var> ad quadratum ipÅ¿ius <var>.a.x.</var> <lb/>hoc eÅ¿t ut productum ipÅ¿ius <var>.k.</var> in <var>.u.i.</var> ad productnm ipÅ¿ius <var>.h.</var> in <var>.x.f</var>. </s> - <s xml:space="preserve">Nunc Å¿i ipÅ¿ius <var>.k.</var> <lb/>ad <var>.h.</var> c<unsure/>Å¿t vt producti ipÅ¿ius <var>.K.</var> in <var>.u.i.</var> ad productum ipÅ¿ius <var>.h.</var> in <var>.f.x.</var> ergo ex .24. Å¿exti, <lb/>& communi conceptu, proportio ipÅ¿ius <var>.k.</var> ad <var>.h.</var> compoÅ¿ita erit ex ea quæ ipÅ¿ius <var>.u.i.</var> <lb/>ad <var>.f.x.</var> & ex ea quæ ipÅ¿ius <var>.k.</var> ad <var>.h</var>. </s> - <s xml:space="preserve">Cum ergo dempta fuerit proportio ipÅ¿ius <var>.k.</var> ad <var>.h.</var> <lb/>(vt Å¿implex) à proportione ipÅ¿ius <var>.k.</var> ad <var>.h.</var> (vt compoÅ¿ita) reliquum nihil erit. </s> - <s xml:space="preserve">Qua-<lb/>re <var>.f.x.</var> æqualis erit ipÅ¿i <var>.u.i</var>.</s> - </p> - <p> - <s xml:space="preserve">Sed quod <var>.f.m.</var> æqualis Å¿it ipÅ¿i <var>.m.i</var>. </s> - <s xml:space="preserve">Videto in Eutotio, quia hoc Å¿atis Å¿ui natura <lb/>facile eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Sed accipe alium modum breuiorem ad probandum <var>.f.x.</var> eſſe æqualem ipÅ¿i <var>.u.i</var>.</s> - </p> - <p> - <s xml:space="preserve">Finge lineam <var>.e.b.g.</var> conting entem in puncto <var>.b.</var> prolungatisqÌue diametris <var>f.<lb/>x.</var> et <var>.u.i.</var> vÅ¿que ad contingentem ipÅ¿am, habebis <var>.f.e.</var> æqualem ipÅ¿i <var>.f.x.</var> et <var>.g.i.</var> ipÅ¿i <var>.u.i.</var> <lb/>Ex .35. primi Pergei, producta poÅ¿tea <var>.x.u.</var> habeb is ex .2. Å¿exti Eucli <var>.x.u.</var> parallelam <lb/>ipÅ¿i <var>.a.c.</var> Å¿ed <var>.e.g.</var> parallela eÅ¿t ipÅ¿imet <var>.a.c.</var> ex quinta Å¿ecundi ipÅ¿ius Pergei, </s> - <s xml:space="preserve">quare ex .30 <lb/>primi Euclid <var>.e.g.</var> parallela erit ipÅ¿i <var>.u.x.</var> & ex .34. eiuÅ¿dem æqualis erit <var>.e.x.</var> ipÅ¿i <var>.u.g.</var> <lb/>vnde <var>.f.x.</var> etiam æqualis erit <var>.u.i.</var> ex communi conceptu.</s> - </p> - <p> - <s xml:space="preserve">Sed ne quid deÅ¿ideres probabo <var>.f.m.</var> æqualem eſſe ipÅ¿i <var>.m.i</var>. </s> - <s xml:space="preserve">Iam igitur Å¿cis quod - <pb o="383" rhead="EPISTOLAE." n="395" file="0395" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0395"/> - cum Å¿it <var>.f.x.</var> æqualis ipÅ¿i <var>.u.i.</var> vt tibi probaui, & inuicem parallelæ ideo <var>.f.i.</var> parallela <lb/>erit ipÅ¿i <var>.x.u.</var> ex .33. primi Euclidis. </s> - <s xml:space="preserve">Vnde ex .30. eiuÅ¿dem, parallela erit etiam ipÅ¿i <var>.a.<lb/>c.</var> Å¿ed cum <var>.x.u.</var> diuiÅ¿a Å¿it ab <var>.d.b.</var> per æqualia, eo quod diuidit <var>.a.c.</var> eodem modo, quÄ™ <lb/>ipÅ¿i parallela eÅ¿t ex .2. Å¿exti. </s> - <s xml:space="preserve">Reliqua tibi conÅ¿ideranda relinquo. </s> - <s xml:space="preserve">cum verò ambæ <var>.f.<lb/>x.</var> et <var>.u.i.</var> parallelæ Å¿int ipÅ¿i <var>.b.d.</var> Å¿equitur quod cum ex .34. primi <reg norm="vnaquæque" type="simple">vnaquæq;</reg> <var>.f.m.</var> et <var>.m.<lb/>i.</var> æqualis Å¿it medietati ipÅ¿ius <var>.x.u.</var> erunt inuicem æquales.</s> - </p> - <figure position="here"> - <image file="0395-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0395-01"/> - </figure> - <p> - <s xml:space="preserve">Minime dubitabam tibi non Å¿atisfacere Eutocium in .3. propoÅ¿itione Å¿ecundi <lb/>lib. de centris Grauium Archimedis, cum citet .6. librum de elementis conicis, ad-<lb/>de quod Å¿i aliud in ipÅ¿o .6. libro ab eo citato non eſſet magis ad propoÅ¿itum, quà m <lb/>ca quæ ab ipÅ¿o citata Å¿unt, nihilominus adhuc irreÅ¿olutus maneres.</s> - </p> - <p> - <s xml:space="preserve">ConÅ¿idera igitur eandem ipÅ¿am figuram præcedentem; </s> - <s xml:space="preserve">pro alia verò parabola Å¿i <lb/>mili dictæ, accipe Å¿ecundam figuram ipÅ¿ius tertiæ dictæ propoÅ¿itionis. </s> - <s xml:space="preserve">Deinde ima <lb/>ginabis duo latera <var>.o.x.</var> et <var>.o.p.</var> diuiÅ¿a eſſe per æqualia in punct is <var>.g.</var> et <var>.K.</var> <reg norm="protractisque" type="simple">protractisqÌ;</reg> <lb/>diametris <var>.g.y.</var> et <var>.K.u.</var> quæ, vt in præcedenti probaui, Å¿unt inuicem æquales, Å¿cire <lb/>debes quod Å¿imiles parabolæ inuicem aliæ non poſſunt eſſe, niÅ¿i eæ quæ diametros <lb/>proportionales Å¿uis baÅ¿ibus habeant, <reg norm="Å¿imiliterque" type="simple">Å¿imiliterqÌ;</reg> poÅ¿itæ, hoc eÅ¿t, ut proportio ipÅ¿ius <lb/><var>b.d.</var> ad <var>.a.c.</var> Å¿it eadem quæ ipÅ¿ius <var>.o.r.</var> ad <var>.x.p.</var> & quod anguli ad <var>.r.</var> Å¿int æquales angulis <lb/>circa <var>.d</var>. </s> - <s xml:space="preserve">Notentur ergo primum puncta communia ip Å¿ius <var>.o.g.</var> cum <var>.y.t.</var> & ipÅ¿ius <var>.b.</var> x <lb/>cum <var>.f.m.</var> characteribus. ω<unsure/>. et <var>.n</var>. </s> - <s xml:space="preserve">Nunc igitur Å¿cimus <var>.f.m.</var> æqualem eſſe <var>.m.i.</var> tota <reg norm="mque" type="simple">mqÌ;</reg> <var>.f.<lb/>i.</var> parallelam eſſe ipÅ¿i <var>.a.c</var>. </s> - <s xml:space="preserve">Idem dico de <var>.y.t.u.</var> <reg norm="triangulique" type="simple">trianguliqÌ;</reg> <var>.x.f.n.</var> et <var>.g.y.</var> ω<unsure/>. eſſe Å¿imiles <lb/>triangulis <var>.n.m.b.</var> et. ω<unsure/> <var>.t.o.</var> quod ita probatur, nam ex .15. primi Euclid. anguli ad <var>.n.</var> <lb/>Å¿unt inuicem æquales, ex .29. verò eiuÅ¿dem anguli <var>.f.x.n.</var> et <var>.n.b.m.</var> Å¿imiliter æquales <lb/>ita etiam <var>.n.f.x.</var> et <var>.n.m.b</var>.</s> - </p> - <p> - <s xml:space="preserve">Idem dico in Å¿ecunda figura, vnde ex .4. Å¿exti Eucli. proportio <var>.n.f.</var> ad <var>.m.n.</var> erit ea <lb/>dem quÄ™ <var>.f.x.</var> ad <var>.b.m.</var> & ipÅ¿ius <var>.n.f.</var> ad <var>.x.f.</var> vt <var>.n.m.</var> ad <var>.m.b.</var> ex .16. quinti. </s> - <s xml:space="preserve">Quare ex .11. <lb/> - <anchor type="figure" xlink:label="fig-0395-02a" xlink:href="fig-0395-02"/> - <pb o="384" rhead="IO. BAPT. BENED." n="396" file="0396" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0396"/> - eiuÅ¿dem erit vt <var>.a.d.</var> ad <var>.d.b</var>. </s> - <s xml:space="preserve">Idem etiam dico in Å¿ecunda parabola, Å¿ed ipÅ¿ius <var>.x.o.</var> ad <lb/><var>o.r.</var> eÅ¿t vt <var>.a.b.</var> ad <var>.b.d.</var> ex .6. Å¿exti Eucli. </s> - <s xml:space="preserve">vnde ex .11. quinti <var>.n.f.</var> ad <var>.f.x.</var> erit vt <var>.ω.y.</var> <lb/>ad <var>.y.g</var>. </s> - <s xml:space="preserve">Sed in precedenti iam tibi dixi <var>.a.b.</var> mediam proportionalem eſſe inter <var>.h.</var> <lb/>et <var>.b.d</var>. </s> - <s xml:space="preserve">Sit nunc <var>.z.</var> pro Å¿ecunda parabola, ita ut <var>.h.</var> eÅ¿t pro prima, vnde <var>.o.x.</var> crit media <lb/>proportionalis inter <var>.z.</var> et <var>.o.r.</var> & ex .11. quinti ita erit <var>.h.</var> ad <var>.a.b.</var> vt <var>.z.</var> ad <var>.x.o.</var> & ex .22. <lb/>h. ad <var>.a.x.</var> ut z. ad <var>.x.g.</var> & quia ex .16. Å¿exti <var>.a.x.</var> media proportionalis eÅ¿t inter <var>.h.</var> et <var>.f.<lb/>x.</var> cum Å¿upponatur productum <var>.h.</var> in <var>.f.x.</var> æquale eſſe quadrato <var>.a.x</var>. </s> - <s xml:space="preserve">Idem dico <var>.x.g.</var> <lb/>mediam eſſe proportionalem inter <var>.z.</var> et <var>.g.y.</var> </s> - <s xml:space="preserve">quare ex .11. iam dicta, ita erit <var>.a.x.</var> ad <var>.f.<lb/>x.</var> vt <var>.y.g.</var> ad <var>.x.o.</var> & ex eadem, ita erit ipÅ¿ius <var>.f.n.</var> ad <var>.a.b.</var> ut <var>.y.ω.</var> ad <var>.x.o.</var> & Å¿ic <var>.f.n.</var> ad <var>.d.a.</var> <lb/>vt <var>.y.ω.</var> ad <var>.x.r.</var> Å¿ed <var>.f.m.</var> ad <var>f.n.</var> eÅ¿t vt <var>.y.t.</var> ad <var>.y.ω.</var> ex .18. quinti vnde <var>.f.m.</var> ad <var>.a.d.</var> erit vt <lb/><var>y.t.</var> ad <var>.x.r</var>. </s> - <s xml:space="preserve">Idem dico de eorum duplis.</s> - </p> - <div type="float" level="5" n="3"> - <figure xlink:label="fig-0395-02" xlink:href="fig-0395-02a"> - <image file="0395-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0395-02"/> - </figure> - </div> - <p> - <s xml:space="preserve">Ex ijÅ¿dem rationibus dico ita eſſe <var>.b.d.</var> ad <var>.b.m.</var> vt <var>.o.r.</var> ad <var>.o.t.</var> & ex .17. quinti <var>.d.m.</var> <lb/>ad <var>.b.m.</var> vt <var>.r.t.</var> ad <var>.t.o</var>. </s> - <s xml:space="preserve">Reliqua tibi conÅ¿ideranda relinquo.</s> - </p> - <figure position="here"> - <image file="0396-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0396-01"/> - </figure> - <p> - <s xml:space="preserve">In reliquis verò propoÅ¿itionibus illius lib. nullo pacto poteris dubitare: </s> - <s xml:space="preserve">Verum ne <lb/>in .4. aliquid tibi noui exurgat, te Å¿cire volo <ref id="ref-0025">corollarium .20. in libr. de quadratuÂ<lb/>ra parabolÄ™</ref> docere poſſibile eſſe inÅ¿criptionem rectilineæ, ea tamen conditione <reg norm="quam" type="context">quã</reg> <lb/>dicit Archimedes.</s> - </p> - <p> - <s xml:space="preserve">In quinta poÅ¿tea animaduertendum eÅ¿t, quod prima pars, probat tantummodo de <lb/>centro trianguli, et .2. pars probat de centro pentagoni, à te ipÅ¿o deinde potes pro-<lb/>bare de centro nonanguli: </s> - <s xml:space="preserve">& Å¿ic de cæteris: </s> - <s xml:space="preserve">eo quod cum probatum fuerit de centro <lb/>figuræ in medio locatæ Å¿i conÅ¿titutæ poÅ¿tea fuerint Å¿imiles figuræ in portionibus la-<lb/>teralibus habebitur propoÅ¿itum in infinitum.</s> - </p> - <p> - <s xml:space="preserve">Idem intelligendum eÅ¿t in .3. propoÅ¿itione quamuis exemplum vlterius non ex-<lb/>tendatur quam ad pentagonos.</s> - </p> - <p> - <s xml:space="preserve">Sexta verò <reg norm="propoÅ¿itio" type="simple">ê“poÅ¿itio</reg> tibi Å¿acilis erit, quæ nihilominus <reg norm="pont" type="context">põt</reg> <reg norm="demonÅ¿trari" type="context">demõſtrari</reg> hoc <reg norm="mon" type="context">mõ</reg> Å¿ciliÂ<lb/>cet. </s> - <s xml:space="preserve">Sint .4. <reg norm="quantitates" type="context">quãtitates</reg> <var>.a.b.c.d.</var> ipÅ¿ius Archimedis <reg norm="Å¿upponendo" type="context">Å¿upponẽdo</reg> <var>.a.</var> pro figura rectilinea <lb/>inÅ¿cripta in parabola, et <var>.b.</var> pro reÅ¿iduo ipÅ¿ius parabolÄ™ et <var>.c.</var> pro triangulo <var>.a.b.c.</var> in me <lb/>dio ipÅ¿ius parabolÄ™ et <var>.d.</var> pro triangulo <var>.r</var>. </s> - <s xml:space="preserve">Nunc cum <var>.a.</var> maior Å¿it <var>.c.</var> prout totum ma-<lb/>ius eÅ¿t Å¿ua parte, ideo ex .8. quinti maior proportio habebit <var>.a.</var> ad <var>.b.</var> quam <var>.c.</var> ad <var>.b.</var> <lb/>Cum autem <var>.b.</var> minor Å¿it <var>.d.</var> ex Å¿uppoÅ¿ito, ideo ex eadem dicta, maior proportio habe <lb/>bit <var>.a.</var> ad <var>.b.</var> quam <var>.c.</var> ad <var>.d.</var> cum verò centrum cuiuÅ¿uis figuræ plenæ neceſſariò Å¿it intra <lb/>ipÅ¿am figuram, idcirco centrum reÅ¿idui ipÅ¿ius parabolÄ™ intra ipÅ¿am reperietur. </s> - <s xml:space="preserve">quod <lb/>ita <reg norm="clarum" type="context">clarÅ©</reg> <reg norm="per" type="simple">ê‘</reg> Å¿e eÅ¿t, <reg norm="quemadmodum" type="wordlist">quẽadmodÅ©</reg> quoduis aliud axioma, & quia <reg norm="dictum" type="context">dictÅ©</reg> <reg norm="centrum" type="context">centrÅ©</reg> ex .8. primi <lb/>de centris, neceſſariò eÅ¿t in linea <var>.b.h.</var> inter <var>.b.</var> et <var>.h</var>. </s> - <s xml:space="preserve">Sit igitur <var>.g.</var> vnde ex eadem .8. ita <lb/>erit <var>.g.h.</var> ad <var>.h.e.</var> vt <var>.a.</var> ad <var>.b.</var> ergo <var>.g.h.</var> ad <var>.h.e.</var> maior proportio erit <reg norm="quam" type="context">quã</reg> <var>.c.</var> ad <var>.d.</var> hoc eÅ¿t <lb/>quam <var>.b.h.</var> ad <var>.f.</var> ex .12. quinti. </s> - <s xml:space="preserve">Sed <reg norm="cum" type="context">cÅ©</reg> <var>.h.b.</var> maior Å¿it ipÅ¿a <var>.h.g.</var> prout omne totum ma-<lb/>ius eÅ¿t Å¿ua parte, ideo maior proportio habebit <var>.h.b.</var> ad <var>.h.e.</var> quam <var>.h.g.</var> ad <var>.h.e.</var> vnde <lb/>multo <reg norm="maiorem" type="context">maiorẽ</reg> <reg norm="quam" type="context">quã</reg> <var>.h.b.</var> ad <var>.f.</var> ex <reg norm="coni" type="context">cõi</reg> <reg norm="conceptu" type="context">cõceptu</reg>, </s> - <s xml:space="preserve">quare <var>.h.e.</var> erit minor ipÅ¿a <var>.f.</var> ex .10. <reg norm="quinti" type="context">quÄ©ti</reg>.</s> - </p> - <p> - <s xml:space="preserve">Septima verò et .8. propoÅ¿itio nullius tibi erit difficultatis.</s> - </p> - <pb o="385" rhead="EPISTOL AE." n="397" file="0397" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0397"/> - <p> - <s xml:space="preserve">Quamuis Eutotius Å¿cribat Å¿uper duas vltimas lib. Å¿ecundi de centris <reg norm="grauium" type="context">grauiÅ©</reg>, nihil <lb/>miror ipÅ¿um tibi non Å¿atisfacere. </s> - <s xml:space="preserve">Accipe igitur quod ego nunc tibi mitto.</s> - </p> - <p> - <s xml:space="preserve">Archimedes eo in loco <reg norm="primum" type="context">primÅ©</reg> Å¿upponit in penultima dicti libri quatuor lineas <lb/>proportionales <var>.a.b</var>: <var>c.b</var>: <var>d.b</var>: et <var>.e.b</var>: Å¿upponit etiam quod proportio quæ eÅ¿t ipÅ¿ius <var>.<lb/>e.b.</var> ad <var>.e.a.</var> <reg norm="eadem" type="context">eadẽ</reg> Å¿it quæ ipÅ¿ius <var>.f.g.</var> ad tres quintas ipÅ¿ius <var>.a.d.</var> & quod proportio com <lb/>poÅ¿iti dupli ipÅ¿ius <var>.a.b.</var> cum quadruplo ipÅ¿ius <var>.b.c.</var> cum Å¿excuplo ipÅ¿ius <var>.b.d.</var> cum triplo <lb/> - <anchor type="figure" xlink:label="fig-0397-01a" xlink:href="fig-0397-01"/> - <pb o="386" rhead="IO. BAPT. BENED." n="398" file="0398" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0398"/> - ipÅ¿ius <var>.b.e.</var> ad compoÅ¿itum quintupli ipÅ¿ius <var>.a.b.</var> cum decuplo ipÅ¿ius <var>.c.b.</var> cum decuplo <lb/>ipÅ¿ius <var>.b.d.</var> cum quintuplo ipÅ¿ius <var>.b.e.</var> eadem Å¿it quæ ipÅ¿ius <var>.g.h.</var> ad <var>.a.d.</var> & vult proba-<lb/>re <var>.f.h.</var> eſſe duas quintas ipÅ¿ius <var>.a.b</var>.</s> - </p> - <div type="float" level="5" n="4"> - <figure xlink:label="fig-0397-01" xlink:href="fig-0397-01a"> - <image file="0397-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0397-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Cum autem dicit proportionem ipÅ¿ius <var>.a.c.</var> ad <var>.c.d.</var> & ipÅ¿ius <var>.c.d.</var> ad <var>.d.e.</var> eſſe vt ipÅ¿ius <lb/><var>a.b.</var> a<unsure/>d.b.c. & cÄ™tera, verum dicit ex .19. quinti Eucli. eo quod cum ex hypotheÅ¿i Å¿it <lb/>ipÅ¿ius, <var>a.b.</var> totalis ad <var>.c.b.</var> totalom vt ipÅ¿ius <var>.c.b.</var> partialis (Å¿umptæ vt pars abÅ¿ciÅ¿a ab <var>.a.<lb/>b.</var> pro nunc) ad <var>.d.b.</var> partialem (abÅ¿ciÅ¿am ab <var>.c.b.</var>) erit ex .19. dicta ipÅ¿ius <var>.a.c.</var> (reÅ¿idui <lb/>ex <var>.a.b.</var>) ad <var>.c.d.</var> (reÅ¿iduum ex <var>.c.b.</var>) vt ipÅ¿ius <var>.a.b.</var> ad <var>.c.b.</var> & ita probabitur de pro-<lb/>portione ipÅ¿iuas <var>.c.d.</var> ad <var>.d.e.</var> eadem ratione.</s> - </p> - <p> - <s xml:space="preserve">Cum verò ex .18. quinti Å¿it ipÅ¿ius <var>.a.b.</var> cum <var>.c.b.</var> ad <var>.c.b.</var> vt ipÅ¿ius <var>.a.d.</var> ad <var>.d.e.</var> ergo ex <lb/>22. eiuÅ¿dem, ita erit ipÅ¿ius <var>.a.b.</var> cum <var>.c.b.</var> ad <var>.d.b.</var> vt <var>.a.d.</var> ad <var>.d.e.</var> & exijÅ¿dem rationibus <lb/>eadem proportio erit ipÅ¿ius <var>.c.b.</var> cum <var>.d.b.</var> ad <var>.b.e.</var> vt <var>.a.d.</var> ad <var>.d.e.</var> quod inquit Archi. <lb/></s> - <s xml:space="preserve">Verum etiam erit (ex .13. quinti) cum dicit eandem proportionem eſſe ipÅ¿ius <var>.a.d.</var> <lb/>ad <var>.d.e.</var> quÄ™ dupli primi antecedentis cum Å¿implo Å¿ecundi antecedentis ad duplum <lb/>primi conÅ¿equentis cum Å¿implo Å¿ecundi conÅ¿equentis, hoc eÅ¿t dupli ipÅ¿ius <var>.a.b.c.</var> <reg norm="cum" type="context">cÅ©</reg> <lb/>Å¿implo <var>.c.b.d.</var> ad duplum ipÅ¿ius <var>.d.b.</var> cum Å¿implo <var>.e.b.</var> hoc eft dupli <var>.a.b.</var> cum triplo ip-<lb/>Å¿ius <var>.b.c.</var> cum Å¿implo <var>.d.b.</var> ad duplum ipÅ¿ius <var>.d.b.</var> cum Å¿implo <var>.e.b</var>. </s> - <s xml:space="preserve">Nunc duplum <var>.a.b.</var> <lb/>cum triplo <var>.b.c.</var> cum Å¿implo <var>.b.d.</var> Å¿ignatum Å¿it charactere <var>.D.</var> Å¿uum verò conÅ¿equens, <lb/>hoc eÅ¿t duplum <var>.d.b.</var> <reg norm="cum" type="context">cÅ©</reg> Å¿implo <var>.e.b.</var> Å¿ignificetur à charactere <var>.B.</var> hinc proportio ipÅ¿ius <lb/> - <anchor type="note" xlink:label="note-0398-01a" xlink:href="note-0398-01"/> - <var>a.d.</var> ad <var>.d.e.</var> erit vt <var>.D.</var> ad <var>.B</var>.</s> - </p> - <div type="float" level="5" n="5"> - <note xlink:label="note-0398-01" xlink:href="note-0398-01a" position="left" xml:space="preserve">M</note> - </div> - <p> - <s xml:space="preserve">Inquit nunc Archimedes, Å¿i quis Å¿umeret aliquod maius antecedens æquale Å¿ci-<lb/>licet duplo ipÅ¿ius <var>.a.b.</var> cum quadruplo ipÅ¿ius <var>.b.e.</var> cum quadruplo ipÅ¿ius <var>.b.d.</var> cum du-<lb/>plo ipÅ¿ius <var>.b.e.</var> <reg norm="compararetque" type="simple">compararetqÌ;</reg> illud cum <reg norm="conÅ¿equente" type="context">cõſequente</reg> <var>.B.</var> clarum eſſet ex .8. quinti quod <lb/>tale antecedens maiorem proportionem haberet ad <var>.B.</var> quam ad <var>.D.</var> hoc eÅ¿t maiorem <lb/>quà m ipÅ¿ius <var>.a.d.</var> ad <var>.d.e.</var> ex .12. quinti.</s> - </p> - <p> - <s xml:space="preserve">Nunc Å¿i Å¿umpta fuerit aliqua linea, puta <var>.d.o.</var> cui <var>.a.d.</var> <reg norm="dictam" type="context">dictã</reg> habeat proportionem <lb/>maiorem, larum erit ex Å¿ecunda parte decimÄ™ quinti quod <var>.d.o.</var> minor erit ipÅ¿a <var>.d.e.</var> <lb/>Corrige igitur impreſſionem BaÅ¿ileÄ™ locando characterem <var>.o.</var> inter <var>.d.</var> et <var>.e.</var> eo quod <lb/>ibi poÅ¿itum non fuit.</s> - </p> - <p> - <s xml:space="preserve">Volo nunc quod dictum maius antecedens æquale Å¿cilicet duplo ipÅ¿ius <var>.a.b.</var> cum <lb/>quadruplo ipÅ¿ius <var>.b.c.</var> cum quadruplo ipÅ¿ius <var>.b.d.</var> cum duplo ipÅ¿ius <var>.b.c.</var> Å¿ignificetur à <lb/>charactere <var>.A</var>. </s> - <s xml:space="preserve">Hinc habebimus proportionem ipÅ¿ius <var>.a.d.</var> ad <var>.d.o.</var> ut <var>.A.</var> ad <var>.B</var>.</s> - </p> - <note position="left" xml:space="preserve">β</note> - <!-- note-0398-02 --> - <p> - <s xml:space="preserve">Ex .18. quinti poÅ¿tea habebimus <var>.A.B.</var> ad <var>.B.</var> vt <var>.a.o.</var> ad <var>.d.o.</var> & proportionalitate <lb/>euerÅ¿a in .19. dicti ita erit <var>.A.B.</var> ad <var>.A.</var> vt <var>.a.o.</var> ad <var>.a.d</var>. </s> - <s xml:space="preserve">Sed hoc vltimum antecedens in <lb/> - <anchor type="note" xlink:label="note-0398-03a" xlink:href="note-0398-03"/> - Å¿e continet id quod Archimedes Å¿cribit, hoc eÅ¿t duplum ipÅ¿ius <var>.a.b.</var> <reg norm="quadruplum" type="context">quadruplÅ©</reg> ipÅ¿ius <lb/><var>b.c.</var> Å¿excuplum ipÅ¿ius <var>.b.d.</var> & triplum ipÅ¿ius <var>.b.e</var>. </s> - <s xml:space="preserve">ConÅ¿equens verò <var>.A.</var> continet du <lb/>plum ipÅ¿ius <var>.a.b.</var> quadruplum ipÅ¿ius <var>.b.c.</var> quadruplum ipÅ¿ius <var>.b.d.</var> & duplum ipÅ¿ius <var>.b.e</var>.</s> - </p> - <div type="float" level="5" n="6"> - <note xlink:label="note-0398-03" xlink:href="note-0398-03a" position="left" xml:space="preserve">T</note> - </div> - <p> - <s xml:space="preserve">Ex Å¿uppoÅ¿ito deinde ipÅ¿ius Archimedis & ex conuerÅ¿a proportionalitate in .19. <lb/>dicta, verum eÅ¿t id quod dicit Archimedes, videlicet quod eadem proportio eÅ¿t <lb/>ipÅ¿ius <var>.a.d.</var> ad <var>.g.h.</var> quod quintupli ipÅ¿ius <var>.a.b.</var> cum quintuplo ipÅ¿ius <var>.b.e.</var> cum decuplo <lb/>ipÅ¿ius <var>.b.c.</var> cum decuplo ipÅ¿ius <var>.b.d.</var> (quod quidem antecedens Å¿ignificetur per <var>.V.</var>) <lb/>ad duplum ipÅ¿ius <var>.a.b.</var> cum quadruplo ipÅ¿ius <var>.b.c.</var> cum Å¿excuplo ipÅ¿ius <var>.b.d.</var> cum triplo <lb/>ipÅ¿ius <var>.b.e.</var> hoc eÅ¿t ad <var>.A.B</var>.</s> - </p> - <p> - <s xml:space="preserve">Erit igitur <var>.V.</var> ad <var>.A.B.</var> vt ipÅ¿ius <var>.a.d.</var> ad <var>.g.h.</var> Å¿ed Å¿uperius vbi Å¿ignatum eÅ¿t <var>.T.</var> iam <lb/>probatum fuit ita eſſe <var>.A.B.</var> ad <var>.A.</var> vt ipÅ¿ius <var>.a.o.</var> ad <var>.a.d</var>. </s> - <s xml:space="preserve">Ergò ex .23. quinti Archime <lb/>des verum Å¿cribit, hoc eÅ¿t quod ita erit ipÅ¿ius <var>.V.</var> ad <var>.A.</var> vt ipÅ¿ius <var>.a.o.</var> ad <var>.g.h</var>.</s> - </p> - <p> - <s xml:space="preserve">Clarum per Å¿e etiam eÅ¿t, id quod Archimed. dicit hoc eÅ¿t quod <var>.V.</var> ad <var>.A.</var> eÅ¿t vt <lb/> - <anchor type="note" xlink:label="note-0398-04a" xlink:href="note-0398-04"/> - <pb o="387" rhead="EPISTOL AE." n="399" file="0399" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0399"/> - quinque ad duo, cum quodlibet ingredientium in compoÅ¿ito <var>.V.</var> ad quodlibet in-<lb/>gredientium in compoÅ¿ito <var>.A.</var> Å¿it vt quinque ad duo. </s> - <s xml:space="preserve">Quare ex .13. quinti verum <lb/>dicit. </s> - <s xml:space="preserve">Vnde <var>.a.o.</var> ad <var>.g.h.</var> erit vt <reg norm="quinque" type="simple">quinq;</reg> ad duo ex .11. <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> vt inquit Archimedes.</s> - </p> - <div type="float" level="5" n="7"> - <note xlink:label="note-0398-04" xlink:href="note-0398-04a" position="left" xml:space="preserve">ω</note> - </div> - <p> - <s xml:space="preserve">Corrige impreſſionem vbi Å¿criptum eÅ¿t, rurÅ¿us quoniam <var>.o.a.</var> quia oportet dicere <lb/>RurÅ¿us quoniam <var>.o.d</var>.</s> - </p> - <p> - <s xml:space="preserve">Archimedes igitur verum dicit, quod ipÅ¿ius <var>.o.d.</var> ad <var>.d.a.</var> eÅ¿t vt ipÅ¿ius <var>.B.</var> ad <var>.A.</var> ex <lb/> - <anchor type="figure" xlink:label="fig-0399-01a" xlink:href="fig-0399-01"/> - <pb o="388" rhead="IO. BABPT. BENED." n="400" file="0400" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0400"/> - conuerÅ¿a proportionalitate in .19. quinti, cum <var>.a.d.</var> ad <var>.d.o.</var> iam probatum fuit (vbi <lb/>B.) ita eſſe ut <var>.A.</var> ad <var>.B</var>.</s> - </p> - <div type="float" level="5" n="8"> - <figure xlink:label="fig-0399-01" xlink:href="fig-0399-01a"> - <image file="0399-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0399-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Sed in principio huius Å¿peculationis probatum iam fuit ita eſſe ipÅ¿ius <var>.d.a.</var> ad <var>.d.e.</var> <lb/>vt ipÅ¿ius <var>.D.</var> ad <var>.B.</var> vbi notatum eÅ¿t <var>.M</var>. </s> - <s xml:space="preserve">quare ex .23. quinti, Archimedes verum dicit, <lb/>qu od <var>.d.o.</var> ad <var>.d.e.</var> erit vt <var>.D.</var> ad <var>.A</var>.</s> - </p> - <p> - <s xml:space="preserve">Sed cum <var>.d.o.</var> ad <var>.d.e.</var> Å¿e habeat ut <var>.D.</var> ad <var>.A.</var> erit ex conuerÅ¿a proportionalitate iam <lb/>dicta <var>.d.e.</var> ad <var>.d.o.</var> vt <var>.A.</var> ad <var>.D.</var> per euerÅ¿am vero erit <var>.d.e.</var> ad <var>.a.o.</var> vt <var>.A.</var> ad Å¿uum reÅ¿i-<lb/> - <anchor type="note" xlink:label="note-0400-01a" xlink:href="note-0400-01"/> - duum. </s> - <s xml:space="preserve">quod reÅ¿iduum componitur ex Å¿implo <var>.b.c.</var> cum triplo <var>.b.</var> cum duplo <var>.b.o.</var> quod <lb/>à te ipÅ¿o videre poteris detrahendo numeros ipÅ¿arum quantitatum quæ in <var>.D.</var> <lb/>reperiuntur, ex numeris earundem, quæ in <var>.A.</var> quod quidem reÅ¿iduum Å¿igniÅ¿icetur <lb/>à charactere <var>.E</var>. </s> - <s xml:space="preserve">Vnde ex conuerÅ¿a proportionalitate verum dicit Archime. hoc eÅ¿t <lb/>quod ita Å¿e hab ebit <var>.o.e.</var> ad <var>.d.e.</var> vt <var>.E.</var> ad <var>.A</var>.</s> - </p> - <div type="float" level="5" n="9"> - <note xlink:label="note-0400-01" xlink:href="note-0400-01a" position="left" xml:space="preserve">λ</note> - </div> - <p> - <s xml:space="preserve">Cum autem Å¿it <var>.a.b.</var> ad <var>.c.b.</var> vt <var>.c.b.</var> ad <var>.d.b.</var> & ita <var>.d.b.</var> ad <var>.e.b.</var> ex Å¿uppoÅ¿ito, ideo ex <lb/>17. quinti verum dicit Archim. </s> - <s xml:space="preserve">hoc eÅ¿t quod ita erit ipÅ¿ius <var>.d.e.</var> ad <var>.e.b.</var> vt <var>.a.c.</var> ad <var>.c.b.</var> <lb/>& vt <var>.c.d.</var> ad <var>.d.b.</var> & ex .13. eiuÅ¿dem eadem proportio erit tripli ipÅ¿ius <var>.c.d.</var> ad triplum <lb/>ipÅ¿ius <var>.d.b.</var> quæ dupli ipÅ¿ius <var>.d.e.</var> ad. duplum ipÅ¿ius <var>.e.b.</var> vt inquit Archi.</s> - </p> - <p> - <s xml:space="preserve">Ex qua .13. compoÅ¿itum ex <var>.a.c.</var> cum triplo ipÅ¿ius <var>.c.d.</var> cum duplo ipÅ¿ius <var>.d.e.</var> ean-<lb/>dem proportionem habebit ad <reg norm="compoÅ¿itum" type="context">compoÅ¿itÅ©</reg> ipÅ¿ius <var>.c.b.</var> cum triplo <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <var>.d.b.</var> cum duplo <lb/>ipÅ¿ius <var>.e.b.</var> quam ipÅ¿ius <var>.d.e.</var> ad <var>.e.b</var>. </s> - <s xml:space="preserve">Sed horum compoÅ¿itorum primum Å¿ignificetur <lb/>per <var>.H.</var> Å¿ecundum verò Å¿ignificatum fuit per <var>.E.</var> vnde <var>.H.</var> ad <var>.E.</var> Å¿e habebit vt <var>.d.e.</var> ad <lb/><var>e.b.</var> Å¿ed <var>.E.</var> ad <var>.A.</var> iam dictum eÅ¿t eſſe vt <var>.o.e.</var> ad <var>.d.e.</var> vbi Å¿ignatum eÅ¿t <var>.λ</var>. </s> - <s xml:space="preserve">quare ex .23. <lb/>quinti eadem proportio erit ipÅ¿ius <var>.o.e.</var> ad <var>.e.b.</var> quæ <var>.H.</var> ad <var>.A.</var> vt ipÅ¿e inquit.</s> - </p> - <note position="left" xml:space="preserve">X</note> - <!-- note-0400-02 --> - <p> - <s xml:space="preserve">Ex .18. poÅ¿tea eiuÅ¿dem ita erit <var>.o.b.</var> ad <var>.e.b.</var> vt <var>.H.A.</var> ad <var>.A</var>.</s> - </p> - <p> - <s xml:space="preserve">Notandum etiam eÅ¿t quod Å¿i collectæ fuerint omnes partes compoÅ¿iti <var>.H.A.</var> hoc <lb/>eÅ¿t duplum <var>.a.b.</var> cum duplo <var>.b.e.</var> cum quadruplo <var>.b.c.</var> cum quadruplo <var>.b.d.</var> cum Å¿implo <lb/><var>a.c.</var> cum triplo <var>.c.d.</var> cum duplo <var>.d.e.</var> habebitur triplum <var>.a.b.</var> triplum <var>.b.d.</var> & Å¿excuplum <lb/><var>b.c.</var> vt ipÅ¿e dixit. </s> - <s xml:space="preserve">Quod autem hoc verum Å¿it, cum diÅ¿tinctæ fuerint omnes partes, <lb/>vt in Å¿ubÅ¿criptis his lineis videre eÅ¿t, videbis quod Å¿i ex <var>.H.</var> detracta fuerit Å¿implex <var>.a.<lb/>c.</var> quæ quidem poÅ¿tea iuncta vni ex partibus quadrupli <var>.b.c.</var> ipÅ¿ius <var>.A.</var> reÅ¿ultabit nobis <lb/>vna inte gra <var>.a.b</var>. </s> - <s xml:space="preserve">Vnde habebimus triplum ipÅ¿ius <var>.a.b.</var> & in <var>.A.</var> remanebit triplum ip <lb/>Å¿ius <var>.c.b</var>. </s> - <s xml:space="preserve">Deinde Å¿i ex <var>.H.</var> auferatur triplum ipÅ¿ius <var>.c.d.</var> & ipÅ¿um addatur tribus parti-<lb/>bus quadrupli <var>.b.d.</var> ipÅ¿ius <var>.A.</var> habebimus tres vices <var>.b.c.</var> quæ Å¿i iungantur tribus, quæ <lb/>remanebant in <var>.A.</var> vt dixi, habebimus Å¿excuplum ipÅ¿ius <var>.b.c.</var> & in <var>.A.</var> remanebit Å¿im <lb/>plum <var>.b.d.</var> cum duplo ipÅ¿ius <var>.b.e</var>. </s> - <s xml:space="preserve">Vnde Å¿i ex <var>.H.</var> demptum fuerit duplum ipÅ¿ius <var>.d.<lb/>e.</var> quod quidem iungatur cum duplo ipÅ¿ius <var>.b.e.</var> habebimus duplum ipÅ¿ius <var>.b.d.</var> quod <lb/>coniunctum cum Å¿implo <var>.b.d.</var> quod in <var>.A.</var> relictum fuerat, habebimus triplum ipÅ¿ius <lb/><var>d.b</var>. </s> - <s xml:space="preserve">Verum igitur eÅ¿t quod inquit Archimedes, hoc eÅ¿t, quod <var>.H.A.</var> eÅ¿t triplum ip-<lb/>Å¿ius <var>.a.b.</var> Å¿excuplum ipÅ¿ius <var>.b.c.</var> & triplum ipÅ¿ius <var>.b.d</var>.</s> - </p> - <p> - <s xml:space="preserve">Verum etiam dicit ex eo (vt Å¿upra probatum eÅ¿t) quod <var>.a.c</var>: <var>c.d</var>: et <var>.d.e.</var> Å¿e <reg norm="habebant" type="context">habebãt</reg> <lb/>in continua proportionalitate, </s> - <s xml:space="preserve">quare ex conuerÅ¿a proportionalitate erunt Å¿ibi inui-<lb/>cem continuæ proportionales.</s> - </p> - <p> - <s xml:space="preserve">Nunc autem cum <var>.a.c</var>: <var>c.d.</var> et <var>.d.e.</var> Å¿int continuæ proportionales in ea proportione <lb/>in qua Å¿unt <var>.a.b</var>: <var>c.b</var>: <var>d.b</var>: et <var>.e.b.</var> vt in principio diximus, erit ex .22. quinti <var>.a.c.</var> ad <var>.d.<lb/>e.</var> vt <var>.a.b.</var> ad <var>.d.b.</var> & Å¿ic etiam <var>.c.b.</var> ad <var>.e.b</var>. </s> - <s xml:space="preserve">Vnde ex .24. eiuÅ¿dem <var>.a.d.</var> ad <var>.d.e.</var> erit vt <var>.a.<lb/>b.</var> cum <var>.b.e.</var> ad <var>.d.b.</var> & vt <var>.c.b.</var> cum <var>.b.d.</var> ad <var>.e.b.</var> & ex .13. dicti vt <var>.a.b.</var> cum <var>.b.e.</var> bis <lb/>Å¿umpto, & cum <var>.b.d.</var> ad <var>.e.b</var>. </s> - <s xml:space="preserve">Quare ex conuerÅ¿a proportionalitate, vt Å¿e habet <var>.e.d.</var> <lb/>ad <var>.d.a.</var> ita Å¿e habebit <var>.e.b.</var> <reg norm="cum" type="context">cÅ©</reg> <var>.d.b.</var> ad <var>d.b.</var> <reg norm="cum" type="context">cÅ©</reg> <var>.b.c.</var> duplicato & <reg norm="cum" type="context">cÅ©</reg> <var>.b.a.</var> vt inquit Archi <lb/>medes. </s> - <s xml:space="preserve">Nunc antecedens vocetur <var>.M.</var> hoc eÅ¿t <var>.b.e.</var> cum <var>.d.b.</var> conÅ¿equens verò, hoc - <pb o="389" rhead="EPISTOLAE." n="401" file="0401" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0401"/> - eÅ¿t <var>.d.b.</var> cum duplo <var>.b.c.</var> cum Å¿implo <var>.b.a.</var> vocetur <var>.N</var>.</s> - </p> - <p> - <s xml:space="preserve">Animaduertendum tamen eÅ¿t quod impreſſio mendoÅ¿a eÅ¿t ubi dicit. <lb/></s> - <s xml:space="preserve"><q open="„" close="">vnaquæque <var>.c.b</var>: <var>b.d.</var> & cætera,</q> <lb/>propterea quod dicendum eÅ¿t ita <lb/><reg norm="vnaquæque" type="simple">vnaquæq;</reg> <var>e.b</var>: <var>b.d</var>.</s> - </p> - <p> - <s xml:space="preserve">Nunc ex .18. quinti, quemadmodum Å¿e habet <var>.a.e.</var> ad <var>.d.a.</var> ita Å¿e habebit <var>.M.N.</var> ad <var>.N.</var> <lb/> - <anchor type="figure" xlink:label="fig-0401-01a" xlink:href="fig-0401-01"/> - </s> - <pb o="390" rhead="IO. BAPT. BENED." n="402" file="0402" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0402"/> - <s xml:space="preserve">Vbiautem Å¿criptum eÅ¿t <lb/><q open="„" close="">ad vtrunque Å¿imul <var>.b.d</var>: <var>d.a.</var> cum dupla <var>.b.c.</var></q> <lb/>dicendum eÅ¿t ita, <lb/>ad vtranque Å¿imul <var>.b.d.b.a.</var> cum dupla <var>.b.c</var>.</s> - </p> - <div type="float" level="5" n="10"> - <figure xlink:label="fig-0401-01" xlink:href="fig-0401-01a"> - <image file="0401-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0401-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Inquit deinde Archi. quod Å¿icut Å¿e haber <var>.e.a.</var> ad <var>.d.a.</var> ita Å¿e habebit duplum <var>.M.N.</var> <lb/>ad duplum <var>.N</var>. </s> - <s xml:space="preserve">Quod quidem verum eÅ¿t ex .13. quinti, huiuÅ¿modi verò antecedons <lb/>& conÅ¿equens, Archi. manifeÅ¿tat ex Å¿uis partibus, Å¿umendo duplum <var>.e.b.</var> cum duplo <lb/><var>b.d.</var> pro duplo <var>.M.</var> & duplum <var>.b.d.</var> cum duplo <var>.a.b.</var> cum quadruplo <var>.b.c.</var> pro duplo <var>.N.</var> <lb/>quÄ™ Å¿imul iuncta æquantur duplo <var>.e.b.</var> cum duplo <var>.a.b.</var> cum quadruplo <var>.b.d.</var> cum qua-<lb/>druplo <var>.b.c.</var> ex quo æquabuntur <var>.A.</var> vocentur igitur hæc omnia <var>.A.</var> potius quà m du-<lb/>plum ipÅ¿ius <var>.M.N</var>.</s> - </p> - <p> - <s xml:space="preserve">Verum etiam Å¿cribit, vbi dicit, quod proportio <var>.e.a.</var> ad tres quintas ipÅ¿ius <var>.a.d.</var> erit <lb/>vt <var>.A.</var> ad tres quintas dupli <var>.N.</var> ex .22. quinti. </s> - <s xml:space="preserve">Sed cum ex Å¿uppoÅ¿ito ita Å¿e habeat <var>.f.<lb/>g.</var> ad tres quintas ipÅ¿ius <var>.a.d.</var> quemadmodum <var>.b.e.</var> ad <var>.e.a.</var> erit ex .16. quinti verum <reg norm="quod" type="simple">ê™</reg> <lb/>dicit Archimed. </s> - <s xml:space="preserve">hoc eÅ¿t, ita Å¿e habere <var>.b.e.</var> ad <var>.f.g.</var> vt <var>.e.a.</var> ad tres quintas ipÅ¿ius <var>.a.d</var>.</s> - </p> - <p> - <s xml:space="preserve">Et per .11. eiuÅ¿dem verum etiam erit quod Å¿icut Å¿e habet <var>.e.b.</var> ad <var>.f.g.</var> ita Å¿e habe-<lb/>bit <var>.A.</var> ad tres quintas dupli <var>.N.</var> quod quidem duplum <var>.N.</var> Å¿ignificetur per <var>.Q</var>.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿uperius iam demonÅ¿tratum fuit (vbi <var>.X.</var>) quod <var>.o.b.</var> ad <var>.b.e.</var> ita Å¿e habebat vt <lb/><var>H.A.</var> ad <var>.A.</var> & <reg norm="nunc" type="context">nÅ©c</reg> demum probatum fuit ita eſſe <var>.A.</var> ad tres quintas ipÅ¿ius <var>.Q.</var> vt <var>.e.b.</var> <lb/>ad <var>.f.g</var>. </s> - <s xml:space="preserve">Quare ex .22. quinti ita erit <var>.H.A.</var> ad tres quintas ipÅ¿ius <var>.Q.</var> vt <var>.o.b.</var> ad <var>.f.g.</var> vt <lb/> - <anchor type="note" xlink:label="note-0402-01a" xlink:href="note-0402-01"/> - idem inquit.</s> - </p> - <div type="float" level="5" n="11"> - <note xlink:label="note-0402-01" xlink:href="note-0402-01a" position="left" xml:space="preserve">Y</note> - </div> - <p> - <s xml:space="preserve">Sed <var>.H.A.</var> ad <var>.Q.</var> (vt ex Å¿uis partibus videre eÅ¿t) ita Å¿e habet vt tres ad duo ex .13. <lb/>quinti, vt inquit Archimedes.</s> - </p> - <p> - <s xml:space="preserve">IpÅ¿e etiam dicit proportionem <var>.H.A.</var> ad tres quintas ipÅ¿ius <var>.Q.</var> eſſe vt quinque <lb/>ad duo. </s> - <s xml:space="preserve">Pro cuius rei euidentia imaginemur tam <var>.H.A.</var> quam <var>.Q.</var> diuiÅ¿a per <reg norm="quinque" type="simple">quinq;</reg> <lb/>partes æquales, vnde ex .16. quinti habebimus quamlibet quintam <reg norm="partem" type="context">partẽ</reg> ipÅ¿ius <var>.Q.</var> <lb/><reg norm="æqualem" type="context">æqualẽ</reg> eſſe duabus tertijs vniuÅ¿cuiuÅ¿que quintæ partis <var>.H.A.</var> vnde tres quintæ ipÅ¿ius <lb/>Q. erunt, ex communi conceptu, Å¿ex tertiæ vnius quintæ ipÅ¿ius <var>.H.A.</var> hoc eÅ¿t duæ <lb/>quintæ. ipÅ¿ius <var>.H.A</var>. </s> - <s xml:space="preserve">Quare <var>.o.b.</var> ita Å¿e habebit ad <var>.f.g.</var> vt quinque ad duo ex commu <lb/>ni <reg norm="conceptu" type="context">cõceptu</reg>, cum <var>.o.b.</var> ad <var>.f.g.</var> probatum fuerit Å¿e habere vt <var>.H.A.</var> ad tres quintas ipÅ¿ius <lb/>Q. (vbi <var>.Y.</var>) Å¿ed iam probatum fuit (vbi. ω) quod <var>.o.a.</var> ad <var>.h.g.</var> erat etiam vt <lb/>quinque ad duo, hoc eÅ¿t quod <var>.f.h.</var> erit duæ quintÄ™ ipÅ¿ius <var>.a.b</var>. </s> - <s xml:space="preserve">Quod eÅ¿t propoÅ¿itum.</s> - </p> - <pb o="391" rhead="EPISTOL AE." n="403" file="0403" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0403"/> - <figure position="here"> - <image file="0403-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0403-01"/> - </figure> - <pb o="392" rhead="IO. BAPT. BENED." n="404" file="0404" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0404"/> - <p> - <s xml:space="preserve">In vltima verò propoÅ¿itione Å¿ecundi lib. de ponderibus Archi. hoc modo intelliÂ<lb/>gendus eÅ¿t, vt Å¿i diceret, <lb/>Sit paraboles <var>.a.</var> cuius baÅ¿is Å¿it <var>.a.c.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.d.e.</var> recta parallela dictæ baÅ¿i <var>.a.c.</var> <reg norm="diameterque" type="simple">diameterqÌ;</reg> <lb/><var>b.f</var>. <lb/></s> - <s xml:space="preserve">Inquit deinde quod linea contingens in <var>.b.</var> parallela erit ipÅ¿i <var>.a.c.</var> et <var>.e.d.</var> quod proba <lb/>bimus hoc modo. <lb/></s> - <s xml:space="preserve">Cum <var>.b.f.</var> diameter Å¿it et <var>.a.c.</var> baÅ¿is, clarum erit ex definitione quod <var>.b.f.</var> diuidet <var>.a.c.</var> <lb/>per æqualia in <var>.g</var>. </s> - <s xml:space="preserve">Vnde ex .7. vel etiam ex .46. primi Pergei <var>.d.e.</var> diuiÅ¿a erit per æqua <lb/>lia à diametro <var>.b.f</var>. </s> - <s xml:space="preserve">Quare verum dicit ex quinta Å¿ecundi ipÅ¿ius Pergei hoc eÅ¿t quod <lb/>dicta contingens in puncto. b parallela erit ambobus <var>.a.c.</var> et <var>.e.d</var>.</s> - </p> - <p> - <s xml:space="preserve">Inquit poÅ¿tea quod diuiÅ¿a cum fuerit pars diametri quÄ™ inter <var>.d.e.</var> et <var>.a.c.</var> poÅ¿ita eÅ¿t <lb/>(hoc eÅ¿t <var>.g.f.</var>) per quinque partes æquales, <reg norm="quarum" type="context">quarÅ©</reg> partium media Å¿it <var>.h.k.</var> diuiÅ¿a etiam <lb/>imaginatione Å¿it in puncto <var>.i.</var> ita quod proportio ipÅ¿ius <var>.h.i.</var> ad <var>.i.K.</var> eadem Å¿it quæ in-<lb/>ter duo Å¿olida quorum vnum (illud Å¿cilicet à quo relatio incipit, hoc eÅ¿t antecedens) <lb/>pro Å¿ua baÅ¿i teneat quadratum ipÅ¿ius <var>.a.f.</var> cuius etiam Å¿olidi altitudo compoÅ¿ita Å¿it ex <lb/> - <anchor type="note" xlink:label="note-0404-01a" xlink:href="note-0404-01"/> - duplo ipÅ¿ius <var>.d.g.</var> cum Å¿implo <var>.a.f</var>. </s> - <s xml:space="preserve">Aliud verò Å¿olidum habeat pro Å¿ua baÅ¿i quadra-<lb/>tum ipÅ¿ius <var>.d.g.</var> eius verò altitudo compoÅ¿ita Å¿it ex duplo ipÅ¿ius <var>.a.f.</var> cum Å¿implo <var>.d.g</var>.</s> - </p> - <div type="float" level="5" n="12"> - <note xlink:label="note-0404-01" xlink:href="note-0404-01a" position="left" xml:space="preserve">R</note> - </div> - <p> - <s xml:space="preserve">Inquit nunc Archi. quod cum ita factum fuerit, oÅ¿tendet punctum <var>.i.</var> centrum eſſe <lb/>portionis abÅ¿ciſſę à tota Å¿ectione, quod <reg norm="fruÅ¿tum" type="context">fruÅ¿tÅ©</reg> <reg norm="nominatur" type="simple">nominat̃</reg> <reg norm="Å¿ignatum" type="context">Å¿ignatÅ©</reg> characteribus <var>.a.d.e.c</var>.</s> - </p> - <p> - <s xml:space="preserve">Sit igitur num@. <var>m.n.</var> inquit, æqualis diametro <var>.b.f.</var> et <var>.n.o.</var> æqualis <var>.b.g.</var> <reg norm="Å¿itque" type="simple">Å¿itqÌ;</reg> <var>.x.n.</var> me <lb/>dia proportionalis inter <var>.n.m.</var> et <var>.n.o.</var> et <var>.t.n.</var> in continua proportionalitate poÅ¿t <var>.o.n.</var> <lb/>hoc eÅ¿t quod ea proportio quæ eÅ¿t ipÅ¿ius <var>.o.n.</var> ad <var>.n.t.</var> eadem Å¿it ipÅ¿ius <var>.x.n.</var> ad <var>.n.o</var>. </s> - <s xml:space="preserve">Hinc <lb/>habebimus .4. lineas in continua proportionalitate Å¿ibi inuicem coniunctas <var>.m.n</var>: <var>x.<lb/>n</var>: <var>o.n.</var> et <var>.t.n</var>.</s> - </p> - <p> - <s xml:space="preserve">Vult etiam quod à linea <var>.i.b.</var> incipiens ab <var>.i.</var> verÅ¿us <var>.g.</var> alia linea abÅ¿ciſſa Å¿it, cui li-<lb/> - <anchor type="note" xlink:label="note-0404-02a" xlink:href="note-0404-02"/> - neæ, ita proportionata Å¿it <var>.f.h.</var> vt <var>.t.m.</var> eÅ¿t ad <var>.t.n.</var> quæ quidem linea Å¿ignata Å¿it <var>.i.r</var>.</s> - </p> - <div type="float" level="5" n="13"> - <note xlink:label="note-0404-02" xlink:href="note-0404-02a" position="left" xml:space="preserve">A</note> - </div> - <p> - <s xml:space="preserve">Dicit poÅ¿tea quod diameter <var>.b.f.</var> erit fortaſſe a xis vel aliqua reliquarum diame-<lb/>trorum, quod quidem in .46. primi Pergei videre eÅ¿t, cum omnes diametri Å¿int in-<lb/>uicem paralleli ipÅ¿i axi.</s> - </p> - <p> - <s xml:space="preserve">Cum poÅ¿tea dicit, quod <var>.a.f.</var> et <var>.d.g.</var> Å¿unt intentæ ductæqÌue, ibi vult id em infer-<lb/>re, quod Pergeus vocat ordinatè, vt ex .11. et .49. primi ipÅ¿ius Pergei videre li-<lb/>cet, vnde ex .20. eiuÅ¿dem proportio <var>.b.f.</var> ad <var>.b.g.</var> erit vt quadrati <var>.a.f.</var> ad quadratum <lb/>ipÅ¿ius <var>.d.g.</var> vt ipÅ¿e dicit.</s> - </p> - <p> - <s xml:space="preserve">Sed ita erit quadrati <var>.m.n.</var> ad qua <reg norm="dratum" type="context">dratÅ©</reg> <var>.x.n.</var> ex .18. Å¿exti Eucli. </s> - <s xml:space="preserve">Quare ex .11. quin-<lb/> - <anchor type="note" xlink:label="note-0404-03a" xlink:href="note-0404-03"/> - ti quadratum ipÅ¿ius <var>.m.n.</var> ad quadratum ipÅ¿ius <var>.n.x.</var> eandem habebit proportionem, <lb/>quam quadratum ipÅ¿ius <var>.a.f.</var> ad quadratum ipÅ¿ius <var>.d.g</var>. </s> - <s xml:space="preserve">Vnde ex .18. & ex communi <lb/><reg norm="Å¿cientia" type="context">Å¿ciẽtia</reg>, eadem proportio erit ipÅ¿ius <var>.m.n.</var> ad <var>.n.x.</var> quÄ™ ipÅ¿ius <var>.a.f.</var> ad <var>.d.g.</var> vt inquit Arch.</s> - </p> - <div type="float" level="5" n="14"> - <note xlink:label="note-0404-03" xlink:href="note-0404-03a" position="left" xml:space="preserve">α</note> - </div> - <p> - <s xml:space="preserve">Quaptopter proportio cubi ipÅ¿ius <var>.m.n.</var> ad cubum ipÅ¿ius <var>.n.x.</var> erit vt cubi ipÅ¿ius <var>.a.<lb/>f.</var> ad cubum ipÅ¿ius <var>.d.g.</var> vt etiam dicit ex communi Å¿cientia, nec non ex .36. vndecimi.</s> - </p> - <p> - <s xml:space="preserve">Inquit poÅ¿tea quod proportio totius Å¿ectionis <var>.a.b.c.</var> ad portionem <var>.d.b.e.</var> eadem <lb/>eÅ¿t quæ cubi ipÅ¿ius <var>.a.f.</var> ad cubum ipÅ¿ius <var>.d.g.</var> quod verum eÅ¿t, vt alià s tibi monÅ¿traui in <lb/>diuiÅ¿ione parabolæ Å¿ecundum aliquam propoÅ¿itam proportionem.</s> - </p> - <p> - <s xml:space="preserve">Quando autem dicit quod proportio cubi ipÅ¿ius <var>.m.n.</var> ad cubum ipÅ¿ius <var>.n.x.</var> eadem <lb/> - <anchor type="note" xlink:label="note-0404-04a" xlink:href="note-0404-04"/> - eÅ¿t quæ ipÅ¿ius <var>.m.n.</var> ad <var>.n.t.</var> verum dicit ex .36. vndecimi. </s> - <s xml:space="preserve">Vnde ex .11. quinti ita Å¿e <lb/>habebit totalis Å¿ectio <var>.a.b.c.</var> ad portionem <var>.d.b.c.</var> vt <var>.m.n.</var> ad <var>.n.t.</var> & ex .17. eiuÅ¿dem ita <lb/>erit ipÅ¿ius <var>.m.t.</var> ad <var>.t.n.</var> vt fruÅ¿ti <var>.a.d.e.c.</var> ad Å¿ectionem <var>.d.b.e.</var> quemadmodum ipÅ¿e di-<lb/>cit. </s> - <s xml:space="preserve">Sed quia Å¿uperius, vbi <var>.A.</var> ipÅ¿a <var>.f.h.</var> (quæ eÅ¿t tres quintæ ipÅ¿ius <var>.f.g.</var>) ad <var>.i.r.</var> ita rela- - <pb o="393" rhead="EPISTOL AE." n="405" file="0405" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0405"/> - ta fuit vt <var>.m.t.</var> ad <var>.t.n.</var> idcirco ex .11. quinti ita erit ipÅ¿ius fruÅ¿ti <var>.a.e.</var> ad Å¿ectionem <var>.d.b.<lb/>e.</var> vt tres quintÄ™ ipÅ¿ius <var>.f.g.</var> ad <var>.i.r</var>.</s> - </p> - <div type="float" level="5" n="15"> - <note xlink:label="note-0404-04" xlink:href="note-0404-04a" position="left" xml:space="preserve">β</note> - </div> - <p> - <s xml:space="preserve">Inquit deinde quod proportio corporis iam Å¿upradicti, quod pro Å¿ua baÅ¿i habeat <lb/>quadratum ipÅ¿ius <var>.a.f.</var> altitudinem verò compoÅ¿itam ex duplo ipÅ¿ius <var>.d.g.</var> cum Å¿implo <lb/><var>a.f.</var> ad cubum ipÅ¿ius <var>.a.f.</var> eadem erit quæ dupli ipÅ¿ius <var>.d.g.</var> cum Å¿implo <var>.a.f.</var> ad <var>.a.f</var>. </s> - <s xml:space="preserve">Quod <lb/>quidem verum eÅ¿t ex .33. vndecimi & ex prima Å¿exti.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿uperius (vbi. α.) iam probauimus eandem proportio nem eſſe inter <var>.m.n.</var> & <lb/><var>n.x.</var> quæ inter <var>.a.f.</var> et <var>.d.g.</var> ideo ex conuerÅ¿a pro portion alitate ita erit ipÅ¿ius <var>.x.n.</var> ad <var>.n.<lb/>m.</var> vt ipÅ¿ius <var>.d.g.</var> ad <var>.a.f.</var> Å¿ed dupli <var>.x.n.</var> ad Å¿implum <var>.x.n.</var> eÅ¿t vt dupli <var>.d.g.</var> ad <var>.d.g</var>. </s> - <s xml:space="preserve">Qua <lb/>re ex .22. quinti dupli <var>.x.n.</var> ad <var>.m.n.</var> erit vt dupli <var>.d.g.</var> ad <var>.a.f.</var> & ex .18. eiuÅ¿dem ita erit <lb/>dupli <var>.x.n.</var> cum Å¿implo <var>.m.n.</var> ad <var>.m.n.</var> vt dupli <var>.d.g.</var> cum Å¿implo <var>.a.f.</var> ad <var>.a.f</var>. </s> - <s xml:space="preserve">Quare Å¿olidi <lb/> - <anchor type="figure" xlink:label="fig-0405-01a" xlink:href="fig-0405-01"/> - <pb o="394" rhead="IO. BAPT. BENED." n="406" file="0406" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0406"/> - iam dicti ad cubum inÅ¿ius <var>.a.f.</var> ex .11. quinti erit vt dupli <var>.x.n.</var> <reg norm="cum" type="context">cÅ©</reg> Å¿implo <var>.m.n.</var> ad <var>.m.n</var>.</s> - </p> - <div type="float" level="5" n="16"> - <figure xlink:label="fig-0405-01" xlink:href="fig-0405-01a"> - <image file="0405-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0405-01"/> - </figure> - </div> - <note position="left" xml:space="preserve">δ</note> - <!-- note-0406-01 --> - <p> - <s xml:space="preserve">Superius autem vbi. β. demonÅ¿tratum fuit ita eſſe ipÅ¿ius <var>.m.n.</var> ad <var>.n.t.</var> vt cubi <var>.m.n.</var> <lb/>ad cubum <var>.x.n.</var> & inter. α et. β probatum fuit ita eſſe cubi <var>.a.f.</var> ad cubum <var>.d.g.</var> vt <lb/>cubi <var>.m.n.</var> ad cubum <var>.x.n</var>. </s> - <s xml:space="preserve">Vnde ex .11. quinti <var>.m.n.</var> ad <var>.n.t.</var> erit vt cubi <var>.a.f.</var> ad cubum <lb/><var>d.g</var>.</s> - </p> - <p> - <s xml:space="preserve">Dicit poÅ¿tea quod eadem proportio erit inter cubum <var>.d.g.</var> & corpus illud quod <lb/>pro baÅ¿i habeat quadratum inÅ¿ius <var>.d.g.</var> altitudinem verò vt dictum eÅ¿t, quæ eÅ¿t inter <lb/><var>d.g.</var> & compoÅ¿itum ex duplo <var>.a.f.</var> cum Å¿implo <var>.d.g.</var> quod compoÅ¿itum eÅ¿t altitudo di <lb/>cta, & <reg norm="verum" type="context">verÅ©</reg> dicit ex ratione Å¿uperius allegata pro reliquo corpore & cubo ipÅ¿ius <var>.a.f</var>. <lb/></s> - <s xml:space="preserve">Quare etiam quemadmodum <var>.t.n.</var> Å¿e habet ad duplum ipÅ¿ius <var>.o.n.</var> cum Å¿implo <var>.t.n.</var> <lb/>ex ijÅ¿dem rationibus Å¿upradictis, vbiloquuti Å¿umus de <var>.x.n.</var> cum <var>.m.n</var>.</s> - </p> - <p> - <s xml:space="preserve">DiÅ¿ponantur <reg norm="nunc" type="context">nÅ©c</reg> omnia tali ordine, ita vt <var>.u.</var> primum Å¿it corpus quod pro Å¿ua ba <lb/>Å¿i habeat quadratum ipÅ¿ius <var>.a.f.</var> & c.</s> - </p> - <p> - <s xml:space="preserve">Et <var>.y.</var> Å¿it cubus ipÅ¿ius <var>.a.f.</var> et <var>.s.</var> Å¿it cubus ipÅ¿ius <var>.d.g.</var> et <var>.z.</var> Å¿it corpus quod baÅ¿im ha-<lb/>bet quadratum ipÅ¿ius <var>.d.g.</var> altitudinem verò vt Å¿upradictum eÅ¿t, et <var>.p.</var> Å¿it compoÅ¿itum <lb/>dupli <var>.n.x.</var> cum Å¿implo <var>.m.n.</var> et <var>.l.</var> Å¿it compoÅ¿itum dupli ipÅ¿ius <var>.n.o.</var> cum Å¿implo <var>.t.n.</var> <lb/>Sed <var>.u.</var> locata Å¿it è regione <var>.p.</var> et <var>.y.</var> è regione <var>.m.n.</var> et <var>.s.</var> è regione <var>.n.t.</var> et <var>.z.</var> è regione <var>.l.</var> <lb/>& habebimus proportionem ipÅ¿ius <var>.u.</var> ad <var>.y.</var> vt <var>.y.</var> ad <var>.m.n.</var> & ipÅ¿ius <var>.y.</var> ad <var>.s.</var> vt <var>.m.n.</var> ad <var>.<lb/>n.t.</var> quod Å¿uperius iam demonÅ¿tratum fuit, vbi, δ. et <var>.s.</var> ad <var>.z.</var> ita Å¿e habebit vt <var>.n.t.</var> ad <var>.<lb/>l.</var> vt vltimò probatum fuit. </s> - <s xml:space="preserve">Quare ex .22. quinti ita Å¿e habebit <var>.u.</var> ad <var>.z.</var> vt <var>.p.</var> ad <var>.l.</var> <lb/>quemadmodum dicit Archi.</s> - </p> - <p> - <s xml:space="preserve">Et quia vt Å¿e habet <var>.u.</var> ad <var>.z.</var> ita facta fuit <var>.h.i.</var> ad <var>.i.K.</var> vbi <var>.R.</var> ideo ex .11. quinti vt Å¿e <lb/>habet <var>.h.i.</var> ad <var>.i.K.</var> ita Å¿e habebit <var>.p.</var> ad <var>.l.</var> vt ipÅ¿e dicit: </s> - <s xml:space="preserve">Et ex .18. quinti ita erit <var>.h.K.</var> <lb/>ad <var>.K.i.</var> vt <var>.p.l.</var> ad <var>.l.</var> & ex communi conceptu <var>.g.f.</var> Å¿e habebit ad <var>.h.K.</var> vt quintuplum <lb/>ipÅ¿ius <var>.p.l.</var> ad <var>.p.l.</var> & ex .22. eiuÅ¿dem ita Å¿e habebit <var>.f.g.</var> ad <var>.i.k.</var> vt quintuplum ipÅ¿ius <var>.p.<lb/>l.</var> ad <var>.l.</var> quintuplum autem ipÅ¿ius <var>.p.l.</var> compoÅ¿itum eÅ¿t ex quintuplo ipÅ¿ius <var>.n.m.</var> cum <lb/>decuplo ipÅ¿ius <var>.n.x.</var> cum quintuplo ipÅ¿ius <var>.n.t.</var> cum decuplo ipÅ¿ius <var>.n.o.</var> vt à te facilè <lb/>computare potes.</s> - </p> - <p> - <s xml:space="preserve">Verum etiam erit ex communi Å¿cientia quod <var>.g.f.</var> ad <var>.f.k.</var> eÅ¿t ut quintuplum ipÅ¿ius <lb/><var>p.l.</var> ad duplum ipÅ¿ius <var>.p.l.</var> eo quod Å¿uperius Å¿uppoÅ¿itum fuit <var>.h.K.</var> eſſe <reg norm="quintam" type="context">quintã</reg> mediam, <lb/>vnde <var>.k.f.</var> relinquebatur pro duabus quintis inferioribus, duplum autem <var>.p.l.</var> com-<lb/>poÅ¿itum eÅ¿t ex duplo ipÅ¿ius <var>.m.n.</var> cum duplo ipÅ¿ius <var>.n.t.</var> cum quadruplo ipÅ¿ius <var>.n.x.</var> & <lb/>cum quadruplo ipÅ¿ius <var>.x.o</var>.</s> - </p> - <p> - <s xml:space="preserve">Ex conuerÅ¿a proportionalitate deinde ita Å¿e habet, <var>i.K.</var> ad <var>.i.k.</var> ad <var>.f.g.</var> vt <var>.l.</var> ad quin-<lb/>tuplum ipÅ¿ius <var>.p.l.</var> et <var>.k.f.</var> ad <var>.f.g.</var> vt duplum ipÅ¿ius <var>.p.l.</var> ad quintuplum ipÅ¿ius <var>.p.l</var>. </s> - <s xml:space="preserve">Vnde <lb/>ex .24. quinti <var>.i.f.</var> Å¿e habebit ad <var>.f.g.</var> vt <reg norm="duplum" type="context">duplÅ©</reg> ipÅ¿ius <var>.p.l.</var> cum Å¿implo <var>.l.</var> ad quintuplum <lb/>ipÅ¿ius <var>.p.l</var>. </s> - <s xml:space="preserve">Deinde ex conuerÅ¿a proportionalitate quintuplum ipÅ¿ius <var>.p.l.</var> Å¿e habebit <lb/> - <anchor type="note" xlink:label="note-0406-02a" xlink:href="note-0406-02"/> - ad duplum ipÅ¿ius <var>.p.l.</var> cum Å¿implo <var>.l.</var> vt <var>.f.g.</var> ad <var>.f.i</var>. </s> - <s xml:space="preserve">Sed compoÅ¿itum dupli ipÅ¿ius <var>.p.l.</var> <lb/>cum Å¿implo <var>.l.</var> æquale eÅ¿t duplo ipÅ¿ius <var>.m.n.</var> cum quadruplo ipÅ¿ius <var>.x.n.</var> cum Å¿excuplo <lb/>ipÅ¿ius <var>.o.n.</var> cum triplo ipÅ¿ius <var>.n.t.</var> vt per te computare potes.</s> - </p> - <div type="float" level="5" n="17"> - <note xlink:label="note-0406-02" xlink:href="note-0406-02a" position="left" xml:space="preserve">θ</note> - </div> - <p> - <s xml:space="preserve">Superius enim Å¿umpta fuit <var>.i.r.</var> ad quam ita Å¿e haberet <var>.f.h.</var> hoc eÅ¿t tres quintæ ip-<lb/>Å¿ius <var>.f.g.</var> vt <var>.m.t.</var> ad <var>.t.n</var>. </s> - <s xml:space="preserve">Quare ex conuerÅ¿a proportionalitate ita Å¿e habebit <var>.i.r.</var> ad tres <lb/>quintas ipÅ¿ius <var>.f.g.</var> vt <var>.t.n.</var> ad <var>.t.m</var>. </s> - <s xml:space="preserve">Et quia <var>.o.n.</var> Å¿umpta fuit æqualis ipÅ¿i <var>.b.g.</var> et <var>.m.n.</var> ipÅ¿i <lb/><var>b.f.</var> ideo <var>.m.o.</var> ex communi Å¿cientia æ qualis erit ipÅ¿i <var>.g.f</var>. </s> - <s xml:space="preserve">Vnde proportio <var>.r.i.</var> ad tres <lb/>quintas ipÅ¿ius <var>.m.o.</var> erit vt <var>.n.t.</var> ad <var>.t.m.</var> vt inquit Archi.</s> - </p> - <p> - <s xml:space="preserve">Sed vbi. θ. iam probauimus ita Å¿e habere <var>.i.f.</var> ad <var>.f.g.</var> vt duplum <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> <var>.p.l.</var> cum Å¿im-<lb/>plo <var>.l.</var> Å¿e habet ad quintuplum ipÅ¿ius <var>.p.l.</var> hoc eÅ¿t <var>.i.f.</var> ad <var>.m.o.</var> vt duplum ipÅ¿ius <var>.p.l.</var> cum <lb/>Å¿implo <var>.l.</var> ad quintuplum ipÅ¿ius <var>.p.l</var>.</s> - </p> - <pb o="395" rhead="EPISTOL AE." n="407" file="0407" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0407"/> - <p> - <s xml:space="preserve">Habemus igitur <reg norm="nuncomnem" type="context">nuncomnẽ</reg><unsure/>s illas conditiones quas Archimedes in præcedenti <lb/>propoÅ¿itione Å¿upponit. </s> - <s xml:space="preserve">Vnde ex rationibus ibi allegatis Å¿equitur <var>.f.r.</var> eſſe duas quin-<lb/>tas ipÅ¿ius <var>.m.n.</var> hoc eÅ¿t ipÅ¿ius <var>.f.b</var>. </s> - <s xml:space="preserve">Quapropter punctum <var>.r.</var> centrum erit ponderis to-<lb/>tius Å¿ectionis parabolæ ex .8. Å¿ecundi lib. de ponderibus eiuÅ¿dem Archimedis.</s> - </p> - <p> - <s xml:space="preserve">Inquit nunc Archimedes, quod exiÅ¿tente <var>.q.</var> centro ponderis ipÅ¿ius parabolæ <var>.d.<lb/>b.e.</var> partialis, centrum fruÅ¿ti erit in linea recta <var>.q.r.f.</var> ita remotum à centro <var>.r.</var> quod <lb/>proportio <var>.q.r.</var> ad partem illam ipÅ¿ius <var>.r.f.</var> quæ reperitur inter centrum <var>.r.</var> & centrum <lb/>huius fruÅ¿ti æqualis eÅ¿t proportioni totius parabolæ ad partialem. </s> - <s xml:space="preserve">Quod quidem ve <lb/>rum eÅ¿t ex .8. primi libri eiuÅ¿dem.</s> - </p> - <p> - <s xml:space="preserve">Inquit etiam punctum <var>.i.</var> illud eſſe, eo quod cum probatum Å¿it <var>.f.r.</var> duas quintas eÅ¿-<lb/>Å¿e ipÅ¿ius <var>.f.b.</var> ideo <var>.b.r.</var> tres quintas erit ipÅ¿ius <var>.b.f.</var> vt ipÅ¿e dicit.</s> - </p> - <figure position="here"> - <image file="0407-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0407-01"/> - </figure> - <pb o="396" rhead="IO, BAPT. BENED." n="408" file="0408" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0408"/> - <p> - <s xml:space="preserve">Sed <var>.q.b.</var> Å¿imiliter tres quintæ eÅ¿t ipÅ¿ius <var>.d.b.</var> ex .8. prædicta. </s> - <s xml:space="preserve">Quare <var>.q.r.</var> tres quintæ <lb/>erit ipÅ¿ius <var>.f.g.</var> ex .19. quinti.</s> - </p> - <p> - <s xml:space="preserve">Dicamus igitur hoc modo cum <var>.f.b.</var> totum ad totum <var>.b.r.</var> ita Å¿e habeat vt abÅ¿ciÅ¿-<lb/>Å¿um <var>.b.g.</var> ad abÅ¿ciſſum <var>.q.b.</var> ex .7. et .8. dicti primi libri eiuÅ¿dem ideo reÅ¿iduum <var>.f.g.</var> ex <lb/><var>f.b.</var> ad reÅ¿iduum <var>.r.q.</var> ex <var>.r.b.</var> erit vt totum <var>.f.b.</var> ad. totum <var>.r.b.</var> ex .19. quinti Eucli.</s> - </p> - <p> - <s xml:space="preserve">Sed iam Å¿ub. β. probauimus ita Å¿e habere fruÅ¿tum <var>.a.d.e.c.</var> ad parabolam <var>.d.b.e.</var> vt <lb/><var>m.t.</var> ad <var>.t.n.</var> Å¿ed vt <var>.m.t.</var> ad <var>.t.n.</var> ita aſſ umpta fuit (vbi <var>.A.</var>). <var>i.r.</var> ad quam Å¿ic Å¿e haberet <var>.f.<lb/>h.</var> hoc eÅ¿t tres quintæ ipÅ¿ius <var>.f.g.</var> hoc eÅ¿t <var>.q.r</var>. </s> - <s xml:space="preserve">quare ex .11. quinti prop ortio fruÅ¿ti <var>.a.<lb/>d.e.c.</var> ad parabolam partialem erit vt <var>.q.r.</var> ad <var>.r.i</var>. </s> - <s xml:space="preserve">ExiÅ¿tente igitur <var>.r.</var> centro totius pa <lb/>rabolæ et <var>.q.</var> centro partialis, ergo <var>.i.</var> centrum erit fruÅ¿ti propoÅ¿iti.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i nullo Å¿olido intercedente, voluerimus centrum <var>.i.</var> fruÅ¿ti <var>.a.e.</var> citius inuenire, <lb/>inueniemus primò centrum <var>.r.</var> totius figuræ ex .8. Å¿ecundi eiuÅ¿dem conÅ¿tituendo <var>.b.r.</var> <lb/>tres quintas totius axis <var>.b.f.</var> & centrum <var>.q.</var> parabolæ <var>.d.b.e.</var> partialis Å¿imiliter.</s> - </p> - <p> - <s xml:space="preserve">Nunc igitur manifeÅ¿tum eÅ¿t nobis, eandem proportionem fore ipÅ¿ius <var>.q.r.</var> <lb/>ad <var>.r.i.</var> quæ fruÅ¿ti <var>.a.e.</var> ad portionem <var>.d.b.e.</var> ex .8. dicta. </s> - <s xml:space="preserve">Vnde ex coniuncta pro-<lb/>portionalitate ita Å¿e habebit <var>.q.i.</var> ad <var>.i.r.</var> vt <var>.a.b.c.</var> ad <var>.d.b.e.</var> Å¿ed vt <var>.a.b.c.</var> ad <var>.d.b.e.</var> ita Å¿e <lb/>habet <var>.m.n.</var> ad <var>.n.t.</var> eo quod vnaquæque harum duarum proportionum Å¿eÅ¿quialtera <lb/>eÅ¿t proportioni <var>.f.b.</var> ad <var>.b.g.</var> eo. quod <var>.f.b.</var> ad <var>.b.g.</var> ita Å¿e habet. vt <var>.m.n.</var> ad <var>.o.n.</var> </s> - <s xml:space="preserve">quare <lb/><var>m.n.</var> ad <var>.t.n.</var> ita Å¿e habebit vt <var>.g.i.</var> ad <var>.r.i.</var> vnde diÅ¿iunctim <var>.m.t.</var> ad <var>.t.n.</var> ita Å¿e habebit vt <lb/><var>q.r.</var> ad <var>.r.i</var>. </s> - <s xml:space="preserve">Iungatur igitur <var>.r.i.</var> quæ quidem <var>.r.i.</var> ita Å¿e habeat ad <var>.r.q.</var> vt <var>.t.n.</var> ad <var>.t.m.</var> vt <lb/>habeatur centrum fruÅ¿ti.</s> - </p> - <pb o="397" rhead="EPISTOL AE." n="409" file="0409" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0409"/> - <figure position="here"> - <image file="0409-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0409-01"/> - </figure> - </div> - </div> - <div type="section" level="3" n="43"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DEFENSIO NOSTRA CONTRA ANTONIVM <lb/>Bergam, & Alexandrum Piccolhomineum.</head> - <head style="it" xml:space="preserve">IlluÅ¿tri Domino Horatio Muto.</head> - <p> - <s xml:space="preserve"><emph style="sc">INter</emph> ea quæ olim contra Antonium Bergam, Å¿ermone Italico Å¿cripÅ¿i, <lb/>hoc vnum erat, quod ip Å¿e Berga non viderat quendam notatu dignum <lb/>errorem ipÅ¿ius Pi ccolhominei, vbi ipÅ¿e Alexander arguit quendam au-<lb/>thorem in tractatu de magnitudine terræ & aquæ pag .37. linea .26. ita di <lb/>cens, & erit maior aqua.</s> - </p> - <pb o="398" rhead="IO. BAPT. BENED." n="410" file="0410" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0410"/> - <p> - <s xml:space="preserve">Quo in loco clare videtur ipÅ¿um putare eandem proportionem inter diametros, <lb/>quæ inter Å¿phæras ipÅ¿as eſſe, nec amplius recordari eius quod Å¿cripÅ¿erat pag .24.</s> - </p> - <p> - <s xml:space="preserve">Piccolhom. igitur ibi <reg norm="Å¿upponens" type="context">Å¿upponẽs</reg> centrum <var>.D.</var> eſſe magnitudinis aquæ, & intra Å¿phæ <lb/>ram terreÅ¿trem, putat omnino cauÅ¿am eſſe vt terra Å¿uperet aquam magnitudine, qua-<lb/>Å¿i quod Å¿i punctum <var>.D.</var> vt centrum Å¿phæræ aquæ, vnum <reg norm="idemque" type="simple">idemqÌ;</reg> eſſet cum puncto <var>.E.</var> <lb/>extremo diametri ipÅ¿ius terræ, Å¿phæra <var>.A.G.H.</var> Å¿phæræ <var>.A.B.E.</var> dupla eſſe deberet, <lb/>quod quidem nullo pacto fieri poteÅ¿t, quamuis etiam proportio <var>.A.H.</var> ad <reg norm="diametrum" type="context">diametrÅ©</reg> <lb/><var>A.E.</var> Å¿uperbipartiensÅ¿eptimas exiÅ¿teret, quæ minor eſſet quam Å¿eÅ¿quitertia, ita quod <lb/>quando etiam <var>.D.E.</var> maior medietate ipÅ¿ius <var>.D.H.</var> fuiſſet, nihilominus tamen <lb/>terra minor eſſet aqua, eo quod proportio dupla minor eÅ¿t, quam tripla ad <reg norm="propor- tionem" type="context">propor-<lb/>tionẽ</reg> Å¿uperbipartientenÅ¿eptimas, & maior <reg norm="quam" type="context">quã</reg> tripla ad proportionem Å¿eÅ¿quiquar-<lb/>tam. </s> - <s xml:space="preserve">Vnde Å¿i Piccolhom. Å¿uppòſuiſſet proportionem ipÅ¿ius <var>.D.H.</var> ad <var>.C.E.</var> eſſe <lb/>Å¿eÅ¿quiquartam, rectè profectò dixiſſet, Å¿ed dicere quod ubicunque exiÅ¿tat <reg norm="punctum" type="context">punctÅ©</reg> <var>.<lb/>D.</var> intra Å¿phæram terreÅ¿trem, Å¿equitur ipÅ¿am eſſe maiorem aquea, verum non eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">ScripÅ¿i etiam quod Piccoloho. decipiebatur vbi loquitur de diaphaneitate aquæ <lb/>pag .40. ita dicens.</s> - </p> - <quote> - <s xml:space="preserve">Et cum rationabiliter aliquis exiÅ¿timare non poteÅ¿t, quod vmbra quæ facit ori-<lb/>ri e cclipÅ¿es Lunæ, producta Å¿it à terra, & ab aqua Å¿imul, vt ab vno corpore aggre-<lb/>gato exijs duobus elementis, & ad vnam communem Å¿phæreceitatem reductis, pro <lb/>pterea quod cum vmbra produci debeat à corporibus opacis, quorum opacitas effi-<lb/>cit illa corpora vmbroÅ¿a, aqua autem, Å¿it corpus diaphanum, & tranÅ¿parens, nullam <lb/>vmbram poterit à ſe eminus producere.</s> - </quote> - <p> - <s xml:space="preserve">Hic enim decipitur Piccolhom. duabus rationibus, quarum prima eÅ¿t, quod ra-<lb/>dius luminoÅ¿us non poteÅ¿t multum in profundum mergi, vt probaui in .8. epiÅ¿tola ad <lb/>Vimercatum, altera verò eÅ¿t, quod cum Å¿phærica Å¿it aqua maris, Å¿upponatur etiam <lb/>quod Å¿ub ea nulla terræ portio eſſet, & quod radij Å¿olares ipÅ¿am, non Å¿ecus ac pilam <lb/>ex criÅ¿tallo fabrÄ™factam penetrarent, cum autem ipÅ¿i radij, tam ab una, quam ab alia <lb/>parte Å¿up erficiei huiuÅ¿modi globi <reg norm="frangantur" type="context">frãgantur</reg>, ob diſſimilem diaphaneitatem inter ae <lb/>rem & aquam, ipÅ¿i Å¿einuicem interÅ¿ecarent, vt poÅ¿t pilam criÅ¿tallinam videre eÅ¿t, de-<lb/>inde procedentes, diÅ¿gregarentur, <reg norm="diÅ¿ciparenturque" type="simple">diÅ¿ciparenturqÌ;</reg> quouÅ¿que nullam vim illuminatio <lb/>nis haberent, quod quilibet experiri poterit mediante aliquo vaÅ¿e uitreo Å¿phærico, <lb/>aqua pleno, cuiuÅ¿uis magnitudinis, Å¿oli expoÅ¿ito.</s> - </p> - <p> - <s xml:space="preserve">Rationes etiam quas eodem loco Piccolho. adducit ad probandum quod Å¿i quis <lb/>in fundo maris exiÅ¿teret, nullum uideret lumen, nihil ualent. </s> - <s xml:space="preserve">Quarum prima eÅ¿t, <lb/>ubi ita dicit.</s> - </p> - <quote> - <s xml:space="preserve">Ille qui Å¿e in aquam mergit, cum maiorem lucem, quæ Å¿upra aquam eÅ¿t, relin-<lb/>quat, iudicat pro magno temporis Å¿patio locum illum obÅ¿curum, quemadmodum <lb/>accidit quando per multum temporis Å¿patium fixis oculis in corpore Solis intuiti Å¿u <lb/>mus, ab eodem poÅ¿tea eoÅ¿dem amouentes, omnia obÅ¿cura nobis videntur.</s> - </quote> - <p> - <s xml:space="preserve">IpÅ¿e autem non conÅ¿iderat quod talis obÅ¿curitas quæ Å¿equitur viÅ¿ionem maioris <lb/>luminis, parum durat, immo cito euaneÅ¿cit, Å¿ed in aqua nunquam reuertimur ad vi-<lb/>dendum, ne que veÅ¿tigium aliquod luminis ibi videtur, in fundo maris dico, quem-<lb/>admodum nobis nuntiauerunt hi qui margaritas expiÅ¿cantur in imis partibus ingen <lb/>tium æquorum indicorum.</s> - </p> - <p> - <s xml:space="preserve">Secunda uerò ratio ipÅ¿ius Piccolhom. eÅ¿t ubi ita dicit.</s> - </p> - <quote> - <s xml:space="preserve">Altera cauÅ¿a quod nobis obÅ¿curus appareat locus Å¿ub aqua, eſſe poteÅ¿t obÅ¿tacu-<lb/>lum quod aquæ habent ab opacitate terræ Å¿ub eorum fundo, etenim Å¿icut <reg norm="chriÅ¿tallum" type="context">chriÅ¿tallÅ©</reg> - <pb o="399" rhead="EPISTOLAE." n="411" file="0411" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0411"/> - quamuis <reg norm="perÅ¿picuum" type="context">perÅ¿picuÅ©</reg> Å¿iue <reg norm="tranÅ¿parens" type="context">tranÅ¿parẽs</reg> Å¿it, nihilominus propter obÅ¿taculum plumbi Å¿ub <lb/>ipÅ¿o poÅ¿iti, efficit vt radij viÅ¿uales repercuſſi reuertantur. ita etiam quamuis aqua Å¿it <lb/>corpus tranÅ¿parens, nihilominus propter obÅ¿taculum terræ opacæ, quæ Å¿ubÅ¿idet in <lb/>fundo maris efficere poteÅ¿t obÅ¿curas partes illas Å¿ub aqua, illis hominibus qui in <lb/>ipÅ¿a aqua mèrguntur.</s> - </quote> - <p> - <s xml:space="preserve">In hac Å¿ecunda ratione decipitur Piccolhom. </s> - <s xml:space="preserve">Primum quia Å¿i vÅ¿que ad imam par <lb/>tem maris, Solis radius ferri poſſet, ille qui ibi eſſet, attollens oculos Å¿urÅ¿um Solem <lb/>cerneret, </s> - <s xml:space="preserve">deinde aÅ¿piciendo ipÅ¿um fundum Maris, videret illum, ratione reflexio-<lb/>nis luminis ab ipÅ¿o fundo, & ex eadem ratione Å¿peculi ab ipÅ¿o adducta, quæ contra <lb/>ipÅ¿um eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Decipitur etiam cum dicatradios viÅ¿uales à ſpeculo Å¿eu plumbo repercuti, eo <reg norm="quod" type="simple">ê™</reg> <lb/>non radij viÅ¿uales Å¿unt hi qui reflectuntur, Å¿ed Å¿unt radij luminoÅ¿i primarij, Å¿eu Å¿ecun <lb/>darij qui non ab oculis exeunt Å¿ed à corpore lucido.</s> - </p> - <p> - <s xml:space="preserve">ScripÅ¿i etiam quod Å¿i verum eſſet proportionalitatem continuam <reg norm="quantitatum" type="context">quãtitatum</reg> ele-<lb/>mentorum ex proportione decupla conÅ¿tare, ignem pro maximo, terram verò pro <lb/>minimo terminorum Å¿umentes, totum aggregatum ex terra, aqua, aere, & igne, ita <lb/>eſſet maius terra, quemadmodum mille centum & vndecim ad vnum, vnde Å¿emidia <lb/>meter regionis elementaris eſſet quaÅ¿i aut paulo maior decuplo Å¿olum Å¿emidiame-<lb/>tro terræ, vnde inter conuexum ignis, & concauum minimi, Å¿eu inferioris orbis lu-<lb/>naris, relinqueretur quidam orbis vacuus Å¿piſſitudinis vnius interualli plus quam vi-<lb/>ginti terræ Å¿emidiametrorum, quod Å¿patium vacuum orbiculariter, maius exi-<lb/>Å¿teret ipÅ¿a totali regione elementari plus quam trigeÅ¿ies millies, immo Å¿i Å¿emidia <lb/>meter dicti primi orbislunaris maior eſſet terreÅ¿tri vt trigintanouem ad unum, <reg norm="dictus" type="simple">dictê°</reg> <lb/>orbis vacuus maior eſſet elementari regione plus quam .58208. ad vnum, proportio <lb/>nalitatem igitur continuam quæ ex decupla proportionalitate reÅ¿ultat in elementis <lb/>eſſe putare eÅ¿t maximus error.</s> - </p> - <p> - <s xml:space="preserve">Subdit deinde Berga, hoc voluiſſe Platonem neceſſario requiri, vt extrema ele-<lb/>menta, nempeignis & terra cum duobus medijs aere, & aqua coniungerentur, cum in <lb/>corporibus Å¿olidis (quaÅ¿i BergÄ™ Å¿int quædam corpora quæ Å¿olida non extent) poſſit <lb/>dari medium æquale in geometrica proportione.</s> - </p> - <p> - <s xml:space="preserve">Sed vbi Plato ad Å¿ermonem de numero elementorum Å¿e confert, poÅ¿tquam ra-<lb/>tione creationis ignis, & terrÄ™ Å¿e propoÅ¿uiſſe putat, vt <reg norm="idem" type="context">idẽ</reg> de alijs duobus corporibus <lb/>medijs probet, comparatione proportionalitatis continuæ geometricæ in tribus ter-<lb/>minis, ratione rerum Å¿uperficialium primò, deinde in quatuor, ratione corporearum <lb/>vtitur, ita dicens.</s> - </p> - <quote> - <s xml:space="preserve">Vinculorum verò ideÅ¿t aptiſſimum atque pulcherrimum quod exÅ¿e, & ex ijs quÄ™ <lb/>aÅ¿tringunt, quam maximè vnum efficit, <reg norm="&c." type="unresolved">&c.</reg></s> - </quote> - <p> - <s xml:space="preserve">Quo in loco Plato inÅ¿erre vult de proportionalitate geometrica trium termino-<lb/>rum, in qua ijdem ita Å¿e habent, vt medius, primi, <reg norm="vltimique" type="simple">vltimiqÌ;</reg> vice fungatur, ita vt vtriuÅ¿-<lb/>que ipÅ¿orum extremorum particeps fiat, cum productum quod à medio termino in <lb/>Å¿eipÅ¿o progignitur idem Å¿it ei quod ab extremis fuit, vnde medius, potentia idem eÅ¿t <lb/>quod productum ab extremis.</s> - </p> - <p> - <s xml:space="preserve">Subdit deinde Plato dicens.</s> - </p> - <quote> - <s xml:space="preserve">Quando enim in tribus numeris, aut molibus, aut viribus, medium ita Å¿e habet <lb/>ad poÅ¿tremum vt primum ad medium, <reg norm="viciſſimque" type="simple">viciſſimqÌ;</reg> vt poÅ¿tremum cum medio, ita me-<lb/>dium cum primo congruit, </s> - <s xml:space="preserve">tunc quod medium eÅ¿t, & primum fit & <reg norm="poÅ¿tremum" type="context">poÅ¿tremÅ©</reg>, po-<lb/>Å¿tremum quoque, & primum & media fiunt.</s> - </quote> - <pb o="400" rhead="IO. BAPT. BENED." n="412" file="0412" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0412"/> - <p> - <s xml:space="preserve">Hic <reg norm="animaduertendum" type="context">animaduertendÅ©</reg> eÅ¿t omnes interpretes falli, qui hoc loco Platonem de omni-<lb/>bus proportionalitatibus continuis quæ ternario numero (alia enim Arithmetica, <lb/>alia geometrica, alia harmonica dicitur) continentur, intelligendum eſſe cenÅ¿ent, <lb/>quia de numeris, magnitudinibus, <reg norm="viribusque" type="simple">viribusqÌ;</reg>, aut ut dici Å¿olet, virtutibus mentionem <lb/>fecerit. </s> - <s xml:space="preserve">Plato enim nihil aliud inferre voluit, quam eandem paſſionem (ut ipÅ¿e reci-<lb/>tat) inter medium <reg norm="extremaque" type="simple">extremaqÌ;</reg> vnius proportionalitatis continuæ geometricæ, tam <lb/>in quantitate, quam in qualitate <reg norm="reÅ¿ultaturam" type="context">reÅ¿ultaturã</reg>, cum tres termini eiuÅ¿dem eſſent Å¿pe-<lb/>ciei, & quia quantitas in duas principes <reg norm="primariasque" type="simple">primariasqÌ;</reg> partes, ideÅ¿t in continuam, & <lb/>diÅ¿cretam diuiditur, hanc ob cauÅ¿am Plato hoc præcipuè Å¿ignificat numerorum ma-<lb/><reg norm="gnitudinisque" type="simple">gnitudinisqÌ;</reg> vocabulis vtens, quibus vniuerÅ¿um quantitatis genus complectitur.</s> - </p> - <p> - <s xml:space="preserve">Cum verò ait vires, uniuerÅ¿um qualitatis genus inferre uult. </s> - <s xml:space="preserve">Quia proportio & <lb/>proportionalitas tam continua quam diÅ¿creta, non Å¿olum interterminos quanti, Å¿ed <lb/>inter eos etiam qui quali attribuuntur elucet.</s> - </p> - <p> - <s xml:space="preserve">Sed quod eo loco de harmonica proportionalitate quæ <reg norm="cum" type="context">cÅ©</reg> geometrica magis Å¿im <lb/>bola eÅ¿t quam cum Arithmetica Plato minime intelligat, ex eiuÅ¿dem uerbis cum ita <lb/>Å¿cribit manifeÅ¿tè patet.</s> - </p> - <quote> - <s xml:space="preserve">Quando enim medium ita Å¿e habet ad poÅ¿tremum ut primum ad medium, <reg norm="uiciÅ¿- Å¿imque" type="simple">uiciÅ¿-<lb/>Å¿imqÌ;</reg> ut poÅ¿tremum cum medio ita medium cum primo congruit.</s> - </quote> - <p> - <s xml:space="preserve">Id enim in harmonica proportionalitate non cernitur in qua primus terminus ad <lb/>poÅ¿tremum, & non ad medium, ita Å¿e habet geometricè ut differentia inter primum <lb/>& medium ad differentiam inter medium & ultimum.</s> - </p> - <p> - <s xml:space="preserve">Quod Å¿i clarum eÅ¿t ipÅ¿um de harmonica proportionalitate nullo modo intellige-<lb/>re, quanto minus de Arithmetica, quæ cum geometrica nihil habet commune.</s> - </p> - <p> - <s xml:space="preserve">Cum uerò Plato ait.</s> - </p> - <quote> - <s xml:space="preserve">Tunc quod medium eÅ¿t & primum fit & poÅ¿tremum, poÅ¿tremum quoque, & pri-<lb/>mum media fiunt, <reg norm="&c." type="unresolved">&c.</reg></s> - </quote> - <p> - <s xml:space="preserve">Nihil aliud oÅ¿tendere uult, quam Å¿imilitudinem quæ inter huiuÅ¿modi medium & <lb/>extrema intercedit, cum ipÅ¿um medium ad poÅ¿tremum, quem primus ad Å¿eipÅ¿um, <lb/>eundem reÅ¿pectum habeat, in quo eÅ¿t Å¿imilis primo, & contra ad primum <reg norm="terminum" type="context">terminÅ©</reg>, <lb/>eundem reÅ¿pectum, quem poÅ¿tremum ad Å¿eipÅ¿um habet, </s> - <s xml:space="preserve">unde hac ratione ultimum <lb/>repręſentat, uolens Plato inferre de conuenientia quÄ™ inter media elementa, & ex-<lb/>trema intercedit, ut aquæ inter aerem, & terram, cum aqua, ratione Å¿uæ frigiditatis, <lb/>terrÄ™, ratione uero Å¿uæ humiditatis aeri Å¿imilis euadat. </s> - <s xml:space="preserve">Aer uero qui inter ignem, <lb/><reg norm="aquamque" type="simple">aquamqÌ;</reg> ponitur quod ad caliditatem attinet cum igne, quod uero ad humidita-<lb/>tem Å¿pectat cum aqua communicet.</s> - </p> - <p> - <s xml:space="preserve">Sed quia Plato multis in rebus doctrinam Pythagoricam Å¿equutus eÅ¿t, Pythago-<lb/>rici aut em omnia numeris metiebantur, & de omnire Å¿ecundum numerorum ratio <lb/>nem diſſerebant, <reg norm="uidensque" type="simple">uidensqÌ;</reg> Plato quod inter duos numeros Å¿uperficiales, <reg norm="inuicemque" type="simple">inuicemqÌ;</reg> <lb/>Å¿imiles exiÅ¿tentes, unum tantum numerum medium in proportionalitate continua <lb/>geometrica cadere poteÅ¿t, ideo Å¿ubiungit.</s> - </p> - <quote> - <s xml:space="preserve">Quod Å¿i uniuerÅ¿i corpus latitudinem habere debuiſſet, nullam uerò profundita-<lb/>tem, unum Å¿anè, tum ad Å¿eipÅ¿um, tum ad extrema uincienda interiectum medium <lb/>Å¿uffeciſſet.</s> - </quote> - <p> - <s xml:space="preserve">Sequitur poÅ¿tea Å¿ic.</s> - </p> - <quote> - <s xml:space="preserve">Sed cum Å¿oliditatem mundus requireret, Å¿olida uerò non uno, Å¿ed duobus Å¿em-<lb/>per modis copulentur, inter ignem, & terram, Deus, Aerem, AquamqÌue loca-<lb/>uit, <reg norm="&c." type="unresolved">&c.</reg></s> - </quote> - <pb o="401" rhead="EPISTOLAE." n="413" file="0413" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0413"/> - <p> - <s xml:space="preserve">Volens inferre, quod quemadmodum inter duos numeros Å¿olidos, & inuicem <lb/>Å¿imiles, <reg norm="vnus" type="simple">vnê°</reg> <reg norm="tantum" type="context context">tãtÅ©</reg> medius proportionalis intercedere <reg norm="non" type="context">nõ</reg> poteÅ¿t, Å¿ed duo neceſſariò re <lb/>quiruntur (vt exijs quæ Euclid .8. lib. 16. 17. 18. et .19. propoÅ¿itione proponit videnÂ<lb/>tur) ita <reg norm="dictante" type="context">dictãte</reg> ratione inter igneum, <reg norm="terreumque" type="simple">terreumqÌ;</reg> corpus duo corpora interiecta <reg norm="eÅ¿sent" type="context">eÅ¿sẽt</reg>, <lb/>non ratione proportionalitatis continuæ in quantitate <reg norm="eorundem" type="context">eorÅ©dem</reg> corporum, Å¿ed pro <lb/>pter Å¿imilitudinem connexionis, cum productum ex duobus medijs proportionali-<lb/>bus æquale Å¿it producto ab extremis, & idem reÅ¿pectus, quem primum ipÅ¿orum qua <lb/>tuor ad Å¿ecundum habet, Å¿ecundi ad tertium extet, vnde Å¿ecundum primo Å¿imile <lb/>euadit, & contra, reÅ¿pectus qui eÅ¿t quarti ad tertium, Å¿it etiam tertij ad Å¿ecundum, vn-<lb/>de ipÅ¿um tertium, ratione vltimi Å¿ubit, & eius imaginem induit, & hanc ob cauÅ¿am <lb/>Å¿ic Å¿cribit Plato.</s> - </p> - <quote> - <s xml:space="preserve">Propterea ex huiuÅ¿modi rebus numero quaternario concluÅ¿is, mundi corpus con <lb/>flatum eÅ¿t, ea connexum comparatione qua dixi. </s> - <s xml:space="preserve">Ex quo Å¿eipÅ¿um amicitia concor-<lb/>di complectitur, <reg norm="&c." type="unresolved">&c.</reg></s> - </quote> - <p> - <s xml:space="preserve">Vbi Platonem, elementa maiora, <reg norm="minoranue" type="context">minorãue</reg> in proportionalitate continua, nec <lb/>geometrica, nec alterius cuiuÅ¿uis generis eſſe noluiſſe, clarè perÅ¿picitur, Å¿ed huiuÅ¿mo <lb/>di Å¿imilitudine, in eo quod media elementa cum extremis conueniunt eÅ¿t vÅ¿us, quæ <lb/>quidem conuenientia, nullibi maior, quam in proportionalitate continua geome-<lb/>trica reperitur. </s> - <s xml:space="preserve">Sed etiam Å¿i Plato de huiuÅ¿modi corporea elementorum magnitu-<lb/>dine Å¿eipÅ¿um intelligi voluiſſet, Å¿i Å¿emidiameter regionis elementaris ex Ä™quo vt .39 <lb/>ad vnum, reÅ¿pectu Å¿emidiametri terræ fuiſſet, aqua, ipÅ¿am terram, magis quam tri-<lb/>geÅ¿ies, & octies, non Å¿olum decies, & aer quoque eandem magis quam .1500. & <lb/>ignis magis quam .55000. partibus magnitudine Å¿uperaret.</s> - </p> - <p> - <s xml:space="preserve">SubÅ¿tantia vero rerum quas Å¿cripÅ¿eram circa finem illius conÅ¿iderationis talis fuit.</s> - </p> - <p> - <s xml:space="preserve">Nunc autem tempus eſſe videtur, vt ego etiam, ne tantum deÅ¿truxiſſe, Å¿ed etiam <lb/>conÅ¿truxiſſe videar aliquid pro veritate diſſeram.</s> - </p> - <p> - <s xml:space="preserve">Non eÅ¿t igitur dubium, Å¿olidæ doctrinæ viris, quin præſtantiſſimus Piccolo. Å¿e-<lb/>cutus Å¿it tutam viam ad explorandum, quod terra maior Å¿it quam aqua, metiendo <lb/>vtriuÅ¿que horum corporum Å¿uperficiem detectam. </s> - <s xml:space="preserve">Omittamus autem compenÅ¿a-<lb/>tionem illam curuitatis, & concauitatis vallium, & montium, <reg norm="&c." type="unresolved">&c.</reg> quam ipÅ¿e Piccolo. <lb/>propè finem Å¿exti cap. vellet dare fluminibus, Å¿tagnis, fontibus, & eiuÅ¿modi aquis. <lb/></s> - <s xml:space="preserve">eo enim in loco labitur Piccolo. vbi non conÅ¿iderat, quod eiuÅ¿modi obliquis Å¿uper-<lb/>ficiebus non reÅ¿pondent anguli Å¿olidi centri Å¿phæræ, qui reÅ¿piciunt eorum baÅ¿im ad <lb/>rectos angulos. </s> - <s xml:space="preserve">Sed poÅ¿tquam Piccolo. comperit Å¿uperficiem terrÄ™ detectam, eſſe <lb/>maiorem apparente Å¿uperficie Å¿phærica aquæ, proculdubio poterat concludere ter-<lb/>ram eſſe maiorem aqua, Å¿icuti fecit, <reg norm="etiam" type="context">etiã</reg> Å¿i aqua profunda eſſet pyramidaliter <reg norm="vÅ¿que" type="simple">vÅ¿q;</reg> <lb/>ad mundi centrum, ideÅ¿t .3500. milliaria, Å¿upponendo tantum eſſe huius globi Å¿emi <lb/>diametrum.</s> - </p> - <p> - <s xml:space="preserve">Verum quia poſſet aliquis dubitare circa diligentiam Piccolo. in hiÅ¿cæ duabus Å¿u <lb/>perficiebus dimetiendis, viÅ¿um eÅ¿t mihi non alienum Å¿equi aliam viam pro hac veri <lb/>tate probanda, Å¿upponendo verum eſſe, quod non vnus Å¿olus metitus fuerit, Å¿ed mul <lb/>ti, ideÅ¿t Å¿upponendo <reg norm="verum" type="context">verÅ©</reg> eſſe quod maris profunditas menÅ¿urari poſſit, & præterea, <lb/>quod non modo ipÅ¿ius maris maxima profunditas non perueniat ad quingentos paÅ¿ <lb/>Å¿us, Å¿icuti refert Piccolo. in fine Å¿ui tractatus, & mihi aſſeruerunt HiÅ¿pani multi, & <lb/>LuÅ¿itani præſtantiſſimi nautæ, tum Venetijs, tum Parmæ, in Aula Sereniſſimæ quon <lb/>dam Principis, inter quos, Venetijs fuit IlluÅ¿tris Rodericus Guzmanus, Dominus <lb/>FranciÅ¿cus Lopes, Dominus Garzias de Seuilia, <reg norm="multique" type="simple">multiqÌ;</reg> alij. </s> - <s xml:space="preserve">Parmæ autem varij - <pb o="402" rhead="IO. BAPT. BENED." n="414" file="0414" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0414"/> - quos omnes <reg norm="recenÅ¿erem" type="context">recenÅ¿erẽ</reg> moleÅ¿tum eſſet. </s> - <s xml:space="preserve">Sed etiam Å¿upponendo quod maxima <lb/>pelagi profunditas Å¿it, non modo .500. paſſuum, Å¿ed etiam .500. millium paſſuum, vt <lb/>dixi, & quod mare Å¿it huius profunditatis, non vno in loco tantum, aut multis, Å¿ed <lb/>quod Å¿upra totam etiam faciem terræ, mare tantÄ™ profunditatis ipÅ¿am terram vn-<lb/>dique operiret, ideÅ¿t, quod vbicunque nunc terra detecta eÅ¿t, eſſet aqua, Å¿piſſitudi-<lb/>nis .500. millium paſſuum. </s> - <s xml:space="preserve">Atque vt planius intelligar Å¿upponendo quod Å¿icuti to-<lb/>tus huius globi Å¿emidiameter eÅ¿t <reg norm="milliarium" type="context">milliariÅ©</reg> .3500. </s> - <s xml:space="preserve">TerreÅ¿tris partis Å¿emidiameter <lb/>eſſet <reg norm="tantum" type="wordlist">tm̃</reg> .3000. & reliquum Å¿emidiametri, id eÅ¿t quingenta milliaria eſſet craſſitudo <lb/>Å¿iue profunditas orbis aquei, in quo nihil neceſſe eſſet laborare in dimetiendis fon-<lb/>tibus, fluminibus, lacubus, Å¿tagnis, paludibus, & huiuÅ¿modi particulis nullius momen <lb/>ti apud peritos, nec curare Å¿ubterraneas aquas cauernarum, aut aliorum terræ cauo-<lb/>rum, Å¿eu terræ porroÅ¿itatum, quæ omnia Å¿unt circa ipÅ¿ius terræ Å¿uperficiem. </s> - <s xml:space="preserve">Quia ve <lb/>riÅ¿imile non eÅ¿t naturam eiuÅ¿modi caua Å¿iue Å¿pong oÅ¿itates produxiſſe demiſſius li-<lb/>bramenti maris. </s> - <s xml:space="preserve">Supponendo igitur ea quæ nunc dicta Å¿unt, terra tamen eſſet ferè <lb/>duplo maior aqua, hoc eÅ¿t, vt .12. ad .7. </s> - <s xml:space="preserve">Quod quidem, cuiuis mathematicæ philoÅ¿o-<lb/>phiæ mediocriter perito, Å¿upputatu facillnnum eÅ¿t. </s> - <s xml:space="preserve">Cum proportio diametrorum, <lb/>Å¿eu Å¿emidiametrorum, tertia pars exiÅ¿tat proportionis eorundem Å¿phærarum. </s> - <s xml:space="preserve">Sed <lb/>vt parum periti minore labore Å¿upputare poſſint.</s> - </p> - <p> - <s xml:space="preserve">Primum Å¿ciendum eÅ¿t, quod Å¿upponendo diametrum globi, ex terra, & aqua com <lb/>poÅ¿iti, eſſe .3500. milliarium, & Å¿emidiametrum puræ terreÅ¿tris partis eſſe .3000. tan <lb/>tum, eiuÅ¿modi proportio erit ut .7. ad .6. quia communis maior numerator horum <lb/>duum Å¿emidiametrorum erit .500. qui in maiorem ingredietur Å¿epties, in minorem <lb/>a utem Å¿exies. </s> - <s xml:space="preserve">Et eiuÅ¿modi proportio Å¿uperparticularis, vocatur Å¿eÅ¿quiÅ¿exta, cuius <lb/>triplum erit vt .57. cum Å¿exta parte ad .36. & idem erit inter dictum globum compo-<lb/>Å¿itum, & partem terreÅ¿trem Å¿implicem. </s> - <s xml:space="preserve">Quare Å¿ubtrahendo puram, Å¿eu Å¿implicem <lb/>partem terreÅ¿trem, ex compoÅ¿ito, reliqua pars erit, vt .21. cum Å¿exta, pro quantitate <lb/>aquei orbis, ad quam, terreÅ¿tris quantitas .36. erit ferè in <reg norm="eadem" type="context">eadẽ</reg> proportione, quæ .12. <lb/>ad .7.</s> - </p> - <p> - <s xml:space="preserve">Nunc fortaſſe alienum non erit videre quanto ferè maior eſſet terra, quam tota <lb/>aqua, non dico <reg norm="autem" type="context">autẽ</reg> Å¿olum de parte illa maximæ e ius profunditatis, quæ nuÅ¿quam <lb/>ad quingentos paſſus peruenit, Å¿ed de ficto illo orbe aqueo, profunditatis .500. paÅ¿-<lb/>Å¿uum, qui totum terreÅ¿trem orbem circundaret, & tegeret, Å¿upponendo quod per <lb/>quingentos paſſus profunditatis, quidquid eÅ¿t terra, eſſet aqua, ideÅ¿t Å¿uppoÅ¿ito quod <lb/>ex totius orbis compoÅ¿iti Å¿emidiametro exiÅ¿tente .3500. milliarium, purÄ™ terræ Å¿emi <lb/>diameter eſſet milliarium .3499. cum dimidio. </s> - <s xml:space="preserve">Supponendo igitur, vt Å¿upradixi. <lb/></s> - <s xml:space="preserve">Comperietur quod terra eſſet maior aqua amplius quam .2333. vicibus. </s> - <s xml:space="preserve">Sed quia <lb/>partes terræ detectæ rumpunt eiuÅ¿modi fictum orbem aqueum, quæ quidem partes, <lb/>Å¿unt ampliores Å¿uperficiÄ™ aquæ, vt obÅ¿eruauit Piccolo. atque alij <reg norm="præſtantes" type="context">præſtãtes</reg> viri, ideo <lb/>Å¿equetur, vt terra Å¿it maior aqua amplius .4666. vicibus imo amplius quinquies mil-<lb/>lecuplo. </s> - <s xml:space="preserve">Si autem quis diceret, in quantitate aquæ computari etiam illam, quæ gi-<lb/>gnatur ex vaporibus, qui globum hunc compoÅ¿itum circundant: </s> - <s xml:space="preserve">reÅ¿pondeo quod <lb/>non modò ei concedo computari eiuÅ¿modi aquam, Å¿ed Å¿upponendo etiam quodto <lb/>tus locus à vaporibus occupatus, qui attolluntur .52. milliaria Å¿upra Å¿uperficiem <lb/>huius globi, vt iam Å¿upradictum eÅ¿t, totus eſſet aqueus, & amplius, Å¿upponendo quod <lb/>orbis hic aqueus eſſet Å¿piſſitudinis, Å¿iue altitudinis quingentorum milliarium Å¿upra <lb/>totum ipÅ¿um globum compoÅ¿itum. </s> - <s xml:space="preserve">Tamen terra eſſet maior ipÅ¿a aqua ferè duplo; <lb/></s> - <s xml:space="preserve">qua dere, quiÅ¿que eiuÅ¿modi Å¿upputationum peritus certior fieri poterit. </s> - <s xml:space="preserve">Vnde iti- - <pb o="403" rhead="EPISTOLAE." n="415" file="0415" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0415"/> - dem affirmare poſſemus, terram non Å¿olum maiorem eſſe aqua, Å¿ed aqua & præte-<lb/>rea aere, Å¿i aer non tam altè pertingit, quam multi alij præter Piccolo. Å¿entiunt, qui <lb/>dicuntinde euenire quod aerea humiditas non tam altè aÅ¿cendere poteÅ¿t, quoniam <lb/>humiditas ipÅ¿a grauitatem Å¿ecum affert, præterquam quod nubium Å¿itus oÅ¿tendit Å¿u <lb/>pra eas materiam eſſerariorem quam Å¿int ipſę nubes, infra vero denÅ¿iorem. </s> - <s xml:space="preserve">Corpo-<lb/>ra enim eouÅ¿que aÅ¿cendunt donec inueniunt conÅ¿titutionem mediam formæ æqua-<lb/>lis (vt ita dicam) Å¿uis. </s> - <s xml:space="preserve">Quare materia illa quæ impropriè ignis vocatur (non enim <lb/>eÅ¿t ignis) incipit carere humiditate (qua mediante aer definitur) circa quinquage-<lb/>Å¿imum Å¿ecundum milliarium Å¿upra Å¿uperficiem terræ, vt iam Å¿upradixi à Vitellione <lb/>demonÅ¿tratum fuiſſe. </s> - <s xml:space="preserve">AriÅ¿to. autem affert <reg norm="rationem" type="context">rationẽ</reg> quare nubes altius <reg norm="non" type="context">nõ</reg> <reg norm="tranÅ¿cendant" type="context context">tranÅ¿cẽdãt</reg>. <lb/></s> - <s xml:space="preserve">Vnde apparet tertiam aeris regionem impropriè aerem appellari, Å¿i humiditate ca-<lb/>ret, vt ait AriÅ¿t. qua mediante aer definitur, immo potius retinet ignis naturam, vt <lb/>etiam aſſerunt interpretes AriÅ¿totelis in primum Meteororum. </s> - <s xml:space="preserve">Qui AriÅ¿to. in locis <lb/>Å¿upra citatis itidem oÅ¿tendit Å¿e etiam huius modi eſſe opinionis.</s> - </p> - <p> - <s xml:space="preserve">Quod autem attinet ad probandum quod Å¿uperficies terrÄ™ detecta Å¿it altior quam <lb/>Å¿uperficies detecta aquæ, id tam clarum eÅ¿t Å¿ua Å¿ponte philoÅ¿ophis, qui Å¿ciunt quid Å¿it <lb/>altum, quidue demiſſum, quod Å¿uperfluum eſſet quidquid Å¿uper hoc dicerem præ <lb/>terquam, quòd conÅ¿tat ex demonÅ¿tratione ab AriÅ¿to. Å¿acta textu 31. li .2. de cÅ“lo, in <lb/>quo agit de corporibus in aqua poÅ¿itis, vnde eiuÅ¿modi veritas planiſſimè aperitur. <lb/></s> - <s xml:space="preserve">Omittimus etiam quod præſtantes Moderni omnes, eam pro manifeÅ¿tiſſima <reg norm="ponunt" type="context">ponÅ©t</reg>, <lb/>Å¿icutiapud omnes Å¿ani iudicij homines reuera exiÅ¿timatur.</s> - </p> - <p> - <s xml:space="preserve">Hæc enim Å¿unt quæ in fine illius conÅ¿iderationis Å¿cripÅ¿eram.</s> - </p> - <p> - <s xml:space="preserve">Anno autem præterito editus in lucem fuit tractatus quidam Pulcherrimus, ab Ex <lb/>cellentiſſimo, nec non Doctiſſimo viro AuguÅ¿tino Michele, Patritio Veneto, ad cor <lb/>roborandam opinionem antiquorum, vbi tot authoritates, <reg norm="totque" type="simple">totqÌ;</reg> rationes adducit, <lb/>vt nil amplius dici poſſit. </s> - <s xml:space="preserve">Atego Å¿enÅ¿um, <reg norm="rationemque" type="simple">rationemqÌ;</reg>, & non authoritatem <reg norm="aliquam" type="context">aliquã</reg> <lb/>Å¿equutus Å¿um: </s> - <s xml:space="preserve">cum verò dico Å¿enÅ¿um, de Å¿enÅ¿u illorum intelligo, qui profunditatem <lb/>maris metiti Å¿unt, vt non mihi Å¿olum, Å¿ed, & Piccolo. & alijs permultis retulerunt, <lb/>de ratione vero à me adducta, aliorum Å¿it iudicium.</s> - </p> - <quote> - <s xml:space="preserve">SediÅ¿te mirabilis & Excellentiſſimus vir, verba mea non accepit in eo Å¿enÅ¿u, vt <lb/>ego Å¿cripÅ¿i, ita vt omnino alienas conÅ¿equentias Å¿ibi confingat, <reg norm="quemadmodum" type="context">quemadmodÅ©</reg> pag. <lb/>3. Å¿ui tractatus inquit, me non concedere naturam produxiſſe in magna quantitate, <lb/>atque immenÅ¿a, id totum, quod bonum, & neceſſarium eÅ¿t. </s> - <s xml:space="preserve">Hanc enim conÅ¿equen <lb/>tiam ipÅ¿e colligit ex eo, quod ego pag .19. meæ conÅ¿iderationis contra Antonium <lb/>Bergam Å¿cripÅ¿eram, quod videntur multa corpora alijs nobiliora, nihilominus mi-<lb/>nora, eo quod quantitas non Å¿equitur nobilitatem, neque ab ea pender, ita vt res <lb/>illa quæ nobilior eÅ¿t, neceſſarium Å¿it vt etiam maior exiÅ¿tat. </s> - <s xml:space="preserve">Sed Excellentiſſimus <lb/>iÅ¿tæ vir Å¿cribit ita me dixiſſe.</s> - </quote> - <p> - <s xml:space="preserve">Multa immo infinita corpora Å¿unt nobilia, & neceſſaria, nihilominus Å¿unt paruÄ™ <lb/>molis.</s> - </p> - <p> - <s xml:space="preserve">Vide igitur quantum hoc diÅ¿tat ab illo.</s> - </p> - <p> - <s xml:space="preserve">Præterea cap .12. aliam conÅ¿equentiam facit, quam ego non tam amplam facio. <lb/></s> - <s xml:space="preserve">„ IpÅ¿e enim me inferre vult in alijs terrÄ™ partibus cauernas non reperiri, eo quod Mon <lb/>„ tes Å¿int cauernoÅ¿i. </s> - <s xml:space="preserve">AÅ¿pice quæſo. pag .29. meæ conÅ¿iderationis, & clarè videbis me <lb/>nullo modo negare illas concauitates Å¿eu porroÅ¿itates terræ extra montana loca, <lb/>circa Å¿uperficiem terræ, vÅ¿que ad æquilibrium, orbiculariter, infimæ profunditatis <lb/>maris.</s> - </p> - <p> - <s xml:space="preserve">Sed putare inferius has porroÅ¿itates reperiri, cum nulla ratio nobis perÅ¿uaÅ¿ibilis - <pb o="404" rhead="IO. BAPT. BENED." n="416" file="0416" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0416"/> - adhuc ab aliquo prodita Å¿it, idoneum nullo pacto eſſet. </s> - <s xml:space="preserve">Rationes autem ab ipÅ¿o Ex <lb/>cellentiſſimo AuguÅ¿tino adductas circa huiuÅ¿modi rem, alij dijudicent, de authori-<lb/>tatibus verò, nihil dicam, quia ab illis petendæ Å¿unt, qui profitentur tales facultates, <lb/>quorum vnius tantummodo authoritas præualere deberet, contra omnes alias <reg norm="eorum" type="context">eorÅ©</reg> <lb/>qui nunquam attigerunt Å¿ummis labris orificia harum <reg norm="Å¿cientiarum" type="context">Å¿cientiarÅ©</reg>. </s> - <s xml:space="preserve">Vt Å¿i exempli gra-<lb/>tia non Å¿olum authoritas illorum virorum, quos ipÅ¿e recenÅ¿uit, Å¿ufficiens eſſet vt pu <lb/>ta Pioccolo. </s> - <s xml:space="preserve">Naibodæ, Bordini, Clauij, <reg norm="reliquorumque" type="simple">reliquorumqÌ;</reg> fautorum verÄ™ opinionis, <lb/>Å¿ed FranciÅ¿ci Maurolici tantummodo, qui in primo Dialogo Å¿uæ coÅ¿mographiæ ita <lb/>inquit.</s> - </p> - <quote> - <s xml:space="preserve">ExiÅ¿timo autem totum terræ corpus rigidum eſſe Å¿axum, nam Å¿i arena eſſet, aue <lb/>gleba fragilis, ita humorem imbiberet, vt cum eo quaÅ¿i confunderetur; </s> - <s xml:space="preserve">huc ac-<lb/>cedit, quod Å¿i mineræ, ac rupes, quæ Å¿unt grauiſſimæ partes in ipÅ¿a plerunque Å¿uper-<lb/>ficie comperiuntur, multo magis apud centrum eſſe debent. </s> - <s xml:space="preserve">Videtur ita ratio exi-<lb/>gere, vt grauiora centro quoque Å¿int propinquiora.</s> - </quote> - <p> - <s xml:space="preserve">Hæcigitur Å¿ola authoritas, inÅ¿tar reliquarum omnium Å¿ufficere poſſet. </s> - <s xml:space="preserve">Verum <lb/>de authoritatibus minime curandum eÅ¿t, vbi Å¿enÅ¿us, <reg norm="ratioque" type="simple">ratioqÌ;</reg> vera illis opponuntur.</s> - </p> - <p> - <s xml:space="preserve">Quod autem numerus animalium aquatilium maior exiÅ¿tat numero terreÅ¿trium, <lb/>Å¿atis reÅ¿pondimus pag .41. noÅ¿træ conÅ¿iderationis.</s> - </p> - <p> - <s xml:space="preserve">Sed in cap .14. Excellentiſſimus AuguÅ¿tinus ita inquit (vt etiam Å¿uperius dixerat) <lb/></s> - <quote> - <s xml:space="preserve">quod certiorem cognitionem homo non habet illa, quæ à ſenÅ¿u prouenit. </s> - <s xml:space="preserve">Et quod <lb/>nemo eÅ¿t qui aÅ¿piciat terram, & aquam, quod hanc maiorem illa non iudicet, & <reg norm="non" type="context">nõ</reg> <lb/>exiÅ¿timet.</s> - </quote> - </p> - <p> - <s xml:space="preserve">Quod autem certiorem cognitionem homo non habeat illa, quæ à ſenÅ¿u proue-<lb/>nit, concedendum non cenÅ¿eo. </s> - <s xml:space="preserve">Nam omnis cognitio mathematica (cum primum <lb/>gradum certitudinis obtineat) ab ipÅ¿o Å¿enÅ¿u fieret, quod omnino alienum <lb/>eÅ¿t à veritate. </s> - <s xml:space="preserve">SenÅ¿us enim nunquam vidit incommenÅ¿urabilitates magnitudinum, <lb/>vel incoincidentias linearum non tangentium cum curuitate hyperbolica, aut angu <lb/>lum contingentiæ aliquem, nec (vt vno verbo dicam) aliquam concluÅ¿ionem ma-<lb/>thematicam, quam volueris. </s> - <s xml:space="preserve">Neque per Å¿enÅ¿um eÅ¿t Å¿cire, inquit AriÅ¿toteles. </s> - <s xml:space="preserve">Co-<lb/>gnitio igitur Å¿enÅ¿itiua, certior non eÅ¿t illa, quæ per habitum Å¿cientiÅ¿icum acquiritur.</s> - </p> - <p> - <s xml:space="preserve">Ad reliqua verò, Å¿upponamus nos tunc fuiſſe in Arca Noe, <reg norm="cum" type="context">cÅ©</reg> aquæ cooperiebant <lb/>omnia cacumina montium, vbi nullum terræ veÅ¿tigium videbatur, </s> - <s xml:space="preserve">quare proculdu <lb/>bio aquam iudicaremus, atque exiÅ¿timaremus maiorem terra, <reg norm="dum" type="context">dÅ©</reg> nulla aliare vtere-<lb/>mur niÅ¿i Å¿enÅ¿u abÅ¿que alio diÅ¿curÅ¿u intellectuali, ut reliqua illa animalia irrationalia, <lb/>quæ nobiÅ¿cum erant in dicta arca. </s> - <s xml:space="preserve"><reg norm="Non" type="context">Nõ</reg> Å¿ufficit igitur Å¿uperficiem aquæ tantummodo <lb/>aÅ¿picere, quia neque tunc temporis, aqua erat maior terra, etiam Å¿i non Å¿olum tot <lb/>cubitis attolleretur Å¿upra cacumina montium, Å¿ed quingenta milliaria, vt Å¿upradi-<lb/>ximus.</s> - </p> - <p> - <s xml:space="preserve">Ratio autem illa, ex infinitis, ab ipÅ¿o, eo in loco adducta, talis eÅ¿t.</s> - </p> - <quote> - <s xml:space="preserve">Aqua eÅ¿t eccentrica ad terram, & pro <reg norm="centro" type="context">cẽtro</reg> habet centrum grauitatis terræ, aqua <lb/>igitur maioris eÅ¿t amplitudinis ipÅ¿a terra.</s> - </quote> - <quote> - <s xml:space="preserve">Hanc etiam conÅ¿equentiam alijs relinquo PhiloÅ¿ophis dijudicandam.</s> - </quote> - <p> - <s xml:space="preserve">SubÅ¿equitur poÅ¿tea dicens.</s> - </p> - <quote> - <s xml:space="preserve">Præterea proprius locus terræ, eÅ¿t Å¿uperÅ¿icies aquæ, igitur terram oportet ab <lb/>aqua tegi.</s> - </quote> - <p> - <s xml:space="preserve">Ad hoc etiam aliquis poſſet quærere, quis nam erit locus illius partis terræ de-<lb/>tectæ ab aqua? </s> - <s xml:space="preserve">nulli dubium erit quin Å¿uperficies aeris, & non aquæ exiÅ¿tet.</s> - </p> - <pb o="405" rhead="EPISTOL AE." n="417" file="0417" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0417"/> - <p> - <s xml:space="preserve">Nune autem Å¿i locus terræ eÅ¿t Å¿ub aqua, ergo locus aquæ proprius eÅ¿t Å¿ub aere, & <lb/>non Å¿ub terra, vnde non erit rationabile putare maiorem copiam aquarum exiÅ¿tere <lb/>in cauernis Å¿ubterraneis, quam Å¿upra Å¿uperficiem terræ. </s> - <s xml:space="preserve">Adde quod locus illarum <lb/>aquarum non eſſet Å¿uperficies aeris, Å¿ed terræ, vnde non minus locus aquÄ™ eſſet ter-<lb/>ra, quam locus terræ, aqua. </s> - <s xml:space="preserve">Sed miſſa faciamus hæc.</s> - </p> - <p> - <s xml:space="preserve">Cap. verò .20. ita inquit.</s> - </p> - <p> - <s xml:space="preserve">Materia elementorum æqualis eÅ¿t. </s> - <s xml:space="preserve">Ergo aqua maior eÅ¿t terra.</s> - </p> - <quote> - <s xml:space="preserve">Hæc enim conÅ¿equentia veriſſima eſſet. </s> - <s xml:space="preserve">Sed nullus vnquam PhiloÅ¿ophus (vt Phi-<lb/>loÅ¿ophus dico) concedet totam materiam elementarem, in quatuor æquales partes <lb/>eſſe diuiÅ¿am.</s> - </quote> - <p> - <s xml:space="preserve">Cap. verò .21. inquit me dixiſſe non Å¿uffecturam paucam Å¿piſſitudinem. </s> - <s xml:space="preserve">Eo enim <lb/>in loco pag .26. mei tractatus contradicens ipÅ¿i Bergæ, dixi, quod Å¿ecundum ipÅ¿um <lb/>Bergam non Å¿ufficeret pauca Å¿piſſitudo.</s> - </p> - <p> - <s xml:space="preserve">Similiter etiam dixi, quod Å¿ecundum ipÅ¿um, quanto remotius diffunditur lumen <lb/>fortaſſe tantò magis illuminat. </s> - <s xml:space="preserve">Putans ipÅ¿e Berga quod in propinquo debilius exi-<lb/>Å¿teret dictum lumen. </s> - <s xml:space="preserve">Et propter ea dixi, quod apud ipÅ¿um fortaſſe nihil valet illa <lb/>propoÅ¿icio, quæ dicit. </s> - <s xml:space="preserve">Agens in propinquo, fortius agit quam in remoto.</s> - </p> - <p> - <s xml:space="preserve">Cap. autem .22. vbi Excellentiſſimus AuguÅ¿tinus inquit, vnum tantummodo ele <lb/>mentum non Å¿ufficere ad generationem miÅ¿torum. </s> - <s xml:space="preserve">Hoc enim concedo, Å¿ed hoc ni-<lb/>hil ad me Å¿pectat, eo quod meum reÅ¿ponÅ¿um ad Bergam, erat circa tranÅ¿itum lumi-<lb/>nis, & non circa generationem elementorum.</s> - </p> - <p> - <s xml:space="preserve">Cap. demum .23. pag .20. linea .10. vbi Å¿cribit me dixiſſe, iudicare, oportebat <lb/>Å¿cribere, dubitare.</s> - </p> - <p> - <s xml:space="preserve">Puto tamen hoc vocabulum eſſe errorem Thypographi, quamuis in correctione <lb/>illud non inuenerim, quia vt ego multoties expertusÅ¿um, difficillimum omnes Thy <lb/>pographi errores corrigere, neque (vt fertur) Argi oculi Å¿ufficerent.</s> - </p> - <p> - <s xml:space="preserve">Hactenus enim in mei defenÅ¿ionem hæc Å¿ubiungere volui.</s> - </p> - <p> - <s xml:space="preserve">Ad defenÅ¿ionem autem Piccolo. <reg norm="aliorumque" type="simple">aliorumqÌ;</reg> virorum meæ opinionis, nec non de <lb/>proportione duplicata profunditatis maris ad Å¿uam amplitudinem, ex conÅ¿equentia <lb/>pyramidali: </s> - <s xml:space="preserve"><reg norm="alijsque" type="simple">alijsqÌ;</reg> Å¿imilibus rationibus, prodeant alij. </s> - <s xml:space="preserve">HuiuÅ¿modi tamen Doctiſſi-<lb/>mi viriingenium, memoriam, nec non doctrinam valde admiror, atque obÅ¿eruo.</s> - </p> - </div> - </div> - <div type="section" level="3" n="44"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE METHODO PRODVCTIONIS FRACTORVM <lb/>qua vtuntur Pedemontani AgrimenÅ¿ores.</head> - <head style="it" xml:space="preserve">AnÅ¿elmo RoÅ¿emburg AgrimenÅ¿ori CaÅ¿areo.</head> - <p> - <s xml:space="preserve"><emph style="sc">MEthodvs</emph> quà m mihi Å¿cribis in Prouincia tua maximè in vÅ¿u eſſe, nimis <lb/>longa atque prolixa eÅ¿t, Pedemontani verò AgrimenÅ¿ores in produ-<lb/>ctione fractorum, valde breui methodo vti Å¿olent, quam libenter tibi <lb/>Å¿cribo, eo maxime, vt videas quam rationabiliter operentur.</s> - </p> - <p> - <s xml:space="preserve">Scire igitur primum te oportet illos, maximam eorum communem menÅ¿uram <lb/>vocare Trabucum, cuius Å¿extam partem vocant Pedem, duodecimam verò pe-<lb/>dis, Vnciam, <reg norm="duodecimam" type="context">duodecimã</reg> <reg norm="autem" type="context">autẽ</reg> vnciæ <reg norm="punctum" type="context">punctÅ©</reg>, <reg norm="duodecimam" type="context">duodecimã</reg> demum puncti; </s> - <s xml:space="preserve">Attomum.</s> - </p> - <p> - <s xml:space="preserve">QuotieÅ¿cunque igitur multiplicant trabucum, per trabucum nulli dubium eÅ¿t <lb/>quin producant trabucum Å¿uperficialem Å¿cilicet.</s> - </p> - <pb o="406" rhead="IO. BAPT. BENED." n="418" file="0418" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0418"/> - <p> - <s xml:space="preserve">Similiter multiplicando pedes, vncias, puncta, & attoma per trabucum, produ-<lb/>cunt pedes, vncias, puncta, & attoma Å¿uperficialia rectangula oblonga, quorum lon <lb/>gitudo eÅ¿t ipÅ¿ius trabuchi, latitudo vero lineæ dictarum Å¿pecierum.</s> - </p> - <p> - <s xml:space="preserve">Dum vero multiplicant pedem per pedem, nulli dubium eÅ¿t quin producant pe-<lb/>dem quadratum, Å¿ed apud ipÅ¿os non vocatur quadratum, quamuis reuera ita Å¿it, Å¿ed <lb/>illud vocant duas vncias, quæ quidem Å¿unt rectangula oblonga iam hic Å¿upradicta, <lb/>quarum vniuÅ¿cuiuÅ¿que longitudo Å¿it vnius trabuchi, latitudo vero vnius duodecimæ <lb/>partis ipÅ¿ius pedis linearis.</s> - </p> - <p> - <s xml:space="preserve">Productum autem pedis per vnciam, vocant duo puncta, quæ etiam Å¿unt duo re-<lb/>ctangula oblonga, vt Å¿upra.</s> - </p> - <p> - <s xml:space="preserve">Productum deinde vnciæ per vnciam, vocant duos attomos, qui <reg norm="etiam" type="context">etiã</reg> Å¿unt duo re-<lb/>ctangula oblonga, vt dictum eÅ¿t, quæ omnia Å¿cientificè videbimus.</s> - </p> - <p> - <s xml:space="preserve">Pro cuius rei cognitione, Å¿it, exempli gratia <var>.a.e.</var> vnus Trabuchus linearis <var>.e.i.</var> ve-<lb/>ro vnus pes <var>.i.o.</var> autem vna vncia, <var>o.u.</var> poÅ¿tea vnum punctum, et <var>.u.t.</var> vnus At-<lb/>tomus.</s> - </p> - <p> - <s xml:space="preserve">Vnde <var>.e.i.</var> erit Å¿exta pars ipÅ¿ius <var>.a.e.</var> et <var>.i.o.</var> duodecima ipÅ¿ius <var>.e.i.</var> et <var>.o.u.</var> duodecima <lb/>ipÅ¿ius <var>.i.o.</var> et <var>.u.t.</var> duodecima ipÅ¿ius <var>.o.u</var>. </s> - <s xml:space="preserve">Sit etiam <var>.a.b.</var> æqualis <var>.a.e.</var> lineæ & Å¿ic <var>.e.d</var>: <var>i.<lb/>f</var>: <var>o.g</var>: <var>o.n.</var> & c. <reg norm="terminenturque" type="simple">terminenturqÌ;</reg> parallelogramma <var>.b.e</var>: <var>d.i</var>: <var>f.o</var>: <var>g.u.</var> et <var>.c.t.</var> vnde <var>.b.e.</var> erit <lb/>trabuchum quadratum, et <var>.d.i.</var> pes rectangulus oblongus vt Å¿upra, et <var>.f.o.</var> vncia rectan <lb/>gula oblonga, et <var>.g.u.</var> punctum rectangulum oblongum, et <var>.c.t.</var> attomus rectangu-<lb/>lus oblongus.</s> - </p> - <p> - <s xml:space="preserve">De producto igitur trabuchi per <reg norm="trabuchum" type="context">trabuchÅ©</reg>, nulli dubium eÅ¿t quin Å¿it quadratum <var>.<lb/>a.d.</var> vt Å¿uperius diximus.</s> - </p> - <p> - <s xml:space="preserve">Productum autem trabuchi cum pede erit <var>.d.i.</var> Å¿exta pars ipÅ¿ius <var>.a.d.</var> cum <var>.e.i.</var> Å¿it Å¿ex <lb/>ta ipÅ¿ius <var>.a.e.</var> ex prima Å¿exti vel .18. aut .19. Å¿eptimi, Å¿iue etiam ex .15. quinti Eucli.</s> - </p> - <p> - <s xml:space="preserve">Productum autem pedis cum pede erit <var>.e.K.</var> quadratum, quod probandum eÅ¿t <lb/> - <anchor type="figure" xlink:label="fig-0418-01a" xlink:href="fig-0418-01"/> - <pb o="407" rhead="EPISTOL AE." n="419" file="0419" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0419"/> - duplum eſſe rectangulo <var>.f.o.</var> <reg norm="Nam" type="context">Nã</reg> <var>.K.i.</var> Å¿exta pars eÅ¿t ipÅ¿ius <var>.f.i.</var> ex Å¿uppoÅ¿ito, et <var>.i.o.</var> duo-<lb/>decima ipÅ¿ius <var>.e.i.</var> proportio igitur <var>.e.i.</var> ad <var>.o.i.</var> dupla eÅ¿t proportioni ipÅ¿ius <var>.f.i.</var> ad <var>.K.i</var>. <lb/></s> - <s xml:space="preserve">quare <var>.K.e.</var> duplo maius eÅ¿t ipÅ¿ius <var>.f.o.</var> eo quod Å¿i <var>.i.o.</var> vel <var>.f.g.</var> (quod idem eÅ¿t) duplo <lb/>maius eſſet ipÅ¿o latere pręſenti <var>.o.i.</var> vel <var>.f.g.</var> </s> - <s xml:space="preserve">tunc <var>.f.o.</var> æquale eſſet ipÅ¿i <var>.K.e.</var> ex .15. Å¿exti <lb/>vel .20. Å¿eptimi quod quidem <var>.f.o.</var> duplo maius eſſet ipÅ¿o præſenti <var>.f.o</var>. </s> - <s xml:space="preserve">Rectè igitur <lb/>inquiunt dicentes productum pedis cum pede eſſe duas vncias, vel Å¿i mauis, ita dicas <lb/><var>e.K.</var> Å¿exta pars eÅ¿t ipÅ¿ius <var>.d.i.</var> ex iam dictis propoÅ¿itionibus <var>.f.o.</var> autem eÅ¿t duodecima <lb/>ipÅ¿ius <var>.d.i.</var> ex ijÅ¿dem, cum exÅ¿uppoÅ¿ito <var>.i.o.</var> duodecima Å¿it ipÅ¿ius <var>.e.i.</var> </s> - <s xml:space="preserve">quare <var>.e.K.</var> <reg norm="duplum" type="context">duplÅ©</reg> <lb/>erit ipÅ¿ius <var>.f.o.</var> ex commu ni notione.</s> - </p> - <div type="float" level="5" n="1"> - <figure xlink:label="fig-0418-01" xlink:href="fig-0418-01a"> - <image file="0418-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0418-01"/> - </figure> - </div> - <p> - <s xml:space="preserve">Productum verò pedis cum vncia. Å¿it <var>.K.o.</var> quod probabimus ex ijÅ¿dem rationibus <lb/>duplum eſſe ipÅ¿ius <var>.g.u.</var> puncti rectanguli oblongi. </s> - <s xml:space="preserve">Nam <var>.l.o.</var> Å¿exta pars Å¿imiliter eÅ¿t <lb/>ipÅ¿ius <var>.g.o.</var> et <var>.o.u.</var> duodecima ipÅ¿ius <var>.o.i.</var> </s> - <s xml:space="preserve">quare proportio <var>.i.o.</var> ad <var>.o.n.</var> dupla eÅ¿t propor <lb/>tioni <var>.g.o.</var> ad <var>.o.l.</var> Å¿equitur ergo ex prædictis rationibus <var>.k.o.</var> duplum eſſe ipÅ¿ius <var>.g.u.</var> <lb/>vel Å¿ic, vtlin præcedenti, cum <var>.K.o.</var> Å¿it Å¿exta pars ipÅ¿ius <var>.f.o.</var> ex dictis propoÅ¿itionibus <var>.<lb/>g.u.</var> verò duodecima eiuÅ¿dem <var>.f.o.</var> ex ijÅ¿dem, nam <var>.o.u.</var> duodecima eÅ¿t ipÅ¿ius <var>.o.i.</var> ergo <lb/><var>K.o.</var> duplo maius eÅ¿t ipÅ¿o <var>.g.u</var>.</s> - </p> - <p> - <s xml:space="preserve">Ex ijÅ¿demmet rationibus productum <var>.l.u.</var> pedis cum puncto duplum eÅ¿t ipÅ¿ius <var>.c.t.</var> <lb/>attomi rectanguli oblongi.</s> - </p> - <p> - <s xml:space="preserve">Probandum nunc relinquitur productum <var>.o.n.</var> vnciæ cum vncia, quod eÅ¿t quadra-<lb/>tum, duplum eſſe ipÅ¿ius <var>.c.t.</var> attomi rectanguli oblongi. </s> - <s xml:space="preserve">Nam <var>.i.n.</var> eÅ¿t pars vna ex .72. <lb/>ipÅ¿ius <var>.c.u.</var> et <var>.u.t.</var> pars vna ex .144. ipÅ¿ius <var>.o.i.</var> ex Å¿uppoÅ¿ito, </s> - <s xml:space="preserve">quare proportio <var>.i.o.</var> ad <var>.u.t.</var> <lb/>dupla eÅ¿t proportioni ipÅ¿ius <var>.c.u.</var> ad <var>.n.i.</var> ex dictis igitur rationibus <var>.o.n.</var> duplo maius <lb/>eÅ¿t ipÅ¿o <var>.c.t</var>. </s> - <s xml:space="preserve">Vel Å¿i placet dicas <var>.n.o.</var> eÅ¿t vna pars ex .72. ipÅ¿ius <var>.f.o.</var> exÅ¿upradictis, eo <lb/>quod <var>.n.i.</var> ita Å¿e habet ad <var>.f.i.</var> vt vnitas ad .72. Å¿ed ex ijÅ¿dem rationibus <var>.c.t.</var> pars vna ex <lb/>144. eÅ¿t ipÅ¿ius <var>.f.o.</var> eo quod ita Å¿e habet <var>.u.t.</var> ad <var>.o.i.</var> </s> - <s xml:space="preserve">quare <var>.o.n.</var> duplo maius erit ipÅ¿o <var>.<lb/>c.t</var>.</s> - </p> - <p> - <s xml:space="preserve">PropoÅ¿itum Å¿it nobis nunc, exercitij gratia, quærere Å¿uperficiem alicuius rectan <lb/>guli, cuius vnum latus Å¿it <reg norm="trabuchorum" type="context">trabuchorÅ©</reg> .3. pedum .2. & vnciarum .3. aliud vero latus Å¿it <lb/>trabuchorum .2. pedum .3. vnciarum vero .2.</s> - </p> - <p> - <s xml:space="preserve">HuiuÅ¿modi autem methodo mediante, multiplicando primum latus <reg norm="dictum" type="context">dictÅ©</reg> .3. 2. 3. <lb/>per numerum trabucorum Å¿ecundi lateris .2. Å¿cilicet producentur nobis primò trabu <lb/>cha Å¿uperficialia .6. pedes .4. & vnciæ .6. omnia rectagula, vt dictum eÅ¿t. </s> - <s xml:space="preserve">Multiplican-<lb/>do deinde idem primum latus .3. 2. 3. per pedes .3. Å¿ecundi lateris. </s> - <s xml:space="preserve">Ex trabuchis .3. <lb/>primi lateris cum .3. pedibus Å¿ecundi, producentur .9. pedes rectanguli, hoc eÅ¿t <lb/>vnus trabuchus cum tribus pedibus rectangulis. </s> - <s xml:space="preserve">Ex pedibus autem huius .2. cum <lb/>ijÅ¿dem alterius lateris .3. producentur .12. vnciæ rectangulæ ideÅ¿t vnus pes rectangu-<lb/>lus. </s> - <s xml:space="preserve">ExijÅ¿dem pedibus .3. Å¿ecundi lateris, cum .3. vncijs primi lateris producentur <num value="18">.<lb/> - <anchor type="table" xlink:label="table-0419-01a" xlink:href="table-0419-01"/> - <pb o="408" rhead="IO. BAPT. BENED." n="420" file="0420" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0420"/> - 18.</num> puncta rectangula, hoc eÅ¿t vna vncia cum .6. punctis rectangulis. </s> - <s xml:space="preserve">Deinde ex <lb/>multiplicatione vnciarum .2. Å¿ecundi lateris, cum .3. trabuchis primi lateris, produ-<lb/>centur .6. vnciæ. </s> - <s xml:space="preserve">Ex multiplicatione poÅ¿tea dictarum .2. vnciarum Å¿ecundi lateris <lb/>cum .2. pedibus primi, producentur .8. puncta.</s> - </p> - <div type="float" level="5" n="2"> - <xhtml:table xlink:label="table-0419-01" xlink:href="table-0419-01a"> - <xhtml:tr> - <xhtml:th xml:space="preserve">Trabucha.</xhtml:th> - <xhtml:th xml:space="preserve">pedes.</xhtml:th> - <xhtml:th xml:space="preserve">vnciæ.</xhtml:th> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">3.</xhtml:td> - <xhtml:td xml:space="preserve">2.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">2.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - <xhtml:td xml:space="preserve">2.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">6.</xhtml:td> - <xhtml:td xml:space="preserve">4.</xhtml:td> - <xhtml:td xml:space="preserve">6.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">1.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - <xhtml:td xml:space="preserve">1.</xhtml:td> - <xhtml:td xml:space="preserve">6.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td xml:space="preserve">1.</xhtml:td> - <xhtml:td xml:space="preserve">6.</xhtml:td> - <xhtml:td xml:space="preserve">8.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td xml:space="preserve">1.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">8.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - <xhtml:td xml:space="preserve">2.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - </xhtml:tr> - </xhtml:table> - </div> - <p> - <s xml:space="preserve">Demum ex ijÅ¿dem .2. vncijs Å¿ecundi lateris cum .3. primi, producentur .12. atto-<lb/>mi, ideÅ¿t vnum punctum. </s> - <s xml:space="preserve">Quæ omnia collecta facient trabucha .8. pedes .3. uncias <lb/>2. & attomi .3. omnes rectanguli oblongi. </s> - <s xml:space="preserve">Pulcherrima profecto operatio.</s> - </p> - <xhtml:table> - <xhtml:tr> - <xhtml:th xml:space="preserve">Trabucha.</xhtml:th> - <xhtml:th xml:space="preserve">pedes.</xhtml:th> - <xhtml:th xml:space="preserve">vnciæ.</xhtml:th> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">3.</xhtml:td> - <xhtml:td xml:space="preserve">2.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">2.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - <xhtml:td xml:space="preserve">2.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">6.</xhtml:td> - <xhtml:td xml:space="preserve">4.</xhtml:td> - <xhtml:td xml:space="preserve">6.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">1.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - <xhtml:td xml:space="preserve">1.</xhtml:td> - <xhtml:td xml:space="preserve">6.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td xml:space="preserve">1.</xhtml:td> - <xhtml:td xml:space="preserve">6.</xhtml:td> - <xhtml:td xml:space="preserve">8.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td xml:space="preserve">1.</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">8.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - <xhtml:td xml:space="preserve">2.</xhtml:td> - <xhtml:td xml:space="preserve">3.</xhtml:td> - </xhtml:tr> - </xhtml:table> - <p> - <s xml:space="preserve">Videamus nunc exercitij cauÅ¿a, vt dixi, quomodo conueniat calculus iÅ¿te cum <lb/>calculo ordinario communi?</s> - </p> - <p> - <s xml:space="preserve">Nam quotieÅ¿cunque dicta latera, fracta fuerint in vncias, primum latus erit <lb/>vnciarum .243. Å¿ecundum autem .182. productum vero vnius in alterum erit vn-<lb/>ciarum quadratarum .44 226. quod quidem productum cum diuiÅ¿um fuerit per <num value="5184">.<lb/>5184.</num> vncias quadratas vnius trabuchi quadrati, prouentus erit .8. trabucho-<lb/>rum, reliquus verò numerus, Å¿iue fractus, erit vnciarum quadratarum .2754. qui <lb/>cum diuiÅ¿us fuerit per numerum .144. vnciarum vnius pedis quadrati, prouenient <lb/>pedes .19. quadrati cum vncijs .18. Å¿uperabundantibus, dicti autem pedes .19. Å¿ignifi-<lb/>cant tres pedes rectangulos oblongos cum vno pede quadrato, hoc eÅ¿t cum duabus <lb/>vncijs rectangulis oblongis, vt Å¿upra.</s> - </p> - <p> - <s xml:space="preserve">Videndum nunc eÅ¿t, vtrum illæ .18. vnciæ æquipolleant tribus punctis rectangu-<lb/>lis oblongis: </s> - <s xml:space="preserve">Å¿ed hoc manifeÅ¿tè videre eÅ¿t, ex hoc, quia quÄ™libet vncia rectangula <lb/>oblonga componitur ex .72. quadratis, punctum autem rectangulum oblongum, <reg norm="cum" type="context">cÅ©</reg> <lb/>Å¿it duodecima pars ipÅ¿ius vnciæ rectangulæ oblongæ, ipÅ¿um componetur ex .6. vn-<lb/>cijs quadratis .18. igitur vncijs quadratis, triplum erit ipÅ¿ius puncti rectanguli dicti. <lb/></s> - <s xml:space="preserve">Vnde clarè patet, quod, quotieÅ¿cunque voluerimus Å¿cire proportionem ipÅ¿arum vn <lb/>ciarum quadratarum Å¿uperabundantium, ad punctum rectangulum oblongum, Å¿i <lb/>dixerimus ex regula de tribus, Å¿i .72. (vncia rectangula oblonga) dat .18. quid <reg norm="dabunt" type="context">dabÅ©t</reg> <lb/>12? </s> - <s xml:space="preserve">puncta rectangula oblonga, quarum vnaquæque eÅ¿t duodecima pars ipÅ¿ius vn-<lb/>ciæ rectangulæ oblongæ, in præſenti autem caÅ¿u prouenient .3. pro quarto termino <lb/>quæſito, & habebimus propÅ¿itum.</s> - </p> - <pb o="409" rhead="EPISTOL AE." n="421" file="0421" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0421"/> - </div> - </div> - <div type="section" level="3" n="45"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">SOLVTIO CVIVSDAM QVÆSITI.</head> - <head style="it" xml:space="preserve">Magnifico Ludouico Fauzzoni amico cariβimo.</head> - <p> - <s xml:space="preserve">TVI quæſiti Å¿olutio quam neÅ¿cio quis te docuit, valde diuerÅ¿a eÅ¿t à vera. <lb/></s> - <s xml:space="preserve">quæſitum enim tale fuit.</s> - </p> - <p> - <s xml:space="preserve">Reperiuntur quatuor Å¿ocij, Ludouicus, Hieronymus, FranciÅ¿cus, & Lau <lb/>rentius quorum primus, Ludouicus Å¿cilicet, poÅ¿uit aureos .6000. Hierony <lb/>mus verò aureos .5000. FranciÅ¿cus autem .2000. & Laurentius .1000. quorum Å¿um-<lb/>ma faciebat aureos .14000. </s> - <s xml:space="preserve">interim tamen de tali Å¿umma Ludouicus recepit aureos <lb/>2000. Hieronymus verò .1000. FranciÅ¿cus autem .900. & Laurentius .800. quapro-<lb/>pter in Å¿umma reÅ¿idua Ludouicus non habebat niÅ¿i aureos .4000. Hieronymus <reg norm="etiam" type="context">etiã</reg> <lb/>4000. FranciÅ¿cus .1100. & Laurentius .200. quorum Å¿umma erat .9300. </s> - <s xml:space="preserve">Nunc au-<lb/>tem iÅ¿ti Å¿ocij cupiunt augere hanc Å¿ummam per aureos .20000. tali tamen conditio-<lb/>ne quod quilibet tantum tribuat vt in totali Å¿umma, tantam partem unus habeat, <lb/>quantam alter.</s> - </p> - <p> - <s xml:space="preserve">Hoc autem problema tam Å¿acile eÅ¿t, & cum Å¿uo theoremate ita coniunctum, quod <lb/>miror amicum noÅ¿trum illud illico non vidiſſe.</s> - </p> - <p> - <s xml:space="preserve">Accipe igitur illos aureos .20000. & eos collige cum Å¿umma .9300. vnde habebis <lb/>aureos .29300. pro <reg norm="summa" type="context">sÅ©ma</reg> totali, cuius quarta pars erit .7325. <reg norm="quam" type="context">quã</reg> <reg norm="vnuÅ¿quisque" type="simple">vnuÅ¿quisq;</reg> poÅ¿tea <lb/>habebit in dicta Å¿umma. </s> - <s xml:space="preserve">Sed ut reperias quantitatem aureorum quam quilibet <lb/>prius debet contribuere, vt poÅ¿tea habeat aureos .7325. in dicta Å¿ocietate. </s> - <s xml:space="preserve">Iubeo, <lb/>vt Ludouicus demat illos aureos .4000. quos demum habebat, ex .7325. reliquum <lb/>autem erit .3325. qui quidem numerus erit aureorum nunc contribuendorum ipÅ¿ius <lb/>Ludouici. </s> - <s xml:space="preserve">Demptis Å¿imiliter aureis .4000. ex dictis .7325. <reg norm="remanebunt" type="context">remanebÅ©t</reg> .3325. pro con <lb/>tributione ipÅ¿ius Hieronymi. </s> - <s xml:space="preserve">Deinde Å¿i ex .7325. extracti fuerint aurei .1100. relin-<lb/>quent .6225. pro contributione FranciÅ¿ci. </s> - <s xml:space="preserve">Demptis demum .200. ex .7325. reÅ¿idui <lb/>erunt .7125. pro contributione Laurentij, & Å¿ic quilibet habebit æqualem portio-<lb/>nem in totaliÅ¿umma.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">Speculatio cuiuÅ¿dam Methodire ductionis numiÅ¿matum <lb/>unius Å¿peciei in aliam.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">MIrum tibi videtur quo pacto verum Å¿it, quod Å¿umma <reg norm="mendietatis" type="context">mẽdietatis</reg> cuiuÅ¿uis <lb/>numeri illorum numiÅ¿matum, quæ hic vocantur Blanci, cum Å¿exta parte eiuÅ¿ <lb/>dem medietatis, Å¿emper Å¿it numerus florenorum huius prouinciæ. </s> - <s xml:space="preserve">Vt exempli gra <lb/>tia, quotieÅ¿cunque reducere voluerimus .48. Blancos in Florenos, Å¿i medietati ip-<lb/>Å¿ius .48. hoc eÅ¿t .24. adiecta fuerit Å¿exta pars ipÅ¿ius medietatis, quæ eÅ¿t .4. </s> - <s xml:space="preserve">tunc habebi <lb/>mus .28. & ita dicemus quod .48. Blanci conÅ¿tituunt Florenos .28. quod quidem <lb/>verum eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">HuiuÅ¿modi autem rei Å¿peculatio ita Å¿e habet. </s> - <s xml:space="preserve">Nam vnuÅ¿quiÅ¿que Blancus diuidi-<lb/>tur in .7. æquales partes, quarum .12. conÅ¿tituunt vnum Florenum, horum verò nu-<lb/>miÅ¿matum communis menÅ¿ura, vocatur Groſſus, vt Å¿cis, ex quo Å¿equitur, quod Å¿i - <pb o="410" rhead="IO. BAPT. BENED." n="422" file="0422" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0422"/> - 28. Floreni æquantur Blancis .48. tot Groſſi erunt in .28. Florenis quot in .48. Blan-<lb/>cis. </s> - <s xml:space="preserve">Fingamus igitur, mente, noÅ¿tram figuram .79. Theorematis Arithmetici <var>.x.u.o.<lb/>e.n.</var> Å¿upponendo ambo producta <var>.u.x.</var> et <var>.n.e.</var> inuicem Ä™qualia exiÅ¿tere, & vnumquod-<lb/>que eſſe groſſorum .336. Å¿it etiam <var>.o.x.</var> vnus Florenus .12. groſſorum <var>.o.n.</var> verò Blan-<lb/>cus .7. eorundem groſſorum <var>.o.e.</var> autem Blancorum .48. </s> - <s xml:space="preserve">Nunc certi erimus ex .15. <lb/>Å¿exti vel .20. Å¿eptimi Euclidis eandem fore proportionem <var>.o.u.</var> ad <var>.o.e.</var> quæ <var>.o.n.</var> ad <var>.o.<lb/>x.</var> Å¿ed <var>.o.n.</var> eÅ¿t Å¿umma medietatis ipÅ¿ius <var>.o.x.</var> cum Å¿exta parte dictæ medietatis, ita igi-<lb/>tur erit <var>.o.u.</var> ipÅ¿ius <var>.o.e.</var> hoc eÅ¿t Å¿umma medietatis <var>.o.e.</var> <reg norm="cum" type="context">cÅ©</reg> Å¿exta parte medietatis eiuÅ¿-<lb/>dem, quæ Å¿umma in præſenti exemplo erit .28.</s> - </p> - <p> - <s xml:space="preserve">Hac enim Å¿peculatione mediante, poteris methodum inuenire conuertendi Flo-<lb/>renos in Blancos. </s> - <s xml:space="preserve">Vt Å¿i nobis propoÅ¿iti fuerint Floreni .28. </s> - <s xml:space="preserve">VoluerimusqÌue inuenire <lb/>quot Blancos faciant, Å¿uppoÅ¿ita menÅ¿ura communi, iam Å¿upradicta. </s> - <s xml:space="preserve">Nam duplica-<lb/>bimus numerum Florenorum, à quo duplo detrahemus Å¿eptimam partem, <reg norm="reliquum" type="context">reliquÅ©</reg> <lb/>verò erit numerus quæſitus.</s> - </p> - <p> - <s xml:space="preserve">HuiuÅ¿modi autem rei ratio eÅ¿t, quia, cum in Å¿upradicta figura, proportio <var>.o.e.</var> ad <lb/><var>o.u.</var> Ä™qualis exiÅ¿tat ei, quæ <var>.o.x.</var> ad <var>.o.n.</var> atque etiam <var>.o.x.</var> Å¿it minor duplo ipÅ¿ius <var>.o.n.</var> <lb/>per Å¿eptimam partem ipÅ¿ius dupli <var>.o.n.</var> minor erit <var>.o.e.</var> duplo ipÅ¿ius <var>.o.u.</var> per <reg norm="Å¿eptimam" type="context">Å¿eptimã</reg> <lb/>partem eiuÅ¿dem dupli ipÅ¿ius <var>.o.u</var>.</s> - </p> - <p> - <s xml:space="preserve">Idem affirmo de quauis conuerÅ¿ione aliorum numiÅ¿matum, quorum Å¿emper <var>.o.x.</var> <lb/>maior Å¿it <var>.o.n.</var> verò minor. </s> - <s xml:space="preserve">Vt Å¿i <var>.o.x.</var> æquiualeret .7: et <var>.o.n.</var> valeret .4. et <var>.o.e.</var> valeret <lb/>42. quæ quidem <var>.o.e.</var> menÅ¿uraretur ab <var>.o.n</var>.</s> - </p> - <p> - <s xml:space="preserve">Si cuperemus Å¿cire quot <var>.o.x.</var> Å¿int in <var>.o.n</var>. </s> - <s xml:space="preserve">Primo dicemus in <var>.o.n.</var> reperiri Å¿ummam <lb/>medietatis Å¿ex Å¿eptimorum ipÅ¿ius <var>.o.x.</var> collectæ cum vna Å¿eptima parte ipÅ¿ius <var>.o.x.</var> Å¿eu <lb/>(vt ita dicam) cum tertia ipÅ¿ius medietatis. </s> - <s xml:space="preserve">Vnde dempta Å¿eptima parte ipÅ¿ius .42. <lb/>quæ eÅ¿t .6. <reg norm="collectaque" type="simple">collectaqÌ</reg> cum medietate reÅ¿idui, quæ eÅ¿t .18. habebimus .24. res, quarum <lb/>vnaquæque æqualis erit ipÅ¿i <var>.o.x</var>.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i quis cupiat reperire <var>.o.e.</var> dato <var>.o.u.</var> duplicet <var>.o.u.</var> à quo demat quartam <reg norm="par- tem" type="context">par-<lb/>tẽ</reg> ipÅ¿ius <var>.o.u.</var> & habebit propoÅ¿itum. </s> - <s xml:space="preserve">Nam ita Å¿e habere oportet <var>.o.e.</var> ad <var>.o.u.</var> quemad <lb/>modum <var>.o.x.</var> ad <var>.o.n</var>.</s> - </p> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">De lucro mercantili.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve"><emph style="sc">QVod</emph> demum Å¿cire à me deÅ¿ideras, eÅ¿t, quod cum vendideris libram vnam <lb/>mercis pro .4. Å¿olidis, & lucratus fueris .2. cum quarta parte vnius pro Å¿ingu-<lb/>lis decem libris, Å¿cire velles quantum lucri facturus eſſes in libris <reg norm="decem" type="context">decẽ</reg> dan-<lb/>do Å¿ingulam libram pro .6. Å¿olidis.</s> - </p> - <p> - <s xml:space="preserve">Nulli dubium eÅ¿t quin decima pars de .2. cum quarta vnius Å¿it lucrum libræ vnius. <lb/></s> - <s xml:space="preserve">Quæ decima pars Å¿unt <reg norm="nouem" type="context">nouẽ</reg> quadrageÅ¿imæ partes, & hæc Å¿ubducta à ſolidis .4. reli-<lb/>qui erunt Å¿olidi .3. cum .31. quadrageÅ¿imis partibus pro Å¿orte vnius libræ. </s> - <s xml:space="preserve">QuÄ™ Å¿ors <lb/>Å¿ubtracta à ſolidis .6. remanebunt Å¿ol .2. cum .9. quadrageÅ¿imis lucri pro libra, quod <lb/>multiplicatum per .10. proueniunt Å¿ol .22. cum quarta parte vnius, & tantum aÅ¿cen-<lb/>deret lucrum, quod fieri poſſet in libris decem Å¿i quamlibet, Å¿ol .3. cum .31. quadra <lb/>geÅ¿imis nobis conÅ¿taret.</s> - </p> - <p> - <s xml:space="preserve">Vel Å¿ic multiplicemus Å¿ortem vnius libræ per .10. productum erit .37. cum tribus - <pb o="411" rhead="EPISTOL AE." n="423" file="0423" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0423"/> - quartis, iterum multiplicemus per .10. Å¿ortem cum lucro vnius librÄ™ quod eÅ¿t .4. pro-<lb/>ductum erit .40. differens à primo Å¿ol .2. cum quarta parte, multiplicemus pariter <lb/>per .10. precium .6. Å¿olidorum proueniens erit .60. à quo deducendo productum Å¿or-<lb/>tis librarum .10. quod erat Å¿ol .37. cum tribus quartis Å¿upererunt Å¿ol .22. cum quar-<lb/>ta parte, vt Å¿upra.</s> - </p> - </div> - </div> - <div type="section" level="3" n="46"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE DIGNITATIBVS PLANETARVM.</head> - <head style="it" xml:space="preserve">Adriano Panetio.</head> - <p> - <s xml:space="preserve"><emph style="sc">QVod</emph> eam diÅ¿tinctionem orbium, quæiam inualuit, nonteneas, Å¿ed putes <lb/>totum eſſe quoddam continuum excipiens corpora Å¿tellarum, nouum <reg norm="non" type="context">nõ</reg> <lb/>eÅ¿t, nam nonnulli Å¿olidæ doctrinæ PhiloÅ¿ophi idem cenÅ¿uerunt. </s> - <s xml:space="preserve">Sed <lb/>quod attinet ad dignitates planetarum in Å¿ignis zodiaci, Å¿cias huiuÅ¿mo-<lb/>di ordinem me comprÄ™hendere eſſe deÅ¿umptum ab ordine antiquo orbium <reg norm="ipÅ¿orum" type="context">ipÅ¿orÅ©</reg> <lb/>planetarum, quiquidem ordo erat, vt Å¿tatim poÅ¿t Lunam Å¿uccederet Sol, poÅ¿t So-<lb/>lem Mercurius, tum Venus deinde Mars, poÅ¿tea Iupiter, & tandem Saturnus per <lb/><reg norm="eoÅ¿demque" type="simple">eoÅ¿demqÌ;</reg> orbes, retro redibant, atque hoc cognoÅ¿citur conÅ¿tituendo Cancrum do <lb/>micilium Lunæ, Leonem, Solis, Virginem, Mercurij, Libram, Veneris, Scorpio-<lb/>nem, Martis, Sagittarium, Iouis, Capricornum, Saturni, Incipientes deinde ab <lb/>Aquario, quiad nos propius accedit <reg norm="eundemque" type="simple">eundemqÌ;</reg> tribuentes Saturno, PiÅ¿ces, Ioui, <lb/>Arietem, Marti, Taurum, Veneri, & Gemellos, Mercurio, Å¿eptem Planetas cum <lb/>duodecim Å¿ignis zodiaci concordes reddebant.</s> - </p> - <p> - <s xml:space="preserve">Quod deinde AriÅ¿toteles in libris de Å¿enÅ¿u & ijs quæ Å¿enÅ¿ibus percipiuntur, dicit <lb/>pupillam oculi eſſe nigram, non ita Å¿e habet, nam idem eÅ¿t, ac Å¿i quis diceret <reg norm="nigrum" type="context">nigrÅ©</reg> <lb/>eſſe illud medium, quod permitteret tranÅ¿itum lumini per Å¿uam diaphaneitatem, nul <lb/>lum lumen à ſeipÅ¿o reflectens, & etiam ac Å¿i quis diceret nigrum eſſe aerem alicuius <lb/>cubiculi vndequaque clauÅ¿i tenebroÅ¿i.</s> - </p> - <p> - <s xml:space="preserve">Quod etiam idem AriÅ¿toteles volens adducere cauÅ¿am, cur oculus magis mate-<lb/>riam aquæ, quam aeris participet, dicensidea ratione fieri, quod aqua magis quam <lb/>aer Å¿eruari poſſit, eodem libro Å¿cribit, eÅ¿t reuera admirandum. </s> - <s xml:space="preserve">Ibi enim clarè de-<lb/>monÅ¿trat Å¿e planè ignorare, & conÅ¿tructionem oculi, & cauÅ¿am diuerÅ¿itatis eorum <lb/>humorum tam in Å¿ubÅ¿tantia, quam in figura, quæ non aliunde dependet quam quod <lb/>diuerÅ¿am refractionem radiorum luminoÅ¿orum producat, qui per pupillam ingre-<lb/>diuntur, vt ad proprios <reg norm="Å¿ibique" type="simple">Å¿ibiqÌ;</reg> deÅ¿tinatos locos dirigantur radij, vt à virtute viÅ¿iua per <lb/>fectius Å¿en tiantur.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">De ratione Frigiditatis locorum umbroÅ¿orum.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">VEra ratio vnde fiat, vt quanto magis Å¿entitur calor in locis expoÅ¿itis Soli, tan-<lb/>to minus Å¿entiatur in vmbra, vbi Solis radius non reflectitur, eÅ¿t quia cum ra <lb/>refactus eÅ¿t aer à vehementi calore radij Å¿olaris, Å¿eipÅ¿um colligit, & condenÅ¿atur in <lb/>locis, à quibus à calore, ratione rarefactionis, non expellitur, & quia naturaliter ca-<lb/>lor Å¿equitur rarum, rarum calorem, & frigidum <reg norm="densum" type="context">densÅ©</reg>, & <reg norm="densum" type="context">densÅ©</reg> frigidum, vt vnicui <lb/>que Å¿anÄ™ mentis patet, hanc ob cauÅ¿am Å¿equitur rem ita Å¿e habere vt diximus. </s> - <s xml:space="preserve">Poſſu <lb/>mus etiam abÅ¿que dubio credere huiuÅ¿modi ratione fieri, vt frigus matutini tempo <lb/>ris, in crepuÅ¿culo maius eſſe eo, quod noctu viguit. </s> - <s xml:space="preserve">Nam materia conÅ¿iÅ¿tens in co-<lb/>no vmbræ terræ, Å¿emper denÅ¿ior eÅ¿t ea, quæ extra reperitur, imo noua materia con <lb/>tinuo condenÅ¿atur, propter motum vmbrÄ™, quæ Å¿emper corpori Å¿olari opponitur. </s> - <s xml:space="preserve">hÄ™c - <pb o="412" rhead="IO. BAPT. BENED." n="424" file="0424" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0424"/> - autem noua condenÅ¿atio dico Å¿emper fit in crepuÅ¿culo matutino, hoc eÅ¿t in parte co <lb/>ni à Sole pulÅ¿a, in parte vero contrari a ipÅ¿ius coni hoc eÅ¿t in parte crepuÅ¿culi ve-<lb/>Å¿pertini, contrarium accidit, quia potius aliquantulum in hac parte materia coni ra <lb/>rificatur, quia extrinÅ¿eca condenÅ¿atur, in parte vero matutina extrinÅ¿eca rarificatur; <lb/></s> - <s xml:space="preserve">& propterea intrinÅ¿eca conde nÅ¿atur.</s> - </p> - </div> - </div> - <div type="section" level="3" n="47"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">QVOD RECTE ARIST. SENSERIT COELVM <lb/>caÅ¿u non eſſe productum.</head> - <head style="it" xml:space="preserve">Hieronymo Condrumerio.</head> - <p> - <s xml:space="preserve">FErunt <reg norm="AriÅ¿tippum" type="context">AriÅ¿tippÅ©</reg> tempeÅ¿tate maris ad incognita littora delatum, cum in are-<lb/>na vidiſſet <reg norm="quaÅ¿dam" type="context">quaÅ¿dã</reg> figuras geometricas delineatas <reg norm="exultantem" type="context">exultantẽ</reg> lÄ™titia dixiſſe: </s> - <s xml:space="preserve">Hæc <lb/>Å¿unt hominum veÅ¿tigia. </s> - <s xml:space="preserve">Nam conÅ¿onum rationi non erat, vt huiuÅ¿modi figuræ ca-<lb/>Å¿u eſſent impreſſæ: </s> - <s xml:space="preserve">neque etiam credendum eÅ¿t ingentem hanc ma chinam tanto or <lb/>dine conÅ¿tantem fortuitò eſſe productam, cum nulla quantumuis minima eiuÅ¿dem <lb/>particula, dummodo nitatur ordine, aliquo modo caÅ¿u effecta fuerit; </s> - <s xml:space="preserve">cum caÅ¿us ni-<lb/>hil producat, quod regulam & ordinem Å¿eruet. </s> - <s xml:space="preserve">Non eÅ¿t igitur producta caÅ¿u admi <lb/>randa correÅ¿pondentia, quæ eÅ¿t obiectorum cum potentijs, luminis cum oculo, Å¿o-<lb/>ni cum auditu, Å¿aporis cum guÅ¿tatu, odoris cum odoratu, qualitatum tangibilium <reg norm="cum" type="context">cÅ©</reg> <lb/>tactu. </s> - <s xml:space="preserve">Si diligenter deinde cuiuÅ¿libet rei naturalis operationem conÅ¿iderabimus, <lb/>eas tanta arte conÅ¿tructas videbimus, vt cogamur fateri aliquam prudentiſſimam, <lb/>& Å¿agaciſſimam mentem eas formaſſe, Å¿i ergo quælibet <reg norm="mundi" type="context">mÅ©di</reg> pars tanta cum ratione <lb/>& ordine eÅ¿t conÅ¿tructa: </s> - <s xml:space="preserve">quomodo fieri poterit, vt de toto ipÅ¿o mundo id in dubium <lb/>vocemus, <reg norm="certiſſimeque" type="simple">certiſſimeqÌ;</reg> non credamus diuiniſſimam aliquam <reg norm="mentem" type="context">mẽtem</reg> eſſe à qua exqui-<lb/>Å¿itiſſima huius vniuerÅ¿i harmonia, quæ ex tot <reg norm="tantisque" type="simple">tantisqÌ;</reg> partibus, maximo ordine ni-<lb/>tentibus conficitur, non dependeat?</s> - </p> - </div> - </div> - <div type="section" level="3" n="48"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">VARIA RESPONSA.</head> - <head style="it" xml:space="preserve">Nicolao Petreio.</head> - <p> - <s xml:space="preserve">AD ea quæ mihi Å¿cribis dico, quod excrementa quæ ex corpore Å¿ano prodeunt <lb/>in Å¿ua <reg norm="ipÅ¿orum" type="context">ipÅ¿orÅ©</reg> qualitate Å¿enÅ¿ibili ita Å¿e <reg norm="habent" type="context">habẽt</reg> ad <reg norm="facultatem" type="context">facultatẽ</reg> illius partis eiuÅ¿dem <lb/>corporis, ut <reg norm="eam" type="context">eã</reg> non lÄ™dant, <reg norm="quemadmodum" type="wordlist">quẽadmodÅ©</reg> efficeret <reg norm="Å¿putum" type="context">Å¿putÅ©</reg>, Å¿i eſſet <reg norm="amarum" type="context">amarÅ©</reg>, aut quod ex <lb/>cernitur naÅ¿o <reg norm="fÄ™tidum" type="context">fÄ™tidÅ©</reg> eſſet. </s> - <s xml:space="preserve">Imagineris igitur <reg norm="quemadmodum" type="wordlist">quẽadmodÅ©</reg> poſſit eſſe <reg norm="verum" type="context">verÅ©</reg> id quod <reg norm="idem" type="context">idẽ</reg> <lb/><reg norm="amicus" type="simple">amicê°</reg> noÅ¿ter ait. </s> - <s xml:space="preserve">PrÄ™terea Å¿i aliquid tibi in <reg norm="oculum" type="context">oculÅ©</reg> inciderit, an neÅ¿cis quomodo Å¿tatim <lb/>affatim affluat humor, vt id foras <reg norm="propellat" type="simple">ê“pellat</reg>, vel abducat? </s> - <s xml:space="preserve">(mirabile opus naturæ.) </s> - <s xml:space="preserve">Dic <lb/><reg norm="etiam" type="context">etiã</reg> eidem non abÅ¿que myÅ¿terio naturam in tot miÅ¿erijs Å¿enectutem poÅ¿uiſſe, cum <lb/>ſæpiſſimè Å¿enex mori deÅ¿ideret, ut huius vitæ calamitatibus liberetur, vnde fit, vt <lb/>cum eius aduentum Å¿entiat, minus affligatur. </s> - <s xml:space="preserve">Dicito etiam eidem, naturam non <lb/>fuiſſe tam Å¿olicitam de quibuÅ¿dam partibus quemadmodum eÅ¿t de toto, vnde ma-<lb/>gis rotunda, & polita poterat eſſe Å¿uperficies terræ, quam nunc eÅ¿t, quia natura ma <lb/>gis reÅ¿picit totum, quam partes, & magis maiores, quam minores.</s> - </p> - <p> - <s xml:space="preserve">Dum tuas legerem, me continere non potui quin riÅ¿erim, id quod Å¿cribis te inter-<lb/>rogaſſe eum PhiloÅ¿ophum naturalem, vnde fit, vt ventus Å¿it frigidus, <reg norm="eumque" type="simple">eumqÌ;</reg> tibi re <lb/>Å¿pondiſſe, quod à remotiſſimis partibus veniat, <reg norm="genereturque" type="simple">genereturqÌ;</reg> à vaporibus terræ frigi-<lb/>dis. </s> - <s xml:space="preserve">( cum ipÅ¿a Å¿it frigida.) </s> - <s xml:space="preserve">Cæterum miror quod ab eo non quæſieris, vnde oriatur <lb/>frigiditas, quæ percipitur ab agitatione aeris, qui quidem à vaporibus terræ non <lb/>proÅ¿ilit, nec à remotiſſimis partibus ad nos accedit. </s> - <s xml:space="preserve">Sed quia de eadem re me in- - <pb o="413" rhead="EPISTOL AE." n="425" file="0425" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0425"/> - rerrogas, Å¿cito <reg norm="naturam" type="context">naturã</reg> coniunxiſſe <reg norm="frigiditatem" type="context">frigiditatẽ</reg> <reg norm="cum" type="context">cÅ©</reg> denÅ¿itate, & <reg norm="caliditatem" type="context">caliditatẽ</reg> <reg norm="cum" type="context">cÅ©</reg> raritate, <lb/>vt Å¿up. diximus, ita vt cum aliquod corpus <reg norm="denÅ¿atur" type="simple">denÅ¿at̃</reg>, <reg norm="frigidum" type="context">frigidÅ©</reg> <reg norm="reddatur" type="simple">reddat̃</reg>, & dum rarefit ma-<lb/>iorem caliditatem acquirat, & Å¿ic econtra fit, vt quanto magis aliquod corpus refri <lb/>geratur, tanto denÅ¿ius reddatur, & quanto calidius fit tanto rarius efficiatur. </s> - <s xml:space="preserve">Quo-<lb/>ties igitur agitabitur aer, aut aliud corpus, quod ratione Å¿uæ Å¿ubtilitatis, velociter <lb/>condenÅ¿ari, & rarefieri poſſit, eius partes denÅ¿iores Å¿emper erunt frigidæ, & hanc <lb/>obrem quilibet ventus, qui per calida loca non tranÅ¿eat, natura Å¿ua frigidus, calidus <lb/>autem per accidens erit. </s> - <s xml:space="preserve">Hinc fit vt vaÅ¿a vitrea, & terrea tam in vehementi frigore, <lb/>quam in magno æſtu frangantur, quia horum vnum fit, ne aliquis locus vacuus rema <lb/>neat, & aliud ob loci neceſſitatem, Å¿ed hoc non Å¿equeretur, Å¿i in materia, qua huiuÅ¿ <lb/>modi vas conÅ¿tat, aliqua aeris portio non contineretur.</s> - </p> - </div> - </div> - <div type="section" level="3" n="49"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE LVMINE LVNÆ, DE FINE LVMINIS, <lb/>de fine motus corporum cÄ™leÅ¿tium, de albedine, <lb/>de Å¿phæra.</head> - <head style="it" xml:space="preserve">Clariβimo Antonio Nauaiero.</head> - <p> - <s xml:space="preserve"><emph style="sc">LVmen</emph> Lunæ etiam Å¿i Å¿it lumen reflexum Solis ab ipÅ¿a Luna, ab ea tamen <lb/>non ita reflectitur, vt à ſuperficie polita Å¿peculi, <reg norm="cum" type="context">cÅ©</reg> eius luminis <reg norm="tantam" type="context">tantã</reg> <reg norm="quan titatem" type="context">quãÂ<lb/>titatem</reg> Å¿uper ipÅ¿um corpus lunare videamus, & eo modo <reg norm="terminatam" type="context">terminatã</reg> quo <lb/>conÅ¿picimus. per Å¿e lumen, cauÅ¿a oculi eÅ¿t effectum, per accidens autem <lb/>puta quod vis. </s> - <s xml:space="preserve">Terra deinde nunquam lunari lumine (<reg norm="quanuis" type="context">quãuis</reg> Å¿olaris reflexio exiÅ¿tat) <lb/>omnino deÅ¿tituta eÅ¿t, dico etiam, neque in ipÅ¿is ecclipÅ¿ibus Å¿olaribus vel lunaribus, <lb/>in Å¿olaribus enim cum Soltot millia vices maior Å¿it Luna, Luna verò minor terra, Å¿e <lb/>quitur, vt terra non omnino priuata remancat lumine Lunæ, in ecclipÅ¿ibus ve-<lb/>rò lunaribus Luna Å¿emper videtur, gratia luminis Å¿olaris, quamuis refracti. </s> - <s xml:space="preserve">Mo-<lb/>tus corporum cÅ“leÅ¿tium fit ratione Å¿itus, & varietatis virtutis Å¿tellæ in diuerÅ¿is locis, <lb/>hæc autem varietas abÅ¿que diuerÅ¿o Å¿itu eiuÅ¿dem Å¿tellæ, nec diuerÅ¿us hic Å¿itus abÅ¿que <lb/>motu fieri poſſet, ita vt motus Å¿tellarum Å¿it ratione diuerÅ¿itatis Å¿ituum ipÅ¿arum, er-<lb/>go motus, & diuerÅ¿itas Å¿ituum, fit, ob diuerÅ¿am influentiam. </s> - <s xml:space="preserve">Quæ autem de albe-<lb/>dine fratri tuo dixeram, erant, quod inter <reg norm="oens" type="context">oẽs</reg> colores albedo, certo quodam modò, <lb/><reg norm="maiorem" type="context">maiorẽ</reg> Å¿imilitudinem habet cum lumine. </s> - <s xml:space="preserve">Primò quia magis coniungitur cum lumi-<lb/>ne. </s> - <s xml:space="preserve">Secundo quia magis afficit Å¿enÅ¿um. </s> - <s xml:space="preserve">Tertiò quia abÅ¿que reÅ¿iÅ¿tentia magis reci-<lb/>pit qualitatem aliorum colorum, quam alij colores. </s> - <s xml:space="preserve">Quartò quia maximus <lb/>eÅ¿t omnium colorum. </s> - <s xml:space="preserve">Quintò quia Å¿implicior eÅ¿t reliquis. </s> - <s xml:space="preserve">Sextò quia diÅ¿gregat vi-<lb/>Å¿um. </s> - <s xml:space="preserve">Septimò quia qualitas quæ in niue alba eſſe videtur, nihil aliud eÅ¿t quam mul-<lb/>titudo quædam luminum reflexorum, & non albedo, Å¿imilis ei, quæ eÅ¿t lactis, aut <lb/>panni, quæ quidem Å¿eptima cauÅ¿a effecit, vt ipÅ¿am albedinem, magis quam alium <lb/>quemuis colorem cum ipÅ¿o lumine compararem, cum nihil Å¿it, quod eſſe Å¿uum <reg norm="tranÅ¿" type="context">trãſ</reg> <lb/>mutans, aut apparenter, aut eſſentialiter, illud ipÅ¿um prius non tranÅ¿mutet in for-<lb/>mam Å¿ibi propin quiorem, vt manifeÅ¿tè patet. </s> - <s xml:space="preserve">EÅ¿t etiam huius rei octaua ratio, <lb/>magni ponderis, quia Å¿cilicet nullus Å¿it color, qui magis reÅ¿iÅ¿tat lumini, aut in quem <lb/>lumen minorem impreſſionem faciat, quam albedo. </s> - <s xml:space="preserve">Vnde Å¿equitur, obiecta alba, <lb/>minus eſſe combuÅ¿tibilia quam alia, cum quælibetres in Å¿uum contrarium quam in - <pb o="414" rhead="IO. BAPT. BENED." n="426" file="0426" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0426"/> - Å¿ibi Å¿imile valentius agat, vtrectè vidit AriÅ¿toteles cum dixit, omne contrarium @ <lb/>Å¿uo contrario patinatum eÅ¿t.</s> - </p> - <p> - <s xml:space="preserve">Inter corpora, multum Å¿implicitatisretinet Å¿phæra.</s> - </p> - <p> - <s xml:space="preserve">Circa quod, præter rationes adductas ab AriÅ¿totele in libris de CÅ“lo, poſſumus <lb/>etiam ratiocinarià facilitate motus <reg norm="vndique" type="simple">vndiqÌ;</reg> ab eo quod violentiæ non reÅ¿iÅ¿tar, ab eo, <lb/>quod apta <reg norm="nataque" type="simple">nataqÌ;</reg> Å¿it quieÅ¿cere Å¿upra quoduis punctum Å¿uÄ™ Å¿uperficiei, ab eo quod ab <lb/>aliqua Å¿uperficie alterius corporis Å¿eÅ¿e tangi non permittat, quæ curuitate concaua <lb/>non adæquetur, niÅ¿i medio vnius puncti. </s> - <s xml:space="preserve">Verum eÅ¿t, quod licet hæc vltima ratio <reg norm="non" type="context">nõ</reg> <lb/>Å¿it propria Å¿phæræ, eÅ¿t tamen cauÅ¿a Å¿implicitatis in eo, in quo reperitur, Å¿ed propriÄ™ <lb/>paſſiones Å¿phæræ Å¿unt Å¿upradictæ, præter quam quod alia eiuÅ¿dem Å¿phæræ eÅ¿t pro-<lb/>prijſſima, quæ eÅ¿t diÅ¿tantia eiustermini ab vno tantummodo puncto ideÅ¿t ab <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> <lb/>centro, & etiam poſſe diuidere corpus aliquod medium, cum æquali reÅ¿iÅ¿tentia circa <lb/>punctum, quod prius in motu reperitur.</s> - </p> - <p> - <s xml:space="preserve">Aequalitas autem rerum, eÅ¿t etiam valde Å¿imilis Å¿implicitati, & vnitati.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">Comparatio uiÅ¿us, & auditus.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">QVodad viÅ¿um & auditum attinet, magis neceſſarium eſſe viÅ¿um, & nobilio-<lb/>rem quam auditum exiÅ¿timo, primò quia Å¿i quis viÅ¿u orbatus eſſet, contra <lb/>frigus, & calorem, contra famen, & Å¿itim nil prouidere poſſet, neque aliud quic-<lb/>quam hoc vocabulum prouidere Å¿ignificat, neque abÅ¿que periculo vitæ ab vno loco <lb/>ad alium ferri poſſet, neque aliquid arte facere.</s> - </p> - <p> - <s xml:space="preserve">Sed Å¿i quis deÅ¿titutus eſſet facultate audiendi, Å¿upradictas tamen operationes <reg norm="prae- Å¿tare" type="simple">prÄ™-<lb/>Å¿tare</reg> poſſet, neque modo careret, quo animi Å¿ui Å¿enÅ¿a abÅ¿que beneficio Å¿oni, Å¿ed <lb/>ope figurarum & characterum alteri aperiret: </s> - <s xml:space="preserve">neque etiam munere Å¿peculandi Å¿cien <lb/>tias (excepta muÅ¿ica) deÅ¿titueretur. </s> - <s xml:space="preserve">Ad Å¿cientiam comparandam, longè magis ne <lb/>ceſſarius eÅ¿t viÅ¿us, quam auditus præterquam, quod viÅ¿us maiorem numerum obie-<lb/>ctorum, & differentiarum rerum percipit, & inter reliquos Å¿enÅ¿us velociſſimè imò <lb/>in inÅ¿tanti operatur, magis remotè quam alij, & exactius Å¿entit, <reg norm="minusque" type="simple">minusqÌ;</reg> quam reli-<lb/>qui afficitur, præterquam quod Å¿emperagit, dummodò non dormiat animal. </s> - <s xml:space="preserve">Præ-<lb/>terea Å¿eÅ¿e magis patefacit, & prodit anima per oculos, quam per aliud, cuiuslibet <lb/>Å¿enÅ¿us, inÅ¿trumentum. </s> - <s xml:space="preserve">Oculo magis quam alia corporis parte, hominis natura co-<lb/>gnoÅ¿citur: </s> - <s xml:space="preserve">& Å¿i aliquid Å¿peculari volumus, quod Å¿ine imaginatiua fieri non poteÅ¿t, <lb/>Å¿tatim imaginamur nos videre huiuÅ¿modirem, ac Å¿i oculo fuiſſet compræhenÅ¿a, & <lb/>ab imagine quæ eÅ¿t vnum ex obiectis oculi, imaginatiua nuncupatur. </s> - <s xml:space="preserve">Beneficio <lb/>oculorum omnes ferè Å¿cientiæ Å¿unt adinuentæ. </s> - <s xml:space="preserve">Auditus nil aliud quam Å¿onum ca-<lb/>pit, auditus nunquam detulit intellectui figuram, corpus Å¿uperficiem, aut lineam, <lb/>materiam, formam, locum, dimenÅ¿ionem, plenum inane, nec innumera alia acci-<lb/>dentia, quæ ab oculo compræhenduntur. </s> - <s xml:space="preserve">Quæ verò viÅ¿ui, & auditui Å¿unt commu-<lb/>nia, Å¿unt etiam tactui communia, vt numerus, motus, maius, & minus, Å¿unt tamen ali <lb/>qua oculo & tactui communia, quæ auditus non poteÅ¿t capere, vt durum, molle, acu <lb/>tum, obtuÅ¿um, aÅ¿perum, lene, planum, curuum, concauum, conuexum, magnum, <lb/>paruum, & Å¿upradicta, ideÅ¿t figura corpus & cÄ™tera, vt ctiam rectum, obliquum, & <lb/>Å¿imilia.</s> - </p> - <pb o="415" rhead="EPISTOL AE." n="427" file="0427" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0427"/> - <p> - <s xml:space="preserve">AriÅ¿toteles circa finem primi capitis libri de Å¿enÅ¿u ait mediante viÅ¿u, magis <reg norm="quam" type="context">quã</reg> <lb/>quolibet alio Å¿enÅ¿u, nos percipere Å¿enÅ¿ibilia communia. </s> - <s xml:space="preserve">Vbi eundem per Å¿e, & <lb/>non per accidens magis neceſſarium eſſe quam auditum, tam in ijs quæ ad victum, <lb/>quam in ijs quæ ad Å¿cientiam pertinent eſſe aſſerit, quia auditus intellectui confert <lb/>per accidens. </s> - <s xml:space="preserve">Vide etiam quod idem Å¿cribit primo metaphyÅ¿icorum. </s> - <s xml:space="preserve">Et Å¿i ad ali-<lb/>quid perfectè cognoÅ¿cendum, oculo Å¿eÅ¿e nobis offerrent ea omnia obiecta, quorum <lb/>Å¿pecies in imaginatiua formamus, ipÅ¿a imaginatiua non egeremus. </s> - <s xml:space="preserve">Sed quia hoc <lb/>fieri non poteÅ¿t, hunc <reg norm="theÅ¿aurum" type="context">theÅ¿aurÅ©</reg> imaginatiuè, Å¿eu memoriæ ad conÅ¿eruandam imagi <lb/>nem omnium obiectorum Å¿enÅ¿ibilium nobis dedit natura, vt ope diÅ¿curÅ¿us <reg norm="intellectus" type="simple">intellectê°</reg> <lb/>circa dictas imagines, rerum veritatem venari poſſimus. </s> - <s xml:space="preserve">Sed vt ad propoÅ¿itum re-<lb/>deamus, beneficio oculi animal liberum eÅ¿t, cum Å¿ine ipÅ¿o locum mutare nequeat, <lb/>vt Å¿it tutum. tenebræ, <reg norm="priuatioque" type="simple">priuatioqÌ;</reg> viÅ¿us Å¿unt ferè vnum, & idem. </s> - <s xml:space="preserve">Neque vllus eÅ¿t@Å¿en-<lb/>Å¿us, qui Å¿it magis Å¿imilis intellectui quam viÅ¿us: </s> - <s xml:space="preserve">neque alij Å¿enÅ¿us habent obiecta vi-<lb/>ciſſim communia, quæ non Å¿int etiam oculo communia, Å¿ed inter oculum, & quem <lb/>libet alium ex Å¿enÅ¿ibus, inuenientur quidem obiecta communia, quæ cum alijs non <lb/>communicabunt, vt inter oculum & tactum, figura, acutum, obtuÅ¿um, & Å¿imilia, <lb/>quæ alijs Å¿enÅ¿ibus non percipiuntur. </s> - <s xml:space="preserve">Mediante viÅ¿u, & auditu etiam, <reg norm="compræhendum" type="context">compræhendÅ©</reg> <lb/>tur variæ diÅ¿tantiæ, <reg norm="Å¿itusque" type="simple">Å¿itusqÌ;</reg> obiectorum, nec non proportiones, & alia quÄ™ ab alijs Å¿en-<lb/>Å¿ibus non compræhenduntur. </s> - <s xml:space="preserve">Multa obiecta deinde Å¿unt Å¿ubiecta guÅ¿tatui, quÄ™ alijs <lb/>accidentibus prÄ™dita Å¿unt, vnde cum fuerint Å¿emel deguÅ¿tata, talia, qualia Å¿unt ab o-<lb/>culo percipiuntur, quod nullus ex alijs Å¿enÅ¿ibus præſtabit. </s> - <s xml:space="preserve">Idem de obiectis odora-<lb/>tus dico. </s> - <s xml:space="preserve">SenÅ¿uum nullus eÅ¿t qui maiorem Å¿imilitudinem gerat cum vigilia & cum <lb/>vita, quam viÅ¿us, neque aliquid eÅ¿t, quod magis repræſentet imaginem Å¿omni, & <lb/>mortis, quà m cÄ™citas.</s> - </p> - <p> - <s xml:space="preserve">Qui Å¿ibi oculos eruit vt melius Å¿pecularetur maxima Å¿tultitia prius obcÄ™catus fuit <lb/>quia Å¿oni magis impediunt Å¿peculationem quà m lumina, imò qui commodè vult <lb/>contemplari, quantum plus poteÅ¿t nititur longius eſſe ab omni Å¿trepitu, magis quà m <lb/>à locis luminoÅ¿is, & animal magis lætatur lumine quam Å¿ono: </s> - <s xml:space="preserve">& ad Å¿peculationem <lb/>nos magis inuitat harmonia luminum variorum colorum & figurarum, quà m har-<lb/>monia Å¿onorum, præterquam quod inÅ¿trumentum viÅ¿us totius corporis partium eÅ¿t <lb/>pulcherrima, & in eminentiori loco locata, Å¿i de inÅ¿trumentis Å¿enÅ¿uum loquamur, & <lb/>veluti fineÅ¿træ animæ. </s> - <s xml:space="preserve">Et Å¿i AriÅ¿toteles dicat oculos & aures in vno <reg norm="eodemque" type="simple">eodemqÌ;</reg> orbe <lb/>exiÅ¿tere, volens inferre quod in eodem æquilibrio Å¿int æqualiter alta non ita Å¿e ha-<lb/>bet, quia (Å¿i de homine loquamur) oculus eÅ¿t altior aure. </s> - <s xml:space="preserve">Beneficio huius Å¿enÅ¿us, eo <lb/>rum quæ abÅ¿unt, & longo iam tempore Å¿unt mortui, animi Å¿enÅ¿a, & conceptus intel-<lb/>ligimus, neque alia ratione rerum omnium memoria Å¿eruari poteÅ¿t. </s> - <s xml:space="preserve">Si cabala un-<lb/>quam vera fuit, nulla alia ratione eÅ¿t deleta, quam quia alicuius Å¿igni viÅ¿ibilis medio <lb/>conÅ¿eruata non fuerit, & quæcunque non Å¿cribuntur, ideÅ¿t oculo non <reg norm="commendantur" type="context">cõmendantur</reg> <lb/>parum durant cito obliuioni <reg norm="traduntur" type="simple">tradunt̃</reg>. </s> - <s xml:space="preserve">In maiori Å¿emper pretio fuit pictura <reg norm="quam" type="context">quã</reg> muÅ¿i-<lb/>ca: </s> - <s xml:space="preserve">obiectis viÅ¿ibilibus magis quam ijs quæ Å¿ub auditu cadunt, affectus animi, <reg norm="atque" type="simple">atq;</reg> <lb/>alia quælibet res naturalis exprimi poſſunt. </s> - <s xml:space="preserve">Aegyptij volentes Å¿ignificare Deum, <lb/>oculi medio id præſtabant.</s> - </p> - <p> - <s xml:space="preserve">Oculus, reÅ¿pectu aliorum inÅ¿trumentorum Å¿enÅ¿uum, eÅ¿t quaÅ¿i epicyclus animæ, <lb/>neque defuit qui crederet oculum eſſe principem animi partem.</s> - </p> - <p> - <s xml:space="preserve">Oculus à Sole, & à Luna ita dependet, vt qui tempore defectus cuiuslibet lumi-<lb/>naris naÅ¿citur, Å¿tatim cæcus euadat, neque aliqua eÅ¿t corporis pars in qua magis ap- - <pb o="416" rhead="IO. BABPT. BENED." n="428" file="0428" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0428"/> - pareat differentia vitæ à morte; </s> - <s xml:space="preserve">quam in oculo. </s> - <s xml:space="preserve">AriÅ¿toteles ad finem cap .15. lib. pri <lb/>mi poÅ¿teriorum ait, clarum eſſe quod Å¿i aliquis Å¿enÅ¿us deficiat, futurum vt aliqua <lb/>quoque Å¿cientia deÅ¿it. </s> - <s xml:space="preserve">ConÅ¿idera, quot Å¿cientijs careret homo, Å¿i viÅ¿u orbaretur.</s> - </p> - <p> - <s xml:space="preserve">Et in tertio de anima ait, eum qui non Å¿entit, nihil intelligere poſſe; </s> - <s xml:space="preserve">id quod in-<lb/>de confirmat, quia nihil Å¿it in intellectu, quod prius non fuerit in Å¿enÅ¿u. </s> - <s xml:space="preserve">Plato in ti <lb/>meo ait, oculos nobis attuliſſe rerum optimarum notitiam, & Å¿i oculus non fuiſſet ni <lb/>hil eorum, quæ ad cÅ“lum Å¿pectant inueniri potuiſſe, & <reg norm="cognitionem" type="context">cognitionẽ</reg> diei ac noctis ab <lb/>oculis ortum duxiſſe, vt reuolutiones menÅ¿ium, & annorum metiri, & tempus co-<lb/>gnoÅ¿cere, & inueÅ¿tigare ordinem naturæ vniuerÅ¿alis poſſemus; </s> - <s xml:space="preserve">quibus <reg norm="philoÅ¿ophiam" type="context">philoÅ¿ophiã</reg> <lb/>nobis comparauimus, ut alia multa omittam, quæ ibi à Platone dicuntur. </s> - <s xml:space="preserve">Addam <lb/>hic & aliam Å¿pecialem differentiam inter auditum & viÅ¿um, quæ eÅ¿t, vt obiectum vi <lb/>Å¿us Å¿it permanens, & obiectum auditus tranÅ¿itorium Å¿iue Å¿ucceſſiuum aut, vt alio mo <lb/>do idem dicamus, obiectum viÅ¿us particpes Å¿it æternitatis, illud autem quod eÅ¿t au-<lb/>ditus non item, nam auditus tempori Å¿ubiectus eÅ¿t, viÅ¿us autem minimè. </s> - <s xml:space="preserve">Vel Å¿i di-<lb/>camus operationem auditus abÅ¿que tempore fieri non poſſe cum Å¿it motio, operatio <lb/>verò viÅ¿us, nullo indiget tempore, cum ip Å¿a Å¿it momentanea, & propterea inÅ¿tan-<lb/>tanea. </s> - <s xml:space="preserve">Nam momentum non eÅ¿t motus, nec inÅ¿tans tempus.</s> - </p> - </div> - </div> - <div type="section" level="3" n="50"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">QVARE HYEME VIDEATVR HALITVS <lb/>animalium non autem æſtate, & de vento.</head> - <head style="it" xml:space="preserve">Pancratio Mellano.</head> - <p> - <s xml:space="preserve"><emph style="sc">VNde</emph> fiat vt hyeme halitum noÅ¿trum videamus, & non æſtate, ratio eÅ¿t ab <lb/>eiuÅ¿dem halitus congelatione, quæ ab extrinÅ¿eco frigore fit. </s> - <s xml:space="preserve">Prius enim <lb/>Å¿cire debes aerem <reg norm="attractum" type="context">attractÅ©</reg> in pulmone, foras deinde erumpere cum alio <lb/>vapore aliquantulum craſſiore humido, & excrementitio expulÅ¿o à natu-<lb/>ra, quæ continuò noÅ¿trum corpus euaporare facit, vnde Å¿equitur dum aer foras à pul <lb/>mone pellitur, maiorem Å¿emper materiæ portionem, ea quæ intus attracta eÅ¿t exire: <lb/></s> - <s xml:space="preserve">vnde Å¿tatim vt dicta materia foras expulÅ¿a, frigidum aerem offendit, cum conÅ¿tet ex <lb/>partibus craſſis, & obnoxiis congelationi, condenÅ¿atur in formam vaporis, ad dif-<lb/>ferentiam aeris ambientis qui in Å¿e eas partes craſſas non habet, à quibus <reg norm="quidem" type="context">quidẽ</reg> par-<lb/>tibus condenÅ¿atis, & redditis opacis reflectitur lumen, atque hanc ob cauÅ¿am æſtate <lb/>hoc non fit, quia calor vim condenÅ¿andi non habet.</s> - </p> - <p> - <s xml:space="preserve">Ventus nihil aliud eÅ¿t quam quidam aeris motus, cum condenÅ¿atur, ob defectum <lb/>caloris, neque (pace AriÅ¿totelis dicam) eÅ¿t exhalatio Å¿icca. </s> - <s xml:space="preserve">Exemplum à Vitruuio <lb/>allatum nil planè valet, quantum Å¿pectatad venti naturam, cuius rationem à mere-<lb/>quiris. </s> - <s xml:space="preserve">Exemplum etiam ventilabri quo tempore æſtate vtimur negligendum pe-<lb/>nitus non eÅ¿t, quia eius beneficio, non Å¿olum arcemus à nobis aerem ambientem <lb/>calidum, Å¿ed alium etiam aerem circa nos condenÅ¿amus: </s> - <s xml:space="preserve">& quia ordo naturæ eÅ¿t hu <lb/>iuÅ¿modi quod quemadmodum calor Å¿equitur raritatem <reg norm="corporum" type="context">corporÅ©</reg>, Å¿ic etiam frigus <lb/>eorundem denÅ¿itatem Å¿equatur. </s> - <s xml:space="preserve">Quod Å¿i vis vt exemplo illuÅ¿trem, diligenter ob-<lb/>Å¿eruato tempore æſtatis cum aliqua nubes nobis Solem adimit, vbiaer qui in eius - <pb o="417" rhead="EPISTOL AE." n="429" file="0429" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0429"/> - vmbra reperitur, tantum quantum defectus caloris radij Å¿olaris fert, qui per vim, <lb/>dictum aerem rarefactum conÅ¿eruabat, Å¿tatim dictum aerem condenÅ¿ari cognoÅ¿ces: <lb/></s> - <s xml:space="preserve">& quia ea condenÅ¿atio homogenea non eÅ¿t, ob diuerÅ¿as rationes, hanc ob cauÅ¿am <lb/>percipimus eam aeris impulÅ¿ionem, & inæqualiter, dum verò eadem vmbra diÅ¿ce-<lb/>dit, ventus, ferè, Å¿tatim ceſſat, & ſæpe ante quam dicta vmbra diÅ¿cedat; </s> - <s xml:space="preserve">cuius rei cau <lb/>Å¿a eÅ¿t longa mora quam trahi vmbra, ita vt prius abÅ¿oluatur reditus aeris ad <reg norm="formam" type="context">formã</reg>, <lb/>quæ ei conuenit in huiuÅ¿modi vmbra, quam faciet nubes dum Sol deregitur.</s> - </p> - <p> - <s xml:space="preserve">Vera non Å¿unt ea, quæ tibi Arnoldus dixit, vt mihi tuis literis Å¿ignificaÅ¿ti. </s> - <s xml:space="preserve">Nam ego <lb/>ita dixi, videlicet, quod quoti eÅ¿cunque aliquis aÅ¿pexerit aliquod punctum in Å¿uper-<lb/>ficie Å¿peculi, </s> - <s xml:space="preserve">tunc imaginem ipÅ¿ius poÅ¿t dictam Å¿uperficie<unsure/>m videbit duplicatam, Å¿i <lb/>verò aÅ¿pexerit imaginem intra Å¿peculum, </s> - <s xml:space="preserve">tunc illud punctum videbit duplicatum, <lb/>huiuÅ¿modi autem rei ratio pendet ab hijs quÄ™ ad FranciÅ¿cum Vimercatum Å¿cri-<lb/>pÅ¿i, quæ Å¿i memoria tenes, nullum tibi dubium remanebit. </s> - <s xml:space="preserve">Nam ea tibi omnia <lb/>oÅ¿tendi.</s> - </p> - <p> - <s xml:space="preserve">Dum verò dicis omnem proportionem rationalem diuidi poſſe duobus numeris <lb/>mediantibus in tres æquas partes, mihi ad memoriam reuocas id quod quidam Vitru <lb/>uij commentator aſſerit Å¿uper primum cap. noni lib. eiuÅ¿dem Authoris, ita dicens.</s> - </p> - <quote> - <s xml:space="preserve">Benè eſſe poteÅ¿t vt diagonalis (quadrati Å¿cilicet) numerorum via reperiatur, Å¿ed <lb/>fortaſſe intercedent fracta.</s> - </quote> - <p> - <s xml:space="preserve">Miror te non memoria tenere quid Å¿int numeri rationales quidúe Å¿urdi, <reg norm="neque" type="simple">neq;</reg> con <lb/>Å¿ideras, non Å¿olum non eſſe diuiÅ¿ibilem in tres æquas partes omnem proportionem <lb/>rationabilem, Å¿ed neque in duas, vt Å¿unt Å¿uperparticulares proportiones, necnon <lb/>aliæ innumeræ, Å¿ed cum talia Å¿cribis te nimis parum verÅ¿atum in iÅ¿tis rebus oÅ¿tendis.</s> - </p> - <p> - <s xml:space="preserve">Id verò quod tibi dicere volebam nudiustertius de Mercurio erat, quod nullo pa <lb/>cto confidendum eÅ¿t calculis qui fiunt de curÅ¿u Mercurij, eo quod eius Å¿itus nullo mo <lb/>do obÅ¿eruabilis eÅ¿t, nam ipÅ¿e nunquam nec vbiuis locorum orbis terrarum viÅ¿ibilis <lb/>eÅ¿t altior .18. gradibus Å¿upra orizontem, Å¿ed neque confidendum eſſet Å¿i <reg norm="etiam" type="context">etiã</reg> ipÅ¿um <lb/>videremus altum .20. gradibus, </s> - <s xml:space="preserve">propterea quod magna refractio <reg norm="radiorum" type="context">radiorÅ©</reg> infra hos <lb/>gradus nos valde fallit, quæ quidem refractio, nec <reg norm="vbique" type="simple">vbiq;</reg>, nec omni tempore vnifor-<lb/>mis eÅ¿t, propter diformem Å¿eu inæqualem craſſiciem vaporum quæ continuò muta <lb/>tur. </s> - <s xml:space="preserve">Imo multoties eum. videre putabimus Å¿upra orizontem, exiÅ¿tente ipÅ¿o Å¿ub <lb/>orizonte.</s> - </p> - </div> - <div type="letter" level="4" n="2"> - <head style="it" xml:space="preserve">Quod Ouidius tr anÅ¿currit à motu diurno, ad motum annuum <lb/>prater rem.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">TVus etiam Ouidius ceÅ¿pitauit, cum pro itinere vnius diei efficiat, vt Phaeton à <lb/>patre edoctus Å¿it etiam de itinere annuali.</s> - </p> - <p> - <s xml:space="preserve">Nam, quod Phaeton petat pro curÅ¿u vnius diei, clarè patet <reg norm="ex" type="conjecture">ẽx</reg> diuerÅ¿is locis, & pri-<lb/>mò vbi ita Å¿cribit Ouidius.</s> - </p> - <quote> - <s xml:space="preserve">Currus petit ille paternos. <lb/></s> - <s xml:space="preserve"><reg norm="Inque" type="simple">Inq;</reg> diem alipedum ius & moderamen equorum.</s> - </quote> - <p> - <s xml:space="preserve">Deinde vbi Pater ita loquitur.</s> - </p> - <quote> - <s xml:space="preserve">Ardua prima via eÅ¿t, & qua vix <reg norm="manem" type="context">manẽ</reg><unsure/> recentes. </s> - <pb o="418" rhead="IO. BAPT. BENED." n="430" file="0430" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0430"/> - <s xml:space="preserve">Enituntur equi medio eÅ¿t altiſſima cÄ™lo. <lb/></s> - <s xml:space="preserve">Vnde mare, & terras ipÅ¿i mihi ſæpe videre. <lb/></s> - <s xml:space="preserve">Fit timor & pauida trepidat formidine pectus. <lb/></s> - <s xml:space="preserve">Vltima prona via eÅ¿t & eget moderamine certo.</s> - </quote> - <p> - <s xml:space="preserve">Etiam vbi dicit.</s> - </p> - <quote> - <s xml:space="preserve"><reg norm="Dumque" type="simple">DumqÌ;</reg> ea magnanimus <reg norm="Phaenton" type="context">Phaẽton</reg> miratur, <reg norm="opusque" type="simple">opusqÌ;</reg> <lb/>PerÅ¿picit, ecce vigil nitido patefecit ab ortu. <lb/></s> - <s xml:space="preserve">Purpureas aurora fores, & plena roÅ¿arum. <lb/></s> - <s xml:space="preserve">Atria, diffugiunt Å¿tellæ, quarum agmina cogit. <lb/></s> - <s xml:space="preserve">Lucifer, & coeli Å¿tatione nouiſſimus exit.</s> - </quote> - <p> - <s xml:space="preserve">Necnon vbi ita inquit.</s> - </p> - <quote> - <s xml:space="preserve">Et Å¿i (modo credimus) vnum <lb/>Iſſe diem Å¿ine Sole ferunt, incendia lumen Præbebant.</s> - </quote> - <p> - <s xml:space="preserve">Quod autem à Patre inÅ¿truatur etiam de curÅ¿u annuali, <lb/>videbitur vbi ita dicit.</s> - </p> - <quote> - <s xml:space="preserve">Nitor in aduerÅ¿um, nec me, qui cætera vincit. <lb/></s> - <s xml:space="preserve">Impetus, & rapido contrarius euehor orbi.</s> - </quote> - <p> - <s xml:space="preserve">Et vbi ita loquitur.</s> - </p> - <quote> - <s xml:space="preserve">ForÅ¿itan & lucos illic, <reg norm="vrbesque" type="simple">vrbesqÌ;</reg> Deorum. <lb/></s> - <s xml:space="preserve">Concipias animo <reg norm="delubraque" type="simple">delubraqÌ;</reg> ditia donis <lb/>Eſſe per inÅ¿idias iter eÅ¿t, <reg norm="formasque" type="simple">formasqÌ;</reg> ferarum. <lb/></s> - <s xml:space="preserve"><reg norm="Vtque" type="simple">VtqÌ;</reg> viam teneas, <reg norm="nulloque" type="simple">nulloqÌ;</reg> errore traharis. <lb/></s> - <s xml:space="preserve">Per tamen aduerÅ¿i, gradieris cornua Tauri. <lb/><reg norm="Aemoniosque" type="simple">AemoniosqÌ;</reg> arcus, <reg norm="violentique" type="simple">violentiqÌ;</reg> ora Leonis. <lb/><reg norm="Sæuaque" type="simple">SæuaqÌ;</reg> circuitu curuantem brachia longo. <lb/>Scorpion atque aliter curuantem brachia cancrum. <lb/></s> - <s xml:space="preserve">Nec tibi quadrupedes animoÅ¿os ignibus illis. <lb/></s> - <s xml:space="preserve">Quos in pectore habent quos ore & naribus efflant, & c.</s> - </quote> - <p> - <s xml:space="preserve">Sed lucidius etiam hoc videre eÅ¿t inferius vbi ita loquitur.</s> - </p> - <quote> - <s xml:space="preserve">Nec tibi directos placeat via quinque per arcus. <lb/></s> - <s xml:space="preserve">Sectus in obliquum eÅ¿t lato curuamine limes. <lb/></s> - <s xml:space="preserve"><reg norm="Zonarumque" type="simple">ZonarumqÌ;</reg> trium contentus fine, <reg norm="polumque" type="simple">polumqÌ;</reg> <lb/>Effugit auÅ¿tralem <reg norm="iunctamque" type="simple">iunctamqÌ;</reg> aquilonibus arcton. <lb/></s> - <s xml:space="preserve">Hac Å¿it iter, manifeÅ¿ta rotæ veÅ¿tigia cernes.</s> - </quote> - <p> - <s xml:space="preserve">Et vbi etiam dicit.</s> - </p> - <quote> - <s xml:space="preserve">Neute dexterior tortum declinet ad anguem. <lb/></s> - <s xml:space="preserve">Ne ve Å¿iniÅ¿terior preſſam rota ducat ad aram.</s> - </quote> - </div> - <div type="letter" level="4" n="3"> - <head style="it" xml:space="preserve">De Å¿upputatione quinque corporum regularium. <lb/>De aliquibus etiam eorum Å¿ympathijs.</head> - <head xml:space="preserve">AD EVNDEM.</head> - <p> - <s xml:space="preserve">ID quod à me deÅ¿ideras, ab alijs etiam factum eÅ¿t, Å¿ed ne me putes laborem euita <lb/>re, non præter mittam aliquid tibi Å¿cribere, earum rerum quæ ab Euclide colle - <pb o="419" rhead="EPISTOLAE." n="431" file="0431" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0431"/> - gi, methodo etiam qua vtebar dum in iÅ¿tisrebus me aliquo modo exercebam.</s> - </p> - <p> - <s xml:space="preserve">QuotieÅ¿cunque igitur Å¿cire volueris quantitatem corpulentiæ <reg norm="cuiuÅ¿que" type="simple">cuiuÅ¿q;</reg> <reg norm="quinque" type="simple">quinq;</reg> cor-<lb/>porum regularium ab vna <reg norm="eademque" type="simple">eademqÌ;</reg> Å¿phæra terminatorum Å¿eu <reg norm="circunÅ¿criptibilium" type="context">circunÅ¿criptibiliÅ©</reg> cu-<lb/>rabis primum, cognoÅ¿cere quantitatem lateris <reg norm="cuiusque" type="simple">cuiusqÌ;</reg> eorum, talium partium, qua-<lb/>lium Å¿emidiameter dictæ Å¿phæræ Å¿it .100000. extabulis Å¿inuum Nicolai Copernici. <lb/></s> - <s xml:space="preserve">Propone igitur tibiante oculos figuram Å¿emicircularem vltimæ propoÅ¿itionis .13. <lb/>lib. Eucli. & inuenies <var>.c.d.</var> tertiam partem Å¿emidiametri <var>.d.b.</var> eſſe partium .33333. æ-<lb/>qualem Å¿inui arcus <var>.f.e.</var> graduum .19. mi .28. qui quidem arcus <reg norm="demptus" type="context">dẽptus</reg> <reg norm="cum" type="context">cÅ©</reg> fuerit à tota <lb/>quarta <var>.b.f.</var> remanebitarcus <var>.e.b.</var> gra .70. mi .32. cuius corda erit latus exaedri, quod<unsure/> <lb/>latus ita cognoÅ¿ces, Å¿umendo Å¿cilicet Å¿inum medietatis <var>.b.e.</var> hoc eÅ¿t Å¿inum gra .35. mi <num value="16">.<lb/>16.</num> qui erit partium .57738. cuius duplum erit partium .115476. pro latere cubi.</s> - </p> - <p> - <s xml:space="preserve">Dempto poÅ¿tea quadrato lateris exaedri, & quadrato totius diametri <var>.a.b.</var> reÅ¿i-<lb/>dui radix quadrata, erit <var>.a.e.</var> latus Tetraedri. </s> - <s xml:space="preserve">Vel Å¿i duplicaueris Å¿inum dimidij ar-<lb/>cus <var>.a.e.</var> qui quidem arcus, componitur ex quarta <var>.a.f.</var> & ex arcu <var>.f.e.</var> iam inuento, Å¿iue, <lb/>vt reÅ¿iduus totius dimidij circuli, dempto <var>.b.e.</var> iam Å¿upra inuento, habebimus idem <lb/>latus <var>.a.e.</var> partium .163294.</s> - </p> - <p> - <s xml:space="preserve">Pro latere verò Octaedri accipere potes radicem quadratam dupli quadrati ip-<lb/>Å¿ius <var>.d.b.</var> & habebis <var>.f.b.</var> latus quæſitum. </s> - <s xml:space="preserve">Vel Å¿i malis accipe duplum Å¿inus medietatis <lb/>arcus <var>.b.f.</var> quod duplum erit <var>.f.b.</var> partium .14142.</s> - </p> - <p> - <s xml:space="preserve">Pro latere verò Duodecaedri, diuide latus Exaedri ex methodo .11. Å¿ecundi <lb/>Eucli. cuius maior pars erit latus quæſitum, partium .71368.</s> - </p> - <p> - <s xml:space="preserve">Sed pro latere IcoÅ¿aedri, te primum oportebit inuenire quantitatem anguli <var>g.d.<lb/>a.</var> hoc eÅ¿t ipÅ¿ius arcus <var>.b.n.</var> qui tali angulo Å¿ubiacet, quod cum pluribus modis inue-<lb/>niri poſſit, nihilominus, hunc Å¿eruabis, inuenies primò quantitatem <var>.d.g.</var> quæ eÅ¿t ra <lb/>dix quadrata Å¿ummæ duorum quadratorum hoc eÅ¿t <var>.d.a.</var> et <var>.a.g.</var> quæ <var>.a.g.</var> æqualis eÅ¿t <lb/>diametro <var>.a.b.</var> vt Å¿cis, dices poÅ¿tea, Å¿i <var>.d.g.</var> correÅ¿pondet ipÅ¿i <var>.g.a.</var> cui correÅ¿pondet <var>.d.<lb/>h.</var> Å¿emidiametro Å¿phæræ? </s> - <s xml:space="preserve">tibi veniet <var>.h.k.</var> Å¿inus arcus <var>.a.h.</var> hoc eÅ¿t <var>.b.n.</var> graduum .63 -<lb/>min .26. cuius medietas gra .31. mi .43. pro Å¿inu Å¿uo habet partes .52571. cuius Å¿inus du <lb/>plum eÅ¿t partium .105142. pro latere IcoÅ¿aedri.</s> - </p> - <p> - <s xml:space="preserve">Incipiendo nunc à Tetraedro, Å¿cire debes, quod pars <var>.a.c.</var> totius diametri <var>.a.b.</var> æ-<lb/>qualis eÅ¿t axi ipÅ¿ius Tetraedri, quæ quidem <var>.a.c.</var> vt Å¿ubÅ¿eÅ¿quialtera ipÅ¿ius <var>.a.b.</var> erit par <lb/>tium .13333.</s> - </p> - <p> - <s xml:space="preserve">Quæres poÅ¿tea quantitatem Å¿uperficialem vnius faciei ipÅ¿ius Tetraedri, hac me-<lb/>thodo, inueniendo primum radicem quadratam trium quartarum quadrati <lb/>ipÅ¿ius <var>.a.e.</var> lateris Tetraedri, eo quod latus hoc, Å¿eÅ¿quitertium in potentia eÅ¿t ipÅ¿i perÂ<lb/>pendiculari terminatÄ™ ab vno angulorum trianguli æquilateris & à latere ei oppoÅ¿i-<lb/>to ex .11. tertijdecimi ipÅ¿ius Eucli. quÄ™ quidem perpendicularis, erit <reg norm="partium" type="context">partiÅ©</reg> .141416. <lb/>& hæc multiplicata cum medietate lateris trianguli, hoc eÅ¿t cum .81647. tibi dabit <lb/>Å¿uperficiem quæſitam, hoc eÅ¿t baÅ¿im Tetraedri <reg norm="partium" type="context">partiÅ©</reg> <reg norm="Å¿uperficialium" type="context">Å¿uperficialiÅ©</reg> .11546192152. <lb/><reg norm="Hanc" type="context">Hãc</reg> demum baÅ¿im multiplicando cum tertia parte axis Tetraedri habebis corpu-<lb/>lentiam totius Tetraedri, quæ erit .513158964003488.</s> - </p> - <p> - <s xml:space="preserve">Neque tibi hoc loco occultare volo quandam meam animaduerÅ¿ionem, quæ eÅ¿t, <lb/>quod diameter Å¿eu perpendicularis (Å¿upradicta) faciei ipÅ¿ius Tetraedri, Å¿emper æ-<lb/>qualis eÅ¿t lateri ipÅ¿ius Octaedri circunÅ¿criptibilis ab eadem Å¿phæra, hoc eÅ¿t ipÅ¿i <var>.b.f.</var> <lb/>quapropter quotieÅ¿cunque ipÅ¿am perpendicularem habere voluerimus accipiendo <lb/><var>b.f.</var> habebimus intentum. </s> - <s xml:space="preserve">Et quod hoc verum Å¿it poſſumus ita demonÅ¿trare.</s> - </p> - <p> - <s xml:space="preserve">Primum, notum nobis eÅ¿t, ipÅ¿am perpendicularem, triplam eſſe eius parti, quæ - <pb o="420" rhead="IO. BAPT. BENED." n="432" file="0432" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0432"/> - à centro circuli, ipÅ¿um triangulum circunÅ¿cribentis, terminatur, & à baÅ¿i, vt in tertio <lb/>propoÅ¿ito decimæſeptimæ quartidecimi Eucli. probatur, ex quo Å¿equitur proportioÂ<lb/>nem huiuÅ¿modi perpendicularis ad axem Tetraedri, hoc eÅ¿t ad <var>.a.c.</var> Å¿eÅ¿quioctauam <lb/>eſſe in potentia, ex penultima primi Eucli. </s> - <s xml:space="preserve">Sed cum <var>.d.c.</var> tertia pars Å¿it ipÅ¿ius <var>.d.a.</var> vt <lb/>etiam ex .2. propoÅ¿ito, Å¿eu corollario decimæſeptimæ .14. lib. diÅ¿currere licet, cum ex <lb/>dicto corollario <var>.d.c.</var> Å¿it Å¿exta pars ipÅ¿ius <var>.a.b</var>. </s> - <s xml:space="preserve">Quare <var>.d.c.</var> quarta pars erit ipÅ¿ius <var>.a.c.</var> vn <lb/>de <var>.a.c.</var> Å¿eÅ¿quitertia erit ipÅ¿i <var>.a.d.</var> in longitudine, <reg norm="ideoque" type="simple">ideoqÌ;</reg> quadratum ipÅ¿ius <var>.a.d.</var> ad qua-<lb/>dratum ipÅ¿ius <var>.a.c.</var> erit vt .9. ad .16: </s> - <s xml:space="preserve">& ita duplum quadrati ipÅ¿ius <var>.a.d.</var> hoc eÅ¿t quadra-<lb/>tum ipÅ¿ius <var>.b.f.</var> ad quadratum ipÅ¿ius <var>.a.c.</var> erit, vt .18. ad .16. hoc eÅ¿t Å¿eÅ¿quioctauum, er-<lb/>go <var>.b.f.</var> æqualis erit dictæ perpendiculari, ex .9. quinti.</s> - </p> - <p> - <s xml:space="preserve">Cubus poÅ¿tea ipÅ¿ius <var>.b.e.</var> erit partium .1539838576570176.</s> - </p> - <p> - <s xml:space="preserve">Pro Octaedro deinde, accipies productum diametri in Å¿emidiametrum, quod <lb/>productum, æquale erit quadrato diuidenti per æqualia Octaedron, hocigitur pro-<lb/>ductum, multiplicando per .100000. Å¿emidiametrum Å¿phæræ, tibi dabit columnam <lb/>quadrilateram cuius tertia pars, erit partium .666666666666666. cuius duplum <lb/>erit ipÅ¿um Octaedron partium .1333333333333.</s> - </p> - <p> - <s xml:space="preserve">Pro IcoÅ¿aedro autem, oportet prius quantitatem perpendicularis inuenire, quæ <lb/>perpendicularis, per æqualia diuidit baÅ¿im ipÅ¿ius IcoÅ¿aedri, quæ vt radix quadrata <lb/>trium quartarum quadrati lateris ipÅ¿ius baÅ¿is, erit partium .91055. talium, qualium <lb/>dictum latus erit partium .105142. cuius medietas eÅ¿t .52571. quæ medietas Å¿i mul-<lb/>tiplicata fuerit cum dicta perpendiculari, dabit totam baÅ¿im Å¿uperficialem, hoc eÅ¿t <lb/>Å¿uperficiem vnius trianguli æquilateris partium Å¿uperficialium .4786852405. quo <lb/>facto, accipe quadratum duarum tertiarum ipÅ¿ius, hic Å¿upra dictæ perpendicularis, <lb/><reg norm="ipÅ¿umque" type="simple">ipÅ¿umqÌ;</reg> deme ex quadrato Å¿emidiametri Å¿phæræ, hoc eÅ¿t, ex quadrato <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> .100000 <lb/>radix poÅ¿tea quadrata reÅ¿idui, erit partium .79468. & hæc erit perpendicularis à cen <lb/>tro Å¿phærÄ™ ad vnam baÅ¿im ipÅ¿ius IcoÅ¿aedri, quam volueris, quam perpendicularem <lb/>Å¿i multiplicaueris cum quantitate Å¿uperficiali, hic Å¿uperius reperta, vnius baÅ¿is, con-<lb/>Å¿equeris columnam trilateram partium .380401586920540. cuius tertia pars, erit <lb/>partium .126800528973513. pro vna ex .20. </s> - <s xml:space="preserve">Pyramidibus ipÅ¿um corpus compo-<lb/>nentibus. </s> - <s xml:space="preserve">Breuius tamen hoc efficiens, Å¿i multiplicaueris baÅ¿im dictam, cum tertia <lb/>parte ipÅ¿ius perpendicularis, hanc poÅ¿tea pyramidem multiplicando per .20. habebis <lb/>totam corpulentiam ipÅ¿ius IcoÅ¿aedri partium .2536010579470260.</s> - </p> - <p> - <s xml:space="preserve">Pro Duodecaedro demum, accipe Å¿inum gra .36. qui <reg norm="gradus" type="simple">gradê°</reg> Å¿unt pro dimidio quin <lb/>tæ partis totius gyri circularis, <reg norm="qui" type="simple">ê—</reg> <reg norm="quidem" type="context">quidẽ</reg> Å¿inus, erit partium .58778. cuius quadratum <lb/>Å¿i <reg norm="dem" type="context">dẽ</reg> pÅ¿eris ex quadrato <reg norm="ipÅ¿ius" type="simple">ipÅ¿iê°</reg> .100000. Å¿emidiametri circuli <reg norm="circunÅ¿cribentis" type="context">circũſcribentis</reg> <reg norm="aliquem" type="context">aliquẽ</reg> <reg norm="pen- tago" type="context">pẽ-<lb/>tago</reg> num æquilaterum, & æquiangulum, </s> - <s xml:space="preserve">tunc radix reÅ¿idui, erit perpendicularis du-<lb/>cta à centro dicti circuli ad medium vnius lateris ipÅ¿ius pentagoni, quæ perp endicu <lb/>laris, erit partium .80902. talium qualium medietas lateris dicti fuerit .58778. <lb/>Nunc verò dicendo Å¿i .58778. dat .80902. quid nobis dabit .35684? </s> - <s xml:space="preserve">medietas lateris <lb/>ipÅ¿ius Duodecaedri, vnde da bit .49116. pro perpendiculari, à centro ipÅ¿ius penta-<lb/>goni, ad latus ipÅ¿ius Duodecaedri, quæ multiplicata cum me dietate Å¿upradicta ip-<lb/>Å¿ius lat eris, hoc eÅ¿t cum .35684. producet vnum ex quinque triangulis componenti-<lb/>bus vn um pentagonum, Å¿eu vnam baÅ¿im ipÅ¿ius Duodecaedri, quod quidem triangu <lb/>lum, erit partium .1752655344. Å¿u perficialium, quas Å¿i per quinque multiplicaueris <lb/>habeb is vnam baÅ¿im pentagonam dicti corporis partium .8763276720. </s> - <s xml:space="preserve">Dicendum <lb/>poÅ¿tea eÅ¿t, Å¿i ad .80901. conuenit Å¿emidiameter circularis partium .100000. quid <reg norm="con" type="context">cõ</reg> <lb/>ueniet partibus .49116. dabit .60711. pro tali Å¿emidiametro circulari, cuius quadra- - <pb o="421" rhead="EPISTOL AE." n="433" file="0433" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0433"/> - tum, Å¿i dempÅ¿eris ex quadrato ipÅ¿ius .100000. Å¿emidiametro Å¿phÄ™ræ, </s> - <s xml:space="preserve">tuncradix qua-<lb/>drata reÅ¿idui, erit perpendicularis à centro Å¿phæræ ad centrum pentagoni partium, <lb/>79461. cuius tertia pars, Å¿i multiplicata fuerit cum pentagono Å¿upra reperto dicti cor <lb/>poris producet vnam ex .12. pyramidibus componentibus dictum Duodecaedron, <lb/>quæ pyramis, demum, multiplicata per .12. dabit totam corpulentiam ipÅ¿ius Duo <lb/>decaedri partium .2785354925791680.</s> - </p> - <p> - <s xml:space="preserve">Nunc verò Å¿i experiri voluerimus vtrum iÅ¿ti calculi duorum corporum maiorum <lb/>Å¿int rectè Å¿upputati, <reg norm="dicemus" type="simple">dicemê°</reg> Å¿i ad <reg norm="corpus" type="simple">corpê°</reg> .12. <reg norm="baÅ¿ium" type="context">baÅ¿iÅ©</reg>, <reg norm="quod" type="wordlist">qÄ‘</reg> eÅ¿t <reg norm="partium" type="context">partiÅ©</reg> .2785354925791680 <lb/>conuenit numerus partium .2536010579470260. ipÅ¿ius IcoÅ¿aedri, quid conueniet <lb/>lateri cubi partium .115476. & inueniemus conuenire latus ipÅ¿ius IcoÅ¿aedri partium <lb/>105138. eo quod probatum Å¿it in <ref id="ref-0026">.10. propoÅ¿itione .14. li. Eucl.</ref> eandem <reg norm="proportionem" type="context">proportionẽ</reg> <lb/>eſſe corpulentiæ ipÅ¿ius Duodecaedri ad corpulentiam ipÅ¿ius IcoÅ¿aedri, quæ lateris <lb/>cubi ad latus IcoÅ¿aedri.</s> - </p> - <p> - <s xml:space="preserve">Hæc autem corpora, ita Å¿ibi inuicem, & cum eorum Å¿phæra harmonicè <reg norm="conueniunt" type="context">conueniÅ©t</reg> <lb/>quemadmodum antiqui philoÅ¿ophi inuenerunt, vt <reg norm="mirandum" type="context">mirandÅ©</reg> non Å¿it, ipÅ¿os credidiÅ¿-<lb/>Å¿e omnia quæ natura conÅ¿tant, aliquo pacto exiÅ¿tis corporibus fieri. </s> - <s xml:space="preserve">ConÅ¿idera quæ-<lb/>Å¿o quomodo conueniant inuicem Tetraedron, Octaedron, & IcoÅ¿aedron, cum uniuÅ¿-<lb/>cuiuÅ¿que baÅ¿es Å¿int triangulares æquilateræ intelli gendo Å¿emper hæc corpora ab ea-<lb/>dem Å¿phæra circunÅ¿criptibilia.</s> - </p> - <p> - <s xml:space="preserve">Octaedron, cum Tetraedro etiam in hoc conuenit, quod latus Octaedri æquale <lb/>Å¿it ei perpendiculari, quæ diuidit baÅ¿im Tetraedri per æqualia, vtÅ¿upra demonÅ¿tra-<lb/>uimus.</s> - </p> - <p> - <s xml:space="preserve">Harmonicis etiam interua llis hæc duo corpora inuicem concordantur, cum baÅ¿is <lb/>Tetraedri ad baÅ¿im Octaedri Å¿eruet proportionem Å¿eÅ¿quitertiam, conÅ¿onantiæ dia-<lb/>teſſaron. </s> - <s xml:space="preserve">Et proportio omnium Å¿uperficierum Å¿iue baÅ¿ium Octaedri Å¿imul Å¿umpta-<lb/>rum, ad omnes baÅ¿es ipÅ¿ius Tetraedri Å¿imul Å¿umptas Å¿it Å¿eÅ¿quialtera, conÅ¿onantiæ dia <lb/>pentis. </s> - <s xml:space="preserve">Neque omittendum eÅ¿t, quod proportio Octaedriad triplum Tetraedri Å¿it, <lb/>vt latus Octaedri ad latus Tetraedri.</s> - </p> - <p> - <s xml:space="preserve">Proportio verò lateris Octaedri, ad axem Tetraedri, potentia eÅ¿t Å¿eÅ¿quioctaua, <lb/>vt Å¿upra vidimus interuallum Å¿cilicet harmonicum toni maioris.</s> - </p> - <p> - <s xml:space="preserve">Harmonia verò Tetraedri, & Exaedri <reg norm="cum" type="context">cÅ©</reg> eorum Å¿phæra, talis eÅ¿t, vt proportio dia <lb/>metriÅ¿phæræ, potentia, tripla Å¿it lateri Exaedri, & Å¿eÅ¿quialtera lateri Tetraedri, ex <lb/>quo Å¿equitur latus Tetraedri potentia duplum exiÅ¿tere lateri Exaedri. </s> - <s xml:space="preserve">Interuallum <lb/>enim triplum in harmonicis, componitur ex diapaÅ¿on, & diapente, & Å¿onat Å¿peciem <lb/>diapentis. </s> - <s xml:space="preserve">Duplum verò eÅ¿t diapaÅ¿on, Å¿eÅ¿quialterum autem eÅ¿t di apente, quÄ™ con-<lb/>Å¿onantiæ perfectiſſimæ Å¿unt.</s> - </p> - <p> - <s xml:space="preserve">Proportio verò diametri Å¿phæræ, potentia dupla eÅ¿t lat eri Octaedri, conÅ¿onantiÄ™ <lb/>diapaÅ¿on. </s> - <s xml:space="preserve">Ex quo Å¿equitur proportionem lateris Tetraedri ad latus Octaedri, po-<lb/>tentia, Å¿eÅ¿quitertiam eſſe, hoc eÅ¿t conÅ¿onantiæ diateſſaron, & proportionem lateris <lb/>Octaedri ad latus Exaedri, potentia, Å¿eÅ¿quialteram eſſe, ita quod quatuor iÅ¿tæ poten <lb/>tiæ, ideÅ¿t diametri Å¿phæræ, lateris Tetraedri, lateris Octaedri, & lateris Exaedri con-<lb/>Å¿tituunt harmoniam ferè perfectiſſimam, ijs terminis comprehenÅ¿am .6. 4. 3. 2. (dixi <lb/>ferè, quia ditonus Å¿upra terminum .3. vel Å¿emiditonus Å¿ub termino .2. hoc loco non <lb/>reperitur, cuius quidem terminus eſſet .2. cum duabus quintis.)</s> - </p> - <p> - <s xml:space="preserve">Adde quod diameter Å¿phæræ triplus eÅ¿t longitudine ad <reg norm="perpendicularem" type="context">perpendicularẽ</reg> ductam <lb/>à centro Å¿phæræ ad baÅ¿im Octaedri, quæ proportio, vt Å¿upra dictum eÅ¿t, dicitur dia-<lb/>paÅ¿ondiapente, practici verò eam vocant duodecimam.</s> - </p> - <pb o="422" rhead="IO. BAPT. BENED." n="434" file="0434" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0434"/> - <p> - <s xml:space="preserve">Diameter verò Å¿phæræ Å¿eÅ¿quialter eÅ¿t longitudine axi Tetraedri, conÅ¿onantiæ <lb/>diapentis. </s> - <s xml:space="preserve">Axis autem Tetraedri Å¿eÅ¿quitertius eÅ¿t longitudinis Å¿emidiametro Å¿phæ-<lb/>ræ conÅ¿onantiæ diateſſaron. </s> - <s xml:space="preserve">Ita quod iÅ¿ti tres termini, qui Å¿unt, diameter Å¿phæræ, <lb/>axis Tetraedri, & Å¿emidiameter Å¿phæræ conÅ¿tituunt etiam valde perfectam harmo-<lb/>niam huiuÅ¿modi numeris contentam .6. 4. 3. corpulentia verò Exaedri ad corpu-<lb/>lentiam Tetraedri tripla eÅ¿t, conÅ¿onantiæ iam Å¿upradictæ diapaÅ¿ondiapente. </s> - <s xml:space="preserve">Si ve-<lb/>rò de vniÅ¿ono aliquid videre deÅ¿ideras, conÅ¿idera æqualitatem dupli quadrati dia-<lb/>metri ipÅ¿ius Å¿phæræ, cum omnibus baÅ¿ibus Exaedri, vel potentia diametri Å¿phæræ <lb/>cum duabus potentijs Å¿imul Å¿umptis, quarum vna eÅ¿t lateris Tetraedri, reliqua verò <lb/>lateris Exaedri, vel æqualitatem numerorum laterum Tetraedri, cum baÅ¿ibus Exae <lb/>dri. </s> - <s xml:space="preserve">Nec mihi videtur Å¿ilentio inuoluendum eſſe, antequam vlterius progrediar noÂ<lb/>tabilem Å¿ympatiam inter triangulum æquilaterum, & Tetraedron (<reg norm="quanuis" type="context">quãuis</reg> <reg norm="triangulum" type="context">triangulÅ©</reg> <lb/>corpus non Å¿it) non Å¿olum ob <reg norm="inalterabilitatem" type="context">inalterabilitatẽ</reg> harum duarum figurarum. </s> - <s xml:space="preserve">(nam omnes <lb/>aliæ alterabiles eſſe poſſunt, ijÅ¿dem lateribns exiÅ¿tentibus, cum ex quadrato rom-<lb/>bus, vel ex pentagono Ä™quiangulo, pentagonum non æquiangulum & c. efficiatur) <lb/></s> - <s xml:space="preserve">Å¿ed quod quemadmodum latus trianguli æquilateri Å¿eÅ¿quitertium potentia eÅ¿t per-<lb/>pendiculari ipÅ¿um per æqualia diuidenti, ita latus Tetraedri, Å¿eÅ¿quialterum eÅ¿t po-<lb/>tentia axi ipÅ¿ius Tetraedri, vnde cum dempta fuerit illa proportio Å¿eÅ¿quitertia, ex <lb/>hac Å¿eÅ¿quialtera relinquetur nobis proportio Å¿eÅ¿quioctaua, inter perpendicularem <lb/>trianguli, & axem Tetraedri (quod etiam Å¿upra demonÅ¿trauimus.) </s> - <s xml:space="preserve">TranÅ¿eamus nunc <lb/>hÄ™c, nec omittamus tamen Å¿ympatias quaÅ¿dam inter Exaedron, Octaedron, & Tetra <lb/>edron, hoc eÅ¿t quod eadem proportio Å¿it inter corpulentias Exaedri, & Octaedri, <lb/>quæinter eorum Å¿uperficies, nec non, vt latus Exaedri ad Å¿emidiametrum Å¿phæræ. <lb/></s> - <s xml:space="preserve">Proportio verò baÅ¿is Exaedri ad baÅ¿im Tetraedri, vtlatus Tetraedri ad perpendicu <lb/>larem diuidentem per æqualia eius baÅ¿im.</s> - </p> - <p> - <s xml:space="preserve">Hactenus Å¿atis dictum Å¿it de Tetraedro, Exaedro, & Octaedro cum Å¿phæra. </s> - <s xml:space="preserve"><reg norm="Dicem" type="context">Dicẽ</reg> <lb/>dum nunc cenÅ¿eo aliquid de reliquis duobus mirabilibus corporibus, quamuis ferè <lb/>omnia hæc ab antiquis philoÅ¿ophis inuenta Å¿int, quorum primum eÅ¿t, quod tam ba-<lb/>Å¿is Duodecaedri, quam IcoÅ¿aedri, ab vno <reg norm="eodemque" type="simple">eodemqÌ;</reg> circulo circunÅ¿criptibiles Å¿unt, ve <lb/>rùm, talis paſſio accidit etiam baÅ¿ibus Exaedri & Octaedri. </s> - <s xml:space="preserve">Præterea quemadmo-<lb/>dum in Duodecaedro, quilibet angulus Å¿olidus terminatur tribus angulis pentago-<lb/>norum æquiangulorum ita in IcoÅ¿aedro, quilibet angulus Å¿olidus viceuerÅ¿a termi-<lb/>natur quinque angulis triangulorum æquiangulorum. </s> - <s xml:space="preserve">Et tam vnum, quam alte-<lb/>rum horum corporum, triginta lateribus continetur. </s> - <s xml:space="preserve">Et tot Å¿olidos angulos trian-<lb/>gulares, habet Duodecaedron, quot baÅ¿es triangulares continet IcoÅ¿aedron.</s> - </p> - <p> - <s xml:space="preserve">Et IcoÅ¿aedron, tot Å¿olidos angulos <reg norm="pentagonos" type="context">pẽtagonos</reg>, quot baÅ¿es <reg norm="pentagonas" type="context">pẽtagonas</reg> habet Duo <lb/>decaedron. </s> - <s xml:space="preserve">Et tam vnum quam alterum habet .60. angulos Å¿uperficiales. </s> - <s xml:space="preserve"><reg norm="Eademque" type="context simple">EadẽqÌ;</reg> <lb/>proportio eÅ¿t omnium baÅ¿ium Å¿imul <reg norm="Å¿umptarum" type="context">Å¿umptarÅ©</reg> Duodecaedri ad omnes baÅ¿es Å¿imul <lb/>Å¿umptas ipÅ¿ius IcoÅ¿aedri, quæ corpulentiæ ipÅ¿ius Duodecaedri ad corpulentiam <lb/>IcoÅ¿aedri (quamuis hÄ™c paſſio accidat Exaedro cum Octaedro, vt Å¿pra diximus) quÄ™ <lb/>quidem proportio, eadem etiam eÅ¿t, quÄ™ lateris Exaedri ad latus IcoÅ¿aedri, vt Å¿u-<lb/>pra iam dictum fuit.</s> - </p> - <pb o="423" rhead="EPISTOL AE." n="435" file="0435" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0435"/> - </div> - </div> - <div type="section" level="3" n="51"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">NOVA INVENTIO COMPONENDI ASTROLABIA <lb/>cum Horologijs artificialibus.</head> - <head style="it" xml:space="preserve">Facobo Mayeto IngenioÅ¿ißimo Horologiorum Serenißimi <lb/>Sabaudiæ Ducis Artifici.</head> - <p> - <s xml:space="preserve"><emph style="sc">NOnnvnqvam</emph> conÅ¿ideraui mirabilem pulchritudinem, Å¿imul cum vtili-<lb/>tate coniunctam, illorum horologiorum, quæin Germania conÅ¿truuntur <lb/><reg norm="cum" type="context">cÅ©</reg> mobili Rete, Å¿eu Aranea AÅ¿trolabij <reg norm="Å¿uper" type="simple">Å¿uê‘</reg> <reg norm="Tabulam" type="context">Tabulã</reg> regionis, in <reg norm="quibus" type="simple">ê—bus</reg> <reg norm="conti" type="context">cõti</reg> <lb/>nuo <reg norm="videntur" type="simple">vident̃</reg> oriri, <reg norm="occidereque" type="simple">occidereqÌ;</reg> cæleÅ¿tia Å¿igna, cælum mediare Å¿upra orizon <lb/><reg norm="tem" type="context">tẽ</reg><unsure/>, necnon Å¿ub eo, & vt vno verbo dicam, continuo erecta videtur tota coelifigura. <lb/></s> - <s xml:space="preserve">Sed quia talia horologia omnia eorum limbum diÅ¿tinctum habent in .24. horas, qua <lb/>propter diametrum limbi, minorem duobus palmis, Å¿eu Å¿emipede eſſe non oportet <lb/>neinterÅ¿titia horarum iuÅ¿tò breuiora Å¿eu anguÅ¿tiora efficiantur, etiam ne interualla <lb/>dentium rotæ indicis nimis anguÅ¿ta Å¿int. </s> - <s xml:space="preserve">Sed quia talis magnitudo vt plurimum in-<lb/>commoda exiÅ¿tit. </s> - <s xml:space="preserve">Ideo non inutile fore cogitaui, Å¿i modus aliquis inuentus fuerit, <lb/>vt ea omnia efficiantur in limbo diuiÅ¿o tantummodo in .12. horas æquales, <reg norm="ipÅ¿umque" type="simple">ipÅ¿umqÌ;</reg> <lb/>inueni, qui quidem erit, efficiendo vt Tabula (in qua deÅ¿ignantur cÄ™leÅ¿tes domus, <lb/>cum almicantarat, atque azimut) Reti Å¿ubiectæ, mobilis Å¿it, tardior tamen ipÅ¿o Re-<lb/>te cum indice, pro duplo temporis, hoc eÅ¿t, quod eo tempore, quo Aranea cum in <lb/>dice circunuoluetur Å¿pacio .12. horarum vno gyro perfecto, ipÅ¿a Tabula efficiat tan <lb/>tummodo Å¿exinterÅ¿titia horarum. </s> - <s xml:space="preserve">IdeÅ¿t dum Tabula dicta eſſicit vnam integram re <lb/>uolutionem, Aranea, Å¿eu Zodiacus cum indice, duas efficiat reuolutiones. </s> - <s xml:space="preserve">Ita quod <lb/>Aranea cum indice perficiet vnam reuolutionem Å¿paci o temporis .12. horarum, Ta-<lb/>bula verò perficiet eam Å¿pacio temporis .24. horarum. </s> - <s xml:space="preserve">Vnde Å¿equetur quod Ara-<lb/>nea Å¿eu Zodiacus cum indice, Å¿pacio .24. horarum perfectè circunuoluetur Å¿upra Ta-<lb/>bulam, & ita huiuÅ¿modi horologia, in hoc nihil differrent ab illis Å¿upradictis. </s> - <s xml:space="preserve">Vt au <lb/>tem facias dictam tabulam tardiorem duplo temporis Araneæ cum indice, quamuis <lb/>diuerÅ¿is modis hoc fieri poſſit, pręſtantiorem tamen iudico, Å¿i cum Rota indicis, <reg norm="aliam" type="context">aliã</reg> <lb/>Rotam <reg norm="concentricam" type="context">concentricã</reg> coniunxeris, ita tamen, vt <reg norm="vnaquÄ™que" type="simple">vnaquÄ™q;</reg> liberè poſſit volui, Å¿imiliter <lb/>Å¿i cum ea horologii particula (quÄ™ <reg norm="circumagit" type="context">circÅ©agit</reg> Rotam indicis, quæ Italicè <emph style="sc">rochetto</emph> <lb/>Germanicè verò <emph style="sc">trib</emph> vocatur, Latinè <reg norm="autem" type="wordlist">aÅ©t</reg> ipÅ¿um vocabo, <emph style="sc">colinvm</emph>, qui Å¿ubro-<lb/>ta fuÅ¿i reperitur) coniunxeris alium colinum quem, Å¿ecundum vocabo, <reg norm="concentricum" type="context">concentricÅ©</reg> <lb/>verò cum primo, cum <reg norm="eoque" type="simple">eoqÌ;</reg> conÅ¿olidato, numerum verò dentium, tam RotÄ™ adiunctÄ™ <lb/>quam Å¿ecundi colini, varijs modis poteris inuenire, quorum primus erit, vt numerus <lb/>dentium Å¿ecundæ RotÄ™ duplus exiÅ¿tat numero dentium primÄ™, efficiendo Å¿ecundum <lb/>colinum eiuÅ¿dem numeri dentium quo primum, Å¿ed quia interualla dentium huiuÅ¿-<lb/>modi RotÄ™, nimis anguÅ¿ta fortaſſe reÅ¿ultabunt, </s> - <s xml:space="preserve">propterea alios etiam modos inue-<lb/>ni, quorum vnus erit (dum numerus dentium primi colini par fuerit) <reg norm="efficiendo" type="context">efficiẽdo</reg> Å¿ecun <lb/>dam <reg norm="Rotam" type="context">Rotã</reg> <reg norm="eiuÅ¿dem" type="context">eiuÅ¿dẽ</reg> numeri <reg norm="dentium" type="context">dentiÅ©</reg> cuius eÅ¿t prima. <reg norm="Å¿ecundum" type="context context">Å¿ecÅ©dÅ©</reg> vero colinum, medietatis <lb/>numeri dentium cuius erit primus. </s> - <s xml:space="preserve">Attamen Å¿i primus colinus eſſet .4. dentium, Å¿ecun <lb/>dum oporteret eſſe duorum dentium, vnde motus Å¿ecundÄ™ RotÄ™ non eſſet ita conti-<lb/>nuus. </s> - <s xml:space="preserve">Quapropter alium etiam mòdum excogitaui, hoc eÅ¿t, cupiendo vt Å¿ecundus <lb/>colinus, extribus dentibus exiÅ¿tat, Å¿i primus ex .4. repertus fuerit, oportebit prius ex <lb/>regula de tribus, numerum quendam inuenire quo inuento ipÅ¿um duplicare, & hunc <lb/>duplicatum numerum conueniet Å¿ecundam Rotam habere, vt ipÅ¿a poſſit ab illo co-<lb/>lino <reg norm="trium" type="context">triÅ©</reg> <reg norm="dentium" type="context">dentiÅ©</reg> circunuolui in duplo temporis, quo prima à ſuo colino quatuor den - <pb o="424" rhead="IO. BAPT. BENED." n="436" file="0436" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0436"/> - tium. </s> - <s xml:space="preserve">Exempli gratia, Å¿i prima Rota conÅ¿taret ex .36. dentibus, dicendum eſſet, Å¿i <lb/>4. conuenit cum .36. cum quibus conuenient .3. & inueniemus .27. cum quo numero <lb/>dicta Å¿ecunda Rota circunuolueretur eodem tempore à ſuo colino trium dentium, <lb/>quo prima à ſuo quatuor dentium, </s> - <s xml:space="preserve">quare duplicando .27. haberemus .54. pro nume-<lb/>ro dentium dictÄ™ Å¿ecundæ Rotæ, vt duplo temporis circunuoluatur quo prima. </s> - <s xml:space="preserve">Sed <lb/>Å¿i primus colinus conÅ¿taret ex .6. dentibus, exiÅ¿tente Å¿ua Rota ex .36. <reg norm="vellemusque" type="simple">vellemusqÌ;</reg> <reg norm="quod" type="simple">ê™</reg> <lb/>Å¿ecundus exiÅ¿teret ex .4. </s> - <s xml:space="preserve">tunc Å¿uam Rotam oporteret habere dentes .48. ex dicta re-<lb/>gula. </s> - <s xml:space="preserve">Si autem primus colinus conÅ¿taret ex numero impari, nihil referret, dummo-<lb/>do huiuÅ¿modi numerus impar, Å¿eu par, exiÅ¿teret pars propria numeri dentium, vel <lb/>ipÅ¿ius dupli primæ RotÄ™, hoc eÅ¿t, eſſet pars aliquota numeri dentium ipÅ¿ius primæ <lb/>Rotæ vel ipÅ¿ius dupli. </s> - <s xml:space="preserve">In ijs verò horologiis in quibus duplum numeri dentium di-<lb/>ctÄ™ primÄ™ RotÄ™ non erit multiplex numero dentium primi colini, hoc fieri non pote <lb/>rit. </s> - <s xml:space="preserve">Ratio enim tam clarè, tibi conÅ¿ideranti, patebit, vt nullis verbis indigeat cum <lb/>Å¿emper numerus dentium Å¿ecundÄ™ RotÄ™ multiplex eſſe debeat numero dentium Å¿e-<lb/>cundi colini. </s> - <s xml:space="preserve">Idem autem non dico de prima Rota cum Å¿uo colino, hoc eÅ¿t, vt nu-<lb/>merus primÄ™ multiplex Å¿it numero Å¿ui colini, nam hoc neceſſarium non eÅ¿t. </s> - <s xml:space="preserve">Pona-<lb/>mus exempli gratia primum colinum conÅ¿tare Å¿ex dentibus, Å¿uam vero Rotam den-<lb/>tibus .21. cuius quidem numeri, 6. non eÅ¿t pars aliquota, Å¿ed dupli ipÅ¿ius .21. ipÅ¿e .6. <lb/>eÅ¿t pars aliquota. </s> - <s xml:space="preserve">Nunc verò Å¿i voluerimus numerum dentium Å¿ecundæ RotÄ™ inue-<lb/>nire, cuius colinus ex quinque dentibus exiÅ¿tat (Å¿uppoÅ¿ito primo ex .6. conÅ¿tare) </s> - <s xml:space="preserve">tunc <lb/>ex regula de tribus, diuiÅ¿o producto, quod fit ex .21. in .5. per .6. exibit .17. cum di-<lb/>midio, cuius duplum eſſet .35. qui multiplex eÅ¿t ipÅ¿i quinque. </s> - <s xml:space="preserve">Reperto igitur nume <lb/>ro Å¿ecundÄ™ RotÄ™, cum numero ipÅ¿ius colini, oportet nunc Å¿cire modum compoÅ¿itio-<lb/>nis, Å¿eu coniunctionis harum rerum, hoc eÅ¿t duorum colinorum <reg norm="concentricorum" type="context">concentricorÅ©</reg> (Å¿ed <lb/>de ijs Å¿atis iam Å¿uperius dictum fuit) duarum Rotarum concentricarum cum Tabula, <lb/>cum Zodiaco, & cum indice, Å¿eu OÅ¿tenÅ¿ore, cuius quidem OÅ¿tenÅ¿oris medietas tan <lb/>tummodo nobis Å¿ufficiet. </s> - <s xml:space="preserve">Sciendum igitur nunc eÅ¿t quod cum primus colinus re-<lb/>uoluat totam primam Rotam, Å¿pacio temporis .12. horarum, oportet vt eius axis, Å¿eu <lb/>arbor voluat oÅ¿tenÅ¿orem, <reg norm="Zodiacumque" type="simple">ZodiacumqÌ;</reg>, eodem temporis Å¿pacio, & quia Rota hÄ™c <lb/>inalterabis eÅ¿t, propter eius coniunctionem cum Å¿uo colino, & nos oporteat indicem <lb/><reg norm="Zodiacumque" type="simple">ZodiacumqÌ;</reg>, quotidie ferè, dirigere, <reg norm="Å¿uisque" type="simple">Å¿uisqÌ;</reg> locis collocare, ideo nos oportet, indi <lb/>cem, Zodiacum, & primam Rotam, ita cum axe, Å¿eu arbore coniungere, vt poſſimus <lb/>dicta omnia efficere. </s> - <s xml:space="preserve">Pars igitur Arboris, Å¿eu axis dicti, quæ ingredi debet in prima <lb/>Rota, Å¿it rotunda, & contigua ipÅ¿i Rotæ, non autem continua, vel cum Rota conÅ¿oli-<lb/>data. </s> - <s xml:space="preserve">Pars verò quÄ™ per foramen Zodiaci, Å¿eu AraneÄ™ tranÅ¿ibit, Å¿it quadrata vÅ¿que <lb/>ad <reg norm="Å¿ummitatem" type="context">Å¿ummitatẽ</reg> ipÅ¿ius axis (tali Å¿piſſitudine, vt in claui ipÅ¿ius horologij ingredi poÅ¿-<lb/>Å¿it) & ita foramen ipÅ¿ius AraneÄ™, quadratum Å¿it, OÅ¿tenÅ¿or autem circa axem, com <lb/>poÅ¿itus Å¿it tali ordine, vt circa paruum circulum volui poſſit, qui paruus circulus ha-<lb/>beat quadratum foramen, per quod tranÅ¿eat axis, qui axis aliquantulum emineat <lb/>Å¿upra <reg norm="oÅ¿tenÅ¿orem" type="context">oÅ¿tenÅ¿orẽ</reg>. </s> - <s xml:space="preserve">Sub Aranea vero vel Zodiaco, locata erit Tabula, vt <reg norm="nunc" type="context">nÅ©c</reg> dicemus, <lb/>Å¿ed Å¿ciendum eÅ¿t prius, quod inter Tabulam, & Å¿uam Å¿ecundam Rotam, aliam lami-<lb/>nam immobilem interpoÅ¿itam eſſe oportet, quÄ™ circulare foramen habeat, per <reg norm="quod" type="wordlist">qÄ‘</reg> <lb/>quÄ™dam breuis fiÅ¿tula tranÅ¿eat circundans axem & coniungens <reg norm="Tabulam" type="context">Tabulã</reg> cum Å¿ua Ro-<lb/>ta, cuius quidem fiÅ¿tulæ Å¿uperficies concaua, rotunda Å¿it, Å¿uperficies verò extrinÅ¿e-<lb/>ca, nontota, niÅ¿i ea pars, quÄ™ Å¿ecundam Rotam ingreditur, vt in rotundo foramine <lb/>ipÅ¿ius RotÄ™, dicta fiÅ¿tula volui poſſit, pars vero extrinÅ¿eca quÄ™ Tabulam ingredi de-<lb/>bet, Å¿it quadrata. </s> - <s xml:space="preserve">Tabula vero quatuor paruiſſima foramina habeat in extremitati- - <pb o="425" rhead="EPISTOL AE." n="437" file="0437" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0437"/> - bus linearum, meridianÄ™, & verticalis, vt acu mediante volui poſſit, prout oportebit.</s> - </p> - <p> - <s xml:space="preserve">Perfectum igitur cum fuerit op us hoc, te oportet Å¿cire modum ipÅ¿o vtendi. </s> - <s xml:space="preserve">Qua-<lb/>propter quotieÅ¿cunque volueris, aÅ¿pice Solis locum in Zodiaco, Ephemeridibus me <lb/>diantibus, idem dico de vnoquoque reliquorum planetarum. </s> - <s xml:space="preserve">Inuento poÅ¿tea So-<lb/>lis loco in noÅ¿tro Zodiaco horologij, manu mediante, volue oÅ¿tenÅ¿orem, ita, vt li-<lb/>nea fiduciæ tranÅ¿eat per gradum Solis, deinde, claui ipÅ¿ius horologij mediante, vol-<lb/>ue indicem, ita cum Zodiaco coniunctum, vt linea fiduciÄ™, punctum, Å¿eu partem ho-<lb/>rÄ™ oÅ¿tendat in limbo horologii, quÄ™ quidem hora notanda eÅ¿t Å¿i fuerit ex ijs quÄ™ in-<lb/>cipiunt à meridie vÅ¿que ad mediam noctem, vel à media nocte vÅ¿que ad meri-<lb/>diem, </s> - <s xml:space="preserve">tunc acu Å¿upradicta mediante, poÅ¿ita in aliquo illorum quatuor foraminum, <lb/>circunuoluenda eÅ¿t Tabula, ita, vt extremitas lineÄ™ meridianÄ™ Å¿upra orizontem, ex <lb/>Ä™quo incidat inter duodecimam horam, & lineam fiduciÄ™, computum incipiendo à <lb/>duodecima hora, Å¿i vero dicta indicis hora fuerit ex ijs quÄ™ <reg norm="incipiunt" type="context">incipiÅ©t</reg> à media nocte & <lb/>deÅ¿inunt poÅ¿tea in meridie, oportebit, acu mediante, circunuoluere Tabulam, quo-<lb/>uÅ¿que punctum extremum meridianæ Å¿ub terra, medio loco exiÅ¿tat inter <reg norm="duodecimam" type="context">duodecimã</reg> <lb/>horam, & horam oÅ¿tenÅ¿am à linea fiduciÄ™. </s> - <s xml:space="preserve">Quo facto continuo videbis erectam <reg norm="cae- li" type="simple">cÄ™-<lb/>li</reg> figuram. </s> - <s xml:space="preserve">& quia vidiÅ¿ti loca planetarum in Ephemeridibus, videbis etiam <lb/>eorum loca accidentalia in domibus Å¿cilicetaccidentalibus, Å¿i aliquas fixarum in <lb/>Aranea deÅ¿iderabis, accipere poteris Ocu. ♉, cor. ♌, Å¿pi. â™, Liram, Aquilam, & <lb/>Arcturum, dum locus fuerit capax. </s> - <s xml:space="preserve">Nec te moueat, quod oportebit lineam fiduciÄ™ <lb/>Å¿upra gra. Solis quotidie collocare, quod nihil refert. </s> - <s xml:space="preserve">Nam oportet etiam quoti-<lb/>die cordam fuÅ¿o circunuoluere.</s> - </p> - </div> - </div> - <div type="section" level="3" n="52"> - <div type="letter" level="4" n="1"> - <head xml:space="preserve">DE DEMONSTRATIONIBVS PROPOSITIONVM <lb/>Mathematicarum, nec non de AÅ¿trologia Iudiciaria.</head> - <head style="it" xml:space="preserve">Fu<unsure/>llustriſſ <var>.D.</var> Volfardo AiÅ¿eÅ¿tain.</head> - <p> - <s xml:space="preserve"><emph style="sc">NIhil</emph> mihi gratius & iucundius afferri potuit tuis literis, quibus te cupi-<lb/>dum oÅ¿tendis Å¿ciendi rationem, quare ego non vna methodo ad omnes <lb/>propoÅ¿itiones demonÅ¿trandas vÅ¿us Å¿im, hoc eÅ¿t, </s> - <s xml:space="preserve">quare non omnia ea Eucl. <lb/>Theoremata citem in vnaquaque propoÅ¿itione, quÄ™ ad <reg norm="eam" type="context">eã</reg> demonÅ¿trandam <reg norm="faciunt" type="context">faciÅ©t</reg>, <lb/>quemadmodum in mea Gnomonica vidiÅ¿ti me aliquando omiÅ¿iſſe. </s> - <s xml:space="preserve">ReÅ¿pondeo <reg norm="quod" type="simple">ê™</reg> <lb/>mathematicÄ™ demonÅ¿trationes, hominibus Euclidis Elementa poſſidentibus, non in <lb/>digent aliqua citatione numerorum Theorematum ipÅ¿ius Euclidis, & Å¿i aliquando <lb/>vÅ¿us Å¿um aliqua citatione eorundem, hoc feci propter conÅ¿uetudinem noÅ¿tri tempo <lb/>ris, vel etiam ad faciliorem intelligentiam illorum, quibus Å¿cribebam. </s> - <s xml:space="preserve">Sed omnia <lb/>quamuis minima citare, vt <reg norm="faciunt" type="context">faciÅ©t</reg> nonnulli, mihi, nimis laborioÅ¿um, <reg norm="Å¿uperfluumque" type="simple">Å¿uperfluumqÌ;</reg> <lb/>videtur, preÅ¿ertim ijs (vt dixi) qui memoria tenent prima Elementa. </s> - <s xml:space="preserve">HÄ™c igitur <lb/>eÅ¿t vna ratio. </s> - <s xml:space="preserve">Alia, quia multoties, ita coniuncta eÅ¿t Å¿peculatio cum ipÅ¿a concluÅ¿io <lb/>ne, vt mihi ſępius viÅ¿um Å¿it Å¿uperfluum, aliquid de ipÅ¿a theoria Å¿cribere. </s> - <s xml:space="preserve">In iis <lb/>enim, quÄ™ dum puer eramÅ¿cripÅ¿i, videbis Å¿crupuloÅ¿am illam methodum, Å¿ed po-<lb/>Å¿tea, non niÅ¿i in arduis propoÅ¿itionibus me nihil eſſentiale prÄ™termittere.</s> - </p> - <p> - <s xml:space="preserve">Circa vero id de quo me interrogas, Å¿cilicet, vtrum putem omnia vera eſſe, ea <lb/>quÄ™ Å¿cripta reperiuntur in libris AÅ¿trologæ iudiciariÄ™. </s> - <s xml:space="preserve">ReÅ¿pondeo quod non, imo - <pb o="426" rhead="IO. PAPT. BENED." n="438" file="0438" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0438"/> - puto plurima falÅ¿a eſſe. </s> - <s xml:space="preserve">Nam illa multitudo partium, vt pars vitÄ™, pars Hylech, pars <lb/>futurorum, & reliquÄ™ omnium domorum cÅ“leÅ¿tium, Å¿alua parte fortunÄ™, Å¿unt merÄ™ <lb/>nugÄ™. </s> - <s xml:space="preserve">Idem dico de faciebus, Å¿iue decanis, de terminis, & de gradibus ipÅ¿is, vt pu-<lb/>ta azemenis, puteis, vacuis, fumoÅ¿is, & de reliquis. </s> - <s xml:space="preserve">De Domibus vero, Exaltationi <lb/>bus, nec non triplicitatibus, experientia <reg norm="confirmat" type="context">cõfirmat</reg> ea vera eſſe. </s> - <s xml:space="preserve"><reg norm="Idem" type="context">Idẽ</reg> affirmo de Domi <lb/>bus accidentalibus, rationalibus tamen, non <reg norm="autem" type="context">autẽ</reg> de Domibus Campani, & Gazuli. <lb/></s> - <s xml:space="preserve">ObÅ¿eruationes etiam complexionum Å¿eu inÅ¿luentiarum ipÅ¿orum Planetarum rectè <lb/>factæ Å¿unt, quÄ™ etiam à coloribus ipÅ¿orum Planetarum ferè iudicari poſſunt. </s> - <s xml:space="preserve">Con-<lb/>iunctiones <reg norm="aÅ¿pectusque" type="simple">aÅ¿pectusqÌ;</reg> ipÅ¿orum inuicem, Å¿imiliter mirabilia faciunt, & ex maiori par <lb/>te, ea, quÄ™ de iÅ¿tis Å¿cribuntur vera Å¿unt. </s> - <s xml:space="preserve">Reuolutiones annuÄ™ Å¿imiliter, cum Domino <lb/>anni. </s> - <s xml:space="preserve">Dominum verò orbis <reg norm="DiuiÅ¿oremque" type="simple">DiuiÅ¿oremqÌ;</reg> non approbo, nam hic pendet à termino, <lb/>ille verò ab hora. </s> - <s xml:space="preserve">Nouenarias autem Dodecathemoria, Alfridarias, & multa iis Å¿i <lb/>milia omnia nego. </s> - <s xml:space="preserve">AntiÅ¿cia, vera Å¿unt, ideÅ¿t influunt, malos tamen effectus, alia <lb/>plus alia verò minus, prout aliqua eorum Å¿unt tetragona, alia verò trigona, alia ma-<lb/>gna, alia parua, magna Å¿unt, vt Arietis cum Virgine, & LibrÄ™ cum PiÅ¿cibus, parua ve <lb/>rò, <reg norm="debiliaque" type="simple">debiliaqÌ;</reg> Geminorum cum Cancro, & Sagittarij <reg norm="cum" type="context">cÅ©</reg> Capricorno. </s> - <s xml:space="preserve">Sed difuÅ¿ius <lb/>hÄ™c <reg norm="omnia" type="wordlist">oÄ©a</reg> videbis in meo illo particulari tractatu, de quo tibi alià s dixi, in quo multa <lb/>videbis, quÄ™ omnia ab experientia, ex multis à me obÅ¿eruatis, comprobata Å¿unt, <lb/>quem quidem tractatum cum quibuÅ¿dam alijs meis Å¿peculationibus in lucem prode <lb/>re cupio, Å¿i fieri poterit, antequam ad directionem mei HoroÅ¿copi cum corpore <lb/>Martis AnÄ™retÄ™ perueniam, quÄ™ quidem directio circa annum milleÅ¿imum quin-<lb/>genteÅ¿imum nonageÅ¿imum Å¿ecundum eueniet.</s> - </p> - <head type="footer" xml:space="preserve">FINIS.</head> - </div> - </div> - </div> - </div> - <div type="back" level="1" n="1"> - <div type="errata" level="2" n="1"> - <pb n="439" file="0439" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0439"/> - <head xml:space="preserve">ERRATA CORRIGITO IVXTA INFRASCRIPTAM TABVLAM,</head> - <head xml:space="preserve">Reliquos verò errores, qui orthographiam reÅ¿piciunt, benignus lector corrigat, quiÅ¿ciat multos errores in <lb/>editione irrepÅ¿iſſe, quod non paucos dies morbo fuerim detentus dum præſens opus excuderetur.</head> - <xhtml:table class="it"> - <xhtml:tr> - <xhtml:th xml:space="preserve">Pag.</xhtml:th> - <xhtml:th xml:space="preserve">Lin.</xhtml:th> - <xhtml:th xml:space="preserve">Errata</xhtml:th> - <xhtml:th xml:space="preserve">Correcta</xhtml:th> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">3</xhtml:td> - <xhtml:td xml:space="preserve">29</xhtml:td> - <xhtml:td xml:space="preserve">æqualis</xhtml:td> - <xhtml:td xml:space="preserve">æquali</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">8</xhtml:td> - <xhtml:td xml:space="preserve">35</xhtml:td> - <xhtml:td xml:space="preserve">maius</xhtml:td> - <xhtml:td xml:space="preserve">maior</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">9</xhtml:td> - <xhtml:td xml:space="preserve">15</xhtml:td> - <xhtml:td xml:space="preserve">in vnitate Å¿uperficialis, erit ac</xhtml:td> - <xhtml:td xml:space="preserve">in vnitate, Å¿upreficialis erit, ac</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">11</xhtml:td> - <xhtml:td xml:space="preserve">1</xhtml:td> - <xhtml:td xml:space="preserve">proueuiens</xhtml:td> - <xhtml:td xml:space="preserve">prouenientem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">11</xhtml:td> - <xhtml:td xml:space="preserve">8</xhtml:td> - <xhtml:td xml:space="preserve">futurum</xhtml:td> - <xhtml:td xml:space="preserve">futurus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">11</xhtml:td> - <xhtml:td xml:space="preserve">31</xhtml:td> - <xhtml:td xml:space="preserve">illæ nihil aliud Å¿unt</xhtml:td> - <xhtml:td xml:space="preserve">illud nihil aliud est</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">11</xhtml:td> - <xhtml:td xml:space="preserve">38</xhtml:td> - <xhtml:td xml:space="preserve">diuidemus</xhtml:td> - <xhtml:td xml:space="preserve">diuidamus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">12</xhtml:td> - <xhtml:td xml:space="preserve">1</xhtml:td> - <xhtml:td xml:space="preserve">tertiæ, Å¿int</xhtml:td> - <xhtml:td xml:space="preserve">tertiæ Å¿int</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">12</xhtml:td> - <xhtml:td xml:space="preserve">2</xhtml:td> - <xhtml:td xml:space="preserve">producturn</xhtml:td> - <xhtml:td xml:space="preserve">productus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">19</xhtml:td> - <xhtml:td xml:space="preserve">30</xhtml:td> - <xhtml:td xml:space="preserve">proueniens</xhtml:td> - <xhtml:td xml:space="preserve">prouenientem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">19</xhtml:td> - <xhtml:td xml:space="preserve">30</xhtml:td> - <xhtml:td xml:space="preserve">productum æquale</xhtml:td> - <xhtml:td xml:space="preserve">productus æqualis</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">21</xhtml:td> - <xhtml:td xml:space="preserve">27</xhtml:td> - <xhtml:td xml:space="preserve">eadem</xhtml:td> - <xhtml:td xml:space="preserve">eædcm</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">24</xhtml:td> - <xhtml:td xml:space="preserve">18</xhtml:td> - <xhtml:td xml:space="preserve">eÅ¿t</xhtml:td> - <xhtml:td xml:space="preserve">eſſet</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">26</xhtml:td> - <xhtml:td xml:space="preserve">39</xhtml:td> - <xhtml:td xml:space="preserve">est</xhtml:td> - <xhtml:td xml:space="preserve">eſſet</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">41</xhtml:td> - <xhtml:td xml:space="preserve">24</xhtml:td> - <xhtml:td xml:space="preserve">quæ</xhtml:td> - <xhtml:td xml:space="preserve">quæ</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">41</xhtml:td> - <xhtml:td xml:space="preserve">31</xhtml:td> - <xhtml:td xml:space="preserve">distinguendæ</xhtml:td> - <xhtml:td xml:space="preserve">diÅ¿tinguendo</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">59</xhtml:td> - <xhtml:td xml:space="preserve">5</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ubÅ¿equens</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ubÅ¿equentem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">61</xhtml:td> - <xhtml:td xml:space="preserve">3</xhtml:td> - <xhtml:td xml:space="preserve">hæc via tenendæ</xhtml:td> - <xhtml:td xml:space="preserve">hanc viamtenere</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">61</xhtml:td> - <xhtml:td xml:space="preserve">3</xhtml:td> - <xhtml:td xml:space="preserve">fuit</xhtml:td> - <xhtml:td xml:space="preserve">fuerit</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">61</xhtml:td> - <xhtml:td xml:space="preserve">45</xhtml:td> - <xhtml:td xml:space="preserve">numerum quæſitum</xhtml:td> - <xhtml:td xml:space="preserve">numerus quæſitus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">62</xhtml:td> - <xhtml:td xml:space="preserve">30</xhtml:td> - <xhtml:td xml:space="preserve">quantum est <reg norm="dimidiam" type="context">dimidiã</reg> occupatam</xhtml:td> - <xhtml:td xml:space="preserve">quanta est dimidia occupata</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">64</xhtml:td> - <xhtml:td xml:space="preserve">4</xhtml:td> - <xhtml:td xml:space="preserve">Å¿peculari</xhtml:td> - <xhtml:td xml:space="preserve">conÅ¿iderari</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">64</xhtml:td> - <xhtml:td xml:space="preserve">15</xhtml:td> - <xhtml:td xml:space="preserve"><reg norm="totque" type="simple">totq;</reg> Å¿unt termini</xhtml:td> - <xhtml:td xml:space="preserve"><reg norm="totque" type="simple">totq;</reg> eße terminos</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">64</xhtml:td> - <xhtml:td xml:space="preserve">18</xhtml:td> - <xhtml:td xml:space="preserve">hoc est numerum</xhtml:td> - <xhtml:td xml:space="preserve">hoc est per numerum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">64</xhtml:td> - <xhtml:td xml:space="preserve">19</xhtml:td> - <xhtml:td xml:space="preserve">primo quod vnus est</xhtml:td> - <xhtml:td xml:space="preserve">primo qui vnus est</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">66</xhtml:td> - <xhtml:td xml:space="preserve">18</xhtml:td> - <xhtml:td xml:space="preserve">minimum</xhtml:td> - <xhtml:td xml:space="preserve">minimns</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">67</xhtml:td> - <xhtml:td xml:space="preserve">13</xhtml:td> - <xhtml:td xml:space="preserve">maximum terminum <reg norm="addendum" type="context">addendÅ©</reg></xhtml:td> - <xhtml:td xml:space="preserve">maximus terminus addendus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">72</xhtml:td> - <xhtml:td xml:space="preserve">1</xhtml:td> - <xhtml:td xml:space="preserve">numerum</xhtml:td> - <xhtml:td xml:space="preserve">numerus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">72</xhtml:td> - <xhtml:td xml:space="preserve">8</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ingulos itinere</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ingulos in itinere</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">74</xhtml:td> - <xhtml:td xml:space="preserve">7</xhtml:td> - <xhtml:td xml:space="preserve">itinerarium</xhtml:td> - <xhtml:td xml:space="preserve">itinerant ium</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">78</xhtml:td> - <xhtml:td xml:space="preserve">21</xhtml:td> - <xhtml:td xml:space="preserve">morum</xhtml:td> - <xhtml:td xml:space="preserve">modum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">78</xhtml:td> - <xhtml:td xml:space="preserve">24</xhtml:td> - <xhtml:td xml:space="preserve">noueratius</xhtml:td> - <xhtml:td xml:space="preserve">nouenarius</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">81</xhtml:td> - <xhtml:td xml:space="preserve">10</xhtml:td> - <xhtml:td xml:space="preserve">iuncta</xhtml:td> - <xhtml:td xml:space="preserve">iunctæ</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">88</xhtml:td> - <xhtml:td xml:space="preserve">35</xhtml:td> - <xhtml:td xml:space="preserve">armonicæ</xhtml:td> - <xhtml:td xml:space="preserve">harmonicæ</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">91</xhtml:td> - <xhtml:td xml:space="preserve">6</xhtml:td> - <xhtml:td xml:space="preserve">Quare <reg norm="argumentande" type="context context">argumẽtãde</reg> permut ando</xhtml:td> - <xhtml:td xml:space="preserve">Quare permutando</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">95</xhtml:td> - <xhtml:td xml:space="preserve">27</xhtml:td> - <xhtml:td xml:space="preserve">vitium</xhtml:td> - <xhtml:td xml:space="preserve">vicium</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">97</xhtml:td> - <xhtml:td xml:space="preserve">14</xhtml:td> - <xhtml:td xml:space="preserve">Å¿umma</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ummam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">97</xhtml:td> - <xhtml:td xml:space="preserve">30</xhtml:td> - <xhtml:td xml:space="preserve">illud verò quod con</xhtml:td> - <xhtml:td xml:space="preserve">illud verò con</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">98</xhtml:td> - <xhtml:td xml:space="preserve">22</xhtml:td> - <xhtml:td xml:space="preserve">deÅ¿ideraremus</xhtml:td> - <xhtml:td xml:space="preserve">deÅ¿iderauerimus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">103</xhtml:td> - <xhtml:td xml:space="preserve">23</xhtml:td> - <xhtml:td xml:space="preserve">diÅ¿poÅ¿itis facit</xhtml:td> - <xhtml:td xml:space="preserve">diÅ¿poÅ¿itis, tantum facit</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">104</xhtml:td> - <xhtml:td xml:space="preserve">39</xhtml:td> - <xhtml:td xml:space="preserve">diÅ¿poÅ¿itis, facit</xhtml:td> - <xhtml:td xml:space="preserve">diÅ¿poÅ¿itis, tantum facit</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">107</xhtml:td> - <xhtml:td xml:space="preserve">21</xhtml:td> - <xhtml:td xml:space="preserve">eccedit</xhtml:td> - <xhtml:td xml:space="preserve">excedit</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">110</xhtml:td> - <xhtml:td xml:space="preserve">41</xhtml:td> - <xhtml:td xml:space="preserve">habuerit eius cerebrum</xhtml:td> - <xhtml:td xml:space="preserve">habuerit cerebrum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">113</xhtml:td> - <xhtml:td xml:space="preserve">46</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ufficeret</xhtml:td> - <xhtml:td xml:space="preserve">Å¿uffecißet</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">116</xhtml:td> - <xhtml:td xml:space="preserve">8</xhtml:td> - <xhtml:td xml:space="preserve">quarum Å¿ecundam</xhtml:td> - <xhtml:td xml:space="preserve">quorum Å¿ecundum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">116</xhtml:td> - <xhtml:td xml:space="preserve">8</xhtml:td> - <xhtml:td xml:space="preserve">primæ, tertiam</xhtml:td> - <xhtml:td xml:space="preserve">primi, tertium</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">116</xhtml:td> - <xhtml:td xml:space="preserve">9</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ecundæ</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ecundi</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">130</xhtml:td> - <xhtml:td xml:space="preserve">23</xhtml:td> - <xhtml:td xml:space="preserve">ducenda</xhtml:td> - <xhtml:td xml:space="preserve">ducendus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">133</xhtml:td> - <xhtml:td xml:space="preserve">10</xhtml:td> - <xhtml:td xml:space="preserve">lineam qua</xhtml:td> - <xhtml:td xml:space="preserve">lineam, qua</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">133</xhtml:td> - <xhtml:td xml:space="preserve">11</xhtml:td> - <xhtml:td xml:space="preserve">dratam</xhtml:td> - <xhtml:td xml:space="preserve">dratum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">134</xhtml:td> - <xhtml:td xml:space="preserve">16</xhtml:td> - <xhtml:td xml:space="preserve">inueniet</xhtml:td> - <xhtml:td xml:space="preserve">inuenimus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">137</xhtml:td> - <xhtml:td xml:space="preserve">22</xhtml:td> - <xhtml:td xml:space="preserve">distans</xhtml:td> - <xhtml:td xml:space="preserve">diÅ¿tantius</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">137</xhtml:td> - <xhtml:td xml:space="preserve">45</xhtml:td> - <xhtml:td xml:space="preserve">Notißimum igitur primum</xhtml:td> - <xhtml:td xml:space="preserve">Notißimum primum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">139</xhtml:td> - <xhtml:td xml:space="preserve">15</xhtml:td> - <xhtml:td xml:space="preserve">est</xhtml:td> - <xhtml:td xml:space="preserve">Å¿it</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">139</xhtml:td> - <xhtml:td xml:space="preserve">28</xhtml:td> - <xhtml:td xml:space="preserve">duas</xhtml:td> - <xhtml:td xml:space="preserve">duos</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">141</xhtml:td> - <xhtml:td xml:space="preserve">5</xhtml:td> - <xhtml:td xml:space="preserve">comperuiſſe</xhtml:td> - <xhtml:td xml:space="preserve">comperiſſe</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">142</xhtml:td> - <xhtml:td xml:space="preserve">19</xhtml:td> - <xhtml:td xml:space="preserve"><reg norm="quam" type="context">quã</reg></xhtml:td> - <xhtml:td xml:space="preserve">ac</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">143</xhtml:td> - <xhtml:td xml:space="preserve">17</xhtml:td> - <xhtml:td xml:space="preserve">linea</xhtml:td> - <xhtml:td xml:space="preserve">lineam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">144</xhtml:td> - <xhtml:td xml:space="preserve">23</xhtml:td> - <xhtml:td xml:space="preserve">patebit, Å¿i quis</xhtml:td> - <xhtml:td xml:space="preserve">patebit, quòd Å¿i quis</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">145</xhtml:td> - <xhtml:td xml:space="preserve">18</xhtml:td> - <xhtml:td xml:space="preserve">constante</xhtml:td> - <xhtml:td xml:space="preserve">constantem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">146</xhtml:td> - <xhtml:td xml:space="preserve">20</xhtml:td> - <xhtml:td xml:space="preserve">paÅ¿tam</xhtml:td> - <xhtml:td xml:space="preserve">maſſam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">146</xhtml:td> - <xhtml:td xml:space="preserve">22</xhtml:td> - <xhtml:td xml:space="preserve">pastam</xhtml:td> - <xhtml:td xml:space="preserve">maſſam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">148</xhtml:td> - <xhtml:td xml:space="preserve">13</xhtml:td> - <xhtml:td xml:space="preserve"><reg norm="proponit" type="simple">ê“ponit</reg> <reg norm="non" type="context">nõ</reg> <reg norm="concludit" type="context">cõcludit</reg> melius <reg norm="autem" type="wordlist">aÅ©t</reg></xhtml:td> - <xhtml:td xml:space="preserve">proponit, <reg norm="non" type="context">nõ</reg> concludit, melius <reg norm="autem" type="wordlist">aÅ©t</reg></xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">14</xhtml:td> - <xhtml:td xml:space="preserve">41</xhtml:td> - <xhtml:td xml:space="preserve">prætergradiatur</xhtml:td> - <xhtml:td xml:space="preserve">prætergredietur</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">152</xhtml:td> - <xhtml:td xml:space="preserve">6</xhtml:td> - <xhtml:td xml:space="preserve">videtur</xhtml:td> - <xhtml:td xml:space="preserve">videri</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">153</xhtml:td> - <xhtml:td xml:space="preserve">22 <lb/>23</xhtml:td> - <xhtml:td xml:space="preserve">quia libra</xhtml:td> - <xhtml:td xml:space="preserve">quia cum libræ</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">153</xhtml:td> - <xhtml:td xml:space="preserve">23</xhtml:td> - <xhtml:td xml:space="preserve">materiales, cum Å¿uÅ¿tineantur</xhtml:td> - <xhtml:td xml:space="preserve">materiales Å¿uÅ¿tineantur</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">153</xhtml:td> - <xhtml:td xml:space="preserve">24</xhtml:td> - <xhtml:td xml:space="preserve">existente. vnde aliqua</xhtml:td> - <xhtml:td xml:space="preserve">exiÅ¿tente, aliqua</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">154</xhtml:td> - <xhtml:td xml:space="preserve">23</xhtml:td> - <xhtml:td xml:space="preserve">futurum</xhtml:td> - <xhtml:td xml:space="preserve">effecturam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">155</xhtml:td> - <xhtml:td xml:space="preserve">25</xhtml:td> - <xhtml:td xml:space="preserve">carta</xhtml:td> - <xhtml:td xml:space="preserve">charta</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">156</xhtml:td> - <xhtml:td xml:space="preserve">23</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ufficere</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ufficeret</xhtml:td> - </xhtml:tr> - </xhtml:table> - <xhtml:table class="it"> - <xhtml:tr> - <xhtml:th xml:space="preserve">Pag.</xhtml:th> - <xhtml:th xml:space="preserve">Lin.</xhtml:th> - <xhtml:th xml:space="preserve">Errata</xhtml:th> - <xhtml:th xml:space="preserve">Correcta</xhtml:th> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">158</xhtml:td> - <xhtml:td xml:space="preserve">26</xhtml:td> - <xhtml:td xml:space="preserve">verÅ¿a</xhtml:td> - <xhtml:td xml:space="preserve">verÅ¿am</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">158</xhtml:td> - <xhtml:td xml:space="preserve">26</xhtml:td> - <xhtml:td xml:space="preserve">Å¿it</xhtml:td> - <xhtml:td xml:space="preserve">Å¿int</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">162</xhtml:td> - <xhtml:td xml:space="preserve">22</xhtml:td> - <xhtml:td xml:space="preserve">cindenda</xhtml:td> - <xhtml:td xml:space="preserve">Å¿cindenda</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">163</xhtml:td> - <xhtml:td xml:space="preserve">7</xhtml:td> - <xhtml:td xml:space="preserve">oppeÅ¿itus</xhtml:td> - <xhtml:td xml:space="preserve">oppoÅ¿itum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">164</xhtml:td> - <xhtml:td xml:space="preserve">24</xhtml:td> - <xhtml:td xml:space="preserve">tanta</xhtml:td> - <xhtml:td xml:space="preserve">tantæ</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">164</xhtml:td> - <xhtml:td xml:space="preserve">37</xhtml:td> - <xhtml:td xml:space="preserve">adiunctæ nobis eſſent duæ aliæ</xhtml:td> - <xhtml:td xml:space="preserve">adiuncti nobis eßent duo alii</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">164</xhtml:td> - <xhtml:td xml:space="preserve">39</xhtml:td> - <xhtml:td xml:space="preserve">Å¿estineret</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ustineret</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">164</xhtml:td> - <xhtml:td xml:space="preserve">40</xhtml:td> - <xhtml:td xml:space="preserve">dictæ</xhtml:td> - <xhtml:td xml:space="preserve">dicti</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">164</xhtml:td> - <xhtml:td xml:space="preserve">41</xhtml:td> - <xhtml:td xml:space="preserve">aliquam</xhtml:td> - <xhtml:td xml:space="preserve">aliquem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">165</xhtml:td> - <xhtml:td xml:space="preserve">3</xhtml:td> - <xhtml:td xml:space="preserve">Å¿uffieiant</xhtml:td> - <xhtml:td xml:space="preserve">Å¿ufficiat</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">165</xhtml:td> - <xhtml:td xml:space="preserve">4</xhtml:td> - <xhtml:td xml:space="preserve"> Å¿ubÅ¿eÅ¿quialter<unsure/> - </xhtml:td> - <xhtml:td xml:space="preserve">Å¿ubÅ¿eÅ¿quialterum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">186</xhtml:td> - <xhtml:td xml:space="preserve">32</xhtml:td> - <xhtml:td xml:space="preserve">finit</xhtml:td> - <xhtml:td xml:space="preserve">finitam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">187</xhtml:td> - <xhtml:td xml:space="preserve">29</xhtml:td> - <xhtml:td xml:space="preserve">propinqua</xhtml:td> - <xhtml:td xml:space="preserve">propinqui</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">187<unsure/> - </xhtml:td> - <xhtml:td xml:space="preserve">30</xhtml:td> - <xhtml:td xml:space="preserve">philoÅ¿ophi Å¿upra</xhtml:td> - <xhtml:td xml:space="preserve">philoÅ¿ophi, proximè Å¿upra</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">187<unsure/> - </xhtml:td> - <xhtml:td xml:space="preserve">31</xhtml:td> - <xhtml:td xml:space="preserve">contingeret menÅ¿e</xhtml:td> - <xhtml:td xml:space="preserve">contingeret vt menÅ¿e</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">187</xhtml:td> - <xhtml:td xml:space="preserve">34</xhtml:td> - <xhtml:td xml:space="preserve">regionis) non conÅ¿iderans</xhtml:td> - <xhtml:td xml:space="preserve">regionis) in quo non conÅ¿iderauit</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">190</xhtml:td> - <xhtml:td xml:space="preserve">11</xhtml:td> - <xhtml:td xml:space="preserve">lenas</xhtml:td> - <xhtml:td xml:space="preserve">lenæs</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">190</xhtml:td> - <xhtml:td xml:space="preserve">23</xhtml:td> - <xhtml:td xml:space="preserve">lenis</xhtml:td> - <xhtml:td xml:space="preserve">leuis</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">191</xhtml:td> - <xhtml:td xml:space="preserve">10</xhtml:td> - <xhtml:td xml:space="preserve">aliud quadratum</xhtml:td> - <xhtml:td xml:space="preserve">alius quadratus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">191</xhtml:td> - <xhtml:td xml:space="preserve">15</xhtml:td> - <xhtml:td xml:space="preserve">qualis</xhtml:td> - <xhtml:td xml:space="preserve">qui</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">193</xhtml:td> - <xhtml:td xml:space="preserve">17</xhtml:td> - <xhtml:td xml:space="preserve">cum</xhtml:td> - <xhtml:td xml:space="preserve">tum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">193</xhtml:td> - <xhtml:td xml:space="preserve">36</xhtml:td> - <xhtml:td xml:space="preserve">fit</xhtml:td> - <xhtml:td xml:space="preserve">facit</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">193</xhtml:td> - <xhtml:td xml:space="preserve">40</xhtml:td> - <xhtml:td xml:space="preserve">pleno vbi</xhtml:td> - <xhtml:td xml:space="preserve">pleno poÅ¿uerimus, vbi</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">195</xhtml:td> - <xhtml:td xml:space="preserve">10</xhtml:td> - <xhtml:td xml:space="preserve">valent</xhtml:td> - <xhtml:td xml:space="preserve">valent apud ipÅ¿os</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">196</xhtml:td> - <xhtml:td xml:space="preserve">19</xhtml:td> - <xhtml:td xml:space="preserve">phyloÅ¿ophiæ</xhtml:td> - <xhtml:td xml:space="preserve">philoÅ¿ophiæ</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">196</xhtml:td> - <xhtml:td xml:space="preserve">20</xhtml:td> - <xhtml:td xml:space="preserve">pellendo</xhtml:td> - <xhtml:td xml:space="preserve">plendo</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">198</xhtml:td> - <xhtml:td xml:space="preserve">18</xhtml:td> - <xhtml:td xml:space="preserve">Å¿unt etiam</xhtml:td> - <xhtml:td xml:space="preserve">eße etiam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">204</xhtml:td> - <xhtml:td xml:space="preserve">12</xhtml:td> - <xhtml:td xml:space="preserve">priñcipium</xhtml:td> - <xhtml:td xml:space="preserve">principum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">204</xhtml:td> - <xhtml:td xml:space="preserve">18</xhtml:td> - <xhtml:td xml:space="preserve">indigna</xhtml:td> - <xhtml:td xml:space="preserve">indignas</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">204</xhtml:td> - <xhtml:td xml:space="preserve">21</xhtml:td> - <xhtml:td xml:space="preserve">diffundetur, creÅ¿cat</xhtml:td> - <xhtml:td xml:space="preserve">diffundatur, & creÅ¿cat</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">205</xhtml:td> - <xhtml:td xml:space="preserve">16</xhtml:td> - <xhtml:td xml:space="preserve">aliquando Å¿er</xhtml:td> - <xhtml:td xml:space="preserve">aliquando me Å¿er</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">205</xhtml:td> - <xhtml:td xml:space="preserve">19</xhtml:td> - <xhtml:td xml:space="preserve">anno præter, <reg norm="neceßitatem" type="context">neceßitatẽ</reg>, gignitur</xhtml:td> - <xhtml:td xml:space="preserve">anno, præter <reg norm="neceßitatem" type="context">neceßitatẽ</reg> gignitur</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">206</xhtml:td> - <xhtml:td xml:space="preserve">21</xhtml:td> - <xhtml:td xml:space="preserve">plenilunium, quod</xhtml:td> - <xhtml:td xml:space="preserve">plenilunium fieri, quod</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">207</xhtml:td> - <xhtml:td xml:space="preserve">26</xhtml:td> - <xhtml:td xml:space="preserve">inchoet annus</xhtml:td> - <xhtml:td xml:space="preserve">inchoetur annus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">209</xhtml:td> - <xhtml:td xml:space="preserve">9</xhtml:td> - <xhtml:td xml:space="preserve">Å¿oli, quod punctum</xhtml:td> - <xhtml:td xml:space="preserve">Å¿olis, cuius punctum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">210</xhtml:td> - <xhtml:td xml:space="preserve">10</xhtml:td> - <xhtml:td xml:space="preserve">cælebrandi</xhtml:td> - <xhtml:td xml:space="preserve">celebrandis</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">212</xhtml:td> - <xhtml:td xml:space="preserve">33</xhtml:td> - <xhtml:td xml:space="preserve">Inuentæ</xhtml:td> - <xhtml:td xml:space="preserve">Inuenti</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">212</xhtml:td> - <xhtml:td xml:space="preserve">33</xhtml:td> - <xhtml:td xml:space="preserve">duæ</xhtml:td> - <xhtml:td xml:space="preserve">duo</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">214</xhtml:td> - <xhtml:td xml:space="preserve">9</xhtml:td> - <xhtml:td xml:space="preserve">claßi</xhtml:td> - <xhtml:td xml:space="preserve">claßis</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">214</xhtml:td> - <xhtml:td xml:space="preserve">10</xhtml:td> - <xhtml:td xml:space="preserve">Inter Eximias</xhtml:td> - <xhtml:td xml:space="preserve">Quia inter Eximias</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">214</xhtml:td> - <xhtml:td xml:space="preserve">26</xhtml:td> - <xhtml:td xml:space="preserve">reperiretur</xhtml:td> - <xhtml:td xml:space="preserve">reperietur</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">214</xhtml:td> - <xhtml:td xml:space="preserve">34</xhtml:td> - <xhtml:td xml:space="preserve">neceſſario Å¿it futurum, vt</xhtml:td> - 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<xhtml:td xml:space="preserve">21</xhtml:td> - <xhtml:td xml:space="preserve">progrediuntur</xhtml:td> - <xhtml:td xml:space="preserve">progredi</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">234</xhtml:td> - <xhtml:td xml:space="preserve">15</xhtml:td> - <xhtml:td xml:space="preserve">aliiori cÅ“lo</xhtml:td> - <xhtml:td xml:space="preserve">altius cælum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">239</xhtml:td> - <xhtml:td xml:space="preserve">12</xhtml:td> - <xhtml:td xml:space="preserve">dum</xhtml:td> - <xhtml:td xml:space="preserve">cum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">241</xhtml:td> - <xhtml:td xml:space="preserve">41</xhtml:td> - <xhtml:td xml:space="preserve">afferens</xhtml:td> - <xhtml:td xml:space="preserve">cogitans</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">245</xhtml:td> - <xhtml:td xml:space="preserve">25</xhtml:td> - <xhtml:td xml:space="preserve">diametri</xhtml:td> - <xhtml:td xml:space="preserve">diametros</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">248</xhtml:td> - <xhtml:td xml:space="preserve">42</xhtml:td> - <xhtml:td xml:space="preserve">Å¿it & ob id</xhtml:td> - <xhtml:td xml:space="preserve">Å¿it, ob id</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">249</xhtml:td> - <xhtml:td xml:space="preserve">42</xhtml:td> - <xhtml:td xml:space="preserve">alium numerum</xhtml:td> - <xhtml:td xml:space="preserve">alio numero</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">250</xhtml:td> - <xhtml:td xml:space="preserve">13</xhtml:td> - <xhtml:td xml:space="preserve">um voluerimus</xhtml:td> - <xhtml:td xml:space="preserve">cum inuestigare voluerimus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">251</xhtml:td> - <xhtml:td xml:space="preserve">1</xhtml:td> - <xhtml:td xml:space="preserve">inuenerimus</xhtml:td> - <xhtml:td xml:space="preserve">inuenimus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">252</xhtml:td> - <xhtml:td xml:space="preserve">6</xhtml:td> - <xhtml:td xml:space="preserve">tros</xhtml:td> - <xhtml:td xml:space="preserve">ter</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">252</xhtml:td> - <xhtml:td xml:space="preserve">6</xhtml:td> - <xhtml:td xml:space="preserve">alia verò diametro</xhtml:td> - <xhtml:td xml:space="preserve">alius verò diameter</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">252</xhtml:td> - <xhtml:td xml:space="preserve">8</xhtml:td> - <xhtml:td xml:space="preserve">te id non</xhtml:td> - <xhtml:td xml:space="preserve">te non</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">252</xhtml:td> - <xhtml:td xml:space="preserve">12</xhtml:td> - <xhtml:td xml:space="preserve">duabus</xhtml:td> - <xhtml:td xml:space="preserve">duobus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">255</xhtml:td> - <xhtml:td xml:space="preserve">16</xhtml:td> - 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<xhtml:td xml:space="preserve">quod</xhtml:td> - <xhtml:td xml:space="preserve">nam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">258</xhtml:td> - <xhtml:td xml:space="preserve">15</xhtml:td> - <xhtml:td xml:space="preserve">ictum</xhtml:td> - <xhtml:td xml:space="preserve">ictus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">258</xhtml:td> - <xhtml:td xml:space="preserve">15</xhtml:td> - <xhtml:td xml:space="preserve">eundem eſſe futurum</xhtml:td> - <xhtml:td xml:space="preserve">idem eßet</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">258</xhtml:td> - <xhtml:td xml:space="preserve">27</xhtml:td> - <xhtml:td xml:space="preserve">debet, quanto</xhtml:td> - <xhtml:td xml:space="preserve">debet, in huiuÅ¿modi Å¿itu, quanto</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">258</xhtml:td> - <xhtml:td xml:space="preserve">41</xhtml:td> - <xhtml:td xml:space="preserve">quod</xhtml:td> - <xhtml:td xml:space="preserve">qui</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">262</xhtml:td> - <xhtml:td xml:space="preserve">17</xhtml:td> - <xhtml:td xml:space="preserve">erunt</xhtml:td> - <xhtml:td xml:space="preserve">Å¿upponuntur</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">263</xhtml:td> - <xhtml:td xml:space="preserve">22</xhtml:td> - <xhtml:td xml:space="preserve">a.K. facta Å¿it</xhtml:td> - <xhtml:td xml:space="preserve">a.K. Å¿it</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">263</xhtml:td> - <xhtml:td xml:space="preserve">22</xhtml:td> - <xhtml:td xml:space="preserve">Å¿emidiameter eße vnius</xhtml:td> - <xhtml:td xml:space="preserve">Å¿emidiameter vnius</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">265</xhtml:td> - <xhtml:td xml:space="preserve">4</xhtml:td> - <xhtml:td xml:space="preserve">cognitæ</xhtml:td> - <xhtml:td xml:space="preserve">cogniti</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">272</xhtml:td> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td xml:space="preserve">infigura vbieÅ¿t <var>.P.</var></xhtml:td> - <xhtml:td xml:space="preserve">ponatur. R</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">272</xhtml:td> - <xhtml:td xml:space="preserve">18</xhtml:td> - <xhtml:td xml:space="preserve">E.i. ad <var>.E.i.</var></xhtml:td> - <xhtml:td xml:space="preserve">E.I. ad <var>.E.i.</var></xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">273</xhtml:td> - <xhtml:td xml:space="preserve">3</xhtml:td> - <xhtml:td xml:space="preserve">licet</xhtml:td> - <xhtml:td xml:space="preserve">liceret</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">274</xhtml:td> - <xhtml:td xml:space="preserve">7</xhtml:td> - <xhtml:td xml:space="preserve">te</xhtml:td> - <xhtml:td xml:space="preserve">ibi</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">275</xhtml:td> - <xhtml:td xml:space="preserve">10</xhtml:td> - <xhtml:td xml:space="preserve">dividere</xhtml:td> - <xhtml:td xml:space="preserve">diuidendo</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">276</xhtml:td> - <xhtml:td xml:space="preserve">47</xhtml:td> - <xhtml:td xml:space="preserve">accipiemus</xhtml:td> - <xhtml:td xml:space="preserve">accipiamus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">278</xhtml:td> - <xhtml:td xml:space="preserve">8</xhtml:td> - <xhtml:td xml:space="preserve">Moreta</xhtml:td> - <xhtml:td xml:space="preserve">Moreta</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">280</xhtml:td> - <xhtml:td xml:space="preserve">25</xhtml:td> - <xhtml:td xml:space="preserve">diminutæ, quartæ</xhtml:td> - <xhtml:td xml:space="preserve">diminutæ Å¿eu defectiuæ, quarta</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">281</xhtml:td> - <xhtml:td xml:space="preserve">36</xhtml:td> - <xhtml:td xml:space="preserve">grauißimum</xhtml:td> - <xhtml:td xml:space="preserve">grauißimo</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">281</xhtml:td> - <xhtml:td xml:space="preserve"> </xhtml:td> - <xhtml:td colspan="2" xml:space="preserve">in fine pag. vnamquanque Zifram accommoda Å¿ub vnoquoque al-<lb/>phabeti characterem ad priores Zifras prius addendo .1. Idem dico <lb/>in pag .282. & inter, D. et E. pone <var>.b.</var></xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">285</xhtml:td> - <xhtml:td xml:space="preserve">24</xhtml:td> - <xhtml:td xml:space="preserve">antea</xhtml:td> - <xhtml:td xml:space="preserve">ante</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">289</xhtml:td> - <xhtml:td xml:space="preserve">11</xhtml:td> - <xhtml:td xml:space="preserve">quanta pars</xhtml:td> - <xhtml:td xml:space="preserve">quantam partem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">289</xhtml:td> - <xhtml:td xml:space="preserve">25</xhtml:td> - <xhtml:td xml:space="preserve">quanta pars</xhtml:td> - <xhtml:td xml:space="preserve">quantam partem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">289</xhtml:td> - <xhtml:td xml:space="preserve">42</xhtml:td> - <xhtml:td xml:space="preserve">eÅ¿t</xhtml:td> - <xhtml:td xml:space="preserve">erit</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">290</xhtml:td> - <xhtml:td xml:space="preserve">10</xhtml:td> - <xhtml:td xml:space="preserve">Å¿atile</xhtml:td> - <xhtml:td xml:space="preserve">Å¿atilis</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">290</xhtml:td> - <xhtml:td xml:space="preserve">19</xhtml:td> - <xhtml:td xml:space="preserve">æqualis</xhtml:td> - <xhtml:td xml:space="preserve">æquale</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">290</xhtml:td> - <xhtml:td xml:space="preserve">43</xhtml:td> - <xhtml:td xml:space="preserve">circunÅ¿criptibilis</xhtml:td> - <xhtml:td xml:space="preserve">circunÅ¿cribentis</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">291</xhtml:td> - <xhtml:td xml:space="preserve">1</xhtml:td> - <xhtml:td xml:space="preserve">od</xhtml:td> - <xhtml:td xml:space="preserve">Quod</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">294</xhtml:td> - <xhtml:td xml:space="preserve">33</xhtml:td> - 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<xhtml:td xml:space="preserve">diÅ¿tincta, procedendo</xhtml:td> - <xhtml:td xml:space="preserve">diÅ¿tincta Å¿int, procedendo</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">299</xhtml:td> - <xhtml:td xml:space="preserve">32</xhtml:td> - <xhtml:td xml:space="preserve">refleßum</xhtml:td> - <xhtml:td xml:space="preserve">reflexum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">299</xhtml:td> - <xhtml:td xml:space="preserve">34</xhtml:td> - <xhtml:td xml:space="preserve">fleſſum</xhtml:td> - <xhtml:td xml:space="preserve">flexum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">302</xhtml:td> - <xhtml:td xml:space="preserve">14</xhtml:td> - <xhtml:td xml:space="preserve">eaÅ¿dem</xhtml:td> - <xhtml:td xml:space="preserve">eoÅ¿dem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">304</xhtml:td> - <xhtml:td xml:space="preserve">3</xhtml:td> - <xhtml:td xml:space="preserve">ta</xhtml:td> - <xhtml:td xml:space="preserve">tus</xhtml:td> - 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DefenÅ¿io nostra <lb/>contra Antonium Bergam, & <lb/>Alexandrum Piccol.</xhtml:td> - <xhtml:td xml:space="preserve">dicatur. contra Anto. Bergam & <lb/>Alex Piccol atq, defenÅ¿io nostra <lb/>contra Excell. August. Michelem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">399</xhtml:td> - <xhtml:td xml:space="preserve">32</xhtml:td> - <xhtml:td xml:space="preserve">tione</xhtml:td> - <xhtml:td xml:space="preserve">tionem</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">400</xhtml:td> - <xhtml:td xml:space="preserve">39</xhtml:td> - <xhtml:td xml:space="preserve">vnum tantum numerum medium</xhtml:td> - <xhtml:td xml:space="preserve">unus tantum numerus medius</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">402</xhtml:td> - <xhtml:td xml:space="preserve">19</xhtml:td> - <xhtml:td xml:space="preserve">diametrum</xhtml:td> - <xhtml:td xml:space="preserve">Å¿emidiametrum</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">403</xhtml:td> - <xhtml:td xml:space="preserve">7</xhtml:td> - <xhtml:td xml:space="preserve">Å¿uis</xhtml:td> - <xhtml:td xml:space="preserve">Å¿uæ</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">403</xhtml:td> - <xhtml:td xml:space="preserve">9</xhtml:td> - <xhtml:td xml:space="preserve">iam Å¿upræ</xhtml:td> - <xhtml:td xml:space="preserve">eo tractatu</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">403</xhtml:td> - <xhtml:td xml:space="preserve">17</xhtml:td> - <xhtml:td xml:space="preserve">quidquid</xhtml:td> - <xhtml:td xml:space="preserve">quidquam</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">404</xhtml:td> - <xhtml:td xml:space="preserve">37</xhtml:td> - <xhtml:td xml:space="preserve">milliaria</xhtml:td> - <xhtml:td xml:space="preserve">milliaribus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">406</xhtml:td> - <xhtml:td xml:space="preserve">19</xhtml:td> - <xhtml:td xml:space="preserve">trabuchum quadratum</xhtml:td> - <xhtml:td xml:space="preserve">trabuchus quadratus</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">410</xhtml:td> - <xhtml:td xml:space="preserve">38</xhtml:td> - <xhtml:td xml:space="preserve">quæ Å¿ors</xhtml:td> - <xhtml:td xml:space="preserve">qua Å¿orte</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">410</xhtml:td> - <xhtml:td xml:space="preserve">41</xhtml:td> - <xhtml:td xml:space="preserve">quamlibet</xhtml:td> - <xhtml:td xml:space="preserve">quælibet</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">411</xhtml:td> - <xhtml:td xml:space="preserve">27</xhtml:td> - <xhtml:td xml:space="preserve">dicens</xhtml:td> - <xhtml:td xml:space="preserve">dicat</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">411</xhtml:td> - <xhtml:td xml:space="preserve">28</xhtml:td> - <xhtml:td xml:space="preserve">eodem libro</xhtml:td> - <xhtml:td xml:space="preserve">vt eodem libre</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">412</xhtml:td> - <xhtml:td xml:space="preserve">24</xhtml:td> - <xhtml:td xml:space="preserve">non dependeat?</xhtml:td> - <xhtml:td xml:space="preserve">dependeat?</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">413</xhtml:td> - <xhtml:td xml:space="preserve">2</xhtml:td> - <xhtml:td xml:space="preserve">vt Å¿upra</xhtml:td> - <xhtml:td xml:space="preserve">vt alias</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">413</xhtml:td> - <xhtml:td xml:space="preserve">22</xhtml:td> - <xhtml:td xml:space="preserve">millia vices</xhtml:td> - <xhtml:td xml:space="preserve">millibus viciu<unsure/>m</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">415</xhtml:td> - <xhtml:td xml:space="preserve">32</xhtml:td> - <xhtml:td xml:space="preserve">fineÅ¿træ</xhtml:td> - <xhtml:td xml:space="preserve">fenestræ</xhtml:td> - </xhtml:tr> - <xhtml:tr> - <xhtml:td xml:space="preserve">416</xhtml:td> - <xhtml:td xml:space="preserve">15</xhtml:td> - <xhtml:td xml:space="preserve">operatio</xhtml:td> - <xhtml:td xml:space="preserve">operationem</xhtml:td> - </xhtml:tr> - </xhtml:table> - <head type="footer" xml:space="preserve">FINIS.</head> - <pb n="441" file="0441" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0441"/> - </div> - <div type="cover" level="2" n="1"> - <handwritten/> - <pb n="442" file="0442" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0442"/> - <pb n="443" file="0443" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0443"/> - <pb n="444" file="0444" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0444"/> - <p> - <s xml:space="preserve">Test: Dieser Satz enthält ein Zeichen mit Unicode-Codepoint über FFFF, nämlich ð†‘ (U+10191; D800+DD91). </s> - <s xml:space="preserve">Das gleiche Zeichen innerhalb eines Wortes: vorherð†‘nachher. </s> - <s xml:space="preserve">Das Zeichen wird testweise zu X normalisiert. </s> - <s xml:space="preserve">Das Zeichen ð†’ (U+10192; D800+DD92) wird dagegen nicht normalisiert: vorherð†’nachher. </s> - </p> - </div> - </div> - </text> -</echo> \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-xml/examples/Goerz_2008.xml Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,2563 +0,0 @@ -<?xml version="1.0" encoding="UTF-8"?> -<TEI xmlns="http://www.tei-c.org/ns/1.0"> - <teiHeader> - <fileDesc xml:lang="de"> - <titleStmt> - <title> - <title type="main">Der Deutsche Ptolemaeus - </title> - <title type="subtitle">Eine digitale Edition</title> - </title> - <author>Goerz, Guenther</author> - <respStmt> - <resp>Gesamtverantwortung</resp> - <name>Guenther Goerz</name> - </respStmt> - <respStmt> - <resp>Technische Umsetzung</resp> - <name>Josef Schneeberger</name> - </respStmt> - <respStmt> - <resp>Unterstützung</resp> - <name>Martin Scholz</name> - </respStmt> - <editor><name type="person">Zihairi, Leila</name><seg>Vorlesung DigiDok 2009</seg></editor> - <editor><name type="person">Babinsky, Anja</name><seg>Vorlesung DigiDok 2009</seg></editor> - <editor><name type="person">Bauer, Sandro</name><seg>Sommerakademie Greifswald 2008</seg></editor> - <editor><name type="person">Baumeister, Dirk</name><seg>Vorlesung DigiDok 2009</seg></editor> - <editor><name type="person">Becker, Saskia</name><seg>Sommerakademie Greifswald 2008</seg></editor> - <editor><name type="person">Behr, Martin</name><seg>Vorlesung DigiDok 2009</seg></editor> - <editor><name type="person">Deuerling, Kathrin</name><seg>Vorlesung DigiDok 2009</seg></editor> - <editor><name type="person">Gade, Katharina</name><seg>Vorlesung DigiDok 2009</seg></editor> - <editor><name type="person">Heck, Philip</name><seg>Sommerakademie Greifswald 2008</seg></editor> - <editor><name type="person">Hegewald, Rick</name><seg>Sommerakademie Greifswald 2008</seg></editor> - <editor><name type="person">Holfeld, Benjamin</name><seg>Sommerakademie Greifswald 2008</seg></editor> - <editor><name type="person">Incesu, Reha-Baris</name><seg>Sommerakademie Greifswald 2008</seg></editor> - <editor><name type="person">Krieglstein, Christina</name><seg>Sommerakademie Greifswald 2008</seg></editor> - <editor><name type="person">Kuckuk, Sebastian</name><seg>Vorlesung DigiDok 2009</seg></editor> - <editor><name type="person">Kühne, Julia</name><seg>Sommerakademie Greifswald 2008</seg></editor> - <editor><name type="person">Maucher, Helen</name><seg>Sommerakademie Greifswald 2008</seg></editor> - <editor><name type="person">Pöllmann, Katja Monika</name><seg>Sommerakademie Greifswald 2008</seg></editor> - <editor><name type="person">Preibusch, Sören</name><seg>Sommerakademie Greifswald 2008</seg></editor> - <editor><name type="person">Raschke, Julian</name><seg>Sommerakademie Greifswald 2008</seg></editor> - <editor><name type="person">Schaper, Ulf</name><seg>Sommerakademie Greifswald 2008</seg></editor> - <editor><name type="person">Schewina, Karen</name><seg>Sommerakademie Greifswald 2008</seg></editor> - <editor><name type="person">Schmitz, Philipp</name><seg>Sommerakademie Greifswald 2008</seg></editor> - <editor><name type="person">Winter, Madeleine</name><seg>Vorlesung DigiDok 2009</seg></editor> - </titleStmt> - - <publicationStmt> - <authority></authority> - <address> - <addrLine>Friedrich-Alexander-Universität Erlangen-Nürnberg</addrLine> - <addrLine>Lehrstuhl Informatik 8</addrLine> - <addrLine>Haberstrasse 2</addrLine> - <addrLine>91058 Erlangen</addrLine> - <addrLine>Deutschland</addrLine> - </address> - <availability status="restricted"> - <p>Available for academic research purposes only.</p> - </availability> - <date>2009-11-10</date> - <idno>/experimental/DeutscherPtolemaeus</idno> - </publicationStmt> - - <sourceDesc> - <bibl> - </bibl> - </sourceDesc> - </fileDesc> - - <encodingDesc> - <editorialDecl> - <p>Alle Druckfehler wurden in der Quelle durch die korrekte Form ersetzt. - Die fehlerhafte Form ist codiert mit - <choice> <orig>fehler>/orig< <reg>korr</reg> - <choice>. Generell ist es schwierig, in diesem frühneuhochdeutschen - Druck eindeutig Druckfehler zu identifizieren. Ein sehr häufiger Fall, der - *nicht* annotiert wurde, ist die Verwendung von iv statt w durch - den Drucker (in Ermangelung von w Lettern).</p> - <p>Alle Abkürzungen und Silbentrennungen wurden ebenfalls durch ein - tei:choice-Konstrukt ersetzt. Bei Abkürzungen wird <am> für - die Abkürzung und <ex> für die ausgeschriebene Form benutzt. - Bei Silbentrennungen wurde im <orig>-Teil die Schreibweise im - Original festgehalten, während im <reg>-Teil das zusammengeschriebene - Wort notiert wurde.</p> - <p>pphi ist eine Abkürzung für philosophi.</p> - <p>pphūs ist eine Abkürzung für philosophus.</p> - <p>Diverse <term ref="#Parallelen"></term> müssen noch korrigiert werden.</p> - </editorialDecl> - </encodingDesc> - - <profileDesc> - <langUsage> - <language ident="de"/> - </langUsage> - </profileDesc> - - <revisionDesc> - <change when="2008-09-06">Digitale Edition des deutschen Ptolemaeus. - Realisiert durch die Arbeitsgruppe "digitales Kulturerbe" im Rahmen der Sommerakademie - der Studienstiftung des deutschen Volkes (Greifswald 24.8.-6.9.2008)</change> - <change when="2008-10-01">Korrekturen und Endschliff für die digitale Edition auf - http://www8.informatik.uni-erlangen.de/IMMD8/Services/textfarm/ - durch J. Schneeberger</change> - <change when="2009-07-30">Korrekturen der Transskription im Rahmen der Vorlesung - "Komponenten Digitaler Bibliotheken" an der FAU im Sommersemester 2009. - http://www8.informatik.uni-erlangen.de/IMMD8/Lectures/DIGIDOK/ - G. Görz, J. Schneeberger, M. Scholz</change> - <change when="2009-11-10">Feinschliff und Generierung einer neuen digitalen Edition - in HTML, J. Schneeberger</change> - </revisionDesc> - </teiHeader> - - <text xml:lang="de"> - <body> - <pb n="1" facs="Ptolemaeus_text_01"/> - <div type="kapitel" xml:lang="lat"> - <head> - <pb n="2" facs="Ptolemaeus_text_02"/> - <lb/>Inuitatio lectoris in cosmographiam <persName><forename>claudi</forename> - <lb/><surname>ptolomei</surname> <addName>Alexandrini</addName></persName> nouiter ideomate - <choice><orig>germa-<lb/>no</orig><reg>germano<lb/></reg></choice> contextam incipit foeliciter. - </head> - <p> - <lb/> - <lb/>Pierides musas et amena vireta sororum. - <lb/>Noscere quisquis amas castaleosque lacus. - <lb/>Et maria et terras vario sub sole iacentes. - <lb/>Atque viam solis que arctipholatis onus - <lb/>Octipedis tepido quod phoebus sideris astro. - <lb/>Describet tardos efficietque dies. - <lb/>India quidve potens o quam pulcherime <placeName type="river">Gange</placeName> - <lb/>Alluis et placidi quid tenet vnda <placeName type="river">Tagi</placeName>. - <lb/>Et scitie gentes rigidum porrecta sub axem - <lb/>Quasque tenet Maurus pigmalionis opes. - <lb/>Insula quave iacet rapido circumflua <placeName type="river">Ponto</placeName> - <lb/><placeName type="island">Creta</placeName> iouis magni <placeName type="island">ciprus</placeName> et inde <placeName type="island">Rodes</placeName> - <lb/>Huc tibi nauis eat pollitum hunc posce libellum. - <lb/>Attingat portus panda carina suos. - <lb/><persName><forename>Claudius</forename></persName> edocuit pictas quas arte tabellas. - <lb/>Nomina conseruit tempore nota suo - <lb/>Nomina regnorum nostris quibus vtimur annis - <lb/>Instruit is populos flumina nota simul. - <lb/>Urbes quas <placeName type="river">rhenus</placeName> spumas atque hister harenas - <lb/><placeName type="river">Flauus</placeName> quas liquidis <placeName type="river">Istula</placeName> cingit aquis. - <lb/>Cuncta refert vasto quicquid modo clauditur orbe. - <lb/>Germana quamuis singula voce canit - <lb/>Dicere nec pudeat tandem suauissime lector. - <lb/>Vates qui tantum nobile promsit opes. - </p> - </div> - <div type="kapitel"> - <head> - <pb n="3" facs="Ptolemaeus_text_03"/> - <lb/>Ein einleitung diß buchleins - <lb/>yn die kunst Cosmographia. - </head> - <p> - <lb/> - <lb/> - <hi rend="initial">g</hi>Otelob in ewigkeith. der do - <choice><orig>mil<lb/>diklich</orig><reg>mildiklich<lb/></reg></choice> gibt. sterkt und merth die - <lb/>vornemen todtlicher menschenn. - <lb/>v<choice><ex>nd</ex><am>n̄</am></choice> in zu erkenen gibt sein - <choice><orig>schepf-<lb/>ung.</orig><reg>schepfung.<lb/></reg></choice> auff das si merken vnd - <lb/>brüfen mogen das alle ding auß ime - <lb/>urspringlich anfang vnd end haben. v<choice><ex>on</ex><am>ō</am></choice> im auch - <lb/>endtlich nach seiner vorbeseh<choice><ex>un</ex><am>ū</am></choice>g geregirt vnd - <choice><orig>ge<lb/>ordent</orig><reg>geordent<lb/></reg></choice> werd<choice><ex>en</ex><am>ē</am></choice>. Auch als vnß <persName><forename>aristotiles</forename></persName> - <choice><orig>beschrei<lb/>bet.</orig><reg>beschreibet.<lb/></reg></choice> das dy vndersten von den obersten ir wesen - <choice><orig>ha-<lb/>hen</orig><reg>haben<lb/></reg></choice> alls durch ein eynfluß der hymel vnd - <choice><orig>plane<lb/>ten.</orig><reg>planeten.<lb/></reg></choice> das da<choice><ex>nn</ex><am>n̄</am></choice> geschicht zam durch ein hantgezew - <lb/>gottis aller ding regirer. So aber v<choice><ex>nd</ex><am>n̄</am></choice> m<choice><ex>en</ex><am>ē</am></choice>schliche - <lb/>vornunft allzeit schneller vnnd begreiflicher ist - <lb/>zu vornemen vnd behalten dise ding di sich vor - <lb/>angesicht unser ougen reglich bewegen vnd - <choice><orig>lei-<lb/>plich</orig><reg>leiplich<lb/></reg></choice> erzeigen. dann dise ding dy do ferer vonn - <lb/>vnserz gesicht ob<choice><ex>en</ex><am>ē</am></choice> in ewiger beweglikeit gesaczt - <lb/>se<choice><ex>in</ex><am>ī</am></choice> vnbegreiflich nach vnser leiplikeit wol aber - <lb/>nach vnserm vornemen. von deßwegen vnd wie - <lb/>dann mer geneigtt sein von natur irkante dingk - <lb/>von weisen mennern zu wissen vnd zu <choice><orig>irforsch-<lb/>en.</orig><reg>irforschen.<lb/></reg></choice> die dobei vns vnder d<choice><ex>em</ex><am>ē</am></choice> firmament sein vnd - <lb/>ir wesen haben. da<choice><ex>nn</ex><am>n̄</am></choice> dise ding di vber vns wirk<choice><ex>en</ex><am>ē</am></choice> - <lb/>die do etlicher mosse vnser vornemen vbertret<choice><ex>en</ex><am>ē</am></choice> - <pb n="4" facs="Ptolemaeus_text_04"/> - <lb/>Sulch vnd erste ding zu yrforschen vnd - <choice><orig>begreiff<lb/>en.</orig><reg>begreiffen.<lb/></reg></choice> lernet vnß der yrfarne <persName><roleName>maister</roleName><surname>Ptholomeus</surname> - <lb/><forename>claudius</forename><addName>von alexandria</addName></persName> yn dem buch do er - <choice><orig>be-<lb/>schreibt</orig><reg>beschreibt<lb/></reg></choice> vnd zeigt die gestalt diser inwonenden - <lb/>werlt mit schonen figuren mit rechter mosse der - <lb/>kunst geometrey wo yezlichs teyl der <choice><orig>wert</orig><reg>werlt</reg></choice> vnd - <lb/>dem hymel sey geleg<choice><ex>en</ex><am>ē</am></choice>. welcher egemelter - <persName><surname><choice><orig>ptolo-<lb/>meus</orig><reg>ptolomeus<lb/></reg></choice></surname></persName> nicht mit cleinem <choice><sic>snnder</sic><corr>sunder</corr></choice> mit grossem vleyß - <lb/>vnd mu dy recht vßteylung diser werlt mit aller - <lb/>gerechtigkeyt deß zirkels hat erfunden. welche - <lb/>kunst ym latein genant wirt <term xml:id="cosmo1" ref="#Cosmographia">Cosmographia</term>. Zu - <lb/>teutsch wirt also bedeut <q corresp="#cosmo1">Ein kunst dye do vns - <choice><orig>be<lb/>zeichent</orig><reg>bezeichent<lb/></reg></choice> vnd zuuersten gibt dy gelegenheyt - <choice><orig>vn-<lb/>ser</orig><reg>vnser<lb/></reg></choice> bekanten werlt</q>. das ist. vnsers obirn - <choice><orig>emispe<lb/>ry.</orig><reg>emispery.<lb/></reg></choice> wo iders teil der selben welt gelegen sey. - <choice><orig>v<choice><ex>mb</ex><am>m̄</am></choice>-<lb/>flossen</orig><reg>v<choice><ex>mb</ex><am>m̄</am></choice>flossen<lb/></reg></choice> mit mere. ouch was gutz reichtumer. - <choice><orig>frü-<lb/>chte</orig><reg>früchte.<lb/></reg></choice> vnd wasser. siten der m<choice><ex>en</ex><am>ē</am></choice>schen. yn einem - <choice><orig>ycz-<lb/>lichen</orig><reg>yczlichen<lb/></reg></choice> teyl sein mag. Dorczu vnd welche - <choice><orig>namhaff<lb/>tigsten</orig><reg>namhafftigisten<lb/></reg></choice> wasser durchfliessen vnd durchschneyd<choice><ex>en</ex><am>ē</am></choice> - <lb/>das ertrich. streichend yn yrem alueo. das ist. yn - <lb/>yren stramen. ouch von wanne si vrspringklich - <lb/>komen seyn. vnd wideru<choice><ex>mb</ex><am>m̄</am></choice> yn das mer sich - <choice><orig>ver-<lb/>mischen</orig><reg>vermischen<lb/></reg></choice> vnd vorbergen - <lb/> - <lb/> - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Uon der gemeinen figur <persName><surname>ptolomei</surname></persName> - <lb/>gegen dises büchleins Figur. - </head> - <p> - <pb n="5" facs="Ptolemaeus_text_05"/> - <lb/> - <hi rend="initial">N</hi>Ach ynhalt vnd mein<choice><ex>un</ex><am>ū</am></choice>g der - <choice><orig>gemei-<lb/>nen</orig><reg>gemeinen<lb/></reg></choice> figur <persName><surname>ptolomei</surname> <forename>claudi</forename></persName>. wirt - <choice><orig>auß<lb/>gezogen</orig><reg>außgezogen<lb/></reg></choice> vnd genomen ein - <choice><orig>sunderli-<lb/>che</orig><reg>sunderliche<lb/></reg></choice> gemeine figur dysser bekantenn - <lb/>werlt klein vnd gefug dysem - <choice><orig>buch-<lb/>lein</orig><reg>buchlein<lb/></reg></choice> dienend mit aller notturfft der grad. linien. - <lb/>paralelln. climaten vnd meridian becleidet. U<choice><ex>on</ex><am>ō</am></choice> - <lb/>welcher figur <persName><surname>ptolomei</surname></persName> zumercken ist das dises - <lb/>buchleins figur ein ander gestalt hat. wann dise - <lb/>gantz <q corresp="zirkel1">rund</q> ist nach <term xml:id="zirkel1" ref="#ZirkelsMass">zirkels moß</term>. mit yren - <choice><orig>volko-<lb/>men</orig><reg>volkomen<lb/></reg></choice> graden vßgeteilt. nach welchen graden alle - <lb/>maß gert diser kunst. vnd ouch gancz vber eine - <lb/>trift mit der mein<choice><ex>un</ex><am>ū</am></choice>g des offtgedacht<choice><ex>en</ex><am>ē</am></choice> <persName><surname>ptolomei</surname></persName> - <lb/>wywol er yn seyner figur seczt alle dise land die - <lb/>do menschlicher erfarung kunth sein vnnd den - <lb/>zirkel der erden nicht volkomen vorendet - <choice><orig>sund<lb/></orig><reg>sunder<lb/></reg></choice> beschleust gegen mittag mit dem sibenden - <choice><orig>para-<lb/>lell</orig><reg>paralell<lb/></reg></choice> vnd gegen mitternacht mit dem 6ten paralell. - <lb/>das do geschehen ist von deßwegen. so sich die - <lb/>bekanten teil diser weltmer cziehen yn dy lenge - <lb/>dan yn die breyt. alls von dem abend yn morgen - <lb/>vnd wideru<choice><ex>mb</ex><am>m̄</am></choice>. so hat er dise <choice><sic>fignr</sic><corr>figur</corr></choice> vberlengt. vff - <lb/>das er gerewmer dy drey teil der werlt doreyn - <lb/>hat mugen pringen. vnd waß ym abgieng an der - <lb/>breit das er es brecht <choice><ex>in</ex><am>ī</am></choice> die leng. vnd bedeutet - <choice><orig>dor-<lb/>mit</orig><reg>dormit<lb/></reg></choice> die l<choice><ex>en</ex><am>ē</am></choice>g diser keulit<choice><ex>en</ex><am>n̄</am></choice> welt. so ist n<choice><ex>un</ex><am>ū</am></choice> off<choice><ex>en</ex><am>ē</am></choice>bar das - <lb/>di kaul des ertrichs r<choice><ex>un</ex><am>ū</am></choice>d se<choice><ex>in</ex><am>ī</am></choice> mus wi wol das erttrih - <lb/>vberal mit m<choice><ex>en</ex><am>ē</am></choice>sch<choice><ex>en</ex><am>ē</am></choice> nit besaczt. hieru<choice><ex>mb</ex><am>m̄</am></choice> dise figur - <lb/>dises buchleins nach vnuerzucktes zirkels mos - <pb n="6" facs="Ptolemaeus_text_06"/> - <lb/>aygentlich gantz keulicht gemacht ist ouch ytzlich - <lb/>land vnttir seinem sunderlichen meridian vnnd - <lb/>paralell gesaczt eben wie <persName><surname>ptolomeus</surname></persName> lernet - </p> - <p> - <lb/>Das aber dis teil gegen mittag außwendig dem. - <lb/>zirkel des <choice><orig>stenboks</orig><reg>steinboks</reg></choice> mit lande nicht belegt ist - <lb/>das macht vmbekantheit der selben erden von - <lb/>nyemands erforst. ydoch ist es ein teil der - <choice><orig>rond<lb/>hait</orig><reg>rondhait<lb/></reg></choice> diser kaul das mag man also volfuren. So - <lb/><persName><surname>ptolomeus</surname></persName> die geschiklikeit seiner figur <term ref="#ZirkelsMass">rundt</term> - <lb/>gemacht hat yn orient vnd in occident noch <term ref="#ZirkelsMass"><choice><orig>czir-<lb/>kels</orig><reg>czirkels<lb/></reg></choice></term> maß. volget hierauß das si diß gestalt haben - <lb/><choice><sic>mnß</sic><corr>muß</corr></choice> ym mittag vnd mitternacht. des zu einer - <choice><orig>be<lb/>sern</orig><reg>besern<lb/></reg></choice> bedeutung ist dise figur gancz keulicht - <choice><orig>ge-<lb/>ordent</orig><reg>geordent<lb/></reg></choice> der vrsach halben so alle spern der ober<choice><ex>en</ex><am>ē</am></choice> - <lb/>rechte rondheit deß zirkels halten so geleich - <choice><orig>bil<lb/>det</orig><reg>bildet<lb/></reg></choice> sich <choice><sic>anch</sic><corr>auch</corr></choice> ine di kaul des ertrichs - <lb/> - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Uon inhalt dises - <lb/>buchleins figur. - </head> - <p> - <lb/> - <lb/> - <hi rend="initial">f</hi>Urth zu reden von inhalt diser - <choice><orig>geg<choice><ex>en</ex><am>ē</am></choice>wer<lb/>tigen</orig><reg>geg<choice><ex>en</ex><am>ē</am></choice>wertigen<lb/></reg></choice> figur ist zu wissen das <q corresp="#para1">dise zirkels - <lb/><choice><sic>linieu</sic><corr>linien</corr></choice> di do geen nach der leng von - <choice><orig>ori<lb/>ent</orig><reg>orient<lb/></reg></choice> in occident</q> werden genant <term xml:id="para1" ref="#Paralelli">paralelli</term> yn der - <lb/>gemein vnd selden dy stelle der recht gesaczten - <lb/>paralelln beruren. auch werden <q corresp="#clima1">dise vorgenant<choice><ex>en</ex><am>ē</am></choice> - <lb/>vnd bezeichent<choice><ex>en</ex><am>ē</am></choice> zirkels linien</q> genant <term xml:id="clima1" ref="#climata">climata</term> yn - <lb/>diser figur mit vnderschaid. vnd werden allzeyt - <pb n="7" facs="Ptolemaeus_text_07"/> - <lb/>von dem Equinoxial das ist von dem geleicher - <lb/>tages vnd nacht nach den graden gerechent als - <lb/>hiernach folgen wirt yn dem capitel von den - <choice><orig>cli<lb/>mata</orig><reg>climata<lb/></reg></choice> vnd paralelln. Es werden auch 5 grad - <choice><orig>ge-<lb/>nomen</orig><reg>genomen<lb/></reg></choice> yn der gemein zwischen den gezogen - <choice><orig>ly-<lb/>nien</orig><reg>lynien<lb/></reg></choice> von orient in occident der vrsach halb. das - <lb/>man bald in gemeinheit finden mag wo iczlichs - <lb/>land gelegen ist vnd wie vil grad von dem - <choice><orig>equi<lb/>noxial.</orig><reg>equinoxial.<lb/></reg></choice> durh welche grad dan vßzurechen ist - <choice><orig>wy<lb/>weit</orig><reg>wyweit<lb/></reg></choice> ein land. stät oder gegenheit aus der zonne - <lb/>weg gelegen ist das ist vß d<choice><ex>em</ex><am>ē</am></choice> <term xml:id="zodiac1" ref="#zodiaco">zodiaco</term> zu deutsch - <lb/>genant. <q corresp="#zodiac1">Tiersteig</q>. Dise zirkels linien die do - <lb/>gen nach der breit das ist vom mittag in - <choice><orig>mitter-<lb/>nacht</orig><reg>mitternacht<lb/></reg></choice> baide polos treffend. das sind gesproch<choice><ex>en</ex><am>ē</am></choice> - <lb/>meridians linien durchschneiden alle paralellen - <lb/>vnd climata in crüczweiß <term xml:id="poli1" ref="#Poli">Poli</term> aber werden - <choice><orig>ge<lb/>genant</orig><reg>genant<lb/></reg></choice> <!-- gegenant ?? --><q corresp="#poli1">die axen des hymels</q> vnd der sind czwen - <lb/>articus vnd <placeName type="continent">antarticus</placeName>. der do gen<choice><ex>an</ex><am>ā</am></choice>t wirt <term xml:id="articus1" ref="#articus">articus</term> - <lb/>das heist vff teutsch <q corresp="#articus1">der wagen am himel</q> den - <choice><orig>se<lb/>hen</orig><reg>sehen<lb/></reg></choice> alle die. dy do ligen vßwendig dem czirkel - <lb/>deß kreps gegen mitternacht dorinne die zonne - <lb/>iren stand hat vnd widerker. von dann dieser - <choice><orig>czir<lb/>kel</orig><reg>czirkel<lb/></reg></choice> wirt geheissen <term xml:id="tropi1" ref="#tropicus">Tropicus</term> vff kriechisch czw - <lb/>deutsch <q corresp="#tropi1">ein widerker</q> - </p> - <p> - <lb/>Der do aber genant wirt <placeName type="continent">Antarticus</placeName> den kon - <lb/>wir noch alle die die do sein gegen mitternacht - <lb/>gelegenn nicht gesehenn. do von dann <persName><surname>Uirgilius</surname></persName> - <pb n="8" facs="Ptolemaeus_text_08"/> - <lb/>schreibt berurnd die eygenschafft diser czweier - <lb/>poli vnd spricht - <lb/> - <lb/><q xml:lang="lat">Hic vertex nobis semper sublimisat illum sub - <lb/>Pedibus stix alta vident manesque prof<choice><ex>un</ex><am>ū</am></choice>di </q> - </p> - <p> - <lb/>Domit gibt er zuuerston wie das alweg der polus - <lb/>articus mit seinem hoen irheben sich vns yn - <choice><orig>Eu-<lb/>ropa</orig><reg><placeName type="continent">Europa</placeName><lb/></reg></choice> vnd den gegen mittirnacht tut irczeigen. - </p> - <p> - <lb/>Der ander aber genant <placeName type="continent">antarticus</placeName> wirt - <choice><orig>vnder-<lb/>ligen</orig><reg>vnderligen<lb/></reg></choice> disem artico gegen mittag den do sehen di - <lb/>tiefsten uff dem selben teil gegen vns ligend - <lb/> - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Uon den drey Zirkeln. - <lb/>der Sunnen - </head> - <p> - <lb/> - <lb/> - <hi rend="initial">A</hi>uch ist zuwissen das dise dry zirkel rot - <lb/>verzeichent yn diser figur. sein genantt - <lb/>zirkel der sunnen von deßwegen so sy - <lb/>kompt mit irem louff in diser czirckell - <lb/>eyn. so macht si <choice><sic>vorwandlnng</sic><corr>vorwandlung</corr></choice> der tag. - <choice><orig>vßgenom<lb/>en</orig><reg>vßgenomen<lb/></reg></choice> ym <choice><sic>Eqninoxial</sic><corr>Equinoxial</corr></choice> so macht si geleich tag vnd - <choice><orig>na-<lb/>cht</orig><reg>nacht<lb/></reg></choice> in der ganczen werlt. vnd diser ist der mitelst - <lb/>zirkel. vnd wirt genant ein gleicher. von - <choice><orig>deßwe<lb/>gen</orig><reg>deßwegen<lb/></reg></choice> das er yn aller werlt gleycht. tag vnd nachtt. - <lb/>vnd alle menschen dy vnd disem zirkel <choice><orig>er</orig><reg>der</reg></choice> - <choice><orig>son-<lb/>n<choice><ex>en</ex><am>ē</am></choice></orig><reg>sonn<choice><ex>en</ex><am>ē</am></choice><lb/></reg></choice> wonen die sehen beyde polos an dem hymel - <lb/>das do sunst yn der ganczen werlt nicht gesche<choice><ex>en</ex><am>ē</am></choice> - <lb/>kan. auch teilet diser zirkel das erttrich gleich yn - <pb n="9" facs="Ptolemaeus_text_09"/> - <lb/>zwey teil das ain leit gegen mittag. das ander - <choice><orig>ge<lb/>gen</orig><reg>gegen<lb/></reg></choice> mitternacht. von orient in occident - </p> - <p> - <lb/>Disen zirkel equinoxial durchschneidet ein <choice><orig>and</orig><reg>ander</reg></choice> - <lb/>zirkel winkelrecht gend von einem polo yn den - <lb/>andern vnd ouch von orient in occident. der selb - <lb/>zirkel wirt anders genant <term xml:id="lauf1" ref="#Sonnenlauf">Orison vniuersalis</term> - </p> - <p> - <lb/>das ist <q corresp="#lauf1">ein gemeiner zirkel dorynn die sonn auff - <lb/>vnd nider geet</q>. diser halben kaul des oebersten - <lb/>emispery das ist. vnser bekanten welt. diese czw<choice><ex>en</ex><am>ē</am></choice> - <lb/>zirkel teilen di kaul des ertrichs yn vierteil - <choice><orig>wel-<lb/>cher</orig><reg>welcher<lb/></reg></choice> teil eins. recht mit menschen besaczt ist - <choice><orig>da-<lb/>uon</orig><reg>dauon<lb/></reg></choice> dy Cosmographia schreibt <!-- space --> Der ander - <choice><orig>zir-<lb/>kel</orig><reg>zirkel<lb/></reg></choice> gegen mitternacht wirt genant Tropicus - <choice><orig>c<choice><ex>an</ex><am>ā</am></choice>-<lb/>cri</orig><reg>c<choice><ex>an</ex><am>ā</am></choice>cri<lb/></reg></choice>. dyser zirkel ist gesaczt von dem Equinoxial - <lb/>nach der breite 23 grad so die sunn yn disen - <choice><orig>zirk<lb/>el</orig><reg>zirkel<lb/></reg></choice> kompt kan si nicht weiter gen <term xml:id="zodiac2" ref="#zodiaco">vß dem zodiaco</term> - <lb/>das ist vß <term ref="#zodiaco">dem tiersteig</term> vnd muß widerker<choice><ex>en</ex><am>ē</am></choice> - <choice><orig>zu<lb/>ruk</orig><reg>zuruk<lb/></reg></choice> gegen dem equinoxial vnd so die so<choice><ex>nn</ex><am>n̄</am></choice>e noch - <lb/>pei disem zirkel vmschweifft macht si vns yn - <choice><orig>eu<lb/>ropa</orig><reg><placeName type="continent">europa</placeName><lb/></reg></choice> die lengsten tag ydoch einem land mer da<choice><ex>nn</ex><am>n̄</am></choice> - <lb/>dem andern Dise menschen aber die do ligenn - <lb/>zwischen disen zwei<choice><ex>en</ex><am>ē</am></choice> zirkeln die haben ym Jor. - <lb/>zwir <term xml:id="zenit1" ref="#Zenit">cenit</term> <q corresp="#zenit1">das ist das yn dy zunn gleich vber - <choice><orig>ir<lb/>en</orig><reg>iren<lb/></reg></choice> heuptern stet v<choice><ex>nd</ex><am>n̄</am></choice> kein schatten von yn werff<choice><ex>en</ex><am>ē</am></choice></q>. - <lb/>alls die menschen sein yn der geg<choice><ex>en</ex><am>ē</am></choice>heit <placeName type="island">meroe</placeName> ym - <lb/>flisse <placeName type="river">nylo </placeName>geleg<choice><ex>en</ex><am>ē</am></choice> das es aber zwier geschicht ist di - <lb/>sach so die sonne sich von dem equinoxial <choice><orig>necht</orig><reg>neicht</reg></choice> - <lb/>geg<choice><ex>en</ex><am>ē</am></choice> d<choice><ex>en</ex><am>ē</am></choice> <term ref="#Krebs">tropico c<choice><ex>an</ex><am>ā</am></choice>cri</term> das ist v<choice><ex>on</ex><am>ō</am></choice> <persName><forename>gregori</forename></persName> biß uf viti - <pb n="10" facs="Ptolemaeus_text_10"/> - <lb/>so trifft des czenit ein mal. vnd so sich die zonne - <lb/>wider gibt zu dem equinoxial So trifft es aber - <lb/>ein mal das ist von viti bis vff lamperti alls - <choice><orig>gele<lb/>gen</orig><reg>gelegen<lb/></reg></choice> ist die insul <placeName type="island">meroe</placeName> Ym andern paralell vom - <lb/>equinoxial vnd alle stete die in disem paralell - <choice><orig>li<lb/>gen</orig><reg>ligen<lb/></reg></choice> die haben czwen cenit. die menschen aber di - <lb/>do gleich ligen vnder disem <q corresp="#krebs1">zirkel deß krepsen</q> - <lb/>die haben nur ein mal zenit im iar ein stund eins - <lb/>tags so die sonne stand hat also ist gelegen di stat - <lb/>Siene im 6 paralell dauon <persName><forename>Lucanus</forename></persName> schreibt - </p> - <p> - <lb/>Umbras nusqu<choice><ex>am</ex><am>ā</am></choice> flectente siene. das ist. das in - <lb/>der stat siene der menschen schatten nimer - <choice><orig>gelen<lb/>get</orig><reg>gelenget<lb/></reg></choice> wirt. vorstanden in einer mittags stunden so - <lb/>die zonne im kreps ist. - <lb/> - </p> - <p> - <lb/>Der dritte zirckell wirt genant <term xml:id="steinbock1" ref="#Steinbock">Tropicus <choice><orig>capri<lb/>corni.</orig><reg>capricorni.<lb/></reg></choice></term> das ist <q corresp="#steinbock1">ein widerkerer des ste<choice><ex>in</ex><am>ī</am></choice>boks</q> so dy - <lb/>zonne in disem zirkel schweift. macht si vnns in - <lb/><placeName type="continent">europa</placeName> kurcze tag. dem teil aber gegen mittage. - <lb/>lenger dy tag wie wol nicht vil besaczter lannd - <lb/>do sein von weg<choice><ex>en</ex><am>ē</am></choice> grosser hicze iedoch - <choice><orig>verpring<lb/>er</orig><reg>verpringet<lb/></reg></choice> die sunne eben irenn lauff in disem Steinbok - <lb/>alls im kreps. diser zirkel leit auch eben als vil - <lb/>von dem equinoxial als der ander im 23 grad - <lb/>vnd so di sonne sich nehet czu disem zirkel macht - <lb/>si vns die wir gelegenn sein in dem sübendenn - <lb/>clima die tage kurczer also das der tag - </p> - <p> - <pb n="11" facs="Ptolemaeus_text_11"/> - <lb/>nur viii stund lang wirt. vnd hebt an zu geen v<choice><ex>on</ex><am>ō</am></choice> - <lb/>equinoxial am tag lamperti vnd get bis auff - <choice><orig>Lu-<lb/>cie</orig><reg>Lucie<lb/></reg></choice> so trit si in den zirkel genant capricorni. dorinn - <lb/>sy muß wider ker<choice><ex>en</ex><am>ē</am></choice>. - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Hie nach geschriben sagt es - <lb/>von dem <term xml:id="zodiac3" ref="#zodiaco">zodiaco</term>. - </head> - <p> - <lb/> - <lb/> - <hi rend="initial">D</hi>Ise vorgenante drei zirkel beruren vnd - <lb/>treffenn den zodiacum. an vier endenn - <lb/>vnd widerum. auch wirt er in dise drey - <lb/>zirkel gelegt vber ort. also das er si aller beruret - <lb/>auff teutsch wirt er gesprochen <q corresp="#zodiac3">der tiersteig</q> von - <lb/>deßwegen das die selben zeichenn die er tregtt - <lb/>tiren gegleicht werd<choice><ex>en</ex><am>ē</am></choice> pey vns hyniden wi wol - <lb/>das si an ym selbst sterne sein so haben dy altte<choice><ex>nn</ex><am>n̄</am></choice> - <lb/>astronomi dy selben sterne also figurirt vnd mit - <lb/>linien zuhuff gezogen hy vnden. vnd worzu sich - <lb/>dy selb figur geschickt oder gegleichet hat. hab<choice><ex>en</ex><am>ē</am></choice> si - <lb/>doruon einen namen genomen vnd auch von - <choice><orig>d</orig><reg>der</reg></choice> - <lb/>eygenschafftt desselben gestyrnis alles eczliche - <lb/>wellen der selbenn zeichen ym zodiaco sein 12 - <lb/>von welchen der zodiacus wirt genant <q corresp="#signi1">eyn - <choice><orig>zey-<lb/>chen</orig><reg>zeychen<lb/></reg></choice> trager</q>. Zu latein <term xml:id="signi1" ref="#Signifer">Signifer</term>. dise 12 zeichenn - <lb/>seyn genant vnd gemerkt wye hyenach folget. - </p> - <p> - <lb/>Der stier Der ochs Dy zwiling Der crebs - <lb/>Der lev Die iunckfrav Die wage. Der - <lb/>scorpio Der schücz Der steinbok Der - <choice><orig>wa-<lb/>sserm<choice><ex>an</ex><am>ā</am></choice></orig><reg>wasserm<choice><ex>an</ex><am>ā</am></choice><lb/></reg></choice> Die vische. so nun die sunn durch dise - <pb n="12" facs="Ptolemaeus_text_12"/> - <lb/>12 zeichen lauffet vnd kumpt in das zeichen des - <lb/>stiers vnd der wag. welche zwey zeychen geg<choice><ex>en</ex><am>ē</am></choice> - <lb/>eynander sten so macht si vns gleich tag v<choice><ex>nd</ex><am>n̄</am></choice> nacht - <lb/>durch die gancz werlt. wann si aber kompt yne - <lb/>das zeichen des creps do pleibt si vnd kert - <choice><orig>wider<lb/>umb</orig><reg>widerumb<lb/></reg></choice> von mitternacht yn mittag do macht si den - <lb/>menschen gegen mitternacht ligend grossen tag - <lb/>von danne wirt genant dises zirkels des kreps - <lb/>oberster grad der sunn<choice><ex>en</ex><am>ē</am></choice> stand des sumers. wann - <lb/>si aber kompt in das zeych<choice><ex>en</ex><am>ē</am></choice> des ste<choice><ex>in</ex><am>ī</am></choice>bocks maht - <lb/>si vns kleine tag vnd kan ouch nicht forder vnd - <lb/>get wider gegen mitternacht von deßwegen wirtt - <lb/>genant dises zirkels hohe. zu latein <term xml:id="wende1" ref="#Sonnenwende">Solsticium</term> - <lb/>hyemale das ist <q corresp="#wende1">der sunnen stand deß winters</q> - <lb/> - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Item welche land zu geeygent - <lb/>werden yn sunderheit disen 12 - <lb/>zeichenn. - <lb/> - </head> - <p> - <lb/> - <hi rend="initial">a</hi>Uff ettliche land vnd stett habenn diese 12 - <lb/>zeichen yn sunderheit <choice><sic>zu ueigung</sic><corr>zueigung</corr></choice>. - <choice><orig>Nem-<lb/>lich</orig><reg>Nemlich<lb/></reg></choice> das zeichen wider. Alls haly halbenragell - <lb/>schreibt sicht an das ertrich <placeName type="country">Babilonie</placeName>. <placeName type="country">armeni<choice><ex>en</ex><am>ē</am></choice></placeName> - <lb/><placeName type="country">moßquitten</placeName> <placeName type="country">klein polen</placeName> vnd mitt dem planeten - <lb/>Marte die stat <placeName type="city">Crokav</placeName> Und das zeichen ist von - <lb/>natur. alls die meister der edl<choice><ex>en</ex><am>n̄</am></choice> k<choice><ex>un</ex><am>ū</am></choice>st Astrologie - <choice><orig>sa<lb/>gen.</orig><reg>sagen.<lb/></reg></choice> warm truken vnd feurig vnd wirt - <choice><orig>zugeeig-<lb/>net</orig><reg>zugeeignet<lb/></reg></choice> dem houpt des menschen gut oderlossen. - <pb n="13" facs="Ptolemaeus_text_13"/> - </p> - <p> - <lb/><term xml:id="thaurus1" ref="#Thaurus">Thaurus</term>. <q corresp="#thaurus1">das zeichen des ochsen</q> sicht an das - <lb/>ertrich nichin megir an dy zanzen vnd das - <choice><orig>gepir<lb/>ge</orig><reg>gepirge<lb/></reg></choice> <placeName type="mountain">daleigk</placeName> deßgleichen <placeName type="country">klein polln</placeName> vnd yn - <choice><orig>sunder-<lb/>heit</orig><reg>sunderheit<lb/></reg></choice> mit dem planeten venus. Dy stat <placeName type="city">poßen</placeName> von - <lb/>natur ist das zeichen kalt truken vnd irdisch - <choice><orig>czu<lb/>geeigent</orig><reg>czugeeigent<lb/></reg></choice> dem hals des menschen. bos daryn - <choice><orig>oder<lb/>lossen.</orig><reg>oderlossen.<lb/></reg></choice> - </p> - <p> - <lb/><term ref="#Zwillinge">Gemini</term>. <q corresp="#Zwillinge">das zeichen der zwiling</q> sicht an <placeName type="country">clein - <lb/>armenien</placeName>. <placeName type="country">iurgen</placeName>. <placeName type="countrry">egipt<choice><ex>en</ex><am>ē</am></choice></placeName>. <placeName type="country">barchan</placeName> vnd <placeName type="country">engelland</placeName> - <lb/>mit dem planeten mercurio dy stat <placeName type="city">Neyss</placeName> <choice><sic>ligeng</sic><corr>ligend</corr></choice> - <lb/>vßwendig dem 7 clima yn dem 15 paralell von - <lb/>d<choice><ex>em</ex><am>ē</am></choice> occident 40 grad vnd 1/2 dis zaich<choice><ex>en</ex><am>ē</am></choice> ist warm - <lb/>feucht v<choice><ex>nd</ex><am>n̄</am></choice> lufftig vnd ist zugegeb<choice><ex>en</ex><am>ē</am></choice> den schultern - <lb/>armen vnd henden vnnucz in oderlossen - <lb/> - </p> - <p> - <lb/><term ref="#Krebs2">Cancer</term>. <q corresp="#Krebs2">das zeichen des kreps</q> helt vnder ym - <lb/>das ertrich <placeName type="country">Barbarischken</placeName> yn <placeName type="continent">affrica</placeName>. <placeName type="country">franckrich</placeName> - <lb/>vnd teilhafftig yn <placeName type="country">egipto</placeName> vnd aller ynsuln yn - <choice><orig>ori-<lb/>ent</orig><reg>orient<lb/></reg></choice> vnd das land <placeName type="country">Prevssen</placeName> mit dem planet<choice><ex>en</ex><am>ē</am></choice> <term ref="#Mond"><choice><orig>Lu-<lb/>na</orig><reg>Luna<lb/></reg></choice></term> vff teutsch <q corresp="#Mond">mon</q> genant sicht an di stat <placeName type="city">danczgk</placeName> - <lb/>dis zeichen ist von natur kalt feucht vnd - <choice><orig>wesse-<lb/>rig</orig><reg>wesserig<lb/></reg></choice> vnd wirt zu gegeben der brust der lung<choice><ex>en</ex><am>ē</am></choice> v<choice><ex>nd</ex><am>n̄</am></choice> - <lb/>dem magen wandelbar yn oderlossung - <lb/> - </p> - <p> - <lb/>Lew. Das zeychen dess lawen hatt vnder ym - <lb/><placeName type="country">Turkey</placeName>. <placeName type="country">Lombardey</placeName> das erttrich <placeName type="country">gogk</placeName> vnd - <placeName type="country"><choice><orig>ma-<lb/>gogk</orig><reg>magogk<lb/></reg></choice></placeName> vnd dy ynsul <placeName type="island">Sicilia</placeName> vnd das land <placeName type="country">persie</placeName> - <lb/>vnd <placeName type="country">Behemen</placeName> Mitt dem planeten Sol - <pb n="14" facs="Ptolemaeus_text_14"/> - <lb/>die stat <placeName type="city">Prag</placeName>. Diß zeichen ist warm vnd trucken - <lb/>vnd feurig vnd hat des menschen ruck vnd - <choice><orig>sei<lb/>ten</orig><reg>seiten<lb/></reg></choice> in gewalt vnd ist sorglich oderlassen. - </p> - <p>ertrich nichin megir an dy zanzen vnd das - <lb/><term ref="#Jungfrau">Uirgo</term>. <q corresp="#Jungfrau">das zeichen der iunkfraven</q> hat vnder - <lb/>im gancz <placeName type="country">kriechenland</placeName> die Jnsul <placeName type="island">creta</placeName> oder - <choice><orig>can<lb/>dia.</orig><reg><placeName type="island">candia</placeName>.<lb/></reg></choice> das ertrich <placeName type="country">almaseil</placeName>. vnd <placeName type="country">Slesie</placeName> mit d<choice><ex>em</ex><am>ē</am></choice> - <choice><orig>pla<lb/>neten</orig><reg>planeten<lb/></reg></choice> mercurio die stat <placeName type="city">Breßlav</placeName> Diß zaichen ist - <lb/>von natur kalt truken vnd irdisch besehen dy - <choice><orig>in-<lb/>geweid</orig><reg>ingeweid<lb/></reg></choice> des menschen. mittel in oderlassen - </p> - <p> - <lb/><term ref="#Waage">Libra</term>. <q corresp="#Waage">das zeichen der wag</q> hat vnder ym daß - <lb/>ertrich der Cristen bis an <placeName type="continent">affric<choice><ex>am</ex><am>ā</am></choice></placeName> vnd das ertrich - <lb/><placeName type="country">Cuß</placeName> vnd <placeName type="country">gros alexandriaz</placeName>. ouch das ertrich - <placeName type="country"><choice><orig>oste<lb/>reich.</orig><reg>ostereich.<lb/></reg></choice></placeName> mit dem planeten mercurio. di stat - <placeName type="city"><choice><orig>nüren<lb/>berg.</orig><reg>nürenberg.<lb/></reg></choice></placeName> diß zeichen ist von natur warm feucht vnd - <lb/>lufftig des menschen nabel vnd vndersten leyb - <lb/>bewarend. nutzlich im oderlossen. - </p> - <p> - <lb/>Scorpio das zaichen helt vnder im die erde. - <lb/><placeName type="country">albiges</placeName> <placeName type="country">arabiam</placeName>. vnd <placeName type="country">barbariam</placeName>. <placeName type="country">romaniam</placeName> v<choice><ex>nd</ex><am>n̄</am></choice> - <lb/>das ertrich <placeName type="country">Zemonem</placeName>. auch <placeName type="country">groß arabiam</placeName> Mitt - <lb/>dem planeten marte. die stat <placeName type="city">padwa</placeName>. diß zeichen - <lb/>ist von natur kalt v<choice><ex>nd</ex><am>n̄</am></choice> feucht des menschen scham - <lb/>vorwarnde - </p> - <p> - <lb/><term ref="#Schütze">Sagitarius</term>. <q corresp="#Schütze">das zeichen des schuczen</q> hat - <choice><orig>vn-<lb/>der</orig><reg>vnder<lb/></reg></choice> im das ertrich <placeName type="country">aziud</placeName>. <placeName type="country">hispaniam</placeName> <placeName type="island">ciliciam</placeName>. - <placeName type="island"><choice><orig><choice><sic>sar<lb/>dinaim.</sic><corr>sar<lb/>diniam.</corr></choice></orig><reg><choice><sic>sardinaim.<lb/></sic><corr>sardiniam.<lb/></corr></choice></reg></choice></placeName> mit dem planeten ioui das erdtrich von - <lb/><placeName type="country">hungern</placeName> dartzu <placeName type="city">wien</placeName>. <placeName type="city">ofen</placeName>. <placeName type="city">merhernn</placeName>. <placeName type="city">olmuntz</placeName>. - <lb/>Das zaichenn ist von natur warm trucken vnnd - <lb/>feurig den beinen vber dem knie zugeschriben. - <pb n="15" facs="Ptolemaeus_text_15"/> - </p> - <p> - <lb/><term ref="#Steinbock2">Capricornus</term>. <q corresp="#Steinbock2">das zaichen des steinbocks</q> hat - <lb/>vnder im das ertrich <placeName type="country">azim</placeName>. <placeName type="country">cheif</placeName> <placeName type="country">albabar</placeName>. <placeName type="country">ioazatt</placeName> - <lb/><placeName type="country">yndian</placeName>. <placeName type="country">constantinopolim</placeName> vnd des gantzen - <choice><orig>occi-<lb/>dents</orig><reg>occidents<lb/></reg></choice> port. mit dem land <placeName type="country">lausitz</placeName> vnd dy stat <placeName type="city">loka</placeName> - <lb/>mit dem planeten saturno. diß zaichen ist von - <choice><orig>na<lb/>tur</orig><reg>natur<lb/></reg></choice> kalt <choice><sic>trncken</sic><corr>trucken</corr></choice> vnd irdisch vnd helt die knie dz - <lb/>menschen vnnutz im oderlossen. - </p> - <p> - <lb/><term ref="#Wassermann">Aquarius</term>. <q corresp="#Wassermann">des zaichen das wassermans</q> hatt - <lb/>das ertrich <placeName type="country">azenet</placeName> <placeName type="country">alkuffam</placeName> vnd <placeName type="country">alkot</placeName> in <placeName type="country">egipten</placeName> - <lb/>vnd <placeName type="country">das gelopte land</placeName> dortzu vnd <placeName type="country">Schwabenn</placeName>. - <lb/>mit dem planeten saturno. di stat <placeName type="city">rotbus</placeName>. dis - <choice><orig><choice><sic>zei-<lb/>ehen</sic><corr>zei-<lb/>chen</corr></choice></orig><reg><choice><sic>zeiehen<lb/></sic><corr>zeichen<lb/></corr></choice></reg></choice> ist von natur warm feucht vnd lufftig. die - <lb/>waden des menschen besehend mittel im - <choice><orig>oder-<lb/>lassen.</orig><reg>oderlassen.<lb/></reg></choice> - </p> - <p> - <lb/><term ref="#Fische">Pisces</term>. <q corresp="#Fische">das zaichen der fisch</q> helt die - <choice><orig>gegenn-<lb/>heit</orig><reg>gegennheit<lb/></reg></choice> gegen mitternacht vnd die stat <placeName type="city">Rom</placeName> mit d<choice><ex>em</ex><am>ē</am></choice> - <lb/>lande <placeName type="country">portigal</placeName>. welchem land zw geeignet wirt. - <lb/>Jupiter der planet mit der stat <placeName type="city">großglogav</placeName> Diß - <lb/>zaichen ist kalt feucht vnd wesserig besehen die - <lb/>füß des menschen mitel im oderlassen - </p> - <p> - <lb/>Dise 12 zeichen werden geteilt in zway teil di - <lb/>ersten 6 werden genant Borealia als vil - <choice><orig>gespro<lb/>chen</orig><reg>gesprochen<lb/></reg></choice> von dem wind Boreas der do blest vonn - <lb/>mitternacht vnd sind dise. Stier Ochs Zwiling - <lb/>Kreps Lev Junckfrav Die andern 6 werd<choice><ex>en</ex><am>ē</am></choice> - <lb/>gesproch<choice><ex>en</ex><am>ē</am></choice> australia v<choice><ex>on</ex><am>ō</am></choice> d<choice><ex>em</ex><am>ē</am></choice> w<choice><ex>in</ex><am>ī</am></choice>d auster blos<choice><ex>en</ex><am>ē</am></choice>t <choice><ex>in</ex><am>ī</am></choice> mit - <lb/>tag v<choice><ex>nd</ex><am>n̄</am></choice> sind dise. wag. scorpio. schucz. steibok - <choice><orig>was<lb/>serman.</orig><reg>wasserman.<lb/></reg></choice> fysch welcher 12 zeichen sind alleczeytt - <pb n="16" facs="Ptolemaeus_text_16"/> - <lb/>6 vnder dem erdtrich vnder vns vnd 6 vber vnß - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Uon den linien - <lb/>paralellenn - </head> - <p> - <lb/> - <lb/><hi rend="initial">w</hi>Eiter zu reden von bedeutunge der - <lb/>paralellen ist zu mercken das <persName><surname><choice><orig>ptolo<lb/>meus</orig><reg>ptolomeus<lb/></reg></choice></surname></persName> spricht. wie <term ref="#paralellus">das paralellus</term> - <q corresp="#paralellus"><choice><orig>ni-<lb/>chtis</orig><reg>nichtis<lb/></reg></choice> anders sey da<choice><ex>nn</ex><am>n̄</am></choice> ein vnderschaid - <choice><orig>zwei-<lb/>er</orig><reg>zweier<lb/></reg></choice> linien so si gezogen werden on - <choice><orig>vorhind-<lb/>ernuß</orig><reg>vorhindernuß<lb/></reg></choice> gleich von einander stehende so - <choice><orig>durch<lb/>schneiden</orig><reg>durchschneiden<lb/></reg></choice> si sich nymer yn ewigkeit.</q> Aber - <lb/>also. <term xml:id="parallelus2" ref="#paralellus">paralellus</term> ist <q corresp="#paralellus2">ein spacium odder velde - <lb/>zwischen zweien linien</q>. welhs spacium am - <lb/>anheben vnd am end zwischen disen linien - <lb/>yn einer gleichen weite ist gezogen. diser - <choice><orig>pa<lb/>ralelln</orig><reg>paralelln<lb/></reg></choice> werden gesaczt pey dyser kaul der - <lb/>werlt nach dem vorgenant<choice><ex>en</ex><am>ē</am></choice> <persName><surname>ptolomeo</surname></persName> yne - <lb/><note place="margin left">1</note>zal 21. <space/>Der erst paralel abstet von dem - <choice><orig>E<lb/>quinoxial</orig><reg>Equinoxial<lb/></reg></choice> 4 grad tag habend 12 stund lang - <lb/><note place="margin left">2</note>1/4 Der ander stet von dem equinoxial 8 <choice><ex>gra</ex><am>ḡ</am></choice>d. - <lb/><note place="margin left">3</note>vnd hat tags lang 12 st<choice><ex>un</ex><am>ū</am></choice>d 1/2 <space/>Der drit stett - <lb/>vom Equinoxial 12 grad vnd hat tags leng 12 - <lb/><note place="margin left">4</note>stund 1/2 1/4 <space/>Der vierde stet vom equinoxial - <lb/>16 grad dy grost leng deß tags ist 15 stund. - <lb/>vnd trifft vber eyn mit dem ersten clima. - </p> - <p> - <lb/><note place="margin left">5</note>Der funfft paralell stat vom Equinoxial - <lb/>20 grad vnd hat dy groste leng dest tags 13 - <lb/><note place="margin left">6</note>stund 1/4 <space/>Der sechst paralell hat 23 <choice><ex>grad</ex><am>ḡ</am></choice>. vom - <pb n="17" facs="Ptolemaeus_text_17"/> - <lb/>Equinoxial. dy grost leng des tags ist in - <choice><orig>dy-<lb/><note place="margin left">7</note>sem</orig><reg>dysem<lb/><note place="margin left">7</note></reg></choice> paralell 13 stund 1/2 . <space/>Der 7 paralel stett - <lb/>vom equinoxial 27 grad vnd hat leng dess - <lb/><note place="margin left">8</note>tags 13 stund 1/2 1/4 . <space/>Der achtet paralell hatt - <lb/>30 grad vom equinoxial gerechent yn dye - <lb/>braite die grost leng deß tags ist 14 stund<choice><ex>en</ex><am>n̄</am></choice>. - <lb/>vnd vbergett das drite clima durch <placeName type="city"><choice><orig>alexan<lb/></orig><reg>alexandriam<lb/></reg></choice></placeName><note place="margin left">9</note> - <space/>Der newnd paralell stett vom - <choice><orig>equi<lb/>noxial</orig><reg>equinoxial<lb/></reg></choice> 33 grad vnd hat tag vffs lengst 14 st. - <lb/><note place="margin left">10</note>1/4 <space/>Der zehend paralell stet vom gleycher - <lb/>der nacht das ist vom equinoxial 36 grad. - <lb/><note place="margin left">11</note>die grost leng deß tags ist 14 stund 1/2 <space/>Der - <lb/>eylfft stet vom equinoxial 38 grad dy grost - <lb/>leng des tags yn disem paralell ist 14 stund - <lb/><note place="margin left">12</note>1/2 1/4 Der zwelfft paralell stet vom - <choice><orig>equino-<lb/>xial</orig><reg>equinoxial<lb/></reg></choice> 40 grad. vnd hat langen tag auff 15 Stunden. - </p> - <p> - <lb/><note place="margin left">13</note>Der 13 stet vom equinoxial 43 grad hatt - <lb/>vffs lengst tag 15 st<choice><ex>un</ex><am>ū</am></choice>d vnd 1/4 vnd trifft - <placeName type="city"><choice><orig>c<choice><ex>on</ex><am>ō</am></choice>st<choice><ex>an</ex><am>ā</am></choice><lb/>tinopel</orig><reg>c<choice><ex>on</ex><am>ō</am></choice>st<choice><ex>an</ex><am>ā</am></choice>tinopel<lb/></reg></choice></placeName><note place="margin left">14</note> Der 14 stet vom equinoxial 45 grad. - <lb/><note place="margin left">15</note>hat vffs lengst tag 15 stund 1/2 Der 15 - <choice><orig>para-<lb/>lell</orig><reg>paralell<lb/></reg></choice> stet 48 grad vom equinoxial vnd hatt 16 - <lb/>stund tag vnnd geett durch das fliß - <placeName type="river"><choice><orig>Bori-<lb/>stenes</orig><reg>Boristenes<lb/></reg></choice></placeName><note place="margin left">16</note> yn <placeName type="country">Revssen</placeName> Der 16 paralell stet v<choice><ex>om</ex><am>ō</am></choice> - <lb/>equinoxial 51 grad. vnd hat leng deß tages - <lb/><note place="margin left">17</note>16 stund 1/2 <space/>Der 17 paralell stett vom - <choice><orig>Equi<lb/>noxial</orig><reg>Equinoxial<lb/></reg></choice> 54 grad vnd hat tags leng 17 stund<choice><ex>en</ex><am>ē</am></choice>. - </p> - <p> - <lb/><note place="margin left">18</note>Der 18 paralell stet vom equinoxial 56 grad - <lb/><note place="margin left">19</note>vnd hat langen tag vff 17 stunden 1/2 <space/>Der 19 - <pb n="18" facs="Ptolemaeus_text_18"/> - <lb/>paralell hat 58 grad vom equinoxial vnnd - <lb/>18 stund tag <space/>Der 20 paralell stet vom - <choice><orig>equi<note place="margin right">20</note><lb/>noxial</orig><reg>equinoxial<note place="margin right">20</note><lb/></reg></choice> - 61 grad vnd hat langen tag 18 stund. - <lb/>1/2 Der letste paralell <persName><surname>ptholomei</surname></persName> stett vom<note place="margin right">21</note> - <lb/>equinoxial 63 grad vnd hat tag 19 stunden. - </p> - <p> - <lb/>Uber dise paralelln hat der <persName><roleName>münch</roleName></persName> - <choice><orig>nach<lb/>gefaczt</orig><reg>nachgesaczt<lb/></reg></choice> 8 paralell dy yn dyser figur v<choice><ex>on</ex><am>ō</am></choice> - <choice><orig>eng-<lb/>heit</orig><reg>engheit<lb/></reg></choice> wegen deß zirkels nicht gesaczt werd<choice><ex>en</ex><am>ē</am></choice> - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Uon den climaten. - </head> - <p> - <lb/> - <hi rend="initial">A</hi>yn <term ref="#Clima">clima</term> wirt gesprochen. alls d<choice><am>ā</am><ex>an</ex></choice> - <lb/>schreibt <persName><forename>Johannes</forename> <surname>desacrobusto</surname></persName>. - <lb/><q corresp="#Clima">ein spaci<choice><am>ū</am><ex>um</ex></choice> oder breite deß ertrichs - <lb/>auff welcher breite man merklich - <lb/>spüren mag vorwandlung deß tags nach - <choice><orig>re<lb/>chtem</orig><reg>rechtem<lb/></reg></choice> gericht<choice><am>ē</am><ex>en</ex></choice> seyger yn ab oder zunemung - <lb/>der stund. nach deß tages größ</q>. welcher - <choice><orig>cli-<lb/>mata</orig><reg>climata<lb/></reg></choice> yn der zal süben sind als hiernach - <choice><orig>be-<lb/>deutet</orig><reg>bedeutet<lb/></reg></choice> wirt vnd werden ouch durch - <choice><orig>czyrk<lb/>els</orig><reg>czyrkels<lb/></reg></choice> linien bezeichent auff diese kaul. vnd - <choice><orig>dy-<lb/>se</orig><reg>dyse<lb/></reg></choice> 7 linien werden gen<choice><am>ā</am><ex>an</ex></choice>t forderlich - <choice><orig>paralel<lb/>len</orig><reg>paralellen<lb/></reg></choice> vnd haben vnderschaid yn den namen. - <lb/>wa<choice><am>n̄</am><ex>nn</ex></choice> worumb dise <q corresp="#clima3">7 linien</q> werden genantt - <lb/>yn sunderhayt <term xml:id="clima3" ref="#climata">Climata</term>. von deßwegen daß - <lb/>durch si dy climata geteilt werden vnd - <choice><orig>zwy<lb/>schen</orig><reg>zwyschen<lb/></reg></choice> dise linien werden gesaczt. aber ander - <q corresp="#par1"><choice><orig>li<lb/>nien</orig><reg>linien<lb/></reg></choice> diser spacia oder stell<choice><am>ē</am><ex>en</ex></choice> der climat<choice><am>n̄</am><ex>en</ex></choice> - <choice><orig>teyl<lb/>lend</orig><reg>teyllend<lb/></reg></choice></q> vnd dise werden genant sunderlich <term xml:id="par1" ref="#paralellus"><choice><orig>pa<lb/>ralelln.</orig><reg>paralelln.<lb/></reg></choice></term> auff das man yn eynem clima sunder - <pb n="19" facs="Ptolemaeus_text_19"/> - <lb/>wandlung deß tags mer dan an ain<choice><am>ē</am><ex>en</ex></choice> ortt so secztt - <lb/>man darein paralelln. <space/>Ein exempel. <space/>Yn dem - <lb/>5 clima ist der tag lang 15 stund gleich vndt man - <lb/>fint doch mer verwandlung des tags dan an eyner - <lb/>stell des ertrichs noch gleichwol seczt man das - <lb/>6 clima von danne vmb eyn gancze Stund. wye - <lb/>wol man spürt vorwandlung deß tags zwischen - <lb/>disen zwey<choice><am>ē</am><ex>en</ex></choice> climaten vmb ein halbe stund vnd 1/4 - <lb/>noch seczt man an dise end nicht climata sunder - <choice><orig>pa<lb/>ralelln.</orig><reg>paralelln.<lb/></reg></choice> auff das man eygentlich merken mag wo - <lb/>sich der tag yn dysem clima vorendert <space/>Dise - <choice><orig>cli-<lb/>mata</orig><reg>climata<lb/></reg></choice> alle werden gerechent nach den grad<choice><am>ē</am><ex>en</ex></choice> v<choice><am>ō</am><ex>on</ex></choice> - <lb/>dem equinoxial. <space/>Zuuorston aber was eyn - <lb/><choice><am>ḡ</am><ex>gra</ex></choice>d sey ist zumerk<choice><am>n̄</am><ex>en</ex></choice> so ein zirkel geteilt wirt yn 360 - <lb/>teyl derselben teil eins ist ein grad. ein grad aber - <lb/>geteilt yn 60 teyl der teil eins ist ein minut. ein - <choice><orig>mi<lb/>nut</orig><reg>minut<lb/></reg></choice> hat 60 secund<choice><am>ē</am><ex>en</ex></choice>. ein secund hat 60 tercien. e<choice><am>ī</am><ex>in</ex></choice> - <lb/>tercia hat 60 quart<choice><am>n̄</am><ex>en</ex></choice>. vnd also biß vff ein decima - <lb/>vnd gleicherweyß als ein minut ist eyn teyl eins - <lb/>grads. Also ist ein paralell ein teil eyns climats. - <lb/>dorynne er stet. - <lb/> - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Das erst clima dyameroes - </head> - <p> - <lb/> - <hi rend="initial">D</hi>Is clima wirt genantt dyameroes von - <lb/>der ynsul <placeName type="island">meroe</placeName>. Die do leitt yn <placeName type="country">egypto</placeName> - <lb/>vmbflossen mit dem edlen flyß <placeName type="river">nilo</placeName> - <choice><orig>ko-<lb/>menndt</orig><reg>komenndt<lb/></reg></choice> von mittagk welche ynsul - <placeName type="island"><choice><orig>me-<lb/>roe</orig><reg>meroe<lb/></reg></choice></placeName> ligt zwischen d<choice><am>ē</am><ex>em</ex></choice> driten vnd vierden paralell - <pb n="20" facs="Ptolemaeus_text_20"/> - <lb/>vnd yn dem 60 meridian mit ein<choice><am>ē</am><ex>em</ex></choice> halben. vnd - <lb/>stet von dem Equinoxial 16 grad nach der - <lb/>breyt alls <persName><forename>Hali</forename> <surname>halbenragel</surname></persName> schreibt. vnd an der - <lb/>leng hat es 180 alls ein yczlichs clima. wie wol - <lb/>das. das letste kurczer ist an der geschikligkeitt - <lb/>dan das erst yedoch hatt es eben dyse 180 grad - <lb/>als das allirlengst yn disem clima ist der tag 13 stunden. - <lb/>v<choice><am>n̄</am><ex>nd</ex></choice> 9 minut vorstanden an d<choice><am>ē</am><ex>em</ex></choice> end diss climats an - <lb/>dem anheben abir pey dem equinoxial hat es 12 - <lb/>stund vnd ist groß an der breit dann der andern - <lb/>yndert eins vrsach halb. das die sunn von disem - <lb/>clima auß dem breiten weg 30 diaci das ist dess - <lb/>tiersteigs nicht geweichen kan auff oddir ab - <choice><orig>vb<lb/>er</orig><reg>vber<lb/></reg></choice> 23 grad von dem eq<choice><orig>n</orig><reg>u</reg></choice>inoxial vnd von - <choice><orig>deßwe<lb/>gen</orig><reg>deßwegen<lb/></reg></choice> bescheinet si mer diß teil der erden wan si - <choice><orig>ge<lb/>leicher</orig><reg>geleicher<lb/></reg></choice> doruff niderhengende ist irer strene da<choice><am>n̄</am><ex>nn</ex></choice> - <lb/>auff ein ander clima. vnd so si dan gleich vnd nit - <lb/>seitenhalb dorauff scheinend ist. so kan man nicht - <lb/>also bald verwandlunge des tags alls in einem - <lb/>andern clima gespüren. auch haben die mensch<choice><am>ē</am><ex>en</ex></choice> - <lb/>yn disem clima die su<choice><am>n̄</am><ex>nn</ex></choice> vber iren heuptern gleich - <lb/>sten zweymol ym Jor vnd wirt genant zenit v<choice><am>n̄</am><ex>nd</ex></choice> - <lb/>das wert nicht lenger dan alls lang dy sunn yn d<choice><am>ē</am><ex>em</ex></choice> - <lb/>selben grad des mittags beharrend ist. vnd alle - <lb/>ynwoner des climats die haben zweierley - <choice><orig>schat-<lb/>ten</orig><reg>schatten<lb/></reg></choice> ein gegen mitternacht so die sunn geht zu dem - <lb/>zirkell des Creps. den andern schaten so si - <choice><orig>wid-<lb/>er</orig><reg>wider<lb/></reg></choice> get gegen mittag zu dem equinoxials zirckel - <pb n="21" facs="Ptolemaeus_text_21"/> - <lb/>diser schatten mögen nicht groß sein so die so<choice><am>n̄</am><ex>nn</ex></choice>e - <lb/>etlicher moß gleich vber yn stet von deßwegenn - <lb/>kan man auch nicht also schier spiren dy - <choice><orig>verwan<lb/>dlung</orig><reg>verwandlung<lb/></reg></choice> deß tags allso do. do die sunn grossen - <choice><orig>scha<lb/>ten</orig><reg>schaten<lb/></reg></choice> wirfft alls yn <placeName type="continent">europa</placeName>. vnd ee weiter gegenn - <lb/>mitternacht von dem equinoxial ye weiter vnd - <lb/>lenger die sonne von den stenden materlichenn - <lb/>corpern schate wirffet. wideru<choice><am>m̄</am><ex>mb</ex></choice> alls nahe dem - <lb/>equinoxial deß kleiner vnd neher schatten. Auß - <lb/>disen vorgenanten sachen mag gezogen werden - <lb/>wie das alsobald ein land gefunden wirt ligend - <lb/>außwendig disen dreyen zyrkeln der sunn<choice><am>ē</am><ex>en</ex></choice> - <choice><orig>leich<lb/>tlicher</orig><reg>leichtlicher<lb/></reg></choice> gespurt mag werden groß verwandlung - <lb/>deß tags dan yn disen landen dy do - <choice><orig>vndtirworf<lb/>en</orig><reg>vndtirworfen<lb/></reg></choice> sein dem weg der so<choice><am>n̄</am><ex>nn</ex></choice>e das ist dem Tiersteyg - <lb/>alls vorgesagt ist. von deßwegen hab<choice><am>n̄</am><ex>en</ex></choice> dyse - <choice><orig>men<lb/>schen</orig><reg>menschen<lb/></reg></choice> yn dem 7 clima lenger tag dan dise vnderr - <lb/>dem tropico cancri oder capricorni. dise - <choice><orig>verwan<lb/>dlung</orig><reg>verwandlung<lb/></reg></choice> der tag vorendert sich von dem - <choice><orig>equinoxi-<lb/>al</orig><reg>equinoxial<lb/></reg></choice> bis zu dem <term xml:id="articus2" ref="#articus">zirkel artico</term>. den man nent zu - <choice><orig>teüt-<lb/>sch</orig><reg>teütsch<lb/></reg></choice> <q corresp="#articus2">den wagen am hymell</q> allso das dye selbigen - <lb/>menschen disem gestirn vndderworffen ein halb - <lb/>Jor tag haben. das ist von <persName><forename>Gregori</forename></persName> byß vff - <choice><orig>Lam-<lb/>perti.</orig><reg>Lamperti.<lb/></reg></choice> Jst von dyswegen wan dise menschen - <choice><orig>hab<lb/>en</orig><reg>haben<lb/></reg></choice> ir zenit vnder dem polo artico. So muß v<choice><am>ō</am><ex>on</ex></choice> - <choice><orig>no<lb/>tes</orig><reg>notes<lb/></reg></choice> weg<choice><am>ē</am><ex>en</ex></choice> der <choice><orig>eqnoxial</orig><reg>equinoxial</reg></choice> sein yrer Orison. das ist der - <lb/>zirkl dory<choice><am>n̄</am><ex>nn</ex></choice> di so<choice><am>n̄</am><ex>nn</ex></choice> vff v<choice><am>n̄</am><ex>nd</ex></choice> nider get v<choice><am>n̄</am><ex>nd</ex></choice> so bald di so<choice><am>n̄</am><ex>nn</ex></choice> - <lb/>sich <choice><orig>negt</orig><reg>neigt</reg></choice> von <choice><orig>Eqnox.</orig><reg>Equinox.</reg></choice> geg<choice><am>ē</am><ex>en</ex></choice> mitnacht so get in ir tag - <pb n="22" facs="Ptolemaeus_text_22"/> - <lb/>an vnd so die so<choice><am>n̄</am><ex>nn</ex></choice>e wider von dem <choice><orig>eqnoxial</orig><reg>equinoxial</reg></choice> sich - <lb/>neyget gegen mittemtag. get yre nacht an. alls m<choice><am>ā</am><ex>an</ex></choice> - <lb/>wol sichtigklich dise dyng sehen mag yn corpore - <lb/>sperico <space/>Diß erst clima hat nach der breit 440 - <lb/>meilen alls <persName><forename>leupoldus</forename></persName> seczet vnd dy erhebunge - <lb/>des <term ref="#Poli2">poli</term> das ist <q corresp="#Poli2">der ax des hymells</q> yn disem - <choice><orig>cli-<lb/>ma</orig><reg>clima<lb/></reg></choice> das ist 16 grad von yrem orisonte. dy stet ym - <lb/>ersten clima sind dise vorderlichsten. <placeName type="city">arim</placeName>. <placeName type="city">thana</placeName> - <lb/><placeName type="city">haw</placeName>. <placeName type="city">manna</placeName>. <placeName type="city">gecz</placeName>. terre <placeName type="country">yndie</placeName>. <placeName type="city">affar</placeName>. <placeName type="city">cerugk</placeName>. - <placeName type="city"><choice><orig>ame-<lb/>dia</orig><reg>amedia<lb/></reg></choice></placeName> yn <placeName type="continent">affrica</placeName>. <placeName type="city">Facili</placeName> yn <placeName type="country">yndia</placeName>. <placeName type="city">Zabech</placeName> ym mittell - <lb/>dy stat des <persName><roleName>kunigs</roleName> <forename>alcimbenis</forename></persName>. <placeName type="city">anthimantipeda - <lb/>amenia</placeName>. dy statt des reichs <placeName type="country">magogk</placeName>. <placeName type="city">ceda</placeName> am ent - <lb/> - </p> - </div> - <div type="kapitel"> - <head> - <lb/>das ander clima diasienes - </head> - <p> - <lb/> - <hi rend="initial">D</hi>Js clima wirt genant dyasienes. von der - <lb/>stat <placeName type="city">siene</placeName> dy do leit vnder dem - <choice><orig>Tropi-<lb/>co</orig><reg>Tropico<lb/></reg></choice> cancri dauon <persName><forename>lucanus</forename></persName> also spricht - <lb/><q xml:lang="lat">Umbras nusqu<choice><am>ā</am><ex>am</ex></choice> flectente siene</q>. wie das - <lb/>der schatten der menschen oder ander corpern - <lb/>nymer gelengt oder gebrochen wirt vorstanden - <lb/>ym mittag desselben tags so dy sonne ym kreps - <lb/>ist. dy grost leng deß tags yn disem clima ist 13 stunden. - <lb/>ein 1/2 vnnd 1/4 vnnd hat an der preit 29 grad v<choice><am>n̄</am><ex>nd</ex></choice> - <lb/>12 minut. das bedeut 400 meiln hy vntten bey - <lb/>vns. vnd dyß clima durch get dy gelegenheit zyn - <lb/>vnd <placeName type="country">yndie</placeName> vnd gett halb durch <placeName type="continent">affricam</placeName> dy - <choice><orig>erhe<lb/>bung</orig><reg>erhebung<lb/></reg></choice> poli yn disem clima ist 24 grad. dy - <choice><orig>furder<lb/>lichsten</orig><reg>furderlichsten<lb/></reg></choice> stet yn dem andern Clima sein. <placeName type="city">Mecha - <pb n="23" facs="Ptolemaeus_text_23"/> - <lb/>anburi</placeName>. dy stat des <persName><roleName>künigs von yndia</roleName></persName>. des <persName><roleName>künig - <lb/>es stat von zyn</roleName></persName>. deß <persName><roleName>konigs</roleName></persName> stat der moren. <placeName type="city">iurg<choice><am>ē</am><ex>en</ex></choice></placeName>. - <lb/><placeName type="city">erana</placeName>. <placeName type="city">almansora</placeName> <placeName type="city">almediana</placeName>. <placeName type="city">alre</placeName>. <placeName type="city">thebim</placeName> - <placeName type="city"><choice><orig>cordu-<lb/>la</orig><reg>cordula<lb/></reg></choice></placeName> am end - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Das dritte clima - <lb/>dyalexandrios - </head> - <p> - <lb/> - <lb/> - <hi rend="initial">D</hi>ys clima anhebend vom orient durchget - <lb/>das gelopte land vnd durch dy end - <placeName type="country"><choice><orig>egy-<lb/>pty</orig><reg>egypty<lb/></reg></choice></placeName> begreiffend dy statt <placeName type="city">alexandriam</placeName> dye - <lb/>do ligt <choice><am>ī</am><ex>in</ex></choice> <choice><orig>de</orig><reg>dem</reg></choice> achten paralel v<choice><am>n̄</am><ex>nd</ex></choice> vnder dem - <choice><orig>sechtzigi-<lb/>sten</orig><reg>sechtzigisten<lb/></reg></choice> meridian. dauon dann das clima gen<choice><am>ā</am><ex>an</ex></choice>t wirt - <lb/>dyalexandrios yn <placeName type="country">kriechen</placeName> vß diser statt ist - <choice><orig>gewe<lb/>sen</orig><reg>gewesen<lb/></reg></choice> <persName><surname>ptolomeus</surname> <forename>claudius</forename></persName>. v<choice><am>ō</am><ex>on</ex></choice> deßweg<choice><am>ē</am><ex>en</ex></choice> er dy zall - <lb/>der meridian anhept zu zel<choice><am>ē</am><ex>en</ex></choice> wyweit ein - <choice><orig>yczlich<lb/>er</orig><reg>yczlicher<lb/></reg></choice> meridian gestand<choice><am>n̄</am><ex>en</ex></choice> ist v<choice><am>ō</am><ex>on</ex></choice> <placeName type="city">alexandria</placeName>. dy leng - <lb/>deß tags yn disem clima ist 14 stund v<choice><am>n̄</am><ex>nd</ex></choice> 15 minutt - <lb/>an der breite aber hatt es 33 grad vnd 40 minut - <lb/>das macht 550 meilen auff dem erttrich. dis - <choice><orig>cli-<lb/>ma</orig><reg>clima<lb/></reg></choice> hat dise vorderlichsten stett alls <placeName type="city">wedecz</placeName>. - <placeName type="city"> <choice><orig>Cati<lb/>duquia.</orig><reg>Catiduquia.<lb/></reg></choice></placeName> <placeName type="city">new babiloni<choice><am>ē</am><ex>en</ex></choice></placeName>. <placeName type="city">alkayr</placeName>. <placeName type="city">albertim</placeName>. <placeName type="city">granat<choice><am>n̄</am><ex>en</ex></choice></placeName> - <lb/><placeName type="city">hyspanie</placeName> Jsta dy stat des winden. <placeName type="city">Jor</placeName>. <placeName type="city">fugica</placeName> - <placeName type="country"><choice><orig>egi<lb/>pty.</orig><reg>egipty.<lb/></reg></choice></placeName> <placeName type="city">alexandria</placeName> <placeName type="city">damnata</placeName>. <placeName type="city">alkuffa</placeName>. <placeName type="city">cartago</placeName> - <placeName type="city"><choice><orig>hierusa-<lb/>lem.</orig><reg>hierusalem.<lb/></reg></choice></placeName> <placeName type="city">damast<choice><am>ū</am><ex>um</ex></choice></placeName>. dye erhebung poli yn disem clima - <lb/>ist 30 grad vnd 30 minut. - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Das vierde Clima - <lb/>dyarodus. - </head> - <p> - <pb n="24" facs="Ptolemaeus_text_24"/> - <lb/> - <hi rend="initial">v</hi>On der stat <placeName type="city">rodis</placeName> hatt das clima sein<choice><am>ē</am><ex>en</ex></choice> - <lb/>namen wan es dy gancz insul <placeName type="island">Rhodis</placeName>. - <lb/>vnd <placeName type="island">cabros</placeName> yn ym beschleust vnd hatt - <lb/>an der braite 39 grad gleich macht pey - <lb/>vns auff erden 300 meylen vnd grosse des tags - <lb/>yn disem clima ist 14 stund vnnd eyn halbe - <choice><orig>dys-<lb/>en</orig><reg>dysen<lb/></reg></choice> ynwonenden menschen yn disem teil wirt der - <lb/>Polus erhabenn 36 grad vber yren Orizon - <choice><orig>dy<lb/>ser</orig><reg>dyser<lb/></reg></choice> zirkel orizon ist dorynne dy su<choice><am>n̄</am><ex>nn</ex></choice> vff vnd - <choice><orig>nyder<lb/>get</orig><reg>nyderget<lb/></reg></choice> vnd endet vnser gesicht. also vorstanden so - <lb/>e<choice><am>ī</am><ex>in</ex></choice> mensch stunde auff einer hoe vnd sech vmb sich - <lb/>lantschafft als weit er mocht. so sicht er ein<choice><am>ē</am><ex>en</ex></choice> zirkll - <lb/>dem ertrich gleich an dem firmanent zu ring v<choice><am>m̄</am><ex>mb</ex></choice> - <lb/>welchs zirkels centr<choice><am>ū</am><ex>um</ex></choice> der selb m<choice><am>ē</am><ex>en</ex></choice>sch macht wo - <lb/>er stehet. dy trefflichst<choice><am>ē</am><ex>en</ex></choice> stete yn dissem clima sein - <lb/>dy <placeName type="island">Maiorica</placeName> <placeName type="island">sibilia</placeName> <placeName type="island">sardinia</placeName> ynsul <placeName type="island">sicilia</placeName> ynsul <placeName type="island">p<choice><am>ā</am><ex>an</ex></choice> - <lb/>noric<choice><am>ū</am><ex>um</ex></choice></placeName> dy stat <placeName type="city">persi</placeName>. <placeName type="city">aldt babilonia</placeName> <placeName type="city">balach eduni</placeName> - <lb/>a <placeName type="city">thilus</placeName> ynsula <placeName type="country">yndie</placeName>. - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Das funffte clima - <lb/>dyaromes - </head> - <p> - <lb/>dys clima anheb<choice><am>ē</am><ex>en</ex></choice>d yn orient durchgehtt - <lb/>dy stat <placeName type="city">Rom</placeName> von danne is genant wirtt - <lb/>vnd ander vmbligende stete vnd - <choice><orig>lanth<lb/>schafft.</orig><reg>lanthschafft.<lb/></reg></choice> vnd hat an der breite 43 grad abst<choice><am>ē</am><ex>en</ex></choice>d vom - <lb/><choice><orig>eqnoxial</orig><reg>equinoxial</reg></choice> macht 255 meiln pey vns gerech<choice><am>ē</am><ex>en</ex></choice>t. dy - <lb/>grose aber des tages ist 15 st<choice><am>ū</am><ex>un</ex></choice>nd<choice><am>ē</am><ex>en</ex></choice> vnd polus articus - <lb/>wirt dis<choice><am>ē</am><ex>en</ex></choice> m<choice><am>ē</am><ex>en</ex></choice>sch<choice><am>ē</am><ex>en</ex></choice> erhab<choice><am>ē</am><ex>en</ex></choice> dy <rs type="inhabitants">hyrin<choice><am>ē</am><ex>en</ex></choice></rs> won<choice><am>ē</am><ex>en</ex></choice> sei von - <lb/>yr<choice><am>ē</am><ex>em</ex></choice> oriz<choice><am>ō</am><ex>on</ex></choice> 41 grad v<choice><am>n̄</am><ex>nd</ex></choice> e<choice><am>ī</am><ex>in</ex></choice> tercia das ist e<choice><am>ī</am><ex>in</ex></choice> 1/2 e<choice><am>ī</am><ex>in</ex></choice>s m<choice><am>ī</am><ex>in</ex></choice>uts - <pb n="25" facs="Ptolemaeus_text_25"/> - <lb/>Dy mechtigisten stete yn dissem cl<choice><am>ī</am><ex>im</ex></choice>a sint disse - <placeName type="city"><choice><orig>Ra-<lb/>gos.</orig><reg>Ragos.<lb/></reg></choice></placeName> <placeName type="city">Lisbona</placeName>. <placeName type="city">tholetum</placeName>. <placeName type="city">cesaria augusta</placeName>. <placeName type="city">neapolis</placeName>. - <lb/><placeName type="city">tharent<choice><am>ū</am><ex>um</ex></choice></placeName>. <placeName type="city">roma</placeName>. <placeName type="city">facar</placeName>. ym mittil dis climatz <placeName type="city">Sena</placeName>. - <lb/><placeName type="city">perus</placeName>. <placeName type="city">amesa</placeName>. <placeName type="mountain">tholoza</placeName> der berg. <placeName type="city">pessulanus</placeName>. - <placeName type="city"><choice><orig>ancho<lb/>ra</orig><reg>anchora<lb/></reg></choice></placeName> <placeName type="city">thaurinum</placeName>. <placeName type="city">valentz</placeName> am ende - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Das sechste clima - <lb/>dyaboristenes - </head> - <p> - <lb/> - <hi rend="initial">w</hi>Irt gen<choice><am>ā</am><ex>an</ex></choice>t dis clima van des wassirflisses - <lb/><placeName type="river">boristenes</placeName> das durch <placeName type="country">rewsserlantt</placeName> - <choice><orig>flew-<lb/>sset</orig><reg>flewsset<lb/></reg></choice> vnd felt yn das <placeName type="ocean">schwartze mehr</placeName>. an - <lb/>der preite hat is 47 grad vnd 15 minut - <lb/>gerech<choice><am>ē</am><ex>en</ex></choice>t vam equinoxial gegen miternacht yn d<choice><am>ē</am><ex>em</ex></choice> - <lb/>polo artico dy groest lenge des tages ist 15 st<choice><am>ū</am><ex>un</ex></choice>d<choice><am>ē</am><ex>en</ex></choice> - <lb/>einhalbs vnd ein 1/4 dissen ynwoneren wirt dy als - <lb/>des himils erhaben 47 grad vnd ein 1/4 vnd hath - <lb/>yn ym disse greste stete <placeName type="city">Massilia</placeName>. <placeName type="city">genua</placeName> dy - <choice><orig>gegen-<lb/>heit.</orig><reg>gegenheit.<lb/></reg></choice> <placeName type="city">wyenn</placeName>. <placeName type="city">mediolan</placeName>. <placeName type="city">ferer</placeName> <placeName type="city">b<choice><am>ā</am><ex>an</ex></choice>nomia</placeName>. <placeName type="city">compestel</placeName>. - <lb/><placeName type="country">Portigal</placeName>. <placeName type="city">prix</placeName>. <placeName type="city">venedig</placeName> ym mittil des climats. - <placeName type="city"><choice><orig>c<choice><am>ō</am><ex>on</ex></choice>-<lb/>stantinopul</orig><reg>c<choice><am>ō</am><ex>on</ex></choice>stantinopul.<lb/></reg></choice></placeName> <placeName type="city">Costintz</placeName> <placeName type="city">straßburg</placeName>. <placeName type="city">villac<choice><am>ū</am><ex>um</ex></choice></placeName> disse - <choice><orig>prei-<lb/>te</orig><reg>preite<lb/></reg></choice> des erttrichs ist 212 meiln gerechnet vom - <choice><orig>mit-<lb/>tag</orig><reg>mittag<lb/></reg></choice> yn mittirnacht - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Das sibend clima dyaripheos. - </head> - <p> - <lb/> - <hi rend="initial">G</hi>enant vrspringlich von den pergen <placeName type="mountain">ryffey</placeName> - <lb/>dye do gehn durch <placeName type="country">reyssen</placeName> vnd - <placeName type="country"><choice><orig>mosquitt<lb/>en</orig><reg>mosquitten<lb/></reg></choice></placeName> vnd diß clima ist das aller kurtste v<choice><am>ō</am><ex>on</ex></choice> - <choice><orig>o-<lb/>ri<choice><am>ē</am><ex>en</ex></choice>t</orig><reg>ori<choice><am>ē</am><ex>en</ex></choice>t<lb/></reg></choice> yn occident das macht di engheyt der kaull - <lb/>oder deß zirkels der erden. vnd hat an der breitt - <lb/>50 <choice><am>ḡ</am><ex>grad</ex></choice> v<choice><am>n̄</am><ex>nd</ex></choice> 3 minut vom equinoxial vnd macht yn - <pb n="26" facs="Ptolemaeus_text_26"/> - <lb/>dy preit der meilen 185. dy gröst leng deß tags - <lb/>ist 16 stund 15 minut. dyse vornemlichsten stett - <choice><orig>li<lb/>gen</orig><reg>ligen<lb/></reg></choice> yn dysem clima. <placeName type="city">Paris</placeName>. <placeName type="city">Ulm</placeName>. <placeName type="city">augusta - <choice><orig>vinde-<lb/>licor<choice><am>ū</am><ex>um</ex></choice>.</orig><reg>vindelicor<choice><am>ū</am><ex>um</ex></choice>.<lb/></reg></choice></placeName> <placeName type="city">Munchen</placeName>. <placeName type="city">Salczpurg</placeName>. <placeName type="city">wyen</placeName>. <placeName type="city">Ofen</placeName>. <placeName type="city">Roto - <lb/>magos</placeName>. <placeName type="city">Colln</placeName>. <placeName type="city">Agrippina</placeName>. <placeName type="city">Mencz</placeName>. <placeName type="city">wirrtzpurgk</placeName> - <lb/><placeName type="city">Nurnberg</placeName>. <placeName type="city">Jngelstat</placeName>. <placeName type="city">Ratisbona</placeName>. <placeName type="city">Erford</placeName>. <placeName type="city">Lipczg</placeName> - <lb/><placeName type="city">Prag</placeName>. Mit disem 7 clima beschliessen vil - <choice><orig>astrono<lb/>mi</orig><reg>astronomi<lb/></reg></choice> dy ynwonenden land als diser <persName><surname>ptholomeus</surname> - <lb/><addName>Dyonisius tessalonicenß</addName></persName>. <persName><forename>Anßhelmus</forename></persName> de vergine - <lb/>mundi. vnd ander vil mit ym sprechen alles das do - <lb/>funden wirt außwendig disem climata sol nit - <choice><orig>ge-<lb/>acht</orig><reg>geacht<lb/></reg></choice> werden vor ein clima wan worumb man - <choice><orig>fin<lb/>det</orig><reg>findet<lb/></reg></choice> nicht alls vil besaczter land geg<choice><am>ē</am><ex>en</ex></choice> mitternacht - <lb/>alls gegen mittemtag yn dem vierteyl diser kaull - <lb/>dorumb schaczen si es fur klein ding hermes - <choice><orig>ab-<lb/>er</orig><reg>aber<lb/></reg></choice> der <choice><am>pphūs</am><ex>philosophus</ex></choice> seczt dorczu das achtet clima vnd - <lb/>dyse stat dorynn <placeName type="city">kunigsberg</placeName>. <placeName type="city">Danczgk</placeName>. - <placeName type="city"><choice><orig>magde-<lb/>burg.</orig><reg>magdeburg.<lb/></reg></choice></placeName> <placeName type="city">Braunczweigk</placeName>. <placeName type="city">Crokau</placeName>. <placeName type="city">Breßlav</placeName>. <placeName type="city">Lubeg</placeName>. - <lb/><placeName type="city">dantisten</placeName>. <placeName type="city">Jrland</placeName>. <placeName type="city">Schotten</placeName> vnd <placeName type="city">schweden</placeName> also - <lb/><choice><orig>st</orig><reg>ist</reg></choice> dise bekante werlt geteylt yn 7 teyl alls - <choice><orig>da<lb/>non</orig><reg>dan<lb/></reg></choice> dauon gesagt ist. vnd hat an yrer breit vberal als - <persName><surname><choice><orig>pto<lb/>lomeus</orig><reg>ptolomeus<lb/></reg></choice></surname></persName> seczt 180000 taus<choice><am>ē</am><ex>en</ex></choice>t gewend das macht - <lb/>20000 meylen vnd 40 meilen. ein gewend ist - <lb/>200 schrit zwey gewend machen 1/4 zwey vierteil - <lb/>einhalbs meyl zwu halb machen ein gancze - </p> - </div> - <div type="kapitel"> - <head> - <lb/>von den meridians zirkeln - </head> - <p> - <lb/> - <hi rend="initial">e</hi>s ist zuwissen das <q corresp="#meri1">dise zirkels lini<choice><am>ē</am><ex>en</ex></choice> di do - <lb/>gen von einem polo yn den andern</q>. - <choice><orig>wer-</orig><reg>wer</reg></choice><pb n="27" facs="Ptolemaeus_text_27"/><choice><orig>denn</orig><reg>denn</reg></choice> genant <term ref="#Meridianus">Meridiani</term>. allso schreibt - <persName><surname><choice><orig>ptolome<lb/>us</orig><reg>ptolomeus<lb/></reg></choice></surname></persName> wie <term ref="#Meridianus">das meridianus</term> sey <q corresp="#Meridianus">ein zirkel linia <choice><orig>gleih</orig><reg>gleich</reg></choice> - <lb/>durch vnser cenit v<choice><am>n̄</am><ex>nd</ex></choice> teylet vnsern orison yn zwei - <lb/>teyl beyde polos treffent vnd durchschneidet - <choice><orig>all<lb/>le</orig><reg>alle<lb/></reg></choice> climata vnd paralelln yn kreiczwyß</q>. diser - <choice><orig>me-<lb/>ridian</orig><reg>meridian<lb/></reg></choice> alls <persName><surname>ptolomeus</surname></persName> seczet der sein 36 vnd - <choice><orig>ycz<lb/>licher</orig><reg>yczlicher<lb/></reg></choice> meridian helt 5 grad so kompt es. das do - <lb/>36 mal 5. macht 180 also vil grad seyn yn einem - <lb/>halben czirkel. yn der figur dises buchleins - <choice><orig>fin<lb/>det</orig><reg>findet<lb/></reg></choice> m<choice><am>ā</am><ex>an</ex></choice> nicht mer w<choice><am>ā</am><ex>an</ex></choice> dy meridians lini<choice><am>ē</am><ex>en</ex></choice> halb. das - <lb/>ist 18 das ist gesche<choice><am>ē</am><ex>en</ex></choice> von wegen der kleinheit - <choice><orig>di<lb/>ser</orig><reg>diser<lb/></reg></choice> figur auff das man dy linien nicht sere nahen - <lb/>bey einander seczt sunder yn moß disem wercke - <lb/>dienend. vnd sind doch gleichwol volkom<choice><am>ē</am><ex>en</ex></choice> an - <lb/>den graden allein das die linien nicht geczogen - <lb/>sein so ist zumerken das yn diser gegenwertigen - <choice><orig>fi<lb/>gur</orig><reg>figur<lb/></reg></choice> eyn spaci<choice><am>ū</am><ex>um</ex></choice> 10 grad beschleust vnd so mach<choice><am>ē</am><ex>en</ex></choice> - <lb/>15 grad 3 meridian das bedeut hye yn der figur - <lb/>anderhalb spaci<choice><am>ū</am><ex>um</ex></choice> daz hatt ouch eben alls wol 3 - <lb/>meridian alls <persName><surname>ptolomeus</surname></persName> seczt vnd 3 Meridian - <lb/>mach<choice><am>ē</am><ex>en</ex></choice> allzeyt e<choice><am>ī</am><ex>in</ex></choice> st<choice><am>ū</am><ex>un</ex></choice>d. das wer 15 <choice><am>ḡ</am><ex>grad</ex></choice>. diser meridi<choice><am>ā</am><ex>an</ex></choice> - <lb/>saczung ist von deßwegen das man lychtlicher - <choice><orig>fin<lb/>den</orig><reg>finden<lb/></reg></choice> kan wye weit ein stat oder land vom obend - <lb/>oder morg<choice><am>ē</am><ex>en</ex></choice> lig<choice><am>ē</am><ex>en</ex></choice> mag so vindet m<choice><am>ā</am><ex>an</ex></choice> ouch wy vil - <choice><orig>stun<lb/>d<choice><am>ē</am><ex>en</ex></choice></orig><reg>stund<choice><am>ē</am><ex>en</ex></choice><lb/></reg></choice> ein l<choice><am>ā</am><ex>an</ex></choice>d ee tag hat dann das ander. alls das l<choice><am>ā</am><ex>an</ex></choice>t - <lb/>zu <placeName type="country">klein polen</placeName> haut ee tagk dan <placeName type="country">hispanienn</placeName> czwu - <lb/>st<choice><am>ū</am><ex>un</ex></choice>d 1/2. gleicherweyß <persName><roleName>priester</roleName> <forename>Johann</forename></persName> yn <placeName type="country">yndia</placeName> - <lb/>hatt Ee tagk yn seynem lannde vier stundenn. - <pb n="28" facs="Ptolemaeus_text_28"/> - <lb/>dann der moscobitt also mag man ouch sag<choice><am>ē</am><ex>en</ex></choice> von - <lb/>stetten dy statt <placeName type="city">crakaw</placeName> hatt er tag 1/4 einer stunden - <lb/>d<choice><am>ā</am><ex>an</ex></choice>n <placeName type="city"><choice><sic>nūr<choice><am>ē</am><ex>em</ex></choice>bergk</sic><corr>nürembergk</corr></choice></placeName>. disse geschichte kom<choice><am>ē</am><ex>en</ex></choice> alle von - <lb/>weg<choice><am>ē</am><ex>en</ex></choice> der runtheith des erttrichs wen<choice><am>ē</am><ex>en</ex></choice> worumb - <lb/>dy son<choice><am>ē</am><ex>en</ex></choice> kann nym<choice><am>ē</am><ex>me</ex></choice>r bescheinen das erttreich gantz - <lb/>sunder allein dy helfft von des wegen so dy zonne - <lb/>yn orient awff geht vber den geme<choice><am>ī</am><ex>in</ex></choice>en oriz<choice><am>ō</am><ex>on</ex></choice> zo - <choice><orig>be<lb/>scheinet</orig><reg>bescheinet<lb/></reg></choice> sy ein vierteill yn dem obren emisperio - <choice><orig>vn<lb/>ser</orig><reg>vnser<lb/></reg></choice> weltt vnd ein viertteil yn dem vnttirn welche - <lb/>zweier vierteill mach<choice><am>ē</am><ex>en</ex></choice>t ein halb teil. so mag man - <lb/>nun merken das 18 Meridian sind yne eynem - <lb/>viertel eins zirkls Und 3 meridi<choice><am>ā</am><ex>an</ex></choice> mach<choice><am>ē</am><ex>en</ex></choice>t ei<choice><am>ē</am><ex>ne</ex></choice> st<choice><am>ū</am><ex>un</ex></choice>d - <lb/>so kompt ys das dy yn orient yn der statt. <placeName type="city">kathigara</placeName> - <lb/>ee tag hab<choice><am>ē</am><ex>en</ex></choice> 12 stund<choice><am>ē</am><ex>en</ex></choice> d<choice><am>ā</am><ex>an</ex></choice> disse zv <placeName type="city">lisp<choice><am>ō</am><ex>on</ex></choice></placeName> in <placeName type="country">hyspa<choice><am>ī</am><ex>ni</ex></choice>en</placeName> - <lb/>das mus verst<choice><am>ā</am><ex>an</ex></choice>d<choice><am>ē</am><ex>en</ex></choice> werd<choice><am>ē</am><ex>en</ex></choice> so dy so<choice><am>n̄</am><ex>nn</ex></choice>e ist im <choice><orig>eqnoxial</orig><reg>equinoxial</reg></choice> - <lb/>vnd das ertrich glich von der son<choice><am>ē</am><ex>ne</ex></choice> beschin<choice><am>ē</am><ex>en</ex></choice> wirtt. - <lb/>hab<choice><am>ē</am><ex>en</ex></choice>t wir verstand<choice><am>ē</am><ex>en</ex></choice> das di statt <placeName type="city">kathigara</placeName> 12 st<choice><am>ū</am><ex>un</ex></choice>d<choice><am>ē</am><ex>en</ex></choice> - <lb/>ee tag hat dan <placeName type="city">lispon</placeName>. so mus dy sone abir so l<choice><am>ā</am><ex>an</ex></choice>ge - <lb/>ober daz ertrich strich<choice><am>ē</am><ex>en</ex></choice> bis sy kompt yn occident - <lb/>bey <placeName type="city">lispon</placeName> das sind ouch 12 stund so wirt si yn 12 - <lb/>stund<choice><am>ē</am><ex>en</ex></choice> das ertrich gleich vbergan. vnd vntt<choice><am>ē</am><ex>en</ex></choice> 12 - <lb/>macht 24 stund<choice><am>ē</am><ex>en</ex></choice> das ist ein natturlicher tag. so - <choice><orig>ab<lb/>er</orig><reg>aber<lb/></reg></choice> sich die so<choice><am>n̄</am><ex>nn</ex></choice>e neiget v<choice><am>ō</am><ex>on</ex></choice> d<choice><am>ē</am><ex>em</ex></choice> <choice><orig>eqnoxial</orig><reg>equinoxial</reg></choice> so machet sy - <lb/>den geg<choice><am>ē</am><ex>en</ex></choice> mitternacht grosse tage So aber iemant - <lb/>gern wissen woltt wi vil meilen di sonne liffe auff - <lb/>d<choice><am>ē</am><ex>em</ex></choice> erttrich gerechnet der thu im also Er nem dy grad - <lb/>di der zcirkel der sonn<choice><am>ē</am><ex>en</ex></choice> hatt der sint 360 vnd teille - <lb/>disse durch 24 stunden so komen im 15 das sintt di - <pb n="29" facs="Ptolemaeus_text_29"/> - <lb/>grad also vil loufft si grad eine stund Darnach n<choice><am>ī</am><ex>im</ex></choice> - <lb/>15 vnd multiplicier das ist manchfeldige si durch - <lb/>60 wann ein grad hatt 60 meiln gemeinlich avff - <lb/>der erden gerechnet so koment dir 900 meiln also - <lb/>vil loufft si eine stund. auch ist zw wissen das sich - <lb/>di meridi<choice><am>ā</am><ex>an</ex></choice> in miternacht vnd in mitt<choice><am>ē</am><ex>em</ex></choice>tag sere enge - <lb/>vnd alle vff einen punct komen. Und doch glich - <lb/>wol ire eigenschafft irer grad also volkomen - <choice><orig>be-<lb/>halten</orig><reg>behalten<lb/></reg></choice> am end als im mittil vnd wie nohet das die - <lb/>lineen pei enander stehn pei den zween poli. doch - <lb/>gelichwol hat ein ietzlich spaci<choice><am>ū</am><ex>um</ex></choice> 5 grad also - <choice><orig>vol-<lb/>kom<choice><am>ē</am><ex>en</ex></choice></orig><reg>volkom<choice><am>ē</am><ex>en</ex></choice><lb/></reg></choice> als do si preit sein vnd disse grad eng<choice><am>n̄</am><ex>en</ex></choice> v<choice><am>n̄</am><ex>nd</ex></choice> - <lb/>vorinng<choice><am>ē</am><ex>en</ex></choice> sich also sere das ouch menschliche - <choice><orig>vor-<lb/>nunfft</orig><reg>vornunfft<lb/></reg></choice> dy selben grad an der zal nicht begriff<choice><am>ē</am><ex>en</ex></choice> kan - <lb/> - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Hienach sagt es von - <lb/>den zwelff windenn. - </head> - <p> - <lb/> - <lb/> - <hi rend="initial">i</hi>n disse kawle des erttrichs. wegenn - <lb/>vnd blozent 12 winde in mancherlei - <lb/>weisse etliche mit storm vnd - <choice><orig>vngesti-<lb/>mugkeith.</orig><reg>vngestimugkeith.<lb/></reg></choice> etliche linde vnd sanffte - <choice><orig>et-<lb/>liche</orig><reg>etliche<lb/></reg></choice> trucken vnd durre etliche - <choice><orig>feu-<lb/>chte</orig><reg>feuchte<lb/></reg></choice> vnd naß etliche kalth vnd wesserig. vnd alle - <lb/>nacheinander So ist zu wissen das 4 forderliche - <lb/>winde sein vnd 8 zwfellige di forderlichsten 4 yn - <lb/>der figure bezceichnet. sein genant ym latein - <choice><orig>sub-<lb/>solanus</orig><reg>subsolanus<lb/></reg></choice> fauonius aspericus vnd auster dewtsch - <pb n="30" facs="Ptolemaeus_text_30"/> - <lb/>wirt genant <term xml:id="ost1" ref="#Osten">Subsolanus</term> <q corresp="#ost1">Ost</q>. der ist sanfft vnnd - <lb/>lind blosend von orient <space/>Der ander <term xml:id="west1" ref="#Westen">Fauonius</term> - <lb/>wirt genant <q corresp="#west1">west</q> wehend von occident. vnd ist - <lb/>miltygklich dy bl<choice><am>ū</am><ex>ü</ex></choice>men bringen <space/>Der drit we<choice><am>ē</am><ex>en</ex></choice>d - <lb/>v<choice><am>ō</am><ex>on</ex></choice> mitternacht <term xml:id="nord1" ref="#Norden">Apercias</term> gen<choice><am>ā</am><ex>an</ex></choice>t. wirt zu deutsch - <lb/>geheyssen <q corresp="#nord1">Norden</q> vnd ist von natur kaltt vnnd - <lb/>schneygk <space/>Der vierd mit namen <term xml:id="süd1" ref="#Sueden">auster</term> wirt czu - <lb/>teutsch genant <q corresp="#süd1">Süden</q> vnd kompt vom mittagk. - <lb/>von natur naß vnd warm diß sein dy - <choice><orig>forderlichst<lb/>en</orig><reg>forderlichsten<lb/></reg></choice> viere - </p> - <p> - <lb/>Die andern 8 komen auß disen vieren alls v<choice><am>ō</am><ex>on</ex></choice> - <lb/>Osten komen zwen seytner der ein gegen - <choice><orig>mitter<lb/>nacht</orig><reg>mitternacht<lb/></reg></choice> heyst Ostnord auff latein Eurus vnd pr<choice><am>ī</am><ex>in</ex></choice>gt - <lb/>tunckel wolken. der ander gegen mittag der - <choice><orig>hey<lb/>set</orig><reg>heyset<lb/></reg></choice> Ostsüd ym latein Uulturnus dieser truckett v<choice><am>n̄</am><ex>nd</ex></choice> - <lb/>tawet. <space/>Der ander heist <term xml:id="west2" ref="#Westen">Fauonius</term> auff teutsch - <lb/><q corresp="#west2">west</q>. vnd hatt ouch czwen seytner. der ein gegen - <lb/>mitternacht heyßt <term xml:id="westnord" ref="#Westnord">Chorus</term> yn latein. czu deutsch - <lb/><q corresp="#westnord">westnord</q> diser bringt Nebel yn klarheyt. der - <choice><orig>an<lb/>der</orig><reg>ander<lb/></reg></choice> hat er gegen mittag vnd heyßt <placeName><term xml:id="westsued" ref="#Westsued">affricus</term></placeName>. tutsch - <lb/><q corresp="#westsued">westsüd</q> vnd ist von natur stormig mit regen - </p> - <p> - <lb/>Der dritte heißtt apercias yn mitternachtt vnd - <lb/>hat ouch zwen seytner den einen geg<choice><am>ē</am><ex>en</ex></choice> orient v<choice><am>n̄</am><ex>nd</ex></choice> - <lb/>heyßt <term xml:id="nordost" ref="#Nordost">aquilo oder boreas</term>. teutsch <q corresp="#nordost">Nordost</q> vnd - <lb/>ist von natur kalt vnd trucken. den andern hat er - <lb/>gegen occid<choice><am>ē</am><ex>en</ex></choice>t v<choice><am>n̄</am><ex>nd</ex></choice> heißt <term xml:id="nordwest" ref="#Nordwest">Circius</term>. teutsch <q corresp="#nordwest">nordwest</q> - <lb/>vnd ist von natur schneygk vnd hagelicht <space/>Der - <lb/>vird heyßt auster v<choice><am>n̄</am><ex>nd</ex></choice> hat ouch zwen seytner. der ein - <pb n="31" facs="Ptolemaeus_text_31"/> - <lb/>geg<choice><am>ē</am><ex>en</ex></choice> ori<choice><am>ē</am><ex>en</ex></choice>t v<choice><am>n̄</am><ex>nd</ex></choice> heißt <q corresp="#südost">Sudost</q>. lateynisch <term xml:id="südost" ref="#Suedost">euronoius</term> - <lb/>vnnd ist von natur warm yn mössigkeit. Der ander - <lb/>ist gegen occid<choice><am>ē</am><ex>en</ex></choice>t vnd haist ym latein <term xml:id="südwest" ref="#Suedwest">euroauster</term>. - <lb/>auff deutsch <q corresp="#suedwest">Sudwest</q> vnnd ist von natur gancz - <lb/>varm Diß sind dy zwelff wind dornoch sich d<choice><am>ā</am><ex>an</ex></choice> - <lb/>die wether vorwandlen auff diser werlt. - <lb/> - </p> - <p> - <table rows="19" cols="3"> - <row> - <cell></cell> - <cell></cell> - <cell>Ostnord v<choice><am>ō</am><ex>on</ex></choice> mitternacht</cell> - </row> - <row> - <cell><term xml:id="ost2" ref="#Ost2">Orientalis</term></cell> - <cell><q corresp="#ost2">Ost</q></cell> - <cell></cell> - </row> - <row> - <cell>vom vffgang</cell> - <cell></cell> - <cell></cell> - </row> - <row> - <cell></cell> - <cell></cell> - <cell>Ostsüd vom mittag</cell> - </row> - <!-- Leerzeile --> - <row> - <cell></cell> - <cell></cell> - <cell></cell> - </row> - <row> - <cell></cell> - <cell></cell> - <cell>westnord von mitternacht</cell> - </row> - <row> - <cell><term xml:id="west3" ref="#West2">occid<choice><am>ē</am><ex>en</ex></choice>talis</term></cell> - <cell><q corresp="#west3">west</q></cell> - <cell></cell> - </row> - <row> - <cell>v<choice><am>ō</am><ex>on</ex></choice> nydergang</cell> - <cell></cell> - <cell></cell> - </row> - <row> - <cell></cell> - <cell></cell> - <cell>westsud vom mittemtag.</cell> - </row> - <!-- Leerzeile --> - <row> - <cell></cell> - <cell></cell> - <cell></cell> - </row> - <row> - <cell></cell> - <cell></cell> - <cell>Nordost von orient.</cell> - </row> - <row> - <cell><term xml:id="nord2" ref="#Nord">sept<choice><am>ē</am><ex>em</ex></choice>trionalis</term></cell> - <cell><q corresp="#nord2">nord</q></cell> - <cell></cell> - </row> - <row> - <cell>v<choice><am>ō</am><ex>on</ex></choice> mitternacht</cell> - <cell></cell> - <cell></cell> - </row> - <row> - <cell></cell> - <cell></cell> - <cell>Nordwest von occident</cell> - </row> - <!-- Leerzeile --> - <row> - <cell></cell> - <cell></cell> - <cell></cell> - </row> - <row> - <cell></cell> - <cell></cell> - <cell>Südost von orient</cell> - </row> - <row> - <cell><term xml:id="süd2" ref="#Süd">meridionalis</term></cell> - <cell><q corresp="#süd2">Süd</q></cell> - <cell></cell> - </row> - <row> - <cell>vom myttag</cell> - <cell></cell> - <cell></cell> - </row> - <row> - <cell></cell> - <cell></cell> - <cell>Sudwest von occident</cell> - </row> - </table> - </p> - </div> - <div type="kapitel"> - <head> - <pb n="32" facs="Ptolemaeus_text_32"/> - <lb/>Uon teilung der irdischen werlt - </head> - <p> - <lb/> - <lb/> - <hi rend="initial">g</hi>Enczlich vorklerung diser ding dy vns - <lb/>dienen außzumessen vnd zuteilen dyse - <lb/>vnsere bekante welt genugsam - <choice><orig>gesche-<lb/>hen</orig><reg>geschehen<lb/></reg></choice> ist. beide zu erdtrich vnd zu mere - <lb/>berge vnd fliß wy und wo yczlichs gelegen sey. - <lb/>yn was teyl der werlt. v<choice><am>n̄</am><ex>nd</ex></choice> yn welch<choice><am>ē</am><ex>em</ex></choice> Clima. neb<choice><am>ē</am><ex>en</ex></choice> - <lb/>welchem paralell vnder welchem meridian das - <lb/>d<choice><am>ē</am><ex>em</ex></choice> leser leicht zu finden wirt sein. so er dise - <choice><orig>vor<lb/>geschribne</orig><reg>vorgeschribne<lb/></reg></choice> ding wol vernomen hat vnd si - <choice><orig>merk-<lb/>en</orig><reg>merken<lb/></reg></choice> wirt. So ist zuwissen wie das vnser ober - <choice><orig>emi<lb/>speri<choice><am>ū</am><ex>um</ex></choice></orig><reg>emisperi<choice><am>ū</am><ex>um</ex></choice><lb/></reg></choice> vnser werlt wirt geteiltt yn ym selbstt. - <choice><orig>dor<lb/>noch</orig><reg>dornoch<lb/></reg></choice> wirt es v<choice><am>ō</am><ex>on</ex></choice>ein<choice><am>ā</am><ex>an</ex></choice>der geteilt durch mere - <choice><orig>dorno<lb/>ch</orig><reg>dornoch<lb/></reg></choice> durch perg vnd ouch durch flyß. zu dem erst<choice><am>ē</am><ex>en</ex></choice> - <lb/>wirt es geteilt yn ym selbst das ist yn drey teil als - <lb/>yn <placeName type="continent">Asiam</placeName>. <placeName type="continent">Affricam</placeName> vnd <placeName type="continent">Europam</placeName>. zu dem - <choice><orig>and<lb/>ern</orig><reg>andern<lb/></reg></choice> werden dise drey teyl von einander - <choice><orig>geschay-<lb/>den</orig><reg>geschayden<lb/></reg></choice> durch die mere. das ist alls <placeName type="continent">asia</placeName> wirt - <choice><orig>geschei<lb/>den</orig><reg>gescheiden<lb/></reg></choice> von <placeName type="continent">europa</placeName> durch <placeName type="ocean">pont<choice><am>ū</am><ex>um</ex></choice> eurim<choice><am>ū</am><ex>um</ex></choice></placeName>. das ist das - <lb/><placeName type="ocean">swarcze mer</placeName> v<choice><am>n̄</am><ex>nd</ex></choice> durh das mer <placeName type="ocean">ege<choice><am>ū</am><ex>um</ex></choice></placeName>. z<choice><am>ū</am><ex>um</ex></choice> <choice><am>ā</am><ex>an</ex></choice>dern mal - <lb/>scheydt sich <placeName type="continent">asia</placeName> v<choice><am>ō</am><ex>on</ex></choice> <placeName type="continent">europa</placeName> durch gepurg. das ist - <lb/>zu latein g<choice><am>n̄</am><ex>enan</ex></choice>t <placeName type="mountain">Riffe<choice><am>ū</am><ex>um</ex></choice></placeName>. z<choice><am>ū</am><ex>um</ex></choice> driten wirt <placeName type="continent">europa</placeName> - <choice><orig>geschy<lb/>den</orig><reg>geschyden<lb/></reg></choice> von <placeName type="continent">asia</placeName> mit dem flyß <placeName type="river">tanays</placeName> das do - <choice><orig>entsprin-<lb/>get</orig><reg>entspringet<lb/></reg></choice> yn dem 18 paralelln vnd 70 meridian auß d<choice><am>ē</am><ex>em</ex></choice> - <lb/>perg <placeName type="mountain">Riffei</placeName>. <placeName type="continent">affrica</placeName> wirt geschaiden von <placeName type="continent">europa</placeName>. - <lb/>durch das <placeName type="ocean">mittel meer</placeName> vnd durch di enge - <choice><orig>hercu<lb/>lis</orig><reg>herculis<lb/></reg></choice> zum andern mol wirt <placeName type="continent">affrica</placeName> gescheyden von - <pb n="33" facs="Ptolemaeus_text_33"/> - <lb/><placeName type="continent">Asia</placeName> durch <placeName type="ocean">Sin<choice><am>ū</am><ex>um</ex></choice> arabic<choice><am>ū</am><ex>um</ex></choice></placeName>. Das ist durch das - <choice><orig>ara-<lb/>bische</orig><reg>arabische<lb/></reg></choice> hap deß <placeName type="ocean">rotten meres</placeName> <space/>Zum dritten mal - <lb/>wirt <placeName type="continent">affrica</placeName> geschaiden von <placeName type="continent">asia</placeName> durch den - <choice><orig>aller-<lb/>edelsten</orig><reg>alleredelsten<lb/></reg></choice> fluß <placeName type="river">Nilum</placeName>. - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Uon den meheren - </head> - <p> - <lb/> - <hi rend="initial">A</hi>Uff dieser kaull dess erttrichs. vnd - <choice><orig>dor<lb/>umb</orig><reg>dorumb<lb/></reg></choice> vindet man forderliche dreyerley - <lb/>mehr dy dyse kaul des ertrichs - <choice><orig>vmflie<lb/>ssen</orig><reg>vmfliessen<lb/></reg></choice> <space/>Das erst meer wirt g<choice><am>n̄</am><ex>enan</ex></choice>t <choice><am>ī</am><ex>in</ex></choice> late<choice><am>ī</am><ex>in</ex></choice> - <placeName type="ocean"><choice><orig>ma<lb/>re</orig><reg>mare<lb/></reg></choice> extrane<choice><am>ū</am><ex>um</ex></choice></placeName> oder <placeName type="ocean">extern<choice><am>ū</am><ex>um</ex></choice></placeName>. zu teutsch dy <placeName type="ocean">offne se</placeName>. dyß - <lb/>mor hat eyn hap das do genant wirt Hesperus - <choice><orig>si<lb/>nus.</orig><reg>sinus.<lb/></reg></choice> aber auff teutsch ist ein vsgewaschen hol oder - <lb/>kluns yn das ertrich nicht gleich dem rechten - <choice><orig>v-<lb/>ber</orig><reg>vber<lb/></reg></choice> auff teutsch eyn hap wideru<choice><am>m̄</am><ex>mb</ex></choice> wirt zu zeyten - <lb/>gesehen ein erttrich das sich wirfft vnd streckt in - <lb/>ein groß m<choice><am>ō</am><ex>ö</ex></choice>r. vnd wirt gen<choice><am>ā</am><ex>an</ex></choice>t auf kriechisch <term xml:id="halb1" ref="#Halbinsel"><choice><orig>cher-<lb/>sonesus.</orig><reg>chersonesus<lb/></reg></choice></term> zu teutsch <q corresp="#halb1">eyn vorschuß ein reff oder - <choice><orig>vor<lb/>schus</orig><reg>vorschus<lb/></reg></choice> deß erttrichs</q>. so hat das mör ouch ein - <choice><orig>vorsch<lb/>uß</orig><reg>vorschuß<lb/></reg></choice> g<choice><am>n̄</am><ex>enan</ex></choice>t <placeName type="island">zimbricus</placeName> der sich strekt gegen mitternachtt - <lb/>von orient vnd hat ettliche schene ynsuln - <placeName type="island"><choice><orig>hyber-<lb/>niam</orig><reg>hyberniam<lb/></reg></choice></placeName> das ist <placeName type="island">yrland</placeName>. <placeName type="country">Engelland</placeName>. <placeName type="county">Daciam</placeName>. - <placeName type="country"><choice><orig>Norive<lb/>gen.</orig><reg>Norivegen.<lb/></reg></choice></placeName> <placeName type="country">Sweden</placeName>. <placeName type="island">Gotland</placeName> vnd <placeName type="island">Schön</placeName>. von dan dy - <lb/>schene hering kom<choice><am>ē</am><ex>en</ex></choice> <space/>Das ander mer wirt gen<choice><am>ā</am><ex>an</ex></choice>t - <placeName type="ocean"><lb/>Mediterrane<choice><am>ū</am><ex>um</ex></choice></placeName> vnd hat ein hap das do g<choice><am>n̄</am><ex>enan</ex></choice>t wirt - <lb/>adriaticus welchs hap schaidet <placeName type="country"><choice><orig>kryechnlannd</orig><reg>kryechenlannd</reg></choice></placeName> - <lb/>vnd <placeName type="country">welsch land</placeName> vnd hat dyse vorderlichsten - <choice><orig>yn<lb/>suln</orig><reg>ynsuln<lb/></reg></choice> als <placeName type="island">Ciprus</placeName> <placeName type="island">Rodis</placeName> <placeName type="island">Candi<choice><am>ā</am><ex>am</ex></choice></placeName> <placeName type="island">Sicili<choice><am>ā</am><ex>am</ex></choice></placeName> <placeName type="island">Sard<choice><am>ī</am><ex>in</ex></choice>am</placeName> - <lb/><placeName type="island">Euboeam</placeName>. <placeName type="island">Nigropont</placeName> <placeName type="island">Curson</placeName> vnd <placeName type="island">Cursicam</placeName>. - <pb n="34" facs="Ptolemaeus_text_34"/> - </p> - <p> - <lb/>Das dritt mör wirt gen<choice><am>ā</am><ex>an</ex></choice>t <placeName type="ocean">yndic<choice><am>ū</am><ex>um</ex></choice></placeName>. das ist das - <placeName type="ocean"><choice><orig>yn<lb/>dische</orig><reg>yndische<lb/></reg></choice> mehr</placeName> vnd hat diese hap Bangetic<choice><am>ū</am><ex>um</ex></choice> persic<choice><am>ū</am><ex>um</ex></choice> - <lb/>arabic<choice><am>ū</am><ex>um</ex></choice> vnd das grosse hap yn orient. ouch hatt - <lb/>es ein forderliche ynsul <placeName type="island">Taprobana</placeName> gen<choice><am>ā</am><ex>an</ex></choice>t - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Uon den teylen <placeName type="continent">Europe</placeName> - </head> - <p> - <lb/> - <hi rend="initial">F</hi>Urderlich zu sagen wy dy teyl diser - <choice><orig>yr-<lb/>dischen</orig><reg>yrdischen<lb/></reg></choice> kaul geteylt vnt geschaid<choice><am>ē</am><ex>en</ex></choice> - <choice><orig>wer-<lb/>den</orig><reg>werden<lb/></reg></choice> durch fliß vnd berg yczlichs yn - <choice><orig>s<choice><am>ū</am><ex>un</ex></choice>-<lb/>derhait.</orig><reg>s<choice><am>ū</am><ex>un</ex></choice>derhait.<lb/></reg></choice> So well<choice><am>ē</am><ex>en</ex></choice> wir an heben an d<choice><am>ē</am><ex>em</ex></choice> - <lb/>teyl <placeName type="continent">europe</placeName> vnd dyß teyl also zu gliden vnd - <choice><orig>auß-<lb/>legen</orig><reg>außlegen<lb/></reg></choice> <space/><placeName type="continent">Europa</placeName> wirt gen<choice><am>ā</am><ex>an</ex></choice>t von der tochter dess - <lb/><persName><roleName>kunigs</roleName> <forename>Generois</forename></persName> yn <placeName type="country">Libia</placeName> dy do g<choice><am>n̄</am><ex>enan</ex></choice>t was - <persName><forename><choice><orig>Euro-<lb/>pa.</orig><reg>Europa.<lb/></reg></choice></forename></persName> <space/>Alls v<choice><am>ō</am><ex>on</ex></choice> yr dan dy poet<choice><am>ē</am><ex>en</ex></choice> schymflich - <choice><orig>schrei-<lb/>ben.</orig><reg>schreiben.<lb/></reg></choice> wie das <persName><forename>Jupiter</forename></persName> geg<choice><am>ē</am><ex>en</ex></choice> yr hiczlich yn lieb was - <lb/>entzundet vnd mocht si bey dem <persName><roleName>künig</roleName> <forename>generois</forename></persName> - <lb/>nicht zu willen hab<choice><am>ē</am><ex>en</ex></choice> do nam er an sich ein gestalt - <lb/>eins ochsen vnd schwam vber mör zu dieser - <choice><orig>i<choice><am>ū</am><ex>un</ex></choice>ck<lb/>frawen.</orig><reg>i<choice><am>ū</am><ex>un</ex></choice>ckfrawen.<lb/></reg></choice> vnd si auff seinem ruck fassend. füret er si - <lb/>vber mör vnd pracht si yn <placeName type="continent">europ<choice><am>ā</am><ex>am</ex></choice></placeName> dauon hat dis - <lb/>teil seinem nam<choice><am>ē</am><ex>en</ex></choice>. diß teyl hatt yn yme vil schoner - <lb/>berg dy do alle vrspringlich v<choice><am>ō</am><ex>on</ex></choice> einer worzel - <choice><orig>k<choice><am>ō</am><ex>om</ex></choice>-<lb/>en.</orig><reg>k<choice><am>ō</am><ex>om</ex></choice>en.<lb/></reg></choice> welche hoe der würczell aller geburg yn - <choice><orig>diss<lb/>er</orig><reg>disser<lb/></reg></choice> <placeName type="continent">europa</placeName> ein urspr<choice><am>ū</am><ex>un</ex></choice>g gibt vnd wirt gen<choice><am>ā</am><ex>an</ex></choice>t <placeName type="mountain">Alpes</placeName> - <lb/>auß welchem perg dy grösten fliß yn <placeName type="continent">europa</placeName> - <choice><orig>ent-<lb/>spring<choice><am>ē</am><ex>en</ex></choice>.</orig><reg>entspring<choice><am>ē</am><ex>en</ex></choice>.<lb/></reg></choice> alls der <placeName type="river">rein</placeName> der do fleüst durch - <placeName type="country"><choice><orig>teutsch<lb/>land</orig><reg><lb/>teutschland</reg></choice></placeName>. <placeName type="river">Rodanus</placeName> der do fleust durch <placeName type="counrty">franckreich</placeName> - <lb/>Dy <placeName type="river">thunav</placeName> <placeName type="river">danubius</placeName> g<choice><am>n̄</am><ex>enan</ex></choice>t dy do fleist durch - <placeName type="country"><choice><orig>Pay<lb/>ern</orig><reg>Payern<lb/></reg></choice></placeName> <placeName type="country">Steyern</placeName> <placeName type="country">H<choice><am>ū</am><ex>un</ex></choice>gern</placeName> vnd <placeName type="country">Ostereich</placeName>. Und ouch - <pb n="35" facs="Ptolemaeus_text_35"/> - <lb/>durch dy <placeName type="country">walachey</placeName> vnd felt yn <placeName type="ocean">mare Ponticum</placeName>. - <lb/><placeName type="river">Padus</placeName> flevst durch <placeName type="country">Lumbardey</placeName> vnd <placeName type="country">welschl<choice><am>ā</am><ex>an</ex></choice>d</placeName> - <lb/>Der vorg<choice><am>n̄</am><ex>enan</ex></choice>t berg wirfft von ym drey armen. Der - <lb/>erst get durch <placeName type="country">Swab<choice><am>ē</am><ex>en</ex></choice></placeName>. <placeName type="country">behem<choice><am>ē</am><ex>en</ex></choice></placeName>. <placeName type="country">slesie</placeName>. bis <choice><am>ī</am><ex>in</ex></choice> <placeName type="country">bol<choice><am>ē</am><ex>en</ex></choice></placeName>. - <lb/>do wirt er gen<choice><am>ā</am><ex>an</ex></choice>t <placeName type="river">Carpatus</placeName> das ist krumppach - <lb/>dornoch durch <placeName type="country">revssen</placeName> vnt wirt g<choice><am>n̄</am><ex>enan</ex></choice>t <placeName type="river">Riffeus</placeName> - <choice><orig>dor<lb/>noch</orig><reg>dornoch<lb/></reg></choice> durch <placeName type="continent">aziam</placeName> Der ander arm get durch - <placeName type="country"><choice><orig>swey<lb/>czerland</orig><reg>sweyczerland<lb/></reg></choice></placeName> durch <placeName type="country">Payern</placeName> <placeName type="country">Steyern</placeName>. <placeName type="country">schlauoni<choice><am>ē</am><ex>en</ex></choice></placeName>. - <placeName type="country"><choice><orig>wa<lb/>lachey</orig><reg>walachey<lb/></reg></choice></placeName> vnd durch <placeName type="country">Criecherland</placeName> bis geg<choice><am>ē</am><ex>en</ex></choice> - <placeName type="city"><choice><orig>C<choice><am>ō</am><ex>on</ex></choice>stan<lb/>tinopolim</orig><reg>C<choice><am>ō</am><ex>on</ex></choice>stantinopolim<lb/></reg></choice></placeName> do er sich an dem mör endet Der dritt - <lb/>arm gat durch gancz <placeName type="country">welschland</placeName> vnd teylt das <choice><am>ī</am><ex>in</ex></choice> - <lb/>zwey teil. do mit yn gemeinheit sey gnug gesagtt - <lb/>von den teyln <placeName type="continent">europe</placeName> yn sunderhayt aber also - <lb/> - </p> - </div> - <div type="kapitel"> - <head> - <lb/><placeName type="island">Hybernia</placeName> <placeName type="island">yrland</placeName>. - <lb/><placeName type="island">Albion</placeName> <placeName type="island">Engelland</placeName> 1 - </head> - <p> - <lb/> - <lb/>Hyenach ist zu mercken das dyser Numerus - <lb/>yn der figur ouch verczeichnet ist. - <lb/> - </p> - <p> - <lb/> - <hi rend="initial">D</hi>Jse zwu ynsuln lig<choice><am>ē</am><ex>en</ex></choice> ym occid<choice><am>ē</am><ex>en</ex></choice>t yn dem - <lb/>18 paralell vnd vnder dem 4 meridi<choice><am>ā</am><ex>an</ex></choice> das - <lb/>sind 25 grad vom occident vberal mit - <lb/>dem <placeName type="ocean">eustern mör</placeName> vmgeben vnd gegen - <lb/>mittag hab<choice><am>ē</am><ex>en</ex></choice> si dy land von <placeName type="country">franckreich</placeName>. yn der - <choice><orig>yn<lb/>sull</orig><reg>ynsull<lb/></reg></choice> <placeName type="island">albion</placeName> oder <placeName type="island">brittania</placeName> ligt dy stat <placeName type="city">lundin</placeName> dess - <lb/><persName><roleName>kungs v<choice><am>ō</am><ex>on</ex></choice> Engelland</roleName></persName> der eyn herr ist dyser - <choice><orig>zwey<lb/>er</orig><reg>zweyer<lb/></reg></choice> Jnsuln <placeName type="island">Britanie</placeName> vnd <placeName type="island">hybernie</placeName> - </p> - <p> - <pb n="36" facs="Ptolemaeus_text_36"/> - <lb/>Jn disem <placeName type="island">yrland</placeName> findt man mensch<choice><am>n̄</am><ex>en</ex></choice> dy do sw<choice><am>ē</am><ex>en</ex></choice>cz - <lb/>haben an yren hynderteyln grob sind sy vnd - <choice><orig>vn<lb/>edel</orig><reg>vnedel<lb/></reg></choice> vnd haben yn diser geg<choice><am>ē</am><ex>en</ex></choice>heyt schön vich czu - <lb/>narung py disen ynsuln sind dy orchen oder - <choice><orig>nor-<lb/>wegischen</orig><reg>norwegischen<lb/></reg></choice> clippen 30 yn der zall. auch hat dyse - <lb/>ynsul <placeName type="island">albion</placeName> einen schönen lustig<choice><am>ē</am><ex>en</ex></choice> wald - <choice><orig>Cottidu<lb/>nia</orig><reg>Cottidunia<lb/></reg></choice> g<choice><am>n̄</am><ex>enan</ex></choice>t nuczlich von gutter waid bey dises lands - <lb/>gelegenhait entspringt ein schön flis <placeName type="river">Liua</placeName> genant - <lb/>fallend yn das <placeName type="ocean">teutsch mör</placeName> dise ynsulln haben d<choice><am>ē</am><ex>en</ex></choice> - <lb/>lengsten tag 18 stund vnd ein grad nach der leng - <lb/>helt eyn grad 32 meylln 1/2 . - </p> - </div> - <div type="kapitel"> - <head> - <lb/><placeName type="island">Hyspania</placeName> 2 - </head> - <p> - <lb/> - <placeName type="island"><hi rend="initial">H</hi>Yspania</placeName> leyt ym occident vmflossen v<choice><am>ō</am><ex>on</ex></choice> - <lb/>obend mit dem <placeName type="ocean">ewssern mehr</placeName>. v<choice><am>ō</am><ex>on</ex></choice> - <choice><orig>mitt-<lb/>tag</orig><reg>mittag<lb/></reg></choice> mit dem <placeName type="ocean">yberischen mer</placeName> vnd mit - <choice><orig>d<lb/>enge</orig><reg>der<lb/>enge</reg></choice> herculis auch mit dem - <placeName type="ocean"><choice><orig>Balleary-<lb/>schen</orig><reg>Ballearyschen<lb/></reg></choice> mör</placeName>. von miternacht mit der offenbaren se - <lb/>vom morgen wirt das land beschlossen mit den - <lb/>bergen <placeName type="mountain">Pirrenes</placeName>. vnd wirt beschlossen zwischen - <lb/>dem ersten vnd dritten meridian vnd czwischen - <lb/>dem 11 vnd 14 paralell. dise gegenheytt hat siben - <lb/>kunigkreich alls <placeName type="country">Portigal</placeName> <placeName type="country">Granate</placeName> <placeName type="country">Nauarre</placeName> - <placeName type="country"><choice><orig>Hi<lb/>spanie</orig><reg>Hispanie<lb/></reg></choice></placeName> <placeName type="country">Castelle</placeName> <placeName type="country">Kathelonie</placeName> vnd <placeName type="country">Gallicie</placeName>. yn dysem - <lb/>kunigkreich <placeName type="country">gallicie</placeName> leyt <persName><roleName>Sant</roleName> <forename>Jacob</forename></persName> zu - <placeName type="city"><choice><orig>compe-<lb/>stel</orig><reg>compestel<lb/></reg></choice></placeName> yn dem land <placeName type="country">Bethica</placeName> zugehörig dem <persName><roleName>künig - <lb/>v<choice><am>ō</am><ex>on</ex></choice> hispania</roleName></persName> ligt ein stat <placeName type="city">Cordula</placeName> gen<choice><am>ā</am><ex>an</ex></choice>t von dan ist - <lb/>burtig gewest <persName><forename>Seneca</forename> <roleName>der maister</roleName></persName> ouch ligt eyn - <lb/>stat yn <placeName type="coutry">hyspania</placeName> <placeName type="city">Lispon</placeName> gen<choice><am>ā</am><ex>an</ex></choice>t do ist groß - <choice><orig>handt-</orig><reg>handt</reg></choice><pb n="37" facs="Ptolemaeus_text_37"/><choice><orig>lung</orig><reg>lung</reg></choice> vnd nyderlag <name type="place">europe</name> vnd <name type="place">affrice</name>. dy - <choice><orig>forder<lb/>lichsten</orig><reg>forderlichsten<lb/></reg></choice> flisss yn <placeName type="country"><choice><orig>hyspana</orig><reg>hyspania</reg></choice></placeName> sind dyse <placeName type="river">Tagus</placeName> - <placeName type="river"><choice><orig>Hy-<lb/>berus</orig><reg>Hyberus<lb/></reg></choice></placeName> <placeName type="river">Betis</placeName>. <placeName type="river">doria</placeName>. vnd <placeName type="river">pisarga</placeName>. <placeName type="river">Hiberus</placeName> vnnd - <lb/><placeName type="river">Pysarga</placeName> entspringen auß dem berg <placeName type="mountain">vindio</placeName> - <placeName type="river"><choice><orig>Tag-<lb/>us</orig><reg>Tagus<lb/></reg></choice></placeName> vß dem berg <placeName type="mountain">ortospeda</placeName> dy grest leng deß tags - <lb/>yn <placeName type="country">hyspania</placeName> ym mittl ist 15 stund. ein grad macht - <lb/>nach der leng 47 meiln. - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Gallia <placeName type="country">Franckreich</placeName> 3 - </head> - <p> - <lb/> - <placeName type="country"><hi rend="initial">F</hi>Ranckreich</placeName> das lantd wirt gelobt von - <lb/>fruchtbarkeyt der menschen vnd deß - <lb/>vichs mit guter bequemikeit aller - <choice><orig>nott-<lb/>turfft.</orig><reg>nottturfft.<lb/></reg></choice> vnd ist gelegen vom occid<choice><am>ē</am><ex>en</ex></choice>t das - <lb/>ist von dem gemeinen orison ym obend 18 grad - <lb/>das ist yn dem fierden meridi<choice><am>ā</am><ex>an</ex></choice> v<choice><am>n̄</am><ex>nd</ex></choice> get bys auf den - <lb/>7 meridian v<choice><am>ō</am><ex>on</ex></choice> occident vnd ligt zwischen d<choice><am>ē</am><ex>em</ex></choice> 15 - <lb/>vnd 30 Grad <space/>Yn <placeName type="country">Franckreich</placeName> ist der - <lb/>lengst tag 16 stund. yn dysem <placeName type="country">franckreich</placeName> sind vil - <lb/>löblicher stet dy von kurczwegen hye nit benant - <lb/>werden yn dysem <placeName type="country">franckreich</placeName> ist eyn gegenhaytt - <lb/><placeName type="country">Reynland</placeName> g<choice><am>n̄</am><ex>enan</ex></choice>t dorin <placeName type="city">coln</placeName> leit vnd ander vil - <choice><orig>nam-<lb/>hafftiger</orig><reg>namhafftiger<lb/></reg></choice> stett ouch sein yn dysem vorg<choice><am>n̄</am><ex>enan</ex></choice>ten - <placeName type="country"><choice><orig>frank<lb/>reich</orig><reg>frankreich<lb/></reg></choice></placeName> vorzeyten gelegen <placeName type="country">Picarden</placeName> <placeName type="country">Prabendern</placeName> - <lb/><placeName type="country">Lotringen</placeName> <placeName type="country">Elzezir</placeName> <placeName type="country">Schweiczer westrich</placeName> vnd - <choice><orig>an<lb/>der</orig><reg>ander<lb/></reg></choice> vil nacion czu streyt wol tugendhevtt greme - <lb/>zind mit germania dyß teyl offtgenant hat dy perg - <lb/><placeName type="mountain">Alpes</placeName> doruß entspringen Dye czwen schön flyß - <lb/><placeName type="river">Reyn</placeName> vnd Rhodanus. v<choice><am>n̄</am><ex>nd</ex></choice> das sind ouch dy - <choice><orig>treff<lb/>lichsten</orig><reg>trefflichsten<lb/></reg></choice> wasser yn diss<choice><am>ē</am><ex>em</ex></choice> land. - <pb n="38" facs="Ptolemaeus_text_38"/> - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Germania nyderste <placeName type="country">teutschland</placeName>. 4 - <lb/> - </head> - <p> - <lb/> - <hi rend="initial">D</hi>Js teil wirt beslossen von mitternachtt - <lb/>mit dem <placeName type="ocean">germ<choice><am>ā</am><ex>an</ex></choice>ico mör</placeName> das ist das - <placeName type="ocean"><choice><orig>teut<lb/>sch</orig><reg>teutsch<lb/></reg></choice> mör</placeName> Am andern teil mit dem - <placeName type="ocean"><choice><orig>sarma<lb/>tisch<choice><am>ē</am><ex>en</ex></choice></orig><reg>sarmatisch<choice><am>ē</am><ex>en</ex></choice><lb/></reg></choice> mör</placeName> geg<choice><am>ē</am><ex>en</ex></choice> mitternacht von dem morgen mitt - <lb/>der <placeName type="river">weichsel</placeName> vom ob<choice><am>ē</am><ex>en</ex></choice>d mit dem <placeName type="river">Reyn</placeName> vom mittag mit - <lb/>der <placeName type="river">thunav</placeName> Dise germania leyt v<choice><am>ō</am><ex>on</ex></choice> ab<choice><am>ē</am><ex>en</ex></choice>t 30 - <lb/>grad. das ist der 6 meridian vom ob<choice><am>ē</am><ex>en</ex></choice>d vndt - <lb/>get bys yn den 9 meridian. Und merk alweg - <lb/>das ein spaci<choice><am>ū</am><ex>um</ex></choice> in diser figur 2 meridi<choice><am>ā</am><ex>an</ex></choice> bedeit v<choice><am>n̄</am><ex>nd</ex></choice> leyt - <lb/>zwischen 13 vnd 15 paralelln gegen mitternachtt - <lb/>Dy namhafftigste perg yn germ<choice><am>ā</am><ex>an</ex></choice>ia sind arme - <lb/>v<choice><am>ō</am><ex>on</ex></choice> dem <placeName type="mountain">Swiczer gepurg</placeName>. diser berg lengt vnd - <choice><orig>brei<lb/>tet</orig><reg>breitet<lb/></reg></choice> sich durch <placeName type="country">Peyern</placeName> <placeName type="country">Behemen</placeName> vnd <placeName type="country">slesie</placeName> etc. - <lb/>Und dise germ<choice><am>ā</am><ex>an</ex></choice>ia hat dise gröst<choice><am>ē</am><ex>en</ex></choice> fliß domit si - <choice><orig>be-<lb/>kleidet</orig><reg>bekleidet<lb/></reg></choice> vnd geedert ist als der <placeName type="river">Reyn</placeName> der ouch - <lb/><choice><am>ī</am><ex>in</ex></choice> <placeName type="country">franckreich</placeName> verzeichnet ist. das ist v<choice><am>ō</am><ex>on</ex></choice> deßweg<choice><am>ē</am><ex>en</ex></choice> das - <lb/>sich dy herschafft der selb<choice><am>ē</am><ex>en</ex></choice> land offt verendert hatt - <lb/>ydoch ist der <placeName type="river">Reyn</placeName> ein vnderschaid zwisch<choice><am>ē</am><ex>en</ex></choice> - <placeName type="country"><choice><orig>frank<lb/>reich</orig><reg>frankreich<lb/></reg></choice></placeName> vnd <placeName type="country">teutsch<choice><am>ē</am><ex>en</ex></choice> landen</placeName> Dornoch ist yn - <lb/>germ<choice><am>ā</am><ex>an</ex></choice>ia dy <placeName type="river">thunav</placeName> dy do aber scheidet <placeName type="country">teutschland</placeName> - <placeName type="country"><choice><orig>O-<lb/>stereich</orig><reg>Ostereich<lb/></reg></choice></placeName> vnd <placeName type="country">vngern</placeName> Dornoch dy <placeName type="river">Elbe</placeName>. dy <placeName type="river">ader</placeName>. di - <lb/><placeName type="river">weser</placeName>. dy <placeName type="river">Mora</placeName>. <placeName type="river">Regus</placeName> dau<choice><am>ō</am><ex>on</ex></choice> <placeName type="city">Reg<choice><am>ē</am><ex>en</ex></choice>spurg</placeName> wirtt - <lb/>gen<choice><am>ā</am><ex>an</ex></choice>t. dis land hat 15 stund tag 1/2 vff das lengst. - <lb/>vnd helt yn ym dise Nacion. <placeName type="country">Schwaben</placeName> <placeName type="country">Fr<choice><am>ā</am><ex>an</ex></choice>ck<choice><am>ē</am><ex>en</ex></choice></placeName> - <lb/><placeName type="country">Peyern</placeName>. <placeName type="country">westual<choice><am>ē</am><ex>en</ex></choice></placeName> <placeName type="country">marker</placeName> <placeName type="country">Pomern</placeName>.<placeName type="country">großpoln</placeName> - <placeName type="country"> <choice><orig>kle<choice><am>ī</am><ex>in</ex></choice><lb/>poln</orig><reg>kle<choice><am>ī</am><ex>in</ex></choice>poln<lb/></reg></choice></placeName> <placeName type="country">merher</placeName>. <placeName type="country">slesie</placeName>. <placeName type="country">Behemen</placeName>. <placeName type="country">Frisen</placeName>. <placeName type="country">Hollend</placeName> - <lb/><placeName type="country">Brabender</placeName>. <placeName type="country">Schweiczer</placeName> - </p> - </div> - <div type="kapitel"> - <head> - <pb n="39" facs="Ptolemaeus_text_39"/> - <lb/><placeName type="island">Gotland</placeName>. <placeName type="country">norweg<choice><am>ē</am><ex>en</ex></choice></placeName>. <placeName type="country">schwed<choice><am>ē</am><ex>en</ex></choice></placeName> - <lb/> - </head> - <p> - <hi rend="initial">D</hi>Jse land lig<choice><am>ē</am><ex>en</ex></choice> in der offenbar<choice><am>ē</am><ex>en</ex></choice> se auff allen - <lb/>orten vmfloss<choice><am>ē</am><ex>en</ex></choice> v<choice><am>n̄</am><ex>nd</ex></choice> ligt ser geg<choice><am>n̄</am><ex>en</ex></choice> mitirnacht - <lb/>vnder dem 21 paralell v<choice><am>n̄</am><ex>nd</ex></choice> zwischen dem 9 - <lb/>vnd 10 meridian vnd ligt von d<choice><am>ē</am><ex>em</ex></choice> gemein<choice><am>ē</am><ex>en</ex></choice> - <choice><orig>occi-<lb/>dent</orig><reg>occident<lb/></reg></choice> 40 <choice><am>ḡ</am><ex>grad</ex></choice>. vnd e<choice><am>ī</am><ex>in</ex></choice> zweyteil ouch haben ander vil - <lb/>land grenicz nohe mit disen Alls <placeName type="country">holl<choice><am>ā</am><ex>an</ex></choice>d</placeName> - <placeName type="country"> <choice><orig>engro-<lb/>nen</orig><reg>engronen<lb/></reg></choice> <choice><orig>ladt.</orig><reg>landt.</reg></choice></placeName> <placeName type="country">v<choice><am>ē</am><ex>en</ex></choice>tel<choice><am>ā</am><ex>an</ex></choice>d.</placeName> <placeName type="country">vermel<choice><am>ā</am><ex>an</ex></choice>d</placeName>. <placeName type="country">eml<choice><am>ā</am><ex>an</ex></choice>d</placeName>. <placeName type="country">leifl<choice><am>ā</am><ex>an</ex></choice>d</placeName>. vndt - <lb/>dise l<choice><am>ā</am><ex>an</ex></choice>d hab<choice><am>ē</am><ex>en</ex></choice> tag 21 st<choice><am>ū</am><ex>un</ex></choice>d vffs lengst gegen mitag - <lb/>aber do ist <placeName type="island">sel<choice><am>ā</am><ex>an</ex></choice>d</placeName> <placeName type="island">laul<choice><am>ā</am><ex>an</ex></choice>d</placeName>. <placeName type="island">sicholm</placeName>. <placeName type="island">bernholn</placeName>. <placeName type="island">falster</placeName> - <lb/>vnd ander vil ynsuln dy v<choice><am>ō</am><ex>on</ex></choice> kurczwege nicht - <choice><orig>verczei-<lb/>chnet</orig><reg>verczeichnet<lb/></reg></choice> werden. Es wirt ouch gesaczt v<choice><am>ō</am><ex>on</ex></choice> der Figur - <lb/><persName><surname>ptolomei</surname></persName>. <placeName type="country">Eyßl<choice><am>ā</am><ex>an</ex></choice>d</placeName> v<choice><am>n̄</am><ex>nd</ex></choice> das <placeName type="ocean">geliuert mör</placeName> das ligt als - <lb/>vßwendig dez zirkel artico gegen mitternacht di - <lb/>selben haben tag 3 Monat l<choice><am>ā</am><ex>an</ex></choice>gk - </p> - </div> - <div type="kapitel"> - <head> - <lb/><placeName type="country">P<choice><am>ā</am><ex>an</ex></choice>n<choice><am>ō</am><ex>on</ex></choice>ia</placeName> <placeName type="country">H<choice><am>ū</am><ex>un</ex></choice>gern</placeName> vnd <placeName type="country">Ostereich</placeName> 5 - <lb/> - </head> - <p> - <hi rend="initial">D</hi>Je l<choice><am>ā</am><ex>an</ex></choice>tschaft also g<choice><am>n̄</am><ex>enan</ex></choice>t ligt v<choice><am>ō</am><ex>on</ex></choice> dem gemein - <choice><orig>o<lb/>ccident</orig><reg>occident<lb/></reg></choice> yn yrem anheb<choice><am>ē</am><ex>en</ex></choice> 30 <choice><am>ḡ</am><ex>grad</ex></choice>. das ist der 6 - <lb/>meridi<choice><am>ā</am><ex>an</ex></choice> vnd begreifft in der leng 3 meridi<choice><am>ā</am><ex>an</ex></choice> 1/2 das - <lb/>ist biß yn den 47 <choice><am>ḡ</am><ex>grad</ex></choice>. v<choice><am>n̄</am><ex>nd</ex></choice> wirt beschlossen v<choice><am>ō</am><ex>on</ex></choice> - <choice><orig>miter<lb/>nacht</orig><reg>miternacht<lb/></reg></choice> mit der <placeName type="mountain">tunav</placeName> v<choice><am>ō</am><ex>on</ex></choice> mittag mit dem - <placeName type="ocean"><choice><orig>venedy<lb/>schen</orig><reg>venedyschen<lb/></reg></choice> mor</placeName> v<choice><am>ō</am><ex>on</ex></choice> orient mit der <placeName type="river">theysse</placeName> v<choice><am>ō</am><ex>on</ex></choice> occid<choice><am>ē</am><ex>en</ex></choice>t mitt - <lb/>d<choice><am>ē</am><ex>en</ex></choice> hoh<choice><am>ē</am><ex>en</ex></choice> <placeName type="mountain">alp<choice><am>ē</am><ex>en</ex></choice> perg<choice><am>n̄</am><ex>en</ex></choice></placeName>. von disen <placeName type="mountain">alpen</placeName> k<choice><am>ō</am><ex>om</ex></choice>pt durch - <choice><orig>di<lb/>se</orig><reg>dise<lb/></reg></choice> <placeName type="country">p<choice><am>ā</am><ex>an</ex></choice>n<choice><am>ō</am><ex>on</ex></choice>ia</placeName> e<choice><am>ī</am><ex>in</ex></choice> arm vnt scheidt diwind<choice><am>ē</am><ex>en</ex></choice> das ist das l<choice><am>ā</am><ex>an</ex></choice>t - <lb/><placeName type="country">illiris</placeName> vnt <placeName type="country">liburnia</placeName>. v<choice><am>ō</am><ex>on</ex></choice> beiden teilln <placeName type="country">H<choice><am>ū</am><ex>un</ex></choice>gerl<choice><am>ā</am><ex>an</ex></choice>d</placeName>. Dise - <lb/><placeName type="country">p<choice><am>ā</am><ex>an</ex></choice>n<choice><am>ō</am><ex>on</ex></choice>ia</placeName> wirt geteilt yne czwey teil als <choice><am>ī</am><ex>in</ex></choice> <placeName type="country">pan<choice><am>ō</am><ex>on</ex></choice>iam</placeName> - <lb/>das ober vnd das niderl<choice><am>ā</am><ex>an</ex></choice>d. das do wirt g<choice><am>n̄</am><ex>enan</ex></choice>t das niderteil - <lb/>wirt g<choice><am>ā</am><ex>an</ex></choice>cz <placeName type="country">h<choice><am>ū</am><ex>un</ex></choice>gerl<choice><am>ā</am><ex>an</ex></choice>t</placeName> besloss<choice><am>ē</am><ex>en</ex></choice> mit der <placeName type="country">stirmark</placeName> <placeName type="country">crabat<choice><am>ē</am><ex>en</ex></choice></placeName> - </p> - <p> - <pb n="40" facs="Ptolemaeus_text_40"/> - <lb/>Das ander wirt g<choice><am>n̄</am><ex>enan</ex></choice>t das oberteil. das ist <placeName type="country">ostereich</placeName>. - <lb/>ouch hat es weyte vnd grosse geg<choice><am>ē</am><ex>en</ex></choice>heitt reci<choice><am>ā</am><ex>am</ex></choice> das - <lb/>ist <placeName type="country">corual<choice><am>n̄</am><ex>en</ex></choice></placeName> <placeName type="country">vindeliciam</placeName> das ist <placeName type="country">hochswab<choice><am>ē</am><ex>en</ex></choice></placeName> <placeName type="country">Noricu</placeName> - <lb/>das iß <placeName type="country">bairn</placeName>. <placeName type="country">ylliris</placeName> vnd <placeName type="country">liburnia</placeName> das sind <placeName type="country">w<choice><am>ī</am><ex>in</ex></choice>dische - <lb/>l<choice><am>ā</am><ex>an</ex></choice>d</placeName>. dy grest<choice><am>n̄</am><ex>en</ex></choice> stett hy yn <placeName type="country">swoben</placeName> pey <placeName type="country">Ostereych</placeName>. - <lb/>v<choice><am>n̄</am><ex>nd</ex></choice> steirn ligend sind Augspurg landshut v<choice><am>n̄</am><ex>nd</ex></choice> - <placeName type="city"><choice><orig>reg<lb/>ensburg</orig><reg>regensburg<lb/></reg></choice></placeName> Yn <placeName type="country">ostereich</placeName> dy forderlichste stat ist <placeName type="city">wyen</placeName>. - <lb/>yn diser geg<choice><am>ē</am><ex>en</ex></choice>heit ligt der berg <placeName type="mountain">cecius</placeName>. den <choice><orig>ma</orig><reg>man</reg></choice> n<choice><am>ē</am><ex>en</ex></choice>t - <lb/><placeName type="mountain">kalenberg</placeName>. dy grost<choice><am>ē</am><ex>en</ex></choice> flyß yn <placeName type="country">hungern</placeName> sind dise dy - <lb/><placeName type="river">Thunav</placeName> dye <placeName type="river">saw</placeName> vnnd dy <placeName type="river">teysse</placeName> dy vorg<choice><am>n̄</am><ex>enan</ex></choice>te l<choice><am>ā</am><ex>an</ex></choice>d - <lb/>hab<choice><am>ē</am><ex>en</ex></choice> grosten tag 16 stund dy an der <placeName type="river">thunav</placeName> ligen. - <lb/>dyse land aber dy do ser lige gegen mittag dy - <choice><orig>ha<lb/>ben</orig><reg>haben<lb/></reg></choice> 15 stund vnd 1/2</p> - </div> - <div type="kapitel"> - <head> - <lb/><placeName type="country">Jtalia</placeName> <placeName type="country">welschland</placeName> 6 - <lb/> - </head> - <p> - <placeName type="country"><hi rend="initial">W</hi>elsche land</placeName> sein gelegen yn <placeName type="continent">europa</placeName> v<choice><am>ō</am><ex>on</ex></choice> - <lb/>dem occid<choice><am>ē</am><ex>en</ex></choice>t mit irem anheben 29 grad. - <lb/>das ist yn dem 6 meridian an czwen grad. - <lb/>vnd streket sich yn den morgen bis auff 43 grad - <lb/>das macht v<choice><am>ō</am><ex>on</ex></choice> occident gerechent dy preyte <placeName type="country">ytalie</placeName> - <lb/>gleich drey meridian <space/>Diß land wirt vm - <lb/>slossen vom mittag mit dem <placeName type="ocean">Tirrenische mer</placeName> v<choice><am>ō</am><ex>on</ex></choice> - <lb/>mitternacht mit ein<choice><am>ē</am><ex>em</ex></choice> arm von den <placeName type="mountain">alpenbergen</placeName>. - <lb/>von ori<choice><am>ē</am><ex>en</ex></choice>t mit dem <placeName type="ocean">yonisch<choice><am>ē</am><ex>en</ex></choice> mer</placeName> von occid<choice><am>ē</am><ex>en</ex></choice>t mit - <lb/>d<choice><am>ē</am><ex>em</ex></choice> <placeName type="mountain">sweyczer gepurg</placeName>. dyß land wirt geteylt yn 2 - <lb/>teyl alls yn <placeName type="country">lombardy</placeName> vnd yn <placeName type="country"> welschland</placeName>. yn - <choice><orig>dy-<lb/>ser</orig><reg>dyser<lb/></reg></choice> geg<choice><am>ē</am><ex>en</ex></choice>heyt sind etliche namhafftig stet als - <placeName type="city"><choice><orig>Ge-<lb/>nva.</orig><reg>Genva.<lb/></reg></choice></placeName> <placeName type="city">mediolan</placeName>. <placeName type="city">florencz</placeName> <placeName type="city">Rom</placeName>. <placeName type="city"><choice><sic>Uenedig</sic><corr>Venedig</corr></choice></placeName>. <placeName type="city">neapolis</placeName> - <lb/>vnd ander vil stet dy hye nit g<choice><am>n̄</am><ex>enan</ex></choice>t werd<choice><am>ē</am><ex>en</ex></choice>. ouch hat - <pb n="41" facs="Ptolemaeus_text_41"/> - <lb/>diß <placeName type="country">welschland</placeName> vm sich vil schoner ynsuln dy - <choice><orig>vor<lb/>derlichste</orig><reg>vorderlichste<lb/></reg></choice> v<choice><am>n̄</am><ex>nd</ex></choice> schonste ist <placeName type="island">cursita</placeName>. auff tutsch <placeName type="island">cufr<choice><am>ū</am><ex>um</ex></choice></placeName> - <lb/>Dy <choice><orig>namhaffigist<choice><am>n̄</am><ex>en</ex></choice></orig><reg>namhafftigist<choice><am>n̄</am><ex>en</ex></choice></reg></choice> flyß in <placeName type="country">ytalia</placeName> das sind dy zwen. - <lb/><placeName type="river">Padus</placeName> vnd <placeName type="river">Thiberis</placeName>. <placeName type="river">padus</placeName> entspringt auß den - <lb/>bergen <placeName type="mountain">alpen</placeName> vnd velt yn das <placeName type="ocean">venedisch mer</placeName> - <choice><orig>ge<lb/>n<choice><am>ā</am><ex>an</ex></choice>t</orig><reg>gen<choice><am>ā</am><ex>an</ex></choice>t<lb/></reg></choice> zu late<choice><am>ī</am><ex>in</ex></choice> <placeName type="ocean">sinus adriaticus</placeName>. <placeName type="river">Tiberis</placeName> aber velt yn - <lb/>das <placeName type="ocean">Tyrrenische mer</placeName> geg<choice><am>ē</am><ex>en</ex></choice> mittag geleg<choice><am>ē</am><ex>en</ex></choice> yn disem - <lb/>land <placeName type="country">ytalie</placeName> wechst gutter wein das doryn das pöst - <lb/>ist</p> - </div> - <div type="kapitel"> - <head> - <lb/><placeName type="island">Sicilia</placeName> <placeName type="island">Sardinia</placeName> ynsulln - <lb/> - </head> - <p> - <hi rend="initial">D</hi>Jse zwu ynsuln ligen in dem <placeName type="ocean">mitelmör</placeName> v<choice><am>n̄</am><ex>nd</ex></choice> - <lb/>sind vor zeyt<choice><am>ē</am><ex>en</ex></choice> gestand<choice><am>ē</am><ex>en</ex></choice> an dem <choice><orig>lad</orig><reg>land</reg></choice> <placeName type="country">ytalie</placeName>. - <lb/>sunder durch vngestyme deß mörs sint si - <choice><orig>ab<lb/>gesundert</orig><reg>abgesundert<lb/></reg></choice> von <placeName type="country">welsch<choice><am>ē</am><ex>en</ex></choice> land<choice><am>ē</am><ex>en</ex></choice></placeName> vnt sind geleg<choice><am>ē</am><ex>en</ex></choice> v<choice><am>ō</am><ex>on</ex></choice> - <lb/>d<choice><am>ē</am><ex>em</ex></choice> obend yczliche yn sunderhait wan <placeName type="island">sardinia</placeName> lign - <lb/>vom abend ym 6 meridian das ist yn dem 30 grad - <lb/>vnd <placeName type="island">sicilia</placeName> ligt yn dem 7 meridi<choice><am>ā</am><ex>an</ex></choice> das ist vom - <choice><orig>ob<lb/>end</orig><reg>obend<lb/></reg></choice> 35 grad v<choice><am>n̄</am><ex>nd</ex></choice> ligen zwischen dem 10 vnd 11 - <choice><orig>pa-<lb/>ralell</orig><reg>paralell<lb/></reg></choice> vnd dem 4 clima vnd hab<choice><am>ē</am><ex>en</ex></choice> tag 14 stund 1/2 - <lb/>1/4 In der ynsul <placeName type="island">sicilia</placeName> do ligt der berg <placeName type="mountain">Ethna</placeName> der do - <lb/>ewig print. Das ist v<choice><am>ō</am><ex>on</ex></choice> deßweg<choice><am>ē</am><ex>en</ex></choice>. Dyßer pergk ist - <lb/>gancz löcherig vnd mit sweuelicht<choice><am>ē</am><ex>em</ex></choice> ercz - <choice><orig>durche<lb/>dert</orig><reg>durchedert<lb/></reg></choice> Und so dy wind dysen berg durchblos<choice><am>ē</am><ex>en</ex></choice> - <choice><orig>er-<lb/>weck<choice><am>n̄</am><ex>en</ex></choice></orig><reg>erweck<choice><am>n̄</am><ex>en</ex></choice><lb/></reg></choice> si es vnd wirft v<choice><am>ō</am><ex>on</ex></choice> im ins mer feirig kol<choice><am>ē</am><ex>en</ex></choice> des - <lb/>swebels v<choice><am>ō</am><ex>on</ex></choice> deßweg<choice><am>ē</am><ex>en</ex></choice> ist stet vngestymigkeyt vff - <lb/>d<choice><am>ē</am><ex>em</ex></choice> mer. dise ynsul stet g<choice><am>ā</am><ex>an</ex></choice>cz auff <q corresp="#Topos">ein<choice><am>ē</am><ex>em</ex></choice> swarcz<choice><am>ē</am><ex>en</ex></choice> - <choice><orig>loch<lb/>erigen</orig><reg>locherigen<lb/></reg></choice> stein</q> <term ref="#Topos">Topos</term> gen<choice><am>ā</am><ex>an</ex></choice>t den man hie yn dis<choice><am>ē</am><ex>en</ex></choice> - <choice><orig>l<choice><am>ā</am><ex>an</ex></choice>d<lb/>en</orig><reg>l<choice><am>ā</am><ex>an</ex></choice>den<lb/></reg></choice> landen braucht. yn d<choice><am>ē</am><ex>em</ex></choice> bad<choice><am>ē</am><ex>en</ex></choice> sich domit crawen oder - - <pb n="42" facs="Ptolemaeus_text_42"/> - <lb/>yucken das do gemainer ist bey den froven dan - <lb/>bei den menern ouch vint m<choice><am>ā</am><ex>an</ex></choice> am vber diß mörs - <lb/><placeName type="ocean">tyrrem</placeName> das do <placeName type="island">sicili<choice><am>ā</am><ex>am</ex></choice></placeName> besleüst v<choice><am>ō</am><ex>on</ex></choice> mitternacht den - <lb/>stein b<choice><am>ō</am><ex>om</ex></choice>bes do mit m<choice><am>ā</am><ex>an</ex></choice> pergamen slicht vndt ist - <lb/>scha<choice><am>ū</am><ex>um</ex></choice> des mörs yn derbheit gew<choice><am>ā</am><ex>an</ex></choice>delt v<choice><am>ō</am><ex>on</ex></choice> der - <choice><orig>sun<lb/>nen.</orig><reg>sunnen.<lb/></reg></choice> yn diser ynsul fint m<choice><am>ā</am><ex>an</ex></choice>. alls m<choice><am>ā</am><ex>an</ex></choice> spricht das loch - <lb/><placeName type="island">patriey</placeName> dise ynsul helt der <persName><roleName>kung v<choice><am>ō</am><ex>on</ex></choice> arrogonia</roleName></persName> - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Jtem sarmac<choice><am>ā</am><ex>am</ex></choice> <placeName type="continent">europe</placeName> <placeName type="country">reyssen</placeName> lit<choice><am>n̄</am><ex>en</ex></choice> 8 - <lb/> - </head> - <p> - <hi rend="initial">J</hi>N sarmac<choice><am>ā</am><ex>am</ex></choice> vint m<choice><am>ā</am><ex>an</ex></choice> nitt vill s<choice><am>ū</am><ex>un</ex></choice>derlichs v<choice><am>ō</am><ex>on</ex></choice> - <lb/>schönen steten aber v<choice><am>ō</am><ex>on</ex></choice> sinreichem volk als - <lb/>yn andern landn s<choice><am>ū</am><ex>un</ex></choice>der vil wüst<choice><am>ū</am><ex>un</ex></choice>g grosse - <lb/>haiden. breit l<choice><am>ā</am><ex>an</ex></choice>d. wenigk volk vnd halp Cristen - <lb/>vnd wirt gehalten v<choice><am>ō</am><ex>on</ex></choice> dem <persName><roleName>polnischen kunig</roleName></persName> - <choice><orig>lig<lb/>end</orig><reg>ligend<lb/></reg></choice> v<choice><am>ō</am><ex>on</ex></choice> obend mit se<choice><am>ī</am><ex>in</ex></choice>em anheben der grenicz 44 - <lb/>grad das ist yn dem 9 meridi<choice><am>ā</am><ex>an</ex></choice> vndt streckt sich yn - <lb/>orient bis auff 63 grad das ist bis yn den 14 - <choice><orig>meri<lb/>di<choice><am>ā</am><ex>an</ex></choice></orig><reg>meridi<choice><am>ā</am><ex>an</ex></choice><lb/></reg></choice> dy breit ist deß lands 4 meridi<choice><am>ā</am><ex>an</ex></choice> das ist 20 grad - <lb/>Ein grad nach der leng y<choice><am>n̄</am><ex>nn</ex></choice> disem paralell machtt - <lb/>40 meiln welsch. vnd wirt beschlossen zwischen - <lb/>dem 15 vnd 20 paralell gegen mitternacht ligen - <lb/>yn disem land sein <rs type="inhabitants">reussen</rs>. <rs type="inhabitants">litten</rs>. <rs type="inhabitants">massurn</rs> - <rs type="inhabitants"><choice><orig>mosqui<lb/>rer</orig><reg>mosquiter<lb/></reg></choice></rs> an der grenicz. ouch <rs type="inhabitants">poln</rs>. <rs type="inhabitants">thatern</rs> yn welcher - <lb/>thatarey dy stat <placeName type="city">kaffa</placeName> gelegen ist ouch hat das - <lb/>l<choice><am>ā</am><ex>an</ex></choice>t schöner fliß zwen alls den <placeName type="river">nepper </placeName>vnd den - <placeName type="river"><choice><orig>nest<lb/>er</orig><reg>nester<lb/></reg></choice></placeName> beide fliessen si Ad <placeName type="ocean">pont<choice><am>ū</am><ex>um</ex></choice> eurun<choice><am>ū</am><ex>um</ex></choice></placeName>. das ist yn - <lb/>das <placeName type="ocean">schwarcz mör</placeName>. durch diß l<choice><am>ā</am><ex>an</ex></choice>d get ein arm des - <lb/>gepurgs v<choice><am>ō</am><ex>on</ex></choice> den <placeName type="mountain">alpen</placeName> auß dem <placeName type="mountain">schweiczer - <choice><orig>ge-<lb/>purg.</orig><reg>gepurg.<lb/></reg></choice></placeName> yn disem l<choice><am>ā</am><ex>an</ex></choice>d ist der lengst tag ym sumer 16 - <pb n="43" facs="Ptolemaeus_text_43"/> - <lb/>stund 1/2 Ligent in dem 7 clima. - </p> - </div> - <div type="kapitel"> - <head> - <lb/><placeName type="country">Dacia</placeName> <placeName type="country">walachei</placeName>. 9 - </head> - <p> - <lb/> - <hi rend="initial">D</hi>as land wirt g<choice><am>n̄</am><ex>enan</ex></choice>t di <placeName type="country">groß walachei</placeName>. Ouch - <lb/>wenig namhafter stet haben. - <choice><orig>außgeno-<lb/>men</orig><reg>außgenomen<lb/></reg></choice> di anstossende land di in dissem teill - <lb/><placeName type="continent">europe</placeName> begriffen sein als do ist <placeName type="country">sibenpurge</placeName> Ein - <lb/>fruchtpar land von we<choice><am>ī</am><ex>in</ex></choice> vnd getreid <rs type="inhabitants">teutsch volk</rs> - <lb/>habend vnd wirt genant v<choice><am>ō</am><ex>on</ex></choice> <persName><roleName>ptolomeo</roleName></persName>. Jaziges - <lb/>metanest<choice><am>ū</am><ex>um</ex></choice> vff kriechisch. auch heltt diss teil die - <lb/>gancz <placeName type="country">disputei</placeName> oben vnt niden di <placeName type="country">sirfen</placeName> vnd - <placeName type="country"> <choice><orig>dar-<lb/>dinen</orig><reg>dardinen<lb/></reg></choice></placeName> vnd <placeName type="country">boßnerlant</placeName> das ist <placeName type="country">misia</placeName>. diß land - <choice><orig>sto-<lb/>sset</orig><reg>stosset<lb/></reg></choice> von orient an das <placeName type="ocean">schwartze meher</placeName> v<choice><am>ō</am><ex>on</ex></choice> obend - <lb/>an <placeName type="country">illiris</placeName> das ist <placeName type="country">winden landt</placeName> von mitternacht an - <placeName type="country"><lb/>sarmacia</placeName>. <placeName type="continent">ewrope</placeName> vom mittage an <placeName type="country">macedo<choice><am>ī</am><ex>ni</ex></choice>a</placeName> das - <lb/>ist <placeName type="country">criecher lant</placeName> durch disse g<choice><am>ā</am><ex>an</ex></choice>tze <placeName type="country">wallachei</placeName> - <choice><orig>flew-<lb/>sset</orig><reg>flewsset<lb/></reg></choice> di <placeName type="river">thunaw</placeName> vnd pei <persName><forename>kilian</forename></persName> vorlewsset si den - <choice><orig>na<lb/>men</orig><reg>namen<lb/></reg></choice> vnd heisset <placeName type="river">ister</placeName> vnd fellt in das mehr an sechs - <lb/>enden. ander wasser mer hat disse gegenheit di - <lb/>man klerlicher sehen mag in <persName><surname>ptolomeo</surname></persName> de situ - <choice><orig>or<lb/>bis.</orig><reg>orbis.<lb/></reg></choice> - </p> - </div> - <div type="kapitel"> - <head> - <lb /><placeName type="country">Macedonia</placeName> <placeName type="country">criechenlant</placeName> 10 - <lb/> - </head> - <p> - <placeName type="country"><hi rend="initial">M</hi>acedonia</placeName> wirt gesprochen von dem - <persName><roleName><choice><orig>ku<lb/>nige</orig><reg>kunige<lb/></reg></choice></roleName><forename>macedon</forename></persName> vnd ist gewest das - <choice><orig>vatter<lb/>land</orig><reg>vatterland<lb/></reg></choice> <persName><forename>allexandri</forename> <roleName>magni</roleName></persName> der di grosse - <choice><orig>we-<lb/>rld</orig><reg>werld<lb/></reg></choice> bestritten hat dis landt leit vom - <choice><orig>ab-<lb/>endt</orig><reg>abendt<lb/></reg></choice> 45 grad das ist in dem 9 meridian vnd geht - <lb/>bis in den 11 vnd wirt beschlossen mit den - <placeName type="ocean"><choice><orig>adri-<lb/>atischen</orig><reg>adriatischen<lb/></reg></choice> mehr</placeName> vnd mit dem <placeName type="ocean">mehre ionio</placeName> gegen - <pb n="44" facs="Ptolemaeus_text_44"/> - <lb/>geg<choice><am>ē</am><ex>en</ex></choice> occid<choice><am>ē</am><ex>en</ex></choice>t mit dem <placeName type="ocean">mör Egeo</placeName> vnd <placeName type="island">cret<choice><am>ē</am><ex>en</ex></choice>si<choice><am>ū</am><ex>um</ex></choice></placeName> v<choice><am>ō</am><ex>on</ex></choice> - <lb/>ori<choice><am>ē</am><ex>en</ex></choice>t v<choice><am>ō</am><ex>on</ex></choice> mittage mit d<choice><am>ē</am><ex>em</ex></choice> <placeName type="ocean">mitel mör</placeName> v<choice><am>ō</am><ex>on</ex></choice> mitternacht - <lb/>mit d<choice><am>ē</am><ex>em</ex></choice> <placeName type="country">sirfin</placeName> vnd <placeName type="country">Boßnerl<choice><am>ā</am><ex>an</ex></choice>d</placeName> dys l<choice><am>ā</am><ex>an</ex></choice>d <placeName type="country">Grecia</placeName> yst - <lb/>gewest ein Muter aller k<choice><am>ū</am><ex>un</ex></choice>st wa<choice><am>n̄</am><ex>nn</ex></choice> auß yr ist - <choice><orig>kom<lb/>en</orig><reg>komen<lb/></reg></choice> dy Lateynisch sprach vnd ander vil bucher dye - <lb/>dy <rs type="inhabitants">krieche</rs> gemacht hab<choice><am>ē</am><ex>en</ex></choice> vnd dornoch sein sy - <lb/>procht yn late<choice><am>ī</am><ex>in</ex></choice> von dan wir diese k<choice><am>ū</am><ex>un</ex></choice>st vrspr<choice><am>ī</am><ex>in</ex></choice>glich - <lb/>hab<choice><am>ē</am><ex>en</ex></choice> alls da<choice><am>n̄</am><ex>nn</ex></choice> diser <persName><roleName>münch</roleName></persName> der do yrfar<choice><am>ē</am><ex>en</ex></choice> gewest - <lb/>ist yn kriechischer sprach der dan diß werk in - <choice><orig>la-<lb/>te<choice><am>ī</am><ex>in</ex></choice></orig><reg>late<choice><am>ī</am><ex>in</ex></choice><lb/></reg></choice> pracht hat das do s<choice><am>ū</am><ex>un</ex></choice>st villeicht in ewigkait - <choice><orig>vor<lb/>halt<choice><am>ē</am><ex>en</ex></choice></orig><reg>vorhalt<choice><am>ē</am><ex>en</ex></choice><lb/></reg></choice> wer word<choice><am>ē</am><ex>en</ex></choice> bei d<choice><am>ē</am><ex>en</ex></choice> kriech<choice><am>ē</am><ex>en</ex></choice> vnd dise - <choice><orig>gegen-<lb/>heit</orig><reg>gegenheit<lb/></reg></choice> ist geteilt in 4 teil. als in <placeName type="country">maced<choice><am>ō</am><ex>on</ex></choice>iam</placeName> <placeName type="country">Eupir<choice><am>ū</am><ex>um</ex></choice></placeName> - <lb/><placeName type="country">achai<choice><am>ā</am><ex>am</ex></choice></placeName> vnt <placeName type="country">poleponesi<choice><am>ū</am><ex>um</ex></choice></placeName> in dis<choice><am>ē</am><ex>er</ex></choice> geg<choice><am>ē</am><ex>en</ex></choice>heit vint m<choice><am>ā</am><ex>an</ex></choice> - <lb/>vil schöner stet. als <placeName type="city">athene</placeName>. <placeName type="city">Thebe</placeName>. <placeName type="city">Metropolis</placeName>. - <lb/><placeName type="city">Tessalonita</placeName>. <placeName type="city">causandria</placeName> vnd ander mer die do von - <lb/>kurczwegen hi verswig<choice><am>ē</am><ex>en</ex></choice> bleib<choice><am>ē</am><ex>en</ex></choice>. Bei disem - <placeName type="country"><choice><orig>Mace<lb/>donia</orig><reg>Macedonia<lb/></reg></choice></placeName> ligt in dem mör di Insul <placeName type="island">creta</placeName> oder <placeName type="island">c<choice><am>ā</am><ex>an</ex></choice>dia</placeName>. - <lb/>vnder d<choice><am>ē</am><ex>em</ex></choice> 11 meridi<choice><am>ā</am><ex>an</ex></choice> vnd zwisch<choice><am>ē</am><ex>en</ex></choice> d<choice><am>ē</am><ex>em</ex></choice> 10 vnd 11 - <choice><orig>para<lb/>lell</orig><reg>paralell<lb/></reg></choice> di Insul <placeName type="island">Lempnos</placeName> ligt vnder d<choice><am>ē</am><ex>em</ex></choice> 12 paralall v<choice><am>n̄</am><ex>nd</ex></choice> - <lb/>10 meridi<choice><am>ā</am><ex>an</ex></choice> mit e<choice><am>ī</am><ex>in</ex></choice> 1/4 <!-- 1/3?? -->. vil ander Insul sind in <placeName type="country">grec<choice><am>ā</am><ex>am</ex></choice></placeName> di - <lb/>do v<choice><am>ō</am><ex>on</ex></choice> <choice><orig>kuczweg<choice><am>ē</am><ex>en</ex></choice></orig><reg>kurczweg<choice><am>ē</am><ex>en</ex></choice></reg></choice> hie nicht gemelt werd<choice><am>ē</am><ex>en</ex></choice>. domit - <lb/>sei gnug v<choice><am>ō</am><ex>on</ex></choice> diser <placeName type="continent">europa</placeName> gesagt. wer aber weiter - <lb/>wiss<choice><am>ē</am><ex>en</ex></choice> wolt von m<choice><am>ā</am><ex>an</ex></choice>cherlei stet. perg<choice><am>ē</am><ex>en</ex></choice> wassern v<choice><am>n̄</am><ex>nd</ex></choice> - <lb/>insuln der mag anseh<choice><am>ē</am><ex>en</ex></choice> <persName><surname>ptolome<choice><am>ū</am><ex>um</ex></choice></surname></persName> mit sein<choice><am>ē</am><ex>en</ex></choice> - <choice><orig>figu-<lb/>r<choice><am>ē</am><ex>en</ex></choice></orig><reg>figur<choice><am>ē</am><ex>en</ex></choice><lb/></reg></choice> dori<choice><am>n̄</am><ex>nn</ex></choice> er vindet was er such<choice><am>ē</am><ex>en</ex></choice>t ist nach - <choice><orig>vßweiß-<lb/>ung</orig><reg>vßweißung<lb/></reg></choice> der graden - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Zumerck<choice><am>ē</am><ex>en</ex></choice> was buchstab<choice><am>ē</am><ex>en</ex></choice> oder littera - <lb/><placeName type="continent">europa</placeName> <choice><am>ī</am><ex>in</ex></choice> ieb<choice><am>ū</am><ex>un</ex></choice>g der sproch hatt - </head> - <p> - <pb n="45" facs="Ptolemaeus_text_45"/> - <lb/> - <hi rend="initial">D</hi>ise gancze <placeName type="continent">europa</placeName> braucht geme<choice><am>ī</am><ex>in</ex></choice>lich - <lb/>zweierlei <term xml:id="littera1" ref="#Buchstaben">littera</term> oder <q corresp="#littera1">buchstab<choice><am>ē</am><ex>en</ex></choice></q> alls - <choice><orig>La<lb/>tein</orig><reg>Latein<lb/></reg></choice> vnd kriechisch ouch wenig - <choice><orig>rewß<lb/>isch.</orig><reg>rewßisch.<lb/></reg></choice> Das do kompt vrspringklich auß - <lb/>kriechischer littera. Aber lateinisch littera hat - <choice><orig>ein<lb/>en</orig><reg>einen<lb/></reg></choice> vorg<choice><am>ā</am><ex>an</ex></choice>g vor all<choice><am>ē</am><ex>en</ex></choice> andern wan woru<choice><am>m̄</am><ex>mb</ex></choice> m<choice><am>ā</am><ex>an</ex></choice> magk - <lb/>schreib<choice><am>ē</am><ex>en</ex></choice> mit diser lateinisch<choice><am>ē</am><ex>en</ex></choice> litter allerlei sproch - <lb/>di m<choice><am>ā</am><ex>an</ex></choice> deütlich k<choice><am>ā</am><ex>an</ex></choice> außred<choice><am>ē</am><ex>en</ex></choice>. ydoch forderlichen - <lb/>brauchenn sei <rs type="inhabitants">Francos<choice><am>ē</am><ex>en</ex></choice></rs><rs type="inhabitants"> walb<choice><am>ē</am><ex>en</ex></choice></rs> <rs type="inhabitants">Teutsch</rs> <rs type="inhabitants">Behem</rs>. - <lb/>vnd <rs type="inhabitants">Englisch</rs>. di kriechisch z<choice><am>ū</am><ex>un</ex></choice>g gät <choice><am>ī</am><ex>in</ex></choice> <placeName type="country">Grecia</placeName> - <placeName type="country"><choice><orig>Ma<lb/>cedonia</orig><reg>Macedonia<lb/></reg></choice></placeName> <placeName type="country">Epiro</placeName> <placeName type="country">Achaia</placeName> vnd e<choice><am>ī</am><ex>in</ex></choice> teil in <placeName type="continent">Azia</placeName> minori - <lb/>vnd ouch in <placeName type="country">moßquit<choice><am>ē</am><ex>en</ex></choice></placeName> mit der Revßisch<choice><am>ē</am><ex>en</ex></choice> littera. - <lb/>Yn sproch so endern sich alle l<choice><am>ā</am><ex>an</ex></choice>d vnd naci<choice><am>ō</am><ex>on</ex></choice> auss - <lb/><placeName type="country">irland</placeName> vnd <placeName type="country">engelland</placeName> redenn teutsch. ydoch - <choice><orig>an<lb/>ders</orig><reg>anders<lb/></reg></choice> d<choice><am>ā</am><ex>an</ex></choice> <choice><am>ī</am><ex>in</ex></choice> <placeName type="country">hispania</placeName> vnt in <placeName type="country">gallia</placeName>. <choice><am>ī</am><ex>in</ex></choice> <placeName type="country">norweg<choice><am>ē</am><ex>en</ex></choice></placeName> <placeName type="country">swed<choice><am>ē</am><ex>en</ex></choice></placeName> - <lb/><rs type="inhabitant">Brab<choice><am>ā</am><ex>an</ex></choice></rs> der <rs type="inhabitant">Holl<choice><am>ā</am><ex>an</ex></choice>der</rs> <rs type="inhabitant">Merker</rs> <rs type="inhabitant">Paier</rs>. <rs type="inhabitant">Sachßen</rs> anders - <lb/>dan di <rs type="inhabitant">Merhern</rs> <rs type="inhabitant">slesier</rs> <rs type="inhabitant">hungernn</rs> <rs type="inhabitant">sibenburgen</rs> - <lb/>redenn annders dann die <rs type="inhabitant">Ostereicher</rs> - <rs type="inhabitant"><choice><orig>steier-<lb/>marcker</orig><reg>steiermarcker<lb/></reg></choice></rs> <rs type="inhabitant">schwaben</rs> vnd <choice><am>ī</am><ex>in</ex></choice> ander teutschen naci<choice><am>ō</am><ex>on</ex></choice> - <lb/>vnd sein doch alle v<choice><am>ō</am><ex>on</ex></choice> teutscher sprach ouch - <choice><orig>end<lb/>ern</orig><reg>endern<lb/></reg></choice> sich di <rs type="inhabitant">poln</rs> v<choice><am>ō</am><ex>on</ex></choice> den <rs type="inhabitant">winden Beh<choice><am>ā</am><ex>an</ex></choice>en</rs> <rs type="inhabitant">recht<choice><am>ē</am><ex>en</ex></choice></rs> - <lb/><rs type="inhabitant">breissen</rs> v<choice><am>ō</am><ex>on</ex></choice> d<choice><am>ē</am><ex>en</ex></choice> <rs type="inhabitant">littaven</rs> v<choice><am>ō</am><ex>on</ex></choice> den <rs type="inhabitant">revssen</rs> alle<choice><am>ī</am><ex>in</ex></choice> - <placeName type="country"><choice><orig>h<choice><am>ū</am><ex>un</ex></choice>ger<lb/>l<choice><am>ā</am><ex>an</ex></choice>d</orig><reg>h<choice><am>ū</am><ex>un</ex></choice>gerl<choice><am>ā</am><ex>an</ex></choice>d<lb/></reg></choice></placeName> helt vnvermist sein sproch. Auch ist czuwissen - <lb/>das di lateinisch sproch hat gest<choice><am>ā</am><ex>an</ex></choice>den vber 600 - <lb/>Jor. - </p> - </div> - <div type="kapitel"> - <head> - <lb/>Uon den teilenn <placeName type="continent">Affrice</placeName> - <lb/>oder <placeName type="country">Libie</placeName> 1 - </head> - <p> - <pb n="46" facs="Ptolemaeus_text_46"/> - <lb/> - <hi rend="initial">D</hi>Js dritte teyl der werlt wirt geteilt in 4 - <lb/>teil alls yn <placeName type="country">mauritani<choice><am>ā</am><ex>am</ex></choice></placeName> yn <placeName type="continent">affric<choice><am>ā</am><ex>am</ex></choice> minorem</placeName> - <lb/>yn <placeName type="country">egipten</placeName> vnd yn <placeName type="country">ethyopi<choice><am>ā</am><ex>am</ex></choice></placeName>. das ist yn - <lb/><placeName type="country">morenland</placeName> vnder <placeName type="country">egipten</placeName>. Dyse <placeName type="continent">Affrica</placeName> - <lb/>wirt gen<choice><am>ā</am><ex>an</ex></choice>t von <persName><forename>affer</forename></persName> <persName><forename>Abrahams</forename></persName> sun gewest. der - <lb/>am ersten dy teyl der werlt hat gebavt. ouch wirt - <lb/>es gen<choice><am>ā</am><ex>an</ex></choice>t <placeName type="country">Libia</placeName> von dem winde libius oder - <placeName type="continent"><choice><orig>affri<lb/>cus</orig><reg>affricus<lb/></reg></choice></placeName> auß dem winkel des ob<choice><am>ē</am><ex>en</ex></choice>ds vnd mittags - <choice><orig>blo<lb/>send.</orig><reg>blosend.<lb/></reg></choice> vnd bewegt stetes yn vngestymikeytt daß - <lb/>mör dis lands. v<choice><am>ō</am><ex>on</ex></choice> deßweg<choice><am>ē</am><ex>en</ex></choice> <persName><forename>hercules</forename></persName> lies - <choice><orig>durch<lb/>graben</orig><reg>durchgraben<lb/></reg></choice> das erttrich des bergs <placeName type="mountain">Calpe</placeName> auff das. das - <lb/>das <placeName type="ocean">mittelmör</placeName> möcht hab<choice><am>ē</am><ex>en</ex></choice> ein vßg<choice><am>ā</am><ex>an</ex></choice>g yn das - <placeName type="ocean"><choice><orig>au-<lb/>sser</orig><reg>ausser<lb/></reg></choice> mör</placeName> doselbst seczt er hin sevle vermeynett eyn - <lb/>enden do sein der werlt wa<choice><am>n̄</am><ex>nn</ex></choice> <persName><forename>hercules</forename></persName> waß auff - <choice><orig>dy-<lb/>se</orig><reg>dyse<lb/></reg></choice> zeit eyn herr vber dyse zweyteyl <placeName type="continent">affric<choice><am>ā</am><ex>am</ex></choice></placeName> vnd - <placeName type="continent"><choice><orig>eu<lb/>rop<choice><am>ā</am><ex>am</ex></choice></orig><reg>europ<choice><am>ā</am><ex>am</ex></choice><lb/></reg></choice></placeName> do besorget er sich das dy vngestymigkaitt - <lb/>des mörs ertrinck<choice><am>ē</am><ex>en</ex></choice> mucht vil seines l<choice><am>ā</am><ex>an</ex></choice>ds macht - <lb/>er disem <placeName type="ocean">mittelmör</placeName> ein<choice><am>ē</am><ex>en</ex></choice> ausg<choice><am>ā</am><ex>an</ex></choice>g vnd n<choice><am>ā</am><ex>an</ex></choice>te es - <placeName type="ocean"><choice><orig>Fre<lb/>tumberculi<choice><am>ū</am><ex>um</ex></choice></orig><reg>Fretumberculi<choice><am>ū</am><ex>um</ex></choice><lb/></reg></choice></placeName> . das ist das wütennd mör <persName><forename>herculis</forename></persName>. - <lb/>Das erst teil <placeName type="continent">affrice</placeName> wirt gen<choice><am>ā</am><ex>an</ex></choice>t <placeName type="country">Mauritanea - <choice><orig>cesa-<lb/>riensis</orig><reg>cesariensis<lb/></reg></choice></placeName> auff teutsch <placeName type="country">Marisken land</placeName> Das ander - <lb/>teyl heyst <placeName type="country">mauritania tinganica</placeName> alias - <placeName type="country"><choice><orig>maurenssen<lb/>land</orig><reg>maurenssenland<lb/></reg></choice></placeName> welch leüt doselbst sere schwarcz sein vnd - <lb/>selczamer siten nackend gend alleyn v<choice><am>m̄</am><ex>mb</ex></choice> yre sch<choice><am>ā</am><ex>am</ex></choice> - <lb/>bedecket das ist v<choice><am>ō</am><ex>on</ex></choice> weg<choice><am>ē</am><ex>en</ex></choice> grosser hicze wan - <choice><orig><choice><sic>wor<lb/>nmb</sic><corr>wor<lb/>umb</corr></choice></orig><reg><choice><sic>wornmd<lb/></sic><corr>worumb<lb/></corr></choice></reg></choice> si ligen nohe dem weg der so<choice><am>n̄</am><ex>nn</ex></choice>en von - <choice><orig>deßs<lb/>wegen</orig><reg>deßswegen<lb/></reg></choice> alle doryn won<choice><am>ē</am><ex>en</ex></choice> hab<choice><am>ē</am><ex>en</ex></choice> getruckt <choice><orig>kraußhr</orig><reg>kraußhar</reg></choice> - <lb/>sam is vorprent wer gleich den moren. Yedoch - <lb/>sind si on varbe nicht so swarcz vnd diese barbari - <pb n="47" facs="Ptolemaeus_text_47"/> - <lb/>hab<choice><am>ē</am><ex>en</ex></choice> keyn eeliche ordn<choice><am>ū</am><ex>un</ex></choice>g. sunder gleich wy es dy - <lb/><rs type="inhabitants">Türken</rs> halt<choice><am>ē</am><ex>en</ex></choice> si erkenn<choice><am>ē</am><ex>en</ex></choice> yn elich<choice><am>n̄</am><ex>en</ex></choice> werken - <choice><orig>Swe-<lb/>ster</orig><reg>Swester<lb/></reg></choice> müter töchter vnd andre yre mog<choice><am>n̄</am><ex>en</ex></choice> vnd - <choice><orig>hab<lb/>en</orig><reg>haben<lb/></reg></choice> diß vor einen siten so neher gesip vnt - <choice><orig>freündt-<lb/>schafft</orig><reg>freündtschafft<lb/></reg></choice> Ee bessir dy <choice><orig>gattug</orig><reg>gattung</reg></choice> ist bey yn der man vndt - <lb/>der frawen Diß erst teyl <placeName type="continent">affrice</placeName> ligt v<choice><am>ō</am><ex>on</ex></choice> obendt - <lb/>mit seinem vber 7 grad. das ist. es hebt an yn d<choice><am>ē</am><ex>em</ex></choice> - <lb/>ersten meridi<choice><am>ā</am><ex>an</ex></choice> vnd gat biß yn den 5 vnd ligt - <choice><orig>zwi<lb/>sch<choice><am>ē</am><ex>en</ex></choice></orig><reg>zwisch<choice><am>ē</am><ex>en</ex></choice><lb/></reg></choice> dem 7 vnd 9 paralell. vnt hat lengst<choice><am>ē</am><ex>en</ex></choice> tag 14 - <lb/>stund das gebürg yn disem land wirt genantt - <choice><orig>d<lb/>berg</orig><reg>der<lb/></reg></choice> berg <placeName type="mountain">atlas</placeName> welcher perg zwen sein als der größt - <lb/>vnd klein. der größ get durch dise gancze <placeName type="continent">affrica</placeName> - <lb/>oder <placeName type="country">lybia</placeName> yn der gemein bys yn <placeName type="country">egipt<choice><am>ē</am><ex>en</ex></choice></placeName>. Ouch hat - <lb/>es etliche flyß der namen yn disem vnserm land - <lb/>wenig <choice><orig>bekat</orig><reg>bekant</reg></choice> seyn alls <placeName type="river">mulus</placeName>. <placeName type="river">sizaris</placeName>. <placeName type="river">andus</placeName>. <placeName type="river">Cilo</placeName> - <lb/>nach. <placeName type="river">sala</placeName>. vnd ander wasser mer dy man yn den - <choice><orig>Fi<lb/>guren</orig><reg>Figuren<lb/></reg></choice> <persName><surname>ptholomei</surname></persName> vindet N<choice><am>ū</am><ex>un</ex></choice> furt yn dys<choice><am>ē</am><ex>en</ex></choice> - <choice><orig>nach<lb/>gend<choice><am>ē</am><ex>en</ex></choice></orig><reg>nachgend<choice><am>ē</am><ex>en</ex></choice><lb/></reg></choice> land<choice><am>ē</am><ex>en</ex></choice> werd<choice><am>ē</am><ex>en</ex></choice> vil schöner flyß kom<choice><am>ē</am><ex>en</ex></choice> di vns - <lb/>ouch vnbekant seyn ydoch dy vorderlichst<choice><am>ē</am><ex>en</ex></choice> vnt - <lb/>gemein<choice><am>ē</am><ex>en</ex></choice> dy wir kenn<choice><am>ē</am><ex>en</ex></choice> von hören sag<choice><am>ē</am><ex>en</ex></choice> di werd<choice><am>ē</am><ex>en</ex></choice> - <lb/>ouch hy yn benumpt werd<choice><am>ē</am><ex>en</ex></choice> - </p> - </div> - <div type="kapitel"> - <head> - <lb/><placeName type="continent">Affrica minor</placeName>. Ist <placeName type="continent">klein affrica</placeName> 2 - <lb/> - </head> - <p> - <hi rend="initial">D</hi>Js teil ist geleg<choice><am>ē</am><ex>en</ex></choice> v<choice><am>ō</am><ex>on</ex></choice> d<choice><am>ē</am><ex>em</ex></choice> ob<choice><am>ē</am><ex>en</ex></choice>d 27 <choice><am>ḡ</am><ex>gra</ex></choice>d. das ist - <choice><orig>üb<lb/>er</orig><reg>über<lb/></reg></choice> d<choice><am>ē</am><ex>em</ex></choice> 5 meridi<choice><am>ā</am><ex>an</ex></choice> 2 <choice><am>ḡ</am><ex>gra</ex></choice>d. v<choice><am>n̄</am><ex>nd</ex></choice> get bis <choice><am>ī</am><ex>in</ex></choice> d<choice><am>ē</am><ex>en</ex></choice> 10 - <choice><orig>meri<lb/>di<choice><am>ā</am><ex>an</ex></choice></orig><reg>meridi<choice><am>ā</am><ex>an</ex></choice><lb/></reg></choice> v<choice><am>n̄</am><ex>nd</ex></choice> 2 <choice><am>ḡ</am><ex>grad</ex></choice>. das ist 47 <choice><am>ḡ</am><ex>grad</ex></choice>. v<choice><am>n̄</am><ex>nd</ex></choice> leit vnder d<choice><am>ē</am><ex>em</ex></choice> 7 v<choice><am>n̄</am><ex>nd</ex></choice> 8 paralel - <lb/>v<choice><am>n̄</am><ex>nd</ex></choice> wirt besloss<choice><am>ē</am><ex>en</ex></choice> mit dem <placeName type="ocean">mitilmör</placeName> g<choice><am>n̄</am><ex>enan</ex></choice>t <placeName type="ocean">affric<choice><am>ū</am><ex>um</ex></choice></placeName>. v<choice><am>ō</am><ex>on</ex></choice> - <choice><orig>oc<lb/>cid<choice><am>ē</am><ex>en</ex></choice>t</orig><reg>occid<choice><am>ē</am><ex>en</ex></choice>t<lb/></reg></choice> stost <choice><am>ā</am><ex>an</ex></choice> <placeName type="country">marischk<choice><am>ē</am><ex>en</ex></choice>l<choice><am>ā</am><ex>an</ex></choice>t</placeName>. v<choice><am>ō</am><ex>on</ex></choice> ori<choice><am>ē</am><ex>en</ex></choice>t an <placeName type="country">egipt<choice><am>ē</am><ex>en</ex></choice></placeName> von - <lb/>mittag <choice><am>ā</am><ex>an</ex></choice> <placeName type="country">morl<choice><am>ā</am><ex>an</ex></choice>d</placeName>. <choice><am>ī</am><ex>in</ex></choice> dis<choice><am>ē</am><ex>em</ex></choice> l<choice><am>ā</am><ex>an</ex></choice>d <placeName type="continent">kle<choice><am>ī</am><ex>in</ex></choice> affrice</placeName> leit di stat - <lb/><placeName type="city">kartago</placeName> vnd ander namhafftig Stett vnd der pergk - <pb n="48" facs="Ptolemaeus_text_48"/> - <lb/><placeName type="mountain">atlas</placeName> gät nach der leng dise tauel durchschneiden. - <lb/>aber am vber dises lants <choice><am>ī</am><ex>in</ex></choice> dem mör do sind zwe - <lb/>sirtes das sein schlingender mör di sich vndertrehn - <lb/>vnd was darein kompt das ist verlorn. schiff holcz - <lb/>mensch tier oder was es sey das get alles zu bodem - <lb/>vnd der mer sind zwai das klein vnd das groß - <lb/>auff disem l<choice><am>ā</am><ex>an</ex></choice>t find ma sandelholcz zinomei holcz - <lb/>das do tregt muschaten. di negel sind dy stengl - <lb/>muschaten sein fruchte ouch vint man manch - <choice><orig>sel<lb/>czam</orig><reg>selczam<lb/></reg></choice> tier in diser gegenheit alls streus. affen. - <choice><orig>ele-<lb/>ph<choice><am>ā</am><ex>an</ex></choice>ten.</orig><reg>eleph<choice><am>ā</am><ex>an</ex></choice>ten.<lb/></reg></choice> tigrides Jn disem land an etliche ende - <lb/>ist es ser fruchtpar vom wein baumöl feige vnd - <lb/>m<choice><am>ā</am><ex>an</ex></choice>del v<choice><am>n̄</am><ex>nd</ex></choice> an etlichen ende ist es so voller <choice><orig>sd</orig><reg>s<choice><am>ā</am><ex>an</ex></choice>d</reg></choice> das - <lb/>zu zeiten v<choice><am>ō</am><ex>on</ex></choice> den wind<choice><am>n̄</am><ex>en</ex></choice> ein perg hoch zu hawff - <lb/>wirtt getriben alls dan ouch in <placeName type="country">egipten</placeName> ist - <choice><orig>gesche<lb/>en</orig><reg>gescheen<lb/></reg></choice> das der wint offt sand hat erhaben vnd gebei - <lb/>vnd mensche getilgt vber den berg <placeName type="mountain">atlas</placeName> gegen - <lb/>mittag do sind vil gifftiger thier v<choice><am>ō</am><ex>on</ex></choice> schl<choice><am>ā</am><ex>an</ex></choice>gen - <choice><orig>lint-<lb/>würm</orig><reg>lintwürm<lb/></reg></choice> tracken cocodrulln vnd ander <choice><sic>ner</sic><corr>mer</corr></choice> . ouch - <lb/>genhalb des bergs vindt m<choice><am>ā</am><ex>an</ex></choice> walt esel di do - <choice><orig>gros<lb/>er</orig><reg>groser<lb/></reg></choice> sind dan kameltier. dise <placeName type="continent">affrica minoris</placeName> hat - <choice><orig>len<lb/>gsten</orig><reg>lengsten<lb/></reg></choice> tag 14 stund vnd di inwoner sind ouch - <rs type="inhabitants"><choice><orig>bar<lb/>barischken</orig><reg>barbarischken<lb/></reg></choice></rs> .</p> - </div> - <div type="kapitel"> - <head> - <lb/><placeName type="country">Marmarica</placeName> <placeName type="country">libia</placeName> <placeName type="country">egipten</placeName>. 3 - <lb/> - </head> - <p> - <hi rend="initial">D</hi>Js lant wirt beschlossen v<choice><am>ō</am><ex>on</ex></choice> mitternacht - <lb/>mit dem <placeName type="ocean">wietend mor</placeName> vnd mit dem - <placeName type="ocean"><choice><orig>E<lb/>giptischen</orig><reg>Egiptischen<lb/></reg></choice></placeName> v<choice><am>ō</am><ex>on</ex></choice> mittag mit dem geburgk - <lb/><placeName type="continent">affrice</placeName>. das ist der <choice><orig>brg</orig><reg>berg</reg></choice> <placeName type="mountain">atlas</placeName> maior v<choice><am>ō</am><ex>on</ex></choice> - <pb n="49" facs="Ptolemaeus_text_49"/> - <lb/>occident mit der blei<choice><am>ē</am><ex>en</ex></choice> <placeName type="continent">affrica</placeName>. v<choice><am>ō</am><ex>on</ex></choice> orient mit dem - <lb/><placeName type="ocean">roten mör</placeName> aber mit dem arabisch<choice><am>ē</am><ex>en</ex></choice> hap. vnd fügt - <lb/>yn seinem anheb<choice><am>ē</am><ex>en</ex></choice> vom occident 47 grad. das ist - <lb/>yn dem 10 meridian on 1/2 vnd strecket sich bis in - <lb/>den 65 grad das ist v<choice><am>ō</am><ex>on</ex></choice> dem abend 13 Meridian - <lb/>vnd 1/2 . Dis land ist an etlichen <choice><orig>ndenn</orig><reg>endenn</reg></choice> fruchtpar - <lb/>v<choice><am>ō</am><ex>on</ex></choice> edlem holcz als Cipres sandl muscat<choice><am>ē</am><ex>en</ex></choice>. yn - <choice><orig>dy-<lb/>sem</orig><reg>dysem<lb/></reg></choice> land<placeName type="country"> Marmarica</placeName> vint man vil men<choice><am>ū</am><ex>un</ex></choice>g der - <choice><orig>le<lb/>ben</orig><reg>leben<lb/></reg></choice> vnd der vögel Sittich. der thyre Seraphen - <lb/>gen<choice><am>ā</am><ex>an</ex></choice>t. Zygen mit grossen preiten oren vnd - <choice><orig>plem<lb/>eln</orig><reg>plemeln<lb/></reg></choice> das sind thyr sich den m<choice><am>ē</am><ex>en</ex></choice>schen ser gleichend - <lb/>allein das si rauch an yrem leib sind. Yn disem l<choice><am>ā</am><ex>an</ex></choice>t - <lb/>regenet es gar selten aber nymer. Sunder das - <choice><orig>E<lb/>dle</orig><reg>Edle<lb/></reg></choice> fliß <placeName type="river">nilus</placeName> den dis land hat ergeusset vnd - <choice><orig>feu-<lb/>chtet</orig><reg>feuchtet<lb/></reg></choice> das ertrich <placeName type="country">egipti</placeName> vnd macht es fruchtpar - <lb/>nur soweit es das l<choice><am>ā</am><ex>an</ex></choice>d begeust sein yrgissunng - <choice><orig>ge<lb/>schith</orig><reg>geschith<lb/></reg></choice> nicht v<choice><am>ō</am><ex>on</ex></choice> regen oder schne als etlich - <choice><orig>sprech<lb/>en.</orig><reg>sprechen.<lb/></reg></choice> sunder v<choice><am>ō</am><ex>on</ex></choice> verbandl<choice><am>ū</am><ex>un</ex></choice>g der grossen menyg des - <lb/>sands den der wind erhept vnd pringt sterklich - <lb/>yn dy ynfliß <placeName type="river">nili</placeName> yn das mör vnd verhelt dy wasser - <lb/>yn samlung also das si vff dring<choice><am>ē</am><ex>en</ex></choice> v<choice><am>n̄</am><ex>nd</ex></choice> sich durch g<choice><am>ā</am><ex>an</ex></choice>cz - <lb/><placeName type="country">egipten</placeName> breiten. so bald aber dy starcken winde - <lb/>auffhören zu blosen. So gewint der <placeName type="river">nilus</placeName> durch - <lb/>stete mer<choice><am>ū</am><ex>un</ex></choice>g des wasserss ober den newen sand - <lb/>seyn<choice><am>ē</am><ex>en</ex></choice> gang vnd so er vberhand nympt vnd mitt - <lb/>storme get furet er alle disen sand yn das mör vntt - <lb/>get wider yn seinen strome recht. pey disem <placeName type="river">nilo</placeName> - <lb/>sind vil der Cocodruln auch dy statt <placeName type="city">Alexandria</placeName> - <pb n="50" facs="Ptolemaeus_text_50"/> - <lb/>Ist nicht fer do v<choice><am>ō</am><ex>on</ex></choice> ligent. Ein grosse niderlag der - <lb/>koufleut auß <placeName type="country">yndia</placeName> <placeName type="country">arabia</placeName> <placeName type="country">egipten</placeName> vnd - <placeName type="country"><choice><orig>welschl<choice><am>ā</am><ex>an</ex></choice>-<lb/>den</orig><reg>welschl<choice><am>ā</am><ex>an</ex></choice>den<lb/></reg></choice></placeName> . yn diser geg<choice><am>ē</am><ex>en</ex></choice>heit <placeName type="country">egipti</placeName> ist ouch dy stat - <placeName type="city"><choice><orig>Ba-<lb/>bilonia</orig><reg>Babilonia<lb/></reg></choice></placeName> alias <placeName type="city">alkair</placeName> do dann der <persName><roleName>Soldann</roleName></persName> - <choice><orig>wo-<lb/>net</orig><reg>wonet<lb/></reg></choice> vnd ein herr ist diser land. auß welch<choice><am>ē</am><ex>en</ex></choice> landen - <lb/>vns vil specerey komend. vnd der lengst tag yn - <lb/>diser <placeName type="country">marmarica</placeName> <placeName type="country">egipti</placeName> ist 14 stund. - <lb/> - </p> - </div> - <div type="kapitel"> - <head> - <lb/><placeName type="country">Ethiopia</placeName> <placeName type="country">morland</placeName> vnder <placeName type="country">egipten</placeName> geleg<choice><am>ē</am><ex>en</ex></choice> 4 - <lb/> - </head> - <p> - <lb/> - <hi rend="initial">D</hi>as 4 teil <placeName type="continent">affrice</placeName> wirt beslossen v<choice><am>ō</am><ex>on</ex></choice> - <choice><orig>mitt-<lb/>ernacht</orig><reg>mitternacht<lb/></reg></choice> mit mancherlei mör als mit d<choice><am>ē</am><ex>em</ex></choice> - <lb/><placeName type="ocean">egiptisch<choice><am>ē</am><ex>en</ex></choice></placeName> mit d<choice><am>ē</am><ex>em</ex></choice> <placeName type="ocean">wietenden</placeName> v<choice><am>n̄</am><ex>nd</ex></choice> mit d<choice><am>ē</am><ex>em</ex></choice> - <lb/><placeName type="ocean">adriatisch<choice><am>ē</am><ex>en</ex></choice> mör</placeName> von dem morgen wirt - <lb/>es begriffen mit dem <placeName type="ocean">rotten mör</placeName> vnd v<choice><am>ō</am><ex>on</ex></choice> dem - <choice><orig>o-<lb/>bend</orig><reg>obend<lb/></reg></choice> mit dem evssern. gegen mittag aber - <choice><orig>beslev<lb/>set</orig><reg>beslevset<lb/></reg></choice> <persName><surname>ptolomeus</surname></persName> dise bekante werlt v<choice><am>n̄</am><ex>nd</ex></choice> spricht wie - <lb/>das fort nicht bek<choice><am>ā</am><ex>an</ex></choice>t ertrich sey den menschen yn - <lb/>disem firtel der kaul<choice><am>ē</am><ex>en</ex></choice> ligend Dise <placeName type="country">ethiopia</placeName> vnder - <lb/><placeName type="country">egipten</placeName> hept sich an in dem 5 <choice><am>ḡ</am><ex>grad</ex></choice>. v<choice><am>n̄</am><ex>nd</ex></choice> in dem 1 - <choice><orig>meri<lb/>dian</orig><reg>meridian<lb/></reg></choice> vnd get bis yn den 85 grad. geg<choice><am>ē</am><ex>en</ex></choice> orient das - <lb/>sein 16 meridian vnd alweg 5 meridi<choice><am>ā</am><ex>an</ex></choice>s linienn - <lb/>mach<choice><am>ē</am><ex>en</ex></choice> ein stund. wy vorgesagt ist. Yn diser - <choice><orig>gegen<lb/>heyt</orig><reg>gegenheyt<lb/></reg></choice> der <rs type="inhabitants">moren</rs> vnder <placeName type="country">egipten</placeName> entspringt der edl - <lb/>fluß <placeName type="river">Nilus</placeName> auss dem gebyrgk des Mondes - <choice><orig>ge<lb/>n<choice><am>ā</am><ex>an</ex></choice>t</orig><reg>gen<choice><am>ā</am><ex>an</ex></choice>t<lb/></reg></choice> v<choice><am>n̄</am><ex>nd</ex></choice> hoth wenig perg sust dy vns bek<choice><am>ā</am><ex>an</ex></choice>t wern. - <lb/>als do ist der perg <placeName type="mountain">tala</placeName>. <placeName type="mountain">d<choice><am>ā</am><ex>an</ex></choice>chis maschka</placeName>. <placeName type="mountain">Juß</placeName> vnd - <lb/>andere perg. welche devtlich verzeichent werd - - <pb n="51" facs="Ptolemaeus_text_51"/> - <lb/>en yn den taueln <persName><surname>ptholomei</surname></persName>. pey disem l<choice><am>ā</am><ex>an</ex></choice>d yn d<choice><am>ē</am><ex>em</ex></choice> - <lb/><placeName type="ocean">eussern mör</placeName> lig<choice><am>ē</am><ex>en</ex></choice> 7 ynsuln yn welch<choice><am>ē</am><ex>en</ex></choice> mancherley - <lb/>w<choice><am>ū</am><ex>un</ex></choice>derliche gestalt ist der m<choice><am>ē</am><ex>en</ex></choice>schen vnd ouch <choice><orig>d</orig><reg>der</reg></choice> - <lb/>vnuernunfftig<choice><am>ē</am><ex>en</ex></choice> thyre. vnder welchenn ynsulenn - <lb/><choice><orig>ene</orig><reg>eine</reg></choice> genant ist <placeName type="island">canaria</placeName> auß welcher Jnsul (Alls - <lb/><persName><forename>marcianus</forename></persName> sagt) kom<choice><am>ē</am><ex>en</ex></choice> <choice><sic>also also</sic><corr>also</corr></choice> gross h<choice><am>ū</am><ex>un</ex></choice>d das sie - <lb/>ouch dy grösten thier fellen vnd vberwind<choice><am>ē</am><ex>en</ex></choice> alls - <lb/>leben wilde schwe<choice><am>ī</am><ex>in</ex></choice> Behre <choice><am>rc̄</am><ex>etc</ex></choice>. eyn solchen hund - <lb/>hat gehapt <persName><forename>alexander</forename> <roleName>magnus</roleName></persName>. vnd dis l<choice><am>ā</am><ex>an</ex></choice>d wirt - <choice><orig>ge<lb/>teilt</orig><reg>geteilt<lb/></reg></choice> yn dry teyl als in <placeName type="country">lybiam inferior<choice><am>ē</am><ex>em</ex></choice></placeName>. das ist - <choice><orig>vn<lb/>der</orig><reg>vnderste<lb/></reg></choice> <placeName type="country">libia</placeName> vnd yn <placeName type="country">morland</placeName> vnder <placeName type="country">egipt<choice><am>ē</am><ex>en</ex></choice></placeName> vnd - <placeName type="country"><choice><orig>ethi<lb/>opia</orig><reg>ethiopia<lb/></reg></choice> interior</placeName> das ist dy ynnerste gegenheit der - <rs type="inhabitants"><choice><orig>mo<lb/>ren</orig><reg>moren<lb/></reg></choice></rs> vnder dem equinoxial ligend. Yn dem nidern - <lb/><placeName type="country">libia</placeName> vnd in <placeName type="country">morlad</placeName> vnder <placeName type="country">egipten</placeName>. hab<choice><am>ē</am><ex>en</ex></choice> dy - <choice><orig>men<lb/>schen</orig><reg>menschen<lb/></reg></choice> dy s<choice><am>ō</am><ex>on</ex></choice>ne zwir vber yren heuptern vnd - <choice><orig>mu-<lb/>ssen</orig><reg>mussen<lb/></reg></choice> leid<choice><am>ē</am><ex>en</ex></choice> grosse hicz der sone vnd haben - <choice><orig>gew<choice><am>ō</am><ex>on</ex></choice>-<lb/>lych</orig><reg>gew<choice><am>ō</am><ex>on</ex></choice>lych<lb/></reg></choice> yre won<choice><am>ū</am><ex>un</ex></choice>g zu haben vnder der erden vnd vil - <lb/>auß yn dy sich erner<choice><am>ē</am><ex>en</ex></choice> vnd aufhalt<choice><am>ē</am><ex>en</ex></choice> v<choice><am>ō</am><ex>on</ex></choice> den - <choice><orig>wurc<lb/>zeln</orig><reg>wurczeln<lb/></reg></choice> der erden vnd alls dy lerer sagen so se<choice><am>ī</am><ex>in</ex></choice> alle - <lb/>creuter doselbst wachsend yn voller krafft der - <choice><orig>na<lb/>ture</orig><reg>nature<lb/></reg></choice> dy in zugeeigent ist diser wurczeln vnd - <choice><orig>krü<lb/>ter.</orig><reg>krüter.<lb/></reg></choice> ist bei in onezal vnd kom<choice><am>ē</am><ex>en</ex></choice> ouch gegen - <placeName type="city"><choice><orig>allex<choice><am>ā</am><ex>an</ex></choice><lb/>dria</orig><reg>allex<choice><am>ā</am><ex>an</ex></choice>dria<lb/></reg></choice></placeName> auff dem <placeName type="river">nilo</placeName> auch sind in disem l<choice><am>ā</am><ex>an</ex></choice>d - <choice><orig>selcza-<lb/>mer</orig><reg>selczamer<lb/></reg></choice> gestalt der m<choice><am>ē</am><ex>en</ex></choice>schen. alls etliche on hevpt di - <lb/>ougen an der brust tragend. etliche mit einem fuz - <lb/><choice><sic>vud</sic><corr>vnd</corr></choice> anderlei geschlecht vil Alls dan - <persName><forename><choice><orig>Anßhel<lb/>mus</orig><reg>Anßhelmus<lb/></reg></choice></forename></persName> schreibt de ymagine mundi. Diß land hat stets - <lb/>tag vber das gancz Jor 12 stund<choice><am>n̄</am><ex>en</ex></choice> - <pb n="52" facs="Ptolemaeus_text_52"/> - </p> - </div> - <div type="kapitel"> - <head> - <lb/>was buchstab<choice><am>ē</am><ex>en</ex></choice> der sprach. - <lb/><placeName type="continent">affrica</placeName> yn ubung hatt. - </head> - <p> - <lb/> - <lb/> - <hi rend="initial">G</hi>Ancz <placeName type="continent">affrica</placeName> helt dy barbarischen - <choice><orig>Litt-<lb/>ern.</orig><reg>Littern.<lb/></reg></choice> welch <term xml:id="littera2" ref="#Buchstaben">litern</term> oder <q corresp="#littera2">Buchstaben</q> eyn - <lb/>Junckfrav fund<choice><am>ē</am><ex>en</ex></choice> hat yn barbarei eyns - <lb/><persName><roleName>kunigs</roleName></persName> tochter. welcher <persName><roleName>kung</roleName></persName> g<choice><am>n̄</am><ex>enan</ex></choice>t was - <lb/><persName><forename>Cirrenes</forename></persName>. vnd dornoch yn seyner Ere ließ er bau<choice><am>ē</am><ex>en</ex></choice> - <lb/>stat yn seynem nam<choice><am>ē</am><ex>en</ex></choice> <placeName type="city">Cirrene</placeName> g<choice><am>n̄</am><ex>enan</ex></choice>t v<choice><am>ō</am><ex>on</ex></choice> d<choice><am>ā</am><ex>an</ex></choice>ne dan das - <lb/>land auch gen<choice><am>ā</am><ex>an</ex></choice>t was <placeName type="country">rogio arenaica</placeName> yn welcherr - <lb/>geg<choice><am>ē</am><ex>en</ex></choice>hait vil gufftiger wurm vnd selczame tyr - <choice><orig>ge<lb/>fund<choice><am>ē</am><ex>en</ex></choice>?</orig><reg>gefund<choice><am>ē</am><ex>en</ex></choice><lb/></reg></choice> werden. Ouch halten etlich dy litern der - <rs type="inhabitants"><choice><orig>a<lb/>raben</orig><reg>araben<lb/></reg></choice></rs> ym lateyn litere arabice. welcher <rs type="inhabitants">arabes</rs> in - <lb/><placeName type="country">egipten</placeName> vil sein dy sich neren v<choice><am>ō</am><ex>on</ex></choice> Rabary vnd - <choice><orig>dy<lb/>berey</orig><reg>dyberey<lb/></reg></choice> . yr zelt oufslah<choice><am>ē</am><ex>en</ex></choice> yn der wust<choice><am>ū</am><ex>un</ex></choice>g vndt wen - <lb/>si begreiff<choice><am>ē</am><ex>en</ex></choice> dy muss<choice><am>ē</am><ex>en</ex></choice> sich v<choice><am>ō</am><ex>on</ex></choice> yn lösen vnd so si och - <lb/>geleit hetten v<choice><am>ō</am><ex>on</ex></choice> dem <persName><roleName>Soldan</roleName></persName> der ein herr ist yn - <lb/><placeName type="country">Egipten</placeName>. sunst ist ouch eyne in <placeName type="country">arabia</placeName> <persName><forename>Felix</forename></persName> gen<choice><am>ā</am><ex>an</ex></choice>t - <lb/>doryn man ouch braucht diser arabisch<choice><am>ē</am><ex>en</ex></choice> litter zu - <lb/>vorauß yn den ampten götlicher maiestat. Ouch - <lb/>hat <placeName type="continent">affrica</placeName> gemengte sproch. v<choice><am>ō</am><ex>on</ex></choice> <choice><sic>Yudeiseh<choice><am>ē</am><ex>en</ex></choice></sic><corr>Yudeisch<choice><am>ē</am><ex>en</ex></choice></corr></choice> litern - <lb/>caldeischen litern dy do brauch<choice><am>ē</am><ex>en</ex></choice> dye <rs type="inhabitants">suriani</rs> - <rs type="inhabitants"><choice><orig>ma-<lb/>chometi</orig><reg>machometi<lb/></reg></choice></rs> vnd <rs type="inhabitants">yuden</rs>. vnd das nur yn <placeName type="country">egipt<choice><am>ē</am><ex>en</ex></choice></placeName></p> - </div> - <div type="kapitel"> - <head> - <lb/>Uon den teilen <placeName type="continent">asie</placeName> - </head> - <p> - <lb/> - <placeName type="continent"><hi rend="initial">A</hi>sia</placeName> wirt furderlich geteilt yn zveitell als in - <lb/><placeName type="continent">azia</placeName> das klein vnd das groß. Di klein <placeName type="continent">asia</placeName> hat - <lb/>yn dem ersten teyl 12 kunigkreich vnndt - <lb/>wirt geteilt yn 6 teyl alls hernach folgtt. - </p> - <p> - <pb n="53" facs="Ptolemaeus_text_53"/> - <lb/>Das ander teyl wirt gen<choice><am>ā</am><ex>an</ex></choice>t groß <placeName type="continent">azia</placeName> vnd ist - <lb/>nicht also wol mol mit volk besacztt als dy klein - <lb/><placeName type="continent">azia</placeName> dy der Thurk helt. vil reicher an gelt vnd - <choice><orig>E-<lb/>delgestein</orig><reg>Edelgestein<lb/></reg></choice> vnd hat ouch 6 Taueln so machtt das - <lb/>dritteil 12 tauel. welcher 12 tafeln ytliche - <choice><orig>sunder-<lb/>lich</orig><reg>sunderlich<lb/></reg></choice> landt vnd gegenheitt ynhelt. dyß dritteyl der - <lb/>weltheit yn ym das geburg <placeName type="mountain">thauri</placeName> vnd <placeName type="mountain">Ymaus</placeName> - <lb/>welch perg sich strecken v<choice><am>ō</am><ex>on</ex></choice> occid<choice><am>ē</am><ex>en</ex></choice>t yn orient v<choice><am>n̄</am><ex>nd</ex></choice> - <lb/>teyln dis teil <placeName type="continent">asia</placeName> yn zweyteil. Eins gegen - <choice><orig>mitter-<lb/>nacht</orig><reg>mitternacht<lb/></reg></choice> das ander gegen mittag doryn <placeName type="country">yndia</placeName> leit. - <lb/>Auß disem gepurg entspring<choice><am>ē</am><ex>en</ex></choice> dy grosten wasser - <lb/>in <placeName type="country">yndia</placeName> als <placeName type="river"><choice><orig>gages</orig><reg>ganges</reg></choice></placeName> vnd <placeName type="river">yndus</placeName> mit dem flyß <placeName type="river">tigris</placeName> - <lb/>das do entspringt auß dem geburg <placeName type="mountain">niphates</placeName> - <placeName type="river"><choice><orig>Eu-<lb/>frates</orig><reg>Eufrates<lb/></reg></choice></placeName> aber das fliß yn <placeName type="country">armenia</placeName> der grossenn - <choice><orig>ent-<lb/>springt</orig><reg>entspringt<lb/></reg></choice> auß den pergen <placeName type="mountain">periardes</placeName>. diß sei dy - <choice><orig>furd-<lb/>lichsten</orig><reg>furderlichsten<lb/></reg></choice> flis vnd berg yn <placeName type="continent">azia</placeName>. - </p> - </div> - <div type="kapitel"> - <head> - <lb/><placeName type="continent">Azia</placeName> minor klein <placeName type="continent">asia</placeName>. 1 - </head> - <p> - <lb/> - <hi rend="initial">D</hi>Js teil der welt ligt vom occid<choice><am>ē</am><ex>en</ex></choice>t 55 grad - <lb/>das ist yn dem 11 meridi<choice><am>ā</am><ex>an</ex></choice> vnd strecket sich - <lb/>yn ori<choice><am>ē</am><ex>en</ex></choice>t bis yn den 71 grad das ist yn den - <lb/>14 meridian vnd wirt vmgeb<choice><am>ē</am><ex>en</ex></choice> mit mör von - <choice><orig>miter<lb/>nacht</orig><reg>miternacht<lb/></reg></choice> das do gen<choice><am>ā</am><ex>an</ex></choice>t ist <placeName type="ocean">pontus euximus</placeName>. das ist das - <lb/><placeName type="ocean">swarcz mör</placeName>. Uom occid<choice><am>ē</am><ex>en</ex></choice>t mit dem <placeName type="ocean">egeischen mör</placeName> - <lb/>vom mitag mit dem <placeName type="ocean">krichisch<choice><am>ē</am><ex>en</ex></choice> mör</placeName> von orient mit - <lb/>dem schon<choice><am>ē</am><ex>en</ex></choice> fliß <placeName type="river">evfrates</placeName>. dise klein <placeName type="continent">asia</placeName> hat 12 - <choice><orig>kun<lb/>nigreich</orig><reg>kunnigreich<lb/></reg></choice> als das reich <placeName type="country">Bittimie</placeName> <placeName type="country">Paflag<choice><am>ō</am><ex>on</ex></choice>ie</placeName><placeName type="country"><choice><orig>palla-<lb/>cie.</orig><reg>pallacie.<lb/></reg></choice></placeName> <placeName type="country">capadocie</placeName>. <placeName type="country">meonie</placeName>. <placeName type="country">lidie</placeName> <placeName type="country">p<choice><am>ā</am><ex>am</ex></choice>philie</placeName> <placeName type="country">licie</placeName> <placeName type="country">cilicie</placeName> - <pb n="54" facs="Ptolemaeus_text_54"/> - <lb/><placeName type="country">klein armeni<choice><am>ē</am><ex>en</ex></choice></placeName> <placeName type="country">frigie</placeName> vnd <placeName type="country">carie</placeName>. vber alle reich ist - <lb/>der turk ein hermy pye diser clein<choice><am>ē</am><ex>en</ex></choice> <placeName type="continent">azia</placeName> <choice><am>ī</am><ex>in</ex></choice> - <placeName type="ocean"><choice><orig>Carpati<lb/>schen</orig><reg>Carpatischen<lb/></reg></choice> mor</placeName> ligt dy ynsul <placeName type="island">Rhodis</placeName> dy do halt<choice><am>ē</am><ex>en</ex></choice> dye - <persName><roleName><lb/>Joh<choice><am>ā</am><ex>an</ex></choice>niter Ritter</roleName></persName>. Diß ynsul ist ein fart belegerdt - <lb/>worden v<choice><am>ō</am><ex>on</ex></choice> dem Turken zu gewyn<choice><am>ē</am><ex>en</ex></choice>. Ym yor alls - <lb/>m<choice><am>ā</am><ex>an</ex></choice> schreibt 1480 Und nichtz dauon bracht dan - <lb/>laster vnd sch<choice><am>ā</am><ex>an</ex></choice>d Diß ynsul ligt von ocident gleich - <lb/>yn dem 58 grad nohe dem 12 meridi<choice><am>ā</am><ex>an</ex></choice> vom - <choice><orig>ab-<lb/>end</orig><reg>abend<lb/></reg></choice> gerechnt in orient. Dis volk yn disem l<choice><am>ā</am><ex>an</ex></choice>d - <lb/><placeName type="continent">azia</placeName> minori sind <rs type="inhabitants">Turken</rs> <rs type="inhabitants">Crichen</rs> <rs type="inhabitants">Jacobit<choice><am>n̄</am><ex>en</ex></choice></rs> - <rs type="inhabitants"><choice><orig>Arme<lb/>nien</orig><reg>Armenien<lb/></reg></choice></rs> vnd ander vil naci<choice><am>ō</am><ex>on</ex></choice> v<choice><am>ō</am><ex>on</ex></choice> dem gloub<choice><am>ē</am><ex>en</ex></choice> - <choice><orig>macho<lb/>meti.</orig><reg>machometi.<lb/></reg></choice> Ein grad nach der leng auf disem l<choice><am>ā</am><ex>an</ex></choice>d macht - <lb/>47 meiln. Dy grost leng diß tags yn disem teil ist - <lb/>15 st<choice><am>ū</am><ex>un</ex></choice>d yn dieser <placeName type="continent">azia</placeName> minori leyt dy stat <placeName type="city">Troya</placeName> v<choice><am>ō</am><ex>on</ex></choice> - <lb/>den <rs type="inhabitants"><choice><orig>krchen</orig><reg>krichen</reg></choice></rs> zu stört. vil ander groß stet hat - <lb/>dis l<choice><am>ā</am><ex>an</ex></choice>t dy <persName><surname>ptolomeuß</surname></persName> yn seinen Figur<choice><am>ē</am><ex>en</ex></choice> devtet - </p> - </div> - <div type="kapitel"> - <head> - <lb/><placeName type="country">Sarmatia</placeName> <placeName type="continent">Aziatica</placeName> <placeName type="country">Mosquiten</placeName> 2 - </head> - <p> - <lb/> - <hi rend="initial">V</hi><choice><am>ō</am><ex>on</ex></choice> ori<choice><am>ē</am><ex>en</ex></choice>t wirt diß teyl besloss<choice><am>ē</am><ex>en</ex></choice> mit dem groß<choice><am>ē</am><ex>en</ex></choice> - <lb/>flis <placeName type="river">rha</placeName>. das do entspringt auß den perg<choice><am>ē</am><ex>en</ex></choice> - <placeName type="mountain"> <choice><orig>hiperbo<lb/>rei</orig><reg>hiperborei<lb/></reg></choice></placeName> vnd felt yn des <placeName type="ocean">hyrcanisch mer</placeName>. v<choice><am>ō</am><ex>on</ex></choice> occid<choice><am>ē</am><ex>en</ex></choice>t - <choice><orig>gre<lb/>niczt</orig><reg>greniczt<lb/></reg></choice> dis teyl mit <rs type="inhabitants"><choice><sic>litāveu</sic><corr>litāven</corr></choice></rs> vnd <rs type="habitants">revss<choice><am>ē</am><ex>en</ex></choice></rs>. vom mittag - <lb/>mit dem berg <placeName type="mountain">caucaso</placeName> vnd mit dem <placeName type="ocean">hircaaische - <lb/>mor</placeName> vnd hat an der breit 4 meridi<choice><am>ā</am><ex>an</ex></choice> vnd 2 grad - <lb/>v<choice><am>ō</am><ex>on</ex></choice> occid<choice><am>ē</am><ex>en</ex></choice>t ligt es 64 grad v<choice><am>n̄</am><ex>nd</ex></choice> get bys yn den 87 - <lb/>grad. gegen miter<choice><am>ā</am><ex>na</ex></choice>cht preitet sich es bis yn das - <choice><orig><choice><am>ū</am><ex>un</ex></choice><lb/>bek<choice><am>ā</am><ex>an</ex></choice>t</orig><reg><choice><am>ū</am><ex>un</ex></choice>bek<choice><am>ā</am><ex>an</ex></choice>t<lb/></reg></choice> teil der werlt diß land helt der <rs type="inhabitants">mosquit</rs> der - <lb/>do vil wusts l<choice><am>ā</am><ex>an</ex></choice>ds hat wenig v<choice><am>ō</am><ex>on</ex></choice> steten vnd dorff - <pb n="55" facs="Ptolemaeus_text_55"/> - <lb/>s<choice><am>ū</am><ex>un</ex></choice>der ein wolbeslossen l<choice><am>ā</am><ex>an</ex></choice>d hat er mit gepirg vnt - <lb/>mit wasser. in diser <placeName type="country">mosquitey</placeName> hat <persName><forename>alex<choice><am>ā</am><ex>an</ex></choice>der</forename> <roleName>magnus</roleName></persName> - <lb/>gesaczt. ein altar seul <choice><sic>zn</sic><corr>zu</corr></choice> sein<choice><am>ē</am><ex>em</ex></choice> gedechtnuß das er - <choice><orig>al<lb/>so</orig><reg>also<lb/></reg></choice> weit het gestriten yn der werlt Dise seul stat - <choice><orig><choice><am>ū</am><ex>un</ex></choice><lb/>der</orig><reg><choice><am>ū</am><ex>un</ex></choice>der<lb/></reg></choice> dem 80 grad v<choice><am>ō</am><ex>on</ex></choice> occid<choice><am>ē</am><ex>en</ex></choice>t vnd yn dem 16 - <choice><orig>pa-<lb/>ralel.</orig><reg>paralel.<lb/></reg></choice> Dise <rs type="inhabitants">Mosquitn</rs> sind torstig zu streit als v<choice><am>n̄</am><ex>nd</ex></choice> - <lb/>ouch alle dy do lig<choice><am>ē</am><ex>en</ex></choice> yn disem teil <placeName type="country">Sarmacia</placeName> - <choice><orig>leng-<lb/>sten</orig><reg>lengsten<lb/></reg></choice> tag haben si 17 st<choice><am>ū</am><ex>un</ex></choice>d - </p> - </div> - <div type="kapitel"> - <head> - <lb/><placeName type="country">Armenia maior</placeName> <placeName type="country">groß armenien</placeName> 3 - </head> - <p> - <lb/> - <hi rend="initial">E</hi>S ist zu wissen das <placeName type="country">Arm<choice><am>̄</am><ex>en</ex></choice>ia</placeName> wirt beslo - <lb/>ssen v<choice><am>ō</am><ex>on</ex></choice> ori<choice><am>ē</am><ex>en</ex></choice>t mit dem <placeName type="ocean">hircanischen mör</placeName> - <lb/>vom occid<choice><am>ē</am><ex>en</ex></choice>t mit dem<placeName type="ocean"> swarcz<choice><am>ē</am><ex>en</ex></choice> mör</placeName> v<choice><am>ō</am><ex>on</ex></choice> - <lb/>mitternacht mit den <placeName type="country">mosquit<choice><am>ē</am><ex>en</ex></choice></placeName> greniczt - <lb/>hab<choice><am>ē</am><ex>en</ex></choice>d vom mittag hat es grenicz mit dem l<choice><am>ā</am><ex>an</ex></choice>d - <placeName type="country"><choice><orig>a-<lb/>zirie</orig><reg><lb/>azirie</reg></choice>.</placeName> yn ym flewst das flis <placeName type="river">evfrates</placeName> teyl<choice><am>ē</am><ex>en</ex></choice>d v<choice><am>ō</am><ex>on</ex></choice> ym - <lb/><placeName type="country">klein armenien</placeName> vnd <placeName type="continent">klein azi<choice><am>ā</am><ex>am</ex></choice></placeName> ouch hat diß l<choice><am>ā</am><ex>an</ex></choice>t yn - <lb/>ym streich<choice><am>ē</am><ex>en</ex></choice>d das fliß <placeName type="river">Tigris</placeName> das do vrspr<choice><am>ū</am><ex>un</ex></choice>g hat aus - <lb/>den perg<choice><am>ē</am><ex>en</ex></choice> <placeName type="mountain">niphates</placeName> als vorhyn gesagt ist v<choice><am>n̄</am><ex>nd</ex></choice> leit - <lb/>vom occid<choice><am>ē</am><ex>en</ex></choice>t 72 grad mit seinem anheben das ist - <lb/>yn dem 14 meridi<choice><am>ā</am><ex>an</ex></choice> vnd 2 grad v<choice><am>n̄</am><ex>nd</ex></choice> strekt sich bis - <lb/>yn d<choice><am>ē</am><ex>en</ex></choice> 81 grad das ist yn d<choice><am>ē</am><ex>en</ex></choice> 16 meridi<choice><am>ā</am><ex>an</ex></choice> vom obend - <lb/>gecelt. dis teil <placeName type="continent">asie</placeName> hat <choice><am>ī</am><ex>in</ex></choice> ym dy ynwoner. <rs type="inhabitants">colchos</rs> - <lb/><rs type="inhabitants">yberos</rs>. <rs type="inhabitants">abamos</rs>. das ist dy groß Thatarey. vnd - <lb/>hat dy mechtig stat <placeName type="city">tarpes<choice><am>ō</am><ex>on</ex></choice></placeName> als tal<choice><am>ī</am><ex>in</ex></choice>a dise Tafeln - <lb/>diß l<choice><am>ā</am><ex>an</ex></choice>ts helt der <persName><roleName>turkisch keiser</roleName></persName>. vnd si hab<choice><am>n̄</am><ex>en</ex></choice> ym - <lb/>l<choice><am>ā</am><ex>an</ex></choice>t l<choice><am>ē</am><ex>en</ex></choice>gst<choice><am>ē</am><ex>en</ex></choice> tag 15 st<choice><am>n̄</am><ex>un</ex></choice>d ouch wurcht man in <placeName type="country">ormenia - <lb/>maiori</placeName> vnd <choice><am>ī</am><ex>in</ex></choice> <placeName type="city">media</placeName> vil seid<choice><am>ē</am><ex>en</ex></choice> gw<choice><am>ā</am><ex>an</ex></choice>d tebich vnd - <choice><orig><choice><am>ā</am><ex>an</ex></choice>-<lb/>der</orig><reg><choice><am>ā</am><ex>an</ex></choice>der<lb/></reg></choice> m<choice><am>ā</am><ex>an</ex></choice>cherley gw<choice><am>ā</am><ex>an</ex></choice>d - </p> - </div> - <div type="kapitel"> - <head> - <pb n="56" facs="Ptolemaeus_text_56"/> - <lb/><placeName type="country">Siria</placeName> <placeName type="country">mesopotanea</placeName> <placeName type="country">babilonia</placeName>. 4 - <lb/> - </head> - <p> - <hi rend="initial">I</hi>N dem 4 teyl <placeName type="continent">azie</placeName> vint m<choice><am>ā</am><ex>an</ex></choice>n dise l<choice><am>ā</am><ex>an</ex></choice>tschafft - <lb/>alls <placeName type="country">siriam</placeName> <placeName type="country">Mesopoteneam</placeName>. <placeName type="country">Babil<choice><am>ō</am><ex>on</ex></choice>iam</placeName>. - <lb/>dass l<choice><am>ā</am><ex>an</ex></choice>t <placeName type="country">palestina</placeName> das Judisch l<choice><am>ā</am><ex>an</ex></choice>t <placeName type="country">Petrea</placeName>. - <lb/>vnd die wuste arabia. dise alle regi<choice><am>ō</am><ex>on</ex></choice>es werden - <lb/>beslossen bey eynander zwisch<choice><am>ē</am><ex>en</ex></choice> orient vnd - <choice><orig>occi-<lb/>dent</orig><reg>occident<lb/></reg></choice> anzuheben an dem 64 grad. das ist der 13 - <lb/>meridian vom abend bis yn den 80 grad das ist - <lb/>der 16 meridi<choice><am>ā</am><ex>an</ex></choice>. dise l<choice><am>ā</am><ex>an</ex></choice>t werdn beslossen v<choice><am>ō</am><ex>on</ex></choice> - <choice><orig>mitt-<lb/>nacht</orig><reg>mittnacht<lb/></reg></choice> myt <placeName type="country">kle<choice><am>ī</am><ex>in</ex></choice> armenien</placeName> greniczent. v<choice><am>ō</am><ex>on</ex></choice> occident - <lb/>haben dise l<choice><am>ā</am><ex>an</ex></choice>d ein anstossend mör g<choice><am>n̄</am><ex>enan</ex></choice>t <placeName type="ocean">siriac<choice><am>ū</am><ex>um</ex></choice> - <choice><orig>ma<lb/>re</orig><reg>mare<lb/></reg></choice></placeName> das ist das <placeName type="ocean">mör sirie</placeName> sicht. gemeilich ist es - <lb/>g<choice><am>n̄</am><ex>enan</ex></choice>t das <placeName type="ocean">mitell mör</placeName>. vom mitag greniczen - <choice><orig>d<lb/>mit</orig><reg>dise<lb/></reg></choice> land mit <placeName type="country">arabia felici</placeName> das ist mit dem seligen oder - <choice><orig>lobsa-<lb/>men</orig><reg>lobsamen<lb/></reg></choice> l<choice><am>ā</am><ex>an</ex></choice>d <placeName type="country">arabie</placeName> dorin vil k<choice><am>ū</am><ex>un</ex></choice>st der <choice><sic>erczuei</sic><corr>ercznei</corr></choice> f<choice><am>ū</am><ex>un</ex></choice>den - <lb/>ist v<choice><am>ō</am><ex>on</ex></choice> orient wirt es beslossen mit dem wasser - <placeName type="river"><choice><orig>Ti<lb/>gris</orig><reg>Tigris<lb/></reg></choice></placeName> bei disen l<choice><am>ā</am><ex>an</ex></choice>den ligt ouch das glopt l<choice><am>ē</am><ex>en</ex></choice>d - <placeName type="country"><choice><orig>Jud<lb/>ea</orig><reg>Judea<lb/></reg></choice></placeName> <placeName type="country">paliest<choice><am>ī</am><ex>in</ex></choice>a</placeName> g<choice><am>n̄</am><ex>enan</ex></choice>t in welchm <placeName type="country">iudea</placeName> 12 geslecht israel - <lb/>s<choice><am>ū</am><ex>un</ex></choice>derlich l<choice><am>ā</am><ex>an</ex></choice>d gehapt haben. in diser <placeName type="country">iudea</placeName> ist das - <lb/>flis <placeName type="river">iordanus</placeName> der do entspringt uß zwaien born - <lb/>Der ain ist gen<choice><am>ā</am><ex>an</ex></choice>t <placeName type="river">Jor</placeName>. der ander <placeName type="river">dan</placeName>. in diser <placeName type="country">iudea</placeName> - <lb/>leit <placeName type="city">ierusal<choice><am>m̄</am><ex>em</ex></choice></placeName> di namhafftige stat dise l<choice><am>ā</am><ex>an</ex></choice>d haben - <choice><orig>geme<lb/>Ólich</orig><reg>geme<choice><am>ī</am><ex>in</ex></choice>lich<lb/></reg></choice> tage vff das lengst 14 st<choice><am>ū</am><ex>un</ex></choice>d dise tauel oder - <choice><orig>ge<lb/>samelt</orig><reg>gesamelt<lb/></reg></choice> lānd helt in der husahesa <choice><orig>bs</orig><reg>bis</reg></choice> vff <placeName type="country">iude<choice><am>ā</am><ex>am</ex></choice></placeName> - <placeName type="country"><choice><orig>pale<lb/>stinam</orig><reg>palestinam<lb/></reg></choice></placeName> das do helt der <persName><roleName>soldan</roleName></persName>. auch helt dise - <placeName type="country"><choice><orig>pa-<lb/>lestina</orig><reg>palestina<lb/></reg></choice></placeName> <placeName type="country">iudaica</placeName> das <placeName type="ocean">tote mör</placeName> das do ist - <choice><orig>vnteilhaff<lb/>tig</orig><reg>vnteilhafftig<lb/></reg></choice> cleberig pei en<choice><am>ā</am><ex>an</ex></choice>der als ein gemachter leim - </p> - <p> - <pb n="57" facs="Ptolemaeus_text_57"/> - <lb/>diß meres materia kan m<choice><am>ā</am><ex>an</ex></choice> mit nicht geteilee alle<choice><am>ī</am><ex>in</ex></choice> - <lb/>mit einem faden der do gezogen ist durch rot - <choice><orig>d<lb/>frowen</orig><reg>der<lb/></reg></choice> frowen das ist menstr<choice><am>ū</am><ex>um</ex></choice>. Dise ding schreibt - <persName><forename><choice><orig>an-<lb/>ßhelmus</orig><reg>anßhelmus<lb/></reg></choice></forename></persName> de imag<choice><am>ī</am><ex>in</ex></choice>e m<choice><am>ū</am><ex>undi</ex></choice> ouch sagt ein ander Lerer - <lb/>wi daß m<choice><am>ā</am><ex>an</ex></choice> pei disem <placeName type="ocean">toten mör</placeName> vint den gifftigen - <lb/>slangk Thirus g<choice><am>n̄</am><ex>en</ex></choice><choice><am>n̄</am><ex>an</ex></choice>t dau<choice><am>ō</am><ex>on</ex></choice> m<choice><am>ā</am><ex>an</ex></choice> Thiriaca macht - <choice><orig>dis<lb/>er</orig><reg>diser<lb/></reg></choice> slang ist blind bei disen landen ligt di k<choice><am>ū</am><ex>un</ex></choice>gklih - <lb/>insul <placeName type="island">ciprus</placeName> g<choice><am>n̄</am><ex>enan</ex></choice>t gleich vnder dem 13 meridi<choice><am>ā</am><ex>an</ex></choice> das ist - <lb/>vnder dem 65 grad v<choice><am>ō</am><ex>on</ex></choice> occid<choice><am>n̄</am><ex>ne</ex></choice>t vnd ligt in dem 10 - <lb/>paralell dise insul hat am ersten gebavt <persName><forename>iaphet</forename></persName> - <persName><forename><choice><orig>no<lb/>e</orig><reg>noe<lb/></reg></choice></forename></persName> son nach der sintflucht in diser insull wirtt nach - <lb/>gezeigt <persName><forename>veneris</forename></persName> garten dorin <persName><forename>iuno</forename></persName> <persName><forename>pallas</forename></persName> vnd - <persName><forename><choice><orig>ve<lb/>nus</orig><reg>venus<lb/></reg></choice></forename></persName> ein gezenk heten der schönheit halb auß - <choice><orig>di-<lb/>ser</orig><reg>diser<lb/></reg></choice> insul <placeName type="island">ciprus</placeName> ist gezukt worden <persName><forename>helena</forename></persName> v<choice><am>ō</am><ex>on</ex></choice> - <persName><forename><choice><orig>pa<lb/>ris</orig><reg>paris<lb/></reg></choice></forename></persName> daru<choice><am>m̄</am><ex>mb</ex></choice> dan der troianisch krieg l<choice><am>ā</am><ex>an</ex></choice>gzeit wertt - <lb/>mit dem andern nahmen heist diese cetim in der - <lb/>schrifft oder paph<choice><am>ō</am><ex>on</ex></choice> v<choice><am>ō</am><ex>on</ex></choice> einer stat die <choice><am>ī</am><ex>in</ex></choice> <placeName type="island">cipro</placeName> also - <lb/>heist die hevptstad in dieser insull ist <placeName type="city">Nicosia</placeName>. in - <lb/>diser ynsul haben si gut starck wein. Jtem auff - <lb/>den höchsten pergen yn <placeName type="island">cipro</placeName> ist ein kirch dorin - <lb/>stat das krücz des schachers zu der rechten - <lb/>hant <persName><forename>Hiesu</forename></persName> gehangen hat. - </p> - </div> - <div type="kapitel"> - <head> - <lb/><placeName type="country">Media</placeName> <placeName type="country">persidis</placeName> <placeName type="country">hircania</placeName> <placeName type="country">parthia</placeName>. 5 - <lb/> - </head> - <p> - <hi rend="initial">D</hi>ise land alle mit <choice><sic><choice><am>enandēr</am><ex>enandenr</ex></choice></sic><corr>enander</corr></choice> werden - <choice><orig>zusa-<lb/>men</orig><reg>zusamen<lb/></reg></choice> <choice><sic>genomeu</sic><corr>genomen</corr></choice> vnnd gerechend fur ein - <lb/>gegenheit vnd ligen mit irem <choice><am>ā</am><ex>an</ex></choice>hebenn - <lb/>grenicz habend mit <placeName type="country">Mesopotanea</placeName> vnd - <lb/><placeName type="country"><choice><sic>Babilouia</sic><corr>Babilonia</corr></choice></placeName> in dem achtzigisten grad v<choice><am>ō</am><ex>on</ex></choice> occident - <pb n="58" facs="Ptolemaeus_text_58"/> - <lb/>das ist a<choice><am>ā</am><ex>an</ex></choice> dem 16 meridi<choice><am>ā</am><ex>an</ex></choice> v<choice><am>n̄</am><ex>nd</ex></choice> strekt sich bis yn d<choice><am>ē</am><ex>en</ex></choice> - <lb/>101 grad das ist der 18 meridi<choice><am>ā</am><ex>an</ex></choice> v<choice><am>ō</am><ex>on</ex></choice> occid<choice><am>ē</am><ex>en</ex></choice>t. dise l<choice><am>ā</am><ex>an</ex></choice>t - <lb/>werd<choice><am>ē</am><ex>en</ex></choice> v<choice><am>ō</am><ex>on</ex></choice> miternacht besloss<choice><am>ē</am><ex>en</ex></choice> mit d<choice><am>ē</am><ex>em</ex></choice> <placeName type="ocean">hyrk<choice><am>ā</am><ex>an</ex></choice>isch<choice><am>ē</am><ex>en</ex></choice> - <lb/>mör</placeName> v<choice><am>ō</am><ex>on</ex></choice> occid<choice><am>n̄</am><ex>en</ex></choice>t greniczun sy mit <placeName type="country">Mesopotanea</placeName> - <lb/>v<choice><am>ō</am><ex>on</ex></choice> mitag mit dem hap des <placeName type="ocean">persien mers</placeName> v<choice><am>ō</am><ex>on</ex></choice> ori<choice><am>ē</am><ex>en</ex></choice>t - <lb/>mit dem perg <placeName type="mountain">maßdoranus</placeName> dise l<choice><am>ā</am><ex>an</ex></choice>d helt der - <choice><orig>hu-<lb/>ssaheß</orig><reg>hussaheß<lb/></reg></choice> vnd ist fruchtpar yn guten we<choice><am>ī</am><ex>in</ex></choice>en yn vil - <choice><orig>ty<lb/>rÃ</orig><reg>tyr<choice><am>ē</am><ex>en</ex></choice><lb/></reg></choice> zuouorauß kameltir dy do alt werden 60 Jor - <lb/> - </p> - </div> - <div type="kapitel"> - <head> - <lb/><placeName type="country">Arabia Felix</placeName> <placeName type="country">fruchtpar arabia</placeName> 6 - <lb/> - </head> - <p> - <hi rend="initial">D</hi>Js loblich vnd selig l<choice><am>ā</am><ex>an</ex></choice>d <placeName type="country">arabia</placeName> ein muter - <lb/>aller arczney zu enthalt<choice><am>n̄</am><ex>en</ex></choice>. pey dem wesse - <lb/><choice><sic>tötliehe</sic><corr>tötliche</corr></choice> corper. wirt geteilt yn zwey teyll - <lb/>alls yn <placeName type="country">arabi<choice><am>ā</am><ex>am</ex></choice> felicem</placeName> vnd <choice><am>ī</am><ex>in</ex></choice> <placeName type="country">carmania</placeName>. auß - <choice><orig>dyser<lb/> <placeName type="country">arabia</placeName></orig><reg>dyserr<lb/></reg></choice> arabia k<choice><am>ō</am><ex>om</ex></choice>en dise all specerey dy ma haben muß - <lb/>yn den appoteken. Und v<choice><am>ō</am><ex>on</ex></choice> d<choice><am>ā</am><ex>an</ex></choice> ist komen all k<choice><am>ū</am><ex>un</ex></choice>st - <lb/>der arczney. wan worum es ist gewest e<choice><am>ī</am><ex>in</ex></choice> <persName><roleName>k<choice><am>ū</am><ex>un</ex></choice>ig</roleName></persName> yn - <lb/> <placeName type="country">arabia</placeName> der do gehalt<choice><am>n̄</am><ex>en</ex></choice> hat 12 philozophos oder - <lb/>meister dy do haben vor ym mussen Disputiren - <lb/>v<choice><am>ō</am><ex>on</ex></choice> natur der kreüter vnd diselben beweren vnd - <lb/>so yndert eyner f<choice><am>ū</am><ex>un</ex></choice>den ist wordn der do nicht het - <lb/>konden erkenen dy natur des krauts doru<choice><am>m̄</am><ex>mb</ex></choice> er - <lb/>gefragt wer word<choice><am>n̄</am><ex>en</ex></choice> der must furt mer ewiklich - <choice><orig>swy<lb/>gen</orig><reg>swygen<lb/></reg></choice> v<choice><am>ō</am><ex>on</ex></choice> aller arczney. Do sei auffkomen der - <choice><orig>krüt-<lb/>er</orig><reg>krüter<lb/></reg></choice> natur worzu si nuczlich sind. Dise <placeName type="country">arabia felix</placeName> - <lb/>leyt v<choice><am>ō</am><ex>on</ex></choice> occident mit yrem anheben 68 grad das - <lb/>ist in dem 14 meridi<choice><am>ā</am><ex>an</ex></choice> vnd get bis yn den 104 grad - <lb/>vnt wirt beslossen v<choice><am>ō</am><ex>on</ex></choice> orient mit dem <placeName type="ocean"><choice><am>ī</am><ex>in</ex></choice>dische mör</placeName> - <pb n="59" facs="Ptolemaeus_text_59"/> - <lb/>v<choice><am>ō</am><ex>on</ex></choice> occident mit dem <placeName type="ocean">roten mör</placeName> v<choice><am>ō</am><ex>on</ex></choice> mitnacht mit - <lb/>dem hap des <placeName type="ocean">mörs persici</placeName> v<choice><am>ō</am><ex>on</ex></choice> mittag mit dem - <placeName type="ocean"><choice><orig>in<lb/>dischn</orig><reg>indischn<lb/></reg></choice></placeName> vnd <placeName type="ocean">roten mor</placeName>. Diß l<choice><am>ā</am><ex>an</ex></choice>t ist gewest ein - <choice><orig>va<lb/>terl‚d</orig><reg>vaterl<choice><am>ā</am><ex>an</ex></choice>d<lb/></reg></choice> des <persName><roleName>königs</roleName> <forename>Caspar</forename></persName> der do bracht oppfer - <lb/><persName><forename>Jhesu</forename></persName> v<choice><am>ō</am><ex>on</ex></choice> arabisch<choice><am>n̄</am><ex>en</ex></choice> gold auss der stat <placeName type="city">sabba</placeName> in - <placeName type="country"><choice><orig>ara<lb/>bia</orig><reg>arabia<lb/></reg></choice></placeName> v<choice><am>ō</am><ex>on</ex></choice> occident 74 grad 1/2 das ist vnttir dem 14 - <choice><orig>me<lb/>ridi‚</orig><reg>meridi<choice><am>ā</am><ex>an</ex></choice><lb/></reg></choice> vnd yn dem ersten clima am end das ist 16 - <lb/><choice><am>ḡ</am><ex>gra</ex></choice>d. v<choice><am>ō</am><ex>on</ex></choice> Equinoxial <space/>In dem <placeName type="ocean">mer arabiee</placeName> do ligt e<choice><am>ī</am><ex>in</ex></choice> - <lb/><choice><sic>Insnl</sic><corr>Insul</corr></choice> genannt <placeName type="island">insula magor<choice><am>ū</am><ex>um</ex></choice></placeName>. Do sagt m<choice><am>ā</am><ex>an</ex></choice> das do - <lb/>soll<choice><am>ē</am><ex>en</ex></choice> außkomen sein <persName><forename>Malchior</forename></persName> pringed weiroch - <lb/>vnd mirre welche g<choice><am>ū</am><ex>um</ex></choice>mi wachsen yn der selbe - <choice><orig>in-<lb/>sul</orig><reg>insul<lb/></reg></choice> vnd ouch yn <placeName type="country">arabia</placeName> Der drite <persName><roleName>könig</roleName></persName> sol - <choice><orig>kom<lb/>en</orig><reg>komen<lb/></reg></choice> sein als dy schrifft sagt v<choice><am>ō</am><ex>on</ex></choice> orient <placeName type="country">ethiopie</placeName> das - <lb/>ist <placeName type="country">ethiopia</placeName> vnder <placeName type="country">egipten</placeName>. Diß l<choice><am>ā</am><ex>an</ex></choice>d hat <choice><orig>lengten</orig><reg>lengsten</reg></choice> - <lb/>tag 13 st<choice><am>ū</am><ex>un</ex></choice>d ouch ist diß l<choice><am>ā</am><ex>an</ex></choice>d vberflussig in edln - <choice><orig>ge-<lb/>steinen</orig><reg>gesteinen<lb/></reg></choice> vnd perlen dorin ist der perg <placeName type="mountain">sinai</placeName> do <persName><roleName>s<choice><am>ā</am><ex>an</ex></choice>t</roleName> - <lb/><forename>Katherina</forename></persName> auff leit den <persName><surname><choice><orig>ptolomes</orig><reg>ptolomeus</reg></choice></surname></persName> nent sinoden - </p> - </div> - <div type="kapitel"> - <head> - <lb/><placeName type="country">Scitia</placeName> intra ymaum m<choice><am>ō</am><ex>on</ex></choice>tem. - <lb/><placeName type="country">Scitia</placeName> zwischen dez berg <placeName type="mountain">Ymaus</placeName> - </head> - <p> - <lb/> - <hi rend="initial">Di</hi>Se <placeName type="country">scitia</placeName> wirt g<choice><am>n̄</am><ex>enan</ex></choice>t v<choice><am>ō</am><ex>on</ex></choice> den menschen <placeName type="country">Sciti</placeName> - <lb/>di do grausam sint allen menschen wann - <lb/>alls <persName><surname>ptholomeus</surname></persName> schreipt so essenn si - <choice><orig>men<lb/>schen</orig><reg>menschen<lb/></reg></choice> fleisch vnd blut. dise gewonheit haltenn si - <lb/>bei in auff den hevtigen tag vnd sind vnstet yn - <choice><orig>ir<lb/>em</orig><reg>irem<lb/></reg></choice> wesen wan woru<choice><am>m̄</am><ex>mb</ex></choice> si ligen nimmer an - <choice><orig>einer<lb/>stat</orig><reg>einerr<lb/></reg></choice> stat Sunder wan di s<choice><am>ō</am><ex>on</ex></choice>ne sich nehet gegenn dem - <lb/>Tropico Cancri so ziehenn si gegenn mitternacht - <lb/>vnd so di s<choice><am>ō</am><ex>on</ex></choice>n wider get ad Tropicum capricorni - <pb n="60" facs="Ptolemaeus_text_60"/> - <lb/>so zyeh<choice><am>ē</am><ex>en</ex></choice> si aller der son<choice><am>ē</am><ex>en</ex></choice> nach vnd won<choice><am>ē</am><ex>en</ex></choice> yn d<choice><am>ē</am><ex>en</ex></choice> - <choice><orig>wel<lb/>dÃ</orig><reg>weld<choice><am>ē</am><ex>en</ex></choice><lb/></reg></choice> als dy vnuernyfftigen tyr futern v<choice><am>n̄</am><ex>nd</ex></choice>. das weyl si - <lb/>an der selb<choice><am>ē</am><ex>en</ex></choice> stat futr hab<choice><am>ē</am><ex>en</ex></choice> vnd darnach zieh<choice><am>ē</am><ex>en</ex></choice> si - <choice><orig>a<lb/>ber</orig><reg>aber<lb/></reg></choice> weyther. diß l<choice><am>ā</am><ex>an</ex></choice>d wirt beschlossen v<choice><am>ō</am><ex>on</ex></choice> occid<choice><am>ē</am><ex>en</ex></choice>tt - <lb/>mit dem flyß <placeName type="river">raha</placeName> vnd grenicz mit <placeName type="country">sarmacia</placeName> - <placeName type="continent"><choice><orig>azia-<lb/>tica.</orig><reg>aziatica.<lb/></reg></choice></placeName> v<choice><am>ō</am><ex>on</ex></choice> mitternacht mit vnbek<choice><am>ā</am><ex>an</ex></choice>t<choice><am>ē</am><ex>en</ex></choice> land<choice><am>ē</am><ex>en</ex></choice>. vom - <choice><orig>mi<lb/>tag</orig><reg>mitag<lb/></reg></choice> gremczit es mit <placeName type="country">yndia</placeName> zwischen dem fliß - <placeName type="river"><choice><orig>gan<lb/>gem</orig><reg>gangem<lb/></reg></choice></placeName> vnd mit d<choice><am>ē</am><ex>em</ex></choice> l<choice><am>ā</am><ex>an</ex></choice>de <placeName type="country">parthie</placeName> vnd <placeName type="country">arie</placeName>. v<choice><am>ō</am><ex>on</ex></choice> orient - <lb/>mit d<choice><am>ē</am><ex>em</ex></choice> perge <placeName type="mountain">ymaus</placeName>. vnd mit seyn<choice><am>ē</am><ex>en</ex></choice> anheb<choice><am>ē</am><ex>en</ex></choice>denn - <lb/>grenicz<choice><am>ē</am><ex>en</ex></choice> ligt es v<choice><am>ō</am><ex>on</ex></choice> occid<choice><am>ē</am><ex>en</ex></choice>t ym 85 grad. das ist in - <lb/>dem 16 meridi<choice><am>ā</am><ex>an</ex></choice> vnd get bis yn de 28 meridian - <lb/>das ist 140 grad vnd hat lengst<choice><am>ē</am><ex>en</ex></choice> tag 16 st<choice><am>ū</am><ex>un</ex></choice>d. yne - <lb/>diser geg<choice><am>ē</am><ex>en</ex></choice>heit sei m<choice><am>ā</am><ex>an</ex></choice>cherley Naci<choice><am>ō</am><ex>on</ex></choice>. als <rs type="inhabitants">m<choice><am>ā</am><ex>an</ex></choice>giany</rs> - <lb/><rs type="inhabitants">Bactriani</rs>. dy hab<choice><am>ē</am><ex>en</ex></choice> kamelthur dy nymer yre fuss - <lb/>ap gehn oder tret<choice><am>ē</am><ex>en</ex></choice>. dornoch sein <placeName type="country">sogdi<choice><am>ā</am><ex>an</ex></choice>i</placeName> vnd - <placeName type="country"><choice><orig>co-<lb/>mari.</orig><reg>comari.<lb/></reg></choice></placeName> dis l<choice><am>ā</am><ex>an</ex></choice>t hat bevm alls groß das vnder eins - <choice><orig>bav<lb/>mes</orig><reg>bavmes<lb/></reg></choice> schat<choice><am>ē</am><ex>en</ex></choice> lig<choice><am>ē</am><ex>en</ex></choice> mögen 300 m<choice><am>ē</am><ex>en</ex></choice>sche. vnd ein blatt - <lb/>ist 4 schuch lang auff dy pletir hab<choice><am>ē</am><ex>en</ex></choice> <choice><orig>vorzeyen</orig><reg>vorzeyten</reg></choice> - <choice><orig>ge-<lb/>ten</orig><reg>geten<lb/></reg></choice> geschrib<choice><am>ē</am><ex>en</ex></choice> di alltn <choice><am>pphi</am><ex>philosophi</ex></choice>. E das si papir hettenn - <lb/>ouch fint man yn sachar<choice><am>ū</am><ex>um</ex></choice> der geg<choice><am>ē</am><ex>en</ex></choice>heyt vnt yn - <placeName type="country"><choice><orig>ba-<lb/>ctri<choice><am>ā</am><ex>an</ex></choice>a</orig><reg>bactri<choice><am>ā</am><ex>an</ex></choice>a<lb/></reg></choice></placeName> weinbeym alß gros das eyn traub des - <choice><orig>we<choice><am>ī</am><ex>in</ex></choice><lb/>stoks</orig><reg>we<choice><am>ī</am><ex>in</ex></choice>stoks<lb/></reg></choice> gibt drei fuder we<choice><am>ī</am><ex>in</ex></choice>. di grösten perg hatt si - <lb/>ym<choice><am>ā</am><ex>am</ex></choice> vnd thaur<choice><am>ū</am><ex>um</ex></choice> der hy yn dysem l<choice><am>ā</am><ex>an</ex></choice>d wirt gen<choice><am>ā</am><ex>an</ex></choice>t - <lb/><placeName type="country">coronus</placeName>. das grost fliß yn <placeName type="country">Scitia</placeName> ist g<choice><am>n̄</am><ex>enan</ex></choice>t <placeName type="river">Yasartus</placeName> - <lb/>vnd laufft yn sein<choice><am>ē</am><ex>en</ex></choice> strich vber 300 meyl vnd k<choice><am>ō</am><ex>om</ex></choice>pt - <lb/>auß dem berg <placeName type="mountain">Ymaus</placeName> vnd felt yn das <placeName type="ocean">hyrcanisch - <lb/>mör</placeName> ein grad nach der leng ober diß l<choice><am>ā</am><ex>an</ex></choice>t hat 40 <choice><am>ḡ</am><ex>grad</ex></choice> - </p> - </div> - <div type="kapitel"> - <head> - <pb n="61" facs="Ptolemaeus_text_61"/> - <lb/><placeName type="country">Scitia</placeName> axtra <placeName type="mountain">yma<choice><am>ū</am><ex>um</ex></choice></placeName> montem 8 - <lb/> - </head> - <p> - <lb/> - <hi rend="initial">W</hi>Jrt gen<choice><am>ā</am><ex>an</ex></choice>t auff teutsch <placeName type="country">scitia</placeName> auß dem pergk - <lb/><placeName type="mountain">ymaus</placeName> ein l<choice><am>ā</am><ex>an</ex></choice>d ser wust v<choice><am>ō</am><ex>on</ex></choice> volk vnd von - <lb/>stetten weyt feld hab<choice><am>ē</am><ex>en</ex></choice>d vnd fruchtpar yr - <lb/>grenicz helt si mit dem <choice><sic>erstgegacht<choice><am>ē</am><ex>en</ex></choice></sic><corr>erstgedacht<choice><am>ē</am><ex>en</ex></choice></corr></choice> <placeName type="country">scitia</placeName> anheb<choice><am>ē</am><ex>en</ex></choice> - <lb/>v<choice><am>ō</am><ex>on</ex></choice> occid<choice><am>ē</am><ex>en</ex></choice>t 145 grad das ist yn dem 27 meridian - <lb/>vnd get bis an das end der bekant<choice><am>ē</am><ex>en</ex></choice> werlt das ist - <lb/>biß yn 180 grad des letsten meridi<choice><am>ā</am><ex>an</ex></choice>s. dis l<choice><am>ā</am><ex>an</ex></choice>t wirt - <lb/>mit keinem mör vmgeben s<choice><am>ū</am><ex>un</ex></choice>der gegen mitnacht - <lb/>vnd gegen dem vßg<choice><am>ā</am><ex>an</ex></choice>g do begreifft si dy end - <choice><orig>vn-<lb/>ser</orig><reg>vnser<lb/></reg></choice> bekanten werlt. vom occident greniczt si mitt - <lb/>der ynnern <placeName type="country">scitia</placeName> vom mittag mit <placeName type="country">yndia</placeName> dy gröstn - <lb/>fliß yn dysem l<choice><am>ā</am><ex>an</ex></choice>d seyn dyse zwen <placeName type="river">b<choice><am>ā</am><ex>an</ex></choice>tisis</placeName> vnd der - <lb/><placeName type="river">chadus</placeName>. dy gröste perg sind <placeName type="mountain">m<choice><am>ō</am><ex>on</ex></choice>tes casy</placeName> vnd - <placeName type="mountain"><choice><orig>m<choice><am>ō</am><ex>on</ex></choice><lb/>serici</orig><reg>m<choice><am>ō</am><ex>on</ex></choice>serici<lb/></reg></choice></placeName> . yn diser gegenheyt habn dy mensch<choice><am>ē</am><ex>en</ex></choice> e<choice><am>ī</am><ex>in</ex></choice> - <choice><orig>sel-<lb/>czame</orig><reg>selczame<lb/></reg></choice> gew<choice><am>ō</am><ex>on</ex></choice>heit. als wen yndert e<choice><am>ī</am><ex>in</ex></choice> mensch kr<choice><am>ā</am><ex>an</ex></choice>k - <lb/>wirt so sind do seyn fre<choice><am>ū</am><ex>un</ex></choice>d vnd töten vnd essent <choice><am>ī</am><ex>in</ex></choice> - <lb/>sprechen es ist besser das er v<choice><am>ō</am><ex>on</ex></choice> vns gessen werd - <lb/>dan von den worme der erden. vnd welcher der - <choice><orig>neh<lb/>est</orig><reg>nehest<lb/></reg></choice> fr<choice><am>ū</am><ex>un</ex></choice>d ist <choice><orig>d</orig><reg>der</reg></choice> nympt den hyrnschedl vnt late ym - <lb/>yn gold <choice><sic>fasssn</sic><corr>fassen</corr></choice> vnd braucht yn dorauß zu trinken - <lb/>zum gedechtnuß seyns fre<choice><am>ū</am><ex>un</ex></choice>ds. yn disem l<choice><am>ā</am><ex>an</ex></choice>d sint - <lb/>vil greiffe dy do mit mensche streit haben diß l<choice><am>ā</am><ex>an</ex></choice>t - <lb/>lengstn tag 16 st<choice><am>ū</am><ex>un</ex></choice>d im mitel des l<choice><am>ā</am><ex>an</ex></choice>ds. am end - <choice><orig>ab<lb/>er</orig><reg>aber<lb/></reg></choice> gegen mitnacht haben si tag 20 st<choice><am>ū</am><ex>un</ex></choice>d. - </p> - </div> - <div type="kapitel"> - <head> - <pb n="62" facs="Ptolemaeus_text_62"/> - <lb/><placeName type="country">Aria</placeName> <placeName type="country">paropanisis</placeName> <placeName type="country">drangi<choice><am>ā</am><ex>an</ex></choice>a</placeName> <placeName type="country">arachosia</placeName> 9 - </head> - <p> - <lb/> - <hi rend="initial">D</hi>yse yczgemelte l<choice><am>ā</am><ex>an</ex></choice>d liged vom occid<choice><am>ē</am><ex>en</ex></choice>t ym - <lb/>100 grad mit yren anheb<choice><am>n̄</am><ex>en</ex></choice>. das ist yn. dem - <lb/>20 meridi<choice><am>ā</am><ex>an</ex></choice> vnd streckt sich yn 120 <choice><am>ḡ</am><ex>grad</ex></choice>. - <lb/>macht den 24 meridian Dis l<choice><am>ā</am><ex>an</ex></choice>d geg<choice><am>ē</am><ex>en</ex></choice> mittnacht - <lb/>gremczt mit dem land <placeName type="country">Bactri<choice><am>ā</am><ex>an</ex></choice>a</placeName> doryn der groß - <lb/>kam siczt. vom occident mit <placeName type="country">carmānia</placeName> vom - <choice><orig>mit-<lb/>tag</orig><reg>mittag<lb/></reg></choice> mit dem <placeName type="ocean">yndischen mer</placeName>. v<choice><am>ō</am><ex>on</ex></choice> orient mit <placeName type="country">Yndia</placeName> - <lb/>In disem land sind zigen mit grossen oren - <choice><orig>Sup<lb/>til</orig><reg>Suptil<lb/></reg></choice> har trag<choice><am>ē</am><ex>en</ex></choice>. v<choice><am>ō</am><ex>on</ex></choice> welchn zig<choice><am>ē</am><ex>en</ex></choice>harn man den lotzam - <lb/>macht. ouch s<choice><am>ū</am><ex>un</ex></choice>st hat das l<choice><am>ā</am><ex>an</ex></choice>d vil v<choice><am>ō</am><ex>on</ex></choice> seidem gew<choice><am>ā</am><ex>an</ex></choice>t - <lb/>vnd v<choice><am>ō</am><ex>on</ex></choice> edlem gestein vnd perln. yn dis<choice><am>ē</am><ex>em</ex></choice> l<choice><am>ā</am><ex>an</ex></choice>de ist - <lb/>das gebirg <placeName type="mountain">Becius</placeName> vnd <placeName type="mountain">paropanuis</placeName> ser hoch - <choice><orig>all<lb/>so</orig><reg>allso<lb/></reg></choice> das m<choice><am>ā</am><ex>an</ex></choice> meynt es rür dy wolkn. der l<choice><am>ē</am><ex>en</ex></choice>gst tagk - <lb/>yn disen landn ist 14 st<choice><am>ū</am><ex>un</ex></choice>d 1/2 - <lb/> - </p> - </div> - <div type="kapitel"> - <head> - <lb/><placeName type="country">Yndia</placeName> zwischen dem flis <placeName type="river">g<choice><am>ā</am><ex>an</ex></choice>ges</placeName>. - <lb/> - </head> - <p> - <lb/> - <placeName type="country"><hi rend="initial">I</hi>Ndia</placeName> wirt g<choice><am>n̄</am><ex>enan</ex></choice>t v<choice><am>ō</am><ex>on</ex></choice> dem flis <placeName type="river">yndo</placeName> dasdo felt yn das - <lb/><placeName type="ocean">yndisch mer</placeName> mit 7 flissen. vnd leyt czwischen - <placeName type="river"><choice><orig>yn<lb/>do</orig><reg>yndo<lb/></reg></choice></placeName> vnd <placeName type="river">gange</placeName> den zweyenn flissenn. vom obend - <lb/>leit es yn grenicz mit <placeName type="country">arachosia</placeName> vnd <placeName type="country">gerodosia</placeName> in - <lb/>dem 105 grad das ist ym 23 meridia vnd geht <choice><am>ī</am><ex>in</ex></choice> - <lb/>di leng v<choice><am>ō</am><ex>on</ex></choice> occident yn ori<choice><am>ē</am><ex>en</ex></choice>t bis yn den 148 <choice><am>ḡ</am><ex>rad</ex></choice>. das - <lb/>ist yn den 30 meridia v<choice><am>ō</am><ex>on</ex></choice> miternacht wirt es - <choice><orig>besl<lb/>ossen</orig><reg>beslossen<lb/></reg></choice> mit dem gebirg <placeName type="mountain">ymaus</placeName> v<choice><am>ō</am><ex>on</ex></choice> mittag mit d<choice><am>ē</am><ex>em</ex></choice> - <placeName type="ocean"><choice><orig>yn-<lb/>dischen</orig><reg>yndischen<lb/></reg></choice> mor</placeName> v<choice><am>ō</am><ex>on</ex></choice> orient mit dem flis <placeName type="river">g<choice><am>ā</am><ex>an</ex></choice>ge</placeName>. v<choice><am>ō</am><ex>on</ex></choice> - <choice><orig>occi<lb/>dent</orig><reg>occident<lb/></reg></choice> mit dem flis <placeName type="river">yndo</placeName>. v<choice><am>ō</am><ex>on</ex></choice> dyser <placeName type="country">yndia</placeName> schreib<choice><am>ē</am><ex>en</ex></choice> - <pb n="63" facs="Ptolemaeus_text_63"/> - <lb/>Doctores Poeten Philozophi vnd ander vil - <choice><orig>may<lb/>ster</orig><reg>mayster<lb/></reg></choice> w<choice><am>ū</am><ex>un</ex></choice>derliche vnd vngeleuplich d<choice><am>ī</am><ex>in</ex></choice>g. Sagt - <persName><forename><choice><orig>stra<lb/>bo</orig><reg>strabo<lb/></reg></choice></forename></persName> wy daß yn dem <placeName type="ocean">yndische mör</placeName> walfisch sein an - <lb/>der lang hab<choice><am>ē</am><ex>en</ex></choice>d 4 ackerl<choice><am>ā</am><ex>an</ex></choice>d das macht ein hub yn - <lb/>disem mör wechst ror alls hoch vnd groß das m<choice><am>ā</am><ex>an</ex></choice> - <lb/>dau<choice><am>ō</am><ex>on</ex></choice> heuser baven mag vnd v<choice><am>ō</am><ex>on</ex></choice> einem glid eins - <lb/>rores machen dy <rs type="inhabitants">yndi</rs> ein schiffle<choice><am>ī</am><ex>in</ex></choice> doryn drei m<choice><am>ā</am><ex>an</ex></choice> - <lb/>siczen mögen. Ouch fint m<choice><am>ā</am><ex>an</ex></choice> yn diesm mer - <choice><orig>schne<lb/>ken</orig><reg>schneken<lb/></reg></choice> das m<choice><am>ā</am><ex>an</ex></choice> auß eim schneknhauß magk mach<choice><am>ē</am><ex>en</ex></choice> - <lb/>ein ge<choice><am>ā</am><ex>ma</ex></choice>ch fur czway mensch. vnd vil auß den - <rs type="inhabitants"><choice><orig>yn-<lb/>di<choice><am>ā</am><ex>an</ex></choice>en</orig><reg>yndi<choice><am>ā</am><ex>an</ex></choice>en<lb/></reg></choice></rs> won<choice><am>ē</am><ex>en</ex></choice> vnder disen schneken. vnd sind - <choice><orig>swar<lb/>cz</orig><reg>swarcz<lb/></reg></choice> an yrm leib vnd wenig essen si v<choice><am>ō</am><ex>on</ex></choice> fleysch speis - <lb/>S<choice><am>ū</am><ex>un</ex></choice>der s<choice><am>ū</am><ex>un</ex></choice>st hab<choice><am>ē</am><ex>en</ex></choice> si vil edel frucht der erden vnd - <lb/>der bevm dy do vß der maß naturlichs schmack - <lb/>sint. Ouch sagt der vorg<choice><am>n̄</am><ex>enan</ex></choice>t Lerer wy das ein - <choice><orig>recht-<lb/>er</orig><reg>rechter<lb/></reg></choice> <rs type="inhabitants">yndy<choice><am>ā</am><ex>an</ex></choice></rs> außwendig seinem l<choice><am>ā</am><ex>an</ex></choice>d nicht lang leben - <lb/>mag. das ist v<choice><am>ō</am><ex>on</ex></choice> wegen der subtyll vnd nuczlich<choice><am>n̄</am><ex>en</ex></choice> - <lb/>frucht dy si brauch<choice><am>ē</am><ex>en</ex></choice> yn yrem l<choice><am>ā</am><ex>an</ex></choice>d. Und so sie diser - <lb/>frucht nicht hab<choice><am>ē</am><ex>en</ex></choice>. werdn si an irer natur gesweht - <lb/>ouch ist das l<choice><am>ā</am><ex>an</ex></choice>d ser fruchtbar v<choice><am>ō</am><ex>on</ex></choice> mensch<choice><am>ē</am><ex>en</ex></choice> wan - <choice><orig>wo<lb/>ru<choice><am>m̄</am><ex>mb</ex></choice></orig><reg>woru<choice><am>m̄</am><ex>mb</ex></choice><lb/></reg></choice> so eyn <choice><sic>frenlin</sic><corr>freulin</corr></choice> 5 yor alt ist si gebirt si kinder yn - <lb/>disem l<choice><am>ā</am><ex>an</ex></choice>d ist der <persName><roleName>briester</roleName> <forename>Johan</forename></persName> alls yn <placeName type="continent">ewropa</placeName>. - <lb/>der <persName><roleName>Bapst</roleName></persName>. vnd ist gesalbt v<choice><am>ō</am><ex>on</ex></choice> <persName><roleName>Sancto</roleName> <forename>Thoma</forename></persName>. v<choice><am>n̄</am><ex>nd</ex></choice> - <lb/>all sein volk ist kristen vnd hat vnder ym 500000 - <lb/>gemaurther stet. vnd hat vnder ym 24 - <choice><orig>kungkrei<lb/>ch</orig><reg>kungkreich<lb/></reg></choice> vnd so yndert ein l<choice><am>ā</am><ex>an</ex></choice>d ein kung erwelt hat So - <lb/>mus er gekront werden vom <persName><roleName>priester</roleName> <forename>Johan</forename></persName> - <pb n="64" facs="Ptolemaeus_text_64"/> - <lb/>vnd diser <persName><roleName>Briester</roleName> <forename>Johan</forename></persName> seczt auff dy kron d<choice><am>ē</am><ex>en</ex></choice> - <lb/>kung mit sein<choice><am>ē</am><ex>en</ex></choice> fussen. zu bedeuten das si ym - <choice><orig>vnder<lb/>tenig</orig><reg>vndertenig<lb/></reg></choice> sollen wesen dy weil si leben <persName><forename><choice><sic>pliuius</sic><corr>plinius</corr></choice></forename></persName> vnd - <lb/><persName><forename>solinus</forename></persName> dise baide maister sag<choice><am>ē</am><ex>en</ex></choice> wye das vnd - <lb/>den pergen <placeName type="mountain">thauri</placeName> der pfeffer wechst yn mossen wye. - <lb/>Jachandilber zwischen welch<choice><am>ē</am><ex>en</ex></choice> stauden man vint - <lb/>lang<choice><am>ē</am><ex>en</ex></choice> pfeffer. vnder disen stauden sein vil der - <choice><orig>schl<choice><am>ā</am><ex>an</ex></choice><lb/>gen</orig><reg>schl<choice><am>ā</am><ex>an</ex></choice>gen<lb/></reg></choice> dy do fligell hab<choice><am>ē</am><ex>en</ex></choice> vnd dy m<choice><am>ē</am><ex>en</ex></choice>schen ser - <choice><orig>leidig<lb/>en</orig><reg>leidigen<lb/></reg></choice> v<choice><am>ō</am><ex>on</ex></choice> deßweg<choice><am>ē</am><ex>en</ex></choice> dy <rs type="inhabitants">yndi</rs> mussen fewr vndt rauch - <lb/>machen dorvor dyse wormer flieh<choice><am>ē</am><ex>en</ex></choice> also Clauben - <lb/>si den pfeffer. <space/>Sagt <persName><forename>Anßhelmus</forename></persName> de ymagine - <lb/>mundi. das in <placeName type="country">yndia</placeName> sind groß Eleph<choice><am>ā</am><ex>an</ex></choice>ten czu dem - <lb/>streyt wol tugend ouch hatt das landt eyn bestia - <lb/>mocer<choice><am>ō</am><ex>on</ex></choice> genat Sittich vögl vnd ander selczame tyr - <lb/>vil schöner edler steyn tregt diß land mit - <choice><orig>vberflu<lb/>ssikeyt</orig><reg>vberflussikeyt<lb/></reg></choice> des golds man vint Carfunckel. dyamantt - <lb/>berilln vnd grosse berlen <space/>Jtem man vint - <choice><orig>omess<lb/>en</orig><reg>omessen<lb/></reg></choice> als dy schoff groß vnd werd<choice><am>ē</am><ex>en</ex></choice> gespeyst mit - <choice><orig>gul<lb/>den.</orig><reg>gulden.<lb/></reg></choice> Jtem öln 300 schuch l<choice><am>ā</am><ex>an</ex></choice>g. Jtem hohe bevm - <lb/>dy kein armbrust mit schuss<choice><am>ē</am><ex>en</ex></choice> mag erlang<choice><am>ē</am><ex>en</ex></choice>. Jtem - <lb/>der <persName><roleName>kung</roleName> <forename>botras</forename></persName> yn <placeName type="country">yndia</placeName> der vormagk alczeyt. - <lb/>30000 <choice><sic>wolgezvap<choice><am>ē</am><ex>en</ex></choice>ter</sic><corr>wolgewap<choice><am>ē</am><ex>en</ex></choice>ter</corr></choice> Elephant<choice><am>ē</am><ex>en</ex></choice> zu strit. S<choice><am>ū</am><ex>un</ex></choice>st - <lb/>eyn yczlicher kung dorynn vermag yr 8000 Es ist - <lb/>ein volk dy do g<choice><am>n̄</am><ex>enan</ex></choice>t sein <rs type="inhabitants">brachmāni</rs> vnd haben ir - <lb/>r<choice><am>ē</am><ex>en</ex></choice>t v<choice><am>n̄</am><ex>nd</ex></choice> zinß. das si nichcz anders t<choice><am>ū</am><ex>un</ex></choice> wan das si got - <choice><orig>pit<lb/>en</orig><reg>piten<lb/></reg></choice> v<choice><am>m̄</am><ex>mb</ex></choice> kelt w<choice><am>ā</am><ex>an</ex></choice> woru<choice><am>m̄</am><ex>mb</ex></choice> di groß <choice><sic>gicz</sic><corr>hicz</corr></choice> der s<choice><am>ō</am><ex>on</ex></choice>ne zwingtt - <lb/>si zumol ser w<choice><am>ā</am><ex>an</ex></choice> si lig<choice><am>ē</am><ex>en</ex></choice> noch dem weg der sonnen - <pb n="65" facs="Ptolemaeus_text_65"/> - <lb/>Jtem alle schiffleuthe yn <placeName type="country">yndia</placeName> sech<choice><am>ē</am><ex>en</ex></choice>d nicht meh - <lb/>wann eyn<choice><am>ē</am><ex>en</ex></choice> stern. genant Canapos dornach sy sich - <lb/>richt<choice><am>n̄</am><ex>en</ex></choice>. vnd hab<choice><am>ē</am><ex>en</ex></choice>d grosse uinsternis des mones - <lb/>vnd der sonn<choice><am>ē</am><ex>en</ex></choice> so si eclipsiret werd<choice><am>n̄</am><ex>en</ex></choice> das ist - <choice><orig>besthe<lb/>tiget.</orig><reg>besthetiget.<lb/></reg></choice> disse <placeName type="country">yndia</placeName> ym mittil hat si tag awffs lengste - <lb/><choice><orig>31</orig><reg>13</reg></choice> stunden vnd 1/2 - </p> - </div> - <div type="kapitel"> - <head> - <lb/><placeName type="country">Jndia</placeName> extra <placeName type="river">gangem</placeName> awssen - <lb/>dem flis <placeName type="river">gange</placeName> <space/> 1 1 - </head> - <p> - <lb/> - <hi rend="initial">D</hi>Jsse andre <placeName type="country">yndia</placeName> grantzet kegen occident - <lb/>mit der vorgedachten. gegen mitternacht - <lb/>hat sy das gepirg <placeName type="mountain">ymaus alias</placeName>. <placeName type="mountain">emody</placeName> - <choice><orig>pe<lb/>rgk.</orig><reg>pergk.<lb/></reg></choice> <placeName type="mountain">sericy</placeName> pergk. s<placeName type="mountain">em<choice><am>ā</am><ex>an</ex></choice>tini</placeName> perge vnd vntter diss<choice><am>n̄</am><ex>en</ex></choice> - <lb/>pergen allen wechßt pheffer yn mass als - <choice><orig>vorgesa<lb/>gt</orig><reg>vorgesagt<lb/></reg></choice> ist. von orientt endet disses land vnnßer - <choice><orig>bek<choice><am>ā</am><ex>ann</ex></choice><lb/>ten</orig><reg>bek<choice><am>ā</am><ex>ann</ex></choice>ten<lb/></reg></choice> werld. vom mittāge wirt dyß l<choice><am>ā</am><ex>an</ex></choice>d vmbgeben - <lb/>mit dem <placeName type="ocean">yndischen mehre</placeName> yn welch<choice><am>m̄</am><ex>em</ex></choice> mer vil - <choice><orig>yn<lb/>suln</orig><reg>ynsuln<lb/></reg></choice> sein selzcam volk habend nach der natur - <choice><orig>vn<lb/>tter</orig><reg>vntter<lb/></reg></choice> welchen ynsulen sind etliche gnandt. <placeName type="island">maiorica</placeName> - <lb/>vnd ynsula <placeName type="island">guttis glucks</placeName> awff latin <placeName type="island">guttisbone fortune</placeName> - <lb/>pey dyssenn ynsulnn tar kein schiff nohe hynpey - <lb/>gehn das do eyßere negel hatt. wenn worumb das - <lb/>selbe mer hat yn yme g<choice><am>ā</am><ex>an</ex></choice>tze felße des steines - <persName><forename><choice><orig>he<lb/>rculei</orig><reg>herculei<lb/></reg></choice></forename></persName> der an sich zcewcht das yssen dorumbe sint - <lb/>yre schiff eiss<choice><am>ē</am><ex>en</ex></choice> darbende dis land anhebend vom - <lb/>ocid<choice><am>ē</am><ex>en</ex></choice>t 140 grad gehet hyn bys zw dem ende der - <lb/>werld das ist biß yn den 180 grad das ist der letzte - <lb/>meridian. das anheben dis l<choice><am>ā</am><ex>an</ex></choice>des von occid<choice><am>ē</am><ex>en</ex></choice>t ligt - <lb/>in dem 29 meridi<choice><am>ā</am><ex>an</ex></choice>. das grosse flis yn <placeName type="country">yndia</placeName> extra - <pb n="66" facs="Ptolemaeus_text_66"/> - <lb/><placeName type="river">gangem</placeName> ist gnant damas entspringend aüß denn - <lb/>pergen <placeName type="mountain">emody</placeName> vnd <placeName type="mountain">sericy</placeName> yn disem land ligt dy stat - <lb/><placeName type="city">kathigara</placeName> am ende der werld. yn disser <placeName type="city">kathigara</placeName> - <lb/>vnd <choice><sic>vmbligeuden</sic><corr>vmbligenden</corr></choice> land dy habend volk yn. yn dy - <lb/>do schwentze habend gelich als dy yn <placeName type="city">lisbon</placeName> <placeName type="river">ader</placeName> - <lb/> <placeName type="city"><choice><sic>eugilland</sic><corr>engilland</corr></choice></placeName> disse czwen stete ligent von enander 25 306 - <lb/>meilen welsch. der lengste tag yn disser <placeName type="city">yndia</placeName> ist - <lb/>13 stunden vnd 1/2 - </p> - </div> - <div type="kapitel"> - <head> - <lb/><placeName type="island">Taprobana</placeName> ynsula ym - <lb/> <placeName type="ocean">yndischen mehre</placeName> - <lb/> - </head> - <p> - <hi rend="initial">D</hi>J edle <placeName type="island">taprobana</placeName> avff allen seyten - <choice><orig>vmb<lb/>stossen</orig><reg>vmbflossen<lb/></reg></choice> mit mehr treitt vil edler ste<choice><am>ī</am><ex>in</ex></choice>e. - <choice><orig>sch<lb/>on</orig><reg>schon<lb/></reg></choice> gold grosse thier. alte m<choice><am>ē</am><ex>en</ex></choice>schen Als - <lb/><persName><forename>strabo</forename></persName> schreibt das die mensche yn <placeName type="island">Taprobana</placeName> - <choice><orig>yn<lb/>sula</orig><reg>ynsula<lb/></reg></choice> leb<choice><am>ē</am><ex>en</ex></choice>t alz lang das si bytt<choice><am>n̄</am><ex>en</ex></choice> yre kinder das si - <choice><orig>geto-<lb/>tet</orig><reg>getotet<lb/></reg></choice> suld<choice><am>n̄</am><ex>en</ex></choice> werd<choice><am>ē</am><ex>en</ex></choice> aber awß der ynsul gefüret - <choice><orig>wer-<lb/>den</orig><reg>werden<lb/></reg></choice> awff das si mocht<choice><am>ē</am><ex>en</ex></choice>d sterb<choice><am>n̄</am><ex>en</ex></choice>. disse mensch<choice><am>ē</am><ex>en</ex></choice> sein - <lb/>sere groß alz dy rissen vorczeiten gewest sein. yn - <lb/>diser ynsel fellt kein plat von keinem bowme yn - <lb/>keinerley weiße Avß disser ynsul kompt vil - <choice><orig>spe-<lb/>cerey</orig><reg>specerey<lb/></reg></choice> Als pheffer ynbir mußkat<choice><am>m̄</am><ex>um</ex></choice> Thadiln v<choice><am>n̄</am><ex>nd</ex></choice> das - <lb/>holtz ebanus das yn dem fewer nicht vorprent - <choice><orig>su-<lb/>nder</orig><reg>sunder<lb/></reg></choice> nur gluet vnd herter wirt. dysse specerey v<choice><am>n̄</am><ex>nd</ex></choice> - <lb/>ander kowffm<choice><am>ā</am><ex>an</ex></choice>schatz pring<choice><am>n̄</am><ex>en</ex></choice> sy awff das <placeName type="ocean">yndischē - <lb/>mehre</placeName> zw d<choice><am>ē</am><ex>em</ex></choice> flis <placeName type="river">gange</placeName> v<choice><am>n̄</am><ex>nd</ex></choice> widervmb komenn - <lb/>dy indischen kowfflewthe ouch awff dissem - <choice><orig>wa-<lb/>ssem</orig><reg>wassern<lb/></reg></choice> entkegen setzende ire guttur gegen dissenn - <lb/>awß <placeName type="island">taprobana</placeName> v<choice><am>n̄</am><ex>nd</ex></choice> reden nicht mitenander. - <choice><orig>sund</orig><reg>sund</reg></choice><pb n="67" facs="Ptolemaeus_text_67"/><choice><orig>der</orig><reg>der</reg></choice> do ware hat yn synem schiffe. zw stotzen - <choice><orig>ad-<lb/>der</orig><reg>adder<lb/></reg></choice> weschßiln vmb andere war. der steket awff - <lb/>ein fenleyn vnd gibt yn bedewtenn daß er daz - <lb/>seine stotzen will an ein ander schiff dornach - <choice><orig>ni-<lb/>mpt</orig><reg>nimpt<lb/></reg></choice> einer awß zween kawffleuten das ander fenlin - <lb/>vnnd legt ys vff seine war. So der ander sein - <choice><orig>zce<lb/>ichen</orig><reg>zceichen<lb/></reg></choice> sucht vnd <choice><orig>vndet</orig><reg>vindet</reg></choice> awff einer ware dy ym - <choice><orig>ge<lb/>flt</orig><reg>gefelt<lb/></reg></choice> so nimt er sein zceichen vnd legt yß wider awff - <lb/>sein war. zw bedewt<choice><am>ē</am><ex>en</ex></choice> das ys yn beden gefelt - <choice><orig>dor<lb/>nach</orig><reg>dornach<lb/></reg></choice> niht der kawffm<choice><am>ā</am><ex>an</ex></choice> awß <placeName type="country">yndia</placeName> das schiff awß - <lb/><placeName type="island">Taprobana</placeName> vnd der ander widervmb also - <choice><orig>kaw-<lb/>ffschlag<choice><am>n̄</am><ex>nd</ex></choice></orig><reg>kawffschlage<choice><am>n̄</am><ex>nd</ex></choice><lb/></reg></choice> sy mit enander nicht redendt wen - <choice><orig>wor-<lb/>rumb</orig><reg>worrumb<lb/></reg></choice> disse awß <placeName type="island">taprobann</placeName> dy hab<choice><am>ē</am><ex>en</ex></choice>d kein - <choice><orig><choice><sic>gem-<lb/>eiuschafft</sic><corr>gem-<lb/>einschafft</corr></choice></orig><reg><choice><corr>gemeinschafft<lb/></corr><sic>gemeiuschafft<lb/></sic></choice></reg></choice> yn sproch mit yndirt ein ander naci<choice><am>ō</am><ex>on</ex></choice> - <choice><orig>vnt<lb/>er</orig><reg>vnter<lb/></reg></choice> der s<choice><am>ō</am><ex>on</ex></choice>nen. Mit dissen vnd ander dingen sein - <lb/>genug gesagt v<choice><am>ō</am><ex>on</ex></choice> ynhalt disser vnßer bek<choice><am>ā</am><ex>an</ex></choice>ten - <choice><orig>we<lb/>rld</orig><reg>werld<lb/></reg></choice> was volk waß sitt<choice><am>ē</am><ex>ev</ex></choice> sy hab<choice><am>ē</am><ex>en</ex></choice> vntter welchenn - <lb/>teil der werld si geleg<choice><am>n̄</am><ex>en</ex></choice> sein vnd mit was mer - <choice><orig>v<choice><am>n̄</am><ex>nd</ex></choice><lb/>t</orig><reg>v<choice><am>n̄</am><ex>nd</ex></choice>t<lb/></reg></choice> gepirg sy vmgeben sein vnd waß flisse awß den - <lb/>selb<choice><am>n̄</am><ex>en</ex></choice> entspringgen. das m<choice><am>ā</am><ex>an</ex></choice> d<choice><am>ā</am><ex>an</ex></choice>n yn der - <choice><orig><choice><corr>offtgeda-<lb/>chten</corr><sic>offegedo-<lb/>chten</sic></choice></orig><reg><choice><corr>offtgedachten<lb/></corr><sic>offegedochten<lb/></sic></choice></reg></choice> figur wol vinde magk yn einer <choice><orig>gemenn</orig><reg>gemeinn</reg></choice> so - <lb/>man recht merket awff dy grad vom orient yn - <choice><orig>oc-<lb/>cident</orig><reg>occident<lb/></reg></choice> awff dy meridian. awff dy paralelln vnd - <lb/>andere nottorfftige dingk dorzv dien<choice><am>ē</am><ex>en</ex></choice>d so <choice><orig>sid</orig><reg>sind</reg></choice> - <lb/>dy <choice><am>ḡ</am><ex>gra</ex></choice>d gesaczet vntten an der figur vnßer bek<choice><am>ā</am><ex>an</ex></choice>ten - <lb/>werld v<choice><am>ō</am><ex>on</ex></choice> occid<choice><am>ē</am><ex>en</ex></choice>t gezcallt 180 dorauß d<choice><am>ā</am><ex>an</ex></choice> alle - <choice><orig>fig-<lb/>ur<choice><am>ē</am><ex>en</ex></choice></orig><reg>figur<choice><am>ē</am><ex>en</ex></choice><lb/></reg></choice> der l<choice><am>ā</am><ex>an</ex></choice>de off<choice><am>ē</am><ex>en</ex></choice>bar werden so man zceucht ynn - <lb/>dem meridi<choice><am>ā</am><ex>an</ex></choice> avff geg<choice><am>ē</am><ex>en</ex></choice> mitternacht dar <choice><corr>ynne</corr><sic>yune</sic></choice> d<choice><am>ā</am><ex>an</ex></choice> - <pb n="68" facs="Ptolemaeus_text_68"/> - <lb/>ein stat eyn l<choice><am>ā</am><ex>an</ex></choice>d bezceichnet wirt. dysse alle - <choice><orig>obges-<lb/>chrib<choice><am>ē</am><ex>en</ex></choice></orig><reg>obgeschrib<choice><am>ē</am><ex>en</ex></choice><lb/></reg></choice> d<choice><am>ī</am><ex>in</ex></choice>g werden dem leßer wol <choice><orig>bekat</orig><reg>bekant</reg></choice> werden - <lb/>So er avffmerk<choice><am>ē</am><ex>en</ex></choice> hat awff ynhalt der figuren <choice><am>ī</am><ex>in</ex></choice> - <choice><orig>dy-<lb/>ssem</orig><reg>dyssem<lb/></reg></choice> buchlin - <lb/> - <lb/> - <lb/> - </p> - <p> - <hi rend="initial">D</hi>Js gantze dritte teyl der werld <placeName type="continent">azia</placeName> - <lb/>genanth hat yn prawch<choice><am>ū</am><ex>un</ex></choice>ge. dy - <choice><orig>Calid-<lb/>isch<choice><am>ē</am><ex>en</ex></choice></orig><reg>Calidisch<choice><am>ē</am><ex>en</ex></choice><lb/></reg></choice> . arabischen v<choice><am>n̄</am><ex>nd</ex></choice> <choice><sic>yndischeu</sic><corr>yndischen</corr></choice> litern - <lb/>vnd e<choice><am>ī</am><ex>in</ex></choice> teil auch hebreischen. Jtem - <lb/>der husse heze pravcht yn allen <choice><sic>siuen</sic><corr>sinen</corr></choice> l<choice><am>ā</am><ex>an</ex></choice>den der - <lb/>caldeyschen <term xml:id="littera3" ref="#Buchstaben"><choice><sic>litteru</sic><corr>littern</corr></choice></term> adder <q corresp="#littera3">puchstaben</q> yn m<choice><am>ā</am><ex>an</ex></choice>gerlei - <lb/>wiße mit schriben vnd avß sprechen Jtem der - <lb/><persName><roleName>Soldan</roleName></persName> hat ym vbunge am meisten teil di - <choice><orig>arabi-<lb/>schen</orig><reg>arabischen<lb/></reg></choice> littern vormischt mit caldeischen vnd - <choice><orig>he-<lb/>breischen</orig><reg>hebreischen<lb/></reg></choice> Jtem der <persName><roleName>prist</roleName> <forename>iohan</forename></persName> hath yn sinem - <choice><orig>l<choice><am>ā</am><ex>an</ex></choice><lb/>de</orig><reg>l<choice><am>ā</am><ex>an</ex></choice>de<lb/></reg></choice> vbunge der yndischen littern yn schreib<choice><am>ē</am><ex>en</ex></choice> vnd - <lb/>außsprechen Sunst in der gemein ist turcksch. - <choice><orig>ar<lb/>menisch</orig><reg>armenisch<lb/></reg></choice> . indisch vnd schitthiß sprach yn - <choice><orig>gew<choice><am>ō</am><ex>on</ex></choice><lb/>heit</orig><reg>gew<choice><am>ō</am><ex>on</ex></choice>heit<lb/></reg></choice> der m<choice><am>ē</am><ex>en</ex></choice>schen in <placeName type="continent">azia</placeName> vil selczamer sitten der - <lb/>m<choice><am>ē</am><ex>en</ex></choice>schen vnd gewonheithen sind in der werld <choice><am>ā</am><ex>an</ex></choice> - <lb/>sprach tracht vnd geschicklichkeit der mensch<choice><am>ē</am><ex>en</ex></choice> - <lb/>dy yn zcvgeeigenth sein von natur mit grossenn - <lb/>vnttirscheid van enander geteilet nach gestalt der - <lb/>corper yres leichnams. Als do sein Satiri dy do - <lb/><choice><sic>habēv</sic><corr>habēn</corr></choice> ein horn an der stirnen. pignei sein an - <choice><orig>ir-<lb/>er</orig><reg>irer<lb/></reg></choice> statur eins elpogenn hoch <choice><sic>cmocephali</sic><corr>cenocephali</corr></choice> haben - <lb/>h<choice><am>ū</am><ex>un</ex></choice>dis hewpter Blemny yn libia haben dy oug<choice><am>ē</am><ex>en</ex></choice> - <pb n="69" facs="Ptolemaeus_text_69"/> - <lb/>An der prust vnd kein hewbt. pigmei einen fus - <lb/><choice><orig>habed</orig><reg>habend</reg></choice> streiten mit den cranchen Masagitte so - <choice><orig><choice><sic>ei<lb/>uer</sic><corr>ei<lb/>ner</corr></choice></orig><reg><choice><corr>einer<lb/></corr><sic>eiuer<lb/></sic></choice></reg></choice>crank wirt so fressen si in. avff das er <choice><sic>vou</sic><corr>von</corr></choice> den - <lb/>wurmen der erden nicht fressen werd. padei - <choice><orig>ha<lb/>ben</orig><reg>haben<lb/></reg></choice> 4 weibir so der man stirbt mus sich der - <choice><orig>lip-<lb/>ste</orig><reg>lipste<lb/></reg></choice> mit im vorprennen <rs type="inhabitants">Arabes</rs> halten kein - <choice><orig>sch-<lb/>wein.</orig><reg>schwein.<lb/></reg></choice> <rs type="inhabitants">Schitti asperi</rs> di essen m<choice><am>ē</am><ex>en</ex></choice>schen fleisch. vnd - <lb/>so si einen erschlagen hab<choice><am>ē</am><ex>en</ex></choice> trincken si das plut - <lb/>auß seinen wunden. <rs type="inhabitants">yndiani</rs> haben <choice><sic>schöue</sic><corr>schöne</corr></choice> <choice><orig>eiber</orig><reg>weiber</reg></choice> - <lb/>vnd <choice><orig>groe</orig><reg>große</reg></choice> kinder. Disse alle nacion vnd anderr - <lb/>nacion <rs type="inhabitants">Cristen</rs> vnd <rs type="inhabitants">heiden</rs> <rs type="inhabitants"><choice><sic>tnrken</sic><corr>turken</corr></choice></rs> vnd <rs type="inhabitants">iuden</rs> - <lb/><rs type="inhabitants">rewßen</rs> vnd <rs type="inhabitants">criehen</rs>. Di alten vnd iungen sein - <choice><orig>phli<lb/>chtig</orig><reg>phlichtig<lb/></reg></choice> zw loben <choice><sic>deu</sic><corr>den</corr></choice> scheppir aller dinge dem do - <lb/>ere vnd dangksagung sei in ewigkeith. - </p> - </div> - <figure facs="Ptolemaeus_karte"></figure> - </body> - </text> -</TEI> \ No newline at end of file
--- a/software/mpdl-services/mpiwg-mpdl-xml/examples/generateId.xsl Wed Dec 14 13:57:09 2011 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,34 +0,0 @@ -<?xml version="1.0"?> - -<xsl:stylesheet version="2.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform"> - -<xsl:output method="xml" encoding="utf-8"/> - -<xsl:template match="@*|node()"> - <xsl:copy> - <xsl:apply-templates select="@*|node()"/> - </xsl:copy> -</xsl:template> - -<xsl:template match="*:s|*:head"> - <xsl:variable name="elemName" select="name()"/> - <xsl:variable name="docPos"> - <xsl:choose> - <xsl:when test="$elemName = 's'"> - <xsl:value-of select="count(./preceding::*:s) + 1"/> - </xsl:when> - <xsl:when test="$elemName = 'head'"> - <xsl:value-of select="count(./preceding::*:head) + 1"/> - </xsl:when> - <xsl:otherwise></xsl:otherwise> - </xsl:choose> - </xsl:variable> - <xsl:copy> - <xsl:attribute name="xml:id"> - <xsl:value-of select="concat($elemName, $docPos)"/> - </xsl:attribute> - <xsl:apply-templates select="@*|node()"/> - </xsl:copy> -</xsl:template> - -</xsl:stylesheet>
Binary file software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/.DS_Store has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/util/StringUtils.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,526 @@ +package de.mpg.mpiwg.berlin.mpdl.util; + +import java.io.UnsupportedEncodingException; +import java.net.URLEncoder; +import java.text.CharacterIterator; +import java.text.StringCharacterIterator; +import java.util.Hashtable; +import java.util.regex.Matcher; +import java.util.regex.Pattern; + +public class StringUtils { + private static Pattern xmlEntitiesPattern; + private static Pattern nlTabBlankPattern; + private static Hashtable<String, String> xmlEntitiesReplacements; + static { + xmlEntitiesPattern = Pattern.compile("&|<|>|"|'"); + nlTabBlankPattern = Pattern.compile("\n|[ \t]+"); + xmlEntitiesReplacements = new Hashtable<String, String>(); + xmlEntitiesReplacements.put("&", "&"); + xmlEntitiesReplacements.put("<", "<"); + xmlEntitiesReplacements.put(">", ">"); + xmlEntitiesReplacements.put(""", "\""); + xmlEntitiesReplacements.put("'", "'"); + } + + /** + * Puts a zwsp between two ideographic characters (e.g. in CJK Unified Ideographs) + * @param str + * @return + */ + public static String zwsp(String str) { + // based on Unicode 3.2 + String ideographic = "[\u3300-\u33ff\u3400-\u4dbf\u4e00-\u9fff\uf900-\ufaff]"; + String regex = "(" + ideographic + ")(" + ideographic + ")"; + String retStr = str.replaceAll(regex, "$1\u200b$2"); + retStr = retStr.replaceAll(regex, "$1\u200b$2"); + return retStr; + } + + public static String deleteSpecialXmlEntities(String inputStr) { + inputStr = inputStr.replaceAll("<", ""); + inputStr = inputStr.replaceAll(">", ""); + inputStr = inputStr.replaceAll("&lt;", ""); + inputStr = inputStr.replaceAll("&gt;", ""); + return inputStr; + } + + public static String resolveXmlEntities(String inputStr) { + StringBuffer sb = new StringBuffer(inputStr.length()); + Matcher m = xmlEntitiesPattern.matcher(inputStr); + int end = 0; + while (m.find()) { + String wordToReplace = m.group(); + String replacement = xmlEntitiesReplacements.get(wordToReplace); + m.appendReplacement(sb, replacement); + end = m.end(); + } + sb.append(inputStr.substring(end)); + return sb.toString(); + } + + public static String removeNlTabBlanks(String inputStr) { + StringBuffer sb = new StringBuffer(inputStr.length()); + Matcher m = nlTabBlankPattern.matcher(inputStr); + int end = 0; + while (m.find()) { + String wordToReplace = m.group(); + if (wordToReplace.contains(" ") || wordToReplace.contains("\t")) + m.appendReplacement(sb, " "); // make the blanks and Tabs to one blank + end = m.end(); + } + sb.append(inputStr.substring(end)); + return sb.toString(); + } + + private static String resolveXmlEntities_OldSlowMethod(String inputStr) { + inputStr = inputStr.replaceAll("&", "&"); + inputStr = inputStr.replaceAll("<", "<"); + inputStr = inputStr.replaceAll(">", ">"); + inputStr = inputStr.replaceAll(""", "\""); + inputStr = inputStr.replaceAll("'", "'"); + return inputStr; + } + + public static String deresolveXmlEntities(String inputStr) { + StringBuilder buf = new StringBuilder(); + for (int i = 0; i < inputStr.length(); i++) { + char c = inputStr.charAt(i); + String replace = new String(); + switch (c) { + case '&': replace = "&"; break; + case '<': replace = "<"; break; + case '>': replace = ">"; break; + case '"': replace = """; break; + // case '\'': replace = "'"; break; // causes problems in XmlTokenizerContentHandler + default: replace += c; break; + } + buf.append(replace); + } + return buf.toString(); + } + + /** + * Escape characters for text appearing in HTML markup. + * + * <P>This method exists as a defence against Cross Site Scripting (XSS) hacks. + * The idea is to neutralize control characters commonly used by scripts, such that + * they will not be executed by the browser. This is done by replacing the control + * characters with their escaped equivalents. + * See {@link hirondelle.web4j.security.SafeText} as well. + * + * <P>The following characters are replaced with corresponding + * HTML character entities : + * <table border='1' cellpadding='3' cellspacing='0'> + * <tr><th> Character </th><th>Replacement</th></tr> + * <tr><td> < </td><td> < </td></tr> + * <tr><td> > </td><td> > </td></tr> + * <tr><td> & </td><td> & </td></tr> + * <tr><td> " </td><td> "</td></tr> + * <tr><td> \t </td><td> 	</td></tr> + * <tr><td> ! </td><td> !</td></tr> + * <tr><td> # </td><td> #</td></tr> + * <tr><td> $ </td><td> $</td></tr> + * <tr><td> % </td><td> %</td></tr> + * <tr><td> ' </td><td> '</td></tr> + * <tr><td> ( </td><td> (</td></tr> + * <tr><td> ) </td><td> )</td></tr> + * <tr><td> * </td><td> *</td></tr> + * <tr><td> + </td><td> + </td></tr> + * <tr><td> , </td><td> , </td></tr> + * <tr><td> - </td><td> - </td></tr> + * <tr><td> . </td><td> . </td></tr> + * <tr><td> / </td><td> / </td></tr> + * <tr><td> : </td><td> :</td></tr> + * <tr><td> ; </td><td> ;</td></tr> + * <tr><td> = </td><td> =</td></tr> + * <tr><td> ? </td><td> ?</td></tr> + * <tr><td> @ </td><td> @</td></tr> + * <tr><td> [ </td><td> [</td></tr> + * <tr><td> \ </td><td> \</td></tr> + * <tr><td> ] </td><td> ]</td></tr> + * <tr><td> ^ </td><td> ^</td></tr> + * <tr><td> _ </td><td> _</td></tr> + * <tr><td> ` </td><td> `</td></tr> + * <tr><td> { </td><td> {</td></tr> + * <tr><td> | </td><td> |</td></tr> + * <tr><td> } </td><td> }</td></tr> + * <tr><td> ~ </td><td> ~</td></tr> + * </table> + * + * <P>Note that JSTL's {@code <c:out>} escapes <em>only the first + * five</em> of the above characters. + */ + public static String forHTML(String aText){ + final StringBuilder result = new StringBuilder(); + final StringCharacterIterator iterator = new StringCharacterIterator(aText); + char character = iterator.current(); + while (character != CharacterIterator.DONE ){ + if (character == '<') { + result.append("<"); + } + else if (character == '>') { + result.append(">"); + } + else if (character == '&') { + result.append("&"); + } + else if (character == '\"') { + result.append("""); + } + else if (character == '\t') { + addCharEntity(9, result); + } + else if (character == '!') { + addCharEntity(33, result); + } + else if (character == '#') { + addCharEntity(35, result); + } + else if (character == '$') { + addCharEntity(36, result); + } + else if (character == '%') { + addCharEntity(37, result); + } + else if (character == '\'') { + addCharEntity(39, result); + } + else if (character == '(') { + addCharEntity(40, result); + } + else if (character == ')') { + addCharEntity(41, result); + } + else if (character == '*') { + addCharEntity(42, result); + } + else if (character == '+') { + addCharEntity(43, result); + } + else if (character == ',') { + addCharEntity(44, result); + } + else if (character == '-') { + addCharEntity(45, result); + } + else if (character == '.') { + addCharEntity(46, result); + } + else if (character == '/') { + addCharEntity(47, result); + } + else if (character == ':') { + addCharEntity(58, result); + } + else if (character == ';') { + addCharEntity(59, result); + } + else if (character == '=') { + addCharEntity(61, result); + } + else if (character == '?') { + addCharEntity(63, result); + } + else if (character == '@') { + addCharEntity(64, result); + } + else if (character == '[') { + addCharEntity(91, result); + } + else if (character == '\\') { + addCharEntity(92, result); + } + else if (character == ']') { + addCharEntity(93, result); + } + else if (character == '^') { + addCharEntity(94, result); + } + else if (character == '_') { + addCharEntity(95, result); + } + else if (character == '`') { + addCharEntity(96, result); + } + else if (character == '{') { + addCharEntity(123, result); + } + else if (character == '|') { + addCharEntity(124, result); + } + else if (character == '}') { + addCharEntity(125, result); + } + else if (character == '~') { + addCharEntity(126, result); + } + else { + //the char is not a special one + //add it to the result as is + result.append(character); + } + character = iterator.next(); + } + return result.toString(); + } + + + /** + * Escape all ampersand characters in a URL. + * + * <P>Replaces all <tt>'&'</tt> characters with <tt>'&'</tt>. + * + *<P>An ampersand character may appear in the query string of a URL. + * The ampersand character is indeed valid in a URL. + * <em>However, URLs usually appear as an <tt>HREF</tt> attribute, and + * such attributes have the additional constraint that ampersands + * must be escaped.</em> + * + * <P>The JSTL <c:url> tag does indeed perform proper URL encoding of + * query parameters. But it does not, in general, produce text which + * is valid as an <tt>HREF</tt> attribute, simply because it does + * not escape the ampersand character. This is a nuisance when + * multiple query parameters appear in the URL, since it requires a little + * extra work. + */ + public static String forHrefAmpersand(String aURL){ + return aURL.replace("&", "&"); + } + + /** + * Synonym for <tt>URLEncoder.encode(String, "UTF-8")</tt>. + * + * <P>Used to ensure that HTTP query strings are in proper form, by escaping + * special characters such as spaces. + * + * <P>It is important to note that if a query string appears in an <tt>HREF</tt> + * attribute, then there are two issues - ensuring the query string is valid HTTP + * (it is URL-encoded), and ensuring it is valid HTML (ensuring the + * ampersand is escaped). + */ + public static String forURL(String aURLFragment){ + String result = null; + try { + result = URLEncoder.encode(aURLFragment, "UTF-8"); + } + catch (UnsupportedEncodingException ex){ + throw new RuntimeException("UTF-8 not supported", ex); + } + return result; + } + + /** + * Escape characters for text appearing as XML data, between tags. + * + * <P>The following characters are replaced with corresponding character entities : + * <table border='1' cellpadding='3' cellspacing='0'> + * <tr><th> Character </th><th> Encoding </th></tr> + * <tr><td> < </td><td> < </td></tr> + * <tr><td> > </td><td> > </td></tr> + * <tr><td> & </td><td> & </td></tr> + * <tr><td> " </td><td> "</td></tr> + * <tr><td> ' </td><td> '</td></tr> + * </table> + * + * <P>Note that JSTL's {@code <c:out>} escapes the exact same set of + * characters as this method. <span class='highlight'>That is, {@code <c:out>} + * is good for escaping to produce valid XML, but not for producing safe + * HTML.</span> + */ + public static String forXML(String aText){ + final StringBuilder result = new StringBuilder(); + final StringCharacterIterator iterator = new StringCharacterIterator(aText); + char character = iterator.current(); + while (character != CharacterIterator.DONE ){ + if (character == '<') { + result.append("<"); + } else if (character == '>') { + result.append(">"); + } else if (character == '\"') { + result.append("""); + } else if (character == '\'') { + result.append("'"); + } else if (character == '&') { + result.append("&"); + } else { + // the char is not a special one: add it to the result as is + result.append(character); + } + character = iterator.next(); + } + return result.toString(); + } + + /** + * Return <tt>aText</tt> with all <tt>'<'</tt> and <tt>'>'</tt> characters + * replaced by their escaped equivalents. + */ + public static String toDisableTags(String aText){ + final StringBuilder result = new StringBuilder(); + final StringCharacterIterator iterator = new StringCharacterIterator(aText); + char character = iterator.current(); + while (character != CharacterIterator.DONE ){ + if (character == '<') { + result.append("<"); + } + else if (character == '>') { + result.append(">"); + } + else { + //the char is not a special one + //add it to the result as is + result.append(character); + } + character = iterator.next(); + } + return result.toString(); + } + + + /** + * Replace characters having special meaning in regular expressions + * with their escaped equivalents, preceded by a '\' character. + * + * <P>The escaped characters include : + *<ul> + *<li>. + *<li>\ + *<li>?, * , and + + *<li>& + *<li>: + *<li>{ and } + *<li>[ and ] + *<li>( and ) + *<li>^ and $ + *</ul> + */ + public static String forRegex(String aRegexFragment){ + final StringBuilder result = new StringBuilder(); + + final StringCharacterIterator iterator = + new StringCharacterIterator(aRegexFragment) + ; + char character = iterator.current(); + while (character != CharacterIterator.DONE ){ + /* + * All literals need to have backslashes doubled. + */ + if (character == '.') { + result.append("\\."); + } + else if (character == '\\') { + result.append("\\\\"); + } + else if (character == '?') { + result.append("\\?"); + } + else if (character == '*') { + result.append("\\*"); + } + else if (character == '+') { + result.append("\\+"); + } + else if (character == '&') { + result.append("\\&"); + } + else if (character == ':') { + result.append("\\:"); + } + else if (character == '{') { + result.append("\\{"); + } + else if (character == '}') { + result.append("\\}"); + } + else if (character == '[') { + result.append("\\["); + } + else if (character == ']') { + result.append("\\]"); + } + else if (character == '(') { + result.append("\\("); + } + else if (character == ')') { + result.append("\\)"); + } + else if (character == '^') { + result.append("\\^"); + } + else if (character == '$') { + result.append("\\$"); + } + else { + //the char is not a special one + //add it to the result as is + result.append(character); + } + character = iterator.next(); + } + return result.toString(); + } + + /** + * Escape <tt>'$'</tt> and <tt>'\'</tt> characters in replacement strings. + * + * <P>Synonym for <tt>Matcher.quoteReplacement(String)</tt>. + * + * <P>The following methods use replacement strings which treat + * <tt>'$'</tt> and <tt>'\'</tt> as special characters: + * <ul> + * <li><tt>String.replaceAll(String, String)</tt> + * <li><tt>String.replaceFirst(String, String)</tt> + * <li><tt>Matcher.appendReplacement(StringBuffer, String)</tt> + * </ul> + * + * <P>If replacement text can contain arbitrary characters, then you + * will usually need to escape that text, to ensure special characters + * are interpreted literally. + */ + public static String forReplacementString(String aInput){ + return Matcher.quoteReplacement(aInput); + } + + /** + * Disable all <tt><SCRIPT></tt> tags in <tt>aText</tt>. + * + * <P>Insensitive to case. + */ + public static String forScriptTagsOnly(String aText){ + String result = null; + Matcher matcher = SCRIPT.matcher(aText); + result = matcher.replaceAll("<SCRIPT>"); + matcher = SCRIPT_END.matcher(result); + result = matcher.replaceAll("</SCRIPT>"); + return result; + } + + // PRIVATE // + + private StringUtils(){ + //empty - prevent construction + } + + private static final Pattern SCRIPT = Pattern.compile( + "<SCRIPT>", Pattern.CASE_INSENSITIVE + ); + private static final Pattern SCRIPT_END = Pattern.compile( + "</SCRIPT>", Pattern.CASE_INSENSITIVE + ); + + private static void addCharEntity(Integer aIdx, StringBuilder aBuilder){ + String padding = ""; + if( aIdx <= 9 ){ + padding = "00"; + } + else if( aIdx <= 99 ){ + padding = "0"; + } + else { + //no prefix + } + String number = padding + aIdx.toString(); + aBuilder.append("&#" + number + ";"); + } + }
--- a/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/util/Util.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/util/Util.java Tue Nov 27 12:35:19 2012 +0100 @@ -3,8 +3,14 @@ import java.io.File; import java.io.FileInputStream; import java.io.IOException; +import java.text.DateFormat; +import java.text.ParseException; +import java.text.SimpleDateFormat; +import java.util.Date; import java.util.Properties; +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; + public class Util { public Properties getProperties(String fullFileName) { @@ -14,8 +20,47 @@ FileInputStream in = new FileInputStream(file); props.load(in); } catch (IOException e) { + e.printStackTrace(); } return props; } + + public String toYearStr(String inputStr) { + String retYearStr = inputStr.trim(); + int index = inputStr.indexOf("-"); + if (index > 0) { + retYearStr = inputStr.substring(0, index); + retYearStr = retYearStr.trim(); + } + return retYearStr; + } + + public Date toDate(String xsDateStr) throws ApplicationException { + Date retDate = null; + if (xsDateStr == null) + return null; + try { + DateFormat dateFormat = new SimpleDateFormat("yyyy-MM-dd'T'HH:mm:ss.SSS'Z'"); + retDate = dateFormat.parse(xsDateStr); + } catch (ParseException e) { + throw new ApplicationException(e); + } + return retDate; + } + + public String toXsDate(Date date) { + DateFormat dateFormat = new SimpleDateFormat("yyyy-MM-dd'T'HH:mm:ss.SSS'Z'"); + String xsDateStr = dateFormat.format(date); + return xsDateStr; + } + public Double getSecondWithMillisecondsBetween(Date begin, Date end) { + long beginMS = begin.getTime(); + long endMS = end.getTime(); + long elapsedSeconds = (endMS - beginMS) / 1000; + long elapsedMilliSecondsAfterSeconds1 = (endMS - beginMS) - (elapsedSeconds * 1000); + Double seconds = new Double(elapsedSeconds + "." + elapsedMilliSecondsAfterSeconds1); + return seconds; + } + }
Binary file software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/.DS_Store has changed
--- a/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/test/TestLocal.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/test/TestLocal.java Tue Nov 27 12:35:19 2012 +0100 @@ -1,8 +1,18 @@ package de.mpg.mpiwg.berlin.mpdl.xml.test; +import java.io.File; import java.net.URL; +import org.apache.commons.io.FileUtils; + +import net.sf.saxon.s9api.Axis; +import net.sf.saxon.s9api.QName; +import net.sf.saxon.s9api.XdmNode; +import net.sf.saxon.s9api.XdmSequenceIterator; + import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.xml.transform.BasicTransformer; +import de.mpg.mpiwg.berlin.mpdl.xml.xquery.Hits; import de.mpg.mpiwg.berlin.mpdl.xml.xquery.XQueryEvaluator; public class TestLocal { @@ -16,6 +26,7 @@ try { TestLocal test = new TestLocal(); test.xqueries(); + // test.testTransform(); } catch (Exception e) { e.printStackTrace(); } @@ -25,14 +36,57 @@ xQueryEvaluator = new XQueryEvaluator(); } - private void xqueries() throws ApplicationException { + private void testTransform() throws ApplicationException { + try { + BasicTransformer basicTransformer = new BasicTransformer(); + String srcUrl = "file:/Users/jwillenborg/tmp/blablabla/Benedetti_1585.xml"; + String xslUrl = "file:/Users/jwillenborg/tmp/blablabla/replaceAnchors.xsl"; + String result = basicTransformer.transform(srcUrl, xslUrl, null, null); + File outFile = new File("/Users/jwillenborg/tmp/blablabla/Benedetti_1585-New.xml"); + FileUtils.writeStringToFile(outFile, result, "utf-8"); + } catch (Exception e) { + throw new ApplicationException(e); + } + } + + private void test() throws ApplicationException { String result = null; try { - URL srcUrl = new URL("http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/getDoc?doc=/tei/en/Test_1789.xml"); - result = xQueryEvaluator.evaluateAsString(srcUrl, "declare namespace TEI=\"http://www.tei-c.org/ns/1.0\"; //TEI:s"); + URL srcUrl = new URL("http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/getDoc?doc=/tei/de/dt-ptolemaeus-tei-merge2.xml"); + srcUrl = new URL("file:/Users/jwillenborg/tmp/errorFile.xml"); + XdmNode docNode = xQueryEvaluator.parse(srcUrl); + String docType = getNodeType(docNode); String bla = ""; } catch (Exception e) { throw new ApplicationException(e); } } + + private void xqueries() throws ApplicationException { + String result = null; + try { + // URL srcUrl = new URL("http://mpdl-system.mpiwg-berlin.mpg.de/mpdl/getDoc?doc=/tei/de/dt-ptolemaeus-tei-merge2.xml"); + URL srcUrl = new URL("file:/Users/jwillenborg/test/dt-ptolemaeus-tei-merge2.xml"); + Hits hits = xQueryEvaluator.evaluate(srcUrl, "//*[@xml:id != '1']", 0, 9); + String bla = ""; + } catch (Exception e) { + throw new ApplicationException(e); + } + } + + private String getNodeType(XdmNode node) { + String nodeType = null; + XdmSequenceIterator iter = node.axisIterator(Axis.CHILD); + if (iter != null) { + while (iter.hasNext()) { + XdmNode firstChild = (XdmNode) iter.next(); + if (firstChild != null) { + QName nodeQName = firstChild.getNodeName(); + nodeType = nodeQName.getLocalName(); + } + } + } + return nodeType; + } + }
--- a/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/transform/BasicTransformer.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/transform/BasicTransformer.java Tue Nov 27 12:35:19 2012 +0100 @@ -104,6 +104,29 @@ return retStr; } + public String transform(StringReader inputStrReader, String xslUrl, String parametersStr, String outputPropertiesStr) throws ApplicationException { + String retStr = null; + try { + StreamSource xslStreamSource = new StreamSource(xslUrl); + XsltExecutable xsltExecutable = xsltCompiler.compile(xslStreamSource); + xsltTransformer = xsltExecutable.load(); + StreamSource xmlDoc = new StreamSource(inputStrReader); + Serializer serializer = new Serializer(); + StringWriter stringWriter = new StringWriter(); + serializer.setOutputWriter(stringWriter); + setOutputProperties(serializer, outputPropertiesStr); + xsltTransformer.setSource(xmlDoc); // needs some time for bigger documents + xsltTransformer.setDestination(serializer); + setParameters(xsltTransformer, parametersStr); + xsltTransformer.transform(); // needs some time for bigger documents + StringWriter out = (StringWriter) serializer.getOutputDestination(); + retStr = out.toString(); + } catch (SaxonApiException e) { + throw new ApplicationException(e); + } + return retStr; + } + private String getEncoding(String outputPropertiesStr) { String str = outputPropertiesStr.toLowerCase(); String encoding = null; @@ -228,7 +251,7 @@ // String pageFragment = serializer.getOutputDestination().toString(); // System.out.println(pageFragment); } catch (SaxonApiException e) { - throw new ApplicationException(e); + throw new ApplicationException(e); } }
--- a/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/transform/FragmentTransformer.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/transform/FragmentTransformer.java Tue Nov 27 12:35:19 2012 +0100 @@ -3,7 +3,6 @@ import java.io.IOException; import java.io.StringWriter; import java.net.URL; -import java.util.Date; import javax.xml.transform.stream.StreamSource; @@ -24,25 +23,18 @@ private XsltCompiler xsltCompiler; private XsltTransformer xsltFragmentTransformer; - public static void main(String[] args) { - try { - FragmentTransformer fragmentTransformer = new FragmentTransformer(); - String goerzDoc = "file:/Users/jwillenborg/java/mpiwg-mpdl-xml/examples/Goerz_2008.xml"; - String benDoc = "file:/Users/jwillenborg/java/mpiwg-mpdl-xml/examples/Benedetti_1585.xml"; - String pageFragment = fragmentTransformer.getFragment(goerzDoc, "pb", 1, "pb", 2); - Date before = new Date(); - for (int i=1; i<=50; i++) { - pageFragment = fragmentTransformer.getFragment(benDoc, "pb", i, "pb", i+1); - } - Date after = new Date(); - System.out.println("Needed time: " + (after.getTime() - before.getTime()) + " ms"); - } catch (Exception e) { - e.printStackTrace(); - } + public FragmentTransformer() throws ApplicationException { + init(); } - public FragmentTransformer() throws ApplicationException { + public FragmentTransformer(String xmlFileName) throws ApplicationException { init(); + try { + StreamSource xmlDoc = new StreamSource(xmlFileName); + xsltFragmentTransformer.setSource(xmlDoc); // needs some time for bigger documents + } catch (SaxonApiException e) { + throw new ApplicationException(e); + } } private void init() throws ApplicationException { @@ -69,8 +61,6 @@ serializer.setOutputProperty(Serializer.Property.INDENT, "yes"); xsltFragmentTransformer.setSource(xmlDoc); // needs some time for bigger documents xsltFragmentTransformer.setDestination(serializer); - QName xmlFileNameQName = new QName("xmlFileName"); - XdmValue xmlFileNameXdmValue = new XdmAtomicValue(xmlFileName); QName ms1NameQName = new QName("ms1Name"); XdmValue ms1NameXdmValue = new XdmAtomicValue(ms1Name); QName ms1PositionQName = new QName("ms1Position"); @@ -79,7 +69,6 @@ XdmValue ms2NameXdmValue = new XdmAtomicValue(ms2Name); QName ms2PositionQName = new QName("ms2Position"); XdmValue ms2PositionXdmValue = new XdmAtomicValue(ms2Pos); - xsltFragmentTransformer.setParameter(xmlFileNameQName, xmlFileNameXdmValue); xsltFragmentTransformer.setParameter(ms1NameQName, ms1NameXdmValue); xsltFragmentTransformer.setParameter(ms1PositionQName, ms1PositionXdmValue); xsltFragmentTransformer.setParameter(ms2NameQName, ms2NameXdmValue); @@ -91,4 +80,33 @@ } return pageFragment; } + + public String getFragment(String ms1Name, int ms1Pos, String ms2Name, int ms2Pos) throws ApplicationException { + String pageFragment = null; + try { + Serializer serializer = new Serializer(); + serializer.setOutputWriter(new StringWriter()); + serializer.setOutputProperty(Serializer.Property.INDENT, "yes"); + xsltFragmentTransformer.setDestination(serializer); + QName ms1NameQName = new QName("ms1Name"); + XdmValue ms1NameXdmValue = new XdmAtomicValue(ms1Name); + QName ms1PositionQName = new QName("ms1Position"); + XdmValue ms1PositionXdmValue = new XdmAtomicValue(ms1Pos); + QName ms2NameQName = new QName("ms2Name"); + XdmValue ms2NameXdmValue = new XdmAtomicValue(ms2Name); + QName ms2PositionQName = new QName("ms2Position"); + XdmValue ms2PositionXdmValue = new XdmAtomicValue(ms2Pos); + xsltFragmentTransformer.setParameter(ms1NameQName, ms1NameXdmValue); + xsltFragmentTransformer.setParameter(ms1PositionQName, ms1PositionXdmValue); + xsltFragmentTransformer.setParameter(ms2NameQName, ms2NameXdmValue); + xsltFragmentTransformer.setParameter(ms2PositionQName, ms2PositionXdmValue); + xsltFragmentTransformer.transform(); // needs some time for bigger documents + pageFragment = serializer.getOutputDestination().toString(); + } catch (Exception e) { + throw new ApplicationException(e); + } + return pageFragment; + } + + }
--- a/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/transform/getFragment.xsl Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/transform/getFragment.xsl Tue Nov 27 12:35:19 2012 +0100 @@ -12,7 +12,6 @@ <xsl:output method="xml" encoding="utf-8"/> <!-- ms1Name and ms2Name have to be given without namespace: e.g. "pb" --> -<xsl:param name="xmlFileName"></xsl:param> <xsl:param name="ms1Name"></xsl:param> <xsl:param name="ms1Position"></xsl:param> <xsl:param name="ms2Name"></xsl:param> @@ -25,15 +24,29 @@ <xsl:variable name="ms1Ancestors" select="$ms1/ancestor::*"/> <xsl:variable name="ms2Ancestors" select="$ms2/ancestor::*"/> +<!-- all elements between the milestone elements and also the ancestor elements (for filling the node path to the root) --> <xsl:template match="element()[local-name() != $ms1Name and local-name() != $ms2Name]"> <xsl:choose> <xsl:when test="(. >> $ms1 or fn:contains($ms1Ancestors, .)) and ($ms2 >> . or fn:contains($ms2Ancestors, .))"> - <xsl:element name="{local-name(.)}"><xsl:apply-templates/></xsl:element> + <xsl:element name="{local-name(.)}"><xsl:copy-of select="@*"></xsl:copy-of><xsl:apply-templates/></xsl:element> </xsl:when> <xsl:otherwise><xsl:apply-templates/></xsl:otherwise> </xsl:choose> </xsl:template> +<!-- the two milestone elements --> +<xsl:template match="element()[local-name() = $ms1Name or local-name() = $ms2Name]"> + <xsl:choose> + <xsl:when test=". is $ms1"> + <xsl:element name="{local-name(.)}"><xsl:copy-of select="@*"></xsl:copy-of></xsl:element> + </xsl:when> + <xsl:when test=". is $ms2"> + <xsl:element name="{local-name(.)}"><xsl:copy-of select="@*"></xsl:copy-of></xsl:element> + </xsl:when> + <xsl:otherwise></xsl:otherwise> + </xsl:choose> +</xsl:template> + <xsl:template match="attribute()|text()|comment()|processing-instruction()"> <xsl:choose> <xsl:when test=". >> $ms1 and $ms2 >> ."> @@ -43,16 +56,4 @@ </xsl:choose> </xsl:template> -<xsl:template match="element()[local-name() = $ms1Name or local-name() = $ms2Name]"> - <xsl:choose> - <xsl:when test=". is $ms1"> - <xsl:element name="{local-name(.)}"><xsl:copy-of select="@*"></xsl:copy-of></xsl:element> - </xsl:when> - <xsl:when test=". is $ms2"> - <xsl:element name="{local-name(.)}"><xsl:copy-of select="@*"></xsl:copy-of></xsl:element> - </xsl:when> - <xsl:otherwise></xsl:otherwise> - </xsl:choose> -</xsl:template> - </xsl:stylesheet>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/xquery/Hit.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,57 @@ +package de.mpg.mpiwg.berlin.mpdl.xml.xquery; + +public class Hit { + public static int TYPE_ELEMENT = 1; + public static int TYPE_ATTRIBUTE = 2; + public static int TYPE_ATOMIC_VALUE = 3; + private String name; + private int type; + private String content; + private int page = -1; + private int hitPagePosition = -1; + + public Hit(String content) { + this.content = content; + } + + public String getName() { + return name; + } + + public void setName(String name) { + this.name = name; + } + + public int getType() { + return type; + } + + public void setType(int type) { + this.type = type; + } + + public String getContent() { + return content; + } + + public void setContent(String content) { + this.content = content; + } + + public int getPage() { + return page; + } + + public void setPage(int page) { + this.page = page; + } + + public int getHitPagePosition() { + return hitPagePosition; + } + + public void setHitPagePosition(int hitPagePosition) { + this.hitPagePosition = hitPagePosition; + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/xquery/Hits.java Tue Nov 27 12:35:19 2012 +0100 @@ -0,0 +1,58 @@ +package de.mpg.mpiwg.berlin.mpdl.xml.xquery; + +import java.util.ArrayList; + +public class Hits { + private ArrayList<Hit> hits; + private String query; + private int from; + private int to; + private int size = 0; + + public Hits(ArrayList<Hit> hits, int from, int to) { + this.hits = hits; + this.from = from; + this.to = to; + } + + public int getSize() { + return size; + } + + public void setSize(int size) { + this.size = size; + } + + public String getQuery() { + return query; + } + + public void setQuery(String query) { + this.query = query; + } + + public ArrayList<Hit> getHits() { + return hits; + } + + public void setHits(ArrayList<Hit> hits) { + this.hits = hits; + } + + public int getFrom() { + return from; + } + + public void setFrom(int from) { + this.from = from; + } + + public int getTo() { + return to; + } + + public void setTo(int to) { + this.to = to; + } + +}
--- a/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/xquery/XQueryEvaluator.java Wed Dec 14 13:57:09 2011 +0100 +++ b/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/xquery/XQueryEvaluator.java Tue Nov 27 12:35:19 2012 +0100 @@ -3,19 +3,30 @@ import java.io.StringReader; import java.io.StringWriter; import java.net.URL; +import java.util.ArrayList; import javax.xml.transform.stream.StreamSource; +import org.xml.sax.SAXParseException; + import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import net.sf.saxon.s9api.Axis; +import net.sf.saxon.s9api.DocumentBuilder; import net.sf.saxon.s9api.Processor; +import net.sf.saxon.s9api.QName; import net.sf.saxon.s9api.SaxonApiException; import net.sf.saxon.s9api.Serializer; import net.sf.saxon.s9api.XQueryCompiler; import net.sf.saxon.s9api.XQueryExecutable; +import net.sf.saxon.s9api.XdmAtomicValue; +import net.sf.saxon.s9api.XdmEmptySequence; import net.sf.saxon.s9api.XdmItem; +import net.sf.saxon.s9api.XdmNode; +import net.sf.saxon.s9api.XdmNodeKind; import net.sf.saxon.s9api.XdmSequenceIterator; import net.sf.saxon.s9api.XdmValue; +import net.sf.saxon.trans.XPathException; public class XQueryEvaluator { private Processor processor; @@ -26,37 +37,79 @@ xQueryCompiler = processor.newXQueryCompiler(); } + public XdmNode parse(URL srcUrl) throws ApplicationException { + try { + DocumentBuilder docBuilder = processor.newDocumentBuilder(); + StreamSource xmlDoc = new StreamSource(srcUrl.toString()); + XdmNode docNode = docBuilder.build(xmlDoc); + return docNode; + } catch (SaxonApiException e) { + String message = e.getMessage(); + Throwable t = e.getCause(); + if (t instanceof XPathException) { + XPathException xpathException = (XPathException) t; + Throwable t2 = xpathException.getException(); + if (t2 instanceof SAXParseException) { + SAXParseException saxParseException = (SAXParseException) t2; + int lineNumber = saxParseException.getLineNumber(); + int columnNumber = saxParseException.getColumnNumber(); + message = "Line: " + lineNumber + ", Column: " + columnNumber + ": " + e.getMessage(); + } + } + throw new ApplicationException("Your source file (" + srcUrl.toString() + ") is not valid: " + message); + } + } + + public Hits evaluate(URL srcUrl, String xqueryStr, int from, int to) throws ApplicationException { + Hits result = (Hits) evaluate(srcUrl, xqueryStr, from, to, "hits"); + return result; + } + public XdmValue evaluate(String srcXmlStr, String xqueryStr) throws ApplicationException { - XdmValue result = (XdmValue) evaluate(srcXmlStr, xqueryStr, null); + XdmValue result = (XdmValue) evaluate(srcXmlStr, xqueryStr, 0, 9, "XdmValue"); return result; } public XdmValue evaluate(URL srcUrl, String xqueryStr) throws ApplicationException { - XdmValue result = (XdmValue) evaluate(srcUrl, xqueryStr, null); + XdmValue result = (XdmValue) evaluate(srcUrl, xqueryStr, 0, 9, "XdmValue"); return result; } public String evaluateAsString(String srcXmlStr, String xqueryStr) throws ApplicationException { - Object result = evaluate(srcXmlStr, xqueryStr, "asString"); - return (String) result; + Object result = evaluate(srcXmlStr, xqueryStr, 0, 9, "string"); + if (result == null) + return null; + else + return (String) result; } public String evaluateAsString(URL srcUrl, String xqueryStr) throws ApplicationException { - Object result = evaluate(srcUrl, xqueryStr, "asString"); - return (String) result; + Object result = evaluate(srcUrl, xqueryStr, 0, 9, "string"); + if (result == null) + return null; + else + return (String) result; } - public String evaluateAsStringValue(String srcXmlStr, String xqueryStr) throws ApplicationException { - XdmValue val = (XdmValue) evaluate(srcXmlStr, xqueryStr, null); - return toStringValue(val); + public String evaluateAsStringValueJoined(String srcXmlStr, String xqueryStr) throws ApplicationException { + return evaluateAsStringValueJoined(srcXmlStr, xqueryStr, " "); } - public String evaluateAsStringValue(URL srcUrl, String xqueryStr) throws ApplicationException { - XdmValue val = (XdmValue) evaluate(srcUrl, xqueryStr, null); - return toStringValue(val); + public String evaluateAsStringValueJoined(URL srcUrl, String xqueryStr) throws ApplicationException { + return evaluateAsStringValueJoined(srcUrl, xqueryStr, " "); } - public Object evaluate(String srcXmlStr, String xqueryStr, String optionsStr) throws ApplicationException { + public String evaluateAsStringValueJoined(String srcXmlStr, String xqueryStr, String separator) throws ApplicationException { + XdmValue val = (XdmValue) evaluate(srcXmlStr, xqueryStr, 0, 9, "XdmValue"); + return stringJoin(val, separator); + } + + public String evaluateAsStringValueJoined(URL srcUrl, String xqueryStr, String separator) throws ApplicationException { + XdmValue val = (XdmValue) evaluate(srcUrl, xqueryStr, 0, 9, "XdmValue"); + return stringJoin(val, separator); + } + + public Object evaluate(String srcXmlStr, String xqueryStr, int from, int to, String resultType) throws ApplicationException { try { StringReader srcXmlStrReader = new StringReader(srcXmlStr); StreamSource xmlDoc = new StreamSource(srcXmlStrReader); @@ -67,12 +120,12 @@ xQueryEvaluator.setSource(xmlDoc); XdmValue val = xQueryEvaluator.evaluate(); Object result = val; - if (optionsStr != null && optionsStr.contains("asString")) { - int size = val.size(); - if (size <= 0) - result = ""; - else - result = toString(val); + if (resultType != null && resultType.equals("string")) { + result = toString(val); + } else if (resultType != null && resultType.equals("XdmValue")) { + result = val; + } else if (resultType != null && resultType.equals("hits")) { + result = toHits(val, from, to); } return result; } catch (SaxonApiException e) { @@ -80,7 +133,7 @@ } } - public Object evaluate(URL srcUrl, String xqueryStr, String optionsStr) throws ApplicationException { + public Object evaluate(URL srcUrl, String xqueryStr, int from, int to, String resultType) throws ApplicationException { try { StreamSource xmlDoc = new StreamSource(srcUrl.toString()); XQueryExecutable xQueryExecutable = xQueryCompiler.compile(xqueryStr); @@ -89,13 +142,13 @@ net.sf.saxon.s9api.XQueryEvaluator xQueryEvaluator = xQueryExecutable.load(); xQueryEvaluator.setSource(xmlDoc); XdmValue val = xQueryEvaluator.evaluate(); - Object result = val; - if (optionsStr != null && optionsStr.contains("asString")) { - int size = val.size(); - if (size <= 0) - result = ""; - else - result = toString(val); + Object result = null; + if (resultType != null && resultType.equals("string")) { + result = toString(val); + } else if (resultType != null && resultType.equals("XdmValue")) { + result = val; + } else if (resultType != null && resultType.equals("hits")) { + result = toHits(val, from, to); } return result; } catch (SaxonApiException e) { @@ -103,18 +156,141 @@ } } - private String toString(XdmValue xdmValue) { - String result = ""; + private Hits toHits(XdmValue xdmValue, int from, int to) throws ApplicationException { + Hits result = null; + if (xdmValue instanceof XdmAtomicValue) { + XdmAtomicValue av = (XdmAtomicValue) xdmValue; + String avStr = av.getStringValue(); + ArrayList<Hit> hits = new ArrayList<Hit>(); + Hit hit = new Hit(avStr); + hit.setType(Hit.TYPE_ATOMIC_VALUE); + hits.add(hit); + result = new Hits(hits, from, to); + result.setSize(1); + return result; + } XdmSequenceIterator iter = xdmValue.iterator(); - while (iter.hasNext()) { - XdmItem item = iter.next(); - String itemStr = item.toString(); - result += itemStr; + int size = xdmValue.size(); + if (size > 0) { + int counter = 0; + ArrayList<Hit> hits = new ArrayList<Hit>(); + while (iter.hasNext() && counter <= to) { + XdmItem item = iter.next(); + if (counter >= from) { + XdmNode n = (XdmNode) item; + ArrayList<XdmNode> precNodes = getPreceding(n, "pb"); + int page = precNodes.size(); + String nodeName = n.getNodeName().getLocalName(); + String nodeKindName = n.getNodeKind().name(); + ArrayList<XdmNode> precNodesUntilPB = getPrecedingUntil(n, nodeName, "pb"); + int hitPagePosition = precNodesUntilPB.size() + 1; + String itemStr = item.toString(); + // itemStr = itemStr.replaceAll("[ \n\t]+xmlns.*?\".*?\"", ""); // remove the namespace + itemStr = itemStr.replaceAll("[ \n\t]+xmlns", " xmlns"); // remove the blanks before the namespace + Hit hit = new Hit(itemStr); + hit.setPage(page); + hit.setName(nodeName); + if (nodeKindName != null && nodeKindName.equals("ELEMENT")) + hit.setType(Hit.TYPE_ELEMENT); + else if (nodeKindName != null && nodeKindName.equals("ATTRIBUTE")) + hit.setType(Hit.TYPE_ATTRIBUTE); + if (hit.getType() == Hit.TYPE_ELEMENT) + hit.setHitPagePosition(hitPagePosition); + hits.add(hit); + } + counter++; + } + result = new Hits(hits, from, to); + result.setSize(size); + } + return result; + } + + private String toString(XdmValue xdmValue) { + if (xdmValue instanceof XdmAtomicValue) { + XdmAtomicValue av = (XdmAtomicValue) xdmValue; + String avStr = av.getStringValue(); + return avStr; + } + String result = null; + XdmSequenceIterator iter = xdmValue.iterator(); + int size = xdmValue.size(); + if (size > 0) { + result = ""; + while (iter.hasNext()) { + XdmItem item = iter.next(); + String itemStr = item.toString(); + result += itemStr; + } } return result; } - private String toStringValue(XdmValue xdmValue) { - return xdmValue.itemAt(0).getStringValue(); + private String stringJoin(XdmValue xdmValue, String separator) { + if (xdmValue == null || xdmValue instanceof XdmEmptySequence) + return null; + if (xdmValue instanceof XdmAtomicValue) { + XdmAtomicValue av = (XdmAtomicValue) xdmValue; + String avStr = av.getStringValue(); + return avStr; + } + String result = ""; + XdmSequenceIterator iter = null; + if (xdmValue instanceof XdmNode) { + XdmNode xdmNode = (XdmNode) xdmValue; + iter = xdmNode.axisIterator(Axis.CHILD); + } else if (xdmValue instanceof XdmValue) { + iter = xdmValue.iterator(); + } + while (iter.hasNext()) { + XdmNode node = (XdmNode) iter.next(); + XdmNodeKind nodeKind = node.getNodeKind(); + if (nodeKind == XdmNodeKind.TEXT) { + String nodeStr = node.getStringValue(); + String trimmedNodeStr = nodeStr.trim(); + result = result + trimmedNodeStr; + } else if (nodeKind == XdmNodeKind.ELEMENT) { + result = result + stringJoin(node, separator) + separator; // put a separator between child element nodes + } + } + return result; + } + + private ArrayList<XdmNode> getPreceding(XdmNode startNode, String ofNodeName) { + ArrayList<XdmNode> retNodes = null; + XdmSequenceIterator iter = startNode.axisIterator(Axis.PRECEDING); + if (iter != null) { + retNodes = new ArrayList<XdmNode>(); + while (iter.hasNext()) { + XdmNode n = (XdmNode) iter.next(); + QName qName = n.getNodeName(); + if (qName != null) { + String name = qName.getLocalName(); + if (name != null && name.equals(ofNodeName)) + retNodes.add(n); + } + } + } + return retNodes; + } + + private ArrayList<XdmNode> getPrecedingUntil(XdmNode startNode, String ofNodeName, String untilNodeName) { + ArrayList<XdmNode> retNodes = null; + XdmSequenceIterator iter = startNode.axisIterator(Axis.PRECEDING); + if (iter != null) { + retNodes = new ArrayList<XdmNode>(); + while (iter.hasNext()) { + XdmNode n = (XdmNode) iter.next(); + QName qName = n.getNodeName(); + if (qName != null) { + String name = qName.getLocalName(); + if (name != null && name.equals(ofNodeName)) + retNodes.add(n); + if (name != null && name.equals(untilNodeName)) + return retNodes; + } + } + } + return retNodes; } }